Analytical solution for an acoustic boundary layer around an oscillating rigid sphere

Abstract

Analytical solutions in fluid dynamics can be used to elucidate the physics of complex flows and to serve as test cases for numerical models. In this work, we present the analytical solution for the acoustic boundary layer that develops around a rigid sphere executing small amplitude harmonic rectilinear motion in a compressible fluid.The mathematical framework that describes the primary flow is identical to that of wave propagation in linearly elastic solids, the difference being the appearance of complex instead of real valued wave numbers. The solution reverts to well-known classical solutions in special limits: the potential flow solution in the thin boundary layer limit, the oscillatory flat plate solution in the limit of large sphere radius and the Stokes flow solutions in the incompressible limit of infinite sound speed. As a companion analytical result, the steady second order acoustic streaming flow is obtained. This streaming flow is driven by the Reynolds stress tensor that arises from the axisymmetric first order primary flow around such a rigid sphere. These results are obtained with a linearization of the non-linear Navier-Stokes equations valid for small amplitude oscillations of the sphere. The streaming flow obeys a timeaveraged Stokes equation with a body force given by the Nyborg model in which the above mentioned primary flow in a compressible Newtonian fluid is used to estimate the time-averaged body force. Numerical results are presented to explore different regimes of the complex transverse and longitudinal wave numbers that characterize the primary flow.

Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

“…However, k can be complex, and equation (1) then represents an acoustic wave in a medium with absorption. As an example we can mention the description of acoustic boundary layers around a sphere in a viscous liquid [40], which was based on the Navier equation ( 23), but with complex kʼs. There are other types of variations of the Helmholtz equation as well.…”

“…x = (n • e x ), t 1x = (t 1 • e x ) and t 2x = (t 2 • e x ).If we replace the terms in brackets in equation(44) with equation(42) and equation(40) then:…”