Bizon C., Werne J., Predtechensky A. A., Julien K., McCormick W. D., Swift J. B., Swinney Harry L.
Center for Nonlinear Dynamics and Department of Physics, University of Texas, Austin, Texas 78712.
Chaos. 1997 Mar;7(1):107-124. doi: 10.1063/1.166243.
We have studied turbulent convection in a vertical thin (Hele-Shaw) cell at very high Rayleigh numbers (up to 7x10(4) times the value for convective onset) through experiment, simulation, and analysis. Experimentally, convection is driven by an imposed concentration gradient in an isothermal cell. Model equations treat the fields in two dimensions, with the reduced dimension exerting its influence through a linear wall friction. Linear stability analysis of these equations demonstrates that as the thickness of the cell tends to zero, the critical Rayleigh number and wave number for convective onset do not depend on the velocity conditions at the top and bottom boundaries (i.e., no-slip or stress-free). At finite cell thickness delta, however, solutions with different boundary conditions behave differently. We simulate the model equations numerically for both types of boundary conditions. Time sequences of the full concentration fields from experiment and simulation display a large number of solutal plumes that are born in thin concentration boundary layers, merge to form vertical channels, and sometimes split at their tips via a Rayleigh-Taylor instability. Power spectra of the concentration field reveal scaling regions with slopes that depend on the Rayleigh number. We examine the scaling of nondimensional heat flux (the Nusselt number, Nu) and rms vertical velocity (the Peclet number, Pe) with the Rayleigh number (Ra()) for the simulations. Both no-slip and stress-free solutions exhibit the scaling NuRa() approximately Pe(2) that we develop from simple arguments involving dynamics in the interior, away from cell boundaries. In addition, for stress-free solutions a second relation, Nu approximately nPe, is dictated by stagnation-point flows occurring at the horizontal boundaries; n is the number of plumes per unit length. No-slip solutions exhibit no such organization of the boundary flow and the results appear to agree with Priestley's prediction of Nu approximately Ra(1/3). (c) 1997 American Institute of Physics.
我们通过实验、模拟和分析，研究了在非常高的瑞利数（高达对流起始值的7×10⁴倍）下，垂直薄（赫勒 - 肖）槽道中的湍流对流。在实验中，对流由等温槽道中施加的浓度梯度驱动。模型方程处理二维场，降维通过线性壁面摩擦施加其影响。对这些方程的线性稳定性分析表明，当槽道厚度趋于零时，对流起始的临界瑞利数和波数不依赖于顶部和底部边界的速度条件（即无滑移或无应力）。然而，在有限的槽道厚度δ下，具有不同边界条件的解表现不同。我们对两种边界条件下的模型方程进行了数值模拟。来自实验和模拟的完整浓度场的时间序列显示出大量溶质羽流，这些羽流在薄浓度边界层中产生，合并形成垂直通道，有时通过瑞利 - 泰勒不稳定性在其尖端分裂。浓度场的功率谱揭示了斜率依赖于瑞利数的标度区域。我们研究了模拟中无量纲热通量（努塞尔数，Nu）和均方根垂直速度（佩克莱数，Pe）与瑞利数（Ra()）的标度关系。无滑移和无应力解都表现出标度关系NuRa()≈Pe²，这是我们从涉及远离槽道边界的内部动力学的简单论证中得出的。此外，对于无应力解，第二个关系Nu≈nPe由水平边界处出现的驻点流决定；n是单位长度的羽流数量。无滑移解没有这种边界流的组织，结果似乎与普里斯特利对Nu≈Ra¹/³的预测一致。(c) 1997美国物理研究所。
