Lift Formula Research Articles

Coupling analysis of different variables in airfoil lift formula

In the design process of an aircraft, there is an important criterion to consider when designing a wing: how much lift the wing can provide. The magnitude of the lift directly determines the stability of the aircraft during take-off and whether it can take off smoothly, so it has received extensive attention from scholars at home and abroad. This paper uses the Kutta-Joukowski theorem to quantify the factors that affect wing lift and enable the calculation of lift, and engineers need to analyze how these factors affect wing lift when considering this criterion. So how can engineers analyze the effect of these factors on lift, especially in multivariate situations? Need to calculate complex multivariate equations? are issues worth exploring. In this paper, a coupled analysis method is proposed to analyze the wing lift of an aircraft affected by multiple variables, which will greatly simplify the original analysis process and provide convenience for the calculation of wing lift.

Determination of the Circulation for a Large-Scale Wind Turbine Blade Using Computational Fluid Dynamics

The determination of the circulation for wind turbine blades is an important problem in engineering. In the present study, we develop a specific approach to evaluate the integral that represents mathematically the circulation. First the potentialities of the method are assessed using a two-dimensional NACA64_A17 airfoil as a testbed and evaluating the influence of different integration paths and angles of attack on the circulation value. Then the method is applied to blades with different relative heights in order to provide useful reference data to be used for the optimization and reverse design of wind turbine blades. As shown by the results, the integral value changes with the integral path, and an “optimal circle radius” exists. We calibrate the integral value by comparing its value with the lift formula. In this was we succeed in showing that there is a certain error when the radius is too small. However, the error can increase rapidly when the radius is too large. When the radius of the circle is 1–6 times the chord length, the error of all integral values is less than 5%. The optimal radius varies with the angle of attack.

A simple vortex-loop-based model for unsteady rotating wings

An analytical model is developed for the lift force produced by unsteady rotating wings; this configuration is a simple representation of a flapping wing. Modelling this is important for the aerodynamic and control-system design for bio-inspired drones. Such efforts have often been limited to being two-dimensional, semi-empirical, sometimes computationally expensive, or quasi-steady. The current model is unsteady and three-dimensional, yet simple to implement, requiring knowledge of only the wing kinematics and geometry. Rotating wings produce a vortex loop consisting of the root vortex, leading-edge vortex, tip vortex and trailing-edge vortex, which grows with time. This is modelled as a tilted planar loop, geometrically specified by the wing size, orientation and motion. By equating the angular impulse of the vortex loop to that of the fluid volume driven by the wing, the circulatory lift force is derived. Potential flow theory gives the fluid-inertial lift. Adding these two contributions yields the total lift formula. The model shows good agreement with a range of experimental and computational cases. Also, a steady-state lift model is developed that compares well with previous work for various angles of attack.

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

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 $L=-{\it\rho}_{0}U{\it\Gamma}_{{\it\phi}}$, originally derived by Joukowski in 1906 for inviscid potential flow, and the drag formula $D={\it\rho}_{0}UQ_{{\it\psi}}$, 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, ${\it\Gamma}_{{\it\phi}}$ and $Q_{{\it\psi}}$ 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 ${\it\Gamma}_{{\it\phi}}$ and $Q_{{\it\psi}}$, 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 $Re=6.5\times 10^{6}$ and free Mach number $M\in [0.1,2.0]$. 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.

Unsteady Thin-Airfoil Theory Revisited: Application of a Simple Lift Formula

The physical foundations of unsteady thin-airfoil theory are explored in the general framework of viscous flows. The thin-airfoil lift formula is derived by using the simple lift formula that contains the vortex lift and the lift associated with the fluid acceleration. From a broader perspective, the thin-airfoil lift formula could be applicable even when the flow around an airfoil is moderately separated, from which the classical von Kármán–Sears lift formula can be recovered as a reduced case. The quantitative relationship between boundary layer and lift generation is discussed. Direct numerical simulations of low-Reynolds-number flows over a flapping flat-plate airfoil are conducted to examine the accuracy and limitations of the thin-airfoil lift formula.

Evaluation of Lift Formulas Applied to Low-Reynolds-Number Unsteady Flows

Different lift decompositions into the elemental terms are compared based on direct numerical simulations of a flapping flat plate and a flapping rectangular wing at low-Reynolds-number flows. The simple lift formula is given as a useful approximate form that has the vortex force and local acceleration terms. The accuracy of the simple lift formula in lift estimation is quantitatively evaluated in comparison with the general force formulas based on the fully resolved two- and three-dimensional unsteady velocity fields and the planar velocity fields at several spanwise locations in simulated particle-image-velocimetry measurements. In addition, the mathematical connections between the different force formulas are discussed.

A lift formula applied to low-Reynolds-number unsteady flows

A lift formula for a wing in a rectangular control volume is given in a very simple and physically lucid form, providing a rational foundation for calculation of the lift of a flapping wing in highly unsteady and separated flows at low Reynolds numbers. Direct numerical simulations on the stationary and flapping two-dimensional flat plate and rectangular flat-plate wing are conducted to assess the accuracy of the lift formula along with the classical Kutta-Joukowski theorem. In particular, the Lamb vector integral for the vortex force and the acceleration term of fluid for the unsteady inertial effect are evaluated as the main contributions to the unsteady lift generation of a flapping wing.

Helicopter Toy and Lift Estimation

A$1 plastic helicopter toy (called a Wacky Whirler) can be used to demonstrate lift. Students can make basic measurements of the toy, use reasonable assumptions and, with the lift formula, estimate the lift, and verify that it is sufficient to overcome the toy's weight.

Lift forces on a cylindrical particle in plane Poiseuille flow of shear thinning fluids

Lift forces on a cylindrical particle in plane Poiseuille flow of shear thinning fluids are investigated by direct numerical simulation. Previous works on this topic for Newtonian fluids show that the two-dimensional channel can be divided into alternating regions defined by the stability of the particle’s equilibrium. We observe stability regions with the same pattern in flows of shear thinning fluids and study the effects of shear thinning properties on the distribution of the stability regions. Joseph and Ocando [J. Fluid Mech. 454, 263 (2002)] analyzed the role of the slip velocity Us=Uf−Up and the angular slip velocity Ωs=Ωp−Ωf on migration and lift in plane Poiseuille flow of Newtonian fluids. They concluded that the discrepancy Ωs−Ωse, where Ωse is the angular slip velocity at equilibrium, changes sign across the equilibrium position. In this paper we verify that this conclusion holds in shear thinning fluids. Correlations for lift forces may be constructed by analogy with the classical lift formula L=CUΓ of aerodynamics and the proper analogs of U and Γ in the present context are Us and Ωs−Ωse. Using dimensionless parameters, the correlation is a power law near the wall and a linear relation (which can be taken as a power law with the power of one) near the centerline. The correlations are compared to analytical expressions for lift forces in the literature and we believe that the correlations capture the essence of the mechanism of the lift force. Our correlations for lift forces can be made completely explicit provided that the correlations relating Us and Ωs to prescribed parameters are obtained.

On the Application of Linearized Theory to Multi-Element Aerofoils Part I: Tandem Flat Plate Aerofoils

SummaryA linearized two-dimensional incompressible potential flow theory for two-element uncambered tandem aerofoil sections is developed. It leads to formulas for lift and moment which can be calculated rapidly on a programmable hand calculator, and which reduce, when the two aerofoil elements come together, to the familiar thin-aerofoil formulas for an aerofoil with a simple flap. The theory is shown to give lift and moment predictions which are in good agreement with predictions of numerical potential flow theory.

Simplified formulas for lift and moment in unsteady thin airfoil theory

Simplified formulas for lift and moment in unsteady thin airfoil theory

On the Unsteady Supersonic Cascade With a Subsonic Leading Edge—An Exact First Order Theory—Part 2

Pursuant to Part 1, the analysis of the aerodynamic forces acting on slowly oscillating airfoils in a supersonic cascade with a subsonic leading edge is presented. First the flow field between adjacent airfoils is determined. In the limit of sonic leading edge, the present results for the velocity potential agree with the sonic limit of Lane’s supersonic leading edge analysis. The requirement of the continuity of pressure in the “train” leads to functional equations for the train velocity; their solutions, obtained in closed form, are found to involve arbitrary constants which are related to the back pressure. The effect of the back pressure on the “train” is discussed in detail. For a cascade with zero pressure rise across it, the train velocity is determined completely and the formulas for lift and moment, accurate to the first order of a frequency parameter, are obtained in closed form. Stability criteria for a single-degree-of-freedom motion are examined. A pure bending motion is found to be stable, but a pure torsional motion becomes unstable under certain circumstances. These results are consistent with analogous oscillations of an isolated airfoil. However, the stability boundary for a typical cascade differs significantly from the case of the isolated airfoil, being strongly influenced by such cascade parameters as solidity, blade-to-blade phase difference, and stagger angle.