Twenty-Fourth Symposium on Naval Hydrodynamics (2003) / Chapter Skim
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A Spectral-Shell Solution for Viscous Wave-Body Interactions
A Spectral-Shell Solution for Viscous Wave-Body Interactions
Pages 360-375
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From page 360... ...
ABSTRACT To solve viscous wave-body interaction problems, a domain decomposition technique is used in which a viscous flow in the near field is matched to an outer inviscid flow. A pseudo-spectral solution of the integral equation describing the outer flow is formulated in such a way that the outer solution can be applied to a variety of interior-specific inner problems.
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From page 361... ...
Somewhat surprisingly, when tested in the frequency domain, the technique described here does not seem to demonstrate the usual irregular frequency effects associated with the use of a free-surface Green function. The predictor-corrector matching scheme can be extended to provide an outer boundary condition for the solution of viscous flow in the interior region.
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From page 362... ...
. Because the Green function satisfies the linearized free-surface boundary conditions, the bottom boundary condition, and the far-field conditions, only integrals over the matching surface remain in the integral equation.
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From page 363... ...
Because this outer solution is developed for a fixed cylindrical geometry and a "compute once, use many times" approach is being used, this is not a penalty as it would be in the case where the boundary integral equation method is applied directly to a moving body. With the approximation of linearly varying potential, ~,~ is removed from the time integrals and the remaining time integrals of H can be defined as: ~K,k~p;§ z)
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From page 364... ...
Figure 2: Polar coordinate system for evaluation of Green function coefficients on shell surface. COMPUTATION OF SHELL COEFFICIENTS Because the formulation presented here is intended to be a universal solution of the outer flow that can be applied to a variety of internal problems, it is important to determine the most efficient way to compute and store the various coefficients for re-use.
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From page 365... ...
However, finitedifference and finite-element methods for solving the interior potential-flow problem require a point-wise boundary condition in which the relation between potential and normal velocity is specified at each point on the surface. Unfortunately, the shell method for the outer flow admits only a global relation between 6
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From page 366... ...
Int36; stead, the procedure is to predict the normal velocity on Ss at the next time-step, solve the outer-region flow using the shell function boundary-integral equation with the predicted Neumann boundary condition, and use this result as a Dirichlet boundary condition for the interior region. A follow-up corrector step improves the solution of the normal velocity at 7
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From page 367... ...
An alternate matching procedure is to predict the velocity potential ~ on Ss (instead of ~~) at the next time-step and use the outer solution to find the resulting normal velocity, thus providing a Neumann boundary condition for the inner problem.
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From page 368... ...
Incident-wave problems can be studied with the shell-function method by superposing the incident wave in the outer region only and solving for the total potential in the inner region. This departure from the usual technique of modifying the body boundary condition to reflect the presence of incident waves is done with the aim of including nonlinear effects in the inner region, providing an outer boundary condition that not only absorbs outgoing waves but that can independently supply incoming waves.
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From page 369... ...
1 [v2u _ u _ 2 dv] _ dP APPLICATION TO VISCOUS FLOWS This section applies the shell solution for the outer inviscid flow as an outer boundary condition to a viscous-flow problem in the interior region.
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From page 370... ...
In the original work of Yeung and Yu (1994) , noslip boundary conditions were applied to the outer cylinder at r = rO.
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From page 371... ...
The boundary condition on pressure is found from the free-surface boundary condition and a predictor-corrector scheme in which the pressure equation is solved twice at each time-step. The matching technique developed below follows a similar procedure, using the velocity of the K—1 and K—2 steps on the matching surface to provide boundary conditions on the intermediate ve locity field, then using the pressure at the new time step supplied by the outer flow as a boundary condi tion on the pressure equation.
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From page 372... ...
y o -1 -2 -4 _5 4 3 2 Y n -1 -2 3 4 -5— a -2 n 4 -0.020 -0.018 -0.016 -0.014 -0.01 1 -0.009 -0.007 -0.005 -0.003 -0.001 0.001 0.004 0.006 0.008 0.010 3 2 Y O -1 -2 -3 4 -2 0 x 2 4 -4 -2 X 2 4 Figure 8: Wave-elevation contours ~(x, y) and velocity vectors for a non-axisymmetric Cauchy-Poisson problem in a viscous fluid near a cylinder, Re = 5, 000.
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From page 373... ...
This ap proach is based on the physical idea that in a viscous invisid matching, the inviscid outer flow is expected to be incapable of supporting shear stresses on the matching surface. The predicted radial velocity could be used as the final, required boundary condition for the inner flow but good results with less computa tion are achieved by simply using the most current time-steps radial velocity in the expressions for the boundary conditions on the auxiliary velocity A
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From page 374... ...
CONCLUDING REMARKS A highly effective outer boundary condition based on the use of shell functions for an inviscid outer flow, as initially introduced by Yeung (1985) , is demon15 and outgoing directions.
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From page 375... ...
F and Iafrati, A., "Unsteady free surface waves by domain decomposition approach," Proceedings of the 16th International Workshop on Water Waves and Floating Bodies, Hiroshima, Japan, 2001.
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