Mathematical modelling of a stationary flow around a flying vehicle with allowance for jets and air intakes

We consider the stationary nonlinear problem of a separated incompressible fluid flow around a schematic vehicle with operating propulsion system. We take into account the effect of air intakes and the presence of jets of various density on the aerodynamic characteristics. We give a system of integral equations which specifies the form of free surfaces (vortex sheets, jet boundaries) and the intensity of a double layer that is located on rigid and free surfaces. We give the numerical method of solving the system of integral equations and the calculation results of the aerodynamic characteristics of a modern fighter with the operating propulsion system. We show the effect of the propulsion system operation on the aerodynamic characteristics. In this paper we consider the problem of flow around a vehicle with the operating propulsion system at small angels of attack and slip, when we can neglect the flow separation from the leading edges and free surfaces. We also consider this problem at small subsonic velocities, when we can use nonlinear stationary mathematical models, in which the medium considered is an ideal incompressible fluid [4]. The nonlinear stationary mathematical model and the numerical method described in this paper are the extension of the works [2,3,8,11]. We consider a three-dimensional problem here. The fluid densities in a jet and in an incoming flow are different. The fluid transport by air intake is given. This allows us to simultaneously model the necessary air transport by the air intake of the engines, jet impulses, the internal engine thrust. The fluid flow is considered to be a potential one over the whole space except for the surface of a vehicle, its wake, the jet boundaries. The use of the discrete vortex method for solving the problem implies that distributed double or vortex layers are preliminary located on the surface. They are the solution of the Laplace equation for a potential. Using the boundary conditions, we pass on to a singular integral equation, When numerically solving this equation we replace the distributed layers by a system of discrete singularities. The presence of integral equation occasionally allows us to consider the uniqueness of the solution of the problem, the convergence and stability of the numerical method [5]. Such integral equations and the properties of the equations obtained are given in [5] for the case of a continuous flow around bodies, in which the vortex sheets are absent. The singular integral equation for a stationary linear problem is obtained in [6]. The form of a vortex sheet that detaches from a body was considered to be given and located downstream. The system of singular integral equations for a stationary nonlinear problem is given in [9]. The form of a vortex sheet was specified in the process of the problem solution. t N. E. Zhukovskiy Air Force Engineering Academy, Moscow 125167, Russia 34 V.I. Gaidaenko and I.K. Lifanov In this paper we give the system of integral equations for the problem of a stationary flow around a vehicle with the operating propulsion system. The form of a vortex sheet and the jet boundaries as well as the double layer intensity on the surfaces of a vehicle, the sheet, and the jets are defined when solving the problem. We give the numerical method of solving this system of integral equations. The method turns out to be 3 -10 times faster than the conventional discrete vortex method. We give some examples of the application of the numerical method, in particular, the calculation results of aerodynamic characteristics of the modern fighter with operating propulsion system. 1. POSING THE PROBLEM. THE SYSTEM OF INTEGRAL EQUATIONS We pose the problem of an ideal fluid flow around a schematic vehicle with velocity V0 = gradi/0 (see Fig. 1). The fluid density is pQ. The vehicle consists of a thin wing with piecewise smooth surface σ^ and an engine, which is a thin-walled channel with surface στ There is the section σ3 inside the channel, which is called the Active cross-section', where the total and static pressures, the density and velocity of the flow change stepwise. The condition for the conservation of mass transport is satisfied for each streamline that crosses Cy The free vortex sheet σ4 detaches from the line L1 that is part of the edge of the surface σχ. A jet with boundary σ5, which extends ad infinitum, exhausts from the engine with rear edge L2. The fluids in the jet and in the external flow do not mix. The surfaces σ2, σ3, σ5 divide the whole flow region into two parts: DQ is an external flow region and Dj is a flow region in the engine jet. The problem is stationary, the medium is an ideal incompressible fluid. The fluid densities pQ, PJ in the domains Z>0, Df may be different.