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Twenty-First Symposium on NAVAL HYDRODYNAMICS
as the fully nonlinear simulation. Moreover, the discrepancy has probably nothing to do with the separation bubble at the leading edge. This can be inferred from the zero-Froude number results, for which the inviscid model gives an excellent prediction of the hydrodynamic coefficients, in spite of the recirculation.
The present paper is organized as follows: the mathematical models used in the simulation are briefly described in Sec.1, while in Sec.2 the experimental apparatus is illustrated. Then, numerical results with both the inviscid model and the Navier-Stokes model will be reported in Sec.3, and they will be discussed in comparison with the experiments.
In this section, the mathematical models used in the simulation of the flow past a flat plate are described. The first one is the inviscid flow model with a vortex sheet. The second one is the more complex Navier-Stokes model, with an algebraic turbulence model.
In this model viscous effects are supposed to be confined close to the rigid boundary, where the fluid viscosity determines the generation of vorticity. The vorticity evolution and shedding are described by means of the inviscid fluid mechanics, that is viscous effects are completely neglected and rotational zones are modelled as vorticity layers, emerging from known separation lines (the lower tip and the trailing edge).
The fluid is supposed to fill a domain Ω of infinite depth, bounded by the free surface by the plate surface and by a zero thickness wake W emanating from the tip and the trailing edge.
The total fluid velocity U=(U,V,W)T, with respect to a reference frame fixed with the plate, is written in the form U=U∞+u, U∞ being the undisturbed flow velocity. The flowfield is supposed to be solenoidal, i.e.
·u=0 in Ω (1)
and irrotational, that is
×u=0 in Ω (2)
The perturbation velocity u satisfies the impermeability constraint on the body surface
A kinematic condition in terms of the wave height h is enforced on the free surface
together with a dynamic boundary condition, that fix the pressure p, computed from the Bernoulli theorem, to the atmospheric pressure (set to zero)
Continuity of pressure and normal velocity are required through the wake W
p+=p–u·n+=u·n– at W (6)
Finally, the Kutta condition is enforced along the separation lines and the radiation condition is required far from the body.
In the general case, the free surface and the wake shapes are unknown and therefore are to be determined as a part of the solution.
It may be shown that, under quite general assumptions , the field u(Q) may be expressed by the integral form of the Helmholtz decomposition (Poincaré Formula ), that specializes, when dealing with zero-thickness bodies, in
where the source distribution σ accounts for the free surface effects, is the bound vorticity and represents the vorticity confined in the wake. It is relevant to recall that, by virtue of the Helmholtz theorem applied to a vortex layer, both the bound vorticity and the trailing vorticity satisfy the continuity relation
The unknown source and vortex distributions are computed by enforcing the boundary conditions. The problem is strongly nonlinear and, consequently, very CPU time consuming. In order to make the computation as cheap as possible, the problem is simplified on both the free surface and the wake: free surface boundary conditions are linearized by using the double model flow as basis flow, while the dynamics of the wake is simplified by constrainig the wake “particles” to move only in the horizontal plane, thus preventing the wake roll-up (see [8, 9] for details). In the numerical solution, the source and the vorticity distribution are supposed to be piecewise constant. The enforcement of the boundary condition at control points yields a set of algebraic equations to be used in the computation of the unknown discrete values of σ, and .
Some examples of numerical simulation are reported in figures 1 and 2, where the wave pattern is depicted for two values of the Froude number and