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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS 848 Reynolds-Averaged Modeling of High-Froude-Number Free- Surface Jets D.Walker (ERIM International, Inc., USA) ABSTRACT For turbulent free-surface flows, the main effects to be considered are modification of the near-surface turbulent structure by the free-surface boundary when the local Froude number is low, and the effect of the fluctuating interfacial forces when the local Froude number is high. This study focuses on the development of a consistent approach to calculating turbulent free-surface flows at arbitrary Froude number. A framework, centered on the Reynolds-averaged Navier-Stokes equations and the code CFDSHIP-IOWA, is described where the standard k-ε turbulence model has been augmented with an algebraic Reynolds stress model to capture near-surface stress anisotropy. In addition, turbulence generated waves have been described using a wave-action model with a source term for the energy transfer from the sub- surface turbulence to unsteady free-surface disturbances. The effect of the surface fluctuations on the flow are modeled by an approximate shear-stress boundary condition at the free surface. Results are presented for predictions of the mean flow and turbulence quantities, as well as free-surface fluctuations, with comparison to experimental data. 1 INTRODUCTION Engineering predictions for turbulent flow near a free-surface are of interest in applications ranging from ship hydrodynamics to manufacturing processes. The nonlinear nature of free surface boundary conditions, along with the nonlinearity of the underlying Navier-Stokes equations, make this problem analytically intractable, and computationally challenging as well. A major impediment to the accurate modeling of turbulent free surface flows, i.e. flows with a gas- liquid interface, has been the lack of an appropriate form for the Reynolds-averaged equations. Unsteady surface- elevation fluctuations cause the boundary to be indistinct, or ‘fuzzy', in the context of the averaged set of equations. Recently this issue was addressed in Hong & Walker (2000) where, starting from the general, variable-property Navier- Stokes equations, the authors derive a set of Reynolds-averaged equations which apply across the entire gas-liquid domain. The resulting governing equations, most conveniently expressed in terms of density-weighted averages, are of the level-set form (see e.g. Chang et al. 1996), where the interfacial boundary conditions are effectively embedded in the field equations. In most approaches where the Reynolds-averaged Navier-Stokes equations are applied to problems of ship hydrodynamics, the form of the equations used assume constant fluid properties (see e.g. Tahara & Stern et al. 1996). They also assume that the air-water interface is idealized to water flowing below a void with zero pressure, density and viscosity. The mean free surface is deformable, but is assumed to be steady, and therefore a well-defined, sharp interface, at which appropriate boundary conditions are applied. While assuming that there is a void above the water is a reasonable approximation for this problem, ignoring the free-surface fluctuations can lead to serious errors in the predicted behavior of the flow. For a high-Froude-number free-surface jet, Hong & Walker (2000) showed that surface-elevation fluctuations help drive secondary flows which can change both the qualitative and quantitative features of the flow. Hence, inclusion of free-surface effects is required to obtain accurate predictions. The objective of this study was to develop an approach to predicting turbulent free-surface flows in the context of the conventional Reynolds-averaged Navier-Stokes equations. In the approach developed, the exact equations derived by Hong & Walker (2000) are reduced to an approximate form for small surface fluctuations. This allows high-Froude- number flows with unsteady surface fluctuations to be treated with a conventional RANS code such as CFDSHIP-IOWA (Stern et al. 1996, Tahara & Stern 1996), rather than using a level-set approach. Surface fluctuations are modeled via coupled solution of a wave-action spectrum model and their effect on the mean flow are the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS 850 The resulting equations (3), (5) and (6), have several desirable characteristics. In the limit of vanishing surface- elevation fluctuations, the mean fluid properties and become constant, the density-weighted averages reduce to simple time averages, and the terms on the third line of (5) vanish. The p.d.f. f(G) becomes a delta function indicating the position of the free surface. Hence, the equations reduce to the conventional RANS equations, but of a level-set form similar to that of Chang et al. (1996). In the limit of laminar flow, they reduce to the exact level-set formulation of Chang et al. It should be noted that the set of equations (3), (5) and (6) are mathematically exact, in the sense that no modeling has been introduced; however, they do not represent a closed system of equations. As is the case with the usual Reynolds- averaging process, additional unknowns are introduced due to the loss of information. For the present set of equations, these additional unknowns are the Reynolds stresses the fluctuating strain rate terms and the p.d.f. of the surface elevation f(G). For the unknowns beyond the Reynolds stresses, appropriate turbulence models must be developed to allow the set of equations to be closed. Approximate Equations for Small Surface Fluctuations One would like to have an approximate form of these equations which can be solved using a conventional RANS solver of the type employed in ship hydrodynamics. To obtain this, the derivatives containing the variable mean fluid properties in the momentum equations are expanded, and then the terms involving gradients of the fluid properties are collected together. This yields a momentum equation of the form (7) If G(xi, t) is defined, for a single-valued free surface, as the normalized distance from the mean free surface as (8) where η′ is the r.m.s. surface elevation, then in the limit of η′ → 0, f(G) becomes a delta function and the terms multiplied by f(G) decouple from the field equations. These terms then become boundary conditions, applied at the location indicated by G=0. Under these circumstances, the the fluid properties in the equations become constant. For small but nonzero η′, it will be assumed that the terms multiplied by f(G) will still decouple in this fashion, and for the momentum equation, the result is (9) where the overbar indicates a simple time-average (equal to the density-weighted average when the density is constant). The boundary conditions for the momentum equation arise from the terms in (2) involving f(G), (10) Evaluating (10) for η′ → 0 requires integrating the expression over the near-surface region defined by f(G)≠0. This integration yields the approximate boundary conditions for the three components of the momentum equation, to be applied at For the sake of simplicity, it will be assumed that the mean free surface is essentially flat, but that the surface-fluctuation level varies in space (η′≠0). Under these conditions, integrating (10) yields (11) (12) the authoritative version for attribution. (13) Here, the boundary condition for the z-momentum equation (11) is unchanged from that for the conventional RANS equations. However, for the x and y-momentum equations, instead of zero shear stress, there is now an apparent shear stress acting on the mean free surface. The magnitude of the applied shear stress is proportional to the local gradient of the surface-elevation variance η′2. This result is consistent with the analytical results of Hong & Walker (2000) for high- Froude-number jet spreading.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS 851 Near-Surface Reynolds Stress Modeling The approximate governing equations are solved using the code CFDSHIP-IOWA. It incorporates a standard k-ε turbulence model with the Reynolds stresses specified in terms of an eddy viscosity as (14) where vt=Cµk2/ε, k is the turbulence kinetic energy, ε is the dissipation rate for k, and Cµ=0.08. Since the products of vt and the normal strains are typically small compared to k, the k-ε model will result in essentially isotropic turbulence (near-equal Reynolds normal stresses). For zero, or low, Froude number where the free surface is essentially flat (i.e. where there are no unsteady turbulence-generated waves) the turbluence becomes anisotropic. The transfer of energy from the vertical w-velocity fluctuations to the other two components ( and ) is captured by the near-surface redistribution model of Daly & Harlow (1970) and Shir (1973). This model, originally intended for implementation as a near-surface (actually near-wall) correction to the pressure-strain correlation (Φij) model for the Reynolds-stress transport equations is given by (15) =k3/2/ε and ℓ t where is the local turbulent length scale. Introducing this into the algebraic stress model formalism of Rodi (1984) yields the following expression for the ‘corrected' Reynolds stress in the near-surface region: (16) Here, on the right-hand side of the expression is the value obtained while neglecting the near-surface effects. Far from the free surface f(z)=0 and, hence, as f(z) → ∞ and so as is required. The and stresses are given by (17) and (18) where it is seen that as the free surface is approached, and are each increased by and amount equal to conserving turbulence kinetic energy. The shear stresses are given by (19) (20) (21) Here, is unaffected, but the and shear stresses are reduced to zero at the free surface, again consistent with the boundary conditions. 2.2 Surface Fluctuation Model Wave-Action Spectrum Model The unsteady free-surface fluctuations are modeld using a wave-action-spectrum model like that described by Komen et al. (1994). This type of model can predict the evolution of the surface-wave directional spectrum, including wave generation, and the interaction of the waves witht the near-surface velocity field. The wave-action spectrum is governed by an advection equation with source terms: (22) where x and y are spatial position variables on the free surface, and and θ are wavenumber and direction variables for the action spectrum ( =2π/λ, where λ is the wavelength). The vectors x=(x, y) and s=( , θ) will be used to represent spatial and spectral position, respectively. N(x, s, t) is the action spectral density defined as (23) where E is the energy spectral density. The frequency σ is related to the wavenumber by the dispersion relation (24) the authoritative version for attribution. The x- and y-direction components of the wave-propagation velocities are given by (25) where and are the x- and y-direction mean velocities at the free surface, and the wave group velocity is (26)

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS 852 The other two velocities Cκ and Cθ are energy propagation velocities in the spectral domain caused by spatial variations in the surface-velocity field. They are given by (27) k and (28) The source term S(k, θ) on the right-hand side of (22) is discussed in the next section. This set of equations can be solved for the action spectrum for spatial region subject to appropriate initial conditions N(x, s, t=0) on and boundary conditions on . For portions of the wave spectrum with propagation velocities which carry energy into , the ‘incident' wave spectrum N(x, s, t) on the boundary is set to zero; for the portions of the wave spectrum for which the propagation velocities carry the energy outward from , an effective ‘outflow' condition is used (continuity of the action flux at ). In the spectral domain, similar conditions, zero for inflow and continuity of flux for outflow are applied at the upper and lower wavenumber ( k) boundaries. The boundary conditions in θ are that the spectrum is periodic. The velocity field is an input to the wave spectrum model and is obtained from the RANS solution. Turbulent Source Term The unsteady pressure fluctuations in the subsurface flow cause the free surface to move and, as a result, generate waves. In this process energy is transferred from the subsurface turbulence to the wave field. The two-dimensional wave- generation spectrum S(k , θ), which appears on the right-hand side of the wave action balance equation represents this transfer process. This term is derived from the three-dimensional energy spectrum for isotropic turbulence, combined with the dynamic free surface boundary condition. The isotropic pressure spectrum, for the inertial range is given by (29) where k is the three-dimensional wave vector, and This three-dimensional spectrum is integrated to obtain a two-dimensional spectrum by integrating over one Cartesian coordinate direction. This yields (30) for the two-dimensional spectrum which is a function of only. It is independent of direction θ due to the isotropic nature of the original pressure spectrum. Combining this with the dynamic free surface boundary condition (in inviscid form) yields (31) which is applied for k ≥2π/ℓt; S(k )=0 for k < 2π/ℓt. Here, Frt=(k/gℓt)1/2 and W et=γ/k ℓt are the local turblent Froude and Weber numbers based on k and the turbulent length scale ℓt for the given free surface location, and the constant was determined by matching the peak of the predicted surface elevation variance to experiment. 3 NUMERICAL IMPLEMENTATION The computations for this study were carried out using a modified version of CFDSHIP-IOWA. Changes were made to incorporate the near-surface stress anisotropy, coupled solution of the wave-action spectrum model, and the approximate dynamic free-surface boundary condition which relates the apparent stress on the surface to gradients in the surface fluctuation variance. Modeling of the near-surface stress anisotropy was accomplished in the context of the basic k-ε model which is included in CFDSHIP-IOWA. The ability to treat non-isotropic turbulence was added by including terms in the momentum equations which represent the deviations of the Reynolds-stress gradients from those predicted by the standard k-ε model. The form of the RANS equations then looks like (32) the authoritative version for attribution. where These additional source terms in the momentum equations are implemented using second-order finite difference approximations.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS 853 The solution for the wave-action spectrum was obtained using a new subroutine in the CFDSHIP-IOWA code. The new subroutine implements a discrete approximation of the the advection equation governing the wave action spectrum using a fully-implicit scheme based on first-order upwind differencing for the advective terms. (A third-order upwind scheme is currently being implemented.) The wave spectrum solution requires the surface values of the mean velocities, as well as k and ε and is advanced in time in a coupled fashion with the RANS solution for the sub-surface flow. The wave spectrum is integrated to determine the surface-elevation variance η′2. These data are returned to the RANS solver for use in specifying the apparent stress acting on the free surface. Calculations were carried out for a rectangular volume 48d×20d×26d in the x, y and z directions respectively, where d is the jet-exit diameter. The free surface was located at z=0, y=0 corresponded to the vertical symmetry plane of the jet, and the jet exit was located at z=−2d and y=0 on the plane x=0. The CFDSHIP-IOWA code was implemented on a non- uniform Cartesian grid with 33×32×47 grid points in the x, y and z directions respectively, shown in figure 1. Zero velocity was prescribed on the x=0 boundary, except in the region of the jet orifice (a circle of diameter d centered at x=0, y=0 and z=−2d). The flow was assumed symmetric about the plane y=0. Outflow boundary conditions were used for the plane z=48d. The free surface was treated as a flat, shear-free boundary The remaining boundaries, the bottom at z= −26d and outboard boundary at y=20d are actually noslip boundaries, but are treated as symmetry planes to minimise grid density requirements (the velocities are near zero at the boundaries, anyway). The spatial grid for the wave-action conservation equation corresponded to the RANS grid on the free surface and consisted of 33×32 grid points. The spectral grid consisted of 32 directions equally spaced over 0≤θ<2π and 40 logarithmically spaced wavenumbers. The range of wave numbers was covered physical wavelengths from d/30 to 30d. The wave action equation was advanced in time using local time-stepping, where the time-step was based on the local values of the spatial and spectral advection velocities. The large range of wavenumbers required results in a large range of wave group velocities, and the lower velocity waves converge very slowly without local time stepping. In the computations, and in the results to be presented, all quantiteis are normalized twiththe jet exit diameter d, and the jet-exit velocity Ue. 4 RESULTS In this section, results will be presented for three cases. The first case will be a low-Froude-number jet, Fr=0 and Re=12 700. For this case, the main effect of the free surface is the anisotropy of the near-surface Reynolds stresses. These computational results will be compared to experimental data for Fr=1.0 and Re=12 700 (Hong & Walker 2000), where there is little wave generation. The second case examined will be for Fr=Ue/(gd)1/2=8 and Re=Ued/v=12 700, where the wave-action equation is solved for the unsteady turbulence-generated waves. The surface-fluctuation model in this case is ‘one-way' coupled, in the sense that the surface-fluctuation variance is not ‘fed back' into the freesurface boundary conditions. This shows that the wave-action model produces reasonable results, when compared to surface-fluctutation measurements. The final case is a two-way coupled high-Froude-number prediction, with Fr=8 and Re=12 700. Here the near-surface correction which produces the turbulent-stress anisotropy is disabled, but the gradient of the surface- fluctuation variance is used for the free-surface boundary conditions. These two-way coupled results are compared to experimental data for both the velocity field and the free-surface. In the sections that follow, the discussion of results will center on comparisons of transverse planes of experimental and computational results. Comparisons of computed and experimental results for surface fluctuations in on the free- surface plane will also be presented. For reference, and to orient the reader, figure 2 shows the predicted mean streamwise velocity for the entire computational volume. The vertical symmetry plane plane of the jet is shown, as are the far downstream exit plane for the volume, and the free-surface plane. The origin of the jet at the upstream inlet plane of the volume at x=0 is clearly visible, as is the decay in velocity with streamwis distance and interaction with the free surface. ‘Floating' above the volume in figure 2 is a plane showing the computed energy distribution η′2/d for the free-surface fluctuations. 4.1 Modeling of a Low-Froude-Number Jet The first case to be examined is a low-Froude-number jet, where Fr=0 and Re=12 700. The free surface will be taken to be flat, and devoid of any unsteady surface fluctuations. For this case, the main free-surface effect is the requirement that at the free surface, resulting in anisotropy in the near- the authoritative version for attribution.

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As the solution converged, the downstream portion high-velocity ‘core' decelerated and spread in space to take on the characteristics of a jet. The experimental results for free-surface jets presented in Hong & Walker (2000) show that for Fr= 1.0 there is very little mean deformation of the free surface and small surface fluctuations (both are on the order of a few thousandths of a jet diameter). Hence, their Fr=1.0 results are essentially for a ‘flat' free surface with no unsteady surface fluctuations, and are used here for comparison to the computations for Fr=0. Figure 3 shows results for the a transverse (y, z) plane at x/d=16. Only the portion of the plane corresponding to the region where experimental measurements are available is shown. In the rest of the plane the velocity and turbulence kinetic energy are essentially zero. For the mean streamwise velocity, the maximum level observed in the plane for the experiments (figure 3a) and computations (figure 3b) are comparable, as is the overall shape of the jet cross-section. The cross-stream, or secondary-flow velocity vectors are shown in figures 3(b) and 3(c). The overall pattern of the velocity vectors is simlar; however, the magnitude of the velocities at the free surface for this location is low by about a factor of two. The levels of turbulence kinetic energy from the experiments shown in figure 3(e) are comparable to those from the computations shown in figure 3(f). A similar set of results for a transverse plane at x/d=32 is shown in figure 4. Again, only the portion of the plane corresponding to the region where experimental measurements are available is shown. For the mean streamwise velocity, the maximum level observed in the plane for the experiments (figure 4a) and computations (figure 4b) are comparable. Again, as at x/d=16, the overall shape of the jet cross-section for the computations is similar that observed in the experiments, but the jet appears to have been shifted upwards toward the free surface slightly. At the free surface, the velocity vectors are comparable in magnitude to those in the experimental data. The levels of turbulence kinetic energy from the experiments shown in figure 4(e) are again comparable to those from the computations (figure 4f). From these results, it is clear that the CFDSHIP-IOWA code with the addition of the near-surface correction to the Reynolds stresses described above, does a reasonably good job of predicting the overall evolution of a low-Froude- number turbulent free-surface jet over the range of streamwise positions examined. 4.2 One-Way-Coupled Modeling for Unsteady Waves To predict the characteristics of the turbulence-generated surface fluctuations the wave-action balance model is used. To calculate the surface fluctuations, the mean surface velocity field ( and ), as well at the turbulence kinetic energy k and the dissipation ε, are required. For a fully-coupled approach, the resulting surface-elevation variance would then be used in calculating the apparent stress acting at the free surface. For a one-way-coupled approach, the results from the subsurface flow are used in calculating the surface fluctuations, but the surface fluctuations are not used in calculating the sub-surface flow. In this approach, the sub-surface flow is calculated assuming the flow is at zero Froude number. In this case, the sub-surface flow behaves exactly as that described in the previous section, and the only difference is that now, the surface fluctuations are calculated, as well. This was accomplished by advancing the wave-action conservation equation in time in conjunction the Reynolds-averaged equations. The flow was initialized in a similar fashion as that used above, with the additional provision that the wave spectrum was initialized to zero at all locations. The calculataions, again converged in 5000 time steps; since the wave-action equation in the form used here is linear, it imposes no additional difficulties in obtaining a converged solution. Figure 5 show the root-mean-square (r.m.s.) surface fluctuation level η′/d for the one-way coupled computations (figure 5a) along with those observed experimentally (fivure 5b) for the jet with Fr=8.0 and Re=12 700. For both, the peak in η′/d is located above the jet axis at about x/d=20. The peak levels match exactly because the experimental data was used to determine the constant in the turbulent source term in (31) above. (It sould be noted, however, that the constant just scales the magnitude of η′, the spatial distribution is set by the spectral shape of the source term and the spatial variations in the velocity field and k and ε.) The spatial distribution of η′ from the computations appears to be slightly narrower in the y-direction and sligtly longer in the x-direction than the experimental data indicate, but overall, the agreement is quite good. The r.m.s. surface-fluctuation level η′ is an integral measure of the predicted wave spectrum. While the the authoritative version for attribution.

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This is an area of active investigation. 4.3 Two-Way-Coupled Modeling of a High-Froude-Number Jet The final set of results to be described are for a two-way-coupled computation of a high-Froude-number jet where the surface fluctuation variance is used in specification of the apparent stress acting on the free surface. In this case, the Froude number was set to Fr=8.0 and was assumed to be high-enough that there would be no surface-induced anisotropy in the near-surface turbulence. Hence, the near-surface stress redistribution model was disabled. The computations were carried out in a manner similar to that described above for the other cases, again converging in roughly 5000 time steps. The results for η′ will be presented first followed by comparisons of experimental and computational results for x/d=16 and 32. The two-way coupled computational results for the r.m.s. surface fluctuation level η′ is shown in figure 6, again along with experimental data for Fr=8 and Re=12 700. The results differ from the one-way coupled results discussed above in that the the peak level of η′ is slightly large and the elevated region extends further downstream. This is traced to higher turbulence level initially in the jet and a higher streawise velocity, both due to inaccuracies in the predicted sub- surface flow. In general though, the gradients in η′2, which were used in the specification of the apparent stresses acting on the surface, would be equal to or greater than the correct values. The computational results for transverse planes at x/d=16 are shown in figure 7, compared to experimental data for Re=12 700 and Fr=8.0. (The experimental data used here is different from that used above which was for Fr=1.0). It is seen that the computed mean velocity and turbulence kinetic energy levels (figures 7b and 7f) are higher than those seen in the experiments (figures 7a and 7e). In addition, at x/d=16 the jet appears to be slightly farther away from, and interacting less with the free surface. The most striking difference is in the cross-flow velocity vectors; in the computations there is at most only a small outward flow at the free surface. While at this location the near-surface gradient in η′2 is nonzero, it is apparently not sufficient, or acts over too small of an area to cause the experimentally observed outward flow. Figure 8 shows the computational results for transverse planes at x/d=32, compared to experimental data for Re=12 700 and Fr=8.0. For this streamwise location the computed mean streamwise velocity is slightly elevated (figure 8b), but the turbulence kinetic energy level (figure 8f) is about the same those seen in the experiments (figures 8a and 8e). In addition, the localized spreading of the streamwise velocity distribution at the free surface seen in the experimental data in figures 8(a) is less apparent in the computations shown in figures 8(b). Again the major difference is in the cross-flow velocity vectors; in the computations there is again only a small outward flow at the free surface. The results in this section show that in a two-way-coupled approach, the magnitude of the r.m.s. surface fluctuations, and the location of the peak are reasonably well predicted the streamwise extent of the peak is over-estimated, with the elevated region extending too far downstream. Even so, the spreading of the jet near the free surface observed in the experimental data is not captured accurately. This is most likely related to the high-Froude-number nature of the modeling being used. The underlying assumption of the modeling is that there is no near-surface stress-anisotropy, and the free- surface effects are confined completely to the apparent stress resulting from the gradient in η′2. This may be an over- simplification of the problems, since the results of Hong & Walker (2000) show that, while the levels of anisotropy are smaller in high-Froude-number jets, anisotropy becomes more important with increasing streamwise distance, and can still affect the behavior of the flow. An effective, rational method for blending the high- and low-Froude-number approaches, turning on the anisotropy where appropriate, has yet to be developed. 5 SUMMARY AND CONCLUSIONS The objective of this study was to develop an approach to predicting turbulent free-surface flows in the context of the conventional Reynolds-averaged Navier-Stokes equations. In the approach developed, the exact equations derived by Hong & Walker (2000) are reduced to an approximate form for small surface fluctuations. This allows high-Froude- number flows with unsteady surface fluctuations to be treated with a conventional RANS code such as CFDSHIP-IOWA (Tahara & Stern 1996), rather than using a level-set approach. Surface fluctuations are modeled via coupled solution of a wave-action spectrum model their effect on the mean flow are captured using an approximate stress condition at the free surface. The effects the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS 856 of the free surface on the sub-surface turbulence at low Froude number are captured using a near-surface anisotropy model. The approaches developed were applied to turbulent free-surface jet flows, and comparisons of the results to experimental data were made. First a low-Froude-number jet was calculated, where there was little wave generation and near-surface anisotropy dominates the behavior of the flow; here the approach performed quite well when compared to low-Froude-number jet data. A one-way-coupled prediction of a high-Froude-number jet was then presented, where the surface fluctuations were predicted using the wave-action model, but the surface fluctuations were not used in the boundary conditions for the subsurface flow (i.e. the subsurface flow is treated as a low-Froude-number flow). The predicted surface-fluctuation distribution was in good agreement with the observations. Finally, a two-way coupled calculation was presented, where the surface fluctuations were used in the boundary conditions on the subsurface flow. This agrees less well with the experiments; the magnitude of the outward velocity at the free surface was substantially under-estimated. This is believed to be due mainly to a lack of an effective method for switching between the low-Froude- number and high-Froude-number treatment of the free surface. ACKNOWLEDGMENT This is work was supported by the Office of Naval Research under Contract Nos. N00014–99-M-0082 and N00014– 00-C-0057 monitored by Dr E.P.Rood. REFERENCES [1] CHANG, Y.C., HOU, T.Y., MERRIMAN, B. & OSHER, S. 1996 A level-set formulation of Eulerian interface capturing methods for incompressible fluid flows. J. Comp. Phy. 124, 449–464. [2] HINZE, J.O. 1975 Turbulence. McGraw-Hill, New York. [3] DALY & HARLOW 1970 Transport equations in turbulence. Phys. Fluids 13, 2634–2649. [4] HONG, W.-L. & WALKER, D.T. 2000 Reynolds-averaged equations for free-surface flows with application to high-Froude-number jet spreading. J. Fluid Mech. in press. [5] KOMEN, G.J., CAVALERI, L., DONELAN, M., HASSELMANN, K., HASSELMANN, S. & JANSSEN, P.A.E.M. 1994 Dynamics and Modeling of Ocean Waves, Cambridge. [6] SHIR, C.C. 1973 A preliminary nemerical study of atmospheric turbulent flow in the idealized planetary boundary layer. J. Atmos. Sci. 30, 1327. [7] RODI, W. 1984 Turbulence Models and Their Application in Hydraulics, IAHR. [8] STERN, F., PATERSON, E.G. & TAHARA, Y. 1996 CFDSHIP-IOWA: Computational Fluid Dynamics Method for Surface-Ship Boundary Layers, Wakes, and Wave Fields, IIHR Report No. 381, Iowa Institute of Hydraulic Research, University of Iowa. [9] TAHARA, Y. & STERN, F. 1996 A large-domain approach for calculating ship boundary layers and wakes for nonzero Froude number. J. Comput Physics 127, 398–411. the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Figure 2: Computational volume showing results for the mean streamwise velocity and the surface fluctuation variance η′2. Figure 1: Spatial grid used in computations REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS 857

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858 REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS Figure 4: Comparison of low-Froude-number experimental and computational results for mean streamwise velocity, cross stream velocity vectors and turbulence kinetic energy level at x/d=32: (a) from experiment; (b) from computations; (c) vectors from experiment; (d) vectors from computations; (e) from experiment; from computations; (Experimental results are for Fr=1.0 and computational results are for Fr=0; Re=12 700 for both) (f) Figure 3: Comparison of low-Froude-number experimental and computational results for mean streamwise velocity, cross stream velocity vectors and turbulence kinetic energy level at x/d=16: (a) from experiment; (b) from computations; (c) vectors from experiment; (d) vectors from computations; (e) from experiment; from computations; (Experimental results are for Fr=1.0 and computational results are for Fr=0; (f) Re=12 700 for both) the authoritative version for attribution. lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line

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859 REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS Figure 5: Comparison of one-way-coupled computational results and experimental and results for root-mean-square surface fluctuation η′: (a) computational results for η′ for Fr=8.0 using the mean velocity and turbulence quantities from the low- Froude-numebr results above; (b) experimental results for η′ for Fr=8.0 Figure 6: Comparison of two-way-coupled computational results and experimental and results for root-mean-square surface fluctuation η′: (a) computational results for η′ for Fr=8.0 using the mean velocity and turbulence quantities from the low- Froude-numebr results above; (b) experimental results for η′ for Fr=8.0 the authoritative version for attribution. lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line

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860 REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS Figure 7: Comparison of high-Froude-number experimental and computational results for mean streamwise velocity, cross stream velocity vectors and turbulence kinetic energy level at x/d=16: (a) from experiment; (b) from computations; (c) vectors from experiment; (d) vectors from computations; (e) from experiment; (f) from computations; (Both experimental and computational results are for Fr=8.0 and Re=12 700.) Figure 8: Comparison of high-Froude-number experimental and computational results for mean streamwise velocity, cross stream velocity vectors and turbulence kinetic energy level at x/d=32: (a) from experiment; (b) from computations; (c) vectors from experiment; (d) vectors from computations; (e) from experiment; from computations; (Both experimental and computational results are for Fr=8.0 and Re=12 700.) (f) the authoritative version for attribution. lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS 861 DISCUSSION D.G.Dommermuth Science Application International, USA The author has derived equations for Reynolds-stress terms based on a two-phase approximation. A similar set of equations could be derived based on a mapping that uses the exact instantaneous position of the free surface. Could the author please contrast the two approaches from a numerical perspective? AUTHOR'S REPLY The Reynolds-averaged form of the Navier-Stokes equations used in this study were developed by averaging the instantaneous equations at a point in Eulerian space. The alternative approach suggested would be to transform the equations into a surface-mapped coordinate system, and then average there. While this would allow the use of turbulence modeling approaches based on instantaneous distance from the free surface, additional terms would be introduced into the averaged equations by the coordinate transformation. These additional terms would involve the time-averages of the instantaneous coordinate-transformation metrics. It is not clear what the numerical implications would be for this type of approach, but this could complicate other aspects of the problem significantly. Among the complications would be the fact that the turbulence modeling would then be carried out in a non-inertial reference frame, and establishing the relationship between conventional turbulence measures, and those needed to validate the new modeling approach. DISCUSSION M.C.Hyman Coastal Systems Station, USA This paper documents an excellent attempt at understanding the phenomena occurring at and near the free-surface forced by turbulence. In previous works, the author has shown the macro-scale effects of free-surface/turbulence interaction and successful methods of modeling that interaction for low Froude number jets. By construction, these models neglect energy flux due to wave formation and propagation, an assumption that is not correct for high Froude no. flows. The present paper looks more specifically at high Froude no. flow and includes a mechanism for describing energy transport due to wave formation. To achieve this, a spectral transport model is coupled with a flow solver (with appropriate Reynolds stress modeling). The spectral wave model is forced by pressure fluctuations generated by the underlying turbulent flow and the resulting averaged free surface topology is further modified by the near surface mean flow. The turbulent source term is extremely important. It is obtained by assuming that the pressure spectrum is isotropic and unaltered by the presence of the free surface. The spectral wave model can be coupled back to the mean flow solver via the free surface elevation field. Results are provided first for the low Froude no. jet with essentially zero free surface deformation and negligible wave dissipation. These results show the utility of applying anisotropy to the near-surface turbulence and illustrate the formation of a surface current. The results also suggest that the layer over which the anisotropy exists may be thinner that the model predicts. Next, the model is applied to a high Froude no. jet with both one-way and two-way coupling. Shown in the form of free surface elevation, the results seem to suggest that the coupling is one way. This is counterintuitive and merits some further consideration. The results were obtained by neglecting anisotropy in the turbulence field, assuming that since the free surface is allowed to deform, turbulent fluctuations are uninhibited and anisotropy cannot develop (use of an isotropic pressure spectrum as a forcing function for the wave model is consistent with this assumption). What may be occuring is that the free surface deforms due to larger turbulent length scales but continues to act as a rigid wall for the smaller length scales. The results show that the surface current at high Froude no. remains (although is less energetic) but is not reproduced in the computations. In addition, it seems unlikely that even the larger scales remain isotropic near the surface—the vertical fluctuations are doing work against gravitational forces and must be damped to some degree. Thus, I would like to ask the author if the possibility of imposing a scale dependent anisotropy model has a chance of success. Such a model would act on the spectrum in nearly the opposite manner from equilibrium turbulence in that it would reduce isotropy at the smaller scales. AUTHOR'S REPLY While it is possible that there is a length-scale dependence in the turbulence anisotropy, it is not clear how that could be implemented in a Reynolds-averaged model. The averaging process would effectively ‘blend' the two effects together and result in some intermediate, but representative, level of anisotropy. A more-likely possibility, in the author's opinion, is that even at high Froude number, there are regions of the flow where the turbulence is not the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. affect the development of the mean flow. REYNOLDS-AVERAGED MODELING OF HIGH-FROUDE-NUMBER FREE-SURFACE JETS 862 energetic enough to disturb the free surface. In these regions, the near-surface turbulence will be anisotropic and will