Aerodynamic characteristics of V-shaped wings with shock waves detached from leading edges at hypersonic speeds

Abstract

The direct problem of hypersonic flow past a V-shaped wing with a shock wave detached from the leading edges is solved. The reduced normal force coefficient and the lift-drag (L/D) ratio are calculated for a configuration with a lower part in the shape of a V-wing and a streamwise upper part.1. Consider a symmetrical flow past a V-wing with aperture angle "y and apex angle/3 (Fig. 1) 1 (:x -,f) (1.1) b=tg[3cosq',, h=tg~sinT,, T,= Let the function y=y~ z) represent the shape of the shock wave. Using the conservation laws, we can obtain the following expressions for the velocity components u ~ v ~ w ~ along the x, y and z axes, the pressure p~ and the density p~ behind the shock from divided by the free-stream velocity, double ram pressure and density, respectively:Here, • is the specific heat ratio and M is the flee-stream Mach number. The small parameter e, which is that of thin shock layer theory and equal to the ratio of the densities before and behind the shock wave, is defined by the following relation:We shall assume that (x-1)M 2 sin 2 c~_>O(l) and cos e~=O(1). Then, in accordance with (1.2), u~ v~ Yx = O(e tan o0. We also have an estimate for the scale length of the conical vorticity flow in the compressed layer for uniform flow behind the plane shock attached to the leading edge: z/x=O(.4e tan o0. As follows from this estimate, the transition from the flow regime with a shock attached to the wing leading edges to that with a detached shock takes place when (see Fig. 1) b = o ( ~ tg ~) (1.3)Moreover, the thin shock layer theory determines the limits within which h may vary ( Fig. 1)Then, in accordance with (1.1), (1.3), and (1.4), we can obtain the following estimate: T1 -< O(x/e). Let us assume that the shock is attached to the leading edges, this corresponding to the relation ~ oYx-h-byz.Then, using the impermeability condition on the wing surface v~ ~ =h/b and taking (1.2) into account, we get Yz + b ctg ayz~ + e -h ctg a = 0 (1.5)where only the main terms of the expansion have been kept. It follows from (1.5) that flow with a detached shock wave will occur when the following inequality holds:Moscow.