Three-dimensional thermocapillary–buoyancy flow of a binary mixture with Soret effect in a shallow annular pool

International Journal of Heat and Mass Transfer 90 (2015) 1071–1081

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Three-dimensional thermocapillary–buoyancy flow of a binary mixture with Soret effect in a shallow annular pool Jia-Jia Yu, You-Rong Li ⇑, Chun-Mei Wu, Jie-Chao Chen Key Laboratory of Low-grade Energy Utilization Technologies and Systems of Ministry of Education, College of Power Engineering, Chongqing University, Chongqing 400044, China

a r t i c l e

i n f o

Article history: Received 16 December 2014 Received in revised form 1 June 2015 Accepted 10 July 2015

Keywords: Thermocapillary–buoyancy flow Soret effect Binary mixture Annular shallow pool

a b s t r a c t This paper presented a series of three-dimensional numerical simulations on the thermocapillary–buoy ancy flow with Soret effect in a shallow annular pool which was filled with the n-decane/n-hexane mixture with an initial mass fraction of 50%. The Prandtl number and the Lewis number of this binary mixture were 9.08 and 27.89, respectively. The annular pool was heated at the outer cylinder and cooled at the inner cylinder. An adiabatic solid bottom and free surface were considered. The results indicated that the solute concentration gradient depends strongly on the temperature gradient. Due to the Soret effect in the n-decane/n-hexane mixture, the n-decane with a higher density component gathers at the colder area near the inner cylinder. The critical thermal capillary Reynolds number for the incipience of the three-dimensional oscillatory flow in the n-decane/n-hexane mixture is smaller than that in the n-hexane fluid, and decreases with the increase of the aspect ratio. A solute concentration fluctuation which is similar to the temperature fluctuation is observed when the three-dimensional oscillatory flow happens. The nondimensional fundamental oscillation frequency is related to the aspect ratio and the thermal capillary Reynolds number. Meanwhile, the azimuthal wave number decreases with the increase of the aspect ratio, but it is independent of the thermal capillary Reynolds number. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Thermocapillary–buoyancy flow is frequently encountered in many industrial processes, such as crystal growth, evaporation and thin-film coating etc. [1–3]. In the past few decades, this classic physical phenomenon has attracted many researchers’ interest. Smith and Davis [4,5] predicted two types of three-dimensional (3-D) instabilities in an extended shallow pure fluid layer with a free surface subjected to a constant horizontal temperature gradient, i.e. oblique hydrothermal waves (HTWs) and stationary longitudinal rolls. After that, Schwabe [6], Kamotani et al. [7] and Peng et al. [8] conducted a series of numerical simulations and experimental observations to obtain the critical conditions and the bifurcations of various flow patterns. Based on pioneering works, a complete understanding of the thermocapillary-boundary flow in pure fluids has been established. In binary mixtures, the flow is usually driven by the surface tension gradient and/or buoyancy force, which are dependent on temperature and solute concentration distributions. It is generally known that the solute concentration gradient across a binary

⇑ Corresponding author. Tel.: +86 23 6511 2284; fax: +86 23 6510 2473. E-mail address: [email protected] (Y.-R. Li). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.048 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

mixture can be generated by either the solute concentration difference between two walls or the Soret effect. Bergman [9] performed first numerical simulation on the double-diffusive Marangoni convection in a rectangular cavity with the imposed horizontal temperature and concentration gradients. It was found that when the thermal Marangoni number exceeds a critical value, the flow occurs even at the special condition that thermal and solutal capillary forces are equal but opposite. Zhan et al. [10] extended the research on double-diffusive Marangoni convection by the three-dimensional (3-D) simulations. They observed different symmetry-breaking pitchfork bifurcation phenomena and the flow reverting from temporal chaos to steady state. It was certified that the evolution of the flow structure is strongly influenced by the heat and mass transfer rate. Furthermore, the onset of the convection [11], the transition to chaos [12] and the relative contributions of the Marangoni effect and double-diffusion on the instabilities [13] in a rectangular cavity with horizontal temperature and concentration gradients have been studied by numerical simulations, experiments and linear stability analysis. Both steady and oscillatory flow regimes and the characteristics of the double-diffusive Marangoni convection were observed and the physical mechanisms of the instabilities under some conditions were identified. However, the solute concentration gradient which is generated by the Soret effect is often ignored. Bahloul et al. [14] studied the

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Nomenclature C d D eZ f F g GrT Le m Nu p P Pr r R ReC ReT Rq Rr ST t T v V V

mass fraction depth, m mass diffusivity of species, m2/s z-directional unit vector oscillation frequency, Hz nondimensional oscillation frequency gravitational acceleration, g = 9.8 m/s2 Grashoff number Lewis number azimuthal wave number Nusselt number pressure, Pa nondimensional pressure Prandtl number radius, m nondimensional radius solutal capillary Reynolds number thermal capillary Reynolds number buoyancy ratio capillary ratio Soret coefficient, 1/K time, s temperature, K velocity, m/s nondimensional velocity nondimensional velocity vector

Marangoni convection induced by the Soret effect in a horizontal layer of a binary mixture with a vertical temperature gradient. The Bifurcation diagrams were presented for the cases in which the solutal Marangoni effect acts in either the same or opposite direction with the thermal Marangoni effect. It was found that domains of the existence of different regimes for the Marangoni convection depend on the ratio rate of the thermal and solutal Marangoni numbers and the Lewis number. Saravanan and Sivakumar [15] further studied the onset of the Marangoni convection with throughflow and the Soret effect for different types of thermal and solutal boundary combinations. It was certified that the Soret effect offers an additional destabilizing impact on the existing Marangoni convection when the throughflow is not strong enough. Bergeon and Knobloch [16] simulated the oscillatory Marangoni convection in square and nearly square cavities which were filled with a binary mixture and heated from the upper surface. It was concluded that the formation of oscillatory flow should be attributed to the variation of the surface tension with the concentration due to the Soret effect. When the square’s aspect ratio was 1.5, either a standing wave with left–right reflection symmetry or a discrete rotating wave was observed. Oron et al. [17,18] predicted that both the monotonic and oscillatory long-wavelength instabilities are possible in a binary mixture layer with vertical temperature gradient, which is dependent on the Soret coefficient. Furthermore, the Marangoni convection with the Soret effect in a binary mixture system which was imposed by a vertical temperature gradient has been studied by the linear stability analysis, numerical simulations and experiments [19– 21]. The influence of the Soret effect on the critical temperature difference for the onset of oscillatory instability, the oscillatory frequency [19], and the patterns on the free surface of volatile binary liquids [20,21] have been systematically analyzed. As mentioned above, the existing researches mostly focused on the thermal–solutal capillary flow of a binary mixture in a rectangular cavity or a horizontal liquid layer with a vertical temperature or concentration gradient. However, there is little research devoted

z Z

axial coordinate, m nondimensional axial coordinate

Greek symbols a thermal diffusivity, m2/s bC solutal expansion coefficient bT thermal expansion coefficient, 1/K cC solutal coefficient of surface tension, N/m cT temperature coefficient of surface tension, N/(K m) e aspect ratio h azimuthal coordinate, rad m kinematic viscosity, m2/s l dynamic viscosity, kg/(m s) q density, kg/m3 s nondimensional time w nondimensional stream function Subscripts c critical i inner o outer r, R radial z, Z axial h azimuthal 0 initial

to the thermal–solutal capillary flow or the thermocapillary flow of a binary mixture with a horizontal temperature or concentration gradient in an annular pool. In our previous works [22,23], the thermal–solutal capillary flow in an annular pool with horizontal temperature and solute concentration gradients has been studied by asymptotical analysis and the two-dimensional numerical simulation. This paper aims to understand the influence of the Soret effect on the thermocapillary–buoyancy flow of a binary mixture in a shallow annular pool with a horizontal temperature gradient. 2. Model formulation 2.1. Basic assumption and governing equations The physical model is schematically shown in Fig. 1. The shallow annular pool of depth d, inner radius ri and outer radius ro is filled with the n-decane/n-hexane mixture with an initial mass fraction of 50%. The radius ratio and aspect ratio of the pool are

Fig. 1. Physical model and the coordinate system.

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defined as g = ri/ro and e = d/(ro  ri), respectively. The bottom boundary is rigid solid wall while the top boundary is a non-deformable flat free surface. The inner and outer cylinders are maintained at constant temperatures Ti and To, respectively, where Ti < To. Both the free surface and the bottom wall are considered as adiabatic and impermeable. The following assumptions are introduced in the present model: (1) The n-decane/n-hexane mixture is incompressible Newtonian fluid, and the physical properties are taken as constant except the surface tension r and the density q in the buoyancy term of the momentum equation. (2) The velocity is small and the flow is laminar. (3) Both the inner and outer cylinders are impermeable. (4) The Dufour effect is neglected in binary liquid mixtures [24]. The surface tension r and density q are assumed to be linear functions of temperature and the concentration of the n-decane, that is,

rðT; CÞ ¼ r0  cT ðT  T 0 Þ  cC ðC  C 0 Þ;

ð1Þ

qðT; CÞ ¼ q0 ½1  bT ðT  T 0 Þ  bC ðC  C 0 Þ;

ð2Þ

where cT = (or/oT)C, cC = (or/oC)T, bT = (oq/oT)C/q0 and bC = (oq/oC)T/q0. Using (ro  ri), m/(ro  ri), (ro  ri)2/m and lm/(ro  ri)2 as scale quantities for length, velocity, time and pressure, respectively, the flow, heat and mass transfer equations in the cylindrical coordinate system are expressed in a nondimensional form as follows:

r  V ¼ 0;

ð3Þ

@V þ V  rV ¼ rP þ r2 V þ GrT ðH  Rq UÞeZ ; @s

ð4Þ

the flow is enhanced by the adding the solute buoyancy in this binary mixture [28]. 2.2. Boundary and initial conditions Under above assumptions, the boundary conditions are: At the inner cylinder (R = Ri = ri/(rori) = g/(1  g), 0 6 h < 2p, 0 6 Z 6 e):

V h ¼ V R ¼ V Z ¼ 0;

H ¼ 0;

@U @H  ¼ 0: @R @R

ð8a-cÞ

At the outer cylinder (R = Ro = ro/(ro  ri) = 1/(1  g), 0 6 h < 2p, 0 6 Z 6 e):

V h ¼ V R ¼ V Z ¼ 0;

H ¼ 1;

@U @H  ¼ 0: @R @R

ð9a-cÞ

Eqs. (8c) and (9c) show that the local solute concentration gradients at the inner and outer cylinders are related to the local temperature gradients because of the Soret effect in binary mixtures. At the free surface (g/(1  g) < R < 1/(1  g), 0 6 h < 2p, Z = e):

@V R @H @U ¼ ReT þ ReC ; @Z @R @R @V h @H @U @H @U ¼ ReT þ ReC ; ¼ ¼ 0: @Z R@h R@h @Z @Z

V Z ¼ 0;

ð10a-dÞ

At the bottom (g/(1  g) < R < 1/(1  g), 0 6 h < 2p, Z = 0):

V h ¼ V R ¼ V Z ¼ 0;

@H @U ¼ ¼ 0: @Z @Z

ð11a-bÞ

The initial conditions are:

s ¼ 0; V h ¼ V R ¼ V Z ¼ 0; H ¼ 

ln Rð1  gÞ=g ; ln g

U ¼ 0: ð12a-dÞ

@H 1 þ V  rH ¼ r2 H; Pr @s

ð5Þ

@U 1 þ V  rU ¼ ðr2 U þ r2 HÞ: Le  Pr @s

ð6Þ

ReT ¼ cT ðr o  ri ÞðT o  T i Þ=ðmlÞ;

ReC ¼ cC ðro  ri ÞDC=ðml:Þ ð13a-dÞ

where V(VR, Vh, VZ) is the nondimensional velocity vector, eZ the unit vector in the Z direction, H = (T  Ti)/(To  Ti) the dimensionless temperature and U = (C  Co)/(Ci  Co) the dimensionless concentration of the n-decane. Co and Ci are respectively the minimum and maximum solute concentrations when the solute concentration difference is established. GrT = gbTDT(ro  ri)3/m2 is the thermal Grashoff number and Rq = bCDC/(bTDT) is the ratio of the solute to thermal buoyancy forces. Pr = m/a and Le = a/D are the Prandtl and Lewis numbers, respectively. The Soret effect is described by the second term in the right hand of Eq. (6). Because there exists the Soret effect in the binary mixture, the concentration difference DC of the n-decane is generated, which can be calculated by following equation [25,26]

DC ¼ C o  C i ¼ ST C 0 ð1  C 0 ÞðT o  T i Þ

where ReT and ReC are thermal and solutal capillary Reynolds numbers, which are respectively defined as

ð7Þ

where ST = DT/D is the Soret coefficient and C0 is the initial solute concentration. Platten [27] reported that the Soret coefficient ST may be positive or negative depending on the migrational direction of the reference component. For the n-decane/n-hexane mixture, the Soret coefficient is positive. Therefore, the direction of the solute concentration gradient on the free surface is opposite to that of the imposed temperature gradient. For most binary mixtures, thermal expansion coefficient bT is positive but the solutal expansion coefficient bC depends on the contribution of every component to the density of binary mixtures. It should be noticed that bC is negative for the n-decane/n-hexane mixture and the solute to thermal buoyancy ratio Rq > 0. Therefore,

In general, the capillary ratio Rr is used to measure the relationship between the thermal and solute capillary effects [10]

Rr ¼ ReC =ReT ¼ cC DC=ðcT DTÞ

ð14Þ

Substituting Eq. (7) into Eq. (14), we have

Rr ¼ cC ST C 0 ð1  C 0 Þ=cT :

ð15Þ

2.3. Calculation conditions In the present work, the radius ratio of the annular pool is

g = 0.5. The aspect ratio e ranges from 0.04 to 0.13. The initial mass fraction of the n-decane in the n-decane/n-hexane mixture is 50%. The physical properties at 298.15 K are listed in Table 1. In this case, the buoyancy ratio Rq and the capillary ratio Rr are 0.052 and 0.045, respectively. Both of them are positive, and therefore, the Soret effect in the n-decane/n-hexane mixture can enhance the thermocapillary–buoyancy flow. In addition, the thermal Grashoff number GrT ranges from 1.0  105 to 4.3  106. 2.4. Numerical procedure and validation Finite volume method is used to discretize the fundamental equations. The modified central difference approximation is introduced for the diffusion terms while the QUICK scheme is applied to the convective terms [29]. The velocity and pressure coupling is

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Table 1 Physical properties of n-decane/n-hexane mixture and pure n-hexane at 298.15 K. Property

Unit

Density, q Thermal diffusivity, a Mass diffusivity of species, D Soret coefficient, ST Dynamic viscosity, l Thermal expansion coefficient, bT Solutal expansion coefficient, bC Temperature coefficient of surface tension, cT Solutal coefficient of surface tension, cC Prandtl number, Pr Lewis number, Le

kg/m3 m2/s m2/s 1/K kg/(m s) 1/K – N/(m K) N/m – –

handled by the SIMPLEC algorithm [10,11,29]. The nondimensional time step is about 105. At each time step, the convergence is reached if the residual errors of all the momentum, energy and solute concentration equations are less than 104. In order to verify the grid convergence, a series of simulations with four different meshes for the thermocapillary–buoyancy flow of the binary mixture at e = 0.1 and ReT = 13104 were performed. The wave number m and nondimensional oscillation frequency F at the monitoring point P (R = 1.5, h = 0, Z = 0.1) are shown in Table 2. It can be confirmed that the dependence on the grid is very weak. Therefore, the grid of 80R  30Z  240h is an appropriate choice. In order to validate the present numerical method, we carried out some numerical simulations on the thermocapillary–buoyancy flow of 0.65 cSt silicone oil in a shallow annular pool and double-diffusive Marangoni convection of a binary mixture in a rectangular cavity. Results showed that the isotherms, streamlines,

Table 2 Mesh dependence test. Mesh R

Z

h

70  30  200 80R  30Z  240h 100R  30Z  300h 80R  30Z  500h

Wave number, m

Frequency, F

23 24 24 23

75.9 75.6 76.1 74.8

Table 3 Comparison of the Nusselt number. ReT

10 120 200 340 380 450

Nu

Deviation (%)

Present

Ref. [10]

1.010 1.332 1.496 1.770 1.832 1.933

1.007 1.328 1.497 1.762 1.821 1.953

0.30 0.30 0.07 0.45 0.60 1.02

Table 4 Comparison of the oscillation frequency. ReT

350 410 500 540

f/Hz

Deviation (%)

Present

Ref. [11]

6.94 7.89 9.35 10.14

6.84 7.82 9.26 10.02

1.46 0.90 0.97 1.20

Value

Ref.

n-hexane

n-decane/n-hexane

654.79 8.56  108 – – 2.96  104 1.36  103 – 1.02  104 – 5.29 –

689.82 7.50  108 2.69  109 2.42  103 4.70  104 1.20  103 0.103 6.98  105 5.19  103 9.08 27.89

[25] [25] [25] [25] [25] [25] [25] Exp. Exp. – –

iso-concentration lines and surface temperature fluctuation agree well with the results in Refs. [8,10,11]. The Nusselt numbers and oscillation frequencies at different thermal capillary Reynolds numbers in a rectangular cavity are respectively shown in Tables 3 and 4. Obviously, the maximum deviations of the Nusselt number and oscillation frequency between the present results and Refs. [10,11] are less than 1.1% and 1.5%, respectively. These results prove that the numerical method is accurate enough to simulate thermocapillary–buoyancy flow of a binary mixture in a shallow annular pool.

3. Results and discussion 3.1. Basic flow When the temperature difference between the inner and outer cylinders is very small, axisymmetric and steady thermocapillary– buoyancy flow appears in a shallow fluid layer. This type of flow is called ‘basic flow’. In this case, the velocity field is illustrated in terms of the nondimensional stream function, which is defined as

VR ¼

1 @w ; R @Z

VZ ¼ 

1 @w R @R

ð16a-bÞ

Fig. 2 shows the streamlines, isotherms and iso-concentration lines of the n-decane for the n-decane/n-hexane mixture and n-hexane fluid. It is found that the surface fluid is driven by the surface tension gradient and flows from the outer cylinder to the inner cylinder. A return flow is generated near the bottom as a result of mass conservation. Due to the Soret effect in the n-decane/n-hexane mixture, the concentration of the n-decane near the inner cylinder is higher than that near the outer cylinder. From Eq. (7), it is found that the solute concentration difference is always coupled with the temperature difference. Therefore, there exist large concentration gradients of the n-decane near the inner and outer cylinders, as shown in Fig. 2(a) and (b). The thermal capillary Reynolds number ReT and the aspect ratio e have a slight effect on the distribution of the iso-concentration lines. However, the total solute concentration difference tends to increase with the increase of the thermal capillary Reynolds number [25,26]. At a larger thermal capillary Reynolds number, a second rotating roll cell (see Figs. 2(b), and 4(c) in Ref. [30]) emerges in the liquid layers of the n-decane/n-hexane mixture near the outer cylinder. This phenomenon corresponds to the steady multicellular flow (see Fig. 3 in Ref. [31]). The second rotating roll is strengthened with the increase of the thermal capillary Reynolds number, but it does not extend to the entire fluid layer. In pure fluid layer, the evolution of the basic flow with the thermal capillary Reynolds number was reasonably explained by Peng et al. [8]. In the n-decane/n-hexane mixture, because of the positive action of the

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solutal capillary and solutal buoyancy forces, the thermocapillary– buoyancy flow is enhanced and the value of the maximum nondimensional stream function increases slightly, for example, from 1.81 to 1.89 at ReT = 4368 (see Fig. 2(b) and (c)). On the other hand, compared with the results in the n-hexane fluid layer at the same aspect ratio, the temperature gradients near the inner and outer cylinders in the n-decane/n-hexane mixture increase evidently, and that in the mid-region of the free surface decreases, as shown in Fig. 3. The variation of the temperature gradient on the free surface leads to the decrease of the average radial

(a) n-decane/n-hexane mixture, ReT=2184, ψmin=-1.16

velocity in the mid-region and the increase of the peak values of the radial velocity in the thermal boundary layers, especially near the inner cylinder. When the aspect ratio increases from 0.08 to 0.12, the increasing rates of the radial velocity in the n-decane/n-hexane mixture near the inner and outer cylinders are bigger than those in the n-hexane fluid, as shown in Fig. 3. The solute concentration gradients near the inner and outer cylinders slightly increase with the increase of the aspect ratio, which is originated from the temperature variation on the free surface. The thermal and solutal buoyancy forces near the inner and outer cylinders in the n-decane/n-hexane mixture are enhanced with the increase of the aspect ratio. In the mid-region of the free surface, the radial velocity difference between e = 0.08 and e = 0.12 in the n-decane/n-hexane mixture is larger than that in the n-hexane fluid. In the n-decane/n-hexane mixture, the mean solute concentration on the free surface is close to the initial solute concentration, while large solute concentration gradients appear in the boundary layers. In addition, the solute concentration gradient in the mid-region of the free surface is nearly independent on the aspect ratio. Therefore, the surface temperature gradient difference between the n-decane/n-hexane mixture and n-hexane fluid in the mid-region mainly contributes to the radial velocity variation. Fig. 4 shows the variation of the radial velocity VR along the vertical position Z in the n-decane/n-hexane mixture and n-hexane fluid layers at e = 0.12 and ReT = 4368. It is found that the inward radial flow is in the region of 0.079 < Z 6 0.12 and the return flow is in 0 6 Z < 0.079 at R = 1.5. The locations of the zero radial velocity in the n-decane/n-hexane mixture and n-hexane fluid are nearly the same. The gap of radial velocity in the n-decane/n-hexane

0.12

0.09

Z

(b) n-decane/n-hexane mixture, ReT=4368, ψmin=-1.89

0.06

0.03

n-hexane R=1.5 n-hexane R=1.8 n-decane/n-hexane R=1.5 n-decane/n-hexane R=1.8

0.00 -60

-40

-20

(c) n-hexane, ReT=4368, ψmin=-1.81

0.8

0

n-decane/n-hexane =0.08 n-decane/n-hexane =0.12 n-hexane =0.08 n-hexane =0.12

-150

VR

Θ,Φ

0.6 -300

0.5012

0.4

Φ

0.5004

0.0 1.0

0

-45

-150

-50

-300

-450

-600 1.00

0.5000

1.4

R

1.6

1.8

-60

1.02

1.04

1.06 1.40

n-decane/n-hexane =0.08 n-decane/n-hexane =0.12

0.4996 1.000 1.005 1.010 1.015 1.020

1.2

-55

-450

0.5008

0.2

20

Fig. 4. The variation of the radial velocity VR along the vertical position Z at ReT = 4368.

Fig. 2. Characteristics of the basic flow at e = 0.12. Dw = 0.15.

1.0

0

VR

2.0

-600 1.0

1.2

1.4

R

1.45

1.50

1.55

1.60

n-hexane =0.08 n-hexane =0.12

1.6

1.8

2.0

Fig. 3. The radial distribution of surface temperature H and solute concentration U (dotted lines) (left) and radial velocity VR (right) at ReT = 4368.

J.-J. Yu et al. / International Journal of Heat and Mass Transfer 90 (2015) 1071–1081

mixture between R = 1.5 and R = 1.8 is smaller than that in the n-hexane. Because the second rotating roll cell appears in this region with a decreasing radial velocity, the thermal capillary Reynolds number for the onset of a second rotating roll cell in the n-decane/n-hexane mixture layer is smaller than that in the n-hexane fluid layer.

3.2. Critical condition for flow transition In shallow annular pools, when the thermal capillary Reynolds number exceeds a certain threshold value, three-dimensional disturbances are incubated and their amplitudes are gradually amplified [29]. Finally, the basic flow transits a three-dimensional oscillatory flow. The critical thermal capillary Reynolds number is obtained by the dichotomy. The maximum relative deviation of the critical thermal capillary Reynolds numbers is less than 3%. Fig. 5 depicts the variation of the critical thermal capillary Reynolds number (ReTc) with the aspect ratio e. The experimental results obtained with the infrared technique in our previous work [32] are also shown in Fig. 5. Obviously, the variation tendency of the critical thermal capillary Reynolds number is similar to the experimental results at 0.08 6 e 6 0.13. However, some simplifications in numerical simulation lead to some quantitative differences between experimental and simulation results. Xu and Davis [33] found that heat loss at free surface, such as evaporation heat transfer to the environment, stabilizes the flow. Consequently, the critical thermal capillary Reynolds number obtained by the simulations is slightly smaller than that of the experimental results. Whether the n-decane/n-hexane mixture or the n-hexane fluid, the critical thermal capillary Reynolds number decreases with the increase of the aspect ratio. At a small aspect ratio, the capillary force is a dominant factor of driving flow instability. With the increase of the aspect ratio, the effect of the buoyancy is enhanced. Therefore, the critical thermal capillary Reynolds number decreases, as shown in Ref. [8]. It should be noticed that the critical thermal capillary Reynolds number decreases sharply for the n-decane/n-hexane mixture when the aspect ratio increases from 0.025 to 0.05. On the other hand, because of the Soret effect in the n-decane/n-hexane mixture, the n-decane concentration near inner cylinder is higher than that near outer cylinder. Furthermore, the solutal capillary force has the same direction with the thermal capillary force, which results in an enhanced ther mocapillary–buoyancy flow of the n-decane/n-hexane mixture. Therefore, the critical thermal capillary Reynolds number in the n-decane/n-hexane mixture is always lower than that in the n-hexane fluid.

3.3. Three-dimensional oscillatory flow 3.3.1. Effect of aspect ratio Fig. 6 shows the typical time histories of the radial velocity, temperature and solute concentration in the n-decane/n-hexane mixture at the monitoring point P (R = 1.5, h = 0, Z = 0.12) at e = 0.12 when the thermal capillary Reynolds number exceeds the critical value. It is found that the oscillatory flow is a periodic flow with a nondimensional fundamental frequency of 114.1. Obviously, there is a constant phase lag between the radial velocity and temperature oscillations, which is a basic characteristic of the HTW reported by Smith [5]. Therefore, in a binary mixture with moderate Prandtl number, the flow instability in a shallow annular pool should be referred to the HTW instability, even though the Soret effect is taken into consideration. In addition, there is a phase lag between the temperature and solute concentration oscillations, which is related to the Soret effect. In order to extract spatial oscillation, the fluctuation dX of a physical quantity (X) is introduced, which is expressed by

dX ¼ XðR; h; Z; sÞ 

1

Z sp

sp

0

XðR; h; Z; sÞds;

ð17Þ

where sp is the oscillatory period. Fig. 7 shows the surface temperature fluctuation and a spatiotemporal diagram (STD) of the surface temperature fluctuation along a circumference at R = 1.5 at different aspect ratios when the thermal capillary Reynolds number is slightly above the critical value. Many traveling curved spoke patterns are observed on the free surface, which correspond to the HTWs instability. Nevertheless, the propagation angle of the HTWs in the n-decane/n-hexane mixture, measured at R = 1.5, is slightly bigger than 38° which was predicted by the linear stability analysis in an infinite liquid layer [4]. This difference could be mainly attributed to the solutal capillary and buoyancy forces in the n-decane/n-hexane mixture. In a shallow annular pool, the effect of the solutal capillary force is more evident than that of the buoyancy force. At a small aspect ratio, the HTWs propagating in the clockwise direction are clearly observed near the inner cylinder of the annular pool and fade in the outer region, as shown in Fig. 7(a). Many straight lines which are parallel and tilted to the left are indicated in the STD. This phenomenon is comparable to the HTWs that were observed by Peng et al. [8]. With the increase of the aspect ratio, most region of the annular pool is invaded by the HTWs. Two groups of HTWs which are propagating in opposite direction always coexist, but one of them is much weaker than the other, as shown in Fig. 7(b) and (c). At e = 0.11, a three-dimensional oscillatory flow (3DOF) firstly appears near the outer cylinder, as shown

0.744 -90

0.49978

0.742

0.49976

-100

-110

VR

Φ

0.740 0.49974

Θ

1076

0.738 -120

0.49972

0.49970 0.000

Fig. 5. Variation of the critical thermal capillary Reynolds number.

0.736

Θ

Φ

0.021

0.028

VR 0.007

0.014

τ−τ0

0.734 -130 0.035

Fig. 6. Periodic oscillations of the radial velocity, temperature and solute concentration at a monitoring point P at e = 0.12 and ReT = 30,576.

J.-J. Yu et al. / International Journal of Heat and Mass Transfer 90 (2015) 1071–1081

(a) =0.04 and ReT=32760

(b) =0.09 and ReT=8299

(c) =0.11 and ReT=6770

(d) =0.12 and ReT=6552 Fig. 7. Snapshots of surface temperature fluctuation (left) and STD at R = 1.5 (right).

0.12

R =1

0.06

R= 1.

Z

5

.965

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Fig. 8. Temperature distribution as a function of Z at h = 0, e = 0.12 and ReT = 6552.

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in Fig. 7(c). In this case, there is a counter vertical temperature gradient layer in the region of 0 < Z < 0.04 near the outer cylinder, as shown in Fig. 8. Furthermore, the counter vertical temperature gradient is large enough to generate the Rayleigh–Bénard convection. These Rayleigh–Bénard rolls can be observed on the free surface near the outer cylinder and are driven by the HTWs to propagate in the azimuthal direction. It should be noticed that the tilted lines in the STD are fluctuated, as shown in Fig. 7(c). In the annular pools of e P 0.12, both of the 3DOF and HTWs which are traveling in clockwise or anticlockwise direction coexist on the free surface. The HTWs are indicated by the twenty-three parallel slanted straight lines on the STD, as shown in Fig. 7(d). These results show that the aspect ratio has an important effect on the pattern of the thermocapillary–buoyancy flow of the n-decane/n-hexane mixture when the thermal capillary Reynolds number is slightly above the critical value. At e 6 0.08, the capillary force is dominant. On the free surface, only the HTWs are observed. At e P 0.12, the buoyancy force is comparatively important, and both of the HTWs and 3DOF coexist and travel with the same angular velocity in the same direction. Therefore, there is a transition region of 0.08 < e < 0.12. In this region, the enhancing buoyancy force begins to breach the regime based on the capillary force for three-dimensional oscillatory flow. One of two groups of HTWs propagating in opposite direction gradually becomes weaker with the increase of the aspect ratio. Once the aspect ratio rises to 0.12, only one group of HTWs can be observed. Compared this evolution of the flow pattern transition of the n-decane/n-hexane mixture with that of silicone oil reported by Peng et al. [8], it is found that the existence region of the HTWs in the n-decane/n-hexane mixture is larger than that in 0.65 cSt silicone oil of Pr = 6.7. Due to the Soret effect in the n-decane/n-hexane mixture, the solutal capillary and buoyancy forces near the outer cylinder are in the same direction with the thermal capillary and buoyancy forces, respectively. Therefore, mutual reinforcing forces result in this variation of the flow pattern transition. In the n-decane/n-hexane mixture layer at ReT = 21,840, with the increase of the aspect ratio, the oscillatory flow pattern transits from the HTWs to a combination of the HTWs and 3DOF, as shown in Fig. 9(a) and (b). Furthermore, the 3DOF extends its activity space dR with the increase of the aspect ratio as a result of the enhancing buoyance force. Fig. 9(b) and (c) display the snapshots of the surface temperature fluctuation in the n-decane/n-hexane mixture obtained by the numerical simulation and experiment at e = 0.12. The numerical result shows a good agreement with the experiment. Fig. 10 shows the influence of the aspect ratio on the nondimensional fundamental oscillation frequency F and the azimuthal wave number m in the n-decane/n-hexane mixture at ReT = 13,104 and the n-hexane fluid at ReT = 21,840, respectively. It can be seen that an increase of the aspect ratio leads to a decrease of the oscillation frequency. The variation trend is consistent with the experimental results of 0.65 cSt silicone oil in a long container reported by Daviaud and Vince [34]. When the aspect ratio is increased from 0.08 to 0.13, the azimuthal wave number m in the n-decane/n-hexane mixture decreases from 29 to 16. The variation trend of the azimuthal wave number m in the n-hexane fluid is similar to that in the n-decane/n-hexane mixture. Moreover, the difference of the azimuthal wave number m between the n-decane/n-hexane mixture and n-hexane fluid is within 2 at e P 0.08. When e = 0.08, the oscillatory flow in the n-hexane fluid still belongs to the HTWs. However, 3DOF has appeared near the outer cylinder in the n-decane/n-hexane mixture at the same aspect ratio. The Soret effect in the n-decane/n-hexane mixture has a small impact on the nondimensional fundamental oscillation frequency and wave number.

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(a) simulation, =0.05

(b) simulation, =0.12

(c) experiment, =0.12

Fig. 9. Snapshots of surface temperature fluctuation of the n-decane/n-hexane mixture at ReT = 21,840.

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0

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Fig. 10. Variation of oscillation frequency F (triangle) and wave number m (square) with the aspect ratio e. Solid line: n-decane/n-hexane at ReT = 13,104; dashed line: n-hexane at ReT = 21,840; dotted line: Ref. [34].

3.3.2. Effect of thermal capillary Reynolds number For a shallow annular pool of e = 0.08, the evolution of the three-dimensional oscillatory flow patterns with the thermal capillary Reynolds number is similar to that with the aspect ratio.

With the increase of the thermal capillary Reynolds number, the HTWs and a combination of the HTWs and 3DOF appear orderly, as shown in Fig. 11(a) and (b). When ReT = 21,840, the surface temperature fluctuation in the n-decane/n-hexane mixture is different from that in the n-hexane fluid, as shown in Fig. 11(b) and (c). In the n-decane/n-hexane mixture, the HTWs and 3DOF coexist in the entire liquid pool and travel in the anticlockwise direction. In the n-hexane fluid, the HTWs propagate in the clockwise direction and exhibit much more intense temperature oscillatory near the inner cylinder. Furthermore, they fade in about four-fifths of the radial length. Moreover, the HTWs travel in clockwise or anticlockwise directions. Zhan et al. [10] pointed out that the numerical error maybe result in the preferred direction of the HTWs propagating. When the thermocapillary Reynolds number is large enough, a considerable solute concentration gradient is established in the n-decane/n-hexane mixture. In this case, the solute concentration fluctuation is gradually augmented and observed, as shown in Fig. 11(d) and (e), which is similar to the temperature fluctuation wave. However, the nondimensional fluctuation amplitude of the solute concentration is relatively weak compared with that of the temperature, because the solute concentration fluctuation is

Fig. 11. Snapshots of surface temperature (a–c) and solute concentration (d, e) fluctuations at e = 0.08. (a) n-decane/n-hexane, ReT = 10,920; (b) n-decane/n-hexane, ReT = 21,840; (c) n-hexane, ReT = 21,840. (d) n-decane/n-hexane, ReT = 21,840; (e) n-decane/n-hexane, ReT = 39,312.

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Fig. 12. The circumferential view of flow structure at R = 1.5 in the case of e = 0.08 and ReT = 21,840. (a) the n-decane/n-hexane mixture, azimuthal velocity (upper), Vh(+) = 5.04, Vh() = 5.59, DVh = 0.82, axial velocity (bottom), VZ(+) = 1.41, VZ() = 2.94, DVZ = 0.44; (b) the n-hexane fluid, azimuthal velocity (upper), Vh(+) = 5.53, Vh() = 5.57, DVh = 0.74, axial velocity (bottom), VZ(+) = 1.88, VZ() = 3.19, DVZ = 0.51.

driven by the temperature fluctuation. At ReT = 39,312, a deformation of the solute concentration fluctuation wave near the outer cylinder also appears, like the temperature fluctuation wave, as shown in Fig. 11(e). The circumferential views of the flow structure at R = 1.5 in the case of e = 0.08 and ReT = 21,840 are shown in Fig. 12. The liquid layer is occupied by the slanted roll cells. The numbers of the traveling roll cells in the n-decane/n-hexane mixture and the n-hexane fluid are 26 and 20, respectively. It should be noted that the axial and azimuthal flow at R = 1.5 in the n-hexane fluid is slightly stronger than that in the n-decane/n-hexane mixture. The traveling roll cells in the liquid layer have a suppression effect on the axial and azimuthal flow. In addition, the mean solute concentration in the mid-region is close to the initial solute concentration and the solute concentration gradient in this region is very small, as shown in Fig. 2(a) and (b). The solutal capillary and buoyancy forces have slight effect on the axial and azimuthal velocity in the mid-region. Therefore, the difference of the axial and azimuthal velocity in the mid-region between the n-decane/n-hexane mixture and the n-hexane fluid is mainly related to the number of traveling roll cells. In order to extract the fundamental frequency of the oscillatory flow, the spectral amplitude of the azimuthal oscillatory velocity at the monitoring point P (R = 1.5, h = 0, Z = 0.12) has been calculated by the algorithm of the fast Fourier transform (FFT), as shown in Fig. 13. When the thermal capillary Reynolds number is slightly over the critical value at e = 0.12, a single frequency of F1 = 30.35 is observed, as shown in Fig. 13(a), which means that the steady and axisymmetric basic flow has bifurcated to the periodic oscillation flow. Once the thermal capillary Reynolds number is increased to 7000, two main peaks are easily observed at F1 = 38.12 and F2 = 2F1, as shown in Fig. 13(b). Here, F1 is the fundamental frequency and F2 is the harmonic frequency. At the range of 11,356 6 ReT 6 21,840, there are three large peaks with F1 = 54.92, F2 = 2F1 and F3 = 3F1, as shown in Fig. 13(c). When the thermal capillary Reynolds number is further increased to ReT = 30,576, four dominating peaks are F1 = 114.5, F2 = 2F1, F2 = 3F1 and F4 = 4F1, as shown in Fig. 13(d). All frequencies in the periodic oscillation flow are able to be expressed as a linear combination of the fundamental frequency F1, which is similar to the results of 0.65 cSt silicone oil reported by Li et al. [35]. Compared with the evolution of the frequencies with the thermal capillary Reynolds number in the pure fluid [35], the solutal capillary and

buoyancy forces in the n-decane/n-hexane mixture make the oscillation flow much more complicated than that in a pure fluid. The variations of the nondimensional oscillation frequency F and the azimuthal wave number m of the HTWs in both the n-decane/n-hexane mixture and n-hexane fluid with the thermal capillary Reynolds number at e = 0.11 are shown in Fig. 14. It is found that the fundamental oscillation frequency increases monotonously with the increase of the thermal capillary Reynolds number. Under the same conditions, the nondimensional oscillation

Fig. 13. Spectrum amplitude of the azimuthal velocity at the monitoring point P for the oscillatory flow in the n-decane/n-hexane mixture at e = 0.12.

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Conflict of interest

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Acknowledgment

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This work is supported by National Natural Science Foundation of China (Grant No. 51176209). References

1.0

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Fig. 14. Variations of oscillation frequency F (triangle) and wave number m (square) with the thermal capillary Reynolds number ReT at e = 0.11. Solid line: ndecane/n-hexane; dashed line: n-hexane.

frequency F in the n-hexane fluid is higher than that in the n-decane/n-hexane mixture. The smaller viscosity of the n-hexane fluid should be responsible for this difference. Peng et al. [8] and Teitel et al. [36] also found the similar variation trend of the oscillation frequency in 0.65 cSt and 2 cSt silicone oils in a shallow annular pool and a model of Czochralski growth, respectively. However, the wave numbers in both the n-decane/n-hexane mixture and the n-hexane fluid are almost independent of the thermal capillary Reynolds number, which is similar to the simulation results of 0.65 cSt silicone oil in a shallow annular pool with e = 0.05 reported by Shi et al. [30]. Compared with the trend of wave numbers in Fig. 10, it suggests that the aspect ratio of the shallow annular pool plays an essential role on the wave numbers in both the n-decane/n-hexane mixture and the n-hexane fluid.

4. Conclusions A series of unsteady 3-D numerical simulations on the thermoca pillary–buoyancy flow in the n-decane/n-hexane mixture in an annular shallow pool were performed. The effects of the aspect ratio and thermal capillary Reynolds number were analyzed in detail. From the results, the main conclusions can be drawn as follows: (1) Due to the Soret effect in the n-decane/n-hexane mixture, the increase of the thermal capillary Reynolds number induces a large concentration gradient. When the thermoca pillary–buoyancy flow transits to a three-dimensional oscillatory flow, the solute concentration fluctuation is observed, which is similar to the temperature fluctuation. However, the nondimensional fluctuation amplitude of the concentration is relatively weak compared with that of the temperature. (2) The critical thermal capillary Reynolds number for the incipience of the oscillatory flow decreases with the increase of the aspect ratio. Due to the Soret effect, the critical thermal capillary Reynolds number in the n-decane/n-hexane mixture is always smaller than that in the n-hexane fluid. (3) After the three-dimensional oscillatory flow appears, the fundamental oscillation frequency increases monotonously with the increase of the thermal capillary Reynolds number, but decreases with the aspect ratio. Furthermore, the wave number decreases with the increase of the aspect ratio when the hydrothermal waves and three-dimensional oscillatory flow coexist. However, it is independent of the thermal capillary Reynolds number.

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