普朗特数对对流斑图的分区和扰动成长模式的影响
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摘要
该文采用二维流体力学基本方程组对普朗特数Pr(28)0.72流体对流进行数值模拟。取相对瑞利数r(28)4时,随着水平来流雷诺数的逐渐增大可获得行波、局部行波和水平流三种斑图。分析Pr(28)0.72流体的斑图分区情况,可以得出在水平来流雷诺数和相对瑞利数构成的平面上,他们都被水平来流上下临界雷诺数Re_u和Re_l划分为行波区间、局部行波区间和水平流区间;同时对三种流体的水平来流上下临界雷诺数Re_u和Re_l随相对瑞利数r的变化情况进行了观察,可发现他们都是随r的增大而增大的。在相对瑞利数r(28)3 8、和13时,研究普朗特数Pr对水平来流上下临界雷诺数Re_u和Re_l的影响,可知Re_u和Re_l随Pr增大而减小;且介于Re_u和Re_l之间的局部行波的范围随Pr增大而减小。对普朗特数Pr(28)6.99、0.72和0.0272的行波斑图及局部行波斑图的成长过程进行研究,发现了两种扰动成长模式,其中Pr(28)6.99流体的扰动是从腔体中部开始成长的;Pr(28)0.72和0.0272两种流体的扰动是从腔体右端区域内开始成长。
The two-dimensional numerical simulation of the fully hydrodynamic equations is used to the fluid convection in the Prandtl number Pr(28)0.72.Three kinds of patterns have been obtained when the horizontal flow Reynolds number gradually increases at the reduced Rayleigh number r(28)4,including traveling wave pattern,localized traveling wave pattern and horizontal flow pattern.By analysing the partition of the convection pattern at Pr(28)0.72,in the plane which formed by the horizontal flow Reynolds number and the reduced Rayleigh number,it is acquired that they all are divided into traveling wave zone,localized traveling wave zone and horizontal flow zone by the upper and lower critical Reynolds numbersRe_u,Re_l in horizontal flow,which can be acquired.Moreover,by observing the variation of Re_u,Re_l with the reduced Rayleigh number r for three kinds of fluid,it can be found that they increase with increasing r.When the reduced Rayleigh number r(28)3、8、13,it is studied that the influence of the Prandtl number Pr onRe_u,Re_l,so as to know thatRe_u,Re_l decreases with the increase of Pr;and the range of the localized traveling wave zone which situated betweenRe_u,Re_l decreases with increasing Pr.The growth process of traveling wave pattern,localized traveling wave pattern which are studied at Pr(28)6.99、0.72、0.0272.That two kinds of perturbation growth modes are found,i.e.perturbation of traveling wave pattern and localized traveling wave pattern.At Pr(28)6.99,they start to grow in the middle range of the cavity;the perturbation of traveling wave pattern and the localized traveling wave pattern at Pr(28)0.72、0.0272 all start to grow in the right side of the cavity.
The two-dimensional numerical simulation of the fully hydrodynamic equations is used to the fluid convection in the Prandtl number Pr(28)0.72.Three kinds of patterns have been obtained when the horizontal flow Reynolds number gradually increases at the reduced Rayleigh number r(28)4,including traveling wave pattern,localized traveling wave pattern and horizontal flow pattern.By analysing the partition of the convection pattern at Pr(28)0.72,in the plane which formed by the horizontal flow Reynolds number and the reduced Rayleigh number,it is acquired that they all are divided into traveling wave zone,localized traveling wave zone and horizontal flow zone by the upper and lower critical Reynolds numbersRe_u,Re_l in horizontal flow,which can be acquired.Moreover,by observing the variation of Re_u,Re_l with the reduced Rayleigh number r for three kinds of fluid,it can be found that they increase with increasing r.When the reduced Rayleigh number r(28)3、8、13,it is studied that the influence of the Prandtl number Pr onRe_u,Re_l,so as to know thatRe_u,Re_l decreases with the increase of Pr;and the range of the localized traveling wave zone which situated betweenRe_u,Re_l decreases with increasing Pr.The growth process of traveling wave pattern,localized traveling wave pattern which are studied at Pr(28)6.99、0.72、0.0272.That two kinds of perturbation growth modes are found,i.e.perturbation of traveling wave pattern and localized traveling wave pattern.At Pr(28)6.99,they start to grow in the middle range of the cavity;the perturbation of traveling wave pattern and the localized traveling wave pattern at Pr(28)0.72、0.0272 all start to grow in the right side of the cavity.
引文
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[16]宁利中,胡彪,宁碧波,等.Poiseuille-RayleighBénard流动中对流斑图的分区和成长[J].物理学报,2016,65(21):223-230.NING Li-zhong,HU Biao,NING Bi-bo,et al.Partition and growth of convection patterns in PoiseuilleRayleigh-Bénard flow[J].Acta Physica Sinica,2016,65(21):223-230.
[2]DUBOIS M,SILVA R D,DAVIAUD F,et al.Collective oscillating mode in a one-dimensional chain of convective rolls[J].Europhysics Letters,2007,8(2):135-139.
[3]MERCADER I,ALONSO A,BATISTE O.Numerical analysis of the Eckhaus instability in travelling-wave convection in binary mixtures[J].The European Physical Journal E,2004,15(3):311-318.
[4]KOLODNER P.Observations of the eckhaus instability in one-dimensional traveling-wave convection[J].Physical Review A,1992,46(4):R1739-R1742.
[5]NING Li-zhong,HARADA Y,YAHATA H,et al.FullyDeveloped traveling wave convection in binary fluid mixtures with lateral flow[J].Progress of Theoretical Physics,2001,106(3):503-512.
[6]NING Li-zhong,QI Xin,HARADA Y,et al.A periodically localized traveling wave state of binary fluid convection with horizontal flows[J].Journal of Hydrodynamics,2006,18(2):199-205.
[7]宁利中,王永起,袁喆,等.两种不同结构的混合流体局部行波对流斑图[J].科学通报,2016,61(8):872-880.NING Li-zhong,WANG Yong-qi,YUAN Zhe,et al.Two types of patterns of localized traveling wave convection in binary fluid mixtures with different structures[J].Chinese Science Bulletin,2016,61(8):872-880.
[8]FAUVE S,THUAL O.Solitary waves generated by subcritical instabilities in dissipative systems[J].Physical Review Letters,1990,64(3):282-284.
[9]VAN S W,HOHENBERG P C.Pulses and front in the compex ginzburg-landau equation near a subcritical bifurcation[J].Physical Review Letters,1990,64(64):749-752.
[10]RIECKE H,RAPPEL W J.Coexisting pulses in a model for Binary-Mixture convection[J].Physical Review Letters,1995,75(22):4035-4038.
[11]RIECKE H.Self-trapping of traveling-wave pulses in binary mixtures convection[J].Physical Review Letters,1992,68(3):301-304.
[12]HU Jun,YIN Xie-yuan.Two-dimensional simulation of Poiseuille-Rayleigh-Benard flows in binary fluids with soret effect[J].Progress in Natural Science,2007,17(12):1389-1396.
[13]JUNG C,LUCKE M,BUCHEL P.Influence of through-flow on linear pattern formation properties in binary mixture convection[J].Physical Review E,1996,54(2):1510-1529.
[14]李国栋,黄永念.水平流作用下行波对流的成长及周期性重复[J].物理学报,2004,53(11):3800-3805.LI Guo-dong,HUANG Yong-nian.Growth and periodic repeating of traveling-wave convection with through-flow[J].Acta Physica Sinica,2004,53(11):3800-3805.
[15]李国栋,黄永念.水平流作用下行波对流的成长及周期性重复[J].物理学报,2004,53(11):3800-3805.LI Guo-dong,HUANG Yong-nian.Traveling-wave convection with Eckhaus instability modulation in a binary fluid mixture[J].Acta Physica Sinica,2007,56(8):4742-4748.
[16]宁利中,胡彪,宁碧波,等.Poiseuille-RayleighBénard流动中对流斑图的分区和成长[J].物理学报,2016,65(21):223-230.NING Li-zhong,HU Biao,NING Bi-bo,et al.Partition and growth of convection patterns in PoiseuilleRayleigh-Bénard flow[J].Acta Physica Sinica,2016,65(21):223-230.