A reference point invariant Lamb vector based aerodynamic force breakdown in steady compressible flows

Abstract

Some recent developments in the Lamb vector based aerodynamic force breakdown used the concept of vortex force in order to define the lift and to decompose the drag into lift-induced drag and profile drag. However, the Lamb vector formulation is based on moments and the associated force breakdown may depend on the reference point adopted for their computation. Yet, the force acting on an airplane cannot be dependent on this point. Thus, a systematic method based on the far field flow symmetries is here proposed in order to get rid of this dependence. A new formulation is then defined and tested for a two-dimensional transonic flow around an OAT15A airfoil and a three-dimensional flow around the wing of the NASA Common Research Model.

Definition of an Invariant Lamb-Vector-Based Aerodynamic Force Breakdown Using Far-Field Flow Symmetries

“…Additional results concerning the invariance of F m ρ in two-dimensional flows can be found in Ref. [44].…”

“…This condition is actually fulfilled in the far wake since the wake spreads out because of viscous diffusion and progressively satisfies symmetry properties with respect to the (x, y)-plane. Indeed, Fournis et al [44] considered a simplified two-dimensional example to illustrate the symmetries gradually verified by the Lamb vector ρl and the term u 2…”

“…Besides, in the far wake the direction of the flow is almost parallel to that of the freestream and the region where the Lamb vector is nonzero coincides with a portion of S w where n ≈ e x . Therefore, at first order l x l y , l z (see [44])…”

Compressibility Correction to Kutta–Joukowski and Maskell Formulas Using Vortex-Force Theory

“…(1) involves several terms containing the position vector r which raises the question whether the decomposition depends on the reference point chosen for moment computation. Fournis et al [19] showed that the breakdown sensitivity to r vanishes as S e retreats to infinity and proposed an invariant formulation based on far-field flow symmetries. Former studies numerically highlighted that F ρl + F m ρ provides the total lift and the lift-induced drag in compressible flows while the surface integral F S e yields the profile drag (friction and viscous pressure drag)…”

“…The surface integral¸S e n × ρ∆udS is equal to ρ ∞ Γ * where Γ * is a circulation vector which accounts for compressibility effects. Using the far-field symmetries of the wake introduced by Fournis et al [19], it is possible to show that Γ * • e x = Γ * • e z = 0. Hence it contributes solely to the lift while the second line contributes to the lift and the drag: its contribution to the lift, denoted −ρ ∞ δΓ * , also accounts for compressibility effects and is to be included in the Kutta-Joukowski formula [14].…”