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NASA NTRS1983

Multiple-grid convergence acceleration of viscous and inviscid flow computations

A multiple-grid algorithm for use in efficiently obtaining steady solution to the Euler and Navier-Stokes equations is presented. The convergence of a simple, explicit fine-grid solution procedure is accelerated on a sequence of successively coarser grids by a coarse-grid information propagation method which rapidly eliminates transients from the computational domain. This use of multiple-gridding to increase the convergence rate results is substantially reduced work requirements for the numerical solution of a wide range of flow problems. Computational results are presented for subsonic and transonic inviscid flows and for laminar and turbulent, attached and separated, subsonic viscous flows. Work reduction factors as large as eight, in comparison to the basic fine-grid algorithm, were obtained. Possibilities for further performance improvement are discussed. Previously announced in STAR as N83-21847

Johnson, G. M.↗

NASA NTRS1983

Seismic coupling and uncoupling at subduction zones

Some of the correlations concerning the properties of subduction zones are reviewed. A quantitative global comparison of many subduction zones reveals that the largest earthquakes occur in zones with young lithosphere and fast convergence rates. Maximum earthquake size is directly related to the asperity distribution on the fault plane. This observation can be translated into a simple model of seismic coupling where the horizontal compressive stress between two plates is proportional to the ratio of the summed asperity area to the total area of the contact surface. Plate age and rate can control asperity distribution directly through the horizontal compressive stress associated with the vertical and horizontal velocities of subducting slabs. The basalt to eclogite phase change in the down-going oceanic crust may be largely responsible for the uncoupling of subduction zones below a depth of about 40 km.

Ruff, L.↗

NASA NTRS1984

A vectorized solution for incompressible flow

An algorithm is developed to obtain solutions to the unsteady Reynolds-averaged incompressible Navier-Stokes equations in general curvilinear coordinates on a vector processor. The governing equations are in nonconservative form with the velocity and pressure as dependent variables. Two momentum equations and the Poisson equation for pressure form a set of three governing equations for three flow field unknowns: u, v, and p. The governing equations and boundary conditions are expressed in terms of boundary-conforming curvilinear coordinates, and a checkerboard SOR iteration is used to solve the governing equations. Several possible sequences for a checkerboard SOR iteration are investigated for finding the best overall convergence rate. The efficiency and capability of the present algorithm was assessed using the example of an 18 percent thick NACA 66(3)018 airfoil at zero degree angle of attack for chord Reynolds number range 1000-40,000.

Patel, N. R.↗

NASA NTRS1983

Finite element calculation of buoyancy-driven convection near a melt/solid phase boundary

Two iterative schemes based on the mixed finite element method are developed for analyzing steady natural convection in a melt adjacent to its solid phase. The simplest method decouples the calculation of the field variables and the shape of the melt/solid interface into two interlocked iterations that are performed successively. The second method uses Newton's iteration to solve simultaneously for both types of unknowns and has a quadratic convergence rate. Results for a model problem of melt and solid in a cylindrical ampoule show the Newton algorithm to be a factor of three more efficient.

Chang, C. J.↗

NASA NTRS1984

Accelerated convergence for incompressible flow calculations

Two improved algorithms which solve the steady-state Navier-Stokes equations, PISO and SIMPLER, are studied. Computations were carried out on progressively finer grids for the driven cavity and flow over a backward-facing step. The effects of relaxation factor, number of grid nodes and number of sweeps through the pressure equations are studied to evaluate the performance of the PISO and SIMPLER schemes. Results show that these improved schemes accelerate the convergence rate of the solution generally by a factor of two as compared to the SIMPLE method.

Neely, G. M.↗

NASA NTRS1985

Researched applied to transonic compressors in numerical fluid mechanics of inviscid flow and viscous flow

A streamline Euler solver which combines high accuracy and good convergence rates with capabilities for inverse or direct mode solution modes and an analysis technique for finite difference models of hyperbolic partial difference equations were developed.

Thompkins, W. T., Jr.↗

NASA NTRS1985

Recovering pointwise values of discontinuous data within spectral accuracy

The pointwise values of a function, f(x), can be accurately recovered either from its spectral or pseudospectral approximations, so that the accuracy solely depends on the local smoothness of f in the neighborhood of the point x. Most notably, given the equidistant function grid values, its intermediate point values are recovered within spectral accuracy, despite the possible presence of discontinuities scattered in the domain. (Recall that the usual spectral convergence rate decelerates otherwise to first order, throughout). To this end, a highly oscillatory smoothing kernel is employed in contrast to the more standard positive unit-mass mollifiers. In particular, post-processing of a stable Fourier method applied to hyperbolic equations with discontinuous data, recovers the exact solution modulo a spectrally small error. Numerical examples are presented.

Gottlieb, D.↗

NASA NTRS1984

Pole of rotating analysis of present-day Juan de Fuca plate motion

Convergence rates between the Juan de Fuca and North American plates are calculated by means of their relative, present-day pole of rotation. A method of calculating the propagation of errors in addition to the instantaneous poles of rotation is also formulated and applied to determine the Euler pole for Pacific-Juan de Fuca. This pole is vectorially added to previously published poles for North America-Pacific and 'hot spot'-Pacific to obtain North America-Juan de Fuca and 'hot spot'-Juan de Fuca, respectively. The errors associated with these resultant poles are determined by propagating the errors of the two summed angular velocity vectors. Under the assumption that hot spots are fixed with respect to a mantle reference frame, the average absolute velocity of the Juan de Puca plate is computed at approximately 15 mm/yr, thereby making it the slowest-moving of the oceanic plates.

Nishimura, C.↗

NASA NTRS1984

Modification to shock fitting program

A modified form of the Lepping-Argentiero single spacecraft, shock normal determination procedure is presented. The modified method incorporates a simple predictor-corrector algorithm which allows a faster convergence rate and the use of average values of the parameters for the starting vector.

Acuna, M. H.↗

NASA NTRS1985

Accelerated convergence for incompressible flow calculations

Neely, G. M.↗

NASA NTRS1985

A comparison of finite volume flux vector splittings for the Euler equations

A comparison is made between the computational results of the Steger-Warming (1981) and van Leer (1982) flux splitting methods, which have been applied in generalized coordinates to quasi-one-dimensional transonic flow in a nozzle and two-dimensional subsonic, transonic, and supersonic flow over airfoils. The latter splitting method leads to higher convergence rates and a sharper representation of shocks in the transition region. The second-order accurate, one-sided-difference model is extended to a third-order, upwind-biased model with only small additional computational effort.

Anderson, W. K.↗

NASA NTRS1985

Recent advances in computational transonic aerodynamics

The near-term prospects are discussed for calculation of viscous transonic flow fields about realistic configurations at full-scale Reynolds numbers. Three basic algorithms are considered: The central-difference, three-factor ADI method; the central-difference, explicit, multistep Runge-Kutta method with multigrid acceleration; and the relaxation method for the upwind-differenced, flux-split equations. Each method has distinct advantages and disadvantages regarding stability, convergence rate, and vectorizability. It appears that computation times can be 15 to 60 hours on the latest super computers unless 3D algorithms are improved to perform as well as current 2D algorithms.

South, J. C., Jr.↗

NASA NTRS1985

Convergence acceleration for a three-dimensional Euler/Navier-Stokes zonal approach

A fast diagonal algorithm is coupled with a zonal approach to solve the three-dimensional Euler/Navier-Stokes equations. Transonic viscous solutions are obtained on a 150,000 point mesh for a NACA 0012 wing. The new computational approach yields a speedup by as much as a factor of 40 over the standard Beam-Warming algorithm/zonal method originally coded. A three-order-of-magnitude drop in the L2-norm of the residual requires approximately 500 iterations, which takes about 45 min of CPU time on a Cray-XMP. The numerically computed solutions are in good agreement with experimental results. Effects on convergence rate owing to increasing the zonal boundary overlap regions, different stretching distributions in the viscous regions, and different CFL values are also explored.

Flores, J.↗

NASA NTRS1985

Implicit flux-split schemes for the Euler equations

Recent progress in the development of implicit algorithms for the Euler equations using the flux-vector splitting method is described. Comparisons of the relative efficiency of relaxation and spatially-split approximately factored methods on a vector processor for two-dimensional flows are made. For transonic flows, the higher convergence rate per iteration of the Gauss-Seidel relaxation algorithms, which are only partially vectorizable, is amply compensated for by the faster computational rate per iteration of the approximately factored algorithm. For supersonic flows, the fully-upwind line-relaxation method is more efficient since the numerical domain of dependence is more closely matched to the physical domain of dependence. A hybrid three-dimensional algorithm using relaxation in one coordinate direction and approximate factorization in the cross-flow plane is developed and applied to a forebody shape at supersonic speeds and a swept, tapered wing at transonic speeds.

Thomas, J. L.↗

NASA NTRS1985

Application of TVD schemes for the Euler equations of gas dynamics

First-order, second-order, and implicit total variation diminishing (TVD) schemes are reviewed using the modified flux approach. Some transient and steady-state calculations are then carried out to illustrate the applicability of these schemes to the Euler equations. It is shown that the second-order explicit TVD schemes generate good shock resolution for both transient and steady-state one-dimensional and two-dimensional problems. Numerical experiments for a quasi-one-dimensional nozzle problem show that the second-order implicit TVD scheme produces a fairly rapid convergence rate and remains stable even when running with a Courant number of 10 to the 6th.

Yee, H. C.↗

NASA NTRS1985

Application of a viscous-inviscid interaction method to predict transonic separated flows

A viscous-inviscid interaction numerical procedure is used to predict turbulent transonic separated flows over axisymmetric bodies. The viscous flow solutions are obtained by solving the boundary-layer equations inversely in a coupled manner by a finite-difference scheme; the inviscid flow is determined by a finite-difference solution to the full-potential equation in conservative form. Parameters affecting the convergence rate of the interaction procedure are discussed. The predictions of a new turbulence model proposed by Johnson and King are compared with experimental data and numerical solutions to the Navier-Stokes equations for flows with large regions of separation.

Lee, D. S.↗

NASA NTRS1985

Research on output feedback control

In designing fixed order compensators, an output feedback formulation has been adopted by suitably augmenting the system description to include the compensator states. However, the minimization of the performance index over the range of possible compensator descriptions was impeded due to the nonuniqueness of the compensator transfer function. A controller canonical form of the compensator was chosen to reduce the number of free parameters to its minimal number in the optimization. In the MIMO case, the controller form requires a prespecified set of ascending controllability indices. This constraint on the compensator structure is rather innocuous in relation to the increase in convergence rate of the optimization. Moreover, the controller form is easily relatable to a unique controller transfer function description. This structure of the compensator does not require penalizing the compensator states for a nonzero or coupled solution, a problem that occurs when following a standard output feedback synthesis formulation.

Calise, A. J.↗

NASA NTRS1985

Marching iterative methods for the parabolized and thin layer Navier-Stokes equations

Downstream marching iterative schemes for the solution of the Parabolized or Thin Layer (PNS or TL) Navier-Stokes equations are described. Modifications of the primitive equation global relaxation sweep procedure result in efficient second-order marching schemes. These schemes take full account of the reduced order of the approximate equations as they behave like the SLOR for a single elliptic equation. The improved smoothing properties permit the introduction of Multi-Grid acceleration. The proposed algorithm is essentially Reynolds number independent and therefore can be applied to the solution of the subsonic Euler equations. The convergence rates are similar to those obtained by the Multi-Grid solution of a single elliptic equation; the storage is also comparable as only the pressure has to be stored on all levels. Extensions to three-dimensional and compressible subsonic flows are discussed. Numerical results are presented.

Israeli, M.↗