Far-Field Force Theory of Steady Flow

Abstract

This chapter studies the lift and drag experienced by a body in a viscous, compressible and steady flow. By a rigorous linear far-field theory and the Helmholtz decomposition of velocity field, we prove that the KJ lift formula for 2D inviscid potential flow, Filon’s drag formula for 2D incompressible viscous flow, and Goldstein’s lift and drag formulas for 3D incompressible viscous flow are universally true for the whole field of viscous compressible flow in a wide range of Mach number, from subsonic to supersonic flows. Thus, the steady lift and drag are always exactly determined by the values of vector circulation \(\varvec{\varGamma }_\phi \) due to the longitudinal velocity and inflow \(Q_\psi \) due to the transverse velocity, respectively, no matter how complicated the near-field viscous flow surrounding the body might be. We call this result the unified force theorem (UF theorem for short). However, velocity potentials are not directly testable either experimentally or computationally, and hence neither is the UF theorem. Thus, a testable version of it is also derived, which holds in the linear far field. We call it the testable unified force formula (TUF formula for short). Due to its linear dependence on the vorticity, TUF formula is also valid for statistically stationary flow, including time-averaged turbulent flow. For 2D flow, some careful RANS simulations of the flow over a RAE-2822 airfoil with angle of attack \(\alpha = 2.31^\circ \) and \(5.0^\circ \), Reynolds number \(Re = 6.5\times 10^6\), and incoming flow Mach number \(M\in [0.1,2.0]\) is performed to examine the validity of the TUF formula. The computed Mach-number dependence of L and D and its underlying physics, as well as the physical implication of the theorem, are also addressed. These results strongly support and enrich the UF theorem.