An invariant vortex-force theory related to classical far-field analyses in transonic flows

Abstract

Recent studies about the Lamb vector have led to the development of the vortex-force theory, a formulation able to predict the aerodynamic force in compressible flows and decompose it into lift, lift-induced drag and profile drag. Here, a revised formulation of the vortex-force theory developed at ONERA in collaboration with the University of Naples is presented and applied to steady transonic flows. In the mathematical developments, special care is given to the presence of shock wave discontinuities within the flow field. The equivalence between the new definition, the Kutta-Joukowski lift theorem, Maskell's lift-induced drag formula and Betz's profile drag formula is emphasized. The revised formulation also presents several practical advantages: the decomposition is naturally independent of the reference point chosen for the calculation of moment transformations, and the computation can be performed in a better refined part of the grid. The formulation is finally tested on the NASA Common Research Model (CRM) wing-fuselage configuration in cruise flight conditions.

Wave drag prediction using the Lamb vector in two-dimensional steady transonic flows

“…This issue was solved by Fournis et al [44,45] who developed a reference-point-invariant version of Mele et al's decomposition using far-field flow symmetries. In addition to that, the same authors investigated the physics at stake in the vortex-force theory by emphasizing its links with the Kutta-Joukowski theorem, Maskell's lift-induced drag formula and Betz's profile drag formula in compressible flows [46,47]. In particular, they devised two new mathematically equivalent formulations of the vortex-force theory, which are valid in transonic flows, and bridge the gap between classical incompressible aerodynamics and transonic aerodynamics: one is based on local flow quantities (Lamb vector and density gradient) while the other one is based on global flow quantities (circulation, pressure, density, transverse kinetic energy) [47].…”

“…This formulation is directly applicable to compressible subsonic flows but not directly in transonic flows, in which the shock wave discontinuities must be thoroughly accounted for in order to modify the expression of 𝑭 𝑚 𝜌 [43] (a rigorous derivation is provided in [47,55]).…”

“…A first solution was developed by Fournis et al [44,45]: using a flow symmetrization technique, they devised an equivalent reference-point-invariant version of Mele et al's decomposition. A second solution was developed by the same authors [47,55] very recently, this time by adapting the decomposition of Mele et al in order to build two new decompositions mathematically equivalent to each other: the ONERA and Kutta-Joukowski-Maskell-Betz (KJMB) decompositions.…”

“…Given the complexity of the mathematical expressions defining the terms of Mele et al's formulation [41][42][43], the main reason for the development of those two new decompositions was to investigate the physics involved in 𝑭 𝜌𝑙 , 𝑭 𝑚 𝜌 and 𝑭 𝑆 𝑒 in order to better relate lift and drag to their phenomenological sources [47,55]. The first step consisted in identifying the links between the terms defined in Mele et al's formulation and the classical analyses of Kutta, Joukowski, Maskell and Betz in compressible flows:…”

“…Here 𝑭 KJ is the viscous compressible version of the Kutta-Joukowski theorem providing the lift while 𝑭 MSK is the compressible version of Maskell's formula providing the lift-induced drag. Indeed, considering that 𝑆 𝑒 retreats further enough from the body skin [47,55], it is possible to simplify the expressions as…”