Stern flow modelling in transpiration problem

It is well known that the standard solution of the advective equations as the free surface requires the imposition of Dirichlet conditions at the inlet boundaries.

Experimental analyses reveal that transom stern flow, occurring at a sufficient high speed, shows a local discontinuity in the wave elevation field. In those cases, the standard solution of the free surface equation close to this region is inconsistent with the convective nature of this equation. The trial of direct solving the free surface equation in this case results in instability in the wave height close to the transom region. This instability is found experimentally for low speeds, but the flow at a sufficient high speed is more stable and cannot be reproduced by using the standard techniques.

The conclusion of the above discussion is that it is necessary to determine and apply boundary conditions adequate for the free surface solution on the transom boundary. The obvious solution to this problem is to fix both the free surface elevation β and its derivative along the corresponding streamline. Its values will be approximately given by the transom position and the surface gradient. This option is available in Tdyn through the prescription of the wave elevation in the transom boundary (see Fix Beta conditions). However the direct imposition of the mentioned values can influence the transition between the transom flow and the lateral flow, resulting in inaccurate wave maps.

The solution developed in Tdyn, extends the free surface below the floater as shown in Figure 36. The necessary Dirichlet boundary conditions imposed at the inlet of the domain are sufficient to achieve the well-possessed properties of the problem. It is interesting to note, that this imposition is not ad hoc , since the free surface equation have to be accomplished also in the wetted surface below the floater. Obviously proceeding this way will remain valid both for the wetted transom and for the dry transom flows and may also be applied to floaters with regular stern. Unfortunately, in the latter case, a very fine definition of the mesh is required in some cases, in order to capture the discontinuities that may appear in the wave elevation field.

In order to simplify the application of the above ideas, Tdyn includes a methodology to automatically detect when is necessary extend the free surface below the floater. The method is based on calculating the angle between the floating line and the local velocity. Only if this angle is greater than a given value (inserted in the field SternC Angle ), the corresponding elements of the transon edge will have the necessary boundary condition.