Acoustic boundary layers as boundary conditions

TL;DR

This paper introduces an efficient boundary condition approach to model viscous and thermal boundary layer effects in acoustic wave propagation, reducing computational costs significantly compared to full Navier-Stokes simulations.

Contribution

The authors derive a boundary condition that incorporates boundary layer effects into the Helmholtz equation, simplifying simulations of acoustic problems with viscous and thermal effects.

Findings

Model accurately predicts acoustic power spectra in complex devices.

Significantly reduces computational resources compared to full Navier-Stokes solutions.

Validates the approach with cylindrical duct and transducer examples.

Abstract

The linearized, compressible Navier-Stokes equations can be used to model acoustic wave propagation in the presence of viscous and thermal boundary layers. However, acoustic boundary layers are notorious for invoking prohibitively high resolution requirements on numerical solutions of the equations. We derive and present a strategy for how viscous and thermal boundary-layer effects can be represented as a boundary condition on the standard Helmholtz equation for the acoustic pressure. This boundary condition constitutes an $O(\delta)$ perturbation, where $\delta$ is the boundary-layer thickness, of the vanishing Neumann condition for the acoustic pressure associated with a lossless sound-hard wall. The approximate model is valid when the wavelength and the minimum radius of curvature of the wall is much larger than the boundary layer thickness. In the special case of sound propagation in a cylindrical duct, the model collapses to the classical Kirchhoff solution. We assess the model in the case of sound propagation through a compression driver, a kind of transducer that is commonly used to feed horn loudspeakers. Due to the presence of shallow chambers and thin slits in the device, it is crucial to include modeling of visco-thermal losses in the acoustic analysis. The transmitted power spectrum through the device calculated numerically using our model agrees well with computations using a hybrid model, where the full linearized, compressible Navier-Stokes equations are solved in the narrow regions of the device and the inviscid Helmholtz equations elsewhere. However, our model needs almost two orders of magnitude less memory and computational time than the more complete model.