ASYMPTOTIC FORMULATION FOR AN ACOUSTICALLY DRIVEN FIELD INSIDE A RECTANGULAR CAVITY WITH A WELL-DEFINED CONVECTIVE MEAN FLOW MOTION

Abstract

When small amplitude harmonic pressure waves are introduced inside an enclosure where the classic cavity flow is already established, a strong coupling ensues between the mean and time-dependent fields. In addition to the mean flow influence, the presence of solid boundaries gives rise to a strong coupling between acoustic and vortical waves that prescribes the propagation of disturbances. Based on a well-defined mean flow structure, the purpose of this report is to arrive at closed-form expressions for the two-dimensional, time-dependent velocity and vorticity fields, in order to elucidate the nature and extent of intrinsic flow coupling. On that account, regular perturbation tools are used on the vorticity transport equation derived from the linearised Navier–Stokes equations. After an inviscid expansion is achieved, successive approximations that use viscous correction multipliers are employed to derive the vorticity in a manner to satisfy the existing boundary conditions. At the outset, unsteady vorticity is found to originate at the walls due to the acoustic pressure gradient normal to the inflow direction. In consequence, most intense vortices are initiated in regions that coincide with acoustic pressure nodes. Conversely, zero vorticity streaks emanate from acoustic velocity nodes and are carried downstream by the bulk fluid motion. From the vorticity formulation, both components of the time-dependent velocity vector are derived in a manner to satisfy continuity and momentum conservation. The axial velocity component is found to be the most significant, exhibiting characteristics associated with oscillatory flows. In addition to comparisons with numerical simulations, a limiting process validation against the existing exact solution in the event of no mean flow transmission is carried out. In closing, an error assessment is included to quantify both magnitude and order of the global error accumulated in the final asymptotic expressions.