Flow Past Airfoils Research Articles

High-resolution finite volume computation of turbulent transonic flow past airfoils

High-resolution finite volume computation of turbulent transonic flow past airfoils

Inviscid Batchelor-model flow past an airfoil with a vortex trapped in a cavity

An efficient method of constructing inviscid Batchelor-model flows is developed. The method is based on an analytic continuation of the potential part of the flow into the closed-streamline vortex region. Numerical solutions are presented for Batchelor-model flows past airfoils with cavities. With the airfoil and dividing streamline shape, the eddy vorticity, and the jump in the Bernoulli constant across the eddy boundary given, the program calculates the corresponding cavity shape and the entire flow.

A boundary element approach to the 2D potential flow problem around airfoils with cusped trailing edge

The boundary integral method utilised in [1] for investigating the lifting flow past airfoils with non-vanishing trailing edge angle is adjusted herein (by imposing a different Kutta-Joukovsky condition) in order to study the incompressible inviscid flow past airfoils with cusped trailing edge.Using a linear interpolation for the normal derivative of the perturbation stream function, we reduce by discretization the integral representation of the perturbation stream function to an algebraic system. The solutions of this system are used in order to calculate the distribution of the tangential velocity and pressure coefficient over the airfoil.Comparison between calculated and analytical values of the pressure coefficient for a Joukovsky and a Von Mises profile show very good agreement.

Studies in multigrid acceleration of some equations of interest in fluid dynamics

The paper describes the multigrid acceleration technique to compute numerical solutions of three equations of common fluid mechanical interest; Laplace equation, transonic full potential equation and Reynolds averaged Navier-Stokes equations. Starting with the simple and illustrative multigrid studies on the Laplace equation, the paper discusses its application to the cases of full potential equation and the Navier-Stokes equations. The paper also discusses some elements of multigrid strategies like V- and W-cycles, their relative efficiencies, the effect of number of grid levels on the convergence rate and the large CPU time saving obtained from the multigrid acceleration. A few computed cases of transonic flows past airfoils using the full potential equations and the Navier-Stokes equations are presented. A comparison of these results with the experimental data shows good agreement of pressure distribution and skin friction. With the greatly accelerated multigrid convergence, the full potential code typically takes about 10 seconds and the Navier-Stokes code for turbulent flows takes about 5 to 15 min of CPU time on the Convex 3820 computer on a mesh which resolves the flow quantities to good levels of accuracy. This low CPU time demand, made possible due to multigrid acceleration, on one hand, and the robustness and accuracy on the other, offers these codes as designer’s tools for evaluating the characteristics of the airfoils.

Engineering methods of evaluating the characteristics of transonic flutter of control surfaces

Engineering methods of evaluating the characteristics of flutter of control surfaces with consideration of their interaction with shock waves are proposed on the basis of analyzing the results of theoretical and experimental investigations of transonic flow past airfoils. A comparison of the values of the rudder hinge moments obtained by calculation and in flight experiments showed their satisfactory agreement.

Perspective: Numerical Simulations of Wakes and Blade-Vortex Interaction

A method for simulating incompressible flows past airfoils and their wakes is described. Vorticity panels are used to represent the body, and vortex blobs (vortex points with their singularities removed) are used to represent the wake. The procedure can be applied to the simulation of completely attached flow past an oscillating airfoil. The rate at which vorticity is shed from the trailing edge of the airfoil into the wake is determined by simultaneously requiring the pressure along the upper and lower surface streamlines to approach the same value at the trailing edge and the circulation around both the airfoil and its wake to remain constant. The motion of the airfoil is discretized, and a vortex is shed from the trailing edge at each time step. The vortices are convected at the local velocity of fluid particles, a procedure that renders the pressure continuous in an inviscid fluid. When the vortices in the wake begin to separate they are split into more vortices, and when they begin to collect they are combined. The numerical simulation reveals that the wake, which is originally smooth, eventually coils, or wraps, around itself, primarily under the influence of the velocity it induces on itself, and forms regions of relatively concentrated vorticity. Although discrete vortices are used to represent the wake, the spatial density of the vortices is so high that the computed velocity profiles across a typical region of concentrated vorticity are quite smooth. Although the computed wake evolves in an entirely inviscid model of the flowfield, these profiles appear to have a viscous core. The computed spacing between the regions of concentrated vorticity in the wake and the circulations around them are in good agreement with the experimental results. As an application, a simulation of the interaction between vorticity in the oncoming stream and a stationary airfoil is also discussed.

Analysis of flexible-membrane and jet-flapped airfoils using velocity singularities

A method for predicting the flow past thin airfoils in incompressible potential flow is presented. This method makes use of special singularities that directly represent the complex conjugate perturbation velocity in the plane of the airfoil. The method is developed first for rigid cambered airfoils and is then extended to problems for which it is particularly suitable, namely, the flow past flexible impervious membranes and past airfoils with jet flaps.

Navier-Stokes calculations using Cartesian grids. I - Laminar flows

A finite-volume formulation for the Navier-Stokes equations using Cartesian grids is used to study flows past airfoils. In addition to the solution of the complete equations, solutions for two simplified versions of the governing equations were obtained and compared with those using body-fitted grids. Results are presented for two airfoil sections, NACA 0012 and RAE 2822, for a range of Mach numbers, angles of attack, and Reynolds numbers. It is shown that the results are highly dependent on the smoothness of the surface grid. Without such smoothness, the skin friction and pressure converge to nonuniform distributions. On the other hand, when surface cells with smoothly varying areas are used, the results compared favorably with calculations employing body-fitted grids.

Diagonal implicit multigrid algorithm for the Euler equations

A multigrid implementation of the Alternating Direction Implicit algorithm has been developed to solve the Euler equations of inviscid, compressible flow. The equations are approximated using a finite-volume spatial approximation with added dissipation provided by an adaptive blend of second and fourth differences. For computational efficiency, the equations are diagonalized by a local similariity transformation so that only a decoupled system of scalar pentadiagonal systems need be solved along each line. Results are computed for transonic flows past airfoils and include pressure distributions to verify the accuracy of the basic scheme and convergence histories to demonstrate the efficiency of the method.

Approximate solution of inverse shock-free transonic airfoil problem at zero incidence

The present work extends the approximate analytical solution of Niyogi [9] for shock-free transonic flow past thin symmetric airfoils at zero incidence, to the inverse problem for which the velocity distribution is prescribed and the airfoil shape is to be determined. The first step in the solution of the inverse problem leads to a singular integral equation of Betz's type solution of which has been improved iteratively using the transonic integral equation of Oswatitsch and satisfying the tangency boundary condition on the profile contour. Computational results have been presented for Nieuwland airfoils and a NACA 0012 airfoil, all of which indicate very good agreement with exact solutions.

Unsteady transonic flows past airfoils using the Euler equations

An explicit finite difference solver for the Euler equations was developed and used to calculate the flow past two of the AGARD standard aeroelastic configurations, an NACA 64A010 and an NLR 7301. The algorithm employed uses a modified four step Runge-Kutta time stepping scheme, boundary conditions determined from the time-dependent theory of characteristics and an unsteady automatic grid generation procedure. In general, the calculated results are in good agreement with available experiment. Moreover, they demonstrate the importance of using the Euler equations for super-critical airfoils and in situations where the transonic small disturbance theory results in poor agreement with experiment.

Application of a variational method for generating adaptive grids

The application of variational methods for generating adaptive grids is not as straightforward as one is led to believe. Proper scaling, suitable weight functions and appropriate clustering on boundaries must be employed to obtain a satisfactory grid. This work, which is based on the framework developed by Brackbill and Saltzman, provides simple methods for determining scaling and investigates possible options for selecting the weight function and clustering points on the boundaries. The concepts developed here are applied to two two-dimensional problems: a model problem based on Burger's equation which contains two length scales and, transonic flow past airfoils using the Euler equations.

Thin Airfoil Theory for Planar Inviscid Shear Flow

Steady, inviscid, planar and constant density shear flows past thin airfoils have strongly coupled thickness and camber flowfields which can be simply analyzed using some invariant properties of Euler’s equations. In this paper, both the nonlinear vorticity generation term in Poisson’s disturbance stream-function equation and the Bernoulli constant (which varies from streamline to streamline) are explicitly evaluated in terms of the known plane parallel flow far upstream, recognizing that vorticity convects unstretched and unchanged along streamlines; this produces a governing differential equation with an explicitly available right side plus a Bernoulli integral with a true constant fixed throughout the entire flow field. Using these results, the inverse problem, which solves for the shape that induces a prescribed pressure, is formulated as a nonlinear boundary value problem with mixed Dirichlet and Neumann surface conditions; here, trailing edge shape constraints are directly enforced by controlling jumps in the stream-function or its streamwise derivative through the downstream wake (analogies with potential flows past axisymmetric ringwings showing similar source and vortex interactions are also described). For flows with arbitrary profile curvature, we show that the linearized stream-function equation can be put in conservation form, thus leading to new and more general Cauchy-Riemann conditions which extend the notion of the velocity potential; the analysis problem, which determines the flow past a known shape subject to Kutta’s condition, is shown to satisfy a simple boundary value problem for a “super-potential” identical to the irrotational planar formulation except for an axisymmetric-like change to the Cartesian form of Laplace’s equation. For flows with uniformity vorticity, closed-form solutions are given which clearly illustrate the interaction between thickness and camber; also, for arbitrary thin airfoils, a particularly simple expression shows that the lift due to thickness varies directly as the product of vorticity and enclosed area. With more general shears, numerical approaches are especially convenient; these require only simple code changes to existing algorithms, for example, established potential function methods readily available for analysis problems, or “direct stream-function methods” for inverse problems recently developed by this author. Numerical methods and results that cross-check and extend derived analytical solutions are given for both analysis and design problems with vorticity. Also, extensions of the basic theory to three-dimensional constant density and planar compressible shear flows are given.

Transonic flow calculations using the Euler equations

An implicit finite difference method with implicit boundary conditions is employed to solve the steady Euler equations for flows past arbitrary geometries/The resulting code is used to investigate in a systematic way various aspects of flow past airfoils at transonic speeds such as method of solution, boundary conditions, grid stretching and generation, shock and sonic point operators, the Kiitta condition, and smoothing. Results ob- tained are in good agreement with results of other codes. Moreover, it appears that the method of solution employed is such that Kutta's condition need not be invoked. This statement appears to be valid for other existing schemes employed in the solution of Euler equations. HE primitive variable form of the system of partial- differential equations governing the steady motion of an inviscid fluid (Euler equations) are first order and of mixed elliptic and hyperbolic type. Because of this, two approaches are used in obtaining steady-state solutions for subcritical and supercritical flows past bodies. The first seeks temporally asymptotic solutions of the time-dependent equations1'7 while the second exploits relaxation methods developed for the solution of second-order equations by embedding Euler equations in a second-order system8 or by using a stream function formulation.9 The purpose of this investigation is to develop an efficient algorithm for the steady Euler equations in primitive variable form and use it to carry out numerical experiments designed to investigate in a systematic way the various aspects of flows past airfoils at transonic speeds. These include method of solution, i.e., Peaceman-Rachford10 (PR) vs Douglas-Gunn1 1 (DG); boundary conditions; grid generation and stretching; shock and sonic point operators; smoothing; and the Kutta condition. The resulting algorithm solves the strong con- servation form of the Euler equations, with the energy equation replaced by the statement that the total enthalpy is constant, using an ADI finite difference scheme and implicit boundary conditions. To study arbitrary geometries, the algorithm is combined with the automatic grid solver (GRAPE) of Steger and Sorenson.12 With this capability, the code provides a general and efficient method for the solution of flowfields past arbitrary geometries for all speed ranges. The achieved efficiency of this code can be traced to two main reasons: the resulting block tridiagonal matrix for two- dimensional flows is 3 x 3 and not 4 x 4; moreover, the code incorporates a procedure that selects an optimum relaxation factor for each iteration. Formulation of the Problem

Numerical Study of Separated Turbulent Flow over Airfoils

A numerical scheme for the solution of two-dimensional time-dependent Reynolds equations is developed and applied to the flow past airfoils. The airfoil is mapped conformally into a circular cylinder, and the equations are solved in the cylinder plane. The turbulence models used in this study are the mixing-length model and a hybrid model consisting of mixing-length model and the two-equation k-€ model. An explicit integral relation is used to determine the kinematics of the problem. After the method was calibrated using simpler geometries, it was applied to the case of a 12% thick Joukowski airfoil at 15 deg angle of attack, at a Reynolds number of 3.6 X10. It is observed from the results that the turbulent flow past an airfoil at high angles of attack shows oscillatory behavior (with the lift and drag coefficients continuously varying with time) similar to the behavior in laminar flows.

Computational Model for Low Speed Flows Past Airfoils with Spoilers

A computer model has been developed to simulate low-speed flow past an airfoil with a spoiler. An outer solution calculates the potential flow around an effective, closed-wake body. This body is formed by adding to the original airfoil and spoiler geometry: the boundary-layer displacement thickness; a closed wake behind the spoiler; and a trapped vortex at the spoiler hinge. An inner flow solution uses a turbulent jet mixing analysis and conservation of mass and momentum to simulate the time average flow within the wake. The final solution is obtained by iterative matching of the outer and inner solutions.

Supercritical flow past symmetrical airfoils

A numerical method is developed for computing steady supercritical flow about an ellipse at zero angle of attack. The flow is assumed to be two-dimensional, inviscid, isentropic and irrotational. The free-stream Mach number lies in the high subsonic range so that a shock wave occurs locally near the body. The full potential equations are solved by Telenin's method and the ‘method of lines’. Smooth interpolating functions are assumed for the unknown flow variables in selected co-ordinate directions. The resulting set of ordinary differential equations is then integrated away from or along the body depending upon whether the flow is smooth or discontinuous. Jump conditions of the governing equations are applied across the shock wave so that it is perfectly sharp. A doublet solution for flow past a closed body is used as the farfield boundary condition. Supercritical flow calculations have been performed for ellipses with thickness ratio of 0·2 and 0·4 at various free-stream Mach numbers. The present results are compared with the shock-capturing method, and good agreement is obtained.

Unsteady compressible flow - A computational method consistent with the physical phenomena

A methodology for the numerical prediction of 2-D and 3-D unsteady, in viscid, compressible flows is presented. The primitive Euler equations are recast in terms of compatibility equations on characteristic surfaces. In such a way the evolution in time of the flow properties is described explicitly as the interaction of signals corresponding to the physical wave-propagation phenomenon. The equations are discretized through a finite-difference method where the proper domain of dependence of each computed point is preserved, by approximating the space-derivatives with one-sided differences, according to the velocity of propagation of signals along bicharacteristics. Results of the application of the proposed method to subsonic and transonic flow past airfoils are shown and compared with different methods.

A quasisteady theory for incompressible flow past airfoils with oscillating jet flaps

Quasisteady concepts are used for an approximate analysis of incompressibl e flow past airfoils with harmonically oscillating jet flaps. The instantaneous flowfield is considered as a sequence in the streamwise direction of steady flows with a properly enforced tangency condition between the jet and the external flow. The jet kinematics are found from experimentally determined jet decay characteristics, and the time-frozen jet is modeled by using Spence's steady jet flap analysis. Computed lift response shows substantial agreement with available measurements.

Integral Equation Method for Subsonic Flow Past Airfoils in Ventilated Wind Tunnels

The method of contour distribution of sources and vortices, used to calculate potential flows about twodimensional airfoils in free air, is extended to flows around airfoils under the constraint of ventilated wind tunnel walls. Based on the concept of fundamental solutions, the source and vortex singularities are modified to satisfy the boundary conditions at the walls. The application of the airfoil boundary condition reduces the problem to a Fredholm integral equation of the second kind for the source density function. The numerical solutions obtained by a finite element technique are in good agreement with experimental results. I. Introduction A LARGE part of established wall interference theory is based on the simplifying assumption that the wall effects can be evaluated from an estimated far flowfield of the model, considering only the wind tunnel wall boundary condition. The result is interpreted in terms of modifications to the freestream. If the model is small enough, these can be expressed as corrections to the direction and speed of the stream, constant over the model (the angle of attack and Mach number corrections). If the model has appreciable length, a more refined interpretation, involving the streamline curvature and buoyancy (pressure gradient) effects, is required. The model boundary condition enters the problem only in the final stage, if there is a need to evaluate the wall effects in terms of model quantities, such as the surface pressure distribution, aerodynamic forces, moments, etc. A comprehensive account of this approach, its applicability and limitations is given in Ref. 1. It is perhaps worth emphasizing that the above concept