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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

The small amplitude magnetohydrodynamic Riemann problem

The small-amplitude MHD Riemann problem is studied using the Cohen-Kulsrud-Burgers equations. Unlike the coplanar Riemann problem, the evolution of noncoplanar Riemann problems is not self-similar and its flow structures could change in time. But its large-time behavior is very simple and a time-dependent 2 - 3 intermediate shock is always involved for the noncoplanar field rotations. The time-dependent 2 - 3 intermediate shock has a well-defined structure and exists for any degree of field rotation.

Wu, C. C.↗
NASA NTRS1986

A Riemann solver based on a global existence proof for the Riemann problem

Godunov's method and several other methods for computing solutions to the equations of gas dynamics use Riemann solvers to resolve discontinuities at the interface between cells. A new method is proposed here for solving the Riemann problem based on a global existence proof for the solution to the Riemann problem. The method is found to be very reliable and computationally efficient.

Dutt, P.↗
DOE OSTI2024

Vanishing adsorption limit of Riemann problem solutions for the polymer model

Here we examine the vanishing adsorption limit of solutions of Riemann problems for the Glimm–Isaacson model of chemical flooding of a petroleum reservoir. A contact discontinuity is deemed admissible if it is the limit of traveling waves or rarefaction waves for an augmented system that accounts for weak chemical adsorption onto the rock. We prove that this criterion justifies the admissibility criteria adopted previously by Keyfitz–Kranzer, Isaacson–Temple and de Souza–Marchesin, provided that the fractional flow function depends monotonically on chemical concentration. We also demonstrate that the adsorption criterion selects the undercompressive contact discontinuities required to solve the general Riemann problem in an example model with non-monotone dependence.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1989

The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations

Simple inequalities for the Riemann problem for a Hamilton-Jacobi equation in N space dimension when neither the initial data nor the Hamiltonian need be convex (or concave) are presented. The initial data is globally continuous, affine in each orthant, with a possible jump in normal derivative across each coordinate plane, x sub i = 0. The inequalities become equalities wherever a maxmin equals a minmax and thus an exact closed form solution to this problem is then obtained.

Osher, Stanley↗
NASA NTRS1995

Magnetohydrodynamic Riemann problem and the structure of the magnetic reconnection layer

We present a complete solution for a set of magnetohydrodynamic (MHD) Riemann problems in which the upstream and downstream states have the same total pressure, and in which the normal component of the magnetic field is very small. These solutions are pertinent to subfast flows in the earth's magnetic tail and near the magnetopause. In a coplanar situation a family of solutions exists that depend on two parameters as well as on dissapation mechanisms. In the parallel case the transverse magnetic field either does not change direction or changes the direction twice by involving two intermediate shocks. In the antiparallel case an intermediate shock is always required, except when the solution consists of two switch-off shocks. In a noncoplanar case the solution is not self-similar as a function of x/t, but continues to evolve. At early times the evolution is similar to the coplanar case. In general two time-dependent intermediate shocks are required to rotate the magnetic fields. The velocity shear has a strong effect on the Riemann solution. In some cases no Riemann solution can exist because of the cavitation caused by the slow refraction waves. The calculated magnetopause structure resembles the observed structure for northward interplanetary magnetic field (IMF). However, for southward IMF, the MHD result shows the existence of a depletion layer, which is not supported by observations. We also show that on the magnetosheath side, the Walen relation, which is exact for a rotational discontinuity, can also be well satisfied by a slow shock, an intermediate shock, or the head of a slow rarefaction wave.

Wu, C. C.↗
NASA NTRS1991

The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations

Simple inequalities are presented for the viscosity solution of a Hamilton-Jacobi equation in N space dimensions when neither the initial data nor the Hamiltonian need be convex (or concave). The initial data are uniformly Lipschitz and can be written as the sum of a convex function in a group of variables and a concave function in the remaining variables, therefore including the nonconvex Riemann problem. The inequalities become equalities wherever a 'maxmin' equals a 'minmax', and thus a representation formula for this problem is obtained, generalizing the classical Hopi formulas.

Bardi, Martino↗
NASA NTRS1993

Flux-difference split parabolized Navier-Stokes algorithm for non-equilibrium chemically reacting flows

A flux-difference split explicit finite-difference algorithm is presented for solving the parabolized form of the equations governing three-dimensional nonequilibrium chemically reacting flows. The algorithm is based on an explicit noniterative, upwind space-marching scheme developed by Korte, but differs in that the unsteady Riemann problem, rather than the steady Riemann problem, is solved. The algorithm allows either a second or an approximately third-order accurate upwind treatment of the convection terms by employing the unsteady approximate Riemann solver of Roe. The source terms of the species transport equations are treated in either an explicit or implicit manner, and the species diffusion terms are modeled with either a Fickian or a multicomponent model. A validation of the algorithm is performed by comparing computational results with the 2-D Mach 14, 15 degree compression-corner data of Holden. The three-dimensional capability of the algorithm is demonstrated by computing Mach 2.7 flow over a swept wedge scramjet fuel injector, and three-dimensional reacting flow capability is demonstrated by a computing a shock-jet interaction concept for mixing and combustion enhancement.

White, J. A.↗
NASA NTRS2017

Feature Detection and Curve Fitting Using Fast Walsh Transforms for Shock Tracking: Applications

Walsh functions form an orthonormal basis set consisting of square waves. Square waves make the system well suited for detecting and representing functions with discontinuities. Given a uniform distribution of 2p cells on a one-dimensional element, it has been proven that the inner product of the Walsh Root function for group p with every polynomial of degree < or = (p - 1) across the element is identically zero. It has also been proven that the magnitude and location of a discontinuous jump, as represented by a Heaviside function, are explicitly identified by its Fast Walsh Transform (FWT) coefficients. These two proofs enable an algorithm that quickly provides a Weighted Least Squares fit to distributions across the element that include a discontinuity. The detection of a discontinuity enables analytic relations to locally describe its evolution and provide increased accuracy. Time accurate examples are provided for advection, Burgers equation, and Riemann problems (diaphragm burst) in closed tubes and de Laval nozzles. New algorithms to detect up to two C0 and/or C1 discontinuities within a single element are developed for application to the Riemann problem, in which a contact discontinuity and shock wave form after the diaphragm bursts.

Gnoffo, Peter A.↗
NASA NTRS1992

Structure and evolution of time-dependent intermediate shocks

A quantitative description of time-dependent intermediate shocks is formulated using the Cohen-Kulsrud-Burgers equations. In noncoplanar Riemann problems, time-dependent two-three transition intermediate shocks evolve in time as a localized self-similar structure whose strength decreases as 1/the square root of t, and whose width expands as the square root of t. Time-dependent intermediate shocks offer a way of solving the noncoplanar MHD Riemann problem.

Wu, C. C.↗
NASA NTRS1995

Time accurate application of the MacCormack 2-4 scheme on massively parallel computers

Many recent computational efforts in turbulence and acoustics research have used higher order numerical algorithms. One popular method has been the explicit MacCormack 2-4 scheme. The MacCormack 2-4 scheme is second order accurate in time and fourth order accurate in space, and is stable for CFL's below 2/3. Current research has shown that the method can give accurate results but does exhibit significant Gibbs phenomena at sharp discontinuities. The impact of adding Jameson type second, third, and fourth order artificial viscosity was examined here. Category 2 problems, the nonlinear traveling wave and the Riemann problem, were computed using a CFL number of 0.25. This research has found that dispersion errors can be significantly reduced or nearly eliminated by using a combination of second and third order terms in the damping. Use of second and fourth order terms reduced the magnitude of dispersion errors but not as effectively as the second and third order combination. The program was coded using Thinking Machine's CM Fortran, a variant of Fortran 90/High Performance Fortran, and was executed on a 2K CM-200. Simple extrapolation boundary conditions were used for both problems.

Hudson, Dale A.↗
NASA NTRS2014

A Walsh Function Module Users' Manual

The solution of partial differential equations (PDEs) with Walsh functions offers new opportunities to simulate many challenging problems in mathematical physics. The approach was developed to better simulate hypersonic flows with shocks on unstructured grids. It is unique in that integrals and derivatives are computed using simple matrix multiplication of series representations of functions without the need for divided differences. The product of any two Walsh functions is another Walsh function - a feature that radically changes an algorithm for solving PDEs. A FORTRAN module for supporting Walsh function simulations is documented. A FORTRAN code is also documented with options for solving time-dependent problems: an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the usage of the Walsh function module including such features as operator overloading, Fast Walsh Transforms in multi-dimensions, and a Fast Walsh reciprocal.

Gnoffo, Peter A.↗
NASA NTRS2003

Uncertainty Propagation for Turbulent, Compressible Flow in a Quasi-1D Nozzle Using Stochastic Methods

This paper describes a fully spectral, Polynomial Chaos method for the propagation of uncertainty in numerical simulations of compressible, turbulent flow, as well as a novel stochastic collocation algorithm for the same application. The stochastic collocation method is key to the efficient use of stochastic methods on problems with complex nonlinearities, such as those associated with the turbulence model equations in compressible flow and for CFD schemes requiring solution of a Riemann problem. Both methods are applied to compressible flow in a quasi-one-dimensional nozzle. The stochastic collocation method is roughly an order of magnitude faster than the fully Galerkin Polynomial Chaos method on the inviscid problem.

Zang, Thomas A.↗
NASA NTRS2015

Unsteady Solution of Non-Linear Differential Equations Using Walsh Function Series

Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.

Gnoffo, Peter A.↗
NASA NTRS1978

Lagrangian computation of inviscid compressible flows

A Lagrangian method is developed to solve the Euler equations of gas dynamics. The solution of the equations is obtained by a numerical computation with the well-known Flux-Corrected-Transport (FCT) numerical method. This procedure is modified so that the boundary treatment is accurate and relatively simple. Shock waves and other flow discontinuities are captured monotonically without any type of fitting procedures. The Lagrangian method is employed so that the problem of mesh generation is completely avoided. The method is applicable to all Mach numbers except the low subsonic range where compressibility effects are small. The method is applied to a one-dimensional Riemann problem (shock tube) and to a two-dimensional supersonic channel flow with reflecting shock waves.

Klopfer, G. H.↗
DOE OSTI2025

Advection algorithms for quantum neutrino moment transport

Neutrino transport in compact objects is an inherently challenging multidimensional problem. Here, this difficulty is compounded if one includes flavor transformation—an intrinsically quantum phenomenon requiring one to follow the coherence between flavors and thus necessitating the introduction of complex numbers. To reduce the computational burden, simulations of compact objects that include neutrino transport often make use of momentum-angle-integrated moments (the lowest order ones being commonly referred to as the energy density and flux) and these quantities can be generalized to include neutrino flavor, i.e., they become quantum moments. Numerous finite-volume approaches to solving the moment evolution equations for classical neutrino transport have been developed based on solving a Riemann problem at cell interfaces. In this paper we describe our generalization of a Riemann solver for quantum moments, specifically decomposing complex numbers in terms of a (signed) magnitude and phase instead of real and imaginary parts. We then test our new algorithm in numerous cases showing a neutrino fast flavor instability, varying from toy models with analytic solutions to snapshots from neutron star merger simulations. Compared to previous algorithms for neutrino transport with flavor mixing, we find uniformly smaller growth rates of the flavor transformation along with concomitantly larger length-scales, and that the results are a better match with the growth rates seen from multiangle codes.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Stochastic finite volume method for uncertainty quantification of transient flow in gas pipeline networks

We develop a weakly intrusive framework to simulate the propagation of uncertainty in solutions of generic hyperbolic partial differential equation systems on graph-connected domains with nodal coupling and boundary conditions. The method is based on the Stochastic Finite Volume (SFV) approach and can be applied for uncertainty quantification (UQ) of the dynamical state of fluid flow over actuated transport networks. The numerical scheme has specific advantages for modeling intertemporal uncertainty in time-varying boundary parameters, which cannot be characterized by strict upper and lower (interval) bounds. We describe the scheme for a single pipe, and then formulate the controlled junction Riemann problem (JRP) that enables the extension to general network structures. In conclusion, we demonstrate the method's capabilities and performance characteristics using a standard benchmark test network.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Computational shock formation & development: An arbitrary Lagrangian-Eulerian characteristics approach [Slides]

We propose a new computational shock formation-development algorithm. We use a “good” geometry adapted to the evolving solution, along with a good set of variables defined in this geometry. We are able to: (1) Accurately capture the pre-shock; (2) Track distinguished characteristics and capture weak discontinuities; (3) Approximate solutions to classical Riemann problems; and (4) Accurately solve challenging problems for which standard methods fail.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1974

Elementary solutions of coupled model equations in the kinetic theory of gases

The method of elementary solutions is employed to solve two coupled integrodifferential equations sufficient for determining temperature-density effects in a linearized BGK model in the kinetic theory of gases. Full-range completeness and orthogonality theorems are proved for the developed normal modes and the infinite-medium Green's function is constructed as an illustration of the full-range formalism. The appropriate homogeneous matrix Riemann problem is discussed, and half-range completeness and orthogonality theorems are proved for a certain subset of the normal modes. The required existence and uniqueness theorems relevant to the H matrix, basic to the half-range analysis, are proved, and an accurate and efficient computational method is discussed. The half-space temperature-slip problem is solved analytically, and a highly accurate value of the temperature-slip coefficient is reported.

Kriese, J. T.↗