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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
SESSION 6
WAVY/FREE-SURFACE FLOW: VISCOUS-INVISCID INTERACTION

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
Displacement Thickness of a Thick 3-D Boundary Layer
L.Landweber, A.Shahshahan, and R.A.Black
(University of Iowa and Loras College, USA)
ABSTRACT
Thin-boundary-layer solutions for displacement thickness were published in the fifty's. More recently, semi-empirical methods of computing displacement thickness were developed in connection with interactive methods of computing viscous flows about bodies. In the present work, the determination of the displacement thickness is recognized as a member of the class of “ill-posed problems” (IPP), in which small or random errors in the data defining a problem result in much larger errors in its solution. Otherwise, the displacement thickness of a given viscous flow can be well-defined mathematically by generalizing the thinboundary-layer solution. Rational solutions can then be obtained by applying the suggestions for controlling growth of errors given in the IPP literature.
The theory assumes that, exterior to a boundary layer, the flow is irrotational, that this irrotational flow can be continued into the boundary-layer region, and that this flow is singularity-free in that region. That is accomplished by means of a Fredholm integral equation of the first kind which, appropriately is a classical example of an IPP. The procedure is illustrated by applying it to a Wigley form.
INTRODUCTION
We suppose that a given double body of a ship form is at rest in a uniform stream U∞ of an incompressible fluid parallel to its centerplane, and that the vector velocity field v of the mean flow within the boundary layer and wake (BLW) of the ship form is known. Outside the BLW, we assume that the mean flow v is irrotational and coincides there, with a small error, with an irrotational vector V which may be continued as an irrotational velocity field V into the BLW. Let S3 be a member of the family of viscous-flow stream surfaces, lying exterior to the BLW, of which the body itself, S0 is the defining member. Then S3 also defines a different family of irrotational flow stream surfaces of V extending into BLW. The displacement-thickness surface S1 is defined as that member of the latter family which satisfies the flux condition that the flux of v between S0 and S3 is equal to the flux of V between S1 and S3. See Fig. 1.
If S2 is another surface, not necessarily a stream surface, but closer to the edge of BLW, S2 may replace S3 in the above definition of S, since we assume v=V between S2 and S3. The above definition implicitly assumes that V is singularity-free in the space between S1 and S2, except for a surface distribution on the centerplane of the wake. If that condition is not satisfied, then an exact solution for S 1 does not exist, although useful, approximate solutions may be found.
If the body has a well-defined stagnation point, the dividing streamline would be, by symmetry, a straight line, generating the given body in the viscous flow and S1 in the irrotational flow. This is seen to satisfy the flux condition, since, far upstream, the velocity fields of v between S0 and S3 and of V between S1 and S3, coincide. See Fig. 1.
Moore [1] and Lighthill [2] have treated the present subject for thin boundary layers. Both defined the displacement-thickness surface S1 as a stream surface, but neither introduced a flux condition. Moore terminated his analysis with a first-order partial differential equation (PDE) for the displacement thickness, derived from the streamsurface condition, the equations of continuity, and the thin boundary-layer approximations. Lighthill rederived Moore's equations, but also obtained explicit solutions for the displacement thickness and the equivalent source distribution on the body surface. Lighthill assumed that the stream surface of V which passes through the stagnation point(s) is the displacement-thickness surface. This assumption which is also made in the present work, can be

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justified for symmetric flows about double models without a free surface, because then a free-surface boundary layer and wavebreaking are not present ahead of the bow and the stagnation points are well-defined.
In Stern, Yoo and Patel [3] an interactive method of computing the viscous flow about a 3-D body, involving displacement thickness, was developed. Beginning with an assumed first approximation, a succession of displacement bodies and the corresponding irrotational- and viscous-flow fields were computed iteratively. In that work, each transverse section of the S1-surface was assumed to be an ellipse of dimensions satisfying approximately a mean, local flux condition. Nevertheless, their viscous-flow results were in good agreement with experimental data and those computed by a large-domain method.
At the request of Fred Stern (FS), the present work was undertaken by Landweber (LL). It will be seen that an important phase of the project is the calculation of the continued irrotational flow into the boundary layer and wake (BLW). For that purpose, a method using a Fredholm integral equation of the first kind for determining the source distribution on the surface of the body equivalent to the displacement effect of the boundary layer, was proposed by LL and FS assigned an M.S. candidate, RA Black, to work with LL on the validation of the procedure. After verifying that the equivalent source distributions could be obtained with sufficient accuracy, the next step was to calculate the irrotational velocity field of the sources. For that purpose, a complicated Hess-Smith computer program [4] for the velocity field of a source distribution of constant strength on a flat quadrilateral panel was available; but, by transforming the integrals to ones over the centerplane of the body the projected panels became rectangular, and the integrals could be expressed more simply in closed form in terms of elementary functions. Since these results may be new and useful, they will be presented here.
Equivalent source distributions are of interest not only for calculating the effects of waves on the boundary layer of a ship, but also for the effects of the boundary layer on the wavemaking of a ship form. This was shown for the Weinblum very thin form by Kang [5] and by Shahshahan and Landweber [6] for the Wigley form, both using equivalent centerplane distributions. It will be of interest to calculate the wave-making resistance of the Wigley form using the present results for the equivalent source distribution on the body surface, although this is not done in the present work.
Here we shall derive a generalized version of Moore's partial differential equation, without applying the thin-boundary-layer approximations. Since the only application planned was to the Wigley form, it seemed most convenient to work in rectangular Cartesian coordinates, although the generalized PDE had also been derived in nonorthogonal coordinates. The PDE was then solved by a method of finite differences, using a method suggested by Lax [7].
The plan of the present work is as follows:
Obtain a source distribution on the body surface y0 and the centerplane of the wake as the numerical solution of an integral equation of the first kind, using given data for the viscous-flow velocity field exterior but close to the edge of the boundary layer.
Apply that source distribution to compute the irrotational velocity field within the boundary layer and wake.
Apply the irrotational and viscous-flow velocity fields to compute the auxiliary displacement thicknesses α and β defined in Eq. (4). Also compute the derivatives of the velocity components at y0 occurring in Eq. (11).
Use method of finite differences to solve the partial differential equation (10) and (11) for δ1, defined in (4).
NATURE OF THE PROBLEM
Let us suppose that the edge of the BLW is defined by a contour surface at H=0.990 H∞, where H denotes the total head at the contour and H∞ the asymptotic value of the total head at great distances upstream or lateral to the body. Had H been defined by H=0.995 H∞, the edge of the BLW would be much farther from the centerplane and the error in assuming irrotational flow there would be reduced, but the accuracy of analytical continuation of that flow into the BLW region would be greatly diminished. Indeed, the procedure adopted here to continue the potential flow strongly suggests that the present work is on an “ill-posed problem,” in which small or random errors in the data defining a problem results in much larger errors in its solution, as defined by Tikhonov and Arsenin in[8].
The simplest case of an ill-posed problem is a set of linear, algebraic equations, Ax=b, where A is a matrix and x and b are vectors. This has an exact solution when the A is nonsingular (i.e. its determinant is not zero); but if A is nearly singular and the right member is subject to errors, the resulting errors in the solution would be amplified. Another classic case is the Fredholm integral equation of the first kind, which, in general does not

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have an exact solution. If discretized by means of a quadrature formula, this case reduces to the previous one, yielding approximate solutions with possibly large errors.
The literature on ill-posed problems presented in [8] indicates that it is possible to get useful results by use of supplementary information, such as requiring smoothness of the solution, or by a process called “regularization” developed by Tikhonov in a series of six papers from 1963–65. The method of constructing approximate solutions is called the “regularization method.” This method requires finding or constructing a regularization operator which transforms the equation of the ill-posed problem so that approximate, stable solutions (i.e. solutions without error amplification) could be found. Construction of this operator appears to be a difficult task. Twelve years prior to Tikhonov's work on this concept, Landweber [9] had shown that the integral operator of an integral equation of the first kind transforms the equation into one with a symmetric kernel with which stronger convergence properties of an iteration formula could be proved. In that sense, the original operator of the ill-posed problem could serve as a regularization operator. Reference [9] is not included among the 221 papers listed in the Bibliography of [8].
FORMULATIONS OF STREAM-SURFACE EQUATIONS
The equation of continuity and the stream-surface equation will now be applied to generalize the thin boundary-layer treatments of Moore [1] and Lighthill [2] for a double ship form. Let (x,y,z) denote a rectangular Cartesian coordinate system with origin at the forward stagnation point, the x-axis parallel to the uniform stream and positive in the downstream direction, with the x- and z-axes lying in the vertical centerplane, and the z-axis positive upwards. Let (u,v,w) and (U,V,W) denote the components of v and V. Also let y=y0(x,z), y =y1(x,z), y=y2(x,z) denote the equations of a given hull surface S0, its displacement thickness surface S1 and a surface S2, near but exterior to BLW, respectively. At S0, the velocity components will be designated by
where , , denote unit vectors in the x,y,z-directions; and there, by the nonslip condition,
(1)
Since V is irrotational, we have
×V=0, 2V=0 (2)
and on S2, we assume
u2=U2,v2=V2,w2=W2 (3)
We also define the three auxiliary displacement thicknesses
(4)
These definitions of α and β assume that V and W are singularity-free for y0≤y≤y2. We also assume that α and β are of the order O(δ1) for a body of unit length.
The equations of continuity for v and V are
(5)
Then, integrating equations (5) with respect to y from y0 to y2, taking their difference and applying (1) and (3), we obtain
(6)
or, applying (4) and the Leibnitz rule for the derivative of an integral, we get
V0=U∞(αx+βz)+U0y0x+W0y0z (7)
where subscripts x and z denote partial differentiation with respect to the indicated variable.
The condition that the y1 surface be a stream surface is
(8)
where
V1=V(x,y1,z)

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Since the y1 surface is unknown, we transfer (8) to the known surface y0 by means of the Taylor expansions
which, substituted into (8), gives
(9)
By substituting for V0 from (7), replacing Vy by-UX-WZ, Vyy by -VXX-VZZ and neglecting the terms, we obtain
(Px+Qz)0=R(x,z,δ1) (10)
where
(11)
and
P(x,z)=Uδ1−U∞α, Q(x,z)=Wδ1−U∞β
Here the left member is clearly of order 0(δ1). The first term of the right member also seems to be of order 0( δ1); but, as will be seen, computationally, its effect is that of a second-order term, and the remaining terms are of even higher order, at least for the Wigley form. We also observe that the homogeneous form of (11) when R=0, is essentially Moore's partial differential equation expressed in rectangular, Cartesian coordinates.
CONTINUATION OF OUTER IRROTATIONAL FLOW INTO BOUNDARY LAYER
Source Distribution on Body
Let Φ denote the velocity potential of the irrotational flow about the displacement body in a uniform stream and that of the disturbance potential. Then
Φ=+U∞x (12)
and V, the y-component of V, is given by in the exterior of BLW according to (3). Since v is given, grad Φ is known on y2, but we shall require only V.
We assume that potential flow is generated by a source distribution σ(xQ,zQ) on the hull surface yo and the centerplane of the wake may be written as
(13)
where P denotes a point exterior to BLW and rPQ the distance between points P and Q,
Then, differentiating with respect to yP, we obtain
(14)
a Fredholm integral equation of the first kind.
By applying a quadrature formula of order N, (14) can be reduced to a set of N linear equations for σQ, i=1, 2…N which can be solved by a computer using available software. It is well known, however, that small changes in VP in (14) may cause large changes in σQ when the set of linear equations is nearly singular; i.e. when the determinant of the coefficients is nearly zero. This was verified in several test cases in which the velocity was computed at points exterior to a body due to an assumed source distribution on the surface of the

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body. Then the relation between the source distribution and the velocity was treated as an integral equation of the first kind to try to recover the source distribution. This preliminary work yielded the conclusions that double precision should be used and that an iteration formula for integral equations of the first kind due to Landweber [9] yielded useful results even when a linear-equation solver had failed.
In order to derive the convergence properties of the iteration formula of [9], the original kernel of the integral equation was transformed into a symmetric one. Since that requires a large number of additional integrations for a 3-D problem, it is customary to assume that the iteration formula with the original kernel would have similar convergence properties. In general, integral equations of the first kind do not have exact solutions, but a sequence of approximate solutions can be found which converges in the mean, i.e. the integral of squares of the errors becomes very small. In practice, the successive approximations are monitored so that the sequence can be stopped when the errors are as small as desired, or, in case of initial convergence and then divergence, when the errors begin to grow. An important difference between the preliminary tests of the integral equation method on a Wigley form and a calculation of the unknown function is that, in the former case, the exact solution was known, and, in the latter case, the existence of a solution was uncertain.
The equation of the Wigley form for which results were computed is
(15)
Here the body length is L=2ℓ, b is half the beam and h is the draft. We take L=1, b=0.10ℓ and h =0.125ℓ. The limits of integration of the integral in (14) are taken to extend over the double body and over the centerplane of the wake for an additional ship length L and for z varying from −h to +h. The integral in (14) may then be written as
(16)
Since μ(Q) is the same in the four quadrants of a transverse section, the four terms with yQ≠0(0<x<L) and the two terms with yQ=0 (L≤x≤2L) can be collected into a single expression for integration over the first quadrant of Q's for any point P in the first quadrant,
(17)
where
with λ=1 when yQ≠0 and λ=1/2 when yQ=0, and
The integral in (17) may be interpreted as extending over the centerplane; although y0(x,z) is not replaced by zero, i.e. the unknown source distribution is still on the body. Adopting the iteration formula of [9], but with the original kernel, we obtain
(18)
Here P and Q have the coordinates (xP, yP, zP) and (XQ,Y0(xQ,ZQ),ZQ), and VP and the integral are functions of the coordinates of P. Hence, in the present coordinate system we must take (xp,zp) and (xQ, zQ) as the same array of numbers and interpret the iteration as giving a corrected source distribution at the same points Q. “C” is a constant, selected so as to accelerate the convergence. To initiate the iteration, the selected value of , suggested by slender-body theory is

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(19)
Equation (18), discretized by the panel method of the next section, becomes
(20)
results, obtained with 70 values of x and 13 values of z, with C=1, are shown in Figures 2, 3a and 3b.
Velocity Field of a Source Distribution
The basic PDE given in (10) and (11) requires that U,V,W, and their first two derivatives with respect to y at y0, be calculated. Also, α(x,y0,z), and β(x,y0,z) defined in (4), and αx and βz at y0 must be computed. To obtain the latter quantities, the irrotational flow field within the region bounded by y0 and y2 is needed.
Hess, J.L. and Smith, A.M.O. [9] have furnished a formulation for calculating the velocity field for a flat quadrilateral panel on which a source distribution of constant strength is distributed. For forms such as Wigley's for which the potential can be expressed in terms of rectangular panels on the centerplane, it was possible to derive a much simpler set of formulas for U, V and W, by making one additional approximation. As in [4], we assume that the panels are flat; so that the direction cosines of the normals are constant on each panel, and that σQ'S on each panel are also constant. The additional approximation is that y0Q is replaced by y0-value corresponding to the center of the rectangle, y0c; i.e. a constant for each panel.
The velocity potential for a rectangular panel of dimensions 2a and 2b may now be written as
(21)
The results for U,V,W can be readily obtained by operating on the double indefinite integral, with ξ= xQ−xP, μ=y0Q−yP,ζ=zQ−zP,
(22)
Then
(23)
the constant of integration vanishing when the integration limits of (21) are introduced. Similarly
(24)
We also have
(25)
Then, introducing the integration limits in (23), we get
(26)
where
Similarly, from (25),
(27)

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and from (24),
(28)
These expressions must be summed over all the panels for a fixed point P. Here we can again take advantage of symmetry in the y-z plane for 0<xc<L to collect four terms and to collect two terms for L<xc<2L with the same μc.
The approximation of replacing yQ in the integrations introduces appreciable errors in panels for points P close to a panel on y0. Since these constitute a small percentage of the total number of panels usually used, that error may be acceptable.
Velocity distributions computed from (26, 27, 28) are shown in Fig. 4.
SOLUTION OF STREAM-SURFACE EQUATION
We can now consider a procedure for solving (10) and (11) for δ1. Initially, the case R=0, equivalent to Moore's PDE for thin boundary layers, was treated, i.e., with U∞=1,
Px+Qz=R
P=U0δ1−α Q=W0δ1−β (29)
with the boundary conditions
δ1(0,z)=0 at the bow
δ1z(x,0)=0 at the waterline by symmetry
δ1z(x,h)=0 at the keel by symmetry
Equation (29) was discretized by the method suggested by Lax [7]. The finite-difference formulation of (29) is
(30)
where
Here i and j denote indices for increasing values of x and z respectively. This gives a marching procedure in the x-direction. Using this formulation and the aforementioned boundary conditions, the case R=0 was solved. Typical results are shown in Figure 5. Since R was found to be very small, its effect on the calculated values of 1 was too small to show graphically. Thus, the thin-boundary-layer approximation is very good for the Wigley form.
ACKNOWLEDGMENT
The viscous-flow data for a Wigley form were furnished by Y.Tahara, computed for a turbulent boundary layer by a large-domain method. These data will be published among the papers generated by F.Stern 's viscous-flow program. For this and other assistance the authors are grateful to Tahara and Stern.
REFERENCES
1. Moore, F.K., “Displacement Effect of a Three-Dimensional Boundary Layer,” NACA Report 1124, 1953.
2. Lighthill, M.J., “On Displacement Thickness,” Journal of Fluid Mechanics, Vol. 4, Part 4, 1958, pp. 383–392.
3. Stern, F., Yoon, S.Y. and Patel, V.C., “Viscous-Inviscid Interaction with Higher-Order Viscous-Flow Equations, ” IIHR Report No. 304.
4. Hess, J.L. and Smith, A.M.O., “Calculation of Nonlifting Potential Flow About Arbitrary Three-Dimensional Bodies,” Douglas Aircraft Report E.S. 40622, 1962.
5. Kang, S.-H., “Viscous Effects on the Wave Resistance of a Thin Ship,” Ph.D. Thesis, The University of Iowa, July 1978.
6. Shahshahan, A. and Landweber, L., “Boundary-Layer Effects on the Wave Resistance of a Ship Model,” Journal of Ship Research Vol. 34, No. 1, March 1990, pp. 29–37.
7. Lax, P.P., “Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computations,” Communications on Pure and Applied Mathematics, Vol. 7, 1954.
8. Tikhonov, A.M. and Arsenin, V.Y., “Solutions of Ill-Posed Problems,” Published by V.H.Winston & Sons, Distributed by John Wiley & Sons, New York, 1977.
9. Landweber, L., “An Iteration Formula for Fredholm Integral Equations of the First Kind,” American Journal of Mathematics, Vol. 73, No. 3, July 1951.

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Figure 1. Sketch of Wigley hull showing stream surfaces S0, S1 and S3, and a surface S2 exterior to the BLW.

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Fig. 2 Source distributions on Wigley form at various iterations at z/h=1/26

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caused by the inadequate grid resolutions noted earlier. For completeness, the bare-hull potential flow wave elevations are also shown in Figure 17(a) to quantity the viscous effects. It is seen that the viscosity plays a dominant role in eliminating the short wavelength, small amplitude waves at low speeds.
For high Froude number (Fr=0.316) case, it is seen from Figure 17(b) that the wave elevations obtained by interactive RANS/SLAW method are in very good agreement with the measured values over the entire ship. The bare-body potential flow solutions overpredicted the wave elevation in the stern region, but underpredicted the amplitude of the primary wave in the bow region. The overprediction of stern waves is obviously due to the negligence of viscous effects due to thick boundary layer and wake flows. The underprediction of the bow wave, however, must be attributed to the use of linearized free surface conditions in SLAW bare-body calculations. It is noted that the wave elevation changes by as much as 0.03 of ship length or 50% of the ship draft. Although this wave amplitude may still be considered small compared to wavelength, it is quite clear that the viscous boundary layer and wake flows will be greatly influenced by the nonlinearity in the free surface wave field. Consequently, it is necessary to adjust the RANS solution domain in order to properly account for the nonlinear wave effects due to local changes in underwater geometry at each transverse section.
It should be emphasized that both the viscous and wave effects have been included directly in the present RANS calculations, even though the surface tension and viscous stresses on the free surface had been neglected due to a lack of understanding on the physical processes and mathematical modeling of the free surface turbulence. The inclusion of both the viscous and wave effects enables us to capture the complete interaction between the wavemaking and viscous boundary layer and wake in the near field. In particular, the wave-induced effects were formally included by requiring the dynamic free surface boundary conditions to be satisfied on the exact free surface. On the other hand, the influence of viscous boundary layer and wake flows on the free surface waves was captured automatically when the kinematic boundary conditions were used to update the free surface wave profiles.
CONCLUSIONS
An interactive numerical method which combines the Reynolds-Averaged Navier-Stokes (RANS) method and the free-surface potential flow calculations has been developed for the prediction of ship flows including both the viscous and nonlinear free-surface wave effects. In this zonal approach, the RANS method is employed to resolve the viscous boundary layer, wake and the nonlinear free-surface waves around the ship hull while the potential flow method is used to provide the wave information outside the viscous region. For high Froude number cases where the wave effects are significant, the near-field RANS solution is matched with the SLAW solution. Iterations between the two solutions are performed to capture the interaction between the viscous and the wave effects. Only one to two iterations is needed to achieve a converged solution. At low Froude numbers, the wave effects outside the RANS domain are insignificant and the RANS solutions are matched with the SLAW solution without iteration.
In the present RANS calculations, the dynamic free surface boundary conditions are satisfied on the exact free surface and, therefore, it is possible to capture the strong interaction between the nonlinear waves, the boundary layer, and wake flows underneath the exact free surface. With further modifications in the adaptive grid generation procedure, the present method can be readily generalized for the prediction of nonlinear wave effects arose from arbitrary ship motions.
Calculations have been performed for the Series 60, CB=0.6 parent hull for both low and high Froude number causes. The numerical solutions clearly demonstrated the feasibility of coupling the RANS and SLAW methods in an interactive approach for detailed resolution of both the viscous and the free-surface wave effects. This enables us to use a rather small RANS solution domain for efficient and accurate resolution of the interaction between wavemaking and viscous boundary layer and wake.
All of the RANS calculations presented here were performed on CRAY YMP supercomputers. The CPU time required is about 440 CPU seconds for every 100 time steps. Normally, 200 to 300 times steps are needed to achieve complete convergence for a non-interactive solution. In the subsequent interactive calculations, 50 to 100 more time steps

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were used for each interactive RANS/SLAW coupling solution. A complete RANS calculation for the near-field viscous flow with three interactive coupling cycles takes about 30 to 45 CPU minutes. The SLAW calculations for double-body and Fr=0.316 cases were performed on an IRIS R4000 –50 GTX workstation, while the Fr= 0.160 results were obtained on a IBM RS6000/550 workstation. Depending on the Froude Number, the SLAW calculations of the potential flow waves take about 5 to 80 (Fr=0.160 case) CPU minutes for one solution on the workstations. In general, a complete RANS and potential flow coupling solution may be obtained in less than 1 CPU hour on a CRAY YMP. This demonstrates the efficiency of the method presented in this paper and shows the feasibility of using this type of advanced numerical method for evaluation of practical problems.
The method developed in the present study is very general. With the present velocity/pressure coupling approach, it is possible to use the most suitable and efficient solver for each different flow region to provide maximum flexibility, accuracy and efficiency. With some modifications, the method can be readily generalized for the study of nonlinear unsteady interaction of ship and waves, effect of propulsor on stern flows, and other near-field ship flow problems.
ACKNOWLEDGEMENT
This work has been supported by the Office of Naval Research under the Nonlinear Ship Motion Program Grant N00014–90–C–0031, monitored by Mr. James A.Fein. Some of the results reported in this paper have been obtained under the support of the Advanced Research Project Agency (ARPA) Submarine Technology Program monitored by Mr. Gary Jones. Most of the computations were performed on the CRAY Y-MP8e/8128 –4 of Cray Research Inc. at Eagen, Minnesota under the sponsorship of Mr. Chris Hempel. Preliminary developments of the computer codes were done on the CRAY Y-MP2/116 at the Texas A & M Supercomputer Center through a Cray Grant provided by Cray Research Inc.. We are grateful to Cray Research Inc. for the use of these computing resources.
References
[1] Miyata, H. and Nishimura, S., “Finite-Difference Simulation of Nonlinear Waves Generated by Ships of Arbitrary Three-Dimensional Configuration,” Journal of Computational Physics, Vol. 60, No. 3, pp. 391–436, 1985.
[2] Miyata, H., Zhu, M. and Watanabe, O., “Numerical Study on a Viscous Flow with Free-Surface Waves About a Ship in Steady Straight Course by a Finite-Volume Method,” Journal of Ship Research, Vol. 36, No. 4, pp. 332–345, 1992.
[3] Hino, T., “Computation of a Free Surface Flow around an Advancing Ship by the Navier-Stokes Equations,” Proc. 5th International Conference on Numerical Ship Hydrodynamics, 1989, Hiroshima, Japan.
[4] Shahshahan, A. and Landweber, L., “Interactions Between Wavemaking and the Boundary Layer and Wake of a Ship Model,” IIHR Report No. 302, 1986, Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, IA.
[5] Stern, F., “Effects of Waves on the Boundary Layer of a Surface-Piercing Body, ” Journal of Ship Research, Vol. 30, No. 4, pp. 256–274, 1986.
[6] Tahara, Y., Stern, F. and Rosen, B., “An Interactive Approach for Calculating Ship Boundary Layers and Wakes for Nonzero Froude Number,” Proc. 18th ONR Symposium on Naval Ship Hydrodynamics, 20–24 August, 1990, Ann Arbor, MI.
[7] Toda, Y., Stern, F. and Longo, J., “Mean-Flow Measurements in the Boundary Layer and Wake and Wave Field of a Series 60 CB =.6 Ship Model for Froude Numbers .16 and .316,” IIHR Report 352, 1991, Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, Iowa.
[8] Toda, Y., Stern, F. and Longo, J., “Mean-Flow Measurements in the Boundary Layer and Wake and Wave Field of a Series 60 CB =.6 Ship Model—Part 1 : Froude Numbers .16 and .316,” Journal of Ship Research, Vol. 36, No. 4, pp. 360–377, 1993.

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[9] Lighthill, M.J., “On Displacement Thickness,” J. of Fluid Mechanics, Vol. 4, Part .4, 383–392, 1958.
[10] Rosen, B., “SPLASH Free Surface Code : Theoretical/Numerical Formulation,” South Bay Simulation Inc., Babylon, NY, 1989.
[11] Chen, H.C. and Lin, W.M., “Interactive RANS/LAMP Coupling Schemes and Their Applications to Ship Flows,” SAIC-91/1021, February 1991, Science Applications International Corporation, Annapolis, MD.
[12] Chen, H.C. and Korpus, R., “A Multi-block Finite-Analytic Reynolds Averaged Navier-Stokes Method for 3D Incompressible Flows,” Proc. ASME Fluid Engineering Conference, Washington, D.C., June 20–24, 1993.
[13] Letcher, J., Weems, K., Oliver, C., Shook, D., and Salvesen, N., “SLAW: Ship Lift and Wave, Theory, Implementation, and Numerical Results , SAIC Technical Report 89/1196, 1989.
[14] Lin, W.M. and Yue, D.K.P., “Numerical Solutions for Large-Amplitude Ship Motions in the Time-Domain ”, Proc. 18th Symp. Naval Hydro., The University of Michigan, Ann Arbor, MI, USA, 1990.
[15] Dawson, C., “A Practical Computer Method for Solving Ship-Wave Problems” Proc, 2nd International Conference on Numerical Ship Hydrodynamics, University Extension Publishers, Berkeley, CA, 1977, pp 30–38.
[16] Hess, J., and Smith, A., “Calculation of Nonlifting Potential Flow about Arbitrary Three-Dimensional Bodies”, Journal of Ship Research, Vol 8, 1964, pp 20–24.
[17] Jensen, P., “On the Numerical Radiation Condition in the Steady-State Ship Wave Problem”, Journal of Ship Research, Vol 31, pp 14–22, 1987.
[18] Morino, L., Chen, L.-T., and Suciu, E., “ Steady and Unsteady Oscillatory Subsonic and Supersonic Aerodynamics around Complex Configurations” , AIAA Journal, Vol 13, pp 368–374, 1975.
[19] Campana, E., Di Mascio, A., Esposito, P.G., and Lalli, F., “A Multidomain Approach to Free Surface Viscous Flows”, Abstract submitted to the Eighth International Workshop on Water Waves and Floating Bodies, St. John's, Newfoundland, Canada, May 1993.
[20] Chen, H.C. and Lin, W.M., “Interactive RANS/LAMP of Ship Flows Including Viscous ans Wake Effects, ” SAIC Technical Report 92/1049, 1992.
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[24] Hubbard, B, and Chen, H.C., “A Chimera Reynolds-Averaged Navier-Stokers Methods for 3D Incompressible Flows,” Texas A & M Research Foundation, COE Report No. 328, 1993.
[25] Steinbrenner, J.P., Chawner, J.R. and Fouts, C.L., “The GRIDGEN 3D Multiple Block Grid Generation System,” Vols. I & II, WRDC-TR -90–3022, 1990, Wright Patterson AFB, OH.

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Fig. 1: SLAW Potential Flow Panel Model.
Fig. 2: SLAW Zonal Test Calculation.
Fig. 3: Partial View of RANS Grid.
Fig. 4: Convergence History; Double-Body.
Fig. 5: (a) Non-interactive and (b) Interactive Pressure Contours, (c) Non-interactive and (d) Interactive Vertical Velocity Contours; Fr=0.

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Fig. 6: Convergence History; Fr=0.316.
Fig. 7: Influence of SLAW Matching Boundary.
Fig. 8: Fr=0.316 Wave Patterns: (a) Non-interactive, (b) Interactive.
Fig. 9: Fr=0.316 Pressure (left) and Axial Velocity (right) Contours: (a) Non-interactive; ξ2 =1, (b) RANS/SLAW 1; ξ2=30, (c) RANS/SLAW 2; ξ2=32.

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Fig. 10: Fr=0.160 Solutions: (a) Numerical Grid, (b) Bare-Hull SLAW Wave Patterns, (c) RANS Axial Velocity Contours, (d) RANS/SLAW Pressure Contours.
Fig. 11: Pressure (left) and Velocity (right) Contours: (a) Interactive Double-Body, (b) Non-interactive Fr =0.160, (c) Non-interactive Fr=0.316, (d) Interactive Fr=0.316.

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Fig. 12: Pressure Contours: (a) Double-Body, (b) Fr=0.160, (c) Fr=0.316.

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Fig. 13: Axial Velocity Contours: (a) Double-Body, (b) Fr=0.160, (c) Fr=0.316.

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Fig. 14: Crossplane Velocity Vectors: (a) Double-Body, (b) Fr=0.160, (c) Fr=0.316.

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Fig. 15: Nonlinear Waves (ζ/Fr2) in RANS Solution Domain.
Fig. 16: Perspective View of Free Surface Waves.
Fig. 17: Comparison of Wave Profiles on Hull Surface and Along Wake Centerline.

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