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Twenty-Second Symposium on Naval Hydrodynamics Incipient Breaking of Steady Waves M.Miller, T.Nennstiel, L.Fialkowski, S.Pröstler, J.Duncan, A.Dimas (University of Maryland, USA) Abstract The effect of free-surface drift layers on the maximum height that a steady wave can attain without breaking is explored through experiments and numerical simulations. In the experiments, the waves are generated by towing a two-dimensional fully submerged hydrofoil at constant depth, speed, and angle of attack. The drift layer is generated by towing a plastic sheet on the water surface ahead of the hydrofoil. It is found that the presence of this drift layer (free surface wake) dramatically reduces the maximum nonbreaking wave height and that this wave height correlates well with the surface drift velocity. Direct numerical simulations of the two-dimensional Euler equations are used for non-breaking and incipient breaking conditions. Initially symmetric wave profiles are superimposed on a parallel drift layer whose mean flow characteristics match those in the experiments. It is found that for large enough initial wave amplitudes a bulge forms at the crest on the forward face of the wave and the small scale vorticity fluctuations just under the surface in this region grow dramatically in time. This behavior is taken as a criterion to indicate impending wave breaking. The maximum nonbreaking wave elevations obtained in this way are in good agreement with the experimental findings. For breaking conditions, large wave simulations of the corresponding filtered Euler equations are performed. The results exhibit a vortex formation on the forward the face of the breaker. This feature of the flow is qualitatively similar to experimental observations. 1 Introduction The steady free surface flow field generated by a ship moving at constant speed in calm water typically includes breaking waves at the bow and the stern. The bow wave and the parts of the stern wave far from the ship’s track propagate in undisturbed water; however, the parts of the stern wave near the ship track propagate in a flow with a free surface shear layer due to the boundary layer of the hull and/or the surface wakes of upstream breaking waves. (Surface wakes of steady breaking waves have been studied by Battjes and Sakai  and Duncan  and .) A comprehensive theoretical or numerical model of wave breaking in the presence of surface wakes must include information on incipient wave breaking conditions. An incipient breaking wave is defined as a nonbreaking wave for which even a slight increase in its steepness would cause breaking. The fact that upstream surface wakes affect the incipient breaking conditions of downstream waves is demonstrated by observations that the breaking stern wave crest frequently extends out past the side of the ship to a width equal to the width of the breaking bow wave, even when the stern wave steepness appears to be very low in terms of calm water incipient breaking conditions. These effects are illustrated in the photograph shown in Parker . The incipient breaking condition for steady waves in calm water was explored theoretically by Stokes . In this work, it was assumed that at incipient breaking the velocity of the fluid particles at the crest of the wave approach the phase speed of the wave. Thus, using Bernoulli’s equation for a streamline on the free surface and assuming constant pressure on this streamline, the incipient breaking amplitude of a steady wave is given by: (1) where ζmax is the height of the crest above the mean water level, U∞ is the wave phase speed, and g is the acceleration of gravity. Stokes also found, using irrotational flow theory, that the incipient breaking wave would have a sharp crest with an included angle of 120°. Subsequent studies have shown that it is nearly impossible to obtain the above wave form either in nature or in the laboratory due to instabilities which set in at smaller wave amplitudes. The effect of a surface wind drift layer on the incipient breaking condition was investigated by Banner and Phillips . A steady theory was used in which, in the same manner as Stokes, incipient breaking was assumed to occur when the fluid velocity at the crest equaled the wave phase speed. However, in
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Twenty-Second Symposium on Naval Hydrodynamics the presence of a wind drift layer with the surface drift velocity (relative to the fluid at infinite depth) in the same direction as the wave phase speed, the incipient breaking condition occurs at smaller wave amplitudes than predicted by Stokes: (2) where q=(U∞−U(0))/U∞, where U(0) is the fluid velocity at the water surface in the reference frame of the wave crest but with no wave present. The result of Stokes is reproduced when q=0. For the case of ship waves, the wind drift layer would be replaced by the wake from an upstream breaker or the ship hull. Computations of waves propagating in the presence of free surface shear layers have been reported by Simmen and Saffman  and Teles Da Silva and Peregrine . In Simmen and Saffman , waves on a fluid with constant vorticity and infinite depth were considered, while in Teles Da Silva and Peregrine  waves on a layer of fluid with constant vorticity and finite depth were considered. In both cases wave profiles, extreme wave heights and wave propagation speeds were presented. Experimental studies of the incipient breaking conditions for steady two-dimensional waves generated in calm water have been reported by Salvesen and von Kerczek  and Duncan . In both studies, the waves were generated with a submerged hydrofoil moving at constant speed, depth, and angle of attack in a towing tank. In Salvesen and von Kerczek , the incipient breaking condition was determined by fixing the depth and angle of attack of the foil and varying the foil speed from one experimental run to another. At low speeds, the wave steepness was small and no breaking occurred. As the speed was increased, the wave became steeper and, for a high enough speed, the wave broke. If the foil speed was increased past this point, the breaking eventually stopped. Thus, speeds just less than the speed for which breaking started and just high enough for breaking to stop were chosen, and the wave slope was measured at each of these incipient breaking conditions. This procedure was repeated for several depths of submergence. The maximum surface slope of these incipient breaking waves varied from 11 to 25 degrees and did not show any consistent trend. Duncan  found that for fairly steep nonbreaking waves, breaking could be triggered by dragging a cloth for 1 or 2 seconds on the water surface ahead of the wave. For small enough wave steepnesses, when the cloth was removed, the wave would stop breaking. However, for higher wave steepnesses the wave would continue to break after the cloth was removed. The wave profiles measured at the incipient breaking condition determined by whether or not the wave would continue breaking when the cloth was removed were very consistent. The maximum slope of each profile was found to be about 16°; this value increased slowly with towing speed. Even though the cloth was used momentarily to trigger breaking, the above defined incipient breaking condition is for a wave in calm water. In the present paper, the effect of a steady surface wake on the incipient breaking condition of a steady wave is examined experimentally and numerically. In the experiments, a plastic sheet is dragged along the water surface at a fixed distance ahead of the steady wave created by a towed hydrofoil. Unlike the experiments of Duncan , in the present experiments the plastic sheet was always present in front of the wave. With the hydrofoil at a fixed depth of submergence (one for which it produces a nonbreaking wave in calm water), the distance, Δx, between the trailing edge of the plastic sheet and the hydrofoil was varied to obtain the incipient breaking condition. For small Δx, the local surface drift near the wave crest, q, is high and the wave tends to break even when its amplitude is small. For large Δx, q is small and the wave does not break, as if it were propagating in calm water. The incipient breaking wave was taken as the nonbreaking wave for which breaking will start if Δx is decreased by a small amount. Wave profile measurements are taken at the incipient breaking conditions and the wakes of the plastic sheets are characterized through measurements of the mean horizontal velocity distributions. These measurements are used to quantify the effect of q and the wake momentum thickness on the incipient breaking conditions. Direct numerical simulations of a similar flow are performed using a fully nonlinear, inviscid, two-dimensional, free-surface flow code for incipient breaking waves, and large wave simulations for breaking waves. The incipient breaking conditions found in the experiments are compared to the theory of Banner and Phillips  and to the results of the numerical simulations. The experimental data and the numerical results are further used to explore the physics of the instability processes at the incipient breaking condition. The remainder of this paper is divided into five sections. In Section 2, the details of the experimental setup and measurement techniques are presented. This is followed in Section 3 by a description of the experimental results. In Section 4, the numerical model is presented along with some typical results. The experimental and numerical results are compared and discussed in Section 5. Finally, the conclusions are presented in Section 6.
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Twenty-Second Symposium on Naval Hydrodynamics Figure 1: Side view of the towing tank. 2 Experimental Details 2.1 The towing tank The experiment was performed in a towing tank with dimensions of 14.8 m long, 1.22 m wide and 1.0 m deep, see Figure 1. The sidewalls of the tank are made of glass to allow for flow visualization and optical measurements. The tank contains both below-surface and above-surface towing systems. The below-surface towing system includes two fully submerged ‘L’-shaped tracks that are mounted from the bottom of the tank near each of the long sidewalls. Objects are towed along the tracks by two stainless steel wire ropes which enter the water at one end of the tank and leave from the other end. Thus, no part of the towing system breaks the water surface in the vicinity of the towed object. The wire ropes are driven by a servo motor mounted at one end of the tank, see Figure 1. The above-surface towing system uses two tracks mounted above the tank, one near each sidewall. The above-surface towing system includes an instrument carriage which rides on the tracks via four hydrostatic oil bearings. When high-pressure oil is supplied to the bearings, a thin film of oil is forced between the bearings and the tracks, thereby greatly reducing vibration and friction levels of the carriage. The carriage is driven by two separate wire ropes which are powered by the same servo motor that powers the below-surface towing system. Precise towing speeds are obtained by means of a computer-based feedback control system. In the present experiments, steady waves were generated with a hydrofoil mounted to the below-surface towing system. The hydrofoil is an aluminum NACA 0012 airfoil with a 20-cm chord which is operated at a 9° angle of attack. This foil spans the width of the tank with a small clearance of 1.4 cm between the edges of the foil and the walls of the tank. The foil is mounted to two stainless steel plates which, in turn, are mounted to two Delrin blocks, each with a groove cut into it. These grooves provide a low friction bearing surface to slide along the submerged L-shaped tracks. The surface wake was created with sheets of Mylar dragged along the surface of the water at a fixed distance ahead of the hydrofoil. The Mylar sheets have a thickness of 0.13 mm and a specific gravity of 1.25. Although these sheets are heavier than water, the contact angle at the Mylar-air-water interface around the edges of the sheet allowed the sheets to remain on the water surface. Two Mylar sheets were utilized, each having the physical dimensions as shown in Table 1. Table 1. Physical dimensions of the short and long Mylar sheets. short long length, cm 63.5 101.6 width, cm 101.6 114.3 contact length, cm 56 94 The mounting assembly used to hold the Mylar sheets in place was attached to the above-surface carriage via two linear motion slides to allow for vertical positioning of the entire assembly. The relative position between the above-surface carriage and hydrofoil is adjustable. This allows the separation distance between the trailing edge of the Mylar sheet and the leading edge of the hydrofoil to be varied. Towed by itself, the sheet created a wake at the water surface and, as discussed below, a train of small-amplitude waves. In order to control water clarity and surfactants, a recirculating skimmer system was used. This system
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Twenty-Second Symposium on Naval Hydrodynamics Figure 2: Mean horizontal velocity versus depth below the mean free water surface at a distance of 38 cm behind the trailing edge of the short Mylar sheet (no hydrofoil): (a) Raw velocity data; (b) Velocity data after subtracting Uw(z). The solid line is a least squares fit of Equation 5. Short Mylar sheet, X=38 cm. includes two surface skimmers located at one end of the tank. The water from the skimmers was sent to a diatomaceous-earth filter and then sent back to the tank through a port at the opposite end of the tank from the skimmers. When fresh water was needed, tap water was sent through a separate filter before entering the tank. The skimming system was run for at least 2 hours before any measurements were taken. 2.2 Mean velocity measurements A rake of three pitot tubes was used for the mean velocity measurements in the wakes of the Mylar sheets. The rake had a horizontal spacing between tubes of 10 cm and a vertical spacing of 10 mm. The large horizontal spacing was chosen to minimize the interaction between the individual tubes. The rake was mounted to a telescoping arm which was attached to the instrument carriage, allowing the stream-wise distance from the trailing edge of the Mylar sheet to the tips of the tubes to be varied. The Pitot tube rake was attached to the arm via a linear traverser to allow vertical positioning. Each Pitot tube was connected through transparent Tygon tubing to a separate differential, diaphragm-type pressure transducer (Validyne Model P305D), each having a range of ±0.2 psi. The three pressure transducers were mounted to the end of the telescoping arm at equal heights above the water surface. The analog voltage output of each pressure transducer was connected to a 12-bit analog-to-digital (A/D) converter operating at 1200 samples per second and the digitized output was stored in the memory of a PC. The signal taken during the steady state part of each run was then averaged. Division of the full pressure range of the transducer by the resolution of the A/D converter yields an accuracy of 0.1 cm/s at an average speed equal to the towing speed, 80.5 cm/s. However, during runs made with the Pitot tubes moving through the undisturbed water in the towing tank, a pressure fluctuation corresponding to a root-mean-square velocity fluctuation of 0.2 cm/s was observed. The above-described equipment was used to measure the vertical distribution of mean horizontal velocity at three streamwise locations behind each Mylar sheet. These measurements were performed without the presence of the hydrofoil. In performing these experiments, it was observed that, like the hydrofoil, the Mylar sheets generated a train of two-dimensional surface waves. Though the amplitudes of these waves were small (at most 0.17 cm), they were found to have a noticeable effect on the mean velocity distributions (see below). In order to use the Pitot tubes in locations where the fluid velocity was known to be horizontal, the velocity distributions were measured at the streamwise locations of the troughs of the following wavetrain. Visual examination of the wavetrains showed the wavelength of the waves to be about equal to the difference in length of the two Mylar sheet (≈40 cm). The measurement locations were taken as 38, 80, and 120 cm behind the trailing edge of the short Mylar sheet and 40, 80 and 120 cm behind the trailing edge of the long Mylar sheet. A mean velocity distribution at one streamwise
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Twenty-Second Symposium on Naval Hydrodynamics location was typically measured during the period of one day. A series of calibration runs was performed before and after each set measurements. During the calibrations, the Mylar sheet was removed, the carriage was run at a series of known speeds, and the resulting output from the pressure transducers was recorded. During each experimental run with the Mylar sheet, the streamwise and vertical positions of the probes were held fixed. For a given streamwise location, the measurement depths were varied in a random manner from run to run until measurements at enough depths were taken to ensure a well resolved profile of the mean velocity; runs at most measurement depths were repeated three times. The finite size of the Pitot tubes resulted in an inability to make fluid velocity measurements closer than about 5 mm from the water surface. A sample of a raw mean velocity distribution is shown in Figure 2(a). In this figure, a depth of zero is the local free surface elevation. As can be seen in the plot the mean velocity increases slowly as the depth decreases from 10 cm to about 2 cm. Thereafter, the velocity decreases more rapidly to about 65 cm/s near the free surface. The increase in velocity between 10 and 2 cm in depth is an effect due to the small-amplitude wavetrain generated by the Mylar sheet. The horizontal velocity distribution in a potential flow wave at its trough can be described by linear wave theory (see for example ): (3) where z is the depth (positive down) relative to the mean water level, U∞ is the carriage velocity, a is the wave amplitude, k is the wavenumber given by in linear theory, and z is the depth below the mean water surface. The free surface is at z=−a. The above equation was fitted to the data by determining the wave amplitude that minimized the mean squared error over the depth range between 2<z<10. The resulting velocity profile, Uw, was then subtracted from all the data (0≤z≤10 cm). Several of the velocity distributions were also displaced by an amount Uos, where Uos≈−0.2 cm/s at most, to account for the fact that the distributions after subtracting out Uw did not asymptote to the known towing speed, U∞, see Section 3. The reason for this discrepancy between the asymptote and U∞ is not known. The final distribution after the above processing and displacing the profile upward by the wave amplitude, a, was called the wake velocity distribution and given the symbol U(z): where is the measured data. The wake velocity profile (U(z)) corresponding to the raw data in Figure 2(a) is given in Figure 2(b). In order to obtain the surface velocity from the data and to have a mathematical form for the wake velocity profile, the final data U(z) was fitted to an equation given by (4) where a1 and a2 are fitting parameters. The best set of constants (U(0), a1 and a2) for the given velocity data set were determined by minimizing the sum of the squared deviations. (The tanh(a1z2) profile was used by  to fit the velocity profiles in the near wake of a hydrofoil and the tanh(a1z2+a2) profile was used by  in the near wake of a cylinder. In the present work, it was found that the latter function gives the best fit to the data.) A curve with form of Equation 5 with the computed constant set has been overlaid on the data in Figure 2(b). The above data processing was repeated for the three measurement distances for both Mylar sheets giving six velocity profiles (see Section 3). 2.3 Wave-height measurements To measure the height of the incipient breaking waves created by the combination of the hydrofoil and the Mylar sheet, it was not possible to use wire gauges fixed to the tank because, in the towing tank, the Mylar sheet would collide with the gauge during the run. Thus, an optical wave-height gauge was used (). In this device, the beam of a 5-Watt Argon-ion laser (Spectra Physics, model 2017) was pointed vertically down on the water surface at a fixed location in the wave tank. A pair of cylindrical lenses was used to convert the laser beam in to a light sheet that was one-mm thick and 8-cm long as it entered the water with the normal to the light sheet directed in the cross-stream direction. The water in the tank was mixed with Fluorescene dye at a concentration of about 1.5 ppm so that the water illuminated by the laser glowed with a greenish yellow color. The intersection of the laser beam and the water surface was observed via a digital linescan camera (Dalsa Lines-can Digital Camera Model CL-C4 2048A STDJ with a Nikon 200-mm lens). The camera was attached to a vertically oriented linear slide mounted onto a tripod that was fixed to the floor outside the tank. The camera viewed the wave from the side of the tank and
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Twenty-Second Symposium on Naval Hydrodynamics Figure 3: Wave-height profiles, (a) Profiles for four experimental runs with the same experimental condition, (b) Average of wave-height profiles in (a). Short Mylar sheet, Δxi=40 cm, d=26.4 cm. above the waterline at approximately a 25° down angle; the plane defined by the line of sight of the camera and the single line of 2048 CCD elements (pixels) was oriented normal to the water surface and perpendicular to the center plane of the tank. At the plane of the light sheet the array of 2048 pixels covered a physical vertical distance of about 16 cm (0.08 mm per pixel). The CCD elements received little light from the air above the water surface but much more light from the glowing dye at the intersection of the laser light sheet and the water surface. The boundary between the poorly and brightly illuminated CCD elements was taken as the water surface. The camera was set up to record a single line of 2048 eight-bit pixels every 0.004 seconds. A total of 1.2 seconds of data was recorded during each experimental run creating an image (vertical distance versus time) of 2048 by 300 pixels. In each image, the water surface was located by an intensity-based thresholding technique. Before any images were taken, a calibration set was created by recording the position of the flat water surface with the camera set at five different heights above the water surface. These heights were known and repeatable through the linear positioner upon which the camera was mounted and gave the relationship between pixel position of the surface image and measured height. Before each set of experimental runs in measuring the waves, a single height image was taken with no wave present to determine the mean water level. In this run and in the calibration runs, it was found that the root-mean-square surface height fluctuation due to mechanical, electronic and data processing noise was about ±0.006 cm. Each incipient breaking wave profile measurement began by taking an image of the undisturbed water surface to determine its position on the CCD array. Then the profile of the incipient breaking wave was measured in three independent experimental runs. A sample of the three wave profiles for a single experimental condition and the average of the three profiles is given in Figure 3. The wave height, ζ, was taken from the average profile as the vertical distance from the undisturbed water level to the wave crest. 3 Experimental Results In this section the location of the incipient breaking conditions in terms of the external parameters of the experiment (foil depths and separations distances between the foil and the Mylar sheets) and the wake characteristics as a function of distance behind the trailing edge of the sheet are presented. These results along with the incipient breaking wave amplitudes are discussed in light of the numerical results and existing theory in Section 5. 3.1 Incipient breaking conditions Incipient breaking conditions were determined visually through the following procedure. First the depth of submergence of the foil was fixed at a value for which, with no Mylar sheet, the wave did not break. Then the Mylar sheet was put in place and the horizontal separation distance, Δx, between the trailing edge of the Mylar sheet and the leading edge of the hydrofoil was set at a value small enough to cause
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Twenty-Second Symposium on Naval Hydrodynamics wave breaking. Over a series of experimental runs, Δx was increased. For large Δx the wake of the sheet at the location of the wave crest is very thick and has a very small surface drift. In this case, the wave behaves much like a wave in calm water, i.e., it does not break. Subsequent runs where used to locate the incipient breaking value of Δx, denoted as Δxi, which is defined such that for all Δx<Δxi the wave breaks. A plot of Δxi nondimensionalized by c (the chord of the foil) versus the depth of submergence, d, of the trailing edge of the foil (also nondimensionalized by c) is given in Figure 4. One curve for each Mylar sheet and six data points for each curve are given in the plot. Figure 4. Incipient breaking conditions: separation distance between the trailing edge of the Mylar sheet and the leading edge of the hydrofoil, Δxi, versus the depth of submergence of the trailing edge of the hydrofoil, d, where c is the chord of the hydrofoil. Filled circles: short Mylar sheet, open circles: long Mylar sheet. 3.2 Wake characteristics The wake characteristics (for instance the wake thickness and surface drift velocity) at the location of the incipient breaking wave crest must be determined from the wake velocity distributions which were measured at only three locations in each wake. Thus, characteristics of the mean velocity profiles must be determined by interpolation at places other than the three measurement locations. Three important characteristics of the wakes are the surface drift velocity, (5) the wake half-thickness, b1/2, where, (6) and the momentum thickness, (7) In the later analysis, the average values of θ for each Mylar sheet, θ=0.145 cm for the short sheet and θ=0.210 cm for the long sheet, are used. (The standard deviation for θ from the three measurements in each wake was only 3%.) For a self-similar wake (8) Using a non-linear least squares method, the above equation was fitted to the six data points from the present data set. The resulting constants were C1=3.07 and x1=−47.98 and this curve is plotted along with the data in Figure 5(a). The variation of wake thickness can also be described by a similar power law defined by: (9) The constants resulting from a non-linear least squares fit to the data were C2=0.34 and x2=−74.78 and the data and the curve are shown in Figure 5(b). 4 Numerical Simulations The interaction between the surface wake of the Mylar sheet and the gravity wave generated by the submerged hydrofoil is also studied numerically considering the following model of the process: the sheet wake is modeled as a two-dimensional, parallel shear flow, the hydrofoil wave is modeled as a plane, gravity wave, and the time evolution of their interaction is followed by numerical simulations of the Euler equations. Direct numerical simulations (DNS) are used in non-breaking and incipient breaking conditions, while large wave simulations (LWS) are used for breaking conditions. In breaking conditions, the appearance of small scale free-surface fluctuations and overturnings on the face of spilling breakers prohibits the use of DNS. In LWS, only the large scale variables of the flow (velocities, pressure, free-surface elevation) are resolved, while the effect of small scales is modeled. In general, the resolved large scale, of a flow variable, f, is obtained by a spatial filtering operation,
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Twenty-Second Symposium on Naval Hydrodynamics Figure 5: Surface drift velocity, q, (a) and Wake thickness, b1/2/θ, (b) versus measurement distance, x/θ. which produces the following decomposition for every flow variable: (10) where f′ corresponds to the unresolved subgrid scale. The velocity profile of the parallel shear flow is identical to the mean velocity profile measured in the wake of the Mylar sheet at a streamwise distance corresponding to the location where the free surface crosses the mean water level just upstream of the wave crest. As discussed above, the velocity profile of the shear flow is given by Equation 5. The plane gravity wave in the numerical model is a second-order Stokes wave with the appropriate wavelength, λ, according to linear theory: (11) The Froude number Fr of the flow, defined as (12) is related to the dimensionless wavenumber k of the gravity wave according to the following: (13) where the characteristic length scale b1/2 is the half-width of the velocity profile. For the DNS of a two-dimensional, incompressible, inviscid, free-surface flow, the governing equations are the continuity equation (14) and the Euler equations (15) (16) subject to the dynamic and kinematic free-surface boundary conditions, respectively, (17) where t is time, x, z are the cartesian coordinates (x is the horizontal coordinate, z is positive in the opposite direction of gravity, and z=0 corresponds to the mean free-surface level), u, w are the velocity components, p is the dynamic pressure, defined as the pressure P minus the hydrostatic pressure is the Froude number of the flow, and η is the free-surface elevation. In the above equations lengths are nondimensionalized by b1/2 and velocities by U∞. The free-surface elevation is an unknown function of time, which renders the flow domain time-dependent. Boundary fitted coordinates are introduced according to the following transformations: (18)
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Twenty-Second Symposium on Naval Hydrodynamics (19) where xi are the coordinates and ui are the velocity components in the transformed domain. According to the above transformation, the continuity and the Euler equations, respectively, become (20) (21) where (22) while the dynamic and kinematic free-surface boundary conditions, respectively, become (23) For the LWS, the governing equations are obtained by the spatial filtering of equations (20) and (21): (24) (25) where the additional sub-grid scale (SGS) stresses contain the effect of the unresolved subgrid scales. The eddy SGS stress, τij, depends only on velocity interactions, while the wave SGS stress, depends on the free-surface slope as well. The eddy SGS stress is the only term present in large eddy simulations (LES) of unbounded flows, while the wave SGS stress incorporates the free-surface influence. Both the eddy and wave SGS stresses are modeled using eddy-viscosity models (for details, see Dimas ). The dynamic and kinematic free-surface boundary conditions, respectively, become (26) For both the DNS and the LWS of the Euler equations, an operator-splitting scheme is employed for the temporal integration and spectral methods for the spatial discretization with Fourier modes along the x1-direction, and Chebyshev polynomials along the x2 direction (). Specifically for the DNS, 64 Fourier modes are used in the x1-direction, and 64 Chebyshev modes in the x2-direction, while the time step is 0.00025. For the LWS, a sharp Fourier cutoff filter is used to perform the filtering operation (10) where 24 Fourier modes are retained to represent the large scales of the flow; the other parameters are kept the same as in DNS. Periodicity boundary conditions are applied in the x1-direction, while the length of the computational domain in the x1-direction is equal to the wavelength of the gravity wave. Throughout the computation, Fast-Fourier-Transform algorithms are used to transform between physical and spectral space, and a spectral preconditioning technique is used on the pressure step of the splitting scheme, which renders the matrix of the resulting system of linear equations banded, thus dramatically reducing the computation time for its solution. At time t=0, the computation starts with the mean velocity profile and the flow field of the plane gravity wave, while the initial free-surface elevation corresponds to the second-order Stokes wave with an initial wave amplitude ηmax. For this paper, three velocity profiles are considered: θ=1.4 mm, q=0.3, b1/2≈5 mm, Fr=3.60, θ=1.4 mm, q=0.2, b1/2≈7 mm, Fr=3.06, and θ=1.4 mm, q=0.1, b1/2≈13 mm, Fr=2.25. For each velocity profile, the wavelength is evaluated according to (13), while several dimensionless wave amplitudes ηmax are considered. Results for q=0.3 are presented for Hmax=0.30, 0.34, 0.36 and 0.40, where Hmax is defined as: (27) where ηmαx=ζmax/b1/2. For the Stokes limiting wave, Hmax=1 (see Equation 1). The time development of the free-surface elevation for all four cases is shown in a frame of reference moving with the wave phase velocity in Figure 6. In each case, the free-surface shape is plotted every dt=5 time units, while for every new curve, the mean water level is shifted upwards by dη=0.5 from that of the previous curve. In all cases, after about t=20, the free-surface elevation becomes asymmetric about the wave crest, although the initial condition is symmetric. The level of asymmetry increases as Hmax is increased. For
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Twenty-Second Symposium on Naval Hydrodynamics Figure 6: Time development of the free-surface elevation, η, for four different initial wave amplitudes, Hmax. In each case, the free-surface shape is plotted every dt=5 time units, while for every new curve, the mean water level is shifted upwards by dη=0.5 from that of the previous curve. the highest value, Hmax=0.40, the free-surface elevation develops a bulge shape on the forward face. This shape is similar to that found in gentle short-wavelength spilling breakers (see ). The point of maximum upward curvature at the leading edge of the bulge is called the toe. For cases with Hmax≥0.36, vorticity contour plots of the flow around the time of the bulge formation (at about t=80) exhibit an instability of the vorticity field. A typical example is shown in Figure 7 from the DNS of the case with Hmax=0.36. The instability is localized in the area of the toe at x≈−12 and is associated with the sharp variation of the free-surface slope which can not be resolved by a finite number of modes. On the other hand for cases with Hmax≤0.30, there is no bulge formation during the free-surface development and the vorticity distribution remains smooth. As discussed in the following section, these differences in the vorticity field are used to define incipient breaking conditions in the numerical results. For the breaking case with Hmax=0.36, the LWS filters out the SGS instability of the vorticity field (as shown in Figure 7) and successfully continues the computation past the breaking point. In fact at time t=140, the LWS shows the appearance of a vortex at the face of the breaker (see Figure 8) which may be associated with the formation of a shear layer in the wake of steady spilling breakers (Coakley ). To this end, further comparison with experimental data is necessary.
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Twenty-Second Symposium on Naval Hydrodynamics Figure 7: Vorticity contour plots of the flow at a time instant during the bulge formation. The upper plot corresponds to the DNS and the lower to LWS. Solid contours correspond to negative vorticity, broken contours correspond to positive vorticity, and the increment between contours is 0.012.
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Twenty-Second Symposium on Naval Hydrodynamics denoted as h. The wave meters were installed in both parts of the test tank at equal distances from the wave maker. The horizontal distance between the wave meters and the axis of the submerged circular cylinder was x0=30 cm. The output of the wave meter installed at the lee side of the cylinder was treated as the characteristic of transmitted waves. The output of the other wave meter installed in the parallel channel was treated as the characteristic of the incident wave. The cylinder was located at the distance 170 cm from the wave maker. The operation principle of the wave meters is based on the measurement of conductivity of medium between two closely-spaced vertical wires. The output of the wave meters e is related with density distribution ρ(y) as follows: where e0 and e1 are dimensional constants, y1 and y2 are vertical coordinates of the lower and the upper ends of wires. If the vertical displacements of fluid particles η(y) are small, the time-dependent component of the output signal may be written where f(t) is a harmonic function of time, the prime denotes differentiation with respect to y. In experiments the length of wires met the condition (y2−y1)>2δ, so that the following relations were satisfied with high accuracy: ρ(y1)=ρ1; ρ(y2)=ρ2; ρ′(y1)=ρ′(y2)=0. To perform a static calibration, the wave meters were vertically displaced by a known distance in otherwise quiescent fluid. Thus, a resistive wave meter with vertical wires records the following measure of internal wave intensity: (2) This value is the weighted (with ρ'(y) as the weight function) mean value of the amplitudes of vertical displacements of fluid particles in pycnocline. Let us note that for a sharp density variation (free surface or an interface) the derivative ρ'(y) is Dirac delta-function. In this case the definition of according to (2) reduces to usual definition of amplitude. In experiments the wave maker generated the internal waves of the first internal mode. The measure of intensity of these waves defined by (2) is denoted as η0. It should be noted, that the experimental value of η0 was small (η0<0.15 cm). The scattering of incident internal waves is accompanied by the excitation of high internal modes. Moreover, in certain range of parameters non-linear effects manifest themselves in the excitation of the second harmonic at the lee side of a submerged obstacle. The spectral analysis of the output signal of the wave-meter placed at the lee side of the submerged circular cylinder allowed to evaluate the intensities of the first and the second harmonics which are denoted as η1 and η2, respectively. The value η1/η0 characterizes the ratio between the intensities of transmitted and incident waves. This value is, in a limited sense, analogous to the usual definition of the transmission coefficient. However, one should keep in mind the important difference introduced by the integral in (2). It is easy to see that the main contribution to the value is produced by the internal modes having uneven numbers, the input due to each particular mode being difficult to isolate. Nevertheless, the value η1/η0 represents an important measure which reflects the most important features of the wave scattering. In experiments the parameters of internal waves of the first mode satisfied with high accuracy the following dispersion relation  (3) where k=2π/λ is the wave number of the first internal mode. 3. THEORETICAL ANALYSIS It is assumed that the inviscid incompressible fluid occupies the region −∞<x<∞, −Η<y<0 and there are three layers: homogeneous
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Twenty-Second Symposium on Naval Hydrodynamics upper and lower ones and a linearly stratified middle one. Thus, density stratification in an undisturbed state ρ(y) takes the form where H1=h1−δ/2 is the depth of upper layer, Η=h1+h2 is the total depth of the fluid. This three-layer fluid is the model of a smooth pycnocline and approximates the density distribution (1). Let us transfer the origin of the coordinates to the lower boundary of the middle layer by the translation . We investigate the scattering of an internal wave of the given mode, incident on a solid horizontal circular cylinder which is fully immersed in the lower layer. The assumption that the cylinder is positioned outside of the pycnocline is crucial. To our knowledge, the problem for a body which intersects the pycnocline or is placed inside it has not yet been analysed even within the linear approach. The cylinder axis is parallel to the front of the incident wave, so that the problem considered is 2-D one. it is assumed that the upper layer is bounded by a rigid lid. The disturbed oscillating motion of the fluid is assumed to be steady and the flow inside upper and lower layers is potential. In the upper and lower homogeneous layers the total velocity potentials satisfy the equations where H2=H−H1−δ is the depth of lower layer. The internal wave equations for the vertical velocity in the middle layer are written within the Boussinesq approximation where is the buoyancy frequency or Brunt—Vaisala frequency. In the general case . The boundary conditions are the following (4) (5) (6) (7) It is known from the theory of linear internal waves that in such a fluid the existence of free internal waves is possible only with ω<Ν. The wave incident from the right can be an arbitrary internal mode with a vertical velocity (8) The wave number k satisfies the dispersion relation (9) where μ=kχ, t1=tanh kH1, t2=tanh kH2. There exists a countable number of values kj(k1<k2<…), satisfying the given dispersion relation. The eigenfunctions for the vertical velocity of the wave modes are represented in the form where The eigenfunctions are orthogonal (10) The vertical velocity of the incident wave in (8) is conveniently expressed by where
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Twenty-Second Symposium on Naval Hydrodynamics By analogy with Sect. 2 let us define the amplitude of internal wave as the average value of the vertical displacements of fluid particles over the thickness of a middle layer. The vertical displacement of fluid particle is determined from ∂η/∂t=υ0. In this case η0 is the dimensional value of incoming wave amplitude. The total velocity potential in the lower layer can be written as Here the incident wave potential of the l-th internal mode has the form and we seek the diffraction potential as The boundary condition at cylinder surface S: has the form (11) where a=d/2 is the radius of the cylinder, is the distance from the centre of the cylinder to the lower boundary of middle layer, is the inward normal to the cylinder surface. In addition, the radiation condition must be satisfied, implying that only waves outgoing from the body are generated due to the scattering. We can find the solution of this problem by the multipole expansion method. This method was used by many authors for the solution of different radiation and diffraction problems (see, for example, the list of references in ). To write the solution in terms of the multipole expansion, we define the polar coordinates The multipole expansion for the diffraction potential in the upper layer may be written as (12) where (13) (14) The potential in the lower layer is (15) where (16) (17) The vertical velocity of the scattering wave in the middle layer is The multipole expansion for is (18) where (19) (20) Using the relation
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Twenty-Second Symposium on Naval Hydrodynamics and the boundary conditions (4)–(7), we can determine the functions A1,2(k), B1,2(k), C1,2(k), X1,2(k), Y1,2(k). For example, the relations for A1,2(k) and B1,2(k) have the form where At ω<Ν the integrands in (12)–(20) have a countable number of the simple poles kj (j=1, 2,…) which represent the roots of the equation Z(kj)=0. This is tantamount to the dispersion relation (9). Using the radiation relation, equations (16)–(17) then become where the symbol pυ indicates the principal value integration and the superscript j denotes the residue of corresponding function in the point k=kj. To impose the body surface condition, we use known relations Equation (15) then becomes (21) where (22) (23) Differentiating (21) with respect to r and using (11), we obtain the systems of linear equations for determining of pm and qm The solution of these equations may be obtained by truncating the infinite series at a finite number of terms, which depends on the desired accuracy. One of the concerns in the diffraction problem is the exciting forces . They can be computed from
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Twenty-Second Symposium on Naval Hydrodynamics where and are equal to In far field (x → ±∞) the diffraction potential in the lower layer is (24) where C±(k)=T2(k)±T1(k), As the result of solution of the diffraction problem, we can determine backward (x → ∞) and forward (x → −∞) scattering matrices where for each element the row number l corresponds to the incident mode number, and the column number m—to the number of scattered internal wave mode. The wave scattering parameters are conveniently determined with use of energy evaluation (10) The complex values and are the transmission and reflection coefficients of the l-th internal mode, respectively. For the deep lower layer the backward scattering waves are absent because pm=−iqm at H2 → ∞ and hence for exciting forces F2=−iF1. The approximate values of these functions are obtained in  for the body located in the deep lower layer far from the middle layer. 4. RESULTS AND DISCUSSION When obtaining the multipole expansion solution the high accuracy can be achieved by using only several terms in (12), (15), (18). The following numerical results are obtained by using five terms what implies an error less than 10−3. The numerical integration is performed with relative error 10−4. On this basis the number of terms in the infinite sums (22), (23) is determined. In the considered problem it is of interest to investigate the influence of pycnocline thickness on the loads acting on the cylinder. In an undisturbed state the mean line of pycnocline is located on fixed depth h1=a, the total depth of fluid is equal to Η=5a, the distance from the centre of cylinder to the mean line of pycnocline is equal to h=2a, relative difference of density between lower and upper layers is ε=0.03. The incident wave is the first mode (l=1). Fig. 2. The numerical results for non-dimensional exciting forces Πj=|Fj|(2+ε)/ρ2η0aεg depending on Ω=aω2(2+ε)/εg are shown in Fig. 2. The light and dark squares correspond to the pycnocline thickness δ/a=0.5 (H1/a=0.75, H2/a=3.75,), the light and dark triangles correspond to δ/a=1.5 (H1/a=0.25, H2/a=
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Twenty-Second Symposium on Naval Hydrodynamics ). The light (dark) squares and triangles represent the results for the horizontal Π1 (vertical Π2) exciting forces. The value ω=Ν corresponds to Ω=3.94(1.31) at δ/a=0.5(1.5). As the thickness of pycnocline increases the maximal values of exciting forces also increase. But at fixed wave frequency for Ω>0.7, the greater pycnocline thickness corresponds to shorter incident internal wave, with a consequent drop of diffraction loads. In the case being considered the thickness of the lower layer is relatively small and the horizontal exciting forces are greater than the vertical exciting forces. Fig. 3. The backward scattering coefficients are presented in Fig. 3. The input parameters of Fig. 3 and Fig. 2 are the same. The curves 1–3 (4–6) correspond to the reflection coefficient the scattering coefficients of the first mode in the second and in the third ones at δ/a=0.5 (1.5), respectively. From Fig. 3 we notice that the amplitude of scattered waves increase with increased thickness of pycnocline. In the first series of experiments the cylinder was entirely placed in the fluid layer of constant density ρ2, i.e. the density variation over the range of depth within the limits h±a was vanish Fig. 4. ingly small what complies with the assumptions adopted in theoretical formulation of the problem. The comparison of the theoretical and experimental results is performed at H1=12.85 cm, δ=4.3 cm, H2=27.85 cm, ε=0.01051. The absolute values of the transmission coefficient and the forward scattering waves for the first three modes are shown in Fig. 4. The transmission coefficient is just slightly different from 1 despite the fact that the forward scattering wave is pronounced. As different internal modes of given frequency ω have different wavelengths, the inputs in the signal of the wave meter due to each particular mode may be either summed up or subtracted from each other depending on their phase shifts in the point of measurements. Referring to Eq. (24), the output of the wave meters depends not only on the frequency of the incident wave but also on their spatial position. Shown in Fig. 5 are the results of calculations for x0/d=5, 15, 25. The numerical results are compared with experimental data for x0/d=5. As is seen from Fig. 5. the experimental and numerical data agree quite
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Twenty-Second Symposium on Naval Hydrodynamics Fig. 5. well qualitatively. At the same time there is a notable quantitative disagreement. This disagreement may be explained by the effects of viscosity and by the presence of discontinuity of the density gradient in the three-layer system. In the second series of experiments the cylinder was partially submerged in the pycnocline. In this case the cylinder reflects a part of wave energy. The smaller is h/d, the greater is the portion of reflected wave energy and the lower is the value η1/η0. This effect is illustrated in Fig. 6 by the data obtained at h1/d=2.5, h2/d=5 for the following set of parameters: curve 1−h/d=1, δ/d=0.58; curve 2−h/d=0.75, δ/d=0.54; curve 3−h/d=0.6, δ/d=0.58. It easy to see that the effect is most pronounced for the incident waves of small length. The pycnocline plays the role of a waveguide and the cylinder represents an obstacle in this waveguide. The same effect is observed when the submergence of the cylinder is held fixed while the pycnocline thickness increases. The value k is related with frequency ω by Eq. (3) which was experimentally proven. It should be noted that for the experimentally studied range of parameters the dispersion relation for the first internal mode (3) agrees well with the complete dispersion relation (9). Fig. 6. The non-linear effects accompanying the diffraction of waves at a submerged obstacle manifest themselves by the emergence of high-frequency components of wave field at the lee-side of an obstacle. Normally, the second harmonic having doubled frequency compared to the frequency of the incident waves is most pronounced. Under suitable conditions the amplitude of the second harmonic may be equal to the amplitude of the first harmonic . The non-linear effects accompanying the diffraction of internal waves in continuously stratified fluid have not yet been studied in detail. Some qualitative descriptions of the observed wave patterns are given by Gavrilov & Ermanyuk . In continuously stratified fluids the frequency range of high-order effects is limited by the value ω=Nm/2, where Nm is the maximum value of Brunt-Vaisala frequency N(y). The internal waves of second harmonic can not be excited when the frequency of incoming waves ω>Nm/2. It is interesting to determine the frequency range corresponding to most pronounced non-linear effects. The results of experiments are shown in Fig. 7 for h1/d=2.5, h2/d=5, h/d=0.75: curves 1. 2 represent η2/η0; curves 3, 4—η1/η0; curves 5, 6—η0/d. Dark and light symbols correspond to δ/d=0.42 and δ/d=0.55, respectively. For surface waves of sufficiently small amplitude the value η2/η0 is directly proportional to η0/d . The similar dependence should be expected for
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Twenty-Second Symposium on Naval Hydrodynamics internal waves. In experiments the amplitude η0 and the steepness parameter kη0 of the incident waves were small and varied within a narrow range. Thus, the dependence η2/η0 on ω/Nm shown in Fig. 7 gives a clear notion of the frequency range within which the second harmonic is markedly excited. As seen from the Fig. 7, the maximum values of η2/η0 measured at different δ/d take place at the common value ω/Nm≈0.34 while the corresponding non-dimensional lengths of the incident waves are different (kd=0.68 at δ/d=0.42 and kd=0.52 at δ/d=0.55). Using the dispersion relation (3) one can evaluate that under experimental conditions the energy transferred to the second harmonic did not exceed 1.4% of the incident wave energy. However, the presence of the second harmonic at the lee side of the submerged cylinder led to approximately three-fold increase of the local slopes of the curves ρ(y)=const in the middle of the pycnocline compared to the slopes in the incident wave system. Fig. 7. Thus, the perturbations of the local parameters of the diffracted wave field due to non-linear effects may be considerable. On the other hand, the non-linear mechanism of the energy transfer from low to high frequencies due to interaction of internal waves with complicated bottom topography may be of considerable interest on a large scale. CONCLUSIONS The present study continues the investigations started by the authors in . It is experimentally shown that diffraction of internal waves at the circular cylinder is attended with the effects conditioned by complex modal structure of internal wave field. The characteristics of scattered waves essentially depend on position of the cylinder with respect to pycnocline. The portion of wave energy reflected by the cylinder drastically increases as the cylinder approaches the middle of pycnocline. In the case of a cylinder located outside of the pycnocline the theoretical solution of linear diffraction problem is obtained by multipole expansion method. The theoretical solution clarifies a number of qualitative effects observed in experiments. The present results may be useful for the estimation of the accuracy of the numerical algorithms used for a body of an arbitrary form. An effective numerical method of the solution of the linear diffraction problem for a body submerged outside of the pycnocline is the coupled finite-element method (CFEM). This method was applied to study the diffraction of internal waves by a submerged elliptic cylinder in . The comparison of the numerical results obtained for a circular cylinder in a three-layer fluid by CFEM and the solution given in the present report showed fair agreement. In future, the multipole expansion method with point multipoles can be used to study the diffraction of internal waves at a submerged sphere. The reason for quantitative disagreement between theory and experiments necessitates further investigations. To describe the observed nonlinear effects manifested by the emergence of high-frequency components of wave field, high order theory should be developed. ACKNOWLEDGMENT This work was supported in part by the Siberian Division of Russian Academy of Sciences (SD RAS) under the integrate grant no. 43.
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Twenty-Second Symposium on Naval Hydrodynamics REFERENCES 1. Dean, W.R., “On the Reflection of Surface Waves from a Submerged, Circular Cylinder,” Proceedings of the Cambridge Philosophical Society, Vol. 44, 1948, pp. 483–491. 2. Naftzger, R.A., and Chakrabarti, S.K., “Scattering of Waves by Two-Dimensional Circular Obstacles in Finite Water Depth,” Journal of Ship Research, Vol. 23, No. 1, 1979, pp. 32–42. 3. Evans, D.V., and Linton, C.M., “Active Devices for the Reduction of Wave Intensity,” Applied Ocean Research, Vol. 11, 1989, pp. 26–32. 4. Mallayachari, V., and Sunder, V., “Wave Transformation over Submerged Obstacles in Finite Water Depths,” Journal of Coastal Research, Vol. 12, No. 2, 1996, pp. 477–483. 5. Linton, C.M., and McIver, M., “The Interaction of Waves with Horizontal Cylinders in Two-Layer Fluids,” Journal of Fluid Mechanics, Vol. 304, 1995, pp. 213–229. 6. Sturova, I.V., “Plane Problem of Hydrodynamic Rocking of a Body Submerged in a Two-Layer Fluid without Forward Speed,” Fluid Dynamics, Vol. 29, No. 3, 1994, pp. 414–423. 7. Gavrilov, N., Ermanyuk, E., and Sturova, I., “The Forces Exerted by Internal Waves on a Restrained Body Submerged in a Stratified Fluid,” Preprints of 21st Symposium on Naval Hydrodynamics, Trondheim, Norway, 1996, pp. 254–265. 8. Sturova, I.V., “Effect of Anomalous Dispersion Dependencies on Scattering and Generation of Internal Waves,” Journal of Applied Mechanics and Technical Physics, Vol. 35, No. 3, 1994, pp. 366–372. 9. Grue, J., “Nonlinear Water Waves at a Submerged Obstacle or Bottom Topography,” Journal of Fluid Mechanics, Vol. 244, 1992, pp. 455–476. 10. Phillips, O.M., The Dynamics of the Upper Ocean., 2nd ed., Cambridge Univ. Press, Cambridge e.a., 1977. 11. Gavrilov, N.V., and Ermanyuk, E.V., “Effect of a Pycnocline on Forces Exerted by Internal Waves on a Stationary Cylinder,” Journal of Applied Mechanics and Technical Physics, Vol. 37, No. 6, 1996, pp. 825–831. DISCUSSION J.Grue University of Oslo, Norway The paper describes an interesting study on the interaction between internal waves and a submerged cylinder. Main focus is paid to transmission and reflection properties of the waves. The paper combines theoretical modeling and experiments, which is useful. In the theoretical part the linear diffraction problem in a three-layer fluid due to the presence of a cylinder in the lower layer is solved. Results for the wave field are presented. In addition, the induced forces are calculated in a few examples. The experimental part of the study is both as interesting as it is complex. Interesting linear and nonlinear features of the wave field are reported. I think that the authors could have spent more space to describe experimental results which I believe are available. Both theory and experiments are carried out with thicknesses of the pycnocline, leading to results that describe a broader picture. It would, however, also be interesting if results for an infinitely thin pycnocline (δ/α=0) could be included in figures 2 and 3, for comparison. In view of the force computations in figure 2, did you also in the experiments measure the induced forces? The comparison between the wave computations and the wave measurements is quite good in some cases, but less good in other cases. The disagreement between the two approaches is attributed to the effect of viscosity and a discontinuous density gradient in the three-layer system. I wonder if it is possible to calibrate a set of experiments that fit with the assumptions of the theory, where an eventual disagreement between the two approaches can be reduced to a minimum, at least for some parameters? The results on the second harmonic waves are interesting. I wonder if observations in nature of this phenomenon exist?
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Twenty-Second Symposium on Naval Hydrodynamics AUTHORS’ REPLY In the present experiments the measurement of forces has not been conducted. However, in the Proceedings of the 21st SNH (see reference  of our paper) the authors have presented the results of experimental investigation of loads acting on a submerged elliptical cylinder due to internal waves in stratified fluid having pycnocline. It is the authors’ belief that the qualitative physical effects for the circular cylinder are the same, although some quantitative difference must be expected. In more detail the theoretical results are presented by Sturova . Among other things the results for an infinitely thin pycnocline are included. In the authors’ opinion it is possible to make special calibration or some refinement of experimental technique which can allow a fit with theoretical approach. It seems promising to conduct the measurements of the scattered internal wave field by the wave gauge translated horizontally at small constant speed. In this case the inputs due to different modes (having the same physical frequency but different wavelengths) will be separated due to Doppler effect and, as a result, be easily detectable by Fourier analysis. The analysis of inputs due to isolated modes seems to be easier and a better fit with the present theoretical approach based on presentation of internal waves as a sum of different modes. The generation of internal waves of second harmonic was reported in the oceanographic literature in relation to tidal currents. It was observed that tidal oscillations of stratified fluid in the presence of an underwater mountain can generate internal waves having doubled frequency compared to the frequency of tidal motion. Reference 1. Sturova, I.V., “Diffraction and Radiation Problems for the Circular Cylinder in Stratified Fluid,” (submitted to Fluid Dynamics). DISCUSSION C.M.Linton Loughborough University, United Kingdom Over the past five years or so problems associated with wave diffraction in stratified fluids have received considerable attention. There have been three different types of approach—numerical, analytical and experimental—and all three have a role to play in increasing our understanding in this area. Gavrilov et al.’s paper presents both experimental and analytical work, but in this discussion I will restrict my comments to the analytical part of the paper. As far as I am aware this work represents the first analytical attempt to solve an internal wave diffraction problem in which the interface between the two layers of fluid of different density is modeled by a layer of finite thickness in which the density gradient is large (a so-called pycnocline). This is incorporated into a two-dimensional model in which both the upper and lower layers have finite thickness and are bounded by a rigid wall. The authors have analyzed wave scattering for this fluid configuration using the method of multipoles which typically allows problems in which the scatterer is a horizontal cylinder to be solved efficiently and accurately. This is what Gavrilov et al. have done and they confirm that only a small number of terms are required in the multipole expansions to achieve accurate results. The results which are presented are both useful and interesting, particularly those which show the effect of varying the thickness of the pycnocline on the hydrodynamic characteristics. Multipoles for two finite fluid layers separated by a sharp interface and bounded by rigid walls were first derived by S.E.Kassem (Math. Proc. Camb. Phil. Soc. (1982) 91, 323–329), and the derivation of the multipoles in the present paper represents an extension of Kassem’s work to the case of a finite pycnocline. In the experiments that are described, the upper boundary of the upper layer is not a rigid wall but a free surface and the effect of changing this boundary condition in the theoretical analysis is not discussed. It would be interesting to know if the authors have any quantitative estimates of the effect of this approximation. In Linton & McIver (1995) (cited) multipoles were used to examine
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Twenty-Second Symposium on Naval Hydrodynamics scattering problems which contain both a free surface and a sharp interface, in which case energy can be transferred between internal and free surface waves. Presumably a similar analysis could be performed for the finite pycnocline situation and this would have an effect on the calculated results which may bring the experimental and theoretical results closer together. Another aspect of Linton and McIver’s work was the systematic derivation, from Green’s theorem, of identities connecting the various hydrodynamic quantities which arise, such as reflection and transmission coefficients and exciting forces, and which are analogous to those that exist in standard linear water-wave theory. Presumably similar results exist for a finite pycnocline, both with the free surface approximated by a rigid lid and with the linear free-surface boundary condition applied. Such relations are of intrinsic theoretical importance and can also be used as a check on results obtained from any numerical procedure. AUTHORS’ REPLY We agree that the theoretical analysis can be performed with consideration the free surface on the upper boundary. Effect of the free surface on scattering of the internal waves in the two-layer fluid was studied by Sturova . In the experiments described in the present paper the density variation over depth was very small what implies that the transfer of energy from internal modes to surface motion at free surface has been observed in experiments. Indeed, the identities connecting the various hydrodynamic quantities exist for a finite pycnocline and are obtained by Sturova . References 1. Sturova, I.V., “Scattering of Surface and Internal Waves on Submerged Body,” Computational technology, Vol. 2, No. 4, ICT SD RAS, Novosibirsk, 1993, pp. 30–45 (in Russian). 2. Sturova, I.V., “Diffraction and Radiation Problems for the Circular Cylinder in Stratified Fluid,” (submitted to Fluid Dynamics).
Representative terms from entire chapter: