Wave drag due to lift for transonic airplanes

Abstract

SummaryLift dominated pointed aircraft configurations are considered in the transonic range. These are treated as lifting wings of zero thickness with aspect ratio of order one. An inner expansion which starts as Jones' theory is matched to a nonlinear outer transonic theory as in Barnwell's earlier work. New expressions for the wave drag due to the equivalent body are derived. Some examples of numerical calculations for different configurations are presented. IntroductionIn 1946, R. T. Jones (ref. 1) published a paper giving a formula for the lift and induced drag of "low aspect ratio pointed wings below and above the speed of sound". The work presented here, and earlier in the references cited below, represents an extension of Jones' ideas to the transonic range. It is reassuring that, under suitable circumstances, Jones' formula for the lift and induced drag not only continues to hold but is even valid for wings whose aspect ratio is order one. Under these circumstances, also shock waves and wave drag generally appear. The basic ideas of how this type of flow behaves are set out in the report of Barnwell (ref. 2). The principal result is that the lift produces a flow that looks, in the outer region, like the flow past an equivalent axisymmetric body. This physical effect shows up in the inner and outer expansions used by Barnwell. Cheng and Hafez used similar ideas to define the apparent body and general equivalence rule in a series of papers (ref. 3, 4). Cramer (ref. 5) also studied the problem (with zero thickness as is done here) and essentially verified the results of Cheng and Hafez.In this paper and ref. 6, we have also considered wing-like configurations with zero thickness and aspect ratio O(1) as in fig. 1. Thickness effects can be incorporated relatively easily. Inner and outer expansions are defined in essentially the same manner as Barnwell although the asymptotic matching is carried out in a different way using an intermediate limit. Then wave drag associated with the outer expansion is considered. Several computations and an optimization are carried out to show the effect of planform and longitudinal distribution of lift on the transonic wave drag.

Aeroelastic Analysis of Wings in the Transonic Regime: Planform’s Influence on the Dynamic Instability

“…In these cases often the numerical results agree very well with the experimental data [9][10][11]. On the other hand, if the objectives include a realistic estimation of the drag for wing models or for complete aircraft configurations, the research also requires a large amount of human and computational resources [12][13][14]. Today by means of a fluid structure interaction (FSI) technique, it is possible to represent the physics of transonic phenomena which develop around deformable lifting surfaces.…”

Mathematical Fluid Dynamics of Store and Stage Separation

“…(14a): the transforms to N-wave far from the body. 19 The shock normal velocity in this surface is zero on a wall or the wave arises in the theory of the second order, 19 the pressure jump across the shear layer is zero. This pressure jump and the shock are determined by the effect is very important.…”

“…For the porous wall, stability calculations are conducted using the boundary conditions (9), in which the admittance is given by Eqs. (10), (I1) with the dynamic density (18) and compressibility (19). The UAC parameters are specified as: 1,, = 25 pm, s' = 100 pm, the porosity o = 0.2, the porous-layer thickness h' =: 450 p-m. Figure 102 compares theoretical amplification curves (solid lines) with the experimental data (symbols).…”

“…Jo Computational and Parametric Studies and J, are Bessel functions of the arguments k, = r*V/ (i.o10P,/g*) To evaluate-the porous layer effect on the second-mode stability, and k, = k, V(Pr), which measure the ratio of the tube radius to the we solve the eigenvalue problem (2-4) numerically using the adviscous boundary-layer thickness and to the thermal boundary-layer mittance (14) or its limiting form (15) and the thermal admittance thickness on the tube surface, respectively. Using the relation (19). We consider the boundary layer of a perfect gas with Prandtl Jo(x) + J2(x) = 2J 1 (x)/x (7) number Pr = 0.71 and specific heat ratio y = 1.4.…”

“…Separation of a body from a cavity into a subsonic or transonic flow is subdivided into three phases: in Phase I the body drops inside the cavity; in Phase 2 the body crosses the shear layer separating Aircraft-Stores compatibility Symposium XIII, [18][19][20] (1.1a), the flow over the body can be described with slender body theory [5]. According to the inequalities (l.lb), the shear layer can be approximated as a free slip surface by neglecting the flow inside the cavity and considering the cavity wall effect as a small perturbation.…”