Lift and Drag in Two-Dimensional Steady Viscous and Compressible Flow: I. Far-Field Formulae Analysis and Numerical Confirmation

Abstract

This paper studies the lift and drag experienced by a body in a two-dimensional, viscous, compressible and steady flow. We prove formally by linear far-field formulae and Helmholtz decomposition of velocity field that the classic lift formula L = −ρ0UΓφ, originally derived by Joukowski in 1906 for inviscid potential flow and drag formula D = ρ0UQψ for incompressible viscous flow by Filon in 1926, are universally true for the whole field of viscous compressible flow in a wide range of Mach number, from subsonic to supersonic flows. Here, Γφ and Qψ denote the circulation of longitudinal velocity component and inflow of transverse velocity component, respectively. Thus, the steady lift and drag are always exactly determined by the values of Γφ and Qψ, no matter how complicated the near-field viscous flow surrounding the body could be. However, velocity potentials are not directly observable either experimentally or computationally, and hence neither are the JoukowskiFilon formulae (J-F formulae for short). Thus, a testable version of the J-F formulae is also derived, which holds only in the linear far field. These formulae are then examined by a careful numerical simulation of a typical airfoil flow in the range of free Mach number between 0.1 to 2.0. The results strongly support and enrich the J-F formulae. The computed Mach-number dependence of L and D and its underlying physics, as well as the physical implication of the J-F formulae, are addressed. This far-field analysis has been confirmed by a further analytical theory to be reported in a companion paper.