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Boundary layer transition due to the entry of a small particle.

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Boundary layer transition due to the entry of a small particle.

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. __ ® UMI Bell & Howell Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BOUNDARY LAYER TRANSITION DUE TO THE ENTRY OF A SMALL PARTICLE by Paul Hiroshi Taniguchi A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Aerospace Engineering) May 1999 Copyright © 1999 Paul Hiroshi Taniguchi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9933730 Copyright 1999 by Taniguchi, Paul Hiroshi All rights reserved. UMI Microform 9933730 Copyright 1999, hy UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRaOUATE s c h o o l UNIVERSITY PARK LOS ANGELES, CALIFO RNIA 90007 This dissertation, written by ........ under the direction of h.i.-f... Dissertation Committee, and approved by a ll its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of re quirements fo r the degree of DOCTOR O F P H IL O S O P H Y CTV7 O g g a ^ i^ iM ilk a te S tu d i e s Date ..AEril_23,, _1999 DISSERTATION COMMITTEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication This dissertation is dedicated to my parents Yoshio and Michiko who have been a great source of love and support. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments I am deeply indebted to Professors Fred Browand and Ron Blackwelder for their wisdom and their patience. I would like to thank Professors Eckart Meiburg, J.A. Domaradzki, and Paul Ronney for their guidance during the writing of this dissertation. Over the course of this research, I have received a great deal of help from the staff of the Department of the Aerospace Engineering at U.S.C. I greatly appreciate the technical support provided by Thane DeWitt, Ken Richards, Mark Trojanowski, and Dennis Piocher. Also, I want to acknowledge the assistance of Depei Liu, Jared Isaacs. Eric May, and Aaron Tucker. A large part of this research was made possible by the NASA Ames Research Center which graciously provided access to their computers. Funding was provided in the initial stages of this research by the Naval Underwater Systems Center and the Office of Naval Research. I thank the U.S.C. Graduate School for providing a Dissertation Fellowship to me. Finally, I am grateful to Dr. Meng Wang of the Center for Turbulence Research for his assistance during the computational phase of this research. Paul Taniguchi 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Dedication...............................................ii Acknowledgments ..........................................iii List of Figures ..........................................vi List of Tables ............................................xiii ABSTRACT................................................. xiv 1. INTRODUCTION......................................... 1 2. BACKGROUND........................................... 4 2.1. Natural Transition Process .................... 4 2.2. Bypass Transition ............................. 5 2.3. Vortices and Transition.......................15 3. SMALL PARTICLES AND TRANSITION...................... 37 3.1. Previous Experiments with Small Particles and Vortices ............................................37 3.2. Free-Falling Sphere Experiment and Equipment .. 44 3.3. Experimental Results .......................... 49 3.4. Discussion.................................... 58 4. NUMERICAL SIMULATION.................................74 4.1. Hill's Spherical Vortex ....................... 75 4.2. Numerical Code................................ 80 4.3. Vortex Pair Simulation........................83 4.4. Numerical Simulation Parameters .............. 85 4.5. Visualization with Marker Particles .......... 86 5. STRONG CASE.......................................... 93 5.1. Marker Particles .............................. 94 5.2. Vortices present in the boundary layer........ 100 5.3. Velocity Contour Plots ...................... 114 5.4. Wall Shear Stress .............................. 120 5.5. Discussion......................................123 5.6. Comparison of Numerical and Experimental Results ..............................................126 6. WEAK CASE............................................. 154 6.1. Marker Particles ............................... 155 6.2. Vortices Present in the Boundary Layer........ 157 6.3. Velocity Contour Plots ....................... 159 6.4. Wall Shear Stress .............................. 161 6.5. Discussion......................................162 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.6. Comparison of the Strong and Weak Vortex Cases ................................................ 163 6.7. Comparison of Numerical and Experimental Results .............................................. 165 7. CONCLUSION............................................. 176 Appendix A TRANSITION DUE TO FIXED ROUGHNESS ELEMENTS ... 179 Appendix B VORTEX IDENTIFICATION......................... 186 REFERENCES................................................ 191 V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Figure 1 Hairpin and arch vortices in a wall-bounded shear flow.....................................................2 6 Figure 2 Vortex pairs used as the initial conditions in Breuer & Landahl (1990), and Henningson et al. (1993). Contours of Ox vorticity. Contour spacing is 0.01. Negative contours are dashed........................... 26 Figure 3 Evolution of vertical shear layer at z = 0. u- perturbation contours. Contour spacing is 0.02. Dashed contour lines represent negative values . (Breuer S c Landahl, 1990) ......................................... 27 Figure 4 Formation of oblique, spanwise shear layers, u perturbation velocity contours at y/0* = 1.05. Contour spacing is 0.002. Dashed contour lines represent negative values. (Breuer & Landahl, 1990) ............. 28 Figure 5 Fission of vertical shear layer. v perturbation velocity contours at y/5' = 1.05 Contour spacing is 0.002. Negative contours are dashed. (Breuer S c Landahl, 1990) ......................................... 28 Figure 6 u perturbation velocity contours at y/5* = 1.05. Contour spacing is 0.02. Negative contours are dashed. Evolution from negative amplitude disturbance. (from Breuer & Landahl, 1990)............................... 29 Figure 7 u perturbation velocity contours at y = -0.56. Contour spacing is 2.0x10"^. Negative contours are dashed. Evolution of weak (e = 0.0001) disturbance (from Hennings on et al. 1993)......................... 30 Figure 8 u perturbation velocity contours at y = -0.56. Contour spacing 0.025. Evolution of moderate (e = 0.0699) disturbance. (from Henningson et al. 1993)30 Figure 9 u perturbation velocity contours at y = -0.56. Contour spacing is 0.025. Evolution of moderate (e = 0.0699) disturbance rotated 20° about the vertical axis. (from Henningson et al. 1993)................. 30 VI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 10 cùx contours at x = 17, 20, 23 at t = 40 for vortex pair with amplitude e = 0.1399. Negative contours are dashed. Contour spacing is 0.25. (from Henningson et al. 1993) .............................................. 31 Figure 11 u perturbation velocity contours at y = -0.56 at t = 64.2. Contour spacing is 0.025. Turbulent spot created with amplitude s = 0.1399 (from Henningson et al. 1993) .............................................. 31 Figure 12 Breakdown of streamwise streaks into a turbulent spot (from Chambers & Thomas 1983) .................... 32 Figure 13 Perry et al. (1981) model of turbulent spot growth ........................................................ 32 Figure 14 Evolution of a hairpin vortex filament in a mean shear flow. Plan view (left), side view (right) (from Hon S c Walker 1991)..................................... 33 Figure 15 Dye visualization of vortices which developed from the steady wall jet. Plan and side view. Secondary vortices are labeled with a '. (from Acarlar & Smith 1987b) ..................................................33 Figure 16 Hydrogen bubble wire visualization of vortices which developed from the unsteady wall jet. Bubble wire at y = 1.26'. Overhead view. (from Haidari & Smith 1994) .................................................. 34 Figure 17 Hydrogen bubble wire visualization of vortices which developed from the unsteady wall jet. Bubble wire at y = 0.35*. Overhead view. (from Haidari & Smith 1994) .................................................. 34 Figure 18 Hydrogen bubble wire visualization of vortices which developed from the unsteady wall jet. Bubble wire at y = 1.2Ô*. Head-on view. (from Haidari & Smith 1994)34 Figure 19 u perturbation velocity contours at y = 2.41, t = 150.10. Contour spacing is 0.02. (from Singer & Joslin 1994)........................................... 35 Figure 2 0 Plan view of pressure iso-surfaces at t = 153.10. Gray surfaces are p = -0.035, black surfaces are p = 0.02. (from Singer & Joslin 1994)................ 35 V IX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 21 (ûz contours on the z = 0 plane. Contour spacing is 0.1. -0.1<Cùz<0.1 (from Singer 1996) 3 5 Figure 22 Development of system of hairpin vortices over time. These are surfaces where the eigenvalues of the local velocity gradient tensor is complex. (from Zhou et al. 1996)........................................... 36 Figure 23 Smoke visualization pictures of the turbulent spot created by the impact of a falling sphere at two points in its development. Plan view. (from Hall 1967)..... 63 Figure 24 Experimental setup ............................. 63 Figure 25 Ref of the spheres used in the experiment. Solid lines are the Ref of the Hill's spherical vortices in the simulation..............................................64 Figure 2 6 Diagram of camera set-up for oblique angle view 65 Figure 27 Diagram of camera set-up for vertical light sheet65 Figure 2 8 Diagram of camera set-up for overhead view of the flow field..............................................65 Figure 29 Oblique view of boundary layer transition resulting from Ref=5xl0^ sphere (Res.=l. 1x10^ ) . Times are with respect to the time impact and are scaled with respect to freestream velocity and displacement thickness at impact.................................... 66 Figure 3 0 Oblique view of boundary layer interaction with Ref = 2x10^ sphere (Reg- = 1.1x10^) . Times are with respect to the time impact and are scaled with respect to freestream velocity and displacement thickness at impact..................................................67 Figure 31 y-z cross section of wake from Ref=5xl0^ sphere landing at Res- = 1.1x10^ following sphere impact. Light sheet is 185’ away from impact location. Time has been scaled with 5*/Uoo....................................... 68 Figure 32 y-z cross section of wake from Ref=5xl0^ sphere landing at Res- = 1.1x10^ following sphere impact. Light sheet is 465’ away from impact location. Time has been scaled with 5’/U«.......................................69 Vlll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 33 Overhead view of boundary layer transition resulting from Ref=5xlO^ sphere (Res*=l. 1x10^ ) . Times are with respect to the time impact and are scaled with respect to freestream velocity and displacement thickness at impact.....................................71 Figure 3 4 Growth of spanwise width of the dye pattern produced by Ree=5xl0^ acetate sphere (Res»=l. 1x10^) . to is the time of impact of the sphere........................ 72 Figure 3 5 Leading edge position of the dye pattern produced by Reg=5xlO^ sphere (Reg.=l. 1x10^) . Xo and to are the streamwise position and time of impact of the sphere respectively............................................ 72 Figure 3 6 Overhead view of boundary layer interaction with Reg = 2x10^ sphere (Reg. = 1.1x10^) . Times are with respect to the time impact and are scaled with respect to freestream velocity and displacement thickness at impact.................................................. 73 Figure 3 7 Drawing of sphere/wake system and vortex/wake system.................................................. 89 Figure 3 8 Hill's spherical vortex in the Blasius boundary layer. | d ) | vorticity contours on the z = 5.73 plane (-0.035*0 from centerline). (-) contour spacing is 5.0; (--) contour spacing is 0.1............................ 89 Figure 3 9 Diagram of the computational domain.............89 Figure 40 Comparison of growth rates of the s = 0.2 vortex pairs for the Test run and the Henningson et al. (1993) simulation. Time History of total energy normalized by initial energy. (— ) :Test run; (--):Henningson et al. (1993) . Data from Henningson et al. (1993)........... 90 Figure 41 u perturbation velocity contours at y/5* = 1.0. Contour spacing is 0.01. Negative contours are dashed.91 Figure 42 v perturbation velocity contours at y/5* = 1.0. Contour spacing is 0.001. Negative contours are dashed.91 Figure 43 Initial positions of marker particles..........92 Figure 44 Initial position of marker particles in the 'ribbon'. Side view....................................92 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 45 Growth of energy with respect to time for the strong vortex case (a =0.8). v", ( . . . ) : w~ , (--) : ir+v“+w'................................133 Figure 46 Instantaneous (x, z) positions of marker particles which originated in the 'ribbon. ' Side View.........134 Figure 47 Instantaneous (x, y) positions of marker particles in the region 0 . 35<y<l. 0 . Plan View................... 135 Figure 48 Instantaneous (x, y) positions of marker particles in the region 0.0<y<G.35. Plan View 13 6 Figure 49 Trajectories of particles originating in 8.5<x:^9.0, y=0.3 over the time period 0.0^t<94.7......137 Figure 50 Trajectories of particles originating in 15.0<x^l5.5, y=0.3 over the time period 0.0^t<94.7. ..137 Figure 51 Iso-surfaces of regions where the eigenvalues of the local velocity tensor is complex. Arrows show the positions of the planar contour plots 'slices' through the domain............................................. 140 Figure 52 cùx vorticity contours at several streamwise locations at t = 14.8. Strong vortex case. Contour spacing is 0.5. Negative contours are dashed......... 141 Figure 53 cOx vorticity contours at several streamwise locations at t = 3 9.6. Strong vortex case. Contour spacing is 0.2. Negative contours are dashed......... 141 Figure 54 < x » x vorticity contours at several streamwise locations at t = 59.5. Strong vortex case. Contour spacing is 0.2. Negative contours are dashed.........142 Figure 55 Re = 800 vortex pair impacting a wall. Cù contours at different points in time. (from Orlandi 1990).................................................. 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 55 Pressure contours on the z = 5.73 plane (-O.OBô'o from centerline) (a,c,e,g,i,k,m). (a) Contour spacing is 0.02; contour spacing is 0.002 for the rest. cOz vorticity contours on the z = 5.73 plane (b,d,f,h,j,l,n). (b) Contour spacing is 1.0; contour spacing is 0.2 for the rest. Negative contours are dashed. Strong vortex case...........................145 Figure 57 u velocity contours on the z = 5.73 (-0.03ô*o from centerline) plane. Strong vortex case. Contour spacing is 0.1................................................. 148 Figure 58 u velocity contours on the z = 4.77 plane (-5’o from centerline). Strong vortex case. Contour spacing is 0.1................................................. 149 Figure 59 u perturbation velocity contours on the y = 1.0 plane. Strong vortex case. Contours spacing is 0.1. Negative contours are dashed.......................... 150 Figure 50 Perturbation skin friction coefficient on y = 0.0 for the Strong Vortex Case. Contours spacing is 0.001. Negative contours are dashed.......................... 152 Figure 51 Comparison of spanwise widths of the tracer patterns from the Ref=5x10^ sphere and the Strong Vortex case from the simulation, to is the time of impact of the sphere or vortex.......................................153 Figure 52 Comparison of the leading edge of the tracer patterns from the Ref=5x10^ sphere and the Strong Vortex case from the simulation, xq and to are the streamwise position and time of impact of the sphere or vortex. . 153 Figure 63 Growth of energy with respect to time for the weak vortex case (a = 0.4). {— ) : u' , (-.-.): , ( . . . ) : , (--) : 157 Figure 54 Instantaneous (x, z) positions of marker particles in the region 0 . 3 5<y^l. 0 . Plan View...................158 Figure 55 Instantaneous (x, z) positions of marker particles in the region 0.0<y<0.35. Plan View...................158 Figure 65 Instantaneous (x, y) positions of marker particles which originated in the 'ribbon.' Side View.......... 159 XI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 67 Trajectories of particles originating in 9.5<x<10.0, y=0.3 over the time period 0.0<t<100.8. ..170 Figure 68 Trajectories of particles originating in 19.5<x<20.0, y=0.3 over the time period 0.G^t<lG0.8. .170 Figure 69 Iso-surfaces of regions where the eigenvalues of the local velocity tensor is complex..................171 Figure 70 u velocity contours on the z = 5.73 plane (-0.035*0 from centerline). Weak vortex case. Contour spacing is 0.1.........................................173 Figure 71 u perturbation velocity contours on the y = 1.0 plane. Weak vortex case. Contours spacing is 0.1. Negative contours are dashed.......................... 173 Figure 72 Perturbation skin friction coefficient on y = 0.0 for the Weak Vortex Case. Contours spacing is 0.001. Negative contours are dashed.......................... 174 Figure 73 Comparison of growth rates for the strong and weak vortex cases. Time History of total energy normalized by initial energy. (— ) : Strong Vortex Case (a = 0.8); {--) : Weak Vortex Case (a = 0.4).....................175 Figure 74 Contours of u velocity on the y = 1.3 plane, (a) t = 98.74, (b) t = 116.69, (c) t = 134.64, (d) t = 152.6 (from Saiki & Biringen 1997) ..........................184 Figure 75 Transition plot for fixed roughness elements (from Smith & Clutter 1959)................................ 185 Figure 76 Motion of a fluid particle P about O ..........190 Figure 77 Streamline patterns for (a) Spiral (b) Node and two Saddles, (from Soria et al. 1994)................ 190 Figure 78 Contour plot of A in Eqn.9 ....................190 XI1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table 1 Comparison of the size and buoyancy of particles used in transition experiments. (LH): Ladd & Hendricks (1985); (LPS) : Lauchle et al. (1995); (USC) : USC experiment Uœ = 20 cm/s................................. 62 Table 2 Comparison of length scales, velocities, and energies for various disturbances. (BL): Breuer & Landahl (1990) ; (SJ) : Singer & Joslin (1994); (HS) : Haidari & Smith (1994). Lengths are normalized by Ô ; velocity is normalized by ............................88 xiir Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT This dissertation will examine how a laminar, unstable boundary layer can be disturbed by a small particle entering it. This problem is relevant to the performance of small submersibles which operate in oceans that contain plankton which disturb the submersible boundary layer during their missions. An experiment was conducted in a water channel to examine the boundary layer reaction to individual spheres falling through the boundary layer. Flow visualizations showed that the wake left by the falling sphere triggered a turbulent spot only if the sphere had fallen at a sufficiently large velocity. After the sphere impacted the wall, it did not create further turbulent spots. Direct numerical simulations were carried out which modelled the falling sphere as a Hill's spherical vortex of the same size. Like the experiment, the simulation showed that the ability of the Hill's spherical vortex to cause transition depended upon its fall velocity. This fall velocity controlled the severity of the vortex's impact with the wall. When the vortex was sufficiently strong, it produced a disturbance which grew in size and complexity. XIV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. INTRODUCTION Although the majority of boundary layer stability research assumes the absence of heavy, foreign particles in the fluid, small submersibles operate in water laden with particles such as plankton. Plankton are a diverse group of small plants and animals with little or no swimming capability. It has been observed that the presence of small particles in water affected the transition to turbulence in the boundary layers of model submersibles (Ladd & Hendricks 1985; Lauchle, Petrie, & Stinebring 1995). If the particles are smaller than the boundary layer thickness and are travelling fast enough, they are capable of disturbing the boundary layer sufficiently to cause premature transition. This has an adverse impact on small submersibles which rely on maintaining a laminar boundary layer over a significant portion of their hulls in order to minimize drag and fuel consumption. Efforts to stabilize the boundary layer by wall heating or contouring the hull to maintain a favorable pressure gradient (Barker & Gile 1981; Lauchle & Gurney 19 84; Ladd & Hendricks 1985) have been defeated by the effect of the particles. The research described in this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dissertation is a study of the transition due to a single, heavy particle falling through a laminar boundary layer. The first part of this research was an experiment carried out at U.S.C. in which the boundary layer reaction to a single falling sphere was studied. In this experiment, observations of the wake left by the sphere as it fell and the interaction of the wake with the boundary layer were made with the aid of flow visualization. As will be shown in this dissertation, the flow field created by the sphere during its fall and impact played a significant role in the boundary layer transition. The second part of the research was a numerical simulation of the boundary layer interaction with a model of a falling sphere. The numerical simulations were carried out on computers at NASA-Ames Research Center. The falling sphere was modelled as a Hill's spherical vortex travelling through the boundary layer. Under certain conditions, the vortex was able to create a disturbance which grew in size and complexity. It will be shown that the effect of the falling sphere, and the vortex, was to deposit high speed fluid in the lower part of the boundary layer. Similar 'sweeps' of high speed fluid play a role in the bursting process in turbulent boundary layers (Robinson 1991). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In contrast to the disturbance mechanisms used in some previous experiments and numerical simulations on transition, the falling objects described above were smaller than the boundary layer thickness. Also, the falling spheres were not fixed to the wall, like a roughness element or a wall orifice. §2 contains a general discussion of previous boundary layer transition research. §2.1 briefly reviews the transition process in a naturally developing boundary layer. §2.2-2.3 describes research in which transition was caused by the artificial generation of vortices. §3 describes research in which boundary layers were perturbed by particles or vortices moving through them. §3 also includes a description of the experiment conducted at U.S.C. with falling spheres. The numerical simulation with the vortex is described in §4-6. The conclusion is given in §7. This dissertation will borrow some terms described by Wallace, Eckelmann, & Brodkey (1991) for turbulent boundary layers. The terms 'sweep' will refer to the downward movement of high speed fluid and 'ejection' will refer to the upward movement of low speed fluid. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2. BACKGROUND 2.1. Natural Transition Process Under classical stability theory, turbulence in parallel shear flows originates with the linear growth of infinitesimal disturbances into two-dimensional traveling waves. For an inviscid, parallel, two-dimensional flow, the wave characteristics are governed by Rayleigh's equation which is derived from the Euler equation. Growing waves can only occur if the profile has an inflexion point (Drazin & Reid 1981). Both the Rayleigh's inflexion-point theorem and the Fjortoft's theorem state that the growth of these waves is determined by the shape of the mean velocity profile. Because a laminar boundary layer profile contains no inflexion point, it is stable with respect to these waves. The behavior of the infinitesimal, two-dimensional, traveling waves is somewhat different in a viscous fluid, because unstable waves can exist in a laminar boundary layer. In the Blasius boundary layer, the stability of the Tollmien-Schlichting wave depends upon its wave number and the Reynolds number. Res-, where Res- is defined as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R e , =M : V where 5' is the displacement thickness . The range of Reg. and wave numbers for which an unstable wave can exist in the Blasius boundary layer have been experimentally determined by Schubauer & Skramstad (1947) and numerically by Jordinson (1970) . Klebanoff, Tidstrom, & Sargent (1952) have observed that two-dimensional Tollmien-Schlichting waves evolve into an unsteady, three-dimensional, streamwise vortex system. Further downstream, these vortices become hairpin vortices which eventually break down. Hairpin vortices are common structures in the boundary layers during the late stages of transition and in turbulent boundary layers and were described by Head & Bandyopadhyay (1981) . As Figure 1 shows, hairpin vortices consist of a head with spanwise vorticity connected to vortex legs which are tilted with respect to the wall. A similar type of vortex, the arch vortex, has the same structure as a hairpin vortex, however the spacing between the legs is greater for this vortex than for the hairpin vortex. 2.2. Bypass Transition In contrast with the transition, described in the previous section, the following sections will discuss Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transition which does not involve the growth of 2-D waves of infinitesimal amplitude. This has been loosely termed bypass transition, because the transition 'leaps' ahead into the late stages of classical transition when the Tollmien- Schlichting waves have grown into hairpin vortices. This type of transition can be triggered by strong amplitude disturbances introduced into a boundary layer. One mechanism which has been thought to play a role in bypass transition, is the 'lift-up' mechanism. The 'lift-up' mechanism was described by Landahl (1980) as an inviscid instability mechanism which was distinct from those described in §2.1. This mechanism involves the 'lift- up ' of low momentum fluid to a higher elevation in a shear layer. This has been interpreted as the creation of vertical vorticity through the warping of vorticity lines, which were initially aligned in the spanwise direction, into the vertical direction (Henningson, Lundbladh, & Johansson 1993). The 'lift-up' mechanism is quite different from either the inflexional or Tollmien-Schlichting instabilities which are two dimensional in nature. Landahl's (1980) analysis indicated that unlike the exponential growth or decay of the Rayleigh or Orr-Sommerfeld eigenmodes, the total perturbation kinetic energy will have at least a linear Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. growth. rate. The analysis also indicated that the disturbance will grow into streaks of high and low-speed fluid in a shear flow with no inflexion point in the mean velocity profile. As the discussion of the Henningson et al. (1993) simulation will show, a finite amplitude disturbance can grow via the 'lift-up' mechanism in a flow which is subcritical with respect to Tollmien-Schlichting instabilities. There have been several numerical simulations which studied the lift-up mechanism by superposing vortices into a laminar wall-bounded shear flow and observing their subsequent evolution. Breuer & Landahl (1990) conducted a numerical simulation of the Navier-Stokes equations, which used two counter-rotating, streamwise vortex pairs as the initial condition (see Figure 2). This simulation was intended to model an experiment by Breuer & Haritonidis (1990) which studied the response of a boundary layer to a 'pop-up' roughness element. The formula for the vortex disturbance used in this simulation is given below: M = 0 e V = — X f (l - 2z-) (1) w = — xÿ-z{2ÿ^-3) y where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. _ ^ _ y _ z . T : = T-, y-J~' Z =T (x, y, z) are the nondimensional spatial coordinates with respect to the center of the vortex system and are rescaled by the nondimensional numbers dx, ly, 1%) respectively when computing the initial perturbation velocity field. The vorticity field takes the form of vortex loops whose major axes are parallel to the streamwise direction. Fluid is forced up between the downstream vortex pair, and fluid is forced down between the upstream vortex pair for a positive value of e. In Breuer & Landahl's simulation, the vortex disturbance was placed in tandem in a temporally growing Blasius boundary layer whose initial inflow Reynolds number. Res-, was 950. The computational domain was periodic in the streamwise and spanwise directions. s, the nondimensional amplitude of the vortex pairs, was set to 0.2, so |vjmax, the magnitude of the highest v velocity between a vortex pair, was approximately O.OOSUo,. The length scales used in Eqn.(1) were 1% =5, ly = 1-2, 1% = 6. The streamwise scale of the disturbance, based on the distance over which v on the plane y « 1.5 decayed to 0.l|vjm^, was approximately 20S*. This is the same order of magnitude as the wavelength Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of an unstable Tollmien-Schlich.ting wave for Reg* = 950. A similar estimate of the spanwise scale of the disturbance indicates that the spanwise scale was approximately 245". Also, the vortex pairs extended vertically through most of the boundary layer. Because there was nonzero v velocity in the initial velocity field, vertical vorticity was created through the 'lift-up' mechanism, i.e. boundary layer vorticity lines were bent into the vertical direction by the vortex pairs. As a result of the lift-up mechanism, the magnitudes of u and w velocity were found to grow with time. The development of the vortex disturbance in the Breuer & Landahl (1990) simulation was marked by the development of two types of shear layers. The first type was a vertical shear layer shown in Figure 3 which was caused by the deformation of the vortex pairs by the mean shear. Nonlinearity caused the development of oblique high-speed streaks to the sides of the disturbance (Figure 4) . The spanwise shear layers created by the lift-up effect were weaker than the vertical shear layer. In the late stage of simulation, at t = 117, Figure 5 shows the appearance of additional thinner oblique streaks. In addition. Figure 5 shows that the disturbance along the centerline had split into two pieces at t = 117. Breuer & Landahl (1990) wrote Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. that this was the result of a secondary instability in the vertical shear layer. Breuer & Landahl (1990) carried out another simulation in which the sign of the vortex disturbance was reversed, i.e. £ was set to -0.2 in Eqn. (1) . Under this initial condition, fluid was forced down between the downstream vortex pair, and fluid was forced up between the upstream vortex pair. This disturbance structure grew more slowly than the disturbance with the positive amplitude. While there was no strong, vertical shear layer along the centerline, there were weaker vertical shear layers in the side lobes of the structure (see Figure 6). These vertical shear layers were strong enough to form streaks outboard of the disturbance albeit at a less vigorous pace. Thus, the development of the disturbance depended on the strength of these vertical shear layers. Breuer & Landahl (1990) also studied the effect of changing the spanwise length scale of the initial disturbance, 1%, in another set of simulations. Due to continuity, reducing this length scale had the effect of increasing the magnitude of the initial v velocity. It was found that the energy growth rate of the disturbance was quite dependent on the magnitude of the initial v velocity. 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Like Breuer & Landahl (1990), Henningson et al. (1993) carried out a Direct Numerical Simulation using two counter- rotating, horizontal vortex pairs as the initial condition in plane Poiseuille and Blasius boundary layer flows. Henningson et al used Eqn.(l) for the Blasius boundary layer simulations. For the plane Poiseuille simulations, M =0 V = — X (l-y)^(l + y) ^ (2) w = £xz(4y — 4y^)e”^ ' ^ was used. The majority of their article focused on the plane Poiseuille cases. Henningson et al. (1993) looked at three different amplitudes, e, in Eqn. (2) for the plane Poiseuille case. Although Eqn. (2) is different from Eqn.(l), the velocity field is qualitatively similar to the one used by Breuer & Landahl. The length scales used in Eqn. (2) were 1% = 2, 1% = 2. Using the same definitions for disturbance scale as before, this corresponds to a disturbance streamwise length of 8h and a spanwise width of 8h. Henningson et al. also ran additional simulations which investigated the effect of rotating the vortex disturbance about the vertical axis. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The plane Poiseuille flow in their simulation was subcritical, because the Re^ was 3 000 where Re was defined as in which Ud is the centerline velocity and h is the channel half-height. Although all Tollmien-Schlichting waves will decay at this Reynolds number, the disturbance grew in amplitude if the vortex disturbance amplitude, e, was sufficiently large. Linear analysis of the disturbance showed that the 'lift-up' effect was the source of this transient growth. The computational domain in the plane Poiseuille flow was periodic in the streamwise and spanwise directions. Henningson et al. ran a plane Poiseuille flow simulation in which the vortex disturbance amplitude, e, was set to 0.0001. At this particular amplitude, |v|niax of the vortex disturbance was approximately 2xlO~^Uci, and there were negligible nonlinear effects. The disturbance formed an internal shear layer which tilted towards the wall like the disturbance in the Breuer & Landahl (1990) simulation. Later in its evolution, the disturbance formed a wave packet which dissipated as it was convected downstream (Figure 7). 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Henningson et al. also ran a plane Poiseuille flow simulation in which the vortex disturbance had an amplitude of e = 0-0699. This corresponded to |v|max = l-SxlO’^Ud, and the disturbance in this case evolved similarly to the disturbance described by Breuer & Landahl (1990) . Unlike the e = 0.0001 case, the energy of the disturbance increased with time. The initial amplitude was strong enough for the disturbance to form high and low speed streaks (Figure 8) and to generate additional oblique streaks as time progressed. It is important to note that the spanwise width of the disturbance did not change by much over the time period shown in Figure 8. In a plane Poiseuille flow simulation where Henningson et al initially rotated the e = 0.0699 vortex disturbance in the horizontal plane, the disturbance developed in a different manner. The disturbance was asymmetric with respect to the spanwise direction (see Figure 9) . Over the course of its development, the disturbance had a higher perturbation energy level than the simulation in which the vortex pairs had no initial rotation. Henningson et al. ran a plane Poiseuille flow simulation where the vortex disturbance had the amplitude e = 0.1399, which corresponded to |v|max = 0.03Uci. Like the prior cases, high and low speed streaks developed. These 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. high/low speed streaks rolled up into streamwise vortices and were accompanied by the roll-up of the vertical shear layer into a spanwise vortex. These vortices formed a hairpin vortex which was similar to the vortices described in §2.1. Preceding breakdown, secondary vortices developed outboard of the primary vortex pair. The primary and secondary vortices interacted and formed a "twisted vortex pair." (Figure 10) Later on in time, the disturbance broke down into a turbulent spot (Figure 11). It was a turbulent spot in the sense that it was a arrowhead shaped region made up of small scale structures. It is of interest to note that the spanwise width of the turbulent spot was approximately 8h which was the same as the spanwise width of the initial vortex disturbance. Henningson et al. (1993) used the vortex pairs disturbance Eqn.(l) as the initial condition for the Blasius boundary layer simulations. The Reg. of the boundary layer was 950, so it was supercritical with respect to Tollmien- Schlichting wave growth. They observed that the disturbance evolved in qualitatively the same manner as the plane Poiseuille flow cases. Temporal boundary layer growth had little effect on the disturbance development. They were only able to observe the secondary instability that Breuer & Landahl (1990) reported, when the y resolution of the 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. simulation was coarsened. Henningson et al. (1993) concluded that the break up of the vertical shear layer was a numerical artifact. Other transition experiments have been carried out which involve introducing a strong perturbation at the wall. This perturbation would leave a 'slug' of fluid deep in the boundary layer which has a different velocity than the surrounding fluid. One example of such a perturbation is a jet produced by a wall orifice. Another example is the wake left by a roughness element. 2.3. Vortices and Transition There have been experiments in which artificial disturbances were introduced into a boundary layer which developed into vortices prior to breaking down. Mochizuki (19 61) observed the vortex systems created by a single spherical roughness element in a laminar boundary layer. Under certain conditions, hairpin vortices shed from the top of the sphere broke down and formed a turbulent wedge downstream of the sphere. As Appendix A will show, the breakdown was due to the alteration of the local velocity profile caused by the presence of the roughness element. Another example of a wall disturbance is the wall jet. Amini & Lespinard (1982) used an unsteady jet from a wall 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. orifice, |v | max = 1 -3U«, to create turbulent spots in a supercritical flat plate boundary layer (Reg. = 600) . They detected the presence of high and low speed streamwise streaks which they called the 'incipient spot'. These streaks resemble the streaks obseirved in the simulations of Breuer & Landahl and Henningson et al. They also detected a 'defect bridge' which was a region of low speed fluid which straddled the high speed region created by the jet. They speculated that these streaks and the 'bridge' were part of a U-shaped vortex whose two legs pointed upstream. Chambers & Thomas (1983) also used a jet to create turbulent spots in a boundary layer. Flow visualization of the boundary layer showed that the jet created streaky smoke patterns. As these streaks convected downstream, they oscillated, broke down, and formed a turbulent spot (see Figure 12). There has been research into the role vortices play in the late stages of transition. Perry, Teh, & Lim (1981) developed a model for the early growth of a turbulent spot. They modelled the boundary layer as a row of line vortices aligned with the spanwise direction. A perturbation, such as a jet, deformed one of the line vortices. This deformed line vortex induced additional deformations in neighboring line vortices which induced further deformations in line vortices neighboring them (see Figure 13). In this manner, 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a vortical disturbance propagates in both the spanwise and streamwise directions. This type of behavior has been seen in the work of Hon & Walker (1991) . They performed numerical simulations into the development of a hairpin shaped vortex filament in a shear flow. The initial deformation on the vortex filament induced additional deformations ( ' subsidiary hairpins ' ) along the length of the vortex filament due to the Biot-Savart Law. (Figure 14) Hon S c Walker's (1991) simulation neglected any effect that the vortex filament has on the mean flow. Hon & Walker (1991) examined the viscous reaction in the symmetry plane of the mean viscous flow to the presence of the deformed hairpin vortex filament. Upstream of the vortex filament, the boundary layer rapidly thickened and stagnation points formed on the wall. This implies that the vortex was moving low momentum fluid away from the wall. The ability of a hairpin vortex to spawn additional hairpin vortices was observed experimentally by Acarlar & Smith (1987b) and Haidari & Smith (1994) . In Acarlar & Smith's (1987b) experiment, flow visualization was used to observe the development of a low speed streak in a Blasius boundary layer, where the Res» ranged from 200 to 530. A low speed streak was created by a jet issuing from a streamwise slot on the wall. V^, the velocity of the jet 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ranged from 0. 1U« to O.SUoo. Because low speed streaks are a common feature in a turbulent boundary layer, Acarlar & Smith studied the dynamics of one streak in the calmer environment of a laminar boundary layer. Depending on Reg. and Re^w (the Reynolds number based on the jet velocity and the slot width) , a three-dimensional instability developed due to the shear between the streak and the surrounding boundary layer fluid. Eventually, the streak rolled up and formed a train of hairpin vortices as seen in Figure 15. Acarlar & Smith observed two types of secondary structures. One was a streamwise vortex pair which formed outboard of the streak near the wall due to legs of the hairpin vortices pumping fluid down at the sides of the streak. The other secondary structure occurred between the legs of a hairpin vortex upstream of its head. Fluid pumped up between the legs formed a shear layer which rolled up into a vortex. An observer looking at the hairpin vortex from the side would see part of the vortex upstream of the head flatten out (see Figure 15) . Acarlar & Smith attributed this to the legs of the hairpin vortex approaching each together in the spanwise direction due to the effect of image vortices and causing local vorticity annihilation. This reduced the velocity each leg induced on the other and caused the vortex at that location to rotate about the spanwise direction due to mean 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shear. Interestingly, the secondary vortex interacted with the primary vortex in a complicated manner and caused a local ejection of fluid which they likened to the bursting process in turbulent boundary layers (see Figure 15) . Haidari & Smith (1994) conducted a similar experiment in the same facility as Acarlar & Smith (1987b) . But, Haidari & Smith used a pulsed jet to produce, initially, a single hairpin vortex in a laminar boundary layer, 250^eg.<490. They used a similar* range of Re^w as Acarlar & Smith (1987b) . Depending on Reg., Re^,, and 5 ' / t U o o (the nondimensional time the jet was on) , the hairpin vortex either produced secondary structures, or it decayed and had little effect on the surrounding fluid. As seen in the hydrogen bubble wire visualization Figure 16, the hairpin vortex broke down into a turbulent spot under certain conditions. There appeared to be no sign of the secondary streamwise vortices which Acarlar* & Smith (1987b) observed. However, Haidari & Smith did observe the secondary vortices which formed between the legs of the hairpin vortex. They were marked with 'S' in Figure 16. When the hydrogen bubble wire was placed closer to the wall at y = 0.3Ô*, the legs of these vortices were manifest as streaky patterns in Figure 17. These vortices were not limited to being in-line with the hairpin vortex head. Haidari & Smith wrote that the 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. secondary vortices were caused by an inviscid-viscous interaction. Due to the adverse pressure gradient created by the presence of the hairpin vortex, there was an ejection of near wall fluid upward (the viscous action) (Van Domine1en & Crowley 1990) . A shear layer formed between this fluid and the high speed fluid which now surrounded it. This shear layer rolled up into a spanwise vortex (the inviscid action). Haidari & Smith (1994) observed hairpin vortices develop outboard of the primary hairpin vortex's legs which they called 'subsidiary vortices'. The subsidiary vortices were marked with 'SU' in the head-on flow visualization pictures in Figure 18. Haidari & Smith speculated that the 'subsidiary vortices' were formed by the self-deformation of vorticity lines as described by Hon & Walker (1991). Flow visualization also showed the presence of a 'pocket' where the flow was cleared of bubbles released from an upstream bubble wire strung along the spanwise direction (see Figure 16). This 'clearing' was due to the activity of the various vortices. Haidari & Smith compared this 'pocket' to the 'pockets' in turbulent boundary layers (Robinson 1991) . The disturbance continued to spawn new vortices in the manner described above until it became a turbulent spot. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A numerical simulation which was similar to the above two experiments was carried out by Singer & Joslin (1994). They used a spatial simulation of the Navier-Stokes equation on a Blasius boundary layer whose inflow Reg- was 530. The computational domain was periodic in the spanwise direction. During the course of the simulation, the domain was expanded in the streamwise and spanwise directions. A streamwise slot on the wall blew a 'pulsed' jet for 5 nondimensional time units. |v|max of the jet was approximately 0.25Uco. Surfaces of constant pressure were used to visualize the vortices. Like the Haidari & Smith (1994) experiment, the injected fluid rolled up into a hairpin vortex. Unlike the Haidari & Smith experiment, a horseshoe vortex, which was wrapped around the hairpin vortex near the wall with its legs pointed downstream, was observed. They attributed this horseshoe vortex, to a stagnation region at the upstream end of the slot. (While a hairpin vortex is tilted with respect to the wall, the horseshoe vortex lays flat against the wall. The horseshoe vortex is a common structure produced by a roughness element as described in Appendix A. ) The hairpin vortex developed a secondary vortex upstream of the primary vortex head; the hairpin vortex took on a ' flattened ' shape when viewed from the sides and was reminiscent of the dye visualization pictures from Acarlar & 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Smith. (19 87b). Singer & Joslin (1994) also attributed the formation of this secondary vortex to the inviscid-viscous interaction. They observed the formation of a pair of u- shaped vortices underneath the legs of the hairpin vortex. The legs of the u-shaped vortices pointed upstream. The sign of vorticity in these vortices was opposite of the primary hairpin vortex. The presence of the u-shaped vortices made the disturbance wider than the primary hairpin vortex by itself. Singer & Joslin commented that the u- shaped vortices could be the secondary vortices described by Haidari & Smith, but they were more likely artifacts of the injection process. Later on in time, quasi-streamwise vortices develop under the u-shaped vortices. As the flow developed, additional vortices were created by the influence of the existing vortices, while at the same time, older vortices dissipated. Singer & Joslin wrote that the inviscid-viscous interaction accounted for the creation of only some of the vortices. However, they did not observe the formation of the subsidiary vortices that Haidari & Smith observed in their experiment. At the end of the simulation at t=153.10, u perturbation velocity plot at y=2.4l5* (Figure 19) shows that the disturbance developed into a turbulent spot. It was a turbulent spot in the sense that it was an arrowhead shaped region consisting of small 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. scale structures. It is interesting to note that the plan view of the pressure iso-surface plot (Figure 20) at t=153.10 shows a much simpler structure than Figure 19 implies: Figure 2 0 only shows a handful of vortices. Yet as a whole, the flow field created by the combined action of these vortices is very complex. The simulation was continued further along in time in Singer (1996) . Figure 21 shows the shear layer 'backs' of the vortices within the turbulent spot along with the inclined near-wall shear layer at the front of the turbulent spot. The near-wall shear layer overhung a region of undisturbed fluid. These shear layers have been identified as being features of a turbulent boundary layer by Robinson (1991). Another simulation in which a hairpin vortex spawned new vortices was carried out by Zhou, Adrian, & Balachandar (1996) (see Figure 22) . They extracted a streamwise vortex pair from the velocity field of a turbulent channel flow simulation carried out by Kim, Moin, & Moser (1987) . This vortex pair corresponded to an ejection event in the turbulent flow. The vortices were placed into an Reh=33 00 laminar channel flow at y^=49. The mean velocity field from a turbulent channel flow was used as the base flow. The simulation was carried out with a spectral code on a domain which was periodic in the streamwise and spanwise direction. 2 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. visualization of these vortices were carried out using the eigenvalues of the local velocity tensor (see Appendix B) . The fluid pumped up between the vortex pairs rolled up into a spanwise vortex and merged with the vortex pair forming a hairpin vortex. The downstream end of the vortex curled upward and expanded in the spanwise direction and became an arch vortex. Like Haidari & Smith (1994), Zhou et al. observed that a weak hairpin vortex eventually decayed while a strong one formed secondary vortices. The formation of the first secondary vortex proceeded along the lines described by Acarlar & Smith (1987b) . The legs of the arch vortex reconnected at the 'kink' to form a new hairpin vortex. Further along in time, another secondary vortex formed in-line with and downstream of the primary arch vortex. Zhou et al. also made note of a pair of streamwise vortices which formed outboard of the primary arch vortex. They attributed the formation of these vortices to the downwash of fluid outboard of the arch vortex's legs coupled with unsteadiness in the legs' motion. A criterion for transition can be defined based on the results of these experiments and simulations. For the purposes of this criterion, the term 'hairpin vortex' will be used in a generic sense to refer to the structures shown in Figure 1 regardless of the size of spanwise width between 2 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the vortex's legs. If an initial disturbance develops into a hairpin vortex in a boundary layer, it is not sufficient to simply look at a global quantity, like energy. Due to the ' lift-up' effect, even a weak disturbance can grow in terms of size and energy before viscously decaying. If a hairpin vortex has sufficient strength, it can develop additional vortices. The interaction among these vortices will produce a region in the flow which grows in size-and at the same time-generates small scale structures. Thus, the generation of secondary vortices by an initial disturbance will be the criterion for transition used in the discussion of the numerical simulation in this dissertation. 2 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Arch Vortex Hairpin Vortex Figure 1 Hairpin and arch vortices in a wall-bounded shear flow. -10 - 5 10 Figure 2 Vortex pairs used as the initial conditions in Breuer & Landahl (1990), and Henningson et al. (1993). Contours of a>x vorticity. Contour spacing is 0.01. Negative contours are dashed. 2 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VERTICAL SHEAR LAYER r VERTICAL SHEAR LAYER V X /S * Figure 3 Evolution of vertical shear layer at z = 0. u- perturbation contours. Contour spacing is 0.02. Dashed contour lines represent negative values. (Breuer & Landahl, 1990) 2 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ft . / r .4 3 a m Figure 4 Formation of oblique, spanwise shear layers, u perturbation velocity contours at y/5' = 1.05. Contour spacing is 0.002. Dashed contour lines represent negative values. (Breuer & Landahl, 1990) q : - 1 0 -20 - 1 0 -2 0 1 1 7 - 1 0 -20 40 60 x/S* Figure 5 Fission of vertical shear layer. v perturbation velocity contours at y/5* = 1.05 Contour spacing is 0.002. Negative contours are dashed. (Breuer & Landahl, 1990) 2 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / É / Figure 6 u perturbation velocity contours at y/5' = 1.05. Contour spacing is 0.02. Negative contours are dashed. Evolution from negative amplitude disturbance. (from Breuer & Landahl, 199 0) 2 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 ■ 5 - — z 0 - 5 -1 0 - 1 0 - 5 0 5 10 15 X 0 5 10 15 20 X 5 10 15 20 25 10 15 20 25 30 35 X X Figure 7 u perturbation velocity contours at y = -0.56. Contour spacing is 2.0x10'^. Negative contours are dashed. Evolution of weak (s = 0.0001) disturbance (from Henningson et al. 1993). 10 5 0 - 5 - 1 0 - 1 0 - 5 0 5 10 15 0 5 10 15 20 5 10 15 20 25 10 15 20 25 30 35 X X X X Figure 8 u perturbation velocity contours at y = -0.56. Contour spacing 0.025. Evolution of moderate (s = 0.0699) disturbance. (from Henningson et al. 1993) 10 5 z 0 - 5 - 1 0 ( — ___ - 1 0 - 5 0 5 10 15 0 5 10 15 20 X X 5 10 15 20 25 10 15 20 25 30 35 X X Figure 9 u perturbation velocity contours at y = -0.56. Contour spacing is 0.025. Evolution of moderate (e = 0.0699) disturbance rotated 20° about the vertical axis. (from Henningson et al. 1993) 3 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 10 oOx contours at x = 17, 20, 23 at t = 40 for vortex pair with amplitude £ = 0.1399. Negative contours are dashed. Contour spacing is 0.25. (from Henningson et al. 1993) 4 - -4 - 30 35 40 45 50 55 Figure 11 u perturbation velocity contours at y = -0.56 at t = 64.2. Contour spacing is 0.025. Turbulent spot created with amplitude £ = 0.1399 (from Henningson et al. 1993) 3 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 12 Breakdown of streamwise streaks into a turbulent spot (from Chambers & Thomas 1983) » fitKmmt Figure 13 Perry et al. (1981) model of turbulent spot growth 3 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. •' IIIT tM . “ * vorrot wsmm % aw X Figure 14 Evolution of a hairpin vortex filament in a. mean shear flow. Plan view (left) , side view (right) (from Hon & Walker 1991) Top Side (6) x/fi = 30 70 290 Figure 15 Dye visualization of vortices which developed from the steady wall jet. Plan and side view. Secondary vortices are labeled with a ' . (from Acarlar & Smith 1987b) 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ♦ h b w JH B W 1 HBW l i Figure 16 Hydrogen bubble wire visualization of vortices which developed from the unsteady wall jet. Bubble wire at y = 1.25’. Overhead view. (from Haidari & Smith 1994) ♦ HB W ♦ H B W ♦ H B W »HBW Figure 17 Hydrogen bubble wire visualization of vortices which developed from the unsteady wall jet. Bubble wire at y = 0.35’. Overhead view. (from Haidari & Smith 1994) Flow C l) Cross- Bow HBW / = 0 s SÜ1 S U 2 f = 0.083 s Cross-flow ( = 0.25 s / = 0.29 s / = 0 .3 3 s r = 0.166 s t = 0.375 s Figure 18 Hydrogen bubble wire visualization of vortices which developed from the unsteady wall jet. Bubble wire at y = 1.25’. Head-on view. (from Haidari & Smith 1994) 3 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 19 u perturbation velocity contours at y = 2.41, t = 150.10. Contour spacing is 0.02. (from Singer & Joslin 1994). I Low pressure ■ High pressure In-line secondary hairpin vortex Side-lobe secondary hairpin vortex New U-shaped vortex Figure 20 Plan view of pressure iso-surfaces at t = 153.10 Gray surfaces are p = -0.03 5, black surfaces are p = 0.02. (from Singer & Joslin 1994). 3 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G Q ) fc) Figure 21 a>z contours on the z = 0 plane. Contour spacing is 0.1. -0.1<û)z<0.1 {from Singer 199 6) Figure 22 Development of system of hairpin vortices over time. These are surfaces where the eigenvalues of the local velocity gradient tensor is complex. (from Zhou et al. 1996) . 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. SMALL PARTICLES AND TRANSITION 3.1. Previous Experiments with Small Particles and Vortices The experiments and simulations previously discussed had one thing in common: the disturbance which triggered the transition originated from within the boundary layer. They involved either a wall jet, which introduced an unstable disturbance near the wall, or the placing of vortices deliberately within the boundary layer. Roughness elements which locally perturb the surface geometry are discussed in Appendix A. However, a disturbance which originates from outside the boundary layer is just as capable of causing transition as some sort of wall-based disturbance. Numerous experiments (Barker & Gile 1981; Lauchle & Gurney 1984; Ladd & Hendricks 1985; Lauchle et al. 1995) have noted a link between the presence of water-borne particles and changes in the boundary layer transition location on a body travelling through the water. In two similar water channel experiments, Ladd & Hendricks (1985) and Lauchle et al. (1995) carried out velocity measurements on the boundary layer of an ellipsoid when small particles were mixed into the flow. In both of their experiments, the surface of the ellipsoid was heated 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to stabilize the boundary layer against the growth of Tollmien-Schlichting waves. The particles used in both experiments are listed in Table 1. When Ladd & Hendricks added particles which were 80 ^im and larger in diameter to the water, the transition location moved upstream from its position in clean water. Depending on the level of surface heating and the amount of particles added, the transition location changed by as much as 50-70%. Lauchle et al. also noted similar changes in the transition location when they added small particles to the water. However, in contrast to Ladd & Hendrick's results, they observed that particles smaller than 80 pm were capable of changing the transition location. Flow visualization in the Lauchle et al experiment indicated that turbulent spots created by the impact of the small particles originated near the ellipsoid nose. They estimated that the particles were only 0.2-0.65* in size, where 6* was the displacement thickness near the nose. These small particles share a couple of similarities with the plankton a small submersible would encounter in the ocean. As Table 1 shows, these particles were of the same size as microplankton, which is defined as plankton between 0.05 mm and 1.00 mm in size (Halldal 1984). In contrast, the U.S. Navy Mk.48 torpedo is 5.8 m in length and 0 .533 m 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in diameter (Pretty 1980). Some plankton species are heavier than water like the micron-sized particles in Table 1. For example, diatoms have silica shells and sink after their deaths (Halldal 1984). Strickier (1982) observed that copepods do not survive in water which had been treated to have the same density as the copepods. Copepods use their weight with respect to water in order to swim about and seek food. A key difference between the small particles in Table 1 and live plankton is that the small particles cannot move independently. Plankton migrate vertically in response to daylight and food. If a diatom is heavier than the surrounding water and is sinking, it can alter its drag by controlling the configuration of spines on its shell (Groen, Newcombe, & Mero 19 84) . Other species of plankton swim in order to move. Strickler's (1975) measurements of copepods indicate that they can swim at 0.1-0.5 cm/s. Gallager's (1993) measurements of larval molluscs indicate that they have a sinking velocity of 1.3-1.4 cm/s. The terminal velocity of the small particles in Table 1 range from 2x10“^ cm/s to 0.3 cm/s which is in the range of the swimming velocities of these plankton. Thus, these particles can serve as approximate models for the plankton. The velocities of plankton is many times smaller than the 3 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. velocities of small submersibles. For example, the Mk.48 torpedo has a maximum speed of 2 6 m/s (Pretty 1980) . Assuming the micron-sized particles in Table 1 are spherical, the buoyancy of these particles can be described by the Froude number: g(r-\)d where d is the particle diameter, y is the particle specific gravity, and g is the acceleration due to gravity. The denominator of the Froude number is proportional to the square of Vt, the particle terminal velocity: 4 v' = ' 3 Q where Cd is the coefficient of drag of the particle. As Table 1 shows, the Froude numbers of the particles used by Ladd & Hendricks and Lauchle et al. were qnite high due to the smallness of the particles used. It implies that the freestream velocity was more relevant to the particle dynamics than gravity. Although the particle weights were quite small, they were still able to travel from the freestream towards the ellipsoid surface in the experiments of Ladd & Hendricks (1985) and Lauchle et al. (1995). Within the boundary layer, the direction normal to the ellipsoid wall was not 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. parallel to gravity. Even though, the component of particle velocity parallel to gravity was small, the particle would still be capable of reaching the ellipsoid surface at velocities significantly greater than the terminal velocity. The role of gravity in the movement of a small particle was examined in the numerical simulation of Chen, Goland, & Reshotko (1980) . They simulated the trajectory of a small sphere from the freestream to its "capture" by a oncoming larger sphere moving within the flow. The large sphere can be thought of as a model for the blunt nose of a torpedo. The small sphere diameter was 3x10"^ to 1x10"^ times the diameter of the large sphere and had specific gravities between 1.01 and 2.50. The Froude numbers for these small spheres were comparable to those of the particles used by Ladd & Hendricks (1985) and Lauchle et al. (1995) . In the simulation of Chen et al. (1980), the small sphere was initially upstream of the large sphere and in the large sphere path. Depending on the initial position of the small sphere, the small sphere, as it approached the large sphere, either grazed off the large sphere boundary layer or crossed streamlines and entered the large sphere boundary layer. It is important to note that even the small sphere with a specific gravity of 1.01 was able to enter the large sphere boundary layer if it originated from the proper initial 4 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. position (the "capture area"). The simulation did not address how the entry of the small sphere would alter the large sphere boundary layer. As the simulation of Chen et al. (1980) showed, the role of gravity in the experiments of Ladd & Hendricks (1985) and Lauchle et al. (1995) was to shape the particle trajectory towards the ellipsoid. The ability of a single particle to cause transition in a boundary layer by moving within the boundary layer was first explored in the experiment of Hall (1967) . In addition to studying the transition due to individual spheres attached to the wall (see Appendix A) , Hall looked at the effect of lowering a sphere down through the boundary layer until it reached the wall. In Hall's experiment, the boundary layer formed along the wall of a pipe. The sphere was mounted on a wire support so it could not move in the streamwise direction. Rexo, which was defined as Re. » V and where Xg was the streamwise position of the sphere with respect to the pipe inlet, was 1x10®, and the sphere was smaller than the boundary layer thickness at x q . The moving sphere was observed to form a turbulent spot when it hit the wall (Figure 23) . After the sphere reached the wall, it remained fixed to the wall and did not trip the boundary layer. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A different type of experiment involving a disturbance which migrated towards the wall from above was carried out by Chu S c Falco (1988) . In this experiment, laminar vortex rings with different speeds and angles of propagation were released above a moving wall. Because the wall was impulsively started, the boundary layer was the unsteady flow described in Stokes' First Problem/Rayleigh's Problem. The diameter of the vortex rings were 0(10)5 where 5 was the thickness of the boundary layer. Their angle of propagation ranged from -6° to +6°, so the vortex rings were glancing into or off of the wall. The motivation of this experiment was to model the Typical Eddies which were believed to play a role in the bursting process in turbulent boundary layers. 5 was supposed to be analogous to the thickness of the viscous sublayer in a turbulent boundary layer. When the vortex ring was moving towards the wall, it formed 'pockets' as observed in flow visualization. A vortex ring moving towards the boundary layer from above had different effects depending on its propagation speed. A vortex ring whose initial speed, Ur, was 0.35 times the wall velocity, U„, induced streaks in the boundary layer which became unstable and broke down over a long time period. A vortex ring with Ur/U„ > 0.45 caused a lift-up of boundary layer fluid upstream of the vortex ring. This ejected fluid rolled up 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. into hairpin vortices which later broke down in different trials. Thus, disturbances which move through a boundary layer can produce streaks and hairpin vortices like the wall jets previously mentioned in this dissertation. 3.2. Free-Falling Sphere Experiment and Equipment In the experiment carried out at USC as part of this research, small spheres were individually dropped into a transitional flat plate boundary layer at Reg. ranging from 10^ to 1.2x10^. Because there was no pressure gradient in the boundary layer, this represents a more simpler situation than the boundary layer on the nose of a torpedo which does have a favorable pressure gradient. The length scale used in this section is 6% the displacement thickness at the sphere impact location. These spheres had an initial streamwise velocity of Ug and vertical velocity, Vg. Unlike the spheres in Hall's (1967) experiment, these spheres were not constrained in their movement. Once these spheres hit the wall, they continued to roll downstream along the wall. This experiment was performed in the Blue Water Channel facility which has a free surface, a 30x90x500 cm test section, and glass walls in order to allow observation of the flow. The boundary layer developed on a horizontal flat plate in the test section. This flat plate had an 4 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. elliptical leading edge. LDA velocity measurements for X > 0.6 m showed that the boundary layer approximately matched the Blasius boundary layer profile. All of the spheres had the same diameter, but they had three different densities and thus, had different fall velocities. Polystyrene, acetate, and glass spheres were used in this experiment, and their specific gravities were 1.1, 1.5, and 2.5, respectively. Individual spheres were dropped from a computer controlled bead ejector (see Figure 24) . The bead ejector muzzle was 18 cm above the flat plate. The Reynolds number based on the sphere diameter, d, and the free-fall speed, Vf, was v,d Re, where Vf was determined by dropping individual spheres with the bead ejector and filming the descent of the spheres at X = 1 m with the video camera placed to the side of the water channel. Water was flowing through the water channel at a fixed Uœ during the filming. The streamwise and vertical positions of a particular sphere as a function of time was determined from the videotape. Because the spheres had reached terminal velocity in the freestream, a linear 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. least squares fit of the trajectory data was used to obtain the (Us, Vs) velocity of the spheres in the freestream. Using the freestream velocity U« and Eqn.(3), Vf was computed. Vf ranged from 0.30» for the light polystyrene spheres to l.VUa, for the heavy glass spheres when Uoo ~ 20 cm/s. Ref ranged from 2x10^ to 10^ (see Figure 25) . In order to visualize the sphere wakes, some of the spheres were coated with a moustache wax/fluorescein mixture. This coating dissolved slowly enough to dye the wake as the sphere fell through the water. The dyed wake fluoresced when it was exposed to laser light. These coated spheres were released using the a funnel at the free surface. Using the same procedure as before, Vf was obtained for the spheres released by the funnel. Vf ranged from 0.3Uoo for the light polystyrene spheres to 0. 9U« for the acetate spheres when U» ~ 20 cm/s. Because Vf was dependent on the sphere specific gravity, buoyancy effects were implicitly incorporated in Ref. Like Ref, the Froude numbers of the acetate and polystyrene spheres were dissimilar (see Table 1) . The Froude numbers of these spheres were many orders of magnitude lower than the particles used by Ladd & Hendricks (1985) and Lauchle et al. (1995). Because of the size of 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the spheres, gravity made a larger contribution to the sphere velocity than the micron-sized particles in Table 1. If a single sphere was released upstream of the flat plate leading edge like the particles in the simulation of Chen et al. (1980), the flat plate boundary layer would have "captured" that sphere differently than a micron-sized particle. This is not an issue for the USC experiment because the spheres were released above the flat plate and were always "captured" by the boundary layer. For this experiment, the velocity of the sphere during its fall was more important than whether the source of the sphere motion was gravity or the momentum it had acquired in the freestream. Observations of the boundary layer flow and sphere wake were made with the aid of LIE flow visualization. There were dye slots on the plate at x = 0.13 m, 2.5 m, 4.5 m (Figure 24) . Each dye slot is aligned with the spanwise direction and released dye as an undisturbed sheet when the boundary layer in its vicinity was laminar. By observing the dye coming out of a dye slot and measuring the amount of time the flow in that area was turbulent, the turbulent intermittency at that dye slot was determined. Because the turbulent intermittency follows the normal distribution with respect to x (Schubauer & Klebanoff 1955), the X50 position 4 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where turbulent intermittency was 50% was extrapolated for different U»/v. The natural transition Reynolds number was defined as V and was determined to be 0 (10®) . A Spectra-Physics 2016 argon ion laser was used in combination with either an amplifier-driven optical scanner or a combination of a spherical and cylindrical lens to create different types of wedge-shaped light sheets. One type of light sheet was a ~ 1 mm thick light sheet perpendicular to the flat plate which was created with a General Scanning GO612 optical scanner and a General Scanning AX2 00 amplifier. Another type of light sheet was a w 15 cm thick light volume parallel to the flat plate which was created with the spherical and cylindrical lens combination. A third type of light sheet was a ~ 1 mm thick light sheet parallel to the flat plate which was created with a General Scanning IDS 2 512 optical scanner and a General Scanning AX73 0AC amplifier. This horizontal light sheet was approximately one sphere diameter above the water channel flat plate. A Panasonic WV-D5000 video camera and a AG-6750-P VCR were used to film the flow. In order to get an oblique view 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the flow, the camera was placed on a fixed tripod placed to the side of the water channel (Figure 25) or on a platform mounted above the water free surface. The light sheet perpendicular to the flat plate was used to illuminate a vertical-spanwise cross-section of the boundary layer downstream of a given sphere impact position. Using a mirror placed within the water channel test section far downstream of the impact position, this view was filmed by a camera outside of the water channel (see Figure 27) . The light sheet parallel to the flat plate was used to illuminate a plan view of the disturbance which was filmed by a traverse-mounted camera which followed the disturbance as it moved downstream (Figure 28). 3.3. Experimental Results This section will describe the differences between dropping an acetate sphere and a polystyrene sphere into the boundary layer. The acetate sphere had an Re^ of 5x10^; the polystyrene sphere had an Ref of 2x10^. Reg* and d/5" were kept the same between these two types of spheres. While both spheres had the same diameter, the acetate sphere had a larger mass and had a larger fall velocity than the polystyrene sphere. Both types of spheres were dropped into a boundary layer with Reg- = 1.1x10^ which was supercritical 4 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with respect to Tollmien-Schlichting wave growth. The dyed wake left by a moustache wax/fluorescein coated sphere and its effect on the boundary layer were filmed from the oblique view and with the head-on view. Plan views of the disturbance created by the sphere were filmed by a video camera looking down on the water channel flat plate from above. Figure 29 is a sequence of still frames from the flow visualization video of the acetate sphere of diameter 0.65* dropping into a laminar boundary layer with Reg. = 1.1x10^. This is an oblique view of the boundary layer, and the video camera was panned in order to follow the motion of the sphere wake. The dark area in the pictures is a sheet of rhodamine-B dye solution which was released from dye slots upstream of the sphere impact position. In Figure 29(a), the sphere was in the freestream, and the sphere wake was aligned vertically. The sphere wake has a jagged appearance. After the sphere had impacted the wall, the wake was stretched by the mean shear and rotated into the streamwise direction (Figure 29b-c). The sphere which occupied a lower portion of the boundary layer than the wake trailed behind the wake (Figure 29b-c). In Figure 29(d), the sphere was to the right of the field of view. The wake was completely aligned with the streamwise direction. The 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bright streaks near the center of the dye sheet were fluid particles marked with rhodamine-B fluorescing in the light volume (Figure 29d) . After the impact, the wake had split into two threads which moved apart in the spanwise direction (Figure 29b-d). The two threads overhung a region of undisturbed fluid. The bright region upstream of the two thread correspond to a region of mixing. Downstream of the sphere impact position in the time following the impact, the flow broke down and a turbulent spot was formed (Figure 29e-f). Figure 30 is another sequence of still frames from the flow visualization video of a sphere of diameter 0. 5Ô* dropping into a boundary layer with Reg* = 1.1x10^. However, the sphere was made out of polystyrene and had a lower Ref. The video was filmed in the same manner as the previously mentioned video of the acetate sphere. In Figure 30(a), the sphere was in the freestream, and the sphere wake was aligned vertically. Unlike the acetate sphere wake, this wake displayed very little randomness. Like the acetate sphere in Figure 29(b-c), the sphere wake was also stretched by the mean shear and rotated into the streamwise direction (Figure 3 0b-c) . When the sphere hit the wall, the recirculating region above tbe sphere detached and proceeded downstream with two threads trailing behind (Figure 3Oc-e) . 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In contrast to the wake in Figure 29(c), the wake within the boundary layer in Figure 30(c-d) had an undistorted shape and was not as aligned with the streamwise direction as the acetate wake. The low speed fluid marked by the dye sheet was not disturbed by the polystyrene sphere wake (Figure 30d) as it was by the acetate sphere wake (Figure 29d). As Figure 30(d) and Figure 30(e) show, a turbulent spot was not present downstream of the sphere impact position following impact. The sphere rolled downstream trailing behind the wake which was higher in the boundary layer (Figure 3 0f). Head-on views of a vertical-spanwise cut through the boundary layer were filmed for the 0.65* diameter moustache wax/fluorescein coated acetate sphere dropping into a boundary layer with Reg. = 1.1x10^. Figure 31 is a sequence of still frames from the flow visualization video showing the, sphere wake convecting through the vertical-spanwise light sheet which was fixed at 186* downstream of the impact position. In Figure 31(a), the dark band across the bottom of the picture was an acridine-orange dye sheet released form the dye slots upstream of the light sheet. The two dark spots above the dye sheet was a cross-section of the sphere wake showing it had split into two threads (see Figure 29c). These two cores approached the wall as seen in Figure 31(a-c). This indicates that the upstream portion of 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the wake (Figure 31a) was higher in the boundary layer than the downstreaia portion (Figure 31b). Figure 31(c) shows that low speed fluid near the wall and outboard to the cores was induced upward by their influence. This implies that the right core (in Figure 31c) had counter-clockwise rotation and the left core had clockwise rotation. The dyed low speed fluid rolled up, and the dyed cores of the sphere wake become indistinguishable from the surrounding low speed fluid (Figure 3Id) . This rolled-up dye pattern, representing a perturbed portion of the boundary layer, was symmetric with respect to the center of the picture (Figure 31e) . As this dye pattern convected through the light sheet, each half of the dye pattern moved apart slightly in the spanwise direction (Figure 31f ) . The trailing portion of the disturbance was manifest in Figure 31 (g-h) as a rolled-up dye pattern which decreased in size and decayed. Figure 32 shows cross-sections of the dye-marked acetate sphere wake as it passed through the light sheet which was fixed at 465* downstream of the sphere impact position. The light sheet in Figure 3 2 was downstream of the light sheet in Figure 31, and thus shows the disturbance in a latter stage in its development. Like Figure 31(a), the dye in the sphere wake was concentrated into two cores in Figure 32(a) . Unlike Figure 31, the dye in the 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. downstream portion of the sphere wake does not remain confined to two cores : the dye in the sphere wake formed serpentine patterns above the wall (Figure 32b-c). The dyed low speed fluid near the wall formed irregular dye patterns which indicated that the flow was turbulent in Figure 32(d-f). The dye patterns were symmetric with respect to the center of the picture in Figure 3 2 (a-c) but lost its symmetry in Figure 32(d-f). The rolled-up dye pattern in Figure 32(g-h) was similar in appearance to the rolled-up dye pattern in Figure 31(g-h). Figure 33 show a streamwise-spanwise cross-section of the disturbance which developed as a result of a 0.55 diameter uncoated acetate sphere dropping into a boundary layer with Re§- = 1.1x10^. These spheres were released into the water with the computer controlled bead ejector. Figure 33 was filmed by a video camera mounted on a traverse which was pushed downstream in order to keep the disturbance in the field of view. Unlike Figure 29, this video was filmed in a darkened room, and a fluorescein dye solution was released from the dye slots in the flat plate. Because the laser light sheet was at y « 0.65*, the field of view was dark until some disturbance lifted the dyed fluid into the plane of the laser light sheet where the dye would fluoresce. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The lights at the top of Figure 33 were fixed in space and were 5.0 cm apart. Since the camera had traversed downstream, these lights appeared to move from right to left in the videotape. By tracking the passage of successive lights past the center of the field of view, an observer of the videotape can determine the streamwise distance the moving camera had moved from its starting point. After the acetate sphere was released by the bead ejector, the sphere was above the laser light sheet and did not fluoresce. The sphere was invisible to the camera, until it fell into the plane of the laser light sheet and impacted the wall (Figure 33a). In Figure 33(b)-(f), which covered the time period 22<t<65, the disturbance lifted the dyed low speed fluid into the laser light sheet at y » 0.65’ where it was manifest as two parallel streaks separated in the spanwise distance. In some runs not presented in this report, a single streak was observed at this time in the disturbance development. In Figure 33(b-f) there was slow growth in the spanwise direction followed by a sudden growth spurt starting with Figure 33(g-i). The dye streaks developed a spanwise waviness which was asymmetric with respect to the spanwise direction (Figure 33e) . At latter stages, the disturbance rapidly 'bloomed' into a turbulent spot. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 34 is a plot of the growth of the nondimensional spanwise width of the dye pattern with respect to nondimensional time since the impact of an acetate sphere. The width was normalized with ô*i, and the time since impact was normalized with 8'i/Uco where 8"i was the displacement thickness at the sphere impact position. The width was obtained playing the videotape on a VCR, pausing the VCR at a frame, and measuring the width of the widest portion of the dye pattern from the image on the video monitor. The screen to actual size conversion was carried out using a calibration image of tick marks drawn on the water channel flat plate which was filmed earlier. Figure 34 was the ensemble average of the dye pattern width from 15 runs. The error bars in Figure 34 were an average of the standard deviation of the widths for a particular range of time. Figure 34 shows there was a considerable amount of scatter in the spanwise widths of the dye patterns at a fixed point in time. They also show a difference in the spanwise growth rates: the slower growth rates corresponded to early in the disturbance development when the disturbance were streaks ; the larger growth rate corresponded to the breakdown of the streaks into a turbulent spot. Figure 35 is a plot of the nondimens ional streamwise position of the leading edge of the dye patterns from the 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sphere impact position versus the nondimensional time since the impact of an acetate sphere. As before, the streamwise positions were normalized by Ô’i, and the time was normalized by 5‘i/Uao- This was also obtained by playing the videotape on a VCR and pausing the VCR when a reference light was aligned with the center of the screen. Using the still frame, the horizontal distance between the leading edge of the dye pattern to the center of the image of the video monitor was measured. The passage of individual reference light on the top of the screen was used to obtain the streamwise distance of the moving camera to its starting position. Figure 3 5 was also an ensemble average of 15 runs. They show that the leading edge of the disturbance travelled at a constant speed « 0. 6Uœ downstream. The error bars in Figure 3 5 were based on the standard deviation of the leading edge positions. They indicate that there was less scatter in the leading edge positions during the early stages of the disturbance than in the spanwise widths. Figure 3 5 are plan views of the disturbance which developed as a result of a 0.65* diameter uncoated polystyrene sphere dropping into a boundary layer with Reg* = 1.1x10^. With the exception of the sphere, the experimental conditions were the same as those in Figure 33. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Following the impact of the sphere on the wall, there were two parallel streaks which grew in the streamwise direction and were symmetric with respect to the spanwise direction. These dye patterns show an anemic level of development in contrast to the acetate sphere case. Because the low speed fluid was marked by dye. Figure 33 and Figure 36 imply a greater degree of momentum transfer in the vertical direction for the disturbance created by the acetate sphere than the disturbance created by the polystyrene sphere. 3.4. Discussion When the acetate sphere with Ref = 5x10^ and d/6' = 0.5 was dropped into the Reg. = 1.1x10^ boundary layer, a turbulent spot was formed as a result of the sphere impact. Although the boundary layer at the sphere impact position was supercritical with respect to Tollmien-Schlichting wave growth, the boundary layer prior to the impact of the sphere was laminar. This conclusion was reached after observing the undisturbed state of the dye sheet in the time preceding the sphere drop. Also, this Reg. was below the natural transition Reynolds number which was determined through observations of the turbulence intermittency. So, the turbulent spot was not produced by natural causes. The sphere was also much smaller than the wavelength of an 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. unstable Tollmien-Schlichting wave at this Reynolds number (18-375 ) . So, unlike earlier vibrating ribbon experiments, the sphere by itself did not excite these modes. After the sphere hit the flat plate, it rolled downstream, trailing behind the wake. It becomes important to know if the sphere could have caused transition by its presence on the wall. This can be determined by examining the transition characteristics of fixed roughness elements. Several experiments on wall roughness elements are summarized in Appendix A. On the basis of Blackwelder & Browand (1991) experiment, it was determined that if a sphere with height 0.65' was fixed at Reg. = 1.1x10^, it would not cause transition. The roughness Reynolds number Re%, which is defined in Appendix A, quantifies the ability of a roughness element to cause transition. In the experiment with the falling spheres, the roughness Reynolds number. Ret, of a sphere with height 0.65' fixed at Reg* = 1.1x10^, was 2x10^. This was subcritical with respect to the critical Re^ = 585-665 for isolated spheres (Hall, 1967). As the sphere rolled downstream following its impact with the wall, it buried itself deeper in the boundary layer, and its Re^ value effectively decreased below 2x10^. Also, as Appendix A shows, the ability of the fixed roughness element to cause transition is related to the 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shear between the roughness element and the surrounding flow. Thus, the strength of the shear would be weaker for a slowly rolling sphere than a stationary sphere of the same diameter. Under these conditions, it can be concluded that the sphere did not trigger the turbulent spot after the sphere had landed on the plate. When the polystyrene sphere with Ref = 2x10^ and d/5' = 0.6 was dropped into the Reg. = 1.1x10^ boundary layer, the boundary layer remained laminar. Because the acetate and polystyrene spheres both acted as surface roughnesses after they impacted the wall, differences in the way these spheres traversed the boundary layer were responsible for determining whether or not a turbulent spot was formed following impact. The tilted acetate sphere wake behaved differently than the tilted polystyrene wake. Figure 31 and Figure 32 shows that the acetate sphere wake changed its structure as it convected downstream, unlike the polystyrene sphere wake which preserved its hairpin shape as it convected downstream. When the wake reached the light sheet at 465* downstream of the sphere impact point (Figure 32), it took on a loop-like structure and interacted with the boundary layer fluid in a complex manner. As Figure 32(d) shows, the two threads became wavy preceding breakdown to turbulence 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which is reminiscent of the transition observed by Chambers S c Thomas (1983) (see Figure 12). This can be interpreted as an instability in the leg which might have caused it to develop a helical shape. However, the unsteady motion in the undyed portion of the boundary layer could have impressed its motion on the dyed particles in the wake. As will be seen in §7, the latter is the case. When the extreme upstream portion of the wake passed through the light sheet at 465* downstream of the sphere impact position, it returned to the benign roll-up patterns of Figure 31. This portion of the wake was closer to the sphere which was rolling behind the wake, so it would reflect the wake left by a fixed roughness. This roll-up pattern was consistent with one left by the steady horseshoe vortex described by Acarlar & Smith (1987a) and Mochizuki (1961). 6 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Material Diameter (cm) Specific Gravity Froude Number Com Pollen (LH) 8 . 6x10-3 1.04 1.3x10^-6-2x10® Polystyrene DVB (LH) 1.2x10-3 1.11 3.3x103-1.8x10® Black Walnut Pollen (LPS) 3 .7x10-3 1.03 2.0xl0®-2.0x10^ Polystyrene (LPS) 1.4x10-3 1.11 7.4x10^-6.6x10® Polystyrene (LPS) 2 .2x10-3 1.11 3.4xlO*-2.1xlO® Polystyrene (USC) 3 1.1 3 Acetate (USC) 3 1.5 0.3 Glass (USC) 3 2.5 9x10-3 Table 1 Comparison of the size and buoyancy used in transition experiments. (LH): Ladd (1985); (LPS): Lauchle et al. (1995); (USC); Uoo = 20 cm/s. of particles & Hendricks USC experiment 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 23 Smoke visualization pictures of the turbulent spot created by the impact of a falling sphere at two points in its development. Plan view. (from Hall 1967). To Computer Stepper Motor To Water Pump 1 Bead Ejector Dye Slot Figure 24 Experimental setup 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. £ 1400 1200 - ▲ ACETATE • • GLASS • ■ POLYSTYRENE 1000 800 ■ 600 - Strong Vortex 400 - 200 0 - Weak Vortex 0.5 1 1.5 Specific Gravity 2.5 Figure 25 Ref of the spheres used in the experiment. Solid lines are the Ref of the Hill's spherical vortices in the simulation. 6 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CAMERA Uoo X From Laser Spherical +Cylindrical Lenses Figure 26 Diagram of camera set-up for oblique angle view Optical Scanner CAMERA From Laser Mirror Figure 27 Diagram of camera set-up for vertical light sheet TRAVER SE M O UNTED CAMERA oo From Laser Optical Scanner Figure 28 Diagram of camera set-up for overhead view of the flow field. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 02 : 18 : 03 = 00 9 - 04-92 12 31:11 (a) t = -1 02 : 18 : 03 : 05 (b) t = 5 9 - 04-92 12 31:11 02 : 18 : 03 : 1 1 (c) t = 11 9 - 0 4 - 9 2 1 2 : 3 1 : 1 1 02 : 18 : 04 : 08 (d) t = 46 9 - 0 4 - 9 2 12 31 12 = 106 (e) t = 87 (f) Figure 29 Oblique view of boundary layer transition resulting from Ref=5xlG^ sphere (Res.=l. 1x10^) . Times are with respect to the time impact and are scaled with respect to freestream velocity and displacement thickness at impact. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 - 04-92 1 3 : 3 4 . 0 0 03 = 09 = 05=15 9:04^2 03 : 09 : 05 : 24 03:09:06:00 •09=06=15 9 - 0 4 - 9 2 : 1 3 : 3 4 : 0 0 9 - 0 4 - 9 2 03:09:10=25 13 34:05 (e) t = 72 Figure 30 Oblique view of boundary layer interaction with Ref = 2x10^ sphere (Reg. = 1.1x10^). Times are with respect to the time impact and are scaled with respect to freestream velocity and displacement thickness at impact. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - - r;. ^ (a) t = 20 ?Sr-S#§ ■ ■ ' -■•• L # a (c) t = 28 (after impact)(d) t = 38 — — %: (g) t = 54 (h) t = 61 Figure 31 y-z cross section of wake from Ref=5xl0^ sphere landing at Reg. = 1.1x10^ following sphere impact. Light sheet is 186* away from impact location. Time has been scaled with 6*/Uoo. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) t = 53 (after impact) (b) t = d) t = 78 (g) t - 114 (h) t = 125 Figure 32 y-z cross section of wake from Ref=5xl0^ sphere landing at Reg. = 1.1x10^ following sphere impact. Light sheet is 466* away from impact location. Time has been scaled with ô'/U». 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 05 : 1 4 03 : 00 (a) t = 0 6-22-92 18:08-27 05 : 1 4 03 : 1 8 (b) t = 22 6-22-92 18 08 28 05 : 1 4 : 03 : 28 (c) t = 35 6 - 22-92 18 08-28 05 : 1 4 : 04 : 02 (d) t = 40 6 - 22-92 18-08 28 05 : 1 4 : 04 : 1 8 6 - 22-92 18 : 08:29 05 : 14 : 04 22 6 - 22-92 18 : 08:29 (e) t = 60 (f) t = 65 Figure 33(a-f) For caption see p.71. 7 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 05 : 1 4 : 05 1 0 (g) t = 87 6-22-92 18 08:29 05 : 14 : 05 18 (h) t = 97 6-22-92 18 08:29 05 : 1 4 : 06 : 04 (i) t = 117 6 - 22-92 18 : 08:30 05 : 1 4 : 06 : 1 8 (j) t = 134 6 - 22-92 18 08.31 Figure 33 Overhead view of boundary layer transition resulting from Ref=5xl0^ sphere (Res.=l. 1x10^ ) . Times are with respect to the time impact and are scaled with respect to freestream velocity and displacement thickness at impact. 7 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (t-wujs*, Figure 34 Growth of spanwise width of the dye pattern produced by Ref=5xl0^ acetate sphere (Res.=l. 1x10^ ) . to is the time of impact of the sphere. C30 100 120 2 0 40 G O ao 0 ( t - y u j s " , Figure 35 Leading edge position of the dye pattern produced by Ref=5xl0^ sphere (Res.=l. 1x10^) . xq and to are the streamwise position and time of impact of the sphere respectively. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 07 : 06 : 06 : 04 (a) t = 0 6-15-92 16 03.37 07 : 06 : 06 : 1 6 (b) t = 15 6-15-92 16.03 . '3 7 6 - 15-92 16 . 03.38 6 - 15-92 16 . 03:38 07 : 06 : 07 : 1 0 07 : 06 : 06 : 24 07 : 06 : 08 02 (e) t = 70 6 - 15-92 16 . 03.39 07 : 06 : 1 1 : 04 (f) t = 182 6 - 15-92 16 03 42 Figure 36 Overhead view of boundary layer interaction with Ref = 2x10^ sphere (Reg» = 1.1x10^) . Times are with respect to the time impact and are scaled with respect to freestream velocity and displacement thickness at impact. 7 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. NUMERICAL SIMULATION As the falling sphere experiment in §3 showed, the sphere introduced two types of perturbation in the boundary layer. One type of perturbation was the wake left in the boundary layer during the sphere's fall towards the wall. This was manifest as the dyed pathline in the flow visualization videos. In the acetate sphere case, the wake was involved in the transition process. Wu & Faeth's (1993) LDA measurements of spheres travelling in quiescent fluid at Res = 280 and 400 (Res is the Reynolds number based on the sphere velocity and diameter) showed that the mean velocity within the near-wake was approximately equal to the sphere velocity. In the falling sphere experiment, the near wake could be thought of as a 'bag' of fluid which clung to the sphere during the sphere's descent (Figure 37). When the sphere reached the wall, it deposited this fluid in the lower part of the boundary layer. The other type of perturbation was introduced by the sphere after it had impacted the wall and was rolling downstream. The presence of the sphere at the wall altered the boundary layer profile in its presence like a surface roughness element. As noted earlier, this type of 7 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. perturbation did not create turbulent spots in either the acetate or polystyrene sphere cases. Appendix A contains a more in-depth discussion of the way a roughness element can perturb a boundary layer. On the basis of the results of the falling sphere experiment, numerical simulations were carried out which sought to model the way the falling sphere perturbed the boundary layer. These simulations were intended to complement the experimental observations of the interaction between the sphere wake and the boundary layer. In this model, the sphere was eliminated, and the sphere near-wake was modelled with a self-propelled vortex (Figure 37) . The perturbation introduced by the rolling sphere was not incorporated in this model, because of its passive role in the transition process in the experiment. One of the purposes of these simulations was to see if this model was capable of causing transition using the criterion in §2. The second purpose was to see what processes were involved in the early stages of the transition process and to relate them to the observations of the falling sphere experiment. 4.1. Hill's Spherical Vortex Like the sphere used in the experiment, the wake model must be smaller than the boundary layer thickness and must 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be able to move around on its own and leave a wake in the boundary layer. The region surrounding the wake and vortex will be referred to as a 'scar' (Figure 37) . The Hill's spherical vortex moving downwards towards the wall met these qualifications and was chosen as this wake model for the numerical simulation. The Hill's spherical vortex is a toroidal vortex and bears a qualitative resemblance to the near wake of a low Reynolds number sphere as described by Taneda (1955) and Magarvey & Bishop (1961) . There are several parameters which characterize the vortex: radius R, amplitude a, and orientation 0. The velocity field within the vortex is given in the following equations (Lamb 1945) u = -2r'--x'-) + a w = ^x'- + y'-+z'’ <R (4a) where r' = yjy'^+z'^ The velocity field outside of the vortex is 7 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. u = - a 1— 3 - — o r r'^R^ +or v' = —aR X y 5/2 w' = —aR 2 ■Jx'~ + y'-+ z'~ >R (4b) (x',y',z') are the spatial coordinates with respect to the center of the vortex. In this coordinate system, the vortex is moving through fluid which is at rest far away from the vortex. In this simulation, the vortex was given an initial rotation, O, about the z-axis. O determines the initial angle of the path of the vortex (Figure 38) . This rotation is implemented through a coordinate transform = u[x', y \z') cos 0 - v'{x\ y', z') sin 0 ~^off^y-yoff^z-z,^) = u'(x', y', z') sin 0 + v'(x', y ',z ) cos 0 -^off^y-yuff^z-z,ff) = w'{x', y',z') x' = [ x - ) cos 0 + (y - ) sin 0 / = (y -y „ ^ )c o s 0 -(x -x „ ^ )s in 0 w (4c) Z =Z-Z4T where (Xoff, yoff/ Zoff) is the initial position of the vortex with respect to the origin in the computational domain (see Figure 39) . The criterion for choosing a value for a was not to select a value large enough to immediately destabilize the boundary layer at the vortex's initial 7 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. position. Instead, a value for a was selected in order to get the desired initial self-induced velocity of the vortex. The values used for a in the simulations discussed in this paper are shown in Table 2. Although the interface between the vortex and the surrounding fluid was not identical to the boundary layer of a falling sphere, there were sharp velocity gradients at the interface which created the wake within the mean flow as the vortex moved. In Magarvey & Bishop's (1961) experiment, the wakes of liquid spheres in quiescent fluid were shown to behave like solid spheres. In that situation, the role of a solid sphere's no-slip surface is less significant to the development of the sphere wake than the sphere's geometry and velocity. The same conclusion can be made about the role of the falling sphere's no-slip boundary in the transition process. In order to eliminate flow through the domain boundaries, velocity fields from image vortices were superposed onto the velocity field from the vortex. The resulting velocity field Eqn. (4) satisfies the two requirements for a well-posed initial condition to the Navier-Stokes Equations described by Gresho (1991): the velocity field must be solenoidal and there can be no flow through the boundaries. This velocity field was superposed 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. onto the Biasins boundary layer contained within a rectangular computational domain (Figure 39) and used as the initial condition for the simulation of the Navier-Stokes Equations. For the simulations with the Hill's spherical vortex, the radius, R, of the vortex was set to O.BÔ’o, which was the size of the spheres used in the experiment. 5*o was the displacement thickness at x = 0.0, the inflow boundary. The Hill's vortex was initially placed along the domain centerline, 2ô'o downstream of the inflow boundary, 2 . 5ô"o above the wall, i.e., (Xoff, Yott, Zoff) = (2.0, 2.5, 0.0). The vortex was initially at the outer edge of the boundary layer, so the vortex would be able to propagate through most of the boundary layer on its way to the wall. This paper will discuss the results of the simulations where the orientation O was set to -tc/4 . With this orientation, the u and V components of the vortex velocity are the same. In order to match the experimental conditions, Reg.o, the Reynolds number based on 5*o, was set to 1.1x10^. Table 2 shows that the Hill's spherical vortex used in the simulation was much smaller than an unstable Tollmien- Schlichting wave at Reg-o = 1.1x10^, the vortex pairs used by Breuer & Landahl (1990) , and the wall orifices used by 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Haidari & Smith (1994) and Singer & Joslin (1994). In addition to the size and strength differences between the Hill's vortex used in this simulation and Eqn. (1) , there were qualitative differences. The vortex pair used by Breuer & Landahl (1990) and Henningson et al. (1993) had a Gaussian velocity distribution, whereas the Hill's vortex had a concentrated distribution. The Hill's vortex was self-propelled as opposed to the wall orifices which were stationary or the Eqn.(l) vortex pairs which were pushed by the mean flow. 4.2. Numerical Code The simulation described in this paper was run with a finite difference DNS code which was described in Kim & Moin (1985) and Le & Moin (1991) . Wang, Lele, & Moin (1996) utilized this code to observe the development of a wave packet into a lambda-shaped vortex and its subsequent breakdown in a compressible boundary layer. In contrast to the temporal simulations of Breuer & Landahl (1990) and Henningson et al. (1993), this code allows for the spatial development of the boundary layer. The incompressible Navier-Stokes equation is integrated in time using a fractional step method. Between each time step, a "Runge- Kutta type scheme" explicitly computes the convective terms of the Navier-Stokes Equation to third order accuracy in 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. time and implicitly computes the viscous terms to second order accuracy in time. Overall, this results in second order accuracy in time. Spatial derivatives are computed by centered-difference formula for second order accuracy in space (Le & Moin 1991). Because this code uses Chorin's (1958) projection method, the pressure field is computed from the Poisson pressure equation and is used to make the velocity field solenoidal at each time step. The Poisson pressure equation is solved using the Fourier transform (Kim S c Moin 1985) . The grid size and the time step are chosen so as to satisfy the CEL (Courant-Friedrichs-Lewy) condition u(x,y,zj)A t v{x,y,z,t)At w{x,y,z,t)At^ max V < C (5) Ax Ay Az where C is a nondimensional number dependent on the code. This condition dictates how far a fluid particle can travel in one time step without causing numerical instability (Fletcher 1991a) and is required because of the semi explicit nature of the time advancement. Le & Moin determined that C was - \ / J for this program. C was set to a more conservative level of 0.7 based on the recommendation of M. Wang (private communication). The code was capable of independently decreasing At if Eqn. (5) was not true at any time step during a program run. The simulations were run on the Cray C90 and J9 0 supercomputers at the NASA Ames 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Research Center. The runs discussed in §5 -§6 required 25 to 42 Megawords of memory and consumed 68 CPU hours. The program used a rectangular computational domain (see Figure 39). The computational domain employed the staggered grid where the pressure and each component of velocity were defined on separate systems of nodes (Fletcher 1991b). The grid was uniform in the horizontal directions, X & z, while a hyperbolic tangent grid distribution was used in the vertical direction, y (Fletcher 1991b). Since the Fourier transform was used in the Poisson pressure equation solving subroutine, the domain is periodic in z. The Blasius boundary layer with a user-specified Reg^o was enforced at the inflow and outflow boundaries. The outflow boundary condition was a modified version of the Sommerfeld radiation condition, where ü is the velocity and 0^ is the base velocity, d t ” d t (Orlanski 197 6). The v velocity for the Blasius boundary layer was imposed at the top of the boundary layer; the u velocity at the top of the domain was determined by the condition 0)^=0 at that boundary. The no-slip condition was imposed at the bottom. 8 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3. Vortex Pair Simulation In order to validate this code, the vortex pair described by Eqn.(1) was superposed into a Blasius boundary layer and used as the initial condition for the Navier- Stok.es equation. s was set to 0.2, so the vortex pairs were strong enough to go through the early stages of transition. The results of the simulation were compared to the results of the 8 = 0.2 vortex pair simulations of Breuer & Landahl (1990) and Henningson et al. (1993). In this run, the computational domain had an inflow Reg.q = 950. The domain was 128ô’o in the streamwise direction, 205*o in the vertical direction, and 508'o in the spanwise direction. In terms of grid points, the domain was 161x65x129 points and had grid sizes of Ax = 0.80*o and Az = 0.45*0. Through hyperbolic tangent grid stretching. Ay ranged from 0.055*o at the wall to 85*q. Lengths were normalized by ô’ q, and time was normalized by U«/5*o - The simulation was run over the time period 0.0<t<116.6. One way of evaluating the development of the vortex pairs was to examine the perturbation energy, E, which was defined by 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = !!j{u{x,y,z,t)-UXx,y,z)) dxdydz v'(f) = ll!{v{x,y,z,t)-V^{x,y,z)) dxdydz w '(r) =\\\w {x,y,z,t^ dxdydz E{t) = u^{t) + v‘ (r) + w-{t) where (u, v, w) is the velocity field and (Ub, Vb, 0) is the base velocity which was the Blasius boundary layer in this case. Figure 40 is a plot of the perturbation energy normalized by the initial perturbation energy, Eq, for this run and the corresponding run from Henningson et al. (1993) . Figure 40 shows that the normalized perturbation energy for this simulation grew throughout the time period, 0.0<t<115.5, and agreed with the energy growth rate from Henningson et al. (1993). Figure 41-Figure 42 show the u and v perturbation velocity fields on the plane y = 1.0. They show the development of the spanwise shear layers at the leading edge of the disturbance. The disturbance's scale and shape largely agreed with the observations from the e = 0.2 vortex pair simulations of Breuer & Landahl (1990). The secondary instability observed by Breuer & Landahl in the late stages of their simulation, did not appear in Figure 42(c) . Instead, the results of this simulation were more comparable to Henningson's results with the s = 0.2 vortex pair as a 8 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. result of the Ay. Thus, it can be concluded that the program was working properly. 4.4. Numerical Simulation Parameters The dimensions of the computational domain used for the Hill's spherical vortex were different from the dimensions used for the vortex pairs. The computational domain had a streamwise length ranging from 38.45*o to 153.60*o. The domain was extended in the streamwise direction at several instants in time during the simulation. The domain was 200*0 high in the vertical direction and 11.52S*q wide in the spanwise direction. (A series of simulations were carried out with a domain which was 3 . 9ô*o wide, but these will not be discussed in this paper.) The grid points were redistributed in the vertical direction a couple of times in the simulation. In terms of grid points, the domain ranged in size from 514x66x194 to 2050x66x194 points. The size of the initial disturbance influenced the choice of grid resolution. Because the vortex diameter was set to 0.65*o, Ax and Az were set to 0.0750*o and 0.06S'o respectively. Ay ranged from 0.050*o at the wall to 85*o at the freestream. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5. Visualization with Marker Particles Analysis was carried out on instantaneous velocity data as well as by recording the movement of Lagrangian marker particles. Each marker particle was a massless point and had no drag. Interaction between particles was ignored. These particles were buffeted by local flow conditions and acted like the dye used for flow visualization in experiments. The development of vortices within the boundary layer were reflected in the movements of the marker particles from their initial positions and in the patterns these particles accumulated into. Qualitative comparisons will be made between the marker particles and the flow visualizations from the experiments discussed earlier. The motion of these particles were governed by the system of ODE's: dx j where N is the number of particles seeded in the flow, Xp,i is the instantaneous position of particle i, and Up,i is the instantaneous flow velocity at that position Xp,i. The trajectory of an individual particle was controlled by its initial position and by the state of the local flow. This system of ODE's was solved using a fourth order Runge-Kutta method in which the particle positions are obtained at every 8 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. other time step of the Navier-Stokes solver. This scheme was called the RK-4 2X scheme by Darmofal & Haimes (1995) . If the flow-solver time step had changed due to the CFL condition, the routine would automatically switch to a second order Runge-Kutta method until the flow-solver time step had stopped changing. In this simulation, the flow field was initially seeded with 3802 marker particles. 2626 particles were arranged as a 255*o x 65*o sheet at y = 0.35*o in order to simulate the dye sheet used in the experiment (Figure 43) . The rest were arranged in a ribbon which spanned 1.5<x<7 and 0.4<y<3.0 along the centerline of the domain and were intended to visualize the effect of the vortex during its descent (Figure 44). 8 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Disturbance Type Streamwise length Spanwise width 1 ^ 1 max E Unsteüsle T-S wave at Re5*=l.lxl0^ 18-37 « 1 «1 Eqn.ly Vortex Pair, 6=0.2 (BL) 20 24 0.02 0.410 Water Cbannel Experiment Sphere 0.6 0.6 Eqn.4 Hill's vortex, a=0.8, 0=-7C/4 0.6 0.6 1.4 3 .8x10'^ Eqn.4 Hill's vortex, a= 0.4, 0=-7C/4 0 . 6 0.6 0.7 1.9x10"^ Singer slot (SJ) 25 2 0.25 Haidari slot (HS) 13-26 0.4-0 . 8 0 .01-0.2 Table 2 Comparison of length scales, velocities, and energies for various disturbances. (BL): Breuer & Landahl (1990); (SJ): Singer & Joslin (1994); (HS): Haidari & Smith (1994) . Lengths are normalized by 5'; velocity is normalized by U». 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Wake Wake Near Wake -► Scar Sphere Vortex Figure 37 Drawing of sphere/wake system and vortex/wake system. Figure 38 Hill's spherical vortex in the Blasius boundary layer. \â\ vorticity contours on the z = 5.73 plane (-0.035*0 from centerline). Solid line contour spacing is 5.0; dashed line contour spacing is 0.1. Outflow U o o No-Slip Wall Y Inflow Figure 39 Diagram of the computational domain. 8 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 30 Test Case Henningson et ai. (1993) 25 20 s 15 80 60 t 100 40 120 Figure 40 Comparison of growth rates of the s = 0.2 vortex pairs for the Test run and the Henningson et al. (1993) simulation. Time History of total energy normalized by initial energy. (— ) :Test run; (--) : Henningson et al. (1993). Data from Henningson et al. (1993). 9 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o X (b) t (a) t 39.4 87 .4 (c) t = 116.6 Figure 41 u perturbation velocity contours at y/5' = 1.0. Contour spacing is 0.01. Negative contours are dashed. (a) t = 0 . 0 (b) t = 39.4 N 25 100 (d) t 87.4 116.6 Figure 42 v perturbation velocity contours at y/ô* = 1.0. Contour spacing is 0.001. Negative contours are dashed. 9 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Y 35 30 25 20 15 10 Figure 43 Initial positions of marker particles y 3 163 degrees 2 1 0 X 1 2 3 4 5 6 7 8 9 10 Figure 44 Initial position of marker particles in the 'ribbon'. Side view. 9 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5. STRONG CASE This section will deal with the case of the Hill's spherical vortex with the initial amplitude a = 0.8. Basing the Reynolds number of the vortex. Ref, on the amplitude, a, and diameter, d, _ ad Re, = ------ (7) ^ V yields a Ref of 528 for this case. As Table 2 shows, the Hill's vortex with this particular amplitude had a lower perturbation energy, E, than the vortex pairs used by Breuer & Landahl (1990). However, the peak v velocity, |v|max, of the Hill's vortex was many times greater than the vortex pairs. This Ref is approximately the same as the Ref of the acetate sphere in the falling sphere experiment described in §3.2-3 .3 . In this chapter, velocity is normalized by freestream velocity Uoo, space variables are normalized by the displacement thickness at the inflow boundary S*o/ and time is normalized by ô*o/Uco. The transition process can be divided into 5 stages : the fall of the vortex and its impact with the wall; the post-impact stage where fluid was ejected away from the 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. wall; the roll-up of the ejected fluid into vortices; the development of additional secondary vortices which interacted with the older vortices in the flow; and the late Stage in which there was a 'tangle' of vortices. Figure 45 is a plot of the time history of E as defined in Eqn. (6) over the time period 0.0<t<109.6. Figure 45 shows that the energy grew with respect to time over the entire length of the simulation. Over the course of the simulation, u~ was the largest component in E. 5.1.Marker Particles The following is a description of what happened to the marker particles which were initially in the 'ribbon' aligned with the centerline as seen by an obseirver looking in from the side of the domain. Due to the mean flow and to the self-induced motion of the Hill's spherical vortex, the vortex moved towards the wall once the simulation started. The movement of the vortex up until t = 3.5, was manifest as a region cleared of marker particles after the vortex had passed through it (see Figure 46a) . The marker particles which were initially within the vortex interior migrated towards the periphery of the vortex due to the overall motion of the vortex. As the vortex moved through the 'ribbon', the marker particles were 9 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. not entrained within the vortex but moved around its periphery. Initially, the 'ribbon' was at a 163° angle with respect to the wall (Figure 44) . Figure 46 shows that as time progressed, the boundairy layer rotated the 'ribbon' about the spanwise direction towards the wall, because the upper part of the 'ribbon' overtook the lower part. At t = 24.6, the 'ribbon' was at an 8° angle with respect to the wall (see Figure 46b) . Ultimately at t = 109.6, the 'ribbon' was at a 3° angle with respect to the wall. The mean shear also had the effect of stretching the 'ribbon' from a streamwise length of 6Ô'o to 46ô*o- Over the course of the simulation, the remnant of the clear space left by the passage of the vortex was still visible at the top part of the 'ribbon', but the rest of the 'ribbon' was compressed in the streamwise direction. This accumulation represented a border between the disturbance created by the vortex and the undisturbed Blasius boundary layer. In the experiment conducted in USC, a laser light sheet was used with a fluorescing dye in order to visualize flow activity in the plane of the light sheet. To obtain the same effect with the marker particles, particles outside of a designated volume were hidden from view in order to see what patterns those particles within the volume formed. 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These patterns can be compared to the patterns formed by the dye in the light sheet. In Figure 47, spatial filtering was used to hide those marker particles outside of the region 0.35<y<1.0. This was done to show the instantaneous positions of particles which were lifted above the y = 0.3 plane. The 'ribbon' is present in Figure 47 as a line along the centerline downstream of the rest of the disturbance. These figures were from the vantage point of an observer looking down towards the wall from above. Figure 47(a) shows the instantaneous position of the particles at t = 10.0, shortly after the impact of the Hill's spherical vortex. There was an accumulation of particles which formed a arc-shaped pattern whose tip was at x = 10. The dimensions of the arc shaped pattern were 2ô'o in the streamwise direction and 25*o in the spanwise direction. By the time t = 39.6, this arc shaped pattern convected downstream and developed into two parallel streaks which were 75*o in streamwise direction and 35*0 in the spanwise direction. These streaks will be called the primary streaks. At t = 59.5, the two streamwise streaks were now 130*o long in the streamwise direction. In addition, there were two smaller streaks which appeared to the sides at z = 3, 8 at the trailing edge of the primary 9 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. streaks. The distance between these secondary streaks was 8 times the initial size of the Hill's vortex. By the time t = 109.6, these secondary streaks grew to 14ô*o long in the streamwise direction. The spanwise distance between the primary streaks merely increased to 45*o since t = 39 .6 . These streaks bear a resemblance to the streaks produced by a hydrogen bubble wire at y = 0.35* in the experiment of Haidari & Smith's (1994) (see Figure 17). Given that Res-o = 1.1x10^, the final spanwise distance between the primary streaks was 97v/u%. It is interesting to note that the average spacing between low speed streaks in a turbulent boundary layer is lOOv/Ut (Smith & Metzler 1983). In Figure 48, spatial filtering was used to hide those marker particles outside of the region 0.0<y<0.35. This shows how the sheet of particles at y = 0.3 was depleted of particles during the development of the flow. At t = 10 .0 , there was a 'pocket' centered at x = 10. During 10 . 0<t<39.6, this 'pocket ' grew in length in the streamwise direction from 105*o to 125*o- The leading edge of the 'pocket ' coincided with the leading edge of the primary streaks. Except for t = 10.0, the streamwise length of the primary streaks was smaller than the 'pocket. ' The width of the widest part of the 'pocket' also grew in size from 25*o 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to 4Ô'o during this time period. However, the trailing edge did not increase beyond this spanwise width for t>39.6. The streamwise length of the 'pocket' grew to 235*o at t = 69.1. There are regions cleared of particles at z = 3, 8 at X = 27, which corresponded to the secondary streaks. At t = 80.8, the leading edge of the 'pocket ' had already propagated past the downstream boundary of the sheet of particles. If the y = 0.3 sheet of particles was larger in the streamwise and spanwise directions, the primary streaks would have appeared larger in the streamwise direction at this point in time. Additional streaks located further away from the centerline in the spanwise direction might have also appeared with a larger sheet. Figure 47 and Figure 48 show that marker particles in the streaks originated with the y = 0.3 sheet and left behind a region cleared of particles which grew in size as time progressed. These 'pockets' bear some similarity to the 'pockets' observed by Acarlar & Smith (1987b) and Haidari & Smith (1994) . The configuration of streaks followed by a 'pocket' appears similar to the smoke visualization picture (Figure 12) from Chambers & Thomas (1983). In contrast to the filtered plots of instantaneous particle positions. Figure 49 and Figure 50 show the pathlines of select marker particles. Figure 49 shows the 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. particles which originated from 8.5<x<9.0, y = 0.3 over the time period 0.0<t<94.7. These marker particles were at the area where the Hill's spherical vortex impacted the wall at the time of impact. Particles which were at the center of the domain propagated downstream farther than the particles which were near the side walls. It can be inferred that the particles which were near the centerline had been ejected upwards where the mean streamwise velocity was greater. The rest of the particles moved away from the centerline and towards the closest side wall. This clearance of marker particles along the centerline was associated with the impact of the Hill's spherical vortex with the wall. As the disturbance developed and convected downstream, it had a different effect on marker particles further downstream. Figure 50 shows the trajectories of particles which originated from 15.0<x<15.5, y = 0.3. This area was downstream of the impact area and show the effect of the disturbance at a latter stage in its development. The strong movement towards the side walls seen in Figure 49 is not apparent in Figure 50. Instead, many of the particles rose and accumulated into two bands located at z = 3 & 8 which corresponded to the locations of the primary streaks. Of all the marker particles in Figure 50, x = 34 was the farthest position downstream that any of them travelled to. 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Figure 49, the farthest position that any marker particle travelled to was x = 46. Marker particles which originated in 8.5<x<9.0 travelled downstream farther than marker particles which originated in 15.0<x<15.5. 5.2. Vortices present in the boundary layer Because the patterns formed by the marker particles in the previous section were only a 'footprint' of the vortices present in the boundary layer, it was necessary to visualize these vortices. In order to identify the vortices which develop in the boundary layer, the complex eigenvalue method, as described by Chong, Perry, & Cantwell (1990), was used. Appendix B gives a more detailed explanation of this method. In summary, vortices were classified as regions in the computational domain where two of the eigenvalues of the local, instantaneous velocity gradient tensor were complex conjugates. In these regions, the discriminant. A, which was a scalar derived from the velocity field using Eqn.(9), was positive. In Figure 51, the three-dimensional surfaces, where A was at some positive threshold, was plotted at various instances in time. Figure 51(a) shows the Hill's spherical vortex at t = 3.50, before it impacted the wall. Although it was no longer axisymmetric, the vortex was still similar to its 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. initial state in terms of size and shape. It is interesting to note that the vortex wake does not contain the vortex loops seen in a wake of a sphere travelling in quiescent fluid at the same Reg (Shirayama 1992) . Figure 51(b) shows the surfaces of positive A at t = 10.0, which will be shown to be a time shortly after the vortex had hit the wall. Instead of the small, closed-loop seen in Figure 51(a), the vortex was deformed into a vortex pair partially aligned with the streamwise direction. These will be referred to as the primary vortex pair. After the impact with the wall, the Hill's spherical vortex was close enough to the wall for image vortices effects to become significant. The spacing between legs of the vortex pair at 8<x<10 was 0.96'o which was greater than the initial size of the Hill's vortex. This part is marked as 'UP' (upstream part) in Figure 51(b) . The remainder of the vortex pair at 10<x<12 was farther away from the wall, so it did not spread apart in the spanwise direction. This part is marked as 'DP' (downstream part) in Figure 51(b) . The streamwise length of the vortex pain had grown to 40*o due to the mean shear. Further along in time at t = 14.8, the vortex pair had stretched to 65*o in the streamwise direction (Figure 51c) . The vortex pair was pinched inward with respect to 1 0 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the spanwise direction at x = 12. This location was in between the 'UP' and 'DP' parts of the primary vortex pair. The 'DP' portion of the primary vortex pair was at an 24° angle with respect to the wall. The development of the disturbance was not merely limited to an increase in size but was marked by the appearance of a secondary set of vortices outboard of the primary vortex pair. At t = 24.6, two new hairpin vortices centered at x = 16 are present in Figure 51(d) and are displayed with an 'SHV' (secondary hairpin vortices). Like the part of the vortex pair which was pinched inward earlier, the secondary hairpin vortices were located between the 'UP' and 'DP' parts of the vortex pairs. These secondary hairpin vortices were arranged side-by-side at z = 5 and 7 and were separated in the spanwise direction. The 'DP' part of the primary vortex pair for x>15 was at a 9° angle with respect to the wall. This part of the primary vortex pair was rotated towards the wall about the spanwise direction since the impact of the Hill's spherical vortex with the wall. At t = 39.6, there were several different changes in the shape of the vortices. Even though Figure 51(e) uses the same threshold of A as Figure 51(d), the 'DP' part of the primary vortex pair doesn't show up. This implies that 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the topology of the flow motion in the 'DP' portion of the vortex pair corresponded more to straining motion than swirling motion (see Appendix B) . As time progressed, the legs of the secondary hairpin vortices, which flanked the primary vortex pair, were extended in the streamwise direction by the mean shear. The heads of these vortices have connected and foormed a larger u-shaped vortex which straddled the original vortex. This U-shaped vortex interacted with the primary vortex pair in a curious way. In the earliest stages of the interaction at t = 39.6, there was a pair of structures centered at x = 23 which emanated from the primary vortex pair towards the side walls. These structures are marked with a 'BV', which stands for Bridge Vortices in Figure 51(e) . As Figure 51(e-h) show, the vortices marked with 'BV' grew in size, primarily in the spanwise direction, and wound around the legs of the u-shaped vortex. This process caused the system of vortices, as a whole, to grow in the spanwise direction. The interaction between these sets of vortices had another result. At t = 80.8, the area where this vortex-vortex interaction was occurring, 50<x<55, appeared to have become quite convoluted and shows that the vortices were tearing each other apart (Figure 51i). 1 0 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. During the time period, 3 9.6<t<69.1, the leading edge of this system of vortices was also undergoing a major change. As Figure 51 (e-h) shows, the head of the u-shaped vortex pulled away from the primary vortex pair and. was curling upwards. This deformation was due its self-induced motion. This is reminiscent of the behavior of the tip of the hairpin vortex in Hon & Walker's (1991) simulation (see Figure 14) . This tendency for the head of a hairpin vortex to act in this way was also seen before in the Acarlar & Smith's (1987ab) flow visualization pictures. Upstream of the tip of the u-shaped vortex, a secondary spanwise vortex developed. It is marked with a 'SSV' in Figure Sl(g-h). The development of a secondary vortex inline with and upstream of a hairpin vortex was also noted by Acarlar & Smith (1987ab). A necessary feature for the growth of this disturbance is to be able to regenerate certain elements of itself. This regenerative process could be seen in the trailing edge of this system of vortices. At this location at t = 59.5, there was a new inclined streamwise vortex pair at 2 6<x<30. This new vortex pair rose upward at x~30 over the legs of the primary vortex pair. At this threshold of A, there appear to be stubs at the downstream ends of the new vortex pairs. These were marked with a 'THV' (Tertiary Hairpin 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Vortices) in Figure 51(g). As time progressed between 59.5<t<80.8, the vortices marked with 'THV' grew in size and developed into a couple of hairpin-shaped vortices aligned side by side (Figure 51g-i) . These hairpin vortices resembled the secondary hairpin vortices at t=24.6 (see Figure Bid) . Like the secondary hairpin vortices, the tertiary hairpin vortices were initially separated in the spanwise direction, but they eventually moved closer together with respect to the spanwise direction over the time period 59.5<t<80.8. The downstream end of the tertiary hairpin vortices rose over two sets of quasi-streamwise vortex pairs. These vortex pairs linked the tertiary hairpin vortices with the rest of the disturbance. At t=94.7, the tertiary hairpin vortices have connected and formed a new u-shaped vortex at 40<x<55 as marked in Figure 51(j) and resembled the primary disturbance at t=44.4 (Figure 51f) . However this new u-shaped vortex was wider and taller than the original u-shaped vortex at that particular stage in development. The new u-shaped vortex was almost 65' wide in the spanwise direction. Because the vortex-vortex interaction mentioned earlier had widened the disturbance as a whole, the new u-shaped vortex 'inherited' its spanwise scale from the vortices downstream of it. This was because the u-shaped vortex developed from the fluid 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. churned up by the vortices ahead of it. At the same time, the head of the new u-shaped vortex rose to the upper part of the boundary layer and was curling upwards. The head is marked with a 'UH' (u-shaped vortex head) in Figure 51 (j) . Upstream of the tip of the u-shaped vortex, Bridge Vortices, at X = 47, emanated above the central streamwise vortices and are marked with 'BV' in Figure 51(j) . Like the Bridge Vortices in t = 44.4 (Figure 51f) , these new vortices were winding around the legs of the u-vortex. This suggests that the vortex-vortex interaction observed earlier was not an isolated event. Further downstream of the u-shaped vortex, there were two arch-shaped vortices at x ~ 67 which were remnants of the vortex-winding process mentioned in previous paragraphs and were marked with 'AV' (arch vortices) in Figure 51(j). The leading hairpin vortex which was originally the tip of the original u-shaped vortex appeared to be dissipating while it rose higher in the boundary layer. At the final time of the simulation, t=109.6, the flow appeared to be in a more complex state than at previous times. Because of the limits on the amount of memory in the workstation used to generate these surface plots, the surface plots had to be divided into two figures (Figure 51k-l) . Figure 51(1) show the downstream portion of the 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. disturbance and consist of older vortices which were dissipating. The upstream vortices shown in Figure 51 (k) was the more active part of the disturbance. The head of the new u-shaped vortex, marked as 'UH' in Figure 51 (k), pulled ahead from the rest of the vortices and became hairpin-shaped. The disturbance at this point in time was quite different in terms of size and complexity than at its initial stage (Figure 51b). There were some differences between the marker particle patterns and the vortices in the A surface plots. Because the marker particle positions were computed by integration in time, these patterns reflect past activity as well as current activity. This explains why a portion of the streaks were upstream of the constant A surfaces. Also, the marker particles which originated in y = 0.3 were too close to the wall to show very much of the vortex-vortex interaction occurring above the wall over 59.5<t<80.8. The only signs of this spanwise growth of the disturbance were the side-streaks. The side-streaks represent particles which were ejected upward earlier in time when the paired vortices were lower in the boundary layer. For t>80.8, the most disordered part of the flow had propagated past the downstream edge of the y = 0.3 particle sheet, so their influence on the particles could not be completely seen. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In order to verify that the regions identified as vortices by the complex eigenvalue method were indeed vortices, contour plots of pressure and components of vorticity on selected planes of the domain will be examined in the following section. Figure 52-Figure 54 show the ©x vorticity in vertical-spanwise planes at times t = 14.8, 39.6, 59.5. These planes were marked with arrows in the respective surface plots in Figure 51. At t = 14.8, the nonzero regions of ©% vorticity at x = 10 and 15 (Figure 52) show that the surfaces of positive A correspond to a counter-rotating vortex pair which was lowering fluid down between them. They also show that the 'UP' and 'DP' parts of the vortex pairs have the same sign of cOx vorticity. Further on in time at t = 3 9.6, secondary vortices show up in Figure 53 as an additional vortex pair outboard of the primary vortex pair. The sign of ©% vorticity in the outboard vortex pair was opposite to that of the primary vortex pair. The vortex-vortex interaction process which resulted in the widening of the disturbance was manifest in the ©x vorticity in vertical-spanwise planes at several different streamwise positions 3 5<x<39 at t = 59.5 (Figure 54) . These locations correspond to the area where the Bridge Vortices interacted with the u-shaped vortex. Figure 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 show that this interaction took the foriti of a pairing between a vortex near the center-plane of the domain and its outboard neighboring vortex. Figure 54 appears similar to the C O contour plot from Orlandi (1990) (Figure 55) which show a secondary vortex dipole pairing with the primary vortex dipole. As a result of this pairing in Orlandi's simulation, an unsteady flow field was created which increased the horizontal scale of this system of vortices beyond the original size of the vortex ring. The capability of three-dimensional vortices to pair up was hinted at in the Henningson et al (1993) description of a 'twisted vortex pair' (see Figure 10). This implies that pairing is a part of the transition process. An alternate method of looking at the development of the disturbance is to examine the pressure and spanwise vorticity, co^, contours on the streamwise-vertical plane along the centerline of the computational domain. The location of this plane was denoted with arrows in Figure 51. The spanwise vorticity contours have the effect of showing the outline of the disturbance which was created by the vortex. Features, such as the roll-up of vortices, can be observed in these figures and compared to prior observations of transitional flows. 1 0 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 56(a-b) is a cut through the Hill's spherical vortex at t = 5.0 which was just before it hit the wall. Figure 56(a) shows a positive pressure region at the 'tip' of the vortex which coincided with the stagnation region at the head of the vortex, and the negative pressure region which corresponded to the vortex itself. Due to the initial orientation, 0, of the vortex, the stagnation region was downstream of the core of the vortex. Figure 56(b) shows two regions of opposite signed vorticity which corresponded to the cores of the Hill's spherical vortex. There was also a shear layer beneath the cores and inclined at a 29° angle with respect to the wall. As the vortex burrowed through the boundary layer, the fluid approaching the stagnation region at the 'tip' of the vortex did not enter the interior of the vortex but was instead diverted around its periphery. Neither Figure 56(a-b) shows any additional vortices besides the Hill's spherical vortex. Had the Hill's spherical vortex shed vortices during its descent, they would have been apparent in Figure 56(a-b). Figure 56(c-d) is a cut through the primary vortex pair at t = 14.8 which was after the Hill's spherical vortex had hit the wall. Figure 56(d) shows that the region of positive spanwise vorticity which was present earlier at t = 5.0 was no longer present at this time. This region was 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. annihilated by the boundary layer spanwise vorticity which was of the opposite sign. Figure 56(d) also shows that the primary vortex pair was bounded on the bottom by a shear layer which was inclined for x > 12. The region x > 12 corresponded to the 'DP' part of the primary vortex pair. This inclined shear layer persisted in the disturbance throughout its development as seen in Figure 56 (d, f, h, j, 1, n) . At t = 14.8, the inclined shear layer was 35’o long in the streamwise direction, and it made a 21° angle with respect to the wall. At t = 109.6, the inclined shear layer made a 7° angle with respect to the wall. It was difficult to determine the streamwise length of the shear layer, because it interacted with the hairpin vortices at the leading edge. Aside from these changes in size, the inclined shear layer did not change in structure. This shear layer accounts for the accumulation of marker particles at the downstream end of the disturbance. The inclined shear layer was originally manifest as a inclined region of positive pressure near the wall in Figure 56(c) . This region originated as the stagnation region at the tip of the Hill's spherical vortex. As time progressed, there was always a positive pressure region near the wall which coincided with the intersection of the inclined shear layer with the wall (Figure 56c, e, g, i, k, m) . This 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. positive pressure region was a moving stagnation point. As the disturbance propagated downstream, near wall fluid ahead of it was diverted away from the centerline. The roll-up of the spanwise heads of the vortices observed in Figure 51 can be observed in Figure 56. Even though the secondary hairpin vortices were not aligned with the centerline of the domain at t = 24.6, their presence was seen in the negative pressure region centered at x = 17, y = 1.4 in Figure 56(e). In Figure 56(f), there was a weak shear layer centered at x = 19, y = 1.5. As the heads of the secondary hairpin vortices merged into the larger u- shaped vortex, the negative pressure region at the leading edge of the disturbance in Figure 56(g) intensified. The shear layer 'back' of the u-shaped vortex can be seen in Figure 56 (h) . As time progressed, this high shear layer grew in the streamwise and vertical directions (Figure 56h, j, 1, n). The head of the u-shaped vortex, which was marked with a 'U' in Figure 56(i-j), dissipated over this time period. At t = 59.5, a 'kink' could be seen in the high shear layer at x = 40.5, y = 1.8 in Figure 56(j) . This 'kink' corresponded to the head of the secondary spanwise vortex and was marked with a 'SSV' in Figure 56(j) . As time progressed, this 'kink' rolled up in Figure 56(1) . This 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 'kink' was coincident with a new negative pressure region in Figure 56 (i, k) . Ultimately, the head of the secondary spanwise vortex dissipated in Figure 56(m-n) like the head of the u-shaped vortex. This pattern of a 'kink' in a high shear layer rolling up into a spanwise vortex while being accompanied by the coincident development of a low pressure region was observed by Sandham & Kleiser (1992). Upstream of these vortices, the tertiary hairpin vortices were developing. Their state at t = 80.8 can be seen in Figure 56 (k-1) . In Figure 56(1), there was a high shear layer at x = 44 which coincided with a negative pressure region in Figure 56(k). These structures resembled the structures associated with the original u-shaped vortex in Figure 56 (g-h) . Later in time at t = 109.6, the shear layer had rolled up into a spanwise vortex at x = 66 and was developing a 'kink' at x = 63 (Figure 56n) . Negative pressure regions were present at these locations in Figure 56 (m) . This indicated that an additional vortex was developing upstream of the tertiary hairpin vortices. The tertiary hairpin vortices were developing in the same way as the secondary hairpin vortices earlier in the simulation. Figure 56 (m) showed that the flow at this time was very complex and grown considerably from its initial state. 1 1 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparisons can be made between the spanwise vorticity contour plots in Figure 55 and the spanwise vorticity contour plots from Singer's (1996) turbulent spot simulation (Figure 21). Like Singer's turbulent spot, the disturbance created by the Hill's spherical vortex was bounded on the top by high shear layers and on the bottom by an inclined near-wall shear layer. In both simulations, the high shear layers developed 'kinks' and rolled up into spanwise vortices. The high shear layers and the near-wall shear layer were features of a turbulent boundary layer which were identified by Robinson (1991). Since this flow would have become turbulent given enough time, it can be expected that these structures could occur in this flow albeit in a more orderly manner. 5.3.Velocity Contour Plots In this section, streamwise velocity contours along selected planes in the domain will be examined. These planes were marked with arrows in the respective surface plots in Figure 51. This will give information on how the vortices within the disturbance were redistributing momentum within the boundary layer. Figure 57 are plots in the vertical and streamwise planes along the centerline of the domain. At t = 0.3, Figure 57(a) shows the vortex centered at (x = 2.5, y = 2.3) 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. above the wall. Because the simulation had just started, the vortex appeared as a region of high speed fluid smaller than the boundary layer thickness. At t = 5.0, ttais region has grown into an elongated 'scar' of high speed fluid which spanned most of the height of the boundary layer as seen in Figure 57(b). Due to its self-induced motion, the vortex, which occupied the bottom of the 'scar,' burrowed through the boundary layer, and was just above the wall at t = 5.0. The rest of the 'scar' consisted of the wake left by the vortex. Because the vortex was a self-propelled body, there was a momentum surplus in its wake. Since the initial orientation of the vortex was O = -tc/4, the wake trailed behind the vortex with respect to the streamwise direction earlier in the simulation. However, at t = 5.0, the wake had overtaken the vortex below it due to the mean shear. By t = 10.0, the Hill's spherical vortex had. already impacted the wall, and the 'scar' of high speed fluid in the boundary layer was beginning to get stretched in the streamwise direction. Figure 57(c) shows the division of the disturbance at x = 10 between the upstream pairt (marked with 'UP') and the downstream part (marked with 'DP'). The upstream part was the clump of high speed fluid at 7<x<10 and was the manifestation of the Hill's spherical vortex compacted against the wall. The downstream paort was a 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 'sweep' which was above the wall. At the leading edge of the disturbance, beneath the 'sweep', was the near-wall shear layer mentioned previously. The lower edge of this shear layer contacted the wall at t = 14.8 at x = 12 (Figure 57d). As Figure 57(c-j) show, the shear layer was a barrier between the laminar flow beneath it and the vortices which develop above it. The development of secondary vortices in the boundary layer shows up in Figure 57(e-j) as a repeating pattern of 'sweeps' and 'ejections.' At t = 24.6, a new 'sweep,' which was marked in Figure 57(e), occurred at x = 17. This streamwise location corresponded to the region where the secondary hairpin vortices were approaching each other with respect to the spanwise direction. After the heads of these two hairpin vortices reconnected into a single spanwise head, the 'ejection' downstream of the head can be observed at the leading edge of the disturbance. It is marked with a 'U' in Figure 57(f-j). Over the time period 3 9 . 6<t<109 . 6, the 'ejection' rose into the upper portion of the boundary layer. The development of the secondary spanwise vortex over 59. 5<t<109 .6, was also manifest as an 'ejection.' It is marked with a 'SSV' in Figure 57(g-j). At t = 59.5, the effect of the Bridge Vortices is seen in Figure 57(g) as a 'sweep' at x = 35. As Figure 57 (g-j ) 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. show, the 'sweep' increased in size in the streamwise direction and was bounded on the downstream end by the 'U' and 'SSV' vortices aligned with the centerline. During the time period 69.2<t<109.6, Tertiary Hairpin Vortices were developing outboard of the centerline. This was manifest as a 'sweep' and an 'ejection' at the trailing part of the disturbance. This was marked with a 'THV' in Figure 57(i-j). As the Tertiary Hairpin Vortices approached each other in the spanwise direction and formed a single u-shaped vortex, this 'ejection' rose in height. The appearances of additional ejections at x = 52, 52, and 66 in Figure 57(j), correlated to the development of additional vortices. Because the disturbance grew in the spanwise direction, it was necessary to examine activity outboard of the centerline. Figure 58 are contours of streamwise velocity plotted in the streamwise-vertical plane lô'o away from the centerline of the domain. Figure 58(a-b) show the shear layer 'back' of one of the Secondary Hairpin Vortices. The downstream edge of the shear layer rolled up and formed the vortex head. Likewise, the shear layer 'back' of one of the Tertiary Hairpin Vortices can be seen in Figure 58 (c-d) . Figure 58 (c-d) also shows the complex activity due to the vortex-vortex interaction. Like Figure 57, the leading part 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the disturbance in this plane hung over laminar flow. In contrast to Figure 57, however. Figure 58 was dominated by 'ej ections'. In Figure 59, contours of streamwise perturbation velocity are plotted in the streamwise-spanwise plane, y = 1.0. Shortly after the Hill's spherical vortex's impact with the wall, low speed streaks outboard of the centerline were a prominent feature in the y = 1.0 plane (see Figure 59a-d) . Initially, they were due to the upward ' ejection' of low speed fluid during the impact. These streaks grew in size and intensity during the time period 10.0<t<24.6. In Figure 59(d), the location of the low speed streaks coincided with the secondary hairpin vortices. This confirms that the secondary hairpin vortices were a result of the roll-up of low speed fluid outboard of the centerline. Meanwhile, the decay of the Downstream Vortex Pair in 10.0<t<14.8 was manifest as a shrinking high speed region along the centerline. As the secondary hairpin vortices merged, a high speed region appeared along the centerline and grew in intensity (see Figure 59d-h) . As the head of the u-shaped vortex developed into a hairpin-shaped vortex and pulled away from the rest of the disturbance, this high speed region became elongated in the streamwise direction. The high speed 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. region was always accompanied to its sides by low speed streaks. This pattern of high and low speed streaks was reminiscent of the Amini & Lespinard's 'incipient spots.' In Figure 59 (e) , the spanwise shear was greatest at x = 23 where the high speed region was sandwiched between two low speed streaks and were associated with the Bridge Vortices at t = 3 9.6. Because the Bridge Vortices interacted with the legs of the u-shaped vortex. Figure 59(f-h) show that the disturbance grew in the spanwise direction until it spanned the width of the computational domain at t = 109.6. Because periodic boundary conditions were applied at the sidewalls, further spanwise growth was suppressed. At this point, the disturbance could be thought of as interacting with its images to the sides of the domain. This would limit the spanwise movement of the Bridge Vortices and prevent further spanwise growth of the disturbance. The leading part of the disturbance in Figure 59 (h) was dominated by the high speed fluid transported by the hairpin vortices in this area. At the trailing part of the disturbance, a second high speed region developed at the centerline in between two low speed streaks. This was due to the Tertiary Hairpin Vortices growing and merging into a u-shaped vortex. The streamwise velocity contours in Figure 59(f-h) became increasingly 1 1 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. convoluted, and show the progressive development of small scale structures which indicate that the disturbance was breaking down. The final stage of the disturbance appears to be similar to the turbulent wedge generated in Saiki & Biringen's (1997) simulation (see Figure 74). 5.4.Wall Shear Stress The motivation for examining the effect of falling particles on boundary layer was its adverse influence on drag of a submersible. Because drag for a flat plate boundary layer is due to fluid viscosity, it is useful to examine the wall shear stress produced by the disturbance. Figure 60 are contour plots of the perturbation skin friction coefficient, Cf' , which is defined as = where Cfb is the local skin friction coefficient for a Blasius boundary layer .66412 in which When the Hill's spherical vortex was beginning to impact the 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. wall. Figure 60(a) shows a region of negative cg' at x = 7. Because there was no slip at the wall, a transient shear layer at the wall was generated beneath the vortex. After the Hill's spherical vortex had hit the wall, the maximum perturbation skin friction coefficient rose to 2.1x10"^ at the impact area (Figure 60b) . The maximum perturbation skin friction coefficient relaxed to 1.1x10"^ at t = 14.8. At this stage, the Hill's spherical vortex had developed into a vortex pair. The split between the 'DP' and 'UP' parts of the vortex pairs show up in Figure 60(c), respectively, as one maxima on the centerline with two maximas outboard of the centerline. These maximas of cg' occurred away from the centerline, because the legs of the vortex pair were moving apart in the spanwise direction due to image vortices effects. Although the creation of vortices and their interaction occurred away from the wall, their signatures show up in Figure 60 (d-i) as a region of increased cg' which grew in the streamwise direction. It should be noted that the 'sweeps' of fluid from above were related to the maximas of Cg' along the centerline. The region marked with an 'SHV' in Figure 60(e) corresponds to the sweep caused by the merging of the Secondary Hairpin Vortices. The formation of the Tertiary Hairpin Vortices was manifest as the region 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 'THV' in Figure 60(f). Cf' in this region intensified in Figure 60(g-h) as the Tertiary Hairpin Vortices merged. Between 24.6<t<80.8, the sole indication of spanwise growth were streaks of positive Cf' located at z = 4.0, 8.0 and were related to the marker particles side streaks. At t = 109.6, Figure 60(i) regions of negative Cf' were present outboard of the centerline at x = 55 and were due to the vortex-vortex interaction which occurred above. Near the centerline, the largest value of Cf' was 1.0x10'^. Thus, the total Cf was 1.1x10”^ which was 11 times Cfb, the skin friction coefficient for the Reg-o = 1.1x10^ Blasius boundary layer. Because the marker particles which were originally in the y = 0.3 plane were so close to the wall, their movements reflected changes in the wall shear stress. The 'pockets' which were cleared of marker particles coincided with the regions of positive Cf'. Undulations in the instantaneous patterns of marker particles ejected out of the y = 0.3 plane were matched by undulations in the Cf ' contours. The side streaks of marker particles have side streaks of Cf' as their counterpart. On the basis of the similarities in the behavior of the marker particles and wall shear stress, it can be surmised that the patterns formed by the upward ejection of dyed fluid through the horizontal light sheet in 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the experiment also reflected changes in the wall shear stress. 5.5. Discussion The development of the flow field in response to the a = 0.8 Hill ' s spherical vortex can be broken down into five stages. 1.Fall Stage: The vortex traversed the boundary layer and created a 'sweep' of high speed fluid which ultimately contacted the wall. 2.Post Impact Stage : Due to the impact of this 'slug', there was an ejection of low speed fluid around the impact region upwards and away from the centerline. 3.Instability in the Ejected Fluid: The ejected fluid rolled up into hai3rpin vortices like the low speed streak in Acarlar & Smith (1987b) experiment. The hairpin vortices consolidated into a larger u-shaped vortex which was accompanied by a sweep at the centerline. 4.Vortex-Vortex Interaction: Bridge Vortices were formed and wound around the legs of the u-shaped vortex. The head of the u-shaped vortex rose higher in the boundary layer and became elongated. 5.Late Stage: Additional vortices develop in the manner described in Step 3 and 4. The disturbance extended over 1 2 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 310*0 In the streamwise direction and 80*o in the spanwise direction according to Figure 59(h). The flow appeared more disordered and contained small scale structures. Because the Hill's spherical vortex produced new vortices throughout the simulation without the use of artificial forcing, transition, as defined before, can be said to have occurred in this case. At this point, the disturbance was not a turbulent spot, but it was an 'incipient spot'. The flow can be compared to features of a turbulent boundary layer (Robinson 1991) in order to establish whether the flow will develop into a turbulent state given enough time. Features observed in the simulation included the presence of hairpin vortices, the shear layer 'backs' to these vortices, and inclined, near wall shear layers. Other features observed were 'sweeps', 'ejections', and 'pockets' cleared of marker particles. Also, the spacing between the primary streaks created by the marker particles was approximately equal to the average spacing between low speed streaks in a turbulent boundary layer. Throughout the simulation, the center of the vortex system was the fluid which was carried down towards the wall by the Hill's spherical vortex and propagated downstream faster than the undisturbed fluid located off to the sides. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The center of this system was manifest as a 'pocket' cleared of marker particles and as an area of elevated wall shear stress. The vortex system entrained low speed fluid in front of it and lifted this fluid up away from the wall, leaving behind a 'pocket'. The ejected fluid rolled up into vortices which formed downstream and to the sides of the 'pocket'. These vortices rose higher in the boxxndary layer and induced a 'sweep' upstream along the centerline. This 'sweep' caused another 'ejection' of low speed fluid to the sides which rolled up into new vortices. Like the transitional flow in the experiment, the Hill's vortex produced a disturbance which ultimately grew to many times the initial size of the vortex. The increase in the streamwise length of the disturbance over its initial size was due to the mean shear and to the fact that the disturbance spanned the entire height of the boundary layer. At t = 109.5, the spanwise width of the disturbance at y = 1.0 was 13 times larger than the initial size of the Hill's spherical vortex. There seemed to be two causes for the increase in spanwise width. One was the consolidation of two side by side hairpin vortices into a single, wider vortex. The other cause was the unsteady motion caused by pairing between neighboring vortices. This unsteadiness generated small scale structures by tearing apart the 1 2 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vortices involved. Complexity in the flow as a result of interaction between neighboring vortices was seen in Acarlar Sc Smith (1987b). It is possible that interaction between vortices within a turbulent spot was responsible for both the spot's spanwise growth and the generation of small scale structures within it. This implies that the internal vortex dynamics within a turbulent spot may not be as orderly as described by Perry, Lim, & Teh (1981). Because the subsidiary vortices caused some of the spanwise growth of the disturbance in Haidari & Smith's experiment, it was important to note that they were absent in this simulation. It is possible that the limited domain size might have prevented their formation. However, it is also possible that a different configuration of vortices besides subsidiary vortices produced the hydrogen bubble patterns observed by Haidari S c Smith. 5. 6.Comparison of Numerical and Experimental Results Because there were no velocity measurements taken during the experiment, a direct comparison between the flow fields of the simulation and the experiment was not possible. However, comparisons between the experimental flow visualizations and the marker particle trajectories from the simulation can be made. In this section, the term 1 2 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 'tracer' will be used to describe dye and marker particles interchangeably. There were similarities between the wakes left by the Hill's vortex and the sphere during their descents towards the wall. These similarities could be seen in the dye left by the moustache wax covered sphere in the experiment and in the marker particles in the 'ribbon' and in the velocity contour plots of the simulations. Like the sphere wake in the experiment, the vortex wake spanned the thickness of the boundairy layer and was compact in the horizontal directions. After the sphere impacted the wall, the dyed sphere wake was rotated about the spanwise direction by the mean shear, and the upper portion of the wake was visible for many nondimensional time units. The persistence of the dye following the sphere impact showed that the upper portion of the wake didn't dissipate by mixing with the surrounding fluid. When transition occurred, it occurred upstream of this upper portion in the lower part of the boundary layer. In the simulation, the marker particle trajectories and the velocity data showed that the Hill's vortex left a wake in the boundary layer which was also rotated about the spanwise direction by the mean shear. Like the sphere wake, remnants of the vortex wake in the upper part of the boundary layer persisted for many nondimens ional time units. From the 1 2 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. results of the numerical simulation, it can be concluded that the effect of the falling sphere was to transfer some of its momentum to surrounding fluid leaving a wake in the boundary layer. Several common features can be seen in the acetate sphere case and the Strong Vortex Case prior to breakdown. In the time following impact, the disturbances for both the experiment and simulation created tracer patterns which took the form of streamwise streaks. As these streaks propagated downstream, they grew in both the streamwise and spanwise directions. The widest portion of the tracer patterns was upstream of the leading edge. In some runs of the experiment, two parallel streaks separated in the spanwise direction appeared in the laser light sheet prior to breakdown. This was similar to the two streaks formed by the marker particles in the region 0.3 5<y<1.0. One difference between the tracer patterns of the experiment and the simulation was the absence of discrete side streaks in the experimental flow visualizations. Considering that the dye was a scalar field and not a finite group of discrete particles, it is possible that the side streaks would have been smeared out. Another difference was the asymmetry with respect to the spanwise direction which developed at the leading part of the tracer pattern in the experiment. The 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. simulation tracer pattern and the velocity field were symmetric with respect to the centerline of the computational domain along spanwise direction. This was a consequence of the spanwise symmetry of the initial velocity field. Anderson et al. (1990) wrote that the Navier-Stokes equations preserves any symmetry present in the initial condition. In both the experiment and the simulation, the leading edge of the disturbance overhung a region of undisturbed fluid. In the experiment, unsteady motion at the leading part of the disturbance was accompanied by growth in the spanwise direction. Likewise in the simulation, unsteady motion at the leading edge of the disturbance was caused by interaction between vortices and led to the widening of the disturbance. In the simulation, the vortices at the leading part of the disturbance migrated away from the wall to the same height as the remnants of the vortex wake. The marker particles in the 'ribbon' could not capture this unsteadiness because, they were restricted to the centerline. Thus in the experiment, the influence of vortices rising to the same height as the falling sphere wake, and not some instability in the sphere wake, caused the unsteady dye patterns in Figure 32. These vortices were 1 2 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. invisible in Figure 32, because only the sphere wake and the fluid near the wall was dyed. Because the tracer patterns from the experiment and the simulation were both due to ejections of low-speed fluid, a comparison of the sizes of the dye pattern and the particles patterns would give a quantifiable way of comparing the development of the disturbances. Figure 61 is a comparison of the spanwise width of the tracer patterns which were ejected out of the y = constant sheet near the wall for both the experiment and the simulation. The simulation width was plotted using time of the vortex impact, t = 5.0, as the reference time. This impact time was obtained from the wall shear stress data (see §5.4). The width was obtained by finding the largest spanwise distance between any two marker particles in the streaks in Figure 47. As Figure 47 shows, even the outlying marker particles were still clustered near the streaks. The width was not measured for t>80.8 in the simulation, because the disturbance had propagated past the downstream limit of the particle sheet. The sudden jump in width of the simulation tracer patterns at t = 54 was due to the development of the side-streaks. Readily identifiable side-streaks did not occur in the experimental flow visualization at the same time as the side streaks in the simulation. Although both the width of the simulation and 1 3 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. experiment tracer patterns increased with respect to time, the width of tracer pattern in the simulation grew faster than the corresponding tracer pattern in the experiment. Even if the scatter in the width of the experimental tracer patterns was accounted for, the simulation tracer patterns were still wider. The width of the experiment tracer patterns might have been affected by the sphere which rolled behind the turbulent spot. Another possible cause for the disagreement is differences in the wakes left by the falling sphere and the Hill's vortex. As Figure 3 4 shows, there was already quite a bit of scatter in the spanwise width of the disturbances produced by the falling acetate spheres from run to run. The spanwise growth of the disturbance may have been extremely sensitive to its initial condition which, in the experiment, was the wake left by the falling sphere. In contrast to the spanwise width measurements, there was good agreement between the leading edge positions of the tracer patterns from the experiment and simulation (Figure 62) . The impact position of the sphere or vortex was used as the origin of the leading edge positions. The leading edge position of the simulation tracer patterns in the volume 0.3 5<y<1.0 were obtained from Figure 47. The location of negative Cf' at t = 5.0 was used as the impact position for the simulation data (see §5.4) . The marker 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. particles in the 'ribbon' were ignored for this analysis. The times in Figure 61-Figure 62 were with respect to the respective impact times of the sphere and the Hill's vortex. As explained previously, the leading edge position for t>80.8 in the simulation was not measured. Figure 62 shows that the tracer patterns for the experiment and the simulation propagated downstream at the same streamwise velocity. Because the leading edge of the experimental tracer patterns was the farthest position away from the rolling sphere, the sphere's absence in the simulation did not affect the leading edge positions. Because the trailing edge of the tracer patterns in the experiment was affected by the rolling sphere, a comparison between the streamwise lengths of the tracer patterns of the experiment and the simulation was not attempted. In spite of the difference in spanwise growth of the tracer patterns between the simulation and the experiment, there were numerous qualitative similarities between the experimental disturbance fields, as inferred from the flow visualization, and the disturbance field in the simulation. It can be concluded that the simulation captured the triggering mechanism for breakdown in the experiment. 1 3 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 10 U J 60 20 40 0 80 100 120 Figure 45 Growth of energy with respect to time for the strong vortex case (a = 0.8) . ( — ) : u~ , v~ , ( . . . ) : 1 3 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 4 3 2 1 0 20 15 10 5 - G - % 10 (a) t = 3.5 10 30 8 degrees 10 15 20 25 5 (b) t = 24.6 10 15 20 25 30 35 40 (c) t = 39.6 30 40 50 60 7 0 80 90 1 0 0 1 1 0 1 2 0 (d) t = 109.6 Figure 46 Instantaneous (x, y) positions of marker particles which originated in the 'ribbon.' Side View. 1 3 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 10 30 20 25 5 10 15 15 10 10 15 20 25 30 5 (a) t = 10.0 (b) t = 14.8 15 10 5 10 15 20 25 30 (c) t = 24.6 15 10 10 15 20 25 (d) t = 39.6 30 15 10 5 0 15 20 25 30 35 40 45 (e) t = 59.5 50 15 10 5 0 15 20 25 30 35 40 45 50 (f) t = 69.1 15 10 5 0 15 20 25 30 35 40 45 50 (g) t = 80.8 15 10 5 0 - 25 30 35 40 45 50 55 60 ( h . ) t = 109.6 Figure 47 Instantaneous (x, z) positions of marker particles in the region 0.35<y<1.0. Plan View. 1 3 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 10 10 15 20 25 30 (a) t = 10.0 15 10 20 25 30 5 10 15 (b) t = 24.6 15 10 10 (C) 15 t : 20 25 39.6 30 15 10 '"5 0 15 20 25 30 35 40 45 50 (d) t = 69.1 15 10 ■ ' 5 0 ■Æ: ■ î--. 15 20 25 30 35 40 45 50 (e) t = 80.8 Figure 48 Instantaneous (x, z) positions of marker particles in the region 0.0<y<0.35. Plan View. 1 3 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 40 30 20 10 y 10 z Figure 49 Trajectories of particles originating in 8.5<x<9.0, y=0.3 over the time period 0.0<t<94.7. 50 40 30 20 10 Y 10 z Figure 50 Trajectories of particles originating in 15.0<x<15.5, y=0.3 over the time period 0.0<t<94.7. 1 3 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 25 20 A=10 > (a) t 3.5 30 DP 25 20 UP 15 10 A =10 >- (b) t 10.0 30 25 UP 20 15 10 >- (c) t 14.8 30 25 20 15 A = 10 (d) t 24. 6 Figure 51(a-d) For caption see p.140. 1 3 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U-VORTEX / / A =10 15 (e) t = 39.6 U-VORTEX A=10" (g) t = 59.5 Figure 51(e-h.) For caption see p. 140. A =10 30 A = 10 (h) t = 69.1 ssv 1 3 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ssv THV/ 55 50 45 40 A=10‘ 35 12 (i) t 80.8 65 UH AV UH 75 70 BV 65 60 55 50 >- A=10 45 12 40 94.7 (j) t 65 60 55 50 A=10‘ 45 40 12 (k) t 109.6 70 100 95 90 85 80 75 A=10 70 12 109.6 (1) t Figure 51 Iso-surfaces of regions where the eigenvalues of the local velocity tensor is complex. Arrows show the positions of the planar contour plots 'slices' through the domain. 1 4 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) X = 10.0 3 2 1 0 0 2 6 10 12 4 8 (b) X = 15.0 Figure 52 a>x vorticity contours at several streamwise locations at t = 14.8. Strong vortex case. Contour spacing is 0.5. Negative contours are dashed. 3 2 1 0 12 (a) X = 22.0 3 2 1 0 0 2 6 10 4 8 12 (b) X = 25.0 Figure 53 a>x vorticity contours at several streamwise locations at t = 39.6. Strong vortex case. Contour spacing is 0.2. Negative contours are dashed. 1 4 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 2 « 6 e TO t z (a) X = 35 2 0 2 4 6 8 10 12 Z ( b ) X = 36 2 (c) X = 37 (d) X = 38 (e) X = 39 Figure 54 ©x vorticity contours at several streamwise locations at t = 59.5. Strong vortex case. Contour spacing is 0.2. Negative contours are dashed. 1 4 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t =6 t =18 t =8 t =20 t =10 t =22 » * t =12 t =14 t =24 t =26 t =28 t =16 Figure 55 Re = 800 vortex pair impacting a wall. O ) contours at different points in time. (from Orlandi 1990) 1 4 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as s 8 29(Segree9 (a) t = 5.0 Pressure (b) t = 5 .0 C B z DP 21 degrees 25 X (c) t = 14.8 Pressure (d) t = 14.8 c o z »V. « % 25 X X (e) t = 24.5 Pressure (f) t = 24.6 c O z Figure 56(a-f) see p.146 for caption 1 4 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (g) t = 39.6 Pressure (h) t = 39.6 Cùz (i) t = 59.5 Pressure (j) t = 59.5 02 (k) t = 80.8 Pressure (1) t = 80.8 CDz Figure 56(g-l) see p.146 for caption 1 4 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. /degrees (m) t = 109.6 Pressure (n) t = 109.6 C Ù Z Figure 56 Pressure contours on the z = 5.73 plane (-0.035"o from centerline) (a,c,e,g,i,k,m). (a) Contour spacing is 0.02; contour spacing is 0.002 for the rest. ojz vorticity contours on the z = 5.73 plane (b,d,f,h,j,1,n). (b) Contour spacing is 1.0; contour spacing is 0.2 for the rest. Negative contours are dashed. Strong vortex case. 1 4 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X X (a) t = 0.3 (b) t = 5.0 (c) t = 10.0 (d) t = 14.8 (e) t = 24.6 Figure 57(a-f) see (f) t = 39 p. 148 for caption 1 4 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (g) t = 59.5 (h) t = 69.1 (i) t = 80.8 (j) t = 109.6 Figure 57 u velocity contours on the z = 5.73 (-0.035*o from centerline) plane. Strong vortex case. Contour spacing is 0.1. 1 4 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) t = 24.6 (b) t = 39-6 (c) t = 69.1 (d) t = 80.8 Figure 58 u velocity contours on the z = 4.77 plane (-5'o from centerline). Strong vortex case. Contour spacing is 0.1. 1 4 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) t 5.0 (b) t = 10.0 (c) t = 14.8 (d) t = 24.6 (e) t = 39.6 (f) t = 59.5 (g) t = 80.8 (h) t = 109.6 Figure 59 u perturbation velocity contours on the y plane. Strong vortex case. Contours spacing is 0.1. Negative contours are dashed. = 1.0 1 5 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.0 (a) t (c) t = 14.8 (d) t = 24.6 SHV X (e) t = 39.6 (f) t = 59.5 Figure 60(a-f) For caption see p.152. 1 5 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (g) t = 69.1 (h) t = 80.8 ; ° " 4 M P (i) t = 109.5 Figure 60 Perturbation skin friction coefficient on y = 0.0 for the Strong Vortex Case. Contours spacing is 0.001. Negative contours are dashed. 1 5 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 T I a Experim ental Data ' — N um erical Simulation _5 w o i m 2 ■■ 1 60 60 Figure 61 Comparison of spanwise widths of the tracer patterns from the Ref=5xl0^ sphere and the Strong Vortex case from the simulation, to is the time of impact of the sphere or vortex. 80 70 a Experim ental Data — N um erical Simulation 40 o > .30 O ) «20 0 20 40 60 80 100 120 (t-yuj5*, Figure 62 Comparison of the leading edge of the tracer patterns from the Ref=5xl0^ sphere and the Strong Vortex case from the simulation, xq and to are the streamwise position and time of impact of the sphere or vortex. 1 5 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6. WEAK CASE In this simulation, the Hill's spherical vortex was given the initial amplitude a = 0.4 which corresponds to Ref of 264. This was approximately the same Ref as the polystyrene sphere described in §3.2-3.3. Like the a = 0.8 case, the Hill's vortex with this particular amplitude had a lower perturbation energy, E, than the vortex pairs used by Breuer & Landahl (1990) as Table 2 shows. However, the peak V velocity, |v|max, of the a = 0.4 Hill's vortex was many times greater than the vortex pairs. Figure 63 is a plot of the time history of E as defined in Eqn. (6) over the time period 0.0<t<100.8. The perturbation energy, E, rose over the course of the simulation. However, the growth rate of E decreased at t = 15 and at t = 45. As will be seen later on, these times corresponded to the impact of the Hill's vortex with the wall and the formation of a secondary vortex, respectively. Although the perturbation energy, E, and u~ rose throughout the simulation, v~ decayed for t > 15. This coupling between the growth rate of E and was consistent with Breuer & Haritonidis's (1990) observation that a decrease in 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V would cause a decrease in the disturbance's overall growth rate due through the 'lift-up' effect. 6.1. Marker Particles The same initial configuration of marker particles as the Strong Vortex Case was used in this simulation with the Weak Vortex. Figure 64 shows the instantaneous positions of marker particles in the region 0.3 5<y<1.0 as seen by an observer looking down towards the wall from above. Figure 54(a-d) shows that the fall of the Hill's vortex created two parallel streaks which grew in the streamwise direction like the Strong Vortex Case. Throughout this time period, the spanwise distance between the two streaks was 2ô’ ' o • But, the streamwise length of the streaks grew from 3ô’o at t = 24.7 to 145*0 at t = 70.8. The sole change in the patterns formed by the marker particles was a 'break' which was a region cleared of marker particles at the middle of the streaks. This 'break' was on x = 35 at t = 70.8 (Figure 64c) and on x = 42 at t=100.8.( Figure 64d) Unlike the Strong Vortex Case, no additional streaks appeared outboard of the centerline during this time period. Figure 65 shows the marker particles in the region 0.0<y<0.35 as seen from the same viewpoint as Figure 64 over the time period 24. 7<t<100.8. Figure 65(a-d) shows that, 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. like the Strong Vortex Case, the sheet of marker particles was being depleted of particles to create the streaks. The streamwise length of the 'pocket' grew from 30*o to 185*o over the time period 24.7<t<70.8. The counterpart of the 'break' in the streaks was an accumulation of particles along the centerline at x = 33 in Figure 55(c) and at x = 39 at Figure 65(d). Figure 65(d) shows that the disturbance had propagated past the leading edge of the y = 0.3 sheet, so an estimate of the size of the streaks and the 'pocket' was not possible at t = 100.8. Another similarity with the marker particle patterns from the Strong Vortex Case can be seen in Figure 66. Figure 66(a-c) are the instantaneous positions of the marker particles which were originally in the 'ribbon' as seen by an observer looking in from the side. Like the Strong Vortex Case, these particles accumulated at the leading edge of the disturbance. This border was stretched from a length of 125*0 at t = 24.7 to 32ô*o at t = 62.8 in the streamwise direction by the mean shear. Between 24.7<t<62.8, the angle of the border ranged from 9° at t = 24.7 to 4° at t = 62.8. Figure 67 shows the trajectories of marker particles which were originally at 9.5<x<10.0, y = 0.3 over the time period 0.0<t<100.8. These particles were in the area where 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the Hill's vortex impacted the wall. Unlike the Strong Vortex Case, these marker particles did not diverge away from the centerline following the impact of the vortex. Some marker particles were ejected upwards and travelled downstream farther than the lower-lying marker particles. Figure 68 shows the trajectories of marker particles which were initially at 19.5<x<20.0, y = 0.3 during the same time period as Figure 67. Long after the impact of the Hill's vortex, the disturbance grew strong enough to lift the marker particles along the centerline upwards and transport them away from the centerline. This divergence of particles away from the centerline was not as severe as in the Strong Vortex case. 6.2 .Vortices Present in the Boxxndary Layer As in the Strong Vortex case, the complex eigenvalue method was used to visualize the vortices which produced the marker particles patterns before. In Figure 69, three- dimensional surfaces where the scalar A was at a positive threshold were plotted at several instances in time between 14.7<t<70.8. These figures show a considerably more simple structure than the vortices observed in the Strong Vortex case. Figure 69(a) shows that shortly after the impact of the vortex with the wall, the disturbance consisted of a 1 5 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vortex pair tilted about the spanwise direction with respect to the wall. This appeared qualitatively similar to the vortex pair observed in the Strong Vortex case in the time period after the Hill's vortex impact with the wall. However in this case, the part of the vortex pair nearest the wall did not move apart in the spanwise direction as a result of the impact. The spanwise spacing between the vortex pairs was 0.75’o over its length. While the vortex pair was being stretched in the streamwise direction by the mean shear, an additional vortex was developing underneath the vortex pair. At t = 34.7, this vortex was could be seen in Figure 69(b) at x = 26. This corresponded to a split in the vortex pair into an Upstream Part and Downstream Part shown as 'UP' and 'DP' respectively in Figure 69(b). The vortex which separated the Upstream and Downstream parts of the vortex pair was at X = 30 and had grown in size by t = 44.8 (Figure 69c). The rest of the vortex pair had become thinner, and the growth rate of E decreased at this time as noted earlier. Figure 69(b-c) used A = 10"® as the threshold value. The fact that this split in the vortex pair occurred at a much latter time than the Strong Vortex case implies that the disturbance produced by the Weak Vortex developed at a much slower rate. 1 5 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Further along in time at t = 70.8, the vortices showed signs of decay in Figure 69 (d) which used the same threshold of A as Figure 69(b-c) . The 'DP' part of the vortex pair no longer shows up at this threshold of A, and the 'UP' part had diminished in size. The remnant of the vortex which linked the vortex pair was at x = 39. This position was downstream of the 'break' in the marker particles patterns which were remarked on earlier. This was consistent with the observation with the Strong Vortex case that the changes in the marker particle patterns lagged behind the formation of vortices higher up in the boundary layer. 6. 3. Velocity Contour Plots Figure 70 are plots of the streamwise velocity contour in the vertical-streamwise plane along the centerline of the domain. This plane was marked with arrows in the respective surface plots in Figure 69. Figure 70{a-b) shows the progress of the vortex from its initial position towards the wall. The bottom of the 'scar' of high momentum fluid was at (x, y, z) = (7.0, 1.3, 5.76) at t = 4.7 in Figure 70(b). This meant that the vortex took longer to reach the wall than it did in the Strong Vortex Case. 1 5 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Although the 'scar' was almost perpendicular with respect to the wall at t = 4.7, the sweep of high speed fluid in Figure 70(c) was at a 20° angle with respect to the wall by the time t = 14.7. The sharp velocity gradients at the base of the 'scar' which were observed at the impact area of the Strong Vortex were not observed for this case. At t = 3 4.7, the division between the Upstream and Downstream parts of the vortex pair can be seen in Figure 70(d). However, Figure 70(e-g) show little dramatic change in the disturbance between 44 . 8<t<100.8. The vortex which linked the vortex pair corresponded to the border between the Upstream and Downstream parts of the vortex pair. Over time, there was an mild upwelling of low speed fluid at this border. Figure 71 were plots of the streamwise perturbation velocity in the streamwise-spanwise plane y = 1.0. This plane was marked with arrows in the respective surface plots in Figure 69. Figure 71(a) shows the disturbance at t = 14.7 which, as will be seen in §6.4, was the time the vortex impacted the wall. The high speed region which corresponded to the 'scar' is the dominant feature in Figure 71(a) . At t = 24.7, the vortex pair produced a high speed region along the centerline bounded by two low speed streaks to its sides as seen in Figure 71(b). The spanwise spacing 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between the low speed streaks was 2ô’o- Aside from growth in the streamwise direction due to the mean shear, this basic configuration does not change throughout the simulation as seen in Figure 71(c-e). The disturbance remained confined to a narrow 25*o band along the centerline of the domain. At t = 10 0.8, there was a low speed region which was present along the centerline at x = 53 and corresponded to the border between the Upstream and Downstream parts of the vortex pair. The streaks in Figure 71(e) appeared similar to the dissipating disturbance which developed from the s = 0.0001 vortex pairs from Henningson et al. (1993). 6.4. Wall Shear Stress Figure 72 show the perturbation skin friction, Cf', as defined by Eqn.(5) at the wall. When the Hill's vortex was about to impact the wall at t = 14.7, there was a region of negative Cf' at x = 13 (Figure 72a) . This decrease in wall shear stress is a useful diagnostic for determining if an object has impacted a no-slip wall. After the impact of the Hill's vortex. Figure 72(b-e) show that the perturbation skin friction coefficient rose to 8.9x10'^ at t = 19.7 before steadily declining throughout 1 6 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the simulation. This is the clearest sign that the Weak Vortex produced a decaying disturbance. An interesting feature of the skin friction contour plots was that it showed the division of the vortex pair between the Upstream and Downstream parts. A region of negative ce' at x = 30 appeared downstream of a larger region of positive Cf' in Figure 72(c). This configuration of a region of negative Cf' sandwiched between two regions of positive Cf' also occurred with the Strong Vortex case in Figure 60(c). There was a correspondence between the shapes and positions of the regions of positive Cf' and the patterns of the marker particles ejected out of the y = 0.3 plane. At t = 70.8, the region of positive Cf' which tapered at x = 33 was related to the 'break' in the marker particle patterns downstream of it. This again confirms the link between the marker particle patterns and the skin friction patterns seen in Figure 72. 6. 5.Discussion The development of the boundary layer in response to the a = 0.4 Hill's vortex can be summarized in two steps. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.Fall Stage: The vortex traversed the boundary layer and created a 'sweep' of high speed fluid which ultimately contacted the wall. 2.Post-impact stage: There was a mild ejection of low speed fluid in response to the impact of the vortex. The disturbance underwent slow growth and showed signs of dissipation in spite of getting stretched by the mean shear. 6. 6. Comparison of the Strong- and Weak Vortex Cases The disturbance did show some common features with the Strong Vortex Case. There was an inclined near-wall shear layer in Figure 70 which defined a border between the disturbance and the undisturbed fluid beneath it. Like the Strong Vortex case, this shear layer was manifest as an accumulation of marker particles which were initially part of the 'ribbon.' However, there were distinct differences. A comparison of the time histories of perturbation E normalized by the initial perturbation energy, Eo, (Figure 73) between the Strong and Weak Vortex Cases shows that the E/Eo for the Weak Vortex reached a plateau while E/Eo for the Strong Vortex continued to increase. Also, the recurring 'sweeps' and 'ejections' of the disturbance produced by the Strong Vortex did not occur with the Weak Vortex. There was no high shear layer present, and there 1 6 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was no generation of small scale structures. The disturbance did not produce new vortices in a self- sustaining manner, and thus, it cannot be considered a transition under the criterion defined in §2.3. Differences in how the Strong and Weak Vortices impacted the wall, as characterized by Ref, is the factor which determined the occurrence of transition. The Strong Vortex, which had Ref = 528, produced an energetic, growing disturbance, and the Weak Vortex which had the lower Ref = 2 64 produced a laminar, decaying disturbance. These differences were evident in perturbation skin friction in the time period after impact. The appearance of a region of negative Cf' was used as an indicator of when the vortex impacted the wall. Under this definition, t = 5.0 and t = 14.7 were the times of impact for the Strong and Weak Vortex respectively. At t = 10.0, which was 5.0 time units after the Strong Vortex hit the wall, the perturbation skin friction coefficient was 2.1x10"'% At t = 19.7, which was 5.0 time units after the Weak Vortex hit the wall, the perturbation skin friction coefficient was 8.9x10'^. This quantitatively shows that the Strong Vortex made a harder impact with the wall than the Weak Vortex. Differences in the impact between the two cases could be seen in the way the marker particles in the vicinity of the impact area 1 6 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. responded. In the Strong Vortex case, the impact was vigorous enough to eject some of the particles upward and create an area cleared of marker particles. This clearing did not occur with the Weak Vortex. This ejection of low speed fluid was important in the case of the Strong Vortex, because it led to the formation of additional vortices. The streamwise velocity contours on the y = 1.0 plane show that the ejection of low speed fluid during the impact of the Strong Vortex was more intense than the in the Weak Vortex case. Thus, there is a link between Reg of the Hill's spherical vortex and the ejection which resulted from the impact of the vortex with the wall. 6. 7. Comparison of Numerical and Experimental Results A comparison between the polystyrene sphere case and the Weak Vortex case is straightforward. No turbulent spot was created by the impact of the polystyrene sphere, and the Weak Vortex produced a laminar, decaying disturbance. In both the experiment and the simulation, the tracer patterns left by the disturbances showed two parallel streaks which were elongated in the streamwise direction as time progressed. The formation of the streaks in the experiment lagged in time behind the formation of streaks in the simulation, and the streaks in the experiment didn't show the 'break' that appeared in the simulation. A comparison 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between the tracer patterns in these two cases is complicated by the presence of the rolling sphere in the experiment following impact. Even though the sphere was rolling downstream, it produced a system of vortices consistent with a fixed spherical roughness element. One of these vortices included the horseshoe vortex which had wrapped itself around the sphere. (See Appendix A for a description of horseshoe vortices in relation to roughness elements). The influence of the horseshoe vortex might account for the differences between the tracer patterns in the experiment and the simulation. In spite of the differences, the Weak Vortex was able to replicate the polystyrene sphere's inability to cause transition. 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.5 0.45 0.4 0.35 0.3 U J 0.25 0.2 0.15 0.1 0.05 — ■ I- 100 10 20 30 40 60 70 80 0 50 90 Figure 63 Growth of energy with respect to time for the weak vortex case (a = 0.4). ( — ) : v~ , ( . . . ) : w~ , ( — ) : + V" + 1 6 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 15 10 10 5 10 15 20 25 30 (a) t = 24.7 5 10 15 20 25 30 (b) t = 44.8 15 10 15 20 25 30 35 (C ) t = 70.8 40 15 10 25 30 35 40 45 50 ( d ) t = 1 0 0 . 8 Figure 64 Instantaneous (x, z) positions of marker particles in the region 0.3 5<y<1.0. Plan View. 15 10 10 15 20 25 (a) t = 24.7 30 15 10 5 10 15 20 25 30 (b) t = 44.8 15 10 15 20 25 30 35 (C) t = 70.8 40 15 10 25 30 35 40 45 50 (d ) t = 1 0 0 . 8 Figure 65 Instantaneous (x, z) positions of marker particles in the region 0.0<y<0.35. Plan View. 1 6 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y lOr 10 15 20 (a) t = 24.7 2 5 3 0 10 5 G 15 20 25 30 35 40 45 (b) t = 44.8 50 10 5 — 0 2 5 3 0 35 4 0 45 50 5 5 6 0 (c) t = 62.8 Figure 66 Instantaneous (x, y) positions of marker particles which originated in the 'ribbon.' Side View. 1 6 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / J I Figure 67 Trajectories of particles originating in 9 . 5<x<10.0 , y=0.3 over the time period 0.0<t<100.8. 50 45 40 35 30 2 5 X 20 15 10 Y 10 Z Figure 68 Trajectories of particles originating in 19.5<x<20.0, y=0.3 over the time period 0.0<t<100.8 1 7 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 25 20 10 A=10 (a) t 14.7 DP 40 35 UP 30 ■VORTEX 25 20 6=10 15 10 (b) t 34 .7 45 40 35 30 25 >- 20 A=10 15 (c) t 44.8 55 50 45 40 30 (d) t 70.8 60 F i g u r e 69 Iso-surfaces of regions where the eigenvalues of the local velocity tensor is complex. 1 7 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) t = 0 (b) t = 4.7 (c) t = 14.7 (d) t = 34.7 t = 44.8 Figure 70(a-f) For caption see p.173. (f) t = 70.8 1 7 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (g) t = 100.8 Figure 70 u velocity contours on the z = 5.73 plane (-0.035*0 from centerline). Weak vortex case. Contour spacing is 0.1. 25 (a) t = 14.7 (b) t = 24.7 X (c) t = 44.8 (d) t = 70.8 (d) t = 100.8 Figure 71 u perturbation velocity contours on the y = 1.0 plane. Weak vortex case. Contours spacing is 0.1. Negative contours are dashed. 1 7 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2S (a) t = 14.7 (b) t = 19.7 35 (c) t = 44.8 (d) t = 70.8 (e) t = 100.8 Figure 72 Perturbation skin friction coefficient on y for the Weak Vortex Case. Contours spacing is 0.001. Negative contours are dashed. = 0.0 1 7 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 80 70 60 50 40 30 20 0 20 40 60 80 100 120 Figure 73 Comparison of growth rates for the strong and weak vortex cases. Time History of total energy normalized by initial energy. (— ) : Strong Vortex Case (a = 0.8) ; (---) : Weak Vortex Case (a = 0.4) 1 7 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7. CONCLUSION In summary, this project studied how a compact, sphere falling through a laminar boundary layer can cause a transition to turbulence in the boundary layer. This problem is significant to the operation of submersibles in particle-laden water, because the particles interfere with the submersible ' s ability to maintain a low drag. The first part of this project was an experiment which involved letting individual spheres free-fall through a Blasius boundary layer and observing their effect with flow visualization. The second part of this project was a numerical simulation in which the free-falling spheres were modelled with a Hill's spherical vortex. The goal of the simulation was not to duplicate the flow in the experiment in every detail but, to identify the major factors leading to transition. With the exception of Hall (19 67) , the use of falling sphere or free-moving vortices was very different than prior efforts to destabilize a boundary layer. The wall jets used by Singer & Joslin (1994), Acarlar & Smith (1987b), and Haidari & Smith (1994) were ejections of low speed fluid away from the wall; the disturbances used in this project 1 7 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. originated from the upper part of the boundary layer and caused a 'sweep' of high speed fluid towards the wall. The disturbances used in this project were also a fraction of the boundary layer thickness in contrast to the vortex pairs used by Breuer & Landahl (1990) and Henningson et al. (1993) . In the experiment, it was observed that the acetate sphere produced a turbulent spot as a result of its impact, while the slower-falling polystyrene sphere did not. The acetate sphere had a Ref = 5x10^, while the polystyrene sphere had a Ref = 2x10^. Flow visualization of the acetate sphere case showed that the early stages of the transition involved a vigorous ejection of low speed fluid. The dyed disturbance propagated downstream and grew in size but left behind laminar flow in its wake. Unsteadiness was observed in the dye patterns preceding breakdown into a turbulent spot. In contrast, the polystyrene sphere produced a laminar disturbance which was stretched in the streamwise direction but did not undergo any spanwise growth. Under the experimental conditions described in this report, no other factors, such as Tollmien-Schlichting wave breakdown or mean flow modification by the sphere rolling on the wall following impact, were responsible for the transition. 1 7 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the numerical simulation, a Hill's spherical vortex and its image vortices were superposed into a Blasius boundary layer, and the resulting velocity field was used as the initial condition for a finite difference Navier-Stokes equation solver. When the Hill's spherical vortex with Ref = 528 was introduced into the boundary layer, the boundary layer went through the early stages of transition as a result of an ejection of low speed fluid during the impact of the vortex with the wall. There were structures in the boundary layer which were consistent with transitional flows as observed in other experiments and simulations including the experiment using the acetate spheres. When the Hill's spherical vortex with Ref = 264 impacted the wall, the ejection of low speed fluid was less vigorous. Like the polystyrene sphere experiment, a laminar, decaying disturbance was created by the impact of the Ref = 264 vortex. 1 7 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A TRANSITION DUE TO FIXED ROUGHNESS ELEMENTS The structure of the wake behind a fixed roughness element have been observed in several experiments and simulations: Acarlar & Smith (1987a) dealt with hemispheres; Klebanoff, Cleveland, & Tidstrom (1992) dealt with hemispheres and upright cylinders; Mochizuki (1961) dealt with spheres. Saiki & Biringen's (1997) simulation modelled a sphere suspended above the wall as an unsteady force. Acarlar & Smith (1987a) observed hairpin vortices were shed periodically from the top of the hemisphere roughness element in a laminar boundary layer. Boundary layer vorticity lines wrap around the bottom part of the hemisphere to form a horseshoe vortex. The tip of the horseshoe vortex consists of spanwise vorticity at the stagnation point of the roughness element and the vortex's two legs point downstream and consist of streamwise vorticity. Unlike the hairpin vortex mentioned earlier, the horseshoe vortex lies completely flat against the wall. Similar observations have been made by Klebanoff et al. (1992), Mochizuki (1961), and Saiki & Biringen (1997). However, the occurrence of a pair of trailing vortices immediately downstream of the roughness element was observed in these studies but was not noted in Acarlar & Smith 1 7 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1987b). The horseshoe vortex did not occur in Saiki & Biringen's (1997) simulation, because the sphere was suspended above the wall. The ability of a fixed roughness element to cause transition at the roughness location, when its height, k, is smaller than the undisturbed displacement thickness, 5*, has been studied experimental ly by Klebanoff et al. (1992). This ability is dependent on the shape of the roughness element and the roughness element Reynolds number, Re^, where Re^=-^ (8) V u^ is the velocity at the height k in an undisturbed boundary layer. This parameter is proportional to the difference in velocity between the top and bottom of the roughness element. Re^ represents the strength of the shear layer at the edge of the roughness element's wake. They found that when Re^ was below a critical value for roughness elements with a given shape, transition occurred far downstream of the roughness position. As Re^ approached this critical value, the transition location quickly approached the roughness position. For Re^ greater than the critical value, transition occurred at the roughness location. Klebanoff et al. (1992) found that this critical 1 8 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Re^ value was 325 for hemispheres and 450 for upright cylinders. Hall (1957) experimentally determined that the critical Re^ value was between 585 and 665 for a spherical roughness element. Klebanoff et al. (1992) wrote that the velocity profile in the vicinity of a roughness element was inflexional due to the combined effect of the hairpin vortices which lifted fluid up between their legs, and the horseshoe vortex which lowered fluid down between its leg. For subcritical values of Re^, the inflexional instability grew slowly and was eventually damped out. Hall (1967) obseirved that in this situation, the spherical roughness shed laminar vortices which dissipated downstream. In Saiki & Biringen's (1997) numerical simulation of a subcritical spherical roughness element suspended above the wall (Re^ = 322), the hairpin vortices dissipated 24 sphere diameters downstream of the sphere. For supercritical values of Re^, the inflexional instability grew nonlinearly and broke down to turbulence (Klebanoff et al. 1992) . The boundary layer flow prior to breakdown can be seen in Acarlar & Smith's (1987a) experiment. They observed secondary vortices which developed in the hemisphere's wake. One set of secondary hairpin vortices were caused by the ejection of low speed 1 8 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fluid outboard of the horseshoe vortex. The other set of hairpin vortices were formed along the centerline in tandem with the hairpin vortices shed by the hemisphere. Low speed fluid was lifted up between the legs of a hairpin vortex upstream of its head. This fluid rolled up into a new hairpin vortex. Acarlar & Smith (1987a) wrote that the interaction between the hairpin vortices shed by the hemisphere and the in-line secondary hairpin vortices caused an unsteady flow similar to a turbulent burst. In Saiki & Biringen's (1997) simulation of the Reic = 494 suspended flow, transition occurred without the presence of the horseshoe vortex. The trailing vortices and hairpin vortices which were observed for the Re^ = 322 simulation occurred in this case too. The flow breakdown originated with a three-dimensional disturbance which was shed from the sphere at the beginning of the simulation. As time progressed, the sphere wake developed into a turbulent wedge (see Figure 74) . The hairpin vortices shed from the top of the sphere propagated downstream and became incorporated into the turbulent wedge. Re^ is a valid parameter only when k/5' < 1. Smith and Clutter (1959) compiled the data from transition experiments with different types of roughnesses. They found that k/0 and Reg, where 0 is the momentum thickness at the roughness 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. location, determined the capability of a given type of roughness to cause transition. There were combinations of k/0 and Reg for which none of the roughness elements they examined caused transition. These values corresponded to a particular region in k/0 and I/^Rbq space called the 'safe region' (Figure 75). 1 8 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) " 2 it (b) " ( c ) N Sir (d) “ 2n 0 10 20 3 0 40 SO 60 70 80 9 0 100 110 X t sphere Figure 74 Contours of u velocity on the y = 1.3 plane, (a) t = 98.74, (b) t = 116.69, (c) t = 134.64, (d) t = 152.6 (from Saiki & Biringen 1997) 1 8 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T M r O » O lM C N 9 IO N A C o COWL CD •A Ü C W M C » a rACC m OOUCCAS m I N TANOCM'OCUCCAS " ‘ SIT* » C O w iX D 9 tw C M «M V C 9 * COWLCO m o e c i o V ^ IN w iOC-CDw l CD a TACC A r* c C o COWUCO CTUIMMlCAt, a CACCOIW ANO VMftCXCR D L o r r iN # OOOCCAS « SINCLC CTU OOUCCAS a»MCMC9 O « c C B A N o rr CT AL •O^CONCS A c n c c o m r ANO w a c k CA \C N OCCNMcrr it oouccA S R E G IO N 20 Figure 75 Transition plot for fixed roughness elements (from Smith & Clutter 1959) 1 8 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B VORTEX IDENTIFICATION The complex eigenvalue method examines the state of the local velocity gradients at a fixed instant in time. The relative motion of a fluid particle at a point P about point O (see Figure 76) is given by: Dx. A Taylor series approximation of the local fluid velocity is carried out assuming frozen flow. (Perry & Chong 1987) A is the fluid deformation tensor and is defined as Ay = w ,.y Neglecting higher order terms, the trajectory of the particle about point O is given by: x(f) = e^'x(0) Since A is diagonalizable, g ' * ' is : = e^‘' where Àj and u^^’ are the j— eigenvalue and eigenvector respectively of A. Thus, the trajectory about point O is dependent on the eigenvalues of A. In three-dimensional flow, the three eigenvalues are all real or, one eigenvalue is real and the other two are complex conjugates. When two of the eigenvalues are complex conjugates, an observer at 0 will see the fluid particle 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. spiral around the eigenvector of the real eigenvalue as shown in Figure 77(a). This qualitatively corresponds to the influence of a vortex. If the eigenvalues are all real, an observer at O will see streamline patterns corresponding to a node and two saddles as shown in Figure 77(b) (Perry & Chong 1987) . Chong et al. (1990) gave a physical interpretation of the complex eigenvalues definition. They argued that rotational motion is greater than straining motion in a vortex core. When two of the eigenvalues of A are complex conjugates, the discriminant. A, of the characteristic equation \a - i i \ = o is positive. In incompressible flows, the discriminant is defined as A = (9) where Q and R are the second and third invariants of A and are defined as / ? = |A| Q can be rewritten as (Jeong & Hussain 1995) 1 8 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e=v(W-lW) 2 where | | 0 | | and | | 5 | | are the Frobenius norms of the rotation and rate of strain tensors. When rotational motion is greater than straining motion, i.e. | | £ 2 | | > | | 5 | | , A is positive and A has complex eigenvalues at that location. The converse is not true as the next paragraph will show. This definition is independent of frame of reference and is only dependent on the local velocity gradients. This method is implemented by plotting the surfaces corresponding to a fixed positive level of A. (Jeong & Hussain 1995) This was carried out by Blackburn Mansour, & Cantwell (1996) for visualizing vortices in turbulent channel flow. A vortex definition based on the relative strengths of rotational and straining motion is not all-inclusive. As Figure 78 shows, positive A can occur when Q < 0. Under the complex eigenvalue definition, a patch of fluid can still be classified as a vortex even though the straining motion is greater than the rotational motion. Chong et al. (1990) gave the example of a hairpin vortex in a straining field. In this case, Q < 0, but the vortex still acts like a vortex ".. .a s far as the Biot-Savart law is concerned..." There are disadvantages to the complex eigenvalue definition also. Jeong & Hussain (1995) pointed out that 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. small scale regions with A « 1 can be noise and not vortices. An iso-surface plot for one value of A has the potential of excluding vortices with Q > 0 but with different values of A. (Figure 78) Blackburn et al. (1996) speculated that their isosurface plots might have excluded structures with different values of A. Chacin, Cantwell, & Kline (1996) have also found that the A has additional physical significance in a turbulent boundary layer: regions with A > 0 occur near regions with large instantaneous Reynolds stress. This would be expected since the vortices would be involved with the 'sweeps' and 'ejections' in the boundary layer. 1 8 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o Figure 76 Motion of a fluid particle P about O ( a ) (b) Figure 77 Streamline patterns for (a) Spiral (b) Node and two Saddles. (from Soria et al. 1994). 10. 1 5 . -5 -S. -10 -10 -s 0 5 10 Figure 78 Contour plot of A in Eqn.9 1 9 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES Acarlar, M.S. & Smith, C.R. 1987a Study of Hairpin Vortices in a Laminar Boundary Layer, Part 1: Hairpin Vortices Generated by a Hemisphere Protuberance. J. Fluid Mech. 175, 1-41. Acarlar, M.S. & . Smith, C.R. 1987b Study of Hairpin Vortices in a Laminar Boundary Layer, Part 2: Hairpin Vortices Generated by a Fluid Injection. J. Fluid Mech. 175, 43-83 . Amini, J. & Lespinard, G. 1982 Experimental Study of an 'Incipient Spot' in a Transitional Boundary Layer. Phys. Fluids 25, 1743-1750. Anderson, C.R., Greengard, C., Greengard, L., & Rokhlin, V. 1990 On the Accurate Calculation of Vortex Shedding. Phys. Fluids A 2, 883-885. Barker, S.J. & Gile, D. 1981 Experiments on Heat-Stabilized Laminar Boundary Layers in Water. J. Fluid Mech. 104, 139-158. Blackburn, H.M., Mansour, N.N., & Cantwell, B.J. 1995 Topology of Fine-Scale Motions in Turbulent Channel Flow. il. Fluid Mech. 310, 269-292 . Blackwelder, R.F. & Browand, F.K. 1991 Effects of Particulates on Boundary Layer Transition. Naval Underwater Systems Center Final Rep. NUSC N66604-91-C-5555. Breuer, K.S. & Haritonidis, J.H. 1990 Evolution of a Localized Disturbance in a Laminar Boundary Layer, Part 1: Weak Disturbances. J. Fluid Mech. 220, 569-594. 1 9 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Breuer, K.S. & Landahl, M.T. 1990 Evolution of a Localized Disturbance in a Laminar Boundary Layer, Part 2, Strong Disturbances. J. Fluid Mech. 220, 569-594. Chambers, F.W. & Thomas, A.S.W. 1983 Turbulent Spots, Wave Packets, and Growth. Phys. Fluids 26, 1160-1162. Chen, C.P., Goland. Y., & Reshotko, E. 1980 Generation of Turbulent Patches in the Laminar Boundary Layer of a Submersible. In Viscous Drag Flow Reduction (ed. G.R. Hough). Progress in Astronautics and Aeronautics, vol. 72, pp.73-89. AIAA. Chacin, J.M., Cantwell, B.J., & Kline S.J. 1996 Study of Turbulent Boundary Layer Structure Using the Invariants of the Velocity Gradient Tensor. Exp. Thermal & Fluid Sci. 13, 3 08-317. Chong, M.S., Perry, A.E., & Cantwell, B.J. 1990 General Classification of Three-dimensional Flow Fields. Phys. Fluids A 2, 765-777 . Chu, C.C. S c Falco, R.E. 1988 Vortex Ring/Viscous Wall Layer Interaction Model of the Turbulence Production Process Near Walls. Exp. Fluids 6, 3 05-315. Darmofal, D.L. & Haimes, R. 1995 Analysis of 3-D Particle Path Integration Algorithms. J. Comput. Phys. 123, 182-195. Drazin, P.G. & Reid, W.H. 1981 Hydrodynamic Stability. Cambridge University Press. Fletcher, C.A.J. 1991a Computational Techniques for Fluid Dynamics 1. Springer-Verlag. Fletcher, C.A.J. 1991b Computational Techniques for Fluid Dynamics 2. Springer-Verlag. 1 9 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Gallager, S.M. 1993 Hydrodynamic Disturbances Produced by Small Zooplankton: Case Study for the Veliger Larva of a Bivalve Mollusc J. Plankton Res. 15, 1277-1296. Gresho, P.M. 1991 Incompressible Fluid Dynamics: Some Fundamental Formulation Issues. Ann. Rev. Fluid Mech. 23, 413-453. Groen, P., Newcombe, C.L., & Mero, J.L. 1984 Oceans and Seas. In Encyclopœdia Britannica 15— Ed., vol. 13 (ed. W.E. Preece), pp.482-504, Encyclopœdia Britannica. Haidari, A.H. & Smith, C.R. 1994 Generation and Regeneration of Single Hairpin Vortices. J. Fluid Mech. 277, 135-162. Hall, G.R. 1967 Interaction of the Wake from Bluff Bodies with an Initially Laminar Boundary Layer. AIAA J. 5, 1386-1392. Halldall, P.H.H. 1984 Plankton. In Encyclopœdia Britannica 15^ Ed., vol. 14 (ed Encyclopœdia Britannica. 15— Ed., vol. 14 (ed. W.E. Preece), pp.494-496, Henningson, D.S., Lundbladh, A., & Johansson, A.V. 1993 Mechanism for Bypass Transition from Localized Disturbances in Wall-Bounded Shear Flows. J. Fluid Mech. 250, 169-207. Hon, T .L . S c Walker, J .D .A . 1991 Evolution of Hairpin Vortices in a Shear Flow. Computers Fluids 20, 343-358. Jeong, J. S c Hussain, F. 1995 On the Identification of a Vortex. J. Fluid Mech. 285, 69-94. Jordinson, R. 1970 Flat Plate Boundary Layer, Part 1 : Numerical Integration of the Orr-Sommerfeld Equation. J. Fluid Mech. 43, 801-811. 1 9 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Kim, J. & Moin, P. 1985 Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations. il. Comput. Phys. 59, 3 08-323. Kim, J., Moin, P. & Moser, R.D. 1987 Turbulent Statistics in Fully Developed Channel Flow at Low Reynolds Number. J. Fluid Mech. 177, 13 3-166. Klebanoff, P.S., Cleveland, W.G. & Tidstrom, K.D. 1992 On the Evolution of a Turbulent Boundary Layer Induced by a Three-Dimensional Roughness Element. J. Fluid Mech. 237, 101-187. Klebanoff, P.S., Tidstrom, K.D. & Sargent, L.M. 1962 Three- Dimensional Nature of Boundary Layer Instability. J. Fluid Mech. 12, 1-3 4. Ladd, D.M. & Hendricks, E.W. 1985 Effect of Background Particulates on the Delayed Transition of a Heated 9:1 Ellipsoid. Exp. Fluids 3, 113-119. Lamb, H. 1945 Hydrodynamics 6^ Ed. Dover. Landahl, M.T. 1980 Note on an Algebraic Instability of Inviscid Parallel Shear Flows. J. Fluid Mech. 98, 243-251. Lauchle, G.C. & Gurney, G.B. 1984 Laminar Boundary Layer Transition on a Heated Underwater Body. J. Fluid Mech. 144, 79-101. Lauchle, G.C., Petrie, H.L., & Stinebring, D.R. 1995 Laminar Flow Performance of a Heated Body in Particle-Laden Water. Exp. Fluids 19, 305-312. Le, H. & Moin, P. 1991 Improvement of Fractional Step Methods for the Incompressible Navier-Stokes Equations. J. Comput. Phys. 92, 3 69-379. 1 9 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Magarvey, R.H. & Bishop, R.L. 1961 Transition Ranges for Three-Dimensional Wakes. Can. J". Phys. 39, 1418-1422. Mochizuki, M. 1961 Smoke Observation on Boundary Layer Transition Caused by a Spherical Roughness Element. J. Phys. Soc. Japan 16, 995-1008. Orlandi, P. 1990 Vortex Dipole Rebound From a Wall. Phys. Fluids A 2, 1429-1436. Orlanski, I. 1976 Simple Boundary Condition for Unbounded Hyperbolic Flows. J. Comput. Phys 21, 251-269. Perry, A.E. & Chong, M.S. 1987 A Description of Eddying Motion and Flow Patterns Using Critical-Point Concepts. Ann. Rev. Fluid Mech. 19, 125-155. Perry, A.E., Lim, T.T. & Teh, E.W. 1981 Visual Study of Turbulent Spots. J. Fluid Mech. 104, 387-405. Pretty, R.T. ed. 1980 Torpedo Mk 48. In Jane's Weapon Systems II— ed. pp.133-134, Jane's Publishing Co. Robinson, S.K. 1991 Coherent Motions in the Turbulent Boundary Layer. Ann. Rev. Fluid Mech. 23, 601-639. Saiki, E.M. & Biringen, S. 1997, Spatial Numerical Simulation of Boundary Layer Transition: Effects of a Spherical Particle. J. Fluid Mech. 345, 133-164. Sandham, N.D. & Kleiser, L. 1992 Late Stages of Transition to Turbulence in Channel Flow. J. Fluid Mech. 245, 319-348. Schlichting, H. 1987 Boundary Layer Theory 7— ed. McGraw- Hill Book Company. 1 9 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Schubuaer, G.B. & Klebanoff, P.S. 1955 Contributions on the Mechanics of Boundary Layer Transition. Na.tiona.1 Bureau of Standards Rep. 3954. Schubauer, G.B. & Skramstad, H.K. 1947 Laminar Boundary Layer Oscillations and Stability of Laminar Flow. J. Aero. Sci. 14, 69-78. Shirayama, S. 1992 Flow Past a Sphere: Topological Transitions of the Vorticity Field. AIAA J. 30, 349-358. Singer, B.A. 1995 Characteristics of a Young Turbulent Spot. Phys. Fluids 8, 509-521. Singer, B.A. & Joslin, R.D. 1994 Metamorphosis of a Hairpin Vortex into a Young Turbulent Spot. Phys. Fluids 6, 3724-3736. Smith, A.M.O. & Clutter, D.W. 1959 Smallest Height of Roughness Capable of Affecting Boundary-Layer Transition. J. Aero. Sci. 26, 229-245. Smith, C.R. & Metzler, P.J. 1983 Characteristics of Low- Speed Streaks in the Near-Wall Region of a Turbulent Boundary Layer. J. Fluid Mech. 129, 27-54. Soria, J., Sondergaard, R., Cantwell, B.J., Chong, M.S., & Perry, A.E. 1994 Study of the Fine-Scale Motions of Incompressible Time-Developing Mixing Layers. Phys. Fluids 6, 871-884. Strickler, J.R. 1975 Swimming of Planktonic Cyclops Species (Copepoda Crustacea): Pattern, Movements and Their Control. In Swimming and Flying in Nature, vol. 2 (ed. T.Y.-T Wu, C.J. Brokaw, & C. Brennan), pp.599-613. Plenum. 1 9 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. strickler, J.R. 1982 Calanoid Copepods, Feeding Currents, and the Role of Gravity. Science 218, 158-160. Taneda, S. 1955 Experimental Investigation of the Wake behind a Sphere at Low Reynolds Numbers. J. Phys. Soc. Jpn. 11, 1104-1108- Van Dommelen, L.L. & Crowley, S.J. 1990 On the Lagrangian Description of Unsteady Boundary Layer Separation. J. Fluid Mech. 210, 593-62 6. Wallace, J.M. , Eckelmann, H., & Brodkey, R.S. 1972 The Wall Region in Turbulent Shear Flow. J. Fluid Mech. 54, 39-48. Wang, M. , Lele, S.K. & Moin, P. 1996 Sound Radiation during Local Breakdown in a Low-Mach-Number Boundary Layer. J. Fluid Mech. 319, 197-218. Wu, J.S. S c . Faeth, G.M. 1993 Sphere Wakes in Still Surroundings at Intermediate Reynolds Numbers. AIAA J. 31, 1448-1455. Zhou, J. , Adrian, R.J. & Balachandar, S. 1996 Autoregeneration of Near-Wall Vortical Structures in Channel Flow. Phys. Fluids 8, 288-290. 1 9 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Asset Metadata

Creator Taniguchi, Paul Hiroshi (author)

Core Title Boundary layer transition due to the entry of a small particle.

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, aerospace,OAI-PMH Harvest,Physics, Fluid and Plasma

Language English

Advisor Blackwelder, Ron (committee chair), Browand, Fred (committee chair), Domaradzki, J.A. (committee member), Meiburg, E.H. (committee member), Ronney, Paul (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-437662

Unique identifier UC11350886

Identifier 9933730.pdf (filename),usctheses-c17-437662 (legacy record id)

Legacy Identifier 9933730.pdf

Dmrecord 437662

Document Type Dissertation

Rights Taniguchi, Paul Hiroshi

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 au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA