Three-dimensional Supersonic Flow Research Articles

Supersonic flows about conical bodies

The numerical solution of three problems of supersonic flow about conical bodies at zero and nonzero angles of attack is given. The generalized method of integral relations is developed for calculating perfect gas flow about a cone at an angle of attack. The shock layer is subdivided into nonoverlapping strips by means of a number of rays and approximations by trigonometric polynomials with respect to the corresponding variable are carried out. The approximating system is integrated along these rays, starting from the shock wave, the coordinates of which are determined according to the condition of vanishing normal velocity on the body. Supersonic flow about cones in the presence of an exothermal combustion reaction is analyzed. The two-component model is considered, in which the kinetics is described by a single concentration of unreacted molecules. The gas is assumed to be perfect with averaged thermodynamic properties, and direct and inverse reactions are taken into account after an induction delay time. In the general three-dimensional case the angular variable connected with the cross flow is eliminated from the governing system with the aid of trigonometric interpolations. The integration of the two-dimensional approximating system in all the meridian planes of interpolation is carried out by the numerical method of characteristics with a network of inverse type. This characteristic computational scheme using two-dimensional compatibility relations is extended to the case of three-dimensional supersonic flows with nonequilibrium chemical processes, taking into account exact kinetics. The flow about blunt-nose inverted cones at an angle of attack in a supersonic stream of nonequilibrium dissociating oxygen is investigated.

Three-dimensional supersonic nozzle flowfield calculations

A second-order numerical method of characteristics has been developed for the solution of three-dimensional supersonic flows. The method has [been programed for the CDC 6500 and IBM 7094 computers for internal flows. Solutions are presented for several heretofore unsolved three-dimensional nozzle flow problems. These include: transition from an axisymmetric inlet to an elliptical cross section, transition from an axisymmetric inlet to a near rectangular cross section, and nonsymmetric inlet flow into an axisymmetric nozzle. The results of these calculations reveal the complexity of the pressure field in flows of this type and show that small deviations from axial symmetry produce gross three-dimensional effects that can only be determined using a truly three-dimensional calculation scheme.

Effect of nonequilibrium dissociation on supersonic three-dimensional flow past inverted cones

Effect of nonequilibrium dissociation on supersonic three-dimensional flow past inverted cones

Three-dimensional supersonic flow past blunt bodies with contour discontinuities, with account for equilibrium and frozen state of the gas in the shock layer

An algorithm is devised for calculating by the finite difference method the supersonic flow region for three-dimensional steady-state flow of a viscous gas past a blunted body with many contour discontinuities. The state of this gas at high hypersonic flight speeds can be characterized by equilibrium or frozen physicochemical processes.

Unsteady supersonic flow around blunt bodies with detached shock wave

The numerical method of calculating the supersonic three-dimensional flow about blunt bodies with detached shock wave presented in [1–3] is applied to the case of unsteady flow. The formulation of the unsteady problem is analogous to that of [4], which assumes smallness of the unsteady disturbances. The paper presents some results of a study of the unsteady flow about blunt bodies over a wide range of variation of the Mach number M∞=1.50−∞ and dimensionless oscillation frequency ωl/V∞=0−1.0. A comparison is made with the results obtained from the Newton theory.

Closure to “Discussions of ‘On the Calculation of Unsteady Nonlinear Three-Dimensional Supersonic Flow Past Wings’” (1968, ASME J. Basic Eng., 90, pp. 594–595)

Closure to “Discussions of ‘On the Calculation of Unsteady Nonlinear Three-Dimensional Supersonic Flow Past Wings’” (1968, ASME J. Basic Eng., 90, pp. 594–595)

Supersonic three-dimensional gas flows with unstable processes

FOR the numerical solution of three-dimensional supersonic problems of gas-dynamics several computational schemes [1–4], based on the use of characteristic compatibility relationships, have been developed and realized with various assumptions in recent times. Of these schemes a particularly simple one is a numerical scheme of the semi-characteristic type [1] in which one independent variable, connected with the transverse flow, is eliminated by means of corresponding approximations and the actual problem is reduced to the integration of a two-dimensional hyperbolic set of equations. In [1] such a semi-characteristic scheme has been developed for calculating the three-dimensional supersonic stable flow of a gas round a body situated at the angle of attack. The application of this scheme, both for blunt bodies [5] and for bodies with a channel of fairly smooth form [6] has shown its effectiveness. The given semi-characteristic scheme was extended [7] (see also [8]) to the case of three-dimensional gas flows with unstable physico-chemical processes. In the present paper a detailed numerical algorithm is put forward for this scheme for such a case and its application to the numerical solution of a concrete problem—the supersonic flow of unstably diffusing oxygen in an asymmetrical nozzle—is considered.

Three-dimensional supersonic gas flow round a blunt body

Three-dimensional supersonic gas flow round a blunt body

Numerical method of characteristics for three-dimensional supersonic flows

The present paper gives a review of different schemes of the numerical method of characteristics for calculating three-dimensional steady supersonic gas flow about bodies moving at incidence. Some other methods for numerical solution of the supersonic flow problem are outlined. The schemes of method of characteristics proposed for calculation of unsteady two-dimensional flows are also presented as far as they can be extended to the case under consideration. Four schemes of the method of characteristics are presented in detail, as developed recently in works by Soviet authors for the calculation of three-dimensional flow about a body and checked in practical applications. Some results of numerical solutions are given for illustration. This paper considers mainly the flow of a perfect gas, and the two last sections deal with the generalization for the case of equilibrium and non-equilibrium flow of real gas.

Three-dimensional flow calculation by the method of characteristics.

Characteristic method computer program for calculating three-dimensional supersonic flow around blunt and pointed bodies of revolution in reasonable time

Numerical calculation of supersonic flow about ducted bodies at an angle of attack

Abstract : To solve the problem of a three-dimensional supersonic gas flow around three-dimensional bodies several effective computation schemes have been developed in recent years, which are based on the use of the finite difference method, the method of characteristics and the semi-characteristics method. Such numerical methods make it possible to determine with the required accuracy both the overall aerodynamic characteristics and the distribution of the various parameters on the surface of the body and across the shock layer. In these numerical computations it is also not difficult to account for physical-chemical processes occuring in the gas at high temperatures. (Author)

Calculation of supersonic flow round sharp asymmetric bodies

THE determination of the three-dimensional supersonic flow round a body that does not possess rotational symmetry is of considerable practical importance. The numerical method described in [1] and developed in [2] allows fairly complex types of flow to be calculated in the case of a gas mixture which is non-viscous, thermally non-conducting, but subject to physico-chemical changes resulting from high temperatures. In order to calculate the supersonic flow round a smooth body by this method, the body surface has to be represented by a continuous differentiable function of point coordinates. The surfaces of flying machines and their parts are very rarely expressible by elementary functions in practice. They are usually specified as a set of point coordinates for a number of body cross-sections. Hence approximate relationships have to be found for them. An approximation algorithm is described below for a class of closed convex surfaces, and the realization of the algorithm is exemplified by the supersonic flow round a sharp rotationally asymmetric body.

Numerical stability of hyperbolic equations in three independent variables

It is shown that nonsimplicial networks proposed for the solution of hyperbolic equations in three dimensions, which satisfy the Courant-Friedrichs-Lewy condition for stability, should be further analyzed using the von Neumann condition, since the latter is a stronger necessary condition than the former for nonsimplicial networks. Several networks proposed for the solution of three-dimensional supersonic flow fields are evaluated on this basis. The stability predictions for one of the proposed networks are compared with the results of numerical experiments. This method of stability analysis also indicates an approximate degree of instability of unstable networks. With the availability of high-speed digital computers, the von Neumann condition can be tested even for complicated difference networks.

Variational problem on three-dimensional supersonic flows

Variational problem on three-dimensional supersonic flows

Three-dimensional supersonic equilibrium gas flow round a body at the angle of attack

Three-dimensional supersonic equilibrium gas flow round a body at the angle of attack

Boundary-layer drag in three-dimensional supersonic flow

A general theorem for drag is given according to which the boundary-layer drag of a body is equal to the inviscid drag of the displacement surface together with a term which is given as an integral involving the ‘streamwise’ momentum and displacement thicknesses taken round the trailing edge. A less accurate result for thin slender wings is that the boundary-layer drag is equal to the line integral$\rho _\infty U^2_\infty \int \Theta d \sigma$taken round the trailing edge, where Θ is the streamwise momentum thickness. This result leads to the possibility of finding boundary-layer drag by means of a traverse round the trailing edge. The extension of the results to wings with swept trailing edges is also given.

FLUTTER ANALYSIS OF FLAT RECTANGULAR PANELS BASED ON THREE-DIMENSIONAL SUPERSONIC POTENTIAL FLOW

Flutter analysis of flat rectangular panels based on three-dimensional supersonic unsteady potential flow

Calculation of three-dimensional rotational supersonic flow of gas in the vicinity of a curve at which occurs a break in streamline

Calculation of three-dimensional rotational supersonic flow of gas in the vicinity of a curve at which occurs a break in streamline

"Characteristic Directions" in Three-Dimensional Supersonic Flows

"Characteristic Directions" in Three-Dimensional Supersonic Flows

“Characteristic directions” in three-dimensional supersonic flows

“Characteristic directions” in three-dimensional supersonic flows