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Effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool
Journal of Crystal Growth 514 (2019) 21–28
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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool
T
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Li Zhanga,b, You-Rong Lia, , Chun-Mei Wua, Lu Zhanga a
Key Laboratory of Low-grade Energy Utilization Technologies and Systems of Ministry of Education, School of Energy and Power Engineering, Chongqing University, Chongqing 400044, China b Chongqing City Management College, Chongqing 401331, China
A R T I C LE I N FO
A B S T R A C T
Communicated by Jun-ichi Nishizawa
This paper presents a series of three-dimensional numerical simulations on the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool. The annular pool with a fixed aspect ratio of 0.05 and radius ratio of 0.5 is filled with 0.65 cSt silicone oil. Its Prandtl number is 6.7. Biot number of surface heat dissipation ranges from 0 to 50. The results show that when Marangoni number is small, thermocapillary convection is the axisymmetric steady flow. When Marangoni number exceeds the critical value, the basic flow will destabilize and transit to the hydrothermal waves that propagate along the clockwise or anticlockwise direction. The critical Marangoni number of the flow destabilization increases with the increase of Biot number. Furthermore, when Biot number increases, the radial temperature gradient on the free surface decreases near the inner cylindrical wall, but increases near the outer cylindrical wall, which results in the thermocapillary convective cell moving gradually from the inner to the outer cylindrical walls. After the flow destabilizes, the temperature fluctuation on the free surface mainly appears near the outer cylinder at a large Biot number.
Keywords: A1. Computer simulation A1. Convection A1. Heat transfer A2. Microgravity conditions
1. Introduction Thermocapillary convection driven by surface tension gradient widely exists in industrial production and natural phenomena such as crystal growth, film coating, droplet and liquid layer evaporation. In order to understand and control the melt flow to improve the quality of crystal growth, researchers have performed a lot of researches on thermocapillary convection and thermocapillary-buoyancy convection by experiments, numerical simulations and theoretical analyses, and achieved fruitful results [1–3]. Smith and Davis [4,5] performed first linear stability analysis on thermocapillary convection in an infinite horizontal liquid layer. It was found that the steady thermocapillary convection transits to two flow patterns with the increase of the tangential temperature gradient, i.e., inclined hydrothermal wave (HTW) propagating along a certain angle and static longitudinal vortices. They also certified that formation mechanism of the HTW is the phase lag between temperature and velocity fluctuations. Garnier and Normand [6] studied thermocapillary convection in an annular pool by linear stability analysis. The results show that the flow pattern after the flow destabilization is the HTW, and its basic characteristics are similar to those predicted by Smith and
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Corresponding author. E-mail address: [email protected] (Y.-R. Li).
https://doi.org/10.1016/j.jcrysgro.2019.02.060
Available online 27 February 2019 0022-0248/ © 2019 Elsevier B.V. All rights reserved.
Davis [4]. However, the HTWs are curved on the azimuthal direction, and the flow destabilizes first near the cold inner cylindrical wall. Garnier and Chiffaudel [7] and Yu et al. [8] reported the experimental results of thermocapillary convection in a shallow annular liquid pool by using silicone oil (Pr = 10), n-hexane (Pr = 5.52) and the binary mixtures as the working fluids respectively. Due to the special structure of the annular pool, new characteristics of the HTWs near the cold inner cylindrical wall were observed. Torregrosa et al. [9] and Hoyas et al. [10] studied the flow pattern evolution characteristics of thermocapillary convection in an annular pool. It was found that aspect ratio, temperature gradient, Biot (Bi) number and Prandtl number have important effects on the flow bifurcation sequence. Li et al. [11–13] carried out three-dimensional numerical simulations of thermocapillary convection and thermocapillary-buoyancy convection of silicon melt and 0.65 cSt silicone oil in an annular liquid pool, and the simulation results were basically consistent with the experimental results of Azami et al. [14] and Schwabe [15]. In above works mentioned, the adiabatic free surface is assumed to simplify. However, because of the interfacial non-equilibrium effect, surface heat dissipation to the environment should be taken into consideration. Jing et al. [16] carried out three-dimensional numerical
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simulations of thermocapillary-buoyancy convection of LiNbO3 melt in an open crucible, taking into account the radiative heat loss on the free surface. The results indicate that when there is heat dissipation, a thin thermal boundary layer will be formed near the free surface, and the reverse temperature gradient in the thermal boundary layer will cause Marangoni instability, which will lead to the formation of the spoke patterns. Sim et al. [17–19] performed a series of numerical simulations on thermocapillary convection in the open cylindrical annular pools for moderate Prandtl number fluids when surface heat dissipation and liquid pool rotation were considered. It was found that there are two different flow destabilization modes: the pulsed wave with wave number of 2 and the rotating wave with wave number of 3 when the free surface is assumed to be plane. Hoyas et al. [20,21] investigated the flow transition in an open cylindrical cavity with bottom heating and surface heat dissipation by linear stability analysis. The results show that the flow patterns and the flow bifurcation routes after the flow destabilization mainly depend on Marangoni number, Biot number and Prandtl number. Because surface heat dissipation has an important influence on thermocapillary convection, the adiabatic free surface assumption is not suitable. In order to reveal the physical mechanism of effect of surface heat dissipation on the flow stability and the flow pattern transition, this paper presented a set of three-dimensional numerical simulations on thermocapillary convection in a shallow annular pool for moderate Prandtl number fluid. The results are compared with those from the existing studies [22,23].
Based on the above physical model, continuity equation, momentum equation and energy equation describing thermocapillary convection in annular liquid pool are expressed as follows
∇ · V= 0
(1)
∂V + V·∇ V= −∇P + ∇2 V ∂τ
(2)
∂Θ 1 2 + V·∇Θ = ∇Θ ∂τ Pr
(3)
In Eqs. (1)–(3), the dimensionless variables V, τ and P represent velocity vector, time and pressure, respectively.Θ represents the dimensionless temperature and is defined as
Θ = (T − Ti )/(To − Ti )
(4)
The initial conditions are:
τ = 0, U = V = W = 0, Θ = −ln[R (1 − η)/η]/lnη The proper boundary conditions are: At the inner cylinder (R = Ri = ri/(ro − ri) = η/(1 − η), 0 ≤ Z ≤ ε):
U = V = W = 0, Θ = 0; At the 0 ≤ Z ≤ ε):
outer
cylinder
(R = Ro = ro/(ro − ri) = 1/(1 − η),
U = V = W = 0, Θ = 1; At the bottom (Z = 0, η/(1 − η) < R < 1/(1 − η)):
U = V = W = 0, ∂Θ/ ∂Z = 0.
2. Problem statement
At the free surface (Z = ε, η/(1 − η) < R < 1/(1 − η)), the radial velocity U and the azimuthal velocity V depend on the balance between the thermocapillary force and the shear stress. Meanwhile, the axial velocity W is small enough to be ignored. Therefore, we have
2.1. Physical and mathematical models The physical model is shown in Fig. 1. The inner and outer radius of the annular liquid pool are ri and ro respectively, and the depth is d. The liquid pool is filled with 0.65 cSt silicone oil with Pr = 6.7. The radius ratio and the aspect ratio of the liquid pool are defined as η = ri/ro and ε = d/(ro − ri), respectively. The bottom is an adiabatic solid wall, noslip and impermeable conditions are applied for all solid-liquid boundaries. The top is a free surface and there is heat dissipation to the environment. The total surface heat dissipation coefficient is h, and the temperatures for the inner and outer cylindrical walls maintain constant Ti and To (To > Ti), respectively. For simplification, the following assumptions are introduced. (1) The fluid is incompressible Newtonian fluid. (2) All physical parameters are constant except surface tension. (3) Flow is laminar, and viscous dissipation is neglected. In order to reduce the number of variables describing the problem, the following dimensionless parameters are introduced by applying (ro − ri), ν/(ro − ri), (ro − ri)2/ν and μν/(ro − ri)2 as scale quantities for length, velocity, time and pressure, respectively,
∂U Ma ∂Θ ∂V Ma ∂Θ =− , =− , W = 0. ∂Z Pr ∂R ∂Z Pr R∂θ where Ma = γTΔT(ro − ri)/(μα), γT, μ and α are respectively surface tension temperature coefficient, dynamic viscosity, and thermal diffusivity. The thermal boundary condition on the free surface is expressed as
− λ ∂T / ∂z = h (T − T0),
(10)
where λ is thermal conductivity. T0 is ambient temperature. Supposing T0 = Ti, Eq. (10) is also expressed in the dimensionless form
− ∂Θ/ ∂Z = BiΘ.
(11)
In which, Bi is surface heat dissipation Biot number, Bi = h (ro − ri)/λ. In this work, the radius ratio and the aspect ratio of the annular pool are respectively fixed at η = 0.5 and ε = 0.05. Considering actual dimensions of the annular pool and the range of convective heat transfer coefficient, the surface heat dissipation Biot number varies from 0 to 50.
(r , z ) (u, v, w ) t , (U , V , W ) = , τ= , P (ro − ri )2 / ν ro − ri ν /(ro − ri ) p = μν /(ro − ri )2
(R, Z ) =
2.2. Solution procedure and validation The governing equations and the corresponding boundary conditions are discretized by the finite volume method, in which the diffusion term adopts the second order central difference and the convection term is QUICK scheme. The pressure-velocity correction is SIMPLE algorithm. According to different Marangoni numbers and Biot numbers, the dimensionless time step is selected in the range of (1 ∼ 4) × 10−5. During the whole iterative process, if the maximum relative error of temperature and velocity is less than 10−5, we considered the solution to be convergent. Because the velocity and temperature gradients near the solid wall and free surface are larger than those in other regions, non-uniform
Fig. 1. Physical model and the coordinate system. 22
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(a) Bi=0, ψmax=2.52
(b) Bi=5, ψmax=5.20
(c) Bi=10, ψmax=7.25
(d) Bi=50, ψmax=10.64 Fig. 2. Streamlines (upper) and isotherms (lower) for steady thermocapillary convection in R-Z plane when Ma = 1.8 × 105. δΘ = 0.05, δψ = ψmax/20.
staggered grid of 102R × 22Z × 240θ with denser meshes near the solid walls and the free surface is applied. This mesh is finer than one in our previous work [11] for the moderate Prandtl number fluids. In order to verify the calculation method, under the same experimental condition as that of Schwabe et al. [15], the critical condition of thermocapillary flow destabilization for 0.65 cSt silicone oil with Pr = 6.7 in the annular pool of ε = 0.125 was simulated numerically. It was found that the critical Marangoni number deviation between the calculated and the experimental values was less than 5%, which indicated that the calculation results in this paper are believable.
so the radial temperature gradient in the middle region is much smaller than that in the inner and outer walls, and the isotherm distribution is more uniform than that in the low Prandtl number fluid. Because there is the large radial flow cross section near the outer cylindrical wall, the center of the flow cell is closer to the outer cylindrical wall. When there is surface heat dissipation, with the increase of Biot number, the heat dissipated into the environment increases gradually. Therefore, the surface average temperature decreases, the temperature gradient near the inner cylindrical wall decreases, and the isotherm becomes sparse. On the contrary, in order to supplement the surface heat dissipation, the temperature gradient near the outer wall increases, so that the flow near the outer wall is enhanced, and the maximum dimensionless flow function ψmax increases gradually, as shown in Fig. 2(b). When Biot number increases to Bi = 10, the thermal boundary layer near the inner cylindrical wall disappears, and the radial temperature gradient mainly exists near the outer cylindrical wall. Therefore, the flow near the inner wall is very weak and the streamlines is very sparse, as shown in Fig. 2(c). When Biot number increases further, although the flow is continually enhanced and dimensionless flow function increases, the increasing trend is slowing down. Meanwhile, when Bi > 20, a flow stagnation region will be formed near the inner cylindrical wall, which will gradually expand to the outer wall with the increase of Biot number. When Bi = 50, the stagnant region occupies more than half of the liquid pool, and the flow is only confined near the outer cylindrical wall, as shown in Fig. 2(d). Fig. 3 reveals the effect of Marangoni number on the distributions of streamlines and isotherms for steady thermocapillary convection at Bi = 10. When Marangoni number is small, such as Ma = 104, the flow is very weak and is mainly concentrated in about 1/3 region near the outer cylindrical wall. Therefore, the radial variation of the temperature mainly occurs near the outer cylindrical wall, and the isotherms are concentrated near the outer wall, while the flow is stagnant near the inner cylindrical wall and almost half of the radial width the fluid is isothermal, as shown in Fig. 3(a). With the increase of Marangoni number, the inward flow along the free surface strengthens, so the flow gradually expands to the inner cylindrical wall. When the Marangoni number increases to Ma = 105, the streamlines occupy the whole pool, as shown in Fig. 3(c). As Marangoni number continues to increase, the flow near the inner wall strengthens gradually and the radial
3. Results and discussion 3.1. Basic flow When the radial temperature gradient applied to the liquid pool is small, thermocapillary convection is stable. This stable thermocapillary convection is called basic flow. Because the radial temperature gradient on the free surface varies with the surface heat dissipation conditions, thus the structure and intensity of steady thermocapillary convection will be affected. Fig. 2 shows streamlines and isotherms in the R-Z plane for steady thermocapillary convection at different Biot numbers when Ma = 1.8 × 105, where ψ is dimensionless stream function. Obviously, the high temperature fluid near the outer cylindrical wall flows to the inner cylindrical wall along the free surface under the action of the surface tension gradient, and then flows back to the high temperature wall near the bottom, which forms a counter-clockwise rotating flow cell. When the surface is adiabatic, the streamlines are almost parallel in the middle of the liquid pool. Due to the adiabatic free surface, the heat transfer to the inner cylindrical wall depends mainly on the convection along the free surface. Therefore, the isotherms are very dense in the upper region near the inner cylindrical wall, and relatively sparse in the lower region; the opposite is true near the outer cylindrical wall, but there is always the thermal boundary layer near the inner and outer cylindrical walls, as shown in Fig. 2(a). It should be noted that, due to the larger flow cross section and lower flow velocity near the outer wall, the radial temperature gradient on the free surface is smaller than that near the inner wall. On the other hand, for 0.65 cSt silicone oil with Pr = 6.7, the momentum diffusivity is greater than the heat diffusivity, 23
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(a) Ma=104, ψmax=0.90
(b) Ma=5×104, ψmax=2.41
(c) Ma=105, ψmax=4.24
(d) Ma=2.0×105, ψmax=7.98 Fig. 3. Streamlines (upper) and isotherms (lower) for axisymmetric steady flow in a R-Z plane at Bi = 10.δΘ = 0.05, δψ = ψmax/20.
unchanged. When Bi ≥ 10, the thermal boundary layer near the inner cylindrical wall disappears and the radial temperature gradient becomes very small, while the temperature gradient near the outer wall is very large, so the flow is mainly concentrated near the outer wall. Although the flow intensity will still increase, the flow region becomes narrow. Therefore, the critical Marangoni number of flow destabilization will increase gradually. On the other hand, when Biot number is very small, such as Bi ≤ 5, the flow pattern after destabilization is the hydrothermal wave (HTW), and its critical wave number is small, and basically remains unchanged. At this time, the temperature fluctuation mainly concentrates near the inner wall. When Bi > 5, due to the increase of the temperature gradient near the outer cylindrical wall, the center of the main flow cell is closer to the outer wall, and the temperature fluctuation increases after the flow destabilization. Therefore, in order to dissipate the temperature fluctuation, the azimuthal wave number will suddenly increase, and then remain almost constant. The maximum temperature fluctuation on the free surface appears near the outer cylindrical wall. Fig. 4 also shows the linear stability analysis results of Shi et al. [24] under the same conditions. Obviously, the critical Marangoni number and the critical wave number of flow destabilization are very close to the numerical results in this paper.
temperature gradient increases, but no thermal boundary layer is formed. Moreover, the center of the flow cell is always located near the outer cylindrical wall, and the secondary small flow cell appears in the large flow cell, as shown in Fig. 3(d). Therefore, the velocity and temperature fields of steady thermocapillary convection depend on the coupling effect of surface heat dissipation and thermocapillary convection. 3.2. Critical condition of flow destabilization When Marangoni number exceeds the critical value, the axisymmetric steady flow will destabilize and transit to three-dimensional time-dependent flow. Fig. 4 shows variations of the critical Marangoni number Macri and the critical wave number mcri with Biot number after the flow destabilization. In this work, the critical Marangoni number is obtained by the dichotomy. The maximum relative deviation of the critical Marangoni numbers is less than 2%. At a smaller Biot number, such as Bi < 10, with the increase of surface heat dissipation Biot number, the radial temperature gradient in the middle of the free surface increases, but the temperature distribution is more uniform, as shown in Fig. 2(a) and (b). Therefore, although the flow is enhanced, the critical Marangoni number Macri of flow destabilization is basically
10
3.3. Effect of Biot number on flow pattern
40
6
30
4
0
20
Ma m
2 0
10
20
Bi
30
40
For moderate Prandtl fluid, the axisymmetric steady flow will directly transit to three-dimensional time-dependent flow after the flow destabilization. When the Biot number is small, the steady flow first evolves into HTWs, while it will be into radial waves at a large Biot number. Fig. 5 shows the variation of the flow pattern with Biot number after the flow destabilization. When the free surface is adiabatic, that is, Bi = 0, the largest radial temperature gradient is located near the inner cylindrical wall. Therefore, the flow first becomes unstable near the inner cylindrical wall where temperature and velocity first start to oscillate. With the increase of Marangoni number, the fluctuations of velocity and temperature move gradually to the outer wall. Finally, the temperature fluctuation is observed on the whole free surface. Meanwhile, the radial position of the maximum temperature fluctuation is gradually shifted to the outer wall. In the shallow liquid layer, the flow
m cri
Ma cri /10 5
8
50
10
Fig. 4. Variations of critical Marangoni number (●) and critical wave number (○) of flow destabilization with Biot number. Triangle: stability analysis result of Shi et al. [24]. 24
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(a) Bi=0, Ma=2.5×105
When Biot number is large, the flow pattern after the flow destabilization is different from that at a small Biot number. For example, when Bi = 20, the temperature fluctuation on the free surface is similar to that at Bi = 5, as shown in Fig. 5(c) and (b). However, because of the large Marangoni number, the region occupied by the HTWs near the inner cylindrical wall is large, the temperature fluctuation region near the outer cylindrical wall is almost unchanged, but the overall temperature fluctuation range is further reduced. With the further increase of Biot number, for example Bi = 30, the flow pattern has an essential change. The HTWs near the inner cylindrical wall disappear and evolve into a radial rolling cell pattern, which originates near the inner wall and propagates radially to the outer wall. However, the formation and the propagation of the radial rolling cell are not synchronous in the azimuthal direction, as shown in Fig. 5(d). When Bi = 50, the radial temperature gradient becomes almost zero near the inner cylindrical wall, and only exists in a small region near the outer wall. Therefore, thermocapillary convection driven by the radial temperature gradient is very stable, the temperature fluctuation near the outer cylindrical wall is very small. The temperature fluctuation near the inner cylindrical wall is mainly caused by Marangoni-Bénard instability. At this time, in a very small region near the inner wall, the radial rolling cells will twist along the azimuthal direction. Fig. 6 shows variations of the maximum surface temperature fluctuation and the oscillatory frequency with Biot number at Ma = 106. At a small Biot number, the flow pattern belongs to the HTWs, so the temperature fluctuation is large and the main frequency has only slight variation. With the increase of Biot number, the temperature fluctuation amplitude gradually decreases, but the main frequency has a sudden increase near Bi = 5, and then basically remains unchanged.
(b) Bi=5, Ma=2.5×105
3.4. Evolution of flow pattern with Marangoni number
(c) Bi=20, Ma=4.0×105
Under different Biot numbers, the evolution of the flow pattern with Marangoni number is different. When the surface is adiabatic, i.e. Bi = 0, the flow pattern after the flow destabilization is the HTW, which originates near the inner cylindrical wall and then extends outward. Therefore, the largest temperature fluctuation is located near the inner cylindrical wall, as shown in Fig. 7(a), which has been confirmed by many theories and experiments [12–14]. With the increase of Marangoni number, the flow is enhanced, and the HTWs will be full of the whole liquid pool. When Marangoni continues to increase, two or more groups of the HTWs propagating along different directions will appear in the liquid pool. For example, when Ma = 3.0 × 105, two groups of coexisting HTWs are respectively located near the inner cylindrical wall and the outer wall. The HTWs near the outer wall propagate along counterclockwise direction, while the HTWs near the inner wall along clockwise direction. Because the flow at R = 1.5 is affected by two sets of HTWs, the STD diagram is composed of two sets of oblique lines overlapping to the upper left and the upper right, respectively, as
(d) Bi=30, Ma=1.0×106
Fig. 5. Snapshots (above) of surface temperature fluctuation and space-time diagram (lower) of surface temperature at R = 1.5 at different Biot numbers.
destabilization type under the action of tangential temperature gradient belongs to the HTW instability. The HTWs may propagate along clockwise or counterclockwise direction [12]. Therefore, the space-time diagram (STD) of the temperature variation with time at R = 1.5 on the free surface consists of a set of slopes inclining upward to the left. The angle ϕ between the propagating direction of the HTWs and the direction of the temperature gradient is 25–29°, which is close to the results of linear stability analysis [4], as shown in Fig. 5(a). When there is heat dissipation on the free surface, the temperature gradient near the inner cylindrical wall decreases, and near the outer wall increases. Therefore, the position of the maximum temperature fluctuation on the free surface will move to the outer cylindrical wall. At the same time, the flow cell near the outer wall strengthens. Therefore, a large temperature fluctuation concentrates near the outer wall, which also propagates along the azimuthal direction driven by the HTWs. Compared with Fig. 5(a) and (b), it is found that the temperature fluctuation caused by the flow cell near the outer cylindrical wall is smaller than that by the HTWs. With the increase of Biot number, the flow near the inner cylindrical wall gradually weakens, and the temperature fluctuation gradually diminishes until eventually disappears. Obviously, no matter what the cause of temperature fluctuation is, with the increase of Biot number, the HTWs gradually diminishes and the surface temperature fluctuation amplitude gradually decreases.
Fig. 6. Variations of the maximum surface temperature fluctuation δΘmax = (Θmax − Θmin) (○) and the oscillatory frequency (●) with Biot number at Ma = 106. 25
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(a) Ma=2.2×105
(b) Ma=3.0×105
(c) Ma=6.0×105
Fig. 7. Evolution of flow pattern with Marangoni number at Bi = 0: snapshots (above) of surface temperature fluctuation and STD (lower) of surface temperature at R = 1.5.
shown in Fig. 8(b). With the further increase of Marangoni number, temperature fluctuation near the inner cylindrical wall becomes clearer. It should be noted that the temperature fluctuations near the inner and outer walls are totally different. The former is caused by the hydrothermal waves, while the latter by the main flow cell oscillatory near the outer cylindrical wall. Moreover, with the increase of Marangoni number, the center of the main flow cell near the outer wall remains basically unchanged, so the temperature fluctuation region is almost independent on Marangoni number, as shown in Fig. 9. When the surface heat dissipation Biot number increases to Bi = 30, the temperature gradient near the inner cylindrical wall is almost zero, so there is no thermocapillary convection. However, due to the large surface heat dissipation Biot number, a large axial reverse temperature gradient near the free surface will trigger Marangoni-Bénard convection. In this case, the surface temperature fluctuation is completely different from that of the HTWs at a small Biot number, as shown in Fig. 10. Because the axis of Marangoni-Bénard flow cells is along the radial direction, the temperature fluctuations are arranged alternately
shown in Fig. 7(b). It should be noted that the clockwise propagating HTWs near the inner cylindrical wall occupy only about 1/4 of the width of the liquid pool, but the inducing temperature fluctuation is much greater than that caused by the counterclockwise propagating HTWs. With the further increase of Marangoni number, the flow pattern becomes more complex. The obvious HTWs can be observed only in the local region, as shown in Fig. 7(c). When there is surface heat dissipation, for example Bi = 10, the temperature gradient near the inner cylindrical wall becomes almost zero, but it near the outer cylindrical wall increases. Therefore, the main flow cell moves to the outer wall. When Ma = 2.3 × 105, the flow is just unstable, temperature fluctuation occurs only in half of the radial direction near the outer cylindrical wall, while no fluctuation appears in half near the inner cylindrical wall, as shown in Fig. 8(a). With the increase of Marangoni number, the inward thermocapillary flow on the free surface is enhanced, which drives the temperature fluctuation near the outer wall to extend inward. When Ma = 4.0 × 105, obvious temperature fluctuation can be observed on the whole free surface, as
(a) Ma=2.3×105
(b) Ma=4.0×105
(c) Ma=8.0×105
Fig. 8. Evolution of flow pattern with Marangoni number at Bi = 10: snapshots (above) of surface temperature fluctuation and STD (lower) of surface temperature at R = 1.8. 26
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(a) Ma=2.3×105, ψmax=8.91
(b) Ma=4.0×105, ψmax=14.40
(c) Ma=8.0×105, ψmax=23.73 Fig. 9. Snapshots of pseudo-streamlines (upper) and isotherms (lower) for three-dimensional oscillatory flow at the R-Z plane when Bi = 10. δΘ = 0.05, δψ = ψmax/ 20.
direction under the influence of the radial temperature fluctuation. Fig. 11 reveals variations of the fundamental oscillatory frequency and the maximum surface temperature fluctuation with Marangoni number. When the free surface is adiabatic, Marangoni number has little effect on the fluctuation frequency, but the maximum temperature fluctuation amplitude on the free surface increases with the increase of Marangoni number. When there is heat dissipation on the free surface, the fluctuation frequency increases rapidly with Marangoni number, but the maximum temperature fluctuation amplitude on the free surface is much lower than that at the adiabatic surface. When Bi = 10, with the increase of Marangoni number, the influence region of thermocapillary convection gradually shrinks. Therefore, the temperature fluctuation amplitude will increase first, then decrease slightly and then increase again. When Bi = 30, the temperature fluctuation amplitude increases monotonously with Marangoni number.
in the azimuthal direction. However, this temperature fluctuation is unstable, which originates near the inner cylindrical wall and then moves outward. When Marangoni number is small, such as Ma = 0.6 × 106 and 1.0 × 106, the movement of the temperature fluctuation along the radial direction is not synchronous, and there is always a phase lag, as shown in Fig. 10(a) and (b). It can also be found from the corresponding STD diagram that the temperature fluctuation in the azimuthal direction is not on the same horizontal line. When Marangoni number increases further, the flow is enhanced stronger and the surface temperature fluctuation becomes more complicated. At this point, the temperature fluctuations still originate near the inner cylindrical wall and then move outward. However, during the radial movement of temperature fluctuations, obvious azimuthal motion of temperature fluctuations can be observed in the local region on the free surface, as shown in Fig. 10(c). On the other hand, due to the large radial temperature gradient near the outer cylindrical wall, thermocapillary convection cells still exist in a very small region near the outer wall. The temperature fluctuation caused by the flow cell will no longer propagate along the azimuthal direction driven by the HTWs as the case of small Biot numbers, but merge and grow alternately in the azimuthal
(a) Ma=0.6×106
4. Conclusions A series of three-dimensional numerical simulations are performed to analyze the effect of surface heat dissipation on thermocapillary
(b) Ma=1.0×106
(c) Ma=2.0×106
Fig. 10. Evolution of flow pattern with Marangoni number at Bi = 30: snapshots (above) of surface temperature fluctuation and STD (lower) of surface temperature at R = 1.5. 27
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China (Grant No. 51776022, 11532015), Chongqing University Postgraduates Innovation Project (Grant No: CYS18041) and the Fundamental Research Funds for the Central Universities (No. 2018CDXYDL0001). References [1] H.G. Levich, V.S. Krylov, Surface tension-driven phenomena, Annual Rev. Fluid Mech. 1 (1969) 293–316. [2] M. Lappa, Thermal Convection: Patterns, Evolution and stability, Wiley & Sons, 2010. [3] W.Y. Shi, M.K. Ermakov, N. Imaishi, Effect of pool rotation on thermocapillary convection in shallow annular pool of silicone oil, J. Cryst. Growth 294 (2006) 474–485. [4] M.K. Smith, S.H. Davis, Instabilities of dynamic thermocapillary liquid layers 1. Convective instabilities, J. Fluid Mech. 132 (1983) 119–144. [5] M.K. Smith, S.H. Davis, Instabilities of dynamic thermocapillary liquid layers 2. Surface-wave instabilities, J. Fluid Mech. 132 (1983) 145–162. [6] N. Garnier, C. Normand, Effects of curvature on hydrothermal waves instability of radial thermocapillary flows, C R Acad Paris, t.2, Seried IV Phys. 2 (2001) 1227–1233. [7] N. Garnier, A. Chiffaudel, Two dimensional hydrothermal waves in an extended cylindrical vessel, Eur. Phys. J. B 19 (2001) 87–95. [8] J.J. Yu, D.F. Ruan, Y.R. Li, J.C. Chen, Experimental study on thermocapillary convection of binary mixture in a shallow annular pool with radial temperature gradient, Exp. Therm. Fluid Sci. 61 (2015) 79–86. [9] A.J. Torregrosa, S. Hoyas, M.J. Pérez-Quiles, J.M. Mompó-Laborda, Bifurcation diversity in an annular pool heated from below: Prandtl and Biot numbers effects, Commun. Comput. Phys. 13 (2013) 428–441. [10] S. Hoyas, P. Fajardo, M.J. Pérez-Quiles, Influence of geometrical parameters on the linear stability of a Bénard-Marangoni problem, Phys. Rev. E 93 (2016) 043105. [11] Y.R. Li, L. Peng, Y. Akiyama, N. Imaishi, Three-dimensional numerical simulation of thermocapillary flow of moderate Prandtl number fluid in an annular pool, J. Cryst. Growth 259 (2003) 374–387. [12] Y.R. Li, N. Imaishi, T. Azami, T. Hibiya, Three-dimensional oscillatory flow in a thin annular pool of silicon melt, J. Cryst. Growth 260 (2004) 28–42. [13] Y.R. Li, L. Peng, S.Y. Wu, D.L. Zeng, N. Imaishi, Thermocapillary convection in a differentially heated annular pool for moderate Prandtl number fluid, Int. J. Therm. Sci. 43 (2004) 587–593. [14] T. Azami, S. Nakamura, M. Eguchi, T. Hibiya, The role of surface-tension-driven flow in the formation of a surface pattern on a Czochralski silicon melt, J. Cryst. Growth 233 (2001) 99–107. [15] D. Schwabe, Buoyancy-thermocapillary and pure thermocapillary convective instabilities in Czochralski systems, J. Cryst. Growth 237–239 (2002) 1849–1853. [16] C.J. Jing, N. Imaishi, S. Yasuhiro, Y. Miyazawa, Three-dimensional numerical simulation of spoke pattern in oxide melt, J. Cryst. Growth 200 (1999) 204–212. [17] B.C. Sim, A. Zebib, Effect of free surface heat loss and rotation on transition to oscillatory thermocapillary convection, Phys. Fluids 14 (2002) 225–232. [18] B.C. Sim, A. Zebib, Thermocapillary convection with underformable curved surfaces in open cylinders, Int. J. Heat Mass Tran. 45 (2002) 4983–4994. [19] B.C. Sim, A. Zebib, D. Schwabe, Oscillatory thermocapillary convection in open cylindrical annuli. Part 2. Simlations, J. Fluid Mech. 491 (2003) 259–274. [20] S. Hoyas, P. Fajardo, A. Gil, M.J. Perez-Quiles, Analysis of bifurcations in a BénardMarangoni problem: gravitational effects, Int. J. Heat Mass Transf. 73 (2014) 33–41. [21] S. Hoyas, P. Fajardo, A. Gil, M.J. Perez-Quiles, Influence of geometrical parameters on the linear stability of a Bénard-Marangoni problem, Phys. Rev. E 93 (2016) 043105. [22] L. Zhang, Y.R. Li, C.M. Wu, Effect of surface heat dissipation on thermocapillary convection of low Prandtl number fluid in a shallow annular pool, Int. J. Heat Mass Transf. 110 (2017) 660–667. [23] L. Zhang, Y.R. Li, C.M. Wu, Q.S. Liu, Flow bifurcation routes to chaos of thermocapillary convection for low Prandtl number fluid in shallow annular pool with surface heat dissipation, Int. J. Therm. Sci. 125 (2018) 23–33. [24] W.Y. Shi, X. Liu, G. Li, Y.R. Li, L. Peng, M.K. Ermakov, N. Imaishi, Thermocapillary convection instability in shallow annular pools by linear stability analysis, J. Supercond. Nov. Magn. 23 (2010) 1185–1188.
Fig. 11. Variations of the fundamental oscillatory frequency (a) and the maximum surface temperature fluctuation (b) with Marangoni number.
convection of moderate Prandtl number fluid in an annular shallow pool. From the above-mentioned analysis, we can summarize the conclusions as follows: (1). At Bi < 10, the surface heat dissipation has little effect on the critical Marangoni number of flow destabilization. At Bi > 10, the critical Marangoni number increases with the increase of surface heat dissipation Biot number. At Bi ≤ 5, the flow pattern after the flow destabilization is the HTW, and its critical wave number is small and basically unchanged; at Bi = 5, the critical wave number has an abrupt increase. (2). When Biot number is small, the flow pattern after the flow destabilization is the HTWs. With the increase of Biot number, the radial rolling cells near the outer cylindrical wall is gradually enhanced, and the flow pattern eventually transits to the radial waves. For the HTWs at a small Biot number, the temperature fluctuation amplitude is large, and the fundamental frequency keeps almost constant. With the increase of Biot number, the temperature fluctuation amplitude decreases gradually. However, the fundamental oscillatory frequency increases sharply at Bi = 5, and then remains unchanged with the increase of Biot number. (3). When there is surface heat dissipation, the oscillatory fundamental frequency increases rapidly with the increase of Marangoni number. The amplitude of the temperature fluctuation on the free surface also increases in general, but is much lower than that on the adiabatic surface. Acknowledgement This work is supported by National Natural Science Foundation of
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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool
T
⁎
Li Zhanga,b, You-Rong Lia, , Chun-Mei Wua, Lu Zhanga a
Key Laboratory of Low-grade Energy Utilization Technologies and Systems of Ministry of Education, School of Energy and Power Engineering, Chongqing University, Chongqing 400044, China b Chongqing City Management College, Chongqing 401331, China
A R T I C LE I N FO
A B S T R A C T
Communicated by Jun-ichi Nishizawa
This paper presents a series of three-dimensional numerical simulations on the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool. The annular pool with a fixed aspect ratio of 0.05 and radius ratio of 0.5 is filled with 0.65 cSt silicone oil. Its Prandtl number is 6.7. Biot number of surface heat dissipation ranges from 0 to 50. The results show that when Marangoni number is small, thermocapillary convection is the axisymmetric steady flow. When Marangoni number exceeds the critical value, the basic flow will destabilize and transit to the hydrothermal waves that propagate along the clockwise or anticlockwise direction. The critical Marangoni number of the flow destabilization increases with the increase of Biot number. Furthermore, when Biot number increases, the radial temperature gradient on the free surface decreases near the inner cylindrical wall, but increases near the outer cylindrical wall, which results in the thermocapillary convective cell moving gradually from the inner to the outer cylindrical walls. After the flow destabilizes, the temperature fluctuation on the free surface mainly appears near the outer cylinder at a large Biot number.
Keywords: A1. Computer simulation A1. Convection A1. Heat transfer A2. Microgravity conditions
1. Introduction Thermocapillary convection driven by surface tension gradient widely exists in industrial production and natural phenomena such as crystal growth, film coating, droplet and liquid layer evaporation. In order to understand and control the melt flow to improve the quality of crystal growth, researchers have performed a lot of researches on thermocapillary convection and thermocapillary-buoyancy convection by experiments, numerical simulations and theoretical analyses, and achieved fruitful results [1–3]. Smith and Davis [4,5] performed first linear stability analysis on thermocapillary convection in an infinite horizontal liquid layer. It was found that the steady thermocapillary convection transits to two flow patterns with the increase of the tangential temperature gradient, i.e., inclined hydrothermal wave (HTW) propagating along a certain angle and static longitudinal vortices. They also certified that formation mechanism of the HTW is the phase lag between temperature and velocity fluctuations. Garnier and Normand [6] studied thermocapillary convection in an annular pool by linear stability analysis. The results show that the flow pattern after the flow destabilization is the HTW, and its basic characteristics are similar to those predicted by Smith and
⁎
Corresponding author. E-mail address: [email protected] (Y.-R. Li).
https://doi.org/10.1016/j.jcrysgro.2019.02.060
Available online 27 February 2019 0022-0248/ © 2019 Elsevier B.V. All rights reserved.
Davis [4]. However, the HTWs are curved on the azimuthal direction, and the flow destabilizes first near the cold inner cylindrical wall. Garnier and Chiffaudel [7] and Yu et al. [8] reported the experimental results of thermocapillary convection in a shallow annular liquid pool by using silicone oil (Pr = 10), n-hexane (Pr = 5.52) and the binary mixtures as the working fluids respectively. Due to the special structure of the annular pool, new characteristics of the HTWs near the cold inner cylindrical wall were observed. Torregrosa et al. [9] and Hoyas et al. [10] studied the flow pattern evolution characteristics of thermocapillary convection in an annular pool. It was found that aspect ratio, temperature gradient, Biot (Bi) number and Prandtl number have important effects on the flow bifurcation sequence. Li et al. [11–13] carried out three-dimensional numerical simulations of thermocapillary convection and thermocapillary-buoyancy convection of silicon melt and 0.65 cSt silicone oil in an annular liquid pool, and the simulation results were basically consistent with the experimental results of Azami et al. [14] and Schwabe [15]. In above works mentioned, the adiabatic free surface is assumed to simplify. However, because of the interfacial non-equilibrium effect, surface heat dissipation to the environment should be taken into consideration. Jing et al. [16] carried out three-dimensional numerical
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simulations of thermocapillary-buoyancy convection of LiNbO3 melt in an open crucible, taking into account the radiative heat loss on the free surface. The results indicate that when there is heat dissipation, a thin thermal boundary layer will be formed near the free surface, and the reverse temperature gradient in the thermal boundary layer will cause Marangoni instability, which will lead to the formation of the spoke patterns. Sim et al. [17–19] performed a series of numerical simulations on thermocapillary convection in the open cylindrical annular pools for moderate Prandtl number fluids when surface heat dissipation and liquid pool rotation were considered. It was found that there are two different flow destabilization modes: the pulsed wave with wave number of 2 and the rotating wave with wave number of 3 when the free surface is assumed to be plane. Hoyas et al. [20,21] investigated the flow transition in an open cylindrical cavity with bottom heating and surface heat dissipation by linear stability analysis. The results show that the flow patterns and the flow bifurcation routes after the flow destabilization mainly depend on Marangoni number, Biot number and Prandtl number. Because surface heat dissipation has an important influence on thermocapillary convection, the adiabatic free surface assumption is not suitable. In order to reveal the physical mechanism of effect of surface heat dissipation on the flow stability and the flow pattern transition, this paper presented a set of three-dimensional numerical simulations on thermocapillary convection in a shallow annular pool for moderate Prandtl number fluid. The results are compared with those from the existing studies [22,23].
Based on the above physical model, continuity equation, momentum equation and energy equation describing thermocapillary convection in annular liquid pool are expressed as follows
∇ · V= 0
(1)
∂V + V·∇ V= −∇P + ∇2 V ∂τ
(2)
∂Θ 1 2 + V·∇Θ = ∇Θ ∂τ Pr
(3)
In Eqs. (1)–(3), the dimensionless variables V, τ and P represent velocity vector, time and pressure, respectively.Θ represents the dimensionless temperature and is defined as
Θ = (T − Ti )/(To − Ti )
(4)
The initial conditions are:
τ = 0, U = V = W = 0, Θ = −ln[R (1 − η)/η]/lnη The proper boundary conditions are: At the inner cylinder (R = Ri = ri/(ro − ri) = η/(1 − η), 0 ≤ Z ≤ ε):
U = V = W = 0, Θ = 0; At the 0 ≤ Z ≤ ε):
outer
cylinder
(R = Ro = ro/(ro − ri) = 1/(1 − η),
U = V = W = 0, Θ = 1; At the bottom (Z = 0, η/(1 − η) < R < 1/(1 − η)):
U = V = W = 0, ∂Θ/ ∂Z = 0.
2. Problem statement
At the free surface (Z = ε, η/(1 − η) < R < 1/(1 − η)), the radial velocity U and the azimuthal velocity V depend on the balance between the thermocapillary force and the shear stress. Meanwhile, the axial velocity W is small enough to be ignored. Therefore, we have
2.1. Physical and mathematical models The physical model is shown in Fig. 1. The inner and outer radius of the annular liquid pool are ri and ro respectively, and the depth is d. The liquid pool is filled with 0.65 cSt silicone oil with Pr = 6.7. The radius ratio and the aspect ratio of the liquid pool are defined as η = ri/ro and ε = d/(ro − ri), respectively. The bottom is an adiabatic solid wall, noslip and impermeable conditions are applied for all solid-liquid boundaries. The top is a free surface and there is heat dissipation to the environment. The total surface heat dissipation coefficient is h, and the temperatures for the inner and outer cylindrical walls maintain constant Ti and To (To > Ti), respectively. For simplification, the following assumptions are introduced. (1) The fluid is incompressible Newtonian fluid. (2) All physical parameters are constant except surface tension. (3) Flow is laminar, and viscous dissipation is neglected. In order to reduce the number of variables describing the problem, the following dimensionless parameters are introduced by applying (ro − ri), ν/(ro − ri), (ro − ri)2/ν and μν/(ro − ri)2 as scale quantities for length, velocity, time and pressure, respectively,
∂U Ma ∂Θ ∂V Ma ∂Θ =− , =− , W = 0. ∂Z Pr ∂R ∂Z Pr R∂θ where Ma = γTΔT(ro − ri)/(μα), γT, μ and α are respectively surface tension temperature coefficient, dynamic viscosity, and thermal diffusivity. The thermal boundary condition on the free surface is expressed as
− λ ∂T / ∂z = h (T − T0),
(10)
where λ is thermal conductivity. T0 is ambient temperature. Supposing T0 = Ti, Eq. (10) is also expressed in the dimensionless form
− ∂Θ/ ∂Z = BiΘ.
(11)
In which, Bi is surface heat dissipation Biot number, Bi = h (ro − ri)/λ. In this work, the radius ratio and the aspect ratio of the annular pool are respectively fixed at η = 0.5 and ε = 0.05. Considering actual dimensions of the annular pool and the range of convective heat transfer coefficient, the surface heat dissipation Biot number varies from 0 to 50.
(r , z ) (u, v, w ) t , (U , V , W ) = , τ= , P (ro − ri )2 / ν ro − ri ν /(ro − ri ) p = μν /(ro − ri )2
(R, Z ) =
2.2. Solution procedure and validation The governing equations and the corresponding boundary conditions are discretized by the finite volume method, in which the diffusion term adopts the second order central difference and the convection term is QUICK scheme. The pressure-velocity correction is SIMPLE algorithm. According to different Marangoni numbers and Biot numbers, the dimensionless time step is selected in the range of (1 ∼ 4) × 10−5. During the whole iterative process, if the maximum relative error of temperature and velocity is less than 10−5, we considered the solution to be convergent. Because the velocity and temperature gradients near the solid wall and free surface are larger than those in other regions, non-uniform
Fig. 1. Physical model and the coordinate system. 22
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(a) Bi=0, ψmax=2.52
(b) Bi=5, ψmax=5.20
(c) Bi=10, ψmax=7.25
(d) Bi=50, ψmax=10.64 Fig. 2. Streamlines (upper) and isotherms (lower) for steady thermocapillary convection in R-Z plane when Ma = 1.8 × 105. δΘ = 0.05, δψ = ψmax/20.
staggered grid of 102R × 22Z × 240θ with denser meshes near the solid walls and the free surface is applied. This mesh is finer than one in our previous work [11] for the moderate Prandtl number fluids. In order to verify the calculation method, under the same experimental condition as that of Schwabe et al. [15], the critical condition of thermocapillary flow destabilization for 0.65 cSt silicone oil with Pr = 6.7 in the annular pool of ε = 0.125 was simulated numerically. It was found that the critical Marangoni number deviation between the calculated and the experimental values was less than 5%, which indicated that the calculation results in this paper are believable.
so the radial temperature gradient in the middle region is much smaller than that in the inner and outer walls, and the isotherm distribution is more uniform than that in the low Prandtl number fluid. Because there is the large radial flow cross section near the outer cylindrical wall, the center of the flow cell is closer to the outer cylindrical wall. When there is surface heat dissipation, with the increase of Biot number, the heat dissipated into the environment increases gradually. Therefore, the surface average temperature decreases, the temperature gradient near the inner cylindrical wall decreases, and the isotherm becomes sparse. On the contrary, in order to supplement the surface heat dissipation, the temperature gradient near the outer wall increases, so that the flow near the outer wall is enhanced, and the maximum dimensionless flow function ψmax increases gradually, as shown in Fig. 2(b). When Biot number increases to Bi = 10, the thermal boundary layer near the inner cylindrical wall disappears, and the radial temperature gradient mainly exists near the outer cylindrical wall. Therefore, the flow near the inner wall is very weak and the streamlines is very sparse, as shown in Fig. 2(c). When Biot number increases further, although the flow is continually enhanced and dimensionless flow function increases, the increasing trend is slowing down. Meanwhile, when Bi > 20, a flow stagnation region will be formed near the inner cylindrical wall, which will gradually expand to the outer wall with the increase of Biot number. When Bi = 50, the stagnant region occupies more than half of the liquid pool, and the flow is only confined near the outer cylindrical wall, as shown in Fig. 2(d). Fig. 3 reveals the effect of Marangoni number on the distributions of streamlines and isotherms for steady thermocapillary convection at Bi = 10. When Marangoni number is small, such as Ma = 104, the flow is very weak and is mainly concentrated in about 1/3 region near the outer cylindrical wall. Therefore, the radial variation of the temperature mainly occurs near the outer cylindrical wall, and the isotherms are concentrated near the outer wall, while the flow is stagnant near the inner cylindrical wall and almost half of the radial width the fluid is isothermal, as shown in Fig. 3(a). With the increase of Marangoni number, the inward flow along the free surface strengthens, so the flow gradually expands to the inner cylindrical wall. When the Marangoni number increases to Ma = 105, the streamlines occupy the whole pool, as shown in Fig. 3(c). As Marangoni number continues to increase, the flow near the inner wall strengthens gradually and the radial
3. Results and discussion 3.1. Basic flow When the radial temperature gradient applied to the liquid pool is small, thermocapillary convection is stable. This stable thermocapillary convection is called basic flow. Because the radial temperature gradient on the free surface varies with the surface heat dissipation conditions, thus the structure and intensity of steady thermocapillary convection will be affected. Fig. 2 shows streamlines and isotherms in the R-Z plane for steady thermocapillary convection at different Biot numbers when Ma = 1.8 × 105, where ψ is dimensionless stream function. Obviously, the high temperature fluid near the outer cylindrical wall flows to the inner cylindrical wall along the free surface under the action of the surface tension gradient, and then flows back to the high temperature wall near the bottom, which forms a counter-clockwise rotating flow cell. When the surface is adiabatic, the streamlines are almost parallel in the middle of the liquid pool. Due to the adiabatic free surface, the heat transfer to the inner cylindrical wall depends mainly on the convection along the free surface. Therefore, the isotherms are very dense in the upper region near the inner cylindrical wall, and relatively sparse in the lower region; the opposite is true near the outer cylindrical wall, but there is always the thermal boundary layer near the inner and outer cylindrical walls, as shown in Fig. 2(a). It should be noted that, due to the larger flow cross section and lower flow velocity near the outer wall, the radial temperature gradient on the free surface is smaller than that near the inner wall. On the other hand, for 0.65 cSt silicone oil with Pr = 6.7, the momentum diffusivity is greater than the heat diffusivity, 23
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(a) Ma=104, ψmax=0.90
(b) Ma=5×104, ψmax=2.41
(c) Ma=105, ψmax=4.24
(d) Ma=2.0×105, ψmax=7.98 Fig. 3. Streamlines (upper) and isotherms (lower) for axisymmetric steady flow in a R-Z plane at Bi = 10.δΘ = 0.05, δψ = ψmax/20.
unchanged. When Bi ≥ 10, the thermal boundary layer near the inner cylindrical wall disappears and the radial temperature gradient becomes very small, while the temperature gradient near the outer wall is very large, so the flow is mainly concentrated near the outer wall. Although the flow intensity will still increase, the flow region becomes narrow. Therefore, the critical Marangoni number of flow destabilization will increase gradually. On the other hand, when Biot number is very small, such as Bi ≤ 5, the flow pattern after destabilization is the hydrothermal wave (HTW), and its critical wave number is small, and basically remains unchanged. At this time, the temperature fluctuation mainly concentrates near the inner wall. When Bi > 5, due to the increase of the temperature gradient near the outer cylindrical wall, the center of the main flow cell is closer to the outer wall, and the temperature fluctuation increases after the flow destabilization. Therefore, in order to dissipate the temperature fluctuation, the azimuthal wave number will suddenly increase, and then remain almost constant. The maximum temperature fluctuation on the free surface appears near the outer cylindrical wall. Fig. 4 also shows the linear stability analysis results of Shi et al. [24] under the same conditions. Obviously, the critical Marangoni number and the critical wave number of flow destabilization are very close to the numerical results in this paper.
temperature gradient increases, but no thermal boundary layer is formed. Moreover, the center of the flow cell is always located near the outer cylindrical wall, and the secondary small flow cell appears in the large flow cell, as shown in Fig. 3(d). Therefore, the velocity and temperature fields of steady thermocapillary convection depend on the coupling effect of surface heat dissipation and thermocapillary convection. 3.2. Critical condition of flow destabilization When Marangoni number exceeds the critical value, the axisymmetric steady flow will destabilize and transit to three-dimensional time-dependent flow. Fig. 4 shows variations of the critical Marangoni number Macri and the critical wave number mcri with Biot number after the flow destabilization. In this work, the critical Marangoni number is obtained by the dichotomy. The maximum relative deviation of the critical Marangoni numbers is less than 2%. At a smaller Biot number, such as Bi < 10, with the increase of surface heat dissipation Biot number, the radial temperature gradient in the middle of the free surface increases, but the temperature distribution is more uniform, as shown in Fig. 2(a) and (b). Therefore, although the flow is enhanced, the critical Marangoni number Macri of flow destabilization is basically
10
3.3. Effect of Biot number on flow pattern
40
6
30
4
0
20
Ma m
2 0
10
20
Bi
30
40
For moderate Prandtl fluid, the axisymmetric steady flow will directly transit to three-dimensional time-dependent flow after the flow destabilization. When the Biot number is small, the steady flow first evolves into HTWs, while it will be into radial waves at a large Biot number. Fig. 5 shows the variation of the flow pattern with Biot number after the flow destabilization. When the free surface is adiabatic, that is, Bi = 0, the largest radial temperature gradient is located near the inner cylindrical wall. Therefore, the flow first becomes unstable near the inner cylindrical wall where temperature and velocity first start to oscillate. With the increase of Marangoni number, the fluctuations of velocity and temperature move gradually to the outer wall. Finally, the temperature fluctuation is observed on the whole free surface. Meanwhile, the radial position of the maximum temperature fluctuation is gradually shifted to the outer wall. In the shallow liquid layer, the flow
m cri
Ma cri /10 5
8
50
10
Fig. 4. Variations of critical Marangoni number (●) and critical wave number (○) of flow destabilization with Biot number. Triangle: stability analysis result of Shi et al. [24]. 24
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(a) Bi=0, Ma=2.5×105
When Biot number is large, the flow pattern after the flow destabilization is different from that at a small Biot number. For example, when Bi = 20, the temperature fluctuation on the free surface is similar to that at Bi = 5, as shown in Fig. 5(c) and (b). However, because of the large Marangoni number, the region occupied by the HTWs near the inner cylindrical wall is large, the temperature fluctuation region near the outer cylindrical wall is almost unchanged, but the overall temperature fluctuation range is further reduced. With the further increase of Biot number, for example Bi = 30, the flow pattern has an essential change. The HTWs near the inner cylindrical wall disappear and evolve into a radial rolling cell pattern, which originates near the inner wall and propagates radially to the outer wall. However, the formation and the propagation of the radial rolling cell are not synchronous in the azimuthal direction, as shown in Fig. 5(d). When Bi = 50, the radial temperature gradient becomes almost zero near the inner cylindrical wall, and only exists in a small region near the outer wall. Therefore, thermocapillary convection driven by the radial temperature gradient is very stable, the temperature fluctuation near the outer cylindrical wall is very small. The temperature fluctuation near the inner cylindrical wall is mainly caused by Marangoni-Bénard instability. At this time, in a very small region near the inner wall, the radial rolling cells will twist along the azimuthal direction. Fig. 6 shows variations of the maximum surface temperature fluctuation and the oscillatory frequency with Biot number at Ma = 106. At a small Biot number, the flow pattern belongs to the HTWs, so the temperature fluctuation is large and the main frequency has only slight variation. With the increase of Biot number, the temperature fluctuation amplitude gradually decreases, but the main frequency has a sudden increase near Bi = 5, and then basically remains unchanged.
(b) Bi=5, Ma=2.5×105
3.4. Evolution of flow pattern with Marangoni number
(c) Bi=20, Ma=4.0×105
Under different Biot numbers, the evolution of the flow pattern with Marangoni number is different. When the surface is adiabatic, i.e. Bi = 0, the flow pattern after the flow destabilization is the HTW, which originates near the inner cylindrical wall and then extends outward. Therefore, the largest temperature fluctuation is located near the inner cylindrical wall, as shown in Fig. 7(a), which has been confirmed by many theories and experiments [12–14]. With the increase of Marangoni number, the flow is enhanced, and the HTWs will be full of the whole liquid pool. When Marangoni continues to increase, two or more groups of the HTWs propagating along different directions will appear in the liquid pool. For example, when Ma = 3.0 × 105, two groups of coexisting HTWs are respectively located near the inner cylindrical wall and the outer wall. The HTWs near the outer wall propagate along counterclockwise direction, while the HTWs near the inner wall along clockwise direction. Because the flow at R = 1.5 is affected by two sets of HTWs, the STD diagram is composed of two sets of oblique lines overlapping to the upper left and the upper right, respectively, as
(d) Bi=30, Ma=1.0×106
Fig. 5. Snapshots (above) of surface temperature fluctuation and space-time diagram (lower) of surface temperature at R = 1.5 at different Biot numbers.
destabilization type under the action of tangential temperature gradient belongs to the HTW instability. The HTWs may propagate along clockwise or counterclockwise direction [12]. Therefore, the space-time diagram (STD) of the temperature variation with time at R = 1.5 on the free surface consists of a set of slopes inclining upward to the left. The angle ϕ between the propagating direction of the HTWs and the direction of the temperature gradient is 25–29°, which is close to the results of linear stability analysis [4], as shown in Fig. 5(a). When there is heat dissipation on the free surface, the temperature gradient near the inner cylindrical wall decreases, and near the outer wall increases. Therefore, the position of the maximum temperature fluctuation on the free surface will move to the outer cylindrical wall. At the same time, the flow cell near the outer wall strengthens. Therefore, a large temperature fluctuation concentrates near the outer wall, which also propagates along the azimuthal direction driven by the HTWs. Compared with Fig. 5(a) and (b), it is found that the temperature fluctuation caused by the flow cell near the outer cylindrical wall is smaller than that by the HTWs. With the increase of Biot number, the flow near the inner cylindrical wall gradually weakens, and the temperature fluctuation gradually diminishes until eventually disappears. Obviously, no matter what the cause of temperature fluctuation is, with the increase of Biot number, the HTWs gradually diminishes and the surface temperature fluctuation amplitude gradually decreases.
Fig. 6. Variations of the maximum surface temperature fluctuation δΘmax = (Θmax − Θmin) (○) and the oscillatory frequency (●) with Biot number at Ma = 106. 25
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(a) Ma=2.2×105
(b) Ma=3.0×105
(c) Ma=6.0×105
Fig. 7. Evolution of flow pattern with Marangoni number at Bi = 0: snapshots (above) of surface temperature fluctuation and STD (lower) of surface temperature at R = 1.5.
shown in Fig. 8(b). With the further increase of Marangoni number, temperature fluctuation near the inner cylindrical wall becomes clearer. It should be noted that the temperature fluctuations near the inner and outer walls are totally different. The former is caused by the hydrothermal waves, while the latter by the main flow cell oscillatory near the outer cylindrical wall. Moreover, with the increase of Marangoni number, the center of the main flow cell near the outer wall remains basically unchanged, so the temperature fluctuation region is almost independent on Marangoni number, as shown in Fig. 9. When the surface heat dissipation Biot number increases to Bi = 30, the temperature gradient near the inner cylindrical wall is almost zero, so there is no thermocapillary convection. However, due to the large surface heat dissipation Biot number, a large axial reverse temperature gradient near the free surface will trigger Marangoni-Bénard convection. In this case, the surface temperature fluctuation is completely different from that of the HTWs at a small Biot number, as shown in Fig. 10. Because the axis of Marangoni-Bénard flow cells is along the radial direction, the temperature fluctuations are arranged alternately
shown in Fig. 7(b). It should be noted that the clockwise propagating HTWs near the inner cylindrical wall occupy only about 1/4 of the width of the liquid pool, but the inducing temperature fluctuation is much greater than that caused by the counterclockwise propagating HTWs. With the further increase of Marangoni number, the flow pattern becomes more complex. The obvious HTWs can be observed only in the local region, as shown in Fig. 7(c). When there is surface heat dissipation, for example Bi = 10, the temperature gradient near the inner cylindrical wall becomes almost zero, but it near the outer cylindrical wall increases. Therefore, the main flow cell moves to the outer wall. When Ma = 2.3 × 105, the flow is just unstable, temperature fluctuation occurs only in half of the radial direction near the outer cylindrical wall, while no fluctuation appears in half near the inner cylindrical wall, as shown in Fig. 8(a). With the increase of Marangoni number, the inward thermocapillary flow on the free surface is enhanced, which drives the temperature fluctuation near the outer wall to extend inward. When Ma = 4.0 × 105, obvious temperature fluctuation can be observed on the whole free surface, as
(a) Ma=2.3×105
(b) Ma=4.0×105
(c) Ma=8.0×105
Fig. 8. Evolution of flow pattern with Marangoni number at Bi = 10: snapshots (above) of surface temperature fluctuation and STD (lower) of surface temperature at R = 1.8. 26
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(a) Ma=2.3×105, ψmax=8.91
(b) Ma=4.0×105, ψmax=14.40
(c) Ma=8.0×105, ψmax=23.73 Fig. 9. Snapshots of pseudo-streamlines (upper) and isotherms (lower) for three-dimensional oscillatory flow at the R-Z plane when Bi = 10. δΘ = 0.05, δψ = ψmax/ 20.
direction under the influence of the radial temperature fluctuation. Fig. 11 reveals variations of the fundamental oscillatory frequency and the maximum surface temperature fluctuation with Marangoni number. When the free surface is adiabatic, Marangoni number has little effect on the fluctuation frequency, but the maximum temperature fluctuation amplitude on the free surface increases with the increase of Marangoni number. When there is heat dissipation on the free surface, the fluctuation frequency increases rapidly with Marangoni number, but the maximum temperature fluctuation amplitude on the free surface is much lower than that at the adiabatic surface. When Bi = 10, with the increase of Marangoni number, the influence region of thermocapillary convection gradually shrinks. Therefore, the temperature fluctuation amplitude will increase first, then decrease slightly and then increase again. When Bi = 30, the temperature fluctuation amplitude increases monotonously with Marangoni number.
in the azimuthal direction. However, this temperature fluctuation is unstable, which originates near the inner cylindrical wall and then moves outward. When Marangoni number is small, such as Ma = 0.6 × 106 and 1.0 × 106, the movement of the temperature fluctuation along the radial direction is not synchronous, and there is always a phase lag, as shown in Fig. 10(a) and (b). It can also be found from the corresponding STD diagram that the temperature fluctuation in the azimuthal direction is not on the same horizontal line. When Marangoni number increases further, the flow is enhanced stronger and the surface temperature fluctuation becomes more complicated. At this point, the temperature fluctuations still originate near the inner cylindrical wall and then move outward. However, during the radial movement of temperature fluctuations, obvious azimuthal motion of temperature fluctuations can be observed in the local region on the free surface, as shown in Fig. 10(c). On the other hand, due to the large radial temperature gradient near the outer cylindrical wall, thermocapillary convection cells still exist in a very small region near the outer wall. The temperature fluctuation caused by the flow cell will no longer propagate along the azimuthal direction driven by the HTWs as the case of small Biot numbers, but merge and grow alternately in the azimuthal
(a) Ma=0.6×106
4. Conclusions A series of three-dimensional numerical simulations are performed to analyze the effect of surface heat dissipation on thermocapillary
(b) Ma=1.0×106
(c) Ma=2.0×106
Fig. 10. Evolution of flow pattern with Marangoni number at Bi = 30: snapshots (above) of surface temperature fluctuation and STD (lower) of surface temperature at R = 1.5. 27
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China (Grant No. 51776022, 11532015), Chongqing University Postgraduates Innovation Project (Grant No: CYS18041) and the Fundamental Research Funds for the Central Universities (No. 2018CDXYDL0001). References [1] H.G. Levich, V.S. Krylov, Surface tension-driven phenomena, Annual Rev. Fluid Mech. 1 (1969) 293–316. [2] M. Lappa, Thermal Convection: Patterns, Evolution and stability, Wiley & Sons, 2010. [3] W.Y. Shi, M.K. Ermakov, N. Imaishi, Effect of pool rotation on thermocapillary convection in shallow annular pool of silicone oil, J. Cryst. Growth 294 (2006) 474–485. [4] M.K. Smith, S.H. Davis, Instabilities of dynamic thermocapillary liquid layers 1. Convective instabilities, J. Fluid Mech. 132 (1983) 119–144. [5] M.K. Smith, S.H. Davis, Instabilities of dynamic thermocapillary liquid layers 2. Surface-wave instabilities, J. Fluid Mech. 132 (1983) 145–162. [6] N. Garnier, C. Normand, Effects of curvature on hydrothermal waves instability of radial thermocapillary flows, C R Acad Paris, t.2, Seried IV Phys. 2 (2001) 1227–1233. [7] N. Garnier, A. Chiffaudel, Two dimensional hydrothermal waves in an extended cylindrical vessel, Eur. Phys. J. B 19 (2001) 87–95. [8] J.J. Yu, D.F. Ruan, Y.R. Li, J.C. Chen, Experimental study on thermocapillary convection of binary mixture in a shallow annular pool with radial temperature gradient, Exp. Therm. Fluid Sci. 61 (2015) 79–86. [9] A.J. Torregrosa, S. Hoyas, M.J. Pérez-Quiles, J.M. Mompó-Laborda, Bifurcation diversity in an annular pool heated from below: Prandtl and Biot numbers effects, Commun. Comput. Phys. 13 (2013) 428–441. [10] S. Hoyas, P. Fajardo, M.J. Pérez-Quiles, Influence of geometrical parameters on the linear stability of a Bénard-Marangoni problem, Phys. Rev. E 93 (2016) 043105. [11] Y.R. Li, L. Peng, Y. Akiyama, N. Imaishi, Three-dimensional numerical simulation of thermocapillary flow of moderate Prandtl number fluid in an annular pool, J. Cryst. Growth 259 (2003) 374–387. [12] Y.R. Li, N. Imaishi, T. Azami, T. Hibiya, Three-dimensional oscillatory flow in a thin annular pool of silicon melt, J. Cryst. Growth 260 (2004) 28–42. [13] Y.R. Li, L. Peng, S.Y. Wu, D.L. Zeng, N. Imaishi, Thermocapillary convection in a differentially heated annular pool for moderate Prandtl number fluid, Int. J. Therm. Sci. 43 (2004) 587–593. [14] T. Azami, S. Nakamura, M. Eguchi, T. Hibiya, The role of surface-tension-driven flow in the formation of a surface pattern on a Czochralski silicon melt, J. Cryst. Growth 233 (2001) 99–107. [15] D. Schwabe, Buoyancy-thermocapillary and pure thermocapillary convective instabilities in Czochralski systems, J. Cryst. Growth 237–239 (2002) 1849–1853. [16] C.J. Jing, N. Imaishi, S. Yasuhiro, Y. Miyazawa, Three-dimensional numerical simulation of spoke pattern in oxide melt, J. Cryst. Growth 200 (1999) 204–212. [17] B.C. Sim, A. Zebib, Effect of free surface heat loss and rotation on transition to oscillatory thermocapillary convection, Phys. Fluids 14 (2002) 225–232. [18] B.C. Sim, A. Zebib, Thermocapillary convection with underformable curved surfaces in open cylinders, Int. J. Heat Mass Tran. 45 (2002) 4983–4994. [19] B.C. Sim, A. Zebib, D. Schwabe, Oscillatory thermocapillary convection in open cylindrical annuli. Part 2. Simlations, J. Fluid Mech. 491 (2003) 259–274. [20] S. Hoyas, P. Fajardo, A. Gil, M.J. Perez-Quiles, Analysis of bifurcations in a BénardMarangoni problem: gravitational effects, Int. J. Heat Mass Transf. 73 (2014) 33–41. [21] S. Hoyas, P. Fajardo, A. Gil, M.J. Perez-Quiles, Influence of geometrical parameters on the linear stability of a Bénard-Marangoni problem, Phys. Rev. E 93 (2016) 043105. [22] L. Zhang, Y.R. Li, C.M. Wu, Effect of surface heat dissipation on thermocapillary convection of low Prandtl number fluid in a shallow annular pool, Int. J. Heat Mass Transf. 110 (2017) 660–667. [23] L. Zhang, Y.R. Li, C.M. Wu, Q.S. Liu, Flow bifurcation routes to chaos of thermocapillary convection for low Prandtl number fluid in shallow annular pool with surface heat dissipation, Int. J. Therm. Sci. 125 (2018) 23–33. [24] W.Y. Shi, X. Liu, G. Li, Y.R. Li, L. Peng, M.K. Ermakov, N. Imaishi, Thermocapillary convection instability in shallow annular pools by linear stability analysis, J. Supercond. Nov. Magn. 23 (2010) 1185–1188.
Fig. 11. Variations of the fundamental oscillatory frequency (a) and the maximum surface temperature fluctuation (b) with Marangoni number.
convection of moderate Prandtl number fluid in an annular shallow pool. From the above-mentioned analysis, we can summarize the conclusions as follows: (1). At Bi < 10, the surface heat dissipation has little effect on the critical Marangoni number of flow destabilization. At Bi > 10, the critical Marangoni number increases with the increase of surface heat dissipation Biot number. At Bi ≤ 5, the flow pattern after the flow destabilization is the HTW, and its critical wave number is small and basically unchanged; at Bi = 5, the critical wave number has an abrupt increase. (2). When Biot number is small, the flow pattern after the flow destabilization is the HTWs. With the increase of Biot number, the radial rolling cells near the outer cylindrical wall is gradually enhanced, and the flow pattern eventually transits to the radial waves. For the HTWs at a small Biot number, the temperature fluctuation amplitude is large, and the fundamental frequency keeps almost constant. With the increase of Biot number, the temperature fluctuation amplitude decreases gradually. However, the fundamental oscillatory frequency increases sharply at Bi = 5, and then remains unchanged with the increase of Biot number. (3). When there is surface heat dissipation, the oscillatory fundamental frequency increases rapidly with the increase of Marangoni number. The amplitude of the temperature fluctuation on the free surface also increases in general, but is much lower than that on the adiabatic surface. Acknowledgement This work is supported by National Natural Science Foundation of
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