On the use of conformal mapping for the computation of hydrodynamic forces acting on bodies of arbitrary shape in viscous flow. Part 2: multi-body configuration

Abstract

Two-dimensional viscous flows around obstacles are considered in an unbounded liquid. The basic idea developed in Part 1 is further extended from a single body to multi-body configurations. This idea follows from the formulation by Quartapelle and Napolitano (AIAA J. 21:991–913, 1983) who proposed computation of the force and moment in incompressible viscous flow without explicitly calculating the pressure. The principle is the projection of Navier–Stokes equations on a set of functions. Surprisingly, these functions have a precise meaning in potential theory. They are the solutions which lead to the added masses and added moment of inertia for the potential flow around the studied arrangement of obstacles. By revisiting this problem, a general identity of the full coupled matrix of added masses and added moment of inertia is formulated. To this end conformal mappings for multi-body configurations are used. Robustness of the proposed algorithms is tested and illustrated. The obtained potential solution is merely a mathematical solution and it does not allow to describe the actual potential flow since the circulation is not accounted for. However, its interest is crucial for implementing the projection technique developed by Quartapelle and Napolitano. The interest of such a method is two-fold. Firstly, it provides a way of computing the force without explicitly calculating the pressure. Consequently and secondly, it offers an alternate way to validate the computation of the loads. In effect, these loads are always available from a direct integration of the Cauchy stress tensor (pressure plus friction). It is worth mentioning that the present technique allows an a posteriori computation of the pressure.