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NASA NTRS1988

Three-dimensional multigrid algorithms for the flux-split Euler equations

The Full Approximation Scheme (FAS) multigrid method is applied to several implicit flux-split algorithms for solving the three-dimensional Euler equations in a body fitted coordinate system. Each of the splitting algorithms uses a variation of approximate factorization and is implemented in a finite volume formulation. The algorithms are all vectorizable with little or no scalar computation required. The flux vectors are split into upwind components using both the splittings of Steger-Warming and Van Leer. The stability and smoothing rate of each of the schemes are examined using a Fourier analysis of the complete system of equations. Results are presented for three-dimensional subsonic, transonic, and supersonic flows which demonstrate substantially improved convergence rates with the multigrid algorithm. The influence of using both a V-cycle and a W-cycle on the convergence is examined.

Anderson, W. Kyle↗

NASA NTRS1988

The evolution of AAOE observed constituents with the polar vortex

One of the difficulties in determining constituent trends from the ER-2 flight data is the large amount of day to day variability generated by the motion of the polar vortex. To reduce this variability, the observations have been transformed into the conservative (Lagrangian) reference frames consisting of the coordinate pairs, potential temperature (PT) and potential vorticity (PV), or PT and N2O. The requirement of only two independent coordinates rests on the assumption that constituent distributions and their chemical processes are nearly zonal in that coordinate system. Flight data is used everywhere for these transformation except for potential vorticity. Potential vorticity is determined from level flight segments, and NMC PV values during flight dives and takeoffs are combined with flight data in a smooth fashion.

Schoeberl, Mark R.↗

NASA NTRS1989

Numerical simulation and comparison of symmetrical/supercritical airfoils for the near tip region of a helicopter in forward flight

Aerodynamic loads on a multi-bladed helicopter rotor in forward flight at transonic tip conditions are calculated. The unsteady, three-dimensional, time-accurate compressible Reynolds-averaged thin layer Navier-Stokes equations are solved in a rotating coordinate system on a body-conformed, curvilinear grid of C-H topology. Detailed boundary layer and global numerical comparisons of NACA-0012 symmetrical and CAST7-158 supercritical airfoils are made under identical forward flight conditions. The rotor wake effects are modeled by applying a correction to the geometric angle of attack of the blade. This correction is obtained by computing the local induced downwash velocity with a free wake analysis program. The calculations are performed on the Numerical Aerodynamic Simulation Cray 2 and the VPS32 (a derivative of a Cyber 205 at the Langley Research Center) for a model helicopter rotor in forward flight.

Badavi, F. F.↗

NASA NTRS1988

An expert system for restructurable control

Work in progrss on an expert system which restructures and tunes control systems online in real-time is presented. The expert system coordinates the different methods involved in redesigning and implementing the control strategies due to plant changes.

Litt, Jonathan↗

NASA NTRS1989

Inverse wing design in transonic flow including viscous interaction

Several inverse methods were compared and initial results indicate that differences in results are primarily due to coordinate systems and fuselage representations and not to design procedures. Further, results from a direct-inverse method that includes 3-D wing boundary layer effects, wake curvature, and wake displacement are represented. These results show that boundary layer displacements must be included in the design process for accurate results.

Carlson, Leland A.↗

NASA NTRS1989

A 3-dimensional finite-difference method for calculating the dynamic coefficients of seals

A method to calculate the dynamic coefficients of seals with arbitrary geometry is presented. The Navier-Stokes equations are used in conjunction with the k-e turbulence model to describe the turbulent flow. These equations are solved by a full 3-dimensional finite-difference procedure instead of the normally used perturbation analysis. The time dependence of the equations is introduced by working with a coordinate system rotating with the precession frequency of the shaft. The results of this theory are compared with coefficients calculated by a perturbation analysis and with experimental results.

Dietzen, F. J.↗

NASA NTRS1989

Parallel processors and nonlinear structural dynamics algorithms and software

A nonlinear structural dynamics finite element program was developed to run on a shared memory multiprocessor with pipeline processors. The program, WHAMS, was used as a framework for this work. The program employs explicit time integration and has the capability to handle both the nonlinear material behavior and large displacement response of 3-D structures. The elasto-plastic material model uses an isotropic strain hardening law which is input as a piecewise linear function. Geometric nonlinearities are handled by a corotational formulation in which a coordinate system is embedded at the integration point of each element. Currently, the program has an element library consisting of a beam element based on Euler-Bernoulli theory and trianglar and quadrilateral plate element based on Mindlin theory.

Belytschko, Ted↗

NASA NTRS1989

Free-molecule-flow force and moment coefficients of the aeroassist flight experiment vehicle

Calculated results for the aerodynamic coefficients over the range of + or - 90 deg in both pitch and yaw attitude angles for the Aeroassist Flight Experiment (AFE) vehicle in free molecule flow are presented. The AFE body is described by a large number of small flat plate surface elements whose orientations are established in a wind axes coordinate system through the pitch and yaw attitude angles. Lift force, drag force, and three components of aerodynamic moment about a specified point are computed for each element. The elemental forces and moments are integrated over the entire body, and total force and moment coefficients are computed. The coefficients are calculated for the two limiting gas-surface molecular collision conditions, namely, specular and diffuse, which assume zero and full thermal accommodation of the incoming gas molecules with the surface, respectively. The individual contribution of the shear stress and pressure terms are calculated and also presented.

Blanchard, Robert C.↗

NASA NTRS1988

A conservative approach for flow field calculations on multiple grids

In the computation of flow fields about complex configurations, it is very difficult to construct body-fitted coordinate systems. An alternative approach is to use several grids at once, each of which is generated independently. This procedure is called the multiple grids or zonal grids approach and its applications are investigated in this study. The method follows the conservative approach and provides conservation of fluxes at grid interfaces. The Euler equations are solved numerically on such grids for various configurations. The numerical scheme used is the finite-volume technique with a three-state Runge-Kutta time integration. The code is vectorized and programmed to run on the CDC VPS-32 computer. Some steady state solutions of the Euler equations are presented and discussed.

Kathong, Monchai↗

NASA NTRS1988

A multidimensional model for aerosols - Description of computational analogs

The numerical algorithms used to simulate the advection, diffusion, sedimentation, coagulation, and condensational growth of atmospheric aerosols are described. The model can be used in one, two, or three spatial dimensions. The continuity equation in a generalized horizontal and vertical coordinate system is developed, which allows the model to be quickly adapted to a wide variety of dynamical models of global or regional scale. Algorithms are developed to treat the various physical processes, and the results of simulations are presented, which show the strengths and weaknesses of these algorithms. Although the emphasis is on the modeling of aerosols, the work is also applicable to the simulations of the transport of gases.

Toon, O. B.↗

NASA NTRS1988

The long term stability of VLBI earth orientation measurements

Tectonic motions will, in general, change the orientation as well as the length of baselines used in Very Long Baseline Interferometry (VLBI), and will thus cause slow divergences between earth orientation results obtained with different VLBI networks, as well as between VLBI results and those obtained by satellite laser ranging and lunar laser ranging. Such drifts (on the order of a milliarcsecond/year) are inherently interesting as well as being significant in combinations of orientation results from different sources. The geodetic study of tectonic motions is also closely connected to research into the nature and causes of systematic errors in data from the modern techniques of space geodesy. Both a special coordinate system found to be of use in the analysis of VLBI data and tectonic motion estimates for a VLBI baseline stretching from California to Australia are described.

Eubanks, T. M.↗

NASA NTRS1988

Dynamic inflow for practical applications

A derivation is presented which gives both the nonlinear dynamic inflow model and the perturbation dynamic inflow model in a general coordinate system. The perturbation model includes perturbations in hub motions, in skew angle, and in yaw angle. The governing equations may be switched to their dimensional forms prior to application.

Peters, David A.↗

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 NTRS1989

An unconditionally stable Runge-Kutta method for unsteady flows

A quasi-three-dimensional analysis was developed for unsteady rotor-stator interaction in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream surface thickness. The Baldwin-Lomax eddy viscosity model is used for turbulent flows. The equations are integrated in time using a four-stage Runge-Kutta scheme with a constant time step. Implicit residual smoothing was employed to accelerate the solution of the time accurate computations. The scheme is described and accuracy analyses are given. Results are shown for a supersonic through-flow fan designed for NASA Lewis. The rotor:stator blade ratio was taken as 1:1. Results are also shown for the first stage of the Space Shuttle Main Engine high pressure fuel turbopump. Here the blade ratio is 2:3. Implicit residual smoothing was used to increase the time step limit of the unsmoothed scheme by a factor of six with negligible differences in the unsteady results. It is felt that the implicitly smoothed Runge-Kutta scheme is easily competitive with implicit schemes for unsteady flows while retaining the simplicity of an explicit scheme.

Jorgenson, Philip C. E.↗

NASA NTRS1989

Numerical simulation of axisymmetric turbulent flow in combustors and diffusers

Numerical studies of turbulent flow in an axisymmetric 45-deg-expansion combustor and bifurcated diffuser are presented. The Navier-Stokes equations incorporating a k-epsilon model were solved in a nonorthogonal curvilinear coordinate system. A zonal-grid method, where the flow field was divided into several subsections, was developed. This approach permitted different computational schemes to be used in the various zones. In addition, grid generation was made a more simple task. Boundary overlap and interpolating techniques were used, and an adjustment of the flow variables was required to assure conservation of mass flux. Three finite-differencing methods (hybrid, quadratic upwind, and skew upwind) were used to represent the convection terms. Results were compared with existing experimental data. In general, good agreement between predicted and measured values was obtained.

Yung, Chain-Nan↗

NASA NTRS1989

Semi-Lagrangian integration of a gridpoint shallow water model on the sphere

A stable, semi-Lagrangian, semi-implicit, two-time-level, gridpoint integration scheme for the shallow water equations on the sphere is presented. A rotated spherical coordinate system is used to integrate the equations of motion at each gridpoint poleward of a certain latitude, thus overcoming problems associated with the polar singularity. The results of medium term integrations of large scale test patterns using a long time step are presented.

Mcdonald, A.↗

NASA NTRS1988

A study of turbulent flow computations in an angled duct with a step

This paper presents a method for effective computation of turbulent flows in an angled duct which has a step at the angled section by using a grid generation technique. An effect of flow curvature was analyzed on the separating and reattaching flows. The computations are based on the Reynolds averaged transport equations which are transformed for use on a generalized curvilinear coordinate system. A special care was taken on the treatment of pressure corrections. Wall function boundary conditions for k-epsilon model were developed.

Maruszewski, J. P.↗

NASA NTRS1989

Shock structure in non-circular jets

The shock-cell structure of supersonic jets with non-circular exit geometry is modeled using a linearized analysis. The model takes into account the finite thickness of the jet shear layer using realistic velocity and density profiles. The effects of the shear layer turbulence are included by incorporating eddy-viscosity terms. A finite-difference numerical method is used to solve the steady linearized equations of motion. A body-fitted coordinate system is used to describe the shear layer. The variation of the pressure fluctuation with downstream distance is given for circular jets and for an elliptic jet of aspect ratio 2.0. Comparisons with experimental data are made. Difficulties with the numerical technique are also discussed.

Morris, Philip J.↗