Calculation of three-dimensional boundary layer on the leeward side of a delta wing of finite length in the hypersonic flow in viscous interaction regime

Abstract

UDC 533.6.011.55:532.526.2-3 in the undisturbed flow, the length of the body, and the viscosity coefficient at the stagnation temperature. However, in the neighborhood of the apex of the wing and its leading edges a strong viscous interaction regime is often realized, if the hypersonic interaction parameter, calculated along the length of the region in question, Xx = M~ Re-1/2 ~> 1. The flow in the hypersonic boundary layer on a delta wing of finite length at an angle of attack was considered in (2) under conditions of strong interaction (X = oo) over the entire wing. It was shown that on the leeward side of the wing the displacement thickness increases considerably with increase in the angle of attack and that this is consistent with the experimental data (3). In (4) in relation to flow over delta wings in the strong viscous interaction regime at an angle of attack c~ ~ = (s/Reo) 1/4 (s is the wing aspect ratio) it was shown that on the leeward side in the neighborhood of the apex of the delta wing back flow transverse velocities develop in the boundary layer. The assumption that the strong viscous interaction regime is realized on the leeward side of a delta wing requires the satisfaction of the condition that M2(O~e/OX ~ - ce~ 2 ~> 1 (5), where ~e is the displacement thici~ess. However, this condition is not always satisfied. Moreover, in individual regions on the leeward side of the wing it is possible to have situations in which OYe/OX ~ < 0 or O~/Ox ~ - e~ ~ < 0 and local expansion conditions are realized. Hypersonic flow over triangular bodies at an angle of attack in the viscous interaction regime is considered. The results of calculating the parameters of the three-dimensional boundary layer on the leeward side of a wing and its aerodynamic characteristics are presented. 1. It is proposed to consider hypersonic viscous gas flow over a thin delta wing of finite length L at a small angle of attack a~ - Re~ -1/4. The coordinate system with origin at the apex of the delta wing is shown in Fig. 1. The x ~ axis is directed along the axis of symmetry of the wing, the z ~ axis along the span, and the y~ axis along the normal to the surface. The components W ~ of the velocity vector u ~ v ~ , are directed along x ~ , y~ z ~ respectively, and the aspect ratio of the wing s tan 8, where /3 is the apex half-angle. It is assumed that in the flow over the wing viscous interaction between the three-dimensional boundary layer and the outer inviscid flow is realized. Solving the complete boundary-value problem involves taking into account the flow in the wake formed behind the wing (6). However, in the present study, in order not to have to consider this flow, the boundary condition is imposed on the trailing edge of the delta wing. Since it is proposed to consider the flow over a noncold plate (gw ~ 0) and the flow is subcritical, the series expansion of the solution in the neighborhood of the leading edge will contain an arbitrary function (6). Therefore in order to obtain a unique solution of the boundary-value problem on the trailing edge it is also necessary to assign a function, as which we will take the distribution of the pressure p~ In accordance with the estimates for the boundary layer in a hypersonic flow (5) we introduce the dimen~sionless variables x~ z~ y~ ~t,~ w~ v~174