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University of Southern California Dissertations and Theses
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Pattern generation in stratified wakes
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Pattern generation in stratified wakes
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PATTERN GENERATION IN STRATIFIED WAKES by Chan-Ye Ohh A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) December 2023 Copyright 2023 Chan-Ye Ohh Acknowledgements I could not have achieved my PhD without the help of those around me. I would like to take this opportunity to recognize a few people whose support was especially influential in the successful completion of my PhD. Throughout my 12 years of studies at USC, from undergraduate to master’s and PhD programs, I have had the privilege of encountering many exceptional mentors. Above all, I am deeply grateful to my PhD advisor, Dr. Geoffrey Spedding. He has been consistently available to support and guide me, always prioritizing my learning. Dr. Spedding never hesitated to assist with hands-on work when needed, whether it was diagnosing a broken laser or helping with clearing out the lab space during the move day. He also made sure that I could continue my studies without financial concerns and arranged TA experiences that would be beneficial to my learning. His passion for fluid dynamics and his willingness to always learn something new are truly inspiring. His insightful feedback, compassion, and sense of humor made every meeting enjoyable. I also thank Dr. Mitul Luhar for his guidance and support throughout my PhD and master’s studies. Even though he was not my official PhD advisor, his role as a mentor within my PhD com- mittee and during the weekly fluid supergroup meetings, as well as his guidance for my master’s thesis, greatly influenced me to become a better researcher. I greatly appreciate his attention to my master’s research, treating it with the same care as his PhD students. He also gave me the op- portunity to present my master’s work at a conference and recommended me for summer funding through a high school mentorship program. ii I would like to thank other committee members, Dr. Ivan Bermejo-Moreno, Dr. Alejandra Uranga, and Dr. Patrick Lynett, for offering valuable advice not only on research, but also on my career. I want to express my gratitude to the collaborators Vamsi and Dr. Jonathan Tu for their efforts in exploring alternative solutions for the wake classification and providing feedback to strengthen the performance and robustness of the pattern classifier. I also appreciate my colleagues in the stratified fluids lab during my undergraduate years, in- cluding Trystan, XJ, Prabu. They patiently trained me to familiarize myself with the experimental setup and PIV systems. They also made the lab feel like a hangout spot. A special thanks to Shilpa, my officemate, pandemic walking buddy, therapist, and secretary. I greatly appreciate being able to discuss research ideas with her without insecurities getting in the way. Despite spending too much time chitchatting in the office, she helped me set realistic expectations for time management. I am grateful to Emma for being there with me during the countless attempts to figure out tomographic PIV systems and for helping set up experiments that required two people when lab access was limited during the pandemic. I also want to thank Madeleine for quickly adapting to the stratified wakes experimental setup, asking many insightful questions, and being flexible, which made it easier to wrap things up before finishing my PhD. Thanks to other members in the fluid supergroup and neighboring lab students–Sarkar, Yohanna, Arturo, James, Michael, Christoph, Andrew, Mark, Idan, Chase, Madeleine, Morgan, Jocelyn, JP, Yangyang, Hanliang, and Jingyi–I will miss our lunches, coffee breaks, and happy hours. To my friends who supported me throughout this journey–Yeojung, Nayeon, Hyomin, Han, Richel, Malika, Erum, Jeonho, Hyunggon, and Dayung– your encouragement meant a lot. My heartfelt gratitude goes to my parents, sisters, and Nick for always encouraging me to pursue my dreams and not to be discouraged by uncertainties in life. Your wisdom helped me think iii logically and recognize what to prioritize in life, while reminding me to view every experience with appreciation. Finally, I thank God for allowing me the opportunity to observe and explore His beautiful and complex creation through my PhD research, and to grow as an individual with the love and support from those around me. Financial support from the Office of Naval Research (Grants: N0014-15-1-2506 and N0014- 20-1-2584) and the USC AME Teaching Assistantship is gratefully acknowledged. iv Table of Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Parameterizing stratified flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Wakes in stratified fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Low Re sphere wake regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 High Fr turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Signature detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.1 Universal late wakes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter 2: Dissertation overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter 3: Wake classification on stratified sphere wake . . . . . . . . . . . . . . . . . 16 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1 Modal decomposition tools . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.1.1 Temporal reconfiguration of the experimental data . . . . . . . . 20 3.2.2 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.3 Standard DMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.4 Total DMD for noise-contaminated datasets . . . . . . . . . . . . . . . . . 24 3.2.5 Streaming DMD for real-time update and large datasets . . . . . . . . . . . 25 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.1 A combined regime diagram for sphere wakes . . . . . . . . . . . . . . . . 26 3.3.2 Distinguishing characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.2.1 V ortex street (VS) . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.2.2 Symmetric, non-oscillatory (SN) . . . . . . . . . . . . . . . . . 31 3.3.2.3 Asymmetric, non-oscillatory (AN) . . . . . . . . . . . . . . . . . 31 3.3.2.4 Planar oscillatory (PO) . . . . . . . . . . . . . . . . . . . . . . . 32 v 3.3.2.5 Spiral mode (SP) . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.3 3D standard DMD based classification . . . . . . . . . . . . . . . . . . . . 33 3.3.3.1 Mode selection criteria . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.3.2 Non-oscillatory regimes: Symmetric or Asymmetric . . . . . . . 36 3.3.3.3 Oscillatory regimes: Planar motion or Spiral Mode . . . . . . . . 36 3.3.3.4 Planar motion: V ortex Street or Planar Oscillation . . . . . . . . 37 3.3.3.5 The success rate of the classifier with 3D numerical DMD input . 37 3.3.4 Reduced information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.4.1 Spatially reduced: 2D centerplanes . . . . . . . . . . . . . . . . 38 3.3.4.2 Real-time analysis: Streaming DMD . . . . . . . . . . . . . . . 43 3.3.5 Noise contaminated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.5.1 TDMD based classifier for noise-contaminated input . . . . . . . 44 3.4 Extrapolation to higher Re and Fr . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.1 Overturning horizontal length scale l h estimation . . . . . . . . . . . . . . 47 3.4.2 Dissipation rateε estimation . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.3 Fr h -G space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.3.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Chapter 4: Patterns from stratified inclined spheroid wakes . . . . . . . . . . . . . . . 56 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1.1 Characteristics of the flow around spheroids . . . . . . . . . . . . . . . . . 56 4.1.2 Spheroids in a stratified ambient . . . . . . . . . . . . . . . . . . . . . . . 60 4.1.3 Scaling law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1.4 Evolution of trailing vortices . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.1.5 Current state of experiment and simulation studies . . . . . . . . . . . . . 65 4.1.6 Open questions and objectives . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2.1 Wake analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.2 Mount effect and trip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.3 Window of opportunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.1 Inclination angleθ = 0 ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.1.1 Evolution of length and velocity scales ofθ = 0 ◦ . . . . . . . . . 74 4.3.2 Inclination angleθ = 10 ◦ ,20 ◦ . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.2.1 The instantaneous structure of inclined wakes . . . . . . . . . . . 77 4.3.2.2 Streamwise vortices . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3.2.3 Wake propagation . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3.2.4 Internal wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.2.5 Evolution of length and velocity scales of non-zeroθ . . . . . . . 85 4.3.2.6 Lateral asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Chapter 5: Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 vi 5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 vii List of Figures 1.1 Schematic to represent (a) Reynolds number Re=ρUL/µ, (b) Froude number Fr= U/NR, (c) Brunt-V¨ ais¨ al¨ a frequency N . . . . . . . . . . . . . . . . . . . . . 2 1.2 (a) The density gradient (light blue) and the squared Brunt-V¨ ais¨ al¨ a frequency N 2 (magenta) in the thermocline of south Indian ocean measured from the Argo (Array for Real-time Geostrophic Oceanography) data [1]. Pycnocline (⋆) is roughly at the depth of∼ 100− 250 m where the density is linear and N 2 ∼ 10 − 4 s − 2 . (b) An artificially colored satellite image (center image [2]) of a wake generated by the Tristan da Cunha island in South Atlantic (an enlarged view of the island in the bottom left image from JPL/NASA) where a coherent pattern is visualized by the clouds. Top right image [3] shows a side view of the island where the clouds are formed in layers. . . . . . . . . . . . . . . . . . . . . . 2 1.3 (a) The decay rate of stratified flow at 3D-NEQ-Q2D evolution stages (b) vertical vorticity of a sphere wake from the body to late wake, Nt =[10− 240] with varying initial Fr=[10− 80] [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Regime diagrams from CH93 (a) and LI92 (b). The 4 main regimes differentiated in (a) are: 2D, quasi-2D; SLW, strong lee wave; T, transition (SKH, without K-H instability; KH, with K-H instability); and 3D, three-dimensional. In (b), LI92 distinguished 6 main regimes: A u , unsteady, attached 2D vortices; V t , 2D vortex shedding; L, lee-wave instability; N, non-axisymmetric attached vortex; S, symmetric vortex shedding; V , non-symmetric vortex shedding; and T , turbulent wake. Note that the axes of LI92 are flipped compared to CH93 and the Fr definition in LI92 is F i = U/ND characterized by the diameter instead of radius, which is used in CH93 and this studies. . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Regimes in{Fr h -Gn} parameter space identified by Bruyn Kops & Riley [25] . . . 10 1.6 At the late wake, the wake follows the universal curve of a scaled effective Fr regardless of the body geometry. Tested geometry includes: a sphere (• ), a slender spheroid (*), a cylinder (△), a disc (⋄ ), a cube (□ ), and a hemisphere (◦ ) [34]. The vertical axis U 0 /U B F 2/3 eff shown in the plot is equivalent to u 0 Fr 2/3 eff . . . . . . . . . . 11 viii 3.1 Schematic showing experimental setup (a) Two 2D planar PIV appended. The annotated scale is based on a sphere with D= 4 cm to provide a relative scale of the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Temporal data reconfiguration where the colored box A is moving in reference to the sphere. Dash lines indicate column vectors of velocity field within the box A . . 21 3.3 Meshing in simulation and the location of refinement levels. The computational domain is (x,y,z)=([-10,50], [-8,8], [-8,8])D and the results in ([1.5,15], [-2,2], [-2,2])D was used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 Regime diagrams from CH93 (a) and LI92 (b). The 4 main regimes differentiated in (a) are: 2D, quasi-2D; SLW, strong lee wave; T, transition (SKH, without K-H instability; KH, with K-H instability); and 3D, three-dimensional. In (b), LI92 distinguished 6 main regimes: A u , unsteady, attached 2D vortices; V t , 2D vortex shedding; L, lee-wave instability; N, non-axisymmetric attached vortex; S, symmetric vortex shedding; V , non-symmetric vortex shedding; and T , turbulent wake. (c) has the trajectory of local Re and Fr of Re= 1000 and Fr= 8 at Nt=[0.13− 11] plotted over combined (a) and (b) and{Re-Fr} test cases. . . . . . 27 3.5 A regime diagram for stratified sphere wakes. The final result in colored dots combines experiments from CH93 (lighter colors) and LI92 (darker colors) with combined laboratory and numerical experiments. The five identified regimes are labeled and color-coded as: V ortex street (VS, purple), Symmetric non-oscillation (SN, red), Asymmetric non-oscillation (AN, yellow), Planar oscillation (PO, green), and Spiral mode (SP, blue) . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.6 A snapshot of the flow field u in vertical and horizontal centerplane (left) and a dominant DMD mode of simulation in 3D (middle) and experiment in 2D centerplane (right) illustrating various regimes observed: (a) R10F0.5 V ortex Street (R2F0.5 for the experiment), (b) R5F1 Symmetric Non-oscillatory, (c) R3F4 Asymmetric Non-oscillatory, (d) R3F8 Planar Oscillation (e) R10F8 Spiral Mode. The green box represents the equivalent window size between simulation and experiment. The isolevel of the middle 3D plot is 30% of the maximum velocity component. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.7 The plots on the left show the real and imaginary parts of λ, and on the right ˜ E(St) is shown for each mode of the case Re= 500 and Fr= 4. The dark red corresponds to the mean mode. The green dot is the most dominant mode selected to be evaluated in Fig. 3.8. The grey area is considered noise, set below ˜ E threshold and above St threshold. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.8 A flow chart of the wake classifier with 3D DMD input. The selected DMD mode from Section 3.3.3.1 is sorted into five regimes in the non-oscillatory branch (left) and oscillatory branch (right) based on the symmetry of the cross-stream velocities in that mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ix 3.9 Success rate of a classifier with 3D numerical DMD input. The pie chart shows the classification type in color. The number above each filled circle is the the percent accuracy of the majority outcome. . . . . . . . . . . . . . . . . . . . . . . 38 3.10 A flow chart of the wake classifier with two, 2D-centerplane flow fields as input. Two dominant DMD modes of each centerplane are evaluated where the mode for the vertical centerplane is selected based on St w closest to St v . The non-oscillatory branch uses the mode from the horizontal centerplane, while the oscillatory branch uses the vertical centerplane for the first step, and horizontal centerplane for the second step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.11 Classifier success rate with (a) 2D numerical (b) 2D experimental DMD input. Cases with Red X have wake length-scales larger than the FOV for regimes to be distinguishable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.12 Performance of sDMD. Left is the dominant DMD mode and on the right is the dominant sDMD. (a) dominant modes from DMD and sDMD of the Re= 500 and Fr= 4 case in the same given space where similar planar oscillation features are shown. (b) an example of processing sDMD on a large data matrix of a high resolution case (Re= 1000 and Fr= 8) where a dominant sDMD mode successfully isolated the spiral feature. In both cases, an SVD truncation of m= 10 was applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.13 Ritz valuesλ (left), ˜ E versus St (center), v(x,y,z= 0) of a dominant oscillatory mode (indicated by an arrow) for the Re= 300, Fr= 2 case computed from (a) standard DMD (E r = 100% of ˜ E) and (b) TDMD (E r = 91.6%). Modes colored in red are energetic modes selected from Section 3.3.3.1 . . . . . . . . . . . . . . . 45 3.14 A sketch illustrating how local overturning length scale l h and l v are estimated. The fluctuating quantities of the lateral vorticity ω ′ y and vertical velocity w are measured at the wake edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.15 An estimate of∆t, u loc , and l h for Fr h of the experiment at initial Re= 1000 and Fr= 8. The local Froude number Fr h follows the weakly stratified wakes scaling, Fr h ∼ Nt − 1 [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.16 comparison of the ˜ ε estimate for 3D3C, 2D3C, 2D2C with a local isotropy assumption. 2D3C and 2D2C has an ˜ ε estimate for horizontal plane (() h ), and vertical plane (() v ). Sample flow is from the simulation at initial Re = 1000 and Fr= 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.17 Evolution of wake in Nt at initial Re= 1000 and Fr= 8. Red dot is simulation; blue dot is experiment; and black line dotted line is from simulation by Bruyn Kops & Riley [25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.18 a snapshot of ˜ ρ for (a) Re= 1000 and Fr= 8 and (b) Re= 500 and Fr= 4 . . . . . 51 x 3.19 local Fr h andG of various sphere wake regimes are mapped on Fr h − G parameter space with the same color code as the Fig. 3.5. The parameters are measured from (a) simulation and (b) experiments of various initial Re and Fr at a constant downstream distance x/R= 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1 (a) Schematic of a slender body from the side view with an aspect ratio of a major axis L to a minor axis D and an inclination angleθ rotated about y. The origin is the center of the slender body where x is the streamwise direction, y is the lateral direction z is the vertical direction. (b) Schematic of a slender body in 3D where the gray yoz plane is a cross-section of a wake where wake height is L V and wake width is L H defined from the streamwise velocity u. . . . . . . . . . . . . . . . . . 56 4.2 V ortex patterns in separation flow of a slender body with a pointed noise and blunt tail from Nelson & Corke [71]. Four regimes are classified as a function of inclination angleθ where highθ is in the range ofθ ={20,50} ◦ . α denoted in the figure is equivalent to θ in this study. . . . . . . . . . . . . . . . . . . . . . . . 57 4.3 (a) A sketch of a pair of vortex sheets observed in the experiments of a 6:1 spheroid at θ = 10 ◦ and Re D = 3.5× 10 5 from Fu et al. [72]. (b) Primary and secondary vortices represented in a streamline of flow around a 6:1 spheroid at θ = 20 ◦ and Re D = 7× 10 5 from Xiao et al. [73] . . . . . . . . . . . . . . . . . . 58 4.4 (a) vortices separated from a 6:1 spheroid atθ = 10 ◦ and Re D = 3.5× 10 5 from Strandenes et al. [74]; (b) streamwise vorticity of yoz plane at various downstream distance of DARPA SUBOFF at pitching angle, α p = 8 ◦ and Re L = 2.4× 10 6 from Ashok et al. [75] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.5 Schematic of various features involved in the evolution of stratified wakes, including the turbulent drag wake, pancake eddies in the late wake, internal waves, and wake vortices from the body at inclination or complex surfaces. . . . . . . . . 60 4.6 Sketch of a spheroid wake of zero θ (top) and non-zero θ (bottom) with peak defect velocity u 0 and wake height L V defined. Note that L V is measured in the direction of gravity and not perpendicular to the tangent line of the wake trajectory. 62 4.7 (a) The centerline defect velocity u 0 decay rate and (b) the wake width L H growth rate along the downstream distance x/D of the spheroid wake from Ortiz-Tarin et al. [70] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.8 Dye visualization of a counter-rotating vortex pair with (a) Crow instability from Leweke & Williamson [85] and (b) elliptic instability from Leweke & Williamson [86] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.9 (a) Sketch of counter-rotating vortex pair where b is defined as the distance between the two vortex core. (b) Sketch of vortex profile to show how the core size 2a is defined. Both sketchs are from Leweke et al. [87] . . . . . . . . . . . . . 64 xi 4.10 (a) The schematic of the experimental setup of the towed spheroid. Four cameras are in a linear-configuration for tomo-PIV and are placed in front of the tank for the vertical plane with lateral depth setup. (b) Dimensions of the 6:1 spheroid where a spheroid of a diameter D (i.e. minor-axis) and major-axis L is towed with four fishing wires at an inclination angle θ. A trip wire of a height k= 0.38mm<δ is placed at x= 0.2L from the nose of the spheroid. . . . . . . . . . . . . . . . . . . 66 4.11 Parameter Space spanning in Re, Fr,θ. Testing parameters (blue bar) for spheroid wakes in the Re D − Fr− θ space. All cases marked with blue circles are repeated for θ ={0,10,20} ◦ . The dotted lines represent the medium with the same stratification N. 6:1 prolate spheroids with two diameters D={4,8}cm are used to satisfy the Re and Fr conditions within the achievable N. . . . . . . . . . . . . . 67 4.12 (a) Comparison of support system impacts on the instantaneous lateral velocity v between the NACA 0012 airfoil strut (top) and the wire suspension (bottom). (b) Velocity profile across y and z at x/R=[3− 7] of Re D = 10 4 , Fr=∞ andθ = 0 ◦ . . 69 4.13 Time series of inclination angleθ as a function of distance travelled x/D where Re= 5,10,20× 10 3 andθ = 0,10,20 are tested. The green shaded area is the field of view, and the gray shaded area is the uncertainty from measurement resolution. . 71 4.14 Instantaneous streamwise velocity u/U of (a) R5∞ (b) R5F16 (c) R10F∞ (d) R10F32. The left edge of each snapshot t n is aligned with x n = Ut n based on the towing velocity in a body reference frame. The last snapshot is late-wakes at a constant Nt∼ 5. Each snapshot is normalized by the local maximum u/0.75|u max |. Note that x range is not continuous. . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.15 Peak streamwise velocity decay u 0 and wake height L V ofθ = 0. (a) u 0 over x/D (b) L V over x/D (c) u 0 Fr 2/3 over Nt (d) L V Fr − 2/3 over Nt. The legends are the following: unstratified (black line) and stratified (blue line) wakes. R5 (solid), R10 (dash). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.16 Instantaneous streamwise velocity of the spheroid at θ = 20 ◦ and (a) R5F∞ (b) R10F∞ (c) R20F∞ (d) R5F16 (e) R10F32 (f) R20F64. . . . . . . . . . . . . . . . . 76 4.17 (a) Instantaneous streamwise vorticity ω x of R5F∞θ10. The vertical plane xoz slice is at y/D=− 0.25 and the xoy plane is inclined at 10 ◦ starting at z/D=− 0.5. (b) iso-surface of Q-criterion at Q= 0.56 and colored lateral vorticityω y (c) yoz slice of average streamwise vorticity ω x at one of the center of mushroom-like structure (x/D= 5.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.18 (a) Time-Averaged streamwise vorticityω x of R5F∞θ10 where yoz slices at every x/D= 2 increments. (b) The trajectory of positive (red) and negative (blue) streamwise vorticity centroids of R10F∞θ20. The dotted line is from Jemison et al. [93] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 xii 4.19 The horizontal xoy plane of the streamwise velocity u (left), lateral velocity v (center), and vertical vorticity ω z (right) of the spheroid wake at θ = 20 ◦ Re D = 5000 and Fr=∞. The dotted blue line is the cross-section of the inclined spheroid intersecting the illuminated horizontal plane. . . . . . . . . . . . . . . . . 81 4.20 Time-averaged streamwise velocity field, u in vertical centerplane of (a) R10F∞ (b) R10F32. Blue dash line is u max and the purple line is the wake edge defined based on 0.5u max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.21 Trajectory of wake centerline, u max scaled in Nt . . . . . . . . . . . . . . . . . . . 82 4.22 Averaged vertical velocity w/U in Nt of R5F16 (top), R10F32 (middle), and R20F64 (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.23 (a) u 0 /U, centerline velocity decay and (b) L V /D, wake height from time-averaged field of axi-symmetric body, θ = 20. (c) Scaled centerline velocity decay based on Fr(u 0 /U)Fr 2/3 in Nt and (d) wake height based on Fr(L V /D)Fr − 2/3 from the time-averaged field of axi-symmetric body, θ = 20. The legends are the following: unstratified (black line) and stratified (blue line) wakes. R5 (solid), R10 (dash), R20(dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.24 (a) Averaged lateral velocity v/U in Nt where R5F16 (top), R10F32 (middle), R20F64 (bottom) and (b) time-averaged lateral velocity of R5F∞θ10 where yoz slices at every x/D= 2 increments. . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.25 A sketch of counter-rotating separation vortex pair in yoz plane where vortex rotating in− x direction is larger than the vortex rotating in+x. Grey shade is the cross-section of the spheroid and the dashed line is the y centerline of the vertical centerplane xoz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 xiii Abstract The upper ocean and the lower atmosphere of Earth have a stable density gradient with height known as stable stratification. When a stably-stratified flow passes over an obstacle (e.g., moun- tain) or when a body passes through the stratified ambient (e.g. aircraft, submarine), there are two dimensionless numbers that describe the balance between inertial, buoyancy, and viscous terms in the governing equations known as the Reynolds number, Re, and Froude number, Fr. In applica- tion, Re of the initial wake is always high, while Fr ranges in the order ofO(1− 10 2 ). However, as the wake decays, local Re and Fr drops falling into the range accessible through experiments and simulations. There is evidence suggesting that stratified wakes have patterns (i.e. coherent structures), which may contain information about the wake generator. Statistical analysis (e.g. mean and fluctuating quantities) of these wakes in previous studies indicated a similarity in the late wake, implying that no memory of the initial condition is found in the late wake. However, if the measurement is of wake structures, perhaps specific pattern information could be revealed. To develop tools to identify patterns, Dynamic Mode Decomposition (DMD) is used as a basis for wake classification. From energetic DMD modes, a custom-designed algorithm automatically classifies wakes into known flow regimes. The success of this approach depends on the quality and dimension of the input data. While the performance of the wake classifier was high in the low {Re-Fr} domain of sphere wakes, how it works at higher{Re-Fr} and other geometry is not yet known. Wakes of naval interest often include those with special initial conditions, such as strong trail- ing vortex structures shed into the wake by wings, fins, or inclined streamlined bodies such as spheroids. When the body is at incidence, the wake is a combination of drag wake and streamwise xiv vortices separated from the body, and this wake geometry can evolve in ways that are measurably different from the zero incidence case in background stratification. In this geometry, the stratifica- tion effect exists in the early wake, as the inclined wake itself generates large-scale internal wave undulations even when the background stratification is not strong. In addition, the pattern and strengths of the primary streamwise vortices are not symmetric. These findings serve as an initial framework for pattern identification in high Re flows mixed with special conditions. xv Chapter 1 Introduction 1.1 Parameterizing stratified flows The behavior of a flow in a wide range of scales depends on the density and viscosity of the fluid. Suppose a small body of fluid (i.e. fluid parcel) is moving at a velocity U where the density and dynamic viscosity of the fluid is ρ andµ respectively as illustrated Fig. 1.1a. The momentum of the fluid parcel is balanced by a viscous force from the fluid resisting the flow by applied shear stress (body force on the parcel and surface force on the parcel boundary). A dimensionless ratio of the two forces is characterized by Reynolds number Re=ρUL/µ= UL/ν, where L is a characteristic length scale (e.g. diameter of the wake-generating body) andν =µ/ρ is the kinematic viscosity, a ratio of dynamic viscosity to density. In homogeneous fluid, Re can indicate whether the flow is laminar (Re L ∼ 10 2 ) or turbulent (Re L > 10 3 ). In stratified fluid where density varies in the gravity direction, the flow behaves differently from the flow in homogeneous fluid due to the buoyancy force affecting the vertical motion of the fluid parcel. Stratified fluid can be found in significant parts of the earth’s atmosphere and ocean where gradients in pressure and temperature (troposphere) or temperature and salinity (thermo- cline) result in stable and persistent gradients in density with height. As shown in Fig. 1.2a, in the ocean pycnocline, roughly the upper 500 m (apart from a thin surface wind-induced mixed 1 Figure 1.1: Schematic to represent (a) Reynolds number Re=ρUL/µ, (b) Froude number Fr= U/NR, (c) Brunt-V¨ ais¨ al¨ a frequency N (a) (b) Figure 1.2: (a) The density gradient (light blue) and the squared Brunt-V¨ ais¨ al¨ a frequency N 2 (ma- genta) in the thermocline of south Indian ocean measured from the Argo (Array for Real-time Geostrophic Oceanography) data [1]. Pycnocline (⋆) is roughly at the depth of∼ 100− 250 m where the density is linear and N 2 ∼ 10 − 4 s − 2 . (b) An artificially colored satellite image (center image [2]) of a wake generated by the Tristan da Cunha island in South Atlantic (an enlarged view of the island in the bottom left image from JPL/NASA) where a coherent pattern is visualized by the clouds. Top right image [3] shows a side view of the island where the clouds are formed in layers. layer) can be modeled with a constant density gradient dρ/dz where the fluid density ρ(z) in- creases linearly with depth (increasing distance from the ocean surface). In stably stratified fluid, due to the balance between gravity and the vertical pressure gradient from hydrostatic relation, ∂ p/∂z =− gρ(z), buoyancy force is applied to vertical perturbations restoring the fluid parcel back to the equilibrium position where g is the gravitational acceleration. Due to the additional 2 force in the vertical direction, the vertical motion is suppressed and the horizontal motion contin- ues to grow. As the small-scale fluctuation dissipates, large vortices merge into persistent coherent structures. Two-dimensional coherent structures are observed in nature. For example, a flow over an island visualized by the clouds in Fig. 1.2b shows that even at high Re, coherent structures sim- ilar to the von K´ arm´ an vortex streets emerge [4]. The Fr of the flow passing over an island may not be low, but a high Re condition did not eliminate the coherent structures, so the persistent coherent structure may be associated with the influence of stratification. These large-scale flows can impact the energy transport and mixing of the atmosphere and ocean that it is important to understand the characteristics of stratified flows. To quantify the relative strength of stratification, consider a fluid parcel traveling at U in a stratified ambient with a linear density gradient ρ(z) where ρ 0 is an equilibrium density and g is the gravitational acceleration as illustrated in Fig. 1.1b. From dimensional analysis, the relative importance of the inertial effects to the buoyancy effect is a dimensionless number Froude number Fr= U/NL. Here, Brunt-V¨ ais¨ al¨ a frequency N=[(− g/ρ 0 )· (dρ/dz)] 1/2 represents the strength of the stratification. Similar to a damped oscillation in stable stability in dynamics, N is a measure of a natural frequency of a restoring wave motion caused by a balance of vertical perturbations and buoyancy forces. When N is constant (linear density gradient), the oscillation period is 2π/N rad. The time scale in the evolution of the stratified flows is commonly described by Nt, a time normalized by buoyancy timescale, which will be used throughout the thesis to describe a flow in terms of the evolution stage. Therefore, Fr can be interpreted as a ratio of a time scaleπR/U for a fluid particle to travel over a body of radius R (i.e. half a circumference) to the buoyancy timescale 2π/N as illustrated in Fig. 1.1c. The combination of two dimensionless parameters, Re and Fr, can be used to determine the characteristic of the stratified flow (e.g. generation of turbulence and internal waves). Stratified flows in geophysical and naval applications such as an island wake (L ∼ 10 3 m, Re∼ 10 9 Fr∼ 1) or a wake of a submerged body (i.e. underwater vehicle, marine swimmers) (L∼ 1m Re∼ 10 7 and Fr∼ 10 3 ), both Re and Fr are initially high, where the flow is highly turbulent and the stratification effect 3 (a) (b) Figure 1.3: (a) The decay rate of stratified flow at 3D-NEQ-Q2D evolution stages (b) vertical vorticity of a sphere wake from the body to late wake, Nt =[10− 240] with varying initial Fr= [10− 80] [8] is weak. As the flow decays, however, Re and Fr computed based on the local scales decreases. For example, following the unstratified high-Re scaling, an initially high Re flow decays in the downstream distance x at U ∼ x − 2/3 , and the local wake width increases at L∼ x 1/3 [5–7], the resulting local Re and Fr decays at Re∼ x − 1/3 and Fr∼ x − 1 . The increase in the relative importance of stratification implies that stratified flows go through evolution stages of various scales locally. 1.2 Wakes in stratified fluids Numerous studies have been done on stratified wakes of a simple body (e.g. axisymmetric slender body, sphere) to understand how turbulence structures interact with stratified ambient as the wake evolves. Early experimental studies on the stratified wakes of a grid and axisymmetric bodies are reviewed in Lin & Pao [9] where the initially growing turbulent wakes in a wide Fr range are suppressed in the vertical height inhibited by buoyancy while the horizontal width continues to expand. As the wake evolves quasi two-dimensionally, the wake eventually becomes unstable resulting in large-scale alternating-signed vortices on the horizontal plane (i.e. pancake vortices). During the evolution of the stratified turbulent wakes, three distinct stages (3D-NEQ-Q2D) are observed by Spedding [8] using DPIV (digitized particle image velocimetry) on towed sphere wakes for a long downstream distance, Nt = 10− 1600 as illustrated in Fig. 1.3a. The initial 3D 4 stage behaves like unstratified wakes and decays following the 3D unstratified turbulent power law, t − 2/3 [5, 6]. The non-equilibrium regime (NEQ) stage starts at around Nt∼ 2 where the buoyancy effect becomes dominant suppressing the vertical motion of the wake. Small-scale fluctuations continue to dissipate and like-signed vortices merge leaving only the large-scale vortices. The decay rate of the NEQ regime (Nt − 1/4 ) is slower due to buoyancy-induced reduction of the mean to turbulence energy transfer [10]. The wake transitions into the quasi-2D (Q2D) stage at around Nt∼ 50 with an increase in the decay rate of Nt − 3/4 . The wake is dominantly in the horizontal motion and vertical variation is minimal resulting in a strong vertical shear with fully developed pancake vortices. As shown in Fig. 1.3b, these persistent patches of vertical vorticity prolongs for many buoyancy time Nt where Spedding [8] and Spedding et al. [11] observed experimentally up to Nt∼ 2000, which scales to approximately 12 days in the ocean pycnocline (N∼ 2× 10 − 3 ). The formation of these patterns is in the navy’s interest as it implies the possibility of pattern detection. 1.3 Low Re sphere wake regime The flow structures in the evolution of stratified wakes are characterized in regimes of the near- wake. Various regimes are mapped out in{Re-Fr} parameter space and described quite compre- hensively in experiments by Lin et al. [12](LI92) and Chomaz et al. [13](CH93). The two sets of classifications are similar, though differing in detail (Fig. 3.4). Over a range of 10 ≤ Re≤ 10 4 and 0.02≤ Fr≤ 20. LI92 described 6 principal regimes from steady attached vortices at low{Re-Fr} to fully turbulent wakes at high{Re-Fr}. In general, increases in Re were marked by transitions from steady to unsteady vortex shedding and the development of finer scales as the wake shear layers themselves become unstable. Increases in Fr governed the transitions from laminar uniform wakes without notable vertical excursions, to intermediate stages dominated by strong lee waves as the flow can traverse over and below the sphere, and then to the turbulent state at high Fr. CH93 described 4 principal regimes over 150≤ Re≤ 5× 10 4 and 0.25≤ Fr≤ 12.7, controlled mainly by Fr. At low Fr∈[0.25,0.8] the wake has two separate lee wave layers separated by a 5 (a) (b) Figure 1.4: Regime diagrams from CH93 (a) and LI92 (b). The 4 main regimes differentiated in (a) are: 2D, quasi-2D; SLW, strong lee wave; T, transition (SKH, without K-H instability; KH, with K-H instability); and 3D, three-dimensional. In (b), LI92 distinguished 6 main regimes: A u , un- steady, attached 2D vortices; V t , 2D vortex shedding; L, lee-wave instability; N, non-axisymmetric attached vortex; S, symmetric vortex shedding; V , non-symmetric vortex shedding; and T , turbu- lent wake. Note that the axes of LI92 are flipped compared to CH93 and the Fr definition in LI92 is F i = U/ND characterized by the diameter instead of radius, which is used in CH93 and this studies. 6 quasi-two-dimensional region where the flow passes horizontally around the sphere. As Fr in- creases from 0.25 to 0.8 there are flow states with attached, oscillating, and then steady vortices as the structure becomes increasingly influenced by the strong lee waves. When Fr ∼ 1, the buoyancy 2π/N and convective timescales πR/U are approximately equal, which can be interpreted as the oscillation period matching the full travel time of half a circumference as illustrated in Fig. 1.1c. CH93 termed this the saturated lee wave regime. As Fr increases from 1.5 to 4.5 a transitional regime marks a gradually decreasing influence of stratification close to the sphere, and there are differences depending on Re just as there are in a homogeneous fluid. For Fr > 4.5 the near wake has 2 modes, a spiral instability mode and Kelvin-Helmholtz modes associated with the wake shear layers. LI92 covered more details at low{Re-Fr}, and CH93 had more at higher values of both parameters. Though there are differences between them, the overall regime diagram for structures and patterns that can emerge into the near wake is quite consistent. Early direct numerical simulations (DNS) [14] were confined to lower Re = 200 for 0.25≤ Fr≤ 200 but the determining influence of the lee waves was noted, just as found in the subse- quent experiments. Body-inclusive simulations were also run by Orr et al. [15] at Re = 200, Fr∈[0.1,0.4] (for comparison with Hanazaki [14]) and at Re = 1000 for Fr∈[1.0,16] so as to include the commencement of small-scale and turbulent motions on wake measures of fluctuating quantities. At Re = 200, a transition Fr between 0,4 and 0.5 was observed where vortex shedding re-emerges at the lower Fr. Significant differences with Hanazaki [14] were found when Fr < 0.5, and though numerical resolution could be a limiting factor when Fr< 0.2, the discrepancy in com- puted time-averaged drag coefficient C D was not resolved. It is suggested that the low Fr cases in the experiments were at a low Re that the wake is not initially turbulent. DNS by Pal et al. [16] for Re= 3700 and Fr∈[0.025,1] showed not only that vortex shedding re-appears at Fr< 0.5 but also that in the combination when Re is sufficiently high (about Re ≫ 500, [17]) and Fr≤ 0.125, small-scale fluctuations re-emerge due to unsteady vortex shedding and secondary turbulence from Kelvin-Helmholtz instability of strongly sheared, strongly stratified layers. These phenomena were further elaborated by Chongsiripinyo et al. [18] who associated the 7 stronger vortex stretching in the near wake of Fr≤ 0.125 with the unsteady, intermittent shedding of the boundary layer from the side of the sphere. A further study [19] at Re= 3700, Fr=[1,2,3] focused on the role of lee waves at moderate Fr contributing to the particular wake dynamics in stratified flows. To summarize, as the stratified wake evolves, the buoyancy effect becomes dominant where the vertical motion is suppressed and a large coherent structure emerges in the horizontal direction. The relative strength of the inertial effect and buoyancy effect can be described by the Re and Fr parameters. Many previous simulations and experiments have shown that the stratified wakes at low{Re,Fr} exhibit qualitatively different flow that can be categorized into regimes depending on the{Re,Fr} condition. Then,{Re,Fr} parameters can be used to understand the evolution of the stratified wakes. 1.4 High Fr turbulence Stratified flows observed in the atmosphere and the ocean typically have high Re, which is chal- lenging to achieve in the laboratory experiment and simulation. There have been many computa- tional [20–25] and experimental [11, 26, 27] efforts to understand how the flow information in the achievable Re and Fr parameters can be scaled up to geophysical applications (i.e. high Re and Fr) and how the turbulence of various length scales evolves with the effect of the stratification. Turbulence in stratified flow evolves with a forward energy cascade where instabilities break up large structures into small-scale eddies and the energy dissipates into heat energy at the smallest scale [22, 23]. The energy cascade is different from the homogeneous turbulence energy cascade as the stratification affects the energy transfer. In stratified energy cascade, the Ozmidov length scale (l o ∼ (ε/N 3 ) 1/2 ) defines the smallest eddies affected by buoyancy that has sufficient kinetic energy to overturn. Here,ε is the energy dissipation rate. While the buoyancy continues to affect the mean flow, the turbulence scale smaller than l o is unaffected by buoyancy that it is statistically isotropic [28]. Beyond this point, the energy transfer follows the isotropic turbulence energy cascade, where 8 the Kolmogorov length scale (η∼ (ν 3 /ε) 1/4 ) is defined as the smallest scale of turbulence where all kinetic energy is transformed into thermal energy (i.e. viscous dissipation). In stratified turbulence, local parameters analogous to Re and Fr can be calculated based on local horizontal and vertical integral length scale, l h and l v respectively. Fr defined based on l v , becomes approximately a constant in stratified wakes, Fr v ∼ O(1) for Nt > 1 [27]. Thus, l h is typically used, which is defined based on the wake spacing of the overturning motion from Kelvin- Helmholtz instability caused by vertical shear [20, 25, 26, 29]. Local Froude number based on horizontal length scale is defined as Fr h = u rms /Nl h where u rms is the local horizontal RMS (root- mean-square) velocity and local vertical Froude number is defined as Fr v = u rms /Nl h . Due to the strong shear between layers when Nt > 1, the vertical length scale decreases to l v ≪ l h that Fr v becomes approximately constant at Fr v ∼ O(1) [27]. Therefore, stratified turbulence can be characterized by Fr h . When the flow is in the weakly stratified turbulence stage (WST), the flow follows the unstratified high-Re scaling where u rms ∼ x − 2/3 and l h ∼ x 1/3 . This leads to Fr h = Nx − 1 or Fr h = Nt − 1 where t= x/U 0 [11]. The limit at which turbulence in stratified flow occurs can be estimated by the local gradient Richardson number, a ratio of stabilizing effect of buoyancy and viscosity to the vertical strain rate, defined as Ri= N 2 S 2 (1.1) where the vertical strain rate is S 2 = ∂u ∂z 2 + ∂v ∂z 2 . (1.2) The flow is turbulent when Ri h < 1, an average of the local Richardson number in the horizontal plane Ri h = N 2 /⟨S 2 ⟩ h [17, 30]. Applying the Richardson number criterion, the buoyancy Reynolds numberR = Re h Fr 2 h can be defined as the parameter to determined when the viscous effect be- comes important (R <O(1)) [21, 31]. In the forward energy cascade, the inertial subrange, which lies between the l o and η length scale, is wider when theR≫ 1 [32]. In this region, turbulence is unaffected by either buoyancy 9 Figure 1.5: Regimes in{Fr h -Gn} parameter space identified by Bruyn Kops & Riley [25] or viscous effects that theε∼ u 3 /l independent ofν can be estimated from the ratio of the kinetic energy of the flow u 2 to the large eddy turnover time scale l/u using the inviscid estimate scaling for high Re. Then, the advection length scale can be l h ∼ u rms 3 /ε, estimated from the spatial average of in the horizontal plane, which has the maximum shear. Applying l h with this assumption, the buoyancy Reynolds numberR can be re-written as the activity parameterG termed by Bruyn Kops & Riley [25], Re= u 4 νε , Fr= ε Nu 2 ,G =ε/νN 2 , (1.3) which is equivalent to the ratio of Ozmidov to Kolmogorov length scale l o /η [24]. The buoyancy numberR and the activity parameterG have previously been used interchangeably in the literature. However, Bruyn Kops & Riley [25] showed that they are similar, but not exactly proportional to each other. WhileR is defined by the horizontal overturning length scale l h and the local horizontal RMS velocity u rms ,G is defined by the turbulence length scale with the total RMS velocity. G is convenient to use as the kinetic energy or shear terms are absent. In the {Fr h − G} parameter space shown in Fig. 1.5, the trajectory of three cases from the simulation of box-filling turbulence with initial Re h =[160,600,2325] and Fr h = 2 passes through regimes in the order of weak stratification-stratified turbulence-viscous effect dominated. The flow is strongly stratified when Fr h <= 0.6. WhenG > 1, the flow enters into a Strongly Stratified 10 Figure 1.6: At the late wake, the wake follows the universal curve of a scaled effective Fr regardless of the body geometry. Tested geometry includes: a sphere (• ), a slender spheroid (*), a cylinder (△), a disc (⋄ ), a cube (□ ), and a hemisphere (◦ ) [34]. The vertical axis U 0 /U B F 2/3 eff shown in the plot is equivalent to u 0 Fr 2/3 eff . Turbulence regime (SST) and whenG < 1, viscous effects dominates, which is similar toR > O(1) also observed by [21, 24]. Bruyn Kops & Riley [25] speculated that the cause of a faster decay rate in Q2D in Fig. 1.3a from Spedding [8] is due to lowG where viscous effects are dominant. In the ocean and atmosphere,G is high and Fr h is low whereG is typicallyG ∼ O(10 2 ), but can be as high asG ∼ O(10 4 ) and Fr h is typically Fr h ∼ O(10 − 3 − 10 − 4 ) [23, 33]. Based on the knowledge of the turbulence evolution from the box-filling simulation, Fr h andG can be estimated for turbulence in stratified wakes to help us understand the phenomena in high Re wakes. In these studies above, the Coriolis effect is not considered, but if the largest structure (e.g. pancake vortices) is in a small scale for the Coriolis force, a deflection of the flow caused by the Earth’s rotation, is negligible, then the scaling laws can be extrapolated to geophysical scales. 11 1.5 Signature detection 1.5.1 Universal late wakes? The wakes of various initial conditions from the numerical and laboratory experiments in the lit- erature are qualitatively different as competing effects of stratification and increasing Reynolds number generate different dynamical balances that then evolve downstream. However, profiles of mean and fluctuating late wake quantities [8, 11] were shown to be self-similar and could be rescaled according to U and D without reference to Fr, as if in the far wake, the initial Fr had no in- fluence. Spedding [8] showed that the wake-averaged defect velocity u 0 of high Fr (Fr∈[10,240]) follows the scaling law u 0 Fr 2/3 ∼ Nt − 2/3 in 3D evolution stage independent of the initial Fr, then following∼ Nt − 1/4 in NEQ stage and∼ Nt − 0.76 in Q2D as shown in Fig. 1.3a. Moreover, a simple modification to this scaling, taking into account the magnitude of the initial streamwise momentum flux, showed that the wakes of bluff (i.e. body that generates a high pressure drag), streamlined and sharp-edged bodies could all be described by the same power-law functions. Given U, D and a drag coefficient (that depends on the streamwise momentum flux, and hence does depend on the shape), the horizontal and vertical growth rates and the decay of mean and fluctuating velocities were all the same [34]. The velocity defect scaling based on the effective diameter, D eff = D p c D /2 fol- lows, u 0 Fr 2/3 eff ∼ Nt − 0.76 in the late wake as shown in Fig. 1.6. Meanwhile, numerical simulations of stratified wakes that were initialised with prescribed mean and turbulence profiles appeared to evolve in much the same way [10, 17, 35–37], supporting the idea that local turbulence or post- turbulence dynamics primarily govern the flow, regardless of details of initial conditions. This lack of sensitivity to initial conditions may be surprising given the very different near- wake regimes detailed in the experiment. It could be that universality is only achieved when values of the governing parameters Re and Fr are high enough, and the observed collapse of experimen- tal data [11] occurred only when Re≥ 4× 10 3 and Fr≥ 4. Perhaps only when turbulence over a 12 range of scales is allowed can the cascades in energy and enstrophy proceed along common tra- jectories scaled by the buoyancy Reynolds numberR that depends on both Re and Fr [24, 25, 38]. There is also evidence from both experiments [39–41] and simulations [42] that the scale similarity proposed [5] for homogeneous wakes and turbulent free-shear flows may not happen. Finally, just because the information from initial conditions is not clearly measured in average wake statistics does not mean that information from the wake-creator is entirely absent, only that it does not affect certain statistical properties. The possibility of pattern detection, and the informa- tion that such a pattern could contain remain of interest [43]. One place to start looking is to find and characterize wake pattern information that might persist downstream, which will then provide what to look for. 13 Chapter 2 Dissertation overview This study aims to test the pattern classification of stratified wakes of various initial conditions and investigate wakes of a special geometry that introduces coherent structures in the initial conditions to understand the evolution of the stratified wakes and potential patterns of the initial conditions that remain in the late wake. Chapter 3 presents an automated pattern classifier that can work in stratified wakes of known properties. Five sphere wake regimes are identified from the near-wake within the moderately low Re and low Fr parameter space measured from the experiments and simulation. A DMD- based classifier includes multiple binary physics-based criteria to classify wakes into one of the five identified regimes. The robustness of this classifier was tested by reduced flow information and noise-contaminated data. To understand the significance of turbulence at various scales when extrapolated to high Re in the geophysical scale, wakes were explored in the local stratified turbu- lence scaling parameters, a local Froude number Fr h based on horizontal integral length scale and an activity parameterG (i.e. buoyancy Reynolds number). Chapter 4 presents a wake with special conditions that consist of more than the turbulence (e.g. embedded streamwise vortices). Departing from conventional axisymmetric bodies studies, inclined slender body wakes studies provide insights into flow conditions relevant to real sub- mersibles of naval interests. The wake evolution of an inclined 6:1 spheroid is explored from experiments in the parameter space of moderately high Re, high Fr, and body inclination angles. 14 The wake observed is presented in the context of the existing understanding of stratified wake evo- lution. The trajectory of inclined spheroid wakes and the emergence of internal waves, as well as the wake structures evolving in 3D space, are examined and presented in relation to the studies on the evolution of a counter-rotating vortex pair. Chapter 5 summarizes the main findings and suggests potential areas for further investigation, aiming to achieve a complete understanding of the evolution of patterns associated with various initial conditions and, ultimately, providing information for potential pattern detection. In this thesis, excerpts from Ohh & Spedding [44] appear in the following sections: Section 1.3 and Section 1.5.1 of the introduction; Section 3.2, with the addition of Section 3.2.5 in Chapter 3; and Section 3.3, with the addition of Section 3.3.4.2 in Chapter 3. Chapter 4 is currently being prepared for publication under the title: “The effects of stratification on the near wake of a 6:1 prolate spheroid” by C. -Y . Ohh and G. R. Spedding (2023). 15 Chapter 3 Wake classification on stratified sphere wake 3.1 Introduction Evidence shows that many aquatic organisms use wake signatures for navigation [43]. These signatures might originate from stratified wakes, given that buoyancy effects in such wakes lead to the formation of persistent coherent structures [8]. The topology of stratified wakes can provide insight into stratified wake evolution and the emergence of patterns. Previous experimental and numerical studies have characterized the topology of stratified wakes as a function of Re and Fr [12, 13, 16, 19] as explained in Section 1.3. 3.1.1 Modal decomposition tools Identifying coherent structures can benefit from data-driven modal decomposition by representing the nonlinear and high dimensional nature of fluid flows in a low-order model. This is useful in finding patterns in stratified flows. Historically, DFT (Discrete Fourier Transform), a Fourier analysis based on a superposition of oscillating modes from a time domain to a frequency domain, has been used to identify a dominant frequency in a spectrum. However, the limitations of DFT are that it assumes the flow structure to be periodic and requires an integer number of oscillation periods to correctly capture the mode at a particular frequency. Otherwise, the DFT of a non- periodic flow shows a slow decay of the modes [45]. This is constraining to pattern identification 16 of stratified flows when multiple frequencies may be of interest or when large-scale flows need to be captured from a small set of data. One of the popular modal decompositions from a low-dimensional model is POD (Proper Or- thogonal Decomposition), which provides orthogonal modes in the order of vector energy from the eigenvalues of SVD (Singular Value Decomposition). POD was used in a stratified flow applica- tion where Diamessis et al. [46] found an association between the angle of internal wave emission to a layered wake core structure using POD on the vorticity fields of stratified turbulent wake sim- ulation. However, POD yields spatially orthogonal modes ordered based on energy, with temporal coefficients that may contain a mix of frequencies [47] and, as a result, only provides modes that are spatially coherent and does not account for features that are growing or decaying in time. The dominant POD mode may not reflect the dynamical importance of the flow as it is lost during the averaging process of obtaining the spatial correlation tensor [48]. Motivated by the criticism of space-only POD, DMD (Dynamic Mode Decomposition) was developed to isolate modes of a specific frequency. DMD can be understood as a temporal orthog- onalization that organizes modes based on the dynamical importance by approximating nonlinear dynamics with a linear operator [48, 49]. DMD computes complex frequencies from eigenvalues that grow and decay, which provides the growth/decay rate of the modes. DMD is especially useful to identify temporally evolving patterns in stratified flows where the effect of the stratification is time-dependent. DMD mode is a linear combination of DFT modes at a given frequency. DMD mode is equivalent to the DFT modes in a periodic flow or a mean-subtracted DMD despite having an unrelated derivation [45, 48]. Previous efforts have successfully implemented DMD to extract coherent structures in stratified wakes [50, 51]. In the experimental studies of stratified wakes by Xiang et al. [50], spatial and temporal DMD modes successfully captured lee waves and Kelvin- Helmholtz instability from grid wakes and in the computational studies of stratified wakes, Nidhan et al. [51] used DMD to extract coherent wake structures at two different Reynolds numbers with varying buoyancy effects (Fr). Therefore, DMD analysis is well-suited for identifying patterns in stratified flows. 17 3.1.2 Objectives This work aims to test a pattern classification procedure that is based on DMD modes of stratified wakes at moderate{Re-Fr} where there are distinct, identifiable regimes, all ultimately based on the balance between momentum and buoyancy (Fr) and inertia and viscosity (Re). To do this, we will first propose a combined and simplified wake regime classification that reconciles the existing literature (LI92, CH93), and selects for information that could, in principle, appear in the later wake (so details of separation patterns on the body surface, for example, are ignored). An automated classification is then attempted on the wake data alone. The tests are conducted on the DMD modes because they contain information based on the flow dynamics and sorting or ranking of these modes can extract the most important physical characteristics. The procedure is tested on 3D data from numerical simulations and 2D data from laboratory experiments. Anticipating that future applications will not be on complete datasets, preliminary tests on reduced data are described, and serve to indicate whether the effort could be extended to regimes of significantly higher Re and Fr. 3.2 Methods To study the wake characteristics of each regime that is identifiable by a wake classifier, both computational and laboratory experiments were conducted in the same{Re-Fr} parameter space: Re= UD/ν =[200,300,500,1000] and Fr= U/(NR)=[0.5,1,2,4,8,16]. The simulation is in the sphere reference frame, while the experiment is in a fixed laboratory frame with a moving sphere. The reference frame of the experiment was changed from a towed sphere to a fixed sphere frame to compare the performance of the wake classifier on an equivalent basis. 18 Figure 3.1: Schematic showing experimental setup (a) Two 2D planar PIV appended. The anno- tated scale is based on a sphere with D= 4 cm to provide a relative scale of the experiment. 3.2.1 Experimental setup A sphere was towed horizontally in a tank of 1 x 1 x 2.5 m, filled with stably stratified fluid as illus- trated in Fig. 3.1. Refractive index matched salt-stratified water with a linear density gradient was achieved by implementing the RIM technique from Xiang et al. [52] to reduce optical distortions. The sphere was submerged to a depth of 0.51 m and attached to a translation stage by three fishing wires. Sphere shells were 3D-printed using polylactic acid (PLA) and filled with steel balls and clay for stability. The buoyancy frequency N =[0.13− 1.0] rad/s, sphere radius R=[0.72− 5.5] cm and tow speed U =[0.37− 43] cm/s were adjusted to obtain the targeted Fr and Re. The flowfield was estimated using a LaVision PIV (Particle Image Velocimetry) system with multiple cameras (LaVision-Imager sCMOS), each having a resolution of 2560× 2160 pixels with f = 56 mm lenses. Two camera configurations were used for varying Re. In the planar 19 configuration, usually used for lower Fr, Re, the cameras were aligned side-by-side and per- pendicular to the center plane with ∼ 5% overlap to obtain an extended field of view (FOV) in the streamwise direction. The FOV of each camera was approximately 35 mm x 300 mm, which is [x/R,y/R]=[6.5− 49,5.5− 42] for the vertical plane and 230 mm x 190 mm, which is [x/R,y/R]=[4.2− 32,3.5− 26] for the horizontal plane. The measurement plane was illuminated with a pulsed laser (Nd: YAG, LaVision NANO L100-50PIV) in either the vertical XOZ center plane or horizontal XOY center plane with a 2 mm thickness sheet. The sphere entered the FOV 70 cm from the starting point, which is [12.7− 97.2]R (depending on the sphere size), to reduce startup effects. The laser repetition rate was 5-20 Hz, scaled on the towing velocity. The fluid was seeded with titanium dioxide particles with an average density of 4.23g/cm 3 and diameter 15µm. Based on Stokes’ law, the sink speed is approximately v TiO 2 = 0.43mm/s, which gives 7.8-10 min for particles to cross the sphere. The average particle image density was 0.04pixel − 1 . A time-series of images were then processed with Davis where image pairs were cross-correlated by multiple levels of interrogation box sizes. The interrogation box size is determined based on the spatial resolution (smaller box sizes) while containing enough information (i.e. number of par- ticles). For example, one of the test cases was processed with two passes of 64× 64 pixels square box and two passes of 32× 32 pixels circle box where interrogation box of all passes scanned the image with a shift of 50% of its size. This resulted in 160× 135 velocity vectors. Other require- ments for a good quality PIV such as particle image diameter, peak-locking errors, and particle image density are considered in the experiment following the guide in Raffel et al. [53]. 3.2.1.1 Temporal reconfiguration of the experimental data For the reference frame to be consistent with the simulation, the reference frame of the experimen- tal data was switched to a sphere-fixed frame (a necessary step for temporal DMD). The timescale, Nt, of the wake evolution can be written as Nt =(x/R)/Fr. Here Nt and x/R will be used in- terchangeably to denote downstream distance. As illustrated in Fig. 3.2, a fixed sphere frame is assigned as half the streamwise span of the FOV at a constant downstream distance marked box 20 Figure 3.2: Temporal data reconfiguration where the colored box A is moving in reference to the sphere. Dash lines indicate column vectors of velocity field within the box A A in Fig. 3.2. As the sphere is towed across the FOV , box A moves with the sphere while the entire box remains within the FOV . The velocity fields inside the box at each snapshot, u t n x A are then organized into column vectors: x 0 = u t 0 x 0 u t 0 x 1 u t 0 x 2 . . . ,x 1 = u t 1 x 0 u t 1 x 1 u t 1 x 2 . . . ,x 2 = u t 2 x 0 u t 2 x 1 u t 2 x 2 . . . , ··· , (3.1) where u t n x j represents a column vector of the velocity field at x/R= x j with respect to the sphere at time t n , and x n is a column vector consisting all u t n x j vectors at t = n. This set of vectors is used as an input to DMD in Eq. (3.5). When the first temporal sequenced set reaches the last available snapshot for a constant streamwise position, the next half of the FOV is available for the next set (marked B in Fig. 3.2). The number of available snapshots for each set is m= x FOV /(2Udt), where x FOV is the FOV width (x A+B ) in downstream direction and dt is the time interval between two images. At a fixed camera FOV with varying sphere sizes, box A may be overly zoomed in to prevent observation of the whole wake structure (especially for the horizontal FOV at high Re and low Fr conditions). It could also be too wide a view and include multiple evolution stages (especially for the small sphere in vertical FOV low Re high Fr conditions). These two scenarios are challenging 21 (a) Mesh for Re = [200,300,500] has four re- finement levels: 2 − [2··· 5] D (b) Mesh for Re= 1000 has six refinement lev- els: 2 − [2··· 7] D Figure 3.3: Meshing in simulation and the location of refinement levels. The computational domain is (x,y,z)=([-10,50], [-8,8], [-8,8])D and the results in ([1.5,15], [-2,2], [-2,2])D was used. for DMD to isolate the coherent structure. Therefore, box size of the temporal sequence is cho- sen based on the streamwise length that is closest to the observation window x/R=[20,30]. The available box sizes are quarter (0.5x A ), half (x A ), and full FOV (x A + x B ). To obtain the tempo- ral reconfiguration of the full FOV , the FOV from two cameras with parallel viewing angles are appended (subtracting the overlap region) to increase the effective FOV in x. However, since the interest is in the near-wake, and not on details immediately behind the sphere (i.e. recirculating re- gion), A (as shown in the figure) is not used, and DMD is only performed on the temporal sequence following A. 3.2.2 Numerical methods The stratified wake problem can be set up from the incompressible continuity equation and applica- tion of the Boussinesq approximation to the Navier-Stokes momentum equation, where the density variation is only non-negligible when multiplied by g in the buoyancy term. The density and pres- sure fields are decomposed into a background state and a perturbation as ρ(x,t)=ρ(z)+ρ ′ (x,t) and p(x,t)= p(z)+ p ′ (x,t) respectively where the background density gradient is in z. After apply- ing the Boussinesq approximation and hydrostatic balance, (∂ p/∂z=− gρ(z)), the dimensionless form of the governing equations are ∇· u= 0, (3.2) 22 ∂u ∂t + u· ∇u=− ∇p ′ + 1 Re · ∇ 2 u+ 1 Fr 2 ρ ′ g, (3.3) ∂ρ ′ ∂t + u· ∇ρ ′ = w· dρ dz + 1 Pr Re · ∇ 2 ρ ′ , (3.4) where u is the velocity field, g =− ge z is the gravitational acceleration in z, w as the vertical velocity in z, and Pr is the Prandtl number. Re and Fr parameters are modified by changing the dimensionless kinematic viscosityν = 1/Re and density gradient dρ/dz with a fixed background flow velocity, U = 1, and sphere diameter D= 1 fixed at the origin. The governing equations are implemented in OpenFOAM using a PimpleFOAM solver, which uses a finite volume method with a pressure-velocity coupling algorithm typically used in a tran- sient incompressible CFD. Both temporal and spatial terms are second-order accurate. Boundary conditions are set to zero-gradient conditions on the y-normal and z-normal boundaries to reduce internal wave reflection from the boundaries. To allow for possible breaks in the axisymmetry of the wake, the sphere is oscillated in-line once in three axes each as the initial condition [54]. The unstructured mesh of the computational domain 60D× 16D× 16D is distributed where the cell size splits at each grid refinement level closer to the sphere as indicated by the orange box in Fig. 3.3. The simulation is sampled at [− 1.5,15]D× [− 2,2]D× [− 2,2]D. The resolution of the coarse mesh used for Re= 200,300,500 is about 3 million cells while the fine mesh used for Re= 1000 is about 17 million cells. More details of the numerical setup are in [55]. 3.2.3 Standard DMD DMD was applied to the temporal series x k m k=0 of experimental data from Eq. (3.1) and the simu- lation, where the fixed sphere reference frame and observation window of x/R=[20,30] is con- sistent. Since two 2D centerplanes of the experimental data were sampled independently, DMD is performed independently for the equivalent planes of the simulation (see Section 3.3.4.1). In this project, the SVD-based, standard DMD method described iteschmid:10,rowley:09,chen:12 is used, where the minimized residual is placed on the last snapshot x m . The algorithm is summarized in the following: 23 1. Construct snapshots{x k } m k=0 into a data matrix K and an index-shifted matrix K p K := x 0 x 1 ··· x m− 1 K p := x 1 x 2 ··· x m (3.5) 2. Compute reduced SVD of K K= UΣ W T (3.6) 3. Perform eigendecomposition on a companion matrix C [45, 48, 49] C= 0 0 ··· 0 c 0 1 0 ··· 0 c 1 0 1 ··· 0 c 2 . . . . . . . . . . . . . . . 0 0 ··· 1 c m− 1 . (3.7) ˜ C := U T CU= U T K p WΣ − 1 (3.8) 4. Obtain the Ritz vector V= UY and Ritz valuesΛ from ˜ C= YΛ Y − 1 (3.9) Ritz valuesλ k j provide the growth rate,β =|λ| 1/∆t R/U, and Strouhal number (See Eq. (3.14)) of each DMD mode and the Ritz vector v j has the velocity field of each mode. 3.2.4 Total DMD for noise-contaminated datasets In the presence of a low signal-to-noise ratio in the observed data, a total DMD (TDMD or total least-squares DMD), introduced by Hemati et al. [56], was used to reach an unbiased result when 24 noise is significant. This method utilizes an augmented matrix of snapshots and time-shifted data in the subspace projection step to prevent asymmetry in the linear fitting process. When noise is minimal in the data as in the simulation, TDMD gives the same result as DMD. To account for noise from both K and K p , an augmented matrix is introduced. Z := K K p (3.10) In this project, velocity fields have a more spatial resolution in each snapshot than a temporal resolution in the number of snapshots, so the DMD problem is under-constrained. As a result, in TDMD, the subspace of Z may retain traces of noise contamination. Therefore, SVD is performed on Z= UΣ W to project the reduced subspace W r onto K and K p . Then Eq. (3.8) and (3.9) is performed where K and K p are now replaced by ˜ K= KW r and ˜ K p = K p W r . A truncation level of W r was chosen based on excluding energy content below the noise threshold, E noise := ˜ E(St > 1), which is defined as E r := 1− ΣE(E < E noise ) ΣE (3.11) 3.2.5 Streaming DMD for real-time update and large datasets DMD requires the K matrix to have a complete time-series for a full modal description. This is eventually impractical when datasets become large and/or when rapid, or real-time DMD analysis is needed. The streaming DMD (sDMD) method introduced by Hemati et al. [57] incrementally updates DMD as each new snapshot becomes available. Moreover, because each snapshot is pro- cessed one at a time without storing previous snapshots, the method can efficiently process large datasets. This is useful for velocity fields of the flow, which typically have a large dataset. For ex- ample, a flow field of an axisymmetric wakes that contains a spiral behavior would require a large matrix of 3D velocity fields with high resolution in space (i.e. grid points) and time (i.e. snapshots) that extends over large distances in x. In many practical applications, therefore, a real-time DMD method could itself be an advantage. 25 In order to update the Ritz vector and Ritz values incrementally, Eq. (3.8) is rewritten in terms of new matrices A and Gx + , which are updated with the velocity field at one snapshot stacked in a column vector ˜ x i as: ˜ C= U T U p AGx + (3.12a) A= ˜ x 1 ˜ x T 0 Gx= ˜ x 0 ˜ x T 0 i= 0 A= A+ ˜ x i+1 ˜ x T y Gx= Gx+ ˜ x i ˜ x T i i= 1,··· ,m (3.12b) where ˜ x i = U T x i and ˜ x i+1 = U T p x i+1 . Since DMD can be sensitive to noise [58], POD com- pression [57] is applied to the Eq. (3.12) to discard low-energy modes in the updating process. POD modes of U and U p are the eigenvectors and eigenvalues of Gx and Gx p =∑ i j=1 ˜ x i+1 ˜ x T i+1 respectively. 3.3 Results 3.3.1 A combined regime diagram for sphere wakes Here five regimes are proposed for stratified sphere wakes based on the findings of experiments (LI92, CH93) and numerical simulations noted previously [14–16, 18, 19]. The final selection of wake categories is then guided by the current combined experimental and numerical program on the near wakes of spheres in a background stratification [59] which in turn is based partly on computations in [60]. The resulting 5 categories are not as numerous as a superposition of all literature results would yield, and reconciliation of small differences between LI92 and CH93 is not pursued further, but at these locations in {Re-Fr}, their categorisation into one of the five possible is unambiguous and repeatable. 26 (a) (b) (c) Figure 3.4: Regime diagrams from CH93 (a) and LI92 (b). The 4 main regimes differentiated in (a) are: 2D, quasi-2D; SLW, strong lee wave; T, transition (SKH, without K-H instability; KH, with K-H instability); and 3D, three-dimensional. In (b), LI92 distinguished 6 main regimes: A u , un- steady, attached 2D vortices; V t , 2D vortex shedding; L, lee-wave instability; N, non-axisymmetric attached vortex; S, symmetric vortex shedding; V , non-symmetric vortex shedding; and T , turbu- lent wake. (c) has the trajectory of local Re and Fr of Re= 1000 and Fr= 8 at Nt =[0.13− 11] plotted over combined (a) and (b) and{Re-Fr} test cases. 27 Figure 3.5: A regime diagram for stratified sphere wakes. The final result in colored dots com- bines experiments from CH93 (lighter colors) and LI92 (darker colors) with combined laboratory and numerical experiments. The five identified regimes are labeled and color-coded as: V ortex street (VS, purple), Symmetric non-oscillation (SN, red), Asymmetric non-oscillation (AN, yel- low), Planar oscillation (PO, green), and Spiral mode (SP, blue) The combined classification scheme is determined at each of the discrete dots in {Re-Fr} of Fig.Fig. 3.5 where both experiments and computations have been run, each on multiple occasions. Samples through these data fields, from both experiment and simulation, will provide the input data for the classification algorithms. Each category is associated with specific physical mechanisms that shape the near wake. Details of variations close to the sphere surface are not taken into account if they do not cause an observable change in pattern that passes into the free wake. The characteristics of any particular DMD mode associated with a physical regime are also noted. The observation window occurs at x/R=[20,30] and for Fr=[0.5,1,2,4,8,16] the window is equivalent to Nt =(x/R)/Fr=[40− 60,20− 30,10− 15,5− 7.5,2.5− 3.75,1.25− 1.875]. If regime boundaries such as 3D-NEQ are fixed in Nt, then the lower Fr data appear firmly in the NEQ regime while the higher Fr data (particularly for Fr = 16) are extracted from a time domain where the adjustment process has only just begun. 28 3.3.2 Distinguishing characteristics 3.3.2.1 Vortex street (VS) When Fr≤ 0.7 the ambient density gradient strongly influences the amplitude of vertical motions over the sphere, and in a central layer fluid particles travel around the obstruction in a horizontal plane where vortex shedding occurs much as in a two-dimensional cylinder wake at low Re in a homogeneous fluid. The wake disturbance is thus primarily characterised by vortex shedding in an equatorial layer, bounded above and below by lee waves that increase in amplitude as Fr→ 1. The quasi-2D vortex wake occurs over all Re examined, Re∈[10 2 ,10 3 ], but the formation mechanism varies. At lower Re, the shedding is directly from the sphere, passing through an oscillatory phase and then to alternating vortices that appear only further downstream in the wake. It is the signature of the fully-developed wake that is of most interest, rather than details of the origin, so the quasi- 2D, 2D oscillating, and 2D steady regimes of CH93 are combined into VS here. (The equivalent regimes in LI92 are the non-axisymmetric and symmetric vortex shedding regimes, a nomenclature that focuses on observations close to the sphere surface.) VS is colored purple in the lower Fr row of Fig. Fig. 3.5. Lin et al. describes that the first lee wave crest was unstable due to the baroclinic effect exhibiting overturning motion. However, Chomaz contradicts Lin et al. ’s interpretation that this periodic vortex shedding is caused by K-H instability from the shear layer thickening under the lee wave crest. In the simulation, while Re= 1000 and Fr= 0.5 showed K-H instability under the first lee wave crest, Fr = 0.5 cases of low Re (Re= 200− 500) observed small disturbances without a visible overturning motion in the first lee wave crest, which suggests that vortex shedding could still occur without triggering K-H instabilities. However, the spatial resolution may not be fine enough to observe these small-scale overturns to identify the cause of the lee wave instability in this study. The most energetic DMD mode is shown in Fig. 3.6a. The mode alternates in sign and is v-shaped in the vertical. Other energetic modes include higher harmonics of the vortex street and a mean mode that shows lee waves. 29 Figure 3.6: A snapshot of the flow field u in vertical and horizontal centerplane (left) and a domi- nant DMD mode of simulation in 3D (middle) and experiment in 2D centerplane (right) illustrating various regimes observed: (a) R10F0.5 V ortex Street (R2F0.5 for the experiment), (b) R5F1 Sym- metric Non-oscillatory, (c) R3F4 Asymmetric Non-oscillatory, (d) R3F8 Planar Oscillation (e) R10F8 Spiral Mode. The green box represents the equivalent window size between simulation and experiment. The isolevel of the middle 3D plot is 30% of the maximum velocity component. 30 3.3.2.2 Symmetric, non-oscillatory (SN) When Fr is defined on the radius, as it is here, then Fr ≈ 1 is when lee waves have their maximum amplitude,ζ = D/2, effectively suppressing vortex shedding and any other oscillatory motions. As detailed in LI92 and CH93, the flow transitions around Fr = 1 lead to intricate separation patterns on the sphere itself. Increasing in x, the wake pulses with a wavelength determined by the lee waves. CH93 and Meunier et al. [61] show that this wavelength, λ/D=πFr. These waves are steady in the sphere reference frame and dominate the near wake, which is therefore symmetric and non-oscillatory. Thus there is a broad range of Fr≥ 1, Re≤ 300 where the mean wake disturbances are similar, as shown in red in Fig. 3.5 The mean DMD mode contains most of the signal energy from the stationary lee waves, and the highest energy oscillatory mode also resembles the mean (Fig. 3.6b). The wake is symmetric in xy0 and x0z and ω x most clearly describes its geometry. If data are available from the experiment in only one plane, then the v velocity component in the horizontal plane is the most sensitive indicator of the overall geometry. 3.3.2.3 Asymmetric, non-oscillatory (AN) As Fr increases, the lee wave amplitudes diminish and other modes can appear in the wake. These modes first appear in the horizontal centerplane, and at particular {Re-Fr} the earliest appearance of departures from SN is in an asymmetric mode. In the vertical plane, the wake is symmetric and non-oscillating, similar to SN, but in the xy0 plane it is asymmetric about the centerline Fig. 3.6c. The AN regime is not clearly or separately defined in the literature, but when classification criteria are built from symmetry properties in DMD modes, it is readily distinguishable, even if at only one{Re-Fr} test case (Re= 300 and Fr= 4, marked yellow in Fig. 3.5). The AN wake is steady so the most energetic DMD mode is again the mean, but the next most energetic modes share the same pattern, which is measurably asymmetric in y 31 3.3.2.4 Planar oscillatory (PO) Generally, as Fr increases above 1, the continued weakening influence of the stratification eventu- ally allows oscillations in both horizontal and vertical planes, becoming fully three-dimensional at higher Fr. The first departure from the SN/AN modes is an oscillatory mode. In the vertical center- plane the internal wavelength is long compared with the body diameter (λ/D∼ Fr) and contributes to a periodic variation in wake height. In the horizontal plane, there are wavy oscillations that are not steady in the sphere reference frame but that have the same wavelength as the vertical varia- tions, which are. This mode is observed at the boundary in{Re-Fr} space between AN/SN and the spiral mode which is fully three-dimensional. In this regime, vortex shedding and roll-up are possible in the horizontal plane, but all vertical motions are subsumed into the dominant lee wave mode. The boundary marking this mode is approximately on fixed Fr Re − 1 , as originally noted in CH93, and PO modes are labelled green in Fig. 3.5. The combined vortex and wave motions lead to a most energetic DMD mode as an angled structure alternating in y, as shown in Fig. 3.6d. 3.3.2.5 Spiral mode (SP) As both Re and Fr increase, the two major instability modes of the neutral wake are recovered – the Kelvin-Helmholtz and spiral modes. The Fr at which K-H modes appear decreases as Re increases, and at sufficiently high {Re-Fr} the K-H shear-layer instabilities are found alongside the truly three-dimensional spiral mode. SP is marked not just by the long spiral wavelength but also by the appearance and persistence of small-scale turbulence. SP appears in blue at the top right corner of Fig. 3.5. SP is shown in Fig. 3.6e. It is the increased activity in the vertical that distinguished this regime from others, and the DMD modes in the vertical velocity component in the vertical x0z plane can be used to uniquely identify it. The spiral wavelength is about 6D in streamwise extent so the DMD modes from simulation and experiment may not contain a full cycle, depending on the window length in x. Since this regime characterises the move to higher Re and Fr, future work will focus on discriminating patterns inside this regime. 32 3.3.3 3D standard DMD based classification (a) Simulation (b) Experiment Figure 3.7: The plots on the left show the real and imaginary parts of λ, and on the right ˜ E(St) is shown for each mode of the case Re= 500 and Fr= 4. The dark red corresponds to the mean mode. The green dot is the most dominant mode selected to be evaluated in Fig. 3.8. The grey area is considered noise, set below ˜ E threshold and above St threshold. The distinguishing characteristics of the five regimes introduced in Section 3.3.2 are accen- tuated when projected onto DMD modes that act like a dynamic shape or structure filter. Then algorithms can be designed to interrogate these modes for classification into the corresponding regimes. DMD is computed on a K matrix, a temporally reconfigured flow data stream, based on Section 3.2.1.1 producing a total number of modes equal to the number of snapshots. The modes are then sorted based on energy as ˜ E j := √ m+ 1|λ| m j ∥v∥ 2 ∥[K x m ]∥ F , (3.13) 33 where the modal energy∥v∥ 2 is scaled to|λ| m ∥v∥ 2 , a contribution of the j th mode in the recon- struction of the last snapshot, x m , normalized by the rms of the data norm,∥[K x m ]∥ F / √ m+ 1. The Strouhal number of each j th mode is then St j := ω j R 2πU = arg(λ j )R 2π∆t U , (3.14) where arg(λ)/∆t is equivalent to a phase angular velocity and∆t is the time interval between two consecutive snapshots. The St range is limited by max(arg(λ)) = π, for a maximum St max = R/(2∆tU). In the simulations, St max = 0.5 for low Re and St max = 1.25 for Re= 1000; for the experiments, St max is generally much higher as smaller∆t are required for the PIV estimates. 3.3.3.1 Mode selection criteria As the stratified wakes evolve, smaller-scale vortices merge into large-scale pancake eddies. Low St modes associated with these objects then contain most energy while high St modes associated with small scales, noise, or high-order harmonics have much lower energy, as shown in the fre- quency spectra of DMD modes in Fig. 3.7. When DMD is computed from a full field, the highest energetic mode resembles the mean flow with St ≃ 0 (marked in red atλ≃ 1 in Ritz value Fig. 3.7 and omitted in the plots of ˜ E(St)). Most modes are temporally neutrally stable as they are near the unit circle. Minor modes are filtered out by setting ˜ E and St thresholds. The combined effects of vis- cosity and/or an energy cascade mean that most energy in the wakes resides in low-St (or even mean) modes. A robust classifier ought to depend primarily on these modes, so thresholds on ˜ E and St are imposed. In simulations a simple threshold of ˜ E > 10 − 3 was found, empirically, to reduce the number considered to 10 or fewer. In the experiment, non-physical small-scale noise (ultimately attributable to PIV errors) shows an increase in energy for St≥ 1 and this limit is also applied. These somewhat arbitrarily selected constant thresholds are supported to some extent by their continued utility following changes in the algorithm (e.g. TDMD, explained more in detail 34 Figure 3.8: A flow chart of the wake classifier with 3D DMD input. The selected DMD mode from Section 3.3.3.1 is sorted into five regimes in the non-oscillatory branch (left) and oscillatory branch (right) based on the symmetry of the cross-stream velocities in that mode. in Section 3.3.5.1), and the fact that selections based on the highest-ranked modes only are not sensitive to the existence of large numbers of sub-threshold modes. Consequently, the oscillatory (or non-zero frequency) mode with the highest ˜ E (marked in green in Fig. 3.7) is selected for further interrogation. If there are no oscillatory modes in the viable mode selection pool (following threshold), the flow is in the non-oscillatory regime, and this mean mode is selected for further evaluation in the non-oscillatory branch of the classifier. In all cases, the mode in question is then classified into a regime described in Section 3.3.2, based on criteria that depend on the symmetry and energy distribution along x about y= 0 or z= 0. This classification procedure is summarized in the flowchart in Fig. 3.8, and the sequence of decision points is described below. 35 3.3.3.2 Non-oscillatory regimes: Symmetric or Asymmetric The two non-oscillatory regimes are distinguished based on the symmetry of the energy distribution of the mean mode about y= 0. The degree of symmetry is determined through the difference norm of the streamwise vorticityω x in± y in x and z, ∆|ω x | := |∥ω x (− y)∥−∥ ω x (+y)∥| ∥ω x ∥ . (3.15) When∆|ω x |≤ 20%, the flow is in the symmetric, non-oscillatory regime (SN) and otherwise it is in the asymmetric non-oscillatory (AN) regime. (Full asymmetry would yield∆|ω x | max = 1.) 3.3.3.3 Oscillatory regimes: Planar motion or Spiral Mode In the oscillatory branch on the right side of the decision tree in Fig. 3.8, the symmetry of v(x,z) in the z direction readily distinguishes the spiral mode from regimes where flow is confined close to the horizontal centerplane. In the vertical direction of the planar motion, symmetric pulsation in the PO regime and the V-shaped structure in VS are both spatially invariant with respect to the sphere, and symmetric about the horizontal centerplane. By contrast, the spiral mode has spatial variation in z, asymmetric about z= 0. The symmetry measure is ∆v ± z := ∥v − z − v +z ∥ ∥v∥ , (3.16) where v +z is the lateral velocity of the upper half of the field, v(x,0,+z), about the centerplane of the wake. Though the symmetry can also be estimated from the vertical velocity w, the magnitude of w is relatively small for the planar motion branch so that ∆w ± z would be sensitive to noise. In contrast to interrogating the symmetry of the energy distribution in Eq. (3.15) for the non- oscillatory branch, the oscillatory branch constructs a point-by-point difference mirroring the top and bottom half of the field. For ∆v ± z,min = 0= 0% the DMD mode is perfectly symmetric about the horizontal plane and when ∆v ± z,max = √ 2= 100% the upper and lower half of the field are 36 equal and opposite. A symmetry level tolerance of 20% is set by taking the average of∆v ± z from all available sets in each regime decision box, similar to a cross-validation method that allows small asymmetries from instabilities in the spiral mode. When∆v ± z is less than 20%, the mode is either in VS or PO regime. 3.3.3.4 Planar motion: Vortex Street or Planar Oscillation In the Planar motion branch, a dominant mode in VS has alternating signed structures aligned along the wake centerline, with symmetry in both y and z. Since symmetry in z was satisfied earlier in the decision tree, symmetry in y now distinguishes VS from PO. The symmetry between the left and right half of the v about y= 0 can be calculated in the same way as Eq. (3.16) as a function of v ± y . When∆v ± y is greater than 10%, the mode is considered asymmetric in y, and hence in the planar oscillation regime. The coherence of the quasi-2D vortex street that occurs in the VS regime allows the symmetry tolerance in y to be tighter (∆v ± y = 10%) than in z (∆v ± z = 20%). Since the stratified wake evolves, albeit slowly, in the streamwise direction, all criteria are evaluated based on the local cross-stream information. 3.3.3.5 The success rate of the classifier with 3D numerical DMD input There are 24{Re-Fr} points for the classifier and in each one, the correct category is known (as VS, SN, AN, PO, SP). The accuracy can then be determined and is synonymous with the success rate. The success rate of the classifier described in Fig. 3.8 is illustrated in Fig. 3.9 for each Re − Fr pair. In the numerical study, the full 3D flow field is available, while only two centerplanes (horizontal and vertical) are available from the experiment. The test samples were collected from equally splitting one continuous simulation into 10 (or multiple) sets at different time instances. (The total number of sets was determined based on ensuring the independence of the number of snapshots in a set to minimize aliasing error). The color in the pie chart for each case in the Re− Fr parameter space shows the classification distribution. When the pie chart is filled with one uniform color, the classifier has 100% accuracy. 37 Figure 3.9: Success rate of a classifier with 3D numerical DMD input. The pie chart shows the classification type in color. The number above each filled circle is the the percent accuracy of the majority outcome. Most cases have a 100% success rate. The 6 cases that do not live close to regime borders. The worst case at Re= 500,Fr= 2 with only 30% success rate, when misclassified was always classi- fied in one of three neighboring regimes (SN, AN, and PO), caused by occasional small oscillation in the lateral direction. The small oscillations come from slight unsteadiness in the recirculation zone and the resulting unsteady DMD modes led the classifier to the oscillatory branch in Fig. 3.8, even though the modal energy was very close to the energy threshold, ˜ E > 10 − 3 , in Section 3.3.3.1. 3.3.4 Reduced information 3.3.4.1 Spatially reduced: 2D centerplanes The automated classifier relies on having 3D information of the wake in multiple consecutive snap- shots. However, obtaining high dimensional data from an imagined large-scale stratified wake, and 38 Figure 3.10: A flow chart of the wake classifier with two, 2D-centerplane flow fields as input. Two dominant DMD modes of each centerplane are evaluated where the mode for the vertical centerplane is selected based on St w closest to St v . The non-oscillatory branch uses the mode from the horizontal centerplane, while the oscillatory branch uses the vertical centerplane for the first step, and horizontal centerplane for the second step. 39 even in a laboratory experiment is challenging. In this section, limitations and possible improve- ments of spatially reduced input information will be examined. The information available from the laboratory experiment is in two, 2D centerplane (horizontal and vertical) cuts, where each plane is acquired independently. To compare the classification success of DMD modes restricted to one plane, the mode selection is forced towards modes that are available from that plane. For example, in the presence of strong stratification, the most energetic 3D mode(s) may not be obvious when information is restricted to the vertical direction. To guide the classifier to select a similar mode in both planes, the selected mode in the vertical plane is forced to the mode with St w closest to St v , the St of the dominant mode from the horizontal plane. From these two matched St w,v , the mode that is in the direction of the plane of interest was used The flowchart was modified so the direction of symmetry remains the same but the velocity information is confined to the same plane, as illustrated in Fig. 3.10. Thus, ω x in Section 3.3.3.2 is replaced by v to determine the symmetry in y; v in Section 3.3.3.3 is replaced by w to determine the symmetry in z to distinguish SP from planar motion based on flow variation in the vertical direction only; and v symmetry in y in Section 3.3.3.4 is unchanged as the velocity and line of symmetry are already in the same direction. Success rate of the classifier with 2D numerical DMD input To understand the performance of the classifier with limited spatial information compared with the 3D numerical input and to keep similar constraints as the experiment, the same simulation data set in Section 3.2.2 is modified to replicate the spatial limitation of two independent 2D planes from the experiment. The results are summarized in Fig. 3.11a. While most cases in the VS, PO and SN had a high success rate, SP cases bordering the PO regime had a measurably lower success rate (though above 50%). Since SP is the only regime among the five that has flow variation in the vertical direction, the (comparative) failure could have come from the procedure of choosing the dominant mode in the vertical direction based on St w ≃ St v . Note however that the lower success rate of VS does not come from misclassification to a neighboring regime in{Re-Fr} space. 40 Figure 3.11: Classifier success rate with (a) 2D numerical (b) 2D experimental DMD input. Cases with Red X have wake length-scales larger than the FOV for regimes to be distinguishable. Success rate of the classifier with 2D experimental DMD input The same criteria from Fig. 3.10 were applied to the two 2D planes of experimental results, and results are shown in Fig. 3.11b. The more strict the classification criteria (e.g. ˜ E threshold, symmetry requirement), the more the errors increase. In particular, many non-oscillating regimes were incorrectly classified into the oscil- latory branch for zero success rate. As noted in Section 3.3.3.1, the successful classification of non-oscillatory regimes relies heavily on the energy threshold ˜ E max (St> 1). When the real wake flow is steady with respect to the sphere, the small amplitude time-varying input components, which are either noise or small variations in mean wake position (in y and/or z), come to dominate the DMD modes. The influence of small-scale noise typical of experimental conditions can be reduced by per- forming DMD on an appended vector of the measurement and a time-shifted value, [x T k x T k+1 ] T , known as TDMD (see Eq. (3.10) in Section 3.2.4), when the data matrices K and K p become lin- early consistent [56, 62]. Improvement of the classifier using TDMD is discussed in the following Section 3.3.5.1. In strongly-stratified stratified wakes at lower Fr < 8, the body-fixed lee wave forces much of the fluid mechanical information to be steady in the sphere reference frame. Experiments con- ducted in the laboratory reference frame are converted through a constant shift to this reference 41 Figure 3.12: Performance of sDMD. Left is the dominant DMD mode and on the right is the dominant sDMD. (a) dominant modes from DMD and sDMD of the Re= 500 and Fr= 4 case in the same given space where similar planar oscillation features are shown. (b) an example of processing sDMD on a large data matrix of a high resolution case (Re= 1000 and Fr= 8) where a dominant sDMD mode successfully isolated the spiral feature. In both cases, an SVD truncation of m= 10 was applied. frame but then if the computed shift is not exactly correct, there are small residual apparently un- steady motions which, in a DMD analysis will lead to ’shadows’ of the mean mode in the most energetic oscillatory modes. Finally, when the wake lengthscales in any direction are comparable to or larger than the field of view, the algorithm yields results (marked off in a red X in Fig. 3.11b) that are influenced by the spatial truncation. Therefore, only cases with window width in y that is bigger than 3R are evaluated. 42 3.3.4.2 Real-time analysis: Streaming DMD When the dataset is temporally limited to perform a standard DMD, Streaming DMD (sDMD) in Section 3.2.5 can be used as a real-time DMD analysis where sDMD modes are updated as each new snapshot becomes available. As sDMD modes are updated, the user can see the inter- mediate stage to determine when the solution converges without processing an excessive amount of snapshots to ensure convergence. This makes a real-time classifier possible and computation- ally cost-effective by running the classifier with initial sDMD modes and updating the result until convergence. Specifically, in a real-time classifier, sDMD is applied to the initial flow field data following the eigendecomposition on the modified companion matrix ˜ C in Eq. (3.12) where the obtained Ritz values and Ritz vectors are then fed into the classifier. When the new snapshot is available, sDMD modes are re-generated with the updated A matrix as shown in Eq. (3.12b). The accuracy of sDMD compared to the standard DMD was tested with a velocity field of a sphere wake at{Re,Fr}={500,4} from simulation. sDMD was computed with an SVD trunca- tion of m= 10 number of modes for the same number of snapshots used in the standard DMD. As shown in Fig. 3.12a, the dominant mode generated from sDMD replicated important features of the dominant mode from standard DMD that is essential for the planar oscillation (PO) regime classifi- cation (see Section 3.3.2.4). A further investigation is needed to determine how many snapshots are required for the sDMD modes to converge to DMD modes and obtain a quantitative measurement that describes the accuracy of sDMD modes. However, sDMD dominant mode showing a qualita- tively similar flow structure to the DMD dominant mode indicates potential in sDMD analysis to be used in a real-time classifier. To test the computational efficiency, sDMD was applied on the largest flow field matrix within the testing {Re,Fr} parameter of the sphere wakes. Among the five sphere wake regimes, the spiral mode regime (see Section 3.3.2.5) at high Re and Fr has a more disorganized flow with small-scaled eddies that requires a finer time and spatial resolution to be resolved. In addition, spi- ral oscillation in a sphere wake has a large oscillation period that the velocity field in the data matrix should include a wide downstream range (more than a spiral mode wavelengthλ = 2R/0.175∼ 6D 43 [13]). For example, the 3D velocity field of a sphere wake in an x range x=[0,30]R at{Re,Fr}= {1000,8} from simulation has a large flow field data matrix ( ∼ 3000× 1000). Processing a stan- dard DMD on this large matrix requires higher computational power. The standard DMD (left image of Fig. 3.12b) on{Re,Fr}={1000,8} case was limited by the computational power of the device to process only the shorter downstream range of x=[20,30], which the spiral mode was not apparent. However, sDMD modes (right image of Fig. 3.12b) were produced at a faster rate and the spiral mode features are clearly shown. As a result, a classifier based on sDMD modes can be a potential option for a low-storage and real-time classifier. 3.3.5 Noise contaminated The low success rates from 2D experiments in Fig. 3.11 have low cross-stream velocity fluctuations (i.e. low signal-to-noise ratio) in either centerplane. DMD is sensitive to noise that the dominant modes may not be identified correctly [45, 48, 62]. One way to minimize the sensitivity of DMD to noise is by replacing DMD with TDMD (Total least-squares DMD) with an SVD truncation where biased noise from the asymmetric fitting of a linear model is avoided. 3.3.5.1 TDMD based classifier for noise-contaminated input Following the application of TDMD in Section 3.2.4, the dominant TDMD mode of an example non-oscillating case is compared against the dominant standard DMD mode in Fig. 3.13. The main struggle of the DMD-based classifier lies in dealing with a planar view of the flow field in which the dominant oscillatory mode is not present. Then modes triggered by noise are erroneously selected as dominant modes (e.g. SN, AN regimes, and vertical centerplane of VS and PO regimes). This is shown in a sample SN regime case in the horizontal centerplane at Re= 300 and Fr= 2 in Fig. 3.13a where noisy energetic modes are not filtered out by the energy threshold, ˜ E noise , to yield a success rate of 0% in Fig. 3.11b. The same flow field computed on TDMD is shown in Fig. 3.13b where the truncation level of r= 97 from m= 276 total snapshots was used based on E r = 91.6% from Eq. (3.11). Although most DMD modes lie on the unit circle, appearing neutrally stable, 44 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 10 -2 10 -1 10 0 10 1 10 -4 10 -3 10 -2 10 -1 10 0 14 16 18 20 22 -4 -2 0 2 4 -0.01 -0.005 0 0.005 0.01 (a) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 10 -2 10 -1 10 0 10 1 10 -4 10 -3 10 -2 10 -1 10 0 14 16 18 20 22 -4 -2 0 2 4 -0.01 -0.005 0 0.005 0.01 (b) Figure 3.13: Ritz values λ (left), ˜ E versus St (center), v(x,y,z= 0) of a dominant oscillatory mode (indicated by an arrow) for the Re= 300, Fr= 2 case computed from (a) standard DMD (E r = 100% of ˜ E) and (b) TDMD (E r = 91.6%). Modes colored in red are energetic modes selected from Section 3.3.3.1 45 the TDMD modes are concentrated nearλ = 1 (i.e. mean mode) and otherwise lie inside the unit circle. This shows that non-mean modes are damped out, as is physically realistic in a decaying flow. The energy of higher frequency modes above St > 1 associated with noise is significantly reduced. The most energetic mode of DMD at St∼ 0.02 showed a flow structure similar to the mean mode contaminated with noise no longer appeared in the TDMD spectra, while the dominant sDMD mode at St∼ 0.05 showed a characteristically different flow than the mean mode. This shift in dominant modes from DMD to TDMD may be due to the noise causing a little variation in the mean mode ( ˜ E∼ 1) to be considered an energetic oscillatory mode. Although these TDMD modes once again place the flow (incorrectly) into the PO regime, the noise-contaminated mean mode is no longer the dominant oscillatory mode and the corresponding dominant mode shape is more coherent (Fig. 3.13b). This example case in SN is the toughest condition to obtain a clean result, as TDMD is applied to a flow field that varies little in x. Data with a wide range of spatial variations in the flow can be expected to significantly improve a TDMD-based classifier. 46 Figure 3.14: A sketch illustrating how local overturning length scale l h and l v are estimated. The fluctuating quantities of the lateral vorticity ω ′ y and vertical velocity w are measured at the wake edge. 3.4 Extrapolation to higher Re and Fr As mentioned in Section 1.4, estimating the Froude number based on local horizontal overturning length scale Fr h = 2πu h /Nl h and the activity parameterG of wakes can give insight into how tur- bulence evolves and interact with the stratification in high Re stratified wakes. However, estimating local horizontal length scale l h and energy dissipation rate ε from the experiment is challenging, especially when 3D spatial information is not available. In the next few sections, methods used for each of these values are covered. 3.4.1 Overturning horizontal length scale l h estimation To estimate the horizontal length scale of a overturning motion from an experimental result where density fluctuation information is not available, a method suggest by Xiang et al. [52] is used. The local length scale of local overturning can be computed from zero crossing of the auto correlation function of fluctuating lateral vorticity, ω ′ y and vertical velocity w ′ at the wake edge as illustrated in Fig. 3.14. The resulting l h is computed from the average of the two fluctuating quantities, so for 47 simplicity, the fluctuating lateral vorticity is represented as X j =ω ′ y | edge (t j ) in the time step j, and repeated for w ′ . The autocorrelation function, R xx is: R xx (X j ,X j+1 )= Σ N− 1 j=0 (X j − ¯ X)(X j+1 − ¯ X) Σ N j=0 (X j − ¯ X) . (3.17) The first time difference of positive to negative zero-crossing of R xx is computed as∆t = t 2 (R xx = 0)− t 1 (R xx = 0). Then, a local horizontal length scale is estimated to be l h = u local ∆t (3.18) where local velocity, u local (x)= ¯ u(x,⟨z⟩ wake ) estimated from mean streamwise velocity within the wake height⟨z⟩ wake at that particular Nt. Here,∆t is an average of∆t computed individually from ω ′ y and w ′ . The local length scale of the early wake when Nt < 0.5 (dotted line in Fig. 3.15a) can be ignored, because the flow is in the recirculation region. The local Froude number defined based on horizontal integral scale Fr h = u rms /Nl h where u h is local RMS velocity is calculated. The numerical and experimental results of Fr h in Nt are shown in Fig. 3.15b. Ignoring the recirculation region, Fr h decreases at Nt − 1 consistent with the literature [8]. 3.4.2 Dissipation rateε estimation The kinetic energy dissipation rateε of the 2D experimental flow field is challenging to calculate due to missing out-of-plane velocity gradient information. Therefore, ε in the activity parameter G =ε/νN 2 is estimated by following the local isotropy assumption [63] where the out-of-plane components are replaced with the in-plane gradients with a similar magnitude. The original form ofε is ε =− 2ν⟨s i j s i j ⟩ (3.19) 48 (a) (b) Figure 3.15: An estimate of ∆t, u loc , and l h for Fr h of the experiment at initial Re= 1000 and Fr= 8. The local Froude number Fr h follows the weakly stratified wakes scaling, Fr h ∼ Nt − 1 [11] where⟨⟩ is denoted as an ensemble average. The strain rate tensor s i j is s i j = 1 2 ∂u ′ i ∂x j + ∂u ′ j ∂x i ! (3.20) where u i is the fluctuating velocity along the i th direction. Under the local isotropy assumption, the following components are equal to each other * ∂u ′ i ∂x k 2 + = * ∂u ′ k ∂x i 2 + = * ∂u ′ j ∂x k ! 2 + = * ∂u ′ k ∂x j 2 + = 1 2 * ∂u ′ i ∂x j 2 + ∂u ′ j ∂x i ! 2 + (3.21) * ∂u ′ i ∂x k ∂u ′ k ∂x i + = * ∂u ′ j ∂x k ∂u ′ k ∂x j + = * ∂u ′ i ∂x j ∂u ′ j ∂x i + (3.22) 49 Figure 3.16: comparison of the ˜ ε estimate for 3D3C, 2D3C, 2D2C with a local isotropy assump- tion. 2D3C and 2D2C has an ˜ ε estimate for horizontal plane (() h ), and vertical plane (() v ). Sample flow is from the simulation at initial Re = 1000 and Fr= 8. Then theε estimate of 2D2C (two-dimensional, two components) in x i − x j plane is simplified to ε =ν * 4 ∂u ′ i ∂x i 2 + 4 ∂u ′ j ∂x j ! 2 + 3 ∂u ′ i ∂x j 2 + 3 ∂u ′ j ∂x i ! 2 (3.23) + 4 ∂u ′ i ∂x i ∂u ′ j ∂x j ! + 6 ∂u ′ i ∂x j ∂u ′ j ∂x i !+ where ∂u ′ i /∂x j is fluctuating velocity gradient tensor. Although this estimate provided previous studies in structure in the upper layer of the ocean [64, 65], the stratified wakes of interest becomes strongly anisotropic that even local isotropy assumption should be taken with caution. To see how accurate the ε estimation is for our application, 2D information that replicates the PIV setup was extracted from a 3D simulation of initial Re= 1000 and Fr= 8. This includes models replicating a planar (2D2C) and stereo (2D3C) PIV in the vertical (() v ) and horizontal centerplane (() h ). ˜ ε, the average ofε at the wake edge is plotted along Nt in Fig. 3.16. Within the range of Nt=[1,10], the ε of all five variations of 2D flow information are close from the 3D3C curve and the decay rate is consistent, implying that theε estimate from 2D2C information is reasonable. 50 Figure 3.17: Evolution of wake in Nt at initial Re= 1000 and Fr= 8. Red dot is simulation; blue dot is experiment; and black line dotted line is from simulation by Bruyn Kops & Riley [25] (a) (b) Figure 3.18: a snapshot of ˜ ρ for (a) Re= 1000 and Fr= 8 and (b) Re= 500 and Fr= 4 3.4.3 Fr h -G space The trajectory of experimental and numerical result of Re= 1000 and Fr= 8 in{Fr h − G} space is illustrated in Fig. 3.17. The flow crosses three regimes in time. This is consistent with one of the cases from Bruyn Kops & Riley [25] marked in black dotted line. The inconsistency with experimental results near Nt∼ 9 can be due to the l h estimate failing from lack of true overturning motion. 51 3.4.3.1 Limitations Xiang et al. [52] where the method is taken from warned that the ε estimation only works in the early wake stage before the wake edge definition becomes unclear due to vertical suppression in the far wake. To estimate the accuracy of this method, l h , the overturning horizontal length scale is computed for a flow field from simulation and overlaid on top of the density fluctuation in Fig. 3.18. Two cases are shown at the same downstream distance, x/R. One lies in the 3D regime (Re= 1000 and Fr= 8) where the stratification is weak. The second is in a planar oscillating (PO) regime (Re= 500 and Fr= 4) when vertical motion is more strongly suppressed constraining the motion to a central layer. While the overturning motion is observed in Fig. 3.18a, the flow in the PO regime does not have vertical density variation associated with any overturning motion. Since l h estimate depends on the distinct w ′ andω ′ y fluctuations at the wake edge, a measurable overturning motion is necessary to achieve a good l h estimate. (a) Simulation (b) Experiment Figure 3.19: local Fr h andG of various sphere wake regimes are mapped on Fr h − G parameter space with the same color code as the Fig. 3.5. The parameters are measured from (a) simulation and (b) experiments of various initial Re and Fr at a constant downstream distance x/R= 15. To understand when this limitation occurs, the sphere wake of selected {Re-Fr} parameter space from simulation and experiments are superpositioned on the{Fr h − G} regime space from Bruyn Kops & Riley [25]. Since the local Fr h andG varies with Nt as shown in Fig. 3.17. a single instance at a constant downstream distance x/R= 15 is selected to compare the local Fr h andG 52 parameters of various initial Re and Fr conditions. In Fig. 3.19a, the simulation plot shows that the descending order of high local Fr h andG of the regimes is Spiral mode to Planar oscillation to V ortex street to Asymmetric, non-oscillating to Symmetric, non-oscillating. This is similar to the order of regimes that the trajectory of local Re and Fr crossed in Fig. 3.5c. Among the tested Re and Fr parameters, 3D (blue) cases where the stratification effect is not high (Fr h > 0.6) are at the weak buoyancy effect regime and none of the initial Re triggered turbulence for the decaying flow to enter into the stratified turbulence regime ( G > 1). The experiment shows similar trends for planar oscillation and spiral mode regimes, but once the flow is in the viscous dominated regime, the local Fr h is much higher than what was observed in the simulation. When the viscous effects become important in strongly stratified flows ( G < 1), the density fluctuation is stable without overturning instabilities excited at the wake edge resulting in an incorrect measurement of l h . This suggests that another measurable length scale should be used at low Fr h andG . 3.5 Discussion The main objectives of this chapter were to investigate the feasibility of a DMD-based flow clas- sifier for stratified wakes. The known qualitatively different flow regimes at moderate Fr and Re provide a convenient testbed for such an effort. Given these obvious regime differences it is perhaps unsurprising that initial success rates were very high, as they are for a well-trained fluid mechanician. If this example problem set seems too easy, then at least success can be claimed so as to encourage further exploration in more demanding cases. Most of the low{Re-Fr} cases of sphere wakes lie in the regime where the viscous effect is dominant, while flows typically found in the ocean and atmosphere have strongly stratified turbulence, which lies in the regime that is hard to access experimentally unless a large object is towed fast in a large tank. Representing stratified wakes based on local information of the flow (e.g. the length scale of an overturning motion) can link the experimentally accessible scales to geophysical scales. 53 Looking outward to more realistic applications, it will never be the case that we have full 3D, time-resolved information on an unknown flow. Following the first successes, a start in systematic reduction of the input data dimension was tested and indeed success rates dropped, but not to zero. The failures do not lead inevitably to the conclusion that the classification basis is flawed (although it might be) and refinements in the DMD computation that eliminate the effect of small-scale noise were successful. The classifier failed in a number of cases that could be unique to the low {Re- Fr} domain, and the problem of small fluctuations superimposed on a dominant mean mode is much more challenging at low Fr when body-fixed lee waves enforce the mean mode dominance. Flows typically found in geophysical applications may occur in a strongly stratified turbulence regime, where Fr is low even though Re remains high. Though this domain is difficult to access in experiment, the expected shear instabilities would generate signals that are readily distinguished from noise. It is likely that further extensions and tests of this DMD-based classifier will push to domains of higher Fr and higher Re. There the influence of mean modes will be negligible, and it is less clear how the classifier categories will be compiled and extracted. If these categories cannot be defined a priori then the entire approach of designer-categories may be impractical. Instead, a more robust and extensible method should be based on some kind of data-driven sorting and cat- egorisation. Such classifiers and learning algorithms from multilayer neural network architectures (deep-learning methods) are well-known and we suggest that such a program could be imple- mented, first using the known and simple cases described here, and then extrapolating to higher {Re-Fr}. The exploration of classifier space could be purely data-driven, or constrained on a re- duced order physical basis. A recent collaborative effort has successfully implemented the FROLS (Forward Regression with Orthogonal Least Squares) reconstruction method, capable of producing a sparse model, to test classification on a reduced grid resolution and measurement window [66]. The model obtains orthogonal basis functions through reordering of the model terms by mapping the velocity field to DMD modes and classifies a flow field based on the associated mode weights. 54 This effort serves as a solid starting point for understanding the challenges of reduced input dimen- sions and exploring solutions to a more robust classifier. We note that success (or failure) rates at higher{Re-Fr} in a data-driven program may be completely independent of any pattern of weight matrices learned at lower{Re-Fr}. 55 Chapter 4 Patterns from stratified inclined spheroid wakes 4.1 Introduction 4.1.1 Characteristics of the flow around spheroids Slender bodies have been widely studied for aerospace (fuselage) and marine engineering (under- water vehicle) applications. Among the slender bodies, the wake of a prolate spheroid has been a popular canonical case, as simple geometry produces a wide range of special wakes by chang- ing the aspect ratio, the incidence angle θ, Reynolds number of the flow, and high maneuvering conditions due to 3D complex flow separation patterns. The slender body can be characterized Figure 4.1: (a) Schematic of a slender body from the side view with an aspect ratio of a major axis L to a minor axis D and an inclination angleθ rotated about y. The origin is the center of the slender body where x is the streamwise direction, y is the lateral direction z is the vertical direction. (b) Schematic of a slender body in 3D where the gray yoz plane is a cross-section of a wake where wake height is L V and wake width is L H defined from the streamwise velocity u. 56 Figure 4.2: V ortex patterns in separation flow of a slender body with a pointed noise and blunt tail from Nelson & Corke [71]. Four regimes are classified as a function of inclination angle θ where highθ is in the range ofθ ={20,50} ◦ . α denoted in the figure is equivalent to θ in this study. by its aspect ratio AR= L/D where L is the major-axis and D is the minor axis, and its angle of inclinationθ as illustrated in Fig. 4.1a. In this chapter, the coordinate defined in Fig. 4.1b will be used to describe the wake where the origin is defined as the center of the slender body where x is the streamwise direction, y is the lateral direction z is the vertical direction. The corresponding velocity components in{x, y, z} direction is{u, v, w}. A convenient Reynolds number definition for these slender bodies is based on the diameter D as the characteristic length scale and will be referred to as Re D . At zero angle of inclination, the trailing end exhibits a small recirculating region, which results in lower pressure drag. Unlike bluff-body wakes where the flow is dominated by vortex shedding in the near-wake [67], at even moderate Re, the flow immediately behind a slender body is dominated by axisymmetric turbulence where the near wake is quasi-parallel, characterized by the presence of broadband turbulence [68–70]. As the inclination angle increases, one separation line moves closer to the leading edge. When the spheroid is at a high incidence angle θ, the flow wraps around the body and rolls up into a primary separation vortex on both lateral sides [72, 73, 76, 77]. This is well-illustrated in Fig. 4.3a, where the experimentally observed pair of vortex sheets separating from the body is sketched [72]. 57 (a) (b) Figure 4.3: (a) A sketch of a pair of vortex sheets observed in the experiments of a 6:1 spheroid at θ = 10 ◦ and Re D = 3.5× 10 5 from Fu et al. [72]. (b) Primary and secondary vortices represented in a streamline of flow around a 6:1 spheroid at θ = 20 ◦ and Re D = 7× 10 5 from Xiao et al. [73] (a) (b) Figure 4.4: (a) vortices separated from a 6:1 spheroid at θ = 10 ◦ and Re D = 3.5× 10 5 from Strandenes et al. [74]; (b) streamwise vorticity of yoz plane at various downstream distance of DARPA SUBOFF at pitching angle,α p = 8 ◦ and Re L = 2.4× 10 6 from Ashok et al. [75] 58 The formation of the separation vortex wake varies depending on the Re and the inclination angle. Nelson & Corke [71] classified separation vortex wakes into four flow regimes: vortex- free flow, symmetric vortex shedding, asymmetric vortex shedding, and unsteady vortex wakes, as illustrated in Fig. 4.2. Although this classification is based on the wakes of a slender body with a pointed nose and a blunt tail, similar variations of vortex wakes were observed in other inclined slender geometries, such as a cylinder with a hemispherical nose (AR∼ 8) [78] and a prolate spheroid (AR= 6) [76]. Moreover, Re can influence the turbulence production and the separation line. The generated separation vortices then meet the drag wake, altering the near-wake behavior. Delay in the formation and migration of separation lines for unsteady flows and high inclination angles could lead to different flow topologies [79]. As the strength of the separation vortices increases with increasingθ, unstable separation vor- tices [78] or asymmetric separation can appear as observed in Strandenes et al. [74] and Ashok et al. [75] shown in Fig. 4.4. When there is an imbalance in the vortex strength between the counter- rotating vortex pair, the weaker vortex wraps around the stronger vortex, increasing the asymmetry in the wake and eventually resulting in annihilation of the weaker vortex. During separation, strong primary separation vortices can induce secondary vortices located under the primary vortices, and even additional smaller separation vortices can be generated [72, 79, 80]. Simulations by Xiao et al. [73] (see Fig. 4.3b) showed separation and reattachment of the primary (i.e. vortex sheet) and secondary vortices. Previous studies have used various aspect ratios, AR, of the prolate spheroid. While the influence of AR has not been studied extensively, AR={4,6} have been popular choices in aerospace and naval applications. Wang et al. [77] compared the separation flow of prolate spheroid with various aspect ratio (AR = 2,3,4), where a larger AR (i.e. a more slender body) exhibited more open-type separation (i.e. flow separates and detaches from the body) asθ increased, leading to increased drag. 59 Figure 4.5: Schematic of various features involved in the evolution of stratified wakes, including the turbulent drag wake, pancake eddies in the late wake, internal waves, and wake vortices from the body at inclination or complex surfaces. 4.1.2 Spheroids in a stratified ambient Even though many slender bodies in naval applications are surrounded by density stratification, there are few studies on the stratification effect on slender bodies. The flow around the spheroid is vastly different from any bluff body, especially when vortical structures are shed from a body at high incidence, with complex surfaces (e.g. wingtip, fins), or in maneuvering conditions. As illustrated in Fig. 4.5, the wake generated in the stratified fluid has many features, such as wake vortices separated from the body, a drag wake evolving from turbulence to late wake patterns (i.e. pancake eddies), and internal waves where these features interact with each other and undergo complex evolution. Therefore, the turbulent drag wake seeded with these coherent structures in a stratified environment can introduce special patterns that may be detectable in the late wake. As explained in Section 1.5.1, Meunier & Spedding [34] claimed that all stratified wakes reach a universal state when scaled by the drag coefficient. Among the multiple body geometries, a 6:1 prolate spheroid at θ = 0, Re= 5000 and Fr={8,32} was included. Since their experiment is in the late wake x/D={10 2 − 10 4 } (Nt ={10− 10 3 }), the evolution towards similarity solutions could not be tracked. There are no studies on the wake of an inclined body in a background density gradient. 60 Ortiz-Tarin et al. [81] performed a body-inclusive LES (Large-Eddy Simulation) on the wake of the spheroid with a 4:1 aspect ratio of zero inclination. They defined the critical Fr as Fr c = 2AR/π, where AR represents the body aspect ratio L/D. This critical Fr determines the stratification level at which turbulence suppression and turbulence regeneration occur. For instance, when Fr= Fr c , the flow separation is maximally reduced by the lee wave and the wake is no longer turbulent at the moderate Re= 10 4 . Based on their calculation, the critical Fr of a 6:1 spheroid should be Fr c = 12/π≈ 4. Later, Ortiz-Tarin et al. [82] compared the stratified wake of a disk and a 6:1 spheroid at Re= 10 5 and Fr ={1,5,∞} from the hybrid of body-inclusive and body-exclusive simulation. They observed that the NEQ-Q2D transition of the spheroid wake did not follow Nt boundary from the universal scaling, where the transition to Q2D was early at Fr= 1 and the transition to NEQ was delayed to Fr= 10. This indicates that the spheroid does not follow the universal scaling. For the same condition as Ortiz-Tarin et al. [82], Nidhan et al. [83] shows the pressure field on the inclined 6:1 prolate spheroid in the stratified environment where the high Fr (Fr = 6,∞) cases showed lateral asymmetry, while the stronger stratification (Fr = 1,1.9) did not have any visible asymmetry. However, there are still no studies on the near-wake or far-wake of inclined 6:1 prolate spheroid in the stratified environment. 4.1.3 Scaling law When the time-averaged profile of the turbulent wake is considered as a Gaussian profile, the velocity and length scale of the wake can be characterized based on two simple measure of the peak defect velocity u 0 and wake size. u 0 is measured from the peak velocity from the time- averaged streamwise velocity field u at each downstream distance x. The wake height L V and wake width L H shown in Fig. 4.1 are measured from u field at each x. The sample of obtaining L V from a velocity profile is illustrated in Fig. 4.6. L V = z 2 − z 1 is calculated from two z points at u= u 0 /2 in the velocity profile (i.e. z| u=u 0 /2 ) where z 2 > z| u=u 0 is above and z 1 < z| u=u 0 is below the peak velocity u 0 in z. u 0 /U, is normalized by the free-stream or towed velocity U and L V is normalized 61 Figure 4.6: Sketch of a spheroid wake of zeroθ (top) and non-zeroθ (bottom) with peak defect velocity u 0 and wake height L V defined. Note that L V is measured in the direction of gravity and not perpendicular to the tangent line of the wake trajectory. Figure 4.7: (a) The centerline defect velocity u 0 decay rate and (b) the wake width L H growth rate along the downstream distance x/D of the spheroid wake from Ortiz-Tarin et al. [70] by the diameter D. Distance in all three x, y, z direction are also normalized by D. For simplicity, the normalized defect velocity will be referred to as u 0 and normalized wake height will be referred to as L V in rest of the chapter. To characterize the wake based on the length and velocity scale, many studies have been com- paring it to the self-similar theory. According to the self-similarity theory, when the turbulence has reached local invariance, the velocity and length scales of high Re axisymmetric turbulent wake behave as u∼ x − 2/3 and L∼ x 1/3 [5, 6, 40]. The local Re= ul∼ x − 2/3 · x 1/3 ∼ x − 1/3 decays in x. As the local Re decreases and the viscous term in the mean momentum equation is no longer neg- ligible for the low Re regime, the scaling becomes u∼ x − 1 and l∼ x 1/2 where the local Re∼ x − 1/2 decays at a faster rate. Imposing a Gaussian mean profile into the momentum equation, simplifies the governing equation to three independent variables u 0 , L H and L V from the self-similarity anal- ysis on stratified turbulent wake [84]. This solution can also recover u∼ x − 2/3 and L∼ x 1/3 from Tennekes & Lumley [6] in the 3D regime. 62 In the evolution of stratified wakes according to the universal scaling explained in Section 1.5.1, as the velocity decays and the local Fr (Fr∼ x − 1 ) approaches the order of 1 at Nt∼ 2, buoyancy effects become influential entering the NEQ regime. Based on the similarity analysis in Meunier et al. [84], the transition to NEQ occurs at x/D m = Fr m , where D m = D p c D /2 is the momentum thickness and c D is the body drag coefficient. Once the wake has entered the NEQ regime, the mean wake dynamics are determined by a balance between turbulent diffusion of momentum in the horizontal direction and viscous diffusion of momentum in the vertical direction. [84]. The u 0 decays at u 0 ∼ x − 1/4 , and the L V is at l∼ x 0 , where the vertical velocity component is suppressed. However, there are still wide variations among many experiments and simulations with no agreement. Even without stratification, universality is not seen in practice and is dependent on initial condition [70]. Recently, Ortiz-Tarin et al. [70] investigated the scaling of high-Re slender body wakes by extending the simulation of a 6:1 prolate spheroid at Re D = 10 5 to the far wake (up to x/D= 80). They found that the decay rate of u 0 and the wake size transition from a high- Re regime (u 0 ∼ x − 2/3 and l∼ x 1/3 ) to a newly proposed scaling (u 0 ∼ x − 6/5 and l∼ x 3/5 ) at x/D∼ 20 as shown in Fig. 4.7, where the transition occurs due to helical instability becoming more dominant. Recent stratified wake experiments also showed that the grid and sphere wake at low Fr and Re showed that the mean and fluctuation quantities do not collapse under any universal scaling [52, 59]. Stratified wake simulation also shows that the transition point of three regimes for the slender body is different from the bluff body where the NEQ starts at a later time Nt =π than Nt= 2 for the bluff body [70]. In the result section, Section 4.3.1.1, the u 0 and L V are used as a measure to apply the scaling laws and to link the near-wake measurement to the far wake and extrapolate to the higher Re in the geophysical scale. In addition, such scaling laws are not applicable for an inclined spheroid wake. A scaling of non-axisymmetric wakes is explored from inclined spheroid wake measurements in Section 4.3.2.5. 63 (a) (b) Figure 4.8: Dye visualization of a counter-rotating vortex pair with (a) Crow instability from Leweke & Williamson [85] and (b) elliptic instability from Leweke & Williamson [86] (a) (b) Figure 4.9: (a) Sketch of counter-rotating vortex pair where b is defined as the distance between the two vortex core. (b) Sketch of vortex profile to show how the core size 2 a is defined. Both sketchs are from Leweke et al. [87] 4.1.4 Evolution of trailing vortices Trailing counter-rotating vortices generated during separation is one of the key features of inclined spheroid wakes. Investigating the development of these vortices and the influence of stratification can give us an understanding of how the vortices can shape the evolution of the wake. The vortex structure can persist over a long time or rapidly dissipate depending on both the initial geometry and background conditions. 64 In a homogeneous fluid, parallel counter-rotating vortices can be unstable to three-dimensional perturbations, including a symmetric long-wavelength instability known as the Crow instabil- ity (see Fig. 4.8a), a symmetric and antisymmetric short-wavelength elliptical instability (see Fig. 4.8b), and oscillatory instability for large Re (strongly deformed with large cores) [88]. A counter-rotating vortex pair can be characterized by the distance b between the vortex cores, as de- fined in Fig. 4.9a, and the vortex core radius a, which is determined from the Gaussian distribution of the peak vortex ω x profile as illustrated in Fig. 4.9b. A stability analysis of a pair of parallel vortices predicts that the wavelength of Crow instability is atλ = 8.6b, which is derived from the wavenumber that has the maximum amplification from the strong interaction between long waves [89]. The elliptical instability is strongest when the two Kelvin modes (linear perturbation mode of an axisymmetric vortex, usually neutral or stable) have a similar radial structure [87]. At a high Re, multiple elliptical instabilities emerge leading to a transient turbulent flow. In stratified fluid, when vortex pairs are oriented horizontally with respect to the density gra- dient, the development of Crow and elliptical instabilities is enhanced due to stratification that suppresses the separation distance b of the counter-rotating pairs to be closer [90]. 4.1.5 Current state of experiment and simulation studies While many slender body studies focus on separation and drag force, there are only a few works that look into the wake. In addition, this particular geometry requires a longer downstream analysis because the streamlined body has longer wavelengths in the streamwise direction. As a result, the flow evolves slowly due to the elongated structures present in the wake. While the late wake of a 6:1 prolate spheroid atθ = 0 in Meunier & Spedding [34] studies far downstream (x/D={10 2 − 10 4 }) there is no information about the near-wake x/D < 10 2 . Previous studies of slender body near- wakes (e.g. [68, 69]) reached x/D∼ 20 and Ortiz-Tarin et al. [82] reached x/D∼ 80. In the present study, the experimental time window reaches up to x/D= 200. The equivalent buoyancy time scale is Nt > 8, allowing observation into the NEQ regime. 65 Figure 4.10: (a) The schematic of the experimental setup of the towed spheroid. Four cameras are in a linear-configuration for tomo-PIV and are placed in front of the tank for the vertical plane with lateral depth setup. (b) Dimensions of the 6:1 spheroid where a spheroid of a diameter D (i.e. minor-axis) and major-axis L is towed with four fishing wires at an inclination angle θ. A trip wire of a height k= 0.38mm<δ is placed at x= 0.2L from the nose of the spheroid. 4.1.6 Open questions and objectives This study investigates the effect of stratification on the inclined 6:1 prolate spheroid by exper- imentally varying Re, Fr, and inclination angle θ. This parameter space allows the following: the Re is high enough to be initially turbulent, and the Fr is low enough for stratification effects to become dominant within the measurable buoyancy time scale Nt, the inclination angle is high enough for the separation point to change and separation vortices to be formed. 4.2 Methods To investigate the evolution of spheroid wakes, an inclined spheroid experiment is conducted by towing a 6:1 prolate spheroid at various Re, Fr, and inclination angles. As illustrated in Fig. 4.10a, a 3D printed 6:1 spheroid is towed by four fishing wires to minimize the influence of the wake caused by suspension (details of the mount effect are in Section 4.2.2). Similarly to the sphere in Section 3.2.1, the shell is printed with polylactic acid (PLA) and filled with steel balls and epoxy for stability. The tank is filled with refractive index matched stratified flow with salt and ethanol [52]. 66 Figure 4.11: Parameter Space spanning in Re, Fr, θ. Testing parameters (blue bar) for spheroid wakes in the Re D − Fr− θ space. All cases marked with blue circles are repeated for θ = {0,10,20} ◦ . The dotted lines represent the medium with the same stratification N. 6:1 prolate spheroids with two diameters D={4,8}cm are used to satisfy the Re and Fr conditions within the achievable N. The wake is visualized by stereoscopic (2D3C) and tomographic (3D3C) Particle Image Ve- locimetry (PIV) to access 3D velocity fields. The flowfield was estimated using a LaVision PIV system with multiple cameras (LaVision-Imager sCMOS), each having a resolution of 2560× 2160 pixels. Two camera configurations were used for stereo-PIV where the maximum camera view- ing angle β = 45 ◦ . For tomographic PIV , four cameras are used in a linear formation where β = 60 ◦ , which is within the optimal range (30 ◦ <β < 90 ◦ ), to avoid elongation of the recon- structed 3D particles or introducing extra particles (i.e., ghost particles) [91, 92]. Each camera has a Scheimpflug, lens-tilt adapter, to align the plane of focus to the illuminated region. The measurement plane/volume was illuminated with a pulsed laser (Nd:YAG, LaVision NANO L100- 50PIV). To eliminate noise from the low-intensity light at the edge of the illuminated volume, a knife-edge filter is used to limit the diverging Gaussian-shaped laser volume. The field of view is x/D= 3.6, z/D= 4.5, and y/D= 0.85 depth for tomo-PIV . Then, the 3D velocity field is obtained by correlating reconstructed 3D images from the projection of multiple camera views. 67 The Re D -Fr-θ parameter space tested is shown in Fig. 4.11, where the homogenous fluid and strongest stratification allowed in the current setup are tested. The gray line is a constant buoy- ancy frequency that allows varying Re and Fr by changing the towing speed. Note that the strongest achievable stratification in this experimental setup is at N= 0.417, which sets the lower limit of ac- cessible parameters at{Re D -Fr}=[{5000-16},{10000-8},{20000-64}] as illustrated in Fig. 4.11. For simplicity, the notation RxFy is used to denote Re= x× 10 3 , Fr= y in rest of the chapter. 4.2.1 Wake analysis method To perform a time-averaging analysis, the reference frame of each snapshot (i.e. a velocity field at each discrete time) is shifted from the lab frame to a fixed body frame same as Section 3.2.1.1. A time-averaged velocity field u is calculated by the following: u(x,y,z)= 1 N N ∑ n=1 u(x,y,z,t n ) (4.1) where velocity field u(x,y,z) at each snapshot n were averaged over the total number of snapshots N. 6 runs are used in results from stereo-PIV , which consists of total N={100,400} and 1 run for tomo-PIV consists of N={20,70} depending on Re where R5 has the upper end of the range and R20 has the lower end of the range. The vortical structure can be visualized using the Q-criterion, which is an indication of strength of swirling motion. The Q is calculated from the dominant rotation rate tensorΩ relative to the strain rate tensor S shown in the following: Q= 1 2 (∥Ω ∥ 2 −∥ S∥ 2 ), Ω i j = 1 2 ( ∂u i ∂x j − ∂u j ∂x i ), S i j = 1 2 ( ∂u i ∂x j + ∂u j ∂x i ). (4.2) 68 (a) (b) Figure 4.12: (a) Comparison of support system impacts on the instantaneous lateral velocity v between the NACA 0012 airfoil strut (top) and the wire suspension (bottom). (b) Velocity profile across y and z at x/R=[3− 7] of Re D = 10 4 , Fr=∞ andθ = 0 ◦ . 4.2.2 Mount effect and trip While the cylinder strut at the trailing edge of the spheroid, known as the sting, has commonly been used as a mounting structure, recent simulations by [93] revealed that the sting interfered with the wake of the inclined spheroid, altering its trajectory and wake characteristics. To minimize the mount effect, they proposed a vertical support system with a NACA 0012 airfoil cross-section, which significantly reduced the impact of the support. However, when implementing the NACA 0012 airfoil support in this experimental study, coherent lateral velocity v associated with vortex shedding from the airfoil was observed, as illustrated in the top figure of Fig. 4.12a. The impact of vortex shedding is evident in Fig. 4.12b, where the time-averaged velocity profile is asymmetric with values above zero outside the wake in the positive vertical direction. The emergence of vortex shedding is due to the airfoil being in the laminar regime where Re based on chord c= 1/3L= 2D is Re c = 2Re D , which is below the lower critical Reynolds number (Re cr ∼ 10 5 ) for all three Re test cases. However, the suggestion from Jemison et al. [93] was based on the airfoil strut in the turbulent regime (Re c = 1.4× 10 6 ). However, even in the turbulent regime, there is evidence that practical application of a symmetric airfoil strut may not be the best choice as Jim´ enez et al. [68] 69 reported a persistent asymmetry caused by two symmetric airfoil struts in the wake of a DARPA SUBOFF with no inclination angle. In contrast, the bottom figure of Fig. 4.12a shows that a wire suspension does not have a significant magnitude in the velocity field generated from the mount. Therefore, the wire suspension is used as the mounting mechanism to tow the spheroid across the tank as it minimizes the low Re artifacts within the wake of interest. In this study, a trip wire is added to the 3D-printed spheroid model at x/L= 0.2 as shown in Fig. 4.10b to generate a consistent transition to a turbulent boundary layer. Past experiments (e.g. [72, 79, 80]) have tripped the boundary layer located at 20% downstream of the nose to ensure a fully developed boundary layer and not sensitive to the separation location. The trip has a cross- sectional shape of a square with a trip height k= 0.38mm, which is{13 - 25}% of the boundary layer thicknessδ. The trip wire in Fu et al. [72] had k=δ. 4.2.3 Window of opportunity Due to the finite size of the tank, the time window of the experiment is limited by the start-up effects and the wall reflections that enter the field of view and distort the wake. For example, start-up effects occur when the spheroid is towed from rest to acceleration; the wakes reflect from the frontal wall at the end of its travel; and the horizontal advection reflected from the lateral side wall. To study the evolution of stratified wakes, a long observation time is typically required. Therefore, it is important to estimate the maximum measurement time before the wake becomes contaminated. When the force is applied on the inclined spheroid with the four wire suspension the spheroid can swing like a pendulum. Therefore, during acceleration, drag and lift forces acting on the body change the inclination angle. As a result, the time window is also limited by the duration of when the spheroid is steady-state at a desired angle. For example, as shown in Fig. 4.13, R10 reaches a steady-state at x/D= 6 for all test angles. The time it takes for the wake generated at x/D= 6 to enter into the field of view can be estimated from ∆t = dx/u 0 where u 0 = 0.47x − 0.5 is based on the peak defect velocity of R10 immediately behind the wake obtained from the result later 70 Figure 4.13: Time series of inclination angle θ as a function of distance travelled x/D where Re= 5,10,20× 10 3 and θ = 0,10,20 are tested. The green shaded area is the field of view, and the gray shaded area is the uncertainty from measurement resolution. shown in Fig. 4.15a. This results in∆t = 60s, which corresponds to approximately Nt≈ 24. The maximum duration of the current experiment is Nt < 8, which ensures that measurements are not contaminated by the initial stage of the angle changing. R20 deviates from the desired angle within 25%. 4.3 Results 4.3.1 Inclination angleθ = 0 ◦ The wake structure of an axisymmetric body with no inclination is first examined to compare it to the previous existing literature and to understand the influence of body geometry in the range of Re and Fr. Fig. 4.14 compares the instantaneous streamwise velocity u of unstratified and stratified cases of Re= 5,10× 10 3 in the vertical centerplane xoz. Each sampled snapshot (t n ) is positioned along the x-axis using x n = Ut n , based on the towing velocity in a body reference frame. As the 71 Figure 4.14: Instantaneous streamwise velocity u/U of (a) R5∞ (b) R5F16 (c) R10F∞ (d) R10F32. The left edge of each snapshot t n is aligned with x n = Ut n based on the towing velocity in a body reference frame. The last snapshot is late-wakes at a constant Nt∼ 5. Each snapshot is normalized by the local maximum u/0.75|u max |. Note that x range is not continuous. 72 wake travels slower than the towing velocity, it leads to a discontinuous velocity field in x, causing some repeated wake structures to appear in adjacent snapshots. The contour level in each section is scaled by the local maximum (u/0.75|u max |) for visual purposes to account for the decayed local velocity magnitude further downstream. In Fig. 4.14b,d in the NEQ regimes (Nt >π) shows that the vortical structures are suppressed by stratification, resulting in a smaller wake height, as expected for axisymmetric wakes as stated in Spedding [8]. Though the influence of buoyancy effects is expected for Nt > 2, qualitative differences in stratified and unstratified wake can be seen before this point. In certain Re-Fr cases, the early buoyancy effect can decrease the wake height and increase the oscillating frequency compared to the∞ cases. At R5F16 (Fig. 4.14b), smaller wake height in the early wake compared to R5F∞ (Fig. 4.14a) indicates that buoyancy interacts with the streamlined body shape early on before the turbulent wake develops. Low Re wakes have less inertial force in the wake, which leads to less turbulence; additionally, low Fr wakes have a stronger buoyancy effect on the body during the separation. Therefore, both a decrease in Re and Fr can affect the initial stage of wake instability. Comparing the unstratified and stratified wakes at R10 (Fig. 4.14c-d), the oscillating frequency increases at F32 starting from (Nt = 0.5). The increase in oscillation can be influenced by the buoyancy force applied on the vertical perturbation. The periodic oscillation of vortices can be expressed as a Strouhal number St= f D/U, which is a dimensionless parameter that represents the oscillation time scale (1/ f ) relative to the inertial time scale (D/U). In the early wake of R10F∞, St= D/λ = 0.32± 0.03 where λ is the wavelength between vortices in each snapshots. However, R10F32 has higher frequency at St= 0.4± 0.03. Ortiz-Tarin et al. [70] calculated a spheroid wake at Fr= 0.5 with an energy spectrum peak at St= 0.4 at x/D= 6 in the horizontal azimuthal direction, which is same as St of R10F32 estimated in this study. On the other hand, with higher Fr in their study, the energy peaked at a higher St= 0.59 associated with the wavy oscillation observed in the horizontal plane. It could be a coincidence that St of R10F32 is the same as St of a strong Fr in the horizontal direction. However, this is only a qualitative assessment. 73 (a) (b) (c) (d) Figure 4.15: Peak streamwise velocity decay u 0 and wake height L V ofθ = 0. (a) u 0 over x/D (b) L V over x/D (c) u 0 Fr 2/3 over Nt (d) L V Fr − 2/3 over Nt. The legends are the following: unstratified (black line) and stratified (blue line) wakes. R5 (solid), R10 (dash). The Strouhal number can be characterized based on geometry known as the effective Strouhal number, St eff = f D eff /U, which is scaled by the effective diameter based on the effective drag coefficient, D eff . The effective diameter of the spheroid can be estimated to be D eff = 0.3 [34]. Then St eff of R5F16 is St eff = 0.09± 0.01, similar to the sphere wakes at St eff = 0.08 [34, 94], while higher St of R10F32 is St eff = 0.12± 0.01. Further investigation is necessary to identify how the stratification effect leads to a higher St for R10F32 and whether there is a correlation between St identified here with the St from previous literature. 4.3.1.1 Evolution of length and velocity scales ofθ = 0 ◦ Fig. 4.15 compares the evolution of vertical length scales and streamwise velocity scales over a range of Re and Fr. The mean peak defect velocity u 0 and the wake height L V are defined in Section 4.1.3 and illustrated in Fig. 4.6. Comparing the u 0 and L V of stratified wakes with the 74 known evolution of the velocity and length scales provides an understanding of when the buoyancy effect becomes influential. This is described in a range of Re and Fr to delineate the influence of these two parameters. In Fig. 4.15a-b, the velocity and wake height of the unstratified wakes (black line) at both Re show that there is a Re dependence. The velocity decay rate of u 0 of unstratified cases for both Reynolds numbers shown in Figure Fig. 4.15a decays similarly within the uncertainty, initially decaying quickly (∼ x − 2 ) and reaching x − 2/3 associated with the high Reynolds turbulence decay rate from Tennekes & Lumley [6]. However, the wake height L V /D of unstratified cases shown in Fig. 4.15b grows differently between R5 and R10. At R10F ∞, the wake initially grows similarly to the high Re axisymmetric wake growth rate at x 1/3 and increases to x 3/5 at x/D= 50. However, at R5, the wake height starts the same as R10, then grows faster at x/D= 10. The comparison between the stratified case (blue line) and the unstratified case (black line) at the same Reynolds number allows for the identification of when the buoyancy effect becomes more influential, as indicated by deviations in the evolution of velocity and length scale. When the stratified wake enters the NEQ regime, the velocity u 0 is expected to decay at x − 1/4 and the wake height L V stops growing, x 0 . In Fig. 4.15a-b, both velocity and length scale of R10F32 transition to the NEQ regime at Nt =π, which is similar to when NEQ regime starts in the previous literature. However, the difference between R5F∞ and R5F16 deviates early where u 0 deviates at Nt= 1 and L V deviates immediately as the wake leaves the body. L V of R5F16 decreases initially as observed in Fig. 4.14b. The early suppression of the wake instability hinders the mixing and the energy dissipation, contributing to a slower velocity decay and slower wake growth. u 0 and L V can be scaled by the buoyancy time scale Nt and Fr. Since Nt can be written as Nt=(x/R)/Fr, the turbulence scaling u∼ (x/D) − 2/3 and l∼ (x/D) 1/3 can be rewritten as, u· Fr 2/3 ∼ (Nt) − 2/3 ,l· Fr 1/3 ∼ (Nt) 1/3 . (4.3) In this buoyancy scaled velocity and length evolution, if the wake is independent of Re and Fr, all lines should overlap on top of each other. However, the two stratified cases do not coincide. 75 Figure 4.16: Instantaneous streamwise velocity of the spheroid at θ = 20 ◦ and (a) R5F∞ (b) R10F∞ (c) R20F∞ (d) R5F16 (e) R10F32 (f) R20F64. In u 0 Fr 2/3 over Nt shown in Fig. 4.15c, the transition into NEQ regime occurs at different Nt for different Re. R10F32 enters the NEQ regime at Nt=π consistent with the universal characteristic and R5F16 enters later at Nt ≈ 10. The influence of stratification in the early wake of R5F16 may have resulted in the delay of the NEQ transition for R5. While the velocity and length scale transitioning into the known scale of the NEQ regime (i.e. u 0 ∼ Nt − 1/4 and L V ∼ Nt 0 ) is a good indicator of when the wake is influenced by the buoyancy force, it does not necessarily determine that the stratification effect is not present before the transition. For example, the wake structure of R5F16 shown in Fig. 4.14b clearly shows an early buoyancy effect, while the velocity and length scale differs early from the unstratified wakes, they do not reflect the expected scaling of NEQ regime. Therefore, relying solely on statistical analysis of averaged quantities does not provide a complete understanding of the evolution of stratified wakes. 76 4.3.2 Inclination angleθ = 10 ◦ ,20 ◦ When the spheroid is at an inclination, the wake propagates upwards, which introduces interesting wake dynamics as the wake is traveling against the density gradient. Fig. 4.16a-c shows the un- stratified wakes of three Re cases (R5-20) where unstratified wakes continue to propagate upward. However, under stratified conditions (Fig. 4.16d-f), the wake is deflected. Among the unstratified wakes, as the Re increases, the number of small-scale structures increases as expected. Deflection of the wakes can be seen at all Fr even when Fr = 64. While Fr= 64 is considerably weakly stratified, the deflection indicates that Fr = 64 does not behave the same as unstratified wakes. Currently, there is no evidence of high Fr limit where the wake behaves like unstratified wakes as Spedding [8] observed the formation of late wake vortices even at Fr= 240. The occasional appearance of reverse flow (positive u in Fig. 4.16) is observed. The reverse flow appears intermittently throughout the wake due to turbulent fluctuation or the presence of vortical structures. However, in the case of stratified wakes, this reverse flow is primarily observed at the upper edge of the wake showing patches of coherent structures. 4.3.2.1 The instantaneous structure of inclined wakes Fig. 4.17 shows an example of emergence of vortical structures in the wake. The streamwise vorticity, ω x field of R5F ∞θ10 in Fig. 4.17a shows patches of negative streamwise vorticity with wavelengthλ∼ 1D between vertical protrusions of positive vorticity. The angled plane tilted in y atθ y =− 10 ◦ shows that the same-sign separation streamwise vorticity propagates roughly parallel to the major axis of the spheroid. In the iso-surface of a constant Q= 0.56 shown in Fig. 4.17b, A mushroom-like structure emerges on the upper side of the wake, and the lateral vorticity,ω y shows the front and the backside in x of the mushroom-like structure. The emergence of these structures may be associated with Crow instability or secondary vor- tices in elliptical instability [89, 95]. Characteristic length scales can be estimated to test the 77 Figure 4.17: (a) Instantaneous streamwise vorticity ω x of R5F∞θ10. The vertical plane xoz slice is at y/D=− 0.25 and the xoy plane is inclined at 10 ◦ starting at z/D=− 0.5. (b) iso-surface of Q-criterion at Q= 0.56 and colored lateral vorticityω y (c) yoz slice of average streamwise vorticity ω x at one of the center of mushroom-like structure (x/D= 5.1). 78 possible origins of these secondary structures. To relate ω x to the studies of the Crow and el- liptical instabilities of the counter-rotating trailing vortex pairs in Section 4.1.4, yoz slice of the time-averaged ω x at the peak height of the mushroom structure shown in Fig. 4.17c is measured. The lateral core size is estimated to be 2a= 0.23D based on outlining the vortex at the Gaussian distribution, ω x = 0.607ω x,max following how it was defined Fig. 4.9b. The distance between the core is b= 0.41D measured from the center of the two counter-rotating vortices to the core center. Then the normalized core size based on the core distance is a/b= 0.28. The wavelength of the emergence of mushroom-like structures isλ = 1.15D, which on the normalized scale isλ/a= 10 andλ/b= 2.82. In the previous literature, a Crow-instability has a long-wavelength ofλ/b= 8.6 theoretically [89] and elliptical instability has a broad range of short wavelengths. In a simulation of elliptic instability, the secondary vortices were initially at λ/b= 0.4 followed by λ/b= 1 at Re= 5200 and as Re increases, the wavelengths between two wavelengths become increasingly distinct [87]. The Re of vortices Re=Γ/ν within the mushroom structure is above the critical Re at a/b= 0.28 (Re c (a/b= 0.28)= 10 2 ), where the elliptic perturbation becomes unstable. It is difficult to directly relate the wavelength of these structures to the possible associated instabilities as these structures do not emerge from isolated separation vortex pairs; rather, they are mixed with the unsteady separation and turbulent drag wake, potentially affecting the observed wavelength. When introducing stratification to the vortex wakes, stratification can affect the evolution of these vortical structures by suppressing the propagation of these mushroom-like structures or facilitating merging events. In addition, the stratification can affect the change in the separation distance or the core size, which would enhance or weaken the development of Crow and elliptical instabilities [90, 96]. Describing the streamwise vorticity of the wakes in terms of the characteristic length scales of the instabilities from counter-rotating vortex pair provides a means to comprehend the emergence of vortical structures. 79 (a) (b) Figure 4.18: (a) Time-Averaged streamwise vorticity ω x of R5F∞θ10 where yoz slices at every x/D= 2 increments. (b) The trajectory of positive (red) and negative (blue) streamwise vorticity centroids of R10F∞θ20. The dotted line is from Jemison et al. [93] . 4.3.2.2 Streamwise vortices At high incidence, the spheroid wake has a strong streamwise vorticity generated from separation vortices, as shown in the time-averaged streamwise vorticityω x of R5F∞θ10 in Fig. 4.18a. As the vortices propagate, the positive and negative vortices start to become asymmetric in both magnitude (around x/D= 9) and relative height to each other (around x/D= 17). The negative vortex is vertically higher than the positive vortex. Asymmetric propagation is also observed for R10F∞θ20. In Fig. 4.18b, the centroid of the averaged vortex is traced in each yoz plane. The trajectories of the positive and negative vortices center around the trajectory of the separation vortex pair observed by Jemison et al. [93], then they start to diverge at x/D= 6, the same as observed in Andersson et al. [97]. The asymmetry was also observed in Ashok et al. [75], where pitching a DARPA SUBOFF model showed an initial asymmetry in strength (approximately 20%), which amplified downstream. In the yaw case, the weaker vortex was wrapped around and eventually annihilated by the stronger vortex. Separation vortices resemble the primary separation vortices in Fu et al. [72] and Xiao et al. [73]. Secondary separation is difficult to observe in the vertical plane as the separation vortices are developed mainly along the side of the body as shown in Fig. 4.3b. In the horizontal planar 80 Figure 4.19: The horizontal xoy plane of the streamwise velocity u (left), lateral velocity v (center), and vertical vorticityω z (right) of the spheroid wake atθ= 20 ◦ Re D = 5000 and Fr=∞. The dotted blue line is the cross-section of the inclined spheroid intersecting the illuminated horizontal plane. view (xoy plane at z 0 = L/2sin(θ)) of the wake at R5F∞θ20, coherent structures are visible near the edge of the primary separation vortices, as indicated by the arrow in the ω z field shown in Fig. 4.19. These structures may be associated with vortices arising from secondary separation as observed in Xiao et al. [73]. 4.3.2.3 Wake propagation To compare wake propagation with an unstratified and stratified background of various Re, the wake center and wake edge are outlined in the time-averaged u field in Fig. 4.20. The wake centerline is defined as the location of the peak velocity u max , and the wake edge is defined by the z location where u= 1/2u max (x). All stratified wakes exhibit a higher magnitude of streamwise velocity downstream, indicating that the energy is more concentrated in the center of the wake further downstream than in the unstratified case with the same Re. Initially, the trajectories of both unstratified and stratified wakes ascend vertically. However, the trajectory of stratified wakes deviates from that of unstratified wakes, resulting in deflection and vertical oscillations around a distinct equilibrium height. Even in a weakly stratified wake (Fr= 64), the wake oscillates at a small amplitude. This shows that Fr= 64 does not behave the same as a wake in a homogeneous background fluid. The wake trajectories from the wake centerline of all stratified cases are shown in Fig. 4.21. The wake trajectory oscillates with the same wavelength in the buoyancy time scale Nt. Although as Re and Fr increases, the peak amplitude shifts in phase slightly to a higher Nt, but generally, the 81 Figure 4.20: Time-averaged streamwise velocity field, u in vertical centerplane of (a) R10F∞ (b) R10F32. Blue dash line is u max and the purple line is the wake edge defined based on 0 .5u max Figure 4.21: Trajectory of wake centerline, u max scaled in Nt 82 Figure 4.22: Averaged vertical velocity w/U in Nt of R5F16 (top), R10F32 (middle), and R20F64 (bottom). wake starts to deflect vertically around Nt = 2, which is when the buoyancy effect is expected to become influential. The wake trajectory in z varies with both Re and Fr. Since the vertical excursion shown in Fig. 4.16a-c is not sensitive to Re, the oscillation equilibrium height is likely to be a Fr effect where the wake oscillates at a higher equilibrium height as Fr increases. The oscillation amplitude at each height is independent of Re and Fr where the distance from the peak amplitude to the equilibrium height is approximately∆z∼ 0.5D. If we define the initial peak amplitude as the point at which the wake reaches its maximum potential energy, it becomes evident that the buoyancy effect resulting from the wake’s vertical propagation would be more pronounced compared to axisymmetric wakes. 83 (a) (b) (c) (d) Figure 4.23: (a) u 0 /U, centerline velocity decay and (b) L V /D, wake height from time-averaged field of axi-symmetric body, θ = 20. (c) Scaled centerline velocity decay based on Fr(u 0 /U)Fr 2/3 in Nt and (d) wake height based on Fr(L V /D)Fr − 2/3 from the time-averaged field of axi-symmetric body, θ = 20. The legends are the following: unstratified (black line) and stratified (blue line) wakes. R5 (solid), R10 (dash), R20(dotted). 4.3.2.4 Internal wave As the wake initially propagate upwards, internal waves are excited. In Fig. 4.22, the time-averaged vertical velocity shows a coherent vertical oscillatory motion associated with the internal wave. All stratified wakes generate internal waves even at high Fr. Fr = 64 shows a clear internal wave pat- tern, similar to the lower Fr cases when scaled in Nt, though with a smaller magnitude. The wave- length of the internal wave is at Nt λ /D= 2π, consistent with the natural frequency of buoyancy N. 84 4.3.2.5 Evolution of length and velocity scales of non-zeroθ The evolution of velocity and length scale of the inclined spheroid wake over a range of Re and Fr is compared in Fig. 4.23. u 0 and L V are defined in Fig. 4.6. Note that L V is defined as the vertical height of the wake with respect to gravity and not perpendicular to the wake trajectory. The unstratified wakes have a shorter measurement time window as the wake leaves the field of view as shown in Fig. 4.20, so u 0 and L V of the unstratified wakes are displayed up to x/D∼ 40. The wake height ofθ = 20 ◦ is generally an order of magnitude higher (L V ∼ O(1)) than theθ = 0 wakes (L V ∼ O(10 − 1 )). This is expected as high incidences have vortices generated from the earlier separation location. The u 0 of the unstratified wake in Fig. 4.23a decays similarly to axisymmetric turbulent wakes where u 0 initially at x − 2/3 at x/D< 1 and decays faster at x − 1 at x/D> 1. While u 0 of R5 and R10 have similar magnitude, u 0 of R20 is half the magnitude of the other two lower Re wakes. While self-similarity is expected in turbulent wakes, the unstratified wakes velocity decay rate shows a Re dependence. Even in wake height L V shown in Fig. 4.23b, the R20 wake behaves differently from R5 and R10. While the initial wake height was the same, R5 and R10 grows similarly to each other, R20 grows rapidly at x 3/5 from x/D= 6. The initial energy spectrum has a strong influence on the evolution of the fluctuation statistics [98], thus the increased initial turbulent production of R20 can influence the velocity and length scale quantities. This shows that among unstratified wakes, L V growth rate and initial u 0 magnitude vary depending on Re. In Fig. 4.23a, the u 0 of the stratified wakes (blue lines) deviates from u 0 of the unstratified wakes (black line) at the same downstream distance x/D≈ 20 for all wakes regardless of Fr. For example, x/D= 20 in buoyancy time scale for R5F16 is at Nt = 2, while the same distance for R20F64 is equivalent to an earlier buoyancy time at Nt = 0.6. The early deviation of R20F64 is also reflected in L V as shown in Fig. 4.23b where R20F64 grows in x 1/4 initially from the same wake height. Note that the wake height immediately behind the trailing edge of the body varies before converging to the same height (x/D< 6) where the variation occurs before the merging of the separation vortices and the drag wake. 85 In the buoyancy scale u 0 Fr 2/3 and L V Fr − 1/3 shown in Fig. 4.23c-d, all stratified wakes reach the same value at Nt > 10. Stratified wakes are expected to approach u 0 ∼ Nt − 1/4 and L V ∼ Nt 0 when the NEQ regime starts around Nt = 2 [84]. However, none of the inclined spheroid wakes follow this scaling and there is inconsistency in wakes of various Re and Fr overlapping in u 0 Fr 2/3 and L V Fr − 1/3 over the Nt scale. For example, as shown in Fig. 4.23c, the wake at R20F64 has a slower velocity decay rate u 0 Fr 2/3 ∼ Nt − 1/4 starting at an early buoyancy time at Nt = 0.6 compared to the other two stratified cases (R5F16 and R10F32) that has a velocity decay rate of Nt − 2/3 , which is the same rate as the high Re axisymmetric wake velocity decay rate with no transition to a slower decay rate. If the slower velocity decay rate indicates when the wake is influenced by stratification, the L V growth rate should stop around the same Nt based on the previous scaling of the NEQ regime. However, L V continues to grow at Nt 1/4 until Nt = 10 when the wake height stops growing (Nt 0 ) as shown in Fig. 4.23d. While wakes of a high Fr (Fr= 64) having a slower velocity decay rate at an early buoyancy time indicate that there is an early stratification effect, there is no indication that having a lower Fr directly leads to an early transition to a slower decay rate. Instead, certain Re and Fr conditions overlapped, which were not consistent between the velocity and the length scale. For instance, the velocity decay rates of R5F16 and R10F32 overlaps. On the other hand, the wake height of R10F32 grows with R20F64, while the wake height of R5F16 does not show wake growth. Instead, the wake height fluctuates at the wavelength of the internal waves at Nt λ /D= 2π. The wake height of R10F32 also has a minor fluctuation with Nt 1/4 growth rate. Among the test cases, Fr= 16 is the strongest Fr, thus the flow may not have enough kinetic energy to overcome a high inclination angle against stratification, altering the boundary layer and the separation around the body, leading to an initially high wake height when scaled with Fr. The wake trajectory of R5F16 shown in Fig. 4.21 also oscillates out of phase with the internal wave at the same wavelength. Therefore, as the wake oscillates along with the internal wave, it influences the wake height to also fluctuate in phase with the internal wave. 86 (a) (b) Figure 4.24: (a) Averaged lateral velocity v/U in Nt where R5F16 (top), R10F32 (middle), R20F64 (bottom) and (b) time-averaged lateral velocity of R5F∞θ10 where yoz slices at every x/D= 2 increments. When the body is at high incidence, the velocity and length scale of the wake are not consistent with the model based on the axisymmetric wakes that have been used so far to understand when and how the wakes are influenced by stratification. The presence of streamwise vortices makes the self-similarity, two-parameter (velocity and length scale) model insufficient to understand how the stratification affects the inclined wakes. However, all stratified wakes reached the same velocity and wake height at Nt = 10, implying that inclined wakes may have some universal state beyond Nt > 10. 4.3.2.6 Lateral asymmetry When the inclination angle is high, the separated flow can cause an asymmetric evolution of the wake in the lateral direction. Ashok et al. [75] observed vortex shedding in the leeward side of the DARPA SUBOFF atθ = 20 ◦ and Re L = 4× 10 4 . They showed the weaker vortex in the separation vortices diffuses quickly, and the stronger vortex extends downstream. At a high inclination angle, 87 Figure 4.25: A sketch of counter-rotating separation vortex pair in yoz plane where vortex rotating in − x direction is larger than the vortex rotating in +x. Grey shade is the cross-section of the spheroid and the dashed line is the y centerline of the vertical centerplane xoz. the separation is delayed and results in a lateral force, where the separation wake becomes unsteady and pulsating. Lateral asymmetry is observed in Fig. 4.24a, which shows the time-averaged lateral velocity v/U of three Re stratified wakes. If the streamwise vorticity from separation vortices and induced by the vertical propagation of the wake is laterally symmetric, v/U in the vertical centerplane should be zero. However, Fig. 4.24a shows that positive (into the page) lateral velocity v/U exists on the lower side of the wake and negative (out of the page) v/U on the upper side of the wake. The signs of lateral velocity (negative v(y= 0,+z) and positive v(y= 0,+z)) are consistent with a single vortex in − x direction. In a counter-rotating vortex pair, the difference in circulation magnitude of each vortex pair can cause the vortex pair to rotate around each other [87]. Fig. 4.25 illustrates an example of how a counter-rotating vortex pair of unequal circulation strength evolves downstream. The lateral asymmetry observed in inclined spheroid wakes can be due to the wake itself propagating asymmetrically in the lateral direction or a combination of both. 88 To have a better understanding of the lateral asymmetry in 3D space, v/U of yoz slices of an unstratified wake (e.g. R5F ∞θ10) is shown in Fig. 4.24b. Intially, as the separation vortices are generated (x/D < 3), the magnitude|v/U| is more dominant in the positive y direction. When the wake leaves the body (x/D > 3), the|v/U| linked to positive ω x (positive v(y= 0,+z) and negative v(y= 0,+z)) in the positive y direction becomes stronger, resulting in an asymmetric |v/U| distribution about the wake center in both y and z directions. This observed asymmetric behavior aligns with the trajectory of streamwise vorticity illustrated in Fig. 4.18b. Therefore, the asymmetric distribution of v/U observed in stratified wakes in Fig. 4.24a may stem from one of the vortex pairs traveling close to the centerplane as depicted in Fig. 4.25. The signs of lateral v/U persist even further downstream, with the R5F16 wake exhibiting a more stable and pronounced v/U. As previously shown in Fig. 4.23, inclined spheroid wakes begin to display the effect of stratification as early as Nt = 0.6 for R20F64. Given that the vertical motion of the streamwise separating vortices opposes the buoyancy force, it is expected that the stratification effect on these vortices may be strong at the local scale. Therefore, a persistent sign of v/U even at high Fr may be influenced by the stratification. 4.4 Discussion and conclusion The study explored the wake characteristics of inclined 6:1 prolate spheroids in both uniform and stratified backgrounds. Currently, there is little information on inclined spheroid wakes in an unstratified environment, and nothing on the wakes in a stratified environment. Therefore, this is the first study to document the flow up to x/D < 200 downstream for stratified and unstratified wakes of the inclined spheroid. In an effort to know whether such a complex flow can generate patterns that persist in the late wake, the wake characteristics are explored in the Re-Fr, inclination angleθ space. Based on the previous literature on stratified wakes, it is expected that all wakes reach a univer- sal state [34]. However, the experimental result showed that the wake generated from a spheroid 89 at a zero and non-zero inclination of various Re and Fr did not overlap in the buoyancy scale, indicating that the evolution of stratified wakes of an inclined spheroid does not reach a similarity solution at the same Nt. Instead, the wake structure from a zero inclination angle showed that the stratification had an influence on the emergence of wake instabilities in the near-wake even in the weakly stratified wake. For example, a smaller wake height and a higher frequency of periodic oscillation in stratified wakes were observed compared to unstratified wakes. The NEQ transition depended on Re, which is consistent with previous findings in which the transition length of the NEQ regimes increased with increasing Re [17, 84]. The stratification effect on inclined spheroid wakes did not have universal characteristics, having an earlier NEQ transition than the axisymmet- ric stratified wakes, while having a minimal change in wake height downstream. The wakes of inclined spheroids are indeed measurably different from those of axisymmetric bodies, with strong influence from the background stratification. When the spheroid is inclined, the propagating wake generates large-scale internal wave un- dulations even in weakly stratified wakes (Fr = 64). Due to buoyancy, the stratified wakes are deflected downstream, oscillating vertically at an equilibrium height. The vortical mushroom-like structures that emerge in some cases were represented in characteristic length scales to test their possible association with Crow and elliptic instabilities. This provides a way to link these struc- tures to the behavior of instability and its driving mechanism. Lateral asymmetry was observed in all wakes. The wakes from the experiment are observed up to Nt = 8 in the NEQ regime. The observed stratification effect within the near-wake of an inclined spheroid indicates distinct evolutionary behavior compared to that of an axisymmetric spheroid wake. Expanding the study to the quasi- 2D regime could reveal unique patterns in the late wake, associated with special initial conditions of coherent structures seeded in the turbulent wakes. The study can further extend to find the mapping that can extrapolate lab-achievable flow conditions to high Re and low Fr where strongly stratified turbulence lies. 90 While this study demonstrated the importance of accessing 3D velocity field information to understand the 3D evolution of the complex wakes, there remains a necessity to further inves- tigate methods to evaluate and improve the accuracy of the volumetric visualization technique. Especially, many variables of interest rely on gradient quantities, which are susceptible to spatial resolution and noise. As a result, a more sophisticated denoising procedure is necessary. To care- fully improve the accuracy, a robust physics-based approach (e.g. enforcing continuity) should be explored. 91 Chapter 5 Conclusions and outlook 5.1 Summary In the presence of stratification, bluff body wakes assume qualitatively different patterns, depend- ing on the relative magnitudes of inertial, buoyancy, and viscous terms in governing equations. Pat- terns can be automatically classified, so conditions at the wake generator can be inferred. Regime identification of stratified wakes has been studied in an effort to provide tools for pattern detection. For regime identification, sphere wakes at low {Re-Fr} are good test objects, as a wide variety of different regimes are populated in the low{Re-Fr} space, and a thorough database from laboratory experiments and DNS is available. Characteristics of near-wakes are first categorized into five flow regimes. Then, dominant DMD modes representing important flow features associated with the regimes within the identified regimes are demonstrated. Based on simple binary criteria derived from physics-based features of DMD modes, a custom-designed automatic classifier successfully placed wake information into the corresponding regime given the 3D flow field from DNS. In a realistic application, it is unlikely that 3D fully time-resolved wake data will be available. This study demonstrates how the success rate is impacted when incomplete or noise-contaminated wake information is given. Vertical and horizontal centerplanes were extracted from 3D DNS to match the experimentally accessible 2D flow fields to test the classifier performance with spatially 92 limited wake information. While the success rates from the reduced DNS information were still high, the same reduced field from the experiment had lower success rates, possibly due to relatively high noise. In the presence of noise, which is more likely the case for field data, TDMD on the experimental data improved the robustness of this classifier. Streaming DMD was also suggested as a possible solution when the given initial data does not have sufficient snapshots for DMD to be resolved. While standard DMD was used in this DMD-based classifier, exploring these alternative approaches and understanding the limitations and addressing the limitations of the DMD-based classifier provides insight into building a possible pattern identification. To understand where the presented low{Re-Fr} cases are in terms of buoyancy and viscous terms, two parameters were measured: Fr h , a local Froude number based on the horizontal length scale of an overturning motion, andG , an activity parameter similar to a buoyancy Reynolds num- ber. Most of the low {Re-Fr} cases of sphere wakes lie in low {Fr h -G} space, where viscous effects are dominant. While flows typically found in the ocean and atmosphere have strongly strat- ified turbulence (SST), high G where the SST regime lies is hard to access experimentally unless a large object is towed fast in a large tank. However, representing stratified wakes in the {Fr h -G} space, these two parameters Fr h andG can link experimentally accessible scales to geophysical scales. Although stratified sphere wakes were a well-studied convenient database for testing tools for pattern detection, it is uncertain how the pattern detection tools would perform on flow conditions observed in realistic naval applications. Real submersibles often have other non-generic features (wings, propellers, misaligned, or nonlinear trajectories) that can lead to departures from general learned laws of wake evolution. One example is a streamlined body at high angle of attack when the hydrodynamic disturbance is a mix of coherent structures originating from the body and a turbulent drag wake. The inclined 6:1 prolate spheroid used in this study is a good geometry to study, as the wake produces a drag wake together with coherent streamwise vortices that interact with the drag wake. Currently, there is little information on inclined spheroid wakes in an unstratified environment, and nothing on the wakes in a stratified environment. Therefore, this is the first 93 study to document the flow up to x/D< 200 downstream for stratified and unstratified wakes of an inclined spheroid. In an effort to know whether such a complex flow can generate patterns that persist in the late-wake, wake characteristics are explored in the parameter space of Re, Fr, and inclination angleθ. All stratified wakes are expected to reach a universal state in the late stage, regardless of the initial condition of the wake. Universal characteristics were previously observed in the late wake of spheroids with zero inclination, thus a similar scaling is expected in the near-wake. However, this study demonstrated that stratification alters the vortical structures (e.g., wake height and os- cillating frequency) even in the near wake of spheroid wakes with zero inclination, deviating from universality. In addition, the near wakes exhibited variations in velocity and length scales within the buoyancy scale, dependent on Re and Fr. At high incidence, the wakes behaved distinctly differently from axisymmetric wakes. Stream- wise vortices in inclined spheroid wakes introduced various wake structures, including vortical mushroom-like formations with potential associations to Crow and elliptic instabilities, as well as lateral wake asymmetry. In the presence of stratification, vertically propagating wakes counter- acted buoyancy and induced vertical oscillations that were in phase with large internal waves, even under weakly stratified conditions (Fr = 64). The wakes of inclined spheroids exhibited an early influence from background stratification, and they were indeed measurably different from those of axisymmetric bodies. Despite the early stratification effect, the velocity and length scales of inclined wakes showed no agreement among various Re and Fr conditions within the buoyancy scale until they converged at the same value at Nt = 10. This suggests that the self-similarity solution is inapplicable to wakes generated by spheroids and provides insufficient information about the evolution of inclined spheroid wakes. The present study has contributed to providing information on the unique features of an inclined spheroid wakes in a stratified ambient. It has demonstrated that the characteristics velocity and length of stratified wakes depend not only on the combination of Re and Fr, but also on the body 94 geometry and its incidence angle. These findings bring us closer to discovering whether there are special patterns unique to inclined spheroid wakes in stratified ambient that may be detectable in the late wake. Therefore, these findings can ultimately serve as a basis for testing and improving the DMD-based classifier to search for detectable patterns. 5.2 Outlook In this study, the stratified cases presented were taken at a maximum stratification strength that the experiment could achieve. This is at a constant buoyancy frequency N and along this value, Re and Fr both are varied. Variations in wake dynamics with respect to both Re and Fr along a constant N prompt further investigation to disentangle the influence of Re and Fr independently. Independent variation of Re, Fr is required. This study is a good starting point for understanding the underlying mechanism of when the diverse wake structures emerge and how they behave under stratification. However, more research is needed to determine the cause of each effect. For example, mushroom- like structures do not show a direct correlation with known instabilities, so it is necessary to identify the conditions at which the various instabilities emerge. To isolate the effect of Re, Fr and θ on wake dynamics, such as vortical structures, internal waves, and lateral asymmetric in the wake, it would be advantageous to first map out the parameter space in {Re-Fr-θ}, then categorize the spheroid wakes into simpler regimes and determine the effect of Re, Fr, θ, or a combination of these. The presence of large-scale structures, such as vortex shedding and wake instabilities, and wake-induced internal waves, can cause unsteady motion in stratified wakes. Investigating the evolution of unsteady motion in stratified wakes could be of interest. The stratified wake results in this study were generated from a body traveling in a linear stable stratification, a simplified representation of a pycnocline. However, oceans usually have multiple layers, including a mixed layer above and a deep layer below the pycnocline. Examining wakes propagating through a non- linear density profile would provide insights into understanding how the wake evolves differently 95 from the wakes in a linear density gradient. This exploration can broaden our understanding of stratified wakes in real-world settings. 96 References [1] X. Chen et al. “Eddy-induced pycnocline depth displacement over the global ocean”. In: Journal of Marine Systems 221 (2021), p. 103577. [2] J. Stevens and J. Allen. Two Views of Von K´ arm´ an Vortices. 2017. [3] B. Gratwicke. Tristan da Cunha - a perfect volcanic cone. 2012. [4] S. P. Xie et al. “Far-reaching effects of the Hawaiian islands on the Pacific ocean-atmosphere system”. In: Science 292 (2001), pp. 2057–2060. [5] A. A. Townsend. The structure of turbulent shear flow . 2nd ed. Cambridge University Press, Cambridge, UK, 1976. [6] H. Tennekes and J. L. Lumley. 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Abstract (if available)
Abstract
The upper ocean and the lower atmosphere of Earth have a stable density gradient with height known as stable stratification. When a stably-stratified flow passes over an obstacle (e.g., mountain) or when a body passes through the stratified ambient (e.g. aircraft, submarine), there are two dimensionless numbers that describe the balance between inertial, buoyancy, and viscous terms in the governing equations known as the Reynolds number, Re, and Froude number, Fr. In application, Re of the initial wake is always high, while Fr ranges in the order of 1-100. However, as the wake decays, local Re and Fr drops falling into the range accessible through experiments and simulations. There is evidence suggesting that stratified wakes have patterns (i.e. coherent structures), which may contain information about the wake generator. Statistical analysis (e.g. mean and fluctuating quantities) of these wakes in previous studies indicated a similarity in the late wake, implying that no memory of the initial condition is found in the late wake. However, if the measurement is of wake structures, perhaps specific pattern information could be revealed.
To develop tools to identify patterns, Dynamic Mode Decomposition (DMD) is used as a basis for wake classification. From energetic DMD modes, a custom-designed algorithm automatically classifies wakes into known flow regimes. The success of this approach depends on the quality and dimension of the input data. While the performance of the wake classifier was high in the low {Re-Fr} domain of sphere wakes, how it works at higher {Re-Fr} and other geometry is not yet known.
Wakes of naval interest often include those with special initial conditions, such as strong trailing vortex structures shed into the wake by wings, fins, or inclined streamlined bodies such as spheroids. When the body is at incidence, the wake is a combination of drag wake and streamwise vortices separated from the body, and this wake geometry can evolve in ways that are measurably different from the zero incidence case in background stratification. In this geometry, the stratification effect exists in the early wake, as the inclined wake itself generates large-scale internal wave undulations even when the background stratification is not strong. In addition, the pattern and strengths of the primary streamwise vortices are not symmetric. These findings serve as an initial framework for pattern identification in high Re flows mixed with special conditions.
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Pattern generation in stratified wakes
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