Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Submerged Voriex Pair Influence on Ambient Free Surface Waves S. Fish (David Taylor Research Center, USA) C. van Kerczek (University of Maryland, USA) ABSTRACT This paper examines how free surface gravity waves travelling over a fluid are modified by the presence of a submerged pair of vortex singularities. Greater attention has recently been focused on the problem of vortex flows near a free surface due to its possible importance ill determining the influence of vertical ship wakes on radar intakes of the ocean surface. Although much research has concentrated on studying vortices near a still free surface, a theory for vertical influence on ambient free surface waves has not been addressed. Initial steps in uncovering the subtleties of the ambient wave modification by submerged vortices are taken utilizing first order free surface boundary conditions to demonstrate the importance of the vortex-induced surface current in the interaction of vortices with ambient waves. A parametric study is then conducted showing that the relative strength of the vortices and their geometric configurations are more important in characterizing the forth of surface modification than the relative phase or steepness of the ambient waves. And finally, analysis of simulation data is performed indicating how a vortex pair ship wake model can contribute to the observed dark centerline wakes in radar images of ships at sea. INTRODUCTION Research interest in the behavior of vortex structures near a free surface has recently been revived by the detection of dark regions in radar images of the sea surface. The significance of these long narrow dark streaks is their coincidence with the centerline wake regions of ships travelling on the sea surface at the time of radar exposure. For this reason, they are generally referred to as dark centerlines. Examples and details on the radar operation and image processing can be found in Lyden, et all. and Peltzer et al . Although the cause of the dark centerlines is not well understood, there is widespread agreement that the lower energy radar return in this region is a result of the elimination of surface waves 491 normally responsible for constructive interference of the reflected radar signal. Constructive interference occurs when the component of the surface wavenumber vector in the "look" direction of the radar satisfies the following expression (Wrights: kw = 2 kr sin(~) (1) where: kW = 27~/\w = surface wavenumber k:r = 2~/\r = radar wavenumber = wavelength = vertical incidence angle of the radar (normal to the free surface: 0=0°) Waves satisfying this criterion, termed Bragg scattering waves, are typically wind generated, and form the background signal return level in SAR images. Dark centerlines in the radar images of moving ships at sea suggest the reduction or elimination of these Bragg waves in the ship wake. The elimination of these Bragg waves and the long term maintenance of a Bragg-free region in the wake is the subject of intense research in several hydrodynamic fields, from surfactant distribution to nonlinear ship wave generation. It is likely that many of these hydrodynamic processes are combining to produce the dark centerlines and that their relative importance depends on the ambient environmental conditions. The process being studied here is the effect of the large scale rotations generated in the ship wake on the ambient waves present in the background of the radar image. These longitudinal vortex-like notions in the ship wake have been measured (Lindenmuth4) and calculated (Griffin et al.5) for a variety of ships and are illustrated in figure 1. The origin of this vorticity is the overall boundary layer of the ship and occurs in slight variations regardless of propulsion configuration. The weaker persistence of these vortices in the wake Cabot, by itself, account for the length of the dark centerline but may contribute to both the initial Bragg wave elimination and help sustain the other processes (such as surfactant redistribution) in the far wake. The presence of ambient waves in the real ocean

environment therefore indicates the need to determine how waves may be modified by the wake itself. This work will concentrate on wave modification by the large scale wake vorticity just described. hi VORTICES Fig. 1 Ship Wake Vortex Model with Ambient Waves Prior studies of vortex-free surface interactions focused on the dynamics of an initially flat free surface under the influence of the vortex pair. This problem is relevant only under the limited conditions of a glassy smooth sea surface. The more common wind-generated ambient wave background is of concern here. In this case, the important phenomenon is the modification of the ambient Bragg scattering waves by vortices of relatively weak strength. The term "weak" here refers to circulation strength and speed of the vortex relative to the gravitational forces on the free surface. Weak vortices are representative of the large scale flow measurements in the far wake, and result in minimal vertical surface deflections, and no wave breaking. It will be shown that a first order formulation of the free surface boundary condition retains the dominant nonlinear surface characteristics without the additional computational difficulties associated with nonlinear surface modelling. Linearization of the free surface also allows simulation of simple monochromatic ambient waves. The alteration of these simple waveforms by submerged vortices can therefore be easily identified both visually and using spectral analysis. Because Bragg waves for typical radar wavelengths are much shorter than the depth of the modelled wake vortices, their inviscid influence on the vortex paths is negligible and will be ignored in this study. The resulting paths of the vortex pair correspond to the paths for a flat, rigid free surface. Simulations of the two dimensional vortex pair near a free surface with monochromatic ambient waves are used here to determine which parameters of the governing equations are dominant. The Froude number and initial vortex positions are shown to play the most important roles in modifying ambient waves. Other parameters, such as the ambient wave steepness and relative phase that result from the analytic study as independent quantities, play secondary roles, as might be expected from linear theory. In addition, spectral analysis is used in a form which extracts characteristics in the surface similar to the radar imaging process. This analysis shows that submerged vortices can give images with darker center regions where the ambient waves undergo the modification. This analysis will be performed on a vortex pair with parameters derived from typical ship operating characteristics. PROBLEM FORMULATION AND SOLUTION METHOD In developing a solution technique for the vortex/ambient wave interaction problem, one would like to start with the simplest model possible while retaining the most significant flow interaction terms. The model used here follows the classical assumptions of inviscid, incompressible flow without surface tension. In addition, the three dimensional ship wake will be considered as a quasi-two dimensional (2-D) flow. See figure 2. This modelling hypothesis assumes that velocity gradients in the axial direction are sufficiently small to permit 2-D flow realizations in a plane perpendicular to the ship track to be integrated in time to represent the three dimensional problem. The constant ship speed may then be used to transform time data into axial distance bellied the ship. 9 2A 1 ? ANAL waves D Nisi ~ . ~ J r Fig. 2 2-Di~nensional Flow Model Because simple sinusoidal ambient waves are desired for the interaction problem, a linearized approach is formulated with the small perturbation parameter, c, defined as follows: is= r where D is vortex depth, and g the gravitational acceleration. Although ~ represents the traditional 492 (2)

form of Froude number governing the flow when no ambient waves are present, the use of "Froude number" in this paper will refer to another parameter, defined later, which the reader will find more intuitive for cases including ambient waves. Nevertheless, the value of £ will be kept small since this is representative of the "weak" large scale vertical motion in the ship wake. Following a standard perturbation theory and using a zeroth order solution of the vortex near a rigid surface (corresponding to ~ approaching 0), one can derive the following boundary conditions for the free surface. (See Fish or Fish and von Kerczek7 for details.) Dynamic: Kinematic: where: ~ '+°-StP X + Tl =0 (3) -(I y+ll '+¢oX11 X-ll¢Oyy=o (4) =40~f 1 =~0+~q 1 It should be noted here that these equations include the first order Taylor series expansion of each term about its evaluation on the still water level (corresponding to the rigid surface boundary). Note also that the O.S¢x term in tl~e dynamic boundary condition and the Foxed term in the kinematic boundary condition represent the influence of induced surface currents on the modification of ambient waves. In particular, the t0~q,: term provides a direct coupling of the vortex induced surface current and the ambient wave slope. These terms are excluded in the classical free surface linearizations utilizing the quiescent zeroth order base state. For the problem here, the ambient waves are propagated from the side of the 2-D domain as shown in figure 2. These waves represent ambient waves moving transverse to the ship track. Because of Me quasi-2-D nature of the ship wake model, modelling waves propagating in directions much different from this lateral condition should not be assumed practical. The solution method used to march through the 2D computations utilizes the widely used Boundary Integral Technique (BIT) of Trygvasson8, Telste9 and others. The BIT utilizes the Cauchy Integral Theorem to describe potential flows such as this one by the values of the potential and stream function on the boundaries of the fluid domain. r ~ dz+iaO~`zO t,=0 USE where: - = velocity potential = stream function a0 = exterior angle between boundary tangents on either side of zO The boundary is discretized for numerical integration assuming a continuous linear variation in potential and stream function over each element. In the present study, the boundaries are composed of the free surface, two vertical side walls and a flat bottom. At each time step Cauchy's Integral Theorem is used to determine the equilibrium unknown {i or yri on a boundary element in which the corresponding Hi or (i is prescribed. This is achieved by solving the set of simultaneous equations resulting from the application of equation 5 to the node point, ZOi, of each element of the boundary. The prescribed values of by and ~ for He next time step are determined as follows. First, the (i distribution over the free surface elements is obtained by integrating the dynamic free surface boundary condition. ((t+At) = t(~+~°.Stox+loxt 1 x+ ~ ~ Hi on the free surface is also updated by integrating the kinematic boundary condition. 11~+~) =11~+~§ y-~0xTl X+llto'y) (7) Then the new distribution of Wall on the side and bottom boundaries is prescribed by the undisturbed deep water ambient wave stream function. Wall+ = gaff) Cos(eXt+At)-k~i) (8) where: k = 27~/\ ~ = ~ A comparison of calculation methods for the surface disturbance caused by a single vortex moving at constant speed below the free surface is given in figure 3. One can see that the method used here compares very well with the fully nonlinear calculations of Telste9 while calculations utilizing the classical zero flow base state for linearization are in error. In determining the relative importance of the additional terms in equations 3 and 4 with respect to the classical linearized form, equivalent simulations were conducted with each term deleted. It was found in this study that the terms containing ¢0~`, representing the induced surface current from the vortices were significant contributors to the solution. The 1140',~, value remained small. In addition, the cross term +0x¢1~ provides a direct interaction term between the ambient waves (whose potential will be contained in ~ 1 and the submerged vortices). Without this term calculations were almost equivalent 493

to the superposition of independent calculations of the vortices and waves. Figures 4a and 4b show the influence of the induced current terms on the ambient wave modification. This figure shows free surface profiles at discrete time intervals corresponding to the period of the ambient wave, T. which is propagating frown left to right. A counter rotating pair is positioned in the center region as shown in figure 2. The details of the flow parameters will be described further in the next section. 3 ~ -- Clossicol Uneorbed --- Alternate Uneorized Nonlineor ~elate) -2.0 -l is -1.0 -0.5 o.o X/S o.s 1.0 1 S 2.0 Fig. 3 Free Surface Profiles Above Single Vortex £ = 0.5 Cal _ ~ _ c ~ ~ .= ~ -! A,-~ ~ ~ -rem . ~ ~ Am, ~ ~ ~, _ _ _ _ _ _ 7_ ,_ , ~ ~ ! of of ! of . . ! ACE' ~ ! ~' an , ~ Elf . ! of ~ . ~ ~- - . . ~ . . ~ ~ _ ~ ~ ~f ~ ~ ' -~ -5 -~ -3 - 2 - 1 O 1 2 3 ~ 5 ~ 7 X/} Fig. 4a Surface Profiles with Vortex Pair: Classical Linearization _% cat o u vet c ·o a ·c Nb~ ~~W^~ - I `~ `~ IT NOT -7 -~ -5 -~ -3 - 2 - 1 0 1 2 3 ~ 5 ~ 7 X/x |12T Fig. 4b Surface Profiles with Vortex Pair: First Order Theory with Induced Current Terms NONDIMENSIONAL PARAMETER IDENTIFICATION When ambient waves are included in the vortex/free surface problem, the characteristic length scale should correspond to the ambient wavelength. This wavelength is of primary concern since eventually it will be this scale which determines the character of radar return from the free surface. The problem therefore is nondimensionalized as follows: (where ' indicates a dimensional quantity) x=x 'I y=y'~ M~'~ t=t,\2w ¢=¢,~ (9) The governing inputs to the problem are A, g, A, S. D, and \. Dimensional analysis reduces these six variables to four parameters. The choice of parameter definitions must be made with care to allow meaningful conclusions to be drawn. The ambient waves, for instance, are described by their steepness, Am. Since the paths taken by the vortices will be defined by their "rigid wall" paths and therefore are known a-priori, one choice for a parameter would be the ratio of tile initial separation distance to initial depth, S/D. Small values of S/D indicate that the vortex pair is initiated at greater depth than separation, and will initially propagate upwards toward the free surface. Large values of S/D represent a vortex pair which is widely separated and will move parallel to the free surface under the primary influence of their image vortices above the free surface. One then needs to define either S/\ or D/\ as the other parameter relating the vortex geometric configuration to the ambient wavelength. Most of the cases studied here are for small values of S/D, giving a minimum depth achievable by the vortex of approximately S!2. Since this will correspond to the highest value of induced surface current by the vortices, SiL was chosen as the second parameter. The last parameter gives the relative scales of the vortex strength to the wave velocity, and is commonly called the Froude number. The Froude number for this problem can be defined in several ways depending on one's choice of length scale. Previous works without ambient waves have utilized the initial vortex separation following the definition F=e of equation (21. Since it has been shown that He induced surface current has a dominant influence on the modification of waves, it would appear prudent to define a Froude number based on the ratio of this induced current and the group velocity of the ambient waves. The maximum induced velocity (including image vortex above free surface) is given by: vi=2I~/7rS which occurs at D=S/2 (10) 494

and occurs after the vortices have approached the free surface and separated. The deep water group velocity for the ambient waves is Vg=0.5: 2gh (11) The ratio V'/Vg will be used as the Froude number, F. and can therefore be formed as shown below. 4r V i AS - 4~;7 ~ Vg ~ ~ So (12) 27 with r = ~ r vie So ~ ~ F=r~(gS3~1/2 P=A~ Q=S/\ R=S/D where ~ is the vortex circulation, S is the initial vortex pair separation, D is the initial vortex pair depth, ~ is the ambient wavelength, and A is the ambient wave amplitude. Substituting these relations into the previously defined ALM boundary conditions gives the following results: Dvnamic: Kinematic: fit = -¢o~/2 - A;- 1l/`F2 Q3> (14) ill = 4'y + 11§oyy ~Q~llx (15) The corresponding deepwater wave potential using the above nondimensional parameters takes the form: law = P/(F Q3/2 `271;~1/2' e2nY sin [2= - `2~1/2t/`F Q3/2~ (16) The vortex potential is also nond~mensionalized and given below in complex form: p° = to + into - i 1 in [(Z-Z1~/(Z-Z211 i-1 in [(Z-Z 1~/(Z-z 2~] (17) where the "*" character represents the complex conjugate. FROUDE NUMBER VARIATION The examination of Froude number influences will consist of flow simulations with the vortices started at a relatively deep position (S/D=1/31. This value of S/D was chosen to m~nize surface disturbances associated with the transients of the vortices' impulsive start. The ambient wave modifications would therefore be associated predominantly with the velocity field generated by the slowly moving vortices rather than the initial condition wave motion generated by the vortices. S/D=1/3 also corresponds well with the experimental studies of Willmarth et.al.l°. S/\ is set to 1 to equate Froude numbers based on different length scales. The resulting surface profiles for F ranging between 0.1 and 0.6 are shown in figures 5 to 9. Since the time scale of modification is much slower at F=0.1, the profiles shown in figure 5 coincide with 2T intervals rather than the IT intervals used at the other Froude numbers. The vortex path is shown in figure 10. The surface modification in these profiles involves a stretching of waves in the center region above the vortices at later times. The ambient waves also appear to be shortened somewhat on the left or "windward" side due to the opposite directions of vi and vg. The wave shortening on the right or "leeward" side is much less pronounced since vi and vg are in the same direction. The most obvious effect of varying F in these cases is the change in time scale for the surface disturbance to develop. The occurrence of ambient wave modification at earlier times at higher F is due primarily to the increase in relative speed of the vortex pair moving towards the surface. To minimize this attribute, surface wave profiles can be compared between different F cases when the vortex pairs are at common positions. Two common positions are labeled "A" and "B" in figure 10. The corresponding profiles from each F are collected and shown in expanded vertical scale in figures 11 and 12. Case "A" corresponds to a moderately deep vortex pair. Only the higher F cases corresponding to relatively stronger vortices have an influence at this depth, verifying the notion that stronger vortices will influence the surface from greater depths. A more subtle and interesting difference is observed as the vortex pair draws near the surface at position "B". Here one notices the lengthening of residual ambient waves in the central region above the vortices. This increase in residual wave length is presumably caused by the higher tangential velocity gradients at the surface in cases where F is increased. TO width of the disturbed region above the vortex pair does not depend heavily on the value of F. Figure 13 shows the vortex-induced current for condition B without waves. For any threshold value of v~/vg, one would expect from this figure that increasing F would cause a greater increase in the affected region width than the simulations actually 495

en n 17. .C CL ·c CR C`i lo c u 01 ·C x/x Fig. 6 V~/Vg Variation: V~/Vg = 0.2 . Fig. 7 V~/Vg Variation: V,/Vg = 0.4 x/x V IV_ Variation: V /V. = 0.1 Fig. 9 VI/VF Variation: V~/V`~ = 0.6 12T IT OT 6T AT OT Fig. 8 Vi/Vg Variation: Vi/Vg = 0.5 IT IT OT x/x Fig. 10 Vi/Vg Variation: Vortex Paths show in figure 4b. The reduced dependency of affected region width on F suggests that the time scale of the free surface interaction is relatively short, and that using slowly varying current distribution theories such as the conservation of wave action is inadequate for treatment of problems of this type. This further emphasizes the inherent transient nature of the wave modification by vortices of the scale used to model ship wakes. An additional note on the variation of F is the transition occurring around values of 0.5 of the surface disturbance directly above the vortex pair. At values above 0.5, the creation of a center "humps appears similar to center humps calculated by Telste9 at higher Froude numbers without ambient waves. It should be noted that in this case the vortices have not deviated from their rigid wall paths. This indicates a transition Froude number value of 0.5 above which the vortex pair begins dominating the surface profile. INITIAL GEOMETRIC CONDITIONS The initial position of the vortex pair is equally important to its strength in defining the form of ambient wave modification. As noted previously, the ratio of induced surface velocity to ambient wave group velocity is highly dependent on the depth of the vortex. Time scales for the vortex pair to approach the surface from large depths is also obviously dependent on the initial depth and separation distance. 496

En 02 O ~ -0.2 -7 - _ \/ . ~ b bL r \J ~\J \J ~ _ v L -5 -4 Fig. 11 Vi/Vg no -0.2 V \J \J \J V ~ V Fig. 12 Vi/Vg ~ 74 i~ ~ ~ ~ , Err ;T~ ~ ~ 8 - r _ J V -3 -2 -1 0 X/\ Variation: Position"A" 2 od 4 _ _ ~ r . v -3 -2 VL / V fY \ 1 ~- A ' ~1 -1 o X/) Lx Position A l h.~ I I I F ~ IA h ^/1 W1 1 F NVI F ~ 3 4 5 6 7 \ / 0.4 0.2 0.1 '' F=Vi/Vg ax P \ A ) ~ ~ Vortex Ptadi~n B ~ ~ ~ 1 ~ ^ 1 ^ A ~ ~ red in <i F =~0.6 1 1 1~1 ^1~1~1 1~41 ; 1 IF 30/.2: I I I ~ ~ 1 2 3 4 5 6 7 gSitil Lie J . . . Variation: Position "B", F--V~/Vg it 1 o I -1 + -2 + 1 it= :~ F = 0.6 A F = 0.4 F =0.1 , 1 1 1 1 -7 - 6 - 5 - 4 - 3 -2 - 10 1 2 3 4 5 6 7 X/X Fig. 13 V,/V Variation: Induced Current from Positgon "B" The position of the vortex pair can be identified by two parameters. One parameter has been chosen as S/D specifying the relative position of the vortex on its prescribed path. The particular path desired may then be specified by designating a value for either S or D. S/D Variation The influence of starting at various points on a given path will be examined first. The controlling parameter in this case is S/D, which detennines the initial point on the path. Three values of S/D are examined at various stages of the path corresponding to: deep (S/D=0.5), turning (S/D=2), and shallow (S/D=6~. As shown earlier, the direction of motion of the vortex pair is quite different in each of these cases. The value of F used in these cases was set to 0.125, and the common path chosen corresponded to a final submergence depth of the vortices of Dfinal=~. Figures 14 to 16 show the resulting profiles and figure 17 shows the corresponding trajectories of the vortex pair. . - c~ ~ o 1 .' L~ In ~ ' F ~ 1 Hi o n ._ m ·C Cal On .C 01 .c Cal l -7 -~ ram - V 1 7 r: ~3 1 1 -A -5 - ~, it, ~ ~ ~ 4 I ~ 1 ~ 1 1~ 1~ 1 1 ~ 1 ~ 1 ~ ~ ~ ~ . 1~ 1~1~ rulers I'M r -e -s -. -3 rod TV -s -~ ;r~ 1 1 1 ~ ~ ~ ~ ~ vet ~ ~ 1~ 1~-1-'7~1 ~1~1~71~ . ~-1 ~ 4+ ~ ~ ~ ~ ~ 4 To t~] al ~1~1~1~ TV ~ I I I I ~1 ~1 ~1 ~1 A ~ ~ . 1~ 1 ~ I ~ 1 ~ I ~ 1 ~ 1 ~ 1 ~ Ah ~ Ad, ~T~T~ ~ ~ ^\ ~ ~ ~ 4~ ~ ~ 4 ~1-1-1~1-1~1~1~l~ ~T~l~l~l~l-l ~1- 1 -2 - 1 0 1 2 3 ~ S ~7 X/\ SID Variation: S/D = 0.5 ~ Al r rat 1 ~ . ~ ~ ~ 1_ ~ . HA ~ 1_ ~ Vex l . - 3 - 2 - 1 0 1 2 3 ~5 X/ Fig. 15 S/D Variation: S/I) = 2.0 1 8T AT |OT _ L I | | | :~) 8T e==~-~ vt~ ~ x^t 1 1 ^~l~r~r~ ret ~ ~ r YA] ~ L~-T ~ ~ AM A- ~ ~ ,~ ~,~ ~,~ ~ lT T~I~I~T~T~l~T~I~l~I _ ~ 7 ~L~ L] I. -TV= A, HI ~rl: I I It _ . _ ~ TT1 -iT - ~-3 - 2 - 1 0 1 2 3 ~5 6 7 x/x Fig. 16 S/D Variation: S/D = 6.0 In these figures one can see the influence of the instantaneous vortex pair depth on the surface wave modification. In the case of S/D=0.5, the surface is modified only by a slight shifting in the waves caused by the small vortex-induced surface 497 FIT IT OT

1 2 3 ~ -2 3 _s - ~-3 -2 S/D · 0.5 · 2.0 · 6.0 -1 0 x/) Fig. 17 S/D Variation: Vortex Paths current. This wave alteration is also noticed to be almost symmetric about the line of symmetry for the vortex paths. This indicates little net exchange of energy between the vortices and the waves as they pass through. As the depth is decreased, the vortex induced surface current grows to the value of the group velocity of the ambient waves, and essentially stops the transfer of ambient wave energy from left to right. Two important processes occur which deserve further attention. First, the ambient waves are shortened as they enter the region of influence of the vortex pair by the opposing induced current. The resultant decrease in ambient wavelength is accompanied by a decrease in the absolute group velocity of the waves. The result of this process, is that ambient waves of higher speed may be halted by vortices of smaller magnitude than that predicted from v~/vg=1 . S/X Variation As mentioned in the S/D section above, initial vortex pair location is important in examining the influences on ambient surface waves. In addition to specifying the portion of the path that the pair will be started on, a particular path must be chosen. The parameter S./\ is used as a path selection parameter for its role in scaling the vortex path geometry to the ambient wavelength. Three values of S/\ were simulated with a constant value of S/D=6. S/\ values were set to 3.0, 4.5, and 6.0. The upper limit on S/\ t.... . . . . . . . TV van ~ ~ Wl1 For J VV\/ 1 ~ret ~ ~ ~ lIV\J V r -7 -B -5 -~ -3 - 2 -1 o 1 2 3 4 s ~ 7 x/x Fig. 18 S/\ Variation: Final Profiles t=8T is caused by the size limitation of tile computational domain, and the influence of partial wave reflections on the right hand boundary. F varied between 0.06 and 0.125 to preserve a v'/v at the surface of 0.5. The surface profiles after 818 for each condition are shown in figure 18. Note the growth in modified wave region size as S/\ is increased. An additional process observed in the simulations is the dynamics of waves caught in the central influence region during the initial Notion of the vortex pair. These surviving center waves may be leftover from the initial ambient wave profile in this region in the shorter tinge scales. These left over waves are found in greater numbers as S/\ is increased, as shown in figure 18. In the cases shown, a wave blocking action is rapidly generated due to a relatively large value of v'/vg. Wave energy may also propagate into the center region when the vortex pairs are initialized at low values of S/D. This energy propagation is facilitated by the canceling influence of the induced tangential velocity from each of the counter rotating submerged vortices at low values of S/D. The evolution of these center waves is particularly complex in cases of moderate S/D. Initial waves in this region are typically stretched by the surface current gradient. The resulting increase in wavelength gives rise to an increase in phase and group velocity. Because the gravity wave dispersion relation specifies that longer waves move at higher velocity, die central region may be evacuated over a shorter tine than that predicted by ambient wave group velocity. Simulation results were not carried far enough in time to show complete smoothing of the central region due to the influence of partial wave reflection from the right side boundary. Larger surface grids and greater computer times will therefore be necessary to validate this hypothesis. WAVE STEEPNESS AND PHASE VARIATION The wave steepness and phase with respect to the vortex pair are additional parameters which result from the problem formulation and enter the calculations through the initial boundary condition on the free surface and the time varying stream function prescribed on the side boundaries. Simulation results show no significant influence of these parameters on the form of the interaction. In the case of wave steepness, the waves and their modification can be linearly scaled with AN for values less than 1/10 corresponding to the linearized ambient waves being described. The relative phase of the ambient wave was also found to have no effect on the general forth of the wave modification at the S/X values studied here. 498

SPECTRAL ANALYSIS In order to make the previous findings useful in the study of real ocean flows, a method of analysis must be developed which illuminates the impact of wave modification on radar images. It was shown in the introduction how the redected radar intensity is proportional to the amount of Bragg waves present in the region being illuminated by the radar. This intensity is usually expressed as the backscattering cross section per unit surface area, c,O. This cross section is derived by Valenzuela1 in the well known Bragg scattering expressioll: o0 = 47~4cos48 Feed M(2krsin8) (18) where: kr = radar w~avenumber and ~ = vertical incidence angle of radar. Here Feel is a scattering coefficient dependent on the radar polarization and incidence angle, and M() is the component of the wave spectra corresponding to the Bragg condition. Remember, from the introduction, that the Bragg condition is satisfied when kw (surface wavenumber component in the look direction of the radar) is equal to the argument of M() in the equation above. If ~ and kw are known constants, then DO is proportional to Mel. This property indicates that a relative variation in radar backscatter over different sections of the water surface can be determined by the variations in the Bragg wave spectral component among the respective surface sections. Estimates of the relative radar intensity image can therefore be compiled by performing finite Fourier transforms of the surface elevation over regions corresponding in size to the resolution cell size of the radar. This will be performed here by sweeping a window across the simulated surface data and calculating the finite Fourier transform at various window positions. The Fourier transform component corresponding to the Bragg waves in each window position is then assigned to the midpoint location of the window. The resulting distribution of Bragg wave components over the surface is then contour plotted for comparison with SAR imagery. Method The finite Fourier transform used to determine the Bragg wave spectral content in a data window is given by: L/2 Y(k,L)= | 1l(x)e~i2~k~dx J -L/2 (19) where L is the window width. The application of this equation to discrete data can be performed using an algorithm known as the Fast Fourier Transform, or FFT. This cannon method is described in detail by Bendat and Piersoll2. The For output coefficients are complex in form providing phase information for each wavenu~nber component. Since this phase was shown to be unimportant in the previous chapter, the absolute magnitude of each wavenu~nber component is formed from the modulus of the real and imaginary parts: M(ki=(Y 2rki ~ A. 2`k~ 1/2 (20) - -I--, treat v~, ~ lmag. '''" where: FFT(k) is an output coefficient. Before applying the scanning FFT window to simulation results, several comments must be made here to avoid misinterpretation of the method. The regions used in computation of the spectra will be overlapped here because of the finite extent of the surface domain and the desire to have moderate sized surface patches. If the patches are allowed to be too small, the resolution of low wavenumber components decreases, resulting in undistinguishable movements of wave energy among the wavenumbers of concern. This overlapping is not representative of radar imaging, and therefore the results must not be considered direct simulation of radar operation. The general trends in the resultant component levels in the Bragg wavenu~nber range are believed to be somewhat representative of results obtained with radar processing. Application to Ship Wake The case examined using this teclmique is a simulation approximating the wake velocities from a twin screw destroyer calculated by Sweenl3. It should be noted that in simulating the full ship case, the grid resolution was lowered considerably by constraints in available computer memory. The grid density is 32 points per ambient wavelength. The ambient wavelength is set to 25 cm and corresponds to the Bragg scattering wavelength for an "L band" SAR operating at an incidence angle of 30 degrees. The initial vortex separation and depth are 5 m and 2 m respectively. Gravitational acceleration is equal to 9.8 mls2, and the ambient wave amplitude is set to 2.5 cm. The vortex circulation strength, r, is chosen to give approximately the same maximum surface current reported by Sweenl3 for the given depth of submergence. The resulting r value was 1.0. The simulation domain extended to + 6.25 m in the x direction with a depth of 12.5 m. The simulation was carried out for 8 periods of the ambient wave (3.2 seconds) and the resultant profiles at each half period are shown in figure 19. The corresponding downstream distance spanned by the simulation is 25 m for a ship speed of 15 knots. _ The FFT window size was chosen to be 2 m wide, which is typical of SAR resolution cell dimensions. The resultant spectral resolution is 3.14/cm in wavenumber giving components at wavelengths of 22 cm, 25 cm, and 28 cm. The 499

consecutive FFT windows were overlapped in the analysis by 3 \. This overlap gives denser data for examining the variations in Bragg component, though only non-overlapping sections would be directly comparable to typical radar imaging techniques. The Bragg component of each spectrum is assigned to the location of the FFT window center. The resulting contour plot of the magnitude of the Bragg component versus space and time is given in figure 20. This plot indicates a reduction in Bragg wave component in the center region as time progresses. In order to understand the energy flow path in this region, figure 21 is constructed. This figure shows the progression of the spectra as the window is traversed across the contour plot at the 2 second time frame. The thick line indicates the wavenumber associated with the Bragg wavelength. Figure 21 illuminates the shift in wave energy to lower wavenumbers in the center region. Since these wavenumber components do not satisfy the Bragg scattering criterion, the resultant received energy at the radar is decreased. This phenomenon of wavenumber shifting due to induced surface currents appears to produce a weaker radar return signal in the region between the vortices. In the ocean, the wave field can be described by the superposition of many wavelength waves. Pierson-Moskowitz spectral models of the sea show a decreasing level of energy as wavenumber is increased in the range of the L Band Bragg scattering waves. The implication of this is that though higher wavenumber (shorter) waves will be stretched into the Bragg wavelength, their original energy is lower. Thus one would expect that a larger amount of energy was shifted out of the Bragg sensitivity window than was being shifted in. The inclusion of these higher wavenumbers would however, decrease the magnitude of variation in M() shown in figure 20. The importance of this action is that it produces the proper trend in the surface response in the wake region of a ship and may be a significant contributor to the overall signature production process for the dark centerline wake. _~ . :~ ~ . it_ or. 0 it__ ~ -25 - 20 - 1 S -1 0 - 5 0 5 x/\ 10 15 20 25 Fig. 19 Ship Wake Simulated Surface Profiles IT M(k,~) a 74A is o _ .. _ ,,. ° Time (8ec) 3.2 Fig. 20 Contours of Bragg Wave Content 'it k kBragg Fig. 21 Crosswake Variation in Wave Spectra, t=2 see SUMMA RY The modification of surface waves by submerged vortices has been explored here in a straightforward progression from analytical examination through numerical simulation of a simple vortex ambient wave model. The motivation for study of this problem is the determination of ship wake impacts on synthetic aperture radar images showing dark centerline regions behind ships moving at sea. This process has led to several discoveries in the nature of the interaction process between vortices and ambient waves. A perturbation analysis of the free surface boundary conditions about the zero Froude number base state shows that additional terms (not found in the classical free surface linearization for small amplitude waves) are needed to describe the first order interaction of vortices with the free surface. These terms show the importance of the vortex induced surface current on the modification of ambient waves. Example simulations show that neglect of these induced current terms give much smaller modification of surface waves. The simulation program has been used to examine the specific flow configuration of a vortex pair impulsively started below a free surface containing a sinusoidal ambient wave train. This examination determines which parameters are 500

significant in determiriing the form of the ambient wave modification. It is found that the Proude number of the vortices and their position relative both to each other and the free surface play the dominant roles in modifying ambient waves. The ambient wave steepness and initial relative phase with respect to the vortices have only secondary influence in the surface modification. The simulation is then extended to represent the simple twin vortex wake model of a generic ship wake. The results of this simulation are analyzed using a scanning Fast Fourier Transform window on the surface to obtain spatial distributions of the spectral content of the surface. These spatial distributions are filtered to extract the Bragg scattering wave component and contour plotted. The resulting image shows that the current induced shift in wave energy to longer wavelengths can produce an effect consistent with the observed dark centerline wake regions of moving ships in radar images. ACKNOWLEDGEMENT This work was sponsored by the ONT Independent Research Program administered by the David Taylor Research Center. REFERENCES 1. Lyden, J., D. Lyzenga, R. Shuchman, and E. Kasischke, "Analysis of Narrow Ship Wakes in Georgia Strait SAR Data", ERIM report 155900-20-T, 1985, Ann Arbor, MI. 2. Peltzer, R., and W. Garrett and P. Smith, "A Remote Sensing Study of a Surface Ship Wake", International Journal of Remote Sensing, vol 8, no. 5, 1987, pp. 689-704. 3. Wright, J.W., "Backscattering from Capillary Waves with Application to Sea Clutter", IEEE Transactions, AP-14, 1966, pp. 749-754. 4. Lindenmuth, B., private communication; data from model experiments at DTRC, Aug. 1986. 5. Griffin, O.M. and G.A. Keramidas, T.F. Sween, and H.T. Wang, "Ocean and Ship Wave Modification by a Surface Wake Flow Pattem", NRL Mem. report # 6094, 1988. 6. Fish, S., "Ambient Free Surface Wave Modification by a Submerged Vortex Pair", PhD Thesis, 1989, Univ. of Maryland. 7. Fish, S. and C. van Kerczek, "Development of First Order Free Surface Boundary Conditions for Vortical Flows", being submitted to ASME J. of Appl. Mech. 8. Tryggvason, G., "Deformation of a Free Surface as a Result of Vortical Flows", Physics of Fluids, Vol. 31,No.5,1988,pp.955. 9. Telste, J.G., "Potential Flow About Two Counter- Rotating Vortices Approaching a Free Surface", Journal of Fluid Mechanics, Vol 201, 1989. 10. Willmarth, W.W. and G. Trygg~ason and A. Hirsa and D. Yu, "Vortex Pair Generation and Interaction with a Free Surface", Physics of Fluids A, Vol 1, Feb. 1989, pp. 170-172. 11. Valenzuela, G.R., "Theories for the Interaction of Electromagnetic and Ocean Waves - a Review", Boundary Layer Meteorology, Vol 13, 1978, pp. 61- 85. 12. Bendat, J.S., aIld A.G. Piersol, Random Data: Analysis and Measurement Procedures, Wiley Interscience, New York, 1971, pp. 252-253 and pp. 299-309. 13. Sween, T., "Numerical Simulations of the Wake Downstream of a Twin-Screw Destroyer Model", NRL Mem. report # 6131, 1987. 501