Analytic investigation of friction and heat transfer in the neighborhood of a three-dimensional stagnation point at low and moderate Reynolds numbers

A study is made of hypersonic three-dimensional flow of a viscous gas past blunt bodies at low and moderate Reynolds numbers with allowance for the effects of slip and a jump of the temperature across the surface. The equations of the three-dimensional viscous shock layer are solved by an integral method of successive approximation and a finite-difference method in the neighborhood of the stagnation point. In the first approximation of the method an analytic solution to the problem is found. Analysis of the obtained solution leads to the proposal of a simple formula by means of which the calculation of the heat flux to a three-dimensional stagnation point is reduced to the calculation of the heat flux to an axisymmetric stagnation point. A formula for the relative heat flux obtained by generalizing Cheng's well-known formula [1] is given. The accuracy and range of applicability of the obtained expressions are estimated by comparing the analytic and numerical solutions. Three-dimensional problems of the theory of a supersonic viscous shock layer at small Reynolds numbers were considered earlier in [2–5] in a similar formulation but without allowance for the effects of slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–150, March–April, 1988.     

Asymptotic solution of the two-dimensional equations for a thin viscous rarefied shock layer on a cold surface

Hypersonic rarefied flow past blunt bodies is studied in the continuum-free-molecular transition regime. On the basis of an asymptotic analysis three rarefied gas flow patterns are established depending on the relation between the relevant parameters of the problem. In the first regime corresponding to a cold surface asymptotic solutions of the equations of a thin viscous shock layer are derived at low Reynolds numbers in the axisymmetric and plane cases. Simple analytical expressions for the pressure and the heat transfer and friction coefficients are obtained as functions of the freestream parameters and the body geometry. With decrease in the Reynolds number the coefficients approach the values corresponding to free-molecular flow. In this regime a similarity parameter for the hypersonic rarefied flow past bodies is determined. The asymptotic solutions are compared with numerical solutions and the results of direct statistical simulation by the Monte Carlo method.     

Comparison of approximate analytical and numerical solutions for heat fluxes in viscous supersonic flow past a body

The axisymmetric supersonic laminar flow of a viscous perfect gas past a blunt body is studied over a wide range of Reynolds and Mach numbers. An approximate analytical solution for the distribution of the heat flux, divided by its value at the stagnation point, over the body surface is derived. The expression for the relative heat flux depends only on the body geometry and the pressure distribution over the body surface. To evaluate the accuracy and the range of validity of the formulas obtained, a comparison with numerical solutions of the complete Navier-Stokes equations was carried out for bodies of various shapes, such as spheres, ellipsoids, and paraboloids.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 125–132, January–February, 1996.     

Asymptotic solutions of the thin viscous shock layer equations near the symmetry plane of blunt bodies in hypersonic rarefied gas flow

The hypersonic rarefied gas flow over blunt bodies near the symmetry plane is investigated for the regime transitional from continuum to free-molecular. Three rarefied gas flow regimes are considered depending on the relationship between the determining parameters of the problem. For all regimes, at small Reynolds numbers, asymptotic solutions of the thin viscous shock layer equations near the symmetry plane of blunt bodies are obtained in the form of simple analytical expressions for the heat transfer, skin friction and pressure coefficients as functions of the gas-dynamic parameters of the free-stream flow and the geometric parameters and temperature of the body. With decrease in the Reynolds number these coefficients approach their values in free-molecular flow (with the accommodation coefficient equal to unity). From comparison with the data calculated using the direct simulation Monte Carlo method, the accuracy and applicability limits of the analytical solution are estimated.     

Laminar boundary layer on swept-back wings of infinite span at an angle of attack

A study is made of the flow of a compressible gas in a laminar boundary layer on swept-back wings of infinite span in a supersonic gas flow at different angles of attack. The surface is assumed to be either impermeable or that gas is blown or sucked through it. For this flow and an axisymmetric flow an analytic solution to the problem is obtained in the first approximation of an integral method of successive approximation. For large values of the blowing or suction parameters, asymptotic solutions are found for the boundary layer equations. Some results of numerical solution of the problem obtained by the finite-difference method are given for wings of various shapes in a wide range of angles characterizing the amount by which the wings are swept back and also the blowing or suction parameters. A numerical solution is obtained for the equations of the three-dimensional mixing layer formed in the case of strong blowing of gas from the surface of the body. The analytic and numerical solutions are compared and the regions of applicability of the analytic expressions are estimated. On the basis of the solutions obtained in the present paper and studies of other authors a formula is proposed for the calculation of the heat fluxes to a perfectly catalytic surface of swept-back wings in a supersonic flow of dissociated and ionized air at different angles of attack. Flow over swept-back wings at zero angle of attack has been considered earlier (see, for example, [1–4]) in the theory of a laminar boundary layer. In [5], a study was made of flow over swept-back wings at nonzero angle of attack at small and moderate Reynolds numbers in the framework of the theory of a hypersonic viscous shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–39, May–June, 1980.We thank G. A. Tirskii for a helpful discussion of the results.     

Body shape with minimum total radiant energy flux toward its surface

During a space vehicle's entry into a planet's atmosphere at hypersonic speed one of the important problems is the aerodynamical surface heating due to convective and radiant heat fluxes from the gas after passing through a strong shock wave. Due to the high destructive action of this heating, an important problem is the selection of the aerodynamic shape allowing the minimum heat influx to its surface. The problem of determining the shapes of an axisymmetric body from the condition of minimum total convective heat flux along the lateral face of the body was considered under various assumptions in [1–7]. There are a number of entry conditions (for example, into the earth's atmosphere with a speed of 11 km/ sec at an altitude of about 60 km [12]) during which the radiative component becomes dominant in the total heat flux toward the body. A numerical solution of the problem of hypersonic flow of a nonviscous, non-heat-conducting radiating gas around a body is obtained at this time only for a limited class of bodies and primarily for certain entry conditions (for example, [8–12]). On the basis of these calculations it is impossible to make general conclusions concerning arbitrary body shapes. Therefore, approximate methods were proposed which permit the distribution of radiant heat flux to be obtained for an arbitrary axisymmetric body in a wide range of flight conditions [13–15]. In the present work an expression is derived for the total radiant heat flux over the entire body surface and similarity criteria are found. A variational problem is formulated to determine the shape of an axisymmetric body from the condition of minimum total radiant-heat flux over the entire body surface. It is solved analytically for the class of thin bodies and in the case of a strongly radiating gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 84–89, July–August, 1976.     

Investigation of three-dimensional boundary layers on blunt bodies with permeable surface

Numerical and approximate analytic methods are used to investigate three-dimensional laminar boundary layers on blunt bodies with permeable surface in a supersonic gas stream. In the first approximation of the integral method of successive approximation an analytic solution is obtained to the problem for an impermeable surface, small values of the blowing parameter, and arbitrary suction. For large parameters of the blowing (or suction), whose velocity vector in the general case is directed at a certain angle to the vector of the outer normal to the body, asymptotic expressions are derived for the components of the frictional stress and the heat flux. A numerical solution is obtained to the equations of the three-dimensional boundary layer in a wide range of variation of the blowing (or suction) parameter. The accuracy and region of applicability of the analytic solutions is estimated by comparison with the numerical solutions. On the basis of the solutions obtained in the present paper and the work of other authors an expression is proposed for calculating the heat fluxes to a perfectly catalytic surface of a body in a three-dimensional supersonic flow of dissociated or ionized air. The present paper continues earlier work of the authors [1, 2] on boundary layers in the neighborhood of a symmetry plane and on sweptback wings of infinite span.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 49–58, May–June, 1982.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.     

Construction of Equivalent Axisymmetric Bodies for Calculating Heat Transfer and Viscous Stress on Three-Dimensional Bodies at Incidence

An effective approximate technique for calculating heat transfer, viscous stress, and species components on the windward side of three-dimensional bodies at incidence in hypersonic flow is developed. Using the similarity method, the solution of the three-dimensional problem is reduced to the solution of an axisymmetric problem. For determining the heat flux on a real body, modified two-dimensional equations are solved for equivalent axisymmetric bodies, specially constructed for meridional planes of the original body. For an arbitrary three-dimensional geometry and angle of attack formulas are derived and a conversion program is developed. These make it possible to determine all the parameters of the equivalent body corresponding to a given meridional plane of the original body; then these parameters are used as input data for calculating the viscous flow past the body. The solutions of the two-dimensional equations for the equivalent bodies are in good agreement with more exact solutions of the three-dimensional equations.     

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.     

On the applicability of some approximate similitude laws in hypersonic aerodynamics

The range of applicability of some similitude laws for heat transfer, friction and drag coefficients is discussed on the basis of numerical solutions of the complete viscous shock layer equations describing hypersonic flow past blunt bodies. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 121–130, March–April, 1994.     

Laminar boundary layer in a swirling flow on a permeable surface

Numerical and approximate analytic methods are used to investigate the three-dimensional nonself-similar swirling flow of a uniform gas on an axisymmetric permeable surface. For large values of the injection parameter (in the general case the injection velocity vector forms a nonzero angle with the vector of the outward normal to the flow surface) asymptotic expressions are obtained for the velocity and temperature profiles across the injection layer, the components of the friction stress and the heat flux at the surface. Certain results of a numerical solution of the problem obtained on a broad interval of variation of the injection parameter are presented. By comparing the numerical and asymptotic solutions the accuracy and region of applicability of the latter are estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 28–37, November–December, 1985.In conclusion, the author wishes to thank É. A. Gershbein (deceased) for useful discussion of his results.     

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.     

Axisymmetrical bodies with minimal resistance and with minimal flow toward the surface of the body,with different characters of the flow in the boundary layer

Approximate formulas are obtained which permit calculating the distribution of the friction stress and of the local heat flux over the surface of a body of arbitrary form, with a given pressure distribution, both with laminar and with turbulent flow conditions in the boundary layer. These approximate equations were used to solve the variational problems of determination of the form of axisymmetrical bodies with a minimal resistance or with a minimal total flow of heat to the surface (in the class of bodies consisting of a flat leading edge and a lateral surface) in a hypersonic flow of viscous gas. In the solution of variational problems of determination of the optimal form of a body from the conditions of minimal resistance or of a minimal total heat flux toward the surface, we must be able to determine the distribution of the pressure, the friction stress, and the local heat flux along the surface of a body of arbitrary form. At large Reynolds numbers, the problem of determining the pressure distribution comes down to solving the Euler equation with corresponding boundary conditions. However, at the present time there are no effective methods for solving this problem (at least from the point of view of using the methods for solution of variational problems); in the solution of variational problems, to determine the pressure distribution this forces us to use various approximate methods (for example, the method of tangential wedges or cones, the Newton method, etc.). The use of such approximate formulas renders unfeasible on exact solution of the equations of the boundary layer, for which the distribution of the gas-dynamic parameters at the outer limit of the boundary layer (including also the pressure distribution) must be known previously. This makes it necessary to construct approximate formulas to determine the friction stress, _w, and the local heat flux, q_w, whose accuracy in an arbitrary body will be determined by the accuracy of the assignment of the gas-dynamic parameters at the outer limit of the boundary layer. We give below the derivation of such dependences for laminar and turbulent flow in a boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–102, March–April, 1971.     

基于分子平均自由程的热流计算壁面网格准则

面向壁面热流Navier-Stokes方程数值计算的应用, 研究了壁面法向网格尺度问题.从气体分子物理学角度, 对壁面热流进行了微观统计分析,指出壁面热流与气体分子平均自由程之间的内在联系是壁面法向网格尺度存在最优下限的原因. 据此建立了一个新的壁面法向网格尺度准则, 即MFP准则.该准则简单、实用, 具有清晰的热力学物理意义, 且只依赖于壁面局部参数. 并以多种对比算例验证了MFP准则的准确性和适用性.      

Transport Processes in Nonequilibrium Spherically-Symmetric Gas Outflow into a Vacuum

The transport processes in the nonequilibrium outflow of a monatomic gas into a vacuum are investigated using direct numerical statistical simulation. The quantitative dependence of the macroscopic parameters, such as the density, velocity, temperature, stress, and heat flux, on the distance from the source is determined. In the case of a gasdynamic source the dependence of the stress and the heat flux on the strain rate tensor components and the temperature gradient is established. The applicability range of the well-known hypersonic approximation is determined.     

Influence of the temperature of the surface of an axisymmetrical body and forward radiation on the distribution of a radiant flux on its surface in hypersonic flow-past

A solution is obtained of the flow-past problem for an axisymmetrical body with steady-state hypersonic nonviscous, space-radiating gas flow in a hypersonic approximation. It is shown as illustrated by the example of flow-past of a sphere by an air flow, that the relative distribution of the radiant flux weakly depends on a calculation of surface re-radiation, while the size of the radiant flux substantially depends on body temperature T_W at a critical point. The distributions of radiant flux for sphere flow-past by a CO₂-N₂ gas mixture (at T_W = 0) are calculated using a previously developed method. It is shown that different CO₂ contents in the initial mixture of the incident gas flow weakly affect this distribution. The dependence of the distribution of the radiant flux and departure of the shock wave on the boundary condition for gas enthalpy in the pressure shock, taking into account forward radiation, is investigated. Asymptotic expressions are obtained for sphere flow-past for the case of a strongly radiating gas. Distributions of the radiant flux for different assumptions for the boundary conditions in shocks are calculated.     

Investigation of a hypersonic viscous shock layer on rotating axisymmetric bodies in the presence of blowing

The hypersonic flow of a laminar stream of viscous compressible gas past blunt axisyrametric bodies rotating about the longitudinal axis is considered. It is assumed that gas blows from the surface of the body. The solution of the problem is obtained by a finite-difference method in a wide range of Reynolds numbers and blowing and rotation parameters. Some results of the calculations characterizing the effect of the rotation on the velocity and temperature profiles across the shock layer, on the friction and heat transfer coefficients, and the shock wave separation are given for the neighborhood of the stagnation point. For large Reynolds numbers and strong blowing an analytic solution of the problem is found in an approximation of two inviscid layers separated by a contact surface. The calculations are made for the flow past a sphere and a paraboloid and it is shown that in the presence of rotation the maximum of the heat flux is shifted from the stagnation point onto the side surface of the body. The dependence of the pressure distribution, the heat flux, and the friction coefficient is investigated for cases of constant and variable blowing over the contour of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–114, January–February, 1986.     

Optimization of the shape of a body with respect to two criteria: Radiation heat flux and wave drag

The numerical solution of the two-criteria variational problem of the body contour with minimum radiation heat flux and wave drag is obtained in the class of axisymmetric and plane slender bodies in hypersonic flow. Solutions obtained using the Pareto, ideal point and minimax methods are compared. It is shown that in the class of axisymmetric slender bodies the optimum body gives a decrease in the radiation heat flux as compared with a cone of up to 15% for the Pareto method, up to 13% for the ideal point method, and up to 5% for the minimax method. A solution is also obtained in the subclass of power-law slender bodies and it is shown that the optimum power-law bodies are inferior, as compared with the optimum bodies from the general class of such bodies, in reducing both radiation heating and resistance.     

Asymptotic Investigation of the Thin Viscous Shock Layer Equations in the Neighborhood of the Stagnation Point for Low Reynolds Numbers

Hypersonic three-dimensional viscous rarefied gas flow past blunt bodies is investigated in the neighborhood of the stagnation point. The problem of applicability of the model of a thin viscous shock layer to the regime of transition from continuum to free-molecular flow is considered. In [1], it was shown that at low Reynolds numbers three hypersonic flow regimes can be distinguished and one of those regimes was investigated. In the present study an asymptotic solution of the thin viscous shock layer equations is obtained for another flow regime. With decrease in the Reynolds number the heat transfer coefficient determined by the solution obtained approaches its free-molecular value and the friction coefficient approaches its free-molecular limit, provided that the shock layer thickness is small. The analytical solution is compared with a numerical solution and the results of calculations based on direct Monte Carlo simulation.