Numerical Computation of 2-Dimensional Waves behind a Hydrofoil

Abstract

The Navier-Stokes equation is directly solved by the finite-difference method to simulate the 2-dimensional free-surface flow field generated by a submerged hydrofoil (NACA-0012) in a uniform flow. The staggered-mesh system and a body-and boundary-fitted coordinates system are adopted in the whole domain under consideration.The non-slip condition on the wing and the exact non-linear condition on the free-surface are used as the boundary conditions. Flows under three different submergence depths are simulated with the same Froude and Reynolds numbers and attack angle.In the deepest case, the calculated wave length agrees well with the measured while the wave height is appreciably less mainly due to the difference of the Reynolds number.It needs rather long computing time for the waves to develop enough to reach a stationary condition.The results for the shallowest case where the sub-breaking is suspected to take place show a numerical breaking.The criterion for the sub-breaking is applied to the computed results.It detects the appearance of breakings in the course of computation.

On Turbulent Characteristics and Numerical Simulation of 2-Dimensional Sub-Breaking Waves

“…The boundary value of k is fixed to zero on the free-surface. The pressure on the free-surface is not zero any more and given by ( 4 ) . The length scale 1 is provided by the experimental data of [CASE A] in the region of -0.2 <x'<0.4.…”

Numerical Calculation of Free-Surface Flows and Detection of Sub-Breaking Waves around 2-D Floating Body

“…we have following equations with the forward time differencing: ( 5 ) where, ( 6 ) where the superscript denotes the time step and the last term in ( 5 ) is expected to be zero at every time step.…”

Numerical Computation of 3-D Free Surface Flows by N-S Solver and Detection of Sub-breaking

“…To solve directly the full Navier-Stokes equations for flows past geometrically ship-like bodies, the previously developed 2-dimensional numerical codes by Shin and Mori [1] Differentiating ( 1 ) with respect to x, y and z, we can have (5) The last term in ( 5 ) is expected to be zero and can be solved by a relaxation method. The new free-surface at the (n+ 1) th time-step is calculated by moving the marker particles by (6) The z-coordinate of the free-surface is re-arranged by the bivariate linear interpolation in proportion to the newly calculated projected-area.…”