The Effective Impedance of a Finite-Thickness Viscothermal Boundary Layer Over an Acoustic Lining

Abstract

This paper assesses the importance of viscothermal effects for acoustics calculations in lined ducts, both inside and outside of a finite-thickness compressible boundary layer using a combination of asymptotics and numerics. Viscosity is always present, even at the high Reynolds numbers associated with aeroacoustics. The large majority of aeroacoustic calculations are performed inviscidly, however. Existing inviscid impedance boundary conditions (e.g. Myers) have failed in predicting experimental results, and it is suggested that viscosity is the key to accurate computations. Here, numerical solutions of the Linearized Navier–Stokes equations are compared to inviscid numerics inside a sheared boundary layer to quantify the errors associated with neglecting viscosity. It is found that invisicd errors are strongly dependent on frequency, with normalised errors of over 10% common at low frequencies. It is suggested that errors increase with Mach number, though the dependence is weaker than that of frequency. Viscothermal effects are also shown to be as important as shear. Existing impedance boundary conditions rely on the assumption that the acoustics outside the boundary layer are the same as they would be in a completely uniform inviscid flow. This assumption, that the near-wall effects of shear and viscous dissipation do not penetrate far into the duct, is tested here by comparing analytic expressions for the uniform acoustics with viscous numerics. It is found that errors outside a 99% boundary layer are on average 0.006% for the pressure and 0.1% for the radial velocity, validating this assumption Three existing impedance boundary conditions are tested against full viscous numerics and are found to be inadequate for modelling the possibly unstable surface modes. A new asymptotic boundary condition is derived that combines the regularising effect of a finite-thickness shear layer with viscosity and thermal conduction to accurately capture the physics of a boundary layer over an acoutic lining. Comparisons of the new boundary condition with viscous numerics are extremely positive, and due to the decoupling of the Reynolds number and boundary layer thickness in the derivation the condition may be used for any flow. The new condition correctly predicts the stability of modes as parameters vary. Though an analytic form of the new condition is not found, it is suggested that it could be incorporated into a boundary solver at minor computational cost.