Acoustic Propagation/Refraction Through Diffuse Interface Models

TL;DR

This work develops a high-order DGSEM framework with a modified weak compressibility formulation to model acoustic wave propagation across diffuse interfaces between media with different sound speeds, embedded in a two-phase iNS/CH system. By interpolating density, viscosity, and sound speed across the diffuse interface and employing entropy-stable non-conservative fluxes, the method achieves spectral convergence in 1D, validates Snell’s law in 2D, and demonstrates 3D transmission across air–water with large density ratios, while quantifying interface-width modeling errors that scale as $O( ext{ε}^2)$. The total error comprises numerical and modeling components, with modeling errors dominating when the interface is under-resolved, yet approaching the sharp-interface limit as $ε o 0$. The approach offers a scalable, physically grounded tool for direct acoustic simulations in incompressible multiphase flows, with potential applications in marine aeroacoustics and multiphase acoustics.

Abstract

We present a novel approach for simulating acoustic (pressure) wave propagation across different media separated by a diffuse interface through the use of a weak compressibility formulation. Our method builds on our previous work on an entropy-stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes/Cahn-Hilliard system %\cite{manzanero2020entropyNSCH}% (Manzanero et al. (2020)), and incorporates a modified weak compressibility formulation that allows different sound speeds in each phase. We validate our method through numerical experiments, demonstrating spectral convergence for acoustic transmission and reflection coefficients in one dimension and for the angle defined by Snell's law in two dimensions. Special attention is given to quantifying the modeling errors introduced by the width of the diffuse interface. Our results show that the method successfully captures the behavior of acoustic waves across interfaces, allowing exponential convergence in transmitted waves. The transmitted angles in two dimensions are accurately captured for air-water conditions, up to the critical angle of $13^\circ$. In a final example, we show a three-dimensional wave transmission from air into water to demonstrate the potential of this methodology for addressing general multiphase acoustic problems. This work represents a step forward in modeling acoustic propagation in incompressible multiphase systems, with potential applications to marine aeroacoustics.

Figures (12)

• Figure 1: Problem setup for the 1D problem (not to scale).
• Figure 2: Absolute errors for reflection (\ref{['fig:1D1_R_errors']}), and transmission (\ref{['fig:1D1_T_errors']}), coefficients vs degrees of freedom given by different meshes and polynomial orders. Errors decrease with increased resolution
• Figure 3: Transmission and reflection modeling errors versus $\epsilon$ at $f = 1000Hz$. Errors decay with $\varepsilon^2$.
• Figure 4: Modeling errors in (a) transmission versus $\varepsilon$ for frequencies 1000, 2000, and 5000 Hz, and (b) transmission and reflection versus $\varepsilon$ scaled by the wavelength $\lambda$ for frequencies 1000, 2000, 5000, 8000, and 10000 Hz. Errors decay as $\varepsilon^2$.
• Figure 5: Absolute transmission errors including numerical and modeling contributions. (\ref{['fig:1D6_transmission_error']}) Errors for fixed $\varepsilon = 2 \times 10^{-3}$, which decay exponentially before saturating. (\ref{['fig:1D8_transmission_error']}) Errors with $\varepsilon$ scaled to have $10 N_p$ points within the interface, showing quadratic decay