| We present some exact expressions for the evaluation of time-dependent forces on a body in an incompressible and viscous cross-flow which only require the knowledge of the velocity field (and its derivatives) in a finite and arbitrarily chosen region enclosing the body.; Given a control volume V with external surface S which encloses an arbitrary body, the fluid-dynamic force F on the body can be evaluated from one of the following three expressions (in abbreviated form):{dollar}{dollar}eqalign{lcub}{lcub}bf F{rcub}&=-{lcub}1over {lcub}cal N{rcub}-1{rcub}{lcub}dover {lcub}dt{rcub}{rcub}intsb{lcub}V{rcub}{lcub}rm x{rcub}wedgeomega dV+ointsb{lcub}S{rcub} {lcub}bf n{rcub}cdotUpsilonsb1dS + {lcub}rm body motion terms{rcub},cr{lcub}bf F{rcub}&=-{lcub}dover{lcub}dt{rcub}{rcub}intsb{lcub}V{rcub} {lcub}bf u{rcub}dV+ointsb{lcub}S{rcub} {lcub}bf n{rcub}cdotUpsilonsb2dS + {lcub}rm body motion terms{rcub},cr{lcub}bf F{rcub}&= {lcub}rm no volume integral terms{rcub} + ointsb{lcub}S{rcub} {lcub}bf n{rcub}cdotUpsilonsb3dS +{lcub}rm body motion terms{rcub},cr{rcub}{dollar}{dollar}where {dollar}cal N{dollar} is the space dimension, u is the flow velocity, {dollar}omega{dollar} is the vorticity, x is the position vector, and the tensors {dollar}Upsilonsb1, Upsilonsb2, Upsilonsb3{dollar} depend only on the velocity field u and its (spatial and temporal) derivatives.; The first equation is already known for either simply connected domains or inviscid flows. We re-derive it here for viscous flows in doubly connected domains (i.e. domains which include a body). We then obtain the second and third equation through a simple algebraic manipulation of the first equation.; These expressions are particularly useful for experimental techniques like Digital Particle Image Velocimetry (DPIV) which provide time sequences of 2D velocity fields but not pressure fields.; They are tested experimentally with DPIV on two-dimensional, low Reynolds number circular cylinder flows. Both steady and unsteady motions are studied. |
