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Scarred and Striated Signature of a Vortex Pair on the Free Surface T. Sarpkaya, P.B. Suthon (Naval Postgraduate School, USA) ABsTRAcr This paper describes a numerical and experimental study of three-dimensional flow structures (striations, whirls, and scars) resulting from the interaction of a pair of ascending vortices with the free surface. The characteristics of the flow features at the scar-striation interface (a constellation of whirls or coherent vertical structures, thought to be the boundaries of the dark narrow radar images!, are investigated through the use of an infra-red camera, flow visualization, an image analysis system, and the vortex-element method. The results have shown that the striations are a conse- quence of the short wavelength instability, inherent to the vortex pair itself, and the whirls result from the interaction of stria- tions with the surface vorticity Dice = 2qK, twice the tangential velocity times the curva- ture of the intersection of the free surface with the plane normal to it). The whirl- merging leads to an up-cascading process in size and energy in fewer vortices while the total energy decreases rapidly. INTRODUCTION Just a narrow patch of darkness, bounded by two bright lines, provides the impetus for this investigation partly because it is seen in the synthetic aperture radar (S1\R) images of a ship's wake, partly because it extends many miles directly in the ship's track, partly because the reasons for its exis- tence have not yet been explained, and partly because the surface footprints of sub- surface phenomena can give trace of the generating bodies. This is the basis of the current intense interest in the interaction of internal waves, wakes, and vortices with the free surface. A few facts are known about the SAR images: Their physics is elusive; they are by no means easily accessible to precise mea- surement; they are not related, at least directly, to the Kelvin wake; they do not reflect the incident electro-magnetic waves back to the source (negative spectral pertur- bation); and they can bifurcate. Various pro- posals have been advanced to provide a fea- sible explanation of the dark band: Interac- tion of the wake of a vortex pair with the free surface; turbulence and surface mean flow resulting from the ship's motion; redis- tribution of surface impurities by large-scale vertical motions (as in Langmuir circulations [11 and Reynolds ridges t211; entrained air in the wake; bubble scavenging of surface and subsurface surfactant materials; interaction of Kelvin waves, ambient waves, and momen- tum wake; generation of vorticity-retaining inverse bubbles and drops by a Kelvin- Helmholtz instability [3l, just to name a few of the existing proposals. Each model at- tempts to provide a more feasible explana- tion of the dark narrow band seen in the SAR images. As far as the footprints of a vortex pair on the free surface are concerned, the original observations leading to the present study may be summarized as follows. When a trailing vortex pair, generated by a lifting foil, rises toward the free surface, with or without mutual induction instability and/or vortex breakdown, the vortices and/or the crude vortex rings give rise to surface dis- turbances (whirls, scars, and striations). These were first reported by Sarpkaya and Henderson [4, 51 and Sarpkaya [6, 71. The striations are essentially three-dimensional free-surface disturbances (which appear as ridges as shown in Figs. 1 and 2], normal to the direction of the motion of the lifting Prof. Turgut Sarpkaya and LT Peter B. Suthon, Mechanical Engineering, Code: ME, Naval Postgraduate School, Monterey, California 93943-5000 U.S.A. sol

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surface, and come into existence when the vortex pair is at a distance equal to about one initial vortex separation from the free sur- face. The scars are small free-surface depressions, comprised of many randomly distributed whirls, and come into existence towards the end of the pure striation phase, as in Fig. 2, and when the vortices are at a distance equal to about sixty percent of the initial vortex separation from the free sur- face (Fig. 31. When the vortices migrate large distances upward, the vortex pair usually undergoes both short-wavelength and long-wavelength sinusoidal instability (Fig. 4) and often breaks up into isolated rings (Fig. Al, (for additional details see, e.g., Sarpkaya t811. Fig. 3 Evolution of scars and whirls Fig. 1 Appearance of striations Fig. 2 Transition from striations to scars Sarpkaya and Henderson's [51 and Sarpkaya's t61 theoretical models of the scar cross-section created by the trailing vortices was based on the classical solution of Lamb t9l, assuming the vortices to be two-dimen- sional and the free surface to be a rigid plane. For small Froude numbers Fr (= VO/~O, where VO is the initial mutual induction velocity of the vortex pair), the vortices follow the simple path described by Lamb's potential-flow solution, the free sur- face remains fairly flat, and each scar front Fig. 4 Onset of long wave instability . fig ~ id_ _ Fig. 5 Single inclined rising ring approximately coincides with the stagnation point on the Kelvin oval, formed by one of the pair of the trailing vortices and its image, as shown by Sarpkaya and Henderson t4-71. For Froude numbers larger than about 0.15, not only the deformation of the free surface but also the nonlinear interaction 504

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between the said deformation and the motion of the vortices become significant. The vortices follow Lamb's solution only during the early stages of their rise. Subse- quently, they exhibit paths of varying degrees of complexity, depending on Fr and the Reynolds number Re (see, e.g., Fig. 6 for Fr = 0.6 and Re ~ 4000, reproduced from Leeker's thesis t1011. Here the vortex paths rise vertically upward and, instead of moving away from the center, move initially toward the center line as the vortex is drawn up into the domed area underneath the free surface (Fig. 71. Occasionally, the path of the approximate center of a vortex forms a loop as seen in Fig. 6. It is not clear whether this is due to three-dimensional instabilities along the axis of the vortex, or due to the formation of the "wall vortex", observed by Yamada and Honda [111, or due to the suc- cessive 'rebounding' of the decaying turbu- lent vortex (for rebounding in general, see, e.g., Peace and Riley [1211. Even though it was fully realized at the outset that the problem ultimately to be solved is the understanding of the three- dimensional nature of the phenomenon, the relative ease of the two dimensional coun- terpart has attracted the immediate atten- tion of experimentalists and numerical anal- ysists alike (e.g., Sarpkaya et al. [131 Dahm et al. [141, Willmarth et al. [15], Marcus [16], Marcus and Berger [ 17], Tryggvason 118], Telste [191) and Ohring and Lugt [20], just to name a few). Most of the numerical simula- tions dealt with the inviscid, two-dimen- sional interaction between a pair of counter- rotating line vortices and a free, initially pla- nar, surface. In these calculations the criti- cal time at which the numerical instability manifests itself does not correspond to the instability of the free surface or to its extremum position. The calculations of Ohring and Lugt are for a two-dimensional laminar flow at relatively small Reynolds numbers (Re = VO b O / v = 10 and 501. Recently, Marcus [211 calculated the defor- mation of a density interface for Fr = 1.125, Re = 500, and a density ratio of 5:1 using a second-order finite-difference projection method for the full variable-density (i.e., without invoking the Boussinesq approxima- tion). The resolution of fine structure in the flow is obtained through the use of higher- order Godunov methods in evaluating the nonlinear advective terms. Figure 8 shows sample vorticity and density plots (reproduced here with permission from the original color plots). Even though some insight has been gained through the use of two-dimensional r 5 Fig. 6 Sample vortex-center paths Fig. 7 Sample scar cross sections viscous- or inviscid-flow numerical models, the results are not likely to lead to the understanding of the physics of the dark narrow images. Observations and measure- ments strongly suggest that the three sos

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Fig. 8 Vorticity and density contours, (from Marcus t21] with permission) dimensionality of the phenomenon is essen- tial to the existence and longevity of the scars and striations. The three-dimensional instability of an initially parallel vortex pair has attracted great attention because of its importance in the understanding of the demise mecha- nisms of aircraft trailing vortices. Crow [221 was the first to show that both symmetric and asymmetric modes of instability will develop on the vortices due to the mutual inductance of the sinusoidally perturbed pair. Figure 9, adopted from Widnall [231, shows Crow's stability diagram for the vortex pair. The shaded areas show the stability regions. The solid curve shows the wave 1n \/b~ 4 2 o 0 0.1 0.2 0.3 0.4 0.5 0.6 rctbO Fig. 9 Stability diagram for a vortex pair lengths of maximum amplification for each rc/bo. The upper curve shows the most unstable long wave and the lower curve shows the most unstable short wave. The preferred mode of instability at the longest wavelength is 8.6bo. The wave number ,B for the short wavelength instability varies from about 4 to 17, corresponding to wavelengths of ~ = 0.37bo to 1~57bo. It is in view of the foregoing that it was decided to generate nearly two-dimensional vortex pairs at relatively large Froude num- bers, rather than inclined trailing vortices at relatively small Froude numbers, and to con- struct a numerical model of the three- dimensional surface structures. Such an effort is complicated by the difficulty and expense of quantitative measurements, im- age analysis, and high-resolution numerical simulations. Some of the experimental diff~- culties stem from the occurrence of Helmholtz instability during the roll-up of the vortex sheets, the short-wavelength and the lon~wavelength instabilities of the vor- tex pair, turbulent diffusion of the trailing vortices, and the unknown (and perhaps unknowable) physical condition of the liquid surface. Some of the numerical problems stem from the difficulty of prescribing the initial conditions, including the free-surface characteristics, the three-dimensional nature of the resulting surface disturbances, and the stochastic nature of the whirl con- stellation. EXPERIMENTAL PROCEDURES Experiments were conducted in three different water basins. The first was a long towing tank. It was used previously for the exploration of the characteristics of scars and striations resulting from the trailing vortices generated by lifting bodies (Sarpkaya and Johnson [241, Sarpkaya et al. [131) The second, a multipurpose basin, was used for the study of the scar cross section in a two-dimensional mode (Sarpkaya et al. ~ 131) either through the use of counter- rotating plates or through the use of a streamlined nozzle, mounted on top of a piston chamber. The third water basin was constructed partway through the current investigation when it was realized that the existing basins were insufficient for the type and scope of experiments desired. It was built three times wider than the second tank and allowed the generation of two- dimensional Kelvin ovals (with aspect ratios as large as 18) and the study of two counter rotating stationary circulations at prescribed depths below the free surface, through the use of two counter-rotating 506

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cylinders of identical diameter and angular velocity. The test tanks were covered with ply- wood and the water was recirculated con- tinuously prior to each experiment in order to protect the free surface from airborn con- taminants. Nevertheless, the tanks may have had a contaminated free surface with a sur- face viscosity larger than the corresponding bulk viscosity. The average value and/or the spatial distribution of the surface tension ultimately produced are not easy to quantify. The surface tension always exists at a free surface (due to intermolecular cohesive forces) regardless of whether the interface is contaminated or not by agents foreign to either fluid. Surface tension can affect the motion of the free surface partly by imposing a tangential pressure gradient as a result of spatial variations of the curvature of the air- water interface and partly by applying a tan- gential stress to the liquid beneath as a result of the variations of the surface tension along the interface, brought about by the gradients of surfactant concentration and temperature. The effects of viscosity, curva- ture, surface-velocity, and surface-tension appear to play significant roles in the cre- ation of the surface vorticity and the evolu- tion of surface signatures. Experimental data were collected using the same equipment regardless of which basin the experiments were conducted in. Hero video-recorder systems were used, one with a black-and-white video camera and the other with a color video camera. The output signal of each camera was sent to a screen date/timer before it is recorder on a video- tape. The black-and-white camera was used primarily with the shadowgraph technique (for optimum contrast and clarity of the shadows of the surface disturbances), and the color camera was used primarily for the recording of the striations, visualized through the use of various florescent dyes (introduced into the Kelvin oval at the nozzle exit, not onto the free surfacel. In addition, two still cameras were used for the record- ing of streaklines through the use of surface markers. Thermal-imaging was conducted using an infra-red camera which sent a video signal to a date/timer and then to a video- cassette recorder The video-screen images of whirls, scars, and striations were tracked through the use of a digital image processing system and the data were transferred to a computer for flow structure and motion recognition in whirl constellation {for additional details, see Leeker [ 1011. The stored data could be manipulated to generate detailed informa tion on whirl centers, number of whirls per reference area, angular velocities as function of whirl radius, whirl-migration velocity, whirl-whirl interaction (fusing and cancella- tion, etc). All of the experiments cited above were repeated at least twice to assure that the results could be reproduced within reasonable experimental errors. The exper- imental data corresponding to the earlier stages of the motion were used to prescribe the initial conditions in the numerical calcu- lations. The data at later times were used for comparison with the numerical predic- tions at the corresponding times. NUMERICAL SIMULATION Following extensive observations and measurements, the vortex dynamics or the vortex-element method (see, e.g., Sarpkaya [251) was used to simulate the phenomenon. In doing so, the most important flow fea- tures to be reproduced by this or any other model were identified and a numerical code, with very little or no sensitivity to the per- turbations in the input parameters, was developed. The model is idealized enough for simple calculation but realistic enough to be interesting. The initial separation ho of the vortex pair was chosen as the reference length; the mutual-induction velocity VO of the vortex pair as the reference velocity; bo/Vo as the reference time, and the initial strength F0 of a vortex as the reference vortex strength. The equations describing the motion were non-dimensionalized through the use of the characteristic parameters ho, VO, bo/Vo' and IN where VO = F0 /~2~bol. The numerical model simulates the mutual interaction of the whirls on the free surface, beginning with their creation, through the use of the vortex-element method. The results of the previous investi- Cations (Sarpkaya et al. t61), and the experi- mental results obtained in this investigation helped to define the parameters necessary for the construction of the numerical model. As shown in Fig. l 0, the position of the inboard edge of the scar band is denoted by s, the width of the scar band by w, the aver- age spacing of whirls in the longitudinal direction (a measure of whirl population density per unit length) by c, and the hori- zontal component of the convection velocity of one of the vortices in the original vortex pair by Vb. It has been shown previously that the scars are slaved to and transported by the vortices [4-71. 507

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low = (Rnd3Fm)(-l)Rnd4 (4) v,, v,, so Fig. 10 Schematic of scars and striations A right-handed coordinate system has been defined in the complex plane where the real x-axis is parallel to the scars and the y-axis is perpendicular to the scar bands. Ideally, the length of the scar band should extend from -so to +~. However, this is not numerically possible. Instead, a finite scar length L, defined by L= c iNw] and extend- ing from (-L/2, y) to (L/2, y), is used. Nw is the number of whirls in each scar band. The whirl strength.is given by Ew = 2xrCVt; = 2~r2o ~ 1 ~ where rc is the core radius and the tangen- tial velocity at the core boundary is Vt = rc m. The normalized, time- averaged, angular velocity ce of each whirl core, for a large number of whirls, was determined from the image analysis and was found to vary from about 0.6 to 1.2, the majority of the data falling closer to unity. The core radius was then calculated from rc = where now ce is assigned approximately the size- and time-averaged value of unity. The numerical analysis began by ran- domly placing the whirls into two parallel scar bands, z = [RndiL- 24+iRnd2 w (3) where z is the position of the whirl center and Rnd1 and Rnd2 are random numbers (independently seeded) with a uniform dis- tribution from 0.0 to 1.0, inclusive. Each whirl is also assigned a vortex strength in a random manner, where Ew is the strength of a whirl, I-m is the mean absolute value of the strengths of all whirls, Ends is a random number from a standard normal distribution (with a mean of Em and a standard deviation one-quarter the expected range of the whirl strengths), and Rn d 4 is a random integer from 1 to 10, inclusive. It has been customary to amalgamate two or more vortices into a single vortex, placed at their center of vorticity, (see, e.g., Sarpkaya [251), when their cores touch, or are less than a prescribed critical distance , or overlap by a prescribed amount. Like- signed whirls merge into a larger whirl. Oppositely signed whirls are merged into a smaller whirl, to mimic the cancellation of oppositely signed vorticity (thought to be the major mechanism of enhanced energy dissi- pation in turbulent flow). If the resulting whirl strength is below a prescribed mini- mum (ymin), the whirl is removed from the scar. The amalgamation process does not conserve total vorticity nor is the linear or angular momentum conserved due to the merging of oppositely-signed whirls and the removal of weak whirls. Nevertheless, the conservation of these quantities is not con- sidered important for a number of reasons. First, the purpose of the simulation is not an exact treatment of the viscous diffusion although the removal of very small whirls do in fact accomplish this purpose indirectly. Second, the random nature of the distribu- tion has far greater effect on the mutual interaction of the whirls than the occasional amalgamation of the whirls. Third, the cal- culations with different ranges of the funda- mental parameters, such as the whirl den- sity, minimum survival strength (7min), 2 ~amalgamation distance, etc., have shown that the fundamental nature of the randomly dis tributed vortices within a narrow band is to self-limit the amalgamation process and to reduce the effect of the imposed perturba tions (for sensitivity analysis) on the result ing characteristics of the scar band. For an incremental time step At, the mutual induction velocity of each whirl is calculated using a cutoff scheme similar to that first proposed by Rosenhead [261 Az=u-iv=~21~ z ~(~z-z,,~'+i32) s08

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The cutoff parameter ~ was taken to be pro- portional to the sum of the core sizes on the basis of past experience [251. Each whirl is moved to a new position, after the calcula- tion of the induction velocity, using t + At Zt + he + in + Vb:~At (6) The whirl-energy density or the specific energy in the system is estimated through the use of E ~ Abut Ok ' ~ skulk (7) The diffusion of vorticity due to viscosity is introduced artificially by reducing the strength of each whirl either through the use of a Gaussian vorticity distribution or through the use of a simple percentage, (e.g., 0.5 percent per time step), in a man- ner similar to that done previously by others (for details see, e.g., Sarpkaya [2511. The artificial reduction of circulation is justified on the grounds that it accounts for the three-dimensional deformation of vortex fil- aments, for the cross-diffusion of oppositely- signed vorticity, and for the observed fact that the strength of the vortices continues to decrease with time or downstream distance. Numerical experiments did not show a sig- nificant dependence on the type and magni-; tude of the circulation reduction (0.5 % to 1% per time step). A few additional features of the model need to be explained. These concern the creation of the whirls, sensitivity analysis, and the selection of the numerical parame- ters (c, so, w, Fm. 6, c, Nw, At, and Amino Whirls were created at a particular time in the initial stages of the scar formation in accordance with the observed characteris- tics of the whirl population. The sensitivity of the predictions of the later stages of the scar motion to the parameters involved in the simulation was examined with great care. Accordingly, every significant parame- ter involved in the calculations was varied within reasonable limits. For example, c was varied by 100 percent and w was varied from zero to 0.4. The other parameters, such as Em, 6. , Nw, At, Amine were varied by at least 100 percent.. The initial inboard posi- tion of the scar front so was kept at the experimentally-observed value of 1.6 at the corresponding time and was not varied fur- ther since it naturally increased as a function of time and since it influenced only indi- rectly the mutual interaction of the two scar strips. Several runs were made with identi cal sets of parameters, changing only the seeding of the random numbers involved, in order to ascertain that the insensitivity of the results and basic conclusions to the vari- ation of the parameters selected was not a consequence of the use of a particular set of random numbers. In fact, the randomization was performed by the computer using the system clock time at the commencement of each run. Several plots were created at regular intervals during the execution of the numerical simulation, carried out to times corresponding to the disappearance of the surface structures, in order to compare the predicted results with those obtained exper- imentally. DISCUSSION OF RESULTS Physical Experiments A careful frame-by-frame analysis of the video recordings of literally hundreds of test runs has shown that the sloping of the vor- tex pair relative to the free surface or the generation of vortices by a lifting surface is not necessary for the creation of scars and striations. Nearly identical surface signa- tures can be produced through the use of an initially horizontal vortex pair. Figure 11 and many others like it have shown clearly that the three-dimensional instability lead- ing to striations do not originate at the free Fig. 11 Inception of short wavelength in stability on a Kelvin oval below the free surface son.

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surface but rather on the Kelvin oval, long before the latter 'sees' the free surface. Sub- sequently, the free surface interacts with and is modified by the instability brought to it by the Kelvin oval (see Figs. 12-15), i.e., the striations (nearly uniform corrugations, at least at their inception} are a manifesta- tion of the short wavelength instability of the vortex pair. The flow features on the scar front (Figs. 13 -15) are clearly identifiable and show the creation of whirls. Figure 16 shows the whirls and the whirl pairs (Kelvin-oval like structures at the free sur- facel, resulting from a pair of trailing vor- tices. The only contribution of the sloping of Fig. 12 Rise of corrugated vortex dome the trailing vortices is the divergence of the scar lines (the V-shaped footprints) and the fact that the striations come into existence sequentially rather than simultaneously. As noted previously, the wave number ~ for the short wavelength instability varies from about 4 to 17, corresponding to wave- lengths of ~ = 0.37bo to 1 .57bo . The wave- length ~ of the striations has been deduced Fig. 14 Intensification of scars Fig. 13 Formation of scars and whirls Fig. 15 Later stages of scar formation silo

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Fig. 16 Whirl formation in scars from numerous runs as a function of the ini- tial spacing of the main vortex pair. The his- tograms of the striation wavelength have shown that \/bo varies from about 0.75 to 1.25, depending primarily on the initial roll- up of the vortex sheets as dictated by the Froude number. As far as a vortex pair in an infinite homogeneous medium is concerned, the striational instability corresponds to the short wavelength instability (see Fig. 91. The interaction of a 'corrugated' Kelvin oval with the free surface under the influ- ence of gravitational, centrifugal, and surface tension effects is anything but simple. The surface acquires space and time dependent curvature and velocity and produces vorticity even in the absence of surface tension or surface contamination. This vorticity is Ace = 2qK, twice the tangential surface velocity times the surface curvature [271. The sur- face velocities are induced partly by the motion of scars and striations and partly by the vortex pair. The surface vorticity is largest at the scar-striation intersection because this is where the curvature and the surface velocities are largest. Consequently, the scars are the regions where the surface vorticity is most likely to be converted into whirls with vertical axes. Thus, what begins as a short wavelength instability quickly degenerates into a far more complex three- dimensional free-surface phenomenon (Figs. 14-15) with its own source and mechanism of vorticity generation. The unsteady motion of the striations (particularly those of the ends where strongest surface-tension con- centration and whipping action occurs) Fig. 17 An additional sample of Kelvin oval free-surface interaction Fig. 18 Vortex dome for Fr = 1.125 Fig. 19 Rapid increase of the striations

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provide the stroke necessary to concentrate the surface vorticity into whirls. The hypothesis of vorticity transport to the free surface from the main vortex pair is not necessary simply because the free surface acquires its own curvature and surface veloc- ity even if it were initially flat and free of vorticity. Figures 17 through 19 show addi- tional samples of the evolution of Kelvin oval, striations, and scars at a Froude number close to unity. Experiments with rotating cylinders have shown that even at fairly low angular velocities and at fairly large depths of immersion (as much as 1.5D below the free surfacel, the rotation of the cylinders give rise to striations and whirls. Figure 20 shows the evolution of a whirl and its neigh- borhood during a time interval of 0.25 sec- onds. It is clear that neither a trailing vor- tex pair nor a nearly two-dimensional Kelvin oval, neither a contaminated free surface nor a Reynolds ridge is necessary to create scars and striations. The mere presence of two counter-rotating circulations near a free surface (with curvature and velocity] is suf- ficient to create all the phenomena observed in other scar experiments. There is, obvi- ously, much more to be explored. The only purpose of reference to rotating-cylinder experiments at this time is to point out that conjectures regarding surface contamination and Reynolds ridge are not necessary. There are, obviously, some fundamental dif- ferences as well as strong similarities between the classical Taylor instability, the striational instability, resulting from the counter-rotation of two cylinder, and the three-dimensional instability observed in the actual vortex experiments. These will be explored in some detail in future studies. The sequence of events emerging from the experimental observations and mea- surements may be summarized as follows: Uorten pair and Keluin offal; Short rueuelength instability; Interaction with the free surface; Formation of striations, feeding to: 3-D curvature, Surface velocity, and; Dividing and painug of striations; Surface uorticity, mostly near the seers; Grauitational' centrifugal and surface tension effects; Formation of whirls, whirl pairing, and cascading of uorticity; Self-limiting amalgamation process; As the Kelvin oval rises, the spiralling vortex sheets in each vortex undergo Helmholtz and, subsequently, Rayleigh instability and, eventually, degenerate into turbulent motion. The distance travelled by the oval more or less determines the sequence of events. In the present experi- ments the said distance to the free surface has been kept below 6bo in order to obtain a clearly defined oval near the free surface. The vorticity is initially confined to the Kelvin oval and the motion outside it is irro- tational. With the passage of time, the vor- ticity diffuses over a wider area and some vorticity gets annihilated in the overlapping regions of oppositely-signed vorticity. The distance between the vortices during the initial period of rise of the Kelvin oval remains fairly constant but the core radius and, hence, rc/bo increases due to diffusion. As the short wavelength instability begins to grow, the initial value of \/bo in Figure 9 is in the order of unity. Had the cores touched, \/bo would have been as large as 2.45 (point B in Figure 91. However, the vortex cores do not grow to such large sizes, not at least during the short period of migration of the Kelvin oval toward the free surface. Once the vortices begin to diverge, - - - - Fig 20 Evolution of a single whirl in 025 s. Dissipation of Whirls. si2

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As in the case of parallel line vortices approaching a rigid plane, the distance between them increases. This decreases the mutual interaction of the vortices and the relative wavelength \/bo decreases. It is not. therefore, surprising to see that the stria- tions multiply quickly and concentrate in the regions directly above the vortices and adjacent to the scars (see Figs. 13-151. The dividing and pairing of the striations are accompanied by the stroking and meander- ing of the ends (a fishtail-like motion) of the striations. It is this action that is thought to be responsible for the reconstitution of the surface vorticity into numerous, randomly- sized, randomly-shaped, and randomly-dis- tributed whirls. A number of whirls in close proximity to each other may give rise to a number of complex interactions, such as, amalgama- tion, annihilation (at least partially), or pairing for a brief period and then re-sepa- ration. Thus, it is clear than the interaction of three-dimensional whirls, penetrating only a short distance into the fluid, is not a simple matter and needs further study through the inclusion of viscous effects. The amalgamation as well as annihilation is par- ticularly strong during the formation period of the scars. The amalgamation amounts to cascading of circulation into larger vortices and hence to their longer life-span. It is this process and the generation of surface vortic- ity that are thought to be responsible for the longevity of the SAR images. Evidently, some sort of regeneration process is neces- sary without violating the principles of con- servation of energy and circulation. One could interpret this whirl-growth phe- nomenon as an indication of an inverted energy cascade. However, the total internal energy decreases in spite of the growth in size of a number of whirls. This completes the observed as well as perceived evolution of the surface signatures. It will be interest- ing to discover as to how the scar band, comprised of whirls, tend to increase the radar return in order to provide bright lines in the SAR images. Numerical Experiments Figures 21a through 21c show at So/bo = 1.6 (where time is taken to be ~ = 0 I, the position of the whirls, the streamlines (with respect to a coordinate system moving with the scars), and the streamlines (with respect to a fixed coordinate system). The size of the whirls is drawn proportional to the square root of their strength. A solid circle denotes clockwise circulation and a hollow circle indicates counter-clockwise circulation. Figures 22a through 22c show ~ o~oO. ~ Oo04 to <~ - .-9 .~ 8 ~ ~o~9c 30~0~30~ ~.o'~~0~^a4 Fig. 21a Initial whirl distribution on scars Fig. 21b Streamlines at ~ = 0 (comoving with the scars) Fig. 21c Streamlines at ~ = 0 (relative to a fixed coordinate system} the corresponding plots at ~ = 1.6. Thus, it is seen that the whirls amalgamate as time increases. Figures 21c and 22c show that 5~3

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the wide region between the scar bands is fairly calm whereas the scar bands are a con- stellation of depressions created by the whirls. Additional facts emerging from these figures are as follows: The width of the scar band increases naturally partly because of the transport velocity imposed on them by the main vortex pair and partly because of the mutual-induction of the whirls. It is rather remarkable that the band essentially retains its identity (eventually, a long wave instability is seen to develop). What is more remarkable is the fact that the calculated mean scar-separation, for the example shown here, is within 3 percent of that measured at the corresponding times. O me; 0 O C a; O ~ , O ; O O ~ O O .. c 0 c of ,OO. O O .o. O LIMO ..e Fig. 22a Predicted whirl distribution on scars at I= 1.6 Figure 23 shows a sample plot of the path lines of a number of particles initially situ- ated along a line on the scar. Typical whirl strength distribution is shown in Fig. 24. Clearly, the total strength of the whirl system decreases with time. However, the amalgamation process does lead to a set of larger whirls. The increase of the number of large structures slows down or stops after a short time period. In other words, the whirl system reaches an equilibrium. Fig. 22c Predicted streamlines at ~ = 1.6 (relative to a fixed coordinate system) Fig. 22b Predicted Streamlines at ~ = 1.6 (comoving with the scars) Fig. 23 Marker path lines on scars at I= 2 514

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So! 80- Ul O 70 00 > 60 0 40 1~ 30- ID ~ ~- Z 10 ~0 In 80 U 70 o lo 30 a Z 10 ~4 1.2 jig ~05 flu ~ instability as seen in Fig. 22a. At no time did the scar fronts exhibit a V-shaped diver- gence as observed in the SAR images. This was entirely expected because it is the con- tinuous creation and upwash of the trailing vortices that give rise to the V-shaped scar bands with an included angle of 2a Where a is arctantVO/U)~. It was deemed necessary to apply the numerical model to the trailing vortex case with a few minor differences, partly to further substantiate the predictions of the model and partly to observe the simi- larities or differences between the mea- surements and calculations. Figure 25 Fig. 24 Whirl strength distribution Clearly, the whirl-whirl interaction should contribute most significantly to the evolution of the initial distribution, and such events should occur only if the number of whirls is high enough. In the present calcu- lations the whirl population has been dou- bled several times to explore this very ques- tion. Furthermore, numerical experiments were carried out with different random- number seedings to ascertain that the results concerning the energy-density dis- tribution and the cascading of the energy did not depend on either the number of the whirls or on their statistical distribution. It has been found that the population density and the number of random samplings are sufficiently large to arrive at statistically meaningful conclusions. The time variations of the distributions, therefore, allow one to estimate whether whirl-whirl interactions are important. The results presented above show that the shift in the size distribution toward larger structures and the concentra- tion of energy in these structures are an important ingredient of the scar formation and scar life-span. The numerically-simulated scars were intentionally started parallel, with no further intrusion into their evolution in succeeding steps. As described above, the scars evolved, remaining essentially parallel, with the superposition of a long wavelength sinusoidal \\\\~\\\\\\\\\\\\\\\\\~\~\~ Fig. 25 V-shaped scar streamlines at I= 0 Removing with the scars) shows the instantaneous streamlines at ~ = 0, as they would be seen by an observer moving with the scars, and Fig. 26 shows at ~ = 0-1.2, the instantaneous streamlines as they would be seen by an observer fixed to the coordinate system. Finally, Figures 27a and 27b show a comparison of the experi- mental and numerical results. One half of these figures represent the numerical results and the other half the experimental results. A comparison of the two halves show the remarkable similarity between the observations and predictions at the corre- sponding times. V. CONCLUSIONS The numerical and experimental results presented herein warranted the following con- clusions: 1. The sloping of the vortex pair (as in the case of trailing vortices) is not necessary to produce surface signatures or footprints of vortex wakes in the form of scars and stri s~s

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Fig. 26 Growth of a V-shaped scar in time and space 516 ations. An initially two-dimensional vortex pair yields similar structures. 2. A fully submerged vortex pair is sub- ject to both long- and short-wavelength instabili- ties. The latter is particularly prevalent when the former is suppressed. The experiments have revealed their existence and the role played by them in the generation of striations. 3. The short-wavelength instability has a wavelength in the order of unity, compared with the initial spacing of the main vortex pair. The wavelength decreases as the vortex spacing increases. O O. o. 0. . d. ' ~ . me ~ 0 O ~ ~ O me Too . ~ ~ O O. ~ . _0. - ~ O .e c c, I, .o~o .0~; ~ 0 ~ 08; ,O ~0 o ~ me ~ 0, Cue o Fig. 27a Comparison of measured and pre- dicted V-shaped scars (No. 1)

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o o. ego ~ - oou. -a - b;O _ . O a_ I_. O o ~o.o to .o.~; ~ ooO. 0 ~ OB ; 0~ . O; e 0 'a-o e ~ ~ O ~ ~ do ~ 0 .oO 04e o c Jo C,O.O ~ 0 Cot. Oc-~; . 0 _ ~C> Fig. 27b Comparison of measured and pre- dicted V-shaped scars (No 2) 4. The short wavelength instability leads to striations, surface curvature, surface veloc- ity, and surface vorticity. The strongest sur- face-tension concentration occurs near the tips of the striations. Consequently, these are the regions where the new whirls are created through the use of the surface vorticity. This type of whirl formation appears to be unique to the scar generation. 5. The scar and striation generation may be affected by the degree of surface contamina- tion. However, the changes in surface tension and surface elasticity brought about by contam- ination, beyond that due to intermolecular cohesive forces, are neither necessary nor suff;- cient for the creation of striations and scars. The most indispensable ingredients for the gen- eration of whirls are the viscosity of fluid, gen- eration of surface vorticity due to surface curva- ture, and the unstable motion of the ends of whirls. Also, it is equally important to note that the so-called Reynolds ridge is akin to but not related to the scar formation. The Reynolds ridges do not give rise to whirls. 6. The numerical simulation of the phe- nomenon through the use of vortex dynamics or vortex element method has shown that all of the fundamental characteristics of the scar evolu- tion (e.g., the preservation of scar-band width, whirl distribution, whirl-whirl interaction, energy cascading, mutual annihilation of whirls, self-limiting amalgamation, etc.) are faithfully reproduced. 7. Among the numerous mechanisms proposed to explain the physics of the SAR images, the hypothesis of the interaction of a vortex pair with the free surface emerges as the most viable one in view of the observations and measurements made and the conclusions arrived at in this investigation. 8. The above conclusions are further indicated by the experiments carried out with two rotating cylinders. The analogy between the two types of circulatory motions need to be further explored, partly because such experi- ments will provide an excellent opportunity to discover many new and interesting vortex pat- terns and partly because they will lead to the closer examination of the similarities between the striations, whirls, and scars generated by the two types of circulations. ACKNOVVLEDGMENTS The authors wish to express their sin- cere appreciation to the Office of Naval Research and the Naval Postgraduate School for the support of the investigation. The project was monitored by Dr. Edwin P. Rood. REFERENCES 1. Langmuir, I., "Surface Motion of Water Induced by Wind," Science, Vol. 87, 1938, pp. 1 19-123. 2. Scott, J. C., "Flow Beneath a Stagnant Film on Water, The Reynolds Ridge,"Journal of Fluid Mechanics, Vol. 1 16, 1982, pp. 283- 296. 3. Furey, R. J., "Hydrodynamic Stability and Vorticity in a Ship-Model Wake," Research and Development Report No. DTRC-90/005, 1990, David Taylor Research Center, Bethesda, Maryland. 517

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4. Sarpkaya, T. and Henderson, D. O., "Surface Disturbances due to Trailing Vor- tices," Naval Postgraduate School Technical Report No. NPS-69-84-004, 1984, Monterey, CA. 5. Sarpkaya, T., and Henderson, D., "Surface Scars and Striations," AlAA Paper No. 85-0445, 1985. 6. Sarpkaya, T., 1985, "Surface Signatures of Trailing Vortices and Large Scale Instabil- ities," Proceedings of the Colloquium on Vortex Breakdown (Sonderforschungs- bereich 25), Universitr of Aachen, 1985, pp. 145- 187. 7. Sarpkaya, T., "Trailing-Vortex Wakes on the Free Surface," Proceedings of the 16th Symposium on Naval Hydrodynamics, National Academy Press, 1986, pp. 38-50. 8. Sarpkaya, T., "Trailing Vortices in Homogeneous and Density Stratified Media," Journal of Fluid Mechanics, Vol. 136, 1983, pp. 85-109. 9. Lamb, H. (Sir), Hydrodynamics, Dover Publications, (6th ed.), 1945, pp. 221-224. 10. Leeker, R. E., Jr., "Free Surface Scars due to a Vortex Pair," M.S. Thesis, Naval Postgraduate School, Monterey, CA., March 1988. 11. Yamada, H. and Honda, Y., "Wall Vortex Induced by and Moving with a Confined Vor- tex Pair," Physics of Fluids A, Vol. 1, No. 7, 1989, pp. 1280-1282 12. Peace, A. J. and Riley, N., "A Viscous Vortex Pair in Ground Effect," Journal of Fluid Mechanics., Vol. 129, 1983, pp. 409- 426. 13. Sarpkaya, T., Elnitsky, J., and Leeker, R. E., 1988, "Wake of a Vortex Pair on the Free Surface," Proc. Seventeenth Symposium on Naval Hydrodynamics, National Academy Press, Washington, D. C., pp. 53-60. 14. Dahm, W. J. A., Scheil, C. M., Tryggvason, G., "Dynamics of Vortex Inter- action with a Density Interface," Report No. MSM-8707646-88-0 1, The Univ. of Michigan. 15. Willmarth, W. W., Trygg~ason, G., Hirsa, A., and Yu, D., "Vortex Pair Generation and Interaction with a Free Surface," Physics of Fluids, Vol. Al, 1989, pp. 170-172. 16. Marcus, D. L., "The Interaction Between a Pair of Counter-Rotating Vortices and a Free Boundary," Ph. D. thesis, UC Berkeley, 1988. 17. Marcus, D. L., and Berger, S. A., 1989, "The Interaction Between a Counter-Rotat- ing Vortex Pair in Vertical Ascent and a Free Surface," Physics of Fluids, A-1, Vol. 12, pp. 1 988-2000. 18. Tryggvason, G., "Deformation of a Free Surface as a Result of Vortical Flows," Physics of Fluids, Vol. 31, 1988, pp. 955-957. 19. Telste, J. G., "Potential Flow about Two Counter-Rotating Vortices Approaching a Free Surface," Journal of Fluid Mechanics, Vol. 201, pp. 259-278, 1989. 20. Ohring, S., and Lugt, H. J., 1989, Two Counter-Rotating Vortices Approaching a Free Surface in a Viscous Fluid, Research and Development Report No. DTRC-89/013, David Taylor Research Center, Bethesda, Maryland. 21. Marcus, D. L., (Private communication), March 1990. 22. Crow, S. C., "Stability Theory for a Pair of Trailing Vortices," AIAA Journal, Vol. 8, No. 12, 1970, pp.2172-2179. 23. Widnall, S. E., "The Structure and Dynamics of Vortex Filaments," Ann. Rev. Fluid Mech., Vol.7, 1975, pp.141- 165. 24. Sarpkaya, T., and Johnson, S., K., "Trailing Vortices in Stratified Fluids," Technical Report No. NPS-69-82-003, 1982, Naval Postgraduate School, Monterey, California. 25. Sarpkaya, T., "Computational Methods with Vortices 1988 Freeman Scholar Lec- ture," Journal of Fluids Engineering, Trans- actions of ASME, Vol. 111, No. 1, 1989, pp. 5-52. 26. Rosenhead, L., "The Spread of Vorticity in the Wake Behind a Cylinder," Proc. Roy. Soc., Ser. A., Vol.127,1930, pp.590-612. 27. Batchelor, G. K., An Introduction to Fluid Dynamics, Camb. Univ. Press, 1967. sl8

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DISCUSSION Owen M. Griffin Naval Research Laboratory, USA In your introductory remarks you noted that the surface disturbances and effects are caused by shed vortices which accompany the passage of a submerged body. There is equal, if not more, interest in the wake of a surface ship in terms of the persistent synthetic aperture radar (SAR) signature produced by the ship's passage. Can you comment upon the possible importance of the vortex-free surface interactions discussed in your paper as they might apply to the remotely sensed surface ship wake? AUTHORS' REPLY As I have noted in the Introduction of the written version of the paper, "Just a narrow patch of darkness, bounded by two bright lines, provides the impetus for this investigation partly because it is seen in the synthetic aperture radar (SAR) images of a ship's wake, partly because it extends many miles directly in the ship's track, ..... Dr. Griffin is of course correct in reinforcing this fact. My introductory remarks at the oral presentation of the paper were confined to ascending heterostrophic vortices, generated by submerged bodies, primarily because of the mechanism with which the vortices were created in our experiments. I should have pointed out that a well- known example of such a SAR image is that of the wake of the surface ship USS Quapaw. As far as the possible importance of the vortex/free-surface interactions discussed in the paper to the understanding of the mechanisms leading to the SAR images of surface ship wakes is concerned, the current state of the understanding of either phenomena does not allow one to explain the physics of what relationship could scars and striations have with the SAR images. Among the many proposals made, one that appeals this writer most is the interaction of the vertical fluid motions generated by the boundary layers and propellers of the ship with the free surface. However, such an interaction is not as simple and as relatively clean as that of ascending vortices because of intense turbulence (patches and parcels of vorticity of many scales and intensities) and air-water mixture accompanying the ship's wake. The development of a U-shaped vortex wrapping around the outside of the main vortex core (discussed in 1985 in author's Refl6]), restructuring of vorticity in the wake, the reverse energy cascading, and the self-limiting growth of the surface whirls (all discussed in the present paper) may go long ways towards establishing a relationship between the scars and striations generated by ascending heterostrophic vortices and the SAR images of ship wakes. It is also possible that the real fall-out benefits of the investigation will be in yet-unthought-of areas of experimental and computational fluid dynamics. The investigation has just begun. 519

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