Lift and drag in three-dimensional steady viscous and compressible flow

TL;DR

This paper extends a force theory for steady viscous compressible flow to three dimensions, deriving a universal force theorem linking forces to flow circulation and inflow, with practical testable formulas and principles for lift-drag optimization.

Contribution

It develops a comprehensive three-dimensional force theorem for viscous compressible flow, improving classical formulas and establishing a universal relation applicable from subsonic to supersonic regimes.

Findings

Unified force theorem linking forces to flow circulation and inflow.

Derived a testable version valid in the linear far field.

Proposed a principle to enhance lift-drag ratio.

Abstract

In a recent paper, Liu, Zhu and Wu (2015, {\it J. Fluid Mech.} {\bf 784}: 304) present a force theory for a body in a two-dimensional, viscous, compressible and steady flow. In this companion paper we do the same for three-dimensional flow. Using the fundamental solution of the linearized Navier-Stokes equations, we improve the force formula for incompressible flow originally derived by Goldstein in 1931 and summarized by Milne-Thomson in 1968, both being far from complete, to its perfect final form, which is further proved to be universally true from subsonic to supersonic flows. We call this result the \textit{unified force theorem}, which states that the forces are always determined by the vector circulation $\pGamma_\phi$ of longitudinal velocity and the scalar inflow $Q_\psi$ of transverse velocity. Since this theorem is not directly observable either experimentally or computationally, a testable version is also derived, which, however, holds only in the linear far field. We name this version the \textit{testable unified force formula}. After that, a general principle to increase the lift-drag ratio is proposed.