Supersonic flow past a system of two swept wedges mounted on a preliminary compression surface

The results of a numerical modeling of flow past a configuration consisting of two wedges with swept leading edges, so mounted on a preliminary compression surface that the beveled wedge surfaces deflect the wedge-compressed flows counter to each other, are presented. The calculations are performed on the basis of the averaged Navier-Stokes equations, together with the SST k-ω turbulence model, at the freestream Mach number M = 6. For the configuration geometry chosen the flow pattern is characterized by an irregular interaction between the wedge-induced shocks in the plane of symmetry. These shocks also induce three-dimensional, quasi-conical separations of a turbulent boundary layer on the preliminary compression wedge. In the separation zones the flows are directed toward the plane of symmetry of the configuration and interact with one another with the formation of a typical central “bulged” separation flow zone.     

Interaction of a supersonic jet exhausting into vacuum with an obstacle

A study is made of the interaction between an axisymmetric supersonic jet exhausting into vacuum and an obstacle of a fairly complicated configuration and positioned relative to the nozzle in such a way that in the interaction region behind the detached shock wave there is a three-dimensional flow possessing a symmetry plane. The flow in the interaction region is described by the system of equations of motion of an inviscid perfect gas with boundary conditions on the shock wave (Rankine-Hugoniot relation) and on the surface of the obstacle (no-flow condition). The other boundaries of the region are the symmetry plane of the flow and an arbitrarily chosen surface in the supersonic part of the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti Gaza, No. 1, pp. 156–161, January–February, 1981.     

Boundary-layer investigations on a model of the ELAC 1 configuration at high Reynolds numbers in the DNW

Results of boundary-layer investigations on the leeward side of a 1: 12 scale model of the ELAC 1 configuration of a space transportation system are presented. The configuration has the shape of a thick delta wing with a rounded leading edge. The model length is 6 m; the experiments were carried out in the 8×6 m ² low-speed German-Dutch-Windtunnel at Reynolds numbers up to Re=40·10⁶ . In a first series of experiments mean velocity profiles were determined in the turbulent boundary layer on the leeward side of the model, with a single hot-wire probe in the plane of symmetry at four positions. Comparison calculations with a numerical solution of the boundary-layer equations showed good agreement up to angles of attack α=10° . In a second series of tests the laminar-turbulent transition of the flow and its separation near the rounded leading edge were investigated at three positions with multi-sensor hot-film arrays with 40, 56, and 96 elements. These measurements demonstrated that the flow near the rounded leading edge is markedly influenced by the nose radius.     

Effect of streamwise structures on heat transfer in a hypersonic flow in a compression corner

Coefficients of heat transfer to the surface in a laminar hypersonic flow (M _∞ = 21) over plane and axisymmetric models with a compression corner are presented. These coefficients are measured by an infrared camera. The parameters varied in the experiments are the angle of the compression corner and the distance to the corner point. Characteristics of the flow with and without separation in the corner configuration are obtained. The measured results are compared with direct numerical simulations performed by solving the full unsteady Navier-Stokes equations. Experiments with controlled streamwise structures inserted into the flow are described. A substantial increase in the maximum values of the heat-transfer coefficient in the region of flow reattachment after developed laminar separation is demonstrated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 112–120, July–August, 2009.     

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.     

Nonlinear dynamical stability analysis of the circular three-dimensional frame

The three-dimensional frame is simplified into flat plate by the method of quasiplate. The nonlinear relationships between the surface strain and the midst plane displacement are established. According to the thin plate nonlinear dynamical theory, the nonlinear dynamical equations of three-dimensional frame in the orthogonal coordinates system are obtained. Then the equations are translated into the axial symmetry nonlinear dynamical equations in the polar coordinates system. Some dimensionless quantities different from the plate of uniform thickness are introduced under the boundary conditions of fixed edges, then these fundamental equations are simplified with these dimensionless quantities. A cubic nonlinear vibration equation is obtained with the method of Galerkin. The stability and bifurcation of the circular three-dimensional frame are studied under the condition of without outer motivation. The contingent chaotic vibration of the three-dimensional frame is studied with the method of Melnikov. Some phase figures of contingent chaotic vibration are plotted with digital artificial method.     

Supersonic Inviscid Corner Flows with Regular and Irregular Shock Interaction

The results of a numerical and analytical investigation of steady-state supersonic inviscid flows in corners formed by intersecting compression wedges are presented. The flows considered are symmetric about the corner bisector. The distinctive features of flow pattern formation related with the reflection of wedge-generated shocks from the bisector plane are studied. The wedge angles at which transition from regular to irregular shock reflection occurs are determined both numerically and analytically using the criteria available for plane flows; the data thus obtained are found to be in agreement. Flow patterns with irregular shock reflection, namely, single, transitional, and double Mach, as well as von Neumann reflection, are identified; they are similar to the known types of reflection for plane quasi-steady-state flows. Varieties of these types not observed in the plane flows are found to exist. The effects of the angle of inclination of the plane surfaces of the corner to the freestream direction, the sweep angle of the leading edges, and the dihedral angle are investigated. Some previously unknown parameters of corner configurations for which transition may occur in accordance with the von Neumann criterion are determined.     

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.     

Existence of supersonic zones in three-dimensional separation flows

Up till now the region of three-dimensional separation flows which occur with supersonic flow past obstacles has received insufficient study. Supersonic flow with a Mach number of 2.5 past a cylinder mounted on a plate was studied in [1]. A local zone with supersonic velocities was found in the reverse subsonic flow region ahead of the cylinder. Its presence is explained by the three-dimensional nature of the flow. Similar supersonic zones are not observed in the case of supersonic flow over plane and axisymmetric steps.The present paper presents the results of experimental studies whose objective was refinement of the flow pattern ahead of a cylinder on a plate and the study of the local supersonic zones.The experiments were performed in a supersonic wind tunnel with a freestream Mach number M₁=3.11. The 24-mm-diameter cylinder with pressure taps along the generating line was mounted perpendicular to the surface of a sharpened plate. The distance from the plate leading edge to the cylinder axis wasl ₀=140 mm. The plate was pressure tapped along the flow symmetry axis. The Reynolds number was Rl ₀=u₀ l ₀/v ₁, Rl ₀=1.87.10⁷, where u₁ andv ₁ are the freestream velocity and the kinematic viscosity, respectively. The pressures were measured using a Pilot probe with internal and external diameters of 0.15 and 0.9 mm, respectively.The probe was displaced in the flow symmetry plane at a distance of 1.6 mm from the plate surface and at a distance of 1.1 mm along the leading generator of the cylinder. The flow on the surface of the plate and cylinder was studied with the aid of a visualization composition and the flow past the model was photographed with a schlieren instrument. Typical patterns of the visualization composition distribution and the pressure distribution curves over the plate surface, and also photographs of the flow past the model, are shown in [1].     

Vortical structures behind a transverse jet in a supersonic flow at high jet to crossflow pressure ratios

A three-dimensional supersonic turbulent flow with symmetric normal injection of circular jets from the channel walls is numerically simulated. The initial Favre-averaged Navier–Stokes equations closed by the k–ω turbulence model are solved by an algorithm based on an ENO scheme. The mechanism of the formation of vortical structures due to the interaction of the jet with the free stream is studied for jet to crossflow total pressure ratios ranging from 3 to 50. It is known from experiments reported in the literature that, for n ? 10, mixing of the jet with the high-velocity flow leads to the formation of a pair of vortices and of an additional separation zone near the wall behind the jet. It is demonstrated that the present numerical results are consistent with such findings and that the pressure distribution on the wall ahead of the jet in the plane of symmetry is also in reasonable agreement with available experimental data.     

Multisensor Hot Wire Vorticity Probe Measurements of the Formation Field of Two Corotating Vortices

Experimental evidence is reported, regarding the formation of a pair of co-rotating tip vortices by a split wing configuration, consisting of two half wings at equal and opposite angles of attack. Simultaneous measurements of the three-dimensional vector fields of velocity and vorticity were conducted on a cross plane at a downstream distance corresponding to 0.3 cord lengths (near wake), using an in-house constructed 12-sensor hot wire anemometry vorticity probe. The probe consists of three closely separated orthogonal 4-wire velocity sensor arrays, measuring simultaneously the three-dimensional velocity vector at three closely spaced locations on a cross plane of the flow filed. This configuration makes possible the estimation of spatial velocity derivatives by means of a forward difference scheme of first order accuracy. Velocity measurements obtained with an X-wire are also presented for comparison. In this near wake location, the flow field is dictated by the pressure distribution established by the flow around the wings, mobilizing large masses of air and leading to the roll up of fluid sheets. Fluid streams penetrating between the wings collide, creating on the cross plane flow a stagnation point and an “impermeable” line joining the two vortex centres. Along this line fluid is directed towards the two vortices, expanding their cores and increasing their separation distance. This feeding process generates a dipole of opposite sign streamwise mean vorticity within each vortex. The rotational flow within the vortices obligates an adverse streamwise pressure gradient leading to a significant streamwise velocity deficit characterizing the vortices. The turbulent flow field is the result of temporal changes in the intensity of the vortex formation and changes in the position of the cores (wandering).     

Separated flow over a strongly cambered conical wing at small angles of attack

The interference of supersonic flows on the concave surface of conical wings was experimentally investigated in [1] for various values of the camber and angles of attack. In order to establish the detailed structure of the interference flow the laminar flow past a wing model in the form of half the surface of a circular cone with vertex angle 2_k = 34° was numerically modeled within the framework of the quasiconical approximation for the three-dimensional Navier-Stokes equations [2]. Under this assumption, confirmed by analysis of the experimental data [1], it was found that the displacement of the external inviscid flow as a result of intense flow separation beyond the leading edges leads to flow patterns similar to those realized on V wing's with a break in the transverse contour [3]. At nonzero angles of attack weak secondary separation was detected beneath the flattened regions of primary separation located in the shaded parts of the concave surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 130–136, July–August, 1989.     

Scaling of flow separation on a pitching low aspect ratio plate

We use two different dye injection approaches, in two different water tunnels, to visualize the formation and subsequent evolution of leading-edge vortices and related separated structures, for a pitching low aspect ratio plate. The motion is a smoothed linear pitch ramp from 0° to 40° incidence, brief hold, and return to 0°, executed at reduced pitch rates ranging from 0.1 to 0.35 and about various pivot locations. All cases evince a leading edge vortex with pronounced axial flow, which leads to formation of large-scale, three-dimensional flow structures, culminating in a large vortical structure centered at the wing symmetry plane. Pitch is also compared to plunge, where the plunge-induced angle of attack is taken as the geometric pitch incidence angle, ignoring pitch-rate effects. At successively increasing values of convective time C/U, the three-dimensional patterns of the flow structure are remarkably similar for the pitching and plunging motions. The similarity of these patterns persists, though they are shifted in time, for variation of either the location of the pitching axis or the dimensionless pitch rate.     

Numerical investigation of three-dimensional shock focusing effects by an invariant difference scheme

Symmetry preserving difference schemes which are found on the basis of the Lie-group theory for partial differential equations applied to their first differential approximations are used to simulate the evolution and reflection of shock waves in a cubical cavity with rigid walls and walls with outflow boundary conditions. The explicit difference scheme was used for a shock capturing technique to find and trace the discontinuities in the flow field. The numerical model is based on the strong conservative form of the three-dimensional time-dependent Euler equations with an equation of state for an ideal gas. The three-dimensional time-dependent shock waves which are created by a centered gas volume under high pressure interact with the plane walls of the cavity and lead to focusing effects after the explosion. To follow the propagation of strong plane or spherical shock waves the shock capturing method was proved to be sufficient concerning the spatial resolution of the discontinuities. When interacting with the rigid walls of the cavity the shock waves are assumed to be reflected specularly and to interact with shock waves and contact discontinuities moving from the center to the edges and corners of the cavity. The symmetry preserving character of the invariant difference scheme under use was proved numerically by calculations over long time. No significant deviation of the symmetry character of the imposed initial distribution was found. Additionally, a numerical analysis of the conservation of energy in the cube was performed to find the behavior of the energy in time.     

Supersonic viscous perfect-gas flow past a slender sharp elliptic cone

The supersonic M_∞ = 5 flow past slender elliptic cones with the semi-vertex angle in the plane of the major semi-axis ? _c = 4° and an isothermal surface is investigated under the assumption of the flow symmetry. Calculations on the basis of the time-dependent three-dimensional Navier-Stokes and Reynolds equations are carried out on the Reynolds number and angle of attack ranges 10⁴ ≤ Re ≤ 108 and 0 ≤ α ≤ 15° for cones with ellipticity coefficients 1/32 ≤ δ= b/a ≤ 1. The effect of the relevant parameters of the problem on the flowfield structure and the aerodynamic characteristics of the cones is demonstrated.     

Axisymmetric analogy for three-dimensional viscous flow problems

A method of solving three-dimensional flow problems with the aid of two-dimensional solutions, which can be used for any Reynolds numbers, is proposed. The method is based on the use of similarity relations obtained in the theoretical analysis of the approximate analytic solution of the equations of a three-dimensional viscous shock layer. These relations express the heat flux and the friction stress on the lateral surface of a three-dimensional body in terms of the values on the surface of an axisymmetric body. The accuracy is estimated by comparing the results with those of a numerical finite-difference calculation of the flow past bodies of various shapes. Similar similarity relations were previously obtained in [1] for the plane of symmetry of a blunt body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–118, November–December, 1991.The authors are grateful to G. A. Tirskii for his interest in their work.     

Surface contact of elliptical crack under normally incident tension–compression wave

Surface contact interaction of a plane elliptical crack under normally incident tension–compression wave is solved by the method of boundary integral equations. The contact forces and the displacement discontinuity of the crack edges are examined. The dependence of the mode I stress intensity factor on the wave number is studied. The solution is compared with the results obtained for an elliptical crack without allowance for crack edges contact interaction.     

Three-dimensional turbulent boundary layer on a body of complex shape

Flow and heat transfer problems associated with three-dimensional compressible gas flow past a body of complex shape at a small angle of attack are investigated on the basis of a finite-difference calculation. The results of a numerical solution of the equations of the three-dimensional turbulent boundary layer are presented. The effect of the leading parameters on three-dimensional flow development and heat transfer is analyzed. The characteristic flow regions in the boundary layer are found: lines of divergence and convergence on the surface, separation zones and flow interfaces. The location of the maximum values of the heat flux and friction on the surface is determined, the behavior of the limiting streamlines on the body is described, and the intensity of the secondary flows in the boundary layer is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 25–35, September–October, 1986.     

Marginal separation in three-dimensional flows

The boundary layer in the vicinity of the zero skin-friction point on the leeward symmetry line of a prolate spheroid placed at an angle of attack is considered. The existence of this flow was established by Cebeci et al. (1980) for an angle of attack =40°. The current study is based on the results of Brown (1985) who described the marginal separation in the symmetry plane for a zero skin-friction point and on the results of Zametaev (1989) who included the spatial extension of Brown's solution but without interaction between the boundary layer and the outer flow. It is found that the three-dimensional boundary-layer equations in the vicinity of the zero skin-friction point are reduced to a single nonlinear partial differential equation of hyperbolic type which governs the longitudinal skin-friction component. Smooth solutions of this equation may be found which contain separation lines as well as double-valued regions. It is likely that the latter regions are related to the tip of the separation line obtained as a result of calculations of the full boundary-layer equations. The influence of interaction is also considered, in which case the flow is governed by a partial integro-differential equation. Numerical solutions are given for each of these problems.This study was supported by the United Technologies Research Center     

Contribution to the theory of thin-profile trailing edge separation

The conditions of nonsymmetric trailing edge flow with separation are investigated. Solutions of the equations for the interaction zone in the neighborhood of the trailing edge of a thin profile at an angle of attack of the order O(Re^–1/16) in the separated flow regime are constructed numerically. It is shown that for this zone a solution exists up to a certain angle of attack. In all the regimes the value of the friction on the upper surface at the very end of the trailing edge remains a positive quantity. The solution of the equations in the separated flow regimes is found to be nonunique. The flow over the leading edge is assumed to be unseparated, and the separation at the trailing edge, if present, is assumed to be localized in the interior of the boundary layer. The flow over a Kutta profile at zero angle of attack is taken as an example. In this case the satisfaction of the Chaplygin-Joukowsky condition at the trailing edge ensures smooth flow over both the trailing and leading edges.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–59, July–August, 1989.