Computation of Gas–Dynamic Parameters and Heat Transfer in Supersonic Turbulent Separated Flows near Backward–Facing Steps

Computation results of plane turbulent flows in the vicinity of backward–facing steps with leeward–face angles = 8, 25, and 45° for Mach numbers M_infin = 3 and 4 are presented. The averaged Navier—Stokes equations supplemented by the Wilcox model of turbulence are used as a mathematical model. The boundary–layer equations were also used for the case of an attached flow ( = 8°). The computed and experimental distributions of surface pressure and skin friction, the velocity and pressure fields, and the heat–transfer coefficients are compared.     

One-dimensional diffusional model of heterogeneous (gas-film) detonation

The detonation process in a tube filled with a gaseous oxidant (oxygen) and which has a thick layer of fuel (carbon with a low vapor pressure) deposited over its entire perimeter is examined; the weight ratio of fuel to oxidant considerably exceeds the stoichiometric ratio. It is assumed that the rate of heat release is determined by the diffusional (noninstantaneous) process of mixing of the vaporizing fuel and the oxidant. An estimate is made of the effect on the detonation parameters of heterogeneity in the composition over the cross section of the tube and of friction and heat losses. Dependences of the detonation parameters (propagation velocity, pressure profile in the front, distance to the Chapman—Jouquet plane) on the thermophysical properties of the fuel and oxidant are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoe Fiziki, No. 3, pp. 86–95, May–June, 1974.     

Numerical analysis of simulation accuracy for hypersonic heat transfer in subsonic jets of dissociated nitrogen

One-dimensional problems of the flow in a boundary layer of finite thickness on the end face of a model and in a thin viscous shock layer on a sphere are solved numerically for three regimes of subsonic flow past a model with a flat blunt face exposed to subsonic jets of pure dissociated nitrogen in an induction plasmatron [1] (for stagnation pressures of (10⁴–3·10⁴) N/m² and an enthalpy of 2.1·10⁷ m²/sec²) and three regimes of hypersonic flow past spheres with parameters related by the local heat transfer simulation conditions [2, 3]. It is established that given equality of the stagnation pressures, enthalpies and velocity gradients on the outer edges of the boundary layers at the stagnation points on the sphere and the model, for a model of radius R_m=1.5·10^–2 m in a subsonic jet the accuracy of reproduction of the heat transfer to the highly catalytic surface of a sphere in a uniform hypersonic flow is about 3%. For surfaces with a low level of catalytic activity the accuracy of simulation of the nonequilibrium heat transfer is determined by the deviations of the temperatures at the outer edges of the boundary layers on the body and the model. For this case the simulation conditions have the form: dU_e/dx=idem, p₀=idem, T_e=idem. At stagnation pressuresP ₀2·10⁴ N/m² irrespective of the catalycity of the surface the heat flux at the stagnation point and the structure of the boundary layer near the axis of symmetry of models with a flat blunt face of radius R_m1.5·10^–2 m exposed to subsonic nitrogen jets in a plasmatron with a discharge channel radius R_c=3·10^–2 m correspond closely to the case of spheres in hypersonic flows with parameters determined by the simulation conditions [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 135–143, March–April, 1990.     

Laminar boundary layer of infinitely long elliptic cylinders for an arbitrary yaw angle

Results are presented of the calculation of the laminar boundary layer on infinitely long elliptic cylinders in a supersonic perfect gas flow at an arbitrary angle of attack. It was assumed that the Prandtl number is constant and equal to 0.7, the dynamic viscosity coefficient follows a power-law variation ( T^0.76) with temperature, and there is high heat transfer at the body surface (H_1w=0.05).The calculations showed that a change of the body shape—the ellipticity coefficient =b/a—has a significant effect on the nature of the distribution and the magnitude of the local heat flux.In evaluating the thermal fluxes at the blunt leading edges, swept wings are usually considered as infinitely long yawed cylinders. In studying heat transfer at the surface of bodies of small aspect ratio at high angles of attack, wide use is made of the hypothesis of plane sections, when each section, orthogonal to the longitudinal axis of the body, is considered equivalent to a corresponding yawed infinite cylinder.By now quite detailed studies have been made of the behavior of the boundary layer on an infinitely long yawed circular cylinder with both the laminar and turbulent flow regimes for a compressible gas [1, 2]. However, there are no data on the heat transfer at the surface of a yawed infinite cylinder with arbitrary cross section, although the availability of such data is urgently needed, for example, for the proper selection of the form of the leading edges of the swept wing.This article presents the results of the calculation of the characteristics of the laminar boundary layer on the surface of infinite elliptic cylinders in a supersonic perfect gas flow. The calculations were made over a quite wide range of flight Mach number M, yaw angle , and ellipticity factor . The data presented on the distribution of the relative heat flux along the cylinder directrix may be used also for estimating the heat flux with account for the real properties of air if we know the corresponding value of the heat flux in the vicinity of the stagnation line.     

Boundary Layer Stabilization by “Artificial Turbulence”

The possibility of boundary layer stabilization by artificial turbulence, i.e. spanwise-periodic transverse flows induced by mass forces, is considered. It is shown that as the result of additional momentum transfer by the transverse flow the velocity profile in the boundary layer becomes fuller. This results in the suppression of unstable disturbances and increases the laminar-flow interval by 3–4 times.     

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.     

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     

Boundary layer transition on the surface of a delta wing in supersonic flow

Laminar-turbulent transition on the surface of a delta wing has been experimentally investigated in a supersonic wind tunnel at Mach numbers M_t8=3–5. It is shown that when M,=3, Re_L=6.5·10⁶, and =–5.5° much of the upper surface of the wing in the neighborhood of the line of symmetry is occupied by a wedge-shaped region of turbulent flow. In this region the heat fluxes reach the same values as at the heat transfer maxima induced here by separated flows and may exceed the turbulent heat flux level on the windward surface of the wing. Changing the shape of the under surface of the wing from plane to pyramidal leads to acceleration of the boundary layer transition on the under surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 87–92, May–June, 1989.     

Motion of a nonrectilinear fiber in a viscous fluid flow

The plane problem of dynamic interaction of a laminar viscous fluid flow and an inextensible pliable fiber of finite length is solved using the perturbation method. Two types of rheological twodimensional flows — pure shear and simple shear — are considered. Formulas are obtained for the evolution of the tensile force and the shape of the fiber. Results of asymptotic and numerical calculations are compared.     

Effect of a magnetic field on the flow in the vicinity of the stagnation point of a blunt body with ablation of a protective layer

During hypersonic flow around a blunt-nosed body, the gas which passes through the bow shock is heated to high temperatures, where dissociation, ionization, and inverse phenomena (recombination) take place in the gas. If an ionized gas moves in a magnetic field, the ponderomotive force which is set up changes the nature of its motion close to the stagnation point, decreasing the frictional stress and heat transfer at the wall (at the contact surface of the gas and the body about which the gas flows). In this case, the intense heat fluxes from the strongly heated gas to the body about which the gas flows cause phase changes in the surface of the body (melting, sublimation, etc.). These processes, in turn, affect the flow in the vicinity of the stagnation point due to realization of the heat of phase transition, the conduction of heat from the entrained mass, and the diffusion of evaporating material into the boundary layer. References [1, 2] are devoted to a study of the joint influence of the magneto-gasdynamic and ablation effects. The magnetogasdynamic layers and the wall profile of the external velocity (flow around wedges) are discussed in [1], and special cases of such boundary layers-flow close to the stagnation line (the two-dimensional case) and close to the stagnation point (the axisymmetric case) of a blunt body are considered in [2]. Melting and evaporation are taken into account by setting the longitudinal and the transverse velocity components at the wall not equal to zero-the first taking into account the flow of the molten material and the second pyrolysis of the vapor of the surface material into the gaseous boundary layer. However, the values of these components, also the enthalpy on the wall h_w(in[1, 2] h_W 0), are not known beforehand and must be determined from the boundary conditions at the wall which express the mass and heat balances. The general formulation of the problem given in the gasdynamics case by G. A. Triskii in [3, 4], and elsewhere includes a consideration of the boundary-layer equations in the gas, the boundary-layer equations in the melted zone, and the heat conductivity equations in the solid with boundary conditions at the outer edge of the boundary layer, on the gas-molten zone interface, on the molten zone-solid interface, and inside the solid. This approach to the problem can also be utilized in the magnetogasdynamic case, as it is in this article with certain simplifying assumptions as compared with [3, 4]. In this sense, the present article is an extension of the results of [3, 4] to the field of magnetogasdynamics.In conclusion, the author thanks K. A. Lur'e for proposing the subject and useful discussions.     

Discontinuous deformation gradients in plane finite elastostatics of incompressible materials

Summary This investigation is concerned with the possibility of the change of type of the differential equations governing finite plane elastostatics for incompressible elastic materials, and the related issue of the existence of equilibrium fields with discontinuous deformation gradients. Explicit necessary and sufficient conditions on the deformation invariants and the material for the ellipticity of the plane displacement equations of equilibrium are established. The issue of the existence, locally, of elastostatic shocks—elastostatic fields with continuous displacements and discontinuous deformation gradients—is then investigated. It is shown that an elastostatic shock exists only if the governing field equations suffer a loss of ellipticity at some deformation. Conversely, if the governing field equations have lost ellipicity at a given deformation at some point, an elastostatic shock can exist, locally, at that point. The results obtained are valid for an arbitrary homogeneous, isotropic, incompressible, elastic material.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington D.C.     

On mass transfer in an acoustic field

Chemical processes governed by the laws of diffusion kinetics can be intensified by elastic oscillations. It is also known that the rate of combustion of liquid and solid fuels changes markedly with the onset of acoustic vibrations in the combustion chamber. Despite the extensive application of vibrational processes in technology, the mechanisms of heat and mass transfer in the presence of vibrations are not well known. The aim of this research was to analyze the mass transfer from a sphere in an acoustic field.Notation angular frequency of oscillations - wavelength - R characteristic dimension of axisymmetric body - s amplitude of displacement of fluid particles in a plane acoustic wave - B amplitude of oscillation velocity - x, y longitudinal and transverse coordinates - u, v longitudinal and transverse velocity components - v kinematic viscosity - U — A(x) cos t velocity of potential flow - ⁺ thickness of momentum boundary layer - ^– thickness of diffusion boundary layer - m dimensionless concentration - m_* concentration of diffusing component at surface of vaporization - t time - D diffusion coefficient - average density of mixture - erf error function - r radius of axisymmetric body - R Reynolds number - P diffusion Prandtl number - time average - N, N_d Nusselt numbers based on radius and diameter respectively - pulsating component of velocity or concentration - o stationary component of velocity or concentration In conclusion, the authors wish to thank S. S. Kutateladze and I. A. Yavorskii, who supervised the present work.     

Calculation of the transition region of a turbulent jet

The presently known methods for calculating plane and axisymmetric turbulent jets in a wake flow are based on dividing the flow region into two segments, initial and basic [1–3], Here the matching of the parameters of the initial and basic segments is of an artificial nature, since it permits the existence of a physically impossible discontinuity of the curves of the velocity distribution and the jet width along the axis.The aerodynamic characteristics of the transition segment, extending from the point of convergence of the boundary layers at the end of the initial segment to the section corresponding to the point of inflection of the curve u_m(x), differ significantly from the characteristics of the initial and basic segments. This difference is due not only to the sharp increase of the velocity pulsations, but also the marked deformation of the average longitudinal velocity component profile. Consequently, the calculation of the transition segment, in contrast to the initial and basic segments, cannot be based on the single-parameter method.Generally speaking, the flow development in the transition segment may be calculated with the aid of the method [4], which reduces the solution of the problem to an equation of the heat conduction type and assumes the use of an experimental curve of the velocity distribution along the jet axis. Abramovich has carried out the calculation of the transition segment of a plane submerged jet on the basis of certain assumptions which are based on the results of experimental studies [1].Below is presented an approximate method of calculating the transition segment of plane and axisymmetric turbulent jets in a wake flow in which the velocity profiles obtained for the extreme sections of this segment are used for calculating the flow parameters in the initial and basic segments.     

Self-induced interaction between boundary layers in a plane channel

Interaction between boundary layers formed on the walls of a plane symmetric channel is studied when a supersonic perfect-gas stream, homogeneous in the initial cross-section and having a constant specific ratio, flows along the channel at large characteristic Reynolds numbers. In the case under consideration the longitudinal dimension of the interaction zone L coincides in order of magnitude with the channel widthd, i.e., the boundary layers interact as a result of the transfer of a perturbation from one layer to the other through the irrotational flow core.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 131–141, July–August, 1995.     

Motion of an evaporating drop

The quasi steady evaporation of a spherical drop with internal heat release is studied. Energy is transferred from the drop to the ambient vapor—gas medium by molecular heat conduction, convection, and radiation. The differences of the temperature and the concentration between the surface of the drop and the region far from it are assumed to be small. The Reynolds and Péclet numbers, determined, respectively, using the free stream velocity and the mass-average velocity of the vapor—gas medium on the surface of the drop, satisfy the conditions Re R, Pe P, R Re 1, P Pe 1, Re Pe 1. The aim of the paper is to investigate the influence of asymmetry of the heat flux on the drag of the evaporating drop and to establish the conditions of applicability of the model of spherically symmetric evaporation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–10, January–February, 1984.     

Flow of a viscous fluid in the initial section of channels with intense injection

Problems of flows in the initial sections of plane, circular and annular channels are solved by numerical integration of the Navier-Stokes equations on the interval 10 R_b, 3000. The initial section is considered to be the part of the flow where the local Reynolds number does not exceed the critical value.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 187–190, March–April, 1987.     

Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

Turbulent heat—Concentration plume in the case of high viscosity in a nonstratified medium

In this paper, an approximate self-similar solution for the structure of the heat—concentration plume produced by an instantaneous point source of heat in the presence of multicomponent admixture and when the coefficient of turbulence is large is found on the basis of the turbulent transfer equations of vorticity, energy, and matter. Analytical expressions are obtained for the propagation velocity of the buoyancy core and the toroidal vortex formed. The influence of the source parameters and of the coefficient of turbulence on the structure and dynamics of thermal lift is investigated. A comparison is made with the laminar regime of motion of similar formations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 153–163, November–December, 1984.     

Model of particle deposition from turbulent gas-solid flow in channels with absorbent walls

A nonlocal model of particle deposition is developed without resorting to empirical information on the fluctuating motion of the particles. The effects of particle inertia are described by a system of differential equations for the moments of the dispersed phase velocity. The model is tested on examples of flows in channels with smooth walls and with grassy roughness.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 58–65, January–February, 1992.     

Physical properties of the flow in the separation region for three-dimensional interaction of a boundary layer with a shock wave

In a supersonic stream we consider the three-dimensional flow in the plane of symmetry in the region of interaction of a boundary layer with a shock wave which arises ahead of an obstacle mounted on a plate. The principal characteristic of this flow is the penetration of a filament of the ideal fluid within the separation zone and the formation on the surface of the plate and obstacle of narrow segments with high pressures, high velocity gradients, and large heat transfer coefficients.Pressure distribution measurements were made, shadow and schlieren photos were taken, and photographs of the flow pattern on the surface were made using dye coatings and low-melting models. The basic physical characteristics of the separation flow are established. The independence of the separation zone length of the boundary layer thickness is shown. Local supersonic flows are detected in the separation region, flow regimes are identified as a function of the angle of encounter of the separating flow with the obstacles, characteristic flow zones in the interaction region are identified.Notation s coordinate of separation point on the plate - l length of separation zone - H obstacle height - d obstacle transverse dimension - u freestream velocity - velocity gradient on stagnation line of obstacle - b jet width - compression shock standoff from the body - p static pressure - p_* pressure at stagnation point on obstacle - density - viscosity coefficient - boundary-layer thickness - compression shock angle - effective angle of separation zone - setting angle of obstacle on plate - M Mach number - R Reynolds number - P Prandtl number