Computational Fluid Dynamics Simulationof a Thermoacoustic Refrigerator

Ahmed I. Abd El-Rahman∗ and Ehab Abdel-Rahman†

American University in Cairo, New Cairo 11835, Egypt

DOI: 10.2514/1.T4150

The thermal interactions between the stack plates and their neighboring gas particles within the thermal

penetration depth in a thermoacoustic resonator convert acoustic energy into heat energy in the process of standing

thermoacoustic refrigeration systems. Few numerical approximations describe the flow behavior and energy flux

density in standing devices, but almost no simulation results are available for the fully coupled continuity of

Navier–Stokes and energy equations. Here, we report a two-dimensional computational fluid dynamics simulation of

the nonlinear oscillating flow behavior in a helium-filled half-wavelength thermoacoustic refrigerator. The finite

volume method is used, and the solid and gas domains are represented by large numbers of quadrilateral and

triangular elements. The calculations assume a periodic structure to reduce the computational cost and apply the

dynamic mesh technique to account for the adiabatically oscillating wall boundaries. The simulation uses an implicit

time integration of the full unsteady compressible flow equations with a conjugate heat transfer algorithm (ANSYS

FLUENT). A typical run involves 12,000 elements and a total simulation time of 5 s. Simulation results for drive ratios

range Dr � 0.28%–2% are compared to both linear theory and a low Mach number model, and show good

agreement with the experimental values. A maximum cooling effect of 3° is predicted at a nondimensional wave

number kx � π∕4, measured from the resonator rigid end. This simulation provides an interesting tool for

understanding the bulk and microstructural flow behavior and the associated nonlinear acoustic streaming in

thermoacoustic refrigerators, by characterizing and optimizing their performance and building computational fluid

dynamics models of thermoacoustic devices.

Nomenclature

A = stack spacing cross-sectional area, m2

a = speed of sound, m∕sCp = gas specific heat, J∕�kgK�Ceffp;s = stack effective specific heat, J∕�kgK�

Dr = drive ratio, pa∕pmf = operating resonance frequency, Hzh = stack plates spacing, mm�_Hs = time-averaged heat flux rate, kW∕m2

k = wave number, 2π∕λkg = gas thermal conductivity, W∕�mK�keffs = stack effective thermal conductivity,W∕�mK�L = resonator length, mml = length of the stack plates, mmN = number of time increments per cyclep = gas dynamic pressure, Papa = amplitude of dynamic pressure, Papm = mean pressure, Pa�p = time-averaged dynamic pressure, Pa�Pac = local time-averaged acoustic power,W∕m2

t = simulation time, sTg = gas temperature, K

Tm = gas mean temperature, KTo = gas temperature at time equal to zero, KTs = stack temperature, Ktp = plate thickness, mmu = gas dynamic velocity, m∕sua = amplitude of dynamic velocity, m∕suR = Rayleigh velocity, 3u2a∕16a, m∕s�u = time-averaged dynamic velocity, m∕s�_W = area-weighted average of �Pac,W∕m2

Greek

ΔTs = stack ends temperature difference, Kδk = thermal penetration depth,

�������������������������2kg∕ρmCpω

p, mm

δv = viscous penetration depth,�������������������2μg∕ρmω

p, mm

λ = acoustic wavelength, a∕f, mμg = gas dynamic viscosity, kg∕�ms�ρm = gas mean density, kg∕m3

ρeffs = stack effective density, kg∕m3

τ = acoustic wave period, 1∕f, sφ = stack porosity

I. Introduction

A THERMOACOUSTIC refrigerator [1–3] (TAR) is a deviceconsisting of a resonator, a stack, and two heat exchangers.

Typically, the resonator is a long circular tubemade of copper or steeland filled with helium as a friendly working gas, while the stack hasshort and relatively low thermal conductivity ceramic parallel platesaligned with the direction of the prevailing resonant wave. Theresonator of a standing wave refrigerator has one end closed andis bound by the acoustic driver at the other end, enabling thepropagation of half-wavelength acoustic excitation. The hot and coldheat exchangers aremade of copper to allow for efficient heat transferbetween the working gas and the external heat source and sink,respectively.TARs are interesting because they have no moving parts, unlike

conventional refrigerators, and almost no environmental impactexists as they rely on the conversion of acoustic and heat energies.Their fabrication process is simpler and sizes span a wide variety of

Presented as Paper 2012-4233 at the Joint PropulsionConference&Exhibit10th Annual International Energy Conversion Engineering Conference,Atlanta, USA, 30–1 August 2012; received 19March 2013; revision received31May 2013; accepted for publication 1 June 2013; published online 29 July2013. Copyright © 2013 by the American Institute of Aeronautics andAstronautics, Inc. All rights reserved. Copies of this paper may be made forpersonal or internal use, on condition that the copier pay the $10.00 per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923; include the code 1533-6808/13 and $10.00 in correspondencewith the CCC.

*Assistant Professor, Department of Physics, P.O. Box 74; currentlyFaculty of Engineering, Department of the Mechanical Power Engineering,Cairo University, Giza 12613, Egypt; ahmedibrahim@aucegypt.edu.Nonmember AIAA.

†AssociateDean for Graduate Studies andResearch, School of Science andEngineering, Director of Yousef Jameel Science and Technology ResearchCenter; ehab_ab@aucegypt.edu (Corresponding Author). NonmemberAIAA.

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JOURNAL OF THERMOPHYSICS AND HEAT TRANSFERVol. 28, No. 1, January–March 2014

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length scales. The viscous and thermal interactions between the stackplates, heat exchangers’ plates, and the working gas significantlyaffect the flowfield within the plates’ channels (as explained byPanton [4]) and the energy flux density at the plates’ surfaces,respectively. In typical low-pressure excitation systems, thethermoacoustic behavior is dominated by the Swift [5] linear theory.However, practical TARs fall into the high-pressure amplituderegime of wave propagation, where nonlinear behavior is frequent,and so the need to introduce a computational fluid dynamics (CFD)analysis that solves the fully coupled nonlinear compressible flowequations has considerable importance.Swift [5] was the first to develop a theoretical framework of

thermoacoustic devices based only on thermal interactions betweenthe stack plates and the working fluid. In his quasi-one dimensionaltreatment, Swift ignored the fluid viscosity and the plate’s thermalconductivity in the streamwise direction. He further assumed that theplates are much shorter than the acoustic wavelength and that thepressure oscillations are small compared to the mean pressure. Theapplicability of such an idealized model is limited to linear systemanalysis. More recent research [6–11] showed substantial deviationof most cooling curves from Swift theory. Their attempts havefocused on modeling thermoacoustic devices to predict the generalperformance, capture the thermal and viscous flow losses in thedevice, and help define key variables affecting the design conside-rations and system optimization, such as stack porosity, size andlocation, working gas, and appropriate operating mean pressure.Worlikar and Knio [7] and Worlikar et al. [8] simplified the

governing equations into a low Mach number vorticity-basedformulation. Theywere able to reduce the full computational domaininto a single periodic thermoacoustic couple. However, their modelwas limited to short stack lengths, constant gas properties, and smallpressure amplitudes. Blanc-Benon et al. [9] further reported closeagreement between the low Mach number simulation results and theparticle image velocimetry measurements of the flowfield. TheWorlikar [7] approach was then traced in similar manners by Cao et al.[10], who constructed a first-order temporal and spatial discretizationscheme to solve for the time-averaged energy flux density in the gas.Cao [10] only considered short (compared with the particle displace-ment length) isothermal plates. Ishikawa andMee [11] extendedCao’swork to longer plates and investigated the effects of plate spacing usingthe PHOENICS finite volumemethod; however, their single-precisioncalculations caused significant energy imbalances and restricted theoverall accuracy in their heat fluxes results.Later, Marx and Blanc-Benon [12–14] constructed an explicit

numerical model with symmetric boundary conditions, which enablesthe prediction of the stack heat exchangers coupling in TAR made ofisothermal zero-thickness plates. They considered the time-averagedvelocity field above the stack plate and observed the acoustic flowstreaming above the plate. They further identified the minimum platelength, compared to the particle displacement, enough for thegenerated temperature harmonics to be displayed above the wholeplate surface. Although accurate, this model is too computationallyexpensive to limit its applicability to devices operating at high-frequency ranges (their frequency value f � 20 kHz), which is usefulfor miniaturization purposes.Zoontjens et al. [15] was the first to implement a FLUENT model

to simulate the flow and energy fields in a TAR couple of nonzerothickness at a wide range of drive ratio (1.7%–8.5%) and 200 Hzoperating frequency. They assumed symmetric computationaldomain and investigated the time-averaged heat transfer through thestack material. However, their model used a first-order discretizationscheme,which in turn affected the accuracy of their numerical results.This was recently followed by the work of Tasnim and Fraser [16]who introduced a two-dimensional (2-D) model using the STAR-CDfinite volume solver. He assumed a periodic thermoacoustic coupleand described the laminar flow and thermal fields at the low driveratio of 0.7%.Other numerical simulations for thermoacoustic refrigerators have

been reported [17,18]. However, none of them were able to predictthe refrigerator general performance in the nonlinear regime ofacoustic excitations. To our knowledge, no efficient CFD simulation

has been developed that models the large-amplitude excitations ofstanding TARs. Therefore, it is the objective of the present work todevelop a CFD model of TAR, which helps understand the flowproperties in the nonlinear regime and enables the prediction of therefrigerator performance and the secondary flow effects, such as flowstreaming and microstructural vortices.This work numerically simulates the nonlinear oscillating flow

behavior exhibited in half-wavelength thermoacoustic refrigerators,predicts the thermoacoustic cooling effect between the stackends, and calculates the amount of the converted acoustic power.The current numerical results are compared to the experimentalmeasurements of Atchley et al. [6]. The analysis given next usesthe ANSYS FLUENT second-order finite volume, implicit timealgorithm CFD solver with the conjugate heat transfer algorithm.This work predicts the general behavior of TARs, examines theevolvedmicrostructural vortices and the acoustic flow streaming, andpresents a stepping-stone in building a full three-dimensional (3-D)CFD model.

II. Numerical Simulation

A. Overview

We propose a CFD simulation for the oscillating flow behavior oflarge-scale half-wavelength standing thermoacoustic refrigerators.The developed computational model only considers the workingideal gas, the stack plates, and resonator solid walls. For simplicity,our TAR is assumed to initially work at no load and hence, the twoheat exchangers are omitted in the current runs. The simulationallows the prediction of the bulk flow properties within the heliumgap and at the stack neighborhood, and focuses on the acousticallydriven steady-state temperature gradient across the stack plates. Italso enables us to explore themutual thermal and viscous interactionsbetween the stationary solid plates and the surrounding ideal gas, andexamine the microstructural vortices developed at high-amplitudeacoustic excitations.It is unreasonable to numerically model the entire domain of the

resonator at the macroscopic scale, which may contain a largenumber of uniformly repeated stack plates and gas flow channelsconfined within a small volume of the resonator. The stack plates arealso much shorter than the acoustic wavelength, which creates anadditional scale complexity. Instead, we concentrate on a represen-tative unit cell within the main domain, and impose appropriateboundary conditions at the cell surfaces. We choose a 2-D unit cell,with horizontal edge of length B times the stack length l, and verticaledge of length corresponds to helium gap distance h, as a computa-tional domain. To ensure that the enclosed thermoacoustic coupleclosely behaves as a representative part of the entire refrigerator,periodic boundary conditions are enforced on the cell horizontalsurfaces, while ideal and equivalent acoustic flow conditions aresupplied at the cell vertical surfaces. This approach also allows us toreduce the resonator edge effects.The unit cell is initially filled with helium with ideal gas approxi-

mation at nearly ambient conditions. To generate the thermoacousticeffect, the cell is then subjected to low and high acoustic excitationsthrough its transversal edges. This causes small- and large-amplitudeoscillations to be introduced into the system, respectively. Simula-tions are performed using the nonlinear CFD solver in ANSYSFLUENT. The first advantage of using ANSYS FLUENT is that itallows us to model the system response at either low or high driveratios through its nonlinear package. The second advantage is that theconjugate heat transfer algorithm enables us to account for thethermal interactions between the stack walls and the confined gas,and set the model parameters to accurately capture the temperaturegradient across the stack plates.Both refrigerator solid and gas domains are modeled by a series

of 2-D quadrilateral and triangular computational elements. Thenumber and size of quadrilateral elements can be altered to provideaccurate results at convenient computational cost. Four simulationruns are carried out at drive ratios 0.28%, 0.5%, 1.0%, and 2.0%. Thesystem is perturbed at its equivalent boundaries through theoscillatory motions that carefully match the propagation of an ideal

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standing wave. Each of the four runs is then allowed time until thetemperature difference ΔTs across the stack approaches its quasi-steady-state. We observe that 5 s are sufficient to achieve such limitwithin a reasonable CPU time of about 50 h.The simulation predicts the system steady-state response of the

combined structure in terms of the stack walls static temperature andcharacterizes the periodic flow microstructures. The rest of thissection presents the numerical representation of the TAR geometry,including the stack size and location, and describes the domainmeshing used in the simulation. The following section explains theapplication of the periodic and oscillating boundary conditions.Details of the numerical analysis using ANSYS FLUENT solvercapabilities, such as controlling the time incrementation to maintainstable and fast-converging calculations, are then presented. Wefinally discuss the simulation results in comparison with experimentand previous numerical attempts.

B. Geometry and Domain Meshing

Figure 1 shows the geometry of the half-wavelength thermo-acoustic refrigerator being modeled and tested, as described byAtchley et al. [6]. The resonator tube is made of copper and has alengthL of 1.22m. It has the acoustic driver at one end and is closed atthe other end. The resonator is filled with helium at nearly ambientconditions. The stack plates are 6.85 mm long and composed of0.1016-mm-thick fiberglass laminates, each epoxied to 0.0889-mm-thick stainless steel plates. The spacing h between each pair of thestack plates is 1.338mm and the stack porosityφ � 87.5%. The TARgeometry details and the combined stack material properties,including the effective density ρeffs , the specific heat at constantpressure Ceff

p;s, and the thermal conductivity keffs , are reported byAtchley et al. [6] and used in our simulation.In this work, the stack position is maintained such that the

corresponding nondimensional wave number kx � π∕4, measuredfrom the resonator rigid end to the stack’s middle section k � 2π∕λ.The thermoacoustic energy conversion is theoretically expected topeak at such stack location. Initially, the helium-filled TAR systemhas a uniform temperature distribution of 298.4 K and zero gasvelocity. The simulation unit cell is also plotted in Fig. 1. The plotillustrates the geometry and boundary conditions of the comprised 2-D CFD domain. The cell has two horizontal surfaces with periodicboundary conditions (or, simply referred to as periodic pairs) and twovertical surfaces with acoustic flow conditions that correspond to theidealized standing wave predictions at these locations. Although thisapproximation does not allow for accurate replication of the physicalprocess at these boundaries, it can still suggest appropriate modelingat the far field of the stack plates provided that the distance from thetwo vertical surfaces to the stack ends is big enough. For comparison,the simulation cell length and height are chosen to match values usedby the Worlikar simulation [7,8]. Also, the constant B is 4.64 in thecurrent simulation.The stack domain is discretized into quadrilateral elements,

whereas the gas domain is divided into quadrilateral and tetrahedral

computational elements, particularly in the neighborhood of thesharp corners, as illustrated in the insets Fig. 2. The microstructuralanalysis indicates high smoothing and slow transition meshing atproximities and curvatures. A very fine structured mesh is enforcedvery close to the stack plates to accurately capture the thermoacousticeffect within the developed thermal penetration depths, while acoarser unstructured mesh is generated farther where the thermalinteractions with the working gas are expected to become lessimportant. In our analysis, the thermal and viscous penetrationdepths, defined in terms of the gas thermal conductivity kg and the gasdynamic viscosity μg, respectively, are such that δk �

�������������������������2kg∕ρmCpω

p� h∕4.97 and δv �

�������������������2μg∕ρmω

p� h∕6.02.

The current ANSYSmeshing process requires aminimumnumberof elements of 10 across the computational domain. However, toaccommodate to the thin boundary-layer thickness, the number ofelements between the stack plates is variable in our calculation, and isdetermined by the smallest element size. In the meshing process,we choose minimum element sizes of 0.1, 0.05, 0.025, and 0.0125millimeters. The corresponding numbers of elements and nodes aregiven in Table 1.

C. Boundary Conditions

To create a compact periodic structure, conformal periodic pairsare assigned to the computational solid and gas nodes lying on thehorizontal surfaces of the simulation cell. This allows an equal gasflow, to that leaving the simulation unit cell at one surface, to beintroduced at its corresponding image surface. Similarly, the heatflux, rejected at one surface, is fully gained at its opposite periodicsurface. In the current numerical model, both left and right equivalentsurfaces are replaced by moving rigid walls. A layering dynamic

Thermoacoustic Stack Plates

Rigid End

Acoustic Driver

Resonance Tube

Periodic Pairs

Left Equivalent Surface

CFD DOMAIN

Right Equivalent Surface

a)

b)Fig. 1 a) Schematic of the thermoacoustic refrigerator and b) the representative simulation unit cell is shownwith two periodic surfaces and two surfaceswith equivalent boundary conditions.

Fig. 2 Four different mesh sizes are used for grid generation, aspresented in Table 1.

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mesh is allowed at the left and right vertical walls of the computa-tional domain. The split and collapse factors are both set equal to 0.4,and a cell height of 0.13895 mm is used. A preprocessing program isdeveloped to define the instantaneous horizontal motions of thevertical walls’ centers of gravity following the local ideal standingwave propagation

uwall � ua sin�kx� cos�ωt� (1)

Here, ua represents the amplitude of the dynamic velocity at the leftand right oscillating boundaries, k is the wave number, x refers to thelocation of the equivalent surfaces, measured from the rigid end, andω is the angular velocity. Further, ua is expressed in terms of thedynamic pressure amplitude pa, the mean density ρm, and the speedof sound a as

ua � pa∕ρma (2)

The amplitude of the dynamic pressure pa is the gas mean pressuretimes the acoustic drive ratio. According to the Atchley experimentalstudy [6], the helium mean pressure pm and temperature Tm are114.1 kPa and 298.4 K, respectively, and the frequency is 696 Hz. Inour calculation, heat is not allowed to flow across the equivalentsurfaces and the no-slip shear condition is satisfied. The stack innerwalls are kept stationary with the no-slip shear condition used todescribe the viscous interaction with the surrounding fluid. At thesolid–fluid interfaces, thermally coupled boundary conditions areimposed to simulate the thermal interactions between the solid stackwalls and the surrounding gas particles through the conjugate heattransfer algorithm. In such case, ANSYS FLUENT requires that thetemperature and the heat transfer to the wall boundary from a solidelement match the gas side temperature and heat transfer

Tg;wall � Ts;wall (3)

kg∂Tg∂n� keffs

∂Ts∂n

(4)

where n is the local coordinate normal to the wall.

D. ANSYS FLUENT Analysis and Data Postprocessing

TRL‡’sDell PrecisionWorkStation (Intel®Xeon®X5690) is usedto perform our simulations. ANSYS FLUENT is computationallyefficient for the nonlinear flow analysis with complicated oscillatingand periodic boundary conditions. It solves the governing equationsusing a pressure-based finite volume method [19]. Its implicit timeintegration algorithm usually requires many small time incrementsthat allow the solution to proceedwith fewer numbers of iterations pertime increment. The time advancement process is based on iterativescheme, in which governing equations are solved sequentially, for agiven time increment, until convergence criteria are satisfied.Furthermore, the temporal discretization involves a first-orderaccurate backward-difference scheme. Double precision parallelcomputation is essential to this analysis; therefore, the domain isdivided into four partitions using the bisection method (principalaxes), and the simulation uses four local parallel processors and acorresponding memory usage of 2 GB. The default bisection method

(principal axes) and the smooth optimization method are used formesh partitioning.One crucial key in using ANSYS FLUENT is the time increment

size. This is of extreme importance for two reasons: first, to maintaina convergent simulation with minimal number of iterations (thusminimizing the computational cost); and second, to accuratelycapture the thermoacoustic cycling process. This justifies the use of afixed time increment Δt � τ∕100 of a hundredth of the appliedexcitation load period τ. The total real flow time to reach the quasi-steady-state is almost 5 s and the simulation takes approximately3 × 105 time increments for the system to reach its thermalequilibrium (energy balance), as shown in Fig. 3.In our double precision calculations, convergence is claimedwhen

the absolute variation in each of the flow quantities (residuals) is lessthan 10−4% (convergence criteria) in any two successive iterations.The maximum allowed number of iterations is 100 in each timeincrement; however, approximately 20 iterations are sufficient toreach convergence because of our prior choice of small-size timeincrement. The second-order upwind spatial discretization scheme isused and the high-accuracy pressure staggering scheme (PRESTO) isenabled as the pressure interpolation scheme at the computationalelement faces. This scheme uses the discrete continuity balance for astaggered control volume about a face to compute the facecorresponding pressure. PRESTO is recommended for flowsinvolving porous media and high swirling flows when the pressureprofile has a high gradient at the elements’ faces. The pressure-implicit with splitting of operators (PISO) pressure-velocity couplingscheme is applied.ANSYS FLUENT does not provide a direct way to calculate how

much acoustic power is converted each time increment through thethermoacoustic stack into refrigeration effect. It only reports thedynamic pressure and velocity at each node. To learn more about theconsumed acoustic power, a postprocessing program was written touse the instantaneous and local values of the dynamic pressurepi andx velocity ui and calculate their product at each point of the flowfield.According to Swift [5] and Nijeholt et al. [20], the simplest way tocalculate the local time-averaged acoustic power �Pac is by summingthe preceding product at every time increment and dividing by thenumber of time increments in one cycle N. In practice, in order toreduce the computational cost and focus on the quasi-steady-stateresponse, only the last acoustic cycle (with N � 100) is consideredfor the time-averaging process. An area-weighted average of �Pac isthen estimated by integrating over the stack spacing cross-sectional

area A to get the acoustic power intensity�_W

�Pac � pu �1

N

Xi

piui (5)

�_W � 1

A

Z�Pac dA (6)

For the sake of comparison with previous theoretical studies, thefollowing equation defines the theoretical temperature difference,according to Wheatley et al. [21], as function of geometricalparameters, the thermal penetration depth δk, the viscous penetrationdepth δv, the gas constant γ, and the Prandtl number σ � Cpμg∕kg

ΔTs �p2aδk�1�

��σp� sin 2kx

4ρma��keffs tp�kgh�∕l��1�σ�

1� p2aδk�1−σ

��σp��1−cos 2kx�

4ρmTmlω��keffs tp�kgh�∕l��γ−1��1−σ2�

(7)

III. Results

A. Mesh Size Convergence

Because mesh size has a strong impact on the computational cost,studying the convergence of the numerical simulation with respect tothe smallest allowed mesh size is important. This study builds anunderstanding of the influence of the domain meshing on the coolingresults. Figure 2 shows the dependence of the cooling effect, in terms

Table 1 Meshing details

Mesh Element Size Elements Nodes

1 0.1 2,957 3,2172 0.05 5,303 5,6383 0.025 11,758 12,2274 0.0125 28,065 28,780

‡American University in Cairo’s Thermoacoustic Research Lab.

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of the generated temperature difference between the stack ends atsteady state, on the chosen smallest mesh size. In these runs, the driveratioDr � 2%. Interestingly, a significant reduction is demonstratedin the steady-state temperature difference as the element sizedecreases, indicating convergence in the preceding simulationresults. This is because using a finer domain meshing produces a lessdissipative numerical model, which enables a more accuratedetection of the flow physics.The relative error in the temperature difference between every two

successive mesh sizes is found to decrease from 4.15% to nearly1.4%. This yields an average mesh size of δk∕6.12, with a minimumof δk∕10.71 close to the solid wall, in accordance with the work ofCao et al. [10] and Marx and Blanc-Benon [13]. Therefore, anelement size of 0.025 mm is sufficient to capture the oscillating flowgeneral behavior and will be used in the rest of this work.

B. Cooling Effect

In practical thermoacoustic refrigerators, thermal energy isinitially transported along the x direction by the hydrodynamictransport of entropy carried by the velocity oscillations of theworking gas. The present nonisothermal stack plates allow for atemperature gradient to develop as the simulation proceeds, as shownin Fig. 3a, due to thermoacoustic heat pumping effect. This causes thetemperatures of the stack hot and cold ends to reach their upper andlower limits, respectively, then to further increase at nearly the samerate by viscous heating, while closely maintaining a constanttemperature difference across the stack. This then leads to an increase

in the mean temperature of the stack plates. Figure 3b demonstratesconvergence in the stack ends temperature difference as thecomputation proceeds.To understand this behavior, the numerical time-averaged

temperature distribution along the stack plates, over the last acousticperiod, is calculated and plotted in Fig. 4a for three differentdrive ratios. The theory assumed a linearly constant temperaturedistribution along the plates and ignored both fluid and solid thermalconductivities in the x direction, as well as the viscous dissipation,which typically causes a uniform temperature distribution along thestack plates. However, a smoother slope than that predicted by theory,along with slight curvatures near the plates’ ends, is found in thecurrent simulation. This is possibly because the stack plates’ thermalconductivity allows for the heat flow due to conduction down thestack temperature gradient (heat loss), in addition to the heattransferred by the nonzero mass flux resulted from the acousticstreaming, as discussed next. This causes the temperatures of thestack hot and cold ends to further smooth out to their indicated quasi-steady-state values. This effect is also predicted to become of lesserimportance than that of viscous heating at high drive ratios, especiallywhen using shorter stack plates with smaller thermal conductivities.Figure 4b also illustrates the thermal interaction between the stack

plates and its surrounding gas at drive ratios 0.28%, 1.0%, and 2.0%.The time-averaged heat flux distribution, obtained from thecalculated temperature gradient using the conjugate heat transferalgorithm, is plotted along the plates’ surfaces and edges. The heatpumping effect acting between the stack and the surrounding helium

Fig. 3 a) The variation with time of the stack ends temperature and b) the convergence of ΔTs. The drive ratio in this run is 2%.

Fig. 4 Variations of the stack temperature a) and heat flux b) distributions along the stack length at drive ratios 0.28%, 1.0%, and 2.0%.

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is captured. The thermal interaction is mostly localized near the stackends. The stack’s right end (closer to velocity antinode) gets cooler,while the stack’s left end (closer to the pressure antinode) is heatedup. The figure indicates a higher heat flux at the neighborhood of thehot end than the corresponding value at the cold end. This reflects anonzero mean surface heat flux, which normally contributes to thesystem total power.At the surfaces of the stack plates, the heat flux density in Fig. 4 is

principally results from themutual heat conduction between the stackplates and the surrounding gas. This is due to the no-slip viscousboundary condition, which enforces a zero-velocity profile at theplates’ surfaces. To investigate the effect of nonlinearities associatedwith high-amplitude excitations on the generated temperaturegradient, Fig. 5 compares the temperature difference across the stackas a function of the acoustic drive ratio to Atchley’s [6] experimentalvalues, Wheatley’s [21] linear theory for thermoacoustic devices andWorlikar [8] numerical results. Interestingly, the current numericalmethods replicate the nonlinear behavior captured in the quasi-steadyexperimental measurements. The system behavior is clearly non-linear at drive ratios greater than 1% because of the high-amplitudepressure excitations. As a result, the nonlinear propagation of high-

amplitude excitations promotes the generation of harmonicoscillations and the acoustic flow streaming within the stack andsurrounding the stack ends.

C. Acoustic Power

As discussed in the data postprocessing section, the intensity of theacoustic power is calculated and recorded during the last acousticcycle, in which 100 time steps are used for averaging procedure. Aspecial program is developed to evaluate the time-averagedcomponent of the acoustic power, according to Eq. (5), by calculatingthe product of the gas dynamic pressure and velocity at every node ofthe computational domain. This is repeated for the last 100 time stepsof each simulation run for the time-averaged calculations. ANSYSFLUENT allows us to further integrate the acoustic intensity toevaluate the area-weighted acoustic results at each position along thestack plates, as presented in Eq. (6).The acoustic results, shown in Fig. 6a, distinguish the behavior of

the power intensity into three regions. The prestack and poststackresonator sections are characterized by almost similar decreases inthe power intensity because of the acoustic loss due to both viscousdissipation and thermal relaxation. The stack section, however,exhibits higher degradation in the acoustic power with larger slope.This is mainly due to the acoustic energy conversion into heatpumping between the stack ends, in addition to the accompanyingthermal and viscous losses, which result from the interactions of thegas particles with the stack plates. Negative values of acoustic powerreflect the phasing between the gas dynamic pressure and velocity atthe stack location and thus indicate that the acoustic energy is flowingin the negative direction toward the resonator closed end. Thesimulation also reveals a significant sudden drop in the acousticpower at the left stack end, whereas a corresponding abrupt jump isobserved at the right stack end, which extends few millimeters intothe poststack resonator.This reflects the secondary flow losses associated with the sudden

contraction and expansion of the oscillating flow at the edges of thestack plates, which are caused by the immediate change in theviscousboundary conditions at these locations. Such geometrical constraintsalso provoke the separation of the boundary layers and induce theformation of the vortical motions beyond the stack region. To supportthis result, the time-averaged dynamic pressure is monitored andpresented in Fig. 6b. Here, we notice the smooth drop and rise in thedynamic pressure as the gas velocity in the stack must be higher thanthat outside by the cross-sectional area ratio to maintain a constantvolumetric flow rate.The current analysis is also extended to thicker plates to study the

effect of plate thickness (the stack porosity) on the consumed acoustic

Fig. 5 The effect of the drive ratio on the induced cooling effect. Thesolid line refers to the theoretical temperature difference, as derived byWheatley et al. [21] in Eq. (7). Symbols: Open triangles and squares arethe Atchley experimental values and the Worlikar numerical results,respectively, whereas the crosses represent the current simulation.

Fig. 6 a) Longitudinal variation of the time-averaged power intensity at drive ratio 2%. Closed and open symbols refer to thin and thick plates,

respectively. The gray area points out the stack region. The light gray lines are shape-preserving interpolant fitting for both datasets. Solid black lines arethree separate linear fits for the prestack resonator, stack, andpoststack resonator regions. b)Refers to the correspondingdynamicpressure.The stack leftend is closer to the driver side.

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power and evolving microstructure on top of the plates’ surfaces andnear the stack ends. Here, the drive ratioDr � 2%, and the new platethickness tp has significantly increased such that the spacing hbecomes three times the thermal penetration depth δk � h∕3.0 andfour times the viscous penetration depth δv � h∕4.0, keeping thewidth H of the unit cell fixed, and the corresponding stack porosityφ � 35.0%. The numerical prediction reveals that reducing thespacing by 40% consumes almost triple the acoustic power andproduces a lesser amount of cooling, of about 1.4°C, than thecorresponding thin stack geometry. This comparison qualitativelypresents the effect of the stack blockage ratio on the systemperformance. Such CFD results have not been previously reported.

D. Acoustic Streaming and Vorticity Field

ANSYS FLUENT enables the data sampling for unsteady flowstatistics. This tool is used in our simulation to study the time-averaged (again, only for last acoustic cycle) velocity field in thechannel between the stack plates and at the stack ends and furtherexamine the resulting acoustic flow streaming in the stack region andthe vortical motion at the stack ends. In a linear standing wavepropagation, the time-averaged (mean) of the dynamic velocity isexpected to vanish. However, the numerical results demonstrate theexistence of the nonlinear acoustic streaming and its ever-increasinginfluence on the mean axial velocity profiles, with the increase of thesystem drive ratio. Here, the stack porosity φ � 87.5%.

Information about the values of the mean velocity components atevery single node of the computational domain are passed to apostprocessing MATLAB program, which allows us to draw avector map of the mean dynamic velocity field, as displayed inFig. 7. The current simulation of the velocity field within the stackreveals the presence of toroidal circulations, exhibited by theoscillatory gas motion and spanning the whole stack domain at driveratio 2%. This proves the existence of the nonlinear inner streamingcaused by large-amplitude acoustic excitations at drive ratios largerthan 1%.Figure 8 further investigates the effect of increasing the drive ratio

on the mean axial velocity profiles. In Fig. 8a, the absolute values arepredicted in a cross section locatedmidway between the stack ends atdifferent drive ratios. The y position is normalized using the platespacing h. Clearly, the larger the drive ratio, themore pronounced thenonlinear flow streaming is. At drive ratio 0.28%, the velocity is twoorders of magnitude less than that found at drive ratio 2%. Althoughslightly distorted (numerical discretization error due to finite grid sizeand nonuniform mesh size), the velocity profiles are closelysymmetric about their midplanes.To exclude the effect of the amplitude of the imposed acoustic

wave on the streaming profiles, the acoustic axial velocity results arefurther normalized using the Rayleigh velocity uR � 3u2a∕16a, asshown in Fig. 8b. Interestingly, the velocity profiles superimposeeach other and become almost independent on the drive ratio.Therefore, similar flow microstructures, such as that illustrated inFig. 7 for the 2% drive ratio, do exist at other drive ratios but withdifferent intensities. The flow pattern changes sign once in each halfof the enclosed gap, indicating the existence of two opposite (mirror)vortices between the two plates, which is a characteristic of thespacing between the two stack plates. This supports the results of theMarx and Blanc-Benon [13], when the ratio is approximately1.75 < h∕δv < 9.0.Contours of the normalized mean vorticity �Ω field superimposed

by a vector map of the mean velocity field at the neighborhood of thestack ends is additionally presented in Fig. 9. Here, themeanvorticityis normalized using a∕δv and the drive ratio is 2%. The figureillustrates the boundary-layer separation at the sharp corners of thestack plates, because of the no-slip boundary condition and thesudden flow contraction at the left edge and expansion at the rightedge. The generated vortical motions are shown larger at the left endthan at the right stack end. This supports the results in Fig. 6 thatindicate greater mean dynamic pressure, thus velocity at the hot endof the stack plate, which is closer to the system pressure node at therigid end. Also, as can be verified from Fig. 7, these induced vorticalmotions at the stack ends are coupled through the acoustic streamingin gap between the two stack plates.

Fig. 7 Vector map of the mean velocity field in the gap between twostack plates, showing the inner streaming. Blue dashed-dotted linesdenote to the gap centerline and the stack mid-plane. Solid and dashedlines (in red) indicate counter-clockwise and clockwise vortical motions,

respectively.

Fig. 8 Profiles of the mean axial velocity components at the midplane section. a) Plots the absolute values at three different drive ratios. Open circles,diamonds, and squares refer to drive ratios of 2.0%, 1.0%, and 0.28% respectively. Dashed lines are Fourier fits of eight harmonics. b) Shows the axialvelocity normalized using Rayleigh velocity uR � 3u2a∕16a.

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IV. Conclusions

ANSYS FLUENT is a useful tool that allows us to directlysimulate the gas dynamics of nonlinear oscillating compressible flowof half-wavelength thermoacoustic refrigerators, and enables thedetection of thermoacoustic cooling effect between the stack ends andthe prediction of the consumed acoustic power during the acousticloading through its CFD solver and the conjugate heat transferalgorithm.The simulation results agreewellwithAtchley experimentalvalues [6] and capture the general behavior of the generated vorticalmotions at the stack ends and flow streaming, provided that thecondition of no slipping at the stack plates’ surfaces is satisfied.The computational domain considers one pair of the stack plates

and the oscillating gas confined in-between, and assumes periodicboundary conditions to account for the stack periodicity. The stackplates are uniformly arranged in a parallel configuration, whichsupports the capability of reducing the 3-D geometry of the Atchely[6] thermoacoustic refrigerator into a 2-D model. However, thismodel is inappropriate for other stack geometries, such as the gridtype, and nonuniform stack materials, such as the foam materials.Extension to typical nonuniformmaterials requires a direct numericalmodeling of the stack microstructures and a 3-D simulation of theinvolved flowfield. Geometry processing, meshing, and minimiza-tion of the CPU time are the greatest challenges and are subjected toour future investigations.The present simulation uses a fully implicit time integration

algorithm, which is unconditionally stable with respect to the timeincrement size. The choice of the time discretization scheme islimited, however, by the crucial implementation of the dynamicmeshing technique, to the first-order accurate backward differences,whereas the spatial-discretization applies the unrestricted second-order accurate upwind scheme. ANSYS FLUENT allows us tointroduce the dynamic meshing steps by the specification of thehorizontally oscillating boundaries to account for the correspondingideal acoustic flow conditions, while maintaining adiabatic thermalconditions.Finally, this work accurately predicts the stack temperature

differences as a function of the applied drive ratio and discuss theinfluence of the plates’ thicknesseson the acoustic power dissipation. Itshows that three times the acoustic power are dissipated in the stack fora 40% reduction in plates’ inner spacing. Such qualitative result isinteresting as it helps to improve the system performance and decreasethe effort spent in selecting less dissipative stack configurations. Thepresent CFD model successfully captures the nonlinear streamingeffect and characterizes its intensity as a function of the applied driveratio anddraws the correspondingvorticesbothwithin thegapbetweenthe stack plates and at the stack ends, as illustrated in Figs. 7 and 9.

Acknowledgment

The work presented here is supported by the King AbdullahUniversity for Science and Technology under the Integrated Desert

Building Technologies project grant held by theAmericanUniversityin Cairo.

References

[1] Garrett, S. L., Adeff, J. A., and Hofler, T. J., “ThermoacousticRefrigerator for Space Applications,” Journal of Thermophysics and

Heat Transfer, Vol. 7, No. 4, 1993, pp. 595–599.doi:10.2514/3.466

[2] Bauwens, L., “Stirling Cryocooler Model with Stratified Cylinders andQuasisteady Heat Exchangers,” Journal of Thermophysics and Heat

Transfer, Vol. 9, No. 1, 1995, pp. 129–135.doi:10.2514/3.638

[3] Chen, R., Chen, Y., Chen, C. T., and DeNatale, J., “Development ofMiniature Thermoacoustic Refrigerators,” AIAA Paper 2002-0206,Jan. 2002.

[4] Panton, R. L., Incompressible Flow, 2nd ed., Wiley, New York, 1996,pp. 263–277.

[5] Swift, G. W., “Thermoacoustic Engines,” Journal of the Acoustical

Society of America, Vol. 84, No. 4, 1988, pp. 1145–1180.doi:10.1121/1.396617

[6] Atchley, A. A., Hofler, T. J., Muzzerall, M. L., Kite, M. D., and Ao, C.,“Acoustically Generated Temperature Gradients in Short Plates,”Journal of the Acoustical Society of America, Vol. 88, No. 1, 1990,pp. 251–263.doi:10.1121/1.399947

[7] Worlikar, A. S., and Knio, O. M., “Numerical Simulation of aThermoacoustic Refrigerator: I. Unsteady Adiabatic Flow Around theStack,” Journal of Computational Physics, Vol. 127, No. 2, 1996,pp. 424–451.doi:10.1006/jcph.1996.0185

[8] Worlikar, A. S., Knio, O. M., and Klein, R., “Numerical Simulation of aThermoacoustic Refrigerator: II. Stratified Flow Around the Stack,”Journal of Computational Physics, Vol. 144, No. 2, 1998, pp. 299–324.doi:10.1006/jcph.1997.5816

[9] Blanc-Benon, P., Besnoin, E., and Knio, O., “Experimental andComputational Visualization of the Flow Field in a ThermoacousticStack,” C. R. Mecanique, Vol. 331, No. 1, 2003, pp. 17–24.doi:10.1016/S1631-0721(02)00002-5

[10] Cao, N., Olson, J. R., Swift, G. W., and Chen, S., “Energy Flux in aThermoacoustic Couple,” Journal of the Acoustical Society of America,Vol. 99, No. 6, 1996, pp. 3456–3464.doi:10.1121/1.414992

[11] Ishikawa, H., and Mee, D. J., “Numerical Investigations of Flow andEnergy Fields near a Thermoacoustic Couple,” Journal of the AcousticalSociety of America, Vol. 111, No. 2, 2002, pp. 831–839.doi:10.1121/1.1430687

[12] Marx, D., and Blanc-Benon, P., “Numerical Simulation of Stack-HeatExchangers Coupling in a Thermoacoustic Refrigerator,”AIAA Journal,Vol. 42, No. 7, 2004, pp. 1338–1347.doi:10.2514/1.4342

[13] Marx, D., and Blanc-Benon, P., “Computation of the Mean VelocityField Above a Stack Plate in a Thermoacoustic Refrigerator,” C. R.

Mecanique, Vol. 332, 2004, pp. 867–874.doi:10.1016/j.crme.2004.07.010

[14] Marx, D., and Blanc-Benon, P., “Computation of the TemperatureDistortion in the Stack of a Standing-Wave Thermoacoustic

Fig. 9 Contours of the normalized mean vorticity field superimposed by a vector plot of the mean velocity field in the vicinity of the stack plates. Thevelocity magnitude is indicated by the arrow length, whose maximum corresponds to 2.0 m∕s.

ABD EL-RAHMAN AND ABDEL-RAHMAN 85

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17,

201

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Refrigerator,” Journal of the Acoustical Society of America, Vol. 118,No. 5, 2005, pp. 2993–2999.doi:10.1121/1.2063087

[15] Zoontjens, L., Howard, C. Q., Zander, A. C., and Cazzolato, B. S.,“Numerical Study of Flow and Energy Fields in ThermoacousticCouples of Non-Zero Thickness,” International Journal of Thermal

Sciences, Vol. 48, No. 4, 2009, pp. 733–746.doi:10.1016/j.ijthermalsci.2008.06.007

[16] Tasnim, S. H., and Fraser, R. A., “Computation of the Flow andThermal Fields in a Thermoacoustic Refrigerator,” International

Communications in Heat and Mass Transfer, Vol. 37, No. 7, 2010,pp. 748–755.doi:10.1016/j.icheatmasstransfer.2010.04.006

[17] Zink, F., Vipperman, J. S., and Schaefer, L. A., “AdvancingThermoacoustics ThroughCFDSimulationUsing Fluent,”Proceedingsof the ASME International Mechanical Engineering Congress and

Exposition, Boston, MA, 2008, pp. 101–110.

[18] Piccolo, A., and Pistone, G., “Numerical Study of the Performance ofThermally Isolated Thermoacoustic-Stacks in the Linear Regime,”Proceedings of Acoustics, Paris, France, 2008, pp. 3555–3560.

[19] Versteeg, H. K., and Malalasekera, W., An Introduction to

Computational Fluid Dynamics — The Finite Volume Method,2nd ed., Pearson Education Limited, Harlow, England, 2007,pp. 243–266.

[20] Lycklama à Nijeholt, J. A., Tijani, M. E. H., and Spoelstra, S.,“Simulation of a Traveling-Wave Thermoacoustic Engine UsingComputational Fluid Dynamics,” Journal of the Acoustical Society ofAmerica, Vol. 118, No. 4, 2005, pp. 2265–2270.doi:10.1121/1.2035567

[21] Wheatley, J., Hofler, T., Swift, G.W., andMigliori, A., “An IntrinsicallyIrreversible Thermoacoustic Heat Engine,” Journal of the Acoustical

Society of America, Vol. 74, No. 1, 1983, pp. 153–170.doi:10.1121/1.389624

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