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NASA NTRS1986

Implicit time-marching solution of the Navier-Stokes equations for thrust reversing and thrust vectoring nozzle flows

An implicit finite volume method is investigated for the solution of the compressible Navier-Stokes equations for flows within thrust reversing and thrust vectoring nozzles. Thrust reversing nozzles typically have sharp corners, and the rapid expansion and large turning angles near these corners are shown to cause unacceptable time step restrictions when conventional approximate factorization methods are used. In this investigation these limitations are overcome by using second-order upwind differencing and line Gauss-Siedel relaxation. This method is implemented with a zonal mesh so that flows through complex nozzle geometries may be efficiently calculated. Results are presented for five nozzle configurations including two with time varying geometries. Three cases are compared with available experimental data and the results are generally acceptable.

Imlay, S. T.↗

NASA NTRS1986

Generation of a composite grid for turbine flows and consideration of a numerical scheme

A composite grid was generated for flows in turbines. It consisted of the C-grid (or O-grid) in the immediate vicinity of the blade and the H-grid in the middle of the blade passage between the C-grids and in the upstream region. This new composite grid provides better smoothness, resolution, and orthogonality than any single grid for a typical turbine blade with a large camber and rounded leading and trailing edges. The C-H (or O-H) composite grid has an unusual grid point that is connected to more than four neighboring nodes in two dimensions (more than six neighboring nodes in three dimensions). A finite-volume lower-upper (LU) implicit scheme to be used on this grid poses no problem and requires no special treatment because each interior cell of this composite grid has only four neighboring cells in two dimensions (six cells in three dimensions). The LU implicit scheme was demonstrated to be efficient and robust for external flows in a broad flow regime and can be easily applied to internal flows and extended from two to three dimensions.

Choo, Y.↗

NASA NTRS1987

A linearized Euler analysis of unsteady flows in turbomachinery

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

Hall, Kenneth C.↗

NASA NTRS1986

Development of an explicit time marching procedure for laminar and turbulent flow

A finite-volume based, time-marching method for turbulent and laminar flows is described. Extensions to the computational procedure to allow the handling of shock induced separation and large regions of strong backflow are emphasized. Two test cases are considered, the United Technologies Research Center (UTRC) separated and reattached turbulent boundary layer and the strong shock case in the McDonnell Douglas Research Lab (MDRL) transonic diffuser G. The extended method worked well on the UTRC flow with a boundary layer blockage of 58% and a maximum backflow velocity of 37% of the local maximum free-stream velocity. It also worked well on the MDRL diffuser with a shock Mach number of 1.353 and a maximum backflow velocity of -71.7 m/s. A Mach number dependent interpolation formula for effective pressure was developed for use in density-update time-marching methods.

Source record↗

NASA NTRS1985

Numerical solution of the Euler equation for compressible inviscid fluids

An effort is made to develop a satisfactory numerical method for the calculation of steady solutions of the Euler equations for inviscid compressible gas flows. The intended application is the prediction of the aerodynamic properties of aircraft flying at transonic speeds. Particular consideration is given to the modification of the equations to improve convergence to a steady state; finite volume formulation; adaptive dissipation; schemes designed to improve the resolution of shock waves; hybrid multistage time stepping schemes; residual averaging; a multigrid scheme; and trials with Burgers' equation. Results clearly demonstrate that the convergence of a time-dependent hyperbolic system to a steady state can be substantially accelerated by the introduction of multiple grids.

Jameson, A.↗

NASA NTRS1987

Computational and experimental investigation of cavity flowfields

This paper presents a computational and experimental investigation of supersonic flow past a cavity in a flat plate. The source of the particular interest in this problem is the ongoing study of the aerodynamic interference effects between a separating store and its bay in the parent body. An upwind relaxation scheme, utilizing flux vector splitting and line-Gauss-Seidel iterations, is used to solve Reynolds-averaged Navier-Stokes equations. Spatial discretizations of this two-dimensional analysis are based on implicit and finite-volume methods. Turbulence is modeled and shocks are captured. The flowfield of the symmetry plane at the half-width is computationally visualized and all flow properties are computed. Experimental tests are conducted in the Langley Unitary Plan Wind Tunnel to measure wall pressures and to capture schlieren photographs. Qualitative as well as quantitative data of computations and experiments agree very well. These two vehicles of investigation are merged to show open, closed and transitional cavity flow behaviors.

Baysal, O.↗

NASA NTRS1987

A computational analysis of flow separation over five different airfoil geometries at high angles-of-attack

The flux splitting method of Van Leer (1982) and an implicit, upwind-biased, finite-volume scheme of Rumsey (1985) for solving the two-dimensional Reynolds-averaged Navier-Stokes equations were applied to analyze five airfoil geometries (the NACA 0012, 12-percent Joukowski, NACA 4412, NACA 65-015, and LS1-0417 sections), at high angles of attack. The effect of grid density and grid extent on the Strouhal number of the unsteady flow was computed, as well as the effect of variations in Mach number, Reynolds number, and angle of attack. It was found, that past an angle of attack of 20 deg, Strouhal numbers were independent of angle of attack (at a constant value of 0.115), agreeing well with the experimental findings of Tyler (1931).

Rumsey, Christopher L.↗

NASA NTRS1987

Navier-Stokes computations of vortical flows over low aspect ratio wings

An upwind-biased finite-volume algorithm is applied to the low-speed flow over a low aspect ratio delta wing from zero to forty degrees angle of attack. The differencing is second-order accurate spatially, and a multigrid algorithm is used to promote convergence to the steady state. The results compare well with the detailed experiments of Hummel (1983) and others for a Re(L) of 0.95 x 10 to the 6th. The predicted maximum lift coefficient of 1.10 at thirty-five degrees angle of attack agrees closely with the measured maximum lift of 1.06 at thirty-three degrees. At forty degrees angle of attack, a bubble type of vortex breakdown is evident in the computations, extending from 0.6 of the root chord to just downstream of the trailing edge.

Thomas, J. L.↗

NASA NTRS1987

A 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.

Caughey, David A.↗

NASA NTRS1987

Euler calculations for wings using Cartesian grids

A method is presented for the calculation of transonic flows past wings using Cartesian grids. The calculations are based on a finite volume formulation of the Euler equations. Results are presented for a rectangular wing with a flat tip and the ONERA M6 wing. In general, the results are in good agreement with other computations and available experiment. However, Cartesian grids require a greater number of points than body fitted grids in order to resolve the flow properties near the leading edge of a swept wing.

Gaffney, R. L., Jr.↗

NASA NTRS1987

An L-U implicit multigrid algorithm for the three-dimensional Euler equations

An LU implicit multigrid scheme is developed for the calculation of three-dimensional transonic flow through rotating cascades. This numerical method solves the unsteady Euler equations of gas dynamics in a finite-volume form. The implicit scheme makes it possible to take a much larger time step than is normally permitted in most explicit schemes, while the multigrid method is incorporated to accelerate the convergence rate for steady state calculations. Using this method, computational storage requirements are comparable to those of explicit schemes, while operation counts are considerably less than those found in the more widely-used ADI schemes.

Yokota, Jeffrey W.↗

NASA NTRS1987

Euler solution of the transonic flow for a helicopter rotor

A new method is presented for calculating the quasi-steady transonic flow over a lifting or nonlifting rotor blade in both hover and forward flight by using Euler equations. The approach is to solve the Euler equations in a rotor-fixed frame of reference using a finite volume method. A computer program was developed and was then verified by comparison with wind-tunnel data. In all cases considered, good agreement was found with available experimental data.

Chang, I-Chung↗

NASA NTRS1987

Multigrid solution of inviscid transonic flow through rotating blade passages

A fast Euler solver for three dimensional inviscid transonic flow in rotating domains is described. The time dependent Euler equations are discretized spatially with finite volumes, and are advanced temporally with a multiple stage time stepping scheme. A dramatic increase in the rate of convergence for steady solutions is achieved with a multigrid algorithm that employs the multistage scheme as its smoothing procedure. The effectiveness of the multistage scheme as a multigrid driver is enhanced by the utilization of analytically determined combinations of the governing parameters.

Smith, Wayne A.↗

NASA NTRS1986

Accuracy of schemes with nonuniform meshes for compressible fluid flows

The accuracy of the space discretization for time-dependent problems when a nonuniform mesh is used is considered. Many schemes reduce to first-order accuracy while a popular finite volume scheme is even inconsistent for general grids. This accuracy is based on physical variables. However, when accuracy is measured in computational variables then second-order accuracy can be obtained. This is meaningful only if the mesh accurately reflects the properties of the solution. In addition, the stability properties of some improved accurate schemes are analyzed and it can be shown that they also allow for larger time steps when Runge-Kutta type methods are used to advance in time.

Turkel, Eli↗

NASA NTRS1987

Euler solver for three-dimensional supersonic flows with subsonic pockets

A new finite-difference scheme has been developed to solve efficiently the unsteady Euler equations for three-dimensional inviscid supersonic flows with subsonic pockets. The technique utilizes planar Gauss-Seidel relaxation in the marching direction and approximate factorization n the crossflow plane. An 'infinitely large' time step is used in parts of the flowfield where the component of velocity in the marching direction is supersonic - here the Gauss-Seidel sweeps are restricted to the forward direction only, and the procedure reduces to simple space-marching; a finite time step is used in parts of the flowfield where the marching component of velocity is subsonic - here, backward and forward Gauss-Seidel sweeps are employed to allow for upstream and downstream propagation of signals, and a time-asymptotic steady state is obtained. The discretization formulas are based on finite-volume implementation of high accuracy (up to third-order) total variation diminishing formulations. Numerical solutions are obtained for an analytically defined forebody, a realistic fighter configuration, and the Space Shuttle. The results are in very good agreement with available experimental data and numerical solutions of the full-potential equation.

Chakravarthy, Sukumar R.↗

NASA NTRS1987

Application of an upwind algorithm to the three-dimensional parabolized Navier-Stokes equations

A new computer code for the solution of the three-dimensional parabolized Navier-Stokes equations has been developed. The code employs a state-of-the-art upwind algorithm to capture strong shock waves. The algorithm is implicit, uses finite volumes, and is second-order accurate in the crossflow directions. The new code is validated through application to laminar hypersonic flows past two simple body shapes: a circular cone of 10 deg half-angle, and a generic all-body hypersonic vehicle. Cone flow solutions were computed at angles of attack of 12, 20, and 24 deg and results are in agreement with experimental data. Results are also presented for the flow past the all-body vehicle at angles of incidence of 0 and 10 deg.

Lawrence, Scott L.↗

NASA NTRS1987

Grid generation and inviscid flow computation about cranked-winged airplane geometries

An algebraic grid generation procedure that defines a patched multiple-block grid system suitable for fighter-type aircraft geometries with fuselage and engine inlet, canard or horizontal tail, cranked delta wing and vertical fin has been developed. The grid generation is based on transfinite interpolation and requires little computational power. A finite-volume Euler solver using explicit Runge-Kutta time-stepping has been adapted to this grid system and implemented on the VPS-32 vector processor with a high degree of vectorization. Grids are presented for an experimental aircraft with fuselage, canard, 70-20-cranked wing, and vertical fin. Computed inviscid compressible flow solutions are presented for Mach 2 at 3.79, 7 and 10 deg angles of attack. Conmparisons of the 3.79 deg computed solutions are made with available full-potential flow and Euler flow solutions on the same configuration but with another grid system. The occurrence of an unsteady solution in the 10 deg angle of attack case is discussed.

Eriksson, L.-E.↗

NASA NTRS1987

Displacement surface calculations for a hypersonic aircraft

A method is presented to calculate the three-dimensional displacement surface about a hypersonic aircraft. This calculation requires a flow-field-solution to at least the thin-layer Navier-Stokes equations about the aircraft as input. An upwind, finite-volume code developed at NASA Langley Research Center was used to obtain the flow-field solution. The displacement surface is the three-dimensional counterpart to the two-dimensional displacement thickness. Flow-field solutions, along with the displacement surface calculations, are presented for a generic hypersonic aircraft at a Mach number of 24.5 and angle of attack of zero and one degree.

Richardson, Pamela F.↗