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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
SESSION 5
WAVY/FREE-SURFACE FLOW: VISCOUS FLOW AND INTERNAL WAVES

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
Solid-Fluid Juncture Boundary Layer and Wake with Waves
J.E.Choi and F.Stern
(University of Iowa, USA)
ABSTRACT
Laminar and turbulent solutions are presented for the Stokes-wave/flat-plate boundary -layer and wake for small-large wave steepness, including exact and approximate treatments of the free-surface boundary conditions. The macro-scale flow exhibits the wave-induced pressure-gradient effects described in precursory work. For laminar flow, the micro-scale flow indicates that the free-surface boundary conditions have a profound influence over the boundary layer and near and intermediate wake: the wave elevation and slopes correlate with the depthwise velocity; the streamwise and transverse velocities and vorticity display large variations, including islands of maximum/minimum values, whereas the depthwise velocity and pressure indicate small variations; significant free-surface vorticity flux and complex vorticity transport are displayed; wave-induced effects normalized by wave steepness are larger for small steepness with the exception of wave-induced separation; order-of-magnitude estimates are confirmed; and appreciable errors are introduced through approximations to the free-surface boundary conditions. For turbulent flow, the results are similar, but preliminary due to the present uncertainty in appropriate treatment of the turbulence free-surface boundary conditions and meniscus boundary layer.
NOMENCLATURE
A
=wave amplitude
Ak
=wave steepness
Fr
=Froude number
g
=gravitational acceleration
k
=turbulent kinetic energy
=wave number
L
=body characteristic length
n
=unit normal vector
o( ), ( )
=order of magnitude
p
=piezometric pressure
p*
=static pressure
q
=free-surface vorticity flux (=qx,qy,qz)
qw
=wall vorticity flux (=qwx,qwy,qwz)
Re
=Reynolds number (=UoL/v)
u,v,w
=fluctuating velocities
=reference velocity
−uiuj
=Reynolds shear stresses
V
=mean-velocity vector (= U,V,W)
x,y,z
=Cartesian coordinates
δ
=body (δb) or free-surface (δfs)
boundary-layer or wake (δW) thickness
δ*
=streamwise displacement thickness
Δ
=difference between zero and nonzero wave-steepness values of
ε
=rate of turbulent energy dissipation
=boundary-layer and wake thickness
=transport quantities (= U,V,W,k,ε)
=relevant variable or equation
η
=wave elevation
λ
=wave length
μ
=viscosity
v
=kinematic viscosity (=μ/ρ)
ρ
=density
=wall-shear stress
=fluid stress tensor
=external stress tensor
ξ,η,ζ
=nonorthogonal curvilinear coordinates
ω
=mean vorticity vector (=ωx,ωy,ωz)
INTRODUCTION
Ship boundary layers and wakes (blw's) are unique in that they are influenced by the presence of the free-surface and gravity waves. The wave pattern, breaking, and -induced separations along with turbulence/vortex/free-surface interaction, bubble entrainment, etc. are key issues with regard to performance prediction, signature reduction, and propeller-hull interaction.
In spite of this, until fairly recently, very little detailed experimental or rigorous theoretical work has been done on this problem. In

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
particular, over the past ten years, the Iowa Institute of Hydraulic Research (IIHR) has carried out an extensive experimental and theoretical program of research concerning free-surface effects on ship blw' s: problem formulation and model problem identification [Stokes-wave/flat-plate (Sw/fp) flow field] and calculations [1]; towing-tank experiments for idealized (foil-plate model which simulates the Sw/fp flow field) and practical hull form (Series 60 CB=.6) geometries [2–5]; and the development of computational fluid dynamics (cfd) methods, including validation studies for the foil-plate model [3] and Wigley [6] and Series 60 CB=.6 [7] hull forms. Through this work, significant progress has been made in explicating certain features of the flow physics (e.g., wave/blw interaction, including the role of wave-induced pressure gradients, wave-induced separation, and scale-effects on near-field wave patterns) and identifying issues for further study (e.g., the nature of the flow very close to the free surface, including the role of the free-surface boundary conditions and the structure of turbulence, effects of geometry and turbulence on wave-induced separation, wake bias, and pacesetting issues for cfd advancements).
This paper concerns one of the aforementioned issues for further study, i.e., the nature of the flow very close to the free surface, including the role of the free-surface boundary conditions. Laminar-flow solutions are presented for the Sw/fp flow field, including the exact free-surface boundary conditions. The work presents for the first time solutions to the exact governing Navier-Stokes (NS) equations and boundary conditions for a solid-fluid juncture blw with waves. Some additional turbulent-flow solutions are also presented; however, these are preliminary due to the current uncertainty in prescribing appropriate turbulence free-surface boundary conditions and treatment of the meniscus boundary layer.
The complete results are extensive and provided by Choi [8]. In the following, the most important aspects of the solutions are discussed and example results are presented. First, overviews are given of the physical problem, including order-of-magnitude estimates (ome), and precursory and relevant work, and the computational method. Then the computational conditions, grids, and uncertainty are described and results presented and/or discussed for small, medium, and large wave-steepness Ak (where A and k are the wave amplitude and number, respectively) for laminar and turbulent flow. Lastly, a summary and conclusions are made, including recommendations for future study and implications with regard to practical applications.
PHYSICAL PROBLEM AND PRECURSORY WORK
Consider the development of the blw for a ship moving steadily at velocity Uo in an incompressible viscous fluid (figure 1). Following [1], the flow in the neighborhood of the body blw/free-surface juncture is divided into five regions (figure 2): (I) potential flow; (II) free-surface boundary layer; (III) body blw; (IV) solid-fluid juncture blw with waves; and (V) meniscus boundary layer.
The flow in region I is well known, i.e., ome and analytical and cfd methods are well established. The situation is similar for region II, at least for laminar flow, e.g., the analytical solution provided in Appendix A of [8]. Table 1 of [8] provides inviscid and viscous Stokes-wave solutions for regions I and II. In region III, the effects of the free surface are primarily transmitted through the external-flow pressure field and, here again, ome and cfd methods are available. The precursory work mentioned earlier and described later has been very successful in documenting the nature of the flow in this region. In region IV, the effects of the free surface are due both to the influences of the external-flow pressure field and the kinematic and dynamic requirements of the free-surface boundary conditions, which alters both the mean and turbulent velocity components. Presently, the only available information for region IV is that provided by [1], i.e., ome and preliminary calculations for the Sw/fp flow field, including approximate free-surface boundary conditions. Some relevant work, which is also useful in understanding the flow in region IV is described later. Region IV is the topic of this paper. The flow in region V is presently poorly understood, involving surface-tension and contact-line effects. Region V is neglected in this paper, but, as discussed later, recommended for future study.
Order-of-Magnitude Estimates
[1] provides a discussion of the ome for regions I through III and a derivation for those for region IV in connection with the determination of appropriate small-amplitude-wave and more approximate free-surface boundary conditions. In regions I through III, the important nondimensional parameters are Ak and Reynolds number (Re) or related blw thickness ε=δ/L or δ/λ (where δ is the body or free-surface boundary-layer or wake thickness, δb, δfs, δW, respectively, L is the body characteristic length, and λ=2π/k is the wave length). For sufficiently

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
large Re and slender bodies, the ome for regions I through III in terms of these parameters are provided in table 1. In region IV, the ome were derived in consideration of both those for region III, with the assumption of thin-boundary-layer theory, and the requirements of the free-surface boundary conditions. Additionally, Ak/Ɛ is shown to be an important parameter. However, the assumption of thin-boundary-layer theory for region III led to an error for one of the estimates, i.e., Wz1; therefore, an updated derivation is provided as follows.
In consideration of the flow in regions I and III, the ome for V=(U,V,W), η, ∂/∂x, and ∂/∂y are:
V=(1,ε,Ak)
η=(Ak)
∂/∂x=(1)
∂/∂y=(ε−1) (1)
Next, the normal and tangential dynamic and continuity-equation free-surface boundary conditions (see later), respectively
(2)
(3)
(4)
(5)
are used, i.e., using (3)–(5), respectively, to eliminate Uz, Vz, and Wz in (2), solving for p, and with (1) results in the ome for p:
(6)
Finally, using (3)–(5) with (1) and (6) results in the ome for Uz, Vz, and Wz, respectively:
(7)
(8)
(9)
The ome for region IV are provided in table 1 and, as will be shown later, are confirmed by the present results. Note that the only differences with those provided previously by [1] are for Wz, as mentioned earlier, and that a single estimate is not provided for ∂/∂z.
Thus far, no distinction has been made between the flow in the blw regions, which is not necessary, except for the far-wake (fw) region, i.e., hereafter, the blw refers to the boundary-layer and near- and intermediate-wake in distinction from the fw. The fw requires a different ome derivation. In this case, in consideration of the flow in region I and the asymptotic two-dimensional zero-pressure gradient fw solution [9], the ome for V, ∂U, η, ηx, ηy, ∂/∂x, and ∂/∂y are:
(10)
Next, following the usual derivation for region III both for the blw and a similar derivation as provided earlier for region IV both for the blw, the ome for regions III and IV for the fw are derived. These are also provided in table 1 and, here again, as will be shown later, are confirmed by the present results.
1
Subscripts are used to denote derivatives, as indicated here, or in defining certain variables, as indicated in the Nomenclature.

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
Regions III Calculations and Experiments
[1] identified the model problem of a combination Sw/fp flow field (figure 3), which facilitated the isolation and identification of the most important features of the wave-induced effects. Numerical results were presented for laminar and turbulent flow utilizing first-order boundary-layer equations and both small-amplitude-wave and more approximate zero-gradient free-surface boundary conditions. Subsequently [2], results from a towing-tank experiment were presented utilizing a unique, simple foil-plate model geometry, which simulates the Sw/fp flow field. Mean-velocity profiles in the boundary-layer region and wave profiles on the plate were measured for three wave-steepness conditions. For medium and large steepness, the variations of the external-flow pressure gradients were shown to cause acceleration and deceleration phases of the streamwise velocity component and alternating direction of the crossflow, which resulted in large oscillations of the displacement thickness and wall-shear stress as compared to the zero-steepness condition. The measurements were compared and close agreement was demonstrated with the results from the turbulent-flow calculations with the zero-gradient approximation for the free-surface boundary conditions. Also, wave-induced separation was discussed, which was present in the experiments, and the starting point was predicted by the laminar-flow calculations under certain conditions.
More recently [3], results were presented from extensions of both the previous experimental and theoretical work: the measurement region was extended into the wake where both mean-velocity and wave-elevation measurements were made; and a state-of-the-art cfd method was brought to bear on the present problem, in which the Reynolds-averaged NS (RaNS) equations were solved for the blw region with zero-gradient free-surface boundary conditions. Measurements and calculations were performed for the same three wave-steepness conditions. The trends were even more pronounced for the wake than shown previously for the boundary-layer region. Remarkably, the near and intermediate wake displayed a greater response, i.e., a bias with regard to favorable as compared to adverse pressure gradients [8]. The measurements were compared and close agreement was demonstrated with results from the RaNS calculations. Additional calculations were presented, including laminar-flow results, which aided in explicating the characteristics of the near and intermediate wake, the periodic nature of the fw, and wave-induced separation.
Very little is known about wave-induced separation, i.e., three-dimensional boundary-layer separation near the free-surface induced by waves and accompanied by a large disturbance to the free surface itself. The additional laminar-flow computational results of [3] enabled, for the first time, a detailed study, including the flow pattern in the separation region. A saddle point of separation and a focal point of attachment were indicated on the plate and mean free surface, respectively. As the wave steepness increased, the saddle point moved downwards and towards the trailing edge, whereas the focal point moved downstream and away from the plate surface. The U and W components displayed, respectively, flow reversal and complex S-shaped profiles. A longitudinal vortex was generated in which the vortical motion was counterclockwise with respect to the flow direction and towards the free surface and clockwise with respect to the flow direction and in the main stream direction above (i.e., in the reverse-flow region) and below the saddle point, respectively. The identification of these features of wave-induced separation was considered very significant and invaluable, but with some caution and to some extent preliminary due to the approximate nature of the free-surface boundary conditions used in the calculations. This paper also addresses this issue.
RELEVANT WORK
Relevant work concerns viscous-free-surface flow, i.e., solutions of the viscous-flow equations, including various treatments of the free-surface boundary conditions, for a variety of applications, i.e: free-wave problems, open-channel flow, free-surface jet flow, vortex/free-surface interaction, and ship blw's for nonzero Froude number (Fr). In [8], the critical issues with regard to the implementation of the free-surface boundary conditions and the various treatments utilized are discussed. Also, for the latter applications, the most important results are summarized both with regard to experimental information and physical understanding and computational studies. The discussions are useful for the evaluation and interpretation of the present solutions with regard to their significance and the role of the free-surface boundary conditions. The conclusions with regard to the relevant work are summarized as follows (see [8] for references).
A variety of cfd formulations of viscous-free-surface flow are possible with the ability to predict a wide class of flows; however, none have fully implemented the free-surface boundary

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conditions or resolved the free-surface boundary layer and taken into account complicating factors such as the meniscus boundary layer, etc. Turbulence free-surface interaction has been investigated for certain idealized geometries (i.e., open-channel flow, free-surface jet flow, and a submerged tip vortex) all of which indicate similar free-surface effects: constant turbulent kinetic energy (i.e., zero gradient) with a redistribution of energy between the turbulence velocity components, i.e., the vertical turbulence velocity is damped and the horizontal and streamwise components are increased. Free-surface jet flow also displays effects due to jet-induced waves and free-surface induced lateral spreading. The idealized geometries are different than ship blw's in that the source of turbulence does not pierce the free surface and the role of gravity waves is minimal. Considerable experimental and computational information is available for vortex/free-surface interaction for laminar flow and clean free surfaces, which indicates complex features involving interrelated free-surface deformation, secondary-vorticity generation, and vorticity reconnection; however, a complete understanding of the physics is lacking, i.e., most studies are descriptive and controversey exists as to the physical mechanisms. The role of surfactants and turbulence are insufficiently understood. Although certain progress has been made in the understanding of the practical application of ship blw's for nonzero Fr, the detailed flow structures, including turbulence and the micro-scale flow are poorly understood.
COMPUTATIONAL METHOD
The computational method is based on extensions of [3] for region IV calculations, including the use of a two-layer k-ε turbulence model [10]. [3] is a modified version of the large-domain viscous-flow method of [11] for small-domain calculations and free-surface boundary conditions. Only a brief review of the basic viscous-flow method is provided, but with a detailed description of the present solution domain and boundary conditions. Further details are provided in [8] as well as [10,11] and associated references.
In the viscous-flow method, the RaNS equations are written in the physical domain using Cartesian coordinates (x,y,z). For laminar-flow calculations, the equations reduce to the NS equations by simply deleting the Reynolds-stress terms and interpreting (U,V,W) and p as instantaneous values. The governing equations are transformed into nonorthogonal curvilinear coordinates (ξ,η,ζ) such that the computational domain forms a simple rectangular parallelepiped with equal grid spacing. The transformation is a partial one since it involves the coordinates only and not the velocity components (U,V,W). The transformed equations are reduced to algebraic form through the use of the finite-analytic method. The velocity-pressure coupling is accomplished using a two-step iterative procedure involving the continuity equation based on the SIMPLER algorithm. Both fixed and free-surface conforming grids were used for the calculations. In both cases, a simple algebraic technique was used whereby the longitudinal and transverse sections of the computational domain are surfaces of constant ξ and η, respectively; and, moreover, the three-dimensional grids were obtained by simply “stacking ” the two-dimensional grid for the transverse plane.
The Sw/fp solution domain and coordinate system are shown in figure 3. In terms of the notation of figure 3, the boundary conditions are as follows. On the inlet plane Si, is specified from the Stokes-wave solutions (i.e., table 1 of [8]) and typical free-stream values for (k,ε). On the body surface Sb, the no-slip condition is imposed. On the symmetry planes Scp and Sd, ∂(U,W,p,k,ε)∂y=V=0 and ∂(U,V,p,k,ε)/∂z=W =0, respectively. On the exit plane Se, axial diffusion is negligible so that ∂2/∂x2=px=0. On the outer boundary So, the edge conditions are specified from the Stokes-wave solutions (i.e., table 1 of [8]) and zero-gradient conditions for
On the free-surface Sη(or simply η), there are two boundary conditions
∂η/∂t+V·∇η=0 (11)
(12)
where η(x,y,t) is the wave elevation (interpreted as Reynolds averaged for turbulent flow), and are the fluid- and external-stress tensors, respectively, the latter, for convenience, including surface tension, and is the unit normal vector to η. The kinematic boundary condition expresses the requirement that η is a stream surface and the dynamic boundary condition that the stress is continuous across it. Note that η itself is unknown and must be determined as part of the solution. Boundary conditions are also required for the turbulence parameters (k,ε).
(11) can be put in the form:

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∂η/∂t=W−U ηx−V ηy=0 (13a)
on z=η. (13a) is solved for η using finite differences and two different strategies for regions of unseparated and separated flow. For unseparated flow, (13a) is solved in steady form
0=W−Uηx−Vηy (13b)
using a backward difference for the x-derivative, a central difference for the y-derivative, and a tridiagonal-matrix algorithm. For separated flow, (13a) is solved in unsteady form using backward differences for the t- and x-derivatives, a central difference for the y-derivative, a tridiagonal-matrix algorithm, and an iterative procedure whereby the steady-state solution is obtained. In both cases, a special treatment was required for (y, z)=(0,η), which is singular due to the incompatibility of simultaneously satisfying both the no-slip and free-surface boundary conditions. Furthermore, this point is embedded in the meniscus boundary layer, region V; thus, a rigorous treatment is beyond the scope of the present paper. For laminar flow, the approximation was made that the wave elevation was assumed constant across the first three grid points . For turbulent flow, an interpolation procedure was used to obtain the wave elevation across the first ten grid points, i.e., the value at y=0 was assumed .9 of the value at y+≈10 and intermediate values were obtained using a cubic spline. The number of grid points and the wave elevation value at y=0 were determined based on trial and error to minimize the residuals and error in satisfying the dynamic free-surface boundary condition. The assumption used for laminar flow is satisfactory, i.e., it has a minimal influence over a small portion of the overall region of interest. However, the assumption used for turbulent flow requires further justification since it has a large influence over a significant portion of the region of interest such that, as already mentioned, region V is recommended for future study.
(12) and (5) are used to derive free-surface boundary conditions for V and p in conjunction with the solution for η. The external stress and surface tension were neglected in (12), i.e.
(14)
on z=η.
For laminar flow,
(15)
where p* is the static pressure, i.e., Substituting (15) into (14) results in the normal-and two tangential-stress free-surface boundary conditions, i.e., (2)–(4), on z=η, which can be solved for p, Uz, and Vz to provide the free-surface boundary conditions for p, U, and V, respectively:
(16)
(17)
(18)
For the physical domain, the terminology normal and tangential refers to the mean free surface [i.e., (2)–(4) are the components of the stress in each of the Cartesian coordinate directions (z,x,y), respectively, on z=η]; however, upon transformation into the computational domain, it refers to the actual free surface z=η. Finally, (5) is solved for Wz to provide the free-surface boundary condition for W
(19)
Equations (16)–(19) were implemented in finite-difference form using backward differences for the z-derivatives and central differences for the x-and y-derivatives, for (16) and (19) and a backward and central differences, respectively, for the x- and y-derivatives for (17) and vice versa for (18).
For turbulent flow,
(20)
However, the same conditions (16)–(19) apply for turbulent flow with V interpreted as the mean velocity. The kinematic free-surface boundary condition in terms of velocity fluctuations is
−u ηx−v ηy+w=0 (21)

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Multiplying (21) by u, v, and w and Reynolds averaging results in, respectively
(22)
(23)
(24)
Substituting (20) into (14) and using (22)–(24) to eliminate the Reynolds-stress terms results identically in (16)–(18), but with V interpreted as the mean velocity. Note that this derivation neglects the effects of free-surface fluctuations. Equation (19) is also valid for the mean-velocity components. The finite-difference procedures for turbulent flow were the same as those described earlier for laminar flow.
Reasonable approximations for free-surface boundary conditions for k and ε are simply zero-gradient conditions
(25)
(26)
on z=η, which are implemented in finite-difference form using backward differences for the z-derivatives.
In summary, for laminar flow, the exact free-surface boundary conditions are given by (13) and (16)–(19). The corresponding turbulent-flow approximation are these same conditions and (25)–(26). Approximate treatments of the free-surface boundary conditions are now considered, which are useful in assessing various approximations used in the precursory and relevant work, i.e., flat free-surface, inviscid, and zero-gradient conditions.
The flat free-surface conditions are obtained from the exact conditions under the approximation that ηx=ηy=0 in the dynamic free-surface boundary conditions, whereupon (16) –(18) reduce to
(27)
(28)
(29)
The inviscid conditions are obtained from the flat free-surface conditions under the additional assumption that the normal and tangential gradients of the normal velocity are negligible, whereupon (27)–(29) reduce to
(30)
(31)
(32)
(27–(29) and (30)–(32) in conjunction with (13) and (19) are solved in a similar manner as described earlier for the exact conditions, including, for turbulent flow, (25)–(26). The zero-gradient conditions are obtained from the inviscid conditions under the additional assumption that (30) and (19) are replaced by zero-gradient conditions
(33)
(34)
which in conjunction with (31)–(32) are solved in a similar manner as described earlier for the exact conditions, including, for turbulent flow, (25)– (26); however, in this case, (13) is not required since η is no longer present in the equations.
The exact and approximate free-surface boundary conditions are to be applied on the exact free-surface z=η, which is obtained as part of the solution. However, with the additional assumption that the wave elevation is small, all of the above conditions can be represented by first-order Taylor series expansions about the mean wave-elevation surface (i.e., z=0). In the following, this will be referred to as the Taylor-series approximation.
COMPUTATIONAL CONDITIONS, GRIDS, AND UNCERTAINTY
The computational conditions were based on [1–3], i.e., Ak=(0, .01, .11, .21), Re=105 and 1.64×106 for laminar2 and turbulent flow, respectively, and L=λ=1. Typical values for δb (at the trailing edge), δw (in the near wake), and δfs (at the edge of the blw) are (.015, .02, .0018) and (.02, .03, .0004) for laminar and turbulent
2
The Re=2×104 value used in [1,3] was modified for the present work to the value Re= 105 in conformity with other researchers.

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flow, respectively. The corresponding Ak/ε values for both laminar and turbulent flow are O(1) and O(10) for small and medium and large steepness, respectively.
The laminar-flow calculations were performed for all four Ak values utilizing the exact, flat free-surface, inviscid, and zero-gradient conditions applied both on z=η (exact) and 0 (Taylor-series approximation). For zero steepness, the calculations were begun with a zero-pressure initial condition for the pressure field. For nonzero steepness, the complete zero-steepness solution was used as an initial condition. The solutions were built up in stages starting with small Ak values and achieving partial convergence and then incrementally increasing Ak until reaching the desired value and final convergence. For each Ak>0, initially coarse-grid calculations were performed utilizing the zero-gradient condition applied on z=0. These were then used as the initial guess for the fine-grid calculations utilizing the exact and approximate conditions applied both on z=0 and η. For the cases involving free-surface conforming grids (i.e., conditions applied on z= η), usually three updates (i.e., grid regenerations) were sufficient for convergence. Partial views of the coarse and typical fine grids used in the calculations are shown in figure 4. For the coarse grid, 170 axial, with 50 over the plate and 120 over the wake, 24 transverse, and 9 depthwise grid points were used, i.e., imax was 170×24×9 =36720. For the fine grid, 179 axial, with 5 before the leading edge, 54 over the plate and 120 over the wake, 24 transverse, and 25–27 depthwise, with 16–18 over the free-surface boundary layer, grid points were used, i.e., imax was 179×24×27=115992.
The turbulent-flow calculations were performed for all four Ak values utilizing the zero-gradient conditions applied on z=0 and, for Ak=.01, utilizing the exact and zero-gradient conditions on z=η and 0. The procedure for obtaining the solutions was similar to that for laminar flow. Transition was fixed at x=.05, which corresponds to the turbulence stimulators in the experiments. For the coarse grid, 187 axial, with 49 over the plate and 138 over the wake, 24 transverse, with 15 in the inner layer, and 9 depthwise grid points were used, i.e., imax was 187×24×9=40392. The fine grid was similar, except 28 depthwise, with 12 over the free-surface boundary layer, grid points were used, i.e., imax was 187×24×28=125664.
The detailed grid information, and values of the time, velocity, pressure-correction, and pressure under-relaxation factors and total number of global iterations itl used in obtaining both the laminar and turbulent solutions are provided in [8]. The average job run CRAY hours and central memory were 1.05 and 1.17 hours and 1.5 megawords for 100 global iterations for the fine-grid laminar and turbulent solutions, respectively.
Due to the complexity of the present calculations, it was not possible to carry out extensive grid dependency and convergence tests; however, these were done previously both for the basic viscous-flow method [11] and for other applications. The convergence criterion was based on the residual
(35)
and error(x,y,z) in satisfying the dynamic free-surface boundary conditions (17), (18), and (16), respectively, i.e., that R(it) and the error (x,y,z) be of order 10−4. Typical convergence histories and error-bar charts are provided in [8] and figure 5, respectively.
LAMINAR-FLOW SOLUTIONS
First, the small wave-steepness Ak=.01 results are discussed in detail for both the macro and micro scales: the macro scale corresponds to λ=L=1 and includes region III, whereas the micro scale corresponds to the body blw and free-surface boundary-layer thicknesses and is restricted to region IV. Second, the medium and large wave-steepness Ak=.11 and .21 results are discussed with particular reference to the influences of increasing Ak, including wave-induced separation. In general, only detailed results are presented in which the exact free-surface boundary conditions were utilized; however, in the discussion of the error-bar charts, reference is made to the various approximate treatments discussed earlier.
The discussion focuses on the differences between the Ak=0 or equivalently deep solutions and the nonzero wave-steepness Ak=.01, .11, and .21 solutions through the presentation of dependent variable differences from their deep values
Δ=−(deep) (36)
where is any of the relevant dependent variables of interest, e.g., V, p, ω, etc.; thereby, accentuating the wave-induced effects. The equivalence between the Ak=0 and deep solutions is indicated by the form of the governing equations and free-surface boundary conditions with η=ηx=ηy=W=0 (except for small leading- and trailing-edge effects for the

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former case, which were neglected since only zero-gradient conditions were used). Furthermore, note that both the Ak=0 and deep solutions correspond to the Blasius solution and corresponding two-dimensional wake, both of which are recovered to within a few percent, except for leading- and trailing-edge and near-wake effects. In general, results are presented and/or discussed for both the blw and fw regions.
In order to confirm the ome for regions III and IV and for evaluation of the relative contributions of various terms in the equations of interest, average values over the blw thickness are evaluated and designated with an overbar
(37)
where, in this case, is any relevant dependent variable or equation, i.e., (13b), (16) –(19), and the vorticity difference, vorticity-transport equation, and free-surface vorticity flux, respectively
(38)
(39a)
(39b)
39c)
(40a)
(40b)
(40c)
Figures for are included where appropriate in which the numbered dashed and solid lines designated on the figures correspond to the various terms in the equations with the numbering proceeding term by term from left to right. In most cases, the dashed line corresponds to the term representing the left-hand side of the equation. In the cases of (16)–(18) and the vorticity-transport equation, the dashed line represents the sum of all the terms, which, of course, should be zero. In discussing such figures, is identified in symbol with an overbar or in words with inclusive terms simply referred to in symbol without an overbar or by number. Note that the solutions are for the primitive variables V and p subject to the free-surface boundary conditions (13b) and (16) –(19), whereas equations (38)–(40) are derived, which in conjunction with the integration procedure (37) introduces some error; however, are still useful in evaluating the solutions. are evaluated at z= .1 and η for the macro- and micro-scale flows, respectively. Appendix B in [8] provides a summary of the ome for the various variables or equations of interest, including a listing of the confirmations and exceptions based on the blw averaged values.
Lastly with regard to the presentation of the results, the analysis was facilitated by color graphics through the use of PLOT3D from which certain of the present figures were reproduced in black and white. Note that in such figures—W is shown in conformity with the PLOT3D coordinate system (figure 3), i.e., negative values correspond to downward flow and positive values to upward flow.
Small Steepness
First, the results for the macro-scale flow are discussed. Figure 6 displays the free-surface velocity profiles ΔV/Ak vs. y for various axial locations. The streamwise ΔU and depthwise ΔW components display the pressure-gradient induced acceleration and deceleration phases and alternating direction, respectively. The transverse component ΔV indicates outward flow over most of the plate and inward flow near the trailing edge and over most of the wake. Note that the ome conform to table 1, i.e., V=(1,ε,Ak) and (1,ε/x,Ak) for the blw and fw, respectively. Also, noteworthy for laminar flow, is the broad region of large velocity gradients over δ. The results at larger depths are qualitatively similar, but with reduced amplitudes due to the exponential depthwise decay of the streamwise pex and depthwise pez external-flow pressure gradients.
Figure 7 displays the wall-shear stress wake-centerplane velocities ΔUcp and ΔVcp, and displacement thickness Δδ* vs. x for various depthwise locations. The wave-induced oscillations are evident as is the wake bias. The oscillations persist to large depths, i.e.,

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listed in table 2, selected from tank experiments carried out in our laboratory; the case numbers are the same as used in the experiments.
Table 2: Physical Conditions of the Numerical Simulations (D/L=0.097)
Case No.
h/D
δ/D
Δρ/ρ2
Fh
41(c)
1.28
0.54
0.0034
10.
41(b)
1.28
0.54
0.0034
4.8
28(b)
0.96
0.29
0.0036
5.9
25(c)
0.78
0.39
0.0041
6.5
An internal wave field as long as 80 ship lengths is simulated for case 41(c), figure 6(a) & (b),
Figure 6: Ship Internal Waves, Case 41(c), the Darkened Line Is the Cut for Triple Lobe. (a) Near and Intermediate Field; (b) Near Field, Blown up from (a).

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which covers the near and intermediate internal wave field. It can be clearly seen that: a big displacement appears just under the ship; and starting immediately after the ship, a central peak accumulates gradually which separates and pulls up the down lobe until a typical triple-lobe pattern in the transverse direction forms; then, this deformation relaxes and produces dispersive waves which propagate outwards, with the longest waves in the front.
The kinematic wave pattern, represented by the phase lines corresponding to crests and troughs, has been compared with far field theory (Tulin & Miloh, 1990), and very good agreement is obtained, as shown in figure 7.
Occuring one and a half ship lengths after the stern of the ship, case 41c, where the centerline amplitude reaches its peak, a triple-lobe pattern with a sharp peak and two shallow troughs appears as shown in figure 8(a); its vertical velocities are shown in figure 8(b), which, with a little shift, is similar to the triple-lobe distribution, and has an almost zero velocity at the central point as intended. It was suggested by Tulin & Miloh (1990) that the far field internal wave pattern may be calculated upon the assumption that the far field wave pattern originates from the initial conditions represented in the triple lobed pattern. In that case, they showed that the entire far field may be represented by a complex amplitude function, which is readily calculated from a Fourier transform of the triple lobed pattern, amplitudes and velocities.
This calculation has been carried out here, see figure 9. The real part of the transform of the triple-lobe amplitudes, centers at kh around one and decreases slowly in each direction, case 41c; the imaginary part, originating from the vertical velocities in the triple-lobe pattern, has an infinite value at kh equal to zero and decreases fast when kh increases. The modulus of these two parts composes the amplitude function which can be used together with the far field kinematic pattern to represent the far field internal wave field.
In figure 10, downstream wave cuts at increasing transverse distance, y/D, are shown from two different calculations together with experimental data: the current direct numerical calculation, and the far field calculation using the calculated amplitude function. We get very good agreement between the two kinds of calculations, and generally good agreement with the experimental data.
More simulations are carried out for a densimetric Froude number around five and for different depths of pycnoclines. Results are shown from figure 7 to figure 10. It is seen that, very good agreement is also obtained for case 41(b) between current direct calculation and the calculation using the far field theory for both the kinematic wave pattern and wave amplitudes. With the success of the calculation of the amplitude function and the confirmation of the far field theory, in the case 28(b) and 25(c) we halted the direct calculation at the triple-lobe pattern, and then calculated the amplitude function, used it to predict the far field internal wave pattern using the far field theory.
A detailed comparison of the results is shown in Figure 10(a) to (d). In 10(a) and (b) very good agreement is seen between the present numerical far field calculations (–·–) and predictions made from the analytical theory of Tulin and Miloh based on amplitude functions computed from triple-lobed patterns calculated numerically using the present theory (.......). This agreement suggests that

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Figure 7: Far Field Phase Lines. — , Crests, – , – , Troughs, Both from Direct Numerical Calculations; ....., Crests and Troughs, from Far Field Theory.
Figure 8 : Triple-Lobe Patterns in the Near Field Wake.

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Figure 9 : Far Field Amplitude Functions.
these detailed numerical calculations need only be carried out in the near field aft to the triple-lobe, providing that the waves are sufficiently small.
The experimental results (—) shown for comparison have been obtained in a small stratified tank using a towed model 45cm long (Ma & Tulin, 1992). The effects of the turbulent wake of the model are evident in the roughness of the wave patterns close to the model track and sufficiently aft The amplitudes found in the experiments and predicted theoretically are, in general, comparable — there certainly exists no major discrepancies in magnitudes, although a tendancy for theory to underpredict seems present; it must be kept in mind that the turbulent wake is not modeled here. In general, the agreement improves for cuts at the center of the larger transverse distances. A comparison of both theory and experimental measurements at much larger scales would be highly desireable.
A kh-map is shown in figure 11, in the case of 41b, which gives a clear view of the distribution of wave lengths in the patterns, and is helpful when far field wave amplitudes are analyzed using the amplitude function.
The CPU time needed for calculation in the near field to the triple-lobe pattern is only around 265 seconds, and for far field calculation to 80 ship lengths in case 41c, 371 minutes. All these calculations have been carried out on an IBM 9000.

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Figure 10(a): Longitudinal Wave Cuts at Transverse Distances from the Ship, Case 41c, from y/D=4.35 to y/D=40.35 in intervals of Δ y/D=4.5, Where —, Experimental Data, –·–, Direct Calculation Using Current Theory, ·······, Results From Far Field Theory Using Triple-Lobe Pattern From Direct Calculation

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Figure 10(b) : Longitudinal Wave Cuts at Transverse Distances from the Ship, Case 41b, from y/D=4.35 to y/D=40.35 in intervals of Δ y/D=4.5, Where —, Experimental Data, –·–, Direct Calculation Using Current Theory, ·······, Results From Far Field Theory Using Triple-Lobe Pattern From Direct Calculation

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Figure 10(c): Longitudinal Wave Cuts at Transverse Distances from the Ship, Case 28b, from y/D=5 to y/D=32 in intervals of Δy/D=4.5, Where —, Experimental Data, ·······, Results From Far Field Theory Using Triple-Lobe Pattern From Direct Calculation

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Figure 10(d) : Longitudinal Wave Cuts at Transverse Distances from the Ship, Case 25c from y/D=5 to y/D=32 in intervals of Δy/D=4.5, Where —, Experimental Data, ·······, Results From Far Field Theory Using Triple-Lobe Pattern From Direct Calculation

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Figure 11: a Typical kh-map of the Far Field of Ship Internal Waves Showing the Distribution of Wave Lengths in Wave Pattern and in Space
SUMMARY & CONCLUSIONS
A numerical method has been developed which is appropriate for slender bodies of arbitrary cross section traveling in a stratified ocean at sufficiently large densimetric Froude numbers, Fh ≫1.
This method assumes an inviscid flow field composed of a double model flow about the ship in homogeneous water plus a perturbation flow slowly varying in the x direction and therefore 2D in the cross flow plane. The 2D cross flow is described by the Poisson equation for the cross flow stream function, where the forcing is provided by the down stream vorticity, induced by deflections of the pycnoclines. The resulting solution satisfies the boundary conditions everywhere on the ship hull. The vorticity is calculated by a marching procedure, using an algorithm based on Fridman's Equation.
The method is applicable to arbitrary density profiles in depth and to arbitrary ship cross sections, and is non-linear in the cross flow plane and allows the propagation of solitons, none of which are generated in the examples given here.
The numerical method utilizes higher order Hermite finite elements and has been convergence tested.
Total calculations have been made for four cases for which small model experimental data on

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the far field wave pattern exist. These were taken in our own laboratory.
The present computational method predicts a far field kinematical pattern in very good agreement with the predictions of analytical far field theory.
The computations show the development in the near field of a narrow pattern of deflections behind the ship, arising from the initial reaction of the pycnoclines to the depression by the ship hull. These deflections in the wake on the ship track, are upwards and reach a peak behind the ship, at which station the cross section of the depression pattern is upwards in the center and downwards on either side. The triple lobe pattern was originally discovered by Tulin & Miloh, 1990.
Far field amplitudes deduced from the analytic theory of Tulin & Miloh, using an amplitude function calculated from the triple lobe pattern computed here, are in very good agreement with the far field fully numerical calculations.
The results found here thus tend to confirm the triple lobe pattern as the source of initiation of the far field wave pattern. This places an emphasis on the calculation of the near field, and reduces the computing time by orders of magnitude.
The magnitude of computed wave amplitudes are in good general agreement with our experimental results, with same tendency for theory to underpredict, and for detailed comparisons to improve with increasing transverse distance from the ship.
Considering the influence which the turbulent wake might be expected to have on the flow in the direct wake of the ship, the agreement between small scale theory and these numerical predictions must be regarded as very good.
Additional comparisons with large scale experiments would be highly desireable.
ACKNOWLEDGEMENT
The authors are grateful for support from the Office of Naval Research, Ocean Technology Program, directed by Dr. Thomas Swean.
REFERENCES
Ekman, V.W., 1904. On Dead Water. The Norwegian North Polar Expedition 1893–1896, vol. V, Ch. XV, Christiania
Havelock, T.H. “The Collected Papers of Sir Thomas Havelock on HYDRODYNAMICS,” ONR/ACR-103, pp. 377–389
Keller, J.B. & Munk, W.H., 1970, “Internal Wave Wakes of a Body in a Stratified Fluid”, Physics of Fluid, Vol. 13, pp.1425–1431
Kochin, N.E., Kibel, I.A. & Roze, N.V., “Theoretical Hydromechanics,” 1948
Ma, H & Tulin, M.P., 1993. “Experimental Study of Ship Internal Waves : The Supersonic Case”, Journal of OMAE , Vol. 115, No.1, pp. 16–22.
Ma, H. , 1993. Dissertation, OEL, UCSB
Miloh, T. & Tulin, M.P., 1988. “a Theory of Dead Water Phenomena,” Proc. of the 17th Symposium on Naval Hydrodynamics, National Academy Press, 1988
Miloh, T., Tulin, M.P. & Zilman, G., 1992. “Dead-Water Effects of a Ship Moving in Stratified Seas”, Proc. of the 11th Intl Conf. on OMAE, 1992, Vol I, Part A, pp. 59–67
Phillips, O.M., Dynamics of the Upper Ocean, Cambridge, 1969.
Tulin, M.P. & Miloh, T., 1990. “Ship Internal Waves in a Shallow Thermocline: the Supersonic Case ”, Proc. of the 18th Symposium on Naval Hydrodynamics, National Academy Press, 1990
Wong, H.L. & Calisal S.M., 1992. “Waves Generated by a Ship Travelling in Stratified Water,” Proc. of the 3rd Intl. Offshore & Polar Engr. Conference, 1993
Yih, C.S., “Patterns of Ship Waves,” Engineering Science, Fluid Dynamics, World Scientific Publishers, 1990.

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