Three-dimensional Supersonic Flow Research Articles

Calculation of three-dimensional supersonic flow around air intakes in regimes with a detached shock wave

A calculation is presented of the flow around axially-symmetric and close to axially-symmetric bodies with a circular throat and a tapering central body by a supersonic stream of an ideal gas at an angle of attack and regimes with a detached shock wave. The method of solution is based on a treatment of the flow in the coupled meridional half-planes of the cylindrical coordinate system in each of which the problem is solved using the Godunov finite-difference scheme and an approximation of the dependence of the flow functions on the meridional angle by means of trigonometric polynomials. An algorithm is proposed for isolating the three-dimensional shock wave, the surface of which has a particular cleavage line (the line of points of the ternary interaction of the shock waves). The possibilities of the method are demonstrated by means of examples.

Computation of supersonic flows over three-dimensional configurations

An aerodynamic prediction technique based on the steady form of the full-potential equation has been applied to a variety of three-dimensional supersonic flow problems exhibiting embedded subsonic regions. A conservative switching scheme is employed to transition from the supersonic relaxation algorithm, and vice versa. Numerical solutions are obtained for a number of complex configurations, including advanced tactical fighter, Langley canard-wing fighter configuration, isolated shuttle orbiter, and mated shuttle orbiter configuration with external tank. The computed results are in good agreement with available experimental data.

Calculation of three-dimensional flowfields by the unsteady method of characteristics

A numerical algorithm based on the unsteady three-dimensional method of characteristi cs is presented for calculating unsteady three-dimensional subsonic/transonic inviscid flowfields. The flowfield unit processes are based on a local network consisting of six bicharacteristics and the pathline. The algorithm is an explicit secondorder accurate predictor-corrector procedure. An inverse-march ing method is employed, wherein the solution grid in physical space is prespecified. Steady flow solutions are obtained as the asymptotic solution in time. Results are presented for spherical supersonic source flow, for axisymmetric supersonic flow in a conical nozzle, for three-dimensional supersonic flow in a superelliptical nozzle, and for three-dimensional subsonic/transonic flow in a superelliptical nozzle. Solutions obtained by the present method agree well with solutions obtained by steady flow method of characteristi cs algorithms and experimental data.

Full potential solutions of three-dimensional supersonic flows

A nonlinear aerodynamic analysis technique based on the full potential equation in conservative form has been enhanced to allow analysis of more complex geometric configurations. Solution procedures for the equations do not require any specific form of geometry or physical grid system. This results in the capability to analyze very complex geometries easily, provided the posed problem lies within the isentropic restrictions of the full potential theory. To treat embedded subsonic flow, characteristic signal propagation theory is used to monitor the type-dependent flow and a conservative switching scheme is employed to transition from the supersonic marching algorithm to a subsonic relaxation procedure and vice versa. An implicit approximate factorization scheme is used to solve the finite difference equations. These modifications now permit analysis of fully three-dimensional flowfields including the interference effects due to lifting surface wakes. Results are presented showing very good correlations with experimental surface pressure data and aerodynamic force data at both design and off-design operating points. Configurations examined include several waverider concepts, an arrow wing-body with wake, and a fighter forebody-canard configuration.

A harmonic gradient method for unsteady supersonic flow calculations

An accurate and effective method for calculations of unsteady three-dimensional supersonic flow has been developed. The present method is capable of handling general cases of planar, coplanar, and nonplanar wing planforms in the complete frequency domain. A harmonic-gradi ent potential model is provided for elementary doublet panels to be made compatible with the wave number generated. Consequently the number of panel elements required is least affected by the given Mach number and reduced frequency. Thus, the required panel number can be optimized to as few as 30, a fraction of the number required by the existing methods. To assess the accuracy and effectiveness of the present method, comparison with various available data is given. ACP h L m

Explicit-implicit cruising method for computing supersonic flow past a body

A hybrid two-step explicit-implicit finite-difference scheme of second order of approximation is used for computing by the cruising method the three-dimensional supersonic flow of an ideal perfect gas past a blunt body. The basic system of equations is written in the form of conservation laws. Each step consists of two halfsteps, in the first of which the solution is found by the explicit scheme with one-sided differences; in the second half-step the stability condition inherent in explicit schemes is removed, by transforming the finite-difference equations to implicit form. At each implicit half-step a bidiagonal block system of equations is solved. When Courant's stability condition is satisfied, the implicit half-steps fall out automatically and the scheme becomes explicit.

Investigation of unsteady supersonic flow past conical bodies

A study is made of the three-dimensional supersonic flow of ideal gas past conical bodies executing harmonic oscillations in the plane of the angle of attack about some angle β0 in accordance with the law α = α0 cos ωt, so that the total angle of attack is β = β0 + α0 cos ωt.

Three-dimensional supersonic flow of ideal gas past a body with tail fins

The results are given of numerical and experimental investigations of the steady supersonic flow of an ideal gas past a model of a body with + and X fins. The angle of attack varied from 0 to 20 °. A study is made of the physical structure of the flow, and the pressure distributions, the coefficients of the normal force and the pitching moment of the complete body and the individual fins, and also the increment of the coefficient of the normal force acting on the body due to the influence of the fins are found. A comparison with linear theory and the experimental data is made. The agreement between the calculated results and the experimental data is satisfactory.

Numerical investigation of ?lateral? interaction with an obstacle of an axisymmetric jet exhausting into vacuum

The problem of the interaction of a strongly underexpanded axisymmetric jet with an obstacle for which the normal to the surface makes an angle near π/2 with the jet axis is rather laborious for numerical solution due to the high disequilibrium of the gas-dynamic parameters in the peripheral part of the jet and the three-dimensional nature of the flow in the interaction region. Therefore, the results at present available have mainly been obtained experimentally [1, 2]. Among the theoretical studies made in this direction, it is necessary to mention Ivanov and Nazarov's [3], which gives the results of numerical investigation of “lateral” interaction of a jet with obstacles of various shapes in the case of weakly underexpanded jets when the flow in the interaction region is everywhere supersonic. In the present paper, a study is made of the case when a jet exhausts into vacuum and in front of the obstacle there is a detached shock wave, behind which there is mixed subsonic and supersonic three-dimensional flow.

Three-dimensional boundary conditions in supersonic flow

A theoretical analysis of the flow pattern at a solid surface in three-dimensional supersonic flow is presented. The purpose of the study is to develop the additional information necessary to overcome the nonuniqueness associated with the body tangency condition in three dimensions. The analysis is based on the fact that three-dimensional waves propagate locally exactly as they do in axisymmetric flow when viewed in the osculating plane to streamline. The supersonic flow over an infinite swept corner is examined by both the classical solution and the three-dimensional solution in the osculating plane; the results are shown to be identical. A simple numerical algorithm is proposed which accounts for the three wave surfaces that interact at a solid boundary.

A uniformly valid solution for the three-dimensional inviscid supersonic flow past blunt bodies with strong compression in the shock wave

An analytical uniformly valid approximate solution is developed for the steady threedimensional supersonic flow past blunt bodies. The inverse problem is investigated, i.e. the shock shape is prescribed. Viscosity and heat conduction are neglected. The approximation is based on two main assumptions: i) the density ratio across the shock is very small, ii) the pressure does not change its order of magnitude along a normal to the shock surface. The pattern of the streamlines projected on to the shock surface is calculated from an ordinary differential equation of second order by taking into consideration the pressure gradients. By evaluating two integrals the flow quantities and the streamlines are determined in the shock layer together with the body shape. The solution is also valid for sharp nosed bodies. The method is applied to paraboloidal or hyperboloidal shock shapes of various cross sections. Results are presented for the streamline projections in the entire flow field. The flow quantities and the streamline pattern in the shock layer are calculated in the symmetry plane of the flow field. The streamlines differ from the geodesics, i.e. the solution according to the Newtonian model. The flow quantities on the body and the body shape show good agreement with numerical results of the direct problem by Rusanov.

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.

Floating shock fitting for three-dimensional inviscid supersonic flows

A numerical approach based on marching procedures is presented for three-dimensional supersonic flows involving shock-shock interactions, where the shock waves and the associated contact surfaces are determined as floating discontinuities in a fixed system of computational meshes. In order to calculate the regions having subsonic velocity components in the marching direction, which are embedded locally in a supersonic flowfield, a numerical method is also developed that is easily adaptable to the conventional marching procedures. The method consists of an alternate iteration of time-dependen t and marching schemes, in which an explicit finitedifference algorithm is employed to solve Euler's nonconservative equations. The results of complicated threedimensional flowfields about typical blunt-nosed wing/body combinations at angles of attack are presented to demonstrate the validity as well as the applicability of the present approach.

Closure to “Discussions of ‘Three-Dimensional Supersonic Flow Through a Cascade of Twisted Flat Plates’” (1980, ASME J. Fluids Eng., 102, pp. 519–520)

Closure to “Discussions of ‘Three-Dimensional Supersonic Flow Through a Cascade of Twisted Flat Plates’” (1980, ASME J. Fluids Eng., 102, pp. 519–520)

Thin-layer approximation for three-dimensional supersonic corner flows

The thin-layer approximation is extended to an axial corner that is formed by the intersection of two perpendicular plates, one of which has an inclination angle with respect to the free stream. A computer code developed by Hung and MacCormack (1978) is modified for the thin-layer approximation, and a case with Mach 5.9 and a wedge angle of 6 deg is computed. In addition, it is shown that it is not necessary to solve the complete Navier-Stokes equations for a three-dimensional high-Reynolds-number corner flow.

Behavior of sonic lines in a shock layer behind a detached shock wave

A numerical investigation is made into the formation of local supersonic zones in the subsonic flow region between a detached shock wave and the surface of the body in the case of supersonic three-dimensional flow over conical bodies with opening angle θk = 120 ° of the cone in the range of Mach numbers M∞ = 2.5–15.

Numerical investigation of gasdynamic features of control jets

Numerical methods, based on first order difference schemes are used to investigate features of three-dimensional subsonic and supersonic flows of an inviscid non-heat-conducting gas in control jets. Elements of the nozzle channels considered are axisymmetric, and flow symmetry arises from the nonaxial feature of the prenozzle volume and the subsonic part of the nozzle, or because of nonaxiality of elements of the supersonic part. In the first case the nozzle includes an asymmetric subsonic region in which reverse-circulatory flow is observed, and in the second case it includes a region of sudden expansion of the supersonic flow from the asymmetric stagnation zone.

Three-Dimensional Supersonic Flows through Ducts at Incidence

A numerical technique for the solution of three-dimensional hypersonic flow fields consisting of several regions has been developed. Such configurations occur when the incidence of the free-stream ...

Analysis of three-dimensional ducted and exhaust plume flowfields

Computational procedures are described for analyzing three-dimensional supersonic internal flows and multinozzle exhaust plume flowfields. The computer codes (BIGMAC and CHAR3D) embodying these procedures cater to a broad spectrum of geometric situations via the use of multiple reference plane grid networks in several coordinate systems. Shock capturing techniques are employed to trace the propagation and interaction of multiple shock surfaces. Gas properties consist of combustion products in chemical equilibrium. The computational accuracy of the codes is assessed via comparisons with the results of other codes and experimental data. Results are presented for the flows in two-dimensional ducts, corner flows, flow in a rectangular nozzle, and the plume flowfields for exhausts issuing out of single and multiple rectangular nozzles.

A locally two-dimensional numerical method for calculating three-dimensional supersonic flows

A finite difference method of calculating steady supersonic flow of an inviscid ideal gas is described for the case in which a boundary condition has to be satisfied at the surface of any given complicated body shape. The supersonic flow equations are formulated in an arbitrary non-orthogonal co-ordinate system and an operator splitting method is used to construct discrete representations of the equations of motion. Results are presented to demonstrate that the method is capable of capturing shock waves satisfactorily and that it produces results in agreement with other theoretical results. Comparisons between numerical and experimental results for an intake cowl shape, which has a sharp comer and which is representative of a class of complicated body shapes of practical interest, indicates that the method can be relied upon to produce accurate results for complicated body shapes.