On the evaluation of instantaneous fluid-dynamic forces on a bluff body

Abstract

In this article, we present an expression (first derived in 1952) for the evaluation of instantaneous forces on a bluff body in a cross-flow, when only the velocity field (and, therefore, the vorticity field as well) is known in its vicinity. This expression is particularly useful for experimental methods such as DPIV which do not provide any information about the pressure field, but do yield the velocity and vorticity fields in a finite domain. Several interesting features of this expression are noteworthy: • It does not require the knowledge of the pressure field. • It is valid for incompressible, viscous, rotational, and time dependent flows. • It does not require a knowledge of the velocity field over the whole wake; the control volume can be chosen arbitrarily (it must include the body though). Before we give a derivation of this expression, we will remind the reader of some classical expressions for the evaluation of instantaneous forces and the assumptions underlying these expressions, and we will show how these expressions contain the pressure explicitly. We will then present a generalized Green's transformation, the Burgatti identity, which will be the key to the removal of the pressure terms from the force equations. As a starting point, the Burgatti identity will be first applied to the derivation of the force equation in vortex methods. In particular, it will be shown how the pressure terms drop out naturally from the equation, without any ad hoc assumptions about the pressure field. Then, the general expression will be derived. Again, stress will be placed on the absence of the pressure terms in the resulting equation. As a check, we will show that the force equations used in vortex methods as well as in inviscid, rotational flows do follow from the general equation.