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NASA NTRS1986

Finite-volume and integral-equation techniques for transonic and supersonic vortex-dominated flows

Two computational techniques are developed to calculate the compressible vortex-dominated flows. The first technique is a finite-volume Euler Solver which uses four-Stage Runge-Kutta time stepping with second- and fourth-order dissipation terms. The technique is applied to supersonic conical and three-dimensional flows about sharp- and round-edged delta wings. Attached and separated-flow solutions have been obtained depending on the values of damping coefficients. The second technique is an integral-equation solver of the full potential equation which uses a volume-integral term in addition to the classical surface-integral terms. The technique is applied to transonic three-dimensional flows about sharp-edged delta wings. A hybrid technique which combines the finite-volume and the integral-equation solvers is also presented.

Kandil, O. A.↗

NASA NTRS1979

The lambda-scheme

A method for integrating the Euler equations of gas dynamics for compressible flows in any hyperbolic case is presented. This method is applied to the Mach number distribution over a stretch of an infinite duct having a variable cross section, and to the distribution in a channel opening into a vacuum with the Mach number equalling 1.04. An example of the ability of this method to handle two-dimensional unsteady flows is shown using the steady shock-and-isobars pattern reached asymptotically about an ablated blunt body with a free stream Mach number equalling 12. A final example is presented where the technique is applied to a three-dimensional steady supersonic flow, with a Mach number of 2 and an angle of attack of 5 deg.

Moretti, G.↗

NASA NTRS1982

Computation of blast wave-obstacle interactions

Numerical simulations of the interaction of a planar blast wave with various obstacles are presented. These obstacles are either ground structures or vehicles flying in the atmosphere. For a structure on the ground, the blast wave encounter is side-on, while for the flying vehicles the encounter is either head-on or oblique. Second-order accurate, finite-difference, and shock-capturing procedures are employed to solve the two-dimensional, axisymmetric, and three-dimensional unsteady Euler equations. Results are presented for the flow field consisting of blast wave striking obstacles that are at rest, moving subsonically and moving supersonically. Comparison of the numerical results with experimental data for a configuration at rest substantiates the validity of this approach and its potential as a flow analysis tool.

Champney, J. M.↗

NASA NTRS1982

Computation of transonic viscous-inviscid interacting flow

Transonic viscous-inviscid interaction is considered using the Euler and inverse compressible turbulent boundary-layer equations. Certain improvements in the inverse boundary-layer method are mentioned, along with experiences in using various Runge-Kutta schemes to solve the Euler equations. Numerical conditions imposed on the Euler equations at a surface for viscous-inviscid interaction using the method of equivalent sources are developed, and numerical solutions are presented and compared with experimental data to illustrate essential points.

Whitfield, D. L.↗

NASA NTRS1983

Flux vector splitting and approximate Newton methods

In the present investigation, the basic approach is employed to view an iterative scheme as Newton's method or as a modified Newton's method. Attention is given to various modified Newton methods which can arise from differencing schemes for the Euler equations. Flux vector splitting is considered as the basic spatial differencing technique. This technique is based on the partition of a flux vector into groups which have certain properties. The Euler equations fluxes can be split into two groups, the first group having a flux Jacobian with all positive eigenvalues, and the second group having a flux Jacobian with all negative eigenvalues. Flux vector splitting based on a velocity-sound speed split is considered along with the use of numerical techniques to analyze nonlinear systems, and the steady Euler equations for quasi-one-dimensional flow in a nozzle. Results are given for steady flows with shocks.

Jespersen, D. C.↗

NASA NTRS1983

Computation of transonic viscous-inviscid interacting flow

Whitfield, D. L.↗

NASA NTRS1985

Total pressure loss in vortical solutions of the conical Euler equations

A technique for the solution of the conically self-similar form of the Euler equations is described. Solutions for the flow past a flat-plate delta wing at angle of attack are presented. These solutions show strong leading edge vortices with large total pressure losses in the cores. A study of the effects of various computational parameters on the total pressure loss is made. An explanation for the cause of the total pressure loss is presented. It is shown to be consistent with the results for both a quasi-one-dimensional model problem and the conically self-similar flow past the flat-plate delta wing.

Powell, K. G.↗

NASA NTRS1986

A second-order accurate kinetic-theory-based method for inviscid compressible flows

An upwind method for the numerical solution of the Euler equations is presented. This method, called the kinetic numerical method (KNM), is based on the fact that the Euler equations are moments of the Boltzmann equation of the kinetic theory of gases when the distribution function is Maxwellian. The KNM consists of two phases, the convection phase and the collision phase. The method is unconditionally stable and explicit. It is highly vectorizable and can be easily made total variation diminishing for the distribution function by a suitable choice of the interpolation strategy. The method is applied to a one-dimensional shock-propagation problem and to a two-dimensional shock-reflection problem.

Deshpande, Suresh M.↗

NASA NTRS1987

Adaptive finite element methods for high-speed compressible flows

An adaptive finite element algorithm for solving the unsteady Euler equations is described. The finite element algorithm is based on a Taylor/Galerkin formulation and uses a very fast and efficient data structure to refine and unrefine the grid in order to optimize the approximation. A general version of the method which can be applied to moving grids with sliding interfaces is given, and results for a transient supersonic calculation of rotor-stator interaction are presented.

Oden, J. T.↗

NASA NTRS1988

A procedure based on the Euler equations for correcting transonic wind tunnel wall interference

Rizk, Magdi H.↗

NASA NTRS1988

Unsteady transonic airfoil computation using implicit Euler scheme on body-fixed grid

The unsteady Euler equations have been derived for the flow relative motion with respect to a frame of reference that is rigidly attached to the moving airfoil. The grid is generated once by an elliptic solver without a need for dynamic grid computation. An implicit factored finite-volume scheme has been developed and implemented through a fully vectorized computer program. Implicit second-order and explicit second and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The scheme is applied to steady and unsteady transonic airfoil flows and the results are in good agreement with the experimental data. For forced harmonic airfoil motions, periodic solutions are achieved within the third cycle of oscillation.

Kandil, Osama A.↗

NASA NTRS1989

Euler procedure for three-dimensional transonic wall interference

Based on an optimization formulation, a procedure has been developed to evaluate Mach number and angle-of-attack corrections. The Euler equations are assumed to be the flow governing equations. To obtain efficient solutions for the optimization problem, the iterative solutions for the flow variables and the design parameters are simultaneously updated. In addition to the model lift and geometry, the procedure requires pressure measurements near the tunnel walls. The tunnel boundary conditions are based on the introduction of Reimann invariants for a one-dimensional flow normal to the boundary. Computations are performed to verify that the errors introduced by this approximate boundary-condition formulation are acceptably small. The correction scheme is applied to an aircraft configuration in an open jet. The results indicate that the optimization scheme is highly efficient with the rate of convergence of the flow solution nearly equal to the corresponding rate of a regular analysis problem.

Rizk, Magdi H.↗

NASA NTRS1989

Diagonal implicit multigrid solution of the three-dimensional Euler equations

An attempt is made to apply an alternating direction implicit algorithm to the solution of the three-dimensional Euler equations. Results are presented for the flow past a swept wing mounted on a vertical wall or symmetry plane. The solution agrees well with experimental data.

Yadlin, Yoram↗

NASA NTRS1990

Prediction of unsteady transonic flow around missile configurations

This paper describes the preliminary development of a method for predicting the unsteady transonic flow around missiles at transonic and supersonic speeds, with the final goal of developing a computer code for use in aeroelastic calculations or during maneuvers. The basic equations derived for this method are an extension of those derived by Klopfer and Nixon (1989) for steady flow and are a subset of the Euler equations. In this approach, the five Euler equations are reduced to an equation similar to the three-dimensional unsteady potential equation, and a two-dimensional Poisson equation. In addition, one of the equations in this method is almost identical to the potential equation for which there are well tested computer codes, allowing the development of a prediction method based in part on proved technology.

Nixon, D.↗

NASA NTRS1991

Grid convergence for adaptive methods

The ability of adaptive methods to obtain accurate results is examined using two different Euler solvers for a near-sonic flow containing several important flow features. It is shown that the accuracy obtained can be greatly affected by the lack of resolution of smooth portions of the flow caused by adapting only to the more prevalent flow features such as discontinuities. In particular, common methods of adaptation can lead to results in which shocks are well resolved but whose locations are highly inaccurate due to the lack of resolution of the smoother regions. An explanation for this behavior is given and a correction is proposed.

Warren, Gary P.↗

NASA NTRS1992

Computer Code Generates Homotopic Grids

HOMAR is computer code using homotopic procedure to produce two-dimensional grids in cross-sectional planes, which grids then stacked to produce quasi-three-dimensional grid systems for aerospace configurations. Program produces grids for use in both Euler and Navier-Stokes computation of flows. Written in FORTRAN 77.

Moitra, Anutosh↗

NASA NTRS1993

An aerodynamic model for one and two degree of freedom wing rock of slender delta wings

The unsteady aerodynamic effects due to the separated flow around slender delta wings in motion were analyzed. By combining the unsteady flow field solution with the rigid body Euler equations of motion, self-induced wing rock motion is simulated. The aerodynamic model successfully captures the qualitative characteristics of wing rock observed in experiments. For the one degree of freedom in roll case, the model is used to look into the mechanisms of wing rock and to investigate the effects of various parameters, like angle of attack, yaw angle, displacement of the separation point, and wing inertia. To investigate the roll and yaw coupling for the delta wing, an additional degree of freedom is added. However, no limit cycle was observed in the two degree of freedom case. Nonetheless, the model can be used to apply various control laws to actively control wing rock using, for example, the displacement of the leading edge vortex separation point by inboard span wise blowing.

Hong, John↗

NASA NTRS1990

A comparison of optimization-based approaches for solving the aerodynamic design problem

Three optimization-based methods for solving aerodynamic design problems are compared. The Euler equations for one-dimensional duct flow was used as a model problem, and the three methods are compared for efficiency, robustness, and implementation difficulty. The smoothness of the design problem with respect to different shock-capturing finite difference schemes, and in the presence of grid refinement, is investigated.

Frank, Paul D.↗