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

Abstract

This paper studies the lift and drag experienced by a body in a two-dimensional, viscous, compressible and steady flow. By a rigorous linear far-field theory and the Helmholtz decomposition of the velocity field, we prove that the classic lift formula L = −ρ 0 UΓ φ , originally derived by Joukowski in 1906 for inviscid potential flow, and the drag formula D = ρ 0 UQ ψ , derived 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, Γ φ and Q ψ denote the circulation of the longitudinal velocity component and the inflow of the transverse velocity component, respectively. We call this result the Joukowski-Filon theorem (J-F theorem for short). Thus, the steady lift and drag are always exactly determined by the values of Γ φ and Q ψ , no matter how complicated the near-field viscous flow surrounding the body might be. However, velocity potentials are not directly observable either experimentally or computationally, and hence neither are the J-F formulae. Thus, a testable version of the J-F formulae is also derived, which holds only in the linear far field. Due to their linear dependence on the vorticity, these formulae are also valid for statistically stationary flow, including time-averaged turbulent flow. Thus, a careful RANS (Reynolds-averaged Navier-Stokes) simulation is performed to examine the testable version of the J-F formulae for a typical airfoil flow with Reynolds number Re = 6.5 × 10 6 and free Mach number M ∈ [0.1, 2.0]. The results strongly support and enrich the J-F theorem. The computed Mach-number dependence of L and D and its underlying physics, as well as the physical implications of the theorem, are also addressed.

The vortex-entrainment sheet in an inviscid fluid: theory and separation at a sharp edge

“…That γ v depends on q 1 /q 2 and q v on γ 1 /γ 2 is expected from the coupled sheet strength evolution equations in (24). In particular, we expect that θ 1 ≤ θ v ≤ θ 2 and since q ≥ 0, then entrainment of irrotational fluid acts to decrease the vortex sheet strength of the shed sheet by 'diffusing' the previously existing vorticity in the sheet.…”

“…Evidently the evolution of the sheets are coupled to each other through the quantity µ, which partly represents the jump in dynamic pressure. For the vortex sheet (24) represents the familiar result that tangential pressure gradients generate vorticity components bi-normal to the gradient direction [23]. For the entrainment sheet the pressure jump acts to 'push' or entrain fluid into the sheet.…”

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

“…They found that the lift calculated by K-J formula (1.2), with ∇φ replaced by the measured total disturbance velocity u ′ , is a good approximation to that of the real viscous flow for typical aerodynamic applications: (1.5) in which V is the volume enclosed by S. Moreover, the experiment confirmed that Γ may still be independent on S. In his theoretical explanation, Taylor (1926) points out that these positive answers require two conditions: (a) the intersect of S and the wake has to be a vertical plane ("wake plane", denoted by W ) with normal n = e x ; and (b) the net vorticity flux through W must vanish, which can be proven for steady viscous flow at large Reynolds number (for an improved proof of this issue see Wu, Ma & Zhou 2015). We call these conditions the first and second Taylor criteria (Liu, Zhu & Wu 2015), and (1.4) the approximate Taylor lift formula. Independent of the work of Taylor (1926), Filon (1926) makes a thorough analysis of the lift and drag problem for two-dimensional, viscous, incompressible and steady flow.…”

“…where φ f is the dominant term in the far-field, which may be multi-valued or singular, and φ r is the single-valued regular term, which decays faster than φ f at the far-field but may play a crucial role in the near-field. In particular, for two-dimensional incompressible flow, there is (Liu, Zhu & Wu 2015)…”

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

“…It has now become a challenge to investigate the effects of compressibility on the lift and lift-induced drag. For instance, with a linear far-field analysis, Liu et al [12,13] recently showed that the Kutta-Joukowski theorem still holds in compressible steady viscous flows. More recently, Schmitz proposed an extension of the Kutta-Joukowski theorem in viscous incompressible flows [14].…”