Calculation of the boundary layer of a transparent gas on a radiating surface

Heat transfer in the laminar boundary layer of a transparent gas flowing aroud a plane radiating surface is studied. Radiative heat-transfer processes in gases may be divided into two main groups. The first involves heat transfer in absorbing and radiating media. In this case, the effect of radiation lies in the introduction of new terms into the energy equation, representing internal heat sources and sinks. The second group embraces heat-transfer processes in a transparent gas when the effect of radiation on convection expresses itself solely by way of the boundary conditions. Here we study a case of practical importance belonging to the second group: heat transfer in the laminary boundary layer of a transparent gas flowing around a flat plate with the thermal flux q_w specified on its surface.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 107–110, January–February, 1972.     

Solution of the equations of motion for a selectively radiating gas in a shock layer

Numerical solutions are obtained for the system of integro-differential equations describing the flow of a viscous, heat-conducting, selectively radiating gas in the region between the shock wave and a blunt body. The calculations are made for bodies of radius from 0.1 to 3 m with stagnation temperature from 6000° to 15 000° K. As a result of the calculations the convective and radiative thermal fluxes in the vicinity of the stagnation point are obtained. The effect of injection on convective and radiative heat transfer is studied.The first calculations of radiative thermal fluxes in air were made about 10 years ago in [1,2]. However, the results did not take account of the effects of emission and reabsorption, nor the interaction of the convective and radiative heating processes. These effects have been studied primarily with the use of simplified models of a radiating gas. Most often the approximation used is that of a gray gas with absorption coefficient which is independent of wavelength ([3–6] and others).The appearance in the literature of quite detailed data on the selective spectral absorption coefficients of air over a wide temperature range [7,8] has made it possible to solve the direct problem of calculating the flow field of a selectively radiating gas behind a shock wave with account for all the effects mentioned above.     

Effect of energy supply to a gas on laminar boundary layer separation

A decelerated flow in a supersonic boundary layer containing a heat source modeling an electric discharge is studied numerically. Calculations are performed for a wide range of the source power. The possibility of controlling the boundary layer separation is demonstrated. The boundary layer separation on cooled walls is found to occur substantially later than on thermally insulated walls.     

Transient radiative-conductive heat transfer in a flat layer of a selectively absorbing and radiating medium

Results of a numerical solution of the unsteady boundary-value problem of radiative-conductive heat transfer in a flat layer of a selective nonscattering medium with semitransparent mirrorreflecting boundaries are presented. This problem reduces to a nonlinear integral equation in the unknown temperature with the use of a Green function. The optical properties of the walls are shown to have a strong effect on the formation of a temperature field in the layer. The intensity of heating of the layer depends on the radiative fluxes to a greater extent than on the conductive fluxes. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 1, pp. 105–109. January–February, 1998.     

Convective heat exchange in a radiating gas

The problem of radiating gas flow past a blunt body has been studied by many authors. A quite complete review of these studies is presented in [1, 2]. Basic attention has been devoted to determining the radiant fluxes to the body surface and calculation of the parameter distribution across the shock layer.In many studies, particularly the foreign ones, use has been made of the approximation of bulk luminescence. In this approximation a term is added to the energy equation in order to account for the effect of the radiant heat exchange. This term is equivalent to the heat flux, whose intensity depends on the local thermodynamic state of the gas. With the use of this assumption the gas enthalpy in the inviscid flow vanishes at the stagnation point. The singularity resulting from the use of various approximations was revealed in [3–5]. This singularity is caused by the fact that a gas particle moving along the stagnation streamline to the body is retarded over an infinitely long time. Obviously, if we take into account the dissipative processess (for example, thermal conduction) which really take place, the singularity disappears. For a perfect gas it may be shown that the enthalpy and velocity are equal to zero along the entire surface of the body. Then the use of conventional boundary layer theory becomes impossible. The concept of a viscous and heat-conducting shock layer has been used in [5,6] and also by the present authors to eliminate this difficulty. However, this approach leads to unjustified complication of the problem and forces the introduction of more or less rough assumptions in carrying out the calculations.In the present study we have investigated a form of the boundary layer equations and the corresponding boundary conditions for flows with bulk gas luminescence and approximating flow regimes with small optical thickness of the shock layer. The solution was carried out with the aid of the method of inner and outer expansions (for example, [7]). The form of the equations and the boundary conditions differed depending on which of the dissipative processes-thermal conduction or absorption of the radiation by the gas in a narrow layer cooled by luminescence of the near-wall layer, was dominating. (The existence in the inviscid shock layer with small but finite optical thickness of an absorbing near-wall sublayer was discovered by V. N. Zhigulev, and also in [2].) In the present study we have used the Newtonian approximation, analogous to that considered for the nonradiating gas by Shidlovskii [8], This made it possible to obtain most of the results in a simple, easily visualized form. However, the flow regimes considered and the corresponding parameters do, of course, have general significance.     

Planar jet flow of a radiating gas

Propagation of a laminar jet of a thermodynamically equilibrious gray gas is examined in the boundary layer theory approximation. The one-dimensional radiative energy transport is accounted for in the P ₁ approximation of the spherical harmonic method. Numerical solution of the problem is made under additional simplifying assumptions for various values of the radiation parameters to illustrate the radiation effect. The method and the computational scheme used are applicable to the study of complex jet flows of a radiating gas.The author thanks Yu. P. Lun'kin for his assistance in posing the problem and for his continued interest in the study.     

Stability of contact discontinuity in a strongly radiating gas

A very simple flow—the two-dimensional steady infinite planar contact discontinuity in an inviscid non-heat-conducting gas without chemical reactions—is used as an example to study the effect of irreversible energy transport by radiation on the stability of motion with respect to infinitesimal perturbations of the interface.     

On the initial stage of a point explosion in a radiating gas

A study is made of the initial stage of a point explosion in a radiating gray gas whose absorption coefficient is approximated by the dependenceK=x()e ^–n ,where is the density and e is the internal energy of the gas. It is shown that for n > —1/3 the initial stage of the process differs significantly from the solution of the problem in not only the classical adiabatic case [1, 2] but also in the case of a medium with nonlinear thermal conductivity [2–4]. The supply of energy to the medium at a point leads to instantaneous heating of the complete medium. The form of this heating is found analytically. The method of matched asymptotic expansions is used to investigate the behavior of the solution in the neighborhood of the center. It is found that for definite conditions at the center of the perturbed region there are formed a shock wave and a region of reverse flow of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 75–82, May–June, 1980.I should like to thank V. P. Korobeinikov for interest in the work and a helpful discussion of it.     

One class of self-similar flows of a radiating gas

This article discusses self-similar statements of the problem of the motion of a completely radiating and absorbing gas. The field of radiation is assumed to be quasi-steady-state, and the contribution of the radiation to the internal energy, as well as the pressure and the viscosity of the medium, are not taken into account. The presence of local thermodynamic equilibrium is assumed. The absorption coefficient is approximated by a power function of the pressure and the density. Scattering of the radiation is not taken into account. Under these assumptions, there exist self-similar statements of the problem for one-dimensional unsteady-state flows (a strong detonation, the problem of plug-flow, motion under the effect of a radiation source, and others) and two-dimensional steady-state flows (flow in a diffuser, supersonic flow around a wedge or a cone). It is shown that there exists a non steady-state spherically symmetrical flow depending on four parameters; this flow is adiabatic in spite of the presence of radiation. This article is made up of seven sections. It is shown in the first section that the presence of radiation leads to the appearance of new dimensional constants, entering into the equations of the problem. The second section is devoted to self-similar nonsteady-state one-dimensional flows. The third section contains a detailed study of one class of such flows. In a partial case, adiabatic flows of a radiating gas are obtained. In the fourth and fifth sections, a detailed analysis is made of the initial and boundary conditions from the point of view of dimensionality. The sixth section describes self-similar two-dimensional steady-state flows of a radiating-absorbing gas. The seventh section consists of remarks with respect to approximations of the transfer equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 8–22, July–August, 1970.     

Effect of coarse particles on the heat transfer in a particle-laden turbulent boundary layer

The temperature distribution in particle-laden turbulent flow, in a flume, was investigated both by DNS and experimentally. Simulations were performed at Re=171 and Pr=5.4 in order to study the interaction between the particle motion and flow turbulence. Two-way coupling was used to obtain various turbulence statistics, the grid resolution was sufficiently fine to resolve all essential turbulent scales. The effect of particle diameter on momentum, heat transfer and particle deposition was considered. The details of particle-turbulence interaction depend on the particle Stokes number and the particle Reynolds number.

The spatial structures of instantaneous flow and temperature fields were visualized. Low frequency small oscillations of deposited particles were observed. It was found that these small deviations from the initial position, caused strong changes in the instantaneous temperature field near the particle.

The experiments provided details of the temperature field on the heated wall close to the particle. In the front of the particle, a sharp increase in heat transfer coefficient was observed. The experimental results agree well with the computational predictions.     

Nonstationary (transient) radiative-conductive heat transfer in a plane layer of gray absorbing medium

The kinetics of the heating of a plane layer of gray absorbing medium by radiative-conductive heat transfer are considered. The nonstationary energy equation is reduced to a nonlinear integral equation by means of a Green's function, and this is solved numerically by the Newton method. The results of the solution are presented in the form of the temperature fields in the layer for various values of the defining parameters (optical thickness, radiative-conductive heat-transfer criterion, heat-transfer criterion at the boundaries).Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 156–159, January–February, 1972.     

Effect of energy release in the shock layer on supersonic flight

The presence of an energy release zone in a supersonic stream leads to the formation of perturbations propagated from the source of energy release [1, 2]. The action of such a stream on a body past which it flows is determined by the strength of the source and the configuration and location of the energy release zone. The dependence of the aerodynamic characteristics of a blunt cone on the parameters of the source of energy release is analyzed on the basis of the results of a numerical integration of the equations of motion of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 178–182, July–August, 1987.The author wishes to thank V. P. Stulov for his interest in the work and useful suggestions.     

Radiation interactions in transient energy transfer in fully developed laminar flows

Analysis and numerical procedures are presented to investigate the transient radiative interactions of nongray absorbing-emitting species in laminar fully-developed flows between two parallel black plates. The particular species considered are OH, CO, CO₂, and H₂O and different mixtures of these species. Transient and steady-state results are obtained for the temperature distribution and bulk temperature for different plate spacings, wall temperatures, and pressures. Results, in general, indicate that the rate of radiative heating can be quite high during earlier times. This information is useful in designing thermal protection systems for transient operations.Nomenclature A band absorptance = A(u, ), cm^–1 - A ₀ band width parameter, cm^–1 - C ₀ correlation parameter, atm^–1-cm^–1 - C _p specific heat at constant pressure, kJ/kg-K = erg/gm-K - e Planck's function, (W-cm^–2)/cm^–1 - e ₀ Planck's function evaluated at wave number ₀ - e ₁, e ₂ emissive power of surfaces with temperatures T ₁ and T ₂, W-cm^–2 - H _1i , H ₁ gas property for the large path length limit - k thermal conductivity, erg/cm-s-K - K ₁ gas property for the optically thin limit - M large path length parameter, nondimensional - N optically thin parameter, nondimensional - P pressure, atm - q _R total radiative heat flux, W/cm² - q _Rw spectral radiation heat flux, (W-cm^–2)/cm^–1 - q _w wall het flux, W/cm² - S integrated intensity of a wide band, atm^–1-cm² - t time, s - T temperature, K - T ₁ wall temperature, K; T ₁=T _w - T _b bulk temperature, K - u nondimensional coordinate = SPy/A ₀ - u ₀ nondimensional path length = SPL/A ₀ - y transverse coordinate, cm - line structures parameter - nondimensional temperature - _b dimensionless bulk temperature - spectral absorption coefficient, cm^–1 - nondimensional coordinate = y/L=u/u ₀ - density, kg/m³ - nondimensional time - wave number, cm^–1 - ₀ wave number at the band center, cm^–1     

Calculation of inviscid radiating gas flow past a blunt body

During hypersonic gas flow past a blunt body with a velocity on the order of the escape velocity or more, the gas radiation in the disturbed region behind the shock wave becomes the primary mechanism for aerodynamic heating and has a significant effect on the distribution of the gasdynamic parameters in the shock layer. This problem has been considered from different points of view by many authors. A rather complete review of these studies is presented in [1–4].In earlier studies [5, 6] the approximation of bulk emission was used. In this approximation, in order to account for the effect of radiative heat transfer a term is added in the energy equation which is equivalent to the body efflux, whose magnitude depends on the local thermodynamic state of the gas. However, the use of this assumption to solve the problem of inviscid flow past a blunt body leads to a singularity at the body [7, 8]. To eliminate the singularity, account is taken of the radiation absorption in a narrow wall layer [7], or the concept of a viscous and heat-conductive shock layer is used [8]. A further refinement was obtained by Rumynskii, who considered radiation selectivity and studied the flow of a radiating and absorbing gas in the vicinity of the forward stagnation point of a blunt body.In the present paper we study the distribution of the gasdynamic parameters in the shock layer over the entire frontal surface of a blunt body in a hypersonic flow of a radiating and absorbing gas with account for radiation selectivity.