Motion of a gas shock layer with a free boundary under nonadiabatic conditions

This paper is a study of the effect of heat input (removal) on the characteristics of a shock layer produced by a gas at high supersonic velocity encountering a mobile boundary, which for generality is assumed to be free. We will use the Chernyi method, which was employed previously to solve the problem of a shock layer in an adiabatic flow [1, 2]. The results obtained can be useful for analysis of the effect of radiation (absorption) and processes involving the relaxation of internal degrees of freedom of molecules, condensation, chemical reactions, etc., whose effect on the gasdynamics of the flow in a shock layer may be similar to heat input or removal [3–5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 152–154, May–June, 1976.The author thanks A. K. Rebrov for discussion of the results.     

Study of incident shock interaction with a two-dimensional boundary layer

This paper presents the results of an experimental study of a boundary layer disturbed by an incident shock for parameters which are characteristic of problems for flow about blade profiles in the final stages of high-power steam turbines.     

Theory of the interaction of a shock wave with a boundary layer on a moving surface

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 8–14, September–October, 1990.     

Conical gas flows with shock waves and turbulent boundary layer separation

The results of a theoretical and experimental investigation of nonsymmetric flow around a V-wing with supersonic leading edges are presented. The range of the angles of attack and yaw, on which additional singular lines are formed on the windward cantilever of the wing, are experimentally determined using different techniques of flow diagnostics. These are one convergence line and two divergence lines in the transverse flow which were not previously observable in the calculations of ideal-gas flow around wings. It is established that the appearance of the three new alternating singular lines located between the central chord of the wing and a convergence line, exterior to them and occurring within the framework of the ideal gas model, is associated with the relation between the intensities of two contact discontinuities. One of these proceeds from the branching point of the bow shock above the leeward cantilever, while the second issues out of the triple point of a λ-shaped shock configuration accompanying developed turbulent-boundary-layer separation generated by an internal shock incident on the leeward cantilever surface. If the intensity of the contact discontinuity proceeding from the branching point of the bow shock is large as compared with that of the contact discontinuity of the λ-configuration, then the flow pattern realized on the windward cantilever is analogous to that obtained within the framework of the ideal gas model, that is, it includes one convergence line on the wing surface. Under these conditions, the results of the calculations within the framework of the ideal gas model are applicable for understanding the phenomena occurring in the wing shock layer in a considerable part of the control parameter range, including the regimes with intense internal shocks generating turbulent boundary layer separation from the leeward cantilever. Corrections should be made only for a carachteristic pressure distribution in the separation zone and, as a consequence of separation, for an elevated pressure level in the vicinity of the central chord which is the stagnation line of the transverse flow that has passed across the oblique and terminating shocks of the λ-configuration and possesses a higher stagnation pressure than the flow that has passed in an ideal gas across the internal shock incident normally on the leeward cantilever. This is possible only when the divergence line, at which the stream surface enclosing the turbulent boundary layer separation zone enters, does not go over from the leeward onto the windward cantilever.     

Structure of the heating layer ahead of the front of a strong intensely emitting shock wave

The problem of the structure and brightness of strong shock waves arises in the investigation of such phenomena as the motion of large meteoroids in the atmosphere, optical and electrical discharges, the development of strong explosions, and other similar processes and in the creation of powerful radiation sources based on them. This problem also has a general physics interest. As the propagation velocity of a strong shock wave increases the gas temperature behind its front and the role of emission grow. Part of the radiation emitted by the gas heated and compressed in a shock wave is absorbed ahead of the front, forming the so-called heating layer. The quasisteady structure of a strong intensely emitting shock wave was studied in [1, 2]. In this case a diffusional approximation and the assumption of a gray gas were used to describe the radiation transfer. They introduced the concept of a wave of critical amplitude, when the maximum temperature T_- in the heating layer reaches the temperature T_a determined on the basis of the conservation laws, i.e., from the usual shock adiabat; it is shown that behind a compression shock moving through an already heated gas there is a temperature peak in which the maximum temperature T₊ exceeds T_a. The problem of the quasisteady structure of an emitting shock wave in air of normal density was solved numerically in [3]. The angular distribution of the radiation was approximately taken into account — it was assigned by a simple cosinusoidal law. The spectral effects were taken into account in a multigroup approximation. They introduced 38 spectral intervals, which is insufficient to describe a radiation spectrum with allowance for the numerous lines and absorption bands.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 86–92, September–October, 1978.     

Boundary layer flows behind constant speed shock waves moving into a dusty gas

Laminar boundary layer flows behind constant speed shock waves moving into a dusty gas are analyzed numerically. The basic equations of two-phase flows are derived in shock fixed coordinates and solved by an implicit finite-difference method for the side wall boundary layer in a dusty gas shock tube. The development of the boundary layer and resulting velocity and temperature profiles, respectively, for the gas and particles are given from the shock front to far downstream. The effects of diaphragm pressure ratio, mass loading ratio of particles and particle size upon the flow properties are discussed in detail.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Similar solution of laminar boundary layer equations with a flame front

This paper is devoted to the general question of the effect of chemical reactions taking place in the boundary layer with the release of heat on the heat transfer, mass transfer, and friction. In the most general formulation the problem is associated with extremely complex computations because of the presence of many components in the mixture of gases. Moreover, to date the chemical kinetics have received little study, and there are simply no reliable data that can be used in the calculations. Therefore we consider in the following a simplified case in which the chemical reaction proceeds in an infinitely narrow region-a flame front. Mathematically this front is a surface of discontinuity of the concentration and temperature derivatives. Physically this corresponds to the limiting case of chemical equilibrium, in which equilibrium occurs with complete combustion and is realized at moderate temperatures and quite high pressures.This formulation of the problem simplifies the calculation considerably, since the gas mixture may be considered everywhere binary. In this case it is not necessary to introduce into consideration either the chemical kinetics or the chemical equilibrium constants. Moreover, in the case of this formulation we may make use of a similar solution of the boundary layer equations. The extreme simplification of the computations does not alter the physical essence of the problem-the effect of the heat sources within the boundary layer on friction, heat transfer and mass transfer. A large number of computations was made to permit clarification of the specific nature of the dependences of the coefficients of friction, and heat and mass transfer on the rate of fuel injection.Several studies have been published in which a similar assumption was made. Avduevskii and Obroskova [1] consider the case when the flame front is located right at the wall. Harnett and Eckert [2] calculated the boundary layer for a flame front on the assumption that the physical parameters do not depend on temperature. This assumption gives a picture which is inaccurate both quantitatively and qualitatively. Finally, when the present study was nearly completed, the article of Libby and Pierucci [3] appeared. In that study the external flow (air) is a mixture of oxygen and an inert gas (argon). Therefore there is a three-component mixture in the boundary layer at every point. This significantly complicates the computation. At the same time, the dependence of the physical parameters on temperature and the concentrations of the mixture components is taken in simplified form (an unjustified simplification, in our view). The authors of [3] carried out only a limited number of computations, which did not make it possible to clarify the characteristic features of the variation of the friction, heat transfer, and mass transfer coefficients as a function of the rate of combustible gas injection.Notation x coordinate along plate in flow direction - y coordinate along normal to plate - u, v velocity components - T absolute temperature - c mass concentration - volume concentration - density - c_p specific heat at constant pressure - viscosity coefficient - D diffusion coefficient - h specific enthalpy - heat conductivity - M Mach number in external stream - k adiabatic exponent in external stream The author wishes to thank V. S. Avduevskii for guiding this investigation.     

Asymptotic solutions of the two-dimensional equations for a thin viscous shock layer in a rarefied gas

Two-dimensional hypersonic rarefied gas flow around blunt bodies is investigated for the continuum to free-molecular transition regime. In [1], as a result of an asymptotic analysis, three rarefied gas flow regimes, depending on the relationship between the problem parameters, were detected and one of these regimes was investigated. In the present study, asymptotic solutions of the thin viscous shock layer equations at small Reynolds numbers are obtained for the other two flow regimes. Analytical expressions for the heat transfer, friction and pressure coefficients are obtained as functions of the incident flow parameters and the body geometry and temperature. As the Reynolds number tends to zero, the values of these coefficients approach their values in free-molecular flow. The scaling parameters of hypersonic rarefied gas flow around bodies are determined for different regimes. The asymptotic solutions are compared with the results of direct Monte Carlo simulation.     

Flow around stepped bodies with strong compression in a shock layer

This article discusses plane and axisymmetric flows of a nonviscous ideal gas around bodies of stepped form, forming with a Mach number M= and an adiabatic indexN1. The greatest amount of attention is paid to the case where there is no Newtonian free layer, but the shock layer is detached at great distances from the nose of the body.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 4, pp. 104–112, July–August, 1973.     

The flow structure in the front of a moving surface layer

Summary Behind the frontal wedge of a moving surface layer there is a zone of entrainment in which the flow is turbulent due to the onset of a Kelvin-Helmholtz instability. The entrainment coefficient is inversely proportional to the Richardson number. In the turbulent region the gradient Richardson number approached a constant value less than 1/4. The layer thickness and the mean velocity depend on the salt concentration of the underlying fluid. Finally the turbulence decayed leaving a stable interface and the layer behaved much like a free homogeneous flow over a solid boundary.

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Übersicht Hinter der keilförmigen Front einer fließenden Schicht gibt es eine Mischungszone, in der die Strömung turbulent ist. Die Turbulenz entsteht durch eine Kelvin-Helmholtz Instabilität. Der Ansaugbeiwert der turbulenten Strömung verhält sich umgekehrt zur Richardsonschen Zahl. In der turbulenten Strömung nähert sich die Richardsonsche Zahl einer konstanten Zahl kleiner als 1/4. Die Dicke und die mittlere Geschwindigkeit der Schicht hängt sehr von der Dichte der unteren Flüssigkeit ab. Schließlich stirbt die turbulente Strömung ab, und die Schicht fließt wie eine homogene Strömung längs einer ebenen Platte.

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The work described herein was carried out as part of a research programme of the Hydraulics Research Station, and the paper is published by permission of the Director of Hydraulics Research. The author wishes also to express his thanks to Mr. J. A. Weller for his help in measurement, and to a referee for his helpful comments.     

The interaction of a shock wave with the boundary layer in a reflected shock tunnel

The influence of a nontotal reflection on the interaction of a reflected shock wave with the boundary layer in a reflected shock tunnel has been investigated. The calculating method of the velocity, the temperature and the Mach number profiles in the boundary layer in reflected shock fixed coordinates has been obtained. To account for equilibrium real gas effects of nitrogen, the numerical results show that the minimum Mach number in the boundary layer has been moved from the wall into the boundary layer with the increasing of the incident shock Mach number. The minimum Mach number, the shock angle in the bifurcated foot and the jet velocity along the wall to the end plate are reduced owing to the increasing of the area of nozzle throat. The numerical results are in good agreement with measurements.     

A nonsimilar boundary layer on a moving wall

This paper is concerned with an analysis of viscous flow through a varying cross sectional area, typical of application in a Wankel Engine. The analysis models flow past a variable moving boundary. Dimensionless temperature, velocities, friction factor and Nusselt numbers are obtained and evaluated for the inside of a Wankel engine channel for different fluid properties. Research has indicated such application has not been previously performed.     

Resonance properties of two-dimensional wave disturbances in a boundary layer

The nonlinear development of disturbances of the traveling wave type in the boundary layer on a flat plate is examined. The investigation is restricted to two-dimensional disturbances periodic with respect to the longitudinal space coordinate and evolving in time. Attention is concentrated on the interactions of two waves of finite amplitude with multiple wave numbers. The problem is solved by numerically integrating the Navier-Stokes equations for an incompressible fluid. The pseudospectral method used in the calculations is an extension to the multidimensional case of a method previously developed by the authors [1, 2] in connection with the study of nonlinear wave processes in one-dimensional systems. Its use makes it possible to obtain reliable results even at very large amplitudes of the velocity perturbations (up to 20% of the free-stream velocity). The time dependence of the amplitudes of the disturbances and their phase velocities is determined. It is shown that for a fairly large amplitude of the harmonic and a particular choice of wave number and Reynolds number the interacting waves are synchronized. In this case the amplitude of the subharmonic grows strongly and quickly reaches a value comparable with that for the harmonic. As distinct from the resonance effects reported in [3, 4], which are typical only of the three-dimensional problem, the effect described is essentially two-dimensional.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 37–44, March–April, 1990.     

Spatial structure of two-dimensional wave disturbances in a boundary layer

In [1] on the basis of a numerical integration of the Navier-Stokes equations the authors investigated the nonlinear evolution of two-dimensional disturbances of the traveling wave type in the boundary layer on a flat plate. The process of interaction of two waves with different wave numbers and initial amplitudes was examined. In this article the study of these interactions is continued. Special attention is paid to the spatial structure of the disturbances with respect to the cross-flow coordinate (with respect to the longitudinal coordinate the disturbances are assumed to be periodic) at various moments of time. It is shown that if the initial amplitude of one of the waves is sufficiently large, i.e., exceeds a certain threshold value, an undamped quasisteady regime is established during the interaction process. At lower amplitudes the process degenerates and the waves develop independently. In these two cases the evolution of the spatial distribution of the perturbation amplitudes is qualitatively different. In the first case the shape of the amplitude distribution varies only slightly with time, while in the second it depends importantly on the parameters of the wave numbers and the Reynolds number. When the parameters are such that one of the finite-amplitude waves is damped, its amplitude distribution rapidly evolves into the form characteristic of disturbances of the continuous spectrum of the linear stability problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–24, September–October, 1990.