Instabilities of thermal gravitational convection and heat transfer in the Czochralski model at different Prandtl numbers

The results of the calculations of critical Grash of numbers, at which flowfield and temperature fluctuations originate in the axisymmetric and three-dimensional models of crystal growth by pulling from a melt, are presented. The salient features of the convection and heat transfer structure in the zones of stabilization and changeover of dangerous modes are studied over a wide Prandtl number range under different boundary conditions on the melt surface and compared with the experimental data.     

Unsteady nonaxisymmetric flows in the hydrodynamic Czochralski model at high Prandtl numbers

Thermal-gravitational and thermocapillary convection is numerically modeled in the axisymmetric and three-dimensional approximations for the hydrodynamic model configuration corresponding to technological regimes of oxide crystal growth and taken as the basis in an international test. The salient features of the interaction between the convection and the flow driven by crystal rotation are studied at high Prandtl numbers. The flow and temperature fields occurring upon the generation of axisymmetric oscillations and the loss of axial symmetry are studied, analyzed, and compared with the results of other authors.     

Hydrodymagnetic three-dimensional flow over a stretching surface with heat and mass transfer

A general analysis has been developed to study the combined effect of the free convective heat and mass transfer on the steady three-dimensional laminar boundary layer flow over a stretching surface. The flow is subject to a transverse magnetic field normal to the plate. The governing three-dimensional partial differential equations for the present case are transformed into ordinary differential equation using three-dimensional similarity variables. The resulting equations, are solved numerically by applying a fifth order Runge-Kutta-Fehlberg scheme with the shooting technique. The effects of the Magnetic field Parameter M, buoyancy parameter N, Prandtl number Pr and Schmidt number Sc are examined on the velocity, temperature and concentration distributions. Numerical data for the skin-friction coefficients, Nusselt and Sherwood numbers have been tabulated for various parametric conditions. The results are compared with known from the literature.     

Three-dimensional rotational MHD flows in bounded volumes

Three-dimensional laminar flows of a viscous conducting gas in the neighborhood of a rotating disk are considered. The simultaneous impact of an external magnetic field, suction from the disk surface, and the axial temperature gradient as well as the action of the external axial magnetic field on three-dimensional flows in the neighborhood of rigid permeable surfaces are first studied. An exact analytic solution of the system of the boundary layer equations is obtained. It is found that the direction of the radial flow initiated in the boundary layer can be varied by changing the temperature ratio in the external flow and on the disk for various Prandtl numbers Pr. An approximate solution of the problem of flow in the rotating cylinder in the presence of a retarding cover is constructed on the basis of the approach developed for extended disks.     

外加磁场对熔融半导体CZ法拉晶过程中流动场影响的数值计算

数值分析结果表明,外加磁场可以改变熔融半导体中的流型,几千高斯的磁场可以显著地减小熔体的流动,但对温度场影响不大。     

外加磁场对熔融半导体CZ法拉晶过程中流动场影响的数值计算

数值分析结果表明,外加磁场可以改变熔融半导体中的流型,几千高斯的磁场可以显著地减小熔体的流动,但对温度场影响不大。     

Transient natural convection of low-Prandtl-number fluids in a differentially heated cavity

The transient convective motion in a two-dimensional square cavity driven by a temperature gradient is analysed. The cavity is filled with a low-Prandtl-number fluid and the vertical walls are maintained at constant but different temperatures, while the horizontal boundaries are adiabatic. A control volume approach with a staggered grid is employed to formulate the finite difference equations. Numerically accurate solutions are obtained for Prandtl numbers of 0·001, 0·005 and 0·01 and for Grashof numbers up to 1 × 10⁷. It was found that the flow field exhibits periodic oscillation at the critical Grashof numbers, which are dependent on the Prandtl number. As the Prandtl number is decreased, the critical Grashof number and the frequency of oscillation decrease. Prior to the oscillatory flow, steady state solutions with an oscillatory transient period were predicted. In addition to the main circulation, four weak circulations were predicted at the corners of the cavity.     

Radiation effect on temperature distribution in three-dimensional Couette flow with suction or injection

A theoretical analysis of three-dimensional Couette flow with radiation effect on temperature distribution has been analysed, when the injection of the fluid at the lower stationary plate is a transverse sinusoidal one and its corresponding removal by constant suction through the upper porous plate is in uniform motion. Due to this type of injection velocity, the flow becomes three-dimensional. The effect of Prandtl number, radiation parameter and injection parameter on rate of heat transfer has been examined by the help of graphs. The Prandtl number has a much greater effect on the temperature distribution than the injection or radiation parameter.     

Effect of a Standing Acoustic Wave on the Development of Large-Scale Convection in a Horizontal Layer

The effect of a standing acoustic wave on the development of long-wave convective perturbations in a horizontal layer with thermally insulated boundaries is investigated. The main two-dimensional flow is determined. A nonlinear amplitude equation with spatially-periodic coefficients is derived for investigating the stability of the main flow and secondary convection flows in the neighborhood of the stability threshold. The intensity of the acoustic field is assumed to be low. It is shown that the acoustic action leads to destabilization of the layer. Plane and three-dimensional perturbations are critical at large and small Prandtl numbers, respectively. Nonlinear one-dimensional steady-state solutions of the amplitude equation are obtained and their stability is investigated.     

A thermal LBM-LES model in body-fitted coordinates: Flow and heat transfer around a circular cylinder in a wide Reynolds number range

In order to apply the lattice Boltzmann method (LBM) for modeling passive heat transfer at high Reynolds numbers, a number of models were proposed by introducing the large eddy simulation (LES) into the LBM framework to improve numerical stability. Our study finds that the generalized form of interpolation-supplemented LBM (GILBM), likewise, can locally modify the dimensionless relaxation time, thus enhancing the numerical stability at high Reynolds numbers. Given additional advantages of the GILBM in dealing with complicated geometries and improving computing accuracy, a thermal LBM-LES model in body-fitted coordinates is established in this paper. Numerical validation is performed by investigating the flow and heat transfer around a circular cylinder over a wide range of Reynolds numbers. The obtained results agree well with both experimental and numerical data in the previous work. Meanwhile, the effects of Reynolds number and Prandtl number on thermodynamic features of flow past a circular cylinder are revealed. It is found out that when the Reynolds number exceeds the critical value, the local Nusselt number fluctuates rapidly in a specific region of the rear cylinder surface affected by the Prandtl number. In the near-wake region, the temperature field exhibits significant dependence on the Prandtl number at moderate Reynolds numbers, while such effects turn to be slight at high Reynolds numbers.     

Investigation of the heat transfer mechanism in supersonic turbulent boundary layers

A method is described for calculating turbulent Prandtl numbers from Mach number and total temperature profiles in supersonic boundary layers. The calculations are based on boundary layer measurements in the Mach number range from 3.5 to 5. The investigations clearly indicate that in addition to accurate profile measurements reliable values of shear stress and heat flux at the wall must exist, in order to be able to calculate the turbulent Prandtl number in the viscous regime of the boundary layer. For flow conditions with and without heat transfer, the derived turbulent Prandtl numbers indicate that the turbulent transport of heat decreases much faster towards the wall than the turbulent transport of momentum. The results of the analysis show that only the unequivocal qualitative result of increasing turbulent Prandtl numbers in the viscous region of the boundary layer, can be expected. The variation of the turbulent Prandtl number can be described successfully using a simple approximation, based on the mixing length concept, and is applied to the calculation of total temperature distribution using the law of the wall for compressible flow.     

Heat transfer in magnetohydrodynamic laminar radial channel flow

Heat transfer in the steady axisymetrical laminar source flow of an incompressible electrically conducting fluid between two parallel disks in the presence of a transverse applied magnetic field is analyzed. The energy equation is solved numerically for the temperature distribution, where both Joulean and viscous heating are included. Both local and average Nusselt numbers for the case of constant wall temperature are evaluated. For fluids of moderate and high Prandtl numbers, Nusselt number is seen to be a strong function of both Hartmann number and a heat generation parameter together with a modified Peclet number. However, for fluids of small Prandtl number, Joulean heating and viscous dissipation can be neglected without appreciable error.     

半浮区液桥热毛细振荡流

采用非定常、三维直接数值模拟方法研究大Pr数半浮区液桥热毛细对流从定常流向振荡流的过渡过程．文中详细描述了热毛细振荡流的起振和振荡特征,给出了液桥横截面上振荡流的流场和温度分布．在地面引力场条件下计算的结果与地面实验的结果进行比较,得出液桥水平截面上的流场和温度分布图样以一定的速度旋转,自由表面固定点处流体的环向流速正、负交替变化的一致结论．     

On the Ill-Posedness of the Prandtl Equations in Three-Dimensional Space

In this paper, we give an instability criterion for the Prandtl equations in three-dimensional space, which shows that the monotonicity condition on tangential velocity fields is not sufficient for the well-posedness of the three-dimensional Prandtl equations, in contrast to the classical well-posedness theory of the two-dimensional Prandtl equations under the Oleinik monotonicity assumption. Both linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three-dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work (A well-posedness theory for the Prandtl equations in three space variables. arXiv:1405.5308, 2014) on the well-posedness theory for the three-dimensional Prandtl equations with a special structure.     

On the influences of key modelling constants of large eddy simulations for large-scale compartment fires predictions

An in-house large eddy simulation (LES) based fire field model has been developed for large-scale compartment fire simulations. The model incorporates four major components, including subgrid-scale turbulence, combustion, soot and radiation models which are fully coupled. It is designed to simulate the temporal and fluid dynamical effects of turbulent reaction flow for non-premixed diffusion flame. Parametric studies were performed based on a large-scale fire experiment carried out in a 39-m long test hall facility. Several turbulent Prandtl and Schmidt numbers ranging from 0.2 to 0.5, and Smagorinsky constants ranging from 0.18 to 0.23 were investigated. It was found that the temperature and flow field predictions were most accurate with turbulent Prandtl and Schmidt numbers of 0.3, respectively, and a Smagorinsky constant of 0.2 applied. In addition, by utilising a set of numerically verified key modelling parameters, the smoke filling process was successfully captured by the present LES model.     

Analysis of mixed convection in the Czochralski model in a wide range of Prandtl numbers

The results of the calculations and analysis of the effect of separate and joint rotation of the crystal and the crucible on the flow stability are presented for a wide range of Prandtl numbers (from 0.01 to 10). The regimes with a high stability threshold are determined for different combinations of the rotation velocities. It is shown that for high Prandtl numbers, simultaneous rotation of the crystal and the crucible makes it possible to increase the critical Grashof number in 9–12 times. A resultant diagram (map) of the limiting regimes of natural and mixed convection is constructed. Themethodology of control and analysis of 2D and 3D instability modes is discussed.     

Oscillatory instability of advective flow in a horizontal cylinder in the presence of a rotating magnetic field

The oscillatory instability of advective conducting fluid flow in a horizontal circular cylinder in the presence of a rotating magnetic field is investigated. For Prandtl numbers Pr = 0 and 0.01 calculations showed that in both cases the monotonic instability observed in the absence of a field and in weak fields transforms into oscillatory instability at sufficiently large magnetic Taylor numbers. At Pr ?? 0, oscillatory instability appears at substantially higher magnetic Taylor numbers.     

Effect of crystal and crubicle rotation on the flow stability in the Czochralski model at low Prandtl numbers

The results of a numerical investigation of the joint effect of crystal and crubicle rotation on the flow stability at low Prandtl numbers (Pr=0.01 to 0.07) are presented. The regimes with an elevated stability threshold are determined for various combinations of crystal and crubicle rotations and the heat flux distributions over the growing crystal endface are obtained. The mechanisms of the loss of stability, as the critical values of the Grashof number and the rotation velocities are exceeded, are considered and a new regime of stable nonaxisymmetric flow is established.     

Effect of particles on turbulent thermal field of channel flow with different Prandtl numbers

The direct numerical simulation(DNS) of heat transfer in a fully developed non-isothermal particle-laden turbulent channel flow is performed.The focus of this paper is on the modulation of the particles on turbulent thermal statistics in the particle-laden flow with three Prandtl numbers(P r = 0.71,1.5,and 3.0) and a shear Reynolds number(Reτ = 180).Some typical thermal statistics,including normalized mean temperature and their fluctuations,turbulent heat fluxes,Nusselt number and so on,are analyzed.The results show that the particles have less effects on turbulent thermal fields with the increase of Prandtl number.Two reasons can explain this.First,the correlation between fluid thermal field and velocity field decreases as the Prandtl number increases,and the modulation of turbulent velocity field induced by the particles has less influence on the turbulent thermal field.Second,the heat exchange between turbulence and particles decreases for the particle-laden flow with the larger Prandtl number,and the thermal feedback of the particles to turbulence becomes weak.     

Low-Prandtl-number Bénard–Marangoni convection in a vertical magnetic field

The effect of a homogeneous magnetic field on surface-tension-driven Bénard convection is studied by means of direct numerical simulations. The flow is computed in a rectangular domain with periodic horizontal boundary conditions and the free-slip condition on the bottom wall using a pseudospectral Fourier–Chebyshev discretization. Deformations of the free surface are neglected. Two- and three-dimensional flows are computed for either vanishing or small Prandtl number, which are typical of liquid metals. The main focus of the paper is on a qualitative comparison of the flow states with the non-magnetic case, and on the effects associated with the possible near-cancellation of the nonlinear and pressure terms in the momentum equations for two-dimensional rolls. In the three-dimensional case, the transition from a stationary hexagonal pattern at the onset of convection to three-dimensional time-dependent convection is explored by a series of simulations at zero Prandtl number.