Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 479
Vortex Ring Interaction with a Free Surface M. Song, N. Kachman, I. Kwon, L. Bernal, G. Tryggvason (The University of Michigan, USA) ABSTRACT The results of numerical and experimental studies on the interaction of vortex rings with a free surface are presented. New results are reported on the interaction of a large vortex ring with a clean surface at normal incidence. The early stages of the interaction are well described by a simple axisymmetric vortex filament model. Transition to a fully three-dimensional state is observed at later stages of the interaction. Surface waves are generated at high Froude numbers by these three-dimensional motions. Results are also presented on the interaction of vortex rings with clean and contaminated free surfaces at inclined incidence. The phenomenon of vortex lines breaking and attachment to the free surface is documented. It is shown that small amounts of surface active agents greatly alter the interaction at inclined incidence. The effect differ depending on the local topology of the vortex lines. A Reynolds ridge and secondary vorticity generation are observed during the interaction with a contaminated surface. INTRODUCTION The disturbance on a free surface created by a moving ship is composed of several superimposed and sometimes interacting phenomena. The most dramatic and best understood is the surface wave pattern generated, generally referred to as the Kelvin wake. Not only is the Kelvin wake the more visible mark left by the ship, it contributes also significantly to the drag of the ship, and is therefore of direct economic significance. Although it is, of course, well known that the ship also has a large viscous wake that is turbulent and can last for a long time, traditionally the turbulent wake has only been of interest as it directly relates to the drag of the ship, and in most such considerations the effect of the free surface can be neglected. Furthermore, for the purpose of drag estimation the focus is on the turbulent wake near the ship. It is only recently that the far wake of the ship has generated interest. The motivation is remote sensing technology. In order to process the signal and to determine what is being detected, as well as to be able to reduce the detectability of ships, it is necessary to understand the detailed mechanisms of generation of surface signatures of ship wakes. The surface signature of unsteady vertical motions below a free surface has recently been the subject of several investigations. Sarpkaya & Henderson ~ experimented with a delta wing moved below a free surface. The wing was set at a negative angle of attack, so that the trailing vortices moved upward to the free surface. As the vortices approached the surface a pair of long and narrow marks were observed on the free surface that appear to be directly related to the trailing vortices. These marks, called scars by Sarpkaya, are parallel to the direction of motion and moved outward with the vortices. The scars were accompanied by other features called "striations", perpendicular to the line of motion. A somewhat different setup, two-dimensional vortex pair was investigated by Willmarth et al.2 and by Sarpkaya et al.3 The surface signature of the pairs is similar to the trailing vortices, but the mean motion is now strictly two- dimensional. These experiments were motivated primarily by a desire to understand the surface signature of ship wakes, and the focus was mainly on the large scale motion. Several numerical studies of this problem have followed the experimental work. These have mostly assumed an inviscid, two-dimensional motion. Tryggvason4 presents a brief numerical study of surface deformation due to the roll-up of a submerged vortex sheet using a boundary integral/vortex method. Sarpkaya et ala and TelsteS use a similar technique to follow the motion of a vortex pair toward a free surface. A finite difference simulation of the point vortex problem have been reported by Marcus6 who also discusses linearized aspects of the problem. More realistic vortex structure are used by Willmarth et a[2 who simulate the formation of a vortex pair from an initially flat vortex sheet and the subsequent vortex motion and free surface deformation. A brief 479
OCR for page 480
comparison of experimental and computational results is contained in Reference 2. A thorough discussion of both vortex collision as well as the formation of vortices from a shear layer, and the resulting surface signature, is given by Yu and Tryggvason7, who simulated a large number of cases, and, in particular, explored the limits of high and low Froude numbers. It should be noted that the signal from the ocean surface that is received by remote radar sensors is directly related to the presence of relatively short waves. It is therefore the surface roughness that is observed directly. Generally the large features of more interest such as changes in the ocean depth or currents and the wakes of ships can only be inferred through their modulation of the free surface roughness. For interpretation of these data it is therefore necessary to understand the interplay between the large and small scale features of the flow and their role in the generation of short surface waves. An example of such interplay is the relation between the scars and striations in the vortex wake problem studied by Sarpkaya and coworkersl'3 and more recently by Hirsa.8 Also of primary interest in this regard is the role of smaller scale motions in the turbulent wake in the generation of short waves. To examine the surface response to a turbulent subsurface flow Bernal and Madnia9 studied a turbulent jet parallel to the surface located a few diameters from the surface. Near the jet exit, vortex rings are generated that produce some surface deformations and waves, but the most dramatic signature is produced when these vortex rings open and reconnect with the free surface. To explore this mechanism in more detail, Bernal and Kwon1O and Kwonll experimented with a single ring moving parallel and at inclined incidence relative to the surface, respectively. The above discussed investigations were not concerned with the effects of surface contaminants as such, but it appears that some of those results were influenced by the fact that a free surface is hardly ever hydrodynamically clean. Earlier, Daviesl2 discussed the damping of turbulent eddies at a free surface, and Davies and Driscolll3 experimented with ejecting pulses of colored water to a free surface, specifically addressing the rate of surface renewal and the effect of surface contamination. They found that the spreading of the colored water at the surface is reduced considerably for contaminated surfaces. However, their visualization technique did not allow for a clear explanation of the mechanism responsible for this behavior. Experiments on the collision of two-dimensional vortex pairs with a free surface were reported by Barker and Crowl4 whose main interest was in vortex collision with a rigid surface. The motivation for their experiments was the observed rebounding of aircraft trailing vortices from rigid surfaces. This rebounding of a vortex pair from a solid surface is due to the separation of the ground boundary layer and subsequent formation of secondary vortices. Therefore it is not expected that rebounding will occur if the rigid surface is replaced by a stress free surface. However, Barker and Crowl4 observed rebounding in their free surface experiments, just as the rigid surface case, and suggested that this rebounding might be due to inviscid effects effects such as the deformation of the vortex cores. Saffmanis refuted this suggestion, and showed that for inviscid flow and a flat boundary rebounding can not occur. He suggested that the behavior might be due to surface tension effects. Peace and Rileyl6 performed numerical simulations of the Navier-Stokes equations for a two-dimensional vortex pair colliding with a flat no-slip and stress-free surface, and concluded that even for a stress-free boundary viscous effects could cause rebounding. However, even though their calculations clearly show rebounding, those are for rather low Reynolds numbers, and with increasing Reynolds number, the rebounding decreased significantly. Their results can therefore not explain the behavior in the Baker and Chow experiments, which were conducted at a much higher Reynolds number. The explanation for rebounding from a free surface is clear from recent experiments by Bernal, Hirsa, Kwon and Willmarthl7 who investigated the collision of both vortex rings and pairs with a free surface. They observed that the cleanness of the surface lead to considerable differences in the vortex motion itself. For very clean surfaces sufficiently weak vortices were deflected outward in a manner similar to what inviscid theory predicts (if the surface deforms some rebounding is predicted but most experiments have been conducted under conditions where surface deformation is minimal), but for contaminated surfaces the behavior was more like vortices encountering a rigid wall where secondary vorticity from the wall boundary layer is pulled away by the primary vortex that then rebounds as a result of its interaction with the wall vorticity. Detailed observations using Laser Induced Fluorescence (LIF) flow visualization lead Bernal et all7 to conclude that the surface motion induced by the vortex generated an uneven distribution of contaminant that in turn caused shear stress at the surface, generating secondary vorticity. This vorticity rolls-up into a secondary vortex which results in the rebounding of the primary vortex. This generation and roll-up of secondary vorticity and its subsequent interaction with the primary vortex appears to be the leading effect of the surface contaminants. Although rebound can usually be associated with viscous effects Dahm, Scheil and Tryggvasonl8 have shown that a weak vortex colliding with a weak density interface can engulf a portion of the interface containing baroclinically generated vorticity which then causes the primary vortex to rebound in completely inviscid simulations. Yu and Tryggvason7 also show that a deformable surface can lead to rebounding. However this occurs at much higher Froude number than in the experiments. 480
OCR for page 481
Observations of contaminated free surfaces have been reported on numerous occasions for over a century. One important phenomenon is the Reynolds ridge which appears on the boundary between the contaminated and clean surface regions when the surface flow is stopped by a barrier. This flow configuration develops when vortices collide with a free surface as shown by Hirsa8. The upwelling generated by the vortices pushes the contaminated surface water to the side, thereby compressing the contaminated layer. The surface above the vortices is cleaner and is separated from the contaminated surface by a Reynolds ridge. For a Thorough discussion of the Reynolds ridge with historical perspective see Scottl9. We should note that the occurrence of a Reynolds ridge, although often observed when separation takes place, is not directly related to the generation of secondary vortices. Indeed, a Reynolds ridge is easily generated in the absence of separation (see Scottl9) and separation can take place without the formation of a Reynolds ridge. In what follows an overview of recent results on the interaction of vortex rings with the free surface is presented. The flow geometry is shown schematically in Figure 1. Results on the interaction at normal incidence, Figure la, are presented first. Next the results of Kwon1 1 for inclined incidence are briefly reviewed. Finally the results of recent experiments on the effect of surface contamination on the interaction at inclined incidence are presented and discussed. INTERACTION AT NORMAL INCIDENCE The interaction of a vortex ring with the free surface at normal incidence is perhaps the simplest flow geometry involving the interaction of a vertical flow with the free surface. This type of interaction has been studied in detail in a recent experimental and numerical investigation by Song, Bernal and Tryggvason.20 One objective of the investigation was to study the interaction at a scale substantially larger than previous experiments which were conducted in small water tank facilities.l°~ll A large vortex ring generator with a nozzle exit diameter of 10 cm was used in this study. The general design and operating characteristics are similar to the smaller scale vortex ring generators. The experiments were conducted in Tow Tank facility at the Hydrodynamics Laboratories of the University of Michigan. The water surface was cleaned by a continuous surface current. The current was interrupted and the flow motion allowed to dissipate before each vortex ring test. Because of these precautions the water surface is believed to have been free from contaminants on these tests. Hot film velocity measurements along the axis of the flow were used to determine the vortex ring formation characteristics. The flow field during the interaction was LLl (a) ma_ Figure 1. Schematic diagram of flow geometry. a) Normal incidence. b) Inclined incidence. characterized by flow visualization of the underwater flow using the Hydrogen bubble technique and of the free surface using the shadowgraph technique. The surface signature during the interaction was also characterized by measurements of the free surface elevation using a capacitance probe. The numerical simulations were conducted using a vortex/bounda~y-integral method. A single vortex filament with finite core size was found adequate for the objectives of the study. Comparison with experiments was made by adjusting the vortex ring circulation and core parameter to obtain the same circulation and initial propagation speed as in the tests. For additional details on the experimental setup and numerical simulations the reader is referred to Song et al.20 Flow visualization Typical flow visualization results of the interaction at normal incidence are shown in Figure 2. Figure 2a are visualizations of the surface deformation and underwater flow at a Froude number F/(g R3)1l2 = 0.252 where His the circulation of the vortex ring, g is the gravitational acceleration and R is the radius of the ring before interaction. The corresponding Reynolds number was F/v=15,000. Figure 2b are similar visualizations at a 481
OCR for page 482
(a) (b) Figure 2. Flow visualization of the interaction of a vortex ring with a free surface at normal Incidence. Top is a shadowgraph image of the surface, Bottom, underwater flow visualized by hydrogen bubbles. (a) Froude number 0.252. (b) Froude number Froude number of 0.988. The Reynolds number was 64,700. In both cases it is shown the shadowgraph image of the surface on top and the side view image obtained using the hydrogen bubble technique at the bottom. Both images were obtained on the same realization of the flow. The bubbles in the side view pictures tend to migrate towards the core of the vortices due to their low density. Note that on the side view the mirror image of the underwater flow is observed caused by total reflection of scattered light on the water surface. The flow visualization results at low Froude number, Figure 2a, show a highly coherent axisymmetric surface pattern. The side view picture also shows a coherent axisymmetric core. The surface signature consists of a dark band bounded by two bright regions these features indicate a local depression of the surface. This surface depression is located above the vortex core and moves with it as it stretches outward due to the velocity field induced by the image vorticity above the surface. There were no surface waves generated during this process. At very low Froude number these features were observed during the entire interaction process. 482 Figure 2b shows flow visualization pictures at a higher Froude number as indicated. The shadowgraph visualization of the surface shows small scale three- dimensional structures superposed on a axisymmetric dark band associated with the core. Surface waves are generated by these small scale three-dimensional motions. These waves coalesce to form an axisymmetric wave front propagating away from the interaction region. The visualization of the core in the side view picture show small scale three-dimensional distortions of the core associated with the surface features. The development of the small scale three-dimensional features was found to occur rapidly, through an instability process. Before the instability the surface signature and general flow appearance was similar to the case shown in Figure 2a. After transition the flow features discussed in relation to Figure 2b appeared.
OCR for page 483
Surface signature and subsurface flow The measured and calculated trajectories of the vortex cores at a Froude number of 0.25S and 0.988 are shown in Figure 3. The measured and calculated results are in good agreement. There is no rebounding of the vortex core in the low Froude number case. This is expected since contaminants were not allowed to accumulate on the surface. At the larger Froude number the calculations show a small rebounding of the core. The measurements on the other hand are not accurate enough to confirm this result. An interesting observation shown by the data in Figure 3 is that the vortices attain a different final depth after the interaction for the two cases presented. Studies of many numerical simulations of the flow revealed that the final depth of the vortex core is controlled primarily by the core size parameter. For the cases shown in Figure 3 the core size parameter Rle, where e is the core radius, was 4.9 for the low Froude number case and 2.7 for the high Froude number case. ~ .o o.o -1 .0 z/R -2 .0 -3.0 -4.0 -~.0 ~ o Ids- _ pa d. l . 1 , 1 1 1 , 1 0.0 1.0 2.0 3.0 r/R 4.0 5.0 6.0 Figure 3. Vortex core trajectories for normal incidence. Froude number 0.252: Solid line calculated results, circles measurements. Froude number 0.988: Dashed line calculated results, squares measurements. Several parameters can be used to characterize the surface signature during the interaction of a vortex ring with the free surface. Perhaps the simplest measure of the strength of the interaction is the surface elevation at the center. Figure 4 is a plot of the normalized maximum elevation at the center, hlR, as a function of Froude number, F/(g R3J]/, for all the cases tested in the h/R .1 1 r/(g R3 )1/2 .01 / laboratory. The straight line in this plot has slope 2, the expected scaling behavior derived from considerations of momentum balance of the vertical flow and the surface deformation. .001 ~ 1._ ~ _d A ~ _ 10 Figure 4. Normalized maximum surface elevation at the centerline as a function of Froude number. Only experimental results are presented. The surface signature during the interaction was obtained from the numerical simulations. Figure 5 is a plot of the instantaneous surface shape for the case Fr= 0.252 at nondimensional time tF/R2 = 14.16. Note that the vertical axis is stretched by a factor of ten compared to the horizontal axis. The surface features in this plot are in good qualitative agreement with the shadowgraph flow visualization picture of the surface shown in Figure 2a. The dark band in the photograph corresponds to the surface depression at r/R a 3.0. The bright regions next to the dark band in the photograph correspond to the surface rise at either side of the depression. More detailed comparisons between the measured and calculated results is presented in Figures 6 and 7 for Froude numbers of 0.252 and 0.988 respectively. In both Figures we plot various measures of the position of the vortices and associated surface features as a function of time. The solid lines are the calculated location of the vortex core. The open circles are the measured location of the cores determined from the hydrogen bubbles flow visualization. The broken line is the radial location of the maximum surface depression from the numerical simulations. The cross symbols are the measured location of the dark band in the shadowgraph images. 483
OCR for page 484
0.3 0.2 _ 0.1 _ z/R o.o -o. 1 -0.2 -0 . 3 -0.4 _ -0.5 , I . 1 , I , I , I 0.0 1.0 2.0 3.0 4.0 5.0 6.0 r/R Figure 5. Calculated free surface shape for Froude number 0.252. The results at low Froude number, Figure 6, show good agreement between the numerical calculation and the measurements. The agreement is excellent for the distance 6.0 4.0 r/R to the free surface. The calculation show that the location of the maximum depression moves outwards ahead of the core. The calculated speed of this outward motion is in good agreement with the measured value. Although there are discrepancies in the absolute location of the core. At Later times, however, (~/R2 ~ 15) there appears to be a change in the measured core trajectory. The results at high Froude number, Figure 7, show significant discrepancies between the calculated and measured results. Again there is excellent agreement between the calculations and the measurements for the distance to the free surface. The evolution of the radial location of the core is in good agreement up to tFIR2~ 12. After that time the measurements show a slower outward motion of the vortex core compared to the calculations. The reason for this discrepancy is the three-dimensional instability observed in the flow visualization study. Before the instability the outward motion of the core results from the induced velocity field caused by the image vorticity above the free surface. This induced motion is well characterized by the axisymmetric vortex filament model used in the numerical calculations. As three-dimensional motions develop after the instability, vortex lines break and connect to the free surface, a phenomenon discussed in more detail later, thus destroying the axial coherence of the vorticity distribution. These vortex patches attached to the free surface determine the speed of the outward motion of the core after the three-dimensional instability. A/ i+++ /+~+ + r/R 6.0 4.0 _ 2.0 r ~1 2.0 _ ~o~ 0.0 -2 .0 -4.0 ,/ To l I · I , 1 0.0 10.0 20.0 30.0 40.0 rUR2 Figure 6. Temporal evolution of various flow parameters at a Froude number of 0.252. Solid line calculated core location. Dashed line radial location of minimum surface depression. Circles measured core location. Crosses radial location of dark band. 0.0 -2 .0 z/R + ~ _--P a/ ,~ ~+++ lo:- 40! + + ~ O ~ ~ 00 it -4.0 . 0.0 10.0 I . I . 1 . 1 20.0 30.0 40.0 r dR2 Figure 7. Temporal evolution of various flow parameters at a Froude number of 0.988. For key to the symbols see caption of Figure 6. 484
OCR for page 485
INTERACTION AT INCLINED INC1:DENCE Investigations on the interaction of a turbulent jet with the free surfaced revealed the development of characteristics features on the free surface consisting of dark dimples associated with vortex lines terminating at the free surface. It was also observed that vertical structures in the near field of the jet were directly responsible for the formation of the dimples. Bernal and Kwon10 experimented with vortex rings moving parallel 1~ to the surface. These experiments demonstrated that surface dimples result from vortex lines breaking and reconnecting with the free surface. To further examine this phenomenon a systematic investigation on the interaction of a vortex ring, at an inclined incidence to the free surface, was undertaken (see Figure lb). _ _ (a) (b) (c) Figure 8. Flow visualization pictures of the interaction of a vortex ring with a clean free surface at a Reynolds number of S,OOO and incidence angle of 20°. Shadowgraph visualizations of the surface are shown on top and Cross-section views at the bottom. In each photograph the vortex motion is from right to left. (a) rtlR2 = 18. (b) rtlR2 = 26. (c) rtlR2 = 43. 485
OCR for page 486
In considering these flow processes at a free surface the role of surface contamination is an important parameter. The flow visualization results by Bernal, Hirsa, Kwon and Willmarthl7 showed that vorticity generation at the free surface can alter the evolution of the subsurface flow. The study of this interaction in the more complicated case of inclined incidence was another motivation for the present study. The experiments were conducted in the free surface water tank described by Bernal and Madnia9. Several modifications were added to help control contamination of the free surface. A stand-up drain pipe was installed, and a continuous current of water was allowed to flow into the facility to remove surface contaminants before they could have accumulated on the free surface. During the actual tests the water current was temporarily halted and the remaining turbulence in the tank was allowed to dissipate before generating the vortex rings. This procedure resulted in highly reproducible results for a clean free surface. A vortex ring generator having a 2.5 cm nozzle exit diameter was used in all the experiments at inclined incidence. A short duration water pulse is allowed to flow out of the generator to produce the vortex rings. A pressurized tank and a solenoid valve are used to control the speed and duration of the water pulse. A detailed study was conducted to determine the operating characteristics of the vortex ring generator. These as well other information on the operation of the vortex ring generator can be found in Kwonll. The flow field and surface signature was documented by Laser Induced Fluorescence (LIF) flow visualization of the symmetry plane and by shadowgraph visualization of the free surface. Interaction with a clean free surface Shown in Figure 8 are flow visualization pictures of the interaction at an incidence angle of 20°. On top are the shadowgraph images of the free surfaces, at the bottom are LIP cross-sections through the symmetry plane. The vortex ring motion is from left to right. The flow conditions were Reynolds number Fly = 5,000 and the Froude number Fl~gR3~1/2 =0.81. These photographs show the three stages of the interaction process. The first photographs at l~tlR2 = 18, Figure 8a, show the upper vortex core interacting with the free surface which causes a surface depression. The second set of photographs obtained at l~tlR2= 26, Figure 8b, show a single vortex core in the cross-section while the surface signature shows two dimples on the surface at either side of the symmetry plane. These features indicate that the vortex lines in the ring have opened-up and are now attached to the surface at the dimples. The shadowgraph image also shows that surface waves were generated during the vortex line breaking and reconnection process. At a later time (Figure 8c, FtlR2= 43) the flow visualization of the cross-section shows little evidence of the vortex ring core while the surface shadowgraph shows four dimples on the surface. The top pair propagates away from the lower pair in time. Each individual pair of dimples represents the surface signature of a half vortex ring with vortex lines beginning and terminating at the free surface. Flow visualization studies were conducted to determine the effect of Reynolds number and incidence angle on the interaction. The results are summarized in Figure 9. Vortex line breaking and reconnection of the top and bottom parts of the ring were found for a large range of Reynolds numbers (from 2,000 to 7,000) and incidence angles in the range between 10° and 30°. At a Reynolds number of 5,000 the vortex line breaking and reconnection of the lower part of the core was observed for angles up to 45°. Small variations were observed within this range of parameters in the sense that the half-vortex rings would propagate at different angles after the interaction, and in some cases they will move along converging paths, interact with each other again and form a single open ring. ~ Lower Core Vortex Breakdown em Lower Core Vortex Reconnection 50 40 30 a 20 10 1 l 1] 1111] 1~ i I 2 4 6 8 10 F/v (x10-3) Figure 9. Observed interaction outcome as a function of vortex ring Reynolds number and incidence angle. For vortex rings formed at conditions outside this region, the vortex lines of the upper core break and reconnect to the surface as in the other cases. But when the bottom core reached the surface the vertical region broke down into smaller scale vortical structures. 486
OCR for page 487
The vortex line breaking and reconnection process can be quantified be a characteristic time, tr. This reconnection time was defined in the experiments as the elapsed time between the time when the vortex outline in the LIF image first reaches the surface and the time when the dimples on the surface are first observed. Measurement were conducted of the reconnection time of the upper vortex core at several conditions. The results are shown in Figure 10 where the reconnection time nondimensionalized by the circulation and the core diameter, 8, is plotted as a function of the Reynolds number for different incidence angles. The results show that the normalized reconnection time is independent of the Reynolds number, suggesting that the breaking of vortex lines is by and large an inviscid process. There is a systematic reduction of the normalized reconnection time as the incidence angle is increased suggesting a strong dependence on the details of the vortex line topology as they approach the surface. 20 s . . . . r tr/ I; 2 ~ ~ a - 30° 10~ ~~c`=45'~ u 1000 r/v Figure 10. Effect of vortex ring Reynolds number and incidence angle on vortex line reconnection ~ ~i] ' ~ .~: ~ ~ ~ ~ ~ ~ ~] _,: (a) EM . ~,., ~, ~, ,.,.~.,.,., ,. ~_ _ ~ ~ ~'1~ ~ ~""~ ~ ~ ~ ~ ~ E'_ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ ~ 11~ . ~ ~ ~ · __ '.. ~ ~ ~ _ <."""-' ~ ~ ~ ~ ~ ~'~ '' ~ ~ ~_ (C) Figure 11. Flow visualization photographs of the interaction of a vortex ring with a contaminated surface at a Reynolds number of 5,000, incidence angle of 20° and surface pressure of 12.7 dynes/cm. Shadowgraph visualizations of the surface are shown on top and Cross-section views at the bottom. In each photograph the vortex motion is from left to right. (a) I~tIR2 = 42. (b) 1~tIR2 = 53. (c) Ft/R2 = 65. 487
OCR for page 488
The flow visualization at a nondimensional time Ft/R= 53 shown in Figure 1 lb reveals the formation of a pair of counter rotating vortices trailing behind the lower core. This small pair of vortices is the remanence of the upper vortex core and the opposite sign vorticity generated by surface tension effects at the contaminated surface. The shadowgraph image of the surface shows the dimples due to the reconnection of the top vortex core with the surface and the waves generated in the process of reconnection. The shadowgraph image also shows a distinct Reynolds ridge moving ahead of the reconnected vortex cores. At a later time, l~t/R2 = 65 Figure lie, the cross section view still shows evidence of the counter rotating vortex pair at the surface which has now been entrained by the lower vortex core. Also a large secondary vortex has formed ahead of the lower vortex core. The induced velocity field of the secondary vortex tends to stop the motion of the lower vortex core towards the surface. The surface signature in this case show well defined circular dimples as well as a highly distorted Reynolds ridge. These flow visualization results illustrate the effect that surface active agents have on the dynamics of vorticity near a free surface. The effect vary depending on the local details of the flow. For the upper core the surfactant does not suppress the vortex line breaking and reconnection process. This is consistent with the observed Reynolds number independence of the reconnection time. In contrast the lower core dynamics is strongly influenced by the presence of surface active agents. As the lower core region approaches the free surface secondary vorticity generated at the surface by surfactant action accumulates in a secondary vortex. The velocity field induced by the secondary vortex causes rebounding of the core and prevents vortex line breaking at the surface. ACKNOWLEDGEMENTS This research was sponsored by the Office of Naval Research, Contract no. N000184-86-K-0684 under the U.R.I. Program for Ship Hydrodynamics. REFERENCES 1. Sarpkaya, T. & Henderson, Jr., D.O. "Free surface scars and striations due to trailing vortices generated by a submerged lifting surface," AIAA paper 85-0445, 1986. 2. Willmarth, W.W., Tryggvason, G., Hirsa, A. & Yu, D. "Vortex pair generation and interaction with a free surface," Physics of Fluids A, vol 1, no 2, Feb. 1989, pp 170-172. 3. Sarpkaya, T., Elnitsky II, J., & Leeker Jr., R.E., "Wake of a vortex pair on the free surface." Proc. 17th Symposium on Naval Hydrodynamics, The Hague, The Netherlands, 1989. 4. Tryggvason, G., "Deformation of a free surface as a result of vertical flows," Physics of Fluids, Vol. 31, No. 5, May 1988, pp. 955-957. 5. Telste, J.H., "Potential flow about two counter- rotating vortices approaching a free surface," J Fluid Mech. vol 201, pp 259-278, 1989. 6. Marcus, D.L.,"The interaction between a pair of counter-rotating vortices and a free boundary," PhD Thesis, The University of California at Berkeley, 1988. 7. Yu, D. & Tryggvason, G., "The free surface signature of unsteady two-dimensional flows," Submitted to J Fluid Mech., 1989 8. Hirsa, A., "An experimental investigation of vortex pair interaction with a clean or contaminated free surface," PhD Thesis, The University of Michigan, 1990. 9. Bernal, L.P. & Madnia, K., "Interaction of a turbulent round jet with the free surface," Proc. 17th Symposium on Naval Hydrodynamics, The Hague, The Netherlands, 1989. 10. Bernal, L.P. & Kwon, J.T., "Vortex ring dynamics at a free surface," Physics of Fluids A, vol 1, no 3, March 1989, pp 449-451. 11. Kwon, J.T., "Experimental study of vortex ring interaction with a free surface," PhD Thesis, The University of Michigan, 1989. 12. Davies, J.T., "The effect of surface films in damping eddies at a free surface of a turbulent liquid," Proc. Royal Soc. London A. vol 290, pp 515-526, 1966. 13. Davies, J.T. & Driscoll, J.P. Eddies at free surfaces simulated by pulses of water," Industrial and Eng. Chemistry Fundamentals, vol 13, pp 105-109, 1974. 14. Barker, S.J. & Crow, S.C., "The motion of two- dimensional vortex pair in a ground effect," J Fluid Mech. vol 82, pp 659-671, 1977. 15. Saffman, P.G., "The approach of a vortex pair to a plane surface in inviscid fluid," .J Fluid Mech. vol 82, pp 659-671, 1977. 16. Peace, A.J. & Riley, N., "A viscous vortex pair in ground effect," J Fluid Mech. vol 125, pp 187-202, 1983. 17. Bernal, L.P., Hirsa, A., Kwon, J.T. & Willmarth, W.W., "On the interaction of vortex ring and pairs with a free surface for varying amounts of surface active agent," Physics of Fluids A, vol 1, no 12, Dec 1989, pp 2001- 2004. 18. Dahm, W.J.A., Scheil, C.M. and Tryggvason, G., "Dynamics of vortex interaction with a density interface," J Fluid Mech. vol 205, pp 1-43, 1989. 19. Scott, J.C., "Flow beneath a stagnant film on water: the Reynolds ridge," J Fluid Mech. vol 69, pp 339-351, 1975. 20. Song, M., Bernal, L.P., and Tryggvason, G., to be submitted to Physics of Fluids A, 1990. 488
OCR for page 489
DISCUSSION Daniel Marcus Lawrence Livermore National Laboratory, USA What is the effect of changing the ratio of core radius to ring radius on the reconnection event? AUTHORS' REPLY We have only limited amount of data on the effect of core radius or ring radius on the reconnection event. The data presented in Figure 10 shows that the vortex reconnection time when nondimensionalized by core parameters is approximately independent of the Reynolds number. For these data, the change in Reynolds number is accompanied by a change of the ratio of core radius to ring radius. Kwon (1989) estimates the values of this parameter for the cases plotted in Figure 10. The values changed from a minimum value of 0.23 at the low Reynolds number case to a maximum value of 0.39 at the largest Reynolds number case. On the basis of this evidence, it appears that the nondimensional reconnection time is also independent of the ratio of core radius to ring radius. DISCUSSION Theodore Y. Wu California Institute of Technology, USA The authors deserve our warm thanks for the very enlightening report on their investigations of this fascinating subject. In view of Helmholz's theorem that no vortex filaments can terminate in the midst of a fluid (or fluids), I wonder if the authors have investigated the vertical flow induced in the air by the submerged vortices that have opened up at the interface. I further wonder if such more complete vortex systems may have a bearing on the secondary vortices sequentially generated in the course of the vortex bifurcation. AUTHORS' REPLY We would like to thank Professor Wu for his kind remarks. Yes, Helmholz's theorem requires that vortex lines with a component normal to the free surface continue in the air above the free surface. We have not attempted to measure or in any way characterize the air motion above the free surface during the interaction. Elucidation of the topology of the vortex lines in air is a challenging problem. The Reynolds number for the air flow is one order of magnitude lower than for the water flow and consequently the air vortices are expected to dissapate more rapidly. In any case, these induced vortices in air or other weak air flow disturbances above the surface do not significantly influence the vortex reconnection process because of the large density ratio. Free surface dynamics, including contamination effects, dominate the interaction process. DISCUSSION Richard Yue Massachusetts Institute of Technology, USA In your work, the induced velocity on the surface is a known function (of space and time) since the influence of the free surface on the vortex pair is not considered. This is consistent with the assumption of deep submergence. In this context it seem that the present problem can be addressed by simply considering the dispersion relationship in terms of the known surface advection which can be applied directly to a wave spectrum. AUTHORS' REPLY Although the induced velocity on the surface is a known function of space and time, a solution method based solely on the spectral variation due to this known surface current requires an additional assumption. This assumption is that the variation in surface current be slowly varying in time and space with respect to the length and time scales of the ambient waves. The ambient wavelengths here is allowed to be of the same order of magnitude as the vortex pair separation and depth. The problem is therefore transient in nature rather than quasi-steady as would be assumed in the direct modification of the dispersion relationship. The transient problem is handled in a straightforward manner using the simulation technique. In the limit of small waves under the influence of large scale vortices, such as in the ship example at the end of the paper, the direct method suggested will likely give reasonable agreement. 489
OCR for page 490
Representative terms from entire chapter: