Generalized Basset-Boussinesq-Oseen Equation for Unsteady Forces on a Sphere in a Compressible Flow

Viscous compressible flow around a sphere is considered in the limit of zero Reynolds and Mach numbers. An exact expression for the force on the sphere undergoing arbitrary motion with compressibility effects is presented. Quasisteady, inviscid-unsteady, and viscous-unsteady force components are identified. Numerical results are in excellent agreement with the theory. The present formulation offers an explicit expression for the unsteady force in the time domain and can be considered as a generalization of the Basset-Boussinesq-Oseen equation to compressible flow.

Figure 1

The behavior of $C(\tau )$ that accounts for the compressibility effect on the viscous-unsteady force.

Figure 2

Time evolution of the normalized unsteady force. Theoretical predictions [last two terms of Eq. (18)] are plotted as solid lines for ${\mu }_b=0$. Corresponding simulation results for four different cases are shown as symbols.