多孔介质内黏弹性流体的热对流稳定性研究
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摘要
基于修正的Darcy模型,介绍了多孔介质内黏弹性流体热对流稳定性研究的现状和主要进展.通过线性稳定性理论,分析计算多孔介质几何形状(水平多孔介质层、多孔圆柱以及多孔方腔)、热边界条件(底部等温加热、底部等热流加热、底部对流换热以及顶部自由开口边界)、黏弹性流体的流动模型(Darcy-Jeffrey, DarcyBrinkman-Oldroyd以及Darcy-Brinkman-Maxwell模型)、局部热非平衡效应以及旋转效应对黏弹性流体热对流失稳的临界Rayleigh数的影响.利用弱非线性分析方法,揭示失稳临界点附近热对流流动的分叉情况,以及失稳临界点附近黏弹性流体换热Nusselt数的解析表达式.采用数值模拟方法,研究高Rayleigh数下黏弹性流体换热Nusselt数和流场的演化规律,分析各参数对黏弹性流体热对流失稳和对流换热速率的影响.主要结果:(1)流体的黏弹性能够促进振荡对流的发生;(2)旋转效应、流体与多孔介质间的传热能够抑制黏弹性流体的热对流失稳;(3)在临界Rayleigh数附近,静态对流分叉解是超临界稳定的,而振荡对流分叉可能是超临界或者亚临界的,主要取决于流体的黏弹性参数、Prandtl数以及Darcy数;(4)随着Rayleigh数的增加,热对流的流场从单个涡胞逐渐演化为多个不规则单元涡胞,最后发展为混沌状态.
Based on the modified Darcy model,the status and progress in research of thermal instability of viscoelastic fluids in porous media are reviewed.By using the method of linear stability analysis,the effects of the geometry of porous media(i.e.horizontal porous layer,porous cylinder and porous cavity),thermal boundary conditions(i.e.bottom heated with constant temperature,bottom heated with constant heat flux,bottom with Newtonian heating and open top),flow model of viscoelastic fluids(i.e.modified Darcy-Jeffrey,Darcy-Brinkman-Oldroyd and Darcy-Brinkman-Maxwell models),local thermal non-equilibrium and rotation on the critical Rayleigh number of thermal instability of viscoelastic fluids can be calculated.By using the method of weakly non-linear analysis,the bifurcation from the basic state and the analytical solution of Nusselt number at the neighborhood of critical point can be obtained.By the numerical simulation method,the evolution of flow pattern as well as the variations of Nusselt number at high Rayleigh number can be revealed.It has been found that(1) the elasticity of viscoelastic fluids can destabilize the oscillatory convection;(2) the rotation effect and local thermal non-equilibrium effect can suppress the thermal instability of viscoelastic fluids;(3) at the neighborhood of critical point,the bifurcation from the basic state for stationary convection is supercritical,while the bifurcation for the oscillatory can be supercritical or subcritical,mainly depending on the values of viscoelastic parameters,Prandtl number and Darcy number;(4) with the increasing Rayleigh number,the flow pattern of thermal convection evolve from one-cell pattern into multi-cell roll pattern,and finally a chaotic pattern.
Based on the modified Darcy model,the status and progress in research of thermal instability of viscoelastic fluids in porous media are reviewed.By using the method of linear stability analysis,the effects of the geometry of porous media(i.e.horizontal porous layer,porous cylinder and porous cavity),thermal boundary conditions(i.e.bottom heated with constant temperature,bottom heated with constant heat flux,bottom with Newtonian heating and open top),flow model of viscoelastic fluids(i.e.modified Darcy-Jeffrey,Darcy-Brinkman-Oldroyd and Darcy-Brinkman-Maxwell models),local thermal non-equilibrium and rotation on the critical Rayleigh number of thermal instability of viscoelastic fluids can be calculated.By using the method of weakly non-linear analysis,the bifurcation from the basic state and the analytical solution of Nusselt number at the neighborhood of critical point can be obtained.By the numerical simulation method,the evolution of flow pattern as well as the variations of Nusselt number at high Rayleigh number can be revealed.It has been found that(1) the elasticity of viscoelastic fluids can destabilize the oscillatory convection;(2) the rotation effect and local thermal non-equilibrium effect can suppress the thermal instability of viscoelastic fluids;(3) at the neighborhood of critical point,the bifurcation from the basic state for stationary convection is supercritical,while the bifurcation for the oscillatory can be supercritical or subcritical,mainly depending on the values of viscoelastic parameters,Prandtl number and Darcy number;(4) with the increasing Rayleigh number,the flow pattern of thermal convection evolve from one-cell pattern into multi-cell roll pattern,and finally a chaotic pattern.
引文
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26申德勇.热-化学复合驱提高稠油采收率机理的实验研究.[硕士论文].青岛:中国石油大学(华东),2011(Shen Deyong.Experimental research on mechanism of thermal-chemical flooding for enhancing heavy oil recovery.Qingdao:China University of Petroleum,2011(in Chinese))
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34 Niu J,Fu CJ,Tan WC.Stability of thermal convection of an Oldroyd-B fluid in a porous medium with Newtonian heating.Physics Letters A,2010,374(45):4607-4613
35 Niu J,Shi ZH,Tan WC.The viscoelastic effects on thermal convection of an Oldroyd-B fluid in open-top porous media.Journal of Hydrodynamics,2013,25(4):639-642
36 Niu J,Shi ZH,Tan WC.Numerical simulation of thermal convection of viscoelastic fluids in an open-top porous medium with constant heat flux.Journal of Hydrodynamics,2015,27(1):52-61
37 Niu J,Shi ZH,Tan WC.Thermal instability and heat transfer of viscoelastic fluids in bounded porous media with constant heat flux boundary.Journal of Hydrodynamics,2015,27(5):809-812
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41 Wang SW,Tan WC.Stability analysis of soret-driven doublediffusive convection of Maxwell fluid in a porous medium.International Journal of Heat and Fluid Flow,2011,32(1):88-94
42 Kang JH,Fu CJ,Tan WC.Thermal convective instability of viscoelastic fluids in a rotating porous layer heated from below.Journal of Non-Newtonian Fluid Mechanics,2011,166(1-2):93-101
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45 Yin C,Fu CJ,Tan WC.Stability of thermal convection in a fluidporous system saturated with an Oldroyd-B fluid heated from below.Transport in Porous Media,2013,99(2):327-347
46 Yin C,Niu J,Fu CJ,et al.Thermal convection of a viscoelastic fluid in a fluid-porous system subjected to a horizontal plane Couette flow.International Journal of Heat and Fluid Flow,2013,44(4):711-718
47 Malashetty MS,Shivakumara IS,Kulkarni S,et al.Convective instability of Oldroyd-B fluid saturated porous layer heated from below using a thermal non-equilibrium model.Transport in Porous Media,2006,64(1):123-139
48 Malashetty MS,Swamy M,Heera R.The onset of convection in a binary viscoelastic fluid saturated porous layer.Journal of Applied Mathematics and Mechanics,2009,89(5):356-369
49 Malashetty MS,Tan WC,Swamy M.The onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer.Physics of Fluids,2009,21(8):084101
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55 Akhatov IS,Chembarisova RG.The thermoconvective instability in hydrodynamics of relaxational liquids//Instabilities and Multiphase Flows,New York:Plenum Press,1993:277-287
56 Shivakumara IS,Sureshkumar S.Convective instabilities in a viscoelastic fluid-saturated porous medium with through flow.Journal of Geophysics Engeering,2007,4:104-115
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59 Vafai K,Tien CL.Boundary and inertia effects on flow and heat transfer in porous media.International Journal of Heat and Mass Transfer,1981,24(1):195-202
60 Tan WC,Masuoka T.Stokes'first problem for an Oldroyd-B fluid in a porous half-space.Physics of Fluids,2005,17(2):023101
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2 Rayleigh L.On convection currents in a horizontal layer of fluid,when the higher temperature is on the under side.Philosophical Magazine,1916,32:529-546
3 Reid WH,Harris DL.Some further result on the B'enard problem.Physics of Fluids,1958,1:102-110
4 Chandrasekhar S.Hydrodynamic and Hydromagnetic Stability.Oxford:Oxford University Press,1961
5 Huppert HE,Turner JS.Double-diffusive convection.Journal of Fluid Mechanics,1981,106:299-329
6 Sezai I,Mohamad AA.Double diffusive convection in a cubic enclosure with opposing temperature and concentration gradients.Physics of Fluids,2000,12(9):2210-2223
7 Kuznetsov AV,Nield DA.The onset of double-diusive nanofluid convection in a layer of a saturated porous medium.Transport in Porous Media,2000,85(3):941-951
8 Jones CA,Soward AM,Mussa AI.The onset of thermal convection in a rapidly rotating sphere.Journal of Fluid Mechanics,2000,405:157-179
9 Straughan B.A sharp nonlinear stability threshold in rotating porous convection.Proceedings of the Royal Society Lond A,2001,457(2005):87-93
10 Malashetty MS,Swamy M,Kulkarni S.Thermal convection in a rotating porous layer using a thermal nonequilibrium model.Physics of Fluids,2007,19(5):054102
11 Friedrich J,Lee YS,Fischer B,et al.Experimental and numerical study of Rayleigh-Benard convection affected by a rotating magnetic field.Physics of Fluids,1999,11(4):853-861
12 Dawes JHP.Localized convection cells in the presence of a vertical magnetic field.Journal of Fluid Mechanics,2007,570:385-406
13 Wang CY.Onset of convection in a fluid-saturated rectangular box.bottom heated by constant flux.Physics of Fluids,1999,11(6):1673-1675
14 Bringedal C,Berre I,Nordbotten JM,et al.Linear and nonlinear convection in porous media between coaxial cylinders.Physics of Fluids,2011,23(5):094109
15 Horton CW,Rogers FT.Convection currents in a porous medium.Journal of Appllied Physics,1945,16(6):367-370
16 Lapwood ER.Convection of a fluid in a porous medium.Proceedings of the Cambridge Philosophical Society,1948,44(4):508-521
17 Katto Y,Masuoka T.Criterion for onset of convection in a saturated porous medium.Int.J.Heat Mass Transfer,1967,10:297-309
18 Beck JL.Convection in a box of porous materialsaturated with fluid.Physics of Fluids,1972,15:1377-1383
19 Wang CY.Onset of natural convection in a fluid-saturated porous medium inside a cylindrical enclosure bottom heated by constant flux.International Communication of Heat and Mass Transfer,1998,25(4):593-598
20 Wang CY.Convective stability a rectangular box of fluid-saturated porous medium with constant pressure top and constant flux bottom heating.Transport in Porous Media,2002,46:37-42
21 Zebib A.Onset of natural convection in a cylinder of water saturated porous media.Physics of Fluids,1978,21(4):699-700
22 Haugen KB,Tyvand PA.Onset of thermal convection in a vertical porous cylinder with conducting wall.Physics of Fluids,2003,15(9):2661-2667
23 Bau HH,Torrance KE.Onset of convection in a permeable medium between vertical coaxial cylinders.Physics of Fluids,1981,24(3):382-385
24 Otero PDJ,Dontcheva LA,Johnston H,et al.High-Rayleighnumber convection in a fluid-saturated porous layer.Journal of Fluid Mechanics,2004,500(500):263-281
25 Cherkaoui ASM,Wilcock WSD.Characteristics of high Rayleigh number two-dimensional convection in an open-top porous layer heated from below.Journal of Fluid Mechanics,1999,394(394):241-260
26申德勇.热-化学复合驱提高稠油采收率机理的实验研究.[硕士论文].青岛:中国石油大学(华东),2011(Shen Deyong.Experimental research on mechanism of thermal-chemical flooding for enhancing heavy oil recovery.Qingdao:China University of Petroleum,2011(in Chinese))
27余荔,宁利中,魏炳乾等.Rayleigh-Benard对流及其在工程中的应用.水资源与水工程学报,2008,19(3):52-54(Yu Li,Ning Lizhong,Wei Bingqian,et al.Rayleigh-Benard convection and application in engineering.Journal of Water Resources&Water Engineering,2008,19(3):52-54(in Chinese))
28 Rudraiah N,Kaloni PN,Radhadevi PV.Oscillatory convection in a viscoelastic fluid through a porous layer heated from below.Rheology Acta,1989,28(1):48-53
29 Bertola V,Cafaro E.Thermal instability of viscoelastic fluids in horizontal porous layers as initial value problem.International Journal of Heat and Mass Transfer,2006,49(21):4003-4012
30 Kim MC,Lee SB,Chung BJ.Thermal instability of viscoelastic fluids in porous media.International Journal of Heat and Mass Transfer,2003,46(26):5065-5072
31 Tan WC,Masuoka T.Stability analysis of a Maxwell fluid in a porous medium heated from below.Physics Letters A,2007,360(3):454-460
32 Fu CJ,Zhang ZY,Tan WC.Numerical simulation of thermal convection of a viscoelastic fluid in a porous square box heated from below.Physics of Fluids,2007,19(10):104107
33 Niu J,Fu CJ,Tan WC.Thermal convection of a viscoelastic fluid in an open-top porous layer heated from below.Journal of NonNewtonian Fluid Mechanics,2010,165:203-211
34 Niu J,Fu CJ,Tan WC.Stability of thermal convection of an Oldroyd-B fluid in a porous medium with Newtonian heating.Physics Letters A,2010,374(45):4607-4613
35 Niu J,Shi ZH,Tan WC.The viscoelastic effects on thermal convection of an Oldroyd-B fluid in open-top porous media.Journal of Hydrodynamics,2013,25(4):639-642
36 Niu J,Shi ZH,Tan WC.Numerical simulation of thermal convection of viscoelastic fluids in an open-top porous medium with constant heat flux.Journal of Hydrodynamics,2015,27(1):52-61
37 Niu J,Shi ZH,Tan WC.Thermal instability and heat transfer of viscoelastic fluids in bounded porous media with constant heat flux boundary.Journal of Hydrodynamics,2015,27(5):809-812
38 Zhang ZY,Fu CJ,Tan WC.Onset of oscillatory convection in a porous cylinder saturated with a viscoelastic fluid.Physics of Fluids,2007,19(9):098104
39 Zhang ZY,Fu CJ,Tan WC.Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below.Physics of Fluids,2008,20(8):084103
40 Wang SW,Tan WC.Stability analysis of double-diffusive convection of Maxwell fluid in a porous medium heated from below.Physics Letters A,2008,372:3046-3050
41 Wang SW,Tan WC.Stability analysis of soret-driven doublediffusive convection of Maxwell fluid in a porous medium.International Journal of Heat and Fluid Flow,2011,32(1):88-94
42 Kang JH,Fu CJ,Tan WC.Thermal convective instability of viscoelastic fluids in a rotating porous layer heated from below.Journal of Non-Newtonian Fluid Mechanics,2011,166(1-2):93-101
43 Kang JH,Niu J,Fu CJ,et al.Coriolis effect on thermal convective instability of viscoelastic fluids in a rotating porous cylindrical annulus.Transport in Porous Media,2013,98(2):349-362
44 Yin C,Fu CJ,Tan WC.Onset of thermal convection in a Maxwell fluid-saturated porous medium:the effects of hydrodynamic boundary and constant flux heating conditions.Transport in Porous Media,2012,91(3):777-790
45 Yin C,Fu CJ,Tan WC.Stability of thermal convection in a fluidporous system saturated with an Oldroyd-B fluid heated from below.Transport in Porous Media,2013,99(2):327-347
46 Yin C,Niu J,Fu CJ,et al.Thermal convection of a viscoelastic fluid in a fluid-porous system subjected to a horizontal plane Couette flow.International Journal of Heat and Fluid Flow,2013,44(4):711-718
47 Malashetty MS,Shivakumara IS,Kulkarni S,et al.Convective instability of Oldroyd-B fluid saturated porous layer heated from below using a thermal non-equilibrium model.Transport in Porous Media,2006,64(1):123-139
48 Malashetty MS,Swamy M,Heera R.The onset of convection in a binary viscoelastic fluid saturated porous layer.Journal of Applied Mathematics and Mechanics,2009,89(5):356-369
49 Malashetty MS,Tan WC,Swamy M.The onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer.Physics of Fluids,2009,21(8):084101
50康建宏.旋转效应对多孔介质内热对流稳定性与换热的影响.[博士论文].北京:北京大学,2013(Kang Jianhong.Rotating effects on thermal convective stability and heat transfer in porous media.[PhD Thesis].Beijing:Peking University,2013(in Chinese))
51尹晨.黏弹性流体在多孔介质以及流体-多孔介质双层系统内的热对流稳定性分析.[博士论文].北京:北京大学,2012(Yin Chen.Stability analysis of thermal convection of viscoelastic fluids in porous media and fluid-porous systems.[PhD Thesis].Beijing:Peking University,2013(in Chinese))
52 Khuzhayorov B,Auriault JL,Royer P.Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media.International Journal of Engineering Science,2000,38(5):487-504
53 Alishayev MG,Kh A.Mirzadzhanzadeh.About retardation phenomena in filtration theory.Izvestiia VUZov Neft i Gaz,1975,6:71-74
54 Yoon DY,Kim MC,Choi CK.The onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic liquid.Transport in Porous Media,2004,55(3):275
55 Akhatov IS,Chembarisova RG.The thermoconvective instability in hydrodynamics of relaxational liquids//Instabilities and Multiphase Flows,New York:Plenum Press,1993:277-287
56 Shivakumara IS,Sureshkumar S.Convective instabilities in a viscoelastic fluid-saturated porous medium with through flow.Journal of Geophysics Engeering,2007,4:104-115
57 Nield DA,Bejan A.Convection in Porous Media.2nd edn.Springer,1999
58 Rudraiah N,Radhadevi PV.Convection in a viscoelastic fluidsaturated sparsely packed porous layer.Candian Journal of Physics,1990,68:1446-1453
59 Vafai K,Tien CL.Boundary and inertia effects on flow and heat transfer in porous media.International Journal of Heat and Mass Transfer,1981,24(1):195-202
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