Stability Analysis of Natural Convection with Variable Physical Coefficients Using Linear Stability Theory

Authors

1 Shahid Rajaee Teacher Training University

2 Research Institute of Petroleum Industry

3 Research Inst. of Petroleum Industry

4 Shaihd Rajaee Teacher Training Uni

Abstract

In this research, the effects of variation of physical transport coefficient of fluid, such as kinematic viscosity and thermal diffusivity coefficient, on the primary instability of natural convection has been investigated. Accordingly, the governing equations are calculated for the onset of free convective flow by using the perturbation theory on the variation of temperature and velocity. Using the theory of linear stability and the wave function, it can be defined the minimum Rayleigh number as critical Rayleigh number in assumed eigenvalue problem. It is assumed that the flow variables, such as the thermal diffusivity coefficient and kinematic viscosity, change exponentially relative to the location. The simulation results show that the critical-wave number function is a even function. Also, the dependence of the amplitude and frequency of oscillation of temperature and velocity disturbances on effective parameters on properties changes was evaluated and the results were analyzed. It can be said that critical Rayleigh number behavior is completely inverse in relation to the thermal emission and kinematic viscosity coefficients, and therefore their simultaneous effect in solving the equations will lead to a decrease in the critical interval of the oscillation.

Keywords

Main Subjects

References

[1] Aghanjafi C (2013) Numerical solution of a uniform magnetic field effect on the free convection heat transfer from a vertical plate. Journal of Solid and Fluid Mechanics 3(2):65-75.
[2] Alavi N, Armaghani T, Izadpanah E (2016) Natural convection heat transfer of a nanofluid in a baffle L-Shaped cavity. Journal of Solid and Fluid Mechanics 6(3): 311-21.
[3] Sedaghat MH, Yaghoubi M, Maghrebi MJ (2013) On the natural convective heat transfer from a cold horizontal cylinder over an adiabatic surface. Journal of Solid and Fluid Mechanics 2(3): 19-28.
[4] Yekani Motlagh S, Sarvari P (2017) Large Eddy Simulation of Three Dimensional Mixed Convection Flow Inside the Ventilated Cavity Containing Obstacle and Extraction of Coherent Structures using Proper Orthogonal Decomposition (POD). Journal of Solid and Fluid Mechanics 7(3): 199-212.
[5] Bernard H (1900) Les tourbillons cellulaires dans une nappe liquide [The cellular vortices in a liquid layer]. Rev Gén Sci Pure Appl 11: 1261-1271.
[6] RAYLEIOH L (1916) On convection currents in a horizontal layer of fluid when the. Philos Mag 32:529.
[7] Pearson J (1958) On convection cells induced by surface tension. J Fluid Mech 4(5): 489-500.
[8] Kao P-H, Yang R-J (2007) Simulating oscillatory flows in Rayleigh–Benard convection using the lattice Boltzmann method. Int J Heat Mass Transf 50(17): 3315-3328.
[9] Kelly R, Hu HC (1993) The onset of Rayleigh–Bénard convection in non-planar oscillatory flows. J Fluid Mech 249: 373-390.
[10] Yang K, Mukutmoni D (1993) Rayleigh-benard convection in a small aspect ratio enclosure: Part 1—bifurcation to oscillatory convection. J Heat Transf 115:360-366.
[11] Valori V, Elsinga G, Rohde M, Tummers M, Westerweel J, van der Hagen T (2017) Experimental velocity study of non-Boussinesq Rayleigh-Bénard convection. Phys Rev E 95(5):053113.
[12] Kebiche Z, Castelain C, Burghelea T (2014) Experimental investigation of the Rayleigh–Bénard convection in a yield stress fluid. J Non-Newton Fluid Mech 203: 9-23.
[13] Aurnou J, Olson P (2001) Experiments on Rayleigh–Bénard convection, magnetoconvection and rotating magnetoconvection in liquid gallium. J Fluid Mech 430: 283-307.
[14] Wei Y, Wang Z, Yang J, Dou H-S, Qian Y (2015) A simple lattice Boltzmann model for turbulence Rayleigh–Bénard thermal convection. Comput Fluids 118: 167-171.
[15] Dabbagh F, Trias F, Gorobets A, Oliva A (2016) New subgrid-scale models for large-eddy simulation of Rayleigh-Bénard convection. J Phys: Conference Series IOP Publishing.
[16] Curry JH, Herring JR, Loncaric J, Orszag SA (1984) Order and disorder in two-and three-dimensional Benard convection. J Fluid Mech 147: 1-38.
[17] Hernandez R, Frederick R (1994) Spatial and thermal features of three dimensional Rayleigh-Bénard convection. Int J Heat Mass Transf 37(3): 411-24.
[18] Li Y-R, Zhang H, Zhang L, Wu C-M (2016) Three-dimensional numerical simulation of double-diffusive Rayleigh–Bénard convection in a cylindrical enclosure of aspect ratio 2. Int J Heat Mass Transf 98: 472-483.
[19] Chimanski EV, Rempel EL, Chertovskih R (2016) On–off intermittency and spatiotemporal chaos in three-dimensional Rayleigh–Bénard convection. Adv Space Res 57(6): 1440-1447.
[20] Zhang L, Li Y-R, Zhang H (2017) Onset of double-diffusive Rayleigh-Bénard convection of a moderate Prandtl number binary mixture in cylindrical enclosures. Int J Heat Mass Transf 107: 500-509.
[21] Sakievich P, Peet Y, Adrian R (2016) Statistical convergence and the effect of large-scale motions on turbulent Rayleigh-Bénard convection in a cylindrical domain with 6.3 aspect ratio. APS Meeting Abstracts.
[22] Cattaneo F, Hughes DW (2017) Dynamo action in rapidly rotating Rayleigh–Bénard convection at infinite Prandtl number. J Fluid Mech 825: 385-411.
[23] Weiss S, Wei P, Ahlers G (2016) Heat-transport enhancement in rotating turbulent Rayleigh-Bénard convection. Phys Rev E 93(4): 043102.
[24] Gotoda H, Takeuchi R, Okuno Y, Miyano T (2013) Low-dimensional dynamical system for Rayleigh-Benard convection subjected to magnetic field. J Appl Phys 113(12): 124902.
[25] Tasaka Y, Igaki K, Yanagisawa T, Vogt T, Zuerner T, Eckert S (2016) Regular flow reversals in Rayleigh-Benard convection in a horizontal magnetic field. Phys Rev 93(4): 043109.
[26] Ahlers G, Bodenschatz E, He X (2017) Ultimate-state transition of turbulent Rayleigh-Bénard convection. Phys Rev Fluids 2(5): 054603.
[27] Zhang Y, Zhou Q, Sun C (2017) Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh–Bénard convection. J Fluid Mech 814: 165-184.
[28] Sakievich P, Peet Y, Adrian R (2016) Large-scale thermal motions of turbulent Rayleigh–Bénard convection in a wide aspect-ratio cylindrical domain. Int J Heat Fluid Flow 61: 183-196.
[29] Manneville P (2006) Rayleigh-Bénard convection: thirty years of experimental, theoretical, and modeling work. Dynamics of Spatio-Temporal Cellular Structures: Springer 41-65.
[30] Gelfgat AY (1999) Different modes of Rayleigh–Bénard instability in two-and three-dimensional rectangular enclosures. J Comput Phys 156(2): 300-324.
[31] Kaur P, Singh J (2017) Heat transfer in thermally modulated two-dimensional Rayleigh Bénard convection. Int J Therm Sci 114: 35-43.
[32] Siddheshwar P, Meenakshi N (2017) Amplitude equation and heat transport for Rayleigh–Bénard convection in Newtonian liquids with nanoparticles. Int J Appl Comput Math 3(1): 271-292.
[33] Ma X-R, Zhang L, Li Y-R (2017) Linear stability analysis of Rayleigh-Bénard convection of cold water near its density maximum in a vertically heated annular container. J Mech Sci Tech 31(4):1665-1672.
[34] Yu J, Goldfaden A, Flagstad M, Scheel JD (2017) Onset of Rayleigh-Bénard convection for intermediate aspect ratio cylindrical containers. Phys Fluids 29(2): 024107.
[35] Chandrasekhar S (1970) Hydrodynamic and hydromagnetic stability.
[36] Dominguez‐Lerma M, Ahlers G, Cannell DS (1984) Marginal stability curve and linear growth rate for rotating Couette–Taylor flow and Rayleigh–Bénard convection. Phys Fluids 27(4): 856-860.
[37] Wazzan A, Keltner G, Okamura T, Smith A (1972) Spatial stability of stagnation water boundary layer with heat transfer. Phys Fluids 15(12): 2114-2118.
[38] Busse F (1967) The stability of finite amplitude cellular convection and its relation to an extremum principle. J Fluid Mech 30(4):625-649.
[39] Palm E, Ellingsen T, Gjevik B (1967) On the occurrence of cellular motion in Bénard convection. J Fluid Mech 30(4):651-661.
[40] Hoard C, Robertson C, Acrivos A (1970) Experiments on the cellular structure in Bénard convection. Int J Heat Mass Transf 13(5): 849IN7853-852856.
[41] Somerscales E, Dougherty T (1970) Observed flow patterns at the initiation of convection in a horizontal liquid layer heated from below. J Fluid Mech 42(4):755-768.
[42] Busse F (1978) Non-linear properties of thermal convection. Rep Progr Phys 41(12): 1929.
[43] Busse F, Frick H (1985) Square-pattern convection in fluids with strongly temperature-dependent viscosity. J Fluid Mech 150:451-465.
[44] Palm E (1960) On the tendency towards hexagonal cells in steady convection. J Fluid Mech 8(2):183-192.
[45] Segel LA, Stuart J (1962) On the question of the preferred mode in cellular thermal convection. J Fluid Mech 13(2):289-306.
[46] Richter FM (1978) Experiments on the stability of convection rolls in fluids whose viscosity depends on temperature. J Fluid Mech 89(3): 553-560.
[47] Stengel KC, Oliver DS, Booker JR (1982) Onset of convection in a variable-viscosity fluid. J Fluid Mech 120:411-431.
[48] Capone F, Gentile M (1994) Nonlinear stability analysis of convection for fluids with exponentially temperature-dependent viscosity. Acta Mech 107(1-4): 53-64.
[49] Capone F, Gentile M (1995) Nonlinear stability analysis of the Bénard problem for fluids with a convex nonincreasing temperature depending viscosity. Contin Mech Thermodyn 7(3): 297-309.
[50] Richardson L, Straughan B (1993) A nonlinear energy stability analysis of convection with temperature dependent viscosity. Acta Mech 97(1-2): 41-49.
[51] Severin J, Herwig H (1999) Onset of convection in the Rayleigh-Bénard flow with temperature dependent viscosity: An asymptotic approach. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 50(3): 375-386.
[52] Rajagopal K, Saccomandi G, Vergori L (2009) Stability analysis of the Rayleigh–Bénard convection for a fluid with temperature and pressure dependent viscosity. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 60(4): 739-755.
[53] Papanicolaou N, Christov C, Homsy G (2009) Galerkin technique based on beam functions in application to the parametric instability of thermal convection in a vertical slot. Int J Numer Methods Fluids 59(9): 945-967.
[54] Laun HM (2003) Pressure dependent viscosity and dissipative heating in capillary rheometry of polymer melts. Rheol Acta 42(4): 295-308.
[55] Bair S, Kottke P (2003) Pressure-viscosity relationships for elastohydrodynamics. Tribol Trans 46(3): 289-295.
[56] Martın-Alfonso M, Martınez-Boza F, Partal P, Gallegos C (2006) Influence of pressure and temperature on the flow behaviour of heavy fluid oils. Rheol Acta 45: 357-365.
Journal of Solid and Fluid Mechanics
Volume 8, Issue 1
April 2018
Pages 157-169

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