Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles

Abstract

This paper is concerned with small amplitude vortical and entropic unsteady motions imposed on steady potential flows. Its main purpose is to show that, even in this unsteady compressible and vortical flow, the perturbations in pressure p’ and velocity u can be written as p’ = ρ₀D₀ϕ/Dt and u = ϕ + u^(I) respectively, where D₀/Dt is the convective derivative relative to the mean potential flow, u^(I) is a known function of the imposed upstream disturbance and ϕ is a solution to the linear inhomogeneous wave equation \[ \frac{D_0}{Dt}\bigg(\frac{1}{c^2_0}\frac{D_0\phi}{Dt}\bigg)-\frac{1}{\rho_0}\nabla\cdot(\rho_0\nabla\phi)=\frac{1}{\rho_0}\nabla\cdot\rho_0{\bf u}^{(I)} \] with a dipole source term ρ₀⁻¹ [xdtri ]ρ₀u^(I) whose strength ρ₀u^(I) is a known function of the imposed upstream distortion field. (Here c₀ and ρ₀ denote the speed of sound and density of the background potential flow.) This equation is used to extend Hunt's (1973) generalization of the ‘rapid-distortion’ theory of turbulence developed by Batchelor & Proudman (1954) and Ribner & Tucker (1953). These theories predict changes occurring in weakly turbulent flows that are distorted (by solid obstacles and other external influences) in a time short relative to the Lagrangian integral scale.

The theory is applied to the unsteady supersonic flow around a corner and a closed-form analytical solution is obtained. Detailed calculations are carried out to show how the expansion at the corner affects a turbulent incident stream.