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A Boundary Integral Formulation of an Acoustic Boundary Layer Model in 2D

Jacob Linden, Travis Askham, Jeremy Hoskins

TL;DR

The work addresses incorporating visco-thermal boundary-layer losses into 2D acoustics by enforcing a generalized impedance boundary on the boundary and solving via boundary integrals. The authors derive a combined-field layer-potential representation preconditioned by a surface Helmholtz–Beltrami operator, yielding a Fredholm second-kind system for the boundary density under the condition $c_1\Delta_\Gamma u + c_2 u + \partial_n u = f$. For domain decomposition, impedance-to-impedance maps are used to couple subdomains, remaining Fredholm second-kind up to a bounded invertible operator. Numerical tests on smooth and piecewise-smooth geometries demonstrate high accuracy (approximately $10^{-7}$) and good conditioning with standard discretizations. This framework enables efficient, accurate simulations of acoustic devices with narrow features, such as waveguides and phase plugs, without resolving the full viscous boundary layer.

Abstract

We present a boundary integral formulation of the Helmholtz equation with visco-thermal boundary conditions, in two dimensions. Such boundary conditions allow for the accurate simulation of viscous and thermal losses in the vicinity of the boundary, which are particularly relevant in acoustic devices with narrow features. Using cancellations between hyper-singular operators, a variant of the method of images technique, and analytic pre-conditioners, we derive integral equations that are Fredholm second-kind, up to the application of a boundedly invertible operator. This approach allows for the fast and accurate solution of acoustics problems with boundary layers.

A Boundary Integral Formulation of an Acoustic Boundary Layer Model in 2D

TL;DR

The work addresses incorporating visco-thermal boundary-layer losses into 2D acoustics by enforcing a generalized impedance boundary on the boundary and solving via boundary integrals. The authors derive a combined-field layer-potential representation preconditioned by a surface Helmholtz–Beltrami operator, yielding a Fredholm second-kind system for the boundary density under the condition . For domain decomposition, impedance-to-impedance maps are used to couple subdomains, remaining Fredholm second-kind up to a bounded invertible operator. Numerical tests on smooth and piecewise-smooth geometries demonstrate high accuracy (approximately ) and good conditioning with standard discretizations. This framework enables efficient, accurate simulations of acoustic devices with narrow features, such as waveguides and phase plugs, without resolving the full viscous boundary layer.

Abstract

We present a boundary integral formulation of the Helmholtz equation with visco-thermal boundary conditions, in two dimensions. Such boundary conditions allow for the accurate simulation of viscous and thermal losses in the vicinity of the boundary, which are particularly relevant in acoustic devices with narrow features. Using cancellations between hyper-singular operators, a variant of the method of images technique, and analytic pre-conditioners, we derive integral equations that are Fredholm second-kind, up to the application of a boundedly invertible operator. This approach allows for the fast and accurate solution of acoustics problems with boundary layers.
Paper Structure (14 sections, 5 theorems, 83 equations, 10 figures)

This paper contains 14 sections, 5 theorems, 83 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Description of the geometry
  4. Modeling boundary layers as a boundary condition
  5. Mathematical preliminaries
  6. Integral equation formulation
  7. The Laplace-Beltrami operator and Helmholtz layer potentials
  8. An integral representation on $\Gamma_*^j$
  9. Closed $\Gamma_*^j$
  10. Open $\Gamma_*^j$
  11. Mixed boundary conditions and a method of images approach
  12. Domain decomposition and impedance-to-impedance
  13. Numerics
  14. Conclusion

Key Result

Lemma 3.1

Suppose that $\Gamma$ is a smooth curve and $\sigma \in H^1(\Gamma)$. Then

Figures (10)

  • Figure 1:
  • Figure 2: A zoom-in of the fins added to the geometry in the vicinity of a corner.
  • Figure 3: Magnitude of the solution due to a point source at $(x,y)=(-0.4,0.3)$, for the geometry outlined in blue.
  • Figure 4: On the boundary of the waveguide, visco-thermal boundary conditions are used (blue), while the caps of the waveguide are given a Robin boundary condition (green). Artificial fins extend off of the corners into the exterior of the domain (red).
  • Figure 5: Real part of the solution with incoming impedance on the left boundary due to a point source.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark
  • Theorem 3.4
  • proof
  • Corollary 3.4.1
  • ...and 4 more