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Explicit Role of Viscosity in Generating Lift

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Explicit Role of Viscosity in Generating Lift AIAA JOURNAL Vol. 55, No. 11, November 2017 Technical Notes Explicit Role of Viscosity I. Introduction in Generating Lift HIS Note tries to elucidate the subtle but critical role of the fluid T viscosity in generating the aerodynamic lift. The viscous origin of the lift has not been widely understood or sufficiently highlighted Tianshu Liu∗ in the literature. In classical aerodynamics textbooks [1–4], the lift Western Michigan University, Kalamazoo, Michigan 49008 theory of an airfoil is usually developed based on the classical potential-flow theory, in which the Kutta–Joukowski (K-J) theorem and † ‡ is used as a key element in calculating the lift. However, the airfoil Shizhao Wang and Guowei He circulation in the K-J theorem cannot be automatically determined in Chinese Academy of Sciences, 100190 Beijing, the potential-flow framework alone because the vorticity cannot be People’s Republic of China physically generated in an inviscid flow. This fundamental difficulty is intrinsically related to D’Alembert’s paradox stating that the DOI: 10.2514/1.J055907 integrated pressure force of a body is zero in a steady inviscid irrotational incompressible flow [5,6]. To circumvent this problem, the Kutta condition is imposed at the airfoil trailing edge to determine Nomenclature the circulation. To a great extent, the success of the classical Cd = drag coefficient circulation theory in predicting the lift depends on the clever Cl = lift coefficient application of the Kutta condition and its generalized forms as the Cp = pressure coefficient phenomenological models of the viscous-flow effect on the lift c = chord length, m generation [7]. The important implication of D’Alembert’s paradox is D = drag, N that the generation of the lift and drag of a moving body must be a F = fluid-mechanic force or aerodynamic force, N result of a viscous flow no matter how small the fluid viscosity is. The −2 −3 fΩ = boundary enstrophy flux, kg ⋅ m ⋅ s viscous origin of the lift has been recognized, which has to be L = lift, N essentially found in the viscous-flow framework [8–10]. However, an Lpe = pressure lift, N intriguing question is whether there is a direct and explicit connection Lvor = vortex lift, N between the fluid viscosity and the integrated pressure force l = Lamb vector; u × ω, m ⋅ s−2 contributing the lift. This problem is the central topic investigated N = normal force to the flat plate, N from a new perspective here. p = pressure, Pa ρ 2 ∕ q∞ = freestream dynamic pressure; U∞ 2,Pa II. Force Expression Rec = Reynolds number; U∞c∕ν U x = outer flow velocity, m ⋅ s−1 For an incompressible viscous flow over a stationary surface, an ⋅ −1 exact relation between the surface pressure gradient ∇p and the Uref = reference velocity, m s −1 τ – U∞ = freestream velocity, m ⋅ s skin-friction vector can be derived from the Navier Stokes (NS) u = fluid velocity, m ⋅ s−1 equations [11], and it is written as u x; y = velocity in boundary layer, m ⋅ s−1 τ ⋅∇ μ x, y = coordinates on the flat plate, m p fΩ (1) x = normalized coordinate along the flat plate; x∕c α = angle of attack, rad where the scalar quantity is defined as 2 −1 Γg = generalized circulation, m ⋅ s fΩ μ∂Ω∕∂n − 2μκωΩ (2) ΔCp = pressure coefficient difference across the flat plate Δp = pressure loading the flat plate, Pa − − μ Ω jωj2∕ μ = dynamic viscosity of fluid, kg ⋅ m 1 ⋅ s 1 is the dynamic viscosity of the fluid, 2 is the enstrophy, − ω κ ν = kinematic viscosity of fluid, m2 ⋅ s 1 is the vorticity, ω is the curvature of the boundary vorticity line, − ∂∕∂ n ρ = fluid density, kg ⋅ m 3 and n is the derivative along the unit normal outward vector of τ = skin friction, Pa the surface. Equation (1) holds instantaneously for a stationary − μ∂Ω∕∂ Ω = enstrophy; jωj2∕2, s 2 surface. In Eq. (2), the first term n is the boundary enstrophy ω = vorticity; ∇ × u, s−1 flux (BEF), and the second term is the curvature-induced contribution. In general, fΩ is dominated by the BEF, whereas the Downloaded by INSTITUTE OF MECHANICS - CHINESE ACADEMY SCIENCES on December 17, 2017 | http://arc.aiaa.org DOI: 10.2514/1.J055907 curvature term can be neglected. The BEF is an intriguing quantity that is particularly related to the topological features such as isolated critical points and separation attachment lines in a skin-friction field [11]. Received 15 December 2016; revision received 9 May 2017; accepted for τ publication 9 May 2017; published online 20 July 2017. Copyright © 2017 by Along a skin-friction line (called a line), Eq. (1) is rewritten as ∕ s ⋅∇ μ jτj−1 s τ∕jτj the American Institute of Aeronautics and Astronautics, Inc. All rights dp ds p fΩ , where is the unit vector reserved. All requests for copying and permission to reprint should be along the τ line. Therefore, the surface pressure is given by the path submitted to CCC at www.copyright.com; employ the ISSN 0001-1452 integral along a τ line, i.e., (print) or 1533-385X (online) to initiate your request. See also AIAA Rights Z Z and Permissions www.aiaa.org/randp. P x P x ∂jωj −1 *Professor, Department of Mechanical and Aerospace Engineering; p xμ fΩjτj ds μ ds (3) ∂n [email protected]. Associate Fellow AIAA (Corresponding Author). P0 x0 P0 x0 †Associate Professor, Institute of Mechanics, State Key Laboratory of x x τ Nonlinear Mechanics. where P and P0 0 denote a point and a starting point on a line, ‡ x Professor, Institute of Mechanics, State Key Laboratory of Nonlinear respectively. In principle, a set of the starting points P0 0 from Mechanics. which the τ lines are originated could be selected such that the points 3990 AIAA JOURNAL, VOL. 55, NO. 11: TECHNICAL NOTES 3991 P x on a set of the τ lines can cover densely the whole surface. The where ν is the kinematic viscosity. The BEF is given by derivation of the second equality of Eq. (3) uses the skin-friction τ μω × n n ∂Ω 3∕2 relation , where is the unit normal outward vector of the m 1 a 2 00 7m−3∕2 fΩ μ −μβ a f 0x (6) surface of a body and. According to Eq. (3), the surface pressure is ∂y 2 ν related to the accumulating effect of the viscous diffusion flux of the boundary vorticity magnitude (μ∂jωj∕∂n) along a skin-friction line. where the value of the second derivative f 00 0 at the wall is a Equation (3) symbolically expresses a surface pressure field. Thus, function of m, f 000 0−β from the Falkner–Skan equation is the fluid-mechanic force is formally expressed as the following used, and β 2m∕ m 1 is related to the wedge angle of πβ. surface integral: x When P0 0 is the stagnation point at the leading edge, I substitution of Eqs. (5) and (6) into Eq. (3) yields the surface F −pn τ dS pressure distributions on the upper and lower surface, i.e., p xp ρa2x2mk ∕ k ; p S I Z I k 0 k 2 ( 1 2), where 0 is the pressure at P x ∂jωj the leading edge (x 0). In this two-dimensional (2-D) case, −μ n dS ds μ ω × n dS (4) integration along a τ line is simply done along the x coordinate ∂n S P0 x0 S because the τ line is aligned with the surface coordinate. Interestingly, in this case, the relevant boundary-layer parameters Equation (4) explicitly describes the critical role of the fluid are canceled out in Eq. (3). Therefore, the nondimensional form viscosity in generating the fluid-mechanic force. For a steady inviscid Δ − −1 of the pressure loading p p2 p1 is incompressible flow with μ 0, as long as fΩjτj ∂jωj∕∂n does not become infinite as μ → 0,wehaveF 0 according to Eq. (4), Δ p 2m −1 −2m and thus D’Alembert’s paradox is naturally recovered. The first term ΔCp Rqx 1 − 1 − ΔCp;TERq x 1 (7) in Eq. (4) is the integrated pressure force that is explicitly related to q∞ the viscosity. The main consequence of Eq. (4) is that the lift and drag where x x∕c is the chordwise coordinate normalized by the (including the pressure and skin-friction drags) must coexist as a 2 2m1 chord length c; R q ∕q∞ U ∕U∞ Re is the ratio result of the viscous flow over an airfoil. Physically, the lift cannot be q TE ref c between the trailing-edge and freestream dynamic pressures that generated without the cost of generating the viscous drag at the m 2 2 are given by q ρ∕2 a c 1 and q∞ ρ∕2U∞, same time. TE 1 m1 respectively; U∞ is the freestream velocity; Uref a1 ν∕U∞ is a reference velocity; Rec U∞c∕ν is the Reynolds number; Δ Δ Cp;TE is the value of Cp at the trailing edge; and III. Force of Flat Plate m1 −α∕ π − α.TheΔCp distribution is weakly dependent of j j ≈ α∕π ≪ → To elucidate the connection between boundary layer and lift Rec because m1 1. At the leading edge, as x 0,there Δ → 2m1 −α∕ π − α generation through Eq. (4), the steady low-Reynolds-number flow is a singularity of Cp x with m1 ,andthe α → over a two-dimensional flat plate at a small angle of attack (AOA, singularity is weakened as 0, which seems physically denoted by α) is considered as an example [12–14].
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