Transonic Potential Flow Research Articles

Predicting droplet impingement on yawed wings

^ameson, A., and Caughey, D. A., Calculation of the Transonic Flow Past a Swept Wing, New York Univ. ERDA Rept. COO-3077-140, New York, June 1977. Yadlin, Y., and Caughey, D. A., Diagonal Implicit Multigrid Solution of the Three-Dimensional Euler Equations, Proceedings of the Eleventh International Conference on Numerical Methods for Fluid Dynamics, Vol. 323, Lecture Notes in Physics, Springer-Verlag, Berlin, June 1988, pp. 597-601. Jameson, A., Iterative Solution of Transonic Flows over Airfoils and Wings, Including Flows at Mach 1, Communications on Pure and Applied Mathematics, Vol. 27, 1974, pp. 283-309. Wang, L., and Caughey, D. A., Calculation of the Transonic Potential Flow Past a Swept Wing—An Improved Version of Program Flo-22, Cornell Univ., Sibley School of Mechanical and Aerospace Engineering, Fluid Dynamics and Aerodynamics Rept. FDA 88-17, Ithaca, NY, Aug. 1988. Steckel, D. K., Dahlin, J. A., and Henne, P. A., Results of Design Studies and Wind Tunnel Tests of High Aspect Ratio Supercritical Wings for an Energy Efficient Transport, NASA CR159332, 1980. Numerical Procedure

Improved calculation of transonic potential flow past swept wings

Improved calculation of transonic potential flow past swept wings

Approximate factorization scheme for transonic potential flow computations

The implicit approximate factorization scheme known asaf2 is investigated here for the purpose of application to the solution of two-and three-dimensional transonic full potential equations in conservative form. The artificial viscosity used by different authors has been deduced, and is discussed in detail. A second-order correction to the implicit artificial viscosity is tested for transonic flow past a Korn aerofoil at both design and off-design conditions. The inviscid transonic flow past different aerofoils, wings and wing-body configurations has been computed using theaf2 scheme and the solutions are compared with experimental and other numerical results. It is shown that theaf2 scheme is fast, and is not sensitive to grid stretching.

Boundary Singularities in Steady Potential Compressible Flow Through Plane Two-Dimensional Channels

Boundary singularity solutions are reviewed, which relate to the steady, transonic potential flow of a perfect gas, through plane, two-dimensional convergent channels. Two singularities occur in the physical plane, one at upstream infinity and the other at the concave corner in channels that have a parallel entry section followed by a convergent section that ends at the throat plane. Solutions of these singularities are obtained from the general solution of Chaplygin's stream function equation in the hodograph system of coordinates. Uniform, uni-directional subsonic entry flow, in a finite-length channel, gives rise to a boundary singularity in the hodograph plane. The stream function is multi-valued at this singularity, where it varies between one and zero, while the flow direction is zero. This condition is removed by introducing circular polar coordinates (r, α) with origin at the singular point. For a sufficiently small value of r, a second-order partial differential equation of the stream function is obtained in the new coordinates. A particular solution of this equation is given, when r is small, rs say, in which the stream function gradient in the radial direction is taken as zero for all values of α at rs. The full series solution of the new equation is given in a paper which is to follow. Finally, consideration is given to the problem of matching the subsonic and supersonic flow regions at the sonic surface. A new system of equations is given, which determine the stream function gradient at sonic points from values of the stream function in the expansion region. Computed results are given that relate to the steady, choked flow of a perfect gas through a 15° convergent channel. The overall solution is shown to converge.

Extension of Transonic Flow Computational Concepts in the Analysis of Cavitated Bearings

In this paper, an analogy between the mathematical modeling of transonic potential flow and the flow in a cavitating bearing is described. Based on the similarities, characteristics of the cavitated region and jump conditions across the film reformation and rupture fronts are developed using the method of weak solutions. The mathematical analogy is extended by utilizing a few computational concepts of transonic flow to numerically model the cavitating bearing. Methods of shock fitting and shock capturing are discussed. Various procedures used in transonic flow computations are adapted to bearing cavitation applications, e.g., type differencing, grid transformation, an approximate factorization technique, and Newton’s iteration method. These concepts have proved to be successful and have vastly improved the efficiency of numerical modeling of cavitated bearings.

Local Entropy Conditions in Transonic Potential Flow Problems

AbstractThe transonic potential flow problem is handled as a variational problem over a closed convex set which is given by a bound for the gas velocity and by a local entropy condition. It can be shown that the minimum problem has a solution though the functional need not be convex and the given set is not compact. Furthermore, the convergence of an approximation method (KATCHANOV'S method) for the solution to the corresponding variational inequality is proved.

Computation of transonic potential flow past wing-body configurations

A simple approach has been developed to use the two dimensional grid generation method by solving elliptic partial differential equations to generate the three dimensional grid for wingfuselage configurations. This simple method can be applied to generate grids for arbitrary fuselage fitted with any swept wing with dihedral. Three dimensional transonic analysis code TWING, with approximate factorization (AF2) scheme has been suitably used as a flow solver. As an example, RAE-WING-A with body-B2 configuration has been considered. The results obtained have been compared with available numerical and experimental results. It has been observed in the present computations that AF2 scheme is not sensitive to grid stretching.

Speed-up of the strongly implicit procedure with application to subsonic/transonic potential flows

A modified version of the Strongly Implicit Procedure (SIP) for solving partial differential equations is presented. The capabilities of the modified scheme are demonstrated and analyzed for the case of two-dimensional subsonic or transonic potential flow over airfoils. The improved technique accelerates the convergence rate of the SIP considerably

Physically consistent models for artificial dissipation in transonic potential flow computations

The effects that artificial dissipation has on numerical solutions of the transonic full potential equation (FPE) are investigated by comparing the artificially dissipative FPE to a physically dissipative potential (PDP) equation. Analytic expressions were derived for the variables C and M c that are used in the artificial density formulation. It was shown that these new values generate artificial dissipation, which is equivalent to the physical dissipation which exists in the PDP equation. The new expressions for the variables C and M c can easily be incorporated into the existing full potential codes, which are based either on the artificial density or on the artificial viscosity formulation. Comparisons of physically dissipative potential (PDP) with the artificial density or viscosity (ADV), artificial mass flux (AMF), and ADV with variable C and M c formulation (MCC) illustrate the fact that the ADV dissipation creates excessively large amounts of equivalent entropy. As an alternative, AMF and MCC physically consistent formulations offer considerably lower equivalent entropy jumps across shocks.

Finite elements for transonic potential flows

AbstractA finite element method is used for the computation of entropy solutions to the transonic full potential equation. Physically correct solutions with sharp and correctly placed shocks were obtained. (AMS (MOS) 1980 Mathematics subject classifications: 65N30, 76N15, 35M05, 76H05, 49D10, 35A40, 35L67.)

On the classification of discontinuities by the pattern recognition methods

Abstract It is proposed to use the deterministic discriminant methods of the pattern recognition theory for a solution of the problem of classifying the discontinuities into several types (shock waves, contact discontinuities, etc.) in the numerical solutions of two-dimensional gas dynamic problems obtained by finite-difference shock-capturing schemes. The formulae for the computation of the features are derived. Three different schemes for the classification of singularities in gas flows are presented. The efficiency of these schemes is illustrated by the example of a number of applied two-dimensional problems, including transonic potential flow around an airfoil, the diffraction of a strong shock wave on a wedge and supersonic flow in a wind tunnel with a lower-wall step.

A convergent finite element formulation for transonic flow

A finite element formulation for the full potential equation in the case of two-dimensional transonic flow is presented. The formulation is based on an optimal control approach developed by Glowinski and Pironneau. The solution of the full potential equation is obtained by a minimization problem. Using a new compactness result it is possible to prove convergence for the solutions of the minimization problem. The a priori assumption of existence and uniqueness of a weak solution of the full potential equation satisfying an entropy condition implies that the limit function must be the solution. It is possible to extend the convergence result to the case of three-dimensional transonic potential flow.

Effects of modal symmetry on transonic aeroelastic characteristics of wing-body configurations

To study accurately the transonic aeroelastic characteristics, it is important to model the full aircraft configuration, including asymmetry. Recently, an accurate method of computing unsteady transonic flows on full-span wing-body configurations was developed using the transonic potential flow theory. In this work, the method is further developed to account for the aeroelasticity of full-span wing-body configurations. This is accomplished by simultaneously integrating the unsteady aerodynamic forces and modal structural equations of the wing-body configurations. To validate the method, aeroelastic computations are made for a wing-body configuration with a rectangular wing. The aeroelastic responses of this configuration are correlated with the responses of a similar isolated wing. The comparisons are favorable. Aeroelastic computations associated with symmetric and antisymmetric modes are also made to study the influence of modal asymmetry on responses. This new development is further illustrated by computing the aeroelastic characteristics of a typical fighter aircraft. The results from this study will be useful in accurately computing the transonic flutter boundaries of aircraft, including those associated with asymmetric modes.

On some properties of solutions of transonic potential flow problems

The paper deals with solutions of transonic potential flow problems handled in the weak form or as variational inequalities. Using suitable generalized methods, which are well known for elliptic partial differential equations of the second order, some properties of these solutions are derived. A maximum principle, a comparison principle and some conclusions from both ones can be established.

A method of computing transonic potential flows in turbine cascades

A fast method is devised for computing stationary subsonic, transonic, and supersonic potential flows in flat cascades of turbines and on turbines and on cascades located on a surface of revolution in a layer of variable thickness. The method realizes a modification of the approximate factorization method, whereby second order of accuracy is preserved in the supersonic zones. It is based on numerical integration of the full equation for the velocity potential, written in divergence form for any curvilinear system of coordinates. The present version is shown to have advantages over the popular versions which use artificial compressibility in supersonic zones.

Remarks on the solvability of transonic flow problems

The paper is devoted to the study of the viscosity method and upwinding in the transonic flow problems. We weaken the conditions from [6,7] under which we can obtain a nonviscous compressible transonic flow as a limit of viscous solutions, if the viscosity and heat conductivity tend to zero. Further, we give the justification of the density upwinding applied often in the calculation of transonic potential flows.

The optimisation of approximate-factorisation schemes for solving elliptic partial differential equations in three dimensions, featuring a new two-factor scheme

Current three-dimensional approximate-factorisation schemes do not appear capable of the same degree of optimisation as their two-dimensional counterparts. This is due principally to the greater complexity engendered by the third factor that arises in going from two to three dimensions. A new two-factor scheme is proposed which may be optimised with much greater ease and gives reliable and rapid convergence for all problems so far attempted. Applications include numerical grid generation and the calculation of transonic potential flow.

Two complementary approaches to transonic potential flow about oscillating airfoils

Results of two-dimensional (2D) unsteady transonic full-potential flow calculations for AGARD standard aeroelastic configurations are presented and compared with experimental data. Two complementary methods have been applied: 1) the time-integration TULIPS method, and 2) the time-linearized FTRANC method. The calculations have been performed for the NACA 64A010 (Ames) airfoil and for the supercritical NLR 7301 airfoil, both undergoing sinusoidal pitching oscillations. The results of both methods are in good agreement, and the comparison with experimental data is fair.

Transonic potential flow calculations by two artificial density methods

A steady transonic flow over an airfoil is computed by solving the full potential equation discretized over a contour-fitting grid in full conservation form. Near shock waves, density is corrected to account for the rise in entropy in the region. A multigrid alternating direction method is used to drive the iteration. In supersonic flow regions, artificial dissipation is introduced by retarding the density in the upstream direction. In a new formulation for this artificial density, the amount by which the density is biased is made proportional to the local flux gradients. This new formulation is compared with the standard formulation that is based on density gradients. The new scheme, as the old one, is second-order accurate throughout the flow field except for a small region near shock waves. Results of calculations with the two schemes show that, when the shock itself is weak, the scheme based on flux gradients gives a sharper resolution of the shock than the scheme based on density gradients. As shock strength increases, the shock-capturing abilities of the two schemes become equal. Throughout the range of cases tested, the two schemes exhibited comparable speed and robustness.

Improved finite-difference schemes for transonic potential flow calculations

Engquist and Osher (1980) have introduced a finite difference scheme for solving the transonic small disturbance equation, taking into account cases in which only compression shocks are admitted. Osher et al. (1983) studied a class of schemes for the full potential equation. It is proved that these schemes satisfy a new discrete 'entropy inequality' which rules out expansion shocks. However, the conducted analysis is restricted to steady two-dimensional flows. The present investigation is concerned with the adoption of a heuristic approach. The full potential equation in conservation form is solved with the aid of a modified artificial density method, based on flux biasing. It is shown that, with the current scheme, expansion shocks are not possible.