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NASA NTRS1988

A Navier-Stokes solver for cascade flows

A computer code for solving the Reynolds averaged full Navier-Stokes equations has been developed and applied using sheared H-type grids. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. The integration in time is based on an explicit four-stage Runge-Kutta scheme. Local time stepping, variable coefficient implicit residual smoothing, and a full multigrid method have been implemented to accelerate steady state calculations. Comparisons with experimental data show that the code is an accurate viscous solver and can give very good blade-to-blade predictions for engineering applications in less than 100 multigrid cycles on the finest mesh.

Arnone, A.↗

NASA NTRS1988

Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations

Parameter studies are conducted using the Euler and potential flow equation models for unsteady and steady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux-vector-splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flows results are made as well as with experimental data where available.

Anderson, W. Kyle↗

NASA NTRS1987

Computation and analysis of a cylinder wake flow

The Karman vortex wake of a circular cylinder at low Reynolds number was computed by a time-accurate, two-dimensional compressible Navier-Stokes equation solver which uses the MacCormack predictor-corrector finite-difference scheme and a nonreflecting boundary condition on the outer flow boundary. The results from a large number of time steps were analyzed using Fast Fourier Transform techniques to identify the important frequency components for comparison with published experimental data. A strong low-frequency component was found below the vortex shedding frequency and not harmonically related to it. The experimentally discovered low-frequency fluctuations in the cylinder wake are considered possibly to be precursors to transition from laminar to turbulent flow conditions. The present finding of similar frequencies in a computed wake tends to confirm their existence as a real wake phenomenon. This computational work provides a complementary means to experimental investigations of wake phenomena.

Townsend, J. C.↗

NASA NTRS1987

Model-size reduction for the non-linear dynamic analysis of quasi-symmetric structures

A numerical technique is developed to reduce the size of models describing the nonlinear dynamic response of quasi-symmetric structures (i.e., structures with unsymmetric geometry). The response vectors of the structure are approximated by a linear combination of the symmetric and antisymmetric vectors at each time step. The mathematical formulation and numerical implementation of the method are described in detail, and results for a shallow laminated anisotropic panel of quadrilateral planform are presented in graphs and normalized contour plots.

Noor, Ahmed K.↗

NASA NTRS1987

Adaptive remeshing for compressible flow computations

The present, quality-enhancing adaptive-mesh procedure for two-dimensional Euler equation steady state solutions is implemented by means of linear triangular elements and an explicit time-stepping scheme, in conjunction with a finite element solution algorithm. The meshes thus generated typically take the form of stretched elements in the vicinity of one-dimensional flow features; a considerable variation in element size may thereby emerge which allows the desired high-quality solutions to be obtained with commensurately high computational efficiency.

Peraire, J.↗

NASA NTRS1987

Lake Bonneville - Constraints on lithospheric thickness and upper mantle viscosity from isostatic warping of Bonneville, Provo, and Gilbert stage shorelines

Data collected from three deformed shorelines of Lake Bonneville (the Bonneville, Provo, and Gilbert shorelines) are used to constrain the effective elastic lithospheric thickness to 23 + or - 2 km, the mantle viscosity to (1.2 + or - 0.2) x 10 to the 20th Pa sec, and the depth to a significant viscosity increase to no less than 300 km. A modification of the earth model of Nakiboglu and Lambeck (1982, 1983) is used for the calculations, and the water load is computed at each time step from a digital terrain model and a specified lake elevation. Differences noted between the observed and computed shoreline elevations indicate a regional tilt down to the NE of about 6 x 10 to the -5th, which is suggested to be due to collapse of the peripheral bulge formed by the Laurentide ice sheet.

Bills, Bruce G.↗

NASA NTRS1988

Navier-Stokes solutions for transonic flow over a wing mounted in a tunnel

Three-dimensional viscous flow calculations are performed for a swept, NACA 0012 wing mounted inside a wind tunnel for which detailed experimental data is available. A Runge-Kutta time-stepping scheme is used for obtaining steady-state solutions to the thin-layer Navier-Stokes equations. Free-air comptuations are also performed to assess the wall-interference effects. The effects of grid density and artificial dissipation on the accuracy of numerical results are included. The effect of the wind-tunnel sidewall boundary layer on the flow pattern over the wing surface, particularly in the vicinity of the wing/wall juncture is found to be significant.

Vatsa, V. N.↗

NASA NTRS1988

Some numerical and physical aspects of unsteady Navier-Stokes computations over airfoils using dynamic meshes

An upwind-biased implicit approximate factorization algorithm is applied to several unsteady flows on dynamic meshes. The thin-layer form of the compressible Navier-Stokes equations is used to solve both laminar and turbulent flows over airfoils pitching about the quarter chord. Numerical aspects of the solutions are investigated, including grid and time step effects. Two methods for determining fluxes - flux-vector splitting and flux-difference splitting - are compared. Flux-difference splitting predicts results more accurately than flux-vector splitting on a coarse mesh, but both methods agree on a fine mesh. Physical aspects of the computations are also examined. An equilibrium turbulent boundary layer model computes generally better unsteady results in comparison with experiment than a nonequilibrium model for the transonic case analyzed. Also, the size and location of the primary shed vortex for an airfoil pitching up at a constant rate is calculated in good agreement with experiment for two pitch rates.

Rumsey, Christopher L.↗

NASA NTRS1988

Application of finite element and remeshing technique to shock interference on a cylindrical leading edge

The problem of planar oblique shock impingement on a cylindrical leading edge in hypersonic flow is modeled using a Galerkin-Runge Kutta finite element method. The method utilizes a four stage Runge-Kutta time stepping scheme to solve the compressible Euler equations. Freestream Mach numbers of 6.5, 8.0 and 16.0 are studied. The computed surface pressure distributions consistently agree well with available experimental data. The peak pressure amplification ranges from 5.45 at M = 6.5 to approximately 17.0 at M = 16.0. Stagnation point heat transfer rate amplifications are calculated from the inviscid solution using the method of Fay and Riddell. The value and wall location of the peak pressure and heat transfer rate amplifications are extremely sensitive to the location of the impinging shock/bow shock intersection point.

Stewart, James R.↗

NASA NTRS1987

An explicit finite-volume time-marching procedure for turbulent flow calculations

A method was developed which calculates two-dimensional, transonic, viscous flow in ducts. The finite-volume, time-marching formulation is used to obtain steady flow solutions of the Reynolds-averaged form of the Navier-Stokes equations. The entire calculation is performed in the physical domain. Control volumes are chosen so that smoothing of flow properties, typically required for stability, is not required. Different time steps are used in the different governing equations. A new pressure interpolation scheme is introduced which improves the shock capturing ability of the method. A multi-volume method for pressure changes in the boundary layer allows calculations which use very long and thin control volumes (length/height - 1000). The method is compared with two test cases. Essentially incompressible turbulent boundary layer flow in an adverse pressure gradient is calculated and the computed distributions of mean velocity and shear are in good agreement with the measurements. Transonic viscous flow in a converging diverging nozzle is calculated; the Mach number upstream of the shock is approximately 1.25. The agreement between the calculated and measured shock strength and total pressure losses is good.

Nicholson, Stephen↗

NASA NTRS1987

Flow solution on a dual-block grid around an airplane

The compressible flow around a complex fighter-aircraft configuration (fuselage, cranked delta wing, canard, and inlet) is simulated numerically using a novel grid scheme and a finite-volume Euler solver. The patched dual-block grid is generated by an algebraic procedure based on transfinite interpolation, and the explicit Runge-Kutta time-stepping Euler solver is implemented with a high degree of vectorization on a Cyber 205 processor. Results are presented in extensive graphs and diagrams and characterized in detail. The concentration of grid points near the wing apex in the present scheme is shown to facilitate capture of the vortex generated by the leading edge at high angles of attack and modeling of its interaction with the canard wake.

Eriksson, Lars-Erik↗

NASA NTRS1988

Grid generation and adaptation for the Direct Simulation Monte Carlo Method

A grid generation and adaptation procedure based on the method of transfinite interpolation is incorporated into the Direct Simulation Monte Carlo Method of Bird. In addition, time is advanced based on a local criterion. The resulting procedure is used to calculate steady flows past wedges and cones. Five chemical species are considered. In general, the modifications result in a reduced computational effort. Moreover, preliminary results suggest that the simulation method is time step dependent if requirements on cell sizes are not met.

Olynick, David P.↗

NASA NTRS1988

Incompressible and anelastic flow simulations on numerically generated grids

In the numerical simulation of incompressible and anelastic flows, it is necessary to solve an elliptic equation at each time step. When the boundaries of such flows are nonrectangular, it may be advantageous to solve the equations on a new, numerically generated coordinate grid, in which the property of orthogonality has been preserved. Flow equations in general curvilinear coordinates maintaining the conservative form are given for both anelastic models using the momentum equations, and for incompressible models, using the vorticity equation. The general problem of grid-generation in two dimensions is presented, and a quasi-conformal transformation technique is discussed in detail. Some examples of grids generated by this technique are exhibited. Three examples of the flow of a stratified fluid over obstacles are presented, in which the grid-generation permits some new results to be obtained.

Sharman, R. D.↗

NASA NTRS1988

An improved flux-split algorithm applied to hypersonic flows in chemical equilibrium

An explicit, finite-difference, shock-capturing numerical algorithm is presented and applied to hypersonic flows assumed to be in thermochemical equilibrium. Real-gas chemistry is either loosely coupled to the gasdynamics by way of a Gibbs free energy minimization package or fully coupled using species mass conservation equations with finite-rate chemical reactions. A scheme is developed that maintains stability in the explicit, finite-rate formulation while allowing relatively high time steps. The codes use flux vector splitting to difference the inviscid fluxes and employ real-gas corrections to viscosity and thermal conductivity. Numerical results are compared against existing ballistic range and flight data. Flows about complex geometries are also computed.

Palmer, Grant↗

NASA NTRS1988

Numerical analysis of three-dimensional viscous internal flows

A 3-D Navier-Stokes code has been developed for analysis of turbomachinery blade rows and other internal flows. The Navier-Stokes equations are written in a Cartesian coordinate system rotating about the x-axis, and then mapped to a general body-fitted coordinate system. Streamwise viscous terms are neglected using the thin-layer assumption, and turbulence effects are modeled using the Baldwin-Lomax turbulence model. The equations are discretized using finite differences on stacked C-type grids and are solved using a multistage Runge-Kutta algorithm with a spatially-varying time step and implicit residual smoothing. Calculations have been made of a horseshoe vortex formed in front of a flat plate with a round leading edge standing in a turbulent endwall boundary layer. Comparisons are made with experimental data taken by Eckerle and Langston for a circular cylinder under similar conditions. Computer and measured results are compared in terms of endwall flow visualization pictures and total pressure loss contours and vector plots on the symmetry plane. Calculated details of the primary vortex show excellent agreement with the experimental data. The calculations also show a small secondary vortex not seen experimentally.

Chima, Rodrick V.↗

NASA NTRS1988

Analysis of the convergence history of flow through nozzles with shocks

Acceleration techniques such as Wynn's (1986) epsilon algorithm and analysis techiques such as eigensystem analysis are used here to study numerically the convergence properties of an iterative scheme applied to the quasi-one-dimensional Euler and Navier-Stokes equations for flow through nozzles with shocks. The convergence and stability properties are studied by analyzing the dependence of convergence of the code on the discretization technique, boundary conditions, time-step, number of grid points, and the physics of the problem.

Cheer, A. Y.↗

NASA NTRS1988

Variability of pigment biomass in the California Current system as determined by satellite imagery. II - Temporal variability

Characteristics of temporal variability in the California Current system are analyzed using a 30-month time series of CZCS imagery. About 20-25 percent of the variance is produced by a periodic annual cycle with peak values in winter. Analysis of ship-based chlorophyll measurements indicates that the winter peak is only characteristic of the upper portion of the euphotic zone and that total water column chlorophyll peaks during the spring upwelling season. Satellite studies of intraannual variability are modulated by strong 5- to 6-day oscillation in the availability of usable imagery, resulting from a combination of satellite orbital dynamics, which produces images of the study area roughly 4 out of every 6 days, and an oscillation in cloud cover, which controls the availability of clear imagery. The cloud cover oscillation, which is also present in coastal winds, undoubtedly affects the ocean surface and biases the data obtained by satellites. Analysis of data using a 5-day time step indicates that the predominant mode of nonseasonal variability is characterized by in-phase fluctuations throughout the southern and central California coastal region.

Michaelsen, Joel↗

NASA NTRS1988

Analysis of electrophoresis performance

A flexible efficient computer code is being developed to simulate electrophoretic separation phenomena, in either a cylindrical or a rectangular geometry. The code will computer the evolution in time of the concentrations of an arbitrary number of chemical species, and of the temperature, pH distribution, conductivity, electric field, and fluid motion. Use of nonuniform meshes and fast accurate implicit time-stepping will yield accurate answers at economical cost.

Roberts, Glyn O.↗