Can lift be generated in a steady inviscid flow?

Tianshu Liu

Can lift be generated in a steady inviscid flow?

2023, Advances in Aerodynamics

Abstract

This paper presents a critical evaluation of the physical aspects of lift generation to prove that no lift can be generated in a steady inviscid flow. Hence, the answer to the recurring question in the paper title is negative. In other words, the fluid viscosity is necessary in lift generation. The relevant topics include D' Alembert's paradox of lift and drag, the Kutta condition, the force expression based on the boundary enstrophy flux (BEF), the vortex lift, and the generation of the vorticity and circulation. The physical meanings of the variational formulations to determine the circulation and lift are discussed. In particular, in the variational formulation based on the continuity equation with the first-order Tikhonov regularization functional, an incompressible flow with the artificial viscosity (the Lagrange multiplier) is simulated, elucidating the role of the artificial viscosity in lift generation. The presented contents are valuable for the pedagogical purposes in aerodynamics and fluid mechanics.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Supervisors' Foreword This book represents a significant contribution of Dr. Luoqin Liu to the very foundation of modern theoretical aerodynamics. To help understand its achievement, it is necessary to first briefly recall the history of aerodynamic force theory for steady external flow. The birth and development of classic theory for incompressible, steady, and external-flow aerodynamics were achieved by two mutually enriched and alternative approaches at the fundamental level: far-field and near-field approaches. The former uses far-field linearized equations to derive aerodynamic force formulas that still hold exactly true in near field. It was pioneered by Joukowski in 1906 in the proof of the well-known Kutta-Joukowski's circulation theorem for lift, L ¼ qUC in two-dimensional inviscid flow (KJ theorem for short), and by Filon in 1926 in his derivation of the viscous drag formula D ¼ qUQ (a counterpart of the KJ theorem), where Q is the inflow, a contour integral of rotational stream function. Then, Goldstein in 1931 opened an avenue to extend the KJ theorem and Filon's formula to three dimensions, which was never fully completed yet. By nature, these far-field results are all universally valid for any incompressible and steady flow over a body of arbitrary geometry, but leave some key quantities to be determined by the near-field approach, such as circulation C and inflow Q. Contrary to the far-field approach, the near-field approach works on the fully nonlinear equations under specific boundary conditions. It was pioneered by Prandtl in 1918 who proved that the aerodynamic force in three-dimensional flow simply equals the integral of the Lamb vector, known as the vortex force, of which the KJ theorem is a special case and the famous lifting-line theory is a simplified approximation with the KJ theorem built in as an indispensable ingredient. Since this approach enables determination of aerodynamic forces for specific bodies, it has since become the mainstream of theoretical aerodynamics. The extension of the vortex-force theory to include effects of viscosity and unsteadiness has been done since 1920s and completed now. vii Preface Great advances in the theory of aerodynamics have been made in the first half-century since the Wright brothers successfully flew the first powered, controlled, heavier-than-air airplane in 1903. But, although human beings have realized their dream of flying, even at supersonic and hypersonic speeds, it seems that the theoretical aerodynamics has stopped at its classic content as today still being taught in every aerodynamic book. This embarrassment is mainly due to the following two reasons. On the one hand, classic aerodynamics can only handle simple flows, where the flow fields are solved as analytically as possible. This, however, can hardly be done in modern days since we are facing various complex flows involving boundary-layer separations, vortices and shocks, as well as their evolutions and interactions. On the other hand, owing to the rapid development of computational fluid dynamics (CFD), it has now been routine work to find numerical solutions for complex flows. Since the classic aerodynamics theory is mainly based on the Euler equations, however, it is of little help for reading off the precise key physics from CFD data and unable to guide the advancement of CFD. It is our belief that theoretical aerodynamics has to be developed hand in hand with but can never been replaced by CFD since "ideas and concepts come out of the mind, not out of computers or wind tunnels" (Küchemann 1978). Aiming at developing modern theoretical aerodynamics based on exactly the same Navier-Stokes equations as used in modern CFD for viscous and compressible external flows, this book consists of three innovative breakthroughs in the far-field and near-field theories. First, we construct a universal zonal structure of aerodynamic far field, in which all disturbance flow quantities must decay exponentially if and only if the flow is viscous, unsteady, and compressible. Three commonly used simplified flow models, being inviscid, steady, or incompressible, only work in their respective true subspaces of the free space. Second, we prove rigorously a universal and exact total-force formula as direct extension of and with the same form as the classic lift and drag formulas due to Kutta, Joukowski, and Filon, along with its far-field asymptotic in terms of physically testable variables. Third, we derive a general near-field theory for aerodynamic force and moment, steady or unsteady, which permits detailed complex-flow diagnosis including the xi Mach-number dependence of every physical constituent of flow structures. The predictions of these theories are confirmed by numerical tests. These findings fill some long-standing significant gaps of theoretical aerodynamics. Combined with CFD, they provide a unified foundation for future development of modern aerodynamics. This book is intended to scientists, engineers, and applied mathematicians, who are interested in the physical and theoretical foundations of modern aerodynamics, and graduate students of these fields, as well as general fluid dynamics and physics, and other relevant fields. A first draft of this work was one of the excellent doctoral dissertations of Peking University (2016). The manuscript was read by Prof. J.Z. Wu, who offered many valuable comments. While writing this book, the author received moral support from Prof.

A kinematic model of flow on an infinite flat plate is established, describing how the circulatory flow field is developed on the plate and how the circulation slips off to form wake vortices. The two dimensional model is then extended to rectangular plates, proposing a circulation distribution relationship over the span that describes its dependency on the finite plate geometry. Built on these observations, a quasi-steady vortex impulse analysis method is developed and applied to calculate the linear, potential lift and drag forces on the plate. The results match remarkably well with experimental data. Several key findings are reported in the study: 1) the no-flow condition on the solid surface boundary is determined to be the sole cause for circulatory flows. More precisely, the circulation caused by the reactive flow at centerline yields the exact Kutta-Joukowski circulation expression. Shedding of the leeward circulation leaves the windward half attached to the wing, producing aerodynamic lift. 2) for wings of finite span, due to the presence of side edges, the circulatory flow must emanate in radial directions. As a result, the lifting capability is reduced due to this "circulation diffusion effect". The analytical expressions derived yield very accurate results compared to experiments but differ substantially in representation from the classical Prandtl "Lifting-line Theory". For small aspect ratios, a slightly modified circulation model again yields an accurate prediction consistent with tests, providing the familiar narrow wing formula with a sound theoretical basis. These results bring about questions on the basic premise of the Lifting-line Theory, namely that the reduction in lift is solely a consequence of the induced flow. 3) the circulation generation model entails a symmetry principle stating that a plate would produce the same vortex strength whether it is placed across or along the stream of flow. This is a very reasonable proposition previousely not recognized in the field. 4) the flow induction effect of a vortex pair is applied to the transverse motion of the horseshoe "vortex ring" system, resulting in a "drag-polar" expression. The resulting drag coefficient value is, however, much smaller than experiments, indicating that this aspect requires further scrutiny. The results from this investigation suggest that the quasi-steady vortex impulse analysis provides a well-formed theoretical framework for a new direction in aerodynamics studies. Nomenclature A = wing aspect ratio b, c = wing span and chord C L , C D, C F = lift, drag and force coefficients F = force vector I = impulse vector i, j, k = unit vectors in x, y and z directions l = length n = unit vector (normal to the vortex ring) P, Q = points in space, particularly near the vortex ring contained surface S = a surface area, particularly the vortex ring surface area t = time V = velocity w = velocity normal to the wing surface * Engineer Scientist, Boeing Information Technology, PO Box 3707, Seattle WA, 98124-2207/MS 2R-97, AIAA Senior Member. American Institute of Aeronautics and Astronautics 2 x, y, z = Cartesian coordinates = angle of attack i = induced velocity at trailing vortex = velocity potential function = a contour line = circulation = vortex sheet strength or vortex strength per unit length = air density

The classical calculation of inviscid drag, based on farfield flow properties, is reexamined with particular attention to the nonlinear effects of wake roll-up. Based on a detailed look at nonlinear, inviscid flow theory, the paper concludes that many of the classical, linear results are more general than might have been expected. Departures from the linear theory are identified and design implications are discussed. Results include the following: Wake deformation has little effect on the induced drag of a single element wing, but introduces first order corrections to the induced drag of a multi-element lifting system. Far-field Trefftzplane analysis may be used to estimate the induced drag of lifting systems, even when wake roll-up is considered, but numerical difficulties arise. The implications of several other approximations made in lifting line theory are evaluated by comparison with more refined analyses. Subscripts i induced component n normal component w wake volves several simplifying assumptions, which although not strictly valid, lead to very simple and useful results. Numerous experiments have demonstrated that classical theory is sufficiently accurate to be used in many design applications, but quantitative estimates of the error introduced by some of the theory's approximations have not been established. Recent studies have suggested that these approximations may account for errors in induced drag calculations of five to ten percent. 1 Although a calculation of this small force to within five percent might be considered quite acceptable for some applications, such errors would have significant implications for wing design. Nomenclature b wing span Ct section lift coefficient D drag /_ inviscid force l section lift S area h unit normal vector u, v, w perturbation velocity components U, V, W velocity components U_o freestream velocity 17 local flow velocity y spanwise coordinate e wake deflection angle ¢ velocity potential r circulation, vortex strength p fluid density Recently, much attention has been focussed on the significance of wake shape on the computation of induced drag. 1-4 It has been suggested that the nonplanar geometry of the vortex wake caused by self-induced roll-up or produced as a result of wing planform shape leads to a significant reduction in induced drag. s,6 In this paper, the classical calculation of inviscid drag, based on far-field flow properties, is reexamined with particular attention to the nonlinear effects of wake shape.

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