Evolutionary understanding of airfoil lift

This review attempts to elucidate the physical origin of aerodynamic lift of an airfoil using simple formulations and notations, particularly focusing on the critical effect of the fluid viscosity. The evolutionary development of the lift problem of a flat-plate airfoil is reviewed as a canonical case from the classical inviscid circulation theory to the viscous-flow model. In particular, the physical aspects of the analytical expressions for the lift coefficient of the plate-plate airfoil are discussed, including Newton’s sine-squared law, Rayleigh’s lift formula, thin-airfoil theory and viscous-flow lift formula. The vortex-force theory is described to provide a solid foundation for consistent treatment of lift, form drag, Kutta condition, and downwash. The formation of the circulation and generation of lift are discussed based on numerical simulations of a viscous starting flow over an airfoil, and the evolution of the flow topology near the trailing edge is well correlated with the realization of the Kutta condition. The presented contents are valuable for the pedagogical purposes in aerodynamics and fluid mechanics.

An Explanation and Understanding of Aerodynamic Lift by Triple Deck Theory

An explanation of aerodynamic lift still is under controversial discussion as can be seen, for example, in a recent published article in Scientific American [1]. In contrast to an approach via integral conservation laws we here review an approach via the classical Kutta-Condition and its relation to boundary layer theory. Thereby we summarize known results for viscous correction to the lift coefficient for thin aerodynamic profiles and try to remember the work on triple-deck or higher order Boundary Layer theory, its connection to interactive boundary layer theory, viscous/inviscid coupling as implemented to well-known engineering code Xfoil. Finally we compare its findings to simple 2D numerical solution of full Navier Stokes equations (CFD)models. As a conclusion, a clearer definition of terms like understanding and explanation applied to the phenomenon of aerodynamic lift will be given.

Applying the constant strength doublet method to resolve a steady flow around an airfoil

In the paper hereby, a numerical (panel) method is applied to analyze steady two-dimensional flow of ideal gas around an airfoil. Initially, the airfoil is divided into a finite number of panels. Then the panels are replaced by doublets with constant strength. In addition, a wake panel is added to fulfill Kutta condition at the airfoil trailing edge. In order to implement this, a numerical realization is developed and built by means of Tiny C Compiler. To work out a solution to the linear non-homogeneous algebraic system, direct schemes for lower-upper factorization/decomposition of matrix of coefficients were applied, namely Crout, Doolittle, and Cholesky. The obtained results are validated against exact solution and shown for various values of angle of attack and Reynolds number.

On the Kutta Condition in Compressible Flow over Isolated Airfoils

This paper presents a novel and accurate method to implement the Kutta condition in solving subsonic (subcritical) inviscid isentropic compressible flow over isolated airfoils using the stream function equation. The proposed method relies on body-fitted grid generation and solving the stream function equation for compressible flows in computational domain using finite-difference method. An expression is derived for implementing the Kutta condition for the airfoils with both finite angles and cusped trailing edges. A comparison of the results obtained from the proposed numerical method and the results from experimental and other numerical methods reveals that they are in excellent agreement, which confirms the accuracy and correctness of the proposed method.

Viscous extension of potential-flow unsteady aerodynamics: the lift frequency response problem

The application of the Kutta condition to unsteady flows has been controversial over the years, with increased research activities over the 1970s and 1980s. This dissatisfaction with the Kutta condition has been recently rejuvenated with the increased interest in low-Reynolds-number, high-frequency bio-inspired flight. However, there is no convincing alternative to the Kutta condition, even though it is not mathematically derived. Realizing that the lift generation and vorticity production are essentially viscous processes, we provide a viscous extension of the classical theory of unsteady aerodynamics by relaxing the Kutta condition. We introduce a trailing-edge singularity term in the pressure distribution and determine its strength by using the triple-deck viscous boundary layer theory. Based on the extended theory, we develop (for the first time) a theoretical viscous (Reynolds-number-dependent) extension of the Theodorsen lift frequency response function. It is found that viscosity induces more phase lag to the Theodorsen function particularly at high frequencies and low Reynolds numbers. The obtained theoretical results are validated against numerical laminar simulations of Navier–Stokes equations over a sinusoidally pitching NACA 0012 at low Reynolds numbers and using Reynolds-averaged Navier–Stokes equations at relatively high Reynolds numbers. The physics behind the observed viscosity-induced lag is discussed in relation to wake viscous damping, circulation development and the Kutta condition. Also, the viscous contribution to the lift is shown to significantly decrease the virtual mass, particularly at high frequencies and Reynolds numbers.

Unsteady aerodynamics and vortex-sheet formation of a two-dimensional airfoil

Unsteady inviscid flow models of wings and airfoils have been developed to study the aerodynamics of natural and man-made flyers. Vortex methods have been extensively applied to reduce the dimensionality of these aerodynamic models, based on the proper estimation of the strength and distribution of the vortices in the wake. In such modelling approaches, one of the most fundamental questions is how the vortex sheets are generated and released from sharp edges. To determine the formation of the trailing-edge vortex sheet, the classical steady Kutta condition can be extended to unsteady situations by realizing that a flow cannot turn abruptly around a sharp edge. This condition can be readily applied to a flat plate or an airfoil with cusped trailing edge since the direction of the forming vortex sheet is known to be tangential to the trailing edge. However, for a finite-angle trailing edge, or in the case of flow separation away from a sharp corner, the direction of the forming vortex sheet is ambiguous. To remove any ad hoc implementation, the unsteady Kutta condition, the conservation of circulation as well as the conservation laws of mass and momentum are coupled to analytically solve for the angle, strength and relative velocity of the trailing-edge vortex sheet. The two-dimensional aerodynamic model together with the proposed vortex-sheet formation condition is verified by comparing flow structures and force calculations with experimental results for several airfoil motions in steady and unsteady background flows.