Turbulent boundary layer on a permeable surface

Several theoretical [1–4] and experimental [5–7] studies have been devoted to the study of the effect of distributed injection of a gaseous substance on the characteristics of the turbulent boundary layer. The primary study has been made of flow past a flat plate with gas injection. The theoretical methods are based primarily on the semiempirical theories of Prandtl [1] and Karman [2].In contrast with the previous studies, the present paper proposes a power law for the mixing length; this makes it possible to obtain velocity profiles which degenerate to the known power profiles [8] in the case of flow without blowing and heat transfer. This approach yields analytic results for flows with moderate pressure gradient.Notation x, y coordinates - U, V velocity components - density - T temperature - h enthalpy - H total enthalpy - c mass concentration - , , D coefficients of molecular viscosity, thermal conductivity, diffusion - c_p specific heat - adiabatic exponent - r distance from axis of symmetry to surface - boundary layer thickness - U velocity in stream core - friction - c_f friction coefficient - P Prandtl number - S Schmidt number - S_t Stanton number - M Mach number - j=0 plane case - j=1 axisymmetric case The indices 1 injected gas - 2 mainstream gas - w quantities at the wall - core of boundary layer - 0 flow of incompressible gas without injection - v=0 flow of compressible gas without injection - ^* quantities at the edge of the laminar sublayer - quantities at the initial section - turbulent transport coefficients     

Transport phenomena in a nonequilibrium dissociating gas

A disturbance of a quasiequilibrium distribution due either to a relaxation process or hydrodynamic perturbations can significantly influence the rate of the relaxation process in a gas. The expression for this rate then contains additional terms proportional to the spatial derivatives of the hydrodynamic variables [1–2]. According to the estimates of [2], the effect is clearly manifested in a nonequilibrium dissociating gas. The present authors have estimated [3] some of the additional terms in the initial stage of dissociation in a pure diatomic gas. In the present paper, expressions are obtained for the coefficients of the additional terms of the hydrodynamic equations in a dissociating diatomic gas for all dissociation laws. Estimates made for the example of oxygen show that the contribution of the additional terms to the hydrodynamic equations is comparable with the contribution of the ordinary Navier-Stokes terms.     

Turbulent boundary layer on a mildly curved convex surface

Extensive single point turbulence measurements made in the boundary layer on a mildly curved heated convex wall show that the turbulence heat fluxes and Stanton number are more sensitive to a change in wall curvature than the Reynolds stresses and skinfriction coefficient, and that downstream, as the flow adjusts to new curved conditions, the St/c _f ratio of Reynolds analogy is appreciably lower than in plane wall flow for the same conditions. Details of the turbulence structure in unheated flow have been documented in an earlier paper; temperature field measurements now described comprise mean temperature distributions, the streamwise variation of wall heat flux, profiles of the temperature variance, transverse and streamwise heat fluxes, and triple correlations. Turbulent diffusion of heat flux is drastically reduced even by mild curvature; changes in the heat fluxes are of the same order as changes in the shear stress, that is, an order of magnitude greater than the ratio of boundary layer thickness to wall radius of curvature. The data include plane flow measurements taken in a developed boundary layer upstream of a change in wall curvature.     

Turbulent boundary layer on a mildly curved convex surface

Mean flow and turbulence measurements have been made in a boundary layer which grows first on a flat' wall and then on a convex wall of radius of curvature approximately 100 times the boundary layer thickness. The turbulence data include profiles of the four non-zero components of the Reynolds stress tensor and three triple velocity products obtained at five stream-wise positions. A number of measurements were also made for comparison in the boundary layer on a flat wall under the same conditions. The effects of convex curvature are to reduce turbulent intensities, shear stress and wall friction by approximately 10% of their plane flow values; the triple velocity products are halved in the curved layer. The measurements supplement the small quantity of previously published data available for testing mathematical models of turbulence. The results show the same general trends that have been observed in earlier investigations but there are significant differences in detail, notably in respect of levels of the normal stresses.     

Laminar boundary layer in flow of a nonequilibrium ionized gas

The motion of a hypersonic body is accompanied by an increase in the gas temperature in the boundary layer up to tens of thousands of degrees, which causes the gas to ionize. Under these conditions there are problems in calculating coefficients of viscosity, diffusion, and heat conduction. Investigations have shown that it is invalid to extrapolate the widely used approximations for transport coefficients in the high temperature region [1–3]. This paper considers the laminar boundary layer in the vicinity of the stagnation point of a blunt body in a stream of monatomic nonequilibrium ionized gas. The main thrust is a more accurate calculation of transport coefficients and an investigation of their effect on profiles of the gasdynamic parameters. A specific calculation is performed for argon by way of example.     

Turbulent boundary layer velocity profiles over a smooth surface

Mean velocity measurements made for two-dimensional incompressible turbulent boundary layers, for 3 pressure gradients on a smooth wall have been used to evaluate the pressure gradient parameter, the skin-friction and the roughness function by the method of “curve-fit” to the entire profile. Local skin-friction coefficients so obtained do not agree with the experimental results but the integral parameters are well represented.     

Formulas for the heat and diffusion fluxes to a catalytic surface in a chemically nonequilibrium multicomponent boundary layer

On the basis of an asymptotic expansion of the solution of the equations of a multicomponent chemically nonequilibrium boundary layer for large Schmidt numbers, formulas are obtained for the heat flux and the diffusion fluxes of the reaction products and chemical elements on a surface with arbitrary catalytic activity. The results are compared with well-known analytic and numerical solutions. The comparison reveals the high accuracy of the formulas proposed. The results of calculating the diffusional separation of the mixture due to the selectivity of the catalytic properties of the surface with respect to recombination of oxygen and nitrogen atoms are presented. Values of the reduction of the convective heat fluxes due to the catalytic properties of the surface are obtained over a wide range of conditions in the free stream.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 171–176, March–April, 1996.     

Slip boundary conditions on a catalytic surface in a multicomponent gas flow

The boundary conditions for the velocity slip and temperature and concentration jumps on the surface of a body in a rarefied multicomponent gas flow are obtained. The mathematical treatment is given in detail because of the need to examine critically some previous results which disagree with each other in spite of the fact that the initial premises and the methods of solution were the same. The results of this study, which are given in a convenient form, represent the boundary conditions for both the simplified and the complete Navier-Stokes equations in problems of hypersonic rarefied gas flow past bodies with a catalytically active surface.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 159–168, January–February, 1996.     

Turbulent boundary layer heat transfer for a constant property particle-laden gas flow

Solution of a turbulent boundary layer for a constant property, particle-laden gas flow is obtained by a differential method. A dimensionless analysis shows importance of an interaction parameter in increasing heat flux. Boundary layer analysis is done in usual manner by transforming partial differential equations and solution is started at the leading edge by Runge-Kutta method. Velocity and temperature profiles at downstream planes for gas and particles are calculated by an implicit finite-difference iterative procedure, and numerical results are compared with available experimental data.     

A hypersonic chemically nonequilibrium viscous shock layer oh wings with a catalytic surface

Within the framework of the theory of a hypersonic viscous shock layer a study is made of flow round wings of infinite span with blunt leading edges at various angles of attack and slip. Account is taken of multicomponent diffusion, and homogeneous chemical reactions, including dissociation-recombination reactions and exchange reactions. On the shock wave the generalized Rankine-Hugoniot conditions are given, and on the surface of the body conditions which allow for heterogeneous catalytic reactions of the first order with reaction rate constants depending [1] or not depending [2] on the temperature. The cases of an ideally catalytic and a noncatalytic surface are also considered. The surface of the body is assumed to be heatinsulated. A numerical study was made of the problem in a broad range of variation in the angles of attack and slip for different cases of prescribed constants representing the rates of the heterogeneous reactions. The conditions of the flow corresponded to the motion of a body which possess a lifting force along the trajectory of entry into the Earth's atmosphere [3]. The dependences are given of the equilibrium temperature of the surface along the stagnation line of the wing on the height of the flight and the distribution of this temperature along the surface of wings with parabolic and hyperbolic contours. It is shown that for flow regimes with a relatively high degree of dissociation in cases when the proportion of atoms recombined on the surface of the body is small, the dependences of the heat flow and the temperature of the surface on the angle of slip are of a nonmonotonic nature.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza., No. 6, pp. 127–135, November–December, 1984.     

Turbulent boundary layer with positive pressure gradient on a permeable surface under nonisothermal conditions

It is known that the longitudinal pressure gradient can exert a strong influence on the friction law and the characteristics of a dynamic turbulent boundary layer. The thermal and diffusion boundary layers are more conservative to the effect of the pressure gradient, and, hence, methods of analyzing them are based, in the majority of cases, on the hypothesis of conservativity of the heat- and mass-transfer laws to the longitudinal pressure gradient [1]. This hypothesis is verified by experimental results [2, 3] on heat transfer on an impermeable surface in a turbulent stream with positive pressure gradient under almost isothermal conditions. However, such investigations under nonisothermal conditions are practically nonexistent. An approximate theoretical analysis of the heat transfer in a turbulent boundary layer of a nonisothermal stream with a positive pressure gradient is given in this paper. Experimental results are presented. The experimental investigation was conducted in a burned-out graphite diffuser both with and without injection of an inert gas through the wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 43–49, July–August, 1976.     

Turbulent boundary layer flow with a step change from smooth to rough surface

A direct numerical simulation (DNS) dataset of a turbulent boundary layer (TBL) with a step change from a smooth to a rough surface is analyzed to examine the characteristics of a spatially developing flow. The roughness elements are periodically arranged two-dimensional (2-D) spanwise rods, with the first rod placed 80θ_in downstream from the inlet, where θ_in denotes the inlet momentum thickness. Based on an accurate estimation of relevant parameters, clear evidence for mean flow universality is provided when scaled properly, even for the present roughness configuration, which is believed to have one of the strongest impacts on the flow. Compared to previous studies, it is shown that overshooting behavior is present in the first- and second-order statistics and is locally created either within the cavity or at the leading edge of the roughness depending on the type of statistics and the wall-normal measurement location. Inspection of spatial two-point correlations of the streamwise velocity fluctuations shows a continuous increase of spanwise length scales of structures over the rough wall after the step change at a greater growth rate than that over smooth wall TBL flow. This is expected because spanwise energy spectrum shows presence of much energetic wider structures over the rough wall. Full images of the DNS data are presented to describe not only predominance of hairpin vortices but also a possible spanwise scale growth mechanism via merging over the rough wall.     

Three-dimensional chemically nonequilibrium viscous shock layer on a catalytic surface with allowance for associated heat transfer

A three-dimensional flow of dissociating air past blunt bodies is investigated in the framework of the thin viscous shock layer theory. Multicomponent diffusion and homogeneous chemical reactions, including dissociation, recombination, and exchange reactions, are taken into account. The generalized Rankine-Kugoniot conditions are specified on the shock wave and the conditions which take into account the heterogeneous catalytic reactions, on the surface of the body. The viscous shock layer equations are solved together with the heat equations inside the coating, which is carbon with a deposited thin film of SiO₂, or quartz. The case of a thermally insulated surface is also considered. The problem for the case of the motion of a body along the re-entry trajectory into Earth's atmosphere is investigated numerically. The temperature of the surface and the heat flux toward it are given as a dependence on the height (tine) of the flight for different cases of the specification of the catalytic reactions. It is shown that the difference between the heat fluxes towards the thermally insulated surface and the fluxes toward the heat-conducting surface in the neighborhood of the stagnation point is of the order of 6–12% for all the cases considered. This makes it possible to decouple the solution of the problem of heat conduction in the body.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 140–146, November–December, 1985.deceased     

Nonequilibrium laminar boundary layer of dissociating air on axisymmetric bodies

Results are presented of numerical calculations of a nonequilibrium laminar boundary layer on axisymmetric bodies whose surface has arbitrary catalytic activity using a proposed technique. In the published studies devoted to the exact numerical methods for calculating the boundary layer with chemical reactions, it is assumed that the surface of the body is either noncatalytic or has infinite catalytic activity [1], that thermochemical equilibrium exists at the surface [2], or that the temperature and composition of the gas at the surface are given [3, 4]. This problem has been examined in the approximate formulation in several papers, specifically [5].The authors wish to thank V. V. Lunev and I. N. Murzinov for their counsel and comments.