On the Boussinesq approximation in the problems of convection induced by high-frequency vibration Текст научной статьи по специальности «Физика»

УДК 534-14, 532.529.2

On the Boussinesq Approximation in the Problems of Convection Induced by High-Frequency Vibration

Ilya I. Ryzhkov*

Institute of Computational Modelling SB RAS, Krasnoyarsk, 660036

Russia

Yuri A. Gaponenko

Institute of Computational Modelling SB RAS, Krasnoyarsk, 660036 Russia

Microgravity Research Center, Universite Libre de Bruxelles, CP 165/62, av. F.D.Roosevelt 50, B-1050 Brussels,

Belgium

Received 10.05.2010, received in revised form 10.06.2010, accepted 20.08.2010 The applicability of Boussinesq approximation to the problems of thermovibrational convection in closed volumes is analyzed. The limit of high frequency and small amplitude is considered on the basis of averaging approach. The magnitudes of oscillatory and averaged flow fields are estimated. It is found that the dependence of the Reynolds number for averaged motion on the ratio of Gershuni and Prandtl numbers obeys linear and square root laws for small and large Reynolds numbers, respectively. It provides new essential information about the intensity of averaged flows in a wide range of vibration stimuli. Taking into account the obtained estimations, the basic assumptions of Boussinesq approach are applied to the momentum, continuity, and energy equations for a compressible, viscous heat-conducting fluid. The contribution of viscous energy dissipation and pressure work to the energy balance is also taken into account. The order of magnitude analysis provides a number of dimensionless parameters, the smallness of which guarantees the validity of Boussinesq approximation

Keywords: Boussinesq approximation, vibrational convection, averaged motion.

Introduction

The application of vibrations to a fluid system with density gradient can cause relative flows inside the fluid. If this gradient results from thermal or compositional variations, such flows are known as thermovibrational or solutovibrational convection, respectively. The case of small amplitude and high frequency vibration (when the period is much smaller than the characteristic viscous and heat (mass) diffusion times) is of special interest. In this case, the flow field can be represented as a superposition of ’quick’ part, which oscillates with the frequency of vibration, and ’slow’ time-averaged part, which describes the non-linear response of the fluid to a periodic excitation [1]. In the theory of convection, the averaging approach was first used in [2]. The mathematical jusitfication of this approach for convection problems was given in [3].

The study of vibrational impact on fluids has fundamental and applied importance. Vibrational convection provides a mechanism of heat and mass transfer due to the existence of averaged flows. Such flows show some similarity with gravity-induced convection and might serve as a way to control and operate fluids in space [4]. High-frequency oscillations (g-jitter) onboard

* rii@icm.krasn.ru © Siberian Federal University. All rights reserved

microgravity platforms can disturb the experiments that require purely diffusive heat and mass transfer: crystal growth, measurement of transport coefficients, etc. [5].

There have been extensive theoretical studies of thermovibrational convection in weightlessness and ground conditions. The fundamental treatise [1] comprises a systematic study of convective flows induced by high and finite frequency vibrations in closed and infinite cavities. Thermovibrational convection in square, rectangular, and cubic cavities was widely investigated providing a variety of averaged flow structures and bifurcation scenarios [6-8]. Influence of vibration on double diffusive convection with the Soret effect was analyzed in [9]. The experimental studies of vibrational impact on fluids were performed in a number of works. The considered configurations include vertical and horizontal layers [10], Hele-Shaw cell [11], etc. The influence of vibration on the propagation of heat from a point-like source in microgravity conditions was investigated in [12]. Recent parabolic flight experiments [13-15] provided one of the first quantitative observation of averaged flows and related heat transfer in microgravity environment.

The most part of existing theoretical and numerical studies of vibrational convection were performed in the frame of Boussinesq approximation. The fluid was considered as incompressible and the density variations were taken into account only in the vibrational force term. It should be noted that the applicability of this approach in gravitational convection problems was extensively studied in a number of works [16-19]. Concerning vibrational convection problems, the justification of Boussinesq approach was discussed in [1]. However, in that paper, the contribution of pressure work and viscous energy dissipation to the energy balance was not taken into account. This contribution can become significant at high frequencies. In addition, when the frequency of vibration is increased, the pressure gradients induced by the vibrational force can become so large that the fluid cannot be considered incompressible anymore. In this respect, a complete set of criteria, which describe the validity domain of Boussinesq approximation in vibrational convection problems, is still necessary.

In this paper, we analyze the applicability of Boussinesq approximation to the problems of thermovibrational convection in closed volumes. The study is focused on the limit of high frequency and small amplitude. It is assumed that no other external forces except vibration are present. In the first part, a rigorous derivation of averaged equations is given. Note that the averaging procedure is not straightforward; furthermore, it is not described in sufficient detail in the related literature [1,2]. In the second part, the criteria for the validity of Boussinesq approach are derived. The basic assumptions of this approach are applied to the set of momentum, continuity, and energy equations for a compressible, viscous heat-conducting fluid, where the contribution of viscous energy dissipation and pressure work to the energy balance is taken into account. Two cases, which correspond to averaged flows with small and large Reynolds numbers and provide different estimations of averaged velocity, are considered. The obtained estimations are confirmed by numerical modelling. The order of magnitude analysis provides a number of dimensionless parameters, the smallness of which guarantees the validity of Boussinesq approximation. The main results are summarized in Conclusion.

1. Governing Equations and Boussinesq Criteria

Consider compressible viscous heat-conducting fluid in a closed cavity. It is assumed that the cavity performs translational harmonic oscillations with the angular frequency w and amplitude A in the direction of vector e. There are no other external forces (such as, for example, gravity) imposed on the fluid. The general equations describing the fluid in the coordinate system associated with the cavity have the form [1,19,20]:

p(dtV + (V • V) V) = -VP + ^V2V + (A + Q V(V • V) + pAw2 cos(wt) e, (1)

дtp + V • Vp + pV • V = 0,

pep (^T + V • V T ) - вт (To + T ) (^P + V •VP ) =

(2)

(3)

Here x = (жі, ж2,ж3) and V = (V1; V2, V3) are the coordinate and velocity vectors, T and P are the deviations of full temperature and pressure from the mean values To and P0, respectively, p is the fluid density, p is the dynamic viscosity, A is the second viscosity, cp is the specific heat capacity, к is the thermal conductivity, вт is the thermal expansion coefficient, and Sj is the Kronecker delta. Note that the governing equations are written for the case of constant physical properties p, A, к, cp for simplicity (this requirement will be introduced below).

Let us now formulate the basic assumptions of Boussinesq approximation.

1. The deviations of temperature and pressure from the mean values are small, i.e.

2. The density variation with pressure is much smaller than that with temperature, and the latter is also small:

3. The variation of density with T and P is negligible everywhere except the external force term in Eq. (1), where the density is written as

4. The contribution of viscous energy dissipation and pressure variations to the energy balance (Eq. (3)) is negligible.

5. The variations of temperature and pressure are sufficiently small to assume the constancy of physical properties p, A, k, cp.

Under these assumptions, equations (1)-(3) are reduced to the system describing the motion of incompressible fluid under vibration:

definitions will be given below). Then the method of averaging can be effectively applied for studying vibrational convective flows [1]. Using this method, we will estimate the magnitudes of velocity, pressure, and temperature fields and derive the criteria, under which assumptions 1-5 are satisfied.

|T/To| « 1, IP/Pol « 1.

(4)

Then the equation of state can be written as

P — Po(1 — 3t T + 3p P), where 3T — — p-1dp/dT and 3P — p-1dp/dT (the latter is the isothermal compressibility).

IßpPI « !втt I « 1.

(5)

p = po(1 — вт T ).

(6)

дtV + (V • V) V = —p-1VP + vV2 V + (1 — втT)Aw2 cos(wi) e, ^T + V • V T = xV2 T,

V • V = 0,

(7)

(8) (9)

where v — p/p0 is the kinematic viscosity and x — K/p0cp is the thermal diffusivity.

In this work, we consider vibrations with high frequency and small amplitude (the exact

2. Averaging Approach in Vibrational Convection Problems

2.1. Derivation of Averaged Equations

In this section, we provide a detailed derivation of averaged equations for vibrational convective flows. The description of this procedure in the existing literature [1,2] lacks a number of important details, which will be pointed out and extensively discussed below. We believe that this part will be useful for those readers, who would like to acquaint themselves with averaging approach in convection problems.

Let us denote the period of vibration by t — 2n/w. In the averaging method, each field is decomposed into "slow" time-averaged part (with characteristic time much larger than t) and "fast" oscillatory part (with characteristic time t):

V — V + V', T — T + T', P — P + P'. (10)

Here P is the modified pressure, which is given by

P — P — p0Aw2 cos(wt) e • x. (11)

For a given function of time and space coordinates f (t, x), the averaged and oscillatory components are defined by

i+r/2

f (t, x) — 1 J f (t', x) dr', (12)

t—T/2

f'(t, x) — f (t, x) — f(t, x). (13)

Further, a bar over the quantity will indicate the averaging operator according to (12). If f does not depend on time, then the averaged motion is called stationary, while the full motion is called quasi-stationary. It follows from (12) that

dtf — dtf, Vf — Vf. (14)

We assume that the variation of an averaged quantity with time during the vibration period t is negligible, so the second averaging coincides with the original averaged quantity:

f — f (15)

It follows from (13) and (15) that the oscillatory component has zero average:

f — 0. (16)

We also suppose that

fh — fh, (17)

where h(t, x) is a "fast" oscillatory function of t (with characteristic time t). For quasi-stationary

motion, relation (17) follows from (12). Otherwise, it shoud be postulated as an additional

property of the averaging operator [21].

Let us now apply definitions (12) and (13) to the velocity, pressure, and temperature fields taking into account (14)-(17). Substituting representation (10) into the governing equations

(7)-(9) gives

dtV + dt V' + (V • V) V + (V • V) V' + (V' • V) V + (V' • V) V' —

— —P—1VP — p—1VP' + vV2V + vV2 V' — 3T(T + T')Aw2 cos(wt) e, (18)

dtT + dtT' + V • VT + V • VT' + V' • VT + V' • VT' — xV2T + xV2T', (19)

V • V + V • V' — 0. (20)

Applying the averaging operator to equations (18)-( 19), we obtain the equations for averaged fields

dtV + (V • V) V + (V' • V) V' = —p-1 VP + vV2V - ßTT'Aw2 cos(wt) e, (21)

dtT + V • VT + V' • VT' — xV2T, (22)

V • V — 0. (23)

One can see that this system differs from the classical Navier-Stokes and heat trasnfer equations

by the terms containing averaged products of oscillatory components. Let us now subtract

the averaged equations (21)-(23) from the full equations (18)-(20). It leads to the system for oscillatory components of motion:

dtV' + (V • V) V' + (V' • V) V + (V' • V) V' - (V' • V) V' =

= —p-1VP/ + vV2 V' — ßT(T + T')Aw2 cos(wt) e + ßTT'Aw2 cos(wt) e, (24)

dtT' + V • VT' + V' • VT + V' • VT' — V' • VT' — xV2T', (25)

V • V' — 0. (26)

Note that in the existing literature [1,2], the equations for oscillatory components are obtained by selecting "fast" terms from equations (18)-(19) and equating the result to zero. The obtained system does not contain averaged terms (such as (V' • V) V'), which are present in equations (24)-(26). This approach is incorrect from mathematical point of view.

In what follows, we will need the scales of time, length, and averaged temperature, which are naturally determined by the vibration period t, typical size of the system L, and the applied temperature difference AT, respectively. Concerning the scale of averaged velocity, we note that it must depend on the density inhomogeniety caused by the applied temperature difference (i.e. 3TAT), the amplitude and frequency of vibration, and viscous properties of the fluid. Here we assume that |V| does not exceed v/L by the order of magnitude. This choice will be discussed in details in Section 4.1. The scales of oscillatory components V', P', T' are not known in advance and will be determined later. Note that in the existing derivations [1,2], the authors implicitly assume that |V| ~ v/L without any justification of such choice.

Equations (24)-(26) can be simplified by introducing several assumptions.

1. The period of vibration is much smaller than the reference viscous and thermal times:

t < min(L2/v, L2/x). (27)

It allows us to neglect the terms (V • V) V', (V' • V) V, vV2V' in comparison with dtV' in

equation (24) and the terms V • VT' and xV2T' with respect to dtT' in equation (25).

2. The displacement of oscillating fluid particles is much smaller than the characteristic length scale:

t|V'K L. (28)

It follows that the convective terms (V' • V) V' and V' • VT' are negligible in comparison with dt V' and dtT', respectively. The same is true for the average of the above products, as the latter

cannot exceed the original non-averaged terms by magnitude. Note that in the literature [1], the present assumption is introduced in the form of its consequence (41), which hides the real physical meaning expressed by (28).

3. The oscillations of temperature field are much smaller than the applied temperature difference: |T'| C AT, so the last two terms containing T' can be neglected in equation (24). Under the above assumptions, equations for the oscillatory fields become

dtV ' — —p—1VP' — 3TTAw2 cos(wt)e, (29)

dtT ' — —V ' • VT, (30)

V • V ' — 0. (31)

To solve equation (29), the vector Te is decomposed into the solenoidal W and irrotational V$ parts:

T e — W + V$, (32)

where

V • W — 0, W • n|r — 0, (33)

Here n is the unit normal vector to the impermeable boundary r of the domain with fluid. The above decomposition exists for a wide class of vector fields Te [3]. Substituting (32) into (29), we find

dtV ' + 3TAw2 cos(wt)W — —p—1VP' — 3TAw2 cos(wt)V$. (34)

The right-hand side of this equation represents irrotational vector field, thus the left-hand side

has the same property. Then it can be represented as a gradient of some function

dt V ' + 3T Aw2 cos(wt)W — W. (35)

It follows from (31) and (33) that V2^ — 0. To proceed further, one needs to specify boundary condition for the oscillatory velocity component. Note that the viscous force driving the oscillatory flow has been neglected when deriving equation (29), see Assumption 1. It means that the existence of Stokes boundary layer for the oscillatory flow near the rigid impermeable boundary is not taken into account. So, the non-permeability condition rather than the no-slip one should be imposed on the oscillatory velocity component:

V ' • n|r — 0. (36)

Taking into account (33) and (35), we arrive to the problem V2^ — 0, V^ • n|r — 0, from which it follows that V^ — 0 [22]. Then the left and right-hand sides of (34) are equal to zero, so

dt V' — —3TAw2 cos(wt)W, p—1VP ' — —3TAw2 cos(wt)V$. (37)

Note that the separation of (34) into the above equations becomes possible only after the boundary conditions on W and V are specified by (33) and (36), respectively. This logical order is not followed in the derivation presented in [1], where the solenoidal and irrotational parts of (34) are separately equalized to zero with any further explanation.

Let us now integrate the first equation in (37) from t0 to t, where t0 is fixed and |t —10| < t. Taking into account that W practically does not change with time during the vibration period t, we obtain V'(t) — V'(t0) — —3TAwW(sin(wt) — sin(wt0)). In this relation, one should put

V'(t0) — —3TAwWsin(wt0) in order for V' to have a zero average, see (16). The obtained expression for V is substituted into (30), and the resulting equation is integrated similarly. Further, P is found by integrating the last equation in (37) over the space coordinates. Note that

P is determined accurate to an arbitrary function of time with zero average. This ambiguity can be removed by specifying the full pressure at some point of the domain. Finally, the expressions for the oscillatory components are written as

V ' — —^TAw sin(wt)W, T' — —^TA cos(wt)W • VT, (38)

P ' — —p0^TAw2 cos(wt) $. (39)

It follows from (32) that |W| — |V$| — AT, which allows us to determine the characteristic scales of oscillatory fields:

|V'| - ^TATAw, |T'| - ^TAT2A/L, |P'| - p0^TATAw2L. (40)

Then the criteria introduced in Assumptions 2 (see equation (28)) and 3 become

A « sAt <41>

It provides a limitation on the amplitude range.

The final equations for averaged fields are obtained by substituting expressions (38) and (39) into (21)-(23) and averaging the related terms:

dtV + (V • V)V — —p—1VP + vV2V + (^TAw^ ((Te — V$) • V) V$, (42)

dtT + (V • V)T — xV2T, (43)

V • V — 0, (44)

V2$ — VT • e — 0. (45)

Here the last equation follows from (32). The latter relation was also taken into account when

deriving equation (42).

2.2. Thermovibrational Convection in a Square Cavity

In this paper, we perform numerical simulations of thermovibrational convection, which are complementary to the theoretical results. This study is required for validating the theoretical estimations of velocity, temperature, and pressure magnitudes. These estimations will be used for deriving the criteria for the validity of Boussinesq approximation.

Let us consider a square cavity with rigid impermeable walls of the size L (Fig. 1). The top and bottom walls are kept at constant temperatures Thot and Tcold, respectively, providing the temperature difference AT — Thot — Tcold. On the lateral walls, the linear temperature profile is imposed. The cavity performs translational harmonic oscillations with the frequency w and amplitude A in the X direction, which is perpendicular to the temperature gradient. On the boundaries of the cavity, the no-slip condition for the average velocity is imposed. The boundary condition for function $ is derived from (32) and (33):

(V$ — Te) • n|r — 0.

Now let us introduce the dimensionless variables

V T _ P $

v — v/L " — AT P — P0v2/L2, ^ — ATL.

Fig. 1. Square cavity and coordinate system

The scales of length and time are taken as L and L2/v, respectively. The averaged equations will be solved in terms of stream function ^ and vorticity Z, which are introduced by the formulas

_ ( ) . '°0 °0 v = (u-w> = ( ÔZ-— ox

öw öu öx öz ’

The governing equations (42)-(45) are rewritten in the form

oc + OÍOÍ — OÉOZ = V2 Z + Gs

öt öz öx öx öz Pr

öв + ö0 дв 00 öв 1 v2в

öt öz öx öx öz Pr ’

V i = TT“ ’ öx

0в ö2i дв ö2i \ öz öx2 öx öxöz /

v20 = — Z,

V2p = 2 0^0^ — 2f + G

öx2 öz2 \ öxöz / Pr

(^)2—^ ñr)2

\ öx2 / \ öxöy у

овoV o2i o2в fë_ o2в Oi

+ öx öx2 + öz öxöz + öx2 у öx J öxöz öz The system includes the dimensionless Prandtl and Gershuni numbers:

(Awвт ДTL)2

^ V

Pr = —, X

Gs =

2vx

(46)

(47)

(48)

(49)

+

(50)

The Gershuni number (also known as the vibrational Rayleigh number) can be regarded as the ratio of mean vibrational buoyancy force to the product of momentum and thermal diffusivities. It describes the vibrational mechanism of convection represented by the mean flow.

The boundary conditions of the problem are written as

x = 0,1: 0 = д4 = 0, в = z, ^ = в,

ox ox

z = 0,1: 0 = д0 =0, 0 = 0,1, ^ = 0,

z z

öp ö30 Gs ö0 öi

x x2 z Pr z z

öp

öz

ö30 öxöz 2

while the initial conditions have the form

— ¿ta — eta

t — 0: ^ — Z — p — 0, 0 — z, -^ — 0, TT-—0. (52)

dx dz

We assume that the non-uniform thermal field (linear temperature profile) is established before vibration is applied. The last two conditions follow from (32), where the amplitude of oscillatory velocity component W — 0 at the initial moment of time. It should be noted that in the considered problem, the pressure is determined up to some arbitrary constant. To specify this constant, the minimal pressure in the volume is set to zero (it is enough for our purposes as we are interested in the pressure variation across the volume).

The problem (46)-(52) is solved by a finite-difference method using a regular equally spaced mesh [101 x 101]. The dimensionless time step was 10—5. The time derivatives are forward differenced and the convective and diffusive terms are central approximated. The Poisson equation (50) for pressure p is solved by introducing an artificial iterative term, which is analogous to the time-derivative one. ADI method is used to solve the time-dependent problem for the stream function, vorticity, temperature, pressure, and the amplitude of fast pressure <^. More details about the numerical procedure can be found in [23].

3. Boussinesq Approximation for Vibrational Convection Problems

3.1. Estimation of Averaged Flow Fields

To derive the criteria, under which Boussinesq assumptions 1-5 (see Section 1) are satisfied, we need to estimate the magnitudes of velocity, temperature, and pressure. The characteristic scales of oscillatory fields are given by (40). The time derivatives of oscillatory components are estimated by differentiating (38) and (39) with respect to "fast" time t (note that the functions T, W, $ do not depend on this time).

The averaged fields should be estimated from equations (42)-(45). Let us first consider the case of fully developed stationary mean flow. Then the averaged temperature has the order of the applied temperature difference: |T| — AT. To estimate the averaged velocity, we note that convective or viscous term (or both) must be of the same order of magnitude as the mean vibrational force term in equation (42). In addition, these two terms cannot become large in comparison with the external force term as the latter is the source of motion.

Let us first suppose that

(ßr((Te - V$) • V)V$

|v V2V|

It gives the following magnitude of averaged velocity:

(53)

(MTAwliL. (54)

v

Comparing the magnitude of convective and viscous terms, we find

J<V-PV! — EL . Re — ^ (55)

|v V2V| v Pr v 7

where Re is the Reynolds number. This estimation is valid for small Reynolds numbers (Re < 1). When Re is large, relation (55) predicts that viscous term is much smaller than the convective

one. It is in contradiction with the original assumption that viscous term is comparable with the mean vibrational force term.

Now consider the case when the convective term is comparable with the mean vibrational force term:

(#rAw)2 , —

l(V -V)V|

It provides the following estimation

2

-((Te -V$) -V)V$

(56)

|V| - AT Aw. (57)

Comparison of convective and viscous terms gives

|(V -yJV1 s Re - (58)

|vV2V| v VPr/

Arguing in the same way as above, we conclude that the obtained estimation of velocity is valid for large Reynolds numbers (Re > 1).

To validate estimations (54) and (57), numerical calculations of steady vibrational mean flows in a square cavity have been performed. The calculations have shown that one should introduce a proportionality factor between the Reynolds number and the ratio Gs/ Pr in relations (55) and (58) in order to get a correct estimation of averaged velocity. For Re < 1, we have found that

Re - -,2g, |V|- MrAIAwÜL, (59)

while for Re ^ 1

f Gs \ 1/2 ____

Re - Y ( Pr ) , |V| - AT Aw. (60)

One can see that for large Reynolds numbers, the proportionality factor y represents the ratio of averaged and oscillatory velocity components: y ' - |V| / |V'|, see also (40). The averaged velocity is normally smaller than the oscillatory one, so one can expect that 0 < y < 1 in the general case.

The obtained dependence of the Reynolds number on the ratio Gs/ Pr is presented in Fig. 2a for Pr = 25 (the case of liquid). The results of numerical calculations are shown by points, while the linear and square laws are represented by dashed and solid lines, respectively. The values of Reynolds number obtained from numerical simulation correspond to maximum averaged velocity in the cavity. The proportionality factor is y = 0.0195. The results provide a good agreement between numerical calculations and suggested linear and square root laws. It validates estimations (59) and (60).

According to Fig. 2a, the order of the Reynolds number does not exceed unity in a wide range of Gershuni numbers (0 < Gs < 200 x 103). It means that the order of averaged velocity does not exceed v/L (this assumption was introduced when deriving the equations for oscillatory fields in Section 3.1). The average velocity increases with increasing the Gershuni number. However, for very large values of Gs, the steady state motion transforms into non-damped oscillations and eventually chaotic regime [1].

To estimate the averaged pressure, we note that the pressure gradient term should be comparable with the vibrational force term in equation (42):

(ßTAw)2 ((Te - V$) • V)V$

k-1vp | ^

Gs

2

It gives |P| — po(/3TATAw)2, which is equivalent to |p| — e — in dimensionless form. Here we have introduced a proportionality factor e, which can be determined from numerical simulation.

Fig. 2. The dependence of the Reynolds number (a) and dimensionless pressure (b) on the ratio Gs/ Pr. The results of numerical calculations are shown by points, while the linear and square laws are represented by dashed and solid lines, respectively. The proportionality factors are Y — 0.0195 and e — 0.0792

Figure 2b shows the dependence of maximum pressure in the cavity on the ratio Gs/ Pr. The proportionality factor is e — 0.0792. It confirms the suggested linear law. Note that e may vary with Prandtl number. In what follows, it is assumed that 0 < e < 1, which is expected to hold in a wide range of Prandtl numbers.

Let us now make a note on time-dependent averaged flows. In fact, this work was inspired by the need to analyze the applicability of Boussinesq approximation to describing the transient process in a short-duration microgravity experiment [13-15]. If non-uniform thermal field is established before vibrations are applied, the development of convective flow occurs during 1-2 viscous times. The calculations show that at two viscous times from the start of vibration, the dependence of Reynolds number on Gs/ Pr is similar to that of Fig. 2 with slightly smaller values of the parameter Y. Therefore, one can use previously derived estimations for velocity and pressure. To estimate the time derivatives of averaged components, we note that the transfer of heat and mass in the considered transient period is mainly due to averaged motion, so the characteristic time can be estimated as length scale divided by the magnitude of averaged velocity, i.e. L/|V|.

3.2. Validity of Boussinesq Approximation

Let us now introduce the obtained estimations into the basic assumptions of Boussinesq approximation (see Section 1) taking into account (10) and (11). We will also employ criteria (27) and (41), which ensure the applicability of averaging approach. The latter can be rewritten

as

L2w L2w

Assumption 1 expressed by Eq. (4) leads to

AT

~T~ < ^

To

ßT ATA

L

p0Aw2L

Po

(61)

(62)

The first inequality states that the applied temperature difference AT must be small in comparison with the mean temperature T0. The second inequality requires that the pressure variations

caused by vibrational force with acceleration Aw2 cos(wt) must be small in comparison with the mean pressure P0.

Note that here and below we present only independent criteria and skip the dependent ones (including those obtained by multiplying the existing criteria by y < 1). For example, for the oscillatory temperature component, it follows from (4) that

Pt ATA AT c _

L To C ’

which is satisfied once the last inequality in (61) and the first inequality in (62) hold.

In the averaging approach, the full governing equations are separated into the equations for averaged and oscillatory fields. Therefore, we require that Assumption 2 expressed by Eq.

(5) must be satisfied for averaged and oscillatory components separately (including their time derivatives). It gives the following criteria:

/ l \ 2

pTAT < 1 ppP0 ^< 1 (63)

The first inequality states that the density change induced by the applied temperature difference is much smaller than the mean density p0, see also equation (6). The second inequality requires that the density change due to pressure variations caused by the vibrational force must be smaller than the density change due to applied temperature difference. It should be noted that this inequality is related to the requirement that the acoustic wavelength tc (where c is the speed of sound) must be greater than the characteristic length scale L. Indeed, taking into account that c2 — dp/dp — (pPp0)-1, we can rewrite the above requirement as

PpP0(wL)2 C 1,

This requirement is weaker than the last one in (63) as pTAT C 1.

The conditions, under which Assumptions 3 and 4 of Section 1 are satisfied, are derived in two steps. At the first step, we substitute representation (10) into system (1)-(3). The otained equations are averaged over the period and the result is subtracted from the original (nonaveraged) system. It provides equations for oscillatory fields. After that, we derive the criteria, under which these equations are reduced to (29)-(31). At the second step, we substitute (38), (39) into the averaged equations and calculate the related terms. Finally, the criteria, under which the resulting equations are reduced to (42)-(45), are derived.

One can consider two cases depending on whether the Reynolds number is less or greater than unity (in practice, one should check the value of y2 Gs/ Pr).

1. Flows with small Reynolds number. This case is characterized by relations (59), so we assume that

Re — pA^AwL)2 < 1 (64)

Substituting (6) into the continuity equation (2) and deriving the equation for oscillatory fields, we find

-ßT (dtT' + V •VT' + V' •VT + V' • VT' - V' • VT' + T' V • V) + (65)

+ (1 — Ptt — Ptt ') V • V' + PtT'V • V' — 0.

Note that the terms V ' • VT' and pTT'V • V ' enter into this equation together with their averaged counterparts. One should take into account that the latter cannot exceed the former by absolute value when performing the order of magnitude analysis. In what follows, all such

cases are treated in the same way. Equation (65) is reduced to (31) by requiring that all other terms are negligible in comparison with V • V'.

The energy balance (3) provides the following relation for oscillatory fields:

CpPo(1 - ßrT - ßTT')(dtT + V • VT' + V' • VT + V ' • VT') - cppo(1 - ßTT)V ' • VT'-

-cppoßTT '(ôtT + V • VT) + CppoßT (T 'dtT ' + T 'V • VT ' + T'V ' • VT + T'V ' • VT ') -

-ßT (T0 + T + T ') (dtP ' - poAw3 sin(wt) e • x + V • VP ' + V ' •VP + V ' • VP '+

+(V + V ') • e poAw2 cos(wt)) + ßT (To + T ) ( V ' • VP' + V' • e poAw2 cos(wt)) -

-ßTT ' (dtP + V • P) + ßT (T 'dtP' - T 'poAw3 sin(wt) e • x+ (66)

+T'V • VP ' + T'V' • VP + T'V ' • VP ' + T'(V + V') • e poAw2 cos(wt)) +

kV2 T + ^ V

2 A=y

-j=l

3 r/ dV.' dVj' \ / dVi dV , dV- dVj 4 r ,

—L +------------ 2-- + 2—j +------------------------ +-j - - &,• V • V -

dx, dx. y \ dx, dx. dx, dx. 3 j

_ ( + dVj V

\ dxj dxj y

where we have taken into account that V- V' = 0. This equation is reduced to (30) by comparing all terms with V ' • VT. This term is responsible for convective transport of heat by the oscillatory velocity component. We require that all other terms except dtT' are negligible in comparison with it.

Finally, the momentum equation for the oscillatory components has the form

Po(1 - ßTT - ßTT')(ôtV' + (V • V) V ' + (V' • V) V + (V ' • V) V ') - po(1 - ßTT)(V ' • V)V'-

-poßTT'(dtV + (V • V)V) + PoßT (T'ötV' + T'(V • V)V' + (T'V ' • V)V + T'(V ' • V)V')

— —VP' + pV2 V ' — p0pT(T + T')Aw2 cos(wt) e + p0pTT'Aw2 cos(wt)e. (67)

It is reduced to (29) by comparing all terms with the vibrational force term p0pTT Aw2 cos(wt) e and using the criteria of averaging approach (61). Consideration of equations for oscillatory fields (65)-(67) provides an additional criteria

^ <1. <68)

It is derived by assuming that the term pTT0p0Aw3 sin(wt) e • x is small in comparison with

V ' • VT in equation (66).

One can show that inequality (68) can be derived from the previously obtained criteria. Consider the following thermodynamic relation [24]

dV\ — (dV\ T0 (dV\2

dP / s V dP / t cp V dT / p

where V — p-1 is the specific volume and S is the specific enthropy. Let us rewrite it in the form

Tq — Ppp0 — (P0 V (dV

ßT VßW VdP/S

c

p

The left-hand side of this relation is negative since (dV/dP)S < 0 [24]. Combining this result with the second criterium in (63), one can see that if the latter is satisfied, then (68) also holds.

Let us now substitute decomposition (10) with the expressions for oscillatory components (38) and (39) into the governing equations (1)-(3) and average them over the period. The averaged continuity equation is given by

—Pt(dtT + V • VT) + (1 — pTT) V • V — 0.

We require that all terms except V • V are negligible here, so the continuity equation is reduced to (44). Taking it into account, the averaged energy equations is written as

CpPo(1 — PtT) (dtT + V • VT) — Pt (To + T) (dtP + V • P) — (69)

((Te — V$) • VT) V • (cpPtV((Te — V$) • VT) + Ptw2V$ — w2e) —

- V2T +1 ^ /dV- + dV,V + MßTAw)2 A /dw- + dW x 2

2 I dx, dxW 4 I dx, dx

where W = (W1; W2, W3), see (32). According to Assumption 4 in Section 1, we require that the contribution of viscous energy dissipation and pressure variations to the energy balance is negligible (as well as the contribution of the last three terms in the left-hand side of Eq. (69)). These terms are compared with the term V • VT, which describes convective heat transfer. As a result, the energy equation is reduced to (44). It should be noted that the last term

' qv! , dV,' '

in (69) results from time averaging of the term ^ ^ (ax' + axO in the oscillatory part

ij = 1 '

of governing equations (see (66)). Therefore, the viscous energy dissipation due to oscillatory motion contributes to the time-averaged energy balance. The requirement that this contribution is negligible in comparison with convective heat transfer described by the term V • VT provides an additional criteria

1 /pT ATA\2 T0(wL)2 AT , N

Re x w-" ^ 1 (70)

where the Reynolds number is determined by (64). Note that due to (61), (62), (68), the product of the last three factors in the left-hand side of (70) has the order of 10-4 or lower. It follows that viscous dissipation can be important only for mean flows with small Reynolds number (Re < 10-3). The above criterium can be rewritten as

,,2

< 1. (71)

Y2L2 Cp AT

Note that if we assume that a stronger inequality than (68)

To (wL)2

Y 2cp AT2

is satisfied, then criterium (71) will hold automatically. Indeed, the left-hand side of (71) is obtained by multiplying that of (72) by v2/L4w2 and AT/T0, which are small due to (61) and (62), respectively. It should be noted that in contrast to (68), inequality (72) is not automatically satisfied once the second criterium in (63) holds.

Finally, the averaged momentum equation is written as

Po (1 — PtT)(dtV + (V • V)V) —

Table 1. Physical parameters and characteristics of the system (isopropanol in a cubic cell)

To Po Po ßT ßp v X L AT

K Pa k m CO 1—1 o 1 CO K 1 —0-9 Pa-1 —0-6 m2/s —0-7 m2/s J/kgK m K

313 101325 769 1.095 1.02 1.730 0.623 2735 0.005 20

= -VP + pV2V + ((te - v$) • V) (V$ + AtT(Te - V$)).

It is reduced to (42) on the basis of already derived criteria.

2. Flows with large Reynolds number. Such flows are described by relations (60), so we suppose that

Re „ yAtAJA^L ^ i. (73)

This case is treated similar and provides the same criteria. However, relation (70) with the Reynolds number given by (73) is satisfied automatically as Re > 1.

3.3. Example: Isopropanol in a Cubic Cavity

In this section, we consider an example of a real physical system. Experimental studies of thermovibrational convection in microgravity conditions have been recently performed in [14]. In this work, vibrations were applied to a cubic cell filled with isopropanol at atmospheric pressure. The physical parameters and characteristics of this system are given in Table 1. In the experiments, the temperature difference was chosen as AT = 20 K. It ensures that the first inequalities in (62) and (63) as well as inequality (71) are satisfied.

Applying criteria (61)-(63), (71) to the present system, we obtain the following restrictions on the frequency and amplitude of vibration: 0.011 C f C 787 Hz, A C 0.228 m, A(2nf )2 C 26352 m2/s. Here the lower and upper frequency limits follow from the first inequality in (61) and second inequality in (63), respectively. The amplitude limitation results from the last inequality in (61), while the vibrational acceleration is limited by the second inequality in (62). The above requirements ensure the applicability of Boussinesq approximation and averaging approach to the description of the considered physical system.

Conclusion

In this work, we have analyzed the applicability of Boussinesq approximation to the problems of thermovibrational convection in closed volumes. The study is focused on the limit of high frequency and small amplitude, where the averaging technique can be effectively applied. A rigorous derivation of averaged equations for thermovibrational convection is presented including the details, which are not discussed in the existing literature. Using the order of magnitude analysis, we have estimated the magnitudes of oscillatory and averaged flows fields. Two cases, which correspond to averaged flows with small and large Reynolds numbers Re, were distinguished. It was found that Re ~ Y2Gs/ Pr for Re < 1 and Re ~ y(Gs/ Pr)1/2 for Re > 1, where Gs and Pr are the Gershuni and Prandtl numbers, respectively, while y < 1 represents the ratio of averaged velocity magnitude to the oscillatory velocity magnitude. The obtained estimations correlated with numerical modelling provide new essential information about the intensity of averaged flows in a wide range of vibration stimuli. These estimations are used to derive the criteria for the validity of Boussinesq approximation. The basic assumptions of Boussinesq approach are applied

to the momentum, continuity, and energy equations for a compressible, viscous heat-conducting fluid. The contribution of viscous energy dissipation and pressure work to the energy balance is taken into account. The order of magnitude analysis provides a number of dimensionless parameters, the smallness of which guarantees the validity of Boussinesq approximation. It is found that if the density variation with pressure is much smaller than that with temperature, then the contribution of pressure work to the energy balance can be neglected. The contribution of viscous dissipation due to oscillatory motion to the averaged energy balance can be important only for mean flows with small Reynolds number. An example of a concrete physical system is considered. The results of this work provide important and useful information for further analytical and numerical studies of vibrational phenomena in fluids.

This work is supported by the Russian Foundation for Basic Research Grant 08-01-00762-a and The Russian President Grant MK-299.2009.1 (I. Ryzhkov). The authors are grateful to Prof. V.K.Andreev and Dr. M.Hennenberg for useful comments and discussion.

References

[1] G.Z.Gershuni, D.V.Lyubimov, Thermal Vibrational Convection. Wiley & Sons, 1998.

[2] S.M.Zenkovskaya, I.B.Simonenko, Effect of high-frequency vibration on convection initiation, Fluid Dynamics, 1(1966), no. 5, 51.

[3] I.B.Simonenko, A justification of the averaging method for a problem of convection in a field of rapidly oscillating forces and for other parabolic equations, Math. USSR Sb., 16(1972), 245.

[4] D.Beysens, Vibrations in space as an artificial gravity? Europhysicsnews, 37(1966), no. 3, 22.

[5] R.Savino, R.Monti, Fluid-dynamics experiment sensitivity to accelerations prevailing on microgravity platforms, Physics of Fluids in Microgravity (edited by R.Monti), 178, Taylor & Francis, London, 2001.

[6] K.Hirata, T.Sasaki, H.Tanigawa, Vibrational effects on convection in a square cavity at zero gravity, J. Fluid Mech., 445(2001), 327.

[7] I.Cisse, G.Bardan, A.Mojtabi, Rayleigh Benard convective instability of a fluid under high-frequency vibration, Int. J. Heat Mass Trans., 47(2004), 4101.

[8] R.Savino, R.Monti, M.Piccirillo, Thermovibrational convection in a fluid cell, Computers & fluids, 27(1998), no. 8, 923.

[9] V.Shevtsova, D.Melnikov, J.C.Legros, Y.Yan, Z.Saghir, T.Lyubimova, G.Sedelnikov, B.Roux, Influence of vibrations on thermodiffusion in binary mixture: A benchmark of numerical solutions, Phys. Fluids, 19(2007), 017111.

[10] M.P.Zavarykin, S.V.Zorin, G.F.Putin, On thermoconvective instability in vibrational field, Dokl. AN SSSR, 299(1988), no. 2, 309 (Russian).

[11] I.A.Babushkin, V.A.Demin, Vibrational convection in the Hele-Shaw cell. Theory and experiment, J. Appl. Mech. Tech. Phys., 47(2006), no. 2, 183.

[12] A.V.Zyuzgin, A.I.Ivanov, V.I.Polezhaev, G.F.Putin, E.B.Soboleva, Convective motions in near-critical fluids under real zero-gravity conditions, Cosmic Research, 39(2001), no. 2, 175.

[13] D.E.Melnikov, I.I.Ryzhkov, A.Mialdun, V.Shevtsova, Thermovibrational Convection in Microgravity: Preparation of a Parabolic Flight Experiment, Microgravity Sci. Technol., 20(2008), no. 1, 29.

[14] A.Mialdun, I.I.Ryzhkov, D.E.Melnikov, V.Shevtsova, Experimental evidence of thermal vibrational convection in a non-uniformly heated fluid in a reduced gravity environment, Phys. Rev. Lett., 101(2008), 084501.

[15] V.M.Shevtsova, I.I.Ryzhkov, D.E.Melnikov, Y.A.Gaponenko, A.Z.Mialdun, Experimental and theoretical study of vibration-induced thermal convection in low gravity, J. Fluid Mech., 648(2010), 53.

[16] E.A.Spiegel, G.Veronis, On the Boussinesq approximation for a compressible fluid, Astro-phys. J., 131(1960), no. 2, 442.

[17] J.M.Mihaljan, A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid, Astrophys. J., 136(1962), no. 3, 1126.

[18] D.D.Gray, A.Giorginia, The validity of the Boussinesq approximation for liquids and gases, Int. J. Heat Mass Trans., 19(1976), no. 5, 545.

[19] D.J.Tritton, Physical Fluid Dynamics, Clarendon press, Oxford, 1988.

[20] L.D.Landau, E.M.Lifshitz, Fluid Mechanics, Oxford, Pergamon Press, 1982.

[21] L.G.Loycyansky, Mechanics of Liquid and Gas, Moscow, Nauka, 1987 (Russian).

[22] V.S.Vladimirov, Equations of Mathematical Physics, Dekker, New York, 1971.

[23] Y.A.Gaponenko, J.A.Pojman, V.A.Volpert, S.M.Zenkovskaya, Effect of high-frequency vibration on convection in miscible liquids, J. Appl. Mech. Tech. Phys., 47(2006), no. 1, 190.

[24] L.D.Landau, E.M.Lifshitz, Statistical Physics, 1, Oxford, Pergamon Press, 1980.

О приближении Буссинеска в задачах конвекции, вызванной высокочастотными вибрациями

Илья И. Рыжков Юрий А. Гапоненко

Изучена применимость приближения Буссинеска к задачам термовибрационной конвекции в замкнутых объемах. Рассматриваются вибрации высокой частоты и малой амплитуды, что позволяет использовать метод осреднения. Получены оценки пульсационных и осредненных компонент течения. Показано, что число Рейнольдса для осредненного движения пропорционально отношению числа Гершуни к числу Прандтля в области малых чисел Рейнольдса и корню квадратного из этого отношения в области больших чисел Рейнольдса. Эти результаты дают важную информацию о характеристиках осредненных течений в зависимости от интенсивности вибрации. Принимая во внимание полученные оценки, основные предположения приближения Буссинеска применены к уравнениям сохранения массы, импульса и энергии для сжимаемой вязкой теплопроводной жидкости (газа). Учитывается вклад вязкой диссипации и работы сил давления в уравнение сохранения энергии. Анализ порядка величин различных членов в уравнениях движения приводит к набору безразмерных параметров, малость которых гарантирует справедливость приближения Буссинеска.

Ключевые слова: приближение Буссинеска, вибрационная конвекция, осредненное движение.