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NASA NTRS1981

An alternating-direction implicit algorithm for unsteady potential flow

An implicit finite-difference scheme is presented for the efficient computation of unsteady potential flow about airfoils. The formulation uses density and the velocity potential as dependent variables and is cast in conservation form to assure the theoretically correct determination of shockwave location and speed. To enable boundary conditions to be imposed directly on the airfoil surface, a time-varying sheared-rectilinear coordinate transformation is employed. Calculated time-history solutions on a pulsating airfoil are compared with the results of other unsteady transonic codes, including a previous method of the authors. The present method is demonstrated to be unconditionally stable and to give accurate solutions with sharply resolved shocks.

Chipman, R.↗

NASA NTRS1993

Structure of confined laminar spray diffusion flames: Numerical investigation

The structure of confined laminar spray diffusion flames is investigated numerically by solving the gas-phase conservation equations for mass species, continuity, momentum, and energy and the liquid-phase equations for droplet position, velocity, size, and temperature. A one-step global reaction scheme along with six equilibrium reactions are employed to model the flame chemistry. Monodisperse as well as polydisperse sprays are considered. The numerical results demonstrate that liquid spray flames substantially differ from gaseous flames in their structure, i.e., temperature, concentration, and velocity fields, shape, and dimensions under the same conditions. Spray flames are predicted to be taller and narrower than their counterpart gaseous ones and their shapes are almost cylindrical. This is in agreement with experimental observations. The numerical computations also show that the use of the equilibrium reactions with the one-step reaction scheme decreases the flame temperature compared to the one-step reaction scheme without the equilibrium reactions and more importantly increases the surface area of the flame zone due to a phenomenon termed 'equilibrium broadening.' The spray flames also possess a finite thickness with minimal overlap of the fuel and oxygen species. A case for which a fuel-mixture consisting of 20 to 80 percent gas-liquid by mass is introduced into the combustor is also investigated and compared with predictions using only gaseous or liquid fuel.

Mawid, M. A.↗

NASA NTRS2018

High-resolution Wave Propagation Method for Stratified Flows

The implementation of the multidimensional f-waves Riemann solver for the time-dependent, three-dimensional, nonhydrostatic, meso- and microscale atmospheric flows is described in detail. The Riemann solver employs flux-based wave decomposition (f-waves) for the calculation of Godunov fluxes in which the flux differences are written directly as the linear combination of the right eigenvectors of the hyperbolic system. The scheme incorporates the source term due to gravity without introducing discretization errors which is an important property in the context of atmospheric flows. The resulting flow solver is conservative, accurate, stable, and well-balanced. The implementation of the solver is evaluated using benchmark test cases for atmospheric dynamics.

Riemann problem↗

NASA NTRS1994

A double-moment multiple-phase four-class bulk ice scheme, Part 1: Description

A detailed ice-phase bulk microphysical scheme has been developed for simulating the hydrometeor distributions of convective and stratiform precipitation in different large-scale environmental conditions. The proposed scheme involves 90 distinct microphysical processes, which predict the mixing ratios and the number concentrations of small ice crystals, snow, graupel, and frozen drops/hail, as well as the mixing ratios of liquid water on wet precipitation ice (snow, graupel, frozen drops). The number of adjustable coefficients has been significantly reduced in comparison with other bulk schemes. Additional improvements have been made to the parameterization in the following areas: (1) representing small ice crystals with nonzero terminal fall velocities and dispersive size distributions, (2) accurate and computationally efficient calculations of precipitation collection processes, (3) reformulating the collection equation to prevent unrealistically large accretion rates, (4) more realistic conversion by riming between different classes of precipitation ice, (5) preventing unrealistically large rates of raindrop freezing and freezing of liquid water on ice, (6) detailed treatment of various rime-splintering ice multiplication mechanisms, (7) a simple representation of the Hobbs-Rangno ice enhancement process, (8) aggregation of small ice crystals and snow, and (9) allowing explicit competition between cloud water condensation and ice deposition rates rather than using saturation adjustment techniques. For the purposes of conserving the higher moments of the particle distributions, preserving the spectral widths (or slopes) of the particle spectra is shown to be more important than strict conservation of particle number concentration when parameterizing changes in ice-particle number concentrations due to melting, vapor transfer processes (sublimation of dry ice, evaporation from wet ice), and conversion between different hydrometeor species. The microphysical scheme is incorporated into a nonhydrostatic cloud model in Part 2 of this study. The model performed well in simulating the radar and microphysical structures of a midlatitude-continental squall lines and a tropical-maritime squall system with minimal tuning of the parameterization, even though the vertical profiles of radar reflectivity differed substantially between these storms.

Ferrier, Brad Shoenberg↗

NASA NTRS1987

A generalized procedure for constructing an upwind-based TVD scheme

A generalized formulation for constructing second- and higher-order accurate TVD (total variation diminishing) schemes is presented. A given scheme is made TVD by limiting antidiffusive flux differences with some nonlinear functions, so-called limiters. The general idea of the formulation and its mathematical proof of Harten's TVD conditions is shown by applying the Lax-Wendroff method to a scalar nonlinear equation and constant-coefficient system of conservation laws. For the system of equations, several definitions are derived for the argument used in the limiter function and present their performance to numerical experiments. Then the formulation is formally extended to the nonlinear system of equations. It is demonstrated that use of the present procedure allows easy conversion of existing central or upwind, and second- or higher-order differencing schemes so as to preserve monotonicity and to yield physically admissible solutions. The formulation is simple mathematically as well as numerically; neither matrix-vector multiplication nor Riemann solver is required. Roughly twice as much computational effort is needed as compared to conventional scheme. Although the notion of TVD is based on the initial value problem, application to the steady Euler equations of the formulation is also made. Numerical examples including various ranges of problems show both time- and spatial-accuracy in comparison with exact solutions.

Liou, Meng-Sing↗

NASA NTRS1983

Multi-grid calculation of three-dimensional transonic potential flows

A multi-grid algorithm has been developed to speed the iterative convergence of calculations for the transonic potential flow past swept wings and wing-fuselage combinations. The method is based upon a fully-conservative, finite-volume approximation to the steady potential equation which is second-order accurate everywhere in the flow field except near shock waves. The multi-grid scheme is incorporated within the framework of an alternating successive-line-overrelaxation (SLOR) solver of the difference equations. Computed results confirm the second-order accuracy of the scheme, and demonstrate the effectiveness of the multi-grid procedure.

Caughey, D. A.↗

NASA NTRS1987

Unsteady Euler solution of transonic helicopter rotor flow

A new time-accurate Euler method is presented for calculating the transonic flow over a lifting or nonlifting rotor blade in both hover and forward flight. The approach is to solve the conservative Euler equations in a rotor-fixed frame of reference using a finite-volume method. The discretized equations are solved by using a Runge-Kutta multistage scheme with a new higher-order implicit residual smoothing procedure. Preliminary results were compared with wind-tunnel data. In all the cases considered, good agreement was found with experimental data.

Chang, I-Chung↗

NASA NTRS1988

Computation of steady and unsteady vortex-dominated flows with shock waves

The unsteady Euler equations have been derived in the conservation form for the flow relative motion with respect to a rotating frame of reference. The resulting equations are solved by using a central-difference finite-volume scheme with four-state Runge-Kutta time stepping. For steady flow applications local time stepping is used, and for unsteady applications the minimum global time stepping is used. A three-dimensional fully vectorized computer program has been developed and applied to steady and unsteady maneuvering delta wings. The capability of the three-dimensional program has been demonstrated for a rigid sharp-edged delta wing undergoing uniform rolling in a conical flow and rolling oscillations in a locally conical flow.

Kandil, Osama A.↗

NASA NTRS1991

A detailed numerical investigation of Burke-Schumann gaseous and spray flames

The classical Burke-Schumann gaseous diffusion flame, and the effect of introducing fuel in the form of liquid sprays are investigated numerically. The time-dependent two-dimensional, axisymmetric conservation equations are solved for the gas phase. A Lagrangian approach is used for the dispersed phase. The chemical reactions are modeled through the one-step global reaction scheme. The numerical results show that the computed flame structure is significantly different from that given by the analytical solution. The computed flame is about 15 percent longer and 5 percent narrower than the classical Burke-Schumann flame. The predicted flame shape is, however, in better agreement with experimental observations than the Burke-Schumann flame. The effects of introducing liquid sprays is found to produce thinner and longer flames than its counterpart gaseous flame. A detailed comparison of the structures of gaseous and spray flames is made, and the differences are discussed.

Mawid, M. A.↗

NASA NTRS1985

Evaluate interference in digital channels

Any future mobile satellite service (MSS) which is to provide simultaneous mobile communications for a large number of users will have to make very efficient use of the spectrum. As the spectrum available for an MSS is limited, the system's channels should be packed as closely together as possible, with minimum-width guard bands. In addition the employment of frequency reuse schemes is an important factor. Difficulties regarding these solutions are related to the introduction of interference in the link. A balance must be achieved between the competing aims of spectrum conservation and low interference. While the interference phenomenon in narrowband FM voice channels is reasonably well understood, very little effort, however, has been devoted to the problem in digital radios. Attention is given to work, which illuminates the effects of cochannel and adjacent channel interference on digital FM (FSK) radios.

Davarian, F.↗

NASA NTRS1986

The discrete one-sided Lipschitz condition for convex scalar conservation laws

Physical solutions to convex scalar conservation laws satisfy a one-sided Lipschitz condition (OSLC) that enforces both the entropy condition and their variation boundedness. Consistency with this condition is therefore desirable for a numerical scheme and was proved for both the Godunov and the Lax-Friedrichs scheme--also, in a weakened version, for the Roe scheme, all of them being only first order accurate. A new, fully second order scheme is introduced here, which is consistent with the OSLC. The modified equation is considered and shows interesting features. Another second order scheme is then considered and numerical results are discussed.

Brenier, Yann↗

NASA NTRS1991

Interface conditions for domain decomposition with radical grid refinement

Interface conditions for coupling the domains in a physically motivated domain decomposition method are discussed. The domain decomposition is based on an asymptotic-induced method for the numerical solution of hyperbolic conservation laws with small viscosity. The method consists of multiple stages. The first stage is to obtain a first approximation using a first-order method, such as the Godunov scheme. Subsequent stages of the method involve solving internal-layer problem via a domain decomposition. The method is derived and justified via singular perturbation techniques.

Scroggs, Jeffrey S.↗

NASA NTRS1999

High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.

Barth, Timothy↗

NASA NTRS1975

A fast semidirect method for computing transonic aerodynamic flows

A fast, semidirect, iterative computational method, previously introduced for finite-difference solution of subsonic and slightly supercritical flow over airfoils, is extended both to apply to strongly supercritical conditions and to include full second-order accuracy in computing inviscid flows over airfoils. The nonlinear small-disturbance equations are solved iteratively by a direct, linear, elliptic solver. General, fully conservative, type-dependent difference equations are formulated, including parabolic- and shock-point transition operators that provide consistency with the integral conservation laws. These equations specialize to either first-order or to fully second-order-accurate equations. Various free parameters are evaluated for rapid convergence of the first-order scheme. Resulting pressure distributions and computing times are compared with the improved Murman-Cole line-relaxation method.

Martin, E. D.↗

NASA NTRS1981

Analysis of cracks emanating from a circular hole in unidirectional fiber reinforced composites, part 2

An analytical method is developed for cracks emanating from a circular hole in an off-axis unidirectional fiber-reinforced composite. The method which is formulated by using conservation laws of elasticity and fundamental relationships in anisotropic fracture mechanics, provides a convenient and accurate means to examine the complicated crack behavior, when used in conjunction with a suitable numerical scheme such as the finite element method. The formulation is eventually reduced to a system of linear algebraic equations of mixed-mode stress intensity factors. Fracture parameters, describing crack-tip deformation and fracture in the composite, are obtained explicitly. Effects of material anisotropy and crack/hole geometry are examined also. Of particular interest are the energy release rates associated with crack extension; their values are evaluated for various cases. Results show that mixed-mode stress intensity factors and energy release rates associated with the cracks emanating from a hole change very appreciably with fiber orientation in the composite. K sub 1 and G increase monotonically with increasing theta; but K sub 2 reaches its maximum at theta = 45 deg, and then decreases gradually as theta increases further.

Wang, S. S.↗

NASA NTRS1990

Three-dimensional simulation of vortex breakdown

The integral form of the complete, unsteady, compressible, three-dimensional Navier-Stokes equations in the conservation form, cast in generalized coordinate system, are solved, numerically, to simulate the vortex breakdown phenomenon. The inviscid fluxes are discretized using Roe's upwind-biased flux-difference splitting scheme and the viscous fluxes are discretized using central differencing. Time integration is performed using a backward Euler ADI (alternating direction implicit) scheme. A full approximation multigrid is used to accelerate the convergence to steady state.

Kuruvila, G.↗

NASA NTRS1990

Asymptotic-induced numerical methods for conservation laws

Asymptotic-induced methods are presented for the numerical solution of hyperbolic conservation laws with or without viscosity. The methods consist of multiple stages. The first stage is to obtain a first approximation by using a first-order method, such as the Godunov scheme. Subsequent stages of the method involve solving internal-layer problems identified by using techniques derived via asymptotics. Finally, a residual correction increases the accuracy of the scheme. The method is derived and justified with singular perturbation techniques.

Garbey, Marc↗

NASA NTRS1992

Development of a fractional-step method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases.

Rosenfeld, Moshe↗