Исследование конвективных структур в небуссинесковской жидкости вблизи порога устойчивости. Часть первая -анализ упрощенных моделей Работа посвящена исследованию процесса возникновения и развития неустой-чивости Рэлея-Бенара в горизонтальном, несимметричном по вертикали слое жид-кости. Нарушение симметрии обусловлено внутренним подогревом или зависи-мостью коэффициентов вязкости и теплопроводности от температуры и верти-кальной координаты. В первой части работы на основе амплитудных уравнений и методом разложения по малому параметру изучается влияние числа Прандтля на процесс формирования устойчивых конвективных структур. Полученные ре-зультаты используются во второй части работы для анализа результатов численно-го моделирования конвективной неустойчивости в асимметричном по вертикали слое вязкой несжимаемой жидкости.Ключевые слова: Конвекция Рэлея-Бенара, небуссинесковская жидкость, конвективная неустойчивость, амплитудные уравнения, метод разложения по малому параметру, устойчивые планформы, валы, шестиугольные ячейки, число Прандтля.
Viatcheslav Victorovich Kolmychkov, Olga Semenovna MazhorovaInvestigation of convective structures near the stability threshold in non-Bousssinesq fluid. Part one -analysis of simple models The paper investigates stable convective structures in vertically asymmetric horizontal fluid layer. The asymmetry is caused by internal heat generation, variable thermal diffusion and variable kinematic viscosity (temperature and vertical coordinate dependence is considered). The first part of the paper provides analysis of Prandtl number effect on the planform selection in scope of amplitude equations and perturbation method. The second part of the paper deals with a comparison of theoretical data to results of 3D numerical simulation of convective instability in finite layer of non-Boussinesq fluid.
The paper deals with the hexagonal convective flow near the stability threshold in an internally heated fluid layer. In our previous numerical study of convection near the stability threshold in a square box with internal heat generation [Phys. Lett. A 377, 2111 (2013)]PYLAAG0375-960110.1016/j.physleta.2013.06.013 for a region of large horizontal extent, it has been shown that at small values of Prandtl number (Pr), convection sets in as a pattern of hexagonal cells with upward motion in the center (up-hexagons), whereas at large Pr, a stable flow pattern is formed by hexagonal cells with a downward motion in the center (down-hexagons). Here, we study axisymmetric convection in a cylinder as a model of motion in a single hexagonal cell. The radius of the cylinder matches the size of hexagons observed in our three-dimensional simulation. The lateral boundary of the cylinder is free and heat insulated. Horizontal bounding surfaces are rigid. The upper boundary is maintained at a constant temperature; the lower one is insulated. Two stable, steady-state motions with the upward and downward flow at the cylinder axis have been attained in calculations, irrespective of Pr. Cylindrical motion with the same direction of circulation as in the stable hexagons has a maximum temperature drop measured along the radius at the bottom of the cell. We suggest maximization of the temperature drop as a selection criterion, which determines the preferred state of motion in an internally heated fluid layer. This new selection principle is validated by the comparative analysis of the dominant nonlinear effects in low- and high-Prandtl number convection.
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