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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
SESSION 9
VISCOUS FLOW: APPLICATIONS 2

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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
Numerical Simulation of the Effect of Fillet Forms on Appendage-Body Junction Flow
D.Li and L.-D.Zhou
(China Ship Scientific Research Center, PRC)
ABSTRACT
The possibility of using a fillet form to control the horseshoe vortex flow caused by the turbulent shear flow around wing-body junction has been investigated numerically. A numerical method for the solution of three dimensional incompressible, Reynolds-averaged Navier-Stokes equations with the two-equation (k, ) turbulence model has been developed to evaluate the effect of fillet forms on appendage-body junction vortex flow. The wing investigated is NACA0020. The Reynolds number based on a chordlength is 1.0×105. Three configurations including a baseline, a triangle fillet form, and a constant radius convex arc fillet form along the entire wing/flat-plate junction are presented. It is shown that a suitable convex filet form can significantly improve the stability of junction horseshoe vortex and reduce the strength of vortex and the non-uniformity in the wake velocity profile. It is also demonstrated the capability of the numerical approach in the design of vortex flow control devices.
1.INTRODUCTION
When an laminar or turbulent boundary layer on a surface encounters a wing or other protuberances projecting from that surface, a complex and highly three-dimensional flow results. The most significant feature of this flow is the generation of a horseshoe vortex or a set of horseshoe vortices. This horseshoe vortex forms around the nose of the wing and its legs trail downstream into the wake and forms streamwise vortices in the wake. This phenomena occurs at many places including wing/fuselage intersections in aerodynamics, appendage/hull junctions in hydrodynamics, and blade/hub junctions in propellers and turbomachinery, etc.. As a consequence of the generation of the horseshoe vortices, the drag is in general increased and the wake velocity profile becomes significantly non-uniform. This can be a great nuisance in marine applications where a propeller operating in a non-uniform wake results in significant unsteady forces.
As well known, when a boundary layer encounters a protruded bluff body, the streamwise vortex in the form of the horseshoe vortex cannot be prevented from being generated. However, it is possible to minimize the strength of the horseshoe vortex by some flow control devices such that the resulting wake will be less non-uniform and the unsteady forces will be reduced. Experimental and numerical investigations on the use of fillet forms to reduce the interference effects have been reported before [1–5]. In these papers, the type of fillet forms are all concave. According to observation and analysis of our experimental research on the effect of fillet form[6], the strength and the unsteadiness of the horseshoe vortex and non-uniform of wake may be effectively reduced by designing a suitable convex fillet form. Here, a numerical investigation was carried to evaluate the effectiveness of the convex forms and to demonstrate the capability of the numerical approach in the design of vortex flow control devices.
A general purpose computer code for the solution of the complete three-dimensional Reynolds-averaged Navier-Stokes equation has been developed [7]. In this numerical procedure, the 3D Navier-Stokes equation is solved by finite-volume scheme in a nearly-orthogonal body-fitted coordinate system. The pressure-velocity coupling is treated with SIMPLEC algorithm [8] using a non-staggered grid. The Rhie-Chow algorithm [9] is chosen to avoid the well known problems due to chequerboard oscillations in the pressure and velocity fields which are traditionally associated with the naive use of non-staggered grids. The system of algebraic equations formed by the assembly of the convection-diffusion and pressure equations are solved by the strongly implicit procedure (SIP) algorithm and the preconditioned conjugate gradients (PCG) algorithm [7]. The two

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equation k- model with wall functions is used for the turbulent flow involving separation and vortices.
This numerical procedure is used in the present investigation. Three configurations including a baseline configuration consisting of a wing mounted on a flat-plate junction, a triangle fillet form, and a constant radius convex arc fillet form along the entire wing/flat-plate junction are presented here. The numerical solutions were used to evaluate the relative effectiveness of the three configurations. Particular emphasis is placed on the discussion of effectiveness of convex fillet forms on the stability of the horseshoe vortex.
2.
NUMERICAL METHOD
2.1 Governing Equations
We consider the governing equations in Cartesian coordinates (xi,t)=(x,y,z,t) for unsteady, three-dimensional, incompressible flow. The complete three-dimensional Reynolds-averaged equations of continuity and momentum of the mean flow are
(1)
(2)
where Ui=(U,V,W) and ui=(u,v,w) are, respectively, the Cartesian components of the mean and fluctuating velocities, t is time, p is pressure, ρ is mass density, and μ is dynamic viscosity.
If the Reynolds stresses are related to the corresponding mean rate of strain through an isotropic eddy viscosity vt,
(3)
where is the turbulent kinetic energy. Here, vt is related to the turbulent kinetic energy k, and its rate of dissipation , by the two-equation k- model
μt=ρνt (4)
and k and are obtained from the transport equations
(5)
(6)
where G the rate of production of k is defined by
(7)
and (Cμ,C1,C2,σk,σε) are constants whose values are (0.09, 1.44, 1.90, 1.0, 1.3).
It is convenient to rewrite the equation of continuity and the transport equations(1),(2),(5),(6) for momentum(U,V,W) and turbulence quantities(k, ) in the following general form:
(8)
where again represents any one of the convective transport quantities (U,V,W,k,). The scalar diffusivity and source functions for Ui, k and are, respectively,
(9a)
(9b)
(9c)
2.2 Body-Fitted Coordinate Systems
In order to extend the capabilities of the difference methods to deal with complex geometries, a curvilinear coordinate transformation is used to map the complex flow domain in physical space to a simple (i.e. rectangular) flow domain in computational space. In other words, the Cartesian coordinate system (xi)=(x,y,z) in the physical domain is replaced by a curvilinear coordinate system (ξi)=(ξ,η,ζ) such that boundaries of the flow domain correspond to surfaces ξi=constant.
For the present application to wing-body junction flow with fillet forms, the body-fitted numerical grids were generated by a system of elliptic partial differential (Possion) equations of the form of
(10)

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Here, ▽2 is the Laplacian operator in Cartesian coordinates xi. The nonhomogeneous source functions fi may be assigned appropriate values to yield desirable grid distributions. In practical applications, the inverse transformation of equation(10) is used to obtain the coordinate transformation relations xi=xi(ξi), i.e.,
▽2xi=0 (i=1,2,3) (11)
where ▽2 is the Laplacian operator in the transformed plane. (ξi). Using the transformation relations, the equation(11) can be rewritten as,
(12)
where gjk is the inverse metric tensor[7]. The grid-control functions fi were determined by the specified boundary-node distribution in this paper.
The body-fitted numerical grids generated by above method is a general non-orthogonal coordinate system. In order to simple the calculation and ensure the accuracy of solution, it is important to ensure that the grid is nearly orthogonal at boundaries. In present study, a corrective method[7] is used to generate nearly orthogonal grids.
2.3 Transformation of the Equations
The price that has to be paid for the simplicity of implementing boundary conditions using body-fitted coordinate systems is the increase in complexity of the governing equations when the Cardesian coordinates is transformed to the non-orthogonal coordinate system(ξi). The vector operation in the transformed plane is
(13)
where are the normal flux components of V, as defined in [7]. It is convenient to write equation(8), in the steady state, in the equivalent form:
(14)
where Ii is called the total(i.e. convective+ diffusive) flux, and is given by:
(15)
Using the expression (13) for the covariant divergence, we obtain immediately:
(16)
i.e. exactly the same form as equation(14), with the effective total flux given in contravariant and normal components by:
(17)
Summarizing, the governing equations in computational space are:
(18)
(19)
where the normal flux components are the scalar products of velocity vector U with the area vectors
(20)
and
(21)
Note that the effective diffusion tensor Гij is a symmetric tensor, by virtue of the fact that gij is symmetric. Also, it is diagonal if and only if gij is diagonal, so the effective diffusion tensor is orthotropic if and only if the coordinate transformation is orthogonal, and it is fully anisotropic if and only if the coordinate transformation is strictly non-orthogonal. Due to its importance in the subsequent discretization process, we give the tensor multiplier in (21) a special symbol:
(22)
and we call Gij the geometric diffusion coefficients.
2.4 Discretization of Equations
The discretisation of the advection-diffusion equation is now straightforward. Integrating (18) over a control volume in computational space, we obtain, since computational space control volumes are unit cubes:

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(23)
where nn(nearest neighboring face)=u,d,n,s,e, or w as shown in Figure 1. Using (19), we have:
(24)
or
(25)
where are the convection and anisotropic diffusion coefficients defined by:
(26)
(27)
The standard convection and diffusion coefficients are given by:
The Rhie-Chow algorithm[9] is used for the interpolation of velocity components to control volume faces required for the computation of convection coefficients. We employ hybrid differences, i.e. central differences when mesh Peclet number is less than 2 and upwind differences when mesh Peclet number is greater than 2, for the values of Figure 2). For example, we have:
etc.
Explicitly, we obtain, substituting (25) into the discretisation equation (23):
(28)
where NN are the values of nn are the standard matrix coefficients obtained using hybrid differencing normal to control volume faces, and sm is a mass source term, i.e.
(29)
S' is the extra term arising from the cross-derivatives due to the non-orthogonality of the grid:
(30)
In full, we have:
(31)
and inserting (31) into (28) gives the requires linear equations for
we obtain the linear equations:

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(32)
where
Therefore, the 19-point molecule is reduced to a 7-point molecule, and the use of hybrid differencing to compute the matrix coefficients of this 7-point molecule guarantees diagonal dominance of the resulting matrix if the linearisation of the source term is chosen so that sp≤0.
2.5 Velocity-Pressure Coupling Algorithm
If the pressure is known, equation (32) can be employed to solve equation (18) for U,V,W,k,ε. In practice, however, the pressure is not known a priori and must be determined by requiring the velocity field to satisfy the equation of continuity (1). Here, we derive the pressure-correction equation obtained by applying the SIMPLEC algorithm [8] to the momentum equations (32) for Cartesian velocity components on the non-staggered grid.
Let Ui*, P* denote the most recently updated velocity and pressure fields after the linearised momentum equations have been solved. From the equations (32), we may write the momentum equations in the form:
(33)
where S'Ui are the remaining source terms after the pressure gradient source terms have been removed ( i.e. the non-pressure gradient and the non-orthogonality source terms) divided by the matrix diagonal a p,, and is the matrix multiplier of the computational space pressure gradients:
(34)
Now a solution Ui* of (33) does not in general satisfy the continuity equation; it has a residual mass source:
(35)
The term in (35) involve the values of the normal velocity components on mass control volume faces, and these must be approximated somehow from the velocity components at mass control volume centers. The prescription for doing this is the whole crux of the Rhie-Chow algorithm.
The main idea of the SIMPLEC algorithm is to find update velocity and pressure fields Ui*, P** obeying the discrete momentum equations and the discrete continuity equation:
(36)
(37)
We use the term to minus the bath hand of (36) and (37):
(38)
(39)
Assuming:
p'=p**−p*, Ui'=Ui**−Ui*
and
we obtain the following formulae for the velocity-and pressure-corrections by (39)–(38):
(40)
where

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(41)
Thus we obtain update velocity and pressure fields satisfying exact mass continuity and approximately satisfying the discrete momentum equations.
It follows that the corrected values of the normal velocity flux components are given by:
(42)
where
(43)
The pressure-correction equation is obtained by substituting (42) into the continuity equation (37). We obtain:
(44)
where
(45)
and S' are the additional terms due to non-orthogonality. It can be obtained by (30) for =P and
(46)
2.6 Solution Procedure
The system of algebraic equations is formed by the assembly of the convection-diffusion and velocity-pressure correction equations (28) and (44). This approach ignores the non-linearity of the underlying differential equations. Therefore iteration is used at two levels; an inner iteration to solve for the spatial coupling for each variable and an outer iteration to solve for the coupling between variables. Thus each variables is taken in sequence, regarding all other variables as fixed. By always reforming coefficients using the most recently calculated values of the variables.
In present paper, the U, V, W equations are solved by the SIP algorithm with 5 internal iterations; the P pressure equation is solved by the PCG algorithm with 30 internal iterations; the k, ε equations are solved by the ADI (Alternating-Direction-Implicit) algorithm with 4 internal iterations, different iteration methods. The mass source residual, the error in continuity, is chosen as stopping criteria for the outer iteration.
3. CFD VISUALIZATION
One of the great problems with three-dimensional flow calculations is that of interpreting the sheer amount of outputs of the numerical method. Thus an integrated CFD visualization system VISPLOT [10] is developed to process and analysis the numerical results of this paper. The VISPLOT has a friend user-interface which converts the results into a database and allows the user to dialogue with computer and database. Three forms of data processing are carried out: on one-dimensional grid lines, two-dimensional grid planes, or full three-dimensional graphics. The facilities of VISPLOT include:
Color contours of scalar quantities in grid slices and planes.
Profiles of the variables on arbitrary lines through the computational domain.
Velocity vector plots.
Particle tracks.
Full exploitation of color to bring out the features of the flow, for example, to overlay velocity vector plots with contours of the pressure.
3D plot of grid distribution, geometry of problems, or velocity fields etc..
4. RESULTS AND DISCUSSION
We have selected the same configuration used in our experiments[6]. The wing is 25.9 cm chord length, consisting of a 3:2 semielliptic leading edge and a NACA 0020 aft section. The baseline configuration consists of the wing mounted on the flat-plate(See Fig. 3(a)). The triangle fillet form is a triangle form of side length 0.1C along entire wing/flat-plate junction(See Fig. 3(b)). The Convex arc fillet form is a circular arc form of radius 0.1C along entire wing/flat-plate junction(See Fig. 3(c)). Here, C is chord length of the wing. All computations of turbulent junction flows with and without fillet forms were made on the 50 ×30×30 nearly-orthogonal grids(See Fig. 3). The solution

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domain consisted of a semicircular cylinder of radius R/C=1.5 attached to a rectangular region 0 <X/C<2.5, 0<Y/C<1.5, 0<Z/C<1.5. The first spacing normal to the flat-plate and wing is on the order of 0.001C. The mass source residual less than 10−4 was chosen to judge whether convergence has been achieved. In generally the solution became convergence at about 180 cycle outer iterations using about 8 CPU hours on a PC-486/50.
As mentioned earlier, the main objective of using a fillet form is to weaken the horseshoe vortex generated. A simple method can be used to judge these fillet forms with minimum ambiguity on whether the vortex generated in one form is stronger or weaker than the vortex generated by another form. It is by comparing the sharpness of the kinks in the Cp profiles near the leading edge. Sharper kinks in the Cp profiles imply that a stronger vortex has been generated. Applying this criterion to the Cp profiles of the baseline, the triangle fillet form and the convex arc fillet form shown in Figures 4(a), 4(b), and 4(c), it can be seen that the strong kinks exist in the Cp profile of the baseline. But the kinks also exist in the Cp profiles of the fillet forms. This implies that the strength of vortex generated will not be effectively reduced by unsuited fillet forms.
The velocity vector fields in the symmetry plane in front of the leading edge of the baseline, the triangle fillet form and the convex arc fillet form are shown in Figures 5(a), 5(b), and 5(c). It can be seen that the vortex generated by the baseline is an elliptic form but the vortices generated by the fillet forms are circular forms. It means that the vortex generated by the baseline is unstable and will become low-frequently oscillating vortex but the vortex generated by the fillet forms, specially by the convex fillet form, may be stable and adhere the junctions. It is very similar with the observation of our experiments [6]. It implies that the convex fillet form can control the unstable vortex generated by the junction configuration.
Figures 6(a), 6(b), and 6(c) of velocity vectors in transverse section x/C=2.14 clearly show the streamwise vortices in the wake formed by ahead horseshoe vortices. These vortices cause the wake velocity profiles becoming significantly nonuniform (See Figure 7(a), 7(b), and 7(c)). The velocity fields around the leading edge of wing at z/C=0.01 horizontal plane are shown in Figures 8(a), 8(b), and 8(c). They show the flow reversal in fornt of the wing associated with the ahead vortex.
5. CONCLUSIONS
An improved numerical method for the solution of the complete three-dimensional incompressible, Reynolds-averaged Navier-Stokes equation based on finite-volume scheme in a non-staggered grid has been applied to evaluate the effect of fillet forms on wing-body junction flow. Three configurations including a baseline, a triangle fillet form, and a convex arc fillet form were considered. The computed pressures on the entire flat-plate are first used to discuss the effectiveness of the fillet forms in terms of the ability of each to reduce the strength of the horseshoe vortex. The velocity vector fileds in the symmetry plane ahead leading edge of the wing are used to analysis the ability of each to improve the stability of the horseshoe vortex. It has been demonstrated that the numerical approach is a valuable tool for evaluation of the effectiveness of the fillet forms and the design of vortex flow control devices. It is the main purpose of this paper.
REFERENCES
1. Sung, C.H., Michael, J.G., and Roderick, M.C., “Numerical Evaluation of Vortex Flow Control Devices,” AIAA paper 91–1825, AIAA 22nd Fluid Dynamics, Plasma Dynamics & Lasers Conference, June 24–26, 1991, Hawaii.
2. Kubendran, L.R., Bar-sever, A. and Harvey, W.D., “Juncture Flow Control Using Leading-Edge Fillets,” AIAA Paper 85–4097, 1985.
3. Kubendran, L.R., Bar-sever, A. and Harvey, W.D., “Flow Control in a Wing/Fuselage-type Juncture,” AIAA Paper 88–0614, 1988.
4. Pierce, F.J., Frangistas, G.A. and Nelson, D.J., “Geometry Modification Effects on Junction Vortex Flow,” Symposium on Hydrodynamic Performance Enhancement for Marine Application, Newport, RI, November, 1988.
5. Sung, C.H., and Yang, C.I., “Control of Horseshoe Vortex Juncture Flow Using A Fillet,” Symposium on Hydrodynamic Performance Enhancement for Marine Application, Newport, RI, November, 1988.
6. Li, D., and Zhou, L.D., “The Effect of Juncture Form on Appendage-Body juncture Flow,” Osaka Colloquium '91, Japan.
7. Li, D., “Viscous Flow Around Body-Wing Junctures,” Ph.D. Thesis, China Ship Scientific Research Center, 1992.
8. Von Doormal, J.P. and Raithby, G.D., “ Enhancements of the SIMPLE Method for Predicting Incompressible Fluid Flows,” Numerical Heat Transfer, Vol 7, pp147–163, 1984.
9. Rhie, C.M. and Chow, W.L., “Numerical Study of Turbulent Flow Past an Airfoil with

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Trailing Edge Separation,” AIAA Journal, Vol 21, No. 11, pp 1525–1532, 1983.
10. Li, D. and Zhou, L.D., “Computational Fluid Dynamics Visualization System,” To be Published in Journal of Hydrodynamics, 1993
•=Mass control volume centers
○=Mass control volume face centers
Figure 1. Grid Structure and Numerical Molecule
CDS=Central Difference Scheme.
HDS=Hybrid Difference Scheme.
Figure 2. Difference Schemes for Normal and Cross-Derivatives.
Figure 3. Geometric Configuration of Junction and Computational Grids

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Figure 4. Pressure Coefficient Cp Profiles on The Flat-Plate
Figure 5. Velocity Vectors in The Symmetry Plane in Front of Wings

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Fig. 6 Computational domain for the approximate treatment of a catamaran hull for TUMMAC-IV simulations.
Fig. 7 Boundary conditions for the inner side of a catamaran

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Fig. 8 Comparison of simulated wave contours in the outer region of a catamaran at 40kt, contour interval is 0.01 nondimensional wave height, M-P1, M-P2, M-P3 and M-P4 from above, the contour interval is 0.05.
Fig. 9 Same as Fig. 8, in the inner region of a catamaran, top-side of each figure coincides with the catamaran centerline.

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Fig. 10 Computational domain for the shallow water case.
Fig. 11 Bottom boundary condition.
Fig. 12 Simulated wave contours of the outer ( above) and inner (below) regions, the contour interval is 0.05.

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Fig. 13 Transverse section of grid system
Fig. 14 Pressure contours of M-P4 on hull surface and water plane, the contour interval is 2×10−3, bow field (right ) and stern field (left).

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Fig. 15 Longitudinal distribution of pressure on three lines of a hull.
Fig. 16 Contours of longitudinal vorticity component and wake at three cross-sections of M-P4.

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Fig. 17 Simulated bow (right) and stern (left) wave systems for M-P4 with short stern.
Fig. 18 Contours of longitudinal vorticity component of M-P4 with short stern, simulation with free-surface.
Fig. 19 Comparison of hull profile of M-P1 (upper) shortened model (lower).

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Fig. 20 Comparison of pressure contours on hull surface and waterplene, M-P1 (left) and shortened model (right), the contour interval is 2×10−3
Fig. 21 Comparison of pressure coefficient along waterlines of M-P1 and shortened model.
Fig. 22 Comparison of longitudinal vorticity contours at the stern end of M-P1 and the shortened model.

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Fig. 23 Time-historical variation of forces and moment in the simulation of the oblique tow case.
Fig. 24 Contours of wake (left, contour interval 0.1) and longitudinal vorticity (right, contour interval 5) at three sections near the stern end.

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Fig. 25 Pressure distribution on the hull surface and waterplane of the 5º oblique tow case, the contour interval is 2×10−3
Fig. 26 Longitudinal distribution of pressure on the hull surface.

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Fig. 27 Lateral force vs. oblique tow angle.
Fig. 28 Yaw moment coefficient vs. oblique tow angle.

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DISCUSSION
by Dr. J.Ando and Dr. K.Nakatake
Kyushu Univ., Japan.
We would like to congratulate the authors for developing SSTH by concerted applications of several numerical techniques. We are interested in the flow field in the stern region where the propeller is operating. According to our experience, the free surface effect on the stern flow becomes large at high speeds. A comparison between Fig. 18 with Fig. 16 in your paper indicates the strong effect of the free surface on the vorticity field. You showed the calculated flow field near the stern region without free surface. If the free surface is considered, however, we believe that the flow field will be fairly different. Would you comment on this point?
Author's Reply
As the discussors suggest, the interaction of the free-surface with the viscous motion is most important at the stern. This is especially true for high-speed vessels and has been one of the major objectives of the researches at the author's laboratory. However, we have not yet reached the satisfactory results due partly to the nonlinearity of the free-surface motion and partly to the inadequacy of the turbulence modelling.
DISCUSSION
by Dr. Raven
MARIN
According to your presentation, you used two (2) criteria to select the best hull form from the wave-resistance point of view:
the peak wave heights in between the demi-hulls;
the integrated wave energy in the entire domain.
The former is not necessarily related to the wave resistance. Could you clarify the second criterion?
Is this a wave pattern analysis approach, or anything else representative of radiated wave energy?
Author's Reply
One of the shortcomings of such CFD simulation of waves in a restricted region is that the dispersive spread of wave system is not well realized. Therefore, the estimation of the relative magnitude of wave resistance must be made by either integration of wave energy in the computational domain or integration of the surface pressure distribution. For local modification of the hull form the use of the former is useful and the letter for other cases if pressure integration is carefully performed.
DISCUSSION
by Dr. Marshall P.Tulin
Ocean Engineering Laboratory, UCSB
The authors do not provide section plans for the hulls, which are the subject of the paper. It is therefore difficult to analyze their results. Could they provide a sketch showing section plans?
Author's Reply
The purpose of our paper is to demonstrate the extent the CFD techniques can be applied to very practical problems. Unfortunately we cannot show details of the lines, because it is really practical.