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Measurement and Computations of Vortex Pair Interaction with a Clean or Contaminated Free Surface A. Hirsa, G. Tryggvason, I. Abdollahi-Alibeik, W.W. Willmarth (The University of Michigan, USA) Observations of surface deformations produced by the interaction of trailing vortices with a free surface, behind a submerged delta wing at negative angle of attack, were described by Sarpkaya and Henderson (1985~. The flow field during the laminar interaction with the free surface is studied experimentally using a pair of vertically oriented, computer controlled, counter rotating flaps to generate laminar vortex pairs with the same Reynolds number and Froude number as the trailing vortices. Numerical computations in two-dimensions, flow visualization (laser induced fluorescence and shadowgraph) and particle image velocimetry (PIV) with known surfactants on the surface show that surface contamination has a significant influence on the flow field and surface motion during the interaction. New vorticity produced on either side of the vortex pair when the surface is contaminated initially forms a Reynolds ridge, (the surface signature at the leading edge of a subsurface boundary layer), and then the new vorticity beneath the contaminated surface rolls up to form secondary vortices outboard of the original vortices and with opposite sign. The secondary vortices cause the original vortex pair to rebound from the free surface. INTRODUCTION Synthetic aperture radar images made at high altitudes show that a ship produces a long, narrow signature on the surface in the wake that is detectable long after the passage of the ship (Munk et al. 1987~. Since the wake directly behind a ship contains vertical structures produced by the hull and the propeller blades which interact with the surface, it appears that an explanation of the surface signature in the wake will require a fundamental understanding of the interaction of vorticity with the free surface. When a pair of counter rotating vortices in an ideal fluid approach a flat surface the velocity field due to the image vortices will cause them to diverge (Lamb 1932~. Since the motion of each vortex is a result of the velocity induced by the other vortex as well as the image vortices, the diverging vortices ultimately move along straight lines parallel to the flat surface. When the viscosity of the fluid A. Hirsa Department of Mechanical Engineering, Mechanics, Rensselaer PolytechnicInstitute,Troy NY is considered, the flow field will be different owing to the diffusion of the initial vorticity and the additional no slip boundary condition at the flat surface. The observed surface features produced by the wake of a submerged delta wing at a negative angle of attack, as originally described by Sarpkaya and Henderson (1985), are sketched in Figure 1. Delta Wing Trailing Vortices Propagating at a Negative Toward the Free Surface and Angle of Attack Carrying with them some Fluid \ / ii~Vonex Sheet \ ,~ /~ Roli-Up \/ _~~ BY Z ~Striations 1< /; - \Scars Slightly Deformed X ~c~ O Free Surface .~ - ~~3-Dimensional Structures including Vortex Lines Terminatulg at the Surface Figure 1 Surface deformations resulting from the trailing vortices interacting with a free surface. 521 The subsurface flow field associated with many of the surface features that they described are not yet understood. There are also numerous other experimental and theoretical studies of the interaction of vortex pairs with solid and free surfaces. A survey of the literature for many of these studies has been made by Hirsa (1990~. From the literature it is clear that the interaction of vorticity with a free surface is a very complex problem. The flow may or may not be turbulent and one must consider the Reynolds number. In addition, the momentum of the vortex pair may be great enough to cause vertical displacement of the free surface with rebounding of the vortex pair during the interaction, as Tryggvason has shown through numerical computations [see Willmarth et al. (1989~] and one must consider the Froude number. Finally, it has become apparent (as will be demonstrated in this paper) that an understanding of the interaction of vortex pairs with a free surface at low Aeronautical Engineering and 12180 U.S.A. G. Tryggvason and J. Abdollahi-Alibeik Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor MI 48109 U.S.A. W. W. Willmarth Department of Aerospace Engineering, The University of Michigan, Ann Arbor MI 48109 U.S.A.
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Froude and Reynolds numbers requires that one also must consider the effect of a dimensionless number representing the ratio of the viscous force tangential to the free surface to the tangential surface force exerted by a surface film with a gradient in concentration. When surface contaminating films are not present on the free surface (which in nature is a very rare condition) the basic interaction, at low Froude numbers and for a laminar flow, is approximately similar to the flow that would be observed for an inviscid fluid with a flat surface. However, clean surfaces are a rarity and varying amounts of surface film contaminants are usually present on the surface. When present, as will be shown, surface contaminants can completely change the nature of the flow field during the later stages of the interaction of vortex pairs with the surface. Initial experiments were made with a small delta wing, similar to the one used by Sarpkaya and Henderson (1985), which was towed beneath and parallel to the surface at negative angle of attack. The flow beneath the surface was observed using a fluorescent dye (Fluorescein) injected at the trailing edge of the wing and illuminated from below the surface by a thin sheet of laser light normal to the surface and to the towing direction. These flow visualization experiments were in general agreement with the results described by Sarpkaya and Henderson (1985~. In addition to observations of the wake cross- section, we observed the motion of the surface with the aid of particles placed on the surface after the wing had passed. We found that as the vortex pair approached the surface the particles~on the center line of the wake were swept to either side by the upwelling of fluid carried with the vortex pair. We also discovered that the surface motion was not consistent because the velocity and amount of surface motion observed on either side of the wake was dependent upon the type of particles used as passive markers. In one case, punched circles (chaff) from oiled, paper computer tape placed on the surface after the wing were completely motionless. However, a few pieces of chaff that were completely wet and had sunk slightly below the surface were observed to be moving outward from the wake center line while the drier pieces of chaff just above them and floating on the surface were stationary. After these initial observations, the status of the project was discussed with personnel of the Naval Research Laboratory and the Office of Naval Research. We were alerted to the possibility that surfactants could be present in the Ann Arbor city water used to fill our small towing tank. Jack Kaiser of the Naval Research Laboratory offered to measure the surface tension of our water and found severe surface contamination of the Ann Arbor tap water. He suggested that we investigate methods to clean the water in our tow tank water and referred us to the papers of Scott (1975 and 19821. Scott (1975) describes methods to prepare clean water for fluid mechanical experiments with an uncontaminated surface. In the other paper, Scott ( 1982), experimental measurements are described of the boundary layer flow beneath a contaminated surface in which the convection of surface contaminants is blocked by a barrier which penetrates below the surface. A steady state boundary layer is formed as a result of a balance between viscous shear forces in the viscous flow beneath the surface and the force produced by gradients in the surface tension which are a result of surface active agent concentration gradients in the blocked film of surface active agent. At the leading edge of the contaminated surface, where the surface tension begins to decrease, there is a rapid variation in height of the surface which appears as an easily observable ridge. The ridge is formed when the upstream flow, with high surface tension, first encounters the contaminated surface which has a considerably lower surface tension. The upstream fluid is rapidly decelerated as a result of the upstream "pull" of the clean oncoming flow with a rapid increase and then decrease in surface elevation. At this initial stage of the investigation it was apparent that a study of the interaction of vorticity with a free surface would require both experimental measurements of the flow field and numerical computations of the velocity, vorticity and displacement of the fluid particles in the wake flow field. Experimental measurements of the flow field in the wake of the delta wing would require many measurements for a large number of runs in the tow tank facility. To reduce the time and labor required for the experimental and numerical investigation of the wake flow field, the flow in the wake was approximated by a two-dimensional vortex pair propagating toward and interacting with a free surface. In the experiments the flow field of a laminar vortex pair was produced by a vortex pair generator designed for this investigation. In the numerical computations the time dependent, two- dimensional flow resulting when two line vortices were placed at an initial position beneath the surface was calculated. The line vortices were approximated by two blobs of vorticity, with opposite sign. Both a clean free surface and a free surface contaminated by various known amounts of surface active agents were studied in the experiments and in the numerical computations. In the paper we first present a description of the flow visualization experiments and the quantitative measurements that we have made which serve to outline many fundamental aspects of the problem. This is followed by a description of the numerical computations for vortex blobs interacting with clean and contaminated free surfaces at low Froude and Reynolds numbers. The numerical computations allow the study, at little cost in time and labor, of the effect of many different parameters on the flow variables during the interaction of vorticity with the free surface. EXPERIMENTS 1) Vortex pair Generation The vortex pair generator used for the experiments was designed to produce a two-dimensional pair of counter rotating vortices that propagate upward toward the free surface. A pair of initially vertical flaps, see Figure 2, which were driven by a computer controlled stepping motor were rotated toward each other to produce a pair of vortices without any measurable effect on the free surface caused by the flap motion. The velocity induced by the voriicity in the vortex pair causes the pair to propagate upwards as the flaps close. The flow field produced by the flaps was found to be uniform along the span. This vortex generation scheme also produced little interference between the flaps and the newly formed vortex pair. As the vortices form, the inward rotation of the flaps allows the vorticity to roll up and entrain additional fluid as required for the formation of the Kelvin oval associated with the vortex pair. The Kelvin oval is the oval region bounded by a closed streamline around the vortex pair which can be observed in frame of reference moving with the vortex pair. The triangle formed by the flays after they are closed had an apex angel of less than 74 . See Hirsa (1990) for further information on the 522
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vortex pair generator. : ~ Lmage Vortices :~x Free Surface 1 h Vortex Pair A \ // \\ // \\ // \\ // \\ //\\ 'of ~b~ Mylar Sheet Kelvin Oval Counter Rotating, Stainless Steel Flap / Figure 2 Schematic of the vortex pair approaching a free surface. The dimensions of the vortex pair generator were chosen so that the vortex pair would have a Reynolds number, Re, and Froude number, Fr, matching the vortex pair produced in the wake of the delta wing. The Reynolds number of the vortex pair is defined by: Re r v where r is the circulation about one of the vortices and v is the kinematic viscosity. The Froude number of the vortex pair is defined by: r Fr-, (2) where g is the gravitational acceleration and SO is the vortex pair separation when the vortices are far beneath the surface. 2) Flow Field of the Vortex Pair Observations, using laser induced fluorescence (LIF), of a cross-section of the flow field when vortex pairs interact with the free surface were made for a number of different strength vortex pairs generated with the flaps and for various degrees of surface contamination. The dye used was fluorescein which was injected with a syringe fitted with a slender plastic tube into the water between and around the flaps before flap motion was initiated. The laser light sheet was less than 1 mm thick and illuminated the flow field above the flaps from one side, see Hirsa (1990) for further information. - As described in the introduction, our initial measurements with the delta wing and a contaminated surface demonstrated conclusively that the motion of the surface was inhibited if the surface was contaminated. During the experiments with the vortex pair generated with the flaps the free surface with an area of approximately one half square meter above the vortex generator was cleaned for at least one hour by means of a drain pipe and a fan blowing air toward the drain prior to the first experiment each day. After each vortex pair was generated, the free surface was cleaned first for a minimum of 5 minutes by surface draining with the fan on, then the fan was turned off and the surface drain was continued for at least another 5 minutes. The surface drain was then stopped and the free surface was undisturbed for a minimum of 5 minutes to allow the surface currents to completely decay before the next run. To illustrate the typical flow phenomena observed for low Froude number vortex pairs of various strengths a set of (LIF) photographs of a the flow produced for a typical vortex pair (Re=12,400 and Fr=0.217) interacting with the free surface with and without surfactant added on the surface is shown in Figure 3. The photographs on the left, labeled (A), are for a relatively clean free surface with no oleyl alcohol added. The photographs on the right, labeled (B), are for an initially clean free surface with 1.06 x 10-7 (cm3/cm2) of oleyl alcohol spread on the surface before the vortex pairs were generated. Hirsa (1990) contains a complete description of the method used to determine the surface concentration of the oleyl alcohol . From the concentration and using the state relationship for oleyl alcohol, see Hirsa (1990) Figure 4.5, the initial surface pressure ~ - 60 - 6 (the difference between the surface tension, GO, of the clean and the surface tension, 6, of the contaminated free surface was approximately 2.5 dynes/cm). For the vortex pairs shown in Figure 3, the tips of the flaps (when vertical) were at a depth of 21 (cm). The average velocity and spacing of the vortices when the pairs were well below the surface was vp0 = 2.36 (cm/see) and SO = 6.93 (cm/sec). A dimensionless time I* was defined as: :* = ' 'i (3) 60 / vp O 3 where t is the time since the start of the flap motion and, t1 = 7.55 (sec), is the time elapsed from the start of the 523
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flap motion until the time the vortices are at a depth equal to the vortex pair separation after roll-up. Thus, SO / vpO, is the time it takes the vortex pair to propagate a distance equal to their separation, when they are far from the free surface. The circulation, A, and the Froude number, Fr, for the vortex pairs produced in the experiments was estimated using an inviscid point vortex model. The model treats each of the vortices as a point vortex and consists of two point vortices and the two image vortices associated with the original vortices when they are in the proximity of a free surface. Lamb (~932) describes the trajectory of a vortex pair approaching a flat wall in inviscid fluid. Using the trajectory equations and observations of the position, spacing and velocity of the vortex pairs, the apparent circulation of the vortex pairs was estimated. This is equivalent to assuming an inviscid vortex pair with Fr=O, approaching a free surface. ~ ~ ~ PA ~ ; ~:~ ~ ~ (x-~`x*~ Surfs ~ ~ ~ ~ ~ (~3 ~ ~ ~- . ~ ~it. I :_ _ it: ~:~ ~ ~ ~:~ ~ ~ ~ ~(i) em 'I __ I, :: - :~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ::::::: ~ _ _ t~ :~:(j^,) ~ 'I ~ ~'~ it__ :~::: i: : :: ~ ~ i: ~ ~ ~ ~:~ _ ~ ~ ~11 ~ on ~:~ ~ ~2 ~ ~ ~ n ~ ~ (iY) ~ ~ ~ ~ ~ __ ~ ~ '__ ~ ~ ~ :~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~-7 0 ~ ~ : ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ _ ~ ~ ~ : ~ :__ ~ ~ __ ~ ~ :1_ (V) ~ __' ~ i: ~ : _ ~ ~ -__ :- i: :~ ~ .: ~ ~ ~ : : ~ :: :~ :: it: it: :~ : i: i: ~ ::~ :: ::::: ~ :: : : :~ :~ ~ ~ ~ ~ ~ ~ ~ ~ : ~ :: _~e b ~ __ ~:~ _ Figure 3 LIF photograph of a vortex pair interacting with a free surface with: (A) no surfactant added & (B) with oleyl alcohol added with surface pressure of 2.5 (dynes/cm); initial vortex pair separation bo=6.93 (cm) and propagation speed vpo=2.36 (cm/sec), Re=12,400 and Fr=0.217 (scale: 1 cm on the photograph = 8.4 cm in the flow field). The dimensionless times are: (i) I* = 0.59, (ii) I* =0.93, (iii)* = 1.27, (iv)* = 1.61, (v) I* = 1.95, and (vi) I* =2.29. Figure 3 (i) shows the vortex pair approaching the free surface. The dimensionless time I* is 0.59. The vortices have started to move apart and no difference is apparent between the case without oleyl alcohol (A), and the case with oleyl alcohol (B). The subsequent photographs, (ii) to (vi), show the vortex pair at equal I* intervals of 0.34. The mirror image that appears on the upper portion of each photograph is due to total internal reflection at the free surface. Location of the free surface was determined in each photograph by drawing a bisector through the symmetrical image. There is little visible difference in the flow field between the case (A), and case (B) up to the time ~*=1.61. At ~*=1.61, Figure 3 (iv) shows a marked difference between the two cases. In the case where oleyl alcohol is added, the streamlines near the free surface appear to diverge from the free surface whereas in the case with no oleyl alcohol the streamlines near the free surface stay close to the surface. By ~*=1.95, Figure 3 (v) shows that a secondary vortex is being formed for the case (B), but in case (A) no secondary vorticity is formed. In the last Figure 3 (vi), a pair of secondary vortices are visible in the case (B) with oleyl alcohol which drastically alter the path of the primary vortices as they begin to rebound from the free surface. The case without oleyl alcohol shows no sign of secondary vorticity formation or rebounding of the primary vortices. Free Surface Secondary Vortices V , ,_ ,~-' ~` \.\ 1 1 1 --Free Surface - no surfactant added - " w/ Oleyl alcohol, ~0.3 - 1.1 1 .. ·1 x=9.0 -2.5 .... ....... Solid Wall x/bO Figure 4 Average trajectory of the apparent center of the (right) vortex during the interaction of the vortex pair with a solid wall and a free surface with various amounts of oleyl alcohol whose surface pressure is denoted by it. Re=12,400 and Fr=0.217. A tick mark on the primary vortex trajectory indicates the position of the primary vortex at the beginning of the secondary vortex trajectory. The time at the beginning and the end of each primary and secondary vortex trajectory is as follows: no surfactant; -1.55 5 I* < 2.19, not formed: = 0.3; -1.55 S I* < 2.19, 1.85 < I* < 2.19: = 1.1; -1.55 < I* 5 1.85, 1.51 S I* S 1.85: = 9.0; -1.55 S I* 5 1.51, Turbulent: Solid Wall, -1.55 S I* 5 1.85; 1.17 5 I* 5 1.85. 524
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3) Vortex Pair Trajectories Figure 4 shows the trajectories of the apparent center of the right vortex for various surface conditions for vortex pairs with, Re=12,400 and Fr=0.217. Also plotted, are the apparent center of the secondary vortex which is produced outboard of the right vortex. The five primary vortex trajectories overlap in the beginning, but diverge at later times during the interaction. The primary vortex trajectories all begin at ~*= -1.55. The time at the end of each primary vortex trajectory is given in the caption along with the time at the beginning and the end of secondary vortex trajectories. The secondary vortex for the case with the largest surface pressure, 9.0 (dynes/cm), rapidly became turbulent and no distinct vortex center could be observed. A tick mark on each of the primary vortex trajectories indicates the position of the primary vortex at the time of formation of the secondary vortex. Turbulence in the primary vortex was first observed at time, ~*= 2.19 (+0.24) for all the surface conditions tested. The trajectories plotted in Figure 4 show that when oleyl alcohol is present on the free surface, the trajectory of the primary vortex departs from the trajectory for the case with no surfactant. From these trajectories it is apparent that the greater the surface pressure, the greater the amount of rebounding of the primary vortex from the free surface. When the surface pressure is as high a 9.0 (dynes/cm), the vortex trajectory is very similar to that for vortex interaction with a solid wall. 4) Particle Image Velocimetry Measurements In order to provide quantitative data for the phenomena observed and results obtained using flow visualization the entire velocity field in the cross section was measured for a vortex pair at Re=12,400 and Fr=0.217 using a particle image velocimetry (PIV) technique. The method we used is one of the variations of double-pulse velocimetry used in the past fifteen years, as described in Stetson (1975), Lauterborn and Vogel (1984), and Dudderar et al. (1986~. From a double- exposed transparent photograph of a seeded flow, the velocity at each point in the flow is deduced by measuring the displacement of the seeding particles during the time between two exposures. In the present study, a pulsed copper vapor laser was used to illuminate a thin cross- section of the vortex pair flow field. The Young's fringe method was chosen to determine the particle displacements and direction of displacement. The magnitude of the velocity was determined from the displacement divided by the time between the two exposures. The flow direction was obtained by inspection of the velocity magnitude data. The system and method we used to interrogate double exposure photographs of the particle images was developed by L. P. Bernal and is described by Kwon (1989~. This system consists of a Helium-Neon laser, a set of mirrors, a transform lens, a beam stopper, a video camera with a pair of imaging lenses and a Gould model ED 5000 image analysis system controlled by a Zeos model 286 personal computer. Some time was spent learning how to seed the flow. After a number of trials the seeding particles selected for PIV measurements were 12 to 50 microns particles of titanium dioxide in the rutile crystalline form and 50 to 100 micron diameter micro-balloons. Micro- balloons are hollow glass bubbles with specific gravity of less than one which tend to migrate inward toward the core of the vortices while the much heavier titanium dioxide particles migrate away from the vortex core. The fluid in the vicinity of the flaps was seeded with both glass micro-balloons and titanium dioxide particles and titanium dioxide particles were also continuously deposited on the free surface. Figure 5 shows the velocity vector field of the vortex pair for Re=12,400 and Fr=0.217 as it approaches a clean surface at time ~*= 0.49. It} (cm/see) 1 1 5 (cm' Figure 5 Vortex pair velocity field during the interaction with a free surface with no surfactant added; ~*= 0.49, Re=12,400 and Fr=0.217. The non-zero velocities on the top of the figure show that the free surface is adequately clean and free to move. The lack of data near the center of the vortices is primarily due to the limits in the spatial resolution of the interrogation system. Inadequate seeding is responsible for the missing data near the lower part of the figure. The dynamic range of the interrogation system limits the lowest velocity that could be measured. The flow on the plane of symmetry at the free surface resembles a stagnation region of very low velocity. As a result, the spacing between the Young's fringes is too large to measure. PIV images at later times with a clean surface were also obtained and there was no sign of the development of secondary vorticity near the free surface. From the PIV data the contours of constant vorticity in the primary vortices were determined. The circulation for each side of the vortex pair was also calculated using a line integral around a square contour surrounding the vortex core. The magnitude of circulation for each vortex was found to be 145.9 (cm2/sec). A few of the vorticity contour lines extended outside the domain of the calculation. This implies that the total circulation for each vortex is slightly more than the 145.9 (cm2/sec) measured. In contrast, the apparent circulation, i.e. the circulation based on the measured propagation speed and spacing of the vortex pair and using a point vortex model, was found to be 124 (cm2/sec). In marked contrast to the flow with a clean free surface, Figure 6 shows the velocity vector field of the left primary vortex and the secondary vortex developed outboard of the left primary vortex at a time ~*= 1.54, when the free surface is contaminated with oleyl alcohol with a surface pressure of 9.0 (dynes/cm). 525
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1() (cm/see) ~ ~ ~ . . . . . . ~ . I . . '' ;~ , ~ ,, ,, ,, ~ ~ ~ ~ ~ If ~ J ~ .......... / J J J I ~ J I J ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ ''' _ ~ ~ ~ ~ \ ~ ~ ~ ~ ~ ~ ~,, , 1 1 2 (cm) Figure 6 Velocity field for the left vortex interacting with a free surface with oleyl alcohol with ~=9.0 (dynes/cm); the right edge of the figure is 7.7 (cm, actual) from the centerline; ~*=1.54, Re=12,400 and Fr=0.217. The circulation around the secondary vortices in this PIV vector plot was calculated and found to be approximately -45 (cm2/sec), opposite to and approximately 30% of the circulation in the primary vortex. This clearly illustrates the fact that surface contamination can lead to the generation of new vorticity during interaction of vorticity with the free surface. 5) Free Surface Signatures of a Vortex Pair The Froude number for the vortex pairs in these experiments was not large enough to cause appreciable wave generation when the vortex pairs interact with the free surface. However, a variety of slight surface deformations were observed that were associated with the velocity and pressure field of the primary, secondary and any other vorticity produced by the vortex pair. These surface deformations were visually observed using the shadowgraph method. The shadowgraph system used in the experiments with the vortex pair generator was very simple, a mercury vapor light source beneath the glass bottom tank containing the vortex pair generator and a white cloth screen 6 X 2 feet mounted 3 feet above the free surface. The image on the screen was recorded on video tape using the video camera and recorder. The shadowgraph effect is a result of the refraction of light by the free surface. When an approximately parallel beam of light passes upward from water to air normal to the surface, a depression of the surface will result in the light rays diverging and an upward displacement of the free surface will cause the light rays to converge. The screen placed above the free surface will then show surface depressions as darker regions and elevations of surface as brighter regions. The vortex pair with Re=18,700 and Fr=0.277 produced a surface signature which from our observations of the signatures of various strength vortex pairs (at low Froude number) contains the typical phenomena observed with weaker and stronger vortex pairs before transition to turbulent flow occurs. The surface signature for the right half of a clean surface above a vortex pair is shown in Figure 7. 526 _. _ ~ 1 - 8~ ~3 IB8l 8 Figure 7 Shadowgraph view (of the right half) of the free surface during the interaction of a vortex pair with a free surface with no surfactant added; Re=18,700 and Fr=0.277 (scale: 1 cm on the photograph = 10.0 cm on the surface). The first observable features are the striations that were first described by Sarpkaya and Henderson (1985~. Using LIF and other techniques to observe the flow beneath the surface we have determined that the striations are caused by spanwise vorticity which is stretched in the upwelling flow field of the primary vortex pair, see Hirsa (1990~. A paper describing and summarizing these observations of the flow field of the striations is in preparation. The first photograph in Figure 7, (i), is at time ~*=0.79. The striations, observed to produce narrow dark strips of surface depression, are clearly visible in this photograph. A surface depression which is wider than and normal to the striation depressions can be observed directly above the right hand primary vortex. Just outboard of this depression the initial formation of a Reynolds ridge can be observed (an undulating, narrow, line bright on the left and dark on the right) which becomes stronger in the following photographs, (ii) through (iv). The photograph (ii), taken at ~*=1.16, shows the striations and the Reynolds ridge as well as a surface dimple, a dark region at the top of the photograph. This dimple is caused by a vortex line terminating at the free surface. A similar dimple occurred on the bottom but does not appear in this photograph. The vortex induced at the end wall (see Yamada and Honda 1989) is thought to be responsible for this phenomenon. The third photograph (iii), taken at ~ *= 1.90, shows that the Reynolds ridge has been convected to the right. A few striations are still visible in this photograph. Adjacent to the large dimple near the top of the photograph, a series of smaller dimples appear in this picture. The last photograph (iv), shows the Reynolds ~idge which by this time has moved very much to the right. The photograph was taken at ~*=2.63. There are some last remnants of the striations visible to the left of the Reynolds ridge. The dimples are more numerous by this time. It should be noted that the small dimples all appear on the outboard, i.e. on the contaminated side of the Reynolds ridge. The surface deformations on the ~ight side of the same vortex pair formed when a small amount of oleyl alcohol, surface pressure of 0.3 (dynes/cm), is present on the free surface are shown in Figure 8.
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~ - hi= At= ~ - - ~ ~ - ~ ~ - - - - - - - : - :~ Figure 8 Shadowgraph view (of the right half) of the free surface during the interaction of a vortex pair with a free surface with oleyl alcohol with ~=0.3 (dynes/cm); Re=18,700 and Fr=0.277 (scale: 1 cm on the photograph = 10.0 cm on the surface). The first photograph, (i), at ~*=0.79, shows the striations, an intense, narrow bright and dark wavy line, the Reynolds ridge, and two wide depressions (which will be referred to as scars) parallel to the Reynolds ridge and to the primary vortex. The weaker depression is formed to the left of the Reynolds ridge (above the primary vortex) and a stronger one just to the right of the Reynolds ridge (above the secondary vortex). The next photograph, (ii), at ~*=1.16, shows that the Reynolds ridge has moved slightly to the right. The striations are still visible in this photograph. A series of surface deformations, primarily depressions, are visible between the two scars. The next photograph, (iii), shows the effect of three-dimensional structures along with vortex lines terminating at the free surface (dimples) to the right outboard) of the Reynolds ridge. The last photograph, (iv), at ~*=2.63 again shows the surface signatures of the three-dimensional structures which have been onvected to the right. The Reynolds ridge, which did not move very much since the last photograph appears to be less intense at this time. Figure 9 is for the same vortex pair ~ Re=18,700 and Fr=0.277) as the previous two figures but with a higher concentration of oleyl alcohol. For this case, enough oleyl alcohol was spread on the surface to saturate it so that the surface pressure was equal to the saturation pressure which for oleyl alcohol is approximately 31.5 (dynes/cm). The striations and the scars are visible in this photograph. The photograph shows that no Reynolds ridge is formed. The scar above the secondary vortex observed out board of the primary vortex is more intense than the previous cases. NUMERICAL COMPUTATIONS 1) Problem Formulation and Numerical Method The flow is assumed to be viscous, and confined to two dimensions. In addition to the assumption of two- dimensionality, the major limitation is that the free surface is assumed to remain flat for all tinges. This limits the results presented here to relatively low Froude numbers. However, these are the cases most frequently Figure g Shadowgraph view of the free surface showing the scar during the interaction of a vortex pair with the free surface with ~=31 (dynes/cm); ~*= 0.79, F= 187 (cm2/sec) (Re=18,700), Fr=0.277; (scale: 1 cm on the photograph = 4.0 cm on the surface). studied experimentally, and since the surface deformations are observed to be small the linutation is not as severe as might be thought. In order to avoid any arbitrary modeling of inflow and outflow boundaries we simply take the flow domain to be periodic in the horizontal direction, and to have a flat full slip bottom. The effects of this limited domain size is discussed below in section 21. The flow is governed by the Navier-Stokes equation, which in vorticity form can be written, Ocl,/Ot + J(~r~c~) = He V2(c'~) (4) where Jo= fa~lay'~a~lax'- ra~lax1fa~lay', and a Poisson equation relating the stream function to the vorticity V2~= ~ (5) Here the the Reynolds number is defined as Re = F/v. The free surface boundary condition is very important for the present investigation. Surface contaminants are known to have an effect on the motion of vortices. In a previous investigation Davies (1966) described the damping of turbulent eddies at a free surface and later Davies and Driscoll (1974) experimented with ejecting pulses of colored water towards a free surface, specifically addressing the rate of surface renewal and they found that the spreading of colored water at the free surface is reduced considerably for contaminated surfaces. Their simple visualization technique did not allow a clear explanation of the mechanism responsible for this behavior. The explanation for rebounding from a free surface is clear from recent experiments of Bernal et. al. (1989) who investigated collision of both vortex rings and two-dimensional vortex pairs with a free surface. They observed (as did Davies and Driscoll) that surface contamination led to considerable differences in the vortex motion itself. Using LIF for flow visualization they found that the surface motion induced by vorticity 527
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approaching a contaminated free surface generated an uneven distribution of surface contaminant that in turn caused a shear stress at the surface, thereby generating vortieity with sign opposite to the initial vortieity. The generation of vortieity at a contaminated surface appears to be a primary effect of the surface contaminants since for contaminated surfaces the behavior is similar to vortices encountering a rigid wall. For the numerical computations the boundary condition at the surface requires knowledge of the surface contaminant which is assumed to be conserved, leading to a hyperbolic conservation equation De/Ot +akuSc~lax = 0 (6) Here us is the horizontal velocity at the surface. Notice that since the surface divergence of us is in general not zero and depends on e, this equation allows for the possibility of "contamination shocks" (that is the development of a discontinuity in e). The surface contaminant affects the flow field through shear stresses induced by variations in the surface tension. At the free surface, surface tension gradients induce a shear given by = Rolex (7) Since the surface is flat the vortieity at the surface is ces= Ou/Oy. The surface tension depends on the amount of contaminant, c>= Ares, and the boundary conditions for the vortieity, at the surface, is therefore ce = p-I(36/Oc~Oc/Ox (8) The quantity ~ = co (36/Oc) is usually called the elasticity of the surface. If the contamination, e, is nondimensionalized by its initial value, en, and the vortieity is as before, we end up with the boundary condition COS = cOe/ax in nondimensional units, where C = [L/~] en (96/Oe) (9) The flow is therefore governed by the parameters Re and C as well as the initial vortieity configuration. The dimensionless variables used for the computations were, the dimensionless time = t F/L2, and the dimensionless distances, x/L and y/L. Where ~ is the circulation and L is the half width of the computational box. To solve these equations numerically we have used a rather standard finite difference approximations. Equation (4) is integrated by an explicit second order predietor-eorreetor method in time, and the spatial diseretization is done with second order eentered-differenees. For the Jacobian, J(x,y), Arakaw's conservative stencil is used. The Poisson equation is solved with a fast solver (HWCRT form FISHPACK). For the contaminant we also use a second order predictor corrector in time, and second order differences in space. For stability an artificial viscosity term is added on the right hand side of (6) with viscosity that is small everywhere except where the contaminant value changes rapidly. The surface velocity is found by a one sided, second order differentiation of the stream function. Several of our results have been checked for convergence by repeating the calculation using a different resolution. 2) Results and Parameter Studies Most of our computations have been done for the ease of a two dimensional vortex pair colliding head on with the top surface. Since the problem is symmetric about the centerline it is sufficient to calculate only one of the vortices and use symmetry boundary conditions. The central question that we are addressing is how the contaminants on the surface affect the evolution of the vortieity, and how it differs from the ease when the free surface is clean. In Figure 10, for Re = 2000, we show the evolution of the flow produced by two blobs of primary vortieity with opposite sign, with an initial spatial distribution proportional to r exp(-ar2) where r is the distance from the center of each blob. The boundary conditions on the top surface are, (a) a stress free boundary (also called a full slip boundary) and (b) a contaminated top surface with C =2. The right hand vortex is initially half way between the top and bottom boundary, and the first frame is at the time the motion begins. There is no boundary layer for the full slip surface. A slight boundary layer (not visible at the initial time, t = tip is formed beneath the contaminated surface an instant after the motion begins. In the second frame the upward motion of the vortex has ended, and, due to the image vortieity above the free surface, it is now moving outward. The boundary layer in (b) has grown considerably, and it is clear that separation is about to take place. In the third frame the vortex in (a) continues its outward motion along the full slip boundaries, but in (b) the boundary layer has separated and formed a secondary vortex that deflects the path of the primary vortex away from the surface. This evolution continues in the fourth frame, the vortex in (a) moves out along the wall, but in (b) the primary vortex has moved further away from the wall under the influence of the secondary vortex. At the same time the stronger primary vortex swings the secondary vortex around so it is now almost below the primary one, and thus induces an inward motion. Viscosity now has visible effect on the evolution, the maximum vorticity of both the single vortex in (a), as well as in (b) has decreased compared with the previous frames. In the last frame the vortex in (a) has encountered the outer boundaries of the computational box, and is starting to move downward along the outer wall, and in (b) the primary vortex is actually moving upward again es well asinward. Perhaps the most striking feature of the above sequence is the similarity between the results for the contaminated surface case (b) and results of calculations (not shown in this paper) for the case of a "no slip" rigid wall . The flow field, primary vortex motion and secondary vortex motion for case (b) and the "no slip" wall case are very similar. The rebounding of the primary vortex for the contaminated top surface in (b) is obviously due to the uneven distribution of the contaminant produced after the motion begins. This distribution is shown in Figure 11, at times corresponding to those in Figure 10. In (a) the contaminant is passive, and is simply advected with the flow and this does not lead to any shear stresses on the fluid at the boundaries, as mentioned above. As the vortex collides with the surface the contaminant is swept outward, depleting the region between and above the vortices of contaminant and accumulating it outward of the vortices. This contaminant peak is then pushed outward. Since the computational box is of finite width, the contaminant eventually reaches large values at the 528
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(a) (b) (a) (b) - ,~, ,~, t3 t4 l~ts~ . Figure 10 Computational results showing contours of constant vorticity on the right side of a vortex pair approaching the free surface for a clean surface (a), C = 0 and a contaminated surface (b), C = 2. Re = 2000. The dimensionless time is zero at the top of the Figure and increases in increments of 0.7 dimensionless . . time units. t3 l lll c: tip t'4 tats Figure 11 Contamination profiles for the computations shown in Figure 10. Full slip case is (a) and the C = 2 case is (b). The time between the profiles is the same as in Figure 10 and the profiles all become steeper with time except for the last profile for the case C = 2. l Ill . ~ t'2 arts outer side of the box in the down welling region. Although the finite box size obviously has effects on the final profile, the maximum contamination peak increases rapidly even before the side effects become significant, since the outward velocity decreases outboard of the vortex. In the second frame, (b), C = 2 as in Figure 10. Now the uneven contaminant distribution creates shear stresses on the top surface that opposes the outward motion due to the vortices. This balance---outward motion due to the vortices, and inward motion due to the uneven contaminant distribution---eventually slows down the spreading of the clean region above the vortices. The shear stresses due to the contaminants create vorticity that eventually separates and causes the primary vortex to rebound. As the vortices rebound their effect on the surface diminishes, and the contamination "shock" that separates the clean and contaminated surface starts to move inward again. In Figure 11 at time is the inward motion has just started. The large accumulation of contaminants, seen for the "full slip" case does not take place in the contaminated surface case although the contamination profile behind the shock equilibrates with time. We have made computations for larger values of C and find that the restoring effect of the contaminants is much stronger. As a result for C = 10, only a small clean region forms on the surface between the primary vortices. The vortices then move outboard of the shock, and as they pass under and rebound the "hole" closes rapidly. For the case with C = 50 the vortices only cause a small initial dimple in the contamination profile which disappears rapidly as the vortices move outward. Perhaps the most noticeable feature of the contamination profiles is how different they are when compared to the similarity of the vorticity distributions which the contamination has created. Except for the completely stress free boundary a secondary vortex is formed and the primary vortex rebounds, even though in some cases a clean region is formed, and in others the contaminant distribution is hardly changed at all. The only difference in the vorticity distribution is that the boundary layer at the top starts further away from the center when a clean region is formed. The above runs have all been done in a relatively small computational domain. To assess the influence of the boundaries on the evolution we have repeated one of s29
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the runs with, C = 2, in a domain that is twice as wide. The vorticity distribution appeared almost identical, and only for the latest times were there any significant differences in the contamination profiles. The primary difference was that the value of the contamination concentration was slightly higher behind the shock for the shorter box, and as a result the shock moved slightly faster to close the hole in the contamination profile after the vortices had rebounded. We therefore feel rather confident that boundary effects are of minimal . . significance. (a) ~ _-----_ ~_ l (b) i' Figure 12 The paths of the primary and secondary vortices. In (a), the full slip case with no secondary vortex is a dotted line, the solid line is for C = 2 and the dashed line is the solid wall case. The two large radius curves are the secondary vortex paths. In (by, the primary and secondary vortex paths for increasing Reynolds numbers of See, 1000, 2000, & 4000 are shown for contaminated surface with, C = 2. There are no secondary vortices for Re = 500 and 1000. For Re = 2000 and 4000 the secondary vortex paths have a large radius of curvature as the Reynolds number increases. To efficiently display the similarities and differences in the flow fields caused by the surface boundary condition and/or Reynolds number, we plot in Figure 12 (a), the path of the primary vortices and secondary vortices, at Re = 2000, for the no-stress case, for a contaminated surface with C = 2 and for the rigid boundary case. For the no-stress case there is no secondary vortex and no rebounding. The slight downturn in the primary vortex trajectory is caused by the proximity of the right hand boundary. For the contaminated surface and for the rigid wall there is a secondary vortex and there is rebounding of the primary vortex. In Figure 12 (b) the paths of the primary and secondary vortices are shown for a contaminated surface with C = 2 at various Reynolds numbers of 500, 1000, 2000, and 4000. At the higher Reynolds numbers the diffusion of vorticity is less rapid in comparison to convection. This results in greater outboard motion of the primary and secondary vortices as the Reynolds number Increases. CONCLUSIONS · Experiments and numerical computations for vorticity interaction with a free surface show good qualitative agreement. . Surface contamination, in the experiments and computations, was found to have a strong influence on the nature of the interaction of the vortex pair (or the mailing vortices) in both the surface deformations and the flow field below the surface. · During interaction of a vortex pair (or trailing vortices) with a contaminated free surface, shear stress produced at the free surface causes the production of vorticity. This vorticity can roll-up into a pair of secondary vortices with sign opposite to the adjacent primary vortex. This alters the trajectory of the original vortices. · A Reynolds ridge is formed as a result of the interaction of vortex pair (or the trailing vortices) with a slightly contaminated free surface. · The strong scar (surface depression) was found to be caused by secondary vortices formed when a surfactant was present on the surface. · The circulation of the secondary vortex produced as a result of the vortex pair interacting with a free surface with surfactant was found to be about one third of the circulation of the primary vortex. REFERENCES Davies, J. T., 1966 The effect of surface films in damping eddies at a free surface of a turbulent liquid. Proc. Royal Soc. London A 290, 515-526. Davies, J. T. & Driscoll, J. P., 1974 Eddies at free surfaces, simulated by pulses of water. Industrial and Engr. Chemistry Fundamentals 13, 105-109. 530
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Dudderar, T. D., Meynart, R., & Simpkins, P. G., 1986 Laser speckle velocimetry. The Tenth U. S. National Congress of Appl. Mech., The Univ. of Texas at Austin. Hirsa, A., 1990 An Experimental Investigation of Vortex Pair Interaction with a Clean or Contaminated Free Surface. PhD. Thesis, Dept. of Aerospace Engr. The University of Michigan. Kwon, J. T., 1989 Experimental study of vortex ring interaction with a free surface. PhD. Thesis, Dept. of Aerospace Engr. The University of Michigan. Lamb, H., 1932 Hvdrodynamics 6th ed. Cambridge University Press. Lauterborn, W., & Vogel, A., 1984 Modern optical techniques in fluid mechanics. Ann. Rev. Fluid Mech. 16, 223-244. Sarpkaya, T. & Henderson Jr., D. O., 1985 Free surface scars and striations due to trailing vortices generated by a submerged lifting surface. AIAA paper no. 85-0445, AIAA 23rd Aerospace sciences meeting, Jan. '85 Reno, Nevada. Scott, J. C., 1982 Flow beneath a stagnant film on water: the Reynolds Ridge. J. Fluid Mech. 116, 283- 296. Scott, J. C., 1975 The preparation of water for surface clean fluid mechanics. J. Fluid Mech. 69 pt. 2, 339- 351. Stetson, K. A., 1975 A review of speckle photography and interferometry. Optical Engr. 14 (5), 482-489. Willmarth, W. W., Tryggvason, G., Hirsa, A., & Yu, D., 1989 Vortex pair generation and interaction with a free surface. Phys. Fluids A 1~2), 170-172. Yamada, H. & Honda, Y., 1989 Wall vortex induced by and moving with a confined vortex pair. Phys. Fluids A 1 (7), 128()-1282. DISCUSSION Targut Sarpkaya Naval Postgraduate School, USA Contrary to authors' arguments, contamination does not change the physics of the phenomena. Authors' use of low speeds and scales in their model overly accentuated the effect of surface tension and hence the Weber number. They could have minimized the said effects at relatively larger Froude numbers and thereby gained a clearer understanding of the physics of the phenomenon. As I have shown in my papers, variations in contamination at higher Froude numbers (relatively lower numbers) did not alter the fundamental character of the scars and striations. In ocean environments, the vortex motion and the turbulent wake is such that the relative significance of Weber number is not exaggerated. AUTHORS' REPLY In the near-wake of large, high speed ships, it is possible to encounter very energetic, high Froude number vortices which try to Leap out" of the surface. For such high Froude number vortices, it is true that the effect of surface contamination might not be as great as it is for vortices with lower Froude number. On the other hand, in the far-wake of ships, which are observed in the SAR images, the turbulence is decaying and the Froude number for the eddies is relatively small and therefore surface contamination plays an important role in the interaction of the eddies with the free surface. DISCUSSION Hyong-Tae Kim The University of Iowa, USA (Korea) 1. Besides the secondary vortices identified in your measurement of the velocity for the case of the contaminated surface, could you really resolve the free-surface boundary layer in the measurement? 2. Could you tell how this vortex pair model is related with the persistent trace on the ocean surface of the ship wake? AUTHORS' REPLY 1. The measurements which we made using PIV do not resolve the velocity within the free surface boundary layer. For that information we have to rely on our results from full Navier-Stokes simulations. Information on the finer scales of flow obtained from the computations should be reliable since the agreement on large scale comparisons to the laboratory measurements is very good and the flow is relatively laminar. 2. Although there are vortex pair-like structures in the wake of ships (e.g., bilge vortices and propeller vortices), an exact comparison between the vortex pair flow and the actual ship wake is not possible and in fact is not intended. The vortex pair offers a simple flow which can be studied in order to provide insight into the nonlinear ship wake problem. For example, the mass transport to the surface by a vortex pair and the important effects of surface contamination on this transport process shows the role surfactants can play in the wake of a ship in the ocean. 531
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Representative terms from entire chapter: