Chapter 9Linear and Nonlinear Theories forThermoacoustic Waves in a Gas FilledTube Subject to a Temperature Gradient

Nobumasa Sugimoto and Dai Shimizu

Abstract This article reviews briefly the linear and nonlinear theories of ther-moacoustic waves in a gas filled tube subject to a temperature gradient. In theframework of fluid dynamics, asymptotic theories are developed by two parameterson the basis of a narrow tube approximation. One is a parameter measuring theorder of nonlinearity, while the other is a parameter measuring diffusive effects byviscosity and heat conduction. Making use of these parameters as asymptotic ones,the full system of equations is reduced to compact and spatially one-dimensional (1D)equation(s). It is emphasised that the diffusive effects may be covered substantiallyby two cases where the layers are thin or thick enough in comparison with the tuberadius.

9.1 Introduction

Over the last three decades, much interest has been attracted to thermoacousticinstability of a gas subject to a temperature gradient and ensuing emergence ofself-excited oscillations in view of potential applicability to novel heat engines(Wheatley et al., 1983; Swift, 2017). The phenomena themselves are very interestingbut physical mechanisms behind them are not easily understood. To explain themechanisms, there are two approaches.

One is a thermodynamic view of a gas particle undergoing oscillations, i.e., athermodynamic cycle of the particle in the vicinity of a wall in the presence of thetemperature gradient (Wheatley et al., 1983; Swift, 2017). This view is intuitive and

N. Sugimoto (�)Department of Pure and Applied Physics, Kansai University, Osaka, Japane-mail: sugimoto@me.es.osaka-u.ac.jp

D. ShimizuDepartment of Mechanical Engineering, Fukui University of Technology, Fukui, Japane-mail: shimizu@fukui-ut.ac.jp

© Springer Nature Switzerland AG 2019A. Berezovski, T. Soomere (eds.), Applied Wave Mathematics II, Mathematicsof Planet Earth 6, https://doi.org/10.1007/978-3-030-29951-4_9

187

188 N. Sugimoto and D. Shimizu

consistent with Rayleigh’s criterion of instability (Rayleigh, 1945; Howe, 1998), butit is valid only for infinitesimally small oscillations. As the amplitude becomes large,the cycle is not closed due to drift of the particle by acoustic streaming. Thus thisLagrangian view is not appropriate to quantify the phenomena. In contrast, the otheris a fluid dynamical and Eulerian view, which enables a quantitative description.This approach was initially taken by Rott (1969, 1973) to derive marginal conditionsof Taconis oscillations1 discovered in the operation of liquid helium (Taconis et al.,1949).

In a quiescent gas subject to a steady temperature gradient, heat flows in thebackground and the gas is set in a state of thermal nonequilibrium. The heat flow isalways exposed to various uncontrollable disturbances, and therefore not only the gastemperature but its density and pressure are fluctuating. If there exist eigenmodes ofoscillations in a system containing the gas, it is possible that the disturbances wouldexcite one of the modes to make the system unstable. However, because diffusiveeffects by viscosity and heat conduction are present in the gas, they counteractto suppress growth of the disturbances so that the quiescent state appears to bemaintained.

As the temperature gradient becomes steeper and the magnitude of heat flowbecomes greater, it may be expected that the diffusive effects can no longer suppressthe instability but rather they promote it by taking energy from the heat flow. Usuallythe diffusive effects act to dissipate energy and damp disturbances. This occurs inclosed systems. The present system is open, because the heat flow can supply energyinfinitely to the system from outside of it. In this case, the dissipative effects play anadverse role, to take energy from the heat flow to grow disturbances. This may beunderstood by analogy with instability of a viscous fluid flow. Although Rott (1969,1973) did not take such a view, this seems to be easily acceptable.

Most analytical work undertaken to date is mainly based on an equation derivedby Rott (1969, 1973) for an infinitesimally small, harmonic disturbance. In fact, thisequation, called the Rott equation, is regarded as being appropriate in the light ofexperimental results. It explains successfully marginal conditions of not only Taconisoscillations but also other thermoacoustic oscillations in tubes using a stack or aregenerator. However the Rott equation is limited to the linear harmonic disturbanceand is incapable of describing a process leading to self-excited oscillations.

To analyse fully the phenomena from initial instability in linear regime toemergence of self-excited oscillations in nonlinear regime, the computational fluiddynamics (CFD) is promising. High fidelity simulations by the CFD are carried outby Scalo and his team (Scalo et al., 2015; Gupta et al., 2017). Even by the CFD,

1When a long thin tube with one end open is inserted into a dewar containing a liquid helium, andthe open end approaches a liquid surface at several kelvin, a quiescent, gaseous helium in the tubebegins to oscillate violently. Because the closed end is kept at a room temperature, there occurs asteep temperature gradient along the tube so that a heat flows toward the open end.

9 Linear and Nonlinear Theories for Thermoacoustic Waves in a Gas Filled Tube 189

however, there are limitations. For example, a behaviour of the gas in tortuous flowpassages within a regenerator is so complicated that it must be taken into accountby an appropriate model. In addition, high fidelity simulations are not so easy to usebecause it needs expertise and resources of computers.

As the gap between the Rott equation and the CFD is very wide, the authors’ teamhas developed linear and weakly nonlinear theories, aiming at going beyond the Rottequation to fill the gap. Distinction between the linear and nonlinear theories is made,of course, by the magnitude of a maximum pressure disturbance �p relative to areference pressure p0 in a quiescent state. This is represented by ε (= �p/p0). Thediffusive effects appear significantly in the vicinity of a wall so they are measuredtypically by the thickness of the viscous diffusion layer relative to the tube radius R

represented by δ (= √ν/ω/R), ν and ω being, respectively, a kinematic viscosity

and an angular frequency. Rather the thickness of the thermal diffusion layer√

κ/ω, κbeing a thermal diffusivity, may be appropriate, but both thicknesses are comparablewhen the Prandtl number Pr (= ν/κ) is of order unity, though the thermal one is alittle thicker for air (P r = 0.7).

Table 9.1 overviews the theories classified according to ε, δ and a type ofdisturbance. Target indicates which phenomena are expected to be quantified bythe theories. Because the magnitude of disturbance ε in self-excited oscillations inreality is of order 10−3 to 10−1 at most, the whole process from initial instabilityto emergence of self-excited oscillations will be described by the weakly nonlineartheory for a small but finite value of ε (0 � ε � 1). In this sense, the linear theoryis valid only when the disturbance may be regarded as being infinitesimally small(ε → 0).

The diffusive effects are difficult to be taken for an arbitrary value of δ except inthe linear theory. Thus they are treated separately for the case of thin layers (δ � 1)

or that of thick layers (δ � 1) by asymptotic methods for small or large but finitemagnitudes of δ. At any rate, there appears in between an intermediate range of amoderate value of δ. It turns out, however, that the two cases can cover it beyondtheir original ranges of validity so that the intermediate range is much narrowerthan imagined. This is demonstrated by good agreements of the marginal conditionsderived by the asymptotic equations and those by the Rott equation (Sugimoto,2019).

Section 9.2 describes the linear theory to derive the thermoacoustic wave equationin a general case of δ, from which the Rott equation in the frequency domainis reduced. Section 9.3 outlines the approximation of the thermoacoustic waveequation in the two cases, thin and thick diffusion layers. Section 9.4 extends thelinear approximate theories to the weakly nonlinear regime by asymptotic methodssystematically. Finally a summary of the theories is given with some discussions.

190 N. Sugimoto and D. Shimizu

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9 Linear and Nonlinear Theories for Thermoacoustic Waves in a Gas Filled Tube 191

9.2 Linear Theory

Suppose a gas filled tube of radius R is subject to a nonuniform temperature Tw

on the tube wall axially where no gravity is assumed (Fig. 9.1). Denoting axial andradial coordinates by x and r , respectively, Tw is assumed to depend on x only, sothat Tw is axisymmetric. A gas temperature Te in a quiescent state under a uniformpressure p0 depends on x and also r in general.

As is usually the case, the wall temperature Tw changes along x so gradually thatthe following inequalities may hold:

R2

Tw

∣∣∣∣

d2Tw

dx2

∣∣∣∣� R

Tw

∣∣∣∣

dTw

dx

∣∣∣∣� 1 . (9.1)

In the following analysis, R2T −1w d2Tw/dx2 and also (RT −1

w dTw/dx)2 are neglected,and nonuniform effects of Tw are taken into account up to dTw/dx. Then Te may beset equal to Tw to the same approximation (Sugimoto, 2010).

The heat capacity of the tube wall is assumed to be large enough for the walltemperature to be unchanged even when the gas motion takes place. Since Te isnonuniform axially, so is the density denoted by ρe. The gas is assumed to obey thelaw of ideal gas, so it holds that ρeTe = ρ0T0 by Charles’ law, where the subscript 0designates a value at a reference state. Note that the subscript e denotes the quantitiesin the quiescent state which are functions of x determined by Te(x).

With these settings, consider first a behaviour of infinitesimally small disturbancegiven to the quiescent state. On top of the linearisation, a narrow tube approximationis exploited where typical axial length of the disturbance L is much longer than R,

Fig. 9.1 Illustration of a circular tube of radius R filled with a thermoviscous gas and subject to anonuniform temperature Tw(x) along the tube wall, where x and r denote, respectively, the axialand radial coordinates, and Te denotes the gas temperature in a quiescent state. The symbols p′ andu′ represent, respectively, the excess pressure over a uniform pressure p0 and the axial velocity,while s and q represent, respectively, the shear stress acting on the gas at the wall surface and theheat flux into the gas through it. Profiles with (a) and (b) show qualitatively some instantaneousones of u′ over the cross section in the cases of thin and thick diffusion layers, respectively, andvb denotes the radial velocity directed inward at the edge of a boundary layer in the case of thindiffusion layers.

192 N. Sugimoto and D. Shimizu

i.e., R/L � 1. Thicknesses of viscous and thermal diffusion layers are arbitrary incomparison with the tube radius.

Denoting a disturbance by a prime (·)′, equations of continuity, motion, energy,and equation of state for a thermoviscous gas are linearised around the quiescentstate as follows:

∂ρ′

∂t+ ∂

∂x(ρeu

′) + 1

r

∂

∂r(rρev

′) = 0 , (9.2)

ρe

∂u′

∂t= −∂p′

∂x+ μe

r

∂

∂r

(

r∂u′

∂r

)

, (9.3)

0 = −∂p′

∂r, (9.4)

ρecp

(∂T ′

∂t+ u′ dTe

dx

)

= ∂p′

∂t+ ke

r

∂

∂r

(

r∂T ′

∂r

)

, (9.5)

p′

p0= ρ′

ρe

+ T ′

Te

, (9.6)

where ρ, p, u, v, and T denote, respectively, density, pressure, axial velocity, radialvelocity and temperature of the gas, t being time; cp denotes a specific heat atconstant pressure; and μe and ke denote, respectively, the shear viscosity and heatconductivity at temperature Te, which are assumed to obey a power law of Te through

μe

μ0= ke

k0=

(Te

T0

)β

, (9.7)

where β is a constant between 0.5 and 0.6 for air. Boundary conditions at the wallsurface require no slip condition and the isothermal condition is given by

u′ = v′ = T ′ = 0 at r = R. (9.8)

No boundary conditions are imposed in the x direction, since the tube is so long thatno end effects appear.

From (9.4) it appears that p′ is independent of r so that the pressure is uniformover the cross section of the tube, but the other quantities are dependent on r . Theuniformity of p′ is an advantage for the analysis. Applying the Fourier transformwith respect to t , all quantities may be expressed in terms of p′. In the contextthat follows, all quantities should be understood to be transformed. From (9.3), u′is solved in terms of p′ by the use of the boundary condition. Substituting u′ thusobtained into (9.5), T ′ is solved in terms of p′. Using (9.6), ρ′ is expressed in termsof T ′ and therefore of p′. Substituting u′ and ρ′ thus obtained into (9.2), and solvingfor v′, it follows from the boundary condition that a single equation for p′ mustbe satisfied. This equation is finally transformed inversely to yield an equation forp′(x, t).

9 Linear and Nonlinear Theories for Thermoacoustic Waves in a Gas Filled Tube 193

The procedure mentioned is straightforward in principle but calculations arecumbersome. Rather it is illuminating to circumvent it by averaging each equationover the cross section of the tube with the boundary conditions. These equations arecombined into the following form:

∂2p′

∂t2 − ∂

∂x

(

a2e

∂p′

∂x

)

= 2

R

[a2e

cpTe

∂q

∂t− ∂

∂x(a2

e s)

]

, (9.9)

where ae stands for the local adiabatic sound speed at Te defined by√

γp0/ρe

or, equivalently, by√

(γ − 1)cpTe, γ being the ratio of specific heat cp at constantpressure to the one cv at constant volume. The quantities s and q denote, respectively,the shear stress acting on the gas at the wall and heat flux flowing into the gas throughthe wall surface defined as follows:

s = μe

∂u′

∂r

∣∣∣r=R

and q = ke

∂T ′

∂r

∣∣∣r=R

. (9.10)

Equation (9.9) suggests that the heat flux and the wall friction act, respectively, asacoustic monopole and dipole (Howe, 1998). In the present context, s and q areexpressed in terms of p′ as

s = √νeNν

(∂p′

∂x

)

(9.11)

and

∂q

∂t= cpTe

√νe

{

− γ − 1√Pr

Nκ

(1

a2e

∂2p′

∂t2

)

+

+ 1

Te

dTe

dx

[1

1 − PrNν

(∂p′

∂x

)

− 1

(1 − Pr)√

PrNκ

(∂p′

∂x

)] }

. (9.12)

Here Nν denotes a functional defined by

Nν

(∂p′

∂x

)

=∫ t

−∞�

[νe(t − τ)

R2

]∂p′(x, τ )

∂xdτ . (9.13)

The similar functional Nκ is defined by replacing νe in Nν with a thermal diffusivityκe (= ke/ρecp). The kernel function � is defined by

�

(νet

R2

)

= 1

2π

∫ ∞

−∞(−iω)−

12I1(1/δe)

I0(1/δe)e−iωtdω, (9.14)

with δe = (iνe/ω)12 /R. The symbols I0 and I1 denote, respectively, the modified

Bessel functions of zeroth and first order. As the wall friction and heat flux are given

194 N. Sugimoto and D. Shimizu

in the form of the integrals from the remote past τ = −∞ to the present time τ = t ,the diffusive effects give rise to hereditary (memory) effects. Here it is important tonotice that becauseNν appears in (9.12) accompanied with the temperature gradient,the shear stress affects the heat flux, whereas the heat flux does not affect the shearstress. This holds only when the temperature gradient is present.

With the presented specification of s and q, (9.9) is expressed in the followingform:

∂2p′

∂t2 − ∂

∂x

(

a2e

∂p′

∂x

)

+ 2

R

∂

∂x

[

a2e

√νeNν

(∂p′

∂x

)]

+ 2

R

(γ − 1)√Pr

√νeNκ

(∂2p′

∂t2

)

−

− 2

R

a2e

√νe

Te

dTe

dx

[1

1 − PrNν

(∂p′

∂x

)

− 1

(1 − Pr)√

PrNκ

(∂p′

∂x

)]

= 0. (9.15)

This is the final equation derived from (9.2)–(9.6). It is called the thermoacousticwave equation (Sugimoto, 2010).

If a time harmonic disturbance is considered in the form of p′ = P(x)eiωt , whereP and ω are, respectively, a complex pressure amplitude and an angular frequency,(9.15) becomes:

d

dx

[

(1 − fν)a2e

dP

dx

]

+(

fν − fκ

1 − Pr

)a2e

Te

dTe

dx

dP

dx+ ω2 [1 + (γ − 1)fκ ] P = 0 ,

(9.16)

where fν and fκ are defined, respectively, by

fν = 2I1(ηe)

ηeI0(ηe), fκ = f

(√Pr ηe

)

, with ηe = R

√

iω

νe

. (9.17)

The ordinary differential equation (9.16) of the second order for P is known as theRott equation. As fν and fκ result from Nν and Nκ , respectively, they represent theviscous and thermal diffusion.

Finally, the equation of the acoustic energy can also be derived by taking theaverages of (9.2)–(9.6) over the cross section of the tube. Denoting the average by abar as

u′ = 1

πR2

∫ R

02πru′(x, r, t) dr , (9.18)

the equation becomes (Hyodo and Sugimoto, 2014)

∂

∂t

(1

2ρeu′2 + 1

2

p′2

ρea2e

)

+ ∂

∂x

(

p′u′)

= 2

R

(

u′s + p′qρecpTe

)

. (9.19)

9 Linear and Nonlinear Theories for Thermoacoustic Waves in a Gas Filled Tube 195

If the disturbance is periodic and the average is taken over its period, we have

dI

dx= 2

R

(

˜u′s + p′q

ρecpTe

)

, (9.20)

where the tilde means the mean over the period and I denotes the mean acousticenergy flux density (intensity) p′u′ averaged over the cross section. Integrating(9.20) over a whole domain of x, the left-hand vanishes because of boundary ormatching conditions. Hence the integral of the right-hand should vanish, which givesa condition for neutral oscillations to occur, i.e., a marginal condition of instability.This corresponds to Rayleigh’s criterion of instability, though q here is the heat fluxand not the heat release rate (Howe, 1998).

9.3 Approximation of the Thermoacoustic Wave Equation

As the thermoacoustic wave equation is the integro-differential equation, it is difficultto be solved. The kernel function in the integral on the right-hand side of (9.13)depends on the time interval t − τ between the present time t and a past time τ

relative to the viscous diffusion time R2/νe. If ∂p′/∂x is periodic in t with a period2π/ω, then Nν is also periodic with the same period. Then the functional Nν may beapproximated according to whether the period 2π/ω is long or short in comparisonwith R2/νe. This implies, equivalently, that the typical thickness of the diffusionlayer

√νe/ω is thick or thin relative to R.

9.3.1 Case of Thin Diffusion Layers

When the layer is thin enough relative to R, i.e., |δe| � 1, the thermoacoustic waveequation is approximated as

∂2p′

∂t2− ∂

∂x

(

a2e

∂p′

∂x

)

+

+ 2a2e

√νe

R

[

C∂−1

2

∂t−12

(∂2p′

∂x2

)

+ (C + CT )

Te

dTe

dx

∂−12

∂t−12

(∂p′

∂x

)]

= 0 , (9.21)

where the fractional derivative of minus half order is defined by (Sugimoto, 1989)

∂−12

∂t−12

(∂p′

∂x

)

≡ 1√π

∫ t

−∞1√

t − τ

∂p′

∂x(x, τ )dτ (9.22)

196 N. Sugimoto and D. Shimizu

and C and CT are constants defined by

C = 1 + γ − 1√Pr

, CT = 1

2+ β

2+ 1√

Pr + Pr. (9.23)

In this case, the diffusive effects are confined in thin layers, called a boundary layer2on the tube wall. The acoustic field may be divided into the boundary layer and acore region outside of it. In the core region, the diffusive effects are secondary. Asis shown qualitatively by the profile (a) in Fig. 9.1, the axial velocity u′ is almostuniform over a cross section of the core region, while v′ is much smaller than u′, andit increases linearly with r . At the edge of the boundary layer, v takes −vb defined by

vb = √νe

[

C∂−1

2

∂t−12

(∂u′

∂x

)

+ CT

Te

dTe

dx

∂−12 u′

∂t−12

]

. (9.24)

The shear stress and the heat flux in this case are given, respectively, by

s = √νe

∂−12

∂t−12

(∂p′

∂x

)

(9.25)

and

q = ρecpTe√

νe

[

(C − 1)∂−1

2

∂t−12

(∂u′

∂x

)

+(

CT − 1

2− β

2

)1

Te

dTe

dx

∂−12 u′

∂t−12

]

.

(9.26)

It is worth noting that q/ρecpTe is of the same form as vb. Although the correspondingcoefficients are different (C = 1.47 and CT = 1.39 for γ = 1.4, Pr = 0.72, andβ = 0.5), it implies that vb behaves almost in a similar way to the heat flux, i.e., theheat flux pushes (pulls) the edge of the boundary layer into (out of) the core region.Because q/ρecpTe has the same dimension of speed as vb, the product with p′ meanswork on the gas in the core region and it inputs power into it, if a phasing betweenp′ and q is favourable.

9.3.2 Case of Thick Diffusion Layers

When the layer is thick enough relative to R, i.e., |δe| � 1, the acoustic field isdominated by diffusion. In this case, (9.15) is approximated to be the following

2By the boundary layer we mean both viscous and thermal diffusion layers together.

9 Linear and Nonlinear Theories for Thermoacoustic Waves in a Gas Filled Tube 197

diffusion wave (advection) equation

∂p′

∂t− ∂

∂x

(

αe

∂p′

∂x

)

+ αe

Te

dTe

dx

∂p′

∂x+

+[

4

3γ − (γ − 1)P r

]αe

a2e

∂2p′

∂t2− 1

6(1 + β + Pr)

αeR2

νeTe

dTe

dx

∂2p′

∂t∂x= 0 ,

(9.27)

where the diffusivity αe is defined by

αe = a2eR

2

8γ νe

= p0R2

8μe

. (9.28)

If no dependence of μe on Te (i.e., β = 0) is assumed, αe is a constant independentof x. However, if β = 0, αe decreases as Te increases.

The first three terms in (9.27) represent the lowest order relation and the fourthand fifth terms represent the second order correction. Since δ � 1 (the tube radiusis very narrow), the gas is thermally in a perfect contact with the wall and the gastemperature is equal to the local wall temperature to the lowest order, i.e., T ′ = 0.If no temperature gradient is present, the lowest order equation is simply a diffusionequation, and the disturbance is simply diffused, not propagated. In such a situation,the idea of sound speed loses its physical significance, though it might be thoughtthat the isothermal sound speed would take over adiabatic sound speed. When thetemperature gradient is present, however, the disturbance is not only diffused by thesecond term but also propagated (or advected) by the third term in the direction ofthe temperature gradient. This result is a new finding.

Since the tube is narrow, the viscous effects are primary and the axial velocityresembles that of Poiseuille flow in an incompressible fluid, as is shown by the profile(b) in Fig. 9.1. In fact, u′ is given to the lowest order by a parabolic function as

u′ = − 1

4μe

∂p′

∂x

(

R2 − r2)

. (9.29)

However, since the compressibility is present, v′ is present and is given by

v′ = 1

2p0R2

∂p′

∂t(R2 − r2)r . (9.30)

The higher order terms of T ′ are:

T ′ = 1

4ke

∂p′

∂t

(

R2 − r2)

+ 1

64

Pr

ρeν2e

dTe

dx

∂p′

∂x

(

3R2 − r2) (

R2 − r2)

. (9.31)

198 N. Sugimoto and D. Shimizu

Using these relations, the shear stress and the heat flux on the tube wall are given,respectively, by

s = R

2

∂p′

∂x(9.32)

and

q = −R

2

(∂p′

∂t+ γ

γ − 1

αe

Te

dTe

dx

∂p′

∂x

)

. (9.33)

The two terms on the right-hand side of (9.33) are comparable when ae/ωL ∼ δ.It is found from (9.32) that the wall friction 2πRs balances with the pressure force−πR2∂p′/∂x acting over the cross section. It is found from (9.33) that when thepressure increases temporarily at a certain point, the heat is released to the wall bythe first term on the right-hand side of (9.33). The second term is expressed in termsof the shear stress. For a negative pressure gradient, that is, when ∂p′/∂x < 0, theshear stress s is directed in the negative direction of x. If the temperature gradientis positive, then the heat flows into the gas, whereas if this gradient is negative, theheat flows out of the gas.

Using the approximate equations (9.21) and (9.27), marginal conditions ofinstability have been examined. For the Taconis oscillations, the marginal conditionis available analytically by using (9.21) for a smooth temperature distribution in theform of a quadratic function (Sugimoto and Yoshida, 2007). For the Sondhauss tube,similarly, the marginal conditions are obtained analytically (Sugimoto and Takeuchi,2009). In both cases, the marginal curves have two branches with respect to aminimum temperature ratio and appear to be qualitatively similar to those derivedby the Rott equation with no smooth temperature distributions (Rott, 1973; Rott andZouzoulas, 1976).

For a looped tube with a single stack inserted, the marginal conditions are derivedanalytically by using both (9.21) and (9.27) (Hyodo and Sugimoto, 2014), but nocomparison is made with those derived by the Rott equation. The comparison is madein (Sugimoto, 2019) to verify the anticipation that the two cases cover substantiallythe whole domain of δ except for a narrow interval where the marginal temperatureratio takes a minimum.

9.4 Nonlinear Theory

The approximate equations derived from the linear thermoacoustic wave equationin the cases of thin and thick diffusion layers turn out to be valid up to δ ∼ O(1)

beyond their respective ranges of validity, i.e., δ � 1 and δ � 1. Consequentlyit is found that the intermediate range of a moderate value of δ is narrow and thatit may substantially be covered by limits of the theories for thin and thick layers.

9 Linear and Nonlinear Theories for Thermoacoustic Waves in a Gas Filled Tube 199

This encourages the extension of the approximate equations to the weakly nonlinearregime 0 � ε � 1.

9.4.1 Case of Thin Diffusion Layers

As is the case with the linear theory, the acoustic field may be divided into theboundary layer and the core region outside of it. In the core region, nonlineareffects are now assumed to be primary rather than the diffusive ones. For self-excited oscillations in reality, ε is of order 10−1 at most, while the acoustic Reynoldsnumber a2

0/νω is very large, e.g., 107 for 100 Hz, a0 being a typical sound speed.The tube radius is assumed to be much smaller than a typical axial wavelength, butthe starting point is not set in equations simplified by the narrow tube approximationbut in the full equations. Assuming an almost plane wave, u′ is almost uniform overthe cross-section of the core region, while v′ is much smaller than u′.

Averaging the full fluid dynamical equations over the cross section of the coreregion (not over the cross-section of the tube), the equations of continuity, axialmotion and energy are reduced to 1D equations (Sugimoto and Shimizu, 2008a).Designating the averaged quantity by a attaching a bar (·)m with the subscript m, itfollows that

∂ρm

∂t+ ∂

∂x(ρmum) = 2

Rρmvb , (9.34)

ρm

(∂um

∂t+ um

∂um

∂x

)

= −∂pm

∂x, (9.35)

ρmTm

(∂Sm

∂t+ um

∂Sm

∂x

)

= 0 , (9.36)

where vb denotes the radial velocity at the edge of the boundary layer directed intothe core region. The quantity Sm denotes the entropy, which is related to the pressureand the density through

ρm

ρe

=(

pm

p0

)1/γ

exp

(

− Sm − Se

cp

)

, (9.37)

where Se is the entropy in the quiescent state at p0 and Te. Since the boundary layeris thin, it is described by the linear theory, so vb is taken to be (9.24). With thisrelation, thus, the system of Eqs. (9.34)–(9.37) is closed so it can be solved wheninitial and boundary conditions are imposed.

In fact, the Taconis oscillations in a helium filled tube with one end open andthe other closed (quarter wavelength tube) can be simulated by solving the systemof equations. A smooth temperature distribution which increases from the open end

200 N. Sugimoto and D. Shimizu

0 50 100 150 200

-0.01

0

0.01

t/

p'/p0

0 300 600 900

-0.1

0

0.1

t/

p'/p0

Δ Δ

Fig. 9.2 Simulations of the Taconis oscillations in a helium filled quarter wavelength tube forthe temperature ratio 50 at the hot (closed) end to the cold (open) end of the tube where a smalldisturbance is given initially at t = 0 and temporal profiles of p′ at the closed end are shownover short and long times in the left and right figures, respectively, � being 4L/aL. Initial growthby instability (left) tends to be suppressed by nonlinearity to saturation of the amplitude (right).Reprinted from Sugimoto and Shimizu (2008a), with the permission of AIP Publishing.

monotonously to the closed end is imposed. The boundary condition at the open endis simply given by p = p0 without taking account of radiation into free space.

Given a small disturbance initially, the implicit Crank–Nicolson finite differencescheme is used to solve for evolution of the disturbance. For the temperature ratio 50at the hot (closed) end to the cold (open) end, Fig. 9.2 shows the temporal profiles ofp′ at the closed end over short and long times in left and right figures, respectively.Here � = 4L/aL, where L and aL are, respectively, the tube length and the soundspeed at the closed (hot) end. It is seen that the initial growth by instability (left)tends to be suppressed by nonlinearity to saturation of the amplitude (right). Thanksto the simulations, all field variables are available and related quantities such asfluxes are obtainable. For the details, see the papers (Sugimoto and Shimizu, 2008a;Sugimoto et al., 2008b; Shimizu and Sugimoto, 2010).

9.4.2 Case of Thick Diffusion Layers

Starting from the fluid dynamical equations and using δ = √ν/ω/R � 1, the

systematic asymptotic expansion with respect to ε and δ−1 is made. All fieldvariables are evaluated beyond the relations described in Sect. 9.3.2 up to higher orderterms. The lowest order relations are the same as those derived by the narrow tubeapproximation. To proceed to higher order, relations between the small parametersmust be specified. In addition to δ, a parameter χ defined by a/ωL is introduced,a being a typical adiabatic sound speed. The analysis assumes that χ is large andcomparable with δ (Sugimoto, 2016).

9 Linear and Nonlinear Theories for Thermoacoustic Waves in a Gas Filled Tube 201

Specifying higher order terms step-by-step, the nonlinear diffusion wave(advection) equation is derived as follows:

∂p′

∂t− ∂

∂x

(

αe

∂p′

∂x

)

+ αe

Te

dTe

dx

∂p′

∂x+

[4

3γ − (γ − 1)P r

]αe

a2e

∂2p′

∂t2 −

− 1

6(1 + β + Pr)

αeR2

νeTe

dTe

dx

∂2p′

∂t∂x− p′

p0

∂p′

∂t− αe

p0

(∂p′

∂x

)2

= 0. (9.38)

This is the nonlinear version of (9.27). Unlike in the case of thin diffusion layers,nonlinear terms due to advection in the equation of motion are small. The nonlinearterms in (9.38) stem from the density change in the equation of continuity and theyare free from the temperature gradient (Sugimoto, 2016).

The expressions for the shear stress and heat flux on the tube wall are lengthy andnot reproduced here. If time periodic but unharmonic oscillations such as self-excitedoscillations are concerned with a period τ , then they are expressed as

2

Rs = ∂p′

∂x+ R2

6νep0

⎡

⎣˜

2∂p′∂t

∂p′∂x

+ αe

Te

dTe

dx

(∂p′∂x

)2⎤

⎦ (9.39)

and

2

Rq = −

(γ

γ − 1

)αe

Te

dTe

dx

(

∂p′∂x

+˜p′p0

∂p′∂x

)

, (9.40)

where the tilde (·) implies the mean over the period. Note that q changes its sign,depending on the sign of the temperature gradient.

For the time periodic oscillations, (9.38) averaged over the period becomes

− ∂

∂x

(

αe

∂p′∂x

)

+ αe

Te

dTe

dx

∂p′∂x

− αe

p0

(∂p′∂x

)2

= 0 . (9.41)

The linear higher order terms vanish and (9.41) is expressed as

αe

Te

(∂p′∂x

)2

= − ∂

∂x

(

αe

Te

∂p′∂x

)

p0 . (9.42)

This expression shows that the mean pressure gradient ˜∂p′/∂x does not vanish dueto the nonlinear terms so that the mean pressure varies along the tube. This relationalso indicates that the mean of the pressure gradient squared is derived from the firstorder derivative of the mean pressure gradient.

202 N. Sugimoto and D. Shimizu

Next, by evaluating the mean of (9.38) multiplied by αe∂p′/∂x, we have:

αe

T 2e

˜∂p′∂t

∂p′∂x

= ∂

∂x

⎡

⎣α2

e

2T 2e

(∂p′∂x

)2⎤

⎦ , (9.43)

to the lowest order. Using (9.42), (9.43) is derived from the second order derivative ofthe mean pressure gradient. Using these relations, s is expressed in terms of the meanpressure gradient, while q is expressed in terms of s and the mean of (p′/p0)∂p

′/∂x.For both acoustic and thermoacoustic streamings, the means of the mass flux andenergy flux are also expressed in terms of combinations of them (Sugimoto, 2016).In any event, p′ must be obtained by solving (9.38).

Equation (9.38) is usually solved jointly with (9.34)–(9.37) for δ � 1 forsimulations in usual thermoacoustic devices. A simulation of thermoacousticoscillations in a looped tube with a stack is now under way (Shimizu, 2018). For ashock wave to be captured, the thermoviscous effects neglected in (9.35) and (9.36)must be taken into account.

In this connection, it is noted that the approximate equations are spatially 1Dand derived without taking account of an end of flow passage. When they are used,for example, in the vicinity of an end of the stack, appropriate matching conditionsmust be applied. As far as the linear theory is concerned, the continuity of mass fluxand energy flux (which leads to the continuity of pressure) provide the matchingconditions (Hyodo and Sugimoto, 2014). In the nonlinear case, treatment of theenergy flux needs consideration.

9.5 Summary

The linear and nonlinear theories for thermoacoustic waves in a gas filled tube subjectto a temperature gradient axially have been reviewed. As summarised in Table 9.1,they are classified according to the magnitude of disturbance and the thickness ofviscous and thermal diffusion layers relative to the tube radius. It is surprising butuseful to find that the diffusive effects may substantially be covered by the theoriesfor thin and thick diffusion layers.

The assumptions and limitations of the theories are summarised. The effects ofthe temperature gradient are taken into account up to the first order derivative ofTw, so that the local gas temperature Te in a quiescent state is uniform over thecross-section of the tube and is equal to the wall temperature Tw. The linear theory isbased on the narrow tube approximation, by which the pressure is regarded as beinguniform over the cross-section. This uniformity makes the analysis tractable. Theweakly nonlinear theories start with the full equations and do not exploit the narrowtube approximation explicitly. As far as the lowest order relations are concerned,however, this approximation holds. In the higher order terms, nonuniformity of thepressure appears.

9 Linear and Nonlinear Theories for Thermoacoustic Waves in a Gas Filled Tube 203

The assumption of a large heat capacity of the wall becomes questionable whenthe wall is thin and the amplitude of oscillations becomes large. In experiments, infact, the wall temperature is affected considerably by oscillations. Therefore the heatconduction in the wall should be solved together with the gas oscillations where theheat fluxes are continuous across the wall surface. This is a hard task even in thelinear theory (Swift, 1988; Sugimoto and Hyodo, 2012). When a combination ofgeometry, gas and solid materials satisfies a special condition, the thermoacousticwave equation breaks down by nonuniformity, which may be called a resonance.Coupling between the gas and the solid needs to be clarified.

Acknowledgements The authors wish to thank Grants-in-Aid for Scientific Research (KAKENHINo. 26289036, No. 18H01375, and No. 18K03938) by the Japan Society for the Promotion ofScience.

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