Numerical investigation of the hypersonic viscous shock layer on wings of infinite span at angles of attack and slip

In the framework of the theory of a hypersonic viscous shock layer [1] with modified Rankine-Hugoniot relations [2] at the shock wave a study is made of flow past wings of infinite span with a rounded leading edge. A numerical solution to the problem has been obtained in a wide range of variation of the Reynolds number (5–10⁶), the blowing-suction parameter, the angle of attack (0–45 °), and the angle of slip (0–70 °). Data are given on the influence of the angle of slip on the profiles of the temperature and the velocity across the shock layer. A study is made of the dependence of the distributions of the pressure, the heat flux, and the friction coefficients along the surface of the body on the blowing-suction parameter and the angles of attack and slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–108, March–April, 1984.     

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.     

Multicomponent three-dimensional viscous shock layer on blunt bodies with a catalytic surface at angles of attack and yaw

The three-dimensional supersonic flow of nonequilibrium dissociating air past smooth blunt bodies on whose surface heterogeneous chemical reactions are taking place is investigated within the framework of thin viscous shock layer theory. An economical numerical method of solving the equations with an improved order of approximation with respect to the normal coordinate is employed. This method does not require the preliminary solution of the Stefan-Maxwell relations for the diffusion fluxes and makes it possible to calculate flows that do not possess a plane of symmetry. The effect of the angles of attack and yaw, the catalytic reaction model and a number of other parameters of the problem on the pressure, heat flux and equilibrium surface temperature distributions is analyzed with reference to the example of flow past a triaxial ellipsoid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–150, January–February, 1990.Tha authors are grateful to G. A. Tirskii for useful discussions of their results.     

Unsteady viscous shock layer near the leading edge of a wing of infinite span

The effect of unsteady injection and wall temperature variation on the parameters of the viscous shock layer near the stagnation line of a wing of infinite span at an angle of slip is investigated on the basis of the viscous shock layer model. An analytic solution of the nonstationary problem, valid near the surface of the wing for strong injection, is obtained. A numerical investigation is carried out and some results of calculating the unsteady viscous shock layer equations for various forms of the time dependence of the injection velocity and wing surface temperature are presented. The calculations are based on a finite-difference method of the second order of approximation in the space variable and the first order of approximation in time, which makes use of expression of the equations in divergence form, Newtonian linearization and vector sweeps across the shock layer. In the steady-state case the results of the calculations are in good agreement with the data of [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 90–95, January–February, 1988.     

Hypersonic flow past a wing at large angles of attack with detached shock wave

The thin shock layer method [1–3] has been used to solve the problem of hypersonic flow past the windward surface of a delta wing at large angles of attack, when the shock wave is detached from the leading edge (but attached to the apex of the wing) and the velocity of the gas in the shock layer is of the same order as the speed of sound. A classification of the regimes of flow past a delta wing at large angles of attack has been made. A general solution has been obtained for the problem of three-dimensional hypersonic flow past the wing allowing for nonequilibrium physicochemical processes of thermal radiation of the gas at high temperatures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 149–157, May–June, 1985.     

Hypersonic flow past delta wings of a certain class at angle of attack with an attached shock

We study the hypersonic flow of an inviscid ideal gas past a delta wing of small aspect ratio at a finite angle of attack. Increasing the Mach number M of the approaching flow to infinity for a constant geometric parameter characterizing the stream disturbance (for example, the body relative thickness or angle of attack), we obtain the limiting hypersonic flow pattern about the body, when the very strong compression shock approaches close to the body, forming a thin compressed layer of disturbed gas flow. Such a flow may be studied using the method of the small parameter, which characterizes the density ratio across the compression shock [1, 2].In [3,4] an analysis is made of the flow past conical wings whose aspect ratio is of order unity. In this case the compression shock will be attached to the leading edge. In [5] a study is made of the flow past wings of small aspect ratio which diminishes along with the small parameter in such a way that the wing half-apex angle has the same order of magnitude as the Mach cone angle within the compressed layer.In this case the angle of attack remains finite (of order unity) so that as M the hypersonic law of plane sections for slender bodies at large angles of attack [6] is satisfied, which together with the additional limit passage0 leads to the similarity law established in [5]. In this case both the case of the detached shock (when the similarity parameter <2), considered in [5, 7], and the case of the attached compression shock (>2) are possible.The monograph [2] reproduces the results of these studies with certain extensions, and also considers the direct problem of flow past a flat delta plate with attached shock, whose solution was found to contain several singular points which require further investigation.In the present study, considering the inverse problem, we were able to construct a closed pattern of the flow past wings of a certain class with thickness and with an attached compression shock, where the field of the gas-dynamic parameters and the shape of the wing surface and of the shock wave are everywhere continuous and do not contain any singular points with the exception of the known thin entropy layer near the stagnation point, which shows up only in the higher approximations [2, 4].In conclusion I would like to thank V. V. Sychev and V. Ya. Neiland for discussions of the subject and of the results, and I would also like to thank V. P. Kolgan for assistance in making the calculations.     

Hypersonic viscous shock layer in a swirling gas flow on a permeable surface

A hypersonic swirling flow of viscous compressible gas past rotating axisymmetric blunt bodies is considered, its velocity vector being parallel to the axis of rotation of the body. The body surface is assumed permeable, while, in the general case, the gas is not injected (drawn off) along the normal to the body surface. An analytic solution of the problem, valid at small Reynolds numbers, is found in the first approximation of the integral method of successive approximations. On the basis of the results of the numerical solution, obtained in a wide range of variation of the determining parameters of the problem, we investigate the influence of the swirling of the free-stream flow, the angular velocity of rotation of the body, the Reynolds number and the injection (suction) parameter on the structure of the compressed layer, and the coefficients of friction and heat transfer on the body surface. The influence of the swirling of the flow on the nature of the asymptotic behavior of the viscous shock layer equations at large Reynolds numbers is studied. It is shown that the presence of a nonzero peripheral component for the velocity vector of the gas in the shock layer can lead to a qualitative change in the nature of the flow. Deceased Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 27–37, November–December, 1986. The authors thank G. G. Chernyi for his useful discussion of the results of the work.     

Hypersonic flow over a narrow delta plate at large angles of attack

The problem of hypersonic flow over a flat delta plate with a high sweepback anglex at angles of attack close to /2 is solved using a numerical algorithm based on transition to the conical solution. The existence of conical flow at /2 with the velocity vector directed towards the apex of the plate is established. Values ofC _p/sin² and the thickness of the shock layer in the plane of symmetry of the plate are given as functions of the hypersonic similarity parameterk=tan tanx. A comparison of the calculated and experimental data shows that they are in good agreement.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 183–185, September–October, 1992.     

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.     

Computation of field structure and aerodynamic characteristics of delta wings at high angles of attack

A numerical investigation of the structure of the vortical flowfield over delta wings at high angles of attack in longitudinal and with small sideslip angle is presented. Three-dimensional Navier-Stokes numerical simulations were carried out to predict the complex leeward-side flowfield characteristics that are dominated by the effect of the breakdown of the leading-edge vortices. The methods that analyze the flowfield structure quantitatively were given by using flowfield data from the computational results. In the region before the vortex breakdown, the vortex axes are approximated as being straight line. As the angle of attack increases, the vortex axes are closer to the root chord, and farther away from the wing surface. Along the vortex axes, as the adverse pressure gradients occur, the axial velocity decreases, that is, A is negativee, so the vortex is unstable, and it is possible to breakdown. The occurrence of the breakdown results in the instability of lateral motion for a delta wing, and the lateral moment diverges after a small perturbation occurs at high angles of attack. However, after a critical angle of attack is reached the vortices breakdown completely at the wing apex, and the instability resulting from the vortex breakdown disappears.     

Experimental study of compressible boundary layer on a cone at angles of attack

Experimental study was conducted for boundarylayers on a sharp 5° half-angle cone of 400mm length at angles of attack. The model was tested in the T-326 hypersonic wind tunnel （ITAM） at freestream Mach number M = 5.95. Mean and fluctuation wall characteristics of the boundary layer are measured at 0°, 2°, 3° and 4° angles of attack for different stagnation pressures. Pulsation measurements are carried out by means of ALTP sensor. Pressure and temperature distributions along the model are obtained, and transition beginning and end locations have been found. Boundary layer stabilization with the increase of angle of attack and the decrease of stagnation pressure is observed. High frequency pulsations inherent to hypersonic boundary layer （second mode） have been detected.     

Supersonic flow over delta wings and elements of star-shaped bodies at angles of attack and roll

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 81–87, January–February, 1989.     

Investigation of hypersonic flow of rarefied gas past wings of infinite span

In the framework of the locally self-similar approximation of the Navier-Stokes equations an investigation is made of the flow of homogeneous gas in a hypersonic viscous shock layer, including the transition region through the shock wave, on wings of infinite span with rounded leading edge. The neighborhood of the stagnation line is considered. The boundary conditions, which take into account blowing or suction of gas, are specified on the surface of the body and in the undisturbed flow. A method of numerical solution of the problem proposed by Gershbein and Kolesnikov [1] and generalized to the case of flow past wings at different angles of slip is used. A solution to the problem is found in a wide range of variation of the Reynolds numbers, the blowing (suction) parameter, and the angle of slip. Flow past wings with rounded leading edge at different angles of slip has been investigated earlier only in the framework of the boundary layer equations (see, for example, [2], which gives a brief review of early studies) or a hypersonic viscous shock layer [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 150–154, May–June, 1984.     

Nonequilibrium viscous shock layer on blunt cones

Some results are given of the numerical investigation into the parameters of the nonequilibrium flow of air in a viscous shock layer in the case of blunt circular cones at zero angle of attack; they are also compared with experimental data obtained during re-entry of ballistic objects into the Earth's atmosphere. The calculations were made with allowance for the nonequilibrium processes of dissociation and ionization, and also vibrational relaxation. The influence of viscosity, heat conduction, and diffusion is taken into account in the complete shock layer. The conditions on the shock wave are posed with allowance for its finite thickness. The characteristic profiles of the velocity, temperature, and electron concentration in the shock layer are given. Good agreement is obtained between the calculated and experimental data on the level and the profiles of the electron concentration. The parameters of the shock layer were determined by a method that is a natural extension of the numerical method of [1] to the case of nonequilibrium flow in a viscous shock layer. Because of this, only the main differences of the method when applied to the calculation of nonequilibrium flows of a multicomponent mixture such as dissociated and ionized air are described.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 15–20, November–December, 1979.     

Hypersonic three-dimensional viscous shock layer in a nonuniform gas flow in the neighborhood of the stagnation point

Three-dimensional hypersonic viscous gas flow past smooth blunt bodies in the presence of injection or suction is considered. The effect of the nonuniformity of the approach stream on the shock-wave standoff, the flow structure and the friction and heat transfer coefficients is investigated with reference to the examples of flow from a supersonic spherical source and flow of the far wake type. It is shown that this effect depends importantly on the Reynolds number, the nature of the nonuniformity and the shape of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 136–145, November–December, 1987.     

Large injection rates on unsteady compressible boundary layer flow over cones at angles of attack

Summary Three-dimensional unsteady laminar boundary layer near the planes of symmetry of sharp cones at angles of attack subject to large rates of injection is obtained numerically by using an implicit finite difference scheme in combination with the quasi-linearization technique. Several model gases are considered with Mach numbers, wall-to-total-enthalpy ratios, and cross-flow parameters spanning the ranges of main engineering interest. A detailed study has been made of the solutions in the symmetry plane, in order to increase the understanding of the problem. Various cases are considered, when the free-stream velocity and the surface mass transfer (injection) vary arbitrarily with time. The effects of viscous dissipation and the cross-flow parameter have also been discussed.This research has been partially supported by the Research and Development Centre for Iron and Steel, Steel Authority of India Ltd. The constructive comments of Professor G. Nath and Professor A. K. Lahiri are sincerely appreciated.     

Numerical investigation of three-dimensional laminar boundary layer flow over spheroids at angles of attack

The flow in the laminar boundary layers on spheroids with axial ratios of 611 (prolate ellipsoid of revolution) and 616 (circular wing) at angles of attack of 5 and 10° is investigated numerically. The implicit finite-difference method described in [1] is employed. The results obtained are compared with the measurements reported in [2–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 59–68, November–December, 1990.     

Computation of the viscous shock layer on blunted cones

A numerical method is described for solving the equations of the compressible viscous shock layer on smooth spherically blunted axisymmetric cones at zero angle of attack and flow of a perfect gas. Effective use is made of the scheme of separating the original system of equations into parabolic (second order) and inviscid (first order) subsystems, which are solved by intrinsic methods. The results of the computations are presented. The method is capable of natural generalization to the case of nonequilibrium physical and chemical processes and diffusion. In most published papers dealing with computation of the compressible shock layer, the authors examine either the vicinity of the stagnation point or a certain region of spherical blunting [1–5]. In all the papers except [4, 5], a number of simplified assumptions have been made regarding the flow picture. Very few papers [6–8] have calculated the viscous shock layer on the forward surface of blunted bodies. In [6, 7] an approximate examination was made only of hyperboloids and paraboloids of revolution, which have very favorable geometry. Reference [8] used a approximate Karman—Polhausen integral method for a very simple system of equations. The method proposed here is essentially an accurate numerical method for solution of the viscous shock layer equations.