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NASA NTRS1978

A procedure for analyzing transonic flow over harmonically oscillating airfoils

Finite difference procedures were successfully used to solve the steady transonic flow about airfoils and appear to provide a practical means for calculating the corresponding unsteady flow. The purpose of the paper is to describe a finite difference procedure derived from the equations for the potential flow by assuming small perturbations and harmonic motion. The velocity potential is divided into steady and unsteady parts, and the resulting unsteady equation is linearized on the basis of small amplitudes of oscillation. The steady velocity potential, which must be calculated first, is described by the classical nonlinear transonic differential equation.

Weatherill, W. H.↗

NASA NTRS1982

Investigation of rotational transonic flows through ducts using a finite element scheme

An application of the finite element method is presented in order to study the flow through a two-dimensional channel including the choked and nearly-choked flow conditions. The mathematical procedure provides a combined treatment of potential and Euler equations, where the steady Euler equations are solved through the integration of a pseudo-time system which is equivalent to a relaxation scheme; the isentropic, potential flow equations are embedded in this formulation. To analyze the transonic flow through a channel with near-choked flow conditions, the flow through a parallel channel with a 10 percent circular arc bump is calculated as a sample problem.

Akay, H. U.↗

NASA NTRS1982

Finite element formulation of transonic flow problems

Reference is made to the study by Akay and Ecer (1982), which treated the solution of full Euler equations for transonic, rotational, inviscid flows. Attention is given here to some of the important features of a general finite element formulation for transonic flows. Both rotational and irrotational cases are treated. Transonic flow through a parallel channel with a 4.2 percent thick circular bump is analyzed for an upstream Mach number of 0.85. A figure is included showing the computational grid of 44 x 8 elements. In this case, the distance between the walls of the channel is 2.073 times the chord length of the bump. The pressure distributions over the bump for rotational assumptions are presented.

Akay, H. U.↗

NASA NTRS1982

Computations of transonic flow over an oscillating airfoil with shock-induced separation

A numerical study is presented on unsteady transonic flow over an airfoil in pitching oscillation at a moderately high incidence. The study was focused on the effects of unsteady shock-wave/boundary-layer interactions and subsequent flow separations. An implicit finite-difference scheme was used to solve the full time-dependent compressible Navier-Stokes equations in conservation-law form. The algorithm adopted for this study is the Beam-Warming spatially factored scheme. It is, however, constructed in a body-fitted curvilinear coordinate system for efficient unsteady flow computations. The computational results are compared with experimental data measured in the Ames 11- by 11-Foot Transonic Wind Tunnel.

Chyu, W. J.↗

NASA NTRS1973

Numerical calculation of the three dimensional transonic flow over a yawed wing.

Results are presented of calculations of the three dimensional steady transonic flow over a finite yawed wing. The full potential flow equation is solved in a transformed coordinate system which permits the boundary conditions to be satisfied exactly. The correct differential properties are enforced by rotating the difference scheme to conform with the flow direction, and fast convergence is assured by simulating a time dependent equation designed to settle quickly to a steady state. Computed lift drag ratios are consistent with the results of wind tunnel tests of a yawed wing conducted by R. T. Jones (1972).-

Jameson, A.↗

NASA NTRS1977

Computation of unsteady transonic flows by the indicial method

The indicial method is investigated for the computation of unsteady transonic force and moment coefficients for use in flutter analyses. This approach has the advantage that solutions for all reduced frequencies for a given mode of motion can be obtained from a single finite-difference flowfield computation. Comparisons of indicial and time-integration computations for oscillating airfoil and flap motions help define limits on the motion amplitude for the applicability of the indicial method to transonic flows. Within these limits, solutions for various motion modes can be superposed to obtain solutions for multiple-degree-of-freedom aeroelastic systems. Also, a simple aeroelastic problem is solved by an alternative approach in which the structural motion and flowfield equations are integrated simultaneously using a time-integration finite-difference procedure.

Ballhaus, W. F.↗

NASA NTRS1979

Unsteady airloads in supercritical transonic flows

Results obtained from a simplified theory of unsteady perturbations of supercritical two-dimensional transonic flows, introduced in an earlier paper, are presented. Unsteady loads generated by an oscillating flap and by airfoils oscillating as a whole are given with comparisons to experimental results and finite difference solutions. The theory, which was originally formulated for symmetric airfoils at zero mean angle of attack, is extended to treat asymmetric mean flows.

Williams, M. H.↗

NASA NTRS1980

Analysis of transonic flow about harmonically oscillating airfoils and wings

A finite difference method for analyzing the unsteady transonic flow about harmonically oscillating wings is discussed. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting partial differential equations for small disturbances. Initial solutions were obtained using relaxation procedures, but the solution range proved to be limited in terms of Mach number and reduced frequency. Recent two-dimensional results are presented which have been obtained with direct solution procedures in which the difference equations are solved 'all at once' and these provide reasonable correlation for practical values of Mach number and reduced frequency.

Weatherill, W. H.↗

NASA NTRS1980

Unsteady effects of a control surface in two dimensional subsonic and transonic flow

The experimental results of steady and unsteady pressure measurements, carried out in subsonic and transonic flow on a 16 percent relative thickness supercritical aerofoil, equipped with a trailing edge flap involving 25 percent of the chord, in a sinusoidal motion are given. These experimental results are compared with those obtained by various methods of steady and unsteady inviscid flow calculations. Some calculation results in which viscous effects have been taken into account, for both steady and unsteady flows, are also presented.

Grenon, R.↗

NASA NTRS1981

Aerodynamic interaction of heat transfer with steady transonic flow

An experimental study investigated the interactive effect between the heat transfer and the steady transonic flow in a simple convergent-divergent duct. The results show that the position of the shock is considerably affected by the effect of wall cooling. The physical explanation for the shift is also offered. The results may have implications in the other internal flow problems.

Kurosaka, M.↗

NASA NTRS1981

Test problems for inviscid transonic flow

The paper discusses some results obtained in the solving of test problems for inviscid transonic flow with shock waves using the TRANDES program. The method used employs the full inviscid perturbation-potential flow equation in a Cartesian grid system that is stretched to infinity. The equation is represented by a nonconservative system of finite-difference equations that includes at supersonic points a rotated difference scheme and is solved by column relaxation. Except for the NACA 0012 case with the freestream Mach number equal to 0.95, all test problems were solved straightforwardly and appeared to be converged or close to convergence. The only difficulty was some sensitivity to grid placement, which is typical of the Cartesian formulation

Carlson, L. A.↗

NASA NTRS1958

Thin airfoil theory based on approximate solution of the transonic flow equation

A method is presented for the approximate solution of the nonlinear equations of transonic flow theory. Solutions are found for two-dimensional flows at a Mach number of 1 and for purely subsonic and purely supersonic flows. Results are obtained in closed analytic form for a large and significant class of nonlifting airfoils. At a Mach number of 1 general expressions are given for the pressure distribution on an airfoil of specified geometry and for the shape of an airfoil having a prescribed pressure distribution. Extensive comparisons are made with available data, particularly for a Mach number of 1, and with existing solutions.

Spreiter, John R↗

NASA NTRS1989

An interactive three-dimensional boundary-layer method for transonic flow over swept wings

A three-dimensional laminar/turbulent boundary-layer method is developed for transonic flow over swept wings. The governing equations and curvature terms are written for a nonorthogonal body-oriented curvilinear coordinate system. The viscous method is coupled to the full-potential inviscid code, FLO-30, through a displacement-surface interaction. Typically, for transonic Mach numbers and moderate sweep angles, between 5 and 12 viscous updates are required for convergence. The method is applied to a variety of wing planforms, and the results are also compared to those obtained with the three-dimensional integral inviscid-viscous interaction method TAWFIVE (Streett and Melson, 1983).

Woodson, Shawn H.↗

NASA NTRS1979

Linearization of unsteady transonic flows containing shocks

The problem of determining unsteady airloads on a thin, three-dimensional, planar wing oscillating with infinitesimal amplitude in a transonic flow is considered. The flow is assumed to be governed by the transonic small disturbance equation. The unsteady disturbance is taken to be a small perturbation superposed on a given steady mean flowfield. The equations governing the unsteady field, allowing for induced oscillations of any embedded shocks, are obtained. The linearization is shown to fail, locally, at the intersection of a shock with the wing surface, although the failure has little influence on the sectional characteristics of the wing.

Williams, M. H.↗

NASA NTRS1989

Calculating Transonic Flows About Airfoils

Small disturbances and aeroelastic effects simulated. XTRAN2L computer program used to calculate two-dimensional transonic flows about airfoils. Uses time-accurate, alternating-direction, implicit (ADI) finite-difference scheme to solve two-dimensional complete unsteady transonic small-disturbance equation. Engquist-Osher monotone spatial differencing used in ADI solution algorithm to provide robust and efficient program. Written in FORTRAN V.

Seidel, David A.↗

NASA NTRS1973

Unsteady transonic flow analysis for low aspect ratio, pointed wings.

Oswatitsch and Keune's parabolic method for steady transonic flow is applied and extended to thin slender wings oscillating in the sonic flow field. The parabolic constant for the wing was determined from the equivalent body of revolution. Laplace transform methods were used to derive the asymptotic equations for pressure coefficient, and the Adams-Sears iterative procedure was employed to solve the equations. A computer program was developed to find the pressure distributions, generalized force coefficients, and stability derivatives for delta, convex, and concave wing planforms.

Kimble, K. R.↗

NASA NTRS1977

Efficient solution of unsteady transonic flows about airfoils

An implicit finite difference procedure was developed for the efficient solution of unsteady transonic flow fields. Sample computations illustrate applications of procedures to aerodynamic problems. Solutions are presented that illustrate three types of shock wave motion that can result from airfoil control surface oscillations. The significant effect of wind tunnel wall conditions on these shock wave motions is demonstrated. Solutions are also presented for a simple aeroelastic problem in which the flow field equations and the structural motion equations are integrated simultaneously in time. Both stable and unstable aeroelastic interactions are considered. The procedure is adapted to compute unsteady aerodynamic force coefficients by the indicial method.

Ballhaus, W. F.↗

NASA NTRS1976

Analysis of transonic flow over lifting and oscillating airfoils

Finite element procedures are presented for the numerical solution of transonic flow over lifting and oscillating airfoils based on small disturbance theory. The algorithm is simple and effective in handling the mixed flow problem and, for airfoil in free air, treats the entire flow domain by making use of the far field asymptotic solution. Numerical results are presented and compared with experimental data and those obtained by finite difference relaxation techniques.

Chan, S. T. K.↗