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At least 469 records · Page 26

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NASA NTRS1990

Viscous shock profiles and primitive formulations

Weak solutions of hyperbolic systems in primitive (non-conservation) form for which a consistent conservation form exists are considered. It is shown that primitive formulations, shock relations are not uniquely defined by the states to either side of the shock but also depend on the viscous path connecting the two. Scheme-dependent high order correction terms are derived that enforce consistent viscous shock profiles. The resulting primitive algorithm is conservative to the order of approximation. One dimensional Euler calculations of flows containing strong shocks clearly show that conservation errors in primitive flow calculations are of comparable quality.

Karni, S.↗

NASA NTRS1991

A staggered mesh finite difference scheme for the computation of hypersonic Euler flows

A shock capturing finite difference method for systems of hyperbolic conservation laws is presented which avoids the need to solve Riemann problems while being competitive in performance with other current methods. A staggered spatial mesh is employed, so that complicated nonlinear waves generated at cell interfaces are averaged over cell interiors at the next time level. The full method combines to form a conservative version of the modified method of characteristics. The advantages of the method are discussed, and numerical results are presented for the two-dimensional double ellipse problem.

Sanders, Richard↗

DOE OSTI2021

Mass-conserving implicit–explicit methods for coupled compressible Navier–Stokes equations

Earth system models are composed of coupled components that separately model systems such as the global atmosphere, ocean, and land surface. While these components are well developed, coupling them in a single system can be a significant challenge. Computational efficiency, accuracy, and stability are principal concerns. In this study we focus on these issues. In particular, implicit–explicit (IMEX) tight and loose coupling strategies are explored for handling different time scales. For a simplified model for the air–sea interaction problem, we consider coupled compressible Navier–Stokes equations with an interface condition. Under the rigid-lid assumption, horizontal momentum and heat flux are exchanged through the interface. Several numerical experiments are presented to demonstrate the stability of the coupling schemes. Here, we show both numerically and theoretically that our IMEX coupling methods are mass conservative for a coupled compressible Navier–Stokes system with the rigid-lid condition

42 ENGINEERING↗

DOE OSTI2020

The geometric theory of charge conservation in particle-in-cell simulations

In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods’ gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether’s second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms – gauge-compatible splitting methods – that exactly preserve the momentum map associated with a Hamiltonian system’s gauge symmetry – even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Finally, our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

A numerical method for simulating variable density flows in membrane desalination systems

Here, we present a novel method for simulating unsteady, variable density, fluid flows in membrane desalination systems. By assuming the density varies only with concentration and temperature, the scheme decouples the solution of the governing equations into two sequential blocks. The first solves the governing equations for the temperature and concentration fields, which are used to compute all thermophysical properties. The second block solves the conservation of mass and momentum equations for the velocity and pressure. We show that this is computationally more efficient than schemes that iterate over the full coupled equations in one block. We verify that the method achieves second-order spatial-temporal accuracy, and we use the method to investigate buoyancy-driven convection in a desalination process called vacuum membrane distillation. Specifically, we show that with gravity properly oriented, variations in temperature and concentration can trigger a double-diffusive instability that enhances mixing and improves water recovery. We also show that the instability can be strengthened by providing external heating.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1991

Nonequilibrium thermo-chemical calculations using a diagonal implicit scheme

A recently developed computer program for hypersonic vehicle flow analysis is described. The program uses a diagonal implicit algorithm to solve the equations of viscous flow for a gas in thermochemical nonequilibrium. The diagonal scheme eliminates the expense of inverting large block matrices that arise when species conservation equations are introduced. The program uses multiple zones of grids patched together and includes radiation wall and rarefied gas boundary conditions. Solutions are presented for hypersonic flows of air and hydrogen air mixtures.

Imlay, Scott T.↗

NASA NTRS1991

HANA - A three-dimensional Navier-Stokes code for chemically reacting flows

A recently developed computer program for analyzing chemically reacting flows is described. The program uses a diagonal implicit algorithm to solve the equations of viscous flow for a gas in thermochemical nonequilibrium. The diagonal scheme eliminates the expense of inverting large block matrices that arise when species conservation equations are introduced. The program uses multiple zones of grids patched together and includes a k-epsilon turbulence model. Solutions are presented for hypersonic flows of air, hypersonic flows of Martian atmospheric gases, combusting flows of hydrogen/air mixtures, and combusting flows of methane/air mixtures.

Imlay, Scott T.↗

NASA NTRS1995

Heliospheric Termination Shock Motion Due to Fluctuations in the Solar Wind Upstream Conditions: Spherically Symmetric Model

Large-scale fluctuations in the solar wind plasma upstream of the heliospheric termination shock (TS) will cause inward and outward motions of the shock. Using numerical techniques, we extend an earlier strictly one-dimensional (planar) analytic gas dynamic model to spherical symmetry to investigate the features of global behavior of shock motion. Our starting point is to establish a steady numerical solution of the gasdynamic equations describing the interaction between the solar wind and the interstellar medium. We then introduce disturbances of the solar wind dynamic pressure at an inner boundary, and follow the subsequent evolution of the system, especially the motion of the termination shock. Our model solves spherically symmetric gasdynamic equations as an initial-boundary value problem. The equations in conservative form are solved using a fully implicit Total Variation Diminishing (TVD) upwind scheme with Roe-type Riemann solver. Boundary conditions are given by the solar wind parameters on an inner spherical boundary, where they are allowed to vary with time for unsteady calculations, and by a constant pressure (roughly simulating the effect of the local interstellar medium) on an outer boundary. We find that immediately after the interaction, the shock moves with speeds given by the earlier analogous analytic models. However, as the termination shock propagates it begins to slow down, seeking a new equilibrium position. In addition, the disturbance transmitted through the TS, either a shock or rarefaction wave, will encounter the heliopause boundary and be reflected back. The reflected signal will encounter the TS, causing it to oscillate. The phenomenon may be repeated for a number of reflections, resulting in a "ringing" of the outer heliosphere.

Ratkiewicz, R.↗

NASA NTRS1983

High resolution applications of the Osher upwind scheme for the Euler equations

The 'Osher' scheme was introduced by Osher (1981). It represents an upwind finite-difference method for hyperbolic systems of conservation laws, including the Euler equations. In studies conducted by Osher (1981) and Osher and Solomon (1982), the method was applied to the nonisentropic form of the Euler equations in one dimension and the isentropically restricted form in two spatial dimensions, both in Cartesian coordinates. Chakravarthy and Osher (1982) have shown an approach for extending the Osher scheme to the Euler equations written for general geometries, taking into account the use of mappings to arbitrary curvilinear coordinate systems. The present investigation is concerned with the high resolution extension of the Osher scheme to second-order accuracy. Results are presented for several example problems, giving attention to quasi-one-dimensional Laval nozzle flow, a one-dimensional shock tube problem, and supersonic flow over a cylinder.

Chakravarthy, S. R.↗

NASA NTRS1998

A Second Law Based Unstructured Finite Volume Procedure for Generalized Flow Simulation

An unstructured finite volume procedure has been developed for steady and transient thermo-fluid dynamic analysis of fluid systems and components. The procedure is applicable for a flow network consisting of pipes and various fittings where flow is assumed to be one dimensional. It can also be used to simulate flow in a component by modeling a multi-dimensional flow using the same numerical scheme. The flow domain is discretized into a number of interconnected control volumes located arbitrarily in space. The conservation equations for each control volume account for the transport of mass, momentum and entropy from the neighboring control volumes. In addition, they also include the sources of each conserved variable and time dependent terms. The source term of entropy equation contains entropy generation due to heat transfer and fluid friction. Thermodynamic properties are computed from the equation of state of a real fluid. The system of equations is solved by a hybrid numerical method which is a combination of simultaneous Newton-Raphson and successive substitution schemes. The paper also describes the application and verification of the procedure by comparing its predictions with the analytical and numerical solution of several benchmark problems.

Majumdar, Alok↗

NASA NTRS1975

On the computation of two- and three-dimensional steady transonic flows by relaxation methods

The paper is concerned with the application of the Murman and Cole (1971) relaxation scheme to steady, inviscid transonic flow problems in two and three dimensions. This scheme, which automatically accounts for weak shock waves, uses separate difference operators in elliptic and hyperbolic regions. The details of the scheme are described in terms of the original small disturbance formulation of Murman and Cole. In particular, Murman's recent (1973) introduction of fully conservative difference operators to obtain the correct shock jumps is examined. The extension to treating the exact isentropic equation is then covered with special attention given to Jameson's (to appear) rotated difference scheme for supersonic flow regions. The bulk of the discussion is related to two-dimensional procedures, and some comparisons with experiment are made, with emphasis on the effects of viscosity and wind-tunnel walls. Application of the Murman-Cole scheme is then discussed for small disturbances in three dimensions.

Bailey, F. R.↗

NASA NTRS2002

Analysis of High Order Difference Methods for Multiscale Complex Compressible Flows

Accurate numerical simulations of complex multiscale compressible viscous flows, especially high speed turbulence combustion and acoustics, demand high order schemes with adaptive numerical dissipation controls. Standard high resolution shock-capturing methods are too dissipative to capture the small scales and/or long-time wave propagations without extreme grid refinements and small time steps. An integrated approach for the control of numerical dissipation in high order schemes with incremental studies was initiated. Here we further refine the analysis on, and improve the understanding of the adaptive numerical dissipation control strategy. Basically, the development of these schemes focuses on high order nondissipative schemes and takes advantage of the progress that has been made for the last 30 years in numerical methods for conservation laws, such as techniques for imposing boundary conditions, techniques for stability at shock waves, and techniques for stable and accurate long-time integration. We concentrate on high order centered spatial discretizations and a fourth-order Runge-Kutta temporal discretizations as the base scheme. Near the bound-aries, the base scheme has stable boundary difference operators. To further enhance stability, the split form of the inviscid flux derivatives is frequently used for smooth flow problems. To enhance nonlinear stability, linear high order numerical dissipations are employed away from discontinuities, and nonlinear filters are employed after each time step in order to suppress spurious oscillations near discontinuities to minimize the smearing of turbulent fluctuations. Although these schemes are built from many components, each of which is well-known, it is not entirely obvious how the different components be best connected. For example, the nonlinear filter could instead have been built into the spatial discretization, so that it would have been activated at each stage in the Runge-Kutta time stepping. We could think of a mechanism that activates the split form of the equations only at some parts of the domain. Another issue is how to define good sensors for determining in which parts of the computational domain a certain feature should be filtered by the appropriate numerical dissipation. For the present study we employ a wavelet technique introduced in as sensors. Here, the method is briefly described with selected numerical experiments.

Sjoegreen, Bjoern↗

DOE OSTI2020

Energy conservative SBP discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids

In this work, we develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a Cartesian basis decomposition on rotated grids. The discretization is based on high order Summation-By-Parts (SBP) operators and preserves both symmetry and positive definiteness of the contravariant metric tensor. To improve accuracy and decrease computational cost, we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization. Bounds are derived that yield a point-wise condition that can be evaluated to check for stability of the modified discretization. This condition shows that the interpolation operators should be constructed such that their norm is close to one.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Crystallization of ApoA1 and ApoE4 Nanolipoprotein Particles and Initial XFEL-Based Structural Studies

Nanolipoprotein particles (NLPs), also called “nanodiscs”, are discoidal particles with a patch of lipid bilayer corralled by apolipoproteins. NLPs have long been of interest due to both their utility as membrane-model systems into which membrane proteins can be inserted and solubilized and their physiological role in lipid and cholesterol transport via high-density lipoprotein (HDL) and low-density lipoprotein (LDL) maturation, which are important for human health. Serial femtosecond crystallography (SFX) at X-ray free electron lasers (XFELs) is a powerful approach for structural biology of membrane proteins, which are traditionally difficult to crystallize as large single crystals capable of producing high-quality diffraction suitable for structure determination. To facilitate understanding of the specific role of two apolipoprotein/lipid complexes, ApoA1 and ApoE4, in lipid binding and HDL/LDL particle maturation dynamics, and to develop new SFX methods involving NLP membrane protein encapsulation, we have prepared and crystallized homogeneous populations of ApoA1 and ApoE4 NLPs. Crystallization of empty NLPs yields semi-ordered objects that appear crystalline and give highly anisotropic and diffuse X-ray diffraction, similar to fiber diffraction. Several unit cell parameters were approximately determined for both NLPs from these measurements. Thus, low-background, sample conservative methods of delivery are critical. Here we implemented a fixed target sample delivery scheme utilizing the Roadrunner fast-scanning system and ultra-thin polymer/graphene support films, providing a low-volume, low-background approach to membrane protein SFX. This study represents initial steps in obtaining structural information for ApoA1 and ApoE4 NLPs and developing this system as a supporting scaffold for future structural studies of membrane proteins crystalized in a native lipid environment.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1978

Coefficient matrices for implicit finite difference solution of the inviscid fluid conservation law equations

Although the Navier-Stokes equations describe most flows of interest in aerodynamics, the inviscid conservation law equations may be used for small regions with viscous forces. Thus, Euler equations and several time-accurate finite difference procedures, explicit and implicit, are discussed. Although implicit techniques require more computational work, they permit larger time steps to be taken without instability. It is noted that the Jacobian matrices for Euler equations in conservation-law form have certain eigenvalue-eigenvector properties which may be used to construct conservative-form coefficient matrices. This reduces the computation time of several implicit and semiimplicit schemes. Extensions of the basic approach to other areas are suggested.

Steger, J. L.↗

NASA NTRS1986

Interface procedures for overlapping grids

The solution of partial differential equations by means of either implicit or conservative explicit finite-difference methods, on the one hand, or multicomponent overlapping grid schemes on the other, is presently subjected to interpolation at grid boundaries in an attempt to eliminate the time lag. The accuracy of the transient solution of a hyperbolic or parabolic partial differential equation is dependent on the procedures used to transfer information between grids in a composite grid system. The present techniques reduce numerical solution errors.

Mastin, C. Wayne↗

NASA NTRS1990

Three-dimensional Euler time accurate simulations of fan rotor-stator interactions

A numerical method useful to describe unsteady 3-D flow fields within turbomachinery stages is presented. The method solves the compressible, time dependent, Euler conservation equations with a finite volume, flux splitting, total variation diminishing, approximately factored, implicit scheme. Multiblock composite gridding is used to partition the flow field into a specified arrangement of blocks with static and dynamic interfaces. The code is optimized to take full advantage of the processing power and speed of the Cray Y/MP supercomputer. The method is applied to the computation of the flow field within a single stage, axial flow fan, thus reproducing the unsteady 3-D rotor-stator interaction.

Boretti, A. A.↗

NASA NTRS1990

Numerical simulation of steady and unsteady asymmetric vortical flow

The unsteady, compressible, thin-layer, Navier-Stokes equations are solved to simulate steady and unsteady, asymmetric, vortical laminar flow around cones at high incidences and supersonic Mach numbers. The equations are solved by using an implicit, upwind, flux-difference splitting, finite-volume scheme. Locally conical flows are assumed and the solutions are obtained by forcing the conserved components of the flowfield vector to be equal at two axial stations located at 0.95 and 1.0. Computational examples cover steady and unsteady asymmetric flows around a circular cone and its control using side strakes. Unsteady asymmetric flows are also presented for elliptic- and diamond-section cones, which model asymmetric vortex shedding around round and sharp-edged delta wing flows.

Kandil, O. A.↗