Abstract. The paper deals with the investigation of critical perturbations structure in a horizontal layer of fluid with temperature inversion of density. The lower boundary of the layer is rigid and upper boundary is free and undeformable. The constant vertical heat flux is imposed at both boundaries. The thermocapillarity, evaporation and radiation are neglected. The temperature dependence of the density is assumed to be quadratic. The coordinate of the density inversion point which shows the position of the plane of maximal density in the fluid layer in the conductive state is used for the description of density inversion effect. The influence of the location of density inversion point on structure of critical perturbations is studied. It is shown that, depending on the location of the inversion point, velocity profile of longwave critical perturbations can have two-floor or three-floor structure. Stability to finitewavelength perturbations is studied for the entire range of their existence. The asymptotic formulas for critical values of the Rayleigh number and wave number are obtained, the structure of finite-wavelength critical perturbations .is determined.
The three-dimensional solutions of nonlinear long-wavelength approximations for the problem of Marangoni convection in a thin horizontal layer of a viscous incompressible fluid with a free surface is being considered. The temperature distribution in the liquid corresponds to a uniform vertical gradient distorted by the imposition of a weakly inhomogeneous heat flux localized in the horizontal plane, caused by a lattice of either localized or continuously distributed heat sources and sinks. The lower boundary of the layer is solid and thermally insulated, while the upper one is free and deformable. The statement of the problem is motivated by the search for ways to control convection structures. The problem in long-wave approximation is described by a system of nonlinear transport equations for the amplitudes of temperature distribution and surface deformation. The numerical solution of the problem is based on the pseudospectral method. The dynamics of non-stationary dissipative structures is considered.
Исследуется влияние интенсивности подогрева, выражаемой числом Грасгофа, на надкритические режимы тепловой конвекции талой воды в горизонтальной прямоугольной полости с аспектным отношением, равным двум. На боковых твердых границах выполняются условия теплоизолированности, а на нижней твердой и верхней свободной (горизонтальной и недеформируемой) гранях задан постоянный вертикальный поток тепла. При условии, когда средняя по полости температура близка к температуре инверсии плотности воды, в полости возможно состояние механического равновесия, в котором поверх неустойчиво стратифицированного слоя расположен устойчиво стратифицированный. Для двух случаев положения горизонтальной границы между этими слоями рассмотрена структура стационарной плоской надкритической тепловой конвекции. Расчеты проведены конечно-разностным методом на квадратной сетке с 128 узлами по горизонтальной координате и 64 -по вертикальной. Вычисления показали, что при равной толщине неустойчиво и устойчиво стратифицированных слоев надкритическая конвекция в области примерно до шести надкритичностей имеет в горизонтальном направлении двухячеистую структуру с двумя (большим снизу и более слабым сверху) вихрями в каждой из ячеек. Эта двухячеистая структура при увеличении надкритичности гистерезисным образом переходит в четырехячеистую. Для случая, когда толщина устойчиво стратифицированного слоя в три раза меньше толщины неустойчиво стратифицированного, надкритическое конвективное течение имеет вид вытянутой по горизонтали одновихревой ячейки. С увеличением числа Грасгофа до, примерно, стократной надкритичности течение остается в целом одновихревым и не испытывает бифуркаций.
The supercritical modes of water convection are investigated at room temperature in an elongated horizontal cavityes, with a width-to-height ratios of 2 : 1 and 3 : 1. The Prandtl number is assumed to be equal to seven. A constant heat flux is set at the upper free and lower solid boundaries, and the lateral solid boundaries are assumed to be thermally insulated. Calculations carried out by the finite-difference method for values of the Rayleigh number exceeding the critical one by up to thirty times have shown that in the indicated interval of Rayleigh numbers in both cavities in the supercritical region, a single-vortex steady state is realized
The exact solution of boundary layer equations for laminar convective torch from point source of heat in fluid with power dependence of density on temperature was found.Key words: point source of heat, boundary layer, power dependence of density on temperature, point solution.A peculiar example of convective system with nonuniform stratification is infinite water layer which temperature is close to 4 °С over the horizontal heat-insulated surface with point source of heat (axisymmetric torch). In this area of temperatures, water density ρ nonlinearly depends on temperature T and in wide range of salinity, pressure, and temperature values takes the following form [1]:where ρ m is the maximum density, α is the coefficient of thermal expansion, T i is the temperature of density inversion, γ is the indicator of temperature inversion of density.The problem of convective torch in the boundary layer approximation in fluid with power dependence of density on temperature was numerically solved in [1] for several values of inversion coefficient ⎯ γ = 1.5, 1.81, and 2 that agree with water at different pressure and salinity values. Self-similar transformations of coordinates transforming thermal convection equations in Boussinesq approximation to the system of ordinary differential equations have been found. The problem for cubic dependence (γ = 3) was studied with method of decomposition in terms of small parameter and numerically in the work [2] without the boundary-layer approximation.The objective of this work is the generalization of exact solution obtained in the work [3] for convective torch in fluid with linear density dependence on temperature, i.e., for γ = 1, to the case of power dependence of density on temperature in accordance with (1) when temperature inversion coefficient γ is an arbitrary real number.Let viscous incompressible fluid fill space over solid horizontal heat-insulated plane. Assume that kinematic viscosity ν, thermal diffusivity χ, and gravitational acceleration g are constant. Density dependence on temperature is determined by
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