A quasi-one-dimensional model of thermoacoustics in the presence of mean flow

Abstract

In thermoacoustic regenerators, the interaction of thermo-viscous boundary layers and axial temperature gradients causes a conversion from thermal energy to acoustic power or vice versa. In this paper, an improved analytical model for thermoacoustic boundary layer effects in the presence of mean flow is derived and analyzed. Previous formulations of the thermo-acoustic effect take into account effects of mean flow on acoustic propagation only implicitly, i.e. in as much as mean flow influences the mean temperature field. The new model, however, includes additional terms in the perturbation equations, which describe explicitly the interaction between steady mean flow and acoustics. For a parallel plate pore the three-dimensional thermoacoustic equations are derived and reduced to a transversally averaged system of differential equations by applying Green׳s function technique and suitable assumptions. The resulting one-dimensional perturbation equations are then solved numerically for two sets of boundary conditions to obtain the linear scattering matrix coefficients. The solutions, generated for a wide range of frequencies, can be applied in a low-order “network model” context to study the stability of thermoacoustic devices. The impact of mean flow on the thermoacoustic interaction is investigated and validated against full computational fluid dynamics simulations of laminar, compressible flow for one specific configuration. It is shown that at low frequencies (Womersley number $<1$) the new formulation predicts the acoustic behavior more accurately than the earlier formulations. Finally, the ideas and benefit of further improved and more complex models for higher Mach numbers are discussed.

Introduction

The interest in the interaction of acoustics and heat flow dates back more than 200 years [1], [2]. One of the investigated issues is the thermoacoustic (TA) effect: in narrow geometries thermo-viscous boundary layers interact with axial mean temperature gradients. This interaction causes a conversion of thermal energy to acoustic power or vice versa. Sondhauss [3] observed this effect first. He described the occurrence of spontaneous loud sound emissions in narrow pipes during the glass blowing process. Although Rayleigh [4] had already given the first qualitative explanations for this effect, only incomplete descriptions of the phenomenon were provided by different authors [5], [6]. Rott was the first author to completely describe the phenomenon analytically in a series of publications [7], [8], [9], [10], [11], [12]. Wheatley [13], his successor Swift [14] and finally in׳t panhuis [15] and in׳t panhuis et al. [16] improved the analytical understanding of the TA effect. Arnott et al. [17] enhanced the applicability of this approach for arbitrary pore shapes. Watanabe et al. [18] further improved these models by accounting for nonlinear effects such as drag or heat transfer. The TA effect has also been investigated in various computational fluid dynamics (CFD) simulations (e.g. [19], [20], [21], [22], [23]). Their analytical formulations facilitate the description of the axial propagation of acoustic velocity u₁ and pressure p₁ in terms of their cross-section averaged quantities. These transversally averaged parameters $〈p_1〉,〈u_1〉$ form a set of transport equations in the axial pore direction. Those transport equations are applied in a one-dimensional numerical tool [24] to predict the operating conditions in TA devices.

Previous publications have presented ideas for probable applications of TA devices [13], [14], and some authors claim that the technology is ripe for the market [25]. Nevertheless, almost 30 years after the first attempts of technical realizations of such apparatus, the technology is only applied in niche applications [26], [27], [28]. Thus, the main aim of researchers remains to expand the applicability of TA for cooling and power generation.

All cited authors restrict their descriptions to quiescent mean flow conditions. Until now, only a few publications have tried to deal with mean flow affected TA power conversion [29], [30], [31]. Bammann et al. [32] provide an overview over the published literature on flow-through TA refrigeration. One of the main reasons for the modest interest in their topic is the lack of proper modeling of the TA interaction effects under such conditions. All hitherto referenced publications consider the impact of mean flow only implicitly by adapting the axial mean temperature profile, or do not address it at all. Based on the assumption that the mean flow is due to streaming effects induced by acoustic quantities, a second-order enthalpy flux consideration leads to a distortion of the mean temperature profile [8], [33], [34], [35]. The present paper presents and analyzes an improved quasi-one-dimensional method, which explicitly takes the laminar mean flow into account in the perturbation equations. Using this approach enables the prediction of the performance of TA devices that exploit mean flow as a positive effect, e.g. Reid׳s refrigerator [30]. Due to the resulting one-dimensional form, the computational effort for the calculation of one configuration is comparatively small. Conceivable applications may, for example, arise in exhaust gas streams, where the waste heat enthalpy flux is directly transformed into acoustic power.

The derivation of the quasi-one-dimensional model is based on the method applied by in׳t panhuis et al. [16], [15], who derived TA transport equations that account for almost arbitrary stack pore geometry. Like all other proposed approaches, the model takes into account the so-called long-pore assumption, i.e. the hydraulic diameter $2R$ is much smaller than the characteristic length in axial direction L. Further, in׳t panhuis [15] follows the common approach of neglecting the contribution from mean flow when linearizing the basic set of equations. In the present work, this assumption is not invoked and mean flow is explicitly incorporated.

Taking into account the contribution of mean flow leads to an increase in the number of terms in the considered equations. Hence, for the purpose of demonstrating the impact of explicitly accounting for mean flow, the solution is derived for a generic problem resulting in the least complex system of transport equations. The geometrical conditions and thermo-physical correlations are chosen to keep the equations short. Sticking to the general solution with only a few necessary assumptions is still feasible; however, this complicates the interpretation of the impact of mean flow due to the huge numbers of terms inside the system of equations. Thus, to simplify the model where possible, assumptions will be introduced. These assumptions due to compactness of equations are marked as such.

Apart from spatial distributions of the acoustic quantities for a single frequency, acoustic scattering matrices are used for validating the one-dimensional models against CFD results. The latter validation tool is mostly used to investigate the stability of the system in the linear regime by using 1D network models [36]. Although TA engines generally operate in nonlinear limit cycles, the identified growth rate at the onset of the acoustic instability is a quantitative measure for the achievable pressure amplitudes. Altogether, as the simplicity of the solution is a central request, this publication is restricted to the linear acoustic regime. If nonlinearities like the acoustic feedback into the mean temperature distribution or saturation effects have to be taken into account, all steps can be processed for the nonlinear situation.

The paper is organized as follows: In Section 2, the geometrical and mean flow conditions are presented. In Sections 3 and 4, the one-dimensional acoustic transport equations are derived for a parallel plate stack pore. The flow chart of the derivation is sketched in Fig. 1: First the compressible Navier–Stokes equations (NSE) are non-dimensionalized using narrow geometry assumptions. These dimensionless NSEs experience a simplified modal expansion. The mean flow field is solved analytically from the resulting zeroth-order set of equations. These solutions are used as input to the first-order set of equations, the linearized dimensionless NSEs. This set of equations for the acoustic field is then solved in segregated steps: First Green׳s function (GF) solution technique is applied on the axial momentum and energy equations. This yields formulations for their transversal shapes. Closure assumptions have to be made for two convective terms to obtain an analytical solution of these formulations. Integrating the resulting equations in the transversal direction creates a set of averaged perturbation equations. Finally, eliminating density and temperature perturbations leads to one-dimensional TA transport equations in terms of velocity and pressure oscillations. The resulting set of equations accounts for non-zero mean flow contributions, which appear explicitly in some terms. Thus, models based on this system of equations are named “explicit” in the rest of the paper. The existing model [37] is denoted as “implicit” (IMP), because in that model mean flow influences the system of equations only implicitly through its impact on the mean temperature profile, which is computed from a second-order energy equation that considers enthalpy transport by the mean flow [8]. For the explicit equations two alternative approaches for the necessary closure assumptions are investigated. Prediction accuracy of these two approaches, denoted by M I and M II, is tested for the spatial acoustic propagation for a single frequency and scattering behavior for different test cases against a full CFD simulation and the existing implicit one-dimensional solution method. Finally, an outlook to improved modeling approaches is presented.