Mixed convection flow over a vertical plate with localized heating (cooling), magnetic field and suction (injection)

An analysis is carried out to study the effects of localized heating (cooling), suction (injection), buoyancy forces and magnetic field for the mixed convection flow on a heated vertical plate. The localized heating or cooling introduces a finite discontinuity in the mathematical formulation of the problem and increases its complexity. In order to overcome this difficulty, a non-uniform distribution of wall temperature is taken at finite sections of the plate. The nonlinear coupled parabolic partial differential equations governing the flow have been solved by using an implicit finite-difference scheme. The effect of the localized heating or cooling is found to be very significant on the heat transfer, but its effect on the skin friction is comparatively small. The buoyancy, magnetic and suction parameters increase the skin friction and heat transfer. The positive buoyancy force (beyond a certain value) causes an overshoot in the velocity profiles.A mass transfer constant - B magnetic field - C_fx skin friction coefficient in the x-direction - C_p specific heat at constant pressure, kJ.kg^–1.K - C_v specific heat at constant volume, kJ.kg^–1.K^–1 - E electric field - g acceleration due to gravity, 9.81 m.s^–2 - Gr Grashof number - h heat transfer coefficient, W.m².K^–1 - Ha Hartmann number - k thermal conductivity, W.m^–1.K - L characteristic length, m - M magnetic parameter - Nu_x local Nusselt number - p pressure, Pa, N.m^–2 - Pr Prandtl number - q heat flux, W.m^–2 - Re Reynolds number - Re_m magnetic Reynolds number - T temperature, K - T_o constant plate temperature, K - u,v velocity components, m.s^–1 - V characteristic velocity, m.s^–1 - x,y Cartesian coordinates - thermal diffusivity, m².s^–1 - coefficient of thermal expansion, K^–1 - , transformed similarity variables - dynamic viscosity, kg.m^–1.s^–1 - ₀ magnetic permeability - kinematic viscosity, m².s^–1 - density, kg.m^–3 - buoyancy parameter - electrical conductivity - stream function, m².s^–1 - dimensionless constant - dimensionless temperature, K - w, conditions at the wall and at infinity     

Influence of internal degrees of freedom on flow of a rarefied gas over a flat plate

Flow of a rarefied gas over a flat plate has been investigated numerically by a number of authors, using both the kinetic model equations (e.g., 1, 2]) and the Boltzmann equation [3, 6], In most cases a solution was found for a monatomic gas. The appreciable influence of the molecule structure on local and total aerodynamic characteristics and on the flow field over a flat plate at small angles of attack was noted in [1, 5, 7], where the authors examined various models for the rotational molecular degrees of freedom. In the present paper a two-point repulsion center model with constant collision cross section is used to investigate the influence of internal degrees of freedom of the molecule in flow over a plate, positioned parallel to (angle of attack = 0), and transverse to ( = 90 °) a rarefied gas stream. The data are compared with those calculated for a monatomic gas and from experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 151–156, November–December, 1978.     

Free convection around a vertical isothermically heated plate of finite length at large prandtl numbers

A free convection boundary layer arises because of the appearance of viscosity forces near a solid boundary. For high viscosity fluids the viscosity is significant over the whole flow region, and the thermal boundary layer which forms because of the restriction of heat diffusion from a heated wall by convection is characterized by the ratio between the coefficients of viscosity and thermal diffusivity, i.e., the Prandtl number. The divergence between the theoretical [1–4] and experimental data [5, 6] for the velocity profiles of free convective flow around a vertical surface at large Prandtl nunbers is due to an insufficiently clear distinction between the physical laws mentioned. In the present study the form of the velocity and temperature profiles is determined more accurately on the basis of an asymptotic analysis of the complete Navier-Stokes equations and energy equation with Prandtl number Pr and Grashof numbers of the order of unity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–165, September–October, 1984.     

Unsteady Radiative–Convective Heat Transfer in a Flow Emitting‐Absorbing and Scattering Medium around an Ablating Plate

A conjugate problem of radiative–convective heat transfer in a turbulent hightemperature gasdisperse flow around a thermally thin ablating plate is considered. The plate experiences intense radiative heating by an external source, which is a blackbody. The temperature fields and the distributions of heat fluxes along the plate under unsteady conditions are calculated. The data gained make it possible to examine the effect of the Stark number and phasetransition heat in the plate material on the time evolution of the thermal state of the boundarylayer medium and the plate itself being heated by a hightemperature radiation source.     

Heat transfer to an isothermal flat plate in turbulent flow of a liquid over a wide range of prandtl and reynolds numbers

The study of heat transfer in turbulent flow over a flat plate is very important, not only because this situation frequently arises in practice, but also in that data for an isothermal flat plate are used to calculate heat transfer in more complex cases. In particular, such data are necessary when one uses the limiting relative laws which allow calculation of the effect of compressibility, pressure gradient, blowing, and other perturbing factors [1]. Most papers dealing with heat transfer for an isothermal flat plate refer to comparatively low Re values, when the velocity distribution in the boundary layer over almost its entire thickness can be described by the universal law of the wall. However, as Re increases there is an increasing layer adjacent to the outer boundary in which the velocity distribution cannot be described by the law of the wall, and therefore the results obtained for low Re are inapplicable. In the present paper coefficients of heat transfer from a turbulent flow to an isothermal flat plate have been obtained by numerical integration of the thermal boundary-layer equations over a wide range of the parameters 3 · 10⁵ Re 2.5·10¹², 10² Pr 10³.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 94–100, July–August, 1976.     

Approximate integration op nonstationary equations of a diffusion or thermal boundary layer

An approximate method is proposed for integrating the nonstationary equations of a diffusion or thermal boundary layer using the known steady solution in the planar or axisymmetric case. It is shown that the proposed method is exact in problems involving mass or heat transfer of reacting drops and bubbles in a laminar flow of a viscous incompressible fluid and also particles moving in an ideal fluid. An integral equation is obtained for the local diffusion or heat flux in the case of abrupt activation of a reaction on the surface of a particle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 87–92, September–October, 1982.     

Heat transfer in supersonic flow over a single spherical cavity

The flow and heat transfer on a plate with a single spherical cavity has been experimentally investigated for M=4 and Re_,L=3.1 · 10⁶. The flow pattern over the cavity has been obtained. Zones of enhanced heat transfer have been detected, and the heat transfer coefficients in and near the cavity have been determined. It has been established that a single spherical cavity has almost no effect on the integral heat flux.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 48–52, September–October, 1991.The authors are grateful to V. N. Brazhko for assistance in carrying out the experiments and to T. A. Ershova for assistance in analyzing the results.     

Transients in flow and local heat transfer due to a pressure wave in pipe flow

The effect of a pressure wave on the turbulent flow and heat transfer in a rectangular air flow channel has been experimentally studied for fast transients, occurring due to a sudden increase of the main flow by an injection of air through the wall. A fast response measuring technique using a hot film sensor for the heat flux, a hot wire for the velocities and a pressure transducer have been developed. It was found that in the initial part of the transient the heat transfer change is independent of the Reynolds number. For the second part the change in heat transfer depends on thermal boundary layer thickness and thus on the Reynolds number. Results have been compared with a simple numerical turbulent flow and heat transfer model. The main effect on the flow could be well predicted. For the heat transfer a deviation in the initial part of the transient heat transfer has been found. From the turbulence measurements it has been found that a pressure wave does not influence the absolute value of the local turbulent velocity fluctuations. They could be considered to be frozen.Nomenclature A surface area (m²) - D diameter (m) - h heat transfer coefficient (Wm^–2 K^–1) - p pressure drop (Pa) - P pressure (Pa) - Q heat flow (W) - R tube radius (m) - T bulk temperature (K) - T _s surface temperature (K) - t time (s) - u velocity (m/s) - V voltage (V) - y distance from wall (m) - viscosity (N s m^–2) - kinematic viscosity (m^–2 s^–1) - density (kg m^–3) - _w wall shear stress (N m^–2) - Nu Nusselt number - Re Reynolds number     

Thermal effect of a freely expanding gas jet on a flat plate

An experimental investigation was made into the thermal effect of a single gas jet on a plate at Mach numbers of the nozzles in the range 2–6.1, specific heat ratio = 1.4, total pressure difference up to 6·10⁷, gas temperature 450–520 °K in the forechamber, and pressure in the forechamber (10–20)· 10⁵ Pa. The proposed dimensionless numbers made it possible to obtain generalized dependences of the distribution of the heat flux to the plate on the conditions of the problem. A method of approximate calculation of the heat fluxes is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 119–126, July–August, 1981.     

Influence of variable permeability on combined free and forced convection flow past a semi-infinite vertical plate in a saturated porous medium

Combined free and forced convection flow of viscous incompressible fluid past a semi-infinite vertical plate embedded in a porous medium incorporating the variation of permeability and thermal conductivity are studied. Similarity solutions are obtained, for two cases namely uniform permeability (UP) and variable permeability (VP). Velocity and temperature profiles are shown graphically and the numerical values of the skin friction and the rate of heat transfer are entered in tabular form. The effects of the parameters Gr/Re ² (Gr – Grashof number, Re – the Reynolds number), (coefficient of viscositys of the fluid and porous medium), (the Darcy number), * (ratio of thermal conductivity of the solid to the liquid), Pr (the Prandtl number) and E (the Eckert number) on the flow field are discussed.     

Strong recondensation in a one-component and a two-component rarefied gas for an arbitrary value of Knudsen number

As is known, surface phenomena such as evaporation, absorption, and reflection of molecules from the surface of a body depend strongly on its temperature [1–5]. This leads to the establishment of a flow of a substance between two surfaces maintained at different temperatures (recondensation). The phenomenon of recondensation was studied in kinetic theory comparatively long ago. However, up to the present, only the case of small mass flows in a onecomponent gas has been investigated completely [3,4]. Meanwhile it is clear that by the creation of appropriate conditions we can obtain considerable flows of the recondensing substance, so that the mass-transfer rate will be of the order of the molecular thermal velocity. Such a numerical solution of the problem with strong mass flows along the normal to the surface for small Knudsen numbers for a model Boltzmann kinetic equation was obtained in [7]. In this study we numerically solve the problem of strong recondensation between two infinite parallel plates over a wide range of Knudsen numbers for a one-component and a two-component gas, on the basis of the model Boltzmann kinetic equation [6] for a one-component gas and the model Boltzmann kinetic equation for a binary mixture in the form assumed by Hamel [8], for a ratio of the plate temperatures equal to ten. We also investigate the effect of the relative plate motion on the recondensation flow.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1972.     

Cone in a supersonic flow near a surface with a turbulent boundary layer

The published information about the interaction of incident shocks and a turbulent boundary layer relate to cases of a thin boundary layer ( 1–3 mm) on a flat surface. The present study relates to supersonic flow with Mach number M = 3 and stagnation pressure p₀=1.2 MPa past cones near a surface with a thick boundary layer formed on a plate abutting the lower edge of a plane nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 177–180, July–August, 1991.     

Evolution equation for unsteady shock waves of moderate intensity in a viscous heat-conducting gas

A simple model equation that takes into account the nonisentropicity of the flow is. obtained from the equations of a viscous heat-conducting gas. It differs from the Burgers equation in possessing an additional term with a clear physical significance. This equation is suitable for one-dimensional traveling waves on the Mach number interval 1M1.3. The equation obtained gives the asymptotic laws of damping of weak shock waves correct to small terms of the leading and next order for plane [2], cylindrical [3] and spherical [4] symmetry.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 187–190, September–October, 1989.     

Blowing into a supersonic stream through a permeable surface

Distributed blowing of gas into a supersonic stream from flat surfaces using an inviscid flow model was studied in [1–9]. A characteristic feature of flows of this type is the influence of the conditions specified on the trailing edge of the body on the complete upstream flow field [3–5]. This occurs because the pressure gradient that arises on the flat surface is induced by a blowing layer whose thickness in turn depends on the pressure distribution on the surface. The assumption of a thin blowing layer makes it possible to ignore the transverse pressure gradient in the layer and describe the flow of the blown gas by the approximate thin-layer equations [1–5]. In addition, at moderate Mach numbers of the exterior stream the flow in the blowing layer can be assumed to be incompressible [3]. In [7, 8] a solution was found to the problem of strong blowing of gas into a supersonic stream from the surface of a flat plate when the blowing velocity is constant along the length of the plate. In the present paper, a different blowing law is considered, in accordance with which the flow rate of the blown gas depends on the difference between the pressures on the surface over which the flow occurs and in the reservoir from which the gas is supplied. As in [8, 9], the solution is obtained analytically in the form of universal formulas applicable for any pressure specified on the trailing edge of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 108–114, September–October, 1980.I thank V. A. Levin for suggesting the problem and assistance in the work.     

Study of heat transfer in the separation zones formed in front of supersonic jets in a subsonic carrier flow

The heat transfer taking place between the gas and the surface of the plate in the zone of three-dimensional separation of the turbulent boundary layer in front of a set of supersonic jets injected perpendicularly to a subsonic carrier flow is considered. The aim of this investigation is to establish the main physical characteristics of heat transfer in the separation zones in front of jet obstacles and to obtain the distributions of the local heat-transfer coefficients and the temperature of the thermally insulating wall as functions of the parameters of the carrier flow and the injected jets. Analysis of the experimental results yields certain approximating relationships for the distribution of the local heat-transfer coefficients as functions of the Mach number of the carrier flow M, the Mach number of the jet M_j, the relative boundary-layer displacement thickness _s= _s ^* /d, and the degree of jet superheating T_ojT_o relative to the separation zones in front of supersonic jet obstacles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 68–72, July–August, 1975.     

Density distribution in rarefied gas flow behind a perforated plate

The relative density distribution in the rarefied CO₂ flow field behind a perforated plate has been experimentally investigated on the range of Knudsen numbers 10^–3 Kn 10^–1. The results of these experiments serve to demonstrate the validity of using the molecular velocity distribution function for determining the relative density on the Knudsen number interval in question. This distribution function was previously employed for calculating the parameters in molecular beams formed by capillary sieves in the free-molecular flow regime [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 183–184, May–June, 1989.     

A new approach to studying the dynamics of a thin curved vortex

The paper studies the dynamics of a thin curved vortex in a potential flow of an ideal incompressible fluid. The flow is specified by a number of geometrical restrictions and does not satisfy the Biot–Savart law. The form of the derived equation of the vortex dynamics coincides with the form of the wellknown equation of local induction for selfinduced vortex motion. The parameters of the new equation are simultaneously flow parameters, and in this sense, they do not show uncertainty typical of classical equations. The coefficient of the new equation can take any specified values (not necessarily much greater than unity, as required according to the concept of local induction) and generally is a function of a natural filament parameter.     

Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier–Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (, ). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.     

Effect of a dispersed component on the nature of heat exchange during flow of a heterogeneous-jet around a barrier

Flow in a turbulent nonisothermal heterogeneous jet is characterized by considerable velocity [1, 2] and temperature disequilibrium [3] (u_s u and T_s T, where u_s, T_s and u, T are velocity and temperature of dispersed and gas components). As was shown in [4], an impurity is not passive, and it leads to suppression of jet turbulence (a result of interphase exchange by pulse and heat). Nonetheless, during reaction of a heterogeneous jet with a barrier orientated along the normal to the running flow, a significant increase is observed in heat emission characteristics in the vicinity of the point of deceleration [5] (for a single-phase jet an increase in heat exchange is typical with an increase in the intensity of turbulence [6]). The intensity of the change in heat emission in this case is a result of velocity and temperature disequilibrium for flow in jets, and it depends on a number of factors (temperature, concentration, phase condition of the dispersed impurity, etc.) and on the nature of the reaction of the dispersed component with the barrier surface [7]. There are numerous experimental data devoted to this. Apart from work in [5, 7], attention is drawn to [8] where an increase is also noted in the heat flow (by a factor of 1.4) at the deceleration point for a plane cylindrical end and a hemisphere. The aim of the present work is a study of the effect of a dispersed component on heat exchange with jet flow around a barrier.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 63–68, November–December, 1986.     

Stability of a perturbed conducting gas flow in a magnetic field for arbitrary magnetic Reynolds numbers

A numerical calculation is made which describes the conversion into a T-layer of a finite perturbation in electrical conductivity imposed on a one-dimensional supersonic flow of a compressible medium for a finite value of the magnetic Reynolds number. The development of the injected perturbation is significantly affected by the magnetic Reynolds number of the unperturbed flow, and to each value of this number there corresponds a particular boundary region in which the perturbation is taken up by the magnetic field into an induced T-layer. The stability is investigated in the linear approximation for a minimal perturbation, and the dispersion equation is solved with allowance for gradients in the unperturbed parameters. It is shown that an overheating instability can arise in the system and lead to the formation of a T-layer.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 3, pp. 3–9, May–June, 1973.The authors thank L. M. Degtyarev, L. A. Zaklyaz'minskii, and A. P. Favorskii for useful discussions and advice during the completion of this work.