Stability of buoyant convection in a layer submitted to acoustic streaming

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acoustic streaming

Walid Dridi, Daniel Henry, Hamda Ben Hadid

To cite this version:

Walid Dridi, Daniel Henry, Hamda Ben Hadid. Stability of buoyant convection in a layer submitted to acoustic streaming. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2010, 81, pp.056309. �10.1103/PhysRevE.81.056309�. �hal-00566012�

Stability of buoyant convection in a layer submitted to acoustic streaming

W. Dridi, D. Henry,

*

Laboratoire de Mécanique des Fluides et d’Acoustique, CNRS/Université de Lyon–École Centrale de Lyon/Université Lyon 1/INSA de Lyon–ECL, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France

共Received 24 November 2009; published 11 May 2010兲

The linear stability of the flows induced in a fluid layer by buoyant convection共due to an applied horizontal temperature gradient兲and by acoustic streaming共due to an applied horizontal ultrasound beam兲is studied. The vertical profiles of the basic flows are determined analytically, and the eigenvalue problem resulting from the temporal stability analysis is solved by a spectral Tau Chebyshev method. Pure acoustic streaming flows are found to be sensitive to a shear instability developing in the plane of the flow共two-dimensional instability兲, and the thresholds for this oscillatory instability depend on the normalized widthH_bof the ultrasound beam with a minimum forH_b= 0.32. Acoustic streaming also affects the stability of the buoyant convection. For a centered beam, effects of stabilization are obtained at small Prandtl number Pr for large beam widths H_b 共two- dimensional shear instability兲and for moderate Pr共three-dimensional oscillatory instability兲, but destabiliza- tion is also effective at small Pr for small beam widthsH_band at large Pr with a spectacular decrease of the thresholds of the three-dimensional steady instability. An adequate decentring of the ultrasound beam can enhance the stabilization. Insight into the stabilizing and destabilizing mechanisms is gained from the analysis of the fluctuating energy budget associated with the disturbances at threshold. The modifications affecting the two-dimensional shear instability thresholds are strongly connected to modifications of the velocity fluctuations when acoustic streaming is applied. Concerning the three-dimensional steady instability, the spectacular de- crease of the thresholds is explained by the extension of the zone with inverse stratification in the lower half of the layer.

DOI:10.1103/PhysRevE.81.056309 PACS number共s兲: 47.20.Bp, 47.20.Ft, 47.11.Kb

I. INTRODUCTION

Directional solidification is used in the processing of semiconducting and optoelectronic materials, whose perfor- mance relies on the homogeneity of the crystalline material 关1兴. In the horizontal Bridgman technique, the molten crystal is contained in a crucible which is withdrawn horizontally from a furnace. Thus, the melt is subject to a horizontal temperature gradient, which drives endwall convection. In practice, the flow which is first unicellular evolves with the increase of the temperature gradient, undergoes bifurcations and becomes unsteady, and eventually turbulence sets in for large temperature differences. It is now well known that in- stabilities in the melt phase adversely affect the quality of the crystal, as they impose temperature fluctuations at the solidi- fication front and lead to striations in the crystalline product 关2兴. The control of the flows and of the instabilities in the melt phase has then become an important research objective for the past decade. Microgravity and more commonly mag- netic fields have been used for crystal growth applications as they allow the damping of the convective flow in the melt.

They are, however, costly and heavy technologies. An alter- native could be the use of ultrasound waves which can reor- ganize the flow in the melt through the generation of acoustic streaming. It is this possibility we want to study in this paper.

Acoustic streaming describes a steady flow generated by an ultrasound wave propagating in a fluid. This effect was first observed in 1831 by Faraday 关3兴. It is now well known that it is a nonlinear effect which owes its origin to the action

of Reynolds stresses共mean momentum flux due to the ultra- sound wave兲 and the dissipation of acoustic energy flux.

More precisely, it is the dissipation共or spatial attenuation兲of acoustic energy flux that permits gradients in momentum flux to force acoustic streaming motions 关4,5兴. There are two main types of acoustic streaming: Eckart streaming in which the dissipation takes place in the main body of the fluid关6兴, and Rayleigh streaming in which the dissipation is associated with boundary layers at solid surfaces 关7兴. In our study we will consider Eckart streaming in which the flow, generated inside the ultrasound beam, moves the fluid away from the ultrasound source. Such streaming motions have been used to move fluids in microfluidic devices 关8,9兴, induce chaotic mixing 关10兴, and even improve the quality of crystals ob- tained by directional solidification processes关11兴.

The main studies concerning the action of acoustic streaming on directional solidification processes have been performed experimentally by Kozhemyakin and his co- workers. The first studies 关11–13兴 have shown that acoustic streaming was able to decrease or even eliminate the stria- tions in different single crystals grown by the Czochralski process. Kozhemyakin关14兴then studied the influence of ul- trasound waves on the convective flows thermally induced in distilled water. The advantage of distilled water is that it is transparent and allows easy flow measurements by optical methods, but also that some of its properties, such as sound velocity and acoustic spatial attenuation factor, are similar to that of liquid metals as InSb melts 关15兴. The experiment models a Czochralski configuration used for the growth of GaxIn_1−xSb single crystals. It is shown that the ultrasound waves at high frequency allow to reduce the convection. A strong reduction of the flow oscillations is also observed which is shown to be connected to the acoustic standing

*[email protected]

1539-3755/2010/81共5兲/056309共22兲 056309-1 ©2010 The American Physical Society

waves appearing between the quartz waveguide and the model solid/liquid interface. More recently, Kozhemyakinet al. 关16兴 investigated the influence of ultrasound on the growth striations and electrophysical properties of Ga_xIn_1−xSb single crystals. They show that the use of ultra- sound allows to reduce the striations in the single crystals, which induces an improvement of the crystal properties, e.g., an increase of the carrier mobility and the thermal emf, and a decrease of the resistivity.

Our numerical work on the subject has considered a dif- ferent crystal growth configuration, namely, the horizontal Bridgman configuration. More precisely, the numerical stud- ies have investigated the influence of acoustic streaming on the convective flows induced by a horizontal temperature gradient, which are typical of the flows in horizontal Bridg- man crystal growth configurations. A first study was focused on the stabilizing influence of the Rayleigh streaming on convection induced in a three-dimensional cavity 关17兴. The other studies were concerned by Eckart streaming. The ac- tion of the Eckart streaming on the convection instabilities was first studied in side-heated cavities with thermally insu- lated boundaries, three-dimensional cavities of different lengths 关15兴, or an infinite layer 关18兴. Preliminary results have also been obtained in the case of an infinite layer with thermally conducting boundaries关19,20兴. These studies have shown that Eckart streaming can have a stabilizing effect on the convective flows, but only in some parameter ranges.

The objective of this work is to precise and extend the results in the case of the infinite layer with thermally con- ducting boundaries and to deepen our understanding of the influence of Eckart streaming on these convective flows typi- cal of horizontal Bridgman crystal growth configurations. We will determine the flows induced in such a side-heated fluid layer submitted to an ultrasound beam and study their stabil- ity with respect to the dominant instability modes. We will focus our study on the influence of acoustic streaming on these instabilities, and for that will largely vary the charac- teristic parameters, i.e., the beam width and position, and the intensity of the force created by the ultrasound wave. We will finally analyze the stabilizing or destabilizing effects induced by acoustic streaming through energy analyses.

II. GOVERNING EQUATIONS AND BASIC FLOW We consider an incompressible liquid layer of thicknessh 共in the vertical z direction兲 confined between two infinite horizontal walls共Fig.1兲. This layer is subject to a horizontal

temperature gradient ⵜT˜ along the longitudinal x direction and to a radiation pressure caused by an ultrasound beam generated in the x direction by a transducer. The fluid is assumed to be Newtonian with constant kinematic viscosity

␯ and thermal diffusivity ␬. According to the Boussinesq approximation, density variations are restricted to the buoy- ancy term and taken as a linear variation of the temperature,

␳⁼␳0(1 −␤共T˜−T˜

0兲), where␤ is the thermal expansion coef- ficient and ˜T

0 is a reference temperature. The ultrasound beam, which is applied inside the layer, has a characteristic widthhb共hb⬍h兲in the verticalzdirection and is uniform in the transverseydirection. The divergence of the beam is thus assumed to be small, which has been shown to be a reason- able hypothesis in melt configurations 关15兴. The ultrasound field is also assumed to be a plane wave of frequency f traveling in the positivexdirection. Since Lighthill关4兴, it is well known that the attenuation of an ultrasound wave in a viscous fluid gives rise to a body force acting within the ultrasound beam and equal to the spatial variation of the Reynolds stress. For a plane wave traveling in the positivex direction, following Nyborg 关5兴 and Frampton et al. 关9兴, it can be shown that the body force is oriented along thexaxis and that its intensity is given by F=␳␣^Va

2e^−2␣x, where␣ ^is the amplitude attenuation coefficient for ultrasound, and Va

is the amplitude of the acoustic velocity oscillation 关15兴.

Now, provided the attenuation of the wave is sufficiently weak共estimations in关15兴give a few percents兲, a body force, which is constant共F=␳␣V_a²兲 inside the beam共over a height hb兲 and zero above and below the beam, can be defined 关6,8,15,19兴. Following Lighthill 关4兴, this body force can be introduced in the Navier-Stokes equations which, in our case, are coupled with an energy equation through the buoyancy term. If we considerh,h²/␯^,␯/h,␳0␯²/h², andⵜT˜ has scales for length, time, velocity, pressure, and temperature, respec- tively, the relevant dimensionless equations are

⵱·V= 0, 共1兲

⳵^V

⳵^{t +共V·⵱兲V= −⵱P+⵱2V+ GrTez+A}␦bex, 共2兲

⳵^T

⳵^{t +共V·⵱T兲=} 1

Pr⵱²T, 共3兲

where the dimensionless variables are the velocity vector 关V=共U,V,W兲兴, the pressure P, and the temperature T. In these equations, Gr is the Grashof number 共Gr

=g␤ⵜ˜ hT ⁴/␯²兲, Pr is the Prandtl number共Pr=␯/␬兲, andA is an acoustic streaming parameter defined as A=␣^Va

2h³/␯² 关15兴.共Note that the parameterAdoes not directly depend on the sound wave frequency f. This dependence is nevertheless effective through the acoustic attenuation coefficient␣^which is known to vary asf².兲␦bis a function ofzwhich is 1 inside the acoustic beam and 0 outside. The boundary conditions at the horizontal walls共located atz= −1/2 andz= 1/2兲 are no slip conditions, and perfectly conducting thermal conditions.

Note that we have not introduced the Rayleigh streaming influence in our model, despite the fact that the ultrasound

0

x z y

Ultrasound h

hb

source Cold Hot

O

FIG. 1. Schematic diagram of the laterally heated layer subject to an ultrasound beam.

beam could be close to the boundaries. In fact, according to Framptonet al.关9兴, the boundary layer induced streaming is a very small contributor to the total streaming in large-scale channels 共typically with a size of at least 1 mm兲. More pre- cisely, for a plane traveling wave occupying the whole height of a channel, it was shown by Nyborg关5兴that the ratio of the maximum of the “Rayleigh force” to the “Eckart force” is equal to 0.23k/␣共wherek= 2␲^f/cis the wave number andc is the sound velocity兲, and in water at 20° for a frequency of 1 MHz, the ratiok/␣^{is 1.66}⫻10⁵, indicating a much stron- ger intensity of the Rayleigh force. The “Rayleigh force”

however occurs over a very small portion of the channel height 共less than 1␮m along the boundaries兲, whereas the

“Eckart force” is approximately constant and occurs on the whole height of the channel. The relative ability of these forces to generate streaming flows has to be estimated. As an example, for a plane traveling wave in a channel with open ends, Nyborg关5兴showed that the average Rayleigh stream- ing velocity is aboutVR=V_a²/4c, whereas the average Eckart streaming velocity共parabolic pattern兲is VE=␣^Va2h²/12␯^{. In} water at 20° for a channel heighthof 5 cm and a frequency of 1 MHz, the ratio V_R/V_E is about 2.9⫻10⁻⁵. The results for liquid metals are not very different. The ratio k/␣ is really similar to that for water 共due to similar values of the dynamic viscosity␮and attenuation coefficient␣兲. The ratio VR/VEcan be estimated for gallium with a kinematic viscos- ity ␯^{= 2.87}⫻10⁻⁷ m²/s at 346 K 关21兴 in a channel with a typical height of a few centimeters 共h= 5 cm, for instance兲 and an acoustic frequency of 1 MHz. In such conditions the ratio VR/VE is still very small, about 9.2⫻10⁻⁶. It is thus shown that the streaming flow induced by a traveling wave in not too narrow cavities共as those used for crystal growth applications兲 is predominantly generated by Eckart stream- ing, the Rayleigh streaming contribution being really negli- gible. As pointed out by Framptonet al.关9兴, this is not true in microfluidic devices.

We now consider the layer as infinitely long in both hori- zontalxandydirections, but we also assume that there exist two vertical boundaries 共oriented perpendicular to the x di- rection and located atx=⫾⬁兲 which will allow the flow to return共inducing a zero flow rate in each section of the layer at fixed x兲 without reflecting the ultrasound beam. In that case, a stationnary parallel flow solution depending only on the vertical zcoordinate can be obtained关19兴. This solution is governed by the following equations:

⳵^2U0

⳵^{z2 −}

⳵^P0

⳵^{x +A}␦b= 0, 共4兲

−⳵^P0

⳵^{z + GrT0= 0, 共5兲}

⳵^2T0

⳵^{z2 = PrU0, 共6兲}

withU₀=U₀共z兲,T₀=T₀共x,z兲=x+T_i共z兲, and兰zU₀dz= 0.

In the case of pure acoustic streaming effect, Gr= 0, and from Eq.共5兲,P₀is only a function ofx. Then, from Eq.共4兲,

⳵^P0/⳵^xis equal to terms only depending onzand must then be a constantC. Equation共4兲will then give

⳵^2Uac

⳵^{z2 −C+A}␦b= 0. 共7兲 The solution, quadratic inzinside and outside the beam, can be obtained easily assuming no slip conditions at the walls 共Uac= 0兲, continuity conditions for Uac and ⳵^Uac/⳵^{z at the} beam boundary, and mass conservation. In the case where the beam is centered in the cavity, we get

Uac共z兲= −AHb

8 共z+ 0.5兲关2共Hb

2− 3兲z−共Hb

2+ 1兲兴 for − 1/2 艋z艋−Hb/2,

Uac共z兲= − A

16共Hb− 1兲²关4共Hb+ 2兲z²−Hb兴 for −Hb/2艋z 艋H_b/2,

Uac共z兲= −AHb

8 共z− 0.5兲关2共Hb

2− 3兲z+共Hb

2+ 1兲兴 forHb/2 艋z艋1/2,

where H_b=h_b/h is the normalized width of the acoustic source. In the case where the beam is not centered, but with its center located atz=zb共−1/2⬍zb⬍1/2兲, we obtain

Uac共z兲= −AHb

8 共z+ 0.5兲关2共Hb

2− 3 + 12z_b²兲z−共Hb 2+ 1 + 12z_b²− 8z_b兲兴 for − 1/2艋z艋z_b−H_b/2,

Uac共z兲= − A

16关4„共Hb− 1兲²共Hb+ 2兲+ 12z_b²Hb…z²

+ 16z_b共H_b− 1兲z−„共H_b− 1兲²H_b+ 4z_b²共3H_b− 2兲…兴 forzb−Hb/2艋z艋zb+Hb/2,

Uac共z兲= −AHb

8 共z− 0.5兲关2共Hb2− 3 + 12z_b²兲z +共H_b²+ 1 + 12z_b²+ 8z_b兲兴

forzb+Hb/2艋z艋1/2.

In the case of pure buoyancy, A= 0, and the governing equation

⳵^3Ub

⳵^{z3 − Gr = 0 共8兲}

gives the usual cubic profile Ub共z兲=Gr

24共4z³−z兲.

In the general case, the solution of the problem can be written as U₀=Ub+Uac+U⬘^{. U}⬘ is found to verify Eq. 共7兲

without forcing term, which gives a quadratic profile C共z²

− 1/4兲taking into account no slip boundary conditions. Mass conservation implies thatC= 0, and soU⬘^{= 0 and}

U₀共z兲=Ub共z兲+Uac共z兲. 共9兲 When temperature is applied, it is transported by the flow U₀ according to Eq.共6兲giving rise to Tb andTacsuch that

T₀共x,z兲=x+T_i共z兲=x+T_b共z兲+T_ac共z兲. 共10兲 The partTb induced byUb for thermally conducting bound- aries is given by

Tb共z兲=Gr Pr

5760共48z⁵− 40z³+ 7z兲.

The partTacinduced byUacfor thermally conducting bound- aries is given in the centered case by

Tac共z兲= −PrAHb

96 共z+ 0.5兲³关2共Hb

2− 3兲z−共3Hb

2− 1兲兴 for

− 1/2艋z艋−H_b/2,

Tac共z兲= −PrA

192共Hb− 1兲²

冋

+Hb

4 共2Hb+ 1兲

册

Tac共z兲= −PrAHb

96 共z− 0.5兲³关2共Hb2− 3兲z +共3Hb

2− 1兲兴 forHb/2艋z艋1/2, and when the beam is not centered by

Tac共z兲= −PrAHb

96 共z+ 0.5兲关共z+ 0.5兲²„2共Hb

2− 3 + 12z_b²兲z

−共3Hb2− 1 + 36z_b²− 16zb兲…

+ 4zb共4zb

2+H_b²− 1兲兴 for − 1/2艋z艋zb−Hb/2,

Tac共z兲= −PrA

192

冋

− 1兲z³− 6„共H_b− 1兲²H_b+ 4z_b²共3H_b− 2兲…z²+ 8z_b共H_b

− 1兲„4z_b²+H_b共H_b− 2兲…z+共H_b− 1兲²H_b

4 共2H_b+ 1兲 + 8z_b⁴+ 3z_b²Hb共4Hb− 3兲

册

+Hb/2,

T_ac共z兲= −PrAH_b

96 共z− 0.5兲关共z− 0.5兲²„2共H_b²− 3 + 12z_b²兲z +共3Hb

2− 1 + 36z_b²+ 16zb兲…+ 4zb共4zb 2+H_b²

− 1兲兴 for zb+Hb/2艋z艋1/2.

The temperature profiles Tac calculated with conducting

boundary conditions have zero values at the walls, but also zero second derivatives, in direct connection with Eq.共6兲and the no-slip condition. Moreover, in the case of the centered beam, the first derivative of Tac is also zero at the walls, which means that the expression of T_ac in this case is also valid for adiabatic boundary conditions. This zero heat flux value at the wall is enforced by the symmetry properties of the flow 关dTac共z兲/dz must be odd兴 and heat conservation 关dTac共z兲/dz at z= −1/2 must be equal to dTac共z兲/dz at z

= 1/2兴.

Some typical velocity profiles for the flows generated in the layer are shown in Figs.2and3. We first present in Figs.

2共a兲and2共b兲the velocity profiles created by Eckart stream- ing for a centered beam. They correspond to two different beam widths: H_b= 0.3 and H_b= 0.8. Positive velocities are obtained in the center of the layer because of the acoustic radiation pressure created by the ultrasound beam, and nega- tive velocities corresponding to the return flow are obtained along the walls. The location of the velocity zeros is con- nected to the flow conservation in the section: they are out- side the acoustic beam for narrow beams and inside for large beams. These velocity profiles have curvature changes at the upper and lower boundaries of the beam, and they are sym- metric with respect to the center of the layer. The velocity profile generated through buoyancy by the horizontal tem- perature gradient关Fig.2共c兲兴is the classical cubic profile with an inflection point at the center of the cavity. When the two effects are combined, the velocity profiles are more complex and have no more symmetry关Figs.2共d兲and2共e兲兴. Examples of velocity profiles obtained by Eckart streaming with a non- centered beam are shown in Fig.3. The beam width is fixed 共Hb= 0.3兲, but four different beam positions have been cho- sen. In each case, a velocity profile generated by buoyancy is given for comparison. For the positions z_b= 0.1 关Fig. 3共b兲兴 and zb= 0.35 关Fig. 3共d兲兴, Eckart streaming generates flows which are rather opposite to those created by buoyancy共par- ticularly for zb= 0.35兲so that a kind of braking effect of the buoyant flow is induced by Eckart streaming. On the con- trary, for the positions zb= −0.1 关Fig. 3共a兲兴 and zb= −0.35 关Fig.3共c兲兴, Eckart streaming generates flows which are rather in the same direction as the buoyant flows, so that in these cases the buoyant flow is reinforced by Eckart streaming.

Typical temperature profiles T_b and T_ac are also shown in Fig.4.

III. STABILITY APPROACH

The stability of the basic flow solution 关共9兲and 共10兲兴 is investigated here in a general way by a linear analysis. The solution of the three dimensional problem is written as

共V,P,T兲=共V0,P₀,T₀兲+共v,p,␪兲,

i.e., the sum of the basic flow quantities with perturbations.

Substitution into Eqs. 共1兲–共3兲 and linearization with respect to the perturbations yields

⵱·v= 0, 共11兲

⳵^v

⳵^{t +共V0·⵱兲v+共v·⵱兲V0= −⵱p+⵱2v+ Gr}␪^ez, 共12兲

⳵␪

⳵^{t +V0·⵱}␪^+v·⵱T0= 1

Pr⵱²␪^, 共13兲 whereV₀=共U0, 0 , 0兲.

Only boundary conditions in thez-direction共forz= −1/2 andz= 1/2兲are needed because we will use periodic distur-

bances in the horizontalxandydirections. These conditions are共a兲no-slip boundary conditions:v= 0, and

共b兲conducting thermal boundary conditions:␪^{= 0.}

The linear stability study consists, for fixed values of the Prandtl number Pr and acoustic streaming parameter A, in the determination of Grc, the critical value of Gr at which the basic flow loses its stability. For the normal modes analysis, the set of Eqs.共11兲–共13兲is transformed by using the follow- ing disturbances:

共v,p,␪兲=共v,p,␪兲共z兲e^{i共hxx+hyy兲+␻t}, 共14兲 whereh_xandh_yare real wave numbers in the longitudinal,x, and transverse, y, directions, respectively, and␻⁼␻r+i␻i is a complex eigenvalue. The real part of ␻represents an am- plification rate and its imaginary part an oscillation fre- quency. These modes are elementary perturbations from which a general perturbation is obtained by superposition.

-10 -5 0 5 10 15

-0.4 -0.2 0 0.2 0.4

z U_ac

H_b

(a)

-10 -5 0 5 10 15

-0.4 -0.2 0 0.2 0.4

z U_ac

H_b

(b)

-10 -5 0 5 10 15

-0.4 -0.2 0 0.2 0.4

z U_b

(c)

-10 -5 0 5 10 15

-0.4 -0.2 0 0.2 0.4

z U₀

H_b

(d)

-10 -5 0 5 10 15

-0.4 -0.2 0 0.2 0.4

z U₀

H_b

(e)

FIG. 2. Basic velocity profiles obtained for a centered ultrasound beam: flow due to the Eckart streaming for共a兲H_b= 0.3 and 共b兲H_b

= 0.8共A= 1000兲; flow due to the temperature gradient for共c兲Gr= 500共A= 0兲; flow due to the combined effect of the temperature gradient and the Eckart streaming for共d兲H_b= 0.3 and共e兲H_b= 0.8共Gr= 500,A= 1000兲.

-10 -5 0 5 10 15

-0.4 -0.2 0 0.2 0.4

z U₀

H_b

(a) U_ac

U_b

-10 -5 0 5 10 15

-0.4 -0.2 0 0.2 0.4

z U₀

H_b

(b) U_ac

U_b

-10 -5 0 5 10 15

-0.4 -0.2 0 0.2 0.4

z U₀

H_b

(c) U_ac

U_b

-10 -5 0 5 10 15

-0.4 -0.2 0 0.2 0.4

z U₀

H_b U_ac (d)

U_b

FIG. 3. Basic velocity profiles obtained for a noncentered ultra- sound beam共H_b= 0.3,A= 1000兲; the velocity profiles due to Eckart streaming共U_ac, solid lines兲are given for different beam positions, 共a兲 P₁₋共z_b= −0.1兲, 共b兲 P₁₊共z_b= 0.1兲, 共c兲 P₂₋共z_b= −0.35兲, and 共d兲 P₂₊共z_b= 0.35兲, and compared to the velocity profile due to the tem- perature gradient共U_b, dashed lines兲 共Gr= 500兲.

-0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

-0.4 -0.2 0 0.2 0.4

z T_i

T_b T_ac

FIG. 4. Basic temperature profiles induced for Pr= 0.1 by the buoyant flow共T_b, dashed line兲 共Gr= 500兲and by the Eckart stream- ing flow共T_ac, solid line兲 共H_b= 0.3,A= 1000兲.

An eigenvalue problem is then obtained: LX=␻^{MX, where} X=共v共z兲,p共z兲,␪共z兲兲, Lis a linear operator depending on hx, hy, Pr,Hb,zb,A, and Gr, andMis a constant linear operator.

This generalized eigenvalue problem is solved with the spec- tral Tau Chebyshev method by means of a numerical proce- dure using the QZ eigenvalue solver of the NAG library关22兴.

From the thresholds Gr₀共Pr,Hb,zb,A,hx,hy兲 共values of Gr for which an eigenvalue has a real part equal to zero whereas all the other eigenvalues have negative real parts兲, the critical Grashof number Gr_c can be obtained after minimization along hxandhy. In fact, Grcwas obtained for perturbations where eitherhxorhywas equal to zero, i.e., pure longitudinal or transverse waves. It is known that in the pure thermal case, there is no smaller minimum for perturbations with bothh_x⫽0 andh_y⫽0关23兴. We have verified to our best that it is the same in our situation.

The critical Grashof numbers were determined by ex- panding the variables in thezdirection in a Chebyshev series with 80 collocation points. In fact, 20–30 collocation points were sufficient for an accurate determination of the linear stability characteristics for the pure buoyancy case 共A= 0兲, but more points were necessary when acoustic streaming is effective because of the strong gradients generated at the limits of the acoustic beam. Note that in the pure acoustic streaming situation, the critical parameter is the acoustic streaming parameter A, but the determination of its critical value is similar to what has been explained for Gr. In the following, we will first consider the stability of the pure acoustic streaming flow and then study the effect of the acoustic streaming on the stability of the laterally heated layer.

IV. STABILITY RESULTS FOR THE PURE ACOUSTIC STREAMING FLOW

The flows considered in this section are those created by Eckart streaming in a layer. Typical velocity profiles for these flows共denoted as Uac兲have been shown in Figs.2共a兲 and2共b兲共centered beam兲and Fig. 3共noncentered beam兲. In these cases, the linear stability analysis gives the critical value of the acoustic streaming parameter, Ac, above which the acoustic streaming flow becomes unstable. Results have first been obtained in the case of a centered acoustic beam for different beam widths Hb, and then, at constant beam width, for different positions of the beam across the layer.

The critical stability curve giving the evolution ofA_cas a function of H_b for a centered beam is shown in Fig. 5共a兲. These thresholds correspond to two-dimensional instabilities 共hx⫽0,hy= 0兲developing in the plane of the basic flow pro- file共plane xOz兲. The critical curve separates the stable zone 共small values ofA兲and the unstable zone共large values ofA兲. The influence of the beam width on the thresholds is found to be strong. The critical curve has a minimum, Ac= 5143, for Hb⬇0.32, which means that the acoustic streaming flow ob- tained with such a beam width is the more unstable 共the associated critical parameters areh_x

c= 4.5 and␻c= 21兲.关This value of Hb is close to, but smaller than the values which give either the maximum velocity variation 共U_max−U_min兲 共Hb= 0.388兲or the maximum shear 共Hb= 0.366兲 in the basic

velocity profile.兴 The thresholds increase for both smaller and higher values ofHb. The increase for largeHbis particu- larly strong, indicating that the streaming flows induced by large width beams共close to the height of the layer兲are par- ticularly stable. The two-dimensional instability involved in these streaming flows is hydrodynamic: the basic velocity profile is symmetric with two shear zones at the limits of the acoustic beam共see Fig.2兲which destabilize the flow. These shear zones move toward the walls when H_b is increased.

The instability is oscillatory and associated with a single complex eigenvalue共no complex conjugate eigenvalue兲. The critical angular frequency is positive, indicating a right trav- eling wave. This frequency is weak for small values of Hb

and increases with Hb 关Fig. 5共c兲兴. This increase becomes very strong beyondHb= 0.6, which leads to high oscillation frequencies for the large beam widths. The variation of the critical wave numberhx_cis shown in Fig.5共b兲.hx_cdecreases asHbis increased, which corresponds to an evolution toward larger wavelengths for the instabilities. This decrease is al- most linear for 0艋Hb艋0.6 but becomes steeper whenHbis further increased.

10³ 10⁴ 10⁵ 10⁶

0 0.2 0.4 0.6 0.8 1

H_b A_c

(a)

2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

0 0.2 0.4 0.6 0.8 1

H_b h_x

c

(b)

0 200 400 600 800 1000 1200

0 0.2 0.4 0.6 0.8 1

H_b ω_c

(c)

FIG. 5. 共a兲Critical acoustic streaming parameter A_c,共b兲wave numberh_xc, and共c兲angular frequency␻cas a function of the acous- tic beam width H_bfor an isothermal layer and a centred acoustic beam. In共a兲, the flow is unstable above the curve and stable below the curve.

We now consider that the acoustic beam has a fixed width 共Hb= 0.3兲, but that its position inside the layer is changed.

Figure 6共a兲shows the variation of the critical thresholdsAc

as a function of the position of the beam center zb. We see that for opposite positions of the beam 共zb=⫾k兲, the same thresholds A_c are obtained. This is due to the fact that the flows obtained are symmetric one of the other with respect to the center of the layer and are thus completely equivalent.

For zb= 0, the threshold is that found previously for a cen- tered beam. When the beam is moved away from the center, the thresholds A_cfirst decrease, until the positions兩z_b⬇0.2兩 beyond which they increase. These instabilities are oscilla- tory, and, as shown in Fig. 6共c兲, the critical angular fre- quency␻cevolves with兩zb兩from the positive value obtained for a centered beam toward negative values which are reached for兩zb兩⬎0.18. This change of sign corresponds to a transition from right to left traveling waves. It occurs at po- sitions of the beam which are close to those corresponding to the minimum values of the thresholds. The critical wave number of the instabilities, h_x

c, decreases monotonically

when 兩z_b兩 increases, i.e., when the beam gets closer to the walls.

Note that we have also considered the case where the perturbations are three dimensional共hy⫽0兲. In any case, the thresholds have been found larger than those obtained for h_y= 0. In particular, forh_y⫽0 andh_x= 0共i.e. when the insta- bilities develop in the transverseyOzplane兲, no finite thresh- olds have been found, which corresponds to eigenvalues with always negative real parts. The acoustic streaming flow is then linearly stable with respect to these three-dimensional perturbations withh_x= 0, this result being valid for any posi- tions of the beam.

V. STABILITY RESULTS FOR THE LATERALLY HEATED FLUID LAYER SUBJECT TO ACOUSTIC STREAMING

In this section, we will study the effect of acoustic stream- ing on the linear stability of the flow induced in a fluid layer by a horizontal temperature gradient. In the pure buoyancy situation, i.e., without acoustic streaming 关see the velocity profile in Fig. 2共c兲兴, steady two-dimensional instabilities of dynamical origin 共transverse rolls兲 prevail for very small Prandtl number values 共typically, Pr艋0.14兲, whereas oscil- latory and then steady three-dimensional instabilities共longi- tudinal rolls兲prevail when the Prandtl number is further in- creased. These instability thresholds are shown as thick lines in Fig. 7 for 0.004艋Pr艋1. We will first see how the insta- bility thresholds in this Pr range globally evolve with acous- tic streaming. We will then more precisely analyze the influ- ence of acoustic streaming on the different instabilities at fixed values of Pr.

A. Global effect of acoustic streaming on the thresholds The global effect of acoustic streaming on the thresholds of the different instabilities is presented in Fig.7for a beam width H_b= 0.8 and for two values of the acoustic streaming parameter, A= 5⫻10⁴ and 10⁵. The two-dimensional insta- bilities obtained in the domain of low Prandtl numbers are steady for A= 0, but they become oscillatory as soon as acoustic streaming is applied. For these instabilities, an in-

3500 4000 4500 5000 5500 6000

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 z_b

A_c

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 z_b

h_x

c

-50 -40 -30 -20 -10 0 10 20 30

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 z_b

ω_c

FIG. 6. 共a兲Critical acoustic streaming parameter A_c,共b兲 wave numberh_x

c, and共c兲angular frequency␻cas a function of the po- sition of the beam centerz_bfor an isothermal layer and a fixed beam width H_b= 0.3. Identical results are obtained for a decentring to- wards the top of the layer 共z_b⬎0兲 and a decentring towards the bottom共z_b⬍0兲.

0.1 1 10 100 1000

0.01 0.1 1

10-3Grc

Pr

FIG. 7. Critical Grashof number Gr_c for the three dominant instabilities as a function of the Prandtl number Pr for a centered beam with H_b= 0.8 and different values of the acoustic streaming parameter,A= 0共thick lines兲, 5⫻10⁴共lines with squares兲, and 10⁵ 共lines with circles兲. Solid curves correspond to steady thresholds and dashed lines to oscillatory thresholds.

crease of the thresholds is found for A= 5⫻10⁴. There is, however, a limitation of the acoustic streaming parameter in this case due to the destabilization of the pure acoustic streaming flow forAc= 82 641. The thresholds of the oscilla- tory three-dimensional instabilities also increase with the acoustic streaming parameterA. This increase is quite strong for 0.2艋Pr艋0.4; it decreases for smaller values of Pr and becomes negligible for Pr艋0.05. The effect of acoustic streaming is the strongest on the steady three-dimensional instabilities, but it is now a decrease of the thresholds which is found when A is increased. A monotonous effect of the acoustic streaming is thus found for the three-dimensional instabilities, i.e., an increase of the oscillatory thresholds and a decrease of the steady thresholds. This corresponds to a stabilization of the basic flow with respect to the oscillatory instabilities and a destabilization with respect to the steady instabilities.

B. Effect of acoustic streaming on the different instabilities at fixed Pr

We now focus on the precise evolution of the thresholds Gr_cas we vary the acoustic streaming parameter A and the beam width Hb at fixed values of the Prandtl number. We will calculate the thresholds of the two-dimensional instabil- ity for Pr= 0.01 and those of the three-dimensional instabili- ties for Pr= 0.1. In each case, we will first consider the situ- ation where the acoustic beam, of variable width 共0.1艋H_b 艋0.8兲, is centered, and then, for Hb= 0.3, we will consider the effect obtained when the acoustic beam is decentered.

1. Steady two-dimensional instabilities forPr= 0.01 The action of acoustic streaming on the two-dimensional instabilities is shown for a centered beam in Fig. 8共a兲 through the critical curves giving the evolution of GrcwithA for different values ofH_b. ForA= 0共no acoustic effect兲, the flow is thermally induced, and the instability is steady and appears for Grc= 8076. The evolution of GrcwithAdepends on the acoustic beam widthHb. For values ofHblower than 0.6, Grccontinuously decreases with the increase ofA, indi- cating a destabilizing influence of acoustic streaming, whereas for values ofHbgreater than 0.6, Grcfirst increases with the increase ofA, then reaches a maximum, and even- tually decreases. For large acoustic beam widths, it is then possible to find a range of acoustic streaming parameter val- ues where acoustic streaming has a stabilizing influence on the basic thermally induced flow. The extent of this param- eter range and the stabilizing effect induced increase with the increase of the beam width. In any case, the curves of Grc

eventually decrease to zero. The value Gr_c= 0 is reached for values ofAwhich, as expected, are those already obtained in the pure acoustic streaming situation 关Fig. 5共a兲兴. The insta- bility seems hydrodynamic as it evolves, with the increase of A, from an instability connected to the shear of the thermally induced flow in the center of the layer, toward an instability connected to the shear of the acoustic streaming flow at the limits of the acoustic beam.

The curves giving the evolution of the critical wavelength h_x

c with A are shown in Fig. 8共b兲. An increase of h_x

c is

observed, fromhx_c= 2.68 corresponding to the pure buoyancy situation共A= 0兲to the values ofh_x

ccorresponding to the pure acoustic streaming situation 共A=Ac兲. This increase is con- tinuous, except for H_b= 0.8共case with the strongest stabili- zation兲where a slight initial decrease is found.

Finally, as shown in Fig. 8共c兲, the critical angular fre- quency␻c共which is zero forA= 0兲strongly increases withA until a maximum is reached and then decreases down to the value corresponding to the pure acoustic streaming situation.

The maximum value reached by␻cincreases with the acous- tic beam width Hb. These curves show that the two- dimensional instabilities become oscillatory as soon as acoustic streaming is applied. Moreover, the positive values of ␻c indicate the onset of right traveling waves in these cases.

The influence of the beam position on the thresholds of the two-dimensional instabilities is shown in Fig.9共a兲. This figure displays the variation of Gr_cwith the acoustic stream- ing parameterAfor a beam of fixed widthHb= 0.3 located at

0 2 4 6 8 10 12 14

0 10 20 30 40 50 60 70 80 90 10-3 Grc

10^-3A

(a)

H_b=0.1 H_b=0.3 H_b=0.6 H_b=0.7 H_b=0.8

2.5 3 3.5 4 4.5 5

0 10 20 30 40 50 60 70 80 90 10^-3A

h_x

c

(b) H_b=0.1

H_b=0.3 H_b=0.6 H_b=0.7 H_b=0.8

-50 0 50 100 150 200 250 300

0 10 20 30 40 50 60 70 80 90 10^-3A

ωc

(c)

H_b=0.1 H_b=0.3 H_b=0.6 H_b=0.7 H_b=0.8

FIG. 8. 共a兲Critical Grashof number Gr_c,共b兲wave numberh_x

c, and 共c兲 angular frequency ␻cfor the two-dimensional instabilities as a function of the acoustic streaming parameterAfor a centered beam of different widthsH_band Pr= 0.01. The critical curves in共a兲 intersect the axis Gr= 0 at values ofAwhich are the critical values shown in Fig.5共a兲.

different positions inside the layer. The results obtained for a centered beam关position P₀共zb= 0兲兴are compared with those obtained for symmetric positions with respect to the center of the layer, P₁₋共zb= −0.1兲, P₁₊共zb= 0.1兲 and P₂₋共zb= −0.35兲, P₂₊共zb= 0.35兲. Note that all the thresholds are oscillatory, ex- cept, indeed, for A= 0. Figure 9 highlights the strong influ- ence of the beam position. For the positions P₁₋共zb= −0.1兲 and P₂₋共z_b= −0.35兲, where acoustic streaming rather rein- forces the thermally induced flow 关see Figs. 3共a兲and 3共c兲兴, the thresholds decrease monotonically withAas for the cen- tered position of the beam, but more quickly, and they reach Gr_c= 0 at the values ofA_calready obtained in the pure acous- tic streaming situation 关Fig. 6共a兲兴. For these positions, the acoustic streaming strongly destabilizes the basic flow. On the contrary, for the positions P₁₊共zb= 0.1兲 and P₂₊共zb

= 0.35兲 where acoustic streaming rather opposes the ther- mally induced flow 关see Figs.3共b兲and3共d兲兴, the thresholds

first increase with the increase ofA which indicates a stabi- lization of the basic flow by acoustic streaming. The critical curves then reach a maximum and decrease down to the val- ues ofAcfor pure acoustic streaming关the same values as for the symmetric positions, as already shown in Fig. 6共a兲兴.

These critical curves, however, have a particular shape: in- deed, they evolve until values of A larger than Ac 共up to a limit valueA_l兲so that the last portion of curve is associated with decreasing values of A. Moreover, the stable zone in these cases is the zone delimited by the critical curve and containing the origin. A specific evolution is then induced when Gr is increased for values of A larger than A_c and smaller thanA_l. The flow, first unstable for the small values of Gr, is stabilized beyond a first threshold and becomes again unstable beyond a second threshold. This behavior is depicted in Figs.10and11for the positionP₂₊of the acous- tic beam. Figure10shows the neutral curve Gr₀as a function of the wave number hx for different values of A, A= 8000, 9000, 10 000, and 10 678, the last value being close to the disappearance of the critical curve. For each value ofA, two

0 2 4 6 8 10 12 14 16

0 2 4 6 8 10 12

10-3 Grc

10^-3A

(a)

P_2- P_1- P₀ P₁₊

P₂₊

2 2.5 3 3.5 4 4.5 5

0 2 4 6 8 10 12

10^-3A h_x

c

(b)

P_2- P_1- P₀ P₁₊

P₂₊

-60 -40 -20 0 20 40 60 80 100

0 2 4 6 8 10 12

10^-3A ω_c

(c) P_2-

P_1- P₀ P₁₊

P₂₊

FIG. 9. 共a兲Critical Grashof number Gr_c,共b兲 wave numberh_x

c, and 共c兲 angular frequency␻cfor the two-dimensional instabilities as a function of the acoustic streaming parameter A for different positions of the ultrasound beam 关P₂₋共z_b= −0.35兲, P₁₋共z_b= −0.1兲, P₁₊共z_b= 0.1兲, and P₂₊共z_b= 0.35兲兴. The beam width is equal toH_b

= 0.3 and Pr= 0.01. The stable zone in共a兲is the zone delimited by a critical curve and containing the origin. The critical curves in 共a兲 intersect the axis Gr= 0 at values ofAwhich are the critical values shown in Fig.6共a兲共same values for oppositez_b兲.

0 2 4 6 8 10 12 14 16 18

3 3.5 4 4.5 5 5.5

10-3Gr

h_x

FIG. 10. Neutral curves Gr₀for the two-dimensional instabilities as a function of the wavenumberh_x for the beam position P₂₊共z_b

= 0.35兲 and different values of the acoustic streaming parameter, A= 8000, 9000, 10 000, and 10 678 共following the arrows兲. The beam width is equal toH_b= 0.3 and Pr= 0.01. For each value ofA, two neutral curves are obtained. The black dots indicate the minima or maxima of these neutral curves, i.e., the critical thresholds Gr_c shown in Fig.9共a兲.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

11 12 13 14 15 16

ωr

10^-3Gr

FIG. 11. Real part of the dominant eigenvalue,␻r, as a function of the Grashof number Gr for the beam position P₂₊共z_b= 0.35兲, A

= 10 500, andh= 4.48. The beam width is equal toH_b= 0.3 and Pr

= 0.01. The two Gr values associated to zero ␻rbelong to neutral curves as those shown in Fig.10, and the negative values of␻rin between indicate the stability of the zone between the neutral curves.