Method of generalized similitude in free convection problems with arbitrary distribution of the temperature or heat flux on a vertical wall

The self-similar problem of free convection near a heated vertical plate was solved for the first time in [1] for the simplest case of a constant wall temperature. In [2], Yang proved the existence of a self-similar solution to the problem of free convection for vertical plates and cylinders on the surfaces of which the temperature has a power-law distribution. In [3], Yang's solution was generalized to the case of free convection near a slender figure of revolution, but also only in the self-similar case of a power-law distribution of the temperature on the wall. In [4], this problem was solved in an extended nonsimilar formulation but by an artificial and not general method similar to Gertler's, the convergence of the approximations being slow. The present paper contains the solution to the problem of free convection near a vertical plate with arbitrary distribution of the temperature or heat flux on its surface. Rigorous application of the method of generalized similitude [5] leads in this case to universal equations that present insuperable computational difficulties, which forces one to use a simplified but fairly general method to solve this class of problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 167–170, May–June, 1980.I thank L. G. Loitsyanskii and E. M. Smirnov for discussing the results and for valuable comments.     

Similarity solutions for MHD thermosolutal Marangoni convection over a flat surface in the presence of heat generation or absorption effects

The problem of steady, laminar, thermosolutal Marangoni convection flow of an electrically-conducting fluid along a vertical permeable surface in the presence of a magnetic field, heat generation or absorption and a first-order chemical reaction effects is studied numerically. The general governing partial differential equations are converted into a set of self-similar equations using unique similarity transformations. Numerical solution of the similarity equations is performed using an implicit, iterative, tri-diagonal finite-difference method. Comparisons with previously published work is performed and the results are found to be in excellent agreement. Approximate analytical results for the temperature and concentration profiles as well as the local Nusselt and sherwood numbers are obtained for the conditions of small and large Prandtl and Schmidt numbers are obtained and favorably compared with the numerical solutions. The effects of Hartmann number, heat generation or absorption coefficient, the suction or injection parameter, the thermo-solutal surface tension ratio and the chemical reaction coefficient on the velocity, temperature and concentration profiles as well as quantitites related to the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are presented in graphical and tabular form and discussed. It is found that a first-order chemical reaction increases all of the wall velocity, Nusselt and Sherwood numbers while it decreases the mass flow rate in the boundary layer. Also, as the thermo-solutal surface tension ratio is increased, all of the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are predicted to increase. However, the exact opposite behavior is predicted as the magnetic field strength is increased.     

Radiation effect on mixed convection from a horizontal surface in a porous medium

The effect of thermal radiation on the non-Darcy mixed convection flow over a non-isothermal horizontal surface immersed in a saturated porous medium has been studied. The wall temperature is assumed to have a power-law variation with the distance measured from the leading edge of the plate. The non-linear coupled parabolic partial differential equations governing the flow have been solved numerically using a finite-difference scheme. For some particular cases, the self-similar solution has also been obtained. The heat transfer is found to be strongly influenced by the radiative flux number, buoyancy parameter, variation of wall temperature, non-Darcy parameter and the nature of the free stream velocity.     

Influence of magnetic field and thermal radiation by natural convection past vertical cone subjected to variable surface heat flux

A numerical study is performed to examine the heat transfer characteristics of natural convection past a vertical cone under the combined effects of magnetic field and thermal radiation.The surface of the cone is subjected to a variable surface heat flux.The fluid considered is a gray,absorbing-emitting radiation but a non-scattering medium.With approximate transformations,the boundary layer equations governing the flow are reduced to non-dimensional equations valid in the free convection regime.The dimensionless governing equations are solved by an implicit finite difference method of Crank-Nicolson type which is fast convergent,accurate,and unconditionally stable.Numerical results are obtained and presented for velocity,temperature,local and average wall shear stress,and local and average Nusselt number in air and water.The present results are compared with the previous published work and are found to be in excellent agreement.     

An analytical theory of heated duct flows in supersonic combustors

One-dimensional analytical theory is developed for supersonic duct flow with variation of cross section, wall friction, heat addition, and relations between the inlet and outlet flow parameters are obtained. By introducing a selfsimilar parameter, effects of heat releasing, wall friction, and change in cross section area on the flow can be normalized and a self-similar solution of the flow equations can be found. Based on the result of self-similar solution, the sufficient and necessary condition for the occurrence of thermal choking is derived. A relation of the maximum heat addition leading to thermal choking of the duct flow is derived as functions of area ratio, wall friction, and mass addition, which is an extension of the classic Rayleigh flow theory, where the effects of wall friction and mass addition are not considered. The present work is expected to provide fundamentals for developing an integral analytical theory for ramjets and scramjets.     

Front Regime of Heat and Mass Transfer in a Gas Hydrate Reservoir under the Negative Temperature Conditions

The analytical self-similar solution to the nonlinear problem of the front regime of heatand- mass transfer in a gas hydrate reservoir under the negative temperature conditions is obtained. In the initial state the reservoir is assumed to be saturated with a heterogeneous gas hydrate–ice–gas mixture. In particular cases there may be no ice or/and gas. The ice and gas are formed behind the gas hydrate dissociation front. The calculations are presented for a stable hydrate–gas system. The critical curves are constructed in the well-pressure–reservoir-permeability plane. These curves separate the domains of the front regime and the regime of volume gas hydrate dissociation ahead of the front. The velocity of the gas hydrate dissociation front is investigated as a function of various problem parameters. The characteristic temperature and pressure distributions corresponding to various regimes on the diagram are investigated.     

Stability of self-similar pipe flow with suction

The stability of self-similar flows with various boundary conditions at the wall is investigated. In the region of nonexistence of self-similar solutions an oscillatory regime is detected. The problem of stability with respect to disturbances of general form is studied. The dependence of the critical values of the axial Reynolds number and the Strouhal number on Re is calculated for various suction rates.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 18–25, September–October, 1991.     

Motion of an External Load over a Semi-Infinite Ice Sheet in the Subcritical Regime

The three-dimensional problem of steady-state forced vibrations of fluid and semiinfinite ice sheet under the action of a local external load traveling along the rectilinear sheet edge at a constant velocity is considered. Two cases are analyzed. In the first case the fluid surface outside the ice sheet is free and in the second the fluid is confined by a rigid vertical wall and the ice sheet edge adjacent to the wall can be both clamped and free. The ice sheet is simulated by a thin elastic isotropic plate floating on the surface of fluid of finite depth. The load traveling velocity is assumed to be not higher than the minimum phase velocity of the flexural-gravity waves (subcritical regime). The solution to the linear problem is obtained by means of the integral Fourier transform and matching the expansions of the velocity potential in the vertical eigenfunctions. Examples of the numerical investigation of the ice sheet and fluid displacements are given.     

Transient natural convection in a cavity with heat input and a constant temperature wall on opposite sides

The transient natural convection of a fluid with Prandtl number of order 200 in a two-dimensional square cavity has been numerically studied. One of the vertical walls of the cavity is kept at a constant (ambient) temperature and a constant heat flux is applied on the opposite wall. The other walls are adiabatic. Initially, a boundary layer is formed near the heated wall; subsequently, a large vortical structure is generated, together with an upper intrusion layer. As time progresses, the average temperature in the cavity increases, and a descending boundary layer is formed near the constant temperature wall. During the transition to the steady-state regime, a thermal stratification pattern is formed. The results are compared with the scale analysis presented by Patterson and Imberger (1980).     

Inverse laminar boundary-layer problems with assigned shear. The mechul function revisited

An inverse method is presented which accurately determines the pressure distribution for assigned wall shear in a two-dimensional, laminar, incompressible boundary layer. The method reformulates the mechul function scheme of Cebeci and Keller to produce a stable solution in the marching direction and to increase accuracy in the normal direction. In the reformulation a modified pressure gradient parameter variation in the normal direction is used in conjunction with three-point backward differences for streamwise derivatives and fourth-order accurate splines for normal derivatives. The resulting spline-finite difference equations are solved by Newton-Raphson iteration together with partial pivoting. Numerical solutions are presented for self-similar and non self-similar flows and compared with published results.     

Non-Darcy free convection in a thermally stratified porous medium along a vertical plate with variable heat flux

Free convection along an impermeable vertical plate embedded in a thermally stratified, fluid-saturated porous medium is analyzed. The wall heat flux is varied in a power-law form. The non-Darcian effects, such as solid-boundary viscous resistances, high-flow-rate inertia forces, near wall nonuniform porosity distribution and thermal dispersion, have been considered in the present study. Due to the variation of porosity in the near wall region, the stagnant thermal conductivity also varies accordingly. The nonsimilar system of transformed equations is solved with Keller's Box method. It is shown that the thermal stratification effect and the higher value of the exponent m can increase the local Nusselt number. Also the non-Darcian and thermal dispersion effects significantly influence the velocity and temperature profiles and local Nusselt number.     

Radiation effect on mixed convection along a vertical plate with uniform surface temperature

This paper investigates the effect of radiation on the forced and free convection flow of an optically dense viscous incompressible fluid along a heated vertical flat plate with uniform free stream and uniform surface temperature with Rosseland diffusion approximation. With appropriate transformations, the boundary layer equations governing the flow are reduced to local nonsimilarity equations valid in the forced convection regime as well as in the free convection regime. A group of transformation is, also, introduced to reduce the boundary layer equations to a set of local nonsimilarity equations valid in both the forced and free convection regimes. Solutions of the governing equations are obtained by employing the implicit finite difference methods together with Keller box scheme and are expressed in terms of local shear stress and local rate of heat transfer for a range of values of the pertinent parameters.     

Development of pulsation regimes in one-dimensional unsteady MHD flows with switching off of the electrical conductivity

This paper investigates the gas flow in an electromagnetic field when the conductivity, being a function of the thermodynamic gas parameters, vanishes during the flow (switching off of the conductivity). In the case of steady supersonic flows in an expanding nozzle it was first shown analytically [1] and then confirmed by numerical experiment [2] that stable steady flow is not possible for all the problem parameters (for example, the values of the magnetic field at the exit). Instead of a steady flow a periodic regime is realized when narrow regions of conducting gas with currents flowing through them detach from the conducting region and propagate down the channel. In these papers the conductivity was assumed to be a function of only the temperature, such that for T T* (T) = 0. In [3, 4] the flows of conducting gas in the channels were calculated both with the given dependence of the gas conductivity on the temperature and on the basis of a three-component model by means of the Saha equation. At the same time, the development of periodic regimes in the flow in the nozzle was observed in both cases, but the mechanism of the origin of the current layers was not explained. The self-similar problem of the withdrawal of a nonconducting piston from a half-space occupied by a conducting gas with a magnetic field was investigated in [5] in a linear formulation. At the same time, regions of the problem parameters (the velocity of the piston and the magnetic field on it) were found when, in spite of the self-similar formulation of the problem, there is no self-similar solution. At the same time, regions exist where several solutions are possible. The possibility of the formation of isothermal rarefaction zones with low electrical conductivity when the Joule heating is balanced by the cooling of the gas on expansion (Butler waves) [6] was not taken into account in this paper, since they are unstable with respect to superheating. However, in the case of flow in a nozzle it was shown [2] that precisely the development of instabilities in these zones leads to the formation of the periodic regime. In the present paper the solution of the self-similar problem is constructed in a nonlinear formulation. The reason for the occurrence of regions in which the solution is multiply valued, which is associated with the process of arrival at self-similar boundary conditions, is explained. It is shown that a quasiperiodic regime can arise in the solution, occurring, in particular, in the regions of the problem parameters where there is no self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–122, July–August, 1986.     

Exact Solution of the Heat Transfer Problem for a Rotating Disk under Uniform Jet Impingement

An exact solution of the heat transfer problem for a uniform air stream impinging on a rotating disk is found. By introducing self-similar radial velocity and temperature profiles, the problem is reduced to a system of ordinary differential equations which are solved numerically. The Nusselt numbers are calculated for Prandtl numbers equal to 1 and 0.71 and various ratios of the free-stream velocity to the disk rotation velocity. The limits of the flow regime in which the heat transfer is determined solely by the impact jet parameters are found. The results are compared with experimental data for the stagnation point.     

Finite-difference solution of unsteady natural convection flow past a nonisothermal vertical cone under the influence of a magnetic field and thermal radiation

An analysis is performed to study the heat transfer characteristics of natural convection over a vertical cone under the combined effects of a magnetic field and thermal radiation. The cone surface is subjected to a variable surface temperature. The fluid considered is a gray absorbing/emitting, but non-scattering medium. The boundary layer equations governing the flow are reduced to non-dimensional equations using non-dimensional quantities valid in the free-convection regime. The resulting non-dimensional governing equations are solved by an implicit finite-difference method of the Crank-Nicolson type, which is rapidly convergent and unconditionally stable. Numerical results are obtained for velocity, temperature, local and average skin friction, and local and average Nusselt numbers for various values of parameters occurring in the problem and are presented in the graphical form. Excellent agreement of the results obtained with available data is demonstrated.     

An experimental study of the effect of wall temperature nonuniformity on natural convection in an enclosure heated from the side

An experimental study of natural convection in a parallelepipedal enclosure induced by a single vertical wall is described. The upper half of this wall was warm and the lower half cold. The other enclosure walls were insulated. The temperature and flow measurements were performed in the high Rayleigh number regime (10¹⁰<Ra<5×10¹⁰) by using water as the working fluid. The Rayleigh number was based on the enclosure height and the temperature difference between the warm and the cold part of the driving wall. The flow field featured two flat cells, one filled with warm fluid along the top horizontal wall, and the other filled with cold fluid along the bottom horizontal wall. Each of these cells was surrounded by an additional cell as tall as half the enclosure height. The above flow structure prohibited extensive thermal contact between warm and cold fluid, thus limiting the role of convection on the heat transfer process in the cavity. The findings of this study differ significantly from the findings of previous studies based on the ‘classical’ enclosure model possessing two isothermal vertical walls, the one warm and the other cold, and support the view that the use of ‘more realistic‘ temperature boundary conditions in enclosure natural convection needs careful examination.     

Numerical simulation of laminar natural convection in a laterally heated vertical cylindrical enclosure: application to crystal growth

Laminar natural convection has been studied in a laterally heated vertical cylindrical enclosure with a free insulated surface and a centrally located constant temperature wall at the top. These conditions are a simplification of the conditions existing in a Czochralski crystal pulling system. The laminar, axisymmetric flow of a Newtonian, constant physical properties fluid under Boussinesq’s approximation has been considered. Governing equations in primitive variable form are solved numerically by control volume method. SIMPLE algorithm due to Patankar has been used for the numerical simulation. The effects of the constant wall heat flux boundary condition at the side wall have been investigated whereas the bottom wall is considered to be insulated. Streamlines and isotherms are presented for various Rayleigh numbers and Prandtl numbers. Heat flux vectors through the melt are plotted for selected cases. The axial velocity and temperature variations at different horizontal sections of the crucible have been presented graphically to explain the transport processes inside the crucible. It has been observed that in case of low Pr and high Ra, flow separation occurs at the vertical wall of the crucible which leads to an oscillatory flow as Ra increases. The investigation has been extended to the oscillatory regime of flow in the zone of supercritical Rayleigh numbers and some unsteady results are also presented. Finally a heat transfer correlation has been developed for steady-state case.     

Natural convection flow over a vertical frustum of a cone for constant wall heat flux

Laminar natural convection flow and heat transfer over a vertical frustum of a cone has been studied. The governing boundary layer equations are solved using local non-similarity method for constant wall heat flux. The local similarity and the local non-similarity two and three-equation models are constructed and the resulting equations are solved numerically. Results obtained from two and three-equation models are in good agreement. The numerical values of the flow and temperature functions required to calculate the surface skin friction and heat transfer rate are reported for various values of Prandtl numbers.     

Nonsimilar Solutions for Mixed Convection in Non‐Newtonian Fluids Along a Vertical Plate in a Porous Medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in powerlaw type nonNewtonian fluids along a vertical plate with powerlaw wall temperature distribution. The mixed convection regime is divided into two regions, namely,the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.     

Motion of the interface of two fluids with formation of a region of two-phase flow

A study is made of the problem of the displacement of one fluid by another with the formation of a region of combined flow in the case of an elastic flow regime in the region of the displaced fluid. A self-similar solution is constructed for the flow equations averaged with respect to the vertical coordinate. A numerical algorithm is developed for determining the saturation in the region of the mixture, the pressure, and also the position and shape of the interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 8, pp. 78–83, October–December, 1981.We thank A. A. Barmin for a helpful discussion of the work.