Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Developing tuned anisotropic porous substrates for passive turbulence control
(USC Thesis Other)
Developing tuned anisotropic porous substrates for passive turbulence control
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
Developing Tuned Anisotropic Porous Substrates for Passive Turbulence Control by Christoph Efstathiou A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Aerospace Engineering) August 2020 Copyright 2020 Christoph Efstathiou Acknowledgements I would not have been able to undertake and complete this work without the help, support and encouragement of many wonderful people, so I would like to gratefully acknowledge: Dr. Madison Swayne, for her love and support, My parents and brother, Hilde, Takis and Philipp for their encouragement and love, Dr. Mitul Luhar, for the opportunity to work with him on interesting and fullling projects, patient mentorship, thoughtful guidance and teaching me so much, Dr. Geo Spedding, for ten years of generous advice and guidance, Dr. Felipe de Barros, for insightful conversations and comments and for being on my committee, Andrew, Shilpa, Mark, Vamsi, Chelsea, Michael, Yohanna, Trystan, James, Saakar, Joe, Chris, Anup, and Joel, for camaraderie, sympathy, and coee. ii Table of Contents Acknowledgements ii List Of Tables v List Of Figures vi Abstract xi Chapter 1: Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Contribution and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 2: Mean turbulence statistics in boundary layers over high-porosity foams 7 2.1 Comment and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Previous studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Contribution and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Flow facility and at plate apparatus . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Velocity measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.3 Reynolds number ranges and spatio-temporal resolution . . . . . . . . . . . . 19 2.3.4 Porous Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 Boundary layer development . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.2 Eect of pore size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.3 Eect of substrate thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.4 Velocity spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5.1 Amplitude modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5.2 Logarithmic region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.5.3 Non-monotonic behaviour with pore size . . . . . . . . . . . . . . . . . . . . 39 2.5.4 A note on scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter 3: Development and characterization of anisotropic porous media 45 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Lattice design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 Formlabs printers specications and limitations . . . . . . . . . . . . . . . . . . . . 47 3.4 Finished Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5 Permeability estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 iii 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 4: Resolvent-based design and experimental testing of porous materials for passive turbulence control 57 4.1 Comment and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.2 Previous Theoretical Eorts and Simulations . . . . . . . . . . . . . . . . . . 59 4.2.3 Previous Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2.4 Contribution and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.1 Extended Resolvent Formulation . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4.1 3D Printed Porous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4.2 Channel Flow Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.5.1 Model Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.5.2 Mean Flow and Turbulence Statistics . . . . . . . . . . . . . . . . . . . . . . 75 4.5.3 Flow Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.4 Model Sensitivity to Mean Prole . . . . . . . . . . . . . . . . . . . . . . . . 80 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Chapter 5: Turbulent boundary layers over streamwise-preferential porous mate- rials 85 5.1 Comment and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.1 Drag reduction mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3 Previous experiments over porous materials . . . . . . . . . . . . . . . . . . . . . . 89 5.3.1 Contribution and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4.1 Flow facility and at plate apparatus . . . . . . . . . . . . . . . . . . . . . . 92 5.4.2 2D-2C particle image velocimetry . . . . . . . . . . . . . . . . . . . . . . . . 93 5.4.3 Mean ow estimation at single-pixel resolution . . . . . . . . . . . . . . . . . 94 5.4.4 Porous substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4.5 Friction velocity estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.5.1 Boundary layer development . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.5.2 Fully-developed ow statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.5.3 Velocity spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Chapter 6: Closing Remarks 113 Bibliography 116 iv List Of Tables 2.1 Permeability (k), porosity (), average pore sizes (s), and related dimensionless parameters for tested foams. Permeability and pore size values from Manes et al. (2011) are noted in parenthesis for the 10 and 60 ppi foams. Note that s + =su = and Re k = p ku = are dened using the friction velocity upstream of the porous section. Porosity () was estimated from solid volume displacement in water and permeability was estimated from pressure drop experiments. . . . . . . . . . . . . . 21 2.2 Fitted values for log-law parameters in dimensional and dimensionless form. Listed values of u =, y d , and k 0 are averages of the three estimates shown in gure 2.11, which were obtained for three dierent outer limits in the tting procedure. The displacement height is assumed to be zero for the smooth wall ow. . . . . . . . . . 40 4.1 Dimensionless permeability estimates for the 3D-printed porous materials. H = 6:34 mm is the height of the porous substrates tested in the channel ow experiments. . 70 4.2 Friction velocity estimates at the smooth wall (u s ) and porous interface (u p ). The average friction velocity isu t . For the smooth wall caseu p corresponds to the solid tile placed in the cutout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3 Comparison of the normalized gain for NW resolvent modes computed using the synthetic and tted mean velocity proles. . . . . . . . . . . . . . . . . . . . . . . . 82 5.1 Estimates for friction-related parameters along the porous substrate. smooth wall values available for measurement stations 1, 4, and 5 are shown in parentheses. Typical uncertainties are shown at the bottom. . . . . . . . . . . . . . . . . . . . . 103 v List Of Figures 2.1 Schematics showing at plate apparatus in the wall normal-spanwise plane (yz, top) and the streamwise-wall normal plane (xy, bottom). Measurement positions marked 1-4 were located at x=h = 11; 21; 42; 53, where h = 12:7 mm is the baseline substrate thickness, andx = 0 is dened as the smooth-porous wall transition. The smooth wall reference measurements were made at a location x=h =21:5. Note that the wall-normal coordinate is y. . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Mean velocityU and streamwise turbulence intensityu 2 proles corrected by inverse velocity magnitude and inter-arrival time weighting. The wall-normal coordinate is normalized using the local 99% boundary layer thickness, . . . . . . . . . . . . . . 18 2.3 Prole showing standard error of mean velocity (in %) over upstream smooth wall section. The discontinuity in u betweeny= = 0:01 and 0:02 corresponds to the lo- cation where the measurement duration for each point was reduced from 45 minutes to 10 minutes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Photographs showing thin sheets of the 10, 20, 60, and 100 ppi foams (from left to right). Each image represents a 2 cm 2 cm cross section. . . . . . . . . . . . . . . 21 2.5 Streamwise mean velocity (a) and turbulence intensity (b) measured at streamwise locations x=h =21:5; 11; 21; 42; 53 relative to the transition from smooth wall to porous substrate. These data correspond to the 20 ppi foam withh = 12:7mm. The dashed lines correspond to a linear prole of the form U + = y + in the near-wall region and a logarithmic prole of the form U + = (1=) ln(y + ) +B in the overlap region, with = 0:39 andB = 4:3. The friction velocity was estimated from tting a linear slope to the near-wall measurements. . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Normalized boundary layer thickness =h for the 20 ppi foam measured at stream- wise locations x=h =21:5; 11; 21; 42; 53 relative to the substrate transition point. . 24 2.7 Mean velocity (a) and turbulence intensity (b) proles for the smooth wall and for all the porous foams at x=h = 42. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 Wall-normal proles of skewness (Sk) for the smooth wall and for all the porous foams at x=h = 42. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.9 Mean velocity (a) and turbulence intensity (b) proles for the smooth wall and for the 20ppi foam of varying thickness at the same physical location. . . . . . . . . . . 29 2.10 Contour maps showing variation in premultiplied frequency spectra (normalized by U 2 e ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 vi 2.11 (a) Scaled velocity gradient (y=U e )@U=@y plotted as a function ofy= for the smooth wall prole and porous foam data. Equation (2.8) was tted to these data to esti- mate the normalized displacement height, y d =h, and the friction velocity weighted by the von Karman constant, u =(U e ), shown in (b) and (c), respectively. Using these estimates for y d andu =, the roughness heightk 0 =h, shown in (d), was eval- uated from the velocity proles using equation (2.1). Dotted lines in (a) represent the upper limits ofy= = 0:16; 0:20 and 0:25 employed in the tting procedure. The dashed line represents the minimum lower limit, y=> 0:02. Larger marker sizes in (b,c,d) denote higher values for the upper limit. The red cross in (c) represents the friction velocity estimated via a linear t to the near-wall velocity measurements over the smooth wall. Horizontal error bars in panels (b)-(d) represent uncertainty in pore size, s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1 Lattice parameters governing permeabilities and porosities. s x and s y drive the spanwise facing open area, s y and s z drive the streamwise facing open area, and s x and s z drive the wall-normal facing open area. . . . . . . . . . . . . . . . . . . . . . 47 3.2 Diagram of form3 printer (a), and printing conguration (b). Note that the build happens inverted, i.e. the build platform is lowered into the resin tank, and the new structural layer is cured at the bottom of the tank. Between layers, the wiper removes residue and mixes the resin. The model is attached to the build platform via a sacricial layer (raft) and delicate supports. The tiles are printed with the larger openings in the vertical direction to maximize drainage. . . . . . . . . . . . . 48 3.3 Photographs of prepared materials. The rod diameter isd = 0:4mm, and the lattice has rod spacings of s x = 0:8mm,s y = 3:0mm, and s z = 3:0mm. The lattice was printed as 5 full pores and a solid set of rods on the top and bottom layer, yielding a nal thickness of 15.4mm. The nal material was spraypainted black to minimize laser re ections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 Diagram of permeametry experiment, overview (top) and detailed view of test sec- tion (bottom). The working uid (water) is driven by a submersible pump from a reservoir and regulated by a programmable valve. The ow enters the permeameter test section through a ow-straightening mesh and encounters a cubic sample. The pressure gradient across the sample is measured in three dimensions ( @P @x ; @P @y ; @P @z ) and mass ow rate is measured downstream. . . . . . . . . . . . . . . . . . . . . . 53 3.5 Panel a) (left) shows the raw pressure drop drop data (circle) and linear and quadratic ts (lines). In this case, both linear and quadratic t for permeability agree within 5%. In panel b) (right) 1D permeability estimates for sample with varying pore sizes are shown. The labels refer to the rod separations and rod diam- eter, i.e. (s x ;s y ;s z ;d). Note that the streamwise normal area (frontal open area) is proportional to (s z d)(s y d). The cases labelled (1,1.7,1.7) and (1.7,1,1.7) refer to the same cube, rotated 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 vii 3.6 Permeability estimates are made for porous cubes with s x = 2, s y = 2, s z = 0:8mm and rod size d = 0:4mm and rotation angles of = 0 to 90 in 15 increments. Raw pressure drop data is shown in a). Note that for the larger pore sizes (0 ), the pressure drop is an order of magnitude lower than for the smaller open area (90 ). Permeability estimates for the 7 tested orientations are shown in panel b). Uncertainty estimates are obtained by varying the number of data points from a) used to estimate the permeabilities. The solid line and shaded uncertainty bar indicate predictions made by rotating K 0 and K 90 . . . . . . . . . . . . . . . . . . . 55 4.1 (a) Schematic of the channel ow experiment (not to scale). (b,c) Renderings for the x-permeable and y-permeable material. (d) Image of 3D-printed x-permeable porous tile. Note that the x-permeable material features large openings normal to the streamwise incoming ow. The y-permeable material has large openings in the wall-normal direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 (a,b) Predicted singular value ratios ( ;p = ;s ) for resolvent modes resembling the NW cycle as a function of streamwise and wall-normal permeability length-scales. Color contours show predictions forRe = 120; solid black lines correspond toRe = 360. Predictions in panel (a) are for substrates which have a similar conguration to x-permeable material K = diag(K xx ;K yy ;K zz = K yy ). Predictions in panel (b) are for substrates with a similar conguration to the y-permeable substrate, K = diag(K xx ;K yy ;K zz =K xx ). The () symbols in (a) and (b) correspond roughly to the permeabilities for the x-permeable and y-permeable materials tested in the experiments. (c,d) Amplication of spanwise-constant modes at Re = 120 relative to the smooth wall case as a function of streamwise wavelength and mode speed for (c) the x-permeable substrate and (d) for the y-permeable substrate, i.e., for permeability values labeled with () symbols in (a) and (b). . . . . . . . . . . . . . 73 4.3 PIV results for the channel ow experiment. The smooth wall is located at y = 0, while the interchangeable wall is located at y = h. Panel (a) shows the measured mean velocity proles normalized by the bulk-averaged velocity (calculated as U b = Q=[W (H+h)] in all cases). Panel (b) shows Reynolds shear stress proles normalized by the smooth wall friction velocityu s . Friction velocities at the smooth and porous walls were estimated by extrapolating a linear t to the total stress prole to the wall locations. Panels (c) and (d) show proles of the root-mean-square streamwise uctuations normalized by the friction velocities at the smooth wall and porous interface, u s and u p , respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 The rst two POD modes for the smooth wall case (a,b), x-permeable case (c,d), and the y-permeable case (e,f). The modes are computed using 20,000 PIV frames. The shading represents normalized levels for the streamwise velocity component. The solid and dashed black lines represent positive and negative contours for the wall-normal velocity component. The porous interface is at the top wall. . . . . . . 78 4.5 Comparison of the mean velocity proles used in the construction of the resolvent operator for the smooth wall (a), x-permeable material (b), and y-permeable ma- terial (c). Experimental proles are shown using the circular makers ( ). Synthetic proles generated using the eddy viscosity model of Reynolds and Tiederman (1967) are plotted as dashed lines ( ). Fitted proles computed using eddy viscosity prole determined from experimental data are plotted as solid lines ( ). . . . . . . . . . . 80 viii 4.6 Normalized gain ( ;p = ;s ) for spanwise-constant resolvent modes as a function of streamwise wavelength and mode speed. Panels (a,b) show predictions for the x- permeable case computed using the synthetic eddy viscosity prole (a) and the tted eddy viscosity prole (b). Panels (b,c) show predictions for the y-permeable case computed using the synthetic eddy viscosity prole (c) and the tted eddy viscosity prole (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.7 High-gain spanwise-constant resolvent modes identied from model predictions for the y-permeable substrate. (a) Structure corresponding to the highest-gain mode from the synthetic mean prole predictions in Fig. 4.6(c) with ( + x ;c + ) (2000; 9:5). (b) Structure corresponding to the highest-gain mode from the tted mean prole predictions in Fig. 4.6(d) with ( + x ;c + ) (530; 9). The shading represents nor- malized levels for the streamwise velocity component. Solid and dashed black lines respectively represent positive and negative contours for the wall-normal velocity component. The porous interface is at the top wall (y=h = 1). . . . . . . . . . . . . 81 5.1 Schematic showing experimental setup. The laser, optics and camera system are mounted to a precision traverse (not shown) that can be moved from just upstream of the substrate transition to the end of the plate. The dashed vertical lines indicate the center of the 6 measurement stations at (x=h) = (5; 3; 12; 28; 44; 53), where x = 0 is dened as the start of the porous substrate and h = 15:4 mm is substrate thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Schematic illustration of the single-pixel process. The PIV images are used to create a time-stack showing particle motion in the streamwise direction for each wall normal location, y i . A 2D Fourier transform is used to transform the particle trajectories from tx space into frequency-wavenumber (!k x ) space. A least-squares t identies the best estimate for mean particle velocity at each wall-normal location from the group velocity, U(y i ) =d!=dk x . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3 Comparison of the mean velocity prole obtained using DaVis ( ) and the single pixel routine ( ) for ow at station 5 over the smooth wall. . . . . . . . . . . . . . . 96 5.4 Photographs of the 3D-printed anisotropic porous material. The streamwise mean ow goes into the page for the image shown on the left. . . . . . . . . . . . . . . . . 98 5.5 The mean velocity prole acquired using the single pixel method is presented in decit form in panel (a) and in inner-normalized units in panel (b). Black squares ( ) show measurements made over the smooth wall at station 5 while white squares ( ) show measurements made over the porous substrate at the same location. . . . . 99 5.6 Flow development over the porous substrate ( ) and smooth wall ( ). The evolution of the following parameters is plotted as a function of streamwise location: (a) 99% boundary layer thickness normalized by substrate height, =h; (b) friction velocity normalized by freestream velocity, u =U e ; (c) the additive constant B (5.1) for the logarithmic region. The dashed vertical line indicates the transition from the smooth wall to the cutout for the porous substrate. . . . . . . . . . . . . . . . . . . . . . . . 101 ix 5.7 Friction Reynolds Number (a) and friction coecient (b) plotted as a function of the Reynolds number based on momentum thickness, Re . Smooth wall values are shown as black circles ( ) while porous substrate values are shown as white circles ( ). Dashed lines show empirical relations from Schlatter and Orl u (2010a): Re = 1:13Re 0:843 and C f = 0:024Re 0:25 . . . . . . . . . . . . . . . . . . . . . . . . 103 5.8 Inner-normalized mean velocity proles measured at station 1 upstream of the cutout (a) and at station 5 (x=h = 44) where the ow is fully developed over the porous substrate (b). In both plots, white squares ( ) show measurements from the porous substrate experiments while black squares ( ) show measurements made with the smooth wall insert in place. The solid lines () show mean proles ob- tained in DNS by Schlatter and Orl u (2010a) at Re 250 (a) and at Re 360 (b). The dashed line (- -) in panel (b) shows a shifted linear prole of the form U + = p K xx + +y + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.9 Mean turbulence statistics for smooth and porous cases for station 5 at x=h = 44. Mean velocity proles are shown in (a), proles of the root-mean-square streamwise and wall-normal velocity uctuations are shown in (b) and (c), respectively. The Reynolds shear stress prole is shown in (d). Statistics for the smooth wall and porous substrates are shown as black circles ( ) and white circles ( ) respectively. The black ( ) and white squares ( ) in (a) show the single-pixel mean prole estimates.106 5.10 Premultiplied spectra at station 5 for the streamwise velocity uctuations f + E + uu (a,c) and wall-normal velocity uctuations f + E + vv (b,d). Smooth wall results are shown in (a,b) and porous substrate results are shown in (c,d). The black circles ( ) in (a,c) label a frequency of f + = 0:01 at y + = 15, which corresponds roughly to structures associated with the energetic near-wall cycle. . . . . . . . . . . . . . . 109 x Abstract Manipulating turbulence is desirable to control mixing in turbulent combustion and industrial processes, to enhance heat transfer and mitigate aerodynamic noise, and to reduce skin friction in wall-bounded turbulent ows. For wall-bounded ows, varying the surface microstructure has proven to be an eective means of passive ow control, i.e., control that does not require active energy input. For example, streamwise aligned riblets successfully reduce skin friction drag, while porous metal foams enhance heat transfer around nned heat exchangers. This thesis reports ndings from laboratory boundary layer experiments over commercially available and custom- manufactured porous materials with a view to developing passive ow control techniques. Many aspects of the interaction between porous substrates and near-wall turbulence are not fully understood. For example, it is unclear how the near-wall turbulence is modied by substrate geometry and whether the logarithmic law for the mean ow remains appropriate further away. The rst experimental eort investigated the evolution of a turbulent boundary layer at friction Reynolds numberRe 1700 over commercially available foams in a range of pore sizes and thick- nesses. These experiments showed that the boundary layer adjusts from smooth wall conditions to the porous substrate over a streamwise distance of 40h where h is foam thickness. The fully developed mean velocity proles show a decit relative to the smooth wall case, though slip ve- locities roughly 30% of the free stream velocity are found closer to the interface. Mean turbulence statistics and time series analysis indicate that large scale motions typically found in smooth wall boundary layers are replaced by spanwise coherent rollers resembling Kelvin-Helmholtz vortices that result in a drag penalty. Skewness measurements suggest that these spanwise rollers modulate the amplitude of near-wall turbulence, similar to the eect of the so-called very large-scale motions xi in smooth wall boundary layers. Slip velocities are also integral to the drag reduction produced by riblets. Previous theoreti- cal eorts show that drag reduction over riblets arises from the diering slip lengths felt by the streamwise mean ow and the turbulent cross ows. This slip length theory has recently been extended to the case of anisotropic porous materials to show that streamwise-preferential porous materials also have the potential to reduce turbulent skin friction. Building on these predictions, idealized numerical simulations show that porous substrates with high streamwise permeability (K xx ) and low wall-normal (K yy ) and spanwise (K zz ) permeabilities can reduce drag by as much as 25% in turbulent ows. Currently, there are no commercially available materials that exhibit such anisotropy in permeability. The second set of experiments leveraged advances in additive manufacturing to manufacture a set of porous materials with varying pore sizes in orthogonal directions to create such anisotropic materials. Preliminary experiments in a small-scale channel conrmed that materials with highK xx and lowK yy andK zz were promising candidates for passive drag reduction. Materials with higher wall-normal permeability led to the emergence of energetic spanwise rollers that led to a signicant increase in drag. These initial experiments motivated experiments at higher Reynolds Number. A streamwise-preferential porous material with K xx =K yy 8 and K yy = K zz was 3D-printed on a large scale and tested in a at plate boundary layer at Reynolds Numbers ofRe 290410. Mean velocity proles measured at single pixel resolution in the wall-normal direction show good agreement with model predictions for slip velocity at the porous interface. The measured mean velocity proles also include a logarithmic section for which smooth-wall constants are appropriate. Friction estimates show a small increase in drag (< 5%) over the porous substrate. Spectral analysis suggests that this increase is again linked to the emergence of spanwise coherent rollers resembling Kelvin-Helmholtz vortices. These vortices are thought to emerge as the wall-normal permeability increases beyond a threshold value. This work constitutes the rst turbulent ow experiments over porous substrates with streamwise preferential geometries. Manufacturing limitations precluded xii materials with low enough wall-normal permeability from being printed to fully test previous drag- reduction predictions. However, the experiments show that it is possible to introduce relatively thick porous substrates in turbulent boundary layer ows without incurring an excessive drag penalty. Such materials could be used for passive heat transfer enhancement or ush mounted onto airfoils for separation control and noise mitigation. xiii Chapter 1 Introduction 1.1 Motivation Wall-bounded turbulence is ubiquitous in ows of scientic and engineering interest. In these ows, turbulence plays a major role in mixing mass, momentum, and scalars (Pope, 2001). The ability to control turbulence and modulate mixing is desirable with applications in drag reduction, noise mitigation, separation control and heat transfer enhancement. In particular, the ability to modify the ow without active energy input or sensing (passive ow control) is sought. Wall- bounded turbulence is sensitive to the wall condition such as roughness and porosity, and in practice, most operational and manufacturing deviations from smooth walls result in excess drag, noise and separation. Modifying the wall geometry has been successfully used for passive ow control, for example porous metal foams are used to enhance heat transfer around nned heat exchangers. On current generation aircraft engines, v-shaped cutouts called chevrons on nacelles and nozzles help aircraft manufacturers meet stringent noise regulations by mitigating aerodynamic noise (Zaman et al., 2011). Modifying the surface microstructure has proven to be one of the most promising ow control methods for drag reduction, with sharkskin inspired riblets reducing friction drag by up to 15% in numerical simulations (Garcia-Mayoral and Jimenez, 2011), laboratory experiments (Walsh and Lindemann, 1984; Choi et al., 1993; Bechert et al., 1997) and up to 6% at application scale (Walsh et al., 1989). The mechanism through which these streamwise aligned roughness elements reduce drag is well understood. The additional resistance to spanwise (cross-stream) ows inhibits turbulent mixing while preferentially allowing the streamwise mean ow to pass. This displaces 1 the quasi-streamwise vortices associated with the energetic and self-sustaining near-wall cycle away from the wall (Robinson, 1991; Jim enez and Pinelli, 1999a) and weakens them, in turn reducing turblent mixing near the riblet tips (Luchini et al., 1991; Choi et al., 1993; Jim enez et al., 2001). Since the apparent origin of the mean ow lies within the riblet grooves, the mean velocity at the tips is proportional to a slip length, i.e., U + s l + U . Throughout this document, the superscript + denotes normalization with respect to viscosity and friction velocity u = p w =. The perceived origin of the turbulent uctuations is not as deep (l + t l + U ) due to the higher resistance generated for cross- ows. For small riblets, the drag reduction is expected to be proportional to the dierence between the streamwise and transverse slip lengths, which can be computed using Stokes ow simulations. Jim enez (1994) expanded the slip length theory to any surface that produced slip lengths, and Hahn et al. (2002a) employed a slip length model to impose streamwise slip at the boundary of a turbulent channel ow experiment while maintaining impermeability at the wall. The resulting ow showed skin-friction reductions of up to 28%. Slip velocities are also present at the top of porous substrates. Like roughness, porosity and permeability of substrates also modies the near-wall turbulence. Porosity is dened as ratio of empty volume to total volume of a substrate, while permeability refers to the material's resistance to ow. A high permeability means that more ow will pass through the substrate for a given pressure gradient across it. Darcy's law (rP =K 1 U), is a semi-empirical equation that relates the pressure gradient (rP ) across a porous medium to the ow rates through it. Here, is the dynamic viscosity, U = (U;V;W ) is the velocity vector, and K is the permeability tensor, K = 2 6 6 6 6 4 K xx K xy K xz K yx K yy K yz K zx K zy K zz 3 7 7 7 7 5 : The diagonal components (K xx ,K yy ,K zz ) relate a pressure gradient to a ow in the same orienta- tion, while the o-diagonal terms relate a pressure gradient to velocities in directions perpendicular to the pressure gradient. If the diagonal components are equal, (K xx = K yy = K zz ) then the per- meability is said to be isotropic and can be expressed as a scalar. If the diagonal components of K vary, it is said to be anisotropic. 2 Examples of turbulent ows over permeable substrates abound in nature, for example ows over river beds, aquatic vegetation and canopies (Ghisalberti, 2009; White and Nepf, 2007). In such ows, the porous substrate acts as a resistive medium and creates a region of strong shear between obstructed and unobstructed ows. A shear layer instability triggers spanwise coherent, Kelvin- Helmholtz like rollers. Note that spanwise-coherent rollers also appear in numerical simulations and experiments over porous materials at smaller scales (Breugem et al., 2006; Rosti et al., 2015; Chandesris et al., 2013; Kuwata and Suga, 2017; Suga et al., 2018). The Direct Numerical Simulation (DNS) by Jimenez et al. (2001) employed an eective tran- spiration v 0 = p 0 , where is an admittance coecient and p 0 are pressure ucturations, which triggered the Kelvin-Helmholtz like instabilities and increased friction by over 40% even for small eective permeabilities. They showed that the emergence of spanwise rollers, accompanied by a large drag increase, is due to an inviscid shear layer instability. Incidentally, similar rollers were observed over riblets in the drag-increasing regime by Garcia-Mayoral and Jimenez (2011). Typi- cally, the DNS schemes do not resolve the ows within porous substrates, but rather rely on bulk models to model them. A notable exception is the Lattice-Boltzmann simulation by Kuwata and Suga (2017). Volume Averaged Navier-Stokes simulations by Breugem et al. (2006), Chandesris et al. (2013), and Rosti et al. (2015) consider turbulent channel ows with one or both bounding walls consisting of isotropic, permeable substrates. In these simulations, the wall permeability trig- gered the spanwise rollers and drag increase was observed. These observations are consistent with experiments over isotropic foams Manes et al. (2011) and meshes with wall-normal preferential permeability (Suga et al., 2018). More recently, Abderrahaman-Elena and Garc a-Mayoral (2017) expanded the slip length mod- els for drag reduction over riblets, to porous substrates, rigorously showing l + U / p K xx + and l + t / p K zz + . This implies that a potential drag reducing surface needs higher streamwise than spanwise permeabilities. The DNS of G omez-de-Segura and Garc a-Mayoral (2019) found that such substrates reduced friction drag in turbulent channel ows by up to 25% for such materials. The parametric sweep of p K xx + ; p K yy + ; and p K zz + by G omez-de-Segura and Garc a-Mayoral (2019) found that for small values of p K xx + p K zz + , the drag reduction increases with increas- ing anisotropy, i.e., xy = K xx =K yy and p K xx + p K zz + . They also found that the maximum 3 drag reduction occurs at p K yy + 0:4. For higher wall-normal permeabilities, the emergence of Kelvin-Helmholtz like rollers leads to a drag increase. In this thesis, we address shortcomings in our understanding of how substrate geometry interacts with near-wall turbulence. For example, it is unclear how the pore size aects the energetic near- wall cycle, and whether a logarithmic layer in the mean ow prole exists. Previous DNS studies have relied on bulk parameterizations to model the porous substrate, and do not resolve the pore geometry. Recent simulations such as the Lattice-Boltzmann simulation of Suga et al. (2018) resolve the pore geometry. The DNS studies over porous media are also limited in Reynolds Number and have so far only considered turbulent channel ows. Previous experimental eorts have been limited by available substrates, such as meshes and perforated surfaces, which give wall-normal preferential permeabilities, or quasi-isotropic foams and beds of packed spheres. No commercially available porous media with streamwise preferential geometries exist, and thus no experiments of turbulent ows over such substrates have been made. Additionally, previous experiments focused on thick substrates where substrate thickness does not play an important role in dictating ow properties, and measurements were made far enough downstream to avoid development eects. The work detailed in this thesis is focused on thin substrates, both isotropic and with stream- wise preferential anisotropy, and how they aect the ow. Thin substrates could be retrotted to existing structures such as ship hulls or aircraft fuselages without substantially changing the geometry, for example, porous airfoils have shown improved lift-curve characteristics at moderate Reynolds Numbers (Hanna and Spedding, 2019) and reduced noise (Khorrami and Choudhari, 2003) at larger scales. Despite the applications, the interactions between near-wall turbulence and substrate geometry are not fully understood. For example it remains to be seen how pore size and geometry aect the energetic and self-sustaining near-wall cycle; whether the mean ow prole can be described by similar logarithmic proles as in canonical, smooth wall boundary layers; and what the large-scale structures are like and how they interact with small scales. 1.2 Contribution and outline The following chapters detail a set of experiments that are aimed at answering open questions about the interaction of turbulent boundary layers and the substrate geometry. In §2 we describe an 4 experimental eort that took advantage of the economical availability of highly porous, open cell, reticulated foams that come in a range of thicknesses (h 6 25mm), pore sizes (s 0:2 3mm) and sucient sizes (60x90cm) to study the development of turbulent boundary layers over thin, quasi-isotropic substrates. In contrast to similar materials pursued by Manes et al. (2011), our substrates were thinner and we considered locations near the substrate interface as well as down- stream. Using a streamwise and wall-normal traversing Laser Doppler Velocimeter (LDV), we investigated the adaptation of the turbulent boundary layer when it experiences a step change in substrate from solid to permeable. The boundary layer reaches a new fully developed state by a streamwise distance of x=h 40 downstream of the substrate transition. A mean velocity decit relative to the smooth wall case is found, suggesting a signicant drag increase. We found evidence of structures with length scales characteristic of Kelvin-Helmholtz rollers which replaced the canonical large-scale motions found in smooth wall boundary layers. Through skewness mea- surements, we found support for amplitude modulation by these large structures of small scale uctuation amplitudes near the surface. The majority of this chapter is published as Efstathiou and Luhar (2018). To address the lack of experiments over streamwise preferential, porous materials, we designed anisotropic, porous materials optimized for 3D printing. A family of lattices with variable pore sizes in orthogonal directions was used to design materials with tunable permeability. By rotating the matrix about an axis perpendicular to the incoming ow, it was also possible to modify the o- diagonal terms of the permeability tensor. In addition to manufacturing such materials, a method to characterize their permeability matrix was developed. Design and manufacturing considerations for additive manufacturing and the lessons learned through trial and error are described in §3. Two such materials were then tested in a small-scale turbulent channel ow with a smooth wall on one boundary and porous on the other. Using time-resolved, particle image velocimetry (PIV), we found evidence of Kelvin-Helmholtz like rollers in the spatial proper orthogonal decomposition (POD) modes for the wall-normal permeable (y-permeable) material, accompanied by a drag increase of 70%. Mean ow statistics show the bulk of the mean ow was displaced towards the smooth side. The near-wall peak attributed to the near-wall cycle in smooth walled channel ows was not present. In contrast, over a streamwise permeable substrate, we found this near wall peak, 5 indicating that the ow structures were largely unchanged. For this material, the bulk ow was biased only slightly towards the smooth wall side, and the drag increase was 5%. A comparison to low-order model predictions made by my colleague Andrew Chavarin is included in §4, which has been submitted for publication in Chavarin et al. (2020). In §5, I describe a boundary layer experiment similar to the one described in §2, but for this experimental campaign, I manufactured streamwise preferential lattices on a large scale and ush- mounted them into the at plate boundary layer. A high-speed camera was used to acquire temporally resolved, particle image velocimetry (PIV) images at several streamwise locations over the porous substrate. Traditional PIV was used to acquire mean turbulence statistics, and a novel method to extract mean velocity proles with single-pixel resolution in the wall-normal directions was developed to estimate slip velocities near the interface. The velocity prole near the substrate agreed well with the theory proposed by Abderrahaman-Elena and Garc a-Mayoral (2017), U s p K xx + . Further out into the boundary layer, the mean velocity prole showed a logarithmic layer that appears to be well described by smooth wall constants. Although we found evidence for Kelvin-Helmholtz like structures in premultiplied energy spectra, the drag increase was found to be only 52%. While we were not able to create a drag-reducing material (limited to p K yy + > 0:4 by manufacturing limitations), the results agree well with the numerical simulations by G omez-de-Segura and Garc a-Mayoral (2019), in particular their case which best represents our conditions. These experiments mark the rst experimental datasets over streamwise preferential, porous materials which could be used as multifunctional passive ow control devices. 6 Chapter 2 Mean turbulence statistics in boundary layers over high-porosity foams 2.1 Comment and Summary This chapter has been published in the Journal of Fluid Mechanics as Efstathiou and Luhar (2018). Over the past two decades, larger research facilities such as the Princeton Superpipe (Za- garola et al., 1997), and wind tunnel facilities such as the KTH Minimum Turbulence Level Wind Tunnel have permitted controlled experiments at higher Reynolds numbers than were previously possible. They have yielded new insights into the statistics, structures, and dynamics of canonical smooth-wall turbulent boundary layers at higher Reynolds Numbers (Re). Discoveries include the existence of much larger structures than previously thought (Hutchins and Marusic, 2007) and evidence that they modulate the amplitude of smaller scale turbulence (Marusic et al., 2010). How the presence of a porous boundary aects these has not previously been studied. Numerical sim- ulations over porous substrates are limited by the same computational constraints as smooth wall simulations, with the added caveat that the porous substrates are typically modelled rather than resolved at the pore scale. Experimental eorts studying porous substrates adjacent to turbulent boundary layers are lim- ited by the available choice in substrate material. On a small scale, it may be possible to align meshes, pack spheres, set up rods or bore out holes. To study boundary layers on a larger scale requires a large amount of substrate. Open cell, reticulated foams are readily available in a range of pore sizes from s 0:2 2mm, thicknesses (1-25mm) and are economical to purchase in large quantities. This chapter reports turbulent boundary layer measurements made over a set of such 7 foams with varying pore size and thickness, but constant porosity ( 0:97). The foams were ush-mounted into a cutout on a at plate. A Laser Doppler Velocimeter (LDV) was used to measure mean streamwise velocity and turbulence intensity immediately upstream of the porous section, and at multiple measurement stations along the porous substrate. The friction Reynolds number upstream of the porous section was Re 1690. For all but the thickest foam tested, the internal boundary layer was fully developed by < 10 downstream from the porous transition, where is the boundary layer thickness. Fully developed mean velocity proles showed the presence of a substantial slip velocity at the porous interface (> 30% of the free stream velocity) and a mean velocity decit relative to the canonical smooth- wall prole further from the wall. While the magnitude of the mean velocity decit increased with average pore size, the slip velocity remained approximately constant. Fits to the mean velocity prole suggest that the logarithmic region is shifted relative to a smooth wall, and that this shift increases with pore size until it becomes comparable to substrate thickness h. For all foams, the turbulence intensity was found to be elevated further into the boundary layer to y= 0:2. An outer peak in intensity was also evident for the largest pore sizes. Velocity spectra indicate that this outer peak is associated with large-scale structures resembling Kelvin-Helmholtz vortices that have streamwise length scale 2 4. Skewness proles suggest that these large-scale structures may have an amplitude-modulating eect on the interfacial turbulence. 2.2 Introduction Turbulent ows of scientic and engineering interest are often bounded by walls that are not smooth, solid, or uniform. Manufacturing techniques, operational requirements and natural evo- lution often lead to non-uniform, rough, and porous boundaries. Examples include ows over heat exchangers, forest canopies, bird feathers and river beds (e.g., Finnigan, 2000; Jim enez, 2004; Ghisalberti, 2009; Favier et al., 2009; Manes et al., 2011; Jaworski and Peake, 2013; Chandesris et al., 2013; Kim et al., 2016). Porous boundaries also enable active ow control through suction and blowing for use in drag reduction and delaying transition from laminar to turbulent ow (e.g., Parikh, 2011). 8 Despite the potential applications, relatively little is known about the relationship between turbulent ows and porous substrates. For example, it is unclear how established features of smooth-wall ows, such as the self-sustaining near-wall cycle, the logarithmic region in the mean prole, the larger-scale structures found further from the wall, and the interaction between the inner and outer regions of the ow (e.g. the amplitude modulation phenomenon; Marusic et al., 2010) are modied over porous surfaces. As a result, there are few models that can predict how a porous substrate of known geometry will in uence the mean ow eld and turbulent statistics. A key limitation is that few experimental and numerical datasets exist for turbulent ows over porous media. Further, most previous experimental datasets (motivated primarily by ows over packed sediment beds or canopies) include limited near-wall measurements, while previous numer- ical simulations have been restricted to relatively low Reynolds numbers. In addition, almost all previous studies on turbulent ow over porous substrates have employed relatively thick media, such that ow penetration into the substrate depends only on pore size or permeability (see e.g., Manes et al., 2011). However, in many natural and engineered systems, the porous substrate can be of nite thickness and bounded by a solid boundary. Examples include feathers or fur in natural locomotion (Itoh et al., 2006; Jaworski and Peake, 2013), and heat exchangers employing metal foams (Mahjoob and Vafai, 2008). In such systems, substrate thickness can also have an impor- tant in uence on ow development and the eventual equilibrium state. At the limit where porous medium thickness becomes comparable to pore size, the substrate can essentially be considered a rough wall. The transition from this rough-wall limit to more typical porous medium behaviour is not fully understood. The experimental study described in this paper seeks to address some of the limitations described above. 2.2.1 Previous studies Previous numerical eorts investigating the eect of porous boundaries include the early direct numerical simulations (DNS) performed by Jim enez et al. (2001), which employed an eective admittance coecient linking the wall-pressure and wall-normal velocity to model the porous wall. More recent DNS eorts have either employed the volume-averaged Navier-Stokes equations, where 9 the porous substrate is modeled as a resistive medium (Breugem et al., 2006), or explicitly modeled the porous medium as an array of cubes (Chandesris et al., 2013). In particular, the results of Breugem et al. (2006) showed important deviations from turbulent channel ow over smooth walls. The mean velocity was signicantly reduced across much of the channel for the cases with high porosity ( = 0:8 and 0:95) and this was accompanied by a skin friction coecient increase of up to 30%. The presence of the porous substrate also led to substantial changes in ow structure. Large spanwise rollers with streamwise length scale comparable to the channel height were observed at the porous interface. Given the presence of an in ection point in the mean velocity prole near the porous interface, Breugem et al. (2006) attributed the emergence of these large-scale structures to a Kelvin-Helmholtz type of instability mechanism (see also Jim enez et al., 2001; Chandesris et al., 2013). In addition, the near-wall cycle comprising streaks and streamwise vortices, a staple of smooth wall turbulent boundary layers, was substantially weakened over the porous substrate. This weakening was linked to a reduction in the so-called wall-blocking eect and enhanced turbulent transport across the interface. Breugem et al. (2006) also found that the root mean square (rms) of the wall-normal velocity uctuations did not exhibit outer layer similarity and suggested two potential causes for this (i) more vigorous sweeps and ejections due to the absence of an impermeable wall and (ii) insucient scale separation between the channel half-height and the penetration distance into the porous substrate. Chandesris et al. (2013) noted broadly similar trends in their DNS, which also considered thermal transport. Rosti et al. (2015) performed DNS studies of turbulent channel ow atRe = 180 using a VANS formulation that allowed for the porosity and permeability to be decoupled. These simulations showed that even relatively low wall permeabilities led to substantial modication of the turbulent ow in the open channel. Further, despite a substantial variation in the porosities tested ( = 0:3 0:9), the ow was found to be much more sensitive to permeability. Motlagh and Taghizadeh (2016) performed a VANS Large Eddy Simulation (LES) of channel ow over porous substrates with = 0; 0:8 and 0:95 at bulk Reynolds number 5500. Based on proper orthogonal decomposition (POD) of the ow eld, they showed that large-scale features prevalent in smooth-walled ows were still present but beginning to break down for the lower porosity case. Consistent with the 10 results of Breugem et al. (2006), for the higher porosity case, these structures were replaced by spanwise-elongated counter-rotating vortices. Recently, Kuwata and Suga (2016) and Kuwata and Suga (2017) have performed Lattice- Boltzman simulations over rough walls, staggered arrays of cubes, as well as anisotropic porous media. In particular, Kuwata and Suga (2017) considered a model system in which the streamwise, spanwise, and wall-normal permeabilities could be altered individually. These simulations showed that the streamwise permeability is instrumental in preventing the high- and low-speed streaks as- sociated with the near-wall cycle. The simulations also indicated that, unlike the streamwise and spanwise permeabilities, the wall-normal permeability does not signicantly enhance turbulence intensities. Similar to turbulent ows over smooth and rough walls, previous studies suggest that a loga- rithmic region in the mean proleU(y) can also be expected in turbulent ows over porous media. This logarithmic region is often parametrized as: U + = 1 ln y +y d k 0 = 1 ln (y + +y + d ) +B U + ; (2.1) where andB are the von Karman and additive constants,y is the wall-normal distance from the porous interface, y d is the shift of the logarithmic layer (or zero-plane displacement height), k 0 is the equivalent roughness height, and U + is the roughness function (Jim enez, 2004). A superscript + indicates quantities normalized by the friction velocityu and the kinematic viscosity. Fits to the mean velocity proles obtained in DNS by Breugem et al. (2006) suggest that the von Karman constant decreases from 0:4 for the smooth wall case to = 0:23 for the most porous substrate. However, further tests were recommended at higher Reynolds number to conrm this eect. Experimental eorts in this realm have considered ows over beds of packed spheres, perforated sheets, foams, as well as seal fur (e.g., Ru and Gelhar, 1972; Zagni and Smith, 1976; Kong and Schetz, 1982; Itoh et al., 2006; Suga et al., 2010; Manes et al., 2011; Kim et al., 2016). Interestingly, the seal fur experiments show a reduction in skin friction though the exact mechanism behind this remains to be understood (Itoh et al., 2006). Kong and Schetz (1982) investigated the eect of small scale roughness over smooth, rough, and porous surfaces on turbulent boundary layers over blu bodies. The porous boundaries consisted of perforated sheets and mesh screens. These 11 experiments showed that the turbulent Reynolds stresses increased near the interface, as did the skin friction. However, it is dicult to separate roughness eects from permeability eects for these experiments. The mean velocity prole was shifted by U + 3 4, which was similar to the shift obtained over an impermeable rough wall of similar geometry. Suga et al. (2010) studied laminar and turbulent channel ow over foamed ceramics with porosity 0:8 and varying pore sizes via Particle Image Velocimetry (PIV). These experiments were carried out at relatively low Reynolds numbers (bulk Reynolds number Re b 10; 200). The measurements indicated that the transition to turbulence occurs at lower Reynolds number over the porous media. Further, turbulence intensities were generally enhanced over the porous medium, and the displacement and roughness heights in the modied logarithmic law were found to increase with increasing pore size and permeability. Detert et al. (2010) also used PIV to study the ow over packed spheres and gravel from the river Rhine in an open channel ow with friction Reynolds numbers Re = 1:88 14:7 10 3 . At the relatively low porosities tested ( = 0:26 0:33), the ow exhibited many of the large-scale structures observed in turbulent boundary layers over smooth walls, including hairpin vortex packages. The observed ow patterns were also found to be relatively insensitive to Reynolds number. Manes et al. (2011) made open channel turbulent ow measurements over very porous (> 0:96) polyurethane foam mattresses with 10-60 pores per inch (ppi) at Reynolds number Re > 2000 using an LDV. In contrast to Breugem et al. (2006), the results obtained by Manes et al. (2011) supported the outer layer similarity hypothesis. This suggests that the lack of outer-layer similarity in the DNS carried out by Breugem et al. (2006) resulted from insucient scale separation between the inner and outer layers of the ow, which is analogous to the breakdown of outer-layer similarity in shallow boundary layers over rough walls (Jim enez, 2004). Further, Manes et al. (2011) reported a reduction in the streamwise uctuation intensity near the wall and an increase in the intensity of wall-normal uctuations, which they attributed to reduced wall blocking. For the logarithmic region in the mean prole, a tting procedure similar to the one employed by Breugem et al. (2006) and Suga et al. (2010) yielded 0:3 over the porous media and an equivalent roughness height k 0 that generally increased with increasing pore size. However, the mean proles did not exhibit a clear logarithmic region over the 10 ppi foam. Since the ow is expected to penetrate further 12 into the foam as pore size increases, Manes et al. (2011) suggested that a lack of scale separation between the penetration distance and the water depth may have in uenced the results. Finally, note that there are substantial similarities in turbulent ows over porous media and vegetation or urban canopies (e.g., Finnigan, 2000; Poggi et al., 2004; White and Nepf, 2007). For example, large-scale structures resembling Kelvin-Helmholtz vortices play an essential role in dictating mass and momentum transport at the interface (see e.g., Finnigan, 2000) and a shifted log law of the form shown in equation (2.1) provides a reasonable t for the velocity prole above the canopy. In a comparative study of obstructed shear ows, Ghisalberti (2009) suggested that an in ection point exists in the mean prole if the distance to which the ow penetrates into the porous medium or canopy is much smaller than the height of the medium. This penetration distance is expected to scale as p k for porous substrates (e.g., Battiato, 2012), where k is the permeability, and (C D a) 1 for canopies where, C D is a drag coecient and a is the frontal area per unit volume of the canopy. For densely packed or tall substrates where an in ection point is observed, the interfacial dynamics are dominated by structures resembling Kelvin-Helmholtz vortices. In such cases, the slip velocity at the interface depends primarily on the friction velocity, with little dependence on substrate geometry. Further, the interfacial turbulence also tends to be more isotropic, such that intensity of the wall-normal velocity uctuations is comparable to the intensity of the streamwise uctuations. 2.2.2 Contribution and outline The present study builds on the experiments pursued by Manes et al. (2011) to provide further insight into the near-wall ow physics over high-porosity surfaces. An LDV was used to measure streamwise velocity proles over commercially-available foams with systematically varying pore sizes and thicknesses at moderately high Reynolds number. For reference, the friction Reynolds number over the smooth wall upstream of the porous section was Re = u = 1690, where is the 99% boundary layer thickness. The velocity proles include measurements very close to the porous interface (corresponding to 2-3 viscous units over the smooth wall). Unfortunately, wall-normal velocities were not measured due to instrumentation limitations. 13 Morphologically, the foams tested in this study are similar to those considered by Manes et al. (2011). However, one important distinction is the thickness of the foam layer, h. In an eort to isolate the eects of pore size, s, and permeability, k, Manes et al. (2011) considered very thick porous media withh p k andhs. The scale separation is more limited in the present study, with h=s 4:3 44 and h= p k 36 160. As a result, porous layer thickness does in uence the ow for foams with the largest pore sizes and permeabilities. In other words, the present study provides insight into the transition between conditions in which the shear ow penetrates across the entire porous domain and conditions in which the shear layer only reaches a small distance into the porous medium. In a sense, this is analogous to the transition between sparse and dense canopy behaviour observed in vegetated shear ows (Luhar et al., 2008; Ghisalberti, 2009). In recent years, the amplitude modulation phenomenon observed in smooth- and rough-walled turbulent ows has led to a promising class of predictive models (e.g., Mathis et al., 2009; Marusic et al., 2010; Mathis et al., 2013; Pathikonda and Christensen, 2017). Specically, it has been observed that the so-called very-large-scale motions (VLSMs) prevalent in the logarithmic region of the ow at high Reynolds number (Smits et al., 2011) have a modulating in uence on the intensity of the near-wall turbulence. Further, it has been shown that there is an intrinsic link between this phenomenon and the skewness of the streamwise velocity uctuations (Mathis et al., 2011; Duvvuri and McKeon, 2015). Therefore, by considering the skewness of the streamwise velocity, we also evaluate whether such interactions between the inner and outer region persist in turbulent ows over porous media that may be dominated by a dierent class of large-scale structure, i.e. Kelvin-Helmholtz vortices. The remainder of this paper is structured as follows: §5.4 describes the experiments, providing details on the ow facility, porous substrates, and diagnostic techniques; §5.5.1 shows results on ow development over the porous foams; §2.4.2 illustrates the eect of pore size on mean turbulence statistics; §2.4.3 explores the eect of substrate thickness; §2.4.4 presents velocity spectra over all the dierent foams tested; §2.5 tests whether a shifted logarithmic region exists over the porous surfaces, evaluates the relative eect of pore size and substrate thickness, and also considers how porous substrates may aect the amplitude modulation phenomenon. Conclusions are presented in §5.6. 14 21.5h Water depth: 480 mm Reference measurement 2 Garolite backing ∞ trip U 1 3 Porous substrate 4 h=12.7mm 2400mm H=300mm 890mm Glass sidewall LDV Traverse Foam Insert Spanwise measurement location LDV standoff distance: 500mm Figure 2.1: Schematics showing at plate apparatus in the wall normal-spanwise plane (y z, top) and the streamwise-wall normal plane (x y, bottom). Measurement positions marked 1-4 were located at x=h = 11; 21; 42; 53, where h = 12:7 mm is the baseline substrate thickness, and x = 0 is dened as the smooth-porous wall transition. The smooth wall reference measurements were made at a location x=h =21:5. Note that the wall-normal coordinate is y. 2.3 Experimental methods 2.3.1 Flow facility and at plate apparatus All experiments were conducted in the USC water channel, a free-surface, recirculating facility with glass along both sidewalls and at the bottom to allow for unrestricted optical access. The water channel has a test section of length 762 cm, width 89 cm, and height 61 cm, and is capable of generating free-stream velocities up to 70 cm/s with background turbulence levels < 1% at a water depth of 48 cm. The temperature for all experiments was 230:5 C for which the kinematic viscosity is = 0:93 10 2 cm 2 /s. 15 Figure 2.1 provides a schematic of the experimental setup in the wall normal-spanwise (top) and streamwise-wall normal (bottom) planes. A 240 cm long at plate was suspended from precision rails at a height H = 30 cm above the test section bottom. To avoid free-surface eects, mea- surements were made below the at plate. The nominal free-stream velocity was set at U e = 58 cm/s for all the experiments. The connement between the at plate and bottom of the channel naturally led to a slightly favorable pressure gradient and slight free stream velocity increase along the plate ( 3%), however the non-dimensional acceleration parameter, = U 2 e dUe dx was on the order of 10 7 suggesting any pressure gradient eects are likely to be mild (Patel, 1965; De Graa and Eaton, 2000; Schultz and Flack, 2007). A cutout of length 89 cm and width 60 cm was located 130 cm downstream of the leading edge. Smooth and porous surfaces were substituted into the cutout, ush with the smooth plate around it. The porous test specimens, described in further detail below, were bonded to a solid Garolite TM sheet to provide a rigid structure and prevent bleed through. The setup was designed to accommodate porous substrates of thicknesses h = 6:35; 12:7 and 25:4 mm. Care was taken to minimize gaps and ensure a smooth transition from solid to porous substrate. Velocity proles were measured over the smooth section upstream of the cutout and at four additional locations over the porous walls. The ow was tripped by a wire of 0.5 mm diameter located 15 cm downstream of the leading edge. 2.3.2 Velocity measurement Measurements of streamwise velocity, u, were made at the channel centerline using a Laser Doppler Velocimeter (LDV, MSE Inc.) with a 50 cm stando distance and a measurement volume of 300m by 150m by 1000 m (x by y by z). The LDV was mounted on a precision traverse capable of 16m resolution. Polyamide seeding particles (PSP) with an average size of 5m were used to seed the ow. While the large stando distance enabled measurements at the channel centerline, it also limited data rates to 50 Hz in the free-stream and less than 1 Hz at the stations closest to the wall. A minimum of 800 data points in the regions with lowest velocity and 4000- 25000 data points in regions of higher velocity were collected to ensure fully converged statistics. In all cases, a preliminary coarse velocity prole was measured to determine the boundary layer 16 thickness and the approximate location of the wall. This preliminary prole was used to generate a ner logarithmically-spaced vertical grid for the actual measurements. The nominal smooth- wall location (y 0 = 0) was identied as the position where the data rate dropped to zero. For all the proles, the vertical grid resolution was reduced to 30 m in the near-wall region. Given the 150m measurement volume and the 30m vertical resolution, the nominal estimate for wall- normal location suers from an uncertainty of 75m. LDV measurements suer from two signicant distortions: velocity gradients across the mea- surement volume and velocity biasing (Durst et al., 1976; DeGraa and Eaton, 2001). The former is signicant in regions where large velocity gradients are present across the measurement volume, as is the case near solid walls. The latter occurs because, in turbulent ows, more high velocity par- ticles move through the measurement volume in a given period compared to low-velocity particles (assuming uniform seeding density). To correct for this bias, an inverse velocity weighting factor was used to correct mean statistics (McLaughlin and Tiederman, 1973). The mean streamwise velocity was estimated using the following relationship: U = N P i=1 b i u i N P i=1 b i ; (2.2) whereu i is an individual velocity sample, N is the total number of samples, and b i = 1=ju i j is the weighting factor. Similarly, the weighted streamwise turbulence intensity was estimated as: u 2 = N P i=1 b i [u i U] 2 N P i=1 b i : (2.3) Alternative methods of correcting for velocity biasing are discussed and evaluated in Herrin and Dutton (1993). These methods include using the inter-arrival time (b i =t i t i1 ) as the weighting factor, or the sample-and-hold technique (b i = t i+1 t i ). Figure 2.2 shows how these dierent correction techniques aect a representative mean velocity (U) and streamwise turbulent intensity (u 2 ) prole. Inverse velocity weighting leads to the largest correction relative to the raw data in the near-wall region where the sampling is more intermittent (n.b. this is also consistent with 17 10 -3 10 -2 10 -1 10 0 y/δ 0.2 0.4 0.6 0.8 1 U/Ue 0 0.005 0.01 u 2 /U 2 e unweighted inverse velocity interarrival time sample-and-hold Figure 2.2: Mean velocityU and streamwise turbulence intensityu 2 proles corrected by inverse velocity magnitude and inter-arrival time weighting. The wall-normal coordinate is normalized using the local 99% boundary layer thickness, . 10 -3 10 -2 10 -1 10 0 y/δ 0 0.1 0.2 0.3 0.4 0.5 0.6 σ u Figure 2.3: Prole showing standard error of mean velocity (in %) over upstream smooth wall section. The discontinuity in u between y= = 0:01 and 0:02 corresponds to the location where the measurement duration for each point was reduced from 45 minutes to 10 minutes. previous studies). However, the overall trends remain very similar in all cases and the correction is minimal in the outer region of the ow (see e.g. y= 0:1 in gure 2.2). While the corrections accounting for velocity biasing do introduce uncertainty, this uncertainty is likely to be correlated across all cases. Therefore, for comparison across cases, the uncertainty in mean velocity is taken to be the larger of the instrument uncertainty (0:1%, MSE Inc.) and the standard error, given by u = u = p N, where u is the standard deviation of the measurements. In all cases, the standard error was the larger contributor to uncertainty in the near-wall region due to the limited number of samples acquired. As shown in gure 2.3, the standard error was typically u 1% across the entire boundary layer. 18 The friction velocity, u , over the smooth-wall section upstream of the porous cutout was estimated by tting the following relationship to the near-wall mean velocity measurements: U(y) = (u 2 =)(y 0 +y 0 ), in which y 0 represents an oset from the nominal wall location where the data rate falls to zero,y 0 = 0. The smooth-wall velocity proles reported below correct for this oset. In other words, the true wall-normal distance is assumed to be y = y 0 +y 0 , such that the near-wall velocity prole is consistent with the theoretical relation U + = y + (see gure 2.5). No such correction was made for the porous substrate. For reference, the friction velocity upstream of the porous section was estimated to be u = 2:3 0:05 cm/s. The uncertainty is estimated by tting the relationship in the viscous sublayer to dierent ranges of 5-10 points near the wall. The oset was y 0 = 40m, which translates into approximately one viscous length scale. Note that the above estimate for the friction velocity is also consistent with estimates obtained from ts to the logarithmic region in the mean velocity prole using = 0:39 (Marusic et al., 2013). The estimates for u derived above also agreed with the third method utilizing the velocity gradient. See gure 2.11 and related discussion in §2.5.2 for further detail. 2.3.3 Reynolds number ranges and spatio-temporal resolution The 99% boundary layer thickness, estimated via interpolation, was = 6:83 0:07 cm for the smooth-wall prole, and so the friction Reynolds number was + =u = = 1690 70 upstream of the porous section. Over the porous sections, the boundary layer thickness increased; the maximum measured value was = 11:2 0:1 cm. As a result, the Reynolds number based on the nominal free-stream velocity ranged from Re = U e = = 42600 69900. The estimated friction velocity u = 2:3 0:05 cm/s translates into a viscous length-scale =u 40m and viscous time-scale =u 2 1:7 ms. Thus, the 16 m precision of the traverse provides adequate vertical resolution for proling purposes. However, the LDV measurement volume (150m in y) extends across 3 viscous units in the wall-normal direction, which means that the near-wall measurements reported below suer from distortion due to velocity gradients. In the near-wall region, the LDV sampling frequency was approximately 0.5 Hz, which corresponds to an average sampling time of 1200 viscous units. In the outer region of the ow (y=& 0:05), the sampling frequencies were as 19 high as 50 Hz, which translates into an average sampling time of 12 viscous units. In other words, time-resolved velocity measurements are only expected in the outer region of the ow. 2.3.4 Porous Substrates Boundary layer measurements were made adjacent to four dierent types of open-cell reticulated polyurethane foams. Per the manufacturer, the porosity of all the foams was 0:97. This was conrmed to within 0:5% via measurements that involved submerging the foams in water to measure solid volume displacements. The nominal pore sizes corresponded to 10, 20, 60, and 100 pores per inch (ppi, see gure 2.4). Pore size distributions for each foam were estimated from photographs of thin foam sheets via image analysis routines in Matlab (Mathworks Inc.). The measured average pore sizes ranged from s = 2:1 0:3 mm for the 10 ppi foam to s = 0:29 0:02 mm for the 100 ppi foam (see Table 2.1). These measurements are generally within20% of the nominal pore sizes. Pore size measurements for the 10 and 20 ppi foam were also made using precision calipers. These caliper-based measurements were consistent with the imaging-based estimates to within uncertainty (s = 2:2 0:1mm for 10 ppi and s = 1:7 0:1 mm for 20 ppi, where uncertainties correspond to standard error). Note that all of the pore sizes discussed above and listed in Table. 2.1 correspond to the exposed streamwise-spanwise plane of the foam. Caliper-based measurements suggest that the pore structure may be anisotropic. For the 10 ppi foam, average pore sizes were approximately 14% larger than the listed values in the spanwise-wall normal plane of the foam (2:5 0:1 mm) and 21% larger in the streamwise-wall normal plane (2:7 0:1 mm). Similarly, for the 20 ppi foam, average pore sizes were approximately 10% larger in the spanwise-wall normal plane and 23% larger in the streamwise-wall normal plane. Another important length scale arises from the permeability, k, of the porous medium. Speci- cally, p k determines the distance to which the shear penetrates into the porous medium (Battiato, 2012). Permeabilities were estimated from pressure drop experiments using Darcy's law. These estimates ranged from k = 6:6 0:6 10 9 m 2 for the 100 ppi foam to k = 46 1 10 9 for the 10 ppi foam. 20 Figure 2.4: Photographs showing thin sheets of the 10, 20, 60, and 100 ppi foams (from left to right). Each image represents a 2 cm 2 cm cross section. Foam k(10 9 m 2 ) s (mm) Re k s + h= p k h=s 10 ppi 46 1 0:976 0:003 2:1 0:3 5:3 52 59 6 (160) (3.9) 20 ppi 73 8.5 20 ppi thin 30 2 0:972 0:003 1:5 0:2 4:3 37 37 4.3 20 ppi thick 147 17 60 ppi 7:9 0:6 0:965 0:005 0:40 0:03 2:2 10 143 32 (6) (0.5) 100 ppi 6:6 0:6 0:967 0:005 0:29 0:02 2:0 7 156 44 Table 2.1: Permeability (k), porosity (), average pore sizes (s), and related dimensionless parameters for tested foams. Permeability and pore size values from Manes et al. (2011) are noted in parenthesis for the 10 and 60 ppi foams. Note that s + = su = and Re k = p ku = are dened using the friction velocity upstream of the porous section. Porosity () was estimated from solid volume displacement in water and permeability was estimated from pressure drop experiments. Based on the friction velocity measured upstream of the plate, the inner-normalized pore sizes range from s + 7 for the 100 ppi foam to s + 52 for the 10 ppi foam. Similarly, the Reynolds number based on permeability varies between Re k = p ku = 2:0 for the 100 ppi foam to Re k 5:3 for the 10 ppi foam. The baseline thickness tested for all foams was h = 12:7 mm. For the 20 ppi foam, two additional thicknesses,h = 6:35 mm andh = 25:4 mm, were also considered. This means that the ratio of foam thickness to average pore size ranged between h=s = 4:3 for the thin 20 ppi foam to h=s = 44 for the 100 ppi foam. Finally, keep in mind that despite having the same nominal pore sizes, the 10 and 60 ppi foams tested here are not identical to those tested by Manes et al. (2011). For example, the 10 ppi foam tested by Manes et al. (2011) had a pore size (s = 3:9 mm) approximately twice that of the 10ppi foam used here, and a permeability (k = 160 10 9 m 2 ) almost four times higher. Table 2.1 lists physical properties for all the foams tested in the experiments, along with related dimensionless parameters. 21 2.4 Results 2.4.1 Boundary layer development First, we consider boundary layer development over the porous foams. The results presented below correspond to the 20 ppi foam of thickness h = 12:7mm. Similar trends were observed for all the porous substrates. As illustrated schematically in gure 5.1, the transition from the smooth wall to the porous substrate leads to the development of an internal layer, which starts at the transition point and grows until it spans the entire boundary layer thickness. When this internal layer reaches the edge of the boundary layer, a new equilibrium boundary layer prole is established. This new prole re ects the eects of the porous substrate. Internal layers have been studied extensively in the context of smooth to rough wall transitions in boundary layers (e.g., Antonia and Luxton, 1971; Jacobi and McKeon, 2011). In particular, previous literature suggests that turbulent boundary layers adjust relatively quickly for transitions from smooth to rough walls; the adjustment occurs over a streamwise distance ofO(10). The proles of mean velocity (U=U e ) and streamwise intensity (u 2 =U 2 e ) shown in gure 2.5 suggest that the adjustment from smooth to porous wall velocity proles occurs over a similarly short streamwise distance. Upstream of the porous section, the measured proles are consistent with previous smooth wall literature. In the near-wall region (y=< 5 10 3 ), the measurements agree reasonably well with the tted linear velocity proleU + =y + . The measured mean velocities are higher at the rst two measurement locations, which can be attributed to the bias introduced by velocity gradients across the LDV measurement volume. In the overlap region (0:02y= 0:2), the mean velocity prole is consistent with the logarithmic law: U + = (1=) ln(y + ) +B. The streamwise intensity prole shows the presence of a distinct inner peak at y= 0:006, or y + 10, which is also consistent with previous studies. The presence of the porous substrate substantially modies the mean velocity and streamwise intensity proles. Figure 2.5 shows clear evidence of substantial slip at the porous interface (U(y 0)> 0:3U e ). Farther from the interface, there is a velocity decit relative to the upstream, smooth wall prole. This velocity decit increases with distance along the porous substrate, and appears 22 10 -3 10 -2 10 -1 10 0 0 0.2 0.4 0.6 0.8 1 10 -3 10 -2 10 -1 10 0 0 0.01 0.02 (a) (b) Figure 2.5: Streamwise mean velocity (a) and turbulence intensity (b) measured at streamwise locations x=h = 21:5; 11; 21; 42; 53 relative to the transition from smooth wall to porous substrate. These data correspond to the 20 ppi foam withh = 12:7mm. The dashed lines correspond to a linear prole of the formU + =y + in the near-wall region and a logarithmic prole of the formU + = (1=) ln(y + )+B in the overlap region, with = 0:39 andB = 4:3. The friction velocity was estimated from tting a linear slope to the near-wall measurements. 23 -30 -15 0 15 30 45 60 5 5.5 6 6.5 7 Figure 2.6: Normalized boundary layer thickness =h for the 20 ppi foam measured at streamwise locations x=h = 21:5; 11; 21; 42; 53 relative to the substrate transition point. to saturate for the nal two proles measured at x=h 42. The mean velocity proles collapse together for y= 0:5, suggesting that the outer part of the wake region remains unchanged over the porous substrate. Consistent with the mean velocity proles, the streamwise intensity proles also show a substan- tial departure from the smooth case. Although there is some scatter in the measurements closest to the interface, the inner peak is replaced by an elevated plateau at u 2 =U 2 e 0:01, which extends from the porous interface to y= 0:01. Further, an outer peak appears near y= 0:1. The origin of this outer peak is discussed further in §2.4.2 below. In general, the streamwise intensity proles also converge for x=h 42. The streamwise evolution of the boundary layer thickness is presented in gure 2.6. Boundary layer growth over the rst two measurement locations (x=h 21) past the smooth to porous tran- sition is relatively slow and appears unchanged from the smooth wall boundary layer, suggesting that the internal layer does not yet span the boundary layer thickness at these locations. For the last two measurement locations, x=h 42, the boundary layer thickness grows much faster, suggesting that the ow adjustment is complete and that the eects of the porous substrate ex- tend across the entire boundary layer. These observations are consistent with the mean velocity and streamwise intensity proles shown in gure 2.5. As an example, the proles at measurement locationx=h = 11 show that the internal layer only extends toy= 0:2. Fory= 0:2, the mean velocity and streamwise intensity collapse onto the smooth-wall proles. 24 Development data for the remaining foams which are not presented here for brevity. For all the foams of thickness h 12:7 mm, the velocity measurements and boundary layer thickness data suggest that ow adjustment is complete by the measurement station at x=h = 42. Since the incoming boundary layer thickness is 5:5h, the streamwise adjustment happens over x 8, which is consistent with previous literature on the transition from smooth to rough walls. (Antonia and Luxton, 1971) However, this analogy to ow adjustment over rough-wall ows breaks down for the the thick 20 ppi foam with h = 25:4 mm. For the thick foam, ow adjustment was not complete even at the last measurement location (see results presented in §2.4.3). This observation is still in broad agreement with the results presented above since the last measurement location only yields a dimensionless development length of x=h 26 for the thick foam, while gure 2.6 suggests that x=h 30 is required for adjustment. The specic setup considered here may also be seen as ow over a backward facing step, with the region beyond the step lled with a highly porous ( > 0:96) and permeable (Re k > 1) material. Step Reynolds numbers in the present experiment range fromRe h =U e h= = 41610 3 . DNS by Le et al. (1997) and LDV measurements by Jovic and Driver (1994) at Re h = 5100 for a canonical (i.e., unlled) backward facing step indicate that the mean velocity prole behind the step does not return to a log-law for x=h = 20 beyond the step. This is comparable to the development length observed here over the ush-mounted highly porous substrates (x=h> 30). Thus, it appears that both andh play a role in dictating ow adjustment for the system considered here. Unfortunately, since the present study was limited to four measurement locations along the porous substrate, there is insucient spatial resolution in the streamwise direction to provide further insight into the scaling behaviour of boundary layer adjustment and growth over porous substrates. 2.4.2 Eect of pore size Next, we consider the eect of varying pore size on the fully-developed boundary layer proles measured at x=h = 42 for the foams of thickness h = 12:7 mm. Figure 2.7 shows the measured mean velocity and streamwise intensity for each of the foams tested, together with the smooth-wall prole measured upstream of the porous section. 25 10 -3 10 -2 10 -1 10 0 0 0.2 0.4 0.6 0.8 1 10 -3 10 -2 10 -1 10 0 0 0.01 0.02 (a) (b) Figure 2.7: Mean velocity (a) and turbulence intensity (b) proles for the smooth wall and for all the porous foams at x=h = 42. 26 The mean velocity prole over the porous foams is modied in two signicant ways with respect to the smooth wall prole taken upstream. First, there is a substantial slip velocity near the porous substrate. This slip velocity is approximately 30% of the external velocity across all substrates, with little dependence on pore size. The observed slip velocity is consistent with the DNS results of Breugem et al. (2006), who observed a slip velocity of approximately 30% for their highest porosity case ( = 0:95,Re k = 9:35). Note that the properties for the foams tested here (listed in table 2.1) are similar to the properties of the porous substrate considered in the simulations. Second, there is a mean velocity decit relative to the smooth-wall case from 0:004 y= 0:4. This decit generally increases with increasing pore size, though there is some non-monotonic behaviour. The maximum decit relative to the smooth-wall prole is approximately 15% for the 100 ppi foam (s + = 7) and this increases to almost 50% for the 20 ppi foam (s + = 37). However, the decit for the foam with the largest pore sizes (10 ppi, s + = 52) is smaller than the decit over the 20 ppi foam. In the outer wake region (y= > 0:5), all mean velocity proles collapse onto the canonical smooth-wall prole again. The streamwise turbulence intensity proles, plotted in gure 2.7b, show that the inner peak disappears for all the porous foams. Instead, there is a region of elevated but roughly constant intensity that extends from the porous interface toy= 0:01. Fory=> 0:01, the intensity proles show a strong dependence on pore size. For the smallest pore sizes, the streamwise intensity either decreases slightly (100 ppi) or stays approximately constant until y= 0:1 (60 ppi). For foams with larger pore sizes, the intensity increases and a distinct outer peak appears near y= 0:1. All the proles collapse towards smooth-wall values for y= 0:5. Consistent with the mean velocity measurements, the streamwise intensity proles also show non-monotonic behaviour with pore size. While the streamwise intensity in the outer region of the ow generally increases with pore size, the magnitude of the outer peak is higher for the 20 ppi foam compared to the 10 ppi foam. Velocity spectra, presented in §2.4.4 below, provide further insight into the origin of this outer peak. Figure 2.8 shows skewness proles over the smooth wall and porous foams. Consistent with previous measurements at comparable Reynolds number (e.g., Mathis et al., 2011), the skewness over the smooth wall is negative or close to zero across much of the boundary layer (0:004<y=< 27 10 -3 10 -2 10 -1 10 0 -1 0 1 Figure 2.8: Wall-normal proles of skewness (Sk) for the smooth wall and for all the porous foams at x=h = 42. 1). In contrast, over the foam substrates, the sign of the skewness is positive until y= 0:1. Interestingly, the location of this change in sign for the skewness corresponds to the location of the outer peak in streamwise intensity proles. Further, the magnitude of the skewness generally increases with pore size, which is consistent with the intensity measurements (the 10 and 20 ppi cases again show non-monotonic behaviour). These observations are particularly interesting given the intrinsic link between skewness and amplitude modulation (Schlatter and Orl u, 2010b; Mathis et al., 2011), and suggest that structures responsible for the outer peak in streamwise intensity over the foams may have a modulating eect on the interfacial turbulence. This possibility is discussed in greater detail in §2.5.1 below. 2.4.3 Eect of substrate thickness The thickness of the porous substrate, h, is another important parameter that dictates ow behaviour. Figure 2.9 shows the measured mean velocity and streamwise intensity proles over 20 ppi foams of varying thickness, h = 6:35; 12:7 and 25:4 mm. In dimensionless terms, these thicknesses correspond to h=s = 4:3; 8:5 and 17, respectively, where s = 1:5 0:2 mm is the average pore size (Table 2.1). Note that the measurements shown in gure 2.9 are for the same physical location, x = 53:3cm, which yields normalized distances increasing from x=h = 21 for the thickest foam to x=h = 84 for the thinnest foam. The foam with thickness h = 12:7mm is considered the baseline case, since velocity measurements made over this foam have already been discussed in §5.5.1 and §2.4.2. The foams of thickness h = 6:35mm and h = 25:4mm are referred 28 10 -3 10 -2 10 -1 10 0 0 0.2 0.4 0.6 0.8 1 10 -3 10 -2 10 -1 10 0 0 0.01 0.02 (a) (b) Figure 2.9: Mean velocity (a) and turbulence intensity (b) proles for the smooth wall and for the 20ppi foam of varying thickness at the same physical location. 29 to as the thin and the thick foam, respectively. As noted in §5.5.1, the ow was fully developed for the thin and baseline foams, but still developing for the thick foam. Mean velocity proles for both the thick and thin foam show similar slip velocities at the interface compared to the baseline foam of thickness h = 12:7 mm ( 0:3U e ), though the velocity measured over the thin foam is slightly higher ( 0:35U e ). In general, the mean prole for the thin foam is consistently higher than that for the baseline foam, and collapses onto the smooth- wall prole above y= > 0:2 (black triangles in gure 2.9a). In contrast, the mean prole for the thicker foam (white triangles in gure 2.9a) is closer to the baseline case in the near-wall region y= < 0:005. However, a little farther from the wall, the mean prole for the thick foam diverges from that for the baseline foam. The thick foam mean prole shows a smaller velocity decit in the region y= 0:01 to y= 0:1 compared to the baseline case, and verges on the smooth-wall prole for y= 0:2. The streamwise turbulence intensity proles plotted in gure 2.9b show that u 2 =U 2 e is similar near the interface for all three foams. In fact, the baseline and thin foams show very similar intensity proles through most of the boundary layer, barring two minor dierences. First, the outer peak in streamwise intensity appears closer to the interface for the thin foam. Second, the thin foam prole shows a better collapse onto the smooth-wall prole fory= 0:3. The turbulence intensity prole for the thick foam also collapses onto the smooth-wall prole fory= 0:3. However, closer to the wall,u 2 is much higher over the thick foam, and the outer peak in intensity also appears to move slightly closer to the interface. These features are also seen in the velocity spectra presented below. In summary, for the thin foam, the mean velocity and streamwise intensity measurements both collapse onto the smooth-wall prole for y= 0:3. This suggests that the outer-layer similarity hypothesis proposed by Townsend for rough-walled ows (see e.g., Schultz and Flack, 2007) also holds for thin porous media. The mean prole for the thick foam is similar to that for the baseline foam close to the interface, but moves closer to the thin foam prole further from the interface. However, the streamwise intensities are signicantly higher for the thick foam compared to the thin foam for y= 0:1, even in regions where the mean proles show agreement. These discrepancies 30 between the mean velocity and streamwise intensity proles for the thick foam are consistent with a developing ow. 2.4.4 Velocity spectra The premultiplied velocity spectra shown in gure 5.10 provide further insight into the origin of the outer peak in streamwise intensity observed over the porous substrates. As is customary in the boundary layer literature, the premultiplied spectrum is dened as fE uu , where f is the frequency andE uu is the power spectral density, normalized byU 2 e . This quantity is computed for each wall-normal location, and the results are compiled into the contour plots shown in gure 5.10. Due to the low data rates obtained near the interface, spectra are only shown fory= 0:04. Note that the spectra are expressed in terms of a normalized streamwise wavelength estimated using Taylor's hypothesis, U=(f). For the smooth wall case, there is evidence of weak very-large-scale motions (VLSMs), which is similar to results obtained in previous studies at comparable Reynolds number (Hutchins and Marusic, 2007). The box in the top left panel encompasses the spectral region typically associated with VLSMs, i.e. structures of length 6 10 (U=f = 6 10) located between y= = 0:06 and y + = 3:9 p Re (see Hutchins and Marusic, 2007; Marusic et al., 2010; Smits et al., 2011). This box coincides with a region of elevated spectral density for the measurements. Spectra for the porous substrates are dierent in several ways. For all the foams, the spectra are elevated over the frequency range that corresponds to structures with streamwise length scale 1 5. The spectra are most energetic at, or below, wall-normal location y= 0:1 and remain elevated until y= 0:3. There is a marked increase in the spectral energy density from the 100 ppi foam to the 20 ppi foam, and little dierence between the spectra for the 20 ppi and 10 ppi foams. Together, these features suggest that the outer peak in streamwise intensity observed over the porous foams in gure 2.7b) is associated with large-scale structures of length 1 5 that are distinct from VLSMs. Spectra for the 20ppi foam of dierent thickness, plotted in gure 5.10e-f), show a substantial increase in energy for the thickest foam, which is consistent with the elevated streamwise intensity 31 Figure 2.10: Contour maps showing variation in premultiplied frequency spectra (normalized by U 2 e ) for streamwise velocity as a function of wall-normal distance y= over the smooth wall (a), the 100 ppi foam (b), the 10 ppi foam (c), the thin 20 ppi foam (d), the baseline 20 ppi foam (e), and the thick 20 ppi foam (f). The spectra are plotted against a normalized streamwise length scale, U=f, computed using Taylor's hypothesis. The white box in (a) denotes the region typically associated with VLSMs while the markers () represent the nominal frequency f KH for structures resembling Kelvin-Helmholtz vortices. The spectra refer to the same physical measurement location, corresponding to x=h = 42 for the foams with h = 12:7mm and x=h = 21 and 84 for the thick and thin 20 ppi foams, respectively. 32 proles shown in gure 2.9b). For the thick foam, the energy is also concentrated closer to the interface compared to the thin and baseline foams with identical pore sizes. Note that the spectral features described above are consistent with previous experiments and simulations. Manes et al. (2011) showed that the premultiplied frequency spectra for streamwise velocity peak at wavenumberk x 24 over porous substrates, i.e. corresponding to structures of length 1:53. Similarly, the DNS carried out by Breugem et al. (2006) indicated the presence of spanwise rollers with streamwise length-scale comparable to the total channel height. Since these structures have been linked to a Kelvin-Helmholtz shear instability arising from the in ection point in the mean prole, it is instructive to consider the characteristic frequency associated with this mechanism. Drawing an analogy to mixing layers, White and Nepf (2007) and Manes et al. (2011) suggested that the characteristic frequency for the Kelvin-Helmholtz instability can be estimated as f KH = 0:032U= (Ho and Huerre, 1984), in which U = (U p +U e )=2 is the average velocity in the shear layer with U p being the velocity deep within the porous medium, and is the momentum thickness. Estimates for this characteristic frequency, converted to streamwise length-scale based on the mean velocity at y= = 0:1, KH = =U(0:1)=f KH , are shown in gure 5.10. These estimates assume U p 0, so that U 0:5U e , and that the momentum thickness can be approximated based on the measured velocity prole in the uid domain, i.e. for y 0. With these assumptions, the predicted length scales associated with the instability range from KH = 4:3 for the 100 ppi foam to KH = 3:6 for the 10 ppi foam, which is in reasonable agreement with the location of the peaks in the spectra. There is an evident overprediction of length scale for both the 10 ppi foam (gure 5.10c) and the thick 20 ppi foam (gure 5.10c). This overprediction can be attributed to greater ow penetration into the porous medium for thicker foams with larger pore sizes. Greater ow penetration would create higher velocities inside the porous medium, U p . This would increase U and f KH , resulting in lower KH . Greater ow penetration into the porous medium may also result in the in ection point moving closer to the interface. In this scenario, conversion fromf KH to KH should be based on a lower mean velocity from y= < 0:1. This would also result in lower KH compared to the estimates shown in gure 5.10. 33 To test whether the observed peaks in streamwise intensity and velocity spectra tracked the location of in ection points in the mean prole, y i , nite-dierence approximations of the second derivative of mean velocity, (d 2 U=dy 2 ) y i = 0, were considered. This yielded in ection point lo- cations ranging from y= 0:03 to y= 0:12 for the 10 and 20 ppi foams, which is broadly consistent with the location of energetic peaks in gure 5.10. However, the second derivatives estimated from experimental data were very noisy, and the in ection point locations were highly sensitive to the accuracy (i.e., order) of the nite-dierence approximation and the lower limit in y used for the estimates. As a result, exact in ection point locations are not presented here. 2.5 Discussion 2.5.1 Amplitude modulation As noted earlier, the skewness proles shown in gure 2.8 strongly suggest that the amplitude modulation phenomenon that has received signicant attention in recent smooth and rough wall literature (e.g., Mathis et al., 2009; Marusic et al., 2010; Mathis et al., 2013; Pathikonda and Christensen, 2017) may also be prevalent in turbulent ows over permeable walls. For smooth wall ows at high Reynolds number, it has been shown that the VLSMs prevalent in the logarithmic region of the ow can have a modulating eect on the small-scale uctuations found in the near-wall region. Specically, it has been suggested that the turbulent velocity eld in the near-wall region can be decomposed into a large-scale (or low-frequency) component u L that represents the near-wall signature of the VLSMs, and a `universal' (i.e. Reynolds number independent) small-scale component u S that is associated with the local near-wall turbulence. Analysis of the resulting signals indicates that the ltered envelope of the small-scale activity, E L (u S ), obtained via a Hilbert transform (for details, see Mathis et al., 2011), is strongly correlated with the large-scale signal. In other words, the single-point correlation coecient: R = u L E L (u S ) q u 2 L q E L (u S ) 2 (2.4) tends to be positive in the near-wall region, which suggests that the small-scale signal u S is mod- ulated by the large-scale signal u L . 34 Since the local large-scale signal u L arises from VLSM-type structures centered further from the wall, these observations have given rise to predictive models of the form u =u (1 +u OL ) +u OL ; (2.5) where u(y) is the predicted velocity at a specied location in the near-wall region, u (y) is a statistically universal small-scale signal at that wall-normal location,u OL is the large-scale velocity measured in the outer region of the ow, and (y) and (y) are superposition and modulation coecients, respectively (Marusic et al., 2010). Assuming that the universal small-scale signal can be obtained via detailed experiments or simulations carried out at low Reynolds numbers, equation (2.5) allows for near-wall predictions at much higher Reynolds numbers based only on measurements of u OL in the outer region of the ow. Note that the modulation coecient is similar to the single-point correlation coecient R, but also accounts for changes in phase and amplitude of the large-scale signal from the measurement location to the near-wall region. In other words, also accounts for the relationship between u OL and u L (y), which is thought to be Reynolds-number independent (Marusic et al., 2010). Subsequent studies have shown rigorously that the correlation coecient R in (2.4) is intrinsi- cally linked to the skewness of the velocity (Schlatter and Orl u, 2010b; Mathis et al., 2011; Duvvuri and McKeon, 2015), whereby positive correlations between scales (R> 0) translate into increased skewness. Thus, the increase in skewness observed in gure 2.8 near the porous interface suggests that R> 0 in this region. Near-wall measurements made in the present study do not have sucient time resolution to allow for a quantitative evaluation of the amplitude modulation phenomenon over porous substrates (i.e., a decomposition into small- and large-scale components). However, the increase in the skewness in the near-wall region over the porous substrates suggests that the large-scale structures responsible for the outer peak in streamwise intensity at y= 0:1 may have a modulating eect on the turbulence near the interface. This is particularly interesting given that the large-scale structures found over porous substrates are distinct from the VLSMs found over smooth walls, and that the small-scale turbulence near the porous interface may also be modied from the near-wall cycle (Robinson, 1991; Schoppa and Hussain, 2002) found over smooth walls. 35 10 -2 10 -1 10 0 y/δ -0.2 0 0.2 0.4 y Ue ∂U ∂y smooth 10 ppi 20 ppi 60 ppi 100 ppi 20ppi thin 20ppi thick 0 0.05 0.1 0.15 0.2 0.25 0.3 s/h 0 0.5 1 1.5 2 yd h 0 0.05 0.1 0.15 0.2 0.25 0.3 s/h 0 0.2 0.4 0.6 0.8 uτ κUe 0 0.05 0.1 0.15 0.2 0.25 0.3 s/h 0 0.5 1 1.5 2 k0 h (a) (b) (c) (d) Figure 2.11: (a) Scaled velocity gradient (y=U e )@U=@y plotted as a function of y= for the smooth wall prole and porous foam data. Equation (2.8) was tted to these data to estimate the normalized displacement height, y d =h, and the friction velocity weighted by the von Karman constant, u =(U e ), shown in (b) and (c), respectively. Using these estimates for y d and u =, the roughness height k 0 =h, shown in (d), was evaluated from the velocity proles using equation (2.1). Dotted lines in (a) represent the upper limits of y= = 0:16; 0:20 and 0:25 employed in the tting procedure. The dashed line represents the minimum lower limit, y= > 0:02. Larger marker sizes in (b,c,d) denote higher values for the upper limit. The red cross in (c) represents the friction velocity estimated via a linear t to the near-wall velocity measurements over the smooth wall. Horizontal error bars in panels (b)-(d) represent uncertainty in pore size, s. 2.5.2 Logarithmic region Previous studies have devoted considerable eort to testing whether a modied logarithmic region of the form shown in equation (2.1) exists in turbulent ows over porous substrates. While there is no denitive consensus on how the von Karman constant, , displacement height, y d , and equivalent roughness height, k 0 , vary with substrate properties, experiments and simulations generally show that the displacement and roughness heights increase with increasing permeability (or pore size). Manes et al. (2011) showed that empirical relationships of the form: y + d = 15:1Re k 13:5 (2.6) 36 and k + 0 = 6:28Re k 9:82; (2.7) where Re k = u p k= is the permeability Reynolds number, led to reasonable ts for the experi- mental data obtained by Suga et al. (2010) and Manes et al. (2011), but underestimated y d and k 0 for the simulations performed by Breugem et al. (2006). The von Karman constant decreased relative to smooth wall values but demonstrated a complex dependence on both the permeability Reynolds number and the ratio of displacement height to boundary layer thickness, y d = (Manes et al., 2011). Unfortunately, the present study did not involve independent measurements of the shear stress at the interface (or Reynolds' shear stress in the near-wall region) and so it is not possible to estimate the friction velocity and von Karman constant independently. However, the velocity measurements can still be used to test whether a modied logarithmic region exists, and to estimate u =, y d and k 0 . Taking the partial derivative of equation (2.1) with respect to y and rearranging yields: (y +y d ) @U @y = u : (2.8) Following Breugem et al. (2006) and Suga et al. (2010), we estimated y d as the value that forces (y +y d )@U=@y to be constant over specied ranges of y=. Based on equation (2.8), the resulting constant value for the weighted velocity gradient was assumed to be the friction velocity divided by von Karman constant, u =. Using these estimates for displacement height and friction velocity, the roughness height was estimated directly from the velocity measurements using equation (2.1). Since the tting procedure described above relies on noisy velocity gradient data (see g- ure 2.11a), the uncertainty in the tted values and sensitivity to tting ranges was evaluated as follows. First, the tting procedure was carried out over the range 0:02<y= < 0:16. The t was then repeated with the lower limit sequentially increased by one to four measurement points. The estimates of y d , u =, andk 0 reported in gure 2.11(b-d) represent an average of the ve dif- ferent values obtained via this process and the error bars represent the standard error. This entire procedure was then repeated for outer limits y= = 0:20 and y= = 0:25. A similar process was 37 used to evaluate u = and k 0 for the smooth wall case with the displacement height constrained to be zero, y d = 0. Estimates for y d , k 0 , and u = are reported in table 2.2. Note that the tting procedure employed by Manes et al. (2011) was also considered, wherey d is estimated as the value that minimizes residuals between the measured velocity prole and equation (2.1) over specied ranges of y=. The resulting tted coecients are then used to estimate u = and k 0 . This process led to tted values within the uncertainty ranges shown in gures 2.11b-d. Assuming = 0:39, the tting procedure described above led to a friction velocity estimate of u = 2:31 0:02 cm/s for the smooth wall proles with outer limit y= = 0:15. This is consistent with the value obtained via a linear t to the near-wall mean velocity measurements,u = 2:30:05 cm/s. The additive constantB =(1=) lnk + 0 in equation (2.1) was estimated to beB = 4:80:2, which is slightly higher than the value,B = 4:3, reported in Marusic et al. (2013), but still broadly consistent with previous literature. As expected, the tted log law constants for the porous substrates show a strong dependence on average pore size. Figures 2.11(b-d) show that the displacement height, friction velocity, and roughness height generally increase with increasing pore size, though here is some evidence of saturation at the largest pore sizes. Specically, gure 2.11b suggests that the displacement height levels out abovey d =h 1 for the baseline 20 ppi foam, the 10 ppi foam, and the thin 20 ppi foam, for whichs=h> 0:1. This saturation iny d =h as a function of normalized pore size is accompanied by saturation, or perhaps slight decreases, in normalized friction velocity u =U e (gure 2.11c) and roughness height k 0 =h (gure 2.11d) above s=h> 0:1. While the exact values of the log law constants shown in gure 2.11 and table 2.2 must be treated with some caution due to the uncertainty associated with the tting procedure, the overall trends suggest the following physical interpretation. The displacement height y d represents the level at which momentum is extracted within the porous medium (Jackson, 1981), or alternatively, the distance to which the turbulence penetrates into the medium (Luhar et al., 2008) or the eective plane at which attached eddies are initiated (Poggi et al., 2004). For the least permeable substrates tested in the present study (i.e. the 100 ppi and 60 ppi foams),y d increases approximately linearly with average pore size. In other words, at this low permeability or thick substrate limit with h=s 1, the distance to which turbulence penetrates into the porous medium increases with 38 increasing pore size or permeability. Since the ow does not interact with the entire porous medium, the foam thicknessh does not play a role. However, with further increases in pore size or decreases in porous medium thickness, at some point the displacement height becomes comparable to the foam thickness y d h. At this high permeability or thin substrate limit, turbulence penetrates the entire porous medium and the foam essentially acts as a roughness or obstruction. Results shown in gure 2.11 suggest that the baseline 20 ppi foam, the thin 20 ppi foam, and the 10 ppi foam, for which h=s 10, may be approaching this thin substrate limit. Figures 2.11b-d) show that the normalized friction velocity and roughness height are strongly correlated with each other as well as the displacement height. Physically, the friction velocity is a measure of momentum transfer into the porous medium while the roughness height is a measure of momentum loss, or friction increase, due to the presence of the complex substrate. As a result, correlation betweenk 0 andu is unsurprising for boundary layer experiments carried out at constant free-stream velocity (n.b., for channel or pipe ow experiments, the friction velocity can be controlled independently by setting the pressure gradient). The link between the displacement and roughness heights can be explained by considering the rough-wall literature. For ows over conventional K -type roughness,k 0 has been shown to depend on both the height of the roughness elements and the solidity , which is dened as the total projected frontal area per unit wall-parallel area (Jim enez, 2004). Similarly, for ows over porous media, k 0 can be expected to depend on the displacement height, which represents the thickness of porous medium that interacts with the ow, as well as the porous medium microstructure (see also Jackson, 1981; Manes et al., 2011). In other words, a relationship of the form k 0 =y d = f() may be appropriate for turbulent ows over porous media. Note that the physical link between the roughness and displacement heights is also evident in the empirical relationships for y d andk 0 shown in equations (2.6-2.7). 2.5.3 Non-monotonic behaviour with pore size In many ways, the aforementioned transition from thick substrate behaviour, where turbulence penetration into the porous medium is limited andy d increases with permeability, to thin substrate behaviour, where turbulence penetrates the entire porous medium and y d h, is analogous to the 39 Substrate u = (cm/s) y d (mm) y d =h k 0 (mm) k 0 =y d smooth 5.8 0 0 5:3 10 3 1 10 ppi 27.5 14.2 1.1 7.8 0.55 20 ppi 32.9 15.6 1.2 9.8 0.63 20 ppi thin 22.2 8.7 1.4 3.9 0.44 20 ppi thick 26.9 10.2 0.4 5.5 0.52 60 ppi 16.5 5.3 0.42 1.5 0.28 100 ppi 11.9 3.2 0.25 0.49 0.15 Table 2.2: Fitted values for log-law parameters in dimensional and dimensionless form. Listed values of u =, y d , andk 0 are averages of the three estimates shown in gure 2.11, which were obtained for three dierent outer limits in the tting procedure. The displacement height is assumed to be zero for the smooth wall ow. -dependent transition from dense to sparse canopy behaviour proposed in the vegetated ow literature (Belcher et al., 2003; Luhar et al., 2008; Nepf, 2012). As noted in the introduction, for vegetated ows the distance to which the ow penetrates into the canopy is dependent on the drag length-scale (C D a) 1 , whereC D is a representative drag coecient and a is the frontal area per unit volume. As a result, the ratio of shear penetration to canopy height,h, is given by the dimensionless parameterC D ah =C D , where is the solidity as before. For dense canopies withC D O(1), the shear layer does not penetrate the entire canopy and so an in ection point is expected in the mean prole. This gives rise to large-scale structures resembling Kelvin-Helmholtz vortices. However, this instability mechanism is expected to weaken in sparse canopies. For C D < O(0:1), turbulence penetrates the entire canopy and there is no in ection point in the mean prole (Nepf, 2012). The non-monotonic behaviour in mean velocity and turbulence intensity observed for the 10 ppi foam in the present experiments could be attributed to a similar weakening of the shear layer instability as the turbulence penetrates the entire porous medium, i.e. as y d h. Consistent with this hypothesis, the reduced magnitude of the outer peak in streamwise intensity for the 10 ppi foam relative to the 20 ppi foam (gure 2.7b) suggests that the large-scale structures resembling Kelvin-Helmholtz vortices are weaker over the 10 ppi foam. Since these structures contribute substantially to vertical momentum transfer, a reduction in their strength also translates into a smaller mean velocity decit (gure 2.7a). Note that there is evidence of non-monotonic behaviour as a function of solidity in the rough- wall literature as well. Based on a compilation of experimental data, Jim enez (2004) showed that 40 the normalized roughness height (i.e. ratio ofk 0 to roughness dimension) depends on the solidity, and that there are two regimes of behaviour. For sparse roughness with solidity less than 0:15, the normalized roughness increases with increasing . In other words, an increase in frontal area leads to an increase in roughness drag. However, for densely packed roughness with & 0:15, the normalized roughness decreases with increasing because the roughness elements shelter each other. Assuming that the relevant vertical dimension for turbulent ows over porous media is the displacement height y d , we may expect similar non-monotonic behaviour for the normalized roughness k 0 =y d = f(). The values for k 0 =y d listed in table 2.2 provide some support for this hypothesis: the normalized roughness height increases from k 0 =y d = 0:15 for the 100 ppi foam to k 0 =y d = 0:63 for the baseline 20 ppi foam, before decreasing to k 0 =y d = 0:55 for the 10 ppi foam. Bear in mind that, for identical h, the solidity is expected to increase with decreasing pore size from the `sparsely packed' 10 ppi foam to the `densely packed' 100 ppi foam. Although the solidity is a dicult parameter to measure for porous media, it may be estimated for the foams employed here based on simple geometric assumptions. Consider, for instance, a cubic lattice comprising thin rectangular ligaments of cross-section dd and length s (i.e., the pore size). Each unit cell of volume s 3 in this lattice comprises three orthogonal laments aligned in the x;y; and z directions intersecting in a three-dimensional cross. Neglecting the overlapping volume at the center of the cross, the porosity is approximately 1 3d 2 s=s 3 = 1 3(d=s) 2 for this geometry. For the foams tested here, the porosity is constant, 0:97. So, the above equation implies that the lattice must be geometrically similar with d=s p (1)=3 = 0:1. In other words, the ligament width d increases linearly with pore size s to maintain constant . For this assumed geometry, the frontal area per unit volume for ow in either the x,y, orz directions is a 2ds=s 3 = 2d=s 2 , since there are always 2 ligaments with area ds normal to the ow. This results in solidity = ah 2(d=s)(h=s) = 0:2(h=s). This estimate suggests that the pore size threshold above which non-monotonic behavior is observed in the log-law constants (s=h 0:12 in gure 2.11) corresponds to solidity O(1). Of course, since the foam pore structures do not resemble a cubic lattice (gure 2.4), the numerical factors appearing in the equations above are unlikely to be accurate. However, the linear relationship between and h=s is expected to hold. 41 Finally, keep in mind that the non-monotonic behavior with solidity and pore size described above does not preclude the possibility of monotonic behavior with permeability Reynolds number Re k = u p k=. Although y d and k 0 are shown to decrease with increasing pore size, and hence permeability, from the 20 ppi foam to the 10 ppi foam, this decrease in the displacement and roughness heights is also accompanied by a decrease inu = (Table 2.2). In other words,Re k may be lower for the 10 ppi foam compared to 20 ppi foam, even if p k is higher. Unfortunately, this hypothesis cannot be tested further without independent estimates of u . 2.5.4 A note on scaling Previous studies on turbulent ows over porous media indicate that shear penetration into the porous medium depends on the permeability length scale p k, which determines the eective ow resistance within the porous medium per the well-known Darcy-Forchheimer equation (Breugem et al., 2006; Suga et al., 2010). This is also evident in the empirical relationship shown in equation (2.6), which indicates thaty d 15:1 p k for suciently high permeability Reynolds numberRe k 1. Although the permeability is related to geometric parameters such as pore size (e.g. p k=s 0:08 for the foams tested by Suga et al. (2010)) and frontal area per unit volume, it is essentially a dynamic parameter that is typically estimated from tting the Darcy-Forchheimer law: 1 P x = U v k + C f p k U 2 v ; (2.9) to experimental pressure drop measurements (P=x) across porous media. In the equation above,U v is the volume-averaged velocity andC f is the Forchheimer coecient. Since such pressure drop measurements are usually carried out at low Reynolds number in steady pipe or channel ow with uniformly distributed porous media (essentially a one-dimensional system), there are inherent risks in employing the resulting permeability values for unsteady, three-dimensional, spatially varying ows at higher Reynolds number. Further, the non-linear Forchheimer term that becomes increasingly important at higher speeds requires an additional coecientC f that is not a universal constant. To avoid these issues, the present study presents results primarily as a function of the normalized pore size,s=h. For example, gure 2.11b suggests thaty d /s until it becomes comparable to foam 42 thickness. Another alternative would be to use the frontal area per unit volume, a, and solidity, , as the relevant scales. The frontal area per unit volume is a dicult quantity to measure for complex porous media. However, the discussion presented in the previous section suggests that the solidity increases linearly with (h=s) for geometrically-similar porous media with constant porosity. As a result, the use ofs=h for scaling purposes also allows for greater reconciliation with the canopy ow and rough wall literature. 2.6 Conclusions The experimental results presented in this paper show that turbulent boundary layers over high- porosity foams are modied substantially compared to canonical smooth wall ows. Development data in §5.5.1 suggest that the boundary layer adjusts relatively quickly to the presence of the porous substrate. Specically, for most of the foams tested, the mean velocity prole adjusts to a new equilibrium over a streamwise distance < 10, which is similar to the adjustment length observed in previous literature for the transition from smooth to rough walls. However, this rough- wall analogy does not hold for the thickest foam tested, which suggests that the foam thickness may also provide a bound on development length. Fully-developed mean velocity proles presented in §2.4.2 show the presence of substantial slip velocity (> 0:3U e ) that is relatively insensitive to pore size for foams of constant thickness. Proles in §2.4.3 also show a near constant slip velocity over substrates with constant pore size and varying thickness. These observations remain to be explained fully. Proles of streamwise intensity show the emergence of an outer peak at y= 0:1 over the porous substrates, which is associated with large-scale structures of length 24. Such structures have also been observed in previous simulations and experiments, and are thought to arise from a Kelvin-Helmholtz instability associated with an in ection point in the mean prole. Although the magnitude of the outer peak in streamwise intensity generally increases with pore size, there is some evidence of weakening for the foam with the largest pore size. The log-law ts presented in §2.5.2 provide further insight into this non-monotonic behaviour. Specically, the displacement height increases with normalized pore size,s=h, until it becomes comparable to the foam thickness. Further increases in pore size beyond this point do not lead to an increase in y d . In other words, 43 there is a transition from thick substrate behaviour, in which the thickness of the porous medium interacting with the ow is determined by pore size (y d / s), to thin substrate behaviour, in which the ow penetrates the entire porous medium (y d h). The weakening in the outer layer structures may be attributed to this transition from thick to thin substrate behaviour. Drawing an analogy to sparse canopy behaviour for vegetated ows, at the thin substrate limit, the mean velocity prole becomes fuller with increasing pore size and ultimately loses the in ection point. This results in a weakening of the Kelvin-Helmholtz instability. For canopy ows, the transition from dense-canopy behavior to sparse-canopy behavior occurs as the solidity parameter becomes small, 1. Simple geometric arguments show that /h=s for the foams tested here, and that the transition from thick- to thin-substrate behavior occurs around O(1). Interestingly, the skewness of the near-wall velocity measurements increases substantially over the porous substrates relative to smooth wall values. Further, this increase in skewness is correlated with an increase in the magnitude of the outer peak in streamwise intensity. Given the link between skewness and the amplitude modulation phenomenon, these observations suggest that the large-scale structures that are energetic over porous media may have a modulating in uence on the interfacial turbulence. This is analogous to the interaction between VLSMs and near- wall turbulence in smooth wall ows at high Reynolds number. Unfortunately, the near-wall velocity measurements collected as part of this study were not time resolved, and so did not allow for a quantitative evaluation of this eect. However, given the substantial similarities between turbulent ows over porous media and vegetation or urban canopies, further studies into such scale interactions could lead to the development of promising wall models for a variety of ows. 44 Chapter 3 Development and characterization of anisotropic porous media 3.1 Motivation Chapter §1 motivated the design of materials with streamwise preferential permeabilities. Cur- rently, there is no commercial source for high-porosity materials with customizable permeability tensors that could be used for passive ow control. In this section, a method to tune the perme- ability tensor of a porous medium is detailed. Design considerations and limitations for use with specic 3D printers are discussed, and nal material specications are presented. An experimental procedure to characterize the full permeability matrix, K = 2 6 6 6 6 4 K xx K xy K xz K yx K yy K yz K zx K zy K zz 3 7 7 7 7 5 ; is described and validated. Commercially-available high porosity materials typically come in the form of open-cell, reticulated matrices. They are manufactured by adding a foaming agent to materials such as polyurethane solutions prior to curing. When heated, bubbles expand and burst leaving behind large voids. Efstathiou and Luhar (2018) and Manes et al. (2011) used a range of these high-porosity foams ( > 0:9) for their experiments as substrates to turbulent boundary layers. These materials are not tunable in the sense of being able to design the permeability matrix, although the porosity and pore sized can be varied from 20 97% and s 400nm-4mm by modulating variables such as the curing temperature, bubbling agent, and duration. A number of specialized manufacturing techniques are described in Studart et al. (2006). For example, 45 a ceramic \slurry" can be injected into open pores of a polyurethane mold, which melts when the ceramic is kiln red. Ceramic foams, such as aluminium oxide foams, are used as ltration devices for molten metals, environmental lters, catalytic converters (Ciambelli et al., 2007), and to enhance heat transfer and protect electronics (Behrens and Tucker, 2010). Similar techniques have also been developed to manufacture lightweight composite foams out of carbon (Klett, 2000). ERG Aerospace oers reticualted foams made with metal alloys such as copper, aluminium, and silicon carbide. The manufacturing process retains alloy properties after manufacturing, making such foams ideal surfaces for heat transfer enhancement, high temperature ltering, and industrial applications where corrosion resistance and longevity are vital. The high porosity ensures that the nished material has a low density and increased surface area. Because the material is made by expanding bubbles, well-controlled manufacturing processes typically design for an even size distribution of densely packed spherical bubbles, resulting in quasi-isotropic permeabilities. Additive manufacturing has recently been used to create custom porous materials. For example Zhu et al. (2018) manufactured catalysts for electrochemical processes with pore sizes varying in two dimensions from s 10 1000m. Similarly, Hern andez-Rodr guez et al. (2014) used stereolithography printers to optimize the porous microstructure for solid oxide fuel cells. 3.2 Lattice design Rapid advances in additive manufacturing techniques such as stereolithography 3D printing have made it possible to manufacture highly porous ( > 0:85) lattices with small rod sizes and large openings. To manufacture materials with tunable microstructure, a family of lattices such as those shown in Fig. 3.1 are used. A diagram of the lattices showing front, top, and isometric views is contained in Fig. 3.1. The permeability and porosity are controlled by the rod size, d, and the rod spacings in three orthogonal directions, s x , s y , and s z . In this model, the rods have a square cross section ofdd. The streamwise permeability is expected to be approximately proportional to the streamwise facing open area, K xx / (s y d)(s z d). Similarly, the wall-normal permeability K yy / (s x d)(s z d) and the spanwise permeability K zz / (s x d)(s y d). Optimizing the parameters for drag reduction requires maximizing p K xx p K zz and xy = K xx =k yy while 46 minimizng p K yy . For the lattice, this means minimizing s x , which does not in uence K xx but reduces both K yy and K zz and maximizing s y and s z , which increases all three components. The porosity () is dened as the ratio of empty volume to total volume ( = Ve Vt ). Considering a unit cell of the rods, the empty volume is the total volume less three rods. The average porosity of the material is = sxsyszd 2 (sx+sy+sz)2d 3 sxsysz = 1 d 2 (sx+sy+sz)2d 3 sxsysz . Note that the rod dimension squared is a driving factor in the porosity. The models and numerical experiments in G omez-de- Segura and Garc a-Mayoral (2019) assume high porosities of > 0:8, necessitating small rod sizes relative to the spacings, s x , s y and s z . Figure 3.1: Lattice parameters governing permeabilities and porosities. s x and s y drive the spanwise facing open area, s y and s z drive the streamwise facing open area, and s x and s z drive the wall-normal facing open area. 3.3 Formlabs printers specications and limitations For the current experiments, state of the art stereolithography printers from Formlabs TM are used to manufacture substrates. Both form2 and form3 printers were utilized, but the resolution and manufacturing considerations are similar for both. The printers have a print area of 145mm by 145mm (xy) and a vertical range of 175mm (z) (form2) and 185mm (form3). The printers create structures by inserting the build platform into a resin tank. The build platform is lowered until the platform is z o the bottom of the resin tank, where z is the selected z-axis resolution. The resin tank has a clear, silicone bottom, through which a laser beam shines up vertically, photocuring a thin layer of resin. When successful, the newly cured resin is attached to the previous structures 47 printed on the build platform. The laser beam has a diameter of 145m (form2) and 85m (form3) which sets the minimum resolution for the x and y directions. The biggest dierence between the printer models is the resin tray. The form 3 features a exible resin tray which allows the print to peel o with a lower force (\low force stereolithography"). This allows for more delicate parts to be printed.. The z resolution is set by how close the build platform is brought to the surface, and can be selected from 25; 50; 100m. Increasing the resolution (decreasing the layer height) also increases the build time, because the printer raises the print platform outside the resin tank in order to mix up the resin after every layer. A diagram of the printer and porous material print conguration are provided in Fig. 3.2. Figure 3.2: Diagram of form3 printer (a), and printing conguration (b). Note that the build happens inverted, i.e. the build platform is lowered into the resin tank, and the new structural layer is cured at the bottom of the tank. Between layers, the wiper removes residue and mixes the resin. The model is attached to the build platform via a sacricial layer (raft) and delicate supports. The tiles are printed with the larger openings in the vertical direction to maximize drainage. The laser moves much faster in the x and y directions because it only requires micro adjustments from the mirrors to move the laser spot across the build area, while the stepper motor for the z-axis must physically move a minimum of 30mm to clear the tank, and more once the model height has 48 reached 30mm. Building multiple models at once by placing them next to each other is much faster than individually printing models, provided they t. Since the printer runs autonomously, it is most time ecient to set up the printer to print models in 24 hour runs so the operator can remove and replace the build platform once a run has ended and restart the process. To start the printing process, the printer creates a \raft" which is a disposable structure attached to the build platform. A typical raft arrangement is shown in Fig. 3.2 b). From this, vertical supports then hold the model to be printed. After printing. the supports and raft are removed and only the desirable structure remain. The number of supports, and where they attach to the model are crucial to having a successful print that faithfully replicates the inputted geometry. Supports must be spaced adequately to prevent \drooping", bending, or failed attachments to the raft and supports. Having too many supports, and in particular internal supports, diminishes the surface nish. On a solid part, sanding the surface smooth is possible, but for delicate rods the \touch point" of the supports remains Removing the supports entails cutting or breaking them o, so internal supports in lattices are impossible to reach and remove. Consequently the lattice must be self-supporting while the resin is in its green state. Lattices were printed vertically as shown in Fig. 3.2 b) with the supports touching only the bottom layer. This assured that there was no excess roughness on the top of substrate, and that the bottom could be mounted at on the solid supporting sheet. After printing is complete, the models are washed in isopropyl alcohol (IPA) with > 99% concentration to remove any uncured, liquid resin that failed to drip o. At this point, the resin is still in a \green" state, indicating that it is solid but has not yet reached the nal material properties. The structures are more delicate and prone to deformation if not cured correctly. After washing, the nal curing step is to heat the models to 60 C and cure them under 405nm wavelength light which fully cures the resin. Typically, this is done by removing the model from the build platform after washing, and placing them on a rotating tray in the curing oven. However, the lattices are very delicate in green form, and removing them from the build platform in this state led to unacceptable deformation. It was found that the high porosity allowed sucient light to penetrate the models without rotation, allowing for the entire build platform to be placed in the curing oven, so models were only removed from the platform after full curing and cooling down. 49 Ultimately, material properties and printer resolution limit the permeabilties by limitingd;s x ;s y , and s z . Validation tests showed that rods with cross sections of d = 0:2mm were printable on the form2 (laser spot size nominally 0.145mm), but failure rates in printing straight lines were > 50%. This was acceptable when printing small cubic samples for permeametry tests but unsustainable for printing large amounts of tiles. Rod dimensions of d = 0:4 and 0:8mm were found to be more suitable for printing on form2 and form3. For the lattice family shown in Fig. 3.3 and given rod spacings, minimizing rod diameter as much as possible is key to increasing porosity. A rod diameter of d = 0:4mm was selected as a compromise between manufacturability and porosity. In contrast to fused position (lament) printers, the formlabs printers print the models \upside down", i.e. what would be a free-standing antenna undergoing compression due to its own weight is now a hanging lament in tension. This increases the maximum allowable s y or wall-normal spacing because the free-standing laments don't have to support more than their own weight before the next layers of streamwise and spanwise rods are printed. Printing at z = 25m over 100m helped increase adhesion between layers at the expense of time. When using z = 100m, several printing attempts failed as more layers were added below a bad joint and the increased weight caused the joint to fail. Validation tests showed that s y = 3:0mm was the maximum value for rods of d = 0:4mm to consistently print. The spanwise spacing, s z , is limited by the maximum allowable unsupported span between two upright elements. Initial testing showed that the maximum allowable span recommended by formlabs (21mm span for a 3mm by 5mm cross section beam) would not be printable for thin laments. Moreover, testing with d=0.2, 0.4, and 0.8mm rod sizes showed that s z > 2mm for the 0.2mm rod size resulted in signicant deformation (i.e. sagging) of the horizontal beams which also distorted the vertical laments. Similarly, increasing s z increases the load carried by the unsupported beam and the deformation increases with increasing s z . Additionally, the force that the beam experiences is not just supporting its own weight but has to withstand the peeling o the resin tank. A maximum span of 1.7mm was printable for 0.2mm rod sizes, and 3.0mm for 0.4mm rods. Material property estimates are available, but properties strongly depend on curing procedures (duration in IPA, IPA concentration) and degrade under environmental conditions such as exposure 50 to sunlight, prolonged submersion in water and absorption of water and water temperature changes. Beam theory calculations assuming the Young's modulus E for the uncured material is the same as for the \green material" indicate that bending under its own weight is not responsible for the deformations observed for large separations. 3.4 Finished Product Figure 3.3: Photographs of prepared materials. The rod diameter is d = 0:4mm, and the lattice has rod spacings of s x = 0:8mm,s y = 3:0mm, and s z = 3:0mm. The lattice was printed as 5 full pores and a solid set of rods on the top and bottom layer, yielding a nal thickness of 15.4mm. The nal material was spraypainted black to minimize laser re ections. Photographs of the manufactured materials are shown in Fig. 3.3. Dimensions were veried with precision calipers and prints with unacceptable deviations (> 0:2mm) were discarded. The printing success rate was 90% and failures were predominately full layer failures due to bad layer adhesion rather than deformation. Note that the top surface was spray painted with a thin layer of black Rustoleum TM to minimize laser re ections. The substrate was designed with streamwise and spanwise rods in the top layer. This helps with stability for printing, and minimizing roughness at the surface. In practice, this also reduces the local porosity. Considering a averaging volume that is only d in thickness, the local porosity is local = d(sxd)(szd) d(sxsz) 0:4 for s x = 0:8mm, s y =s z = 3:0mm andd = 0:4mm. This represents a lower limit for local porosity at the interface. As the averaging volume is increased, the bulk porosity of 0.87 is reached. 51 3.5 Permeability estimates Permeability was estimated using a modied constant head permeameter. The diagonal terms of the permeability tensor (K xx ;K yy ; and K yy ) were estimated from Darcy's law, @P @x i = K ii U i | {z } Darcy's law U 2 i : |{z} Forchheimer correction (3.1) Here,i represents the direction,U i is the mean velocity in that direction,P is the pressure, is the Forchheimer coecient andK ii is the permeability. For low ow rates, Darcy's law is appropriate, but for higher ow rates inertial eects become important and Forchheimer's correction term is added (Forchheimer, 1901). The experimental setup is shown in Fig. 3.4. The lattice to be characterized was printed as a cube with dimensions of 25x25x25mm, and inserted in the test section. Pressure transducers (Omega Corp PX409-001DWUI, 1psi range, 0.08% resolution), measured the pressure gradient in the streamwise direction across the sample. The mass ow was measuredd using a downstream ow meter (Omega Corp FLR 1012, 0.5-5L/min3%). Raw data is shown in Fig. 3.5 a), along along with ts to Darcy's law with and without the quadratic correction. For the low ow rates present, inertial eects were negligible and a linear estimate is sucient. Panel b) shows the streamwise permeability estimates for 3D printed cubic samples with varying pore sizes. Note that the cubes labelled (1,1.7,1.7) and (1.7,1.7,1) represent the diagonal terms of the permeability matrix for the cube with rod spacings of s x = 1, s y = 1:7, s z = 1:7mm i.e. K = 2 6 6 6 6 4 K xx K xy K xz K yx K yy K yz K zx K zy K zz 3 7 7 7 7 5 = 2 6 6 6 6 4 19:2 0 0 0 7:1 0 0 0 7:1 3 7 7 7 7 5 10 9 m 2 : K yy = K zz were assumed to be equal because pore sizes were equal and o diagonal terms were assumed to be 0. 52 Figure 3.4: Diagram of permeametry experiment, overview (top) and detailed view of test section (bottom). The working uid (water) is driven by a submersible pump from a reservoir and regulated by a programmable valve. The ow enters the permeameter test section through a ow-straightening mesh and encounters a cubic sample. The pressure gradient across the sample is measured in three dimensions ( @P @x ; @P @y ; @P @z ) and mass ow rate is measured downstream. This procedure can be extended to measure all 9 components of the permeability tensor K = 2 6 6 6 6 4 K xx K xy K xz K yx K yy K yz K zx K zy K zz 3 7 7 7 7 5 : From Darcy's law,rP =K 1 U, and we can dene the resistance matrix R = K 1 , which has 9 components: 2 6 6 6 6 4 @P @x @P @y @P @z 3 7 7 7 7 5 = 2 6 6 6 6 4 R xx R xy R xz R yx R yy R yz R zx R zy R zz 3 7 7 7 7 5 2 6 6 6 6 4 U V W 3 7 7 7 7 5 Using the setup shown in Fig.3.4, there is only mean ow in the streamwise direction (V;W = 0), and consequently the rst column of R (R xx ,R yx , andR zx ) was estimated from the three pressure 53 0 0.005 0.01 0.015 0.02 0 1000 experimental data linear fit quadratic fit 0 0.5 1 1.5 2 2.5 3 3.5 4 0 5 10 15 20 25 30 35 40 45 50 (1,1,1,0.2) (1.7,1.7,1,0.2) (1,1.5,1.5,0.2) (1,1.7,1.7,0.2) (1,2,2,0.2) Figure 3.5: Panel a) (left) shows the raw pressure drop drop data (circle) and linear and quadratic ts (lines). In this case, both linear and quadratic t for permeability agree within 5%. In panel b) (right) 1D permeability estimates for sample with varying pore sizes are shown. The labels refer to the rod separations and rod diameter, i.e. (s x ;s y ;s z ;d). Note that the streamwise normal area (frontal open area) is proportional to (s z d)(s y d). The cases labelled (1,1.7,1.7) and (1.7,1,1.7) refer to the same cube, rotated 90 . drop measurements. The sample was then rotated twice more, which gave estimates for the second and third columns of R respectively. R was then inverted to estimate K. Note that methods for estimating permeability have their origins in soil measurements for construction and oil extraction. Motivated by oil reservoirs, Liakopoulos (1965) measured the permeability (1D) of anisotropic sandstone samples taken at an angle of =0, 30, and 60 with respect to a normal axis, using a constant head permeameter and the measured values matched predictions obtained by rotating the permeability matrix measured at 0 , K 0 = 2 4 K xx K xy K yx K yy 3 5 by an angle using a transformation matrix, T = 2 4 cos sin sin cos 3 5 to give a rotated matrix K 0 (), K 0 = TK 0 T 0 = 2 4 Kxx+Kyy 2 + KxxKyy 2 cos 2 +K xy sin 2 KyyKxx 2 sin 2 +K xy cos 2 kyyKxx 2 sin 2 +K xy cos 2 Kxx+Kyy 2 KxxKyy 2 cos 2K xy sin 2 : 3 5 (3.2) 54 Lei et al. (2015) made similar measurements for higher porosity, sandstone samples by applying a set pressure gradients @P @x = @P @y and measuring the out ow q x ;q y as function of the rotation angle. This experiment conrmed that the permeability matrix is elliptical, and a coordinate transformation is appropriate and sucient to estimate permeability matrices for o-axis ows. To test the 3D permeametry, test cubes of the lattice with s x = 2, s y = 2, s z = 0:8mm and rod size d = 0:4mm were printed with rotation angles of = 0 90 . Figure 3.6 a) shows the raw pressure drop data for the two extreme cases, highlighting the order of magnitude dierence in pressure drop between low and high permeabilities. Permeability estimates for the 7 tested orientations are shown in panel b). Uncertainty estimates are obtained by varying the number of data points from a) used to estimate the permeabilities. The solid line and shaded uncertainty bar indicate predications made by rotating K 0 and K 90 . These results imply that the odiagonal terms are modiable within the constraints suggested by Eqn. 3.2. Figure 3.6: Permeability estimates are made for porous cubes with s x = 2, s y = 2, s z = 0:8mm and rod size d = 0:4mm and rotation angles of = 0 to 90 in 15 increments. Raw pressure drop data is shown in a). Note that for the larger pore sizes (0 ), the pressure drop is an order of magnitude lower than for the smaller open area (90 ). Permeability estimates for the 7 tested orientations are shown in panel b). Uncertainty estimates are obtained by varying the number of data points from a) used to estimate the permeabilities. The solid line and shaded uncertainty bar indicate predictions made by rotatingK 0 andK 90 . 55 3.6 Conclusion Experimental diculties measuring @P @y and @P @z have so far prevented the successful estimation of the full permeability tensor. For large pore sizes, the pressure drop is small, and the pressure transducers selected do not have adequate resolution. Reducing the porosity of the sample would help by increasing the pressure drop across the sample. However, the 3D printed material used to create the permeameter itself expanded through water absorption and deformed under pressure. Gaps around the sample are larger than the pore sizes for low porosity models. I have designed an updated permeameter designed to be machined out of aluminium that will address these lim- itations. Although we did not measure o-diagonal terms for permeability tensors, experiments measuring pressure drop for rotated cubes agree with estimates for rotated permeability matrices, indicating that o-diagonal terms have also been modied. Nonetheless, porous materials with tunable permeability tensors and anisotropies of up to xy = Kxx Kyy = 8 were successfully manufactured, facilitating the rst experiments over such materials which are detailed in the next chapters. Stokes Flow simulations were used to estimate permeabilities of the nal materials used in these experiments, because the high permeabilities yielded pressure gradients that were too small to measure in the current set up. Details of numerical permeametry are contained in §4.4.1. 56 Chapter 4 Resolvent-based design and experimental testing of porous materials for passive turbulence control 4.1 Comment and Summary This chapter has been submitted for publication in a special issue of the International Journal of Heat and Fluid Flow (Chavarin et al., 2020) and was presented at the 11 th annual Turbulence and Shear Flow Phenomena (TSFP) conference. This experiment served as a small-scale validation to test the response of a turbulent channel ow to two anisotropic porous substrates with xy 8 and xy 1=8. The streamwise preferential material (\x-permeable") is designed to maximize p K xx + p K zz + and minimize p K yy in an attempt to reach a drag reducing state. To our knowledge, this data set is the rst over purposefully designed anisotropic media with higher streamwise permeabilities. The wall-normal preferential material is designed to maximize p K yy . Such a material could be used to enhance heat transfer over porous substrates. The experiments detailed below were run under adiabatic conditions. For the selected materials, experimental results (mean turbulence statistics and POD modes obtained from PIV elds) are compared to low-order model predictions. My colleague, Andrew Chavarin, extended the resolvent formulation for channel ow to make predictions for modes reselmbling the near-wall cycle and Kelvin Helmholtz rollers using both synthetic and the ex- perimental mean velocity proles. Since there is good agreement between model predictions and experimental results, the resolvent formulation could be used to design anisotropic porous materials as passive ow control devices in the future. 57 Particle Image Velocimetry (PIV) measurements were used to compute mean turbulence statis- tics and to educe coherent structure via snapshot Proper Orthogonal Decomposition (POD). Fric- tion velocity estimates based on the Reynolds shear stress proles do not show evidence of dis- cernible friction reduction (or increase) over the streamwise-preferential substrate with xy 8 relative to a smooth wall ow at identical bulk Reynolds number. A signicant increase in friction is observed over the substrate with xy 1=8. Coherent structures extracted via POD analysis show qualitative agreement with model predictions. In addition to my advisor, this paper was written and submitted with my colleagues Andrew Chavarin (AC) and Shilpa Vijay (SV). Shilpa designed, manufactured, and benchmarked the water channel facility used in this experiment to investigate heat transfer over porous substrates. She also set up the laser and camera that we used to acquire PIV data. I manufactured the porous substrates designed to t in the channels cutout, assisted with acquiring PIV data and processed raw images into velocity statistics. Andrew computed the numerical permeability estimates to characterize the lattices. Model predictions are a result of his expansion of the resolvent operator to account for a porous wall in turbulent channel ows. 4.2 Introduction 4.2.1 Motivation Functional surfaces such as sharkskin-inspired riblets and denticles are some of the simplest and most eective control techniques tested thus far for turbulent friction reduction. Appropriately shaped and sized riblets have shown the ability to reduce drag up to 10% in laboratory experiments and up to 2% in real world conditions (Luchini et al., 1991; Robert, 1992; Walsh and Lindemann, 1984; Garcia-Mayoral and Jimenez, 2011). It is generally accepted that the drag-reducing ability of such surfaces arises from their anisotropy: they oer much less resistance to streamwise ows compared to spanwise ows (Luchini et al., 1991). The mean ow in the streamwise (x) direction is essentially unimpeded within the riblet grooves, generating high interfacial slip. However, cross- ows in the wall-normal (y) and spanwise (z) directions arising from turbulence are blocked by the riblets and pushed further from the wall. This blocking eect weakens the quasi-streamwise vortices 58 associated with the energetic near-wall (NW) cycle (Robinson, 1991; Smits et al., 2011), and reduces turbulent mixing and momentum transfer above the riblets (Choi et al., 1993). Skin friction reduction initially increases with increasing riblet spacing and height. However, above a certain size threshold, performance deteriorates dramatically. Early studies attributed this deterioration of performance to the NW vortices lodging within the riblet grooves (Choi et al., 1993; Lee and Lee, 2001). More recently, Garcia-Mayoral and Jimenez (2011) have shown that a Kelvin-Helmholtz (KH) instability may also contribute to the deterioration of performance. Recent theoretical eorts and numerical simulations suggest that streamwise-preferential per- meable substrates have the potential to reduce drag in wall-bounded turbulent ows through a similar mechanism as riblets (Abderrahaman-Elena and Garc a-Mayoral, 2017; Rosti et al., 2018; G omez-de-Segura and Garc a-Mayoral, 2019). Simulation results predict that as much as 25% drag reduction may be achievable through the use of anisotropic permeable substrates that have streamwise permeability (K xx ) that is higher than the wall-normal (K yy ) or spanwise (K yy ) per- meabilities (Rosti et al., 2018; G omez-de-Segura and Garc a-Mayoral, 2019). Similar to ow over riblets, a KH instability is also predicted to arise for materials with high wall-normal permeability (Abderrahaman-Elena and Garc a-Mayoral, 2017; G omez-de-Segura and Garc a-Mayoral, 2019). However, these predictions remain to be tested in physical experiments. In this work, we seek to design and fabricate anisotropic porous materials that have the potential to reduction skin friction, and to test these materials in benchtop channel ow experiments. 4.2.2 Previous Theoretical Eorts and Simulations As noted above, streamwise-preferential porous materials have the potential to reduce drag in turbulent ows through a similar mechanism as riblets. With anisotropic porous substrates, high porosity and streamwise permeability contribute to a substantial interfacial slip velocity for the mean ow, while low wall-normal and spanwise permeability limit turbulence penetration into the porous substrate. These eects can also be interpreted in terms of the slip length and virtual origin framework used by Luchini et al. (1991) to characterize ows over riblets. Specically, the interfacial slip velocity for the mean ow can be related to a streamwise slip length l + U that determines the virtual origin perceived by the mean ow below the porous interface. Following 59 standard notation, a superscript + denotes normalization with respect to the friction velocity (u ) and kinematic viscosity (). Similarly, the distance to which the turbulent uctuations penetrate into the porous substrate can be related to a transverse slip length l + t that determines the virtual origin for the turbulent cross- ows. Note that the virtual origin for the turbulent cross- ow can also be interpreted as the location at which the quasi-streamwise NW vortices perceive a non- slipping wall (G omez-de-Segura and Garc a-Mayoral, 2019; Garc a-Mayoral et al., 2019). The initial decrease in drag over riblets of increasing size has been shown to depend on the dierence between the streamwise and transverse slip lengths, D/ l + U l + t . Physically, when there is a positive oset between the virtual origins for the mean ow and the transverse uctuations (l + U > l + t ), the quasi-streamwise NW vortices are pushed into a region of lower mean shear. This weakens the ow induced by the NW vortices and leads to a reduction in turbulent Reynolds stresses and skin friction. The virtual origin concept has recently been extended to the case of anisotropic porous substrates. Specically, using the Brinkman equations, Abderrahaman-Elena and Garc a-Mayoral (2017) established the following relationships between the streamwise and transverse slip lengths and permeabilities: l + U / p K + xx and l + t / p K + zz . These relationships indicate that the initial decrease in drag over anisotropic porous materials is expected to be proportional to the dierence between the streamwise and spanwise permeability length scales, D/l + U l + t / p K + xx p K + zz . Recent Direct Numerical Simulation (DNS) results obtained by G omez-de-Segura and Garc a- Mayoral (2019) for turbulent ow over anisotropic permeable substrates show good agreement with the predictions made by Abderrahaman-Elena and Garc a-Mayoral (2017). Specically, DNS results show the presence of a linear regime in which drag reduction is initially proportional to p K + xx p K + zz . Thus, drag reduction is expected for substrates with high streamwise permeability and low spanwise permeability. Further, DNS snapshots of the ow eld over drag-reducing porous substrates indicate that the NW dynamics are similar to those observed over smooth walls (i.e., characterized by the presence of elongated streaky structures). However, there is a decrease in tur- bulent uctuation intensity and Reynolds shear stress close to the wall (Busse and Sandham, 2012; G omez-de Segura et al., 2018; Rosti et al., 2018). These observations indicate that slip length-based models generate useful predictions for the linear drag reduction regime over anisotropic porous sub- strates. However, these models do not incorporate the eects of wall-normal permeability, which 60 could also aect turbulence penetration into the substrate. Moreover, the maximum achievable drag reduction is thought to be limited by the onset of a KH-type instability. Linear instability analyses predict that the emergence of such instabilities is controlled byK + yy (Abderrahaman-Elena and Garc a-Mayoral, 2017; G omez-de Segura et al., 2018). DNS results obtained by G omez-de- Segura and Garc a-Mayoral (2019) conrm the emergence of energetic spanwise rollers as the wall- normal permeability increases beyond p K + yy 0:4. Momentum balance arguments show that the additional Reynolds shear stress generated by these rollers is responsible for the deterioration of drag reduction performance and, eventually, an increase in drag over the porous substrates. For completeness, we note that spanwise rollers have also been documented in other simulations over permeable walls and isotropic porous substrates (Jim enez and Pinelli, 1999a; Breugem et al., 2006; Busse and Sandham, 2012; Rosti et al., 2015; Kuwata and Suga, 2016, 2017; Rosti et al., 2018). The DNS results of G omez-de-Segura and Garc a-Mayoral (2019) suggest that drag reduction over anisotropic permeable substrates depends on two key factors: the ability of the substrate to weaken the energetic NW cycle, which depends onK + xx andK + zz , and the emergence of KH rollers, which is dictated byK + yy . However, these predictions remain to be tested in physical experiments. 4.2.3 Previous Experiments Previous experimental studies of turbulent ow over porous materials have focused primarily on granular media such as packed beds of spheres (e.g., Zagni and Smith, 1976; Pokrajac and Manes, 2009; Horton and Pokrajac, 2009; Kim et al., 2016) or commercially-available materials such as reticulated foams (Manes et al., 2011; Efstathiou and Luhar, 2018), providing signicant insight into how porous substrates modify the near-wall turbulent mean ow and statistics. For instance, they have characterized the eect of substrate permeability on the interfacial slip velocity and the logarithmic region of the mean ow reasonably well, and documented the emergence of spanwise rollers. Recent work by Kim et al. (2020) also attempts to systematically delineate the eect of porosity and interfacial roughness. However, sphere beds and reticulated foams are approximately isotropic. Few studies have ex- plicitly considered the eect of anisotropic porous materials on turbulent ows. Even fewer studies have considered the eect of streamwise-preferential materials that have the potential to reduce 61 drag. The early experiments of Kong and Schetz (1982) over mesh and perforated sheets considered materials with signicantly higher wall-normal permeability compared to streamwise permeability, i.e., materials with anisotropy ratio xy = K xx =K yy < 1. Similarly, recent experiments by Suga et al. (2018) have also focused on materials with xy < 1. These experiments considered channel ow at bulk Reynolds numbersRe b = 90013600, where one wall was lined with a porous material. The porous substrates consisted of layers of co-polymer nets and the resulting anisotropy ratio for these substrates was xy = 1=190 1=1:5. In all the cases considered by Suga et al. (2018), an increase in friction velocity was observed at the porous wall, and the total friction drag increased by 13-73%. To the best of our knowledge, the only prior experimental evidence of turbulent drag reduc- tion over porous materials comes from the seal fur tests pursued by Itoh et al. (2006). Given the streamwise-preferential nature of sea fur, these observations provide limited support for the theoretical predictions and simulation results discussed in the previous section. However, further verication of these prior results requires a more complete characterization of how streamwise- preferential materials with known permeability aect turbulent ows. Finally, note that canopies of terrestrial or aquatic vegetation and corals reefs can also be consid- ered anisotropic porous materials. However, since such substrates do not exhibit high streamwise permeability, the extensive literature on ows over vegetation canopies and coral reefs is not re- viewed here for brevity. 4.2.4 Contribution and Outline In this paper we seek to design and fabricate anisotropic porous materials that have the potential to reduce skin friction, and to test these materials in laboratory experiments. In particular, we leverage advances in additive manufacturing (3D-printing) to fabricate cellular porous materials that have desirable anisotropy ratios xy > 1, and test the eect of these materials in benchtop channel ow experiments. In order to predict the viability of these substrates for passive drag reduction, we extend the resolvent analysis framework the resolvent framework of McKeon and Sharma (2010) to account for porous substrates. Recent work by Chavarin and Luhar (2020) shows that the resolvent framework can serve as a useful assessment and design tool for riblets. 62 In particular, Chavarin and Luhar (2020) show that the resolvent framework can: (i) predict whether riblets of specied geometry are likely to suppress or amplify the energetic NW cycle, and (ii) be used to test for the emergence of spanwise rollers resembling KH vortices. Here, we use resolvent analysis as a preliminary design tool: we use it to identify streamwise-preferential porous geometries with xy > 1 that are likely to suppress the NW cycle. These geometries are then 3D printed for the laboratory experiments. Measurements made over the material with xy > 1 are compared against measurements made over a geometrically-similar porous material with xy < 1 as well as a solid smooth wall. In addition to serving as a test for whether streamwise- preferential materials can reduce friction, these experiments also provide preliminary insights into how anisotropic materials with xy < 1 and xy > 1 modify the near-wall ow physics. The remainder of this paper is structured as follows. The resolvent-based modeling framework is described further in §4.3. The experimental methods are presented in §4.4. Model predictions and experimental results are discussed together in §5.5. Specically, the resolvent-based predictions used to design the porous materials tested in the experiments are shown in §4.5.1. Experimental measurements for the mean prole and turbulence statistics are discussed in §4.5.2 and the ow features identied via snapshot proper orthogonal decomposition (POD) are shown in §4.5.3. One of the inputs required for resolvent analysis is an estimate of the turbulent mean prole. The model predictions shown in §4.5.1 are obtained using a synthetic mean prole computed using an eddy viscosity formulation. These predictions are compared against those made using the mean proles measured in the experiments in §4.5.4. Brief concluding remarks are presented in §4.6. 4.3 Modeling In this section, we describe the extension to the resolvent framework to account for porous substrates and provide details on numerical implementation. 4.3.1 Extended Resolvent Formulation We utilize a modied version of the resolvent formulation proposed by McKeon and Sharma (2010) to predict the drag performance of anisotropic permeable materials for passive turbulence control. For wall-bounded turbulent ows, the resolvent formulation interprets the Navier-Stokes 63 equations, Fourier-transformed in the (approximately) homogeneous streamwise and spanwise di- rections and in time, as a forcing-response system. For this system the nonlinear convective terms are treated as internal forcing (input) to the system composed from the remaining linear terms of the Navier-Stokes equations. At every wavenumber-frequency combination = ( x ; z ;!) this internal forcing generates a turbulent velocity and pressure response. A gain-based singular value decomposition of the forcing-response transfer function|the resolvent operator|yields a set of highly amplied velocity and pressure response modes (left singular vectors) and the correspond- ing forcing-response gains (singular values). The response modes|termed resolvent modes|are ow structures with streamwise and spanwise wavelength x = 2= x and z = 2= z , respec- tively, traveling at speedc =!= x . The forcing-response gain is a measure of energy amplication in the system, and serves as a metric of control performance. Previous work shows that specic high-gain response modes can serve as useful surrogates for energetic structures such as the NW cycle (Moarref et al., 2013). These resolvent modes can therefore serve as building blocks for the design and optimization control strategies (Luhar et al., 2014b, 2015; Nakashima et al., 2017; Toedtli et al., 2019; Chavarin and Luhar, 2020). Specically, these prior eorts show that suppression of the NW resolvent mode is a useful indicator of drag reduction performance for both active and passive control of wall turbulence. In other words, if a control technique is unable to suppress the surrogate NW resolvent mode, then it is unlikely to yield drag reduction for the full turbulent ow eld. In addition, recent work by Chavarin and Luhar (2020) shows that the resolvent framework is also able to predict the emergence of energetic spanwise rollers over riblets that contribute to the deterioration of drag reduction performance (Garcia-Mayoral and Jimenez, 2011). Building on these prior studies, here we use the resolvent framework to test whether a given porous material can (i) suppress the gain for the resolvent mode that serves as a surrogate for the NW cycle, and (ii) limit the emergence of energetic spanwise rollers resembling KH vortices. 64 For this analysis we formulate the resolvent framework using the volume-averaged Navier-Stokes (VANS) equations in which the eect of anisotropic porous substrates is included via a generalized version of Darcy's law (Breugem et al., 2006): @hui @t + 1 " r "huihui +" = 1 " r("hpi) + 1 "Re r 2 ("hui) " Re K 1 hui; (4.1a) r ("hui) = 0 (4.1b) Here,hui andhpi represent the dimensionless volume averaged velocity and pressure respectively, " represents the porosity, and K is the dimensionless permeability tensor. The equations above have been normalized by the channel height h (see Fig. 4.1) and the friction velocity u and the friction Reynolds number is given by Re = h + = u h=. The sub-lter scale stresses which arise from volume-averaging the Navier Stokes equations are dened as =huuihuihui. The unobstructed uid domain is characterized by porosity " = 1 and innite permeability. For this region, the Darcy term drops out of the governing equations, and Eq. 4.1 reduces to the standard Navier-Stokes equations. Note the the expression above omits the nonlinear Forchheimer term. For the remainder of the paper, we focus on porous substrates for which the permeability tensor is diagonal and has the following form K = diag(K xx ;K yy ;K zz ). Further, since we are primarily considering structures (i.e., NW cycle and KH rollers) that are much larger than the pore scale, we assume that the sub- lter scale stresses are negligible. These assumptions and modeling simplications are consistent with those made in recent numerical simulations of ow over anisotropic porous substrates (Rosti et al., 2018; G omez-de-Segura and Garc a-Mayoral, 2019). To further simplify the expressions in Eq. 4.1, we assume that the porous substrate is spatially homogeneous and has constant porosity. This yields: @u @t +r uu) = rp + 1 Re r 2 u 1 Re "K 1 u; (4.2a) ru = 0; (4.2b) 65 where thehi notation has been omitted for simplicity. Resolvent analysis proceeds as follows. First, we employ a standard Reynolds-averaging procedure such that velocity is decomposed into a mean component (U) and a uctuation about this mean (u 0 ). Next, the governing equations for the uctuations are Fourier-transformed and expressed as 2 4 u p 3 5 =H f : (4.3) Here,u andp represents the Fourier-transformed velocity and pressure uctuations, f represents the nonlinear forcing terms, and H is the resolvent operator representing the linear forcing- response dynamics. At every wavenumber-frequency combination , an SVD of the discretized resolvent operator, i.e., H = X m ;m ;m ;m ; (4.4) yields forcing modes (right-singular vectors, ;m ) and velocity/pressure response modes (left- singular vectors, ;m ) that are ordered based on their forcing-response gain (singular values, ;m ). For our analysis, the resolvent operator is scaled to enforce an L 2 energy norm and so the change in singular value relative to the smooth wall case can be interpreted as a measure of energy amplication or suppression. Importantly, previous work shows that the resolvent operator tends to be low-rank at wavenumber-frequency combinations that are energetic in turbulent ows (Moarref et al., 2013; McKeon and Sharma, 2010). Consequently the resolvent operator can be well approximated using a rank-1 truncation after the SVD, i.e., by only considering the rst singular values, ;1 , and response modes, ;1 . Chavarin and Luhar (2020) show that resolvent analysis with this rank-1 approximation provides useful insight into the eect of riblets on wall turbulence. We retain the rank-1 approximation here as well, and drop the additional subscript 1. For further discussion pertaining to resolvent analysis for wall-bounded turbulent ows and the rank-1 approximation, the reader is referred to several recent studies in this area (McKeon and Sharma, 2010; Luhar et al., 2014a; McKeon, 2017). For the remainder of this work, we focus on modes serving as a surrogate model for the dynam- ically important NW cycle (i.e., with corresponding to + x = 10 3 , + z = 10 2 , and c + = 10) and consider the highest singular value as a measure of performance. If the singular value|or gain|is 66 reduced over the porous material ( ;p ) relative to the smooth wall value ( ;s ), the porous material is likely to suppress the corresponding ow structure. We also test for the emergence of high-gain spanwise-constant rollers resembling Kelvin-Helmholtz vortices (i.e., with z = 0) at the porous interface. 4.3.2 Numerical Implementation Previous theoretical and numerical eorts have primarily considered a symmetric channel ge- ometry, with porous materials at both the upper and lower walls (Rosti et al., 2018; G omez-de- Segura and Garc a-Mayoral, 2019). However, this geometry would have limited optical access for Particle Image Velocimetry (PIV) in the laboratory experiments discussed below. Instead, we consider an asymmetric channel geometry corresponding to the experimental setup shown in Fig. 4.1(a). The unobstructed region of the channel spans y 2 [0;h] and the porous material occupies the region corresponding to y2 (h;H +h), with H =h. We generate model predictions for Re = u h= = 120, which corresponds roughly to the conditions tested in the experiments, and for Re = 360, which corresponds to prior numerical simulations (Rosti et al., 2018). As noted earlier, the spatially-homogeneous porous substrate is dened by its principal permeability components K = diag(K xx ;K yy ;K zz ). For simplicity, the porosity is set to " = 1, though the materials tested in the experiment have a porosity " = 0:87. No-slip boundary conditions are applied at the true walls, which are located at y = 0 and y = H +h. The interface between the porous substrate and the unobstructed domain is located at y =h. The resolvent operator is discretized in the wall-normal direction using the Chebyshev collocation method detailed by Aurentz and Trefethen (2017). This approach allows us to discretize the unobstructed and porous domains independently and couple these two domains through the jump boundary conditions proposed by Ochoa-Tapia and Whitaker (1995). The following boundary conditions are applied at the interface: uj y=h =uj y=h + ; (4.5a) pj y=h = pj y=h + ; (4.5b) 67 @u @y y=h 1 " @u @y y=h + = 0; (4.5c) @w @y y=h 1 " @w @y y=h + = 0; (4.5d) where h and h + refer to y-locations on either side of the interface. Construction of the resolvent operator also requires a mean velocity prole, U = [U(y); 0; 0]. The mean velocity is predicted from the Reynolds-averaged mean ow equation which contains a linear Darcy drag term, with the Reynolds stress term modeled using an eddy viscosity. For the unobstructed region, the eddy viscosity proles for our analysis are generated using the analytical model developed by Reynolds and Tiederman (1967). For the porous layer, the eddy viscosity is set to zero. Since resolvent-based predictions are sensitive to the exact form of the mean prole, in Sec- tion 4.5.4, we compare model predictions obtained using the synthetic mean prole discussed above against predictions obtained using a prole tted to the experimental measurements described be- low. This tted prole is synthesized from the experimental measurements as follows. First, the mean velocity (U(y)) and Reynolds shear stress (=u 0 v 0 ) proles are obtained from the PIV mea- surements described below by averaging in time and in the streamwise direction. These proles are then used to estimate the eddy viscosity prole, T =u 0 v 0 = dU=dy , where the mean shear is dU=dy is approximated using a nite dierence scheme. The points near the maximum in the mean prole, corresponding to dU=dy 0, are removed and a smooth cubic spline is tted to the resulting prole. Finally, the eddy viscosity prole is allowed to smoothly transition to zero in the porous medium and values for which t (y)< 0 are removed and set to 0. This tted eddy viscosity prole is then used to generate predictions for the mean prole (see Fig. 4.5) used in the resolvent operator. For the model predictions presented in this paper, a total of 226 Chebyshev nodes are used for discretization in the wall-normal direction. The nodes are divided evenly between the porous and the unobstructed domain. Further grid renement beyond this point led to changes in singular values smaller than O(10 4 ). 68 4.4 Experimental Methods 4.4.1 3D Printed Porous Materials For this study, two custom anisotropic porous materials were fabricated using a stereo-lithographic 3D printer (formlabs Form 2) based on input from the resolvent-based predictions described below. The porous material microstructure consisted of a cubic lattice of rectangular rods with constant cross-section and varying spacing in the x, y, and z directions. Fabrication constraints (printing resolution, allowable unsupported lengths, resin drainage) limited the maximum anisotropy that could be achieved. The anisotropy was varied by controlling the size of the pores normal to the spanwise and streamwise directions. The minimum pore size was dictated by the printer resolution as the rods fused and the surface became solid if the separation between two rods fell below the minimum resolution. Moreover, the maximum pore size was limited by the maximum overhang between rods because with excessive overhang, the horizontal rod sagged and deviated from the design geometry. Considering these limitations, the following two geometries represented a good compromise between reliable fabrication and anisotropy. The rst case with spacings s x = 0:8 mm ands y =s z = 3:0 mm contained larger pores facing the streamwise direction and small pores facing the wall-normal and spanwise directions (see Fig. 4.1(b)). The second geometrically similar, but rotated, case with s x =s z = 3:0 mm and s y = 0:8 mm contained larger pores facing the wall- normal direction (see Fig. 4.1(c)). These materials are referred to as x-permeable and y-permeable, respectively, for the remainder of this paper. For both geometries, the rod cross-section was a square of size dd, with d = 0:4 mm. The porosity was " 0:87. A 3D-printed sample of the x-permeable material is shown in Fig. 4.1(d). Following the approach of Zampogna and Bottaro (2016), the permeability tensor (K) for these anisotropic porous materials was determined by solving independent forced Stokes ow problems for a unit cell of the cubic lattice in the ANSYS Fluent software package. Due to the symmetric nature of the microstructures tested, only two Stokes ow problems were required to determine the permeability components K xx and K yy ; K zz is equal to either K xx or K yy depending on conguration (i.e., x-permeable or y-permeable). For these two Stokes ow problems, a uniform body forcing of unit amplitude was applied in the direction of the permeability component being 69 " K xx =H 2 K yy =H 2 K zz =H 2 x-permeable 0.87 4:3e3 5:5e4 5:5e4 y-permeable 0.87 5:5e4 4:3e3 5:5e4 Table 4.1: Dimensionless permeability estimates for the 3D-printed porous materials. H = 6:34 mm is the height of the porous substrates tested in the channel ow experiments. evaluated. The permeability was determined from the resulting volume-averaged velocity using Darcy's law. Periodic boundary conditions were applied to the boundaries of the unit cell and a no-slip condition was applied at the solid boundaries. A time-marching scheme was used for each of these simulations. The solutions were determined to be at steady state when the residual in the permeability was less than 10 6 . A mesh independence study conrmed that our results were grid converged. The resulting permeability estimates are shown in Table 4.1. The anisotropic ratio is xy =K xx =K yy 8 for the x-permeable case and xy 1=8 for the y-permeable case. 4.4.2 Channel Flow Experiment h = 6.34mm H = 6.34mm L = 320mm≈ 50h x p = 195≈ 31h 3.5h PIV Flo w (a) channel conguration (b) x-permeable lattice (c) y-permeable lattice (d) x-permeable material Figure 4.1: (a) Schematic of the channel ow experiment (not to scale). (b,c) Renderings for the x-permeable and y-permeable material. (d) Image of 3D-printed x-permeable porous tile. Note that the x-permeable material features large openings normal to the streamwise incoming ow. The y-permeable material has large openings in the wall-normal direction. The anisotropic porous substrates described above were tested in a turbulent channel ow experiment, albeit at very low Reynolds number. A schematic of the experimental setup is shown 70 in Fig. 4.1. A custom test section was machined from acrylic with a cutout of length L = 320 mm designed to hold the porous substrates. The width of the test section was W = 50 mm, and the height of the unobstructed region was h = 6:34 mm. The cutout was located approximately 150mm from the in ow, and allowed for 3D-printed tiles of thickness H = h = 6:34 mm to be mounted ush with the smooth wall upstream of the cutout. Note that the number of pores accommodated over the height of the tiles was limited to 2 for the x-permeable case and 8 for the y-permeable case, indicating limited separation between the pore-scale and outer-scale ow. For a baseline comparison, experiments were also carried out with a solid smooth walled insert placed in the cutout. We recognize that there is insucient scale separation between the pore size and the height of the porous medium and, as such, the volume-averaged representation shown in Eq. 4.1 is not truly valid. The large pore sizes are driven by the minimum pore size and the desire to generate a maximum anisotropy ( xy = K xx =K yy ). Nevertheless, this represents the rst set of experiments over custom-designed, anisotropic porous media. Flow in the channel was generated using a submersible pump placed in a large water tank. The ow rate was controlled using an electronic proportioning valve. The volumetric ow rate was Q = 92 cm 3 /s for the smooth wall and x-permeable cases, and Q = 82 cm 3 /s for the y-permeable case. Thus, the bulk Reynolds number was Re b = Q=(W) = 1840 for the smooth wall and x-permeable case, and Re b = 1640 for the y-permeable case. This corresponds to the lower end of the Re b ranges considered by (Suga et al., 2018) in recent experiments over anisotropic porous materials. A 5W continuous wave laser with integrated optics was used to generate a laser sheet in the streamwise-wall normal direction at mid-span. A high-speed camera (Phantom VEO-410L) was used to capture images near the downstream end of the porous section. Recent turbulent boundary layer experiments over isotropic porous foams show that the ow adjusts to the new substrate over a streamwise distance of 30H 40H, where H is the porous layer thickness (Efstathiou and Luhar, 2018). To provide an adequate development length therefore, the PIV eld of view began 195mm ( 31H) from the leading edge of the porous section and extended 22mm ( 3:5H) downstream. Images were acquired at 2kHz for 10 seconds for a total of 20,000 images. The total 71 duration of the measurements is approximately 100 turnover times, where the turnover time is estimated as (H +h)=U b and the bulk-averaged velocity is dened as U b = Q=[W (H +h)]. The images were processed in PIVlab (Thielicke and Stamhuis, 2014) using the Fast-Fourier transform routine with a minimum box size of 16 pixels and 50% overlap, which yielded 36 (vertical) x 125 (horizontal) data points in the unobstructed section. Based on friction velocities computed from the PIV measurements, the vertical resolution was y + = x + = 3:3 5 in inner units. 4.5 Results and Discussion We rst present resolvent-based predictions for the mode that serves as a surrogate for the NW cycle and for spanwise constant KH-rollers, focusing on the x-permeable and y-permeable cases tested in experiment (Section 4.5.1). We then present experimental measurements for the mean ow, turbulence statistics and ow structure (Sections 4.5.2-4.5.3). A comparison of model predictions made using the synthetic mean proles and those obtained from ts to experimental data is presented at the end (Section 4.5.4). 4.5.1 Model Predictions Figures 4.2(a,b) show the predicted change in singular values for resolvent modes resembling the NW cycle (i.e., resolvent modes with + x = 10 3 , + z = 10 2 , and c + = 10) over anisotropic porous substrates as a function of their streamwise and wall-normal permeability length scales, p K + xx and p K + yy . Note that the contours show the forcing-response gain for the porous material normalized by the smooth wall value at the same Re ; ;p = ;s < 1 indicates mode suppression and ;p = ;s > 1 indicates mode amplication. For all the predictions shown in Fig. 4.2 the mean ow was computed using the synthetic eddy viscosity prole. Consistent with prior simulation results (Rosti et al., 2018; G omez-de-Segura and Garc a- Mayoral, 2019), porous substrates with high streamwise permeability and low wall-normal per- meability are found to suppress the NW mode, which is known to be a useful predictor of drag reduction performance (Chavarin and Luhar, 2020). In general, mode suppression increases as the permeability ratio increases, xy 1, though there are some subtle dierences between the results presented in Fig. 4.2(a) for substrates with K + zz = K + yy and in Fig. 4.2(b) for substrates 72 Figure 4.2: (a,b) Predicted singular value ratios ( ;p = ;s ) for resolvent modes resembling the NW cycle as a function of streamwise and wall-normal permeability length-scales. Color contours show predictions for Re = 120; solid black lines correspond to Re = 360. Predictions in panel (a) are for substrates which have a similar conguration to x-permeable materialK = diag(K xx ;K yy ;K zz =K yy ). Predictions in panel (b) are for substrates with a similar conguration to the y-permeable substrate,K = diag(K xx ;K yy ;K zz = K xx ). The () symbols in (a) and (b) correspond roughly to the permeabilities for the x-permeable and y-permeable materials tested in the experiments. (c,d) Amplication of spanwise-constant modes at Re = 120 relative to the smooth wall case as a function of streamwise wavelength and mode speed for (c) the x-permeable substrate and (d) for the y-permeable substrate, i.e., for permeability values labeled with () symbols in (a) and (b). withK + zz =K + xx . The substrates shown in panel (a) produce greater mode suppression than those shown in panel (b) for xy 1. This is consistent with the virtual origin model proposed in previous studies (Abderrahaman-Elena and Garc a-Mayoral, 2017; G omez-de-Segura and Garc a- Mayoral, 2019), which suggests that turbulence penetration into the porous medium is dictated by the spanwise permeability, and that the initial decrease in drag depends on the dierence between the streamwise and spanwise permeability length scales D/ p K + xx p K + zz . The predictions shown in Figs. 4.2(a,b) do not change substantially from Re = 120 (colored shading) to Re = 360 (solid black lines). In particular, the location of the neutral curve cor- responding to p = s = 1 (i.e., no change in gain) is very similar for both Reynolds numbers. 73 In addition, the trends in the suppression and amplication of the NW-mode remain the same between Re = 120 and Re = 360. For the specic porous materials tested in our experiments, the permeability length scales corre- spond to ( p K + xx ; p K + yy ) = (7:9; 2:8) for the x-permeable case and ( p K + xx + ; p K + yy ) = (2:8; 7:9) for the y-permeable case at Re = 120. These values were computed from the dimensionless perme- ability listed in Table 4.1 assumingH + =Re = 120. These specic permeability ratios are labeled using markers in Figs. 4.2(a,b). Model predictions indicate that the x-permeable substrate sup- presses resolvent modes resembling the NW cycle by approximately 25 30% (see marker in Fig. 4.2(a)). In contrast, the y-permeable substrate leads to signicant mode amplication, with ;p = ;s > 2 (see marker in Fig. 4.2(b)). This is broadly consistent with previous simulations, which indicate that drag reduction is only expected over streamwise-preferential materials. In addition the the suppression or amplication of the NW cycle, the other factor that controls the drag-reduction performance of anisotropic porous substrates is the emergence of KH rollers (G omez-de-Segura and Garc a-Mayoral, 2019; Abderrahaman-Elena and Garc a-Mayoral, 2017; Breugem et al., 2006; Chandesris et al., 2013). Linear stability analysis and simulations suggest that the appearance of such rollers is linked to a relaxation of the wall-normal permeability. Specically, the recent simulations of G omez-de-Segura and Garc a-Mayoral (2019) indicate that the spanwise rollers emerge as the wall normal permeability increases beyond p K + yy 0:4. Unfortunately, due to fabrication constraints, both the x-permeable and y-permeable substrates tested here are expected to have wall-normal permeabilities larger than this threshold value. Figures 4.2(c,d) show the normalized gain for spanwise-constant ( z = 0) resolvent modes over the x-permeable and y-permeable material, respectively. For the y-permeable case a region of high amplication is visible in Fig. 4.2(d) for structures with streamwise wavelength + x 700 1200 and mode speed c + 5 10. The wave speed of these structures indicate that these structures are localized close the the uid-porous interface. The most amplied structure in this region has + x 1000 and c + 8:5. The gain for this structure increases by a factor of approximately 100 relative to the smooth wall case. In contrast, Fig. 4.2(c) shows that there is no localized maximum in relative amplication for spanwise-constant structures for the x-permeable case. Model predictions indicate that spanwise rollers with + x 600 andc + 4 15 are amplied relative to the smooth wall case 74 and amplication generally increases with increasing wavelength. The gain for the most amplied structure in Fig. 4.2(c) increases by a factor of approximately 6 relative to the smooth wall case. Thus, even though the x-permeable material is susceptible to the emergence of spanwise-constant structures, the degree of energy amplication relative to the smooth wall case is more limited compared to that for the y-permeable material. Together, the predictions presented in section indicate that the x-permeable material is likely to suppress the energetic NW cycle but could give rise spanwise rollers resembling KH vortices. The y-permeable material is likely to further amplify the NW cycle and give rise to spanwise rollers that are amplied signicantly relative to the smooth wall case. These predictions are compared against the measurements made in the benchtop channel ow experiments in the following sections. 4.5.2 Mean Flow and Turbulence Statistics Figure 4.3: PIV results for the channel ow experiment. The smooth wall is located aty = 0, while the interchange- able wall is located aty =h. Panel (a) shows the measured mean velocity proles normalized by the bulk-averaged velocity (calculated as U b = Q=[W (H +h)] in all cases). Panel (b) shows Reynolds shear stress proles normal- ized by the smooth wall friction velocity u s . Friction velocities at the smooth and porous walls were estimated by extrapolating a linear t to the total stress prole to the wall locations. Panels (c) and (d) show proles of the root-mean-square streamwise uctuations normalized by the friction velocities at the smooth wall and porous interface, u s and u p , respectively. 75 Figure 4.3 shows the measured mean statistics for the channel ow experiments. These statistics were computed by averaging both in time and in the streamwise direction. Results in the region y=h 0:95 were aected by re ections at the smooth/porous tiles and should be treated with caution. As seen from Fig. 4.3(a), the mean prole remains relatively symmetric across the unobstructed region for the x-permeable material. However, for the y-permeable case, the bulk of the ow in the unobstructed region deviates from the parabolic prole and is shifted towards the smooth wall. The location of the maximum mean velocity is y=h = 0:45 for the x-permeable case as compared to y=h = 0:38 for the y-permeable case. Interestingly, the slip velocity at the porous interface appears to be higher for the y-permeable case despite the substantially lower streamwise permeability. However, this observation could be attributed to the specic porous geometry tested here. For the y-permeable material, the porous interface is characterized by much higher local porosity compared to the x-permeable material. The visibly lower bulk-normalized mean prole for the x-permeable material in the unobstructed region is indicative of greater ow through the porous medium itself. The Reynolds shear stress proles in Fig. 4.3(b) show the presence of an (almost) linear region in the middle of the unobstructed domain. Friction velocities at the porous and smooth walls,u p and u s , were estimated by extrapolating the total stress (i.e., Reynolds shear stress plus viscous stress) from this linear region to y = h and y = 0, respectively (see e.g., Breugem et al., 2006) and are listed in Table 4.2. The average friction velocity was estimated as u t = p ((u s ) 2 + (u p ) 2 )=2. Note that the proles shown in Fig. 4.3(b) are normalized by u s . With this normalization, it is clear that u p is higher than u s for the y-permeable material, indicative of greater friction generated at the porous interface than at the smooth wall. However, for the x-permeable material, the estimated friction velocities at the smooth wall and porous interface are comparable, u s u p . In other words, there is no clear increase or decrease in friction at the porous interface relative for the x-permeable material. Note that friction velocities at both interfaces are approximately equal for the smooth wall case, as expected. Thus, the friction velocity estimates are broadly consistent with the mean proles shown in Fig. 4.3(a); only the y-permeable material shows a signicant dierence in behavior at the porous interface. 76 Case u s [m/s] u p [m/s] u t [m/s] smooth wall 0.0194 0.0198* 0.0196 x-permeable 0.0201 0.0205 0.0203 y-permeable 0.0235 0.0326 0.0284 Table 4.2: Friction velocity estimates at the smooth wall (u s ) and porous interface (u p ). The average friction velocity is u t . For the smooth wall case u p corresponds to the solid tile placed in the cutout. Moreover, relative to the smooth wall case, x-permeable material does not lead to a signicant change in friction velocities; u s values dier by 3% and at the porous wall u p is higher by ap- proximately 4% (note that u p for the smooth wall corresponds to the value at the smooth tile at y=h = 1). In contrast, the y-permeable material leads to a signicant increase in friction velocities relative to the smooth wall case; u s increases by 20% and u p by 70%. Thus, in contrast to the resolvent-based predictions for the NW cycle, no reduction in friction is observed at the porous interface for the x-permeable material. However, the y-permeable material leads to a substantial increase in friction at both the smooth wall and the porous interface. This is qualitatively con- sistent with model predictions, which show a substantial increase in NW cycle gain as well as the emergence of high-gain spanwise rollers over the y-permeable material. Proles for the root-mean-square streamwise velocity uctuations normalized byu s andu p , are plotted in Figs. 4.3(c) and 4.3(d) respectively. When normalized by u s , the near-wall peaks on the smooth wall side collapse together for all cases. Further, the normalized peak value for the uctuations is p u 02 +s 3, which is close to that expected in a canonical turbulent channel ow conguration. However, as per Fig. 4.3(d), only the peaks for the smooth wall and x-permeable case collapse together reasonably well near the porous interface when normalized byu p , with a peak value of p u 02 +p 3. This conrms that the ow physics are qualitatively similar for the smooth wall and x-permeable substrate. The prole for the y-permeable case has no discernible peak near the porous interface and the maximum value for the normalized uctuations p u 02 +p 2:5, is lower than the other two cases. These observations are consistent with previous results for ow over porous materials (Breugem et al., 2006; Suga et al., 2018) and are indicative of a change in ow structure. 77 Figure 4.4: The rst two POD modes for the smooth wall case (a,b), x-permeable case (c,d), and the y-permeable case (e,f). The modes are computed using 20,000 PIV frames. The shading represents normalized levels for the streamwise velocity component. The solid and dashed black lines represent positive and negative contours for the wall-normal velocity component. The porous interface is at the top wall. 4.5.3 Flow Structure To provide further insight into the changes in mean statistics discussed above, snapshot proper orthogonal decomposition (POD) was performed on the uctuating velocity elds obtained from PIV. The streamwise velocity elds associated with the rst 2 spatial modes for the smooth wall, x-permeable, and y-permeable cases are shown in Fig. 4.4. As expected, the most energetic modes for the smooth wall case shown in Figs. 4.4(a,b) resemble long streaky structures that are symmetric across the channel. The rst POD mode for the x- permeable case shown in Fig. 4.4(c) does not have a clear physical interpretation. It bears some resemblance to the second POD mode for the smooth wall shown in Fig. 4.4(b) but also some to the asymmetric rst POD mode over the y-permeable case shown in Fig. 4.4(e). However, the second POD mode for the x-permeable substrate closely resembles the rst mode for the smooth wall case. The streamwise extent of the plots shown in Fig. 4.4 corresponds to the PIV eld of view, which is roughly 3:5h or 22 mm. With this in mind, the rst smooth wall mode and the second x-permeable mode appear to have a streamwise wavelength that is more than twice the PIV 78 eld of view, x > 44 mm (or x > 7h). For the friction velocity estimates shown in Table 4.2, this translates into + x = ( x u t =)> 800, which is consistent with the scale of NW streaks (Robinson, 1991). Unlike the smooth wall and x-permeable cases, POD modes for the y-permeable material have a visibly asymmetric structure in the wall-normal direction (see Fig. 4.4(e,f)). For both modes, the streamwise velocity eld is much more intense near the porous interface. This is in contrast to the symmetric streaky structures observed over both the smooth wall and x-permeable material. Moreover, when the streamwise and wall-normal velocity contours are considered together, the full velocity eld for these POD modes is indicative of a counter-rotating structure. Such rollers have been observed in previous numerical simulations (Breugem et al., 2006; G omez-de-Segura and Garc a-Mayoral, 2019), and are typically associated with a Kelvin-Helmholtz instability mecha- nism. This observation provides qualitative support for the model predictions shown in Fig. 4.2(d) which shows that the y-permeable material is susceptible to the emergence of large high-gain spanwise constant structures. More quantitatively, the rst POD mode over the y-permeable material appears to have a streamwise wavelength more than twice the PIV eld of view, x > 7h 44 mm. Using the u t estimate for the y-permeable material shown in Table 4.2, this translates into + x > 1200. Thus, the size of this structure is larger than the region of peak amplication around + x 1000 predicted in Fig. 4.2(d). However, assuming that the spanwise rollers scale in outer units, + x 1000 for the predictions shown in Fig. 4.2(d) at Re 120, corresponds to x =h = + x =Re 8. This is more consistent with the streamwise wavelength of the rst POD mode for the y-permeable substrate shown in Fig. 4.4(e). Put another way, if the resolvent-predictions had been carried out at the measured friction Reynolds number for the y-permeable substrate, Re =u t h= 180 for the u t estimate shown in Table 4.2, they might show a region of higher amplication at higher streamwise wavelengths. This issue is explored further in Section 4.5.4. The streamwise extent of the second POD mode over the y-permeable substrate (see Fig. 4.4(f)) is slightly larger than the PIV window, x 4h. This corresponds to a streamwise wavelength of + x 700, which is at the lower end of the predicted region of highly-amplied spanwise rollers in Fig. 4.2(d). 79 4.5.4 Model Sensitivity to Mean Prole Figure 4.5: Comparison of the mean velocity proles used in the construction of the resolvent operator for the smooth wall (a), x-permeable material (b), and y-permeable material (c). Experimental proles are shown using the circular makers ( ). Synthetic proles generated using the eddy viscosity model of Reynolds and Tiederman (1967) are plotted as dashed lines ( ). Fitted proles computed using eddy viscosity prole determined from experimental data are plotted as solid lines ( ). Figure 4.6: Normalized gain ( ;p = ;s ) for spanwise-constant resolvent modes as a function of streamwise wave- length and mode speed. Panels (a,b) show predictions for the x-permeable case computed using the synthetic eddy viscosity prole (a) and the tted eddy viscosity prole (b). Panels (b,c) show predictions for the y-permeable case computed using the synthetic eddy viscosity prole (c) and the tted eddy viscosity prole (d). 80 Figure 4.7: High-gain spanwise-constant resolvent modes identied from model predictions for the y-permeable substrate. (a) Structure corresponding to the highest-gain mode from the synthetic mean prole predictions in Fig. 4.6(c) with ( + x ;c + ) (2000; 9:5). (b) Structure corresponding to the highest-gain mode from the tted mean prole predictions in Fig. 4.6(d) with ( + x ;c + ) (530; 9). The shading represents normalized levels for the streamwise velocity component. Solid and dashed black lines respectively represent positive and negative contours for the wall-normal velocity component. The porous interface is at the top wall (y=h = 1). The results presented in the previous sections show that there is no clear evidence of NW-cycle suppression and friction reduction over the x-permeable material. For the y-permeable material, a substantial increase in friction is observed at the porous interface relative to the smooth wall and x-permeable material. In addition, the POD modes shown in Section 4.5.3 suggest that large- scale spanwise rollers are energetic over the y-permeable substrate. These results are in partial agreement with the model predictions shown in Section 4.5.1. Here, we revisit resolvent-based predictions for the NW mode and spanwise rollers, but using the mean prole estimated from the experiments based on the procedure described in Section 4.3.2. Figure 4.5 compares mean velocity proles measured in the experiments ( ) with those generated using the synthetic eddy viscosity prole (Reynolds and Tiederman, 1967) ( ) and the tted eddy viscosity prole ( ). In general, the synthetic and tted proles are in close agreement with the measurements for the smooth wall and x-permeable material, as shown in Figs. 4.5(a,b). However, Fig. 4.5(c) shows that the synthetic mean prole for the y-permeable case does not reproduce the asymmetry observed in the experiments. This is because the synthetic eddy viscosity prole from Reynolds and Tiederman (1967) was developed for smooth wall ows. Therefore, it assumes that the turbulence is symmetric across the unobstructed channel. This is reasonable for the smooth wall case and the x-permeable material but not for the y-permeable materials (see u s and u p estimates in Table 4.2). The tted eddy viscosity prole accounts for the asymmetry in turbulence across the unobstructed region over the y-permeable substrate. As a result, it is able to better reproduce the shape of the measured mean prole. Note that all the experimental measurements are under-predicted by the eddy viscosity models to some extent. This could be attributed to 81 synthetic tted x-permeable, ;p = ;s 0.71 0.80 y-permeable, ;p = ;s 2.55 1.46 Table 4.3: Comparison of the normalized gain for NW resolvent modes computed using the synthetic and tted mean velocity proles. spanwise variation in the mean ow in the nite-width channel (recall that the channel has an aspect ratio ofW=h 8). The mean prole measured at the channel centerline may be larger than the true spanwise average. Table 4.3 shows how the normalized gain for the NW resolvent mode changes with the mean prole for the x-permeable and y-permeable substrates. These normalized singular values are computed for friction Reynolds numbers estimated from the experiments: Re =u t h= 124 for the smooth wall case,Re 129 for the x-permeable substrate, andRe 180 for the y-permeable substrate. In other words, the singular values over the porous substrates ( ;p ) are normalized by the singular values for a smooth wall ( ;s ) at the Re estimated from the experiments. For the x-permeable material, the singular value ratio increases from ;p = ;s = 0:71 for the synthetic mean prole to ;p = ;s = 0:80. In other words, the predicted suppression for the NW resolvent mode decreases from 29% with the synthetic prole to 20% for the tted prole. Despite this slight deterioration, the model still predicts suppression for the NW mode which we consider a necessary, but not sucient, condition for friction reduction. For the y-permeable substrate, the singular value ratio decreases from ;p = ;s = 2:55 for the synthetic prole to ;p = ;s = 1:46 for the tted prole. Thus, the tted prole yields more limited NW mode amplication (46%) compared to the synthetic prole (155%). Interestingly, the measured increase in the average friction velocity for the y-permeable material relative to the smooth-wall case is approximately 45% (seeu t values in Table 4.2), which is close to the predicted increase in NW mode amplication. However, one must be cautious in using NW mode amplication as a direct measure of drag reduction performance. The x-permeable material yields a 4% increase inu t relative to the smooth wall case, even though the resolvent model predicts suppression for the NW mode. Together, the predictions shown in Table 4.3 indicate that: (i) the exact shape of the mean prole can have a signicant eect on resolvent-based predictions, and (ii) the eddy viscosity model from Reynolds and Tiederman (1967) may not be the most appropriate choice for generating mean prole 82 predictions over porous substrates. However, the overarching design guidelines do not change: only materials with high streamwise permeability and low wall-normal/spanwise permeabilities are likely to reduce drag. Next, we evaluate how resolvent-based predictions for the spanwise-constant modes change with the mean prole. For the x-permeable case, there is no localized region of very high amplication for either the synthetic or tted mean proles. This is broadly similar to the results shown in Fig. 4.2(c). However, for the y-permeable case, these new predictions show signicant changes relative to the model predictions in Fig. 4.2(d). With the synthetic prole, the localized region of high amplication shifts to larger wavelengths, from + x 1000 in Fig. 4.2(d) to + x > 1800 in Fig. 4.6(c). This could be partially due to the increase in Reynolds number from the a priori estimate, Re = 120, used to generate the model predictions in Fig. 4.2(d) to the measured value for the y-permeable substrate, Re = 180, in Fig. 4.6(c). In other words, this shift in the peak to larger + x for higher Re may indicate that the energetic spanwise rollers scale in outer units. Interestingly, when the tted mean prole is used to generate predictions for the y-permeable substrate, the amplication map for the spanwise rollers changes substantially (see Fig. 4.6(d)). There is no region of high amplication for + x > 1800 and c + 9 as observed with the synthetic prole (or perhaps this gets pushed beyond the eld of view, + x > 2000). Instead a region of high gain is observed from ( + x ;c + ) (300; 11) to ( + x ;c + ) (1000; 6). It could be argued that the second POD mode for the y-permeable substrate shown in Fig. 4.4(f) belongs in this region. However, there is no clear evidence of a longer structure like resembling the rst POD mode shown in Fig. 4.4(e). The predicted structure for the modes with the highest normalized gain in Fig. 4.6(c) and Fig. 4.6(d) is shown in Fig. 4.7(a) and Fig. 4.7(b), respectively. The streamwise extent of these structures is broadly consistent with the rst and second POD modes shown in Figs. 4.4(e,f). However, there are signicant dierences in ow structure between the predictions and the mea- surements. This discrepancy is the source of ongoing research. 83 4.6 Conclusion Consistent with previous theoretical eorts and numerical simulations, the present experiments show a very dierent ow response over the x-permeable and y-permeable materials. The x- permeable material leads to a marginal increase in friction velocity at the porous interface (see Fig. 4.3(b) and Table 4.2). This is counter to resolvent-based predictions, which suggest that the x-permeable material should lead to a reduction in gain for the NW cycle (Fig. 4.2(a)). Possible explanations for this discrepancy include: the emergence of energetic spanwise rollers, as predicted by the resolvent formulation and previous numerical simulations for materials with p K + yy > 0:4; roughness eects at the porous interface that are neglected in the VANS equations; nonlinear (Forchheimer) eects becoming important in the porous medium; and perhaps most importantly, insucient scale separation between the pore-scale and outer ow. The y-permeable material leads to a signicant increase in friction velocities at both walls relative to the smooth wall and x-permeable cases. This is consistent with resolvent-based predictions, which indicate a substantial increase in NW cycle gain as well as the emergence of large, high-gain spanwise rollers over the y-permeable material. POD conrms the presence of such spanwise rollers over the y-permeable material (Fig. 4.4). Resolvent-based predictions are able to predict the streamwise length scale of the POD modes. However, there are important dierences in structure between the highest-gain resolvent modes and the computed POD modes. The model predictions and experimental results shown here conrm that materials with high streamwise permeability and low spanwise and wall-normal permeability are good candidates for drag reduction. Ongoing work seeks to alleviate some of the weaknesses associated with the current experimental setup (limited development length, insucient scale separation, low Reynolds number) and to identify streamwise-permeable porous materials that could be more eective. 84 Chapter 5 Turbulent boundary layers over streamwise-preferential porous materials 5.1 Comment and Summary This chapter has been submitted to the Journal of Fluid Mechanics for publication. The small- scale channel ow experiment described in §4 provided a convenient way to experimentally com- pare multiple 3D printed substrates (x-permeable, y-permeable) and resolvent model predictions at moderate Reynolds Numbers. The small scale facility was designed with heat transfer experiments in mind and subject to operational constraints that limited the size. The limited development length, lack of scale separation between substrate thickness and channel height as well as low Reynolds Number (Re 120) and resolution (36 points across the channel) motivated a further set of experiments utilizing the boundary layer set up from the rst set of experiments described in §2, and high speed PIV setup from the previous set. Printing tiles for one substrate geometry took 6 weeks, limiting us to testing only one 3D printed geometry. The x-permeable case was selected in order to provide comparisons to numerical simulations by G omez-de-Segura and Garc a-Mayoral (2019). The porous material has a diagonal permeability tensor with normalized streamwise per- meability p K xx + 3:0 and wall-normal and spanwise permeabilities p K yy + = p K zz + 1:1 and was ush mounted in the at plate. In contrast to the experiments presented in §2, the ow velocity was reduced to U e 0:15 m/s to minimize p K yy + . Upstream of the cutout, the friction Reynolds number is 250 while at x=h 44 it has increased to 360 . Measurements made at several locations along the porous substrate provide insight into bound- ary layer development. For fully-developed conditions, the mean proles show the presence of a 85 logarithmic region over the porous material with similar constants to those found over a smooth wall. A technique that estimates the mean prole at single-pixel resolution from the particle im- ages suggests the presence of an interfacial slip velocity ofU + s p K xx + over the porous substrate. Friction velocity estimates obtained from outer layer ts to the mean prole suggest a marginal increase in drag over the porous substrate. PIV measurements show a decrease in the intensity of streamwise velocity uctuations in the near-wall region and an increase in the intensity of wall- normal velocity uctuations. These observations are consistent with simulation results, which suggest that materials with p K yy + > 0:4 are susceptible to the emergence of spanwise rollers similar to Kelvin-Helmholtz vortices that degrade drag reduction performance. Velocity spectra indicate that such structures emerge in the experiments as well. These experiments represent the rst experimental data set in turbulent boundary layers over purposefully designed, anisotropic porous media and provide important validation for numerical and theoretical studies. 5.2 Introduction Functional surfaces have shown signicant promise as methods of passive drag reduction in wall- bounded turbulent ows. Streamwise-aligned riblet surfaces have demonstrated drag reductions of up to 10% in laboratory experiments (Walsh and Lindemann, 1984; Bechert et al., 1997, 2000; Garc a-Mayoral and Jim enez, 2011; Garcia-Mayoral and Jimenez, 2011). More recent work shows that streamwise-preferential porous materials also have the potential to reduce turbulent friction drag through a similar mechanism to riblets (Abderrahaman-Elena and Garc a-Mayoral, 2017). In particular, numerical simulations suggest that drag reductions of up to 25% may be possible over such anisotropic porous substrates (Rosti et al., 2018; G omez-de Segura and Garc a-Mayoral, 2019). The work presented in this paper is a step towards testing these predictions in laboratory experiments. 5.2.1 Drag reduction mechanism The mechanism through which riblets reduce drag can be distilled down to their eect on the near-wall turbulence. By providing greater resistance to turbulent cross- ows compared to the 86 streamwise mean ow, riblets displace the quasi-streamwise vortices associated with the energetic near-wall cycle away from the wall (Robinson, 1991; Jim enez and Pinelli, 1999a), which weakens the vortices and inhibits turbulent mixing near the riblet tips (Luchini et al., 1991; Choi et al., 1993; Jim enez et al., 2001). This mechanism can also be understood in terms of the slip lengths perceived by the streamwise mean ow (l + U ) and the turbulent cross- ows (l + t ) below the riblet tips (Luchini et al., 1991). Following standard notation, a superscript + denotes normalization with respect to viscosity and friction velocity u . Since the streamwise mean ow is impeded to a lesser degree than the turbulence cross- ow, it penetrates to a larger distance into the riblet grooves, i.e., l + U l + t . This oset between the mean ow and the turbulence reduces momentum transfer towards the wall and leads to drag reduction. The length scales l + U and l + t depend on the shape and size of the riblets, and can be estimated by solving the viscous Stokes ow equations in the streamwise and spanwise directions over the riblets. For small riblets, the drag reduction is expected to be proportional to the dierence between the streamwise and transverse slip lengths, DR = (C f C f0 )=C f0 /l + U l + t . Here,C f andC f0 are the skin friction coecients over the riblet surface and smooth wall, respectively. Assuming outer-layer similarity holds (e.g., Flack et al., 2007), further away from the wall the eect of small riblets is limited to an upward (drag decrease) or downward (drag increase) shift in the mean velocity prole, which can be quantied in terms of the so-called Hama roughness function U + . Previous experiments and simulations show that U + =m(l + U l + t ), where m is an O(1) constant (e.g., Bechert et al., 1997; Garcia-Mayoral and Jimenez, 2011; Garcia-Mayoral et al., 2019). Note that the friction drag reduction is related to this shift in the mean prole. For small changes in friction it can be shown that DR p 2C f0 U + . Although drag reduction increases initially with increasing riblet size, there is a shape-dependent optimal size beyond which performance degrades. Early studies attributed this deterioration of performance with increasing riblet size to the near-wall turbulence being able to penetrate into the riblet grooves (Choi et al., 1993; Lee and Lee, 2001). However, the high-delity simulations pursued by Garcia-Mayoral and Jimenez (2011) show that the emergence of energetic spanwise- coherent rollers from a Kelvin-Helmholtz type instability also plays an important role in driving this deterioration of performance. Recent resolvent-based model predictions also show the emergence of energetic spanwise rollers over riblet surfaces (Chavarin and Luhar, 2020). A compilation of 87 previous results from experiments and simulations suggests that the cross-sectional area of the riblet grooves (A g ) is a useful predictive measure for the optimal riblet size beyond which performance degrades (Garc a-Mayoral and Jim enez, 2011). The optimal size across many dierent riblet shapes is found to be approximately l + g = p A g + 10:7 (Garc a-Mayoral and Jim enez, 2011; Garcia- Mayoral and Jimenez, 2011; Chavarin and Luhar, 2020). Note that spanwise-coherent rollers also appear in numerical simulations and experiments over porous materials (Breugem et al., 2006; Rosti et al., 2015; Chandesris et al., 2013; Kuwata and Suga, 2017; Suga et al., 2018). Relevant to the present eort, Abderrahaman-Elena and Garc a-Mayoral (2017) expanded the slip length model proposed by Luchini et al. (1991) to ows over anisotropic porous ma- terials characterized by streamwise permeability K xx , wall-normal permeability K yy , and span- wise permeability K zz . This eort shows that l + U / p K xx + and l + t / p K zz + , which implies that U + / p K xx + p K zz + . In other words, turbulent drag reduction may be possible over streamwise-preferential porous materials with p K xx + > p K zz + . In addition, linear stability anal- yses suggest that the maximum achievable drag reduction is limited by the emergence of spanwise rollers as the wall-normal permeability increases. Results from recent direct numerical simulations (DNSs) support these theoretical predictions. Specically, the results obtained by Rosti et al. (2018) and G omez-de Segura and Garc a-Mayoral (2019) show that drag reduction is possible in turbulent ows over anisotropic permeable materials at Re 180. In particular, the extensive parametric sweep pursued by G omez-de Segura and Garc a-Mayoral (2019) conrms that the ini- tial drag reduction depends on the dierence between the streamwise and spanwise permeabilities, DR/ p K xx + p K zz + . In addition, these simulations show drag reductions of up to 25% before the emergence of energetic spanwise rollers leads to a deterioration of performance. The onset of these Kelvin-Helmholtz type rollers is triggered by wall-normal permeabilities above p K yy + > 0:4. The simulation results described above suggest that streamwise preferential materials with p K xx + p K yy + and p K yy + = p K zz + ) have the potential to reduce drag, as long as the abso- lute value of the wall-normal permeability remains small, p K yy + < 0:4. However, these simula- tions employ idealized models for the ow through the porous material (e.g., the Darcy-Brinkman equation) that do not explicitly resolve ow at the pore-scale. Instead, the eect of the permeable medium is included through the use of simplied eective models involving bulk properties such 88 as permeability. It remains to be seen if the trends observed in the numerical simulations hold for physically-realizable materials that have the same eective properties. For completeness, we note that previous simulation eorts that employ slip-length or admittance boundary conditions to account for the presence of permeable walls also show the possibility of drag reduction over anisotropic surfaces (Jim enez et al., 2001; Hahn et al., 2002b; Busse and Sandham, 2012). 5.3 Previous experiments over porous materials Experiments involving turbulent ows over porous substrates have thus far been limited by the materials available. Experiments motivated by aircraft wings and airfoils have investigated at plates and blu bodies with arrays of holes (Kong and Schetz, 1982; Ru and Gelhar, 1972). These materials have limited substrate thickness and porosity, and are only permeable in the wall- normal direction. However, they can have a signicant benet in terms of lift enhancement or stall delay (Hanna and Spedding, 2019). Motivated by environmental ows, several experiments have considered a porous substrate consisting of packed spheres (e.g., Zagni and Smith, 1976; Blois et al., 2020). Packed sphere beds have limited porosity< 0:7 and are approximately isotropic. However, they are readily available in dierent sizes and also allow for refractive index matching. Recent experiments involving index-matched PIV provide useful insight into the relative eects of porosity and roughness on turbulent ows (Kim et al., 2019, 2020). These index-matched experiments also conrm the existence of the so-called amplitude modulation phenomenon observed for smooth wall ows (Marusic et al., 2010) over, and within, the porous substrates. Other experiments studying turbulent boundary layer and channel ows adjacent to porous sub- strates have utilized high-porosity quasi-isotropic foams (Manes et al., 2011; Efstathiou and Luhar, 2018) or meshes with high wall-normal permeability and low streamwise permeability (Suga et al., 2018). In almost all of these experiments, friction increases substantially over the porous substrates and the velocity elds or energy spectra show the emergence of energetic spanwise-coherent Kelvin- Helmholtz rollers. Laser doppler velocimeter (LDV) measurements made by Efstathiou and Luhar (2018) also show evidence of amplitude modulation over high-porosity reticulated foams with per- meability Reynolds numbers Re k = p Ku = = p K + = 1 9, where K is a representative scalar permeability. Measurements made by Manes et al. (2011) using a 2D LDV over high-porosity 89 materials with low surface roughness suggest the existence of a modied logarithmic region in the mean prole with lower von K arm an constants ranging from 0:2 to 0:4. A large number of numerical and experimental datasets support the universality of = 0:39 0:02 (Marusic et al., 2013) for boundary layer ows over impermeable walls. However, the eects of porous walls on this universality have not been studied in detail. This work does not investigate the divergence of , primarily because 0:39 appears appropriate for the porous materials tested. Very recently, Suga et al. (2018) used PIV to make measurements in channel ows with bulk Reynolds numbers Re b = 900 13600 over anisotropic materials with p K yy + > p K xx + made of layered and oset meshes. Their PIV measurements in the streamwise-spanwise (xz) and streamwise-wall-normal planes (xy) conrm the emergence of spanwise coherent structures and show drag increases of up to 97%. None of the experimental eorts described above test the eects of streamwise-preferential permeable materials on turbulent ows. Interestingly, the experiments pursued by Itoh et al. (2006) show drag reductions of up to 12% over porous seal fur, which could be considered a streamwise-preferential porous material. These experiments tested the eects of both riblets and seal fur in turbulent channel ows at Re 120 600. Pressure drop measurements showed that the seal fur led to drag reductions that were nearly twice as large as the riblets. However, proles of the mean velocity and streamwise uctuations measured by LDV showed no signicant departure from smooth wall proles. Although the seal fur could be considered a streamwise-preferential material, there are few experiments that involve turbulence measurements over permeable materials with carefully-controlled anisotropy. To our knowledge, the preliminary channel ow experiments detailed in Chavarin and Luhar (2020) include the rst dataset of turbulent ow over porous materials designed to have streamwise preferential permeability. These small-scale experiments tested the eect of 3D-printed porous materials with both p K xx + > p K yy + and p K xx + < p K yy + at Re 120. The streamwise-preferential material with p K xx + > p K yy + did not show a signicant departure from smooth wall conditions due to the limited anisotropy. As expected, the material with higher wall-normal permeability triggered the onset of Kelvin-Helmholtz type rollers and led to a signicant increase in drag. 90 5.3.1 Contribution and outline This paper presents results from laboratory water channel experiments that tested the eect of 3D-printed porous materials with streamwise-preferential permeability on turbulent boundary layer ows atRe 360. The porous materials were designed to have a cubic lattice microstructure with large openings in the wall normal-spanwise (yz) plane to limit the resistance felt by the mean ow in the streamwise (x) direction, i.e., to ensure high streamwise permeability K xx . The openings in the remaining two planes were designed to be smaller to yield lower wall-normal and spanwise permeabilities, K yy and K zz , respectively. Due to fabrication constraints, the exact permeability values did not fall in the range that is expected to yield drag reduction per the simulations of G omez-de Segura and Garc a-Mayoral (2019). Specically, the wall-normal permeability of the fabricated materials is larger than the threshold value above which energetic spanwise rollers are expected to emerge, p K yy + 0:4. Indeed, the measurements reported below show a small increase in skin friction over the anisotropic porous material. Nevertheless, the measurements provide useful insight into the eect of anisotropic porous materials on the mean prole and turbulence statistics. The remainder of this paper is structured as follows. The experimental methods are described in §5.4. This includes details on the ow facility, porous substrate design and fabrication, and diagnostic techniques. The results are presented and discussed in §5.5. Flow development over the porous materials is considered in §5.5.1. The mean velocity prole and turbulence statistics for fully-developed conditions at a downstream location over the porous substrate are compared against smooth wall conditions in §5.5.2. Changes in velocity spectra are considered in §5.5.3. Brief concluding remarks are presented in §5.6. 5.4 Experimental methods The ow facility and at plate setup used for the boundary layer experiments are described in §5.4.1. This is followed by a description of the PIV system and processing routines in §5.4.2 and the procedure used to estimate the mean prole at single-pixel resolution in §5.4.3. Design and fabrication of the porous materials is discussed in §5.4.4. The approach used to estimate friction velocities from the mean prole measurements is presented in §5.4.5. 91 Figure 5.1: Schematic showing experimental setup. The laser, optics and camera system are mounted to a precision traverse (not shown) that can be moved from just upstream of the substrate transition to the end of the plate. The dashed vertical lines indicate the center of the 6 measurement stations at (x=h) = (5; 3; 12; 28; 44; 53), where x = 0 is dened as the start of the porous substrate and h = 15:4 mm is substrate thickness. 5.4.1 Flow facility and at plate apparatus The experiment utilizes the same at plate apparatus and water channel facility as described in Efstathiou and Luhar (2018). However, the experiments are carried out at a lower Reynolds number. A schematic is provided in 5.1. The water channel has a test section of length 762 cm, width 89 cm, and height 61 cm, and is capable of generating free-stream velocities up to 70 cm/s with background turbulence levels < 1% at a water depth of 48 cm. For the present experiments, a 240 cm long at plate was suspended from precision rails at a height H = 30 cm above the test section bottom. To avoid free-surface eects, measurements were made below the at plate. The connement between the at plate and bottom of the channel naturally led to a marginal increase ( 4%) in the free stream velocity (U e ) and slightly favorable pressure gradient along the plate. However, the non-dimensional acceleration parameter, = U 2 e dUe dx was of O(10 7 ), suggesting any pressure gradient eects are likely to be mild (Patel, 1965; De Graa and Eaton, 2000; Schultz and Flack, 2007). The water temperature for all experiments was 18 0:5 C for which the kinematic viscosity is = 10 2 cm 2 /s. A cutout of length 90 cm and width 60 cm, located 130 cm downstream of the leading edge, was used to mount the test surfaces. Smooth and porous surfaces were substituted into this cutout, and mounted to be ush with the surrounding smooth plate. The porous materials, described in further detail below, were bonded to a solid Garolite TM sheet to provide a rigid structure and prevent bleed ow. Care was taken to minimize gaps and ensure a smooth transition from the 92 solid wall to the porous substrate. The ow was tripped by a wire of 0.5 mm diameter located 10 cm downstream of the leading edge. In order to generate data at Reynolds numbers similar to the DNS simulations by G omez-de Segura and Garc a-Mayoral (2019), the channel was run at its lowest practical velocity. The freestream velocity wasU e = 14:7 0:1 cm/s immediately upstream of the porous cutout with< 1% in background turbulence. The friction Reynolds numbers ranged from Re 290 to Re 410 depending on measurement location and substrate type. 5.4.2 2D-2C particle image velocimetry Time-resolved velocity elds were acquired using a 2-dimensional, 2-component particle image velocimetry (2D-2C PIV) system in the streamwise-wall normal (xy) plane. The camera, laser, and optical components were mounted to a streamwise traversing cart that moved on precision rails above the water channel. Measurements were made upstream of the cutout and at ve streamwise locations over the porous medium. These measurement locations are termed stations 1-6 (see Table 5.1). As a point of comparison, PIV measurements were also made with a smooth-walled insert placed in the cutout. These baseline measurements we carried out upstream of the cutout (station 1) and at two streamwise locations over the smooth wall (stations 4, 5). The ow was seeded with 5 m polyamide seeding particles (PSP, Dantec Dynamics) with specic gravity 1.03. Illumination was provided by a 5W continuous laser emitting a 532 nm beam, with built-in optics that expanded the beam at 10 . The resulting laser sheet was used to illuminate the ow eld in the streamwise-wall normal (xy) plane in the middle of the water channel. The laser sheet thickness at the wall location was measured to be less than 0:5 0:1 mm. Images were acquired with a Phantom 410L high-speed camera at a rate of 1000 frames per second. Images were acquired for 12.5 seconds and subsequently transferred from the camera to the computer. For each ow condition, three runs were acquired approximately 10 minutes apart. The 99% boundary layer thickness was 4 cm upstream of the cutout, and so the total time series duration ofT = 312:5 = 37:5 s translates into approximatelyTU e = 140 turnover times. The Phantom 410L camera has a resolution of 1280 800 pixels with a pixel size of 20 m. A 50 mm lens with an aperture of f/1.8 was used to acquire images. The resulting eld of view was approximately 125 mm (x) by 170 mm (y). The average particle size was roughly 3 3 pixels. 93 The acquired data were processed using standard procedures for 2D-2C time-resolved PIV in DaVis 10 (LaVision GmbH). The data were processed using a nal box size of 16 pixels with 50% overlap. The PIV analysis was carried out using image pairs separated by 4 frames to ensure that the particles had a displacement of roughly 4 pixels in the free-stream. Generally accepted vector validation routines were used to identify and remove spurious vectors. At each measurement location, the mean turbulence statistics were averaged in the streamwise direction as well as ensemble averaged over the 3 runs. The wall-normal proles of mean turbulence statistics were further analyzed and plotted using in-house routines. After processing, the velocity eld resolution was y = x 1:2 mm. For a representative friction velocity of u 0:7 cm/s (see Table 5.1), this translates into a dimensionless spatial resolution of y + = x + 8. The time step was t = 1 ms, yielding t + 0:05. Standard error estimates for each individual run and correlation uncertainties within DaVis (Wieneke, 2015) were small ( 0:1% for streamwise velocities, 0:5% for wall-normal velocities). Correlation values were above 0.9 in the free-stream and 0.7 in the near-wall region. Averaged over the wall-normal prole, uncertainties for turbulence statistics are estimated to be roughly 0:2% in U, 0:5% foru 2 , 1% forv 2 , and 3% foruv. Here,U is the mean velocity in the streamwise direction whileu andv are the turbulent velocity uctuations in the streamwise and wall-normal directions, respectively. An overbar () denotes a temporal average for each PIV run, a spatial average in x over the PIV window, and an ensemble average over the 3 runs. Maximum uncertainties in the near-wall region (y + < 30) are estimated to be 1% in U, 2% in u 2 , 2% in v 2 , and 5% in uv. Variability in U across the three runs for each case was 0:2% in the freestream and 1:5% in the near-wall region. Variability in u 2 and v 2 was less than 7% averaged across the proles. 5.4.3 Mean ow estimation at single-pixel resolution As noted above, the 2D-2C PIV data were obtained at a spatial resolution of y + = x + 8. This resolution corresponds to the 8-pixel separation between the 16-pixel interrogation windows used for the nal pass. Thus, the actual PIV correlation occurs over windows that are roughly 16=u long in each direction. This leads to substantial spatial averaging in the near-wall region 94 Figure 5.2: Schematic illustration of the single-pixel process. The PIV images are used to create a time-stack showing particle motion in the streamwise direction for each wall normal location, y i . A 2D Fourier transform is used to transform the particle trajectories from tx space into frequency-wavenumber (!k x ) space. A least-squares t identies the best estimate for mean particle velocity at each wall-normal location from the group velocity, U(y i ) =d!=dk x . and does not provide sucient spatial resolution for the evaluation of changes in the near-wall ow. For example, an estimate of the interfacial slip velocity is impossible at this resolution. To improve spatial resolution, and in particular to evaluate the mean velocity prole near the porous interface, a simple routine was implemented to take advantage of the high temporal resolution of the acquired images, f s = 1 kHz or f + s 20. This technique is similar in concept to that proposed by Willert (2015), who employed single-line correlation on images acquired at 2-7 kHz with a very narrow eld of view to resolve the velocity eld inside the viscous sub-layer of a turbulent boundary layer at Re 240. The technique used here is illustrated schematically in Fig. 5.2 and described in greater detail below. First, the images acquired for PIV are transformed into a time-stack by extracting all the data for a given y location. In other words, individual rows from each image are stacked together to create a composite image for each wall-normal location (y i ) that shows particle motion in the x t plane. In this composite image, the particle paths appear as diagonal streaks, akin to characteristics. Next, using a 2D fast Fourier transform (FFT), the xt images are transformed into wavenumber-frequency space (k x and!), yielding data similar to that displayed in the bottom right of Fig. 5.2. Finally, the mean velocity U(y i ) for the given wall-normal location is found from a linear least-squares t to the peak intensity in spectral space, U(y i ) = d!=dk x . The 95 10 -4 10 -3 10 -2 10 -1 0 0.04 0.08 0.12 0.16 Figure 5.3: Comparison of the mean velocity prole obtained using DaVis ( ) and the single pixel routine ( ) for ow at station 5 over the smooth wall. procedure outlined above yields an estimate of the mean velocity prole at a wall-normal resolution of y 0:15 mm or y + 1. Note however, that the average particle size is 3 3 pixels in the images and so the same particle path can in uence mean velocity estimates at 3 dierent y-locations. A representative comparison between the higher-resolution prole and the standard PIV prole obtained from DaVis is provided in Fig. 5.3. There is good agreement between the single-pixel prole and the DaVis processed proles across the boundary layer. Dierences between the single- pixel and DaVis proles are larger in the near-wall region due to the spatial-averaging inherent in the PIV results over the 16-pixel box size in the wall-normal direction. Note that, even though the mean proles in Fig. 5.3 are shown in dimensional terms, it is clear that the single-pixel procedure is able to resolve the mean prole into the viscous sublayer. Below, we show that the single-pixel mean proles are also able to provide an estimate for the slip velocity over the porous substrate. Keep in mind that there is some uncertainty (2 pixels) in estimating the location of the wall in the PIV images for both the smooth wall and porous substrate. 5.4.4 Porous substrates The anisotropic porous materials were designed and fabricated using the method described in Chavarin et al. (2020). Motivated by the simulation results discussed in §5.2.1, this procedure 96 was used to generate an anisotropic material that maximized p K xx + p K zz + while minimiz- ing p K yy + within the fabrication constraints imposed by the 3D printer. The structure of the material consisted of a cubic lattice of rectangular rods with constant cross-section (dd) and varying spacing in the streamwise (s x ), wall-normal (s y ) and spanwise (s z ) directions. For this experiment, a lattice that maximized pore area in the yz plane (i.e., normal to the streamwise ow) and minimized the pore areas in the xy and xz planes (i.e., facing the spanwise and wall-normal ows) was fabricated using a stereolithographic 3D printer (formlabs Form3). Fabri- cation constraints (printing resolution, allowable unsupported lengths, resin drainage) limited the maximum anisotropy that could be achieved. The minimum pore size was dictated by the printer resolution as the rods fused and the surface became solid if the separation between two rods fell below the laser spot size (100m). The maximum pore size was limited by the maximum overhang lengths allowed between rods. With excessive overhang lengths, the rods sagged and deviated from the design geometry. After extensive testing with small samples, a lattice with rod spacings of s x = 0:8 mm ands y =s z = 3:0 mm and a rod diameter ofd = 0:4 mm was selected for the exper- iments. For a representative friction velocity of u 7 mm/s (see Table 5.1), the dimensionless rod spacings are s + x 5:6 and s + y = s + z 21 while the dimensionless rod size is d + 2:8. This geometry represented a good compromise between generating the desired anisotropic permeabil- ity and allowing for reliable manufacturability. Tiles with dimensions of 100mm (x) by 15.4 mm (y) by 100 mm (z) were printed in batches of 5 to reduce manufacturing time. The thickness, h = 15:4 mm (h + 108), was selected to allow for 5 full pores in the wall-normal direction. To ll the entire cutout in the at plate, 90 tiles were manufactured and carefully aligned to preserve streamwise alignment of the pores and minimize gaps. The materials were spray-painted black to reduce re ections from the impinging PIV laser sheet. Sample images of the nished materials are shown in Fig. 5.4. The permeability tensor K of the 3D-printed materials was estimated using Stokes ow simula- tions run in ANSYS Fluent (Ansys Inc.) following the approach of Zampogna and Bottaro (2016). To estimate the streamwise permeability, a body force of unit amplitude was imposed in thex direc- tion and the resulting volume-averaged velocity was used to estimate K xx using Darcy's law. This procedure was repeated with body forces imposed in the y and z directions to estimate K yy and 97 Figure 5.4: Photographs of the 3D-printed anisotropic porous material. The streamwise mean ow goes into the page for the image shown on the left. K zz , respectively. The simulations also conrmed zero o-diagonal components in the permeability tensor and so K = diag(K xx ;K yy ;K zz ). The estimated permeabilities for the 3D-printed materials areK xx = 17210 9 m 2 andK yy =K zz = 2210 9 m 2 . The porosity of the materials is = 0:87. In dimensionless terms, the permeabilities are p K xx + 3:0 and p K yy + = p K zz + 1:1. Thus, the dierence between the streamwise and spanwise permeabilities yields p K xx + p K zz + 1:9. Previous theoretical eorts and numerical simulations suggest that the outward shift in the mean prole is expected to be U + p K xx + p K zz + (Abderrahaman-Elena and Garc a-Mayoral, 2017; G omez-de Segura and Garc a-Mayoral, 2019). Thus, the dierence between the streamwise and spanwise permeabilities is indicative of a marginal decrease in drag. However, the wall-normal permeability exceeds the threshold identied in G omez-de Segura and Garc a-Mayoral (2019) for the emergence of Kelvin-Helmholtz rollers, p K yy + 0:4. 5.4.5 Friction velocity estimation Ultimately, this eort seeks to quantify the change in friction over the porous substrate relative to smooth wall values. Unfortunately, the friction drag is not measured directly (e.g., using a force balance). Instead, the friction is estimated indirectly from the friction velocity, u = p w =, where w is the shear stress at the wall and is density. The friction coecient is dened as C f = w 1=2U 2 e = 2u 2 U 2 e . For canonical smooth wall, zero pressure gradient turbulent boundary layers, a number of methods have been developed to estimate u . These methods use data from various parts of the boundary layer and t them to assumed velocity proles with respective constants (see e.g., Rodr guez-L opez et al., 2015). For smooth wall ows, several analytic or implicit formulations exist 98 10 0 10 1 10 2 10 3 0 5 10 15 20 25 b) 10 -2 10 -1 10 0 0 5 10 15 20 25 a) Figure 5.5: The mean velocity prole acquired using the single pixel method is presented in decit form in panel (a) and in inner-normalized units in panel (b). Black squares ( ) show measurements made over the smooth wall at station 5 while white squares ( ) show measurements made over the porous substrate at the same location. that can predict the form of the mean prole in the viscous, buer, and logarithmic regions of the ow (e.g., Clauser, 1956; Musker, 1979; Kendall and Koochesfahani, 2006). However, it is unclear if these proles remain valid over the porous material. Instead, we make use of the logarithmic and wake region data to estimate friction velocity. In other words, we assume that the outer layer similarity hypothesis holds (Townsend, 1980), such that any changes in the mean prole due to the presence of the porous substrate are restricted to the viscous sublayer and buer region of the ow. Outer layer similarity has been validated extensively for rough walls (e.g., Acharya et al., 1986; Krogstad et al., 1992; Flack et al., 2007). Monty et al. (2016) successfully leveraged outer layer similarity to estimate the friction coecient for bio-fouled ship hulls. Compared to rough wall ows, Manes et al. (2011) and Efstathiou and Luhar (2018) found signicant modication to the mean velocity and streamwise turbulence intensity proles deeper into the boundary layer over high-porosity foams. Nevertheless, both proles collapsed onto the canonical smooth wall proles for y= & 0:3, suggesting that a wake region t remains applicable here. Here, is the 99% boundary layer thickness. This approach also has an additional advantage in that it makes use of logarithmic and wake region data which are more readily available. 99 To estimate friction velocity from the logarithmic and wake regions of the ow, we t the following analytic prole (Coles, 1956; Musker, 1979; Chauhan et al., 2009) to the measured mean velocities: U + = 1 logy + +B + W () + 1 (): (5.1) Here, is the von K arm an constant,B is the additive constant for the logarithmic region, =y= is the outer-normalized wall normal coordinate, W () = 1 cos() is the assumed wake function with strength , and = 2 (1). As before, a superscript + denotes normalization with respect to u and . A least-squares t to the analytic prole in (5.1) is used to estimate u , B, and from mean velocity measurements made in the logarithmic region and beyond, i.e., for y + > 30. The von K arm an constant is assumed to be constant, = 0:39, for the tting procedure. Note that we made use of the single-pixel mean proles for the tting since the additional data points led to more robust ts. Figure 5.6 shows the evolution of the tted parameters u and B over the porous substrate and smooth wall. Friction velocity estimates are listed in Table 5.1. To provide uncertainty estimates for the tted friction velocities, we also attempted ts to just the logarithmic region of the ow as well as the composite proles proposed by Musker and Spalding (Clauser, 1956; Musker, 1979; Kendall and Koochesfahani, 2006; Rodr guez-L opez et al., 2015). The uncertainty estimate listed in Table 5.1 is the standard error across these dierent ts. We also recognize that alternative forms have been proposed for the wake function (Chauhan et al., 2009). A limited sensitivity analysis indicated that the friction velocity estimates obtained were robust to the choice of W (). Figure 5.5 shows the mean proles measured at station 5 over the smooth and porous substrates in outer decit form (a) and with inner normalization (b). These normalized proles make use of the tted friction velocity. The proles shown in Fig. 5.5(a) conrm that outer layer similarity holds over the porous materials tested here, at least in the mean velocity prole. The data from the porous case collapse neatly onto the smooth wall data set for y=& 0:1. Figure 5.5(b) shows that the mean prole over the porous substrate departs from the smooth wall prole in the near- wall region. Specically, the normalized mean velocities are higher over the porous substrate for y + < 10, which is indicative of a slip velocity at the porous interface. However, both proles collapse together for y + & 30, which supports the existence of outer layer similarity. Changes in 100 -10 0 10 20 30 40 50 60 2.5 3 3.5 4 a) -10 0 10 20 30 40 50 60 0.045 0.047 0.049 0.051 b) -10 0 10 20 30 40 50 60 3 3.5 4 4.5 5 c) Figure 5.6: Flow development over the porous substrate ( ) and smooth wall ( ). The evolution of the following parameters is plotted as a function of streamwise location: (a) 99% boundary layer thickness normalized by substrate height,=h; (b) friction velocity normalized by freestream velocity, u =U e ; (c) the additive constantB (5.1) for the logarithmic region. The dashed vertical line indicates the transition from the smooth wall to the cutout for the porous substrate. the mean prole over the porous substrate, including the presence of a potential slip velocity, are discussed in greater detail in §5.5. 5.5 Results and discussion This section is structured as follows. Boundary layer development over the porous substrate in discussed in §5.5.1. Changes in the mean prole and turbulence statistics for fully-developed conditions are considered in §5.5.2. The eect of the porous substrate on velocity spectra is discussed in §5.5.3. 5.5.1 Boundary layer development In this section, we compare ow development over the porous substrate to that over the smooth insert. For this, we primarily make use of the tted parameters u and B, the 99% boundary layer thickness , and derived quantities such as the friction coecient C f . Note that all of these parameters are obtained from the single-pixel mean proles. The rst of our measurement locations (station 1) is just upstream of the substrate transition. Subsequent measurement locations (stations 2-6) are located over the cutout into which the porous or smooth inserts are ush-mounted. PIV measurements were made at all 6 stations for the porous material and at stations 1, 4, and 5 for the smooth wall case. Figure 5.6 provides insight into 101 ow development over the porous substrate relative to smooth wall conditions. As expected, the boundary layer thicknesses for both cases agree within uncertainty upstream of the transition. However, Fig. 5.6(a) shows that, after an initial perturbation immediately downstream of the transition, the boundary layer thickness grows less rapidly over the porous substrate. For example, at station 5 (x=h = 44), the normalized boundary layer thickness over the porous medium is =h 3:3 while that over the smooth wall is =h 3:8. This reduction in boundary layer thickness could potentially be attributed to greater ow penetration into the porous substrate. Figure 5.6(b) shows the streamwise evolution of the friction velocity normalized by the freestream velocity, u =U e . Again, the friction velocities upstream of the transition agree within uncertainty. For the smooth wall case, the normalized friction velocities decrease monotonically in the stream- wise direction. However, friction velocities over the porous substrate show some oscillatory behavior at stations 2-3 downstream of the transition. We attribute this to development eects as the bound- ary layer adjusts to the new surface condition. After this initial variability, the normalized friction velocities decrease monotonically over the porous substrate for stations 4-6 (x=h 28; 44; 53). The streamwise evolution of the additive constant for the logarithmic region B shown in Fig. 5.6(c) is consistent with the friction velocity trends. Once again, the B estimates agree within uncertainty upstream of the cutout. Over the smooth wall, the tted values remain consistent at B 4:5. These smooth wall estimates are a little higher than the typically quoted value of B 4:3 for turbulent boundary layer ows (Marusic et al., 2013), but well within the variability reported in previous literature. Over the porous substrate, there is a sharp decrease in B at stations 3 and 4, with values around B 3:6. Note that this sharp decrease in B coincides with an increase in u . For stations 5 and 6 over the porous substrate (x=h = 44; 53), the estimated values return to B 4:5. Together, the estimates foru =U e andB shown in Fig. 5.6 conrm that the conditions upstream of the cutout are identical (within uncertainty) for the smooth wall and porous substrate exper- iments. The initial ow development over the porous substrate leads to an increase in friction velocities and a decrease inB (n.b., we recognize that the logarithmic law may not remain appro- priate for these non-equilibrium conditions). However, the ow appears to be fully developed by station 5 located at x=h = 44. This observation is in good agreement with earlier experimental 102 Station x=h u [mm/s] B Re C f [10 3 ] 1 -5 7.15 4.5 300 4.80 (7.16) (4.6) (290) (4.70) 2 3 7.08 4.5 320 4.42 3 12 7.58 3.7 330 5.00 4 28 7.27 3.6 350 4.61 (7.05) (4.5) (350) (4.42) 5 44 7.19 4.4 360 4.23 (6.98) (4.5) (410) (4.14) 6 53 6.93 4.5 360 4.23 0:07 0:1 10 0:06 Table 5.1: Estimates for friction-related parameters along the porous substrate. smooth wall values available for measurement stations 1, 4, and 5 are shown in parentheses. Typical uncertainties are shown at the bottom. 600 800 1000 1200 250 300 350 400 450 a) 600 800 1000 1200 4 4.4 4.8 5.2 10 -3 b) Figure 5.7: Friction Reynolds Number (a) and friction coecient (b) plotted as a function of the Reynolds number based on momentum thickness, Re . Smooth wall values are shown as black circles ( ) while porous substrate values are shown as white circles ( ). Dashed lines show empirical relations from Schlatter and Orl u (2010a): Re = 1:13Re 0:843 and C f = 0:024Re 0:25 . results, which suggest that ow development over porous substrates takes place over a streamwise distance of roughly 40h (Efstathiou and Luhar, 2018). At station 5, there is a marginal increase in friction velocity and decrease inB over the porous substrate relative to smooth wall conditions. Estimated values for u , B, Re , and C f for all measurement locations are listed in Table 5.1. Note that the friction coecient at station 5 is approximately 2% higher over the porous medium relative to smooth wall conditions. For completeness, Fig. 5.7 shows friction Reynolds number,Re , and friction coecient,C f , es- timates plotted as as a function of the Reynolds number based on momentum thickness,Re . Note 103 10 0 10 1 10 2 10 3 0 5 10 15 20 25 a) 10 0 10 1 10 2 10 3 0 5 10 15 20 25 b) Figure 5.8: Inner-normalized mean velocity proles measured at station 1 upstream of the cutout (a) and at station 5 (x=h = 44) where the ow is fully developed over the porous substrate (b). In both plots, white squares ( ) show measurements from the porous substrate experiments while black squares ( ) show measurements made with the smooth wall insert in place. The solid lines () show mean proles obtained in DNS by Schlatter and Orl u (2010a) atRe 250 (a) and atRe 360 (b). The dashed line (- -) in panel (b) shows a shifted linear prole of the form U + = p K xx + +y + . that the momentum thickness over the porous substrate was estimated only in the unobstructed domain, i.e., this estimate for does not account for ow penetration into the porous medium. Smooth wall estimates for Re and C f agree within uncertainty with previous empirical relations (Schlatter and Orl u, 2010a). This provides condence in the measurement and tting procedures outlined in the previous sections. Finally, Fig. 5.8 shows normalized mean velocity proles collected at station 1 upstream of the cutout and at station 5, where the ow over the porous substrate is expected to be fully developed. The station 1 mean proles shown in Fig. 5.8(a) show good agreement between the smooth wall and porous substrate experiments. These proles show that the single-pixel procedure generates mean prole estimates into the viscous sublayer (y + < 5). Moreover, the proles are in very good agreement with results obtained from DNS at comparable Re (Schlatter and Orl u, 2010a). The station 5 mean proles show that the mean velocity over the porous substrate is higher than that over the smooth wall in the buer region of the ow. As noted earlier, the mean velocity estimates closest to the porous interface are also indicative of an interfacial slip velocity. These features of the fully-developed mean prole are discussed in §5.5.2 below. In the logarithmic and wake regions 104 of the ow, the smooth wall and porous substrate proles are in agreement. This observation further supports the existence of a fully-developed conditions over the porous substrate for station 5. 5.5.2 Fully-developed ow statistics Figure 5.9 shows inner-normalized mean statistics obtained from the 2D-2C PIV analysis in DaVis for both the porous and smooth wall cases at station 5 (x=h = 44). As noted in the previous section, the ow over the porous substrate is expected to be fully developed at this location. Figure 5.9(a) shows the mean velocity proles obtained from the 2D-2C analysis as well as the single-pixel procedure. In general, the proles obtained using the two dierent techniques are in good agreement with one another. For y + > 30, the mean velocity proles for both the smooth and porous cases agree well with results from the simulations by Schlatter and Orl u (2010a). For the smooth wall case, the data also compare favorably with the DNS data into the viscous sublayer. There are minor discrepancies between the DNS prole and the single-pixel prole over the smooth wall fory + < 15. These discrepancies could be attributed to the2 pixel uncertainty in determining the true location of the wall from the images, which translates into roughly2 viscous units, as well as the uncertainty in the estimate foru . The mean prole over the porous substrate agrees with the smooth wall prole in the logarithmic and wake regions of the ow. For y + < 30, the mean velocity is higher over the porous medium. The DaVis prole does not extend into the viscous sublayer of the ow. However, the mean prole estimated using the single-pixel procedure suggests the presence of a slip velocity with magnitude U + s p K xx + . Specically, for y + < 5 the near-wall mean prole over the porous substrate approaches the curve U + = p K xx + +y + (dashed line in Fig. 5.9(a)). A slip velocity ofU + s = p K xx + is consistent with a slip length ofl + U = p K xx + for the mean ow (Abderrahaman-Elena and Garc a-Mayoral, 2017). Figure 5.9(b) shows estimates for the inner normalized root-mean-square (rms) uctuations in streamwise velocity, u + rms = p u 2 + . Although the turbulence statistics are not resolved below y + 15, the streamwise intensity prole over the smooth wall is consistent with the presence of an inner peak at y + 15. Further, the streamwise intensity measured at this location is comparable in magnitude to that observed in DNS. Estimates for the velocity spectra shown in §5.5.3 below 105 10 0 10 1 10 2 10 3 0 5 10 15 20 25 a) 10 0 10 1 10 2 10 3 0 1 2 3 b) 10 0 10 1 10 2 10 3 0 0.3 0.6 0.9 1.2 c) 10 0 10 1 10 2 10 3 0 0.2 0.4 0.6 0.8 1 d) Figure 5.9: Mean turbulence statistics for smooth and porous cases for station 5 atx=h = 44. Mean velocity proles are shown in (a), proles of the root-mean-square streamwise and wall-normal velocity uctuations are shown in (b) and (c), respectively. The Reynolds shear stress prole is shown in (d). Statistics for the smooth wall and porous substrates are shown as black circles ( ) and white circles ( ) respectively. The black ( ) and white squares ( ) in (a) show the single-pixel mean prole estimates. 106 conrm that this peak is associated with near-wall structures with frequencyf + 0:01 (Robinson, 1991; Jim enez and Pinelli, 1999b). Note that the measured streamwise uctuation intensities over the smooth wall are lower than the DNS values between y + 15 and y + 200. Beyond this location, the DNS and measured proles show reasonable collapse. In contrast to the mean velocity proles, the streamwise intensity prole over the porous wall does not collapse onto the smooth wall data untily + 200 ory= 0:5. Further, the magnitude of the near-wall peak in streamwise intensity is attenuated by approximately 10% relative to that for the smooth wall. This observation is consistent with the measurements reported in Efstathiou and Luhar (2018) for isotropic foams with comparable wall-normal permeabilities, i.e., p K + = p K yy + O(1). This reduction in peak u + rms over the porous substrate is also consistent with previous simulation results (Breugem et al., 2006; Chandesris et al., 2013; G omez-de Segura and Garc a-Mayoral, 2019). Proles for the rms wall-normal velocity uctuations,v + rms = p v 2 + , are presented in Fig. 5.9(c). Relative to the smooth wall case, the maximum wall-normal uctuation intensity is roughly 40% higher over the porous substrate. Note that the smooth wall prole for v + rms is also attenuated by roughly 40% relative to the prole obtained in DNS. As a result, the prole measured over the porous substrate shows much better agreement with the DNS data. The quantitative disagreement between the smooth wall measurements and the DNS results can be attributed to the spatial averaging inherent in the PIV analysis algorithm. Recall that the nal 16 16 pixel interrogation windows used in the PIV analyses correspond to boxes that are approximately 16 viscous units in length. In other words, the PIV measurements cannot properly resolve turbulent ow structures with length scales of O(10=u ). Since such smaller-scale ow features contribute signicantly to the energetic content of the wall-normal velocity uctuations, the PIV measurements are likely to underestimate v + rms substantially. Nevertheless, since both sets of measurements suer from the same spatial resolution issues, we expect the trends observed in Fig. 5.9(c) to remain valid. In other words, the observed increase in v + rms is likely to hold in measurements made at higher spatial resolution. Note that theu + rms proles do not suer from the same attenuation because the streamwise velocity uctuations are typically associated with larger-scale ow features than the wall-normal velocity uctuations. 107 The Reynolds shear stress estimates shown in Fig. 5.9 suer from the same limitations as wall- normal velocity uctuations. It is therefore not surprising that measured proles ofuv + over both the smooth wall and the porous substrate are lower than the DNS data. The Reynolds stress prole over the porous wall has a peak value that is roughly 10% higher than over the smooth wall, which is consistent with the increase in v + rms observed in Fig. 5.9(c). To summarize, the mean statistics shown in Fig. 5.9 suggest that the porous substrate leads to an interfacial slip velocity of U + s p K xx + , a suppression of the near-wall peak in u + rms , and a substantial increase in v + rms across much of the boundary layer. These observations are consistent with simulation results obtained by G omez-de Segura and Garc a-Mayoral (2019) over anisotropic porous materials. Specically, the conditions tested in the experiments here correspond roughly to cases A5 and A6 in G omez-de Segura and Garc a-Mayoral (2019). These cases tested materials with streamwise permeability p K xx + 2:5 3:6 and wall-normal and spanwise permeabilities p K yy + = p K zz + 0:7 1:0 in the numerical simulations. Both substrates led to an increase in skin friction relative to smooth wall conditions. G omez-de Segura and Garc a-Mayoral (2019) attributed this increase in skin friction, and the associated changes in turbulence statistics, to the emergence of energetic spanwise rollers with streamwise wavelengths + x 100 400. We consider the emergence of such rollers by evaluating frequency spectra for the velocity uctuations in the following section. 5.5.3 Velocity spectra Figure 5.10 shows premultiplied frequency spectra for the streamwise and wall-normal velocity uctuations,f + E + uu andf + E + vv , over the smooth wall and porous substrate at station 5 (x=h = 44). Here,f is frequency whileE uu andE vv are the spectral densities for the streamwise and wall-normal velocity uctuations. These spectra were computed from the DaVis time series of u and v using Welch's algorithm in Matlab (Mathworks, Inc.). The computed spectra are somewhat noisy | particularly for the wall-normal velocity uctuations | indicating that the acquisition time may not have been long enough for complete convergence. However, these spectra do provide additional insight into the change in turbulence characteristics over the porous substrate. 108 10 1 10 2 10 3 10 -3 10 -2 10 -1 10 0 a) 0 0.5 1 1.5 2 2.5 10 1 10 2 10 3 10 -3 10 -2 10 -1 10 0 b) 0 0.1 0.2 0.3 0.4 10 1 10 2 10 3 10 -3 10 -2 10 -1 10 0 c) 10 1 10 2 10 3 10 -3 10 -2 10 -1 10 0 d) Figure 5.10: Premultiplied spectra at station 5 for the streamwise velocity uctuationsf + E + uu (a,c) and wall-normal velocity uctuations f + E + vv (b,d). Smooth wall results are shown in (a,b) and porous substrate results are shown in (c,d). The black circles ( ) in (a,c) label a frequency of f + = 0:01 at y + = 15, which corresponds roughly to structures associated with the energetic near-wall cycle. 109 The smooth wall velocity spectra shown in Fig. 5.10(a) foru and in Fig. 5.10(b) forv are broadly consistent with previous observations in wall-bounded turbulent ows (Jimenez and Hoyas, 2008; Jim enez et al., 2010; Krishna et al., 2020). Specically, Fig. 5.10(a) shows the presence of a distinct peak in f + E + uu centered near f + 0:01 and y + 15. This peak corresponds to streak- like structures with streamwise wavelength + x = U + =f + 10 3 that are associated with the energetic near-wall cycle (Robinson, 1991; Smits et al., 2011). Note that U + 10 at this wall- normal location. Figure 5.10(b) shows that the peak in f + E + vv is centered further away from the wall (30 < y + . 100) and at higher frequencies, f + 0:03 0:08. In other words, the wall- normal velocity spectra are dominated by structures in the logarithmic region of the ow that have streamwise length scales + x = U + =f + O(10 2 ), which is in agreement with prior results from numerical simulations (Jim enez et al., 2001; Krishna et al., 2020). Figure 5.10(c) shows some important changes to f + E + uu over the porous substrate. Although the usual near-wall peak remains, another region of high energy emerges for structures with f + 0:02 0:04 (see dashed lines). This region of high f + E + uu extends from the lowest measurement location aty + 10 out toy + 200. We suggest that this is the spectral footprint of the energetic spanwise rollers responsible for drag increases in numerical simulations (G omez-de Segura and Garc a-Mayoral, 2019). Assuming a velocity scale ofU + 10, the frequency rangef + 0:020:04 translates into structures with streamwise wavelength + x 250 400. These length scales are in the range identied by G omez-de Segura and Garc a-Mayoral (2019). The large wall-normal extent is also consistent with previous simulation results, which show that the interfacial Kelvin- Helmholtz type rollers that emerge over porous substrates can extend out into the (nominally) logarithmic region of the ow (e.g., Breugem et al., 2006). Note that a region of high spectral content for f + 0:02 0:04 is also evident in the spectra for wall-normal velocity uctuations over the porous substrate (see Fig. 5.10(d)). However, this region is not as distinct since f + E + vv values are generally elevated over the porous substrate relative to smooth wall conditions. 5.6 Conclusions This paper reports some of the rst turbulence measurements made in boundary layers over streamwise-preferential porous materials that have demonstrated drag reduction capabilities in 110 recent modeling and simulation eorts (Abderrahaman-Elena and Garc a-Mayoral, 2017; Rosti et al., 2018; G omez-de Segura and Garc a-Mayoral, 2019). Models developed in these prior studies show that materials with high streamwise permeability and low spanwise permeability (i.e., mate- rials with p K xx + p K zz + > 0) are promising candidates for passive drag reduction. Driven by these predictions, we designed and 3D-printed a porous substrate with normalized permeabilities p K xx + 3 and p K yy + = p K zz + 1:1. Results presented in §5.5.1 show that the initial development over the porous substrate takes place over a streamwise distance of x=h 40 which is similar to the development length observed in previous experiments over high-porosity foams (Efstathiou and Luhar, 2018). For fully devel- oped conditions, indirect friction estimates show that the 3D-printed porous substrate led to a small (< 5%) increase in drag. Despite the drag increase, the experimental measurements are in broad agreement with previous simulation results. For instance, mean prole estimates ob- tained at single-pixel resolution (see §5.4.3) indicate the presence of a slip velocity over the porous substrate that is consistent with theoretical predictions, U + s p K xx + . Further, PIV-based mea- surements of turbulence statistics (§5.5.2) and velocity spectra (§5.5.3) indicate that the observed drag increase can be attributed to the emergence of energetic spanwise rollers resembling Kelvin- Helmholtz vortices. The simulation results of G omez-de Segura and Garc a-Mayoral (2019) show that such rollers emerge in turbulent ows over anisotropic porous substrates once the wall-normal permeability exceeds p K yy + 0:4. The wall-normal permeability of the material tested here exceeds this threshold value. Together, these observations suggest that streamwise-preferential porous materials continue to be promising candidates for passive drag reduction in wall-bounded turbulent ows. At the very least, the results presented in this paper suggest that streamwise-preferential porous substrates could be used for other ow control applications (e.g., to enhance heat transfer) with minimal frictional penalties. Of course, the experiments reported here do have some important shortcomings. For example, the PIV results shown in §5.5.2 do not include any turbulence measurements below y + 10. A more complete characterization of the interfacial turbulence requires additional measurements 111 made at higher spatial resolution. Such measurements would also help evaluate whether roughness eects due to the presence of the rods of size d + 2:8 at the porous interface are important. Finally, keep in mind that the material tested here has relatively large pore openings (rod spacings ofs + x 5:6 ands + y =s + z 21). Such large pore openings allowed us to create the desired anisotropy in permeability. However, this also means that inertial eects are likely to be important for the pore-scale ow. The numerical simulations of Rosti et al. (2018) and G omez-de Segura and Garc a-Mayoral (2019) make use of idealized models for ow within the permeable substrate. These models do not account for interfacial roughness or inertial eects in the porous medium. Geometry-resolving simulations similar to those pursued by Kuwata and Suga (2017) are needed to evaluate whether streamwise-preferential porous materials are capable of drag reduction once inertial eects become important. 112 Chapter 6 Closing Remarks The experiments described in this thesis explored the interactions between porous substrates and turbulent boundary layers. In particular, they provide the rst experimental data for turbulent boundary layers over anisotropic substrates with streamwise preferential geometry. Key ndings and accomplishments for each chapter are summarized here. In §2, mean turbulence measurements were made in turbulent boundary layers over quasi- isotropic substrates. The eect of thickness, pore size, and development length were explored for substrates with porosity> 0:9 and permeabilities ofRe k 1 10. Slip velocities of 30% ofU e were found across all substrates, accompanied by a signicant mean velocity decit that implies a drag increase. Elevated streamwise turbulence intensities extend further into the boundary layer to y= < 0:4, and time series analysis showed that they are due to spanwise-coherent, Kelvin- Helmholtz rollers. These rollers were found to modulate the amplitude of small scale turbulence similar to large scale motions in smooth wall boundary layers. Despite high permeabilities and penetration of turbulence into the substrate, evidence for outer layer similarity and the existence of a modied log-law was found. Next, a framework to manufacture and characterize anisotropic, porous media was introduced. By modifying the rod spacings and orientations making up a lattice, it was possible to modify all 9 components (6 independent components) of the permeability matrix. Materials were optimized for maximum anisotropy and printability, and two substrates with varying anisotropy ratios of xy (8; 1=8) were manufactured. We tested these materials in a small scale turbulent channel ow experiment and compared them to low-order model predictions. The material with higher wall- normal permeability was found to trigger Kelvin-Helmholtz structures and lead to drag increases 113 of 20% over the smooth wall and 70% over the porous substrate, while the x-permeable material led to a marginal drag increase of 3 4%. Lastly, §5 describes a boundary layer experiment with a substrate consisting of the x-permeable material. Temporally resolved PIV experiments were made along a substrate exploring develop- ment eects over the streamwise preferential substrate at Re 250 360. This experiment conrmed important aspects of numerical simulations by G omez-de-Segura and Garc a-Mayoral (2019) for similar cases. A moderate drag increase ( 5% ) was found, along with the spectral signature of Kelvin-Helmholtz like rollers. These were expected, given the high wall-normal perme- ability p K yy + 1:1 which was above the value of 0.4 thought to trigger spanwise coherent rollers. A novel method to estimate mean velocity proles with single pixel resolution was described and applied to evaluate PIV images. Slip velocitiesU s p K xx + agreed favorably with theoretical pre- dictions by Abderrahaman-Elena and Garc a-Mayoral (2017). We were unable to manufacture a material with suciently low wall-normal permeability due to manufacturing limitations, however our results broadly agreed with numerical simulations and suggested that skin friction reduction would be possible with appropriate materials. While the stream-wise preferential materials tested here did not increase drag, the presence of a porous substrate with minimal increase in friction could be used to moderately increase heat transfer with minimal drag cost, delay stall and mitigate noise. These applications were not studied in this thesis, and many aspects of the substrate/ ow interaction remain unanswered. For example, how the ow behaves and develops within the porous substrates for high stream- wise permeability is not understood, particularly when inertial eects become important. The experiments detailed in this thesis motivated an additional research eort to collect a dataset with increased near-wall resolution over the same substrates, resolving the region y + < 100 with y + 1. We hope that this experiment will provide insights into the interactions between the near-wall turbulence and individual rods and pores. This will help validate boundary conditions at the interface. The bulk models employed by Rosti et al. (2018) and G omez-de-Segura and Garc a-Mayoral (2019) do not resolve the ow within the substrates. These models may not be valid at higher Reynolds numbers as inertial eects within the pores become important. In our experiments, the 114 small spanwise and wall-normal pores inhibited optical and physical access for measurements. To resolve the ow within the porous substrate, endoscopic PIV methods or micro LDV prolometers could be used to validate models. Additional data over the same substrate and measurement locations as detailed in §5 were collected at higher ow rate of nominallyU e 15, 30, 45 and 60 cm/s or friction Reynolds Numbers of Re 400 1600 at the downstream location. As u is expected to increase with increasing U e . it is unlikely that these ows result in actual drag reduction, as this would increase p K yy + . The limiting factor for drag reduction in our experiments is the high wall-normal permeability. As manufacturing techniques improve, similar materials with lower s x or increased d oer the possibility to test this hypothesis. Even without manufacturing modications, testing the current x-permeable materials in the presence of pressure gradients such as airfoils at high angles of attack is of interest. Selective Laser Melting (SLM) advances make it possible to 3D print similar materials out of aluminium alloys. Although the cost is prohibitive on a large scale ( $2000 for a 100mm 100mm tile compared to $15 in resin cost), both the x and y-permeable materials present interesting heat transfer opportunities. In summary, these experiments and procedures detail the manufacturing of anisotropic porous materials with stream-wise preferential permeability. Our experiments represent some of the rst datasets for turbulent channel and boundary layer ow over engineered materials. Although no drag reduction was achieved, the results broadly agree with model predictions and numerical simulations. 115 Bibliography Abderrahaman-Elena, N. and Garc a-Mayoral, R. (2017). Analysis of anisotropically permeable surfaces for turbulent drag reduction. Phys. Rev. Fluids, 2:114609. Abderrahaman-Elena, N. and Garc a-Mayoral, R. (2017). Analysis of anisotropically permeable surfaces for turbulent drag reduction. Physical Review Fluids, 2(11). Acharya, M., Bornstein, J., and Escudier, M. (1986). Turbulent boundary layers on rough surfaces. Experiments in Fluids, 4(1):33{47. Antonia, R. and Luxton, R. (1971). The response of a turbulent boundary layer to a step change in surface roughness part 1. smooth to rough. Journal of Fluid Mechanics, 48(4):721{761. Aurentz, J. and Trefethen, L. (2017). Block operators and spectral discretizations. SIAM Review, 59(2):423{446. Battiato, I. (2012). Self-similarity in coupled Brinkman/Navier{stokes ows. Journal of Fluid Mechanics, 699:94{114. Bechert, D., Bruse, M., and Hage, W. (2000). Experiments with three-dimensional riblets as an idealized model of shark skin. Experiments in uids, 28(5):403{412. Bechert, D., Bruse, M., Hage, W. v., Van der Hoeven, J. T., and Hoppe, G. (1997). Experiments on drag-reducing surfaces and their optimization with an adjustable geometry. Journal of uid mechanics, 338:59{87. Behrens, W. W. and Tucker, A. R. (2010). Ceramic foam electronic component cooling. US Patent 7,742,297. Belcher, S., Jerram, N., and Hunt, J. (2003). Adjustment of a turbulent boundary layer to a canopy of roughness elements. Journal of Fluid Mechanics, 488:369{398. 116 Blois, G., Bristow, N. R., Kim, T., Best, J. L., and Christensen, K. T. (2020). Novel environment enables piv measurements of turbulent ow around and within complex topographies. Journal of Hydraulic Engineering, 146(5):04020033. Breugem, W., Boersma, B., and Uittenbogaard, R. (2006). The in uence of wall permeability on turbulent channel ow. Journal of Fluid Mechanics, 562:35{72. Busse, A. and Sandham, N. D. (2012). In uence of an anisotropic slip-length boundary condition on turbulent channel ow. Physics of Fluids, 24(5):055111. Chandesris, M., d'Hueppe, A., Mathieu, B., Jamet, D., and Goyeau, B. (2013). Direct numerical simulation of turbulent heat transfer in a uid-porous domain. Physics of Fluids, 25(12):125110. Chauhan, K. A., Monkewitz, P. A., and Nagib, H. M. (2009). Criteria for assessing experiments in zero pressure gradient boundary layers. Fluid Dynamics Research, 41(2):021404. Chavarin, A., Efstathiou, C., Vijay, S., and Luhar, M. (2020). Resolvent-based design and experi- mental testing of porous materials for passive turbulence control. Chavarin, A. and Luhar, M. (2020). Resolvent analysis for turbulent channel ow with riblets. AIAA Journal, 58(2):589{599. Choi, H., Moin, P., and Kim, J. (1993). Direct numerical simulation of turbulent ow over riblets. Journal of uid mechanics, 255:503{539. Ciambelli, P., Matarazzo, G., Palma, V., Russo, P., Borla, E. M., and Pidria, M. (2007). Reduction of soot pollution from automotive diesel engine by ceramic foam catalytic lter. Topics in Catalysis, 42(1-4):287{291. Clauser, F. H. (1956). The turbulent boundary layer. Advances in applied mechanics, 4:1{51. Coles, D. (1956). The law of the wake in the turbulent boundary layer. Journal of Fluid Mechanics, 1(2):191{226. De Graa, D. B. and Eaton, J. K. (2000). Reynolds-number scaling of the at-plate turbulent boundary layer. Journal of Fluid Mechanics, 422:319{346. 117 DeGraa, D. and Eaton, J. (2001). A high-resolution laser doppler anemometer: design, quali- cation, and uncertainty. Experiments in uids, 30(5):522{530. Detert, M., Nikora, V., and Jirka, G. (2010). Synoptic velocity and pressure elds at the water{ sediment interface of streambeds. Journal of Fluid Mechanics, 660:55{86. Durst, F., Melling, A., and Whitelaw, J. H. (1976). Principles and practice of laser-doppler anemometry. NASA STI/Recon Technical Report A, 76. Duvvuri, S. and McKeon, B. J. (2015). Triadic scale interactions in a turbulent boundary layer. Journal of Fluid Mechanics, 767:R4. Efstathiou, C. and Luhar, M. (2018). Mean turbulence statistics in boundary layers over high- porosity foams. Journal of Fluid Mechanics, 841:351{379. Favier, J., Dauptain, A., Basso, D., and Bottaro, A. (2009). Passive separation control using a self-adaptive hairy coating. Journal of Fluid Mechanics, 627:451{483. Finnigan, J. (2000). Turbulence in plant canopies. Annual Review of Fluid Mechanics, 32(1):519{ 571. Flack, K., Schultz, M., and Connelly, J. (2007). Examination of a critical roughness height for outer layer similarity. Physics of Fluids, 19(9):095104. Forchheimer, P. (1901). Wasserbewegung durch boden. Z. Ver. Deutsch, Ing., 45:1782{1788. Garcia-Mayoral, R., G omez-de Segura, G., and Fairhall, C. T. (2019). The control of near-wall turbulence through surface texturing. Fluid Dynamics Research, 51(1):011410. Garc a-Mayoral, R. and Jim enez, J. (2011). Drag reduction by riblets. Philosophical transactions of the Royal society A: Mathematical, physical and engineering Sciences, 369(1940):1412{1427. Garcia-Mayoral, R. and Jimenez, J. (2011). Hydrodynamic stability and breakdown of the viscous regime over riblets. Journal of Fluid Mechanics, 678:317{347. Garc a-Mayoral, R., G omez-de Segura, G., and Fairhall, C. T. (2019). Fluid Dynamics Research, (1):011410. 118 Ghisalberti, M. (2009). Obstructed shear ows: similarities across systems and scales. Journal of Fluid Mechanics, 641:51{61. G omez-de Segura, G. and Garc a-Mayoral, R. (2019). Turbulent drag reduction by anisotropic permeable substrates{analysis and direct numerical simulations. Journal of Fluid Mechanics, 875:124{172. G omez-de-Segura, G. and Garc a-Mayoral, R. (2019). Turbulent drag reduction by anisotropic permeable substrates { analysis and direct numerical simulations. Journal of Fluid Mechanics, 875:124{172. G omez-de Segura, G., Sharma, A., and Garc a-Mayoral, R. (2018). Turbulent drag reduction using anisotropic permeable substrates. Flow, turbulence and combustion, 100(4):995{1014. Hahn, S., Je, J., and Choi, H. (2002a). Direct numerical simulation of turbulent channel ow with permeable walls. Journal of Fluid Mechanics, 450:259{285. Hahn, S., Je, J., and Choi, H. (2002b). Direct numerical simulation of turbulent channel ow with permeable walls. Journal of Fluid Mechanics, 450:259{285. Hanna, Y. G. and Spedding, G. R. (2019). Aerodynamic performance improvements due to porosity in wings at moderate re. In AIAA Aviation 2019 Forum, page 3584. Hern andez-Rodr guez, E., Acosta-Mora, P., M endez-Ramos, J., Chinea, E. B., Ferrera, P. E., Canales-V azquez, J., N u~ nez, P., and Ruiz-Morales, J. (2014). Prospective use of the 3d printing technology for the microstructural engineering of solid oxide fuel cell components. computer aided design (CAD), 3:4. Herrin, J. and Dutton, J. (1993). An investigation of LDV velocity bias correction techniques for high-speed separated ows. Experiments in uids, 15(4):354{363. Ho, C.-M. and Huerre, P. (1984). Perturbed free shear layers. Annual Review of Fluid Mechanics, 16(1):365{422. Horton, N. and Pokrajac, D. (2009). Onset of turbulence in a regular porous medium: An experi- mental study. Physics of uids, 21(4):045104. 119 Hutchins, N. and Marusic, I. (2007). Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. Journal of Fluid Mechanics, 579:1{28. Itoh, M., Tamano, S., Iguchi, R., Yokota, K., Akino, N., Hino, R., and Kubo, S. (2006). Turbulent drag reduction by the seal fur surface. Physics of Fluids, 18(6):065102. Jackson, P. (1981). On the displacement height in the logarithmic velocity prole. Journal of Fluid Mechanics, 111:15{25. Jacobi, I. and McKeon, B. (2011). New perspectives on the impulsive roughness-perturbation of a turbulent boundary layer. Journal of Fluid Mechanics, 677:179{203. Jaworski, J. W. and Peake, N. (2013). Aerodynamic noise from a poroelastic edge with implications for the silent ight of owls. Journal of Fluid Mechanics, 723:456{479. Jim enez, J. (1994). On the structure and control of near wall turbulence. Physics of Fluids, 6(2):944{953. Jim enez, J. (2004). Turbulent ows over rough walls. Annual Review of Fluid Mechanics, 36:173{ 196. Jimenez, J. and Hoyas, S. (2008). Turbulent uctuations above the buer layer of wall-bounded ows. Journal of Fluid Mechanics, 611:215{236. Jim enez, J., Hoyas, S., Simens, M. P., and Mizuno, Y. (2010). Turbulent boundary layers and channels at moderate reynolds numbers. Journal of Fluid Mechanics, 657:335{360. Jim enez, J. and Pinelli, A. (1999a). The autonomous cycle of near-wall turbulence. Journal of Fluid Mechanics, 389:335{359. Jim enez, J. and Pinelli, A. (1999b). The autonomous cycle of near-wall turbulence. Journal of Fluid Mechanics, 389:335{359. Jim enez, J., Uhlmann, M., Pinelli, A., and Kawahara, G. (2001). Turbulent shear ow over active and passive porous surfaces. Journal of Fluid Mechanics, 442:89{117. Jimenez, J., Uhlmann, M., PINELLI, A., and KAWAHARA, G. (2001). Turbulent shear ow over active and passive porous surfaces. Journal of Fluid Mechanics, 442:89{117. 120 Jovic, S. and Driver, D. M. (1994). Backward-facing step measurements at low reynolds number, re h = 5000. NASA STI/Recon Technical Report N, 94. Kendall, A. and Koochesfahani, M. (2006). A method for estimating wall friction in turbulent boundary layers. In 25th AIAA aerodynamic measurement technology and ground testing con- ference, page 3834. Khorrami, M. R. and Choudhari, M. M. (2003). Application of passive porous treatment to slat trailing edge noise. Kim, T., Blois, G., Best, J., and Christensen, K. T. (2016). Experimental study of a coarse-gravel river bed: Elucidating the nearwall and pore-space turbulent ow physics. In River Flow 2016, pages 950{955. CRC Press. Kim, T., Blois, G., Best, J. L., and Christensen, K. T. (2019). Piv measurements of turbulent ow overlying large, cubic-and hexagonally-packed hemisphere arrays. Journal of Hydraulic Research. Kim, T., Blois, G., Best, J. L., and Christensen, K. T. (2020). Experimental evidence of amplitude modulation in permeable-wall turbulence. Journal of Fluid Mechanics, 887. Klett, J. W. (2000). Process for making carbon foam. US Patent 6,033,506. Kong, F. and Schetz, J. (1982). Turbulent boundary layer over porous surfaces with dierent surface geometries. In 20th Aerospace Sciences Meeting, page 30. Krishna, C. V., Wang, M., Hemati, M. S., and Luhar, M. (2020). Reconstructing the time evolution of wall-bounded turbulent ows from non-time-resolved piv measurements. Physical Review Fluids, 5(5):054604. Krogstad, P.- A., Antonia, R., and Browne, L. (1992). Comparison between rough-and smooth-wall turbulent boundary layers. Journal of Fluid Mechanics, 245:599{617. Kuwata, Y. and Suga, K. (2016). Lattice-Boltzmann direct numerical simulation of interface turbulence over porous and rough walls. International Journal of Heat and Fluid Flow, 61(Part A):145 { 157. 121 Kuwata, Y. and Suga, K. (2017). Direct numerical simulation of turbulence over anisotropic porous media. Journal of Fluid Mechanics, 831:41{71. Le, H., Moin, P., and Kim, J. (1997). Direct numerical simulation of turbulent ow over a backward- facing step. Journal of uid mechanics, 330:349{374. Lee, S.-J. and Lee, S.-H. (2001). Flow eld analysis of a turbulent boundary layer over a riblet surface. Experiments in uids, 30(2):153{166. Lei, G., Dong, P., Mo, S., Yang, S., Wu, Z., and Gai, S. (2015). Calculation of full permeabil- ity tensor for fractured anisotropic media. Journal of Petroleum Exploration and Production Technology, 5(2):167{176. Liakopoulos, A. C. (1965). Darcy's coecient of permeability as symmetric tensor of second rank. International Association of Scientic Hydrology. Bulletin, 10(3):41{48. Luchini, P., Manzo, F., and Pozzi, A. (1991). Resistance of a grooved surface to parallel ow and cross- ow. Journal of uid mechanics, 228:87{109. Luhar, M., Rominger, J., and Nepf, H. (2008). Interaction between ow, transport and vegetation spatial structure. Environmental Fluid Mechanics, 8(5):423{439. Luhar, M., Sharma, A., and McKeon, B. (2014a). On the structure and origin of pressure uctua- tions in wall turbulence: predictions based on the resolvent analysis. Journal of uid mechanics, 751:38{70. Luhar, M., Sharma, A. S., and McKeon, B. (2015). A framework for studying the eect of compliant surfaces on wall turbulence. Journal of Fluid Mechanics, 768:415{441. Luhar, M., Sharma, A. S., and McKeon, B. J. (2014b). Opposition control within the resolvent analysis framework. Journal of Fluid Mechanics, 749:597{626. Mahjoob, S. and Vafai, K. (2008). A synthesis of uid and thermal transport models for metal foam heat exchangers. International Journal of Heat and Mass Transfer, 51(15):3701{3711. Manes, C., Poggi, D., and Ridol, L. (2011). Turbulent boundary layers over permeable walls: scaling and near-wall structure. Journal of Fluid Mechanics, 687:141{170. 122 Marusic, I., Mathis, R., and Hutchins, N. (2010). Predictive model for wall-bounded turbulent ow. Science, 329(5988):193{196. Marusic, I., Monty, J. P., Hultmark, M., and Smits, A. J. (2013). On the logarithmic region in wall turbulence. Journal of Fluid Mechanics, 716:R3. Mathis, R., Hutchins, N., and Marusic, I. (2009). Large-scale amplitude modulation of the small- scale structures in turbulent boundary layers. Journal of Fluid Mechanics, 628:311{337. Mathis, R., Marusic, I., Chernyshenko, S. I., and Hutchins, N. (2013). Estimating wall-shear-stress uctuations given an outer region input. Journal of Fluid Mechanics, 715:163. Mathis, R., Marusic, I., Hutchins, N., and Sreenivasan, K. (2011). The relationship between the velocity skewness and the amplitude modulation of the small scale by the large scale in turbulent boundary layers. Physics of Fluids, 23(12):121702. McKeon, B. (2017). The engine behind (wall) turbulence: perspectives on scale interactions. Journal of Fluid Mechanics, 817. McKeon, B. J. and Sharma, A. S. (2010). A critical-layer framework for turbulent pipe ow. Journal of Fluid Mechanics, 658:336{382. McLaughlin, D. and Tiederman, W. (1973). Biasing correction for individual realization of laser anemometer measurements in turbulent ows. The Physics of Fluids, 16(12):2082{2088. Moarref, R., Sharma, A. S., Tropp, J. A., and McKeon, B. J. (2013). Model-based scaling of the streamwise energy density in high-reynolds-number turbulent channels. Journal of Fluid Mechanics, 734:275{316. Monty, J., Dogan, E., Hanson, R., Scardino, A., Ganapathisubramani, B., and Hutchins, N. (2016). An assessment of the ship drag penalty arising from light calcareous tubeworm fouling. Biofouling, 32(4):451{464. Motlagh, S. Y. and Taghizadeh, S. (2016). POD analysis of low Reynolds turbulent porous channel ow. International Journal of Heat and Fluid Flow, 61(Part B):665 { 676. 123 Musker, A. (1979). Explicit expression for the smooth wall velocity distribution in a turbulent boundary layer. AIAA Journal, 17(6):655{657. Nakashima, S., Fukagata, K., and Luhar, M. (2017). Assessment of suboptimal control for turbulent skin friction reduction via resolvent analysis. Journal of Fluid Mechanics, 828:496{526. Nepf, H. M. (2012). Flow and transport in regions with aquatic vegetation. Annual Review of Fluid Mechanics, 44:123{142. Ochoa-Tapia, J. A. and Whitaker, S. (1995). Momentum transfer at the boundary between a porous medium and a homogeneous uid|i. theoretical development. International Journal of Heat and Mass Transfer, 38(14):2635{2646. Parikh, P. G. (2011). Passive removal of suction air for laminar ow control, and associated systems and methods. US Patent 7,866,609. Patel, V. C. (1965). Calibration of the preston tube and limitations on its use in pressure gradients. Journal of Fluid Mechanics, 23(1):185{208. Pathikonda, G. and Christensen, K. T. (2017). Inner{outer interactions in a turbulent boundary layer overlying complex roughness. Physical Review Fluids, 2(4):044603. Poggi, D., Porporato, A., Ridol, L., Albertson, J., and Katul, G. (2004). The eect of vegetation density on canopy sub-layer turbulence. Boundary-Layer Meteorology, 111(3):565{587. Pokrajac, D. and Manes, C. (2009). Velocity measurements of a free-surface turbulent ow pene- trating a porous medium composed of uniform-size spheres. Transport in porous media, 78(3):367. Pope, S. B. (2001). Turbulent ows. Reynolds, W. C. and Tiederman, W. G. (1967). Stability of turbulent channel ow, with application to malkus's theory. Journal of Fluid Mechanics, 27(2):253{{272. Robert, J. (1992). Drag reduction: an industrial challenge. Technical report, AIRBUS INDUSTRIE BLAGNAC (FRANCE). Robinson, S. K. (1991). Coherent motions in the turbulent boundary layer. Annual Review of Fluid Mechanics, 23(1):601{639. 124 Rodr guez-L opez, E., Bruce, P. J., and Buxton, O. R. (2015). A robust post-processing method to determine skin friction in turbulent boundary layers from the velocity prole. Experiments in Fluids, 56(4):68. Rosti, M. E., Brandt, L., and Pinelli, A. (2018). Turbulent channel ow over an anisotropic porous wall{drag increase and reduction. Journal of Fluid Mechanics, 842:381{394. Rosti, M. E., Cortelezzi, L., and Quadrio, M. (2015). Direct numerical simulation of turbulent channel ow over porous walls. Journal of Fluid Mechanics, 784:396{442. Ru, J. and Gelhar, L. (1972). Turbulent shear ow in porous boundary. J. Engrg. Mech, 504(98):975. Schlatter, P. and Orl u, R. (2010a). Assessment of direct numerical simulation data of turbulent boundary layers. Journal of Fluid Mechanics, 659:116{126. Schlatter, P. and Orl u, R. (2010b). Quantifying the interaction between large and small scales in wall-bounded turbulent ows: A note of caution. Physics of uids, 22(5):051704. Schoppa, W. and Hussain, F. (2002). Coherent structure generation in near-wall turbulence. Journal of uid Mechanics, 453:57{108. Schultz, M. and Flack, K. (2007). The rough-wall turbulent boundary layer from the hydraulically smooth to the fully rough regime. Journal of Fluid Mechanics, 580:381{405. Smits, A. J., McKeon, B. J., and Marusic, I. (2011). High{Reynolds number wall turbulence. Annual Review of Fluid Mechanics, 43:353{375. Studart, A. R., Gonzenbach, U. T., Tervoort, E., and Gauckler, L. J. (2006). Processing routes to macroporous ceramics: a review. Journal of the American Ceramic Society, 89(6):1771{1789. Suga, K., Matsumura, Y., Ashitaka, Y., Tominaga, S., and Kaneda, M. (2010). Eects of wall permeability on turbulence. International Journal of Heat and Fluid Flow, 31(6):974{984. Suga, K., Okazaki, Y., Ho, U., and Kuwata, Y. (2018). Anisotropic wall permeability eects on turbulent channel ows. Journal of Fluid Mechanics, 855:983{1016. 125 Thielicke, W. and Stamhuis, E. J. (2014). Pivlab { towards user-friendly, aordable and accurate digital particle image velocimetry in matlab. Journal of Open Research Software, 2. Toedtli, S. S., Luhar, M., and McKeon, B. J. (2019). Predicting the response of turbulent channel ow to varying-phase opposition control: Resolvent analysis as a tool for ow control design. Physical Review Fluids, 4(7):073905. Townsend, A. (1980). The structure of turbulent shear ow. Cambridge university press. Walsh, M. and Lindemann, A. (1984). Optimization and application of riblets for turbulent drag reduction. In 22nd Aerospace Sciences Meeting, page 347. Walsh, M. J., Sellers III, W. L., and Mcginley, C. B. (1989). Riblet drag at ight conditions. Journal of Aircraft, 26(6):570{575. White, B. L. and Nepf, H. M. (2007). Shear instability and coherent structures in shallow ow adjacent to a porous layer. Journal of Fluid Mechanics, 593:1{32. Wieneke, B. (2015). Piv uncertainty quantication from correlation statistics. Measurement Sci- ence and Technology, 26(7):074002. Willert, C. E. (2015). High-speed particle image velocimetry for the ecient measurement of turbulence statistics. Experiments in uids, 56(1):17. Zagarola, M., Perry, A., and Smits, A. (1997). Log laws or power laws: The scaling in the overlap region. Physics of Fluids, 9(7):2094{2100. Zagni, A. F. and Smith, K. V. (1976). Channel ow over permeable beds of graded spheres. Journal of the Hydraulics Division, 102(2):207{222. Zaman, K., Bridges, J., and Hu, D. (2011). Evolution from `tabs' to `chevron technology'-a review. International Journal of Aeroacoustics, 10(5-6):685{709. Zampogna, G. A. and Bottaro, A. (2016). Fluid ow over and through a regular bundle of rigid bres. Journal of Fluid Mechanics, 792:5{35. 126 Zhu, C., Qi, Z., Beck, V. A., Luneau, M., Lattimer, J., Chen, W., Worsley, M. A., Ye, J., Du- oss, E. B., Spadaccini, C. M., et al. (2018). Toward digitally controlled catalyst architectures: Hierarchical nanoporous gold via 3d printing. Science advances, 4(8):eaas9459. 127
Abstract (if available)
Abstract
Manipulating turbulence is desirable to control mixing in turbulent combustion and industrial processes, to enhance heat transfer and mitigate aerodynamic noise, and to reduce skin friction in wall-bounded turbulent flows. For wall-bounded flows, varying the surface microstructure has proven to be an effective means of passive flow control, i.e., control that does not require active energy input. For example, streamwise aligned riblets successfully reduce skin friction drag, while porous metal foams enhance heat transfer around finned heat exchangers. This thesis reports findings from laboratory boundary layer experiments over commercially available and custom-manufactured porous materials with a view to developing passive flow control techniques.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Model based design of porous and patterned surfaces for passive turbulence control
PDF
Flow and thermal transport at porous interfaces
PDF
Modeling and analysis of parallel and spatially-evolving wall-bounded shear flows
PDF
Passive flight in density-stratified fluid environments
PDF
Aerodynamics at low Re: separation, reattachment, and control
PDF
Physics-based and data-driven models for bio-inspired flow sensing and motion planning
PDF
Biomimetics and bio-inspiration for moderate Reynolds number airfoils and aircraft
PDF
Investigations of fuel and hydrodynamic effects in highly turbulent premixed jet flames
PDF
Reconstruction and estimation of shear flows using physics-based and data-driven models
PDF
The effect of lattice structure and porosity on thermal conductivity of additively-manufactured porous materials
PDF
Molecular- and continuum-scale simulation of single- and two-phase flow of CO₂ and H₂O in mixed-layer clays and a heterogeneous sandstone
PDF
RANS simulations for flow-control study synthetic jet cavity on NACA0012 and NACA65(1)412 airfoils.
PDF
Biomimetic engineered dolphin skin inspired materials to reduce skin-friction drag
PDF
Machine-learning approaches for modeling of complex materials and media
PDF
Effective flow and transport properties of deforming porous media and materials: theoretical modeling and comparison with experimental data
PDF
Understanding properties of extreme ocean wave runup
PDF
Pattern generation in stratified wakes
PDF
Multiscale and multiresolution approach to characterization and modeling of porous media: From pore to field scale
PDF
Passive rolling and flapping dynamics of a heaving Λ flyer
PDF
Large eddy simulations of turbulent flows without use of the eddy viscosity concept
Asset Metadata
Creator
Efstathiou, Christoph A.
(author)
Core Title
Developing tuned anisotropic porous substrates for passive turbulence control
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Aerospace Engineering
Publication Date
08/01/2020
Defense Date
05/29/2020
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
additive,Control,drag,flow,manufacturing,OAI-PMH Harvest,passive,porous,reduction,turbulence
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Luhar, Mitul (
committee chair
), deBarros, Felipe (
committee member
), Spedding, Geoffrey (
committee member
)
Creator Email
christoph.efstathiou@gmail.com,efstathi@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c89-353318
Unique identifier
UC11666276
Identifier
etd-Efstathiou-8853.pdf (filename),usctheses-c89-353318 (legacy record id)
Legacy Identifier
etd-Efstathiou-8853.pdf
Dmrecord
353318
Document Type
Dissertation
Rights
Efstathiou, Christoph A.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
additive
drag
flow
passive
porous
reduction
turbulence