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Three-Dimensional Numerical Simulation on Thermocapillary Convection of Moderate Prandtl Number Fluid in an Annular Shallow Pool with Surface Heat Dissipation

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Abstract

In order to understand the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in an annular shallow pool, we performed a series of three-dimensional numerical simulations by using the finite volume method. An annular shallow pool is full of 0.65cSt silicone oil with Prandtl number of 6.7. The radius ratio and the aspect ratio of the pool are respectively fixed at 0.5 and 0.1. Surface heat dissipation Biot number is varied from 0 to 50. Results indicate that the critical Marangoni number of flow destabilization mainly depends on the coupling effect of the thermocapillary force and surface heat dissipation on the free surface, which decreases first, and then increases gradually. When Biot number is small, the steady axisymmetric flow after the flow destabilization will bifurcate to the hydrothermal wave. With the increase of Marangoni number, the flow pattern evolution at a small Biot number is similar to that of an adiabatic free surface, and the fundamental oscillatory frequency increases gradually. With the increase of Biot number, the radial roll cells near the outer cylindrical wall gradually extend to the inner wall, and eventually occupy the whole liquid pool. When Biot number is large, the flow pattern after the flow destabilization is a radial rolling cell pattern with alternating azimuthal direction. Then it gradually evolves into azimuthal waves, and the fundamental oscillatory frequency has a slight decrease.

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Abbreviations

Bi :

Biot number

d :

Depth of annular pool, m

F :

Dimensionless frequency

h :

Convective heat transfer coefficient, W/(m2·K)

m :

Wave number

Ma :

Marangoni number

p :

Pressure, Pa

P :

Dimensionless pressure

Pr :

Prandtl number

r :

Radius or radial coordinate, m

R :

Dimensionless radius, dimensionless radial coordinate

t :

Time, s

T :

Temperature, K

u :

Radial velocity, m/s

U :

Dimensionless radial velocity

v :

Azimuthal velocity, m/s

V :

Dimensionless azimuthal velocity

V :

Dimensionless velocity vector

w :

Axial velocity, m/s

W :

Dimensionless axial velocity

z :

Axial coordinate, m

Z :

Dimensionless axial coordinate

α :

Thermal diffusivity, m2/s

ε :

Aspect ratio

γ T :

Temperature coefficient of surface tension, N/(m·K)

λ :

Thermal conductivity, W/(m·K)

η :

Radius ratio

μ :

Dynamic viscosity, kg/(m·s)

ν :

Kinematic viscosity, m2/s

Θ :

Dimensionless temperature

ρ :

Density, kg/m3

τ :

Dimensionless time

ψ :

Dimensionless stream function

0:

Ambient

i:

Inner

o:

Outer

p:

Period

References

  • Bekezhanova, V.B., Shefer, I.A.: Influence of gravity on the stability of evaporative convection regimes. Microgravity Sci. Technol. 30, 543–560 (2018)

    Article  Google Scholar 

  • Burguete, J., Mukolobwiez, N., Daviaud, F., Garnier, N., Chiffaudel, A.: Buoyant-thermocapillary instabilities in extended liquid layers subjected to a horizontal temperature gradient. Phys. Fluids. 13, 2773–2787 (2001)

    Article  Google Scholar 

  • Chen, J.C., Li, Y.R., Yu, J.J., Zhang, L.: Wu, C.M.: flow pattern transition of thermal-solutal capillary convection with the capillary ratio of −1 in a shallow annular pool. Int. J. Heat Mass Transf. 95, 1–6 (2016)

    Article  Google Scholar 

  • Colinet, P., Legros, J.C., Velarde, M.G.: Nonlinear Dynamics of Surface-Tension-Driven Instabilities. Wiley-VCH Verlag Berlin GmbH, Berlin (2001)

    Book  Google Scholar 

  • Daviaud, F., Vince, J.M.: Traveling waves in a fluid layer subjected to a horizontal temperature gradient. Phys. Rev. E. 48, 4432–4436 (1993)

    Article  Google Scholar 

  • Garnier, N., Chiffaudel, A.: Two dimensional hydrothermal waves in an extend cylindrical vessel. Eur. Phys. J. B. 19, 87–95 (2001)

    Article  Google Scholar 

  • Garnier, N., Normand, C.: Effect of curvature on hydrothermal waves instability of radial thermocapillary flows. CRAcad. Paris Seried IV Phys. 2(8), 1227–1233 (2001)

    Google Scholar 

  • Hoyas, S., Fajardo, P., Gil, A., Perez-Quiles, M.J.: Analysis of bifurcations in a Bénard-Marangoni problem: gravitational effects. Int. J. Heat Mass Transf. 73, 33–41 (2014)

    Article  Google Scholar 

  • Hoyas, S., Fajardo, P., Gil, A., Perez-Quiles, M.J.: Influence of geometrical parameters on the linear stability of a Bénard-Marangoni problem. Phys. Rev. E. 93, 043105 (2016)

    Article  Google Scholar 

  • Jing, C.J., Imaishi, N., Yasuhiro, S., Miyazawa, Y.: Three-dimensional numerical simulation of spoke pattern in oxide melt. J. Cryst. Growth. 200, 204–212 (1999)

    Article  Google Scholar 

  • Kamotani, Y., Ostrach, S., Masud, J.: Microgravity experiments on oscillatory thermocapillary flow in cylindrical containers. J. Fluid Mech. 410, 211–233 (2000)

    Article  Google Scholar 

  • Lappa, M.: Thermal Convection: Patterns, Evolution and Stability. Wiley & Sons (2009)

  • Li, Y.R., Peng, L., Shi, W.Y., Imaishi, N.: Convective instability in annular pools. Fluid Dyn. Mater. Process. 2(3), 153–166 (2006)

    MathSciNet  Google Scholar 

  • Li, Y.R., Zhang, L., Zhang, L., Yu, J.J.: Experimental study on Prandtl number dependence of thermocapillary-buoyancy convection in Czochralski configuration with different depths. Int. J. Therm. Sci. 130, 168–182 (2018)

    Article  Google Scholar 

  • Liu, Q.S., Wang, Y., Ji, Y.: Coupling mechanism of evaporation phase change and interfacial flow. J. Eng. Thermophys. 31, 1751–1754 (2010)

    Google Scholar 

  • Peng, L., Li, Y.R., Shi, W.Y., Imaishi, N.: Three-dimensional thermocapillary-buoyancy flow of silicone oil in a differentially heated annular pool. Int. J. Heat Mass Transf. 50, 872–880 (2007)

    Article  Google Scholar 

  • Qin, T.R., Grigoriev, R.O.: The effect of noncondensables on buoyancy -thermocapillary convection of volatile fluids in confined geometries. Int. J. Heat Mass Transf. 90, 678–688 (2015)

    Article  Google Scholar 

  • Qin, T.R., Tukovic, Z., Grigoriev, R.O.: Buoyancy-thermocapillary convection of volatile fluids under their vapors. Int. J. Heat Mass Transf. 80, 38–49 (2015)

    Article  Google Scholar 

  • Schwabe, D.: Buoyant-thermocapillary and pure thermocapillary convective instabilities in Czochralski systems. J. Cryst. Growth. 237-239, 1849–1853 (2002)

    Article  Google Scholar 

  • Schwabe, D., Benz, S.: Thermocapillary flow instabilities in an annulus under microgravity - results of the experiment magia. Adv. Space Res. 29(4), 629–638 (2002)

    Article  Google Scholar 

  • Schwabe, D., Zebib, A., Sim, B.C.: Oscillatory thermocapillary convection in open cylindrical annuli. Part 1. Experiments under microgravity. J. Fluid Mech. 491, 239–258 (2003)

    Article  Google Scholar 

  • Shevtsova, V.M., Nepomnyashchy, A.A., Legros, J.C.: Thermocapillary-buoyancy convection in a shallow cavity heated from the side. Phys. Rev. E. 67, 066308 (2003)

    Article  Google Scholar 

  • Shi, W.Y., Li, Y.R., Ermakov, M.K., Imaishi, N.: Stability of thermocapillary convection in rotating shallow annular pool of silicon melt. Microgravity Sci. Technol. 22, 315–320 (2010)

    Article  Google Scholar 

  • Shi, W.Y., Rong, S.M., Feng, L.: Marangoni convection instabilities induced by evaporation of liquid layer in an open rectangular pool. Microgravity Sci. Technol. 29, 91–96 (2017)

    Article  Google Scholar 

  • Sim, B.C., Zebib, A.: Effect of free surface heat loss and rotation on transition to oscillatory thermocapillary convection. Phys. Fluids. 14, 225–231 (2002)

    Article  Google Scholar 

  • Sim, B.C., Zebib, A., Schwabe, D.: Oscillatory thermocapillary convection in open cylindrical annuli. Part 2. Simulations. J. Fluid Mech. 491, 259–274 (2003)

    Article  Google Scholar 

  • Smith, M.K., Davis, S.H.: Instabilities of dynamic thermocapillary liquid layers. 1. Convective instabilities. J. Fluid Mech. 132, 119–144 (1983)

    Article  Google Scholar 

  • Villers, D., Platten, J.K.: Coupled buoyancy and Marangoni convection in acetone: experiments and comparison with numerical simulations. J. Fluid Mech. 234, 487–510 (1991)

    Article  Google Scholar 

  • Yu, J.J., Ruan, D.F., Li, Y.R., Chen, J.C.: Experimental study on thermocapillary convection of binary mixture in a shallow annular pool with radial temperature gradient. Exp. Thermal Fluid Sci. 61, 79–86 (2015)

    Article  Google Scholar 

  • Yu, J.J., Zhang, L., Li, Y.R., Chen, J.C.: Numerical simulations of thermocapillary flow of a binary mixture with the Soret effect in a shallow annular pool. Microgravity Sci. Technol. 28, 1–10 (2016)

    Article  Google Scholar 

  • Zhang, L., Li, Y.R., Wu, C.M.: Effect of surface evaporation on steady thermocapillary convection in an annular pool. Microgravity Sci. Technol. 28(5), 499–509 (2016)

    Article  Google Scholar 

  • Zhang, L., Li, Y.R., Wu, C.M.: Effect of surface heat dissipation on thermocapillary convection of low Prandtl number fluid in a shallow annular pool. Int. J. Heat Mass Transf. 110(660–667), 460–466 (2017)

    Article  Google Scholar 

  • Zhang, L., Li, Y.R., Wu, C.M., Liu, Q.S.: 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, 23–33 (2018)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No. 51776022) and the Fundamental Research Funds for the Central Universities (No. 2018CDXYDL0001).

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Authors and Affiliations

  1. School of Liberal Education, Chongqing City Management College, Chongqing, 401331, China

    Li Zhang

  2. 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

    You-Rong Li

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Zhang, L., Li, YR. Three-Dimensional Numerical Simulation on Thermocapillary Convection of Moderate Prandtl Number Fluid in an Annular Shallow Pool with Surface Heat Dissipation. Microgravity Sci. Technol. 31, 733–747 (2019). https://doi.org/10.1007/s12217-019-9704-3

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