Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics (1994) / Chapter Skim
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Session 4- Wavy/Free Surface Flow: Field Equation Methods
Session 4- Wavy/Free Surface Flow: Field Equation Methods
Pages 153-212
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From page 155... ...
Hino uses a finite difference scheme expressed in body-fitted curvilinear coordinates to discretize the solution domain on and below the free surface. The computational grid is not allowed to move with the free surface so an am proximation must be employed to model the free surface boundary conditions.
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From page 156... ...
The nonlinear free surface boundary condition is satisfied by an iterative procedure in which the grid is moved with the free surface. Comparisons of numerical predictions with experimental data, for the Wigley hull and Series 60, Cb = 0.6 ship hull, show encouraging results for both waterline profiles and wave drag.
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From page 157... ...
This set of equations shall be solved subject to the following boundary conditions. 2.2 Boundary Conditions 2.2.1 Free Surface When the effects of surface tension and viscosity are neglected, the boundary condition on the free surface consists of two equations.
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From page 158... ...
Each formulation is explicit and uses local time stepping. Both multigrid and residual averaging techniques are used in the bulk Bow to accelerate convergence.
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From page 159... ...
Although this situation would presumably lead to a more accurate solution through the "penalty effect" in the pressure equation, very small time steps would be required to ensure stability. Conversely, for small I', the difference in the maximum and minimum wave speeds may be significantly reduced, but at the expense of accuracy.
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From page 160... ...
3.1.1 Artificial Dissipation This scheme reduces to a second order accurate, nondissipative central difference approximation to the bulk flow equations on sufficiently smooth grids. A central difference scheme permits oddeven decoupling at adjacent nodes which may lead to oscillatory solutions.
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From page 161... ...
, which states that the fastest waves in the system may not be allowed to propagate farther than the smallest grid spacing over the course of a time step. In this work, local time stepping is used such that regions of large grid spacing are permitted to have relatively larger time steps than regions of small grid spacing.
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From page 162... ...
A ~level "W-cycle" is used in the present work for each time step on the fine grid i18~. 3.1.6 Grid Refinement The multigrid acceleration procedure is embedded in a grid refinement procedure to further reduce the computer time required to achieve steady state solutions on finely resolved grids.
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From page 163... ...
After the free surface is updated, its new values are used as a boundary condition for the pressure on the bulk flow for the next time step. The entire iterative process, in which both the bulk Dow and the free surface are updated at each time step, is repeated until some measure of convergence is attained; usually steady state wave profile and wave resistance coefficient.
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From page 164... ...
4.3 Series 60, Cb 0.6 Hull In contrast to the Wigley hull, which is an idealized shape, the Series 60 hull is a practical geometry for an actual ship hull. The only major difference in the method of computing the flow about this hull and the Wigley model is the en fort required to maintain the proper hull shape as the grid is distorted by the moving free surface.
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From page 165... ...
[3] Hino,T., "Computation of Free Surface Flow Around an Advancing Ship by the NavierStokes Equations", Proceedings, Fifth International Conference on Numerical Ship Hydrodynamics, pp.
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From page 166... ...
[12] Kodama,Y., "Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme", Proceedings, Fifth International Conference on Numerical Ship Hydrodynamics, pp.
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From page 167... ...
Figure 2: Fine Grid for Wigley Hull Navier-Stokes Computations (193 x 65 x 49) Figure 3: Fine Grid for Series 60, Cb = 0.6 Navier-Stokes Computations (193 x 65 x 49)
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From page 168... ...
~) \J Navier-Stokes Figure 6: Comparison of Computed Overhead Wave Profiles, Wigley Hull, Fr = 0.250 168
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From page 169... ...
400. soot Multigrid Cycles Navier-Stokes Figure 9: Comparison of Computed Overhead Wave Profiles, Wigley Hull, Fr = 0.289 169
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From page 170... ...
a - - - - - ~ -if 5 ~ ~ ~ _ _ ~ a_ Figure 10: Velocity Vectors, Wigley Hull Stern Region, Fr = 0.289, Re = 3.2E+06 Figure 11: Velocity Vectors, Wigley Hull Stern Region, Fr = 0.289, Re = 3.2E+06 (close up view)
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From page 171... ...
400. soot Multig rid Cycles Navier-Stokes ~.~; \ \~ `\\ \\ Figure 14: Comparison of Computed Overhead Wave Profiles, Series 60, Cb = 0.6, Fr = 0.316 171
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From page 172... ...
Author's Reply (1) In the viscous boundary layer the velocity will tend toward zero due to the no-slip boundary condition s the normal distance to the ship hull tends to zero.
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From page 173... ...
The flow domain around a submerged body is divided into triangular cells, which makes up the unstructured grid system fitted to a free surface boundary. The incompressible Euler equations and the continuity equation with artificial compressibility are discretized by the finite-volume method in the unstructured grid.
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From page 174... ...
and so on. When one solves nonlinear free surface flows around a ship with a boundary-fitted grid, which is common in the recent CFD method, a grid must be generated at each time step, because free surface is dynamic in time.
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From page 175... ...
P is a matrix defined as ~ 32 0 0 ~ P= 0 1 0 (10) O 0 1 Boundary conditions needed for free surface flow problems are a body surface condition, a free surface condition and a far field condition.
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From page 176... ...
Artificial Dissipation Since the evaluation of Eq.~16) described above is the scheme equivalent to the central difference scheme for the Euler equations, this scheme is not stable due to the decoupling of neighboring node unless one adds the artificial dissipation terms to the equations.
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From page 177... ...
qj and Cj can be evaluated by replacing pi, vi and p2 in the above equations by Hi, Vj and pa, respectively. Boundary Conditions Body Boundary Condition Free-slip condition on the body (11)
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From page 178... ...
To prevent the reflection of waves to the solution domain, the outflow conditions must be carefully implemented. The open boundary conditions for free surface problems are treated by the various methods [124.
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From page 179... ...
Then, the free surface kinematic condition (14) is solved in the same manner as the flow equations and the wave height at time step (n + 1)
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From page 180... ...
A local time stepping is the method in which the solution at each point proceeds in time with the time step defined locally from the local stability limit, while a residual smoothing is used to increase the bound of the stability limit of the time stepping scheme itself. A multigrid method is an efficient way to accelerate the convergence, where the time stepping is carried out by using successively coarser grids as well as the original finest grid.
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From page 181... ...
The multigrid cycle employed here is V-cycle in which 181 1 A2 :=73 , ~ f A 2 \ II III Figure 5: Transfer of solution the equations are solved only when the solution mo~res from the fine grid to the coarse one and the interpolation is used in the transfer of correction from the coarse grid to the fine one. In case of the structured grids, the generation of successively coarser grids can be done simply by deleting the alternate points along each grid line.
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From page 182... ...
This feature is important, because the the driving force in the coarse grid comes only from the fine grid residuals due to the forcing function (59~. Finally, Ii+i is for the interpolation of the corrections in the coarse grid to the fine grid.
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From page 183... ...
This is due to the fact that time advancement in one multigrid cycle in the three-level mutigrid is approximately At + 2At ~ 4At = 7At, where At is the time step in the finest grid, because the local time stepping is taken proportional to the cell area. On the other hand, time advancement in the single-grid case in one cycle is just At.
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From page 184... ...
This improvement together with development of the time-accurate scheme makes it possible to simulate transient breaking or overturning waves in the near future. CONCLUSIONS In the present study, a finite-volume method with an unstructured grid method which has been originally developed for transonic flow computations is successfully applied to incompressible flows with a free surface.
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From page 185... ...
[15] Duncan, J.H.: "The Breaking and NonBreaking Wave Resistance of a TwoDimensional Hydrofoil.", J
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From page 186... ...
~ ~ ~ ~ /\ /\ Figure S: Sequence of multigrids around NACA0012.
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From page 187... ...
Figure 9: Magnified view of the finest grid around NACA0012. Figure 10: Computed pressure distribution with the multigrid case.
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From page 188... ...
h/L 0.20 0.10 0.00 -0.lC LONG DOMAIN SHORT DOMAIN -0 2t 1 2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 ~ {.0 ~ ) .0 X/L Figure 12: Comparison of wave profiles with the long and the short domains.
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From page 189... ...
-0.1( -0.2( ; - MEASIJRED s/L=1.034 COMPORTED -2.0 - 1.0 \;1 . 0.0 1.0 2.0 3.0 X/L 7: , _ Figure 13: Comparison of wave profiles at s = 1.034 4.0 5.0 (~ I/ Figure 14: Computed pressure distributioll.
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From page 190... ...
Figure 15: Computed velocity vectors.
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From page 191... ...
To ~ o x ~ o ~ o To · ~ o To ~ v To on A + x .3k,X + x + x + + x + x + x + + + + + + + x x x x x x x x x x x NACA0012 s/L=1.034 Froude no. 0.567 alpha 5.000 Figure 16: Computed Cp distribution.
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From page 192... ...
. Figure 18: Computed wave profile and velocity distributions at s = 0.911 192
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From page 193... ...
The unstructured grid method shown here can cope with the dynamic condition in case of the complex geometry which may occur in the wave breaking process. The implementation of the kinematic free surface condition is crucial for breaking wave simulations.
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From page 195... ...
Calisal (University of British Columbia, Canada) ABSTRACT Potential flow initial-boundary value problems describing fluid structure interaction with fully nonlinear free surface boundary conditions have been studied using a mixed Lagrangian Eulerian formulation.
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From page 196... ...
One of the first attempts to use slender body theory for surface ships was made by Cumminst2] in the solution of the wave resistance problem.
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From page 197... ...
Slender body theory assumes the validity of a two-dimensional cross-sectional flow that is carried along the remaining third dimension through appropriate boundary conditions. In this case it seems advisable to initially develop a robust model that can describe two-dimensional flow with a nonlinear free surface in a reliable and accurate form.
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From page 198... ...
The two dimensional results are subsequently extended to study a mathematically defined slender body, the Wigley hull, advancing with forward speed in otherwise calm water. All concepts and results reported in this paper can be found in greater detail int63~.
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From page 199... ...
describes a boundary value problem which is solved subject to the moving boundary conditions (2)
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From page 200... ...
the free surface remains regular for linear free surface boundary conditions. However, there is qualitative differences in the free surface profiles.
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From page 201... ...
ear free surface profiles corresponding to w = 1.25 and A = 0.5 through 20 cycles of oscillation. The nonlinear free surface tends to be more irregular than that predicted by linear the Ll~R FREE SURFACE ~ _ FRE()
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From page 202... ...
No smoothing was required for this case. Figures 11 and 12 respectively show linear and nonlinear free surface time histories for A = 0.5.
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From page 203... ...
SUBMERGE CYLINDER A~.5 ~ hit) NLINEAR FREE SURFACE Figure 12: Nonlinear free surface time record for submerged cylinder (NI = 10, At = 0.05~.
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From page 204... ...
Nondimensional frequencies for linear and nonlinear free surface boundary conditions are presented. Smoothing in the worst of these cases was applied using c' = 1 every 10th time step.
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From page 205... ...
6.4 Slender body calculations In this section we present results corresponding to the forward motion of a mathematically defined ship hull advancing at zero angle of attack in otherwise calm water. The Wigley parabolic hull, whose main parameters are given in Table 4, is defined as B{1_ (2~2~(1-(Y)
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From page 206... ...
These wiggles were typical of nonlinear free surface profiles and occasionally were the origin of unstable behavior. In order to maintain the size of these wiggles bounded, small time steps Aim were used.
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From page 207... ...
In turn, this suggests that the distribution of forces is predicted with similar accuracy in both formulations. CONCLUSIONS The boundary value problem defined by the | NONLINEAR FREE SURFACE I F,l u 26 ~ Figure 20: Nonlinear free surface elevation for Wigley hull with forward speed, Fn = 0.267.
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From page 208... ...
. 0.15 0.20 O.25 0.30 0.35 FROUDE NUMBER Figure 25: Wave resistance coefficient of Wigley hull versus En flow in two dimensions has been formulated using a Bubnov-Galerkin weighted residual method to obtain the spatial variation of the velocity v.
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From page 209... ...
[17] Troesch A.W., 'S,aray, Roll and Yaw Motion Coefficients Based on a Forward Speed Slender Body Theory: Part 2', Journal of Ship Research, vol.
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From page 210... ...
t41] Cointe R., 'Nonlinear Simulation of Transient Free Surface Flows', Proceedings of the 5th International Conference on Numerical Ship Hydrodynamics, 1989.
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From page 211... ...
t54] Bai K.J., 'A Localized Finite Element Method for Steady, Two Dimensional Free Surface Flow Problems', Proceedings of the 1st International Conference on Numerical Ship Hydrodynamics' 1975.
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From page 212... ...
t74] 'Proceedings of the Second Workshop on Ship Wave Resistance Computations',David Taylor Naval Ship Research and Development Center, 1983.
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