Thermoacoustic Wave Propagation Modeling Using a Dynamically Adaptive Wavelet Collocation Method

Thermoacoustic wave propagation in a two-dimensional rectangular cavity is studied numerically. The thermoacoustic waves are generated by raising the temperature locally at the walls. The waves, which decay at large time due to thermal and viscous diffusion, propagate and reflect from the walls creating complicated two-dimensional patterns. The accuracy of numerical simulation is ensured by using a highly accurate, dynamically adaptive, multilevel wavelet collocation method, which allows local refinements to adapt to local changes in solution scales. Subsequently, high resolution computations are performed only in regions of large gradients. The computational cost of the method is independent of the dimensionality of the problem and is O(N ), where N is the total number of collocation points. INTRODUCTION When a localized region of a solid wall surrounding a compressible medium is subjected to a sudden temperature change, the medium in the immediate neighborhood of that region expands. This expansion generates pressure waves. These thermallygenerated waves are referred to as thermoacoustic (TAC) waves. TAC waves propagate at the local speed of sound of the medium and their amplitudes gradually, over a long time scale, decay due to thermal and viscous diffusion. The main interest in thermoacoustic waves is motivated by their property to enhance heat transfer by inducing convective motion away from the heated area. This property may be utilized in cryogenic engineering applications and under micro-gravity conditions when other modes of transport, such as natural convection, may be weak. Let us briefly review the relevant literature on the subject of thermoacoustic waves. For more comprehensive survey we refer to recent articles by Crocker and Parang (1994) and Hyun (1994). One-dimensional linear thermoacoustic waves were studied by Trilling (1955) by use of Laplace transform to obtain a long-time asymptotic solution for a TAC wave in a semi-infinite medium which is generated by a small step-wise increase in temperature at the surface. Huang and Bau (1995) generalized this approach for general wall heating conditions. In addition, they also solved numerically the nonlinear thermoacoustic wave problem and determined conditions under which the linear approximation is adequate. Huang and Bau (1996) also studied the dynamics of a TAC wave in a one-dimensional confined medium. Practically the same problem was studied experimentally by Brown and Churchill (1993). In order to study the general problem, a number of investigators (Spradley and Churchill, 1975; Ozoe et al., 1980, 1990; Crocker and Parang, 1994) numerically solved two-dimensional TAC waves in confined regions. Unfortunately, all of these finite difference solutions use crude grid spacing relative to the length scale of the thermoacoustic wave at short times. Recently Fusegi et al. (1995) performed numerical simulations of thermoacoustic convection in a two-dimensional cavity using a high-order, non-linear monotone numerical algorithm with the positivity-preserving property. They studied the case when a compressible fluid in a localized region was suddenly energized to generate pressure waves. In the majority of the existing studies on thermoacoustic convection, pressure waves are generated by applying sudden heating (or cooling) to the entire length of one or more walls of the confined region. In this paper we consider the case when TAC waves are generated by raising the temperature in localized regions of the walls. The nonlinear TAC waves propagate away from these regions, interact with each other, and reflect from the walls creating complicated two-dimensional patterns. The process of reflection and diffusion continues until the waves die out and a quiescent thermal conduction condition is achieved. In this work the nonlinear two-dimensional TAC wave problem is solved numerically using a highly accurate dynamically adaptive wavelet collocation