Strong wave–mean-flow coupling in baroclinic acoustic streaming

# Strong wave–mean-flow coupling in baroclinic acoustic streaming

Guillaume Michel1    Gregory P. Chini2 Email address for correspondence: greg.chini@unh.edu
###### Abstract

The interaction of an acoustic wave with a stratified fluid can drive strong streaming flows owing to the baroclinic production of fluctuating vorticity, as recently demonstrated by Chini et al. (J. Fluid Mech., 744, 2014, pp. 329–351). In the present investigation, a set of wave/mean-flow interaction equations is derived that governs the coupled dynamics of a standing acoustic-wave mode of characteristic (small) amplitude and the streaming flow it drives in a thin channel with walls maintained at differing temperatures. Unlike classical Rayleigh streaming, the resulting mean flow arises at rather than at . Consequently, fully two-way coupling between the waves and the mean flow is possible: the streaming is sufficiently strong to induce rearrangements of the imposed background temperature and density fields, which modifies the spatial structure and frequency of the acoustic mode on the streaming time scale. A novel Wentzel–Kramers–Brillouin–Jeffreys analysis is developed to average over the fast wave dynamics, enabling the coupled system to be integrated strictly on the slow time scale of the streaming flow. Analytical solutions of the reduced system are derived for weak wave forcing and are shown to reproduce results from prior direct numerical simulations (DNS) of the compressible Navier–Stokes and heat equations with remarkable accuracy. Moreover, numerical simulations of the reduced system are performed in the regime of strong wave/mean-flow coupling for a fraction of the computational cost of the corresponding DNS. These simulations shed light on the potential for baroclinic acoustic streaming to be used as an effective means to enhance heat transfer.

Key words: acoustics, baroclinic flows, mixing enhancement

## 1 Introduction

Sound waves can drive Eulerian flows that evolve on a slow time scale compared to the period of the waves. The theoretical study of this phenomenon, called acoustic (or, in other contexts, steady) streaming, can be traced back to Rayleigh in the 19th century (Rayleigh 1884). Given that ultrasonic power sources are now routinely used in laboratory experiments, acoustic streaming has been widely observed, often as a source of unwanted flow. Nonetheless, streaming also has been recognized as a practical means to enhance transport and mixing and has, for instance, been used to improve the efficiency of chemical reactions occurring near a catalytic solid phase that otherwise would be controlled by molecular diffusion (Bengtsson & Laurell 2004); to directly mix chemical species (Yaralioglu et al. 2004); and for activated irrigation in medical applications including root-canal procedures (Verhaagen et al. 2014). Heat also can be transported by streaming flows, and acoustic waves therefore can be used to accelerate the cooling of a hot object, as recently reviewed by Legay et al. (2011). Acoustic streaming technologies are of particular interest in the zero-gravity environment, where natural convective flows do not exist and acoustic thermal management systems may provide a reliable, efficient and lightweight alternative to fans.

In a characteristically lucid lecture, Lighthill identified the different regimes of acoustic streaming in a homogeneous medium (Lighthill 1978). Owing to attenuation mechanisms, acoustic waves generate a Reynolds stress divergence capable of driving a mean flow, which is balanced either by viscous forces (termed “Rayleigh streaming” if, in addition, the sound waves are damped in oscillatory boundary layers) or by inertia (“Stuart streaming”). The former regime occurs for small values of the streaming Reynolds number , where is a characteristic streaming speed, is a typical dimension of the system and is the kinematic viscosity of the fluid. In Rayleigh streaming, the (laminar) cellular mean flow that is generated is localized within a few wavelengths of any solid boundary (Nyborg 1958); the streaming can be analytically computed for simple geometries, e.g. in a channel (Rayleigh 1884; Hamilton et al. 2003) or adjacent to a circular cylinder (Holtsmark et al. 1954). This regime has become important in microfluidics, where the vortices induced by acoustic streaming in microchannels can be used to mix chemicals (see references above). In contrast, for large , the streaming flow acquires a jet-like structure and can become turbulent (Stuart 1966; Lighthill 1978).

The presence of an inhomogeneous background temperature (or density) field strongly affects the fundamental mechanics and kinematics of acoustic streaming: streaming velocities are significantly enhanced and the flow patterns are substantially altered. These changes are evident in the early experiments of Fand & Kaye (1960) and in subsequent experiments and numerical simulations; see e.g. Loh et al. (2002), Hyun et al. (2005), Lin & Farouk (2008), Nabavi et al. (2008), Atkas & Ozgumus (2010), Dreeben & Chini (2011) and Karlsen et al. (2018). Consequently, the resulting flow and associated transport cannot be computed simply by coupling the corresponding isothermal (e.g. Rayleigh or Stuart) streaming with the heat or other appropriate transport equation. Instead, new physical phenomena occur, which renders this problem both complex and interesting. Experimental challenges arise because natural convection and acoustic streaming may be difficult to disentangle in the laboratory; numerical challenges result from the need to resolve compressible fluid dynamics on temporal scales ranging from the acoustic wave period to the slow time scale over which the streaming flow evolves; while the primary theoretical challenge is to elucidate the novel phenomenology resulting from fully two-way coupling between the sound waves and the mean flow.

This striking change in the character of the streaming has been observed in various contexts in which an agency other than viscosity generates vorticity in the oscillatory flow (e.g. the acoustic waves). Amin (1988) and Riley & Trinh (2001) noted fundamental changes in the steady streaming driven by a non-conservative body force in their study of fluid flow in the presence of -jitter, i.e. a fluctuating gravitational field in an otherwise gravity-free environment. Motivated by the observation that streaming velocities in high-intensity discharge lamps are two orders of magnitude larger than those predicted by Rayleigh streaming theory (Dreeben & Chini 2011), Chini et al. (2014) derived a theory capable of accounting for both the observed streaming pattern and magnitude. As discussed more fully in § 2, the mechanism underlying this large-amplitude streaming is the baroclinic production of sound-wave vorticity arising from the misalignment of fluctuating isobars and mean isopycnals. Similarly, non-classical streaming phenomena have been observed in microfluidic systems with gradients in density; in particular, Karlsen et al. (2016, 2018) recently obtained a local expression for the acoustic force density driving the streaming flow as a function of the acoustic-wave characteristics.

The primary objective of the present investigation is to systematically extend the recent theory of Chini et al. (2014) to efficiently capture the two-way coupling that can occur in baroclinic acoustic streaming and to quantify the concomitant heat transfer. Accordingly, a novel Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) analysis is performed to enable prediction of the slow evolution of the acoustic wave amplitude, thereby obviating the need to explicitly simulate the fast oscillatory dynamics. We focus on perhaps the most well-documented acoustic streaming configuration: a thin two-dimensional channel with an imposed standing acoustic wave oscillating in the wall-parallel direction. For a homogeneous system, the resulting streaming flow was first described in the pioneering work of Rayleigh (1884) in the limits and , where is the thickness of the oscillatory (Stokes) boundary layers ( is the kinematic viscosity, and is the wave angular frequency), is the channel width and is the wavenumber of the acoustic wave. The streaming flow comprises a wall-parallel array of counter-rotating vortices, stacked in the wall-normal direction and having a characteristic velocity , where is the maximum fluctuating velocity induced by the standing acoustic wave and is the speed of sound. Experiments were first performed in a tube and showed quantitative agreement with the predictions of Rayleigh (Andrade 1931). Hamilton et al. (2003) extended this theoretical study to channels of arbitrary width , the only restrictions being and . When the upper and lower walls of the channel are maintained at fixed but differing temperatures, both experiments and direct numerical simulations of the compressible Navier–Stokes and heat equations indicate a change in the streaming phenomenology: the stacked vortices merge and their characteristic velocity increases (Loh et al. 2002; Lin & Farouk 2008). To date, no theory has correctly predicted the resulting streaming-flow pattern and intensity; our study, which focuses on the regime , fills this gap in the literature.

The remainder of the paper is organized as follows. After formulating the problem for the instantaneous dynamics, we carry out a multiple scale analysis (§ 2) to obtain a reduced but two time-scale system. In § 3, we analyse the wave dynamics to show that this multiscale system can be integrated strictly on the slow time scale. We then consider, in § 4, the limit of weak wave forcing, in which the streaming flow does not produce appreciable feedback on the waves; in particular, we derive an approximate analytical solution and compare it to the streaming flow numerically computed by Lin & Farouk (2008), demonstrating excellent quantitative agreement. In § 5, we perform numerical simulations of our reduced model and characterize the resulting fully-coupled waves and mean flows. We summarize our key findings and suggest possible further extensions in § 6.

## 2 Two time-scale wave/mean-flow system

### 2.1 Flow configuration

The problem we consider is similar to that introduced in Chini et al. (2014). Specifically, we analyze the two-dimensional flow of an ideal gas, with specific gas constant and constant dynamic viscosity and thermal conductivity , in a channel with walls separated by a distance in the coordinate direction (see figure 1). Here and throughout, tildes refer to dimensional variables, while asterisks are used to denote dimensional parameters. Subsequently, overbars will be used to designate dimensionless time-averaged fields, while primes will be reserved for dimensionless oscillatory fields. The gas is presumed to be driven in an approximately time-periodic fashion, with frequency , yielding a standing sound wave with spatial wavenumber . The velocity field is required to satisfy no-slip and zero normal-flow boundary conditions along the channel walls located at and . All dependent fields are required to satisfy a periodicity condition in the horizontal () coordinate. In addition, to fix the spatial phase of the sound wave, we impose a symmetry condition along (or, equivalently, a zero mass-exchange condition across) ; i.e. =0, where is the -velocity component and is the time variable. This additional boundary condition holds for any flow developing from initial conditions that are symmetric with respect to , e.g. for the quiescent diffusive state, since then both the initial conditions and the governing equations are invariant with respect to the transformation , and .

In contrast to the study of Chini et al. (2014), the thermal driving is achieved by fixing the temperatures of the lower and upper walls to be and , respectively, rather than by including a volumetric heat source. For convenience, we take the temperature differential , but note that this restriction is not dynamically significant since we do not consider the influence of buoyancy (gravity) in this investigation. Denoting the density, pressure, temperature and velocity fields by , , and , respectively, where and is the -velocity component, the governing (compressible) Navier–Stokes, continuity and energy equations and the ideal gas equation of state can be written as

 ~ρ[∂~t~u+(~u⋅~∇)~u]=−~∇~p+μ∗[~∇2~u+13~∇(~∇⋅~u)], (2.0)
 ∂~t~ρ+~∇⋅(~ρ~u)=0, (2.0)
 ~ρcv[∂~t~T+(~u⋅~∇)~T]=−~p(~∇⋅~u)+κ∗~∇2~T, (2.0)
 ~p=~ρR∗~T, (2.0)

where the two-dimensional gradient operator . Note that dilatational (or ‘bulk’) viscosity has been neglected in (2.1), and viscous heating has been omitted in (2.1). In practice, the bulk viscosity vanishes for a monatomic gas and, according to early experiments, is smaller than the dynamic (or shear) viscosity for the specific diatomic ideal gas (i.e. nitrogen) studied here (Prangsma et al. 1973). More importantly, although significant variations in the dynamic viscosity may be expected owing to the temperature dependence of this coefficient, these variations are neglected to facilitate the analysis.

The steady-state pressure and temperature fields in the absence of acoustic waves and streaming flow, that is for , are referred to as the background fields (denoted with a subscript ‘’) and are found to be

 ~TB=T∗(1+Γy∗H∗),     ~pB=p∗, (2.0)

where is a constant and the dimensionless temperature differential . Table 1 summarizes the dimensional fields and parameters used in the following analysis.

### 2.2 Scaling and non-dimensionalisation

To facilitate the asymptotic analysis, we non-dimensionalise the governing equations by scaling the dependent and independent variables as outlined in table 2. The -velocity component is scaled with , the sound speed at temperature . This scaling introduces into the dimensionless governing equations the Strouhal number , where (rather than ) is a characteristic oscillatory velocity induced by the standing acoustic wave. The Strouhal number is large ( or larger) in many applications, and therefore we introduce and consider the asymptotic limit with all other dimensionless parameters scaled as appropriate powers of . Since , the leading-order acoustic wave dynamics is linear. Nevertheless, weak wave–wave nonlinearities are crucial for acoustic streaming, as their cumulative effect can be significant over sufficiently many [] acoustic wave periods (i.e. over the slow time scale). The implied temporal scale separation between the wave and streaming dynamics is readily achieved in both laboratory experiments and streaming-enabled technologies.

The and coordinates are scaled with and , respectively, so that the gas lies in the domain defined by and . Time is non-dimensionalised using the inverse reference wave frequency . The vertical (), or wall-normal, velocity component is scaled by . The domain aspect ratio is assumed to be small and, more precisely, is chosen so that , where is a dimensionless parameter of order unity. Although the cross-channel heat flux is not expected to be maximized as (since the streaming flow will be largely horizontal), we follow Chini et al. (2014) and continue to focus on the small aspect-ratio regime for the following reasons. First, as noted in the introduction, most theoretical and computational studies have been performed in this regime, so meaningful comparisons to prior investigations can be made. Secondly, the analysis of the acoustic wave is simplified in a domain that is thin relative to the wavelength of the sound wave. Indeed, the acoustic wave then is dynamically constrained to maintain its first-mode wall-normal structure. Finally, in the small aspect-ratio regime, the leading-order fluctuating pressure gradient is orthogonal to the imposed background density gradient, resulting in a crucial baroclinic contribution to the production of fluctuating vorticity.

The temperature of the lower wall is used to nondimensionalise the temperature field . In the analysis that follows, is fixed, i.e. , as , although the smallness of in acoustic streaming ensures that our multiple scale analysis remains accurate even for , as will be evident in § 4, where we compare our theoretical predictions with the results of direct numerical simulations.

The Reynolds and Péclet numbers characterizing the acoustic waves are denoted and , respectively. Since these parameters are very large compared to unity, both momentum and thermal diffusion can be neglected in the leading-order wave dynamics, at least in the domain interior. Note that in baroclinic acoustic streaming, typical streaming velocities also are of size (in contrast to Rayleigh streaming, where streaming speeds are proportional to ). Consequently, and also would appear to characterize the streaming flow. Because we consider the limit of small aspect ratio, however, diffusion in the wall-normal () direction is enhanced by a factor . We therefore define the streaming Reynolds and Péclet numbers and as in traditional Rayleigh streaming as the proper measure of mean inertia to the dominant mean diffusive effects.

### 2.3 Asymptotic analysis

Using the scalings described above (and summarized in table 2), the governing equations and boundary conditions can be recast in dimensionless form. The occurrence of the small parameter in the dimensionless system prompts a multiple scale asymptotic analysis in which the single time variable characterizing the fast dynamics of the acoustic waves is augmented with a slow time variable to capture the cumulative effect of weakly nonlinear wave dynamics that ultimately drives streaming. Furthermore, we posit the following asymptotic expansions for the various fields:

 (u,v) = ϵ(u1,v1)+ϵ2(u2,v2)+O(ϵ3), (2.0) π = ϵπ1+ϵ2π2+O(ϵ3), (2.0) Θ = Θ0+ϵΘ1+O(ϵ2), (2.0) ρ = ρ0+ϵρ1+O(ϵ2), (2.0)

where and are, respectively, the dimensionless perturbation pressure and temperature fields. Note that, at each order in these expansions, the field variables can have both fluctuating and mean components, which subsequently will be disentangled via the introduction of a fast-time averaging operation. Expansion (2.3) follows from the scaling of the Strouhal number (). The state equation constrains the temperature and density perturbations to be of the same order. In baroclinic acoustic streaming, the streaming flow is sufficiently strong to induce rearrangements of the background temperature and density fields over an time period. Thus, in contrast to other studies of acoustic streaming in the presence of inhomogeneous temperature fields (e.g. Červenka & Bednarřik (2017)), it is crucial that the expansions for both and begin at , as first shown in Chini et al. (2014).

Owing to the large perturbations to the background density field, the natural frequency of an acoustic mode may evolve in time. Here, we extend the analysis of Chini et al. (2014) by employing a WKBJ approximation to properly capture this slow temporal evolution. Specifically, a generic dependent field is re-expressed as , where and are treated as independent variables. The rapidly-varying phase may be written as

 ϕ(t)=Φ(T)ϵ, (2.0)

where is of order unity. We define the instantaneous angular frequency ,

 ω(T)=dϕdt=dΦdT, (2.0)

and expand , so that

 ω=ω0+ϵω1+O(ϵ2). (2.0)

Finally, in order to distinguish the streaming flow from the acoustic wave, we introduce the fast-time average of a function ,

 ¯f(x,y,T)=12nπ∫ϕ+2nπϕf(x,y,s,T)ds (2.0)

for sufficiently large positive integer , so that any function can be decomposed such that

 f(x,y,ϕ,T)=¯f(x,y,T)+f′(x,y,ϕ,T), (2.0)

where . Thus, represents the acoustic wave and the streaming flow.

### 2.4 Leading-order multiscale wave/mean-flow interaction equations

The multiple time-scale governing equations for the coupled acoustic-wave/streaming-flow system were first derived in Chini et al. (2014) and are reproduced here for completeness. The evolution of the streaming fields is governed by the following equations:

 ¯ρ0(∂T¯u1+¯u1∂x¯u1+¯v1∂y¯u1) = −∂x¯π2γ−∂x(¯ρ0¯¯¯¯¯¯u′21)−∂y(¯ρ0¯¯¯¯¯¯¯¯¯¯u′1v′1)+∂yy¯u1Resh2, (2.0) ∂y¯π2 = 0, (2.0) ∂T¯ρ0+∂x(¯ρ0¯u1)+∂y(¯ρ0¯v1) = 0, (2.0) ∂T¯Θ0+¯u1∂x¯Θ0+¯v1∂y(¯Θ0+TB) = (1−γ)(¯Θ0+TB)(∂x¯u1+∂y¯v1)+γ∂yy¯Θ0Pesh2¯ρ0, (2.0) ¯ρ0 = 1¯Θ0+TB. (2.0)

In this slow-time system, the sole – but crucial – effect of the waves arises from the Reynolds stress divergence in the mean -momentum equation (2.4) that drives the streaming flow. For a homogeneous fluid, this wave-induced force can be offset by a pressure gradient in the bulk: a Rayleigh streaming flow is driven at next order in by the Reynolds stress divergence acting in the oscillatory boundary layers that arise near the no-slip channel walls. In contrast, for baroclinic acoustic streaming, the mean temperature gradient causes this wave-induced forcing to be rotational even within the bulk interior of the domain. Consequently, the bulk force that is created cannot be balanced by a mere adjustment of the mean pressure gradient and instead induces a (strong) mean flow even in the absence of diffusive boundary layers. (Further discussion of this distinction is given in § 6.)

To evaluate this force, we employ the equations governing the leading-order acoustic wave dynamics, viz.

 ω0¯ρ0∂ϕu′1+1γ∂xπ′1=0,∂yπ′1=0, (2.0)
 ω0∂ϕρ′1+∂x(¯ρ0u′1)+∂y(¯ρ0v′1)=0, (2.0)
 ω0∂ϕΘ′1+u′1∂x¯Θ0+v′1∂y(¯Θ0+TB)+(γ−1)(¯Θ0+TB)(∂xu′1+∂yv′1)=0, (2.0)
 π′1−ρ′1(¯Θ0+TB)−¯ρ0Θ′1=0. (2.0)

These equations describe the fast dynamics of approximately linear non-dissipative acoustic waves in a medium whose mean density field evolves slowly in time. Conversely, the evolution of depends on the acoustic waves, as can be gleaned from inspection of (2.4)–(2.4). Note further that, here, unlike in Rayleigh streaming, the oscillatory Stokes layers are dynamically passive because the streaming induced by near-wall fluctuating viscous torques is while the streaming flow governed by (2.4)–(2.4) is . Therefore, the leading-order wave field is required to satisfy only a zero normal-flow boundary condition at each wall, and the details of the oscillatory flow within the Stokes (boundary) layers do not have to be determined at this order (see § 6).

Taken together, these two sets of equations form a closed but two time-scale system. As emphasized in Chini et al. (2014), the fully two-way coupling between the waves and mean flow captured by this multiscale system renders it fundamentally distinct from classical Rayleigh streaming theory, in which the acoustic wave field can be computed first and then the response of the streaming flow to the acoustic wave forcing self-consistently determined (i.e. one-way coupling). Subsequently, Karlsen et al. (2016, 2018) also noted this two-way coupling in their computational studies of acoustic streaming in inhomogeneous fluids. Nevertheless, to make analytical progress, Chini et al. (2014) considered a small Prandtl-number limit in which the coupling is effectively one way. A primary contribution of the present investigation is to extend the analysis of Chini et al. (2014) to systematically treat fully two-way wave/mean-flow interactions. This extension requires the derivation of a novel amplitude equation governing the slow evolution of the acoustic waves, which can only be determined by carrying the asymptotic analysis to next order and imposing an appropriate solvability condition. Heuristically, the slow evolution of the waves is controlled by higher-order terms that, e.g., account for energy exchanges with the streaming flow or with the solid boundaries. In the next section, we show how these effects can be self-consistently incorporated.

## 3 Averaging over fast wave dynamics

We now characterize the dynamics of the acoustic waves on both the fast and slow time scales with the aim of eliminating the need to explicitly simulate the fast evolution. Inspection of (2.4)–(2.4) reveals that the wave dynamics directly depends on a single slowly-varying field: the leading-order mean density . For purposes of the analysis described in this section, this field is presumed to be given. Consequently, the fluctuation equations (2.4)–(2.4) comprise a linear homogeneous system, and we henceforth consider a single eigenvector. A generic fluctuation field can be expressed as

 f′1(x,y,ϕ,T)=A(T)2(^f1(x,y,T)eiϕ+c.c.), (3.0)

where stands for any fluctuation variable (, , , , ); is the slowly-evolving modal amplitude (here taken to be real without loss of generality); is a complex function that describes the spatial structure of the mode; and c.c. denotes the complex conjugate. A normalization condition, specified subsequently, must be imposed on to render this decomposition unique. We next describe the determination of the spatial structure of the mode (as defined by the functions , , etc.) and then derive a novel amplitude equation governing the slow evolution of the generally a priori unknown function .

### 3.1 Mode structure

Substituting the decomposition (3) into the fluctuation equations (2.4)–(2.4) yields a linear but non-separable two-dimensional (partial) differential eigenvalue problem for the spatial structure and frequency of the leading-order fluctuation fields. By continuing to exploit the small aspect-ratio limit, we nevertheless are able to reduce the required computation to the solution of a one-dimensional eigenvalue problem – a crucial simplification.

To proceed, we note that, using (3), (2.4)–(2.4) can be combined to deduce

 ^π1=iγω0(∂x^u1+∂y^v1). (3.0)

With this expression for the acoustic wave pressure , the momentum equations (2.4) become two coupled partial differential equations for and :

 ∂x(∂x^u1+∂y^v1)+ω20¯ρ0^u1=0, (3.0)
 ∂y(∂x^u1+∂y^v1)=0. (3.0)

A further reduction to a single ordinary differential equation is possible by formally integrating a linear combination of these equations,

 ∂y(???)+∂x(???)⇒∂y(¯ρ0^u1)=0⇒^u1=q¯ρ0, (3.0)

where is an unknown function of and only. Integration of (3.1) gives

 ∂x^u1+∂y^v1=g, (3.0)

where is a second unknown function of and only. Equations (3.1), (3.1) and (3.1) imply that , where a prime is used to denote partial differentiation of the function with respect to , since the dependence is parametric. The general solution of this system of equations can be obtained using the kinematic boundary condition , viz.

 ^u1=−g′ω20¯ρ0, (3.0)
 ^v1=yg+∂x(g′ω20∫y0dy¯ρ0). (3.0)

Finally, the upper boundary condition provides a constraint on and in the form of the ordinary differential eigenvalue problem

 ddx(g′α)+ω20g=0, (3.0)

where

 α(x,T)=∫10dy¯ρ0. (3.0)

To characterize the function , we first take, without loss of generality, to be a real field: (3.1) and (3.1) then imply that and, thence, are also real-valued fields. Moreover, in order to ensure , the ordinary differential equation (3.1) must be solved subject to the boundary conditions . This requirement leads to an orthogonality condition: let be two eigenvectors and their angular eigenfrequencies; then

 ∫2π0gA(x)gB(x)dx=1ω2A−ω2B[α(gAg′B−gBg′A)]2π0=0. (3.0)

Equation (3.1) provides a convenient scalar product on eigenvectors, and therefore we normalize them according to

 ∫2π0g(x)2dx=1. (3.0)

This normalization condition resolves the ambiguity in the definition of and in (3).

### 3.2 Wave amplitude

As explained in the previous subsection, the acoustic mode shape and frequency can be computed at every time for a given mean density profile . The advection of hot or cold gas by the streaming flow will cause both and to evolve on the slow time scale and induce a two-way coupling between the waves and the streaming flow. The amplitude of the acoustic mode is also expected to evolve on this slow time scale owing to dissipation by viscosity, energy exchanges with the streaming flow and walls and/or external forcing. To obtain an evolution equation for , we proceed as follows, relegating details to the appendices.

1. We collect terms in the dimensionless governing equations at . The resulting equations are reported in Appendix A.

2. We make the ansatz that generic fluctuation field can be represented as

 f′2(x,y,T,ϕ)=B(T)2(^f2(x,y,T)eiϕ+c.c.) (3.0)

and, in direct analogy with the manipulations performed in § 3.1, reduce the fluctuation system for the five unknown fields , , , and to a system of two equations for the two fields and . [Harmonics of the form also exist at this order, but are non-resonant and thus do not contribute to the slow-time dynamics; accordingly, these harmonics need not be explicitly computed.] The resulting system of equations has the form

 ∂x(∂x^u2+∂y^v2)+ω20¯ρ0^u2=F, (3.0)
 ∂y(∂x^u2+∂y^v2)=G. (3.0)

The linear operator acting on the left-hand side of this system is identical to that arising in the leading-order fluctuation equations (3.1)–(3.1). The right-hand side functions and include resonant forcing terms involving the leading-order fluctuations fields (). Analytical expressions for the imaginary parts of and , which are needed for the derivation of the (real) amplitude equation, are given in Appendix B.

3. We derive a solvability condition for the system, which requires determination of the adjoint linear operator, by invoking the Fredholm alternative theorem in the usual manner; see Appendix C.

4. We enforce the solvability condition to obtain an equation for ; cf. Appendix D.

Employing this procedure, we derive the following novel amplitude equation:

 2Aω−10 (3.0) +∫Ddxdy(∂x¯u1+∂y¯v1)[(1−γ)g2+g′2ω20¯ρ0(PesRes−12)],

where refers to definite integration over the spatial domain. (Note that, since the temperature and velocity fluctuations are out of phase, is strictly imaginary.) The lack of terms nonlinear in in (3.0) confirms that phenomena such as shock-wave formation or harmonic generation are sub-dominant dynamical processes in the given parameter regime relative to, for example, heat exchange with the boundaries or energy exchange with the evolving stratified environment. In particular, the first term on the right-hand side of (3.0) accounts for the time-mean heat transfer between the waves and the boundaries. In § 4.5, we demonstrate that this term can be positive: in this scenario, the waves are driven by a process that is loosely akin to the classical thermoacoustic instability in which acoustic waves in a channel can be excited when a temperature gradient is imposed along the channel walls (Swift 1988). Owing to the occurrence of the mean fields , and in (3.0), the amplitude equation is, in fact, nonlinear. A distinguishing feature of (3.0) is that, unlike amplitude equations derived in numerous other contexts, determination of the coefficients requires evaluation of functional derivatives that capture the variations in generic eigenfunction caused by changes in the mean density field that occur on the slow time ; see Appendices B and D for details.

The quantity on the left-hand side of (3.0) is proportional to the square-root of the dimensionless energy of the acoustic wave. Indeed, the leading-order dimensionless kinetic energy of the acoustic wave averaged over the fast time scale is given by

 (3.0)

This expression can be evaluated using (3.1) to obtain

 ¯¯¯¯EK=A(T)24ω40∫2π0dx∫10dyg′2¯ρ0=A(T)24ω40∫2π0dxg′2α. (3.0)

The last integral reduces to following an integration by parts and ulitisation of the differential equation (3.1) and the normalization condition (3.1), yielding

 ¯¯¯¯EK=(A(T)2ω0)2. (3.0)

The amplitude equation therefore can be interpreted as an energy balance for the acoustic wave, with the left-hand side of (3.0) equalling .

An important outcome of this study is that even allowing for two-way wave/mean-flow coupling, the WKBJ analysis enables the streaming flow to be computed without evolving the sound waves over the fast time scale. (Of course, in the absence of two-way coupling, as in classical Rayleigh streaming, this averaging is trivial.) Instead, the evolving spatial structure of the waves can be computed at every coarse time step (in a numerical simulation) by solving the one-dimensional eigenvalue problem (3.1), while the evolution of the amplitude can be determined by integrating (3.0) over the slow time scale. These computations are performed in conjunction with the numerical solution of the streaming equations (2.4)–(2.4).

## 4 One-way coupling

Although the asymptotically-reduced equations constitute a substantial simplification of the full compressible Navier–Stokes equations, they defy analytical solution owing to the occurrence of nonlinearities and two-way coupling between the waves and the streaming flow. For sufficiently weak streaming, however, an approximate steady-state solution can be derived: in this limit, mean advection is weak and the mean density perturbations are small, thereby ameliorating these two difficulties. Here, we derive this approximate analytical solution and demonstrate that it accurately describes results obtained from direct numerical simulations of the compressible Navier–Stokes and heat equations reported in the literature.

### 4.1 Acoustic waves

In this section, we assume that the mean density profile varies little and thus can be accurately approximated by the diffusive solution (2.1), i.e.

 ¯ρ0=1TB=11+Γy. (4.0)

The coefficient defined in (3.1) then does not depend on and is simply equal to . The acoustic-wave eigenfunction characterizing the shape of the acoustic mode follows from the solution of the second-order differential equation (3.1) and, with the prescribed boundary conditions and the normalization condition (3.1), is given by

 g(x)=cos(nx)√π, (4.0)

where the integer is set to unity for this study. The angular frequency of this mode , and the velocity field (, ) follows from (3.1)–(3.1) and reduces to

 ^u1=(1+Γy)sin(x)(1+Γ/2)√π, (4.0)
 ^v1=Γy(1−y)cos(x)2(1+Γ/2)√π. (4.0)

This velocity field is plotted in figure 2 for two different values of . A rotational (vortical) component may be discerned, particularly for the scenario, even though viscous torques are absent since the dynamics in oscillatory boundary layers has been self-consistently omitted. Crucially, the Reynolds stress divergence terms, here denoted , arising in (2.4) and responsible for driving the streaming flow can be explicitly evaluated:

 R(x,y)=−∂x(¯ρ0¯¯¯¯¯¯u′21)−∂y(¯ρ0¯¯¯¯¯¯¯¯¯¯u′1v′1)=−A22π(1+Γ/2)2(1+Γ4+Γy2)sin(2x). (4.0)

If , i.e. for a homogeneous fluid, . Clearly, in that case, the wave-induced Reynolds-stress divergence can be balanced by a mean pressure gradient, so that streaming is not directly driven; instead, the associated Rayleigh streaming flow arises at next order in owing to the action of viscous torques within oscillatory boundary layers. If, however, , then can no longer be reduced to gradient form and, consequently, directly drives a streaming flow.

### 4.2 Streaming flow

We assume that the steady streaming driven by given in (4.0) is sufficiently weak that the streaming equations (2.4)–(2.4) can be linearised. Note that, unlike Rayleigh streaming, the mean flow is compressible even though the streaming Mach number is negligible.

In a steady state, the pressure field can be eliminated from the linearised versions of equations (2.4)–(2.4), yielding

 ∂xyR=−∂yyyx¯u1Resh2, (4.0)

while conservation of mass and internal energy (2.4)–(2.4) imply

 ∂x¯u1=−(1+Γy)ΓPesh2∂yyy¯Θ0. (4.0)

Finally, we obtain from equations (4.0)–(4.0) a single differential equation for :

 (1+Γy)∂yyyyyy¯Θ0+3Γ∂yyyyy¯Θ0=−2A2Γ2ResPesh4π(2+Γ)2cos(2x). (4.0)

This equation is supplemented with the boundary conditions:

1. , since the wall temperatures are held constant;

2. , to enforce at the walls [see (2.4)–(2.4)]; and

3. , to enforce the no-slip boundary condition at each wall, upon using (4.0).

With these boundary conditions, a unique solution can be found. For illustration, we focus on the case , for which

 ¯Θ0(x,y)=−2A2ResPesh49πG(y)cos(2x), (4.0)

where

 G(y)= 11080(−3+log(16))[60(1+y)2log(1+y) +y(94−222log(2)−90y−20y2−5y3(−5+log(64))+3y4(−3+log(16)))].

For the self-consistency of this approximation, the assumptions of one-way coupling and linear streaming dynamics require that the mean density change little during the evolution, i.e. , and that the nonlinear terms be negligible, e.g. . From equations (2.4)–(2.4) and (4.0), these constraints imply (for ) upper bounds on the dimensionless parameter combinations and .

### 4.3 Comparison with previous work

Using the analytical solution (4.0), various properties of the streaming flow can be deduced. First, the maximum dimensional baroclinic streaming velocity obtained here

 ~uB∝(max |G|)A2Resh2U∗ (4.0)

can be compared to the corresponding velocity in Rayleigh streaming resulting from dissipation in the Stokes boundary layers, i.e.

 ~uR∝ϵA2U∗. (4.0)

Clearly, for small values of , the baroclinic streaming dominates any streaming driven by viscous torques acting in near-wall Stokes layers.

The streaming velocity field (, ) computed from (4.0) for is plotted along with the total temperature in figure 3. Clearly, baroclinic streaming differs from Rayleigh streaming not only in intensity but also in spatial structure; specifically, here the streaming cells span the channel while in Rayleigh streaming the cells are stacked in the wall-normal direction. These various distinguishing properties accord with prior experiments and numerical simulations; e.g. see Loh et al. (2002); Hyun et al. (2005); Lin & Farouk (2008); Nabavi et al. (2008); Atkas & Ozgumus (2010); Dreeben & Chini (2011).

Quantitative comparisons can be made with the results of Lin & Farouk (2008), who performed direct numerical simulations of the compressible Navier–Stokes and heat equations specifically to investigate the impact of acoustic streaming in a thin channel on cross-channel heat transport. Thus, the system they considered is very similar to that studied here. In the absence of thermal driving, the streaming flow is accurately predicted by using Rayleigh’s formulation and, accordingly, exhibits a pattern of counter-rotating cells stacked in the direction. When a temperature difference is imposed, however, the stacked cells merge, resulting in counter-rotating cells that span the channel. In particular, for their case 1C, corresponding to the largest imposed temperature differential, the dimensionless parameters used by Lin & Farouk (2008) are approximately

 ϵ=10−2,   γ=1.4,   Γ=0.2,   h=2.3,   Res=5.7,   Pes=4.1. (4.0)

The amplitude of the acoustic waves is not reported, but can reasonably be assumed to be similar to the value arising in the absence of thermal driving, for which . The authors report the values of the (resp. ) component of the dimensional streaming velocity at (resp. ). We compute and for the same parameters in this one-way coupling limit, and then obtain the corresponding dimensional velocities by multiplying by the sound speed and by .

Figure 4 shows the resulting comparison. The evident quantitative agreement – with no adjustable parameters and despite the fact that the numerical simulations of Lin & Farouk (2008) include several physical effects (oscillatory boundary-layer dynamics, viscous heating, inertia and temperature-dependent viscosity and diffusivity coefficients) not incorporated in our analysis – provides strong confirmation of the baroclinic streaming theory developed by Chini et al. (2014) and systematically extended in the present study.

### 4.4 Heat flux enhancement

The streaming flow enhances cross-channel heat transport, as we now demonstrate using the leading-order solutions derived in this section. For this purpose, we introduce the steady-state Nusselt number as the ratio of the (dimensional) total heat flux to the diffusive flux,

 Nu=˙Q∗2πκΔΘ∗(k∗H∗)−1, (4.0)

and compute . The total heat flux is evaluated at the top boundary,

 ˙Q∗=∫2πk−1∗0κ∂~y~T(~x,~y=H∗)d~x=κ(k∗H∗)−1T∗[∫2π0∂y(TB+¯Θ0)(x,y=1)dx+O(ϵ)], (4.0)

so that the Nusselt number is given at leading order by

 Nu=1+12πΓ∫2π0∂y¯Θ0(x,y=1)dx+O(ϵ). (4.0)

Since viscous heating is neglected, the top and bottom heat fluxes are equal in a steady state; hence, the following result holds:

 ∫2π0[(1+Γ)∂yΘ(x,y=1)−∂yΘ(x,y=0)]dx, (4.0) = Γ∫2π0∂y¯Θ0(x,y=1)dx+O(ϵ). (4.0)

Moreover, from equations (2.4)–(2.4), the following equality can be derived,

 ∫2π0dx∫10dyT∂yy¯Θ0=0⇒∫2π0[T∂yΘ]10dx=∫2π0dx∫10dy(∂yT)(∂y¯Θ0)+O(ϵ). (4.0)

Finally, using the boundary condition at the top and bottom walls, we obtain

 Nu−1=12πΓ2∫2π0dx∫10dy(∂yT)(∂y¯Θ0)+O(ϵ)=12πΓ2∫2π0dx∫10dy(∂y¯Θ0)2+O(ϵ). (4.0)

In the limit of one-way coupling and for , we evaluate this quantity with the approximate expression (4.0) for derived in § 4.2,

 Nu−1≃3.2×10−10(A2ResPesh4)2. (4.0)

This expression should be contrasted with that obtained for Rayleigh streaming (i.e. for sufficiently small temperature differences) and computed by Vainshtein et al. (1995):

 (Nu−1)R=6.2×10−5(ϵA2Pesh2)2. (4.0)

This comparison indicates that, although the Nusselt number derived here is very small (as noted in § 4.2, for the validity of the one-way coupling assumption the dimensionless parameter combination cannot be too large), it nevertheless is orders of magnitude larger than resulting from boundary-layer-driven acoustic streaming provided that, numerically, . Equation (4.0) also suggests that significant heat transport enhancement may be achieved in the limit , i.e. when there is strong two-way coupling between the acoustic waves and the streaming flow. This strong coupling scenario is investigated in § 5.

### 4.5 Stability of the quiescent background state

Intriguingly, the amplitude equation (3.0) suggests that an acoustic wave may be amplified via interaction with the solid boundaries. Indeed, the first term on the right-hand side of (3.0) does not depend on the streaming flow, and quantifies how the divergence of the acoustic-wave velocity field (i.e. ) and the net heat flux to/from the solid boundaries [proportional to ] are coupled. In a certain parameter regime, this system may act as a thermoacoustic engine and spontaneously generate an acoustic wave that, in turn, would drive a streaming flow and increase the cross-channel heat transport.

To see this, note that can be determined from the acoustic wave solution derived in § 4.1 and then used to evaluate

 −iω0Pesh2∫D