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Passive flight in density-stratified fluid environments
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Passive flight in density-stratified fluid environments
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PASSIVE FLIGHT IN DENSITY-STRATIFIED FLUID ENVIRONMENTS by Try Lam A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Mechanical Engineering) December 2018 Copyright 2018 Try Lam Dedication To my family ii Acknowledgements I would like to rst thank my advisor, Prof. Eva Kanso, for her all her support and guidance. This work and the completion of my degree would not be possible without her mentorship. I would like to thank my committee members: Professors Georey Spedding, Paul Newton, Mitul Luhar, Patrick Lynett, and Juhi Jang. I am grateful for their guidance, support, and questions, which help to guide the research and the nalization of this manuscript. I would also like to thank people who have motivated and guided me during my study: Aaron Schutte, Firdaus Udwadia, Jon Sims, Nathan Strange, Jordi Casoliva, Martin Lo, and Rodney Anderson. A special acknowledgment to Lionel Vincent, who probably deserves another doctorate for all his help and guidance he has provided. I would like to thank the Jet Propulsion Laboratory, California Institute of Technology, for their nancial support and allowing me to pursue my doctoral work. Finally, I would like to thank my parents, my very understanding and loving wife, and my kids. Their support made this possible. iii Table of Contents Dedication ii Acknowledgements iii List Of Tables vi List Of Figures vii Abstract xiii Chapter 1: Introduction 1 1.1 Density-stratied uid environments . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Passive ight in homogeneous environments . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Terminology and basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter 2: Descent motion in a density-stratied uid 15 2.1 Laboratory setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 3D reconstruction of disc trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Direct shadowgraph for ow visualization . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Experimental observations of disc motion . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 3: Physical mechanisms governing disc motion 32 3.1 Apparent drag enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Buoyancy-driven restoring torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Eect of stratication on horizontal motion . . . . . . . . . . . . . . . . . . . . . . 38 Chapter 4: Quasi-steady model 42 Chapter 5: Radial dispersion of discs' landing sites 47 Chapter 6: Rotational motion in density-stratied uids 50 6.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.2 Flow eld visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.3 Plate rotational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.4 Physical mechanisms governing plate rotation . . . . . . . . . . . . . . . . . . . . . 63 6.5 Quasi-steady model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter 7: Conclusion and discussion 71 Reference List 74 iv Appendix A Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 A.1 Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 A.2 Buoyancy (Brunt-V ais al a) Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A.3 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 A.4 Baroclinic Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 v List Of Tables 2.1 Key dimensional and non-dimensional parameters explored in this thesis. An av- erage uid density value is used to compute quantities for the stratied cases. U is the terminal speed where the weight is balanced by buoyancy and drag forces. I is the dimensionaless moment of inerita, Re is the Reynolds number, G is the Galileo number, Ar is the Archimedes number, Fr is the internal Froude number, Ri is the Richardson number, N is the buoyancy frequency as described inx1.2. . . . . . . . 20 5.1 Distribution parameters of the landing distribution for 500 free-falling uttering discs. 1 and 2 are the standard deviations along the major and minor axes from gure 5.1(a,b). r m and r var are the mean and variance of the radial distribution normalize by the descent heighth, andc:d:f: is the cumulative distribution function as shown in gure 5.1(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A.1 Internal wave phase v p and group v g velocity directions for all four quadrant as a function of k x and k z for !> 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 vi List Of Figures 1.1 Temperature prole of the Earth's atmosphere showing the dierent atmospheric regions for the mean zonal temperature proles for July (solid line) and January (dashed line) from Committee on Space Research (COSPAR) International Refer- ence Atmosphere 1986 (CIRA-86) at latitude of 50 N. Adopted from Seramovich [77]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Temperature T, (in-situ) density, and potential density as a function of depth for the ocean at 180 W and 0 N, where 1000 kg/m 3 needs to be added to and to convert the respective to. Potential density referenced to the sea surface. Most of the in-situ density increase with depth is due to pressure eects, which are removed in the calculation of potential densities, showing how weakly stratied the ocean is at larger depth. Data from NODC (Levitus) World Ocean Atlas 1994 [77]. 3 1.3 Schlieren visualization of internal gravity waves for (a) !=N = 0:9 and for (b) initial disturbance transient waves. Adopted from [60]. . . . . . . . . . . . . . . . . 4 1.4 Sample trajectories from each falling regime reconstructed by Heisinger et al. [28]. (A) steady, (B) uttering, (C) chaotic, and (D) tumbling descents. . . . . . . . . . 6 1.5 Phase diagram (Re,I) that clearly showed the transition (Re 100) from steady to unsteady ( uttering and tumbling) descent motions. Adopted from [102] . . . . 7 1.6 Dye visualization of wake and vortex structure for uttering descent in homoge- neous uid for Re 2200. Vortices consist of primary vortices, secondary vortices, and counter rotating vortex pair (CRVP). Adopted from [107]. . . . . . . . . . . . 9 1.7 Dye visualization of wake and vortex structure for spiraling uttering descent in homogeneous uid for Re < 2000. Adopted from [43]. . . . . . . . . . . . . . . . . 10 1.8 Parameter space (Re,I) showing the transition location from planar uttering regime and spiraling (helical) uttering. Adopted from [43]. . . . . . . . . . . . . . 11 1.9 (top row) Distribution of landing sites for (A) steady, (B) uttering, (C) chaotic and (D) tumbling motions. (bottom row) histograms of the radial distribution of the landing positions. Adopted from Heisinger et al. [28]. . . . . . . . . . . . . . . 11 vii 2.1 (a) Disc of diameter d, thickness e, and density disc at an inclination angle dened as the angle between the vertical direction z-direction and the normal to the disc. (b) and (c) Side and top views of the experimental setup used to record the disc's landing location and 3D trajectory, respectively; the mirror in (c) captures an orthogonal view necessary for 3D reconstruction. (d) Electromagnet release mechanism prior to release with the disc and after the disc is released. (e) Two-tank experimental free-drained setup used to generate a stable linear density prole in the tank. (f ) Schematic of the tank setup with a sample reconstructed trajectory. The coordinate system is centered at the initial release location. The landing location for multiple consecutive drops are shown in red and used to compute the landing distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Discs freely falling in homogeneous uids belong to one of four descent regimes: steady, uttering, chaotic and tumbling. These regimes are mapped onto the pa- rameter space (Re,I) based on the results of [17]. The parameter values explored in this thesis are highlighted: constant density uid () consisting of pure water = w = 1, saltwater density values = w = 1:048 and = w = 1:102, and stratied saltwater uid () at two levels of stratication Fr = 2:34 and Fr = 1:26. For stratied uids, we computed I and Re using the average density values. . . . . . 17 2.3 Example linear density prole ( =d=dz) generated using the two-tank method for water and saltwater for the setup shown in gure 2.1(d), where the 1- uncertainty estimate for the buoyancy frequency 0:02 to 0:05 rad/s. . . . . . . . . . . . . . . 19 2.4 Image processing steps for obtaining the state and orientation of the disc along with the Matlab pseudo-code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Example reconstructed trajectory and orientation of a descending disc released from (x;y;z) = (0; 0; 0) and scaled by the diameter of the disc d. Disc's face color changes as a function of depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Direct shadowgraph technique with the camera out of alignment with the light source. The camera is set at an angle to record the image cast on the screen, otherwise a double image (actual and shadow) would appear. Digital processing is done to restore the image's perspective. . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7 Shadowgrams at various time of a free-falling uttering disc in stratied uid (Fr = 1.31; N = 1.63 rad/s). Note the lack of the secondary vortices at increasing depth compared to gure 1.6 in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 Shadowgrams at various time of a free-falling uttering disc in stratied uid (Fr = 1.31; N = 1.63 rad/s). Note the lack of the secondary vortices compared to gure 1.6 in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.9 (a) Reconstructed examples of a disc uttering in water (gray) and in stratied uid (black), showing enhanced radial drift from the initial drop location and de- crease in side-to-side uttering amplitude. The axis of the disc (normal arrows) indicates its inclination from the vertical. (b) Snapshots of two uttering periods at dierent depths for water and stratied uid (Fr = 1.26), showing the vertical and horizontal contraction of the trajectory and the increase of the uttering period in the stratied case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 viii 2.10 Left column: average descent speed _ z, uttering peak-to-peak amplitude a, peak inclination p , and average radial ranger for various uid densities and uid types: pure water = w = 1 (green), saltwater (blue) of uniform density for two density values = w = 1:048 and = w = 1:102, and stratied saltwater uid (orange) at two levels of stratication Fr = 2:34 and Fr = 1:26. Right column: the linear slope with respect to depth of each variable based on the best linear data t. Mean uid density values are used for stratied uid when computing = w , where w is the density of water. U is the terminal speed, accounting for buoyancy. 1- uncertainty bars are included for each data set. Stratication induces longer descent time, smaller uttering amplitude, smaller peak inclination, and appear to larger radial dispersions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1 Reconstructed top view of uttering descent in water and stratied uid (Fr = 1.26). For the uttering descent in stratied uid two types of motion are observed: (1) planar uttering to spiraling and (2) spiraling descent. . . . . . . . . . . . . . . . . 33 3.2 (a) Depth as a function of time, and (b) time-averaged descent velocity as a function of depth for descents in water (gray) and Fr = 1:26 stratied uid (black), where h is the height of the tank and U is the terminal speed (see table 2.1). Integrated states from (3.1) are overlaid in bold lines. C D = 1:436 (blue) for the water descent with root mean square error RMSE = 0:0015z=h, C D = 1:97 (red) with RMSE = 0:0042z=h andC D = 0:754z=h + 2:18 (orange) with RMSE = 0:0010z=h for the stratied uid descents. The values for C D were found by performing a least-squares t on the respective experimental z data. . . . . . . . . . . . . . . . . 34 3.3 (a) Peak inclination of the disc as a function of depthz=h. Descents in water are in gray and descents in a stratied uid with Fr = 1:26 are in black. Linear ts of individual descents are computed and the average of the t are overlaid for water (bold blue line) and stratied uid (bold red line). The linear slope for descents in water is near zero while descents in stratied uid have a negative slope. (b) - _ phase plot for descents in water () and stratied uid (). A sample descent trajectory plotted in - _ -plane (c) in water and (d) in the stratied uid. . . . . . 36 3.4 (a) Shows the phase space (, _ ) for eq. (3.4). (b) Shows the same phase space, but with an force ofK 1 + _ _ to illustrate the aect of damping. . . . . . . . . . 39 3.5 Fluttering amplitudea versus vertical location of discs descending in (a) water and (b) stratied uid for Fr = 1:26. Linear ts of individual descents are computed and the average of the t are overlaid for (a) water (bold blue line) and (b) stratied uid (bold red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ix 3.6 (a) Schematic representation of the expected eect of the restoring torqueT S on the translational and rotational motion of a uttering disc. During the gliding segment, T S tends to increase the angle of attack, inducing higher lift and drag forces F R and F D , thus shrinking and de ecting the gliding segment upwards. At the turning point, T S moderates the peak disc inclination p . (b) and (c) Free-body diagrams of the forces and moments acting on the falling plate in the quasi-steady model inx4. Entrainment of lighter uid is modelled as an area of constant density in the wake of the disc; the added buoyancy force F S is calculated from the resulting density and pressure jumps. The density jump also amplies the restoring torque T S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 (Colour online) Quasi-steady model: comparison between descent motion in pure water (gray lines) and higher density or stratied uid (black lines). Top row shows the descent trajectories in the (x;z) plane, middle row the inclination angle versus time, and bottom row the uttering motion in thex-direction versus time. (a) pure water versus higher density uid (= w = 1:05), (b) pure water versus stratied uid ( =290 kg/m 4 ) with T S = 0 and F S = 0, (c) pure water versus stratied uid with F S = 0, and (d) pure water versus stratied uid. . . . . . . . . . . . . 46 5.1 Distribution of the landing sites in the (x;y) plane for 500 consecutive drops in water in (a) and (c) and in stratied uid in (b) and (d). Cumulative distribution functions (c:d:f:) are shown in the inset of (d). Discs are initially released from (x;y) = (0; 0). The stratied uid is characterized by Fr = 1.26. . . . . . . . . . . 48 6.1 Experimental setup showing the side view for a rotating plate xed near the center of gravity. Plate cross section length is d = 0:152 m. Tank height and length is 0.45 m and 1.23 m, respectively. For the uid ow investigations we use particle image velocimetry (PIV) and shadowgraph techniques. . . . . . . . . . . . . . . . . 51 6.2 Shadowgram from direct shadowgraph for a plate (a) released at an initial angle of = 40 o and zero initial angular rate in stratied uid N = 1.63 rad/s. (b) - (d) Formation of vortices are seen during the initial release. As the vortices dissipate, (e) internal waves are formed. Fan like transient waves are seen from (e) - (g). . . 53 6.3 Shadowgram from direct shadowgraph for a plate (a) released at an initial angle of = 0 deg and non-zero initial angular rate in stratied uid N = 1.63 rad/s. (b) Formation of vortices are seen during the initial release along with uid entrain- ment. As the vortices dissipate, (d) internal waves are formed. Transient waves are seen from (d) - (f ). Internal waves appear to form and persist at the plate's edge for much longer durations as shown in (h). . . . . . . . . . . . . . . . . . . . 54 6.4 Vorticity eld on the left side of the plate rotating about its center of gravity from PIV data for 0 =20 deg and _ 0 6= 0 in water (left column) and N = 1:63 rad/s stratied uid (right column). In water, the plate rotates much further down than that in stratied uid. Additionally, the vortex strength in water remains fairly constant as compared to that in stratied uid, were it begins to diminish quickly after t = 1 s. We also see the formation of counter-rotating vortices adjacent to the primary vortex in (h) and (i). . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 x 6.5 Particle image velocimetry (PIV) experimental setup showing (a) the illuminated tracer particles, and (b) the setup with the laser, camera, and uid tank with laser sheet illuminating the particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.6 (a) Left tip vortex circulation strength from a plate rotating about its center of gravity in water (blue) and inN = 1:63 rad/s stratied uid (red), where 0 =20 deg and _ 0 6= 0. Circulation strength is normalized by the maximum strength max observed for the time period shown for each of their respective case. (b) Angle of the tip vortex center location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.7 Vorticity eld on the left side of the plate rotating about its center of gravity from PIV data for 0 =20 deg and _ 0 6= 0 inN = 1:63 rad/s stratied uid. Formation of counter-rotating vortices adjacent to the primary vortex are visible. . . . . . . . 58 6.8 An example of vertical velocity eld illustrating the vertical `slice' locations. Slice 1 is located 6.4 cm left of the plate's rotation center, and includes the motion of the plate. Slice 2 is located 15.2 cm left of the plate's rotation center, and is slightly down stream of the plate. For the vertical speed, the black corresponds to _ z > 0 and white corresponds to _ z< 0. Plate's length d = 15:2 cm. . . . . . . . . . . . . . 59 6.9 Vertical slice of the raw image (left), vertical speed (center), and its corresponding frequency! (right) normalized byN = 1:63 rad/s. For the vertical speed, the black corresponds to _ z > 0 and white corresponds to _ z < 0. (a) slice 1 from gure 6.8 located 6.4 cm left of the plate's rotation center, and includes the motion of the plate. (b) slice 2 from gure 6.8 located 15.2 cm left of the plate's rotation center, and is slightly down stream of the plate. y axis location is normalized the plate's length d = 15:2 cm. The gap in the frequency near y=d = 0:4 in (a) is due to the failure of the FFT in nding a dominant frequency. . . . . . . . . . . . . . . . . . . 60 6.10 Time evolution of a plate released from rest at various initial angles in N = 1.62 stratied uid. Plate were released ve times at approximately=3,=4,=6, =6, =4, and =3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.11 Angular velocity ! (mean with standard deviation) of the small amplitude oscil- lations from gure 6.10. Plate were released ve times at approximately=3, =4,=6,=6,=4, and=3. Buoyancy frequencyN = 1:62 rad/s is plotted in dark grey along with its 1- high and low value for N. The mean value of all the cases is <!> = 1.58 0.05 rad/s (red dotted line). . . . . . . . . . . . . . . . . . 62 6.12 Time evolution of a plate released from rest at various initial angles in N = 1.62 stratied uid. Plate were released ve times at approximately=3,=4,=6, =6, =4, and =3. Numerical model using eq. (6.1) is overlaid in black. . . . . . 66 6.13 (a) Phase plot (, _ ) for eq. (6.1) setting T E = 0. (b) A zoomed in near (0,0) showing the eect of the periodic forcing, T W . An example trajectory highlighted in green to illustrate the motion and direction as it approaches (0,0). . . . . . . . . 67 6.14 Example trajectories using eq. (6.2) withT D =T E = 0 for excitation frequency (a) ! = 1.16 rad/s and (b) ! = 1.64 rad/s in eq. (6.3). Location where the poincare sections (c) are taken are highlighted in red. The poincare section shows > 0 values in (, _ ) phase space that crosses the _ = 0 for a range of parameter !. . . . 68 xi 6.15 Example trajectories using eq. (6.2) withT D =T E = 0 for excitation frequency (a) plate = 600 kg/m 3 and (b) plate = 1200 kg/m 3 in eq. (3.4). Location where the poincare sections (c) are taken are highlighted in red. The poincare section shows > 0 values in (, _ ) phase space that crosses the _ = 0 for a range of parameter plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.16 Example trajectories using eq. (6.2) withT D =T E = 0 for excitation frequency (a) A = 0.55 and (a)A = 0.70 rad/s in eq. (6.3). Location where the poincare sections (c) are taken are highlighted in red. The poincare section shows > 0 values in (, _ ) phase space that crosses the _ = 0 for a range of parameter A. . . . . . . . . 70 A.1 Orientation of the group velocity vectorv g , phase velocity vectorv p , and wavenum- ber vector k (propagation direction of internal gravity wave) for an object oscillating vertically at the excitation frequency !, where v p k k and v g v p = 0. Fluid par- ticles move along lines of constant phase. Energy propagates with group velocity v g . Thickness of the beams (solid black lines) is approximately the diameter of the perturbing body for spheres and cylinders. . . . . . . . . . . . . . . . . . . . . . . . 84 A.2 Orientation of the group velocity vectorv g and phase velocity vectorv p for all four quadrants for an object oscillating vertically at the excitation frequency !, where v g v p = 0. Fluid particles move along lines of constant phase. Energy propagates with group velocity v g . Thickness of the beams (solid black lines) is approximately the diameter of the perturbing body for spheres and cylinders. . . . . . . . . . . . 86 A.3 Counter rotating vortices (blue and red circles) perturbing lines of constant den- sities (in grey) creating osets between the density gradient vectorsr and the pressure gradients vectorsrP , which generates vorticity of opposite sign due to the baroclinic torques (bold blue and red curve arrows), as described in eq. (A.19). 89 xii Abstract Leaves falling in air and marine larvae settling in water are examples of unsteady descents due to complex interactions between gravitational and aerodynamic forces. Understanding these de- scent modes is relevant to many branches of engineering and science, ranging from estimating the behaviour of re-entry space vehicles to analyzing the biomechanics of seed dispersion. The mo- tion of regularly-shaped objects falling freely in homogenous uids is relatively well understood. However, less is known about how density stratication of the uid medium aects the falling be- haviour. Here, we experimentally investigate the descent of heavy discs in stably stratied uids for Froude numbers of order 1 and Reynolds numbers of order 1000. We specically consider ut- tering descents, where the disc oscillates from side-to-side as it falls. In comparison to pure water and homogeneous saltwater uid, we nd that density stratication signicantly enhances the radial dispersion of the disc, while simultaneously decreasing the vertical descent speed, uttering amplitude, and inclination angle of the disc during descent. We explain the physical mechanisms underlying these observations in the context of a quasi-steady force and torque model. These ndings could have signicant impact on the understanding and design of uncontrolled vehicle descents and of geological and biological transport where density and temperature variations may occur. xiii Chapter 1 Introduction Stably stratied uids are found throughout nature in lakes, ponds, oceans, the atmosphere, and even in the sun. For example, apart from the upper layer and isolated regions, the ocean is generally stably stratied;the vertical density gradient, measured from the ocean oor, is negative while the vertical temperature gradient, if any, is positive [84]. Figure 1.1 shows a representative temperature prole for the Earth's atmosphere for July (solid line) and January (dashed line) at latitude of 50 N. An example density prole of the ocean is shown in in gure 1.2. This stable stratication is prevalent in isolated environments such as pores and fractures where mixing is negligible, and can lead to intense biological activities and accumulation of particles and organisms [51]. In engineering, stable stratication can be utilized for heat and mass transport problems, such as cooling of nuclear reactors [106] and energy generation from solar ponds [46]. 1.1 Density-stratied uid environments Stratication plays an important role in engineering design and analysis of submerged objects since density variations in the uid may in uence the object's motion. A notable example is the \dead-water" phenomenon [50], where a boat on the surface experiences an increase in drag due to low-pressure build up behind it from internal waves being generated along the interface of two-layer stratied uids [55]. This phenomenon exists when a layer of lower density uid, such 1 Figure 1.1: Temperature prole of the Earth's atmosphere showing the dierent atmospheric regions for the mean zonal temperature proles for July (solid line) and January (dashed line) from Committee on Space Research (COSPAR) International Reference Atmosphere 1986 (CIRA- 86) at latitude of 50 N. Adopted from Seramovich [77]. 2 0 20 40 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 0.02 0.04 0.06 0.02 0.025 0.03 Figure 1.2: Temperature T, (in-situ) density , and potential density as a function of depth for the ocean at 180 W and 0 N, where 1000 kg/m 3 needs to be added to and to convert the respective to . Potential density referenced to the sea surface. Most of the in-situ density increase with depth is due to pressure eects, which are removed in the calculation of potential densities, showing how weakly stratied the ocean is at larger depth. Data from NODC (Levitus) World Ocean Atlas 1994 [77]. 3 () () Figure 1.3: Schlieren visualization of internal gravity waves for (a) !=N = 0:9 and for (b) initial disturbance transient waves. Adopted from [60]. as fresh water, overlays higher density uid, such as sea water at the mouth of a river or near melting glaciers. In swimmers, [22] found that swimming strokes are less ecient in stratied uid and swimming speeds are slower by as much as 15%. Enhanced drag was also observed in horizontally moving spheres by [49] and later by [47, 48], with changes in the drag coecient being a function of the stratication level. While most studies focused on horizontal motion in stratied uids, similar changes in behaviour have been observed in vertically moving objects [91, 90, 105, 26, 9, 12]. For example, [91] found an increase in drag on a sphere settling in a stably stratied uid due to a buoyant jet forming behind the sphere. Some of these observed eects are due to internal gravity waves, which are produced when the excitation frequency! is less than the buoyancy or Brunt-V ais al a frequencyN of the system. For an oscillating object where!N, beams or ray patterns are produced as shown in gure 1.3(a), where the ray like bands are parallel to the group speed and perpendicular to the phase speed. Fluid particles move along lines of constant phase and energy propagates with group velocity [94]. More discussion on the internal waves is inxA.3. 4 For objects starting impulsively, a fan of internal waves called transient gravity waves is emitted [60, 59, 85, 45, 72] (see gure 1.3(b)). [99, 13, 72] modelled the perturbing object as a point source and found the transient waves have a wide range of frequencies. Ray pattern were also discussed in [54], where they showed that counter-rotating vortex pairs generate internal waves that are fan like emanating from the vortex pair. This was again conrm by [62] for vortex pairs, and noted that periodic array of vortices establish a standing wave pattern near the buoyancy frequency. Additionally, for motion in stratied uid that perturbs the density gradient, such as with entrainment or with vortices, causing the pressure and density gradients to not be parallel, baro- clinic torques are generated. This becomes important to the evolution of vortices in the ow eld, especially the dissipation rate of the vortex structure [93, 76, 30, 18, 44]. Experimentally, vortex motion in rotating and non-rotating stratied uid have been investigated in past, see for example [93, 25, 100, 18, 20, 19]. Specically, in [18], monopolar and dipolar vortices were investigated for non-rotating stratied uid. Where they investigated cyclonic and anticyclonic vortices and observed that unstable cyclonic vortices transformed into a tripolar vortex and some anticyclonic vortices transformed into triangular vortices. Later, expanding on the work of Turner et al. on buoyant vortex in rings stratied uid [92] and rising expanding vortices [93], Scase et al. [72, 73, 75, 74] looked at vortex rings and towing points sources in stratied uid. In [33, 34, 32], they showed that baroclinic torque decrease the speed and cause detrainment of the uid for descending vortex pairs. homogeneous 1.2 Passive ight in homogeneous environments Descending discs in homogeneous uid falls into four main descent modes: steady, uttering, chaotic, and tumbling. For steady descents the motion is nearly vertical and the ow eld in the wake is near constant. For uttering descents, the ow eld in the wake becomes unstable and is no longer steady. However, the motion is periodic. For tumbling descents, the motion is no 5 Figure 1.4: Sample trajectories from each falling regime reconstructed by Heisinger et al. [28]. (A) steady, (B) uttering, (C) chaotic, and (D) tumbling descents. longer periodic and the disc rotates or turn over on itself as it descends. Chaotic descent mode is a combination of uttering and tumbling motion. The four type of descent trajectories are shown in gure 1.4. Willmarth et al. [102] were the rst to experimentally construct a phase diagram (Re,I) that clearly showed the transition from steady to unsteady ( uttering and tumbling) near Re 100 for descent motions for falling discs (see gure 1.5). This phase space was later rened in [17] to include the boundaries between the uttering, chaotic and tumbling regimes (see left panel of gure 2.2). [86] performed similar experiments with spheres, cylinders and discs, and computed drag coecients for a wide range of Reynolds numbers. Numerical investigations of discs and cards falling freely in a homogeneous uid were con- ducted by [66, 2, 3, 39, 10, 4]. In addition, [66, 2, 3] formulated quasi-steady force models similar to those in [88, 6]. The associated nonlinear dynamics was analysed in [42], showing the existence of xed points, limit cycles, attractors, and bifurcations. [39] used a moving mesh method for the 6 Figure 1.5: Phase diagram (Re,I) that clearly showed the transition (Re 100) from steady to unsteady ( uttering and tumbling) descent motions. Adopted from [102] . Navier-Stokes equations and showed good agreement between experimental and computational trajectories, while clarifying some discrepancies noted in [2]. [40] and [58] used low-order repre- sentations of the uid in the context of the inviscid vortex sheet and unsteady point vortex models to shed light into the role of vorticity in destabilizing the descent motion. A comprehensive review of both experimental and computational techniques prior to 2012 is found in [16]. [4] explored the parameter space (Ar,I), where the Archimedes number Ar is proportional to Re. Meanwhile, [10] used non-dimensional mass and the Galileo number G, expressed as Ar = p 3=4G.[4] focused on the range Re < 300 (or Ar < 110) and identied non-planar sub- regimes of the uttering and tumbling regimes, which they referred to as hula-hoop (gyrating while uttering) and helical autorotation (helical tumbling). [10] focused on the range of G < 500. In our experiments, G> 750 and Ar> 366. 7 Experimentally, the spiraling or hula-hoop behaviour was observed in [28] and investigated by [43] using dye visualization and particle image velocimetry (PIV) to highlight the uid-structure interactions in these uttering motions. They observed a critical dimensionless moment of inertia I cr where the transition from side-to-side uttering to more spiraling behavior occurs. Figure 1.6) shows the vortex structure of for a uttering descent case in homogeneous uid for Re 2200 using dye visualization conducted by [107]. Vortex structure consist of primary vortices, secondary vortices, and counter rotating vortex pair (CRVP) leading to a hair-pin vortex. Similar, gure 1.7 shows the dye visualization performed by [43] for spiraling uttering descent in homogeneous uid for Re < 2000. In gure 1.7, we note the disappearance of the secondary vortices, the the less turbulent wake structure. The parameters space (Re,I) showing the transition line from planar uttering and spiraling uttering from [43] is reprinted here in gure 1.8. Recently, [98] investigated the falling behaviour of annular discs in the (Re,I) parameter space and found that the central hole stabilizes the descent motion of the disc. [28] dropped discs repeatedly in water to determine the probability density function (p.d.f) associated with the landing positions for each of the four trajectory types (reprinted here in 1.9). They found that the centre, directly beneath the point from which the disc is dropped, is one of the least likely landing sites for unsteady descents. 1.3 Thesis outline The organization of this document is as follows: a description of the experimental methods is given inx2. The resulting experimental observations for freely-falling discs are presented inx2.4, where comparisons are made for descending disc in homogeneous uid and for descending disc in stratied uid. Inx3, we discuss the physical mechanisms underlying these observations. Based on these mechanisms, we formulate inx4 a two-dimensional quasi-steady model for freely-falling disc that include forces and moments that arise from density stratication. Taken together, the 8 Figure 1.6: Dye visualization of wake and vortex structure for uttering descent in homogeneous uid for Re 2200. Vortices consist of primary vortices, secondary vortices, and counter rotating vortex pair (CRVP). Adopted from [107]. 9 Figure 1.7: Dye visualization of wake and vortex structure for spiraling uttering descent in homogeneous uid for Re < 2000. Adopted from [43]. 10 Zhong et al. Figure 1.8: Parameter space (Re,I) showing the transition location from planar uttering regime and spiraling (helical) uttering. Adopted from [43]. Figure 1.9: (top row) Distribution of landing sites for (A) steady, (B) uttering, (C) chaotic and (D) tumbling motions. (bottom row) histograms of the radial distribution of the landing positions. Adopted from Heisinger et al. [28]. 11 experimental results and analytical model explain how density stratication decreases the vertical descent speed, uttering amplitude, and inclination angle of the disc. Inx5, we probabilistically examine the eect of stratication on the radial dispersion of the disc by comparing the probabil- ity distribution function of landing sites in density-stratied uid to that in pure water, where we nd that stratication enhances radial dispersion. To help explain the enhanced dispersion, we then explore the wake structure of the descending disc in stratied uid using direct shadowgraph visualization techniques. To further investigate the eect buoyancy driven restoring torque has on an disc, inx6 we look at the dynamics of a purely rotating rectangular plate (our two-dimensional proxy) in stratied uid. Similarly to the descending disc, we explore the rotation of the plate in both water and stratied uid, and discuss the physical mechanisms responsible for our experi- mental observations. We then formulate a two-dimensional quasi-steady model and compare its predictions to experimental observations. Inx7, we conclude the motion of descending discs by commenting on the relevance of these results to engineering and biological applications. 1.4 Terminology and basic denitions Throughout the manuscript we will assume the uid is unbounded and has uniform stable strat- ication where the density prole is (z) = w + z, where z is the vertical direction (positive upward), w is density at z = 0, and = d=dz is the linear density prole where < 0. The linear density prole generates a constant buoyancy (or Brunt-V ais al a) frequency dened as N = r g w (1.1) whereg is the gravitational constant equal tog = 9:81 m/s 2 . The Brunt-V ais al a or buoyancy fre- quency is the natural frequency of oscillation of a vertically-displaced uid parcel in the stratied uid. For details on the assumption and derivation seexA.2. 12 The relevant dimensionalless parameters for the ow are Reynolds number and internal Froude number, dened as Re = Ud ; Fr = U Nd where d and U are the diameter and terminal velocity of the disc while and are the density and viscosity of the uid, respectively. We dene the disc's terminal speed as the speed at which the disc's weight balances buoyancy and drag forces, dened as U = s 2eg C D disc 1 ; where disc is the density of the disc. In calculating U, we set the drag coecient to C D = 1:2, consistent with the value for a disc normal to a uniform ow [31]. The Froude number re ects the stability and strength of the stratication. A small Froude number means strong, stable stratication, while Fr =1 denotes the absence of stratication (uniform density). If shear forces are important, the Reynolds number Re and Froude number Fr can be combined and expressed in terms of the viscous Richardson number Ri = Re Fr 2 ; which expresses the relative importance of the stabilizing eect from buoyancy to the destabilizing shear forces. Note that this Richardson number is distinct from the one used in oceanography, which accounts for large-scale shear in water columns, and is usually expressed as Ri = N 2 @u @z 2 ; where u is the horizontal uid speed. This term is usually used in oceanography to determine sensitivity to turbulence. If Ri< 0, then density variations will enhance turbulence (i.e., N 2 < 0) 13 and could lead to convective mixing of the water column. If Ri > 0, then N 2 > 0 (stable stratication), and shear force must be large to generate turbulences. In our investigation with the descending disc, the relevant shear forces originate from the vertical descending object. For the disc, the relevant dimensionless term is the moment of inertia and is given by I = disc e 64d : where d is the diameter, e is the thickness, and disc is the density of the disc. 14 Chapter 2 Descent motion in a density-stratied uid We experimentally investigate the motion of rigid discs falling freely in a vertically stratied uid of saltwater solution (gure 2.1). Even without stratication, the uid-structure interactions lead to rich descent dynamics that attracted the attention of scientists since the early observations of Maxwell in 1853 [53]. In homogeneous uids (no density stratication), the descent motion depends on the Reynolds number Re and the dimensionless moment of inertia I of the falling object. Starting from the uttering regime, we systematically vary the uid environment in order to examine the eect of changes in the uid density on the uttering behaviour. In particular, we consider three uid environments: pure water, constant-density saltwater, and stratied saltwater. We choose the parameters carefully so that in all three environments, (Re;I) lies robustly in the uttering regime, as detailed in the inset of gure 2.2, where the parameter space will be discussed later inx2.1. We reconstruct the descent trajectories and orientation of the uttering disc and we analyze the eect of stratication on the descent behaviour. We nd that stratication signicantly decreases the vertical descent speed, uttering amplitude, and inclination angle of the disc, while simultaneously increasing its radial dispersion (horizontal distance from drop location). Our 15 (a) θ d e z Tank Camera 2 x y Mirror Camera 1 x z Tank (b) (c) (e) (f) Tank 1 Tank 2 Tank ρ 1 ρ 1 ρ 2 ρ 2 ρ 2 > ρ 1 x y z θ disc release location h ρ disc (d) Prior to Release After Release Figure 2.1: (a) Disc of diameterd, thicknesse, and density disc at an inclination angle dened as the angle between the vertical directionz-direction and the normal to the disc. (b) and (c) Side and top views of the experimental setup used to record the disc's landing location and 3D trajectory, respectively; the mirror in (c) captures an orthogonal view necessary for 3D reconstruction. (d) Electromagnet release mechanism prior to release with the disc and after the disc is released. (e) Two-tank experimental free-drained setup used to generate a stable linear density prole in the tank. (f ) Schematic of the tank setup with a sample reconstructed trajectory. The coordinate system is centered at the initial release location. The landing location for multiple consecutive drops are shown in red and used to compute the landing distribution. 16 10 2 10 4 10 -3 10 -2 10 -1 1000 1500 2000 4 4.1 4.2 4.3 4.4 4.5 10 -3 Tumbling Chaotic Fluttering Steady Figure 2.2: Discs freely falling in homogeneous uids belong to one of four descent regimes: steady, uttering, chaotic and tumbling. These regimes are mapped onto the parameter space (Re,I) based on the results of [17]. The parameter values explored in this thesis are highlighted: constant density uid () consisting of pure water= w = 1, saltwater density values= w = 1:048 and = w = 1:102, and stratied saltwater uid () at two levels of stratication Fr = 2:34 and Fr = 1:26. For stratied uids, we computed I and Re using the average density values. 17 ndings are consistent but go beyond current numerical and experimental observations of two- dimensions ellipses in stratied uids [36]. 2.1 Laboratory setup Producing density stratication. To create linear density stratication in the lab, we used the two-tank method proposed by [21] and [64]. This method is used widely due to its simplicity and robustness [81, 29, 14]. We use a top lled free-drained approach (gure 2.1e), where the uid from the water reservoir (tank 1) is connected to the saltwater reservoir (tank 2), mixed, and then ows from the saltwater reservoir to the main tank through a water hose. A sponge and oater is placed at the end of the hose to minimize mixing and ensure the uid is lled from bottom to top. Mixing is performed in tank 2 throughout the lling process to ensure that the uid is thoroughly mixed. This process creates a linearly decreasing density prole. Example linear density proles we generated using this two-tank method is shown in gure 2.3. Disc properties and disc release mechanism. In all experiments, we used a single acrylic disc of density disc = 1144 5 kg/m 3 , diameter d = 2:54 cm and thickness e = 2 mm (see gure 2.1a), leading to a dimensionless moment of inertia I = 0:00442 0:00002. We considered a 60-gallon cubic acrylic tank of dimension 0:60 m on each side. We released the disc just below the surface of the uid using an electromagnetic release mechanism. (gure 2.1d). In all experiments, the disc was initially horizontal and was released with zero initial conditions, barring small uncertainty introduced by the release mechanism. To determine this uncertainty, a heavy disc made of steel was released in air ten times. The location of the disc when it reached the bottom of the tank was recorded with a top-mounted camera (gure 2.1b). To prevent the disc from sliding after landing, a grid mesh was added to the bottom of the tank. The standard deviation of the landing position in air was found to be less than 1.5% of the descent height h. In recording these data points, it was convenient to introduce a Cartesian coordinate system (x;y;z) 18 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 x ; water 0 2 4 6 8 10 12 14 16 18 20 tank height (in) =−290 / , =−181 / , =−138 / , =1.7 / =1.3 / =1.2 / / : z / d Figure 2.3: Example linear density prole ( =d=dz) generated using the two-tank method for water and saltwater for the setup shown in gure 2.1(d), where the 1- uncertainty estimate for the buoyancy frequency 0:02 to 0:05 rad/s. 19 Case I Re = w G Ar Fr Ri N (rad/s) U (cm/s) Water 0.00442 1934 1.000 1328 649 1 0 0 6.85 Saltwater-1 0.00422 1608 1.049 1104 539 1 0 0 5.43 Saltwater-2 0.00401 1094 1.102 751 367 1 0 0 3.52 Stratied-1 0.00433 1812 1.020 1244 608 2.34 331 1.06 6.30 Stratied-2 0.00423 1600 1.050 1099 537 1.26 1011 1.69 5.40 Table 2.1: Key dimensional and non-dimensional parameters explored in this thesis. An average uid density value is used to compute quantities for the stratied cases. U is the terminal speed where the weight is balanced by buoyancy and drag forces. I is the dimensionaless moment of inerita, Re is the Reynolds number, G is the Galileo number, Ar is the Archimedes number, Fr is the internal Froude number, Ri is the Richardson number, N is the buoyancy frequency as described inx1.2. with origin located at the site of the release mechanism and z-axis pointing vertically upward (gure 2.1f). In each experimental trial, the disc traveled a vertical distance h = 0:53 m to reach the bottom of the tank. We recorded the descent motion using a high-resolution monochrome digital video camera (Point Grey Grasshopper3) set to a moderate frame rate (3050 fps). A properly- positioned mirror was used to simultaneously capture a side view of the disc for three-dimensional reconstruction purposes (gure 2.1c). The position (x;y;z) of the centre of the disc and its orientation represented by the direction of the unit vector n normal to the disc were reconstructed directly from the high speed photography using an in-house image processing algorithm [28, 98] and described brie y in 2.2. The instantaneous velocity was determined by taking time dierence between consecutive frames. Successive descents are approximately ten minutes apart or until the uid is quiescent, based on visual examination. We considered three uid environments: (i) pure water with uniform density w = 1000 kg/m 3 ; (ii) constant-density saltwater obtained by uniformly increasing the salt concentration. The salt used is99.5% NaCl. We specically examined two constant-density saltwater cases: = w = 1:048 and = w = 1:102; (iii) density-stratied uid obtained by linearly increasing salinity with depth, with density gradient = d=dz < 0. We considered two stratied environments with density increasing linearly from= w = 1 atz = 0 to= w = 1:060 and= w = 1:153, respectively, 20 at z =h, resulting in stratication density gradients of =114 kg/m 4 and =290 kg/m 4 , respectively. The Froude number for the two stratied environments we considered are Fr = 2:34 (N = 1:06 rad/s) and Fr = 1:26 (N = 1:69 rad/s). For Fr = 1:26, the density at the bottom of the tank w h = 1154 kg/m 3 is greater than disc , causing the disc in some experiments to come to a stop before travelling the entire depth h. We therefore limited the trajectory reconstruction analysis to 0:65h (0:35 m). The average uid density ( = w 0:65 h=2) encountered by the disc is given by = w = 1:02 for Fr = 2:34 and = w = 1:05 for Fr = 1:26. Table 2.1 lists the key parameters for our test cases. Based on Reynolds number and the dimensionless moment of inertia, the descent motion of the disc falls into one of four main descent regimes: steady, uttering, tumbling, or chaotic. In gure 2.2, we map our four descent regimes onto the phase space (Re;I) based on the results of [17]. In gure 1.4, sample descent trajectories are shown for the four falling regimes reconstructed by [28]. 2.2 3D reconstruction of disc trajectories Reconstruction of the disc descent is done using Matlab's image processing toolbox. During the image recording process, each individual image contains the front and side view of the disc, as described inx2.1. The image processing algorithm, used by [28, 98], loads the individual images and obtain the locations and orientation angles of the front and side views of the disc using the method described in gure 2.4. The overall procedure is a follows: (1) load the raw image from the camera with a eld-of-view (FOV) that encompasses the disc with ample padding such that the same FOV contains the disc at the next time step; (2) convert the image to greyscale (if needed); (3) convert image to black-and-white; (4) clean up image by removing any outliers; (5) nd the shape boundaries in the image; (6) gather the location and the orientation angle (along the primary axis) of the object. 21 1. load raw image A = imread(image) 2. convert to greyscale A = rgb2gray(A) 3. convert to black-and-white A = im2bw(A,graythresh(A)) 4. removes objects < N pixels A = bwareaopen(A,N) 5. nd boundaries A = bwboundaries(A,`noholes') 6. obtain location and orientation regionprops(BW,...) Figure 2.4: Image processing steps for obtaining the state and orientation of the disc along with the Matlab pseudo-code. 22 -14 -12 -10 -8 -6 -2 -4 -2 0 0 2 2 0 -2 Figure 2.5: Example reconstructed trajectory and orientation of a descending disc released from (x;y;z) = (0; 0; 0) and scaled by the diameter of the discd. Disc's face color changes as a function of depth. The front view is used to reconstruct the x, z, and the projected angle in the xz-plane, xz . The side view is used to reconstruct they,z, and the projected angle in theyz-plane, yz . Scaling of the side view is required to match the dimensions from the front view. xz and yz is used to reconstruct the inclination (nutation) angle and precession angle . A resulting reconstructed trajectory and orientation for a disc descending in a uid released from (x;y;z) = (0; 0; 0) is shown in gure 2.5. 23 Figure 2.6: Direct shadowgraph technique with the camera out of alignment with the light source. The camera is set at an angle to record the image cast on the screen, otherwise a double image (actual and shadow) would appear. Digital processing is done to restore the image's perspective. 2.3 Direct shadowgraph for ow visualization A visualization method used for ow visualization is shadowgraph. Here we used direct shadow- graph technique as described in [56] and [78], where a small bright light source passes through the uid of varying density causing the light ray to focus and unfocus. See gure 2.6 for the setup. Since the screen acts as a surface where the shadow is projected on, the camera location is arbitrary with best results generally obtained when the camera is coincident with the light source [27]. The optimal sensitivity of the direct shadowgraph method is around 0:3<l g =l h < 0:7 [78, 27], wherel g is the length between the screen and the perturbing object, and l h is the length between the screen and the light source, as shown in gure 2.6. Shadowgraph is particularly eective for viewing sharp changes in uid density, e.g., viewing motion of interfacial waves between two-layer uids or breaking waves [41]. If density variations are gradual ow visualization may be harder to visualize. This method has also been eective for large scale ow observations, such as explosions and moving shock waves [15] and tip vortices of 24 helicopters [65]. A review of recent developments in Schlieren and shadowgraph techniques can be found in [79]. For general background on ow visualization, see [56] and [57]. For internal wave visualization and analyses for laboratory experiments see [87] To investigate the uid ow eld produced by the descending disc, shadowgraph visualization were conducted. The setup is described inx2.3. Figure 2.7 shows the time evolution of the falling disc at various time step using the shadowgraph technique for the stratied case of Fr = 1.31. We note that the in the early segment of the descent, the vortex structure is very similar to what was shown by [107] for planar uttering descent in water (shown in gure 1.6) where the wake consist of primary vortices, secondary vortices, and counter rotating vortex pair (CRVP). However, at later times, gure 2.7, show that the secondary vortices disappear. This wake structure is similar to those of hula-looping or spiraling uttering descents, as shown by [43] and repeated here in gure 1.7. Figure 2.8 shows another uttering descent in stratied uid for Fr = 1.31, but for a much longer time span, showing the disappearance of the vortex structure. From gures 2.7 and 2.8 we also see the eect stratication has on the wake: lighter uid is entrained into regions of higher densities then rises back up. This will eventually lead to quicker mixing and dissipation of the wake structure, as compared to descents in homogeneous uid shown in gures 1.6 and 1.7. 2.4 Experimental observations of disc motion In this chapter we compare the descent behaviour of the disc in a homogeneous uid to that in a stratied uid. Figure 2.9 shows two individual trajectories for pure water and stratied uid of Fr = 1:26, respectively. In gure 2.9(b), we compare two segments of the descent trajectories taken shortly after release (top row in gure 2.9b) and at later depth (bottom row in gure 2.9b). Each segment includes a full oscillation cycle of the uttering disc. In pure water, the amplitude and period of oscillation is independent of depth; the oscillation period is t = 1:3 s and the 25 0.3 sec 1.0 sec 1.8 sec 2.8 sec 1 cm Figure 2.7: Shadowgrams at various time of a free-falling uttering disc in stratied uid (Fr = 1.31; N = 1.63 rad/s). Note the lack of the secondary vortices at increasing depth compared to gure 1.6 in water. 2.8 sec 3.4 sec 4.0 sec 4.6 sec 5.2 sec 12.6 sec 1 cm Figure 2.8: Shadowgrams at various time of a free-falling uttering disc in stratied uid (Fr = 1.31; N = 1.63 rad/s). Note the lack of the secondary vortices compared to gure 1.6 in water. 26 (a) (b) Descent in Water Descent in Stratified Fluid -6 -5 -4 -3 -6 -5 -4 -3 -12 -10 -11 -13 -10 -11 -12 -13 t = 1.3 s t = 1.3 s t = 1.4 s t = 1.6 s Descent in Water Descent in Stratified Fluid r r r r z/d z/d z/d z/d Figure 2.9: (a) Reconstructed examples of a disc uttering in water (gray) and in stratied uid (black), showing enhanced radial drift from the initial drop location and decrease in side-to-side uttering amplitude. The axis of the disc (normal arrows) indicates its inclination from the vertical. (b) Snapshots of two uttering periods at dierent depths for water and stratied uid (Fr = 1.26), showing the vertical and horizontal contraction of the trajectory and the increase of the uttering period in the stratied case. 27 0.2 0.3 0.4 0.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.4 -0.2 0 0.2 1 1.05 1.1 0 1 2 3 4 5 1 1.05 1.1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 -4 -2 0 2 Figure 2.10: Left column: average descent speed _ z, uttering peak-to-peak amplitude a, peak inclination p , and average radial range r for various uid densities and uid types: pure water = w = 1 (green), saltwater (blue) of uniform density for two density values = w = 1:048 and = w = 1:102, and stratied saltwater uid (orange) at two levels of stratication Fr = 2:34 and Fr = 1:26. Right column: the linear slope with respect to depth of each variable based on the best linear data t. Mean uid density values are used for stratied uid when computing= w , where w is the density of water. U is the terminal speed, accounting for buoyancy. 1- uncertainty bars are included for each data set. Stratication induces longer descent time, smaller uttering amplitude, smaller peak inclination, and appear to larger radial dispersions. 28 descent speed is about 3:2d per utter cycle or 2:5d=s. In the stratied environment, the speed of the disc decreases with depth. Initially, the disc descends about 2:5d in t' 1:4 s, and near the bottom, it takes t' 1:6 s to cover the same distance. That is, the vertical speed decreases from 1:8d=s to 1:6d=s. To systematically quantify the eect of stratication on the descent behaviour, we consider the three uid environments described inx2 and listed in table 2.1: pure water w , saltwater of uniform density for two density values= w = 1:048 and= w = 1:102, and stratied saltwater uid at two levels of stratication Fr = 2:34 and Fr = 1:26. In each environment, we repeatedly drop the disc ten times and reconstruct the descent trajectories for each drop. For each reconstructed trajectory, we calculate the following quantities: the speed of descent _ z, the peak-to-peak amplitudea of each utter oscillation (see gure 2.9a), and the inclination angle p of the disc at \peak" positions of each oscillation cycle where the disc reverses direction. We also examine the eect of stratication on the horizontal dispersion of the disc, which we dene as the radial coordinate r away from the drop location of the disc (see gure 2.9a). We normalizez by the heighth,a andr by the disc diameterd, p by=2 and _ z by the terminal speed U. Figure 2.10 (left column) depicts the normalized values _ z=U, a=d, 2 p =, r=d averaged over each descent trajectory, as functions of the average density ratio = w of the uid. Values corresponding to each descent trajectory are represented by a lled circle. For each uid medium, the average (open circle) and standard deviation (vertical bar) over all ten descent trajectories are superimposed. Results are depicted in green for pure water, blue for homogeneous saltwater, and red for the stratied uids. To emphasize that _ z=U, a=d, 2 p = and r=d vary with depth z=h, we calculate the slope with respect to depth associated with the best linear t for each trajectory. The slope of each trajectory, as well as the average and standard deviation over all ten trajectories per uid medium, are depicted in the right column of gure 2.10. We make four observations on descending uttering discs in stratied uid based on g- ures 2.10: stratication enhances drag and decreases vertical speed, decreases uttering amplitude, 29 decreases inclination angle, and possibly increases the average radial distance from the initial drop location as the disc descends. We note that the results in gures 2.10(g,h) indicate that the ra- dial dispersion away from the drop location seems to increase in stratied uids. However, these results are not conclusive given that the sample size is small and the variance between trials is large. We address this issue and report a more conclusive observation inx5. Inx3, we explore the physical mechanisms underlying these observations discuss in this chapter, and inx4, we develop a quasi-steady model that reproduces similar results by incorporating the mechanisms proposed inx3. Apparent drag enhancement. The vertical speed of descent is slower in stratied uids compared to constant-density uids. A decrease in speed at= w = 1:02 and then increase in speed at= w = 1:05 with respect to the terminal speedU for the stratied cases, shows that the speed for the weaker stratication (Fr = 2:34) is much slower than that of the theoretical terminal speed. The negative slope of _ z=U indicates that in stratied uids, the disc slows down with increasing depth. In constant-density uids, the speed is nearly constant with depth (gures 2.10 a,b). This slowing down of the vertical motion for the stratied uid case is consistent with intuition and what was observed in literature [91, 90, 105, 26, 9, 12]. As the density increases with depth, the buoyancy-corrected weight decreases and drag increases, thus decelerating the descending disc and increasing the descent time. We explore the physical mechanism of the vertical motion in x3.1. Buoyancy-driven restoring torque. For descents in homogeneous uid, peak inclination p show very little variation. However, the peak inclination of the disc decreases in stratied uids. The negative slope of p indicates that in stratied uids, the peak inclination continues to decrease with increasing depth (gures 2.10 e,f). The hypothesis a torque due to stratication is causing the decrease in the inclination angle, and explore physical mechanism further inx3.2. 30 Eect of stratication on horizontal motion. In constant-density uid, the amplitude of oscillations show some variations for cases with increasing density, but the amplitude does not change with depth. In stratied uids, the uttering amplitude decreases with increasing depth and increasing stratication strength (gures 2.10 c,d). This decrease in the uttering amplitude for the stratied uid cases is consistent with intuition. As the disc descends into regions of higher density (increases with depth), the drag increases, thus decelerating and reducing the horizontal motion. This enhanced drag was also observed in horizontally-moving spheres by [49] and later by [47, 48]. We explore this behavior further inx3.3. 31 Chapter 3 Physical mechanisms governing disc motion In this chapter, we discuss the physical mechanisms causing the vertical speed, uttering amplitude and inclination angle to decrease with depth in stratied uids. We focus our analysis on a subset of the experimental data reported in gure 2.10: ten descents in water and ten descents in stratied uid at Fr = 1:26. We expect the underlying mechanisms to be applicable to all the other cases. 3.1 Apparent drag enhancement Figure 3.2(a) shows the vertical position z of the centre of the disc as a function of time for the ten descents in water and ten descents in stratied uid at Fr = 1:26. Clearly, the descent time is higher in the stratied case: 10 s in stratied uid as compared to 6:5 s in pure water. Furthermore, the slope of dz/dt is not linear in stratied uid, indicating a deceleration in speed as the disc descends. [8] noted similar descent proles for sediments in stratied uids. Figure 3.2(b) depicts the time-averaged descent velocity as a function of depthz=h and clearly shows the vertical deceleration in stratied uids. In pure water, the average velocity is almost constant,h _ zi=U =0:816 0:05 with a small negative slope (linear t of -0.045) with respect to depth. For Fr = 1:26, the average descent speed approachesh _ zi=U =0:715 0:03, with a non-zero slope (linear t of 0.175) indicating deceleration with respect to depth. 32 Figure 3.1: Reconstructed top view of uttering descent in water and stratied uid (Fr = 1.26). For the uttering descent in stratied uid two types of motion are observed: (1) planar uttering to spiraling and (2) spiraling descent. For the descents in water, the observed uttering behavior is purely side-to-side (planar ut- tering), and the spiraling uttering (or hula-hoop motion) was not observed. However, for the descents in stratied uid, we observed both uttering types. Figure 3.1 are example reconstructed trajectories viewed from the top (xyplane) in water and stratied uid. The the descent in water the side-to-side uttering motion is periodic and does not change, which is consistent with the results from gure 2.10. For the stratied descents, two mode exist, the planar side-to-side motion which slowly transitions to a spiraling motion, and the pure spiraling motion. The eect of the two uttering types can also be observed in gure 3.2 with the stratied cases. For side-to-side uttering motion, the disc's vertical motion heaves as it descends, which is evident by the periodic vertical descent prole and vertical speed in gure 3.2. For hula-loop uttering, the motion is similar to that of a hula-loop as viewed from the top (xy view as dened in gure 2.1), as opposed to side-to-side uttering, where the motion is more rectilinear or star-like. The hula-loop motion occurred in two out of the ten Fr = 1:26 stratied cases. As the disc descends, the inclination is fairly constant and the disc does not heave, producing a much atter vertical prole, which is seen in gure 3.2. The near constant inclination is evident in gure 3.3(a). 33 0 2 4 6 8 10 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Figure 3.2: (a) Depth as a function of time, and (b) time-averaged descent velocity as a function of depth for descents in water (gray) and Fr = 1:26 stratied uid (black), where h is the height of the tank and U is the terminal speed (see table 2.1). Integrated states from (3.1) are overlaid in bold lines. C D = 1:436 (blue) for the water descent with root mean square error RMSE = 0:0015z=h,C D = 1:97 (red) with RMSE = 0:0042z=h andC D = 0:754z=h+2:18 (orange) with RMSE = 0:0010z=h for the stratied uid descents. The values forC D were found by performing a least-squares t on the respective experimental z data. The vertical descent motion in gure 3.2 follows closely the dynamics of a particle falling under the in uence of gravity and subject to buoyancy and drag forces z =g + 1 m (z)Vg + 1 2m (z)j _ zj 2 C D S: (3.1) Here,V = Se is the volume of the disc, S = d 2 =4 its area, and m = disc V its mass. We solve eq. (3.1) numerically and superimpose the solution onto gure 3.2 using estimated values of the drag coecient C D that best tted experimental z(t) averaged over all ten trajectories. In pure water, we obtain C D = 1:436 (solid blue lines in gure 3.2), with root mean square error RMSE = 0:0015z=h. For the stratied uid, we considered two models for the drag coecient: a constant drag coecient (best t is shown in red in gure 3.2) and a drag coecient that varies linearly with depth z (best t is shown in orange). We found that for the constant coecient model, C D = 1:97 matches closely the experimental data (RMSE = 0:0042z=h), while for the 34 second model, C D = 0:754 z=h + 2:18 (RMSE = 0:0010z=h); that is to say, the estimated drag coecient decreases linearly with increasing depth (z=h). The non-constant C D (z) is slightly better at estimating the stratied descents (RMSE = 0:0010z=h versus 0:0042z=h), and as shown in gure 3.2. Another advantage of the linearly-varying drag coecient is that it allows us to use eq. (3.1) to predict the motion beyond the experimental data, whereas constant C D may only be valid for the available data. We emphasize that eq. (3.1) ignores the horizontal and orientational motion of the disc and lumps all the dynamics into the drag coecients. We present a more detailed model that takes into account these eects inx4. The constant drag coecient in stratied uid is noticeably larger than that of water, about 40% larger (ratio: 1:97=1:44 = 1:37). Enhanced drag was observed by [90] and [105] for vertical motion of spheres in stratied uid, where they found correlations between C D and the Froude number Fr. This phenomenon was investigated further by [12]. The increase in drag coecient in stratied uid (or, more specically, its linear dependence on depth) has a major implication on the modelling and prediction of the descent motion in stratied uid. 3.2 Buoyancy-driven restoring torque In order to examine the eect of uid stratication on the orientation dynamics of the disc, we plot in gure 3.3(a) the peak inclination angle p as a function of depthz=h. The peak inclination is measured at the instants when the oscillatory motion of the disc reverses direction. Clearly, there is a notable decrease in peak inclination in the stratied uid. The average peak inclination for the descents in water and stratied uid are p = 0:667 rad and p = 0:561 rad, respectively. A linear t of the respective data sets leads to p =0:046 z=h + 0:65 rad for the descents in pure water and p = 0:339z=h + 0:71 rad in stratied uid. In other words, the peak inclination remains fairly constant for the descents in pure water and decreases with depthz=h for the descents in stratied uid. 35 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0 0.25 0.5 -1 -0.5 0 0.5 1 0 0.25 0.5 -1 -0.5 0 0.5 1 0 0.25 0.5 -1 -0.5 0 0.5 1 Figure 3.3: (a) Peak inclination of the disc as a function of depthz=h. Descents in water are in gray and descents in a stratied uid with Fr = 1:26 are in black. Linear ts of individual descents are computed and the average of the t are overlaid for water (bold blue line) and stratied uid (bold red line). The linear slope for descents in water is near zero while descents in stratied uid have a negative slope. (b) - _ phase plot for descents in water () and stratied uid (). A sample descent trajectory plotted in - _ -plane (c) in water and (d) in the stratied uid. 36 In gure 3.3(b), we map the descent data for all ten trials in pure water and ten trials in stratied uid onto the (, _ ) phase plane. For the descents in water (), the motion is almost periodic and centered at (; _ ) (=8; 0); this behaviour can be seen more clearly when examining a single descent trajectory as done in gure 3.3(c). For the descent in stratied uid (), the motion spirals inwards approaching (; _ ) (=8; 0), as seen more clearly in the single descent trajectory shown in Figure 3.3(d). At (; _ ) (=8; 0), the disc descends at a xed inclination angle following either (i) the hula-hoop motion, where the disc's trajectory is helical [4], or (ii) a steady straight-line descent at constant orientation. Taken together, the results in gure 3.3 suggest the existence of a restoring torque in stratied uid that dampens the orientation dynamics of the disc as it descends through the uid. We postulate that this restoring torque is induced by the oset in the centre of gravity and the centre of buoyancy. When the inclination angle is non-zero, the side of the disc closer to the bottom of the tank will experience higher buoyancy force due to the uid's higher density, causing a torque in the direction which minimizes the inclination angle. We derive an expression for the stratication-induced torque T S for a disc undergoing planar oscillations. We rst compute the oset distance ` between the geometric centre of the disc and the centre of buoyancy in stratied uid as follows, ` = R V ( (z)j j sin) dV (z)V : (3.2) Here, the center of the disc is at height z and is the radial distance measured along the disc from the its centre. Integrating over the disc's volumeV yields ` = j jd 2 16 (z) sin(): (3.3) The torque induced by this oset is given by 37 T S = (z)Vg` cos: Substituting eq. (3.3) and using the denition of the Brunt-V ais al a frequency one gets T S = J 2 (z) disc N 2 sin (2); (3.4) where J = disc Vd 2 =16 is the moment of inertia of the disc about its diameter. For heavy discs with disc , this torque has no eect on the rotational motion of the disc. When the disc density is comparable to the uid density, this torque is proportional to N 2 ; that is to say, if the stratication frequency doubles, the stratication-induced torque quadruples. The eect of this torque on the full dynamics of the disc is discussed in details inx4. For heavy discs that is disc , this torque has no eect on the rotational motion ( 0). If the descending disc is near the density of the uid, disc , we havej jN 2 =2. Further, note that the deceleration is maximum at ==4 and it is zero at = 0 or ==2 when the disc is parallel or perpendicular to the horizontal plane, respectively. In addition, we note that eq. (3.4) is very similar to a pendulum equation with half angle, sin (2) instead of sin (). Figure 3.4(a) shows the phase space (, _ ) for eq. (3.4). Figure 3.4(b) shows the phase space for eq. (3.4) if some dissipation term is included, that is, if T S = J 2 (z) disc N 2 sin (2)k 1 + _ _ : 3.3 Eect of stratication on horizontal motion Figure 2.10(c,d) shows that stratication reduces the peak-to-peak uttering amplitude a, with the stronger stratication leading to a two-fold decrease in amplitude. In gure 3.5, we compare the uttering amplitude in pure water (gure 3.5a) and stratied uid (gure 3.5b) as a function 38 Figure 3.4: (a) Shows the phase space (, _ ) for eq. (3.4). (b) Shows the same phase space, but with an force ofK 1 + _ _ to illustrate the aect of damping. 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 Figure 3.5: Fluttering amplitude a versus vertical location of discs descending in (a) water and (b) stratied uid for Fr = 1:26. Linear ts of individual descents are computed and the average of the t are overlaid for (a) water (bold blue line) and (b) stratied uid (bold red line). 39 of depth -z=h. In pure water, the amplitude stays nearly constant with depth, with a linear t of a=d =0:002 z=h + 1:35. In stratied uid, the uttering amplitude decreases with depth at a rate a=d =1:656 z=h + 1:34. We argue that the buoyancy-induced restoring torque in eq. (3.4) is responsible for the re- duction in uttering amplitude. As the disc starts gliding (see gure 3.6a), the restoring torque causes it to pitch up, thus increasing its angle of attack. The later is dened as the angle be- tween the disc and its velocity vector and is denoted by (gure 3.6b). The increase in angle of attack, in turn, increases both lift and drag. Increased lift de ects the trajectory upwards, while increased drag slows down the horizontal motion, causing the gliding segment to end sooner than in pure water. As the disc begins to turn past its horizontal orientation, the restoring torque now reduces the angle of attack, which causes the disc to drift slightly further in the turning segment for the stratied case. However, the increased drift in the turning segment is not sucient to compensate for the decrease in the gliding segment, because the former is much shorter than the latter, resulting in an overall shortening of the uttering amplitude in stratied uid. Our proposed mechanism can be used to explain the behaviour in gure 2.10(d), which shows that the uttering amplitude decreases with depth, with stronger stratication inducing a steeper reduction. As depth increases, the density ratio between the disc and the uid approaches unity. Consequently, the disc's velocity decreases, leading to smaller hydrodynamic forces and moments, since they all depend quadratically on the velocity. In contrast, the restoring torque in eq. (3.4) only depends on the local density ratio (z)= disc , which increases with depth. It is therefore expected that the relative eect of the restoring torque increases signicantly with depth, hence leading to a reduction in uttering amplitude as depth increases. 40 (a) T S induces larger angle of attack no restoring torque with restoring torque T S induces smaller angle of attack θ P θ P Gliding Turning Drag Drag Drag F F D D Lift F R T S T S α Lift F R Drag F D plate θ weight mg (b) added buoyancy F S v ρ ρ(Z-δ) plate lighter fluid entrainment ρ(Z) δ dρ/dz = -γ γδ z p 1 ρ ω p 2 > p 1 F S v Z 0 circulation induced torque T R α plate θ v Dissipative torque T D T T restoring torque T S (c) amplified restoring torque without fluid entrainment with fluid entrainment z weight restoring torque T S buoyancy Z Figure 3.6: (a) Schematic representation of the expected eect of the restoring torque T S on the translational and rotational motion of a uttering disc. During the gliding segment, T S tends to increase the angle of attack, inducing higher lift and drag forces F R and F D , thus shrinking and de ecting the gliding segment upwards. At the turning point, T S moderates the peak disc inclination p . (b) and (c) Free-body diagrams of the forces and moments acting on the falling plate in the quasi-steady model inx4. Entrainment of lighter uid is modelled as an area of constant density in the wake of the disc; the added buoyancy force F S is calculated from the resulting density and pressure jumps. The density jump also amplies the restoring torque T S . 41 Chapter 4 Quasi-steady model In this chapter we develop quasi-steady model incorporating the mechanisms proposed in x3. Quasi-steady force models have been widely used in the context of falling discs and plates in homogeneous uids; see, e.g., [88, 2, 35, 66]. Inspired by this body of literature and based on our experimental observations, we formulate a new quasi-steady model for two-dimensional descent motions in stratied uids. In two-dimensions, we represent the disc as a thin elongated ellipse with major axis d and minor axis e. We introduced a orthonormal frame (b 1 ; b 2 ; b 3 ) centered at the ellipse with b 1 along to the major axis and b 2 along the minor axis. We start from the balance of linear and angular momenta on the rigid disc, written in the disc- xed frame, and we account for lift, drag and buoyancy eects (see gures 3.6b,c). This includes the restoring torque and the added buoyancy force due to uid entrainment. The equations of motion governing the motion of the disc can be written in vector form as follows: _ P =P + F D + F R + F S (mm b )gk; _ = P V + T D + T R + T S : (4.1) 42 Here, P = (mI + M add )V and = (J +J add ) are the linear and angular momenta of the disc expressed in body frame. The symbol I = diagf1; 1g denotes the identity matrix. The mass and moment of inertia of the disc are m = 4 disc ed and J = disc ed(e 2 +d 2 )=64; respectively. The buoyancy-corrected mass is mm b =( disc (z))ed=4. The added mass for an elliptical object is M add =(z) diagfd 2 ;e 2 g=4 and the added moment of inertia is J add =(z)(d 2 e 2 ) 2 =32: The unit vector k = sinb 1 + cosb 2 points vertically up in the z-direction. In eq. (4.1), F D and T D denote the force and torque due to drag. Following [2, 3], we model the drag force as a quadratic function of the angle of attack, which we denote by , F D = 1 2 e(z) C D (0) cos 2 +C D (=2) sin 2 jVjV: (4.2) Here, C D (0) and C D (=2) is the drag coecient at = 0 and = =2, respectively. The dissipative torque is modelled as T D = 1 16 d 4 j _ j _ : (4.3) 43 where is a dimensionless constant. Flow circulation around the falling disc induces a translation force F R = b 3 V; where is the circulation around the disc and b 3 is a unit vector perpendicular to the plane of motion. The circulation depends on both the translational speed and the angular velocity of the disc, = 1 2 C T ejVj sin 2 +C R e 2 _ ; (4.4) whereC T is the dimensionless translational lift coecient and C R is the dimensionless rotational lift coecient. We assume that the circulation-induced torque T R is zero. Finally, F S and T S are the force and torque due to density stratication. We argued in x3.2 that stratication induces a buoyancy-driven restoring torque due to the oset between the center of mass and the center of buoyancy, leading to T S = T S b 3 , where T S is given in eq. (3.4). Stratication also induces an additional buoyancy force F S that arises from the fact that the disc entrains lighter uid into regions of higher density uid as it falls. This phenomenon was acknowledged by previous work on axisymmetric objects moving through stratied uids [90, 105, 12] without quantifying this eect. Here, we model the entrainment as a volume of uid of vertical extent and of constant density(Z) above the disc, whereZ is the current position of the disc (see gure 3.6b). We assume that the uid below the disc is unperturbed, leading to a density jump at height Z. The pressure jump at the disc is computed by integrating the density prole (z) over the area below and above the disc, resulting in p 2 p 1 = 1 2 g 2 (right panel 44 of gure 3.6b). The added buoyancy force due to uid entrainment therefore equals to, in the plate's rotating reference frame, F S = 1 2 g 2 d cos()(sin b 1 + cos b 2 ): (4.5) We solve the system of equations (4.1) numerically. In all simulations, we set d=e = 0:0787, I = 0:0023, C T = 2:0, C R = 0:6, C D (0) = 0:15, C D (=2) = 2:0 and = 0:33. In gure 4.1(a), we compare the descent trajectory in pure water (gray lines) to that in saltwater (black lines) at uniform density = 1:05 w . The two trajectories exhibit the same peak inclination and uttering amplitude and only dier in the descent time. In gure 4.1(b), we compare the trajectories in pure water and stratied uid ( =290 kg/m 4 ) without taking into account the buoyancy-driven restoring torqueT S and added buoyancy force F S . Again, we see little dierence between the two trajectories. This implies that changes in density in the uid medium alone have only a small eect on the orientation and the translational motion of the descending disc. In gure 4.1(c), we take into account the buoyancy-driven restoring torque T S due to stratication, which results in a more prominent eect on the descent trajectory compared to pure water; namely, both the orientation angle () and the uttering amplitude (shown in x=d) decrease as the disc descends, in agreement with our experimental observations. Lastly, in gure 4.1(d), we account for both T S and F S and obtain similar behaviour. In sum, the numerical results based on eq. (4.1), when accounting for the forces and moment due to stratication, exhibit increased descent time and reduced inclination angle and uttering amplitude, consistent with our experimental observations inx2 andx2.4. 45 x/d z/d (a) 0 π/4 θ (rad) π/8 -π/4 -π/8 x/d 0 4 8 12 -2 -12 -10 -8 -6 -4 -2 0 2 0 -2 2 0 t (s) -1 1 0.5 -0.5 0 x/d x/d -2 2 0 t (s) t (s) t (s) x/d -2 2 0 (b) (c) (d ) 0 4 8 12 0 4 8 12 0 4 8 12 0 4 8 12 0 4 8 12 0 4 8 12 0 4 8 12 z/h 0 -0.2 -0.4 -0.6 Figure 4.1: (Colour online) Quasi-steady model: comparison between descent motion in pure water (gray lines) and higher density or stratied uid (black lines). Top row shows the descent trajectories in the (x;z) plane, middle row the inclination angle versus time, and bottom row the uttering motion in the x-direction versus time. (a) pure water versus higher density uid (= w = 1:05), (b) pure water versus stratied uid ( =290 kg/m 4 ) with T S = 0 and F S = 0, (c) pure water versus stratied uid with F S = 0, and (d) pure water versus stratied uid. 46 Chapter 5 Radial dispersion of discs' landing sites We have shown, using a combination of experiments and analytical modelling, that stable vertical stratication in the uid density aects the descent dynamics of freely-falling discs. Specically, stratication decreases the side-to-side uttering amplitude and maximum disc inclination while increasing the descent time. We now revisit the eect of stratication on the radial dispersion away from the disc drop location. We recall that the results presented in gures 2.10(g,h) indicate that the radial dispersion r increases in stratied uids. However, these results are not conclusive given the small sample size (ten drops in each case) and the large variability between drops. To test whether the enhanced dispersion is statistically signicant, we collect the dispersion distance from a larger data set. Following [28], we conduct 500 drops in pure water and 500 drops in stratied uid at Fr = 1:26. Due to the small uncertainty in the drop mechanism, for the same uid and disc parameters, distinct drops result in distinct falling trajectories and landing positions. We recorded the landing position of each drop and constructed the probability distribution functions (p.d.f.s) of the landing sites in the (x;y)-plane for all drops in water and in stratied uid. The p.d.f.s are shown in gures 5.1(a) and (b), respectively. Descent in stratied uid is characterized by a larger dispersion away from the drop location at (0; 0), with average standard 47 0 0.2 0.4 0.6 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0 0.5 1 water stratified 0 0.2 0.4 0.6 0 0.05 0.1 0.15 0.2 0 0.1 0.5 0.5 0.2 0 0 -0.5 -0.5 0 0.1 0.5 0.5 0.2 0 0 -0.5 -0.5 (a) (b) (c) (d ) y/h y/h x/h x/h r/h r/h Figure 5.1: Distribution of the landing sites in the (x;y) plane for 500 consecutive drops in water in (a) and (c) and in stratied uid in (b) and (d). Cumulative distribution functions (c:d:f:) are shown in the inset of (d). Discs are initially released from (x;y) = (0; 0). The stratied uid is characterized by Fr = 1.26. 48 Case Fr 1 /h 2 /h r m =h r var =h 0.9 c:d:f: Water 1 0.055 0.036 0.060 0.002 r=h = 0.11 Stratied 1.26 0.097 0.078 0.084 0.011 r=h = 0.20 Table 5.1: Distribution parameters of the landing distribution for 500 free-falling uttering discs. 1 and 2 are the standard deviations along the major and minor axes from gure 5.1(a,b). r m and r var are the mean and variance of the radial distribution normalize by the descent height h, and c:d:f: is the cumulative distribution function as shown in gure 5.1(d). deviation approximately 1.92 times larger than in water. The corresponding radial distribution is shown in gures 5.1(c) and (d). The results in water (gure 5.1c) are similar to those reported in [28], where they observed a dip near the origin and a tight radial distribution around the origin (standard deviation < 10% of the descent height). The larger radial distribution in stratied uid is also evident when comparing gure 5.1(c) and gure 5.1(d), where the variance is 0.002h for descents in water and 0.011h for descents in stratied uid. The cumulative distribution functions for both pure water and stratied uid are shown in the inset of gure 5.1(d). The two curves deviate quickly after r=h > 0:05. At r=h = 0:11, 90% of the descents in water are accounted for. For the descents in stratied uid, the 90% cumulation is reached at r=h = 0:20. We note that for both the descents in water and stratied uid (gures 5.1c and 5.1d) dips near the origin exist, suggesting that even in stratied uid the origin is the one of the least likely places of landing for uttering descents. A summary of the distribution parameters is listed in table 5.1. These results provide evidence that density stratication enhances radial dispersion. For descents in stratied uid, we hypothesis that stratication acts as a mechanism that amplify disturbances, both from the inherent uncertainties of the release mechanism and from three-dimensional eects that are not modelled inx3.3 or inx4. We discuss this further inx7. 49 Chapter 6 Rotational motion in density-stratied uids In this chapter, we experimentally investigate the motion of a rotating rectangular plate xed at its center of gravity and free to rotate about a single axis in a vertically-stratied uid of saltwater solution (gure 6.1), and comparisons are made to a rotating plate in water. This study allows us to further investigate the eect uid stratication has on the orientation dynamics, albeit in a simplied context of two-dimensional motion. Fromx3.2, in a stratied uid, we expect the buoyancy restoring torque to aect the plate's motion. Similar to the disc, the plate will begin to rotate toward the direction which minimizes . This contrast sharply with a plate in water, where the plate does not rotate without any initial perturbation. Inx6.3, we investigate the motion of the plate by releasing it at various initial inclination angles 0 , and reconstruct the time history of the inclination to analyze the eect stratication has on the rotation motion. We nd that stratication does provide the initial perturbation through the buoyancy restoring torque which decreases inclination angle of the disc. However, periodic oscillations due to the buoyancy restoring torque does not manifest due to the large overdamping eect of drag forces. For steady state motion, we nd that the plate's frequency is near the buoyancy frequency N while the surrounding uid's frequency (away from the plate's wake) oscillate slower due to internal waves. 50 " # = rotate approx. at center of gravity d tank x z () () Figure 6.1: Experimental setup showing the side view for a rotating plate xed near the center of gravity. Plate cross section length is d = 0:152 m. Tank height and length is 0.45 m and 1.23 m, respectively. For the uid ow investigations we use particle image velocimetry (PIV) and shadowgraph techniques. In addition, from ow eld visualization using shadowgraph techniques and particle image velocimetry (PIV), we see existence of transient waves and internal waves, and is discussed in x6.2. SeexA.2 andxA.3, for further discussion on buoyancy frequency, transient waves, internal waves, and their wave dispersion relationship. Following our experimental observations, we explore the physical mechanisms, and develop a quasi-steady model inx6.5. 6.1 Experimental setup To isolate only the rotation in uence, we xed the plate such that it can only rotate near its center-of-gravity about a single axis, as shown in gure 6.1. After testing multiple rotation apparatus (multiple bearing types, bearing and no-bearing, and no-bearing setups), we found that a cylindrical axial appended to the plate's axis of rotation and free to rotate in a housing (without bearing) provided the smoothest and resistant free rotation. To test the rotation mechanism, we placed the setup in water, and conrm that without any initial perturbations the plate does not rotate in homogeneous uid. Similarly, we test the system in stratied uid and noted that it was sensitive to the perturbation from the buoyancy restoring torque described inx3.2 and the plate begins to rotate. The plate's cross sectional length is d = 0:152 m, and the tank height and length is 0.45 m and 1.23 m, respectively. We assume the thickness of the plate e is negligible (e = 0:0063 m = 51 0:0413 d). Linear density stratication of the tank is done following the two-tank procedure describedx2.1. We explore buoyancy frequency of N 1:63 rad/s, unless otherwise noted. A non-intrusive thin string is used to hold the plate xed at pre-specied inclination angles, and the the plate is free to rotate by releasing the tension on the string. If an initial angular velocity is desired, the string can be attached to an electromagnet spring system of varying strength. Both cases are explored here. 6.2 Flow eld visualization To investigate the ow eld produced by the plate, shadowgraph visualization and particle im- age velocimetry (PIV) were conducted. Setup for shadowgraph visualization of the ow eld is described inx2.3. Shadowgrams of rotating plate. Qualitative ow visualization of a rotating plate is done using direct shadowgraph in stratied uid N = 1.63 rad/s. In gure 6.2 a plate is released with a non-zero initial inclination 0 and zero initial angular velocity ( _ 0 = 0), we see that the plate begins to rotate and edge vortices are formed. Entrainment of surrounding uid is visible as the horizontal stratied layers of constant densities are perturbed. As the vortices begin to dissipates, ray-like transient gravity waves are formed, similar to those discussed in [60, 59, 85, 45, 72]. Transient waves and their dispersion relationship are discussed further inxA.3. Although dicult to see in still images, we also see persisting gravity waves in the recorded video and naked eye. In gure 6.3, we perform similar analysis to what was shown in gure 6.2, but with an initial angle of 0 = 0 deg and non-zero initial angular velocity ( _ 0 6= 0). As the plate rotates, edge vortices are formed and uid entrainment is again visible. Mixing and turbulence is much more prominent in gure 6.3 than 6.2 as the system is more energetic due to the imparted initial angular velocity. As the vortices begin to dissipates, a complex system of transient gravity waves are formed originating from the plate's edge and from the turbulent uid. The more traditional 52 (a ) (b ) (c ) (d ) (e ) (f ) (g ) (h ) 0.0 sec 3 cm 0.7 sec 1.0 sec 1.3 sec 2.7 sec 3.3 sec 6.7 sec 13.3 sec Figure 6.2: Shadowgram from direct shadowgraph for a plate (a) released at an initial angle of = 40 o and zero initial angular rate in stratied uid N = 1.63 rad/s. (b) - (d) Formation of vortices are seen during the initial release. As the vortices dissipate, (e) internal waves are formed. Fan like transient waves are seen from (e) - (g). 53 (a ) (b ) (c ) (d ) 3 cm (e ) (f ) (g ) (h ) 0.0 sec 0.3 sec 1.3 sec 3.0 sec 5.0 sec 11.3 sec 25.3 sec 38.3 sec Figure 6.3: Shadowgram from direct shadowgraph for a plate (a) released at an initial angle of = 0 deg and non-zero initial angular rate in stratied uid N = 1.63 rad/s. (b) Formation of vortices are seen during the initial release along with uid entrainment. As the vortices dissipate, (d) internal waves are formed. Transient waves are seen from (d) - (f ). Internal waves appear to form and persist at the plate's edge for much longer durations as shown in (h). 54 ray-shaped transient waves are visible when the turbulence dissipates. Internal gravity waves from the edges of the plate are also more visible here than those in gure 6.2. PIV of rotating plate. For a more quantitate analysis of a rotating plate in stratied uid, we investigate the vortex structure and the uid ow around the plate in water and in stratied uid using particle image velocimetry (PIV) [1, 83, 38, 103, 87]. The visualization method obtains the instantaneous velocity vectors by tracking the illuminated particles in the uid, where quantitate analysis can be made on the ow eld. In general, two dimensional ow elds can be reconstructed using PIV, but complex three dimensional ow structure can also be analyzed [83, 82, 61]. Similar to the setup described in [98] (see gure 6.5), the uid tank is seeded with titanium oxide (TiO) particles of approximately 10 m (density 4.3 g=cm 3 ), and a vertical laser sheet was used to illuminate a vertical cross-section of the tank. For homogeneous uid, the tracer particles are placed directly into the tank and mixed. For stratied uid, the particles are placed in the salt- water tank and is mixed into the test tank during the stratication process described inx2.1. A Phantom M-110 high-speed camera was used to record the digital images. Images were processed with the open-source Matlab application PIVlab [89]. Since the plate does not rotate in water, we applied a small initial angular velocity ( _ 0 6= 0) on the plate to start the plate's motion. Figure 6.4 shows the time evolution of the vorticity for the left side of the plate in water and in N = 1:63 rad/s stratied uid. From the gure, the plate rotates much further down in water than in stratied uid due to buoyancy eects. Additionally, the vortex strength in water remains fairly constant as compared to that in stratied uid, were it begins to diminish quickly after t = 1 sec. In gure 6.6(a), we plot the circulation strength for the two cases, and diminishing circulation strength is evident in stratied iud. Angles of the tip vortex center v are also shown as a function of time in 6.6(b), where the vortex center in stratied uid approaches zero but due to buoyancy eects returns to a negative angle. 55 (a ) (b ) (c ) (d ) (f ) (g ) (h ) (i ) 0 sec (e ) (j ) 0.2 sec 1.0 sec 2.5 sec 4.8 sec 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 -20 0 20 Figure 6.4: Vorticity eld on the left side of the plate rotating about its center of gravity from PIV data for 0 =20 deg and _ 0 6= 0 in water (left column) andN = 1:63 rad/s stratied uid (right column). In water, the plate rotates much further down than that in stratied uid. Additionally, the vortex strength in water remains fairly constant as compared to that in stratied uid, were it begins to diminish quickly after t = 1 s. We also see the formation of counter-rotating vortices adjacent to the primary vortex in (h) and (i). 56 () () camera tank laser Figure 6.5: Particle image velocimetry (PIV) experimental setup showing (a) the illuminated tracer particles, and (b) the setup with the laser, camera, and uid tank with laser sheet illumi- nating the particles. 0 1 2 3 0 0.2 0.4 0.6 0.8 1 0 1 2 3 -20 -15 -10 -5 0 5 10 Figure 6.6: (a) Left tip vortex circulation strength from a plate rotating about its center of gravity in water (blue) and in N = 1:63 rad/s stratied uid (red), where 0 =20 deg and _ 0 6= 0. Circulation strength is normalized by the maximum strength max observed for the time period shown for each of their respective case. (b) Angle of the tip vortex center location. 57 (a ) (b ) (c ) (d ) (e ) (f ) 0.2 sec 1.2 sec 2.0 sec 2.5 sec 3.0 sec 4.0 sec 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 0 5 10 Figure 6.7: Vorticity eld on the left side of the plate rotating about its center of gravity from PIV data for 0 =20 deg and _ 0 6= 0 inN = 1:63 rad/s stratied uid. Formation of counter-rotating vortices adjacent to the primary vortex are visible. For the stratied case in gure 6.4, we see the formation of counter rotating vortices. To further enhance the resolution of these vortices, in gure 6.7, we decrease the range of the vorticity contour from -20 to 20 Hz to -5 to 10 Hz, which makes the counter-rotating vortices more visible. From gure 6.7, we see what appears to be a tripolar vortex structure (b) and triangular vortices (d), similar to what has been observed by [18]. From gures 6.2 and 6.3, we see evidence of internal waves generated by the plate. To observe the internal waves using PIV information we generate vertical velocity contours of the uid ow. Figure 6.8 shows the slice location in the ow eld where the vertical speeds are computed: Slice 1, located 6.4 cm left of the plate's rotation center, and includes the motion of the plate, and Slice 2, located 15.2 cm left of the plate's rotation center. 58 Slice 2 Slice 1 Figure 6.8: An example of vertical velocity eld illustrating the vertical `slice' locations. Slice 1 is located 6.4 cm left of the plate's rotation center, and includes the motion of the plate. Slice 2 is located 15.2 cm left of the plate's rotation center, and is slightly down stream of the plate. For the vertical speed, the black corresponds to _ z> 0 and white corresponds to _ z< 0. Plate's length d = 15:2 cm. By viewing the frequency of the vertical velocity proles, we can analyze the internal gravity waves at various location. The rst column in gure 6.9 shows the raw time history image as seen by the camera where some of the tracer particles can be seen. In gure 6.9(a) we see that the plate motion is in the slice. The second column shows the time history of the vertical velocity for the two slice locations. For slice 1 (gure 6.9a), we see the aect vortices has on the vertical velocity prole in the vertical range near the plate (z=d between 0.3 and 0.9). The angle of the vertical speed prole also provide us information about the internal wave frequency. From the linear dispersion relation (seexA.3), if the excitation frequency is near the buoyancy frequency !N, then the gravity wave propagates horizontally and the vertical velocity prole in 6.9 should be vertical. Since they are not for most of the prole, we can speculate that ! is smaller than N. The third column shows frequency of the vertical speed prole. We see that very near the plate vertical range (z=d between 0.4 and 0.6) the frequency closely match the buoyancy frequency. Away from the plate the frequency decreases, this is consistent with what was expected based on the orientation of vertical velocity prole. 59 0 0.2 0.4 0.6 0.8 0.5 1 1.5 z 10 20 30 10 20 30 (a) (b) 10 20 30 10 20 30 0 0.2 0.4 0.6 0.8 0.5 1 1.5 z Figure 6.9: Vertical slice of the raw image (left), vertical speed (center), and its corresponding frequency ! (right) normalized by N = 1:63 rad/s. For the vertical speed, the black corresponds to _ z > 0 and white corresponds to _ z < 0. (a) slice 1 from gure 6.8 located 6.4 cm left of the plate's rotation center, and includes the motion of the plate. (b) slice 2 from gure 6.8 located 15.2 cm left of the plate's rotation center, and is slightly down stream of the plate. y axis location is normalized the plate's length d = 15:2 cm. The gap in the frequency near y=d = 0:4 in (a) is due to the failure of the FFT in nding a dominant frequency. 60 0 10 20 30 40 50 60 - /3 - /6 0 /6 /3 Figure 6.10: Time evolution of a plate released from rest at various initial angles in N = 1.62 stratied uid. Plate were released ve times at approximately=3,=4,=6, =6, =4, and =3. 6.3 Plate rotational motion In this section we observe the motion of the plate released from rest at various initial inclination angle 0 = -60, -45, -30, 30, 45, and 60 deg. For a plate in water, we observe that the plate does not rotate, therefore only the results for stratied uid are reported here. Five releases were performed for each initial angle in stratied uidN = 1:62 rad/s. The time evolution of the angle versus time are obtain via image processing of the plate center line with respect to the horizontal plane and are plotted in gure 6.10. The following observation can be made from gure 6.10: (1) the plate rotates toward zero as predicted by eq. (3.4), (2) the motion is extremely over-damped, (3) small amplitude oscillations motion are superimposed on top of the slower damped descent, (4), the motion does not reach 61 - /2 - /3 - /6 0 /6 /3 /2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Figure 6.11: Angular velocity! (mean with standard deviation) of the small amplitude oscillations from gure 6.10. Plate were released ve times at approximately=3,=4,=6, =6, =4, and=3. Buoyancy frequency N = 1:62 rad/s is plotted in dark grey along with its 1- high and low value for N. The mean value of all the cases is <!> = 1.58 0.05 rad/s (red dotted line). 62 zero, indicating some imperfection in the experimental mechanism, and (5) a bias in the system exist such that nal inclination is not centered, i.e., 0 does not approach the same nal angle. From gure 6.10, the angular rate ! is estimated for the small amplitude oscillations. The average angular velocity and its standard deviation for small amplitude oscillations are plotted in gure 6.11 based on the ve test cases per initial inclination angle 0 . From gure 6.11 the mean average angular velocity for all cases is <!> = 1.58 0.05 rad/s, approximately 0.98 N. 6.4 Physical mechanisms governing plate rotation Inx6.3, we observed motion on two time scales: a slower over-damped motion which tries to align the plate horizontally, and a faster small amplitude oscillation (<!> N) on top of the slower motion. For the slower damped motion, the motion is initiated by the buoyancy restoring torque eq. (3.4) described inx3.2 andx4, but due to the presence of drag on the plate the rotation is much slower than what is predicted by eq. (3.4), ! S = r plate N 2 = 1:51 rad/s; and is not periodic as _ ! 0. For the high-frequency oscillation, the motion is established early after the disc is released. This motion is not due to the buoyancy restoring torque since eq. (3.4) requires crossing zero for periodic motion. We believe this motion is due to buoyancy waves (that oscillates at frequency N due to corrective buoyancy forces) from entrained uid and is maintained by internal gravity waves generated by the plate's motion. As the plate rotates and entrains uid (especially at the ends of the plate, see for example gure 6.2), buoyancy forces will correct the motion and provide an additional torque on the plate. For example, when the plate rotates and tip vortices are generated, the uid density is associated with uid for that particular depth. As the vortices are entrained to another depth, the buoyancy 63 forces will attempt to return the uid to a neutral position, as shown in 6.6(b). Additionally, in gure 6.9, we note that at height z where vortices are formed the frequency of the vertical speed is that of the buoyancy frequency N, conrming the buoyancy waves. From gure 6.10 we see see that the plate motion does not reach zero and is not centered at zero, indicating a bias on the experimental setup. The former eect is due to inherent resistance of the rotation apparatus, and the latter is due to a small misalignment between the plate's rotation axis and center of gravity. Both of these eects are estimated and accounted for in the next section (x6.5) where a dynamical model for the plate rotating in stratied uid is developed. 6.5 Quasi-steady model Here we formulate a quasi-steady model for two-dimensional rotation of the plate motion in stratied uids similar tox4. In two-dimensions, we represent the plate as a thin plate with length d and ignore the plate's thickness. Starting with the balance of angular momenta we account for drag and buoyancy eects. This includes the bouyancy restoring torque T S , eq. (3.4). The equations of motion governing the motion of the disc can be written in vector form as follows: J _ =T D +T S +T W +T E ; (6.1) where J is the moment of inertia of the plate, J =m d 2 =12 = 0:000653 kg m 2 , mass m = 0.3374 kg and length d = 0.1524 m. Here T D denote torque due to drag and is dened as T D = C M d 4 _ ; (6.2) 64 where C M is constant that is estimated base on the experimental data and = 1062:6 kg=m 3 is the density at the plate's center of gravity. T S is identical to eq. (3.4), and T W is the small amplitude oscillation due from the internal waves estimated from gure 6.11 and of the form T W = A sin!t: (6.3) Equation (6.3) can be viewed as the periodic forcing function on the system with amplitude A. T E is the experimental torque form center-of-gravity oset and the biases from the experimental setup, both of which we estimate from the experimental data. T E = sgn()A 1 2mg cos(=2)A 2 where sgn() is the sign of . From gure 6.10, we nd thatA = 0.124 Nm,A 1 = 4.410 4 Nm,A 2 = - 2.610 5 m, andC M = 0.011. The dynamics of the system is sensitive to the estimation of torque drag coecient C M , where the termA only aect the small amplitude motion andA 1 andA 2 is due to the experimental biases. Integrating equation eq. (6.1) using the parameters above for various initial inclination angles we obtain gure 6.12, which overlays the numerical results over the original dataset for the rotating plate for N = 1.62 rad/s. The trajectories exhibit similar behavior of rotating from its initial inclination to a near zero inclination during the transient and superimposed to the motion is a higher frequency and smaller amplitude oscillation which persist for a much longer duration. In general, the numerical results based on eq. (6.1) exhibits the observed experimental behavior inx6.3. Viewing eq. (6.1), we note that the equation is similar to a spring-damper system with periodic external forcing. Removing the experimental torque biases, settingT E = 0, we can view the phase plot (, _ ) of the system, plotted in gure 6.13. However, the damping of the system (T D , eq. (6.2)) 65 0 10 20 30 40 50 60 - /3 - /6 0 /6 /3 Figure 6.12: Time evolution of a plate released from rest at various initial angles in N = 1.62 stratied uid. Plate were released ve times at approximately=3,=4,=6, =6, =4, and =3. Numerical model using eq. (6.1) is overlaid in black. 66 -50 0 50 -30 -20 -10 0 10 20 30 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Figure 6.13: (a) Phase plot (, _ ) for eq. (6.1) settingT E = 0. (b) A zoomed in near (0,0) showing the eect of the periodic forcing, T W . An example trajectory highlighted in green to illustrate the motion and direction as it approaches (0,0). dominates most of the motion and only when zoomed to near the equilibrium points (e.g., near (, _ ) = (0,0)) we see the aect of the periodic forcing T W (gure 6.13b). Quasi-steady model: without damping From gure 6.13 we see the strong dominance of the damping termT D , eq. (6.2). By removing the eect of damping T D , we can observe the dynamics of stratication T S and the in uence the forcing function T W has on the system. Figure 6.14-6.16 shows the evolution of the poincare section for the trajectory in phase space (, _ ) for trajectories that cross the _ = 0 for > 0 for various change in parameters, external forcing rate!, plate , and forcing amplitudeA, respectively. The forcing amplitude A appears to have the greater in uence on the motion, transitioning from periodic or quasi-periodic orbits to those which appear to be chaotic near A 0:6. 67 - /2 - /4 0 /4 /2 - /2 - /4 0 /4 /2 - /2 - /4 0 /4 /2 - /2 - /4 0 /4 /2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 6.14: Example trajectories using eq. (6.2) with T D =T E = 0 for excitation frequency (a) ! = 1.16 rad/s and (b) ! = 1.64 rad/s in eq. (6.3). Location where the poincare sections (c) are taken are highlighted in red. The poincare section shows > 0 values in (, _ ) phase space that crosses the _ = 0 for a range of parameter !. 68 - /2 - /4 0 /4 /2 - /2 - /4 0 /4 /2 - /2 - /4 0 /4 /2 - /2 - /4 0 /4 /2 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 6.15: Example trajectories using eq. (6.2) with T D =T E = 0 for excitation frequency (a) plate = 600 kg/m 3 and (b) plate = 1200 kg/m 3 in eq. (3.4). Location where the poincare sections (c) are taken are highlighted in red. The poincare section shows > 0 values in (, _ ) phase space that crosses the _ = 0 for a range of parameter plate . 69 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 - /2 - /4 0 /4 /2 - /2 - /4 0 /4 /2 - /2 - /4 0 /4 /2 - /2 - /4 0 /4 /2 Figure 6.16: Example trajectories using eq. (6.2) with T D =T E = 0 for excitation frequency (a) A = 0.55 and (a)A = 0.70 rad/s in eq. (6.3). Location where the poincare sections (c) are taken are highlighted in red. The poincare section shows > 0 values in (, _ ) phase space that crosses the _ = 0 for a range of parameter A. 70 Chapter 7 Conclusion and discussion For freely-falling uttering discs stratication we observed enhanced radial dispersion of the land- ing location from the original drop site. In addition, the descent speed, uttering amplitude, and the inclination angles of the disc's trajectory decrease during the descent in stratied uid. The descent speed and uttering amplitude can be explained by the enhanced drag in stratied uid. We then derived and observed the restoring torque on a the disc due to density gradient in stratied uid, which reduces the inclination angle as the disc descends. This restoring torque further explains the decreasing descent speed and uttering amplitude. Stratication appears to enhance the radial dispersion of the landing location, where the average range drifts from the original drop location. Horizontal drifts were also observed by [36] for freely-falling elliptical cylinders, both numerically and experimentally, with more subdued drifts in the latter. [7] later noted similar drifts in the case of oscillating and levitating spheres in stratied uid, and proposed that strong drifts are a result of a feedback of the nonlinear vortices and lee waves. In weakly stratied turbulent uid, Aartrijk et al. [97] and [96] observed enhanced horizontal dispersions for particles. Quasi-steady models were investigated for both disc and plate motion, and qualitative behav- iors were observed and consistent with experimental data, with the exception of the enhanced dispersion observed for descending disc in stratied uid. We postulate that the origin of such 71 drift is due fundamentally to unsteady eects beyond what is accounted for in the quasi-steady model inx4. The quasi-steady model, while it captures the increase in descent time and decrease in uttering amplitude and peak inclinations of the disc, it does not exhibit enhanced dispersion, even when the disc is initially given a small horizontal velocity (results not shown). The absence of radial drift in the quasi-steady model could be due to (1) unsteady force corrections, (2) eects due to the interaction of the disc and its vortices [2, 3], and (3) three-dimensional eects. The rst eect includes lift generation during acceleration from rest [69, 101] and unsteady forces due to vortex shedding [2, 43, 107]. Investigation of rigid plate rotating in stratied uid conrms the aect of the buoyancy restoring torque, where buoyancy and internal gravity waves are generated by a rotating plate. Using shadowgraph and PIV visualization techniques, the complex uid-body interactions are shown, including transient internal waves evident with fan-like wave structures and gravity internal waves after the transient. In addition, primary shedding vortices are generated for rotating plates. For stratied uid, baroclinic vortices are generated by these primary vortices, which quickly dissipate the vortex structure. Transient waves originating from the vortices and from the entrained uid were also observed. Stratication appears to amplify disturbances of the descent motion, which is evidence of the enhances radial dispersion and the triggering of the transition of planar uttering motion to hula-loop motion for falling discs. This can be seen by viewing the descent trajectories form the top, as shown in the center image gure 3.1. Additionally, the eect can be seen in the vertical z prole and inclination, gure 3.2 and gure 3.3, respectively. Lee et al. [43] claims that the transition from planar uttering motion to three-dimensional spiral motion occurs due to the growth of three-dimensional disturbances for smaller non-dimensional moment of inertia I values. The transition in (Re,I) parameter space, can be seen in gure 1.8. In pure water, [28] showed that enhanced dispersion is usually associated with the chaotic regime (gure 1.9), where the uid acts as a \randomization device" that dissociates successive 72 drops of the disc, resulting in a larger distribution of landing sites away from the drop location and almost equal probability of landing on either side of the disc (head or tail). Stratication appears to achieve similar enhancement of radial dispersion but in the uttering regime, where the disc never ips during its descent. In fact, stratication enhances dispersion while simultaneously reducing the uttering amplitudes and inclination angles, making the descent trajectories similar to steady descents. 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Named after Joseph Valentin Boussinesq, the approximation assumes small variation in background density and tem- peratures about their mean values, except where they appear in terms that multiply by gravity g. The equation provides an approximation of the Navier-Stokes equations for nearly incompressible ow, as in the case of water. Given the general continuity equation for compressible medium by looking at the conservation of mass we have 1 D Dt +ru = 0 (A.1) where u is the velocity vector and is the uid density. For incompressible ow, we have D Dt = 0: (A.2) Here, D=Dt represent the total, particle, or substantial derivative, such that, 80 Du Dt = @u @t +uru = 0 (A.3) where the partial derivative term is the local derivative and the term with the dot product is the convective derivative term. Assuming small density variation, the continuity equation becomes, ru = 0 (A.4) which states that there is no convergence (compression) nor divergence (expansion) of the ow. Eq. (A.4) can now be used to simplied the momentum equation or better known as the Navier-Stokes equations Du Dt =rp +g +r 2 u (A.5) where p is pressure, and is the dynamic viscosity ( = =). Since Boussinesq approximation assumes a density variation only aects the buoyancy term, then we can replace the density with 0 for all terms excluding g. This yields 0 Du Dt =rp +g +r 2 u: (A.6) By dening a reference pressure such that @p 0 (z) @z =g 0 (A.7) and introducing = 0 + and p = p 0 (z) + p, we haverpg =r (p) ()g, and eq. (A.6) becomes Du Dt = r (p) 0 + ()g 0 +r 2 u (A.8) 81 If we assume the ow is inviscid (Euler's equation, that is, = 0) and make the following substitutions, = p= 0 and b = = 0 , we then have simplied Boussinesq equations for an inviscid non-rotating system Du Dt =r +bg ru = 0 Db Dt = 0 (A.9) Linearizing eq. (A.9) by removing all non-linear convective terms (i.e., uru = 0), and linearizing the continuity equation, (A.2), we get the linearized Boussinesq equation @u @t =r +bg ru = 0 @ @t + _ z @ 0 @z = 0 (A.10) Further simplication of eq. (A.10) is discussed in the next section. A.2 Buoyancy (Brunt-V ais al a) Frequency Considering a small displacement z 0 about the equilibrium in the vertical direction, we can use eq. (A.2) and the linear continuity equation from eq. (A.10) (third equation in the set) and assuming small pressure dierences, we get the following z 0 = g 0 = g 0 @ @z z 0 (A.11) The solution to this homogeneous second order dierential equation, eq. (A.11), is z 0 =Ae p N 2 t +Be p N 2 t ; 82 whereA andB are constants determined by the initial conditions, and N is the Brunt-V ais al a or buoyancy frequency, in units of rad/s. Additionally, if N is a positive constant, then the solution can be expressed as a simple harmonic oscillator of the form z 0 =A sinNt +B cosNt; where N = s g 0 @ @z = r g 0 (A.12) where density gradient = d=dz. The corresponding period of the motion is T N = 2=N. For the upper ocean and deep ocean, the periods are minutes long and hours long, respectively. For the atmosphere, they are a few minutes [94]. For N 2 > 0, the uid is statically stable, and a particle in the uid will oscillate around its equilibrium state with frequency of N. IfN 2 < 0 the motion of the particle is amplied and resulting in convection and is statically unstable. We note here, that in some density proles (e.g., the atmosphere), the density equation is expressed as an exponential function, 0 = i e z=H , which simplies N to N = p g=H; where H is a scale height (vertical range where the density decreases by a factor of 1=e) [37]. A.3 Dispersion Relation Now assuming a sinusoidal solution in two-dimensions (horizontal and vertical) with the form =Ae i(kxx+kzz!t) P =Be i(kxx+kzz!t) 83 Figure A.1: Orientation of the group velocity vectorv g , phase velocity vectorv p , and wavenumber vector k (propagation direction of internal gravity wave) for an object oscillating vertically at the excitation frequency!, where v p k k and v g v p = 0. Fluid particles move along lines of constant phase. Energy propagates with group velocity v g . Thickness of the beams (solid black lines) is approximately the diameter of the perturbing body for spheres and cylinders. where A and B is a constant determined by the initial conditions, k x and k z are the horizontal and vertical wavenumber, and ! is the excitation frequency, and substituting it into eq. (A.10), (A.11), and (A.12), we have the dispersion relation for internal waves ! 2 =N 2 k 2 x k 2 x +k 2 z =N 2 cos 2 (A.13) where is the angle between the horizontal (direction of !) and wavenumber k. The ray pattern is illustrated in gure A.1. From eq. (A.13), if the excitation is slow (!N) then cos 0 and =2 and the wave is propagated almost in the vertical direction. If the excitation frequency !N, then cos 1 and 0, and the wave is propagated almost in the same direction as the excitation frequency. We note that eq. (A.13) results in !<N, and that no wave is created when !>N. 84 The phase speed v p is the speed of phase propagation in the direction of k, where k =k x e x +k z e z ; is directed from crest to crest, and written as v p = ! jkj = ! p k 2 x +k 2 z =N k x k 2 x +k 2 z ; where e x is the horizontal direction and e z is the vertical direction. In vector form we have v p = ! k = ! jkj k x jkj e x + k z jkj e z = k x N jkj 3 (k x e x +k z e z ) (A.14) where the magnitude is v p = (N=jkj) cos. For the group speed we have v g = @! @k x e x + @! @k z e z = k z N jkj 3 (k z e x k x e z ) (A.15) where the magnitude is v g = (N=jkj) sin. Examining eq. (A.15) we see that v p and v g are perpendicular to each other (v g v p = 0). Figure A.2 and Table A.1 summarizes the internal wave phase v p and group v g velocity directions for all four quadrants as a function of k x and k z for !> 0. Another usefully relation, is one relatingN,!, and the components of the wavenumber vector k, as follows k x k z = x z = tan = r 1 cos 2 cos 2 k x k z = s N ! 2 1 (A.16) 85 Figure A.2: Orientation of the group velocity vector v g and phase velocity vector v p for all four quadrants for an object oscillating vertically at the excitation frequency !, where v g v p = 0. Fluid particles move along lines of constant phase. Energy propagates with group velocity v g . Thickness of the beams (solid black lines) is approximately the diameter of the perturbing body for spheres and cylinders. k x k z v p v g + + right, up right, down + - right, down right, up - + left, up left down - - left, down left, up Table A.1: Internal wave phase v p and group v g velocity directions for all four quadrant as a function of k x and k z for !> 0. 86 where x = 2=k x and z 2=k z are the wavelength in the x and z directions, respectively. Note, from eqs. (A.14) and (A.15), when !!N, then cos! 1 and sin! 0 and k z ! 0. This leads to v p in the horizontal direction and v g in the vertical direction, i.e., the internal wave rays are parallel to the vertical direction. It is important to note that eq. (A.13) assume an unbounded uid, that is, _ z 0 for the uid at the water surface and bottom of the tank. For the bounded uid, the equation for the dispersion relation eq. (A.13) is modied with k z =n=h, where n is the mode number and h is the depth of the tank. For objects starting impulsively, a fan of internal waves called transient gravity waves is emitted from near its original position [60, 59, 85, 45, 72]. The radiated waves have a wide range of frequencies. From [99, 13, 72, 75], the internal wave eld from an (extended source) object is modelled as a point source. m =m 0 e i!0t (r) (A.17) where is the Dirac delta function, ! 0 is the frequency of oscillation of the source's strength, m =r cot v is the source strength (rate of volume out ow from the source) per unit volume, and r =xe x +ye y +ze z is measured from the origin. For a monopoles point source described in [99, 13, 72, 75], they found that for transient gravity waves ! = Njzj r ; v g = r t ; k = ! v g r r r z e z ; (A.18) where k is the wavenumber vector, ! is the excitation frequency, and v g is the group speed. 87 A.4 Baroclinic Vorticity For objects moving in stratied uid that perturbs surfaces of constant densities, either directly or by vortices it generates, strong non-vertical density gradients are established when lighter uid is transported into regions of heavier uid and vice versa. This cause the pressure and density gradients to not be parallel, which generate vorticities due to the baroclinic torque, as described by the last term in the vorticity equation, @! @t + ur! =!ru + 1 2 rrp: (A.19) where u is the ow velocity, p is the pressure, and vorticity ! =r u. The vorticity equation, eq. (A.19), assuming non-rotating uid ( = 0). Figure A.3 illustrates the formation of counter rotating vortices from the primary vortices due to baroclinic torques. Counter-rotating vortices due to baroclinc torques and its eects on circulation have been thoroughly investigated numerically and experimentally. The evolutions of counter-rotating vortex pairs in two-dimensions in stratied uid due to baroclinic torques were investigated by [76, 30, 80, 23], and noted a reduction of the separation distance between the vortices. Similarly, three- dimensional vortex pair in stratied uid were observed experimentally by [71, 11] and numerically by [70, 23, 80, 62] and also noted the reduction of separation distance and deceleration of the vortex speed. In [33, 34, 32], they showed that baroclinic torque decrease the speed and cause detrainment of the uid for descending vortex pairs. For a general review of instability of vortex pairs, see [44]. Since the circulation is the ux of vorticity ! through some closed surface, then will decrease due to the generation of the baroclinic torque. This decrease in circulation was observed by [62] and [63] for vortices in stratied uid as opposed to those in homogeneous uid, where circulation remain constant. 88 Figure A.3: Counter rotating vortices (blue and red circles) perturbing lines of constant densities (in grey) creating osets between the density gradient vectorsr and the pressure gradients vectorsrP , which generates vorticity of opposite sign due to the baroclinic torques (bold blue and red curve arrows), as described in eq. (A.19). 89
Abstract (if available)
Abstract
Leaves falling in air and marine larvae settling in water are examples of unsteady descents due to complex interactions between gravitational and aerodynamic forces. Understanding these descent modes is relevant to many branches of engineering and science, ranging from estimating the behaviour of re-entry space vehicles to analyzing the biomechanics of seed dispersion. The motion of regularly-shaped objects falling freely in homogenous fluids is relatively well understood. However, less is known about how density stratification of the fluid medium affects the falling behaviour. Here, we experimentally investigate the descent of heavy discs in stably stratified fluids for Froude numbers of order 1 and Reynolds numbers of order 1000. We specifically consider fluttering descents, where the disc oscillates from side-to-side as it falls. In comparison to pure water and homogeneous saltwater fluid, we find that density stratification significantly enhances the radial dispersion of the disc, while simultaneously decreasing the vertical descent speed, fluttering amplitude, and inclination angle of the disc during descent. We explain the physical mechanisms underlying these observations in the context of a quasi-steady force and torque model. These findings could have significant impact on the understanding and design of uncontrolled vehicle descents and of geological and biological transport where density and temperature variations may occur.
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Asset Metadata
Creator
Lam, Try
(author)
Core Title
Passive flight in density-stratified fluid environments
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Publication Date
11/30/2018
Defense Date
05/06/2018
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Aerodynamics,descent,flow–structure,fluids,fluttering,free-falling,interactions,OAI-PMH Harvest,stratified
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Kanso, Eva (
committee chair
), Jang, Juhi (
committee member
), Luhar, Mitul (
committee member
), Newton, Paul (
committee member
), Spedding, Geoffrey (
committee member
)
Creator Email
trylam@gmail.com,trylam@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c89-109902
Unique identifier
UC11676631
Identifier
etd-LamTry-6988.pdf (filename),usctheses-c89-109902 (legacy record id)
Legacy Identifier
etd-LamTry-6988.pdf
Dmrecord
109902
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Lam, Try
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
descent
flow–structure
fluids
fluttering
free-falling
interactions
stratified