Thermovibrational low-mode model of convection in a horizontal layer with longitudinal vibrations Текст научной статьи по специальности «Математика»

УДК 669.86: 536.25

Thermovibrational Low-Mode Model of Convection in a Horizontal Layer with Longitudinal Vibrations

Vadim A. Sharifulin*

Perm National Research Politechnic University Pozdeeva, 11/B, Perm, 614990

Russia

Received 02.10.2016, received in revised form 10.01.2017, accepted 20.02.2017 Thermovibrational convection in a horizontal layer of fluid between isothermal solid boundaries heated to different temperatures in the presence of longitudinal vibrations is considered in this paper. Stability and supercritical bifurcation of convection is investigated in a low-mode approximation. Bifurcation diagrams of supercritical modes are analytically obtained in the area of stability of supercritical convection. The analysis of diagrams shows that vibrations can lead to the rigid type of the occurrence of convection when upper boundary is heated. In addition, the hysteresis between stationary states is observed. The size of hysteresis interval of the Rayleigh numbers increases with the growth of the Gershuni number. A numerical study of the linear stability of the supercritical vibration-convective flows in the interval of Prandtl numbers 1 < Pr < 10 is conducted in the context of the proposed model. The region of flow stability decreases with increasing the Prandtl number. For any value of the Prandtl number from the given interval drastic excitation of stationary vibrational convection with hysteresis is possible.

Keywords: thermovibrational convection, low-mode model, flat layer, hard excitation, hysteresis. DOI: 10.17516/1997-1397-2017-10-2-158-169.

The study of supercritical regimes of thermal convection in a plane horizontal layer between the solid isothermal planes heated to different temperatures and subjected to longitudinal vibration is of considerable interest. It would help to understand the nucleation mechanism in terms of the laws of the bifurcation of complex convective phenomena. It is also of importance in obtaining new materials. In the absence of vibration this problem is the classical problem of Rayleigh convection in a horizontal layer heated from below [1-3]. This is the most studied problem of thermal convection. Supercritical mode of convection is investigated in numerous works both theoretically and experimentally when the Rayleigh number Ra exceeds the critical value Racr = 1708. Supercritical convection under the action of longitudinal vibrations in the case of zero-gravity has been studied in [4]. Nonlinear supercritical vibrational convection was investigated numerically by finite difference method when the Prandtl number is equal to one and the Gershuni number Gs exceeds the critical value Gscr = 2129 [5] (in papers [6,7] the Gershuni number was called the vibrational Rayleigh number).

When vibrational and gravitional mechanisms of the convection are present the linear stability of the state of averaged mechanical equilibrium was studied only theoretically. It follows from the linear theory of stability [5] that the set of Rayleigh and Gershuni numbers, for which the state of averaged mechanical equilibrium is neutrally stable, is defined by the following relation

Ra/Racr + Gs/Gscr = 1. (1)

* vadim.sharifulin@pstu.ru © Siberian Federal University. All rights reserved

When the left side of this equation is less (more) than one, the averaged state of mechanical equilibrium is stable (unstable). Linear relationship was predicted in the experimental work of Putin G.F. et al [8] where the relationship between the Nusselt number and Ra, Gs was determined. Similar to (1) relation was obtained by extrapolation of experimental results for the heat flux through the layer for the value Nu = 1. It is obviously from (1) that for sufficiently large values of Gershuni number (Gs > Gscr) convective instability occurs when heating is from above, i.e. when Ra < 0.

This work presents an attempt to understand the nature of supercritical regimes of thermal vibrational convection in a layer in low-mode approximation that has been never conducted before.

1. Problem formulation

We consider an infinite horizontal layer of a viscous incompressible fluid of thickness h between solid horizontal planes (Fig. 1). The coefficients of kinematic viscosity v, thermal diffusivity x, the average density p, the volume coefficient of thermal expansion ¡3 and the acceleration of gravity g are assumed to be constant. Ideal heat conductivity and constant transverse temperature gradient A are assumed. Walls have different constant temperatures and the temperature difference is Ah. The layer undergoes high frequency harmonic vibration along the unit vector k.

Fig. 1. Geometry of the problem

It is assumed that the vibration frequency is quite high. Therefore, equations of thermal vibrational convection [5] are used to consider the behavior of the fluid. We assume that the liquid moves in the plane. The following characteristic values are introduced: time is h2 v , length is h temperature is Ah, velocity is vh-1, streamfunction is v and pressure is pv2h-2. Dimensionless equations of the thermal vibrational convection, describing the flat convection motion in terms of streamfunction of average motion amplitude of streamfunction of oscillating motion F and average temperature T have the form

1 idAV dV dAV

Pr \ dt dx dz

dV dAV

dz dx )

\

dT dV dT dt dx dz

dT

= AAV + Ra— + Gs dx

dV dT =AT

dz dx '

dTd2 F dT d2F

dx dz2 dz dxdz

AF = - —

dT

dz

(2)

(3)

(4)

It is assumed that the boundaries of the layer are solid and isothermal [4]:

d^

z = 0 : ^ = — = F = 0, T = 1,

z = 1 : ^ = — = F = 0, T = 0.

dz

The solution is dictated by three dimensionless numbers: the gravitational Rayleigh number Ra, the vibration Rayleigh number Gs (the Gershuni number) and the Prandtl number Pr:

Ra = , (6)

vx

Gs = , (7) 2vx

Pr= (8)

x

Equations (2)-(5) have trivial solution that corresponds to the mechanical quasiequilibrium:

^g = 0, (9)

To = 1 - z, (10)

Fo = 1 z (z - 1). (11)

Let us introduce new variables for deviations from this solution. The system of equations (2)-(4) can be rewritten in terms of new variables l and f :

+ didA-dîdAî) = AAe + Ra—+Gs\— f1+ df ) + (1 - — )d2f

Pr V dt dx dz dz dx ) dx dx\ dz2 J V dz) dxdz

, (12)

de + d^dd - d^di - dç =M (13)

dt dx dz dz dx dx '

Af = -£ (14)

Boundary conditions (5) take the form

z = 0, 1: £ = dZ = ° = f = 0- (!5)

2. Low-mode model

Let us apply Galerkin method with a minimal set of basic functions to the analysis of nonlinear modes. The following basic functions are used: sin(kx)p1(z) for £ and cos(kx)p1(z), p2(z) for 0. Here pi(z) and p2(z) are mutually orthogonal polynomials of the form

Pi(z) = 1-z2(z - 1)2, (16)

2 ' ' -\2 . 2,

P2(z) = p1(z) = - (z(z - 1)2 + z2(z - 1)) , (17)

p2(z) = p'i'(z) = 11 ((z - 1)2 + 4z(z - 1) + z2) , (18)

p1V = 2. (19)

Polynomial p1(z) was first proposed to approximate the vertical velocity in the layer with solid boundaries [1]. Let us assume that

£ (x,z,t) = £ (t)sin kx • p1 (z), (20)

e (x,z,t) = e1 (t) cos kxp1 (z) + e2 (t) p2 (z), (21)

f (x,z,t) = e2 (t) f2 (z). (22)

Where functions £, e1 and e2 depend only on time. Substituting (22) into (14), we obtain the equation for f2 (z):

f2" = -p2(z) = -pi". (23)

The following conditions must be satisfied at boundary points:

f2 (0) = f2 (1) = 0. (24)

Substituting (20)-(22) into equations (12)-(14), we obtain equations for £, e1 and e2

k2 + 12 £ = k (Ra + Gs - 121 Gse2 ) ex (25)

(k2 + 12)2 + 360 Pr = £ + (k2 + 12)2 + 360 ' ()

30

0i = - (k2 + 12) ei + k£ + — k£e2, (26)

^ 28" 2 k£ei. (27)

--I/O -

3 3267 Let us introduce new variables:

ei = ^ e2 = -121/30^2, £ ^ t = r/(12 + k2). (28)

Then system (25)-(27) can be written as

c

pr0 = -0 + (rg + rv) tfi + rv^1^2, (29)

1 = -^1 + 0 - 0$2, (30)

2 = -b-&2 + 0ti1. (31)

The following designations are used:

b = , 28 2,, c = (k2 + 22)2 , (32)

3(12 + k2)' (k2 + 12)2 + 360 V 7

rg = Ra/R*, rv = Gs/Ri-m, (33)

Ri-m = (k2 + 12) ((k2 + 12)2 + 360) . (34)

System (29)-(31) with c =1 h rv =0 is well-known Lorentz model [9]. Therefore, this system can be regarded as a generalization of the Lorenz model to the case of longitudinal vibration and more realistic conditions on solid walls (sticking in the Lorentz model is absent). There is generalization of the Lorenz model to the case of transverse vibrations [10].

3. Stability of trivial solution

Equations (29)-(31) have trivial solution that corresponds to the mechanical quasi-equi-librium state:

01 =0, tf i = 0, Zi = 0. (35)

In order to investigate the stability of (38) small perturbations are introduced:

0 = 0 exp(At), tf = tfexp(At), Z = Cexp(At). (36)

Substituting (36) into (29)-(31) and taking into account that we have small-amplitude perturbations, system of homogeneous equations with respect to amplitudes of perturbations is obtained. The homogeneous system of equations has nontrivial solution if its determinant is equal to zero:

A = (A - b) (pA2 - (p + l) A +1 - rg - rv) =0. (37)

This equation has the following roots:

1 + Pr' 11 2

Ai = -b, A23 =--^^ ± ^(1 + Pr')2 +4Pr' (rg + rv - 1). (38)

It is clear that root Ai is always real and negative. Roots A2 and A3 can be either complex conjugate with negative real parts or real.

If rg + rv < 1 then real parts of A2 and A3 are negative and quasi-equilibrium state is stable for such set of parameters (rg ,rv). If rg is negative and sufficiently large in absolute value (it corresponds to strong heating from above) then A2 and A3 are complex conjugate. This points to the oscillatory nature of perturbations. Dumping of perturbations takes place under the condition:

rg < -- rv• (39)

4 Pr c

It is evident that the greater the intensity of the vibration the more intense heating from above is necessary for the occurrence of oscillations.

If rg + rv > 1 then A2 is positive and quasi-equilibrium state becomes unstable. In the case {A2 = 0} we obtain

r0 + r°v = 1. (40)

Taking into account (32), one can obtain that quasi-equilibrium state becomes neutral stable

for

Ra + Gs = Rai-m. (41)

Thus, the dependence (34) for Gs = 0 (Ra = 0) defines in low-mode approximation the critical value of the Rayleigh number Ra* = Rat-m — Gs (the Gershuni number Gs* = Ral-m — Ra ) above which the perturbation wave number k increases. It is clear from (34) that neutral curve Rai-m(k) has a minimum at km = 3.214.

This means that the most dangerous perturbation occurs when the cell size is approximately equal to the thickness of the layer. This value corresponds to the wave number of the critical number:

Ra*-m = 1856. (42)

This value is a good approximation to more accurate values of the critical Rayleigh number Racr and the critical Gershuni number Gscr.

Comment. System (29)-(31) with c =1 h rv =0 becomes Lorentz model [9]. Therefore, this system can be regarded as a generalization of the Lorenz model to the case of longitudinal vibration and more realistic non slip boundary conditions on solid walls. The analysis of linear model shows that the occurrence of both gravitational and vibrational convection is predicted with reasonable accuracy. In this model the critical gravitational Rayleigh number and vibrational Rayleigh numbers coincide and they are equal to 1856. This value differs from the exact values of the critical Rayleigh number 1708 [1] and the critical value of the vibrational Rayleigh number (the Gershuni number) 2129 [5] by about 10 per cent. In this model non-slip conditions on the walls are assumed in contrast to the Lorentz model where viscous stresses at the borders are missed.

4. Stationary finite-amplitude solutions

The stationary solution of system of nonlinear ordinary differential equations (29)-(31) can be obtained in analytical form. From (30) and (31) one can express d and Z in terms of 0:

ê =

b0

b + 02

C =

02

b + 0

2

(43)

These expressions were obtained in [11]. By substituting (43) into (29), we obtain the equation of the fifth degree with respect to 0:

0 (04 + (2(1 - rv) - rg) b^2 + (1 - rv - rg) b2) = 0. This equation, apart from the trivial solution ^i = 0, has four roots:

02,3 = \lr92 + 4rv (rg + rv - 1) + rg + 2 (rv - 1)

04,5 = 2

vfp

,g + 4rv (rg + rv - 1) + rg + 2 ( ^ - 1)

(44)

(45)

(46)

The Gershuni number can not be negative, and the Rayleigh number can be either positive or negative. To analyse equations (44)-(46) it is sufficient to consider the half-plane of parameters (rg,rv) with rv > 0. Results of the analysis on the basis of (45), (46)) are shown in Fig. 2 solutions by two bifurcation curves (solid and dashed lines).

I . Range of values, where the only real solution is 0i = 0:

[(0 < rg < 1) f (rv < 1 - rg )] U [(rg < 0) fl (rv < \ (1 - rg + ^T^^"))] .

(47)

II . Range of values where there are three distinct real solutions - trivial, positive and negative 03 = -02:

(rv > 0) n (rg > 1 - rv ). (48)

III . Range of values where there are five different real solutions - trivial 0i =0 , two positive

02 ,04 and two negative 03 = -02 and 05 = -04 :

(rg < 0) n (rv < 1 - rg ) n [rv >

1 - rg +

(49)

2

r

g

Fig. 2. Map of stationary solutions on the parameter plane (rg,rv). The trivial solution 01 exists throughout the parameter plane. Two pairs of solutions (02,04) and (03,05) appear or disappear in crossing bifurcation curve (51) (dashed line). It depends on the direction of crossing. Pair of solutions (02,03) ((04,0s)) appear or disappear for positive (negative) values of rg in crossing the bifurcation curve rg + rv = 1 (solid line). This is the result of the fork bifurcation

Areas I and II are connected by the line defined by relations

0 < rg < 1, rv = 1 - rg. (50)

On this line 0i = 02 = 03 = 0, it means that the solutions 02 and 03 are branched out from the trivial solution 01 during transition from region I to region II.

Areas I and III are connected by the bifurcation curve defined by relations

rv = 2 (1 - rg + y/T=2r~) , -œ <rg < 0. (51)

On this line the following conditions are satisfied for nontrivial roots:

02 = -03 = 04 = -05 = y//1 - 2rg —1. (52)

Areas II and III are connected by the bifurcation curve defined by relations

œ < rg ^ 0, rv = 1 - rg. (53)

On this line all five roots are real and the following conditions are satisfied

01 = 04 = 05 =0, 02,3 = (54)

Thus, when heating is from above (rg < 0) then for each rv > 1 there is a range of Rayleigh numbers rg defined by system of inequalities (47) wherein system (30)-(31) has 5 different stationary solutions. We have 04 = 05 =0 on the bifurcation curve rg = 1 -rv (solid line in Fig. 2), 04,5 are imaginary above this curve and there are 3 real solutions. The system has only the trivial solution 01 =0 below the bifurcation curve defined by equation (51) (dashed line).

Consider a smooth change of parameters rg and rv. It can be represented as a line in the parameter plane (rg, rv). In crossing these bifurcation curves one or two pairs of solutions appear or disappear. It depends on the direction of crossing.

Typical bifurcation diagrams describing a bifurcation that occurs in crossing bifurcation curves shown in Fig. 2 are presented in Fig. 3 and Fig. 4. Parameters are changed along dotted and dashed-dotted lines. Fig. 3 shows the bifurcation diagram 0 (rg) for the vibrational Rayleigh number rv = 1 (it corresponds to the dash-dotted line in Fig. 3). The occurrence of convection is observed when the Rayleigh number increases. This is the result of the soft fork bifurcation when the Rayleigh number is above zero.

Fig. 3. The diagram of the fork bifurcation that occurs for a smooth variation of the Rayleigh number rg and rv = 1

Fig. 4 shows the bifurcation diagram 0 (rg) for the vibrational Rayleigh number rv =9 (it corresponds to the dotted line in Fig. 2). One can see that with increasing the Rayleigh number convection occurs in a drastic way at the moment of crossing the bifurcation curve (53). Presented bifurcation diagrams are derived from solutions (45), (46). Constant b = 0.4179 is obtained from (32) for k = km = 3.214.

To determine the stability of solutions presented in Fig. 3 and Fig. 4, numerical solution of the original system of equations (29)-(31) is obtained with the use of the Runge-Kutta-Felberg method (RKF45). The required for the numerical integration parameters b and c are determined from (34) for k = km = 3.214 and they are b = 0.4179 and c = 0.5808. The Prandtl number is set to Pr = c = 0.5808. The initial value of the amplitude of the stream function 0 is close to zero, initial values of d and Z are calculated from (43). Calculations were carried out for (—20 <rg < 20) n (rv < 20). They show that solutions 02 and 03 are stable but solutions 04 and 05 are unstable.

Long arrows in Fig. 3 show the drastic transition from quasi-equilibrium state to one of supercritical states 02 or 03. These transitions occur with a continuous stepwise increase in the normalized Rayleigh number rg along the dotted line in Fig. 2 (from left to right). The initial value of the amplitude of stream function 0 is set to the value obtained in the previous step. In the reverse step by step change of the Rayleigh number, sharp damping of the supercritical flow

Fig. 4. The bifurcation diagram for rv =9

is observed at the moment of crossing the dotted line. When the Gershuni number increases the hysteresis depth also increases.

Calculations were also performed for Pr = 1. Bifurcation diagrams are similar to that shown in Figs. 3 and 4.

5. Stability of stationary finite-amplitude solutions

Numerical integration for various Prandtl numbers 0 < Pr < 102 shows that solutions 04 and 05 are always unstable. Solutions 02 and 03 are unstable for sufficiently large values of Prandtl, Rayleigh and Gershuni numbers. Let us consider small perturbations

0' = 02 + 02 exp(At), $' = $2 + $2 exp(At), C' = C2 + C2 exp(At). (55)

Since 03 = -02 it is sufficient to consider only the stability of state 02. Hereinafter the subscript is omitted.

Substituting (55) into (29)-(31), we obtain the system of linear equations for amplitudes of perturbations. Setting the determinant of the system matrix equal to zero, we obtain a cubic equation for A:

A3 + (b + 1 + Pr') A2+Pr'

+Pr'

b + 02 (b + 02) (rs + rv)+ rv (b +1) 02

b + 1 + r - b

Pr' (b + 02)2

b + 02 + b

(rs + rv) (04 - b2) + rv02 (02 - 3b)

A+

(56)

0.

(b + 02)2

The lack of vibrations (rv =0). In this case, the stationary solutions (45) take the form:

0 = b (ra - 1). (57)

Substitution of this solution into (56) gives

A3 + (b + 1 + Pr') A2 + b (Pr' + ra) A + 2Pr'b (ra - 1) = 0. (58)

This equation has one negative and two complex conjugate roots when rg > 1. Real parts of complex conjugate roots for rg > 1 and small or moderate super-criticalities are also negative. Equation (58) is similar to that obtained and analyzed by E. Lorenz [9]. It was shown that by equating the product of coefficients at A2 and A to the constant term, one can get the criteria of neutral stability of oscillating disturbances [9]. In our case such equation has the form

(b + 1 + Pr') (Pr' + rg ) = 2Pr'b (rg - 1) .

(59)

The solution of equation (59) with respect to rg gives a simple expression for the critical value of rg above which the instability of steady convection occurs:

Pr (Pr /c + b + 3) Pr -c (b + 1) '

(60)

Thus if Pr < c (b +1) « 0.8 then not positive value of rg satisfies (60) and steady convection regime (57) is always stable. If Pr > c (b + 1) then steady convection is unstable with respect to oscillating disturbances for sufficiently large Rayleigh numbers.

Numerical study of stability in the general case rg = 0, rv ^ 0. Now we consider the stability of periodic steady solutions with wave number km. We perform the numerical analysis of equations (56) with fixed values of the Prandtl number Pr =1, 3, 5 and 10, incrementing small perturbations A. Analysis of coefficients at A, A2 and constant term in equation (56) shows that they all are non-negative for solutions 02,3.

r

-10 -5 0 5 10 15

r

g

Fig. 5. Areas of stability of supercritical solution 02 with respect to oscillatory disturbances on the parameter plane (rg,rv) for various values of the Prandtl number. Bifurcation curves are marked by thin lines. The oscillatory instability appears in crossing these curves. The solid line and the dashed line correspond to the bifurcation curves shown in Fig. 2

This means that, as in the case above rv = 0, the equating of the product of coefficients at A and A2 to constant term leads to the equation that is similar to (59). This gives the stability criteria of solutions with respect to oscillatory disturbances. The equation is rather cumbersome, and it is not presented here. The results of the numerical analysis of the equation

are presented in Fig. 5 in the form of bifurcation curves (thin solid lines) for various values of the Prandtl number. First we consider the case Pr = 5. In accordance with (60), oscillatory instability occurs when the normalized Rayleigh number rg = 14.4 in the case rv = 0. When rg = 0, oscillatory instability occurs if the normalized Gershuni number rv exceeds the critical value r'v = 5.2. The figure shows that in the case of heating from above, i.e., when rg < 0 , the stability region of supercritical steady solutions 02,3 is defined by the Gershuni number and the Rayleigh number. When 1 < rv < 5 the smooth cyclic variation of the Rayleigh number leads to drastic hysteretic transitions similar to that shown in Fig. 4.

As the normalized Rayleigh number rg undergoes smooth cyclical changes, the corresponding path is a horizontal line on the parameter plane. The normalized Rayleigh number should consistently cross solid and dashed lines but should not cross a thin line. Fig. 5 shows that for all considered values of the Prandtl number such situations are observed for 1 < rv < 4.5 when the heating is at the top (rg < 0). This is supported by the direct numerical integration of the original system of equations (29)-(31). The obtained bifurcation diagrams are similar to that shown in Fig. 4.

Conclusion

The low-mode model of the thermovibrational convection in an infinite plane layer of incompressible viscous fluid confined between horizontal solid isothermal planes has been considered. The layer undergoes longitudinal vibrations. In the limiting case of the lack of vibrations the model is similar to the well-known Lorenz model. The results of linear stability analysis of the model are in good agreement with the results of the linear stability analysis of complete equations of thermal vibrational convection.

It is shown that the model predicts the drastic excitation of stationary supercritical vibrational convection in the case of heating from the top. Cyclic variation of the gravitational Rayleigh number leads to hysteretic transitions between stationary solutions.

The study of linear stability of the supercritical flows shows that for all considered Prandtl numbers the region of stability decreases with increasing the Prandtl number. However, there are regions where drastic hysteretic excitation of supercritical convection is possible.

The work was supported by the Perm National Research Polytechnic University under an internal university grant.

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Маломодовая модель тепловой вибрационной конвекции в горизонтальном слое при наличии продольных вибраций

Вадим А. Шарифулин

Пермский национальный исследовательский политехнический университет

Поздеева, 11/В, Пермь, 614990 Россия

В маломодовом приближении исследованы устойчивость и бифуркации надкритической термовибрационной конвекции в горизонтальном слое жидкости между нагретыми до различной температуры изотермическими твердыми границами при наличии продольных вибраций. В области устойчивости надкритической конвекции аналитически получены бифуркационные диаграммы надкритических режимов, анализ которых показал, что при подогреве сверху вибрации могут приводить к жесткому типу возникновения конвекции, сопровождающемуся гистерезисом между стационарными состояниями. Область гистерезиса по числу Рэлея увеличивается с увеличением числа Гершуни. В рамках предложенной модели проведено численное исследование линейной устойчивости надкритических вибрационно-конвективных течений для интервала чисел Прандтля 1 < Pr < 10. Расчеты показали что хотя область их устойчивости при увеличении числа Прандтля и уменьшается, при всех рассмотренных его значениях из указанного интервала имеются области, где возможно жесткое гистерезисное возбуждение стационарной вибрационной конвекции.

Ключевые слова: дифференциальные уравнения, задача Коши, расщепление, устойчивость, сходимость.