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NASA NTRS2004

On controlling nonlinear dissipation in high order filter methods for ideal and non-ideal MHD

The newly developed adaptive numerical dissipation control in spatially high order filter schemes for the compressible Euler and Navier-Stokes equations has been recently extended to the ideal and non-ideal magnetohydrodynamics (MHD) equations. These filter schemes are applicable to complex unsteady MHD high-speed shock/shear/turbulence problems. They also provide a natural and efficient way for the minimization of Div(B) numerical error. The adaptive numerical dissipation mechanism consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. The numerical dissipation considered consists of high order linear dissipation for the suppression of high frequency oscillation and the nonlinear dissipative portion of high-resolution shock-capturing methods for discontinuity capturing. The applicable nonlinear dissipative portion of high-resolution shock-capturing methods is very general. The objective of this paper is to investigate the performance of three commonly used types of nonlinear numerical dissipation for both the ideal and non-ideal MHD.

Yee, H. C.↗

NASA NTRS2009

Adiabatic Shock Capturing in Perfect Gas Hypersonic Flows

This paper considers the streamline-upwind Petrov/Galerkin (SUPG) method applied to the compressible Euler and Navier-Stokes equations in conservation-variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, is briefly reviewed. Of particular interest is the behavior of the shock capturing operator, which is required to regularize the scheme in the presence of strong, shock-induced gradients. A standard shock capturing operator which has been widely used in previous studies by several authors is presented and discussed. Specific modifications are then made to this standard operator which are designed to produce a more physically consistent discretization in the presence of strong shock waves. The actual implementation of the term in a finite dimensional approximation is also discussed. The behavior of the standard and modified scheme is then compared for several supersonic/hypersonic flows. The modified shock capturing operator is found to preserve enthalpy in the inviscid portion of the flowfield substantially better than the standard operator.

Kirk, Benjamin S.↗

NASA NTRS2009

Axial and Centrifugal Compressor Mean Line Flow Analysis Method

This paper describes a method to estimate key aerodynamic parameters of single and multistage axial and centrifugal compressors. This mean-line compressor code COMDES provides the capability of sizing single and multistage compressors quickly during the conceptual design process. Based on the compressible fluid flow equations and the Euler equation, the code can estimate rotor inlet and exit blade angles when run in the design mode. The design point rotor efficiency and stator losses are inputs to the code, and are modeled at off design. When run in the off-design analysis mode, it can be used to generate performance maps based on simple models for losses due to rotor incidence and inlet guide vane reset angle. The code can provide an improved understanding of basic aerodynamic parameters such as diffusion factor, loading levels and incidence, when matching multistage compressor blade rows at design and at part-speed operation. Rotor loading levels and relative velocity ratio are correlated to the onset of compressor surge. NASA Stage 37 and the three-stage NASA 74-A axial compressors were analyzed and the results compared to test data. The code has been used to generate the performance map for the NASA 76-B three-stage axial compressor featuring variable geometry. The compressor stages were aerodynamically matched at off-design speeds by adjusting the variable inlet guide vane and variable stator geometry angles to control the rotor diffusion factor and incidence angles.

Veres, Joseph P.↗

NASA NTRS2013

Spurious Behavior of Shock-Capturing Methods: Problems Containing Stiff Source Terms and Discontinuities

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The smearing introduces a nonequilibrium state into the calculation. Thus as soon as a nonequilibrium value is introduced in this manner, the source term turns on and immediately restores equilibrium, while at the same time shifting the discontinuity to a cell boundary. The present study is to show that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Moreover, employing finite time steps and grid spacings that are below the standard Courant-Friedrich-Levy (CFL) limit on shockcapturing methods for compressible Euler and Navier-Stokes equations containing stiff reacting source terms and discontinuities reveals surprising counter-intuitive results. Unlike non-reacting flows, for stiff reactions with discontinuities, employing a time step and grid spacing that are below the CFL limit (based on the homogeneous part or non-reacting part of the governing equations) does not guarantee a correct solution of the chosen governing equations. Instead, depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The present investigation for three very different stiff system cases confirms some of the findings of Lafon & Yee (1996) and LeVeque & Yee (1990) for a model scalar PDE. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general.

Yee, Helen M. C.↗

NASA NTRS1986

Numerical simulation of electrophoresis separation processes

A new Petrov-Galerkin finite element formulation has been proposed for transient convection-diffusion problems. Most Petrov-Galerkin formulations take into account the spatial discretization, and the weighting functions so developed give satisfactory solutions for steady state problems. Though these schemes can be used for transient problems, there is scope for improvement. The schemes proposed here, which consider temporal as well as spatial discretization, provide improved solutions. Electrophoresis, which involves the motion of charged entities under the influence of an applied electric field, is governed by equations similiar to those encountered in fluid flow problems, i.e., transient convection-diffusion equations. Test problems are solved in electrophoresis and fluid flow. The results obtained are satisfactory. It is also expected that these schemes, suitably adapted, will improve the numerical solutions of the compressible Euler and the Navier-Stokes equations.

Ganjoo, D. K.↗

NASA NTRS1982

Computation of transonic viscous-inviscid interacting flow

Transonic viscous-inviscid interaction is considered using the Euler and inverse compressible turbulent boundary-layer equations. Certain improvements in the inverse boundary-layer method are mentioned, along with experiences in using various Runge-Kutta schemes to solve the Euler equations. Numerical conditions imposed on the Euler equations at a surface for viscous-inviscid interaction using the method of equivalent sources are developed, and numerical solutions are presented and compared with experimental data to illustrate essential points.

Whitfield, D. L.↗

NASA NTRS1983

Computation of transonic viscous-inviscid interacting flow

Whitfield, D. L.↗

NASA NTRS1994

Boundary Conditions for Unsteady Compressible Flows

This paper explores solutions to the spherically symmetric Euler equations. Motivated by the work of Hagstrom and Hariharan and Geer and Pope, we modeled the effect of a pulsating sphere in a compressible medium. The literature available on this suggests that an accurate numerical solution requires artificial boundary conditions which simulate the propagation of nonlinear waves in open domains. Until recently, the boundary conditions available were in general linear and based on nonreflection. Exceptions to this are the nonlinear nonreflective conditions of Thompson, and the nonlinear reflective conditions of Hagstrom and Hariharan. The former are based on the rate of change of the incoming characteristics; the latter rely on asymptotic analysis and the method of characteristics and account for the coupling of incoming and outgoing characteristics. Furthermore, Hagstrom and Hariharan have shown that, in a test situation in which the flow would reach a steady state over a long time, Thompson's method could lead to an incorrect steady state. The current study considers periodic flows and includes all possible types and techniques of boundary conditions. The technique recommended by Hagstrom and Hariharan proved superior to all others considered and matched the results of asymptotic methods that are valid for low subsonic Mach numbers.

Hariharan, S. I.↗

NASA NTRS1984

Spectral methods for the Euler equations: Fourier methods and shock-capturing

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collocation methods are put in historical context. The basic concepts of Fourier spectral collocation methods are provided. Filtering strategies for shock-capturing approaches are also presented. Fourier shock capturing techniques are evaluated using a one dimensional, periodic astrophysical ""nozzle'' problem.

Hussaini, M. Y.↗

NASA NTRS1985

Spectral methods for the Euler equations. I - Fourier methods and shock capturing

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collociation methods are put into historical context. The basic concepts of Fourier spectral collocation methods are provided. Filtering strategies for shock-capturing approaches are also presented. Fourier shock-capturing techniques are evaluated using a one-dimensional, periodic astrophysical 'nozzle' problem.

Hussaini, M. Y.↗

NASA NTRS1988

On the coupling of hyperbolic and parabolic systems: Analytical and numerical approach

The coupling of hyperbolic and parabolic systems is discussed in a domain Omega divided into two distinct subdomains omega(+) and omega(-). The main concern is to find the proper interface conditions to be fulfilled at the surface separating the two domains. Next, they are used in the numerical approximation of the problem. The justification of the interface conditions is based on a singular perturbation analysis, i.e., the hyperbolic system is rendered parabolic by adding a small artifical viscosity. As this goes to zero, the coupled parabolic-parabolic problem degenerates into the original one, yielding some conditions at the interface. These are taken as interface conditions for the hyperbolic-parabolic problem. Actually, two alternative sets of interface conditions are discussed according to whether the regularization procedure is variational or nonvariational. It is shown how these conditions can be used in the frame of a numerical approximation to the given problem. Furthermore, a method of resolution is discussed which alternates the resolution of the hyperbolic problem within omega(-) and of the parabolic one within omega(+). The spectral collocation method is proposed, as an example of space discretization (different methods could be used as well); both explicit and implicit time-advancing schemes are considered. The present study is a preliminary step toward the analysis of the coupling between Euler and Navier-Stokes equations for compressible flows.

Gastaldi, Fabio↗

NASA NTRS1992

LEWICE/E: An Euler based ice accretion code

A new version of the LEWICE ice accretion computer code was developed which calculates the ice growth on two dimensional surfaces, incorporating the effects of compressibility through the solution of the Euler equations. The code is modular and contains separate stand-alone program elements that create a grid, calculate the flow field parameters, calculate the droplet trajectory paths, determine the amount of ice growth, and plot results. This code increases the applicability of ice accretion predictions by allowing calculations at higher Mach numbers. The new elements of the code are described. Calculated results are compared to experiment for several cases, including a LEWICE example case and a thin airfoil section at a Mach number of 0.58.

Potapczuk, Mark G.↗

NASA NTRS1992

LEWICE/E - An Euler based ice accretion code

Potapczuk, Mark G.↗

NASA NTRS1995

A Hybrid Parallel Preconditioning Algorithm For CFD

A new hybrid preconditioning algorithm will be presented which combines the favorable attributes of incomplete lower-upper (ILU) factorization with the favorable attributes of the approximate inverse method recently advocated by numerous researchers. The quality of the preconditioner is adjustable and can be increased at the cost of additional computation while at the same time the storage required is roughly constant and approximately equal to the storage required for the original matrix. In addition, the preconditioning algorithm suggests an efficient and natural parallel implementation with reduced communication. Sample calculations will be presented for the numerical solution of multi-dimensional advection-diffusion equations. The matrix solver has also been embedded into a Newton algorithm for solving the nonlinear Euler and Navier-Stokes equations governing compressible flow. The full paper will show numerous examples in CFD to demonstrate the efficiency and robustness of the method.

Barth,Timothy J.↗

NASA NTRS1996

Some Experiences with Nonoverlapping Schur Complement Parallel Preconditioning for CFD Calculations

In this work we consider solving matrices which arise from the discretization of advection-diffusion field equations on arbitrary triangulated domains using stabilized numerical methods. The talk will discuss several candidate matrix preconditioning algorithms based on the 2 x 2 block factorization induced by an apriori partitioning of the triangulated domain. Application of the 2 x 2 block preconditioner requires the formation and inversion of the Schur complement submatrix. We consider several strategies for simplifying this task: incomplete Schur complement factorizations, drop tolerance element filling, Schur complement probing, and localized Schur complement inversion. Numerical results will be shown comparing performance and efficiency of these approximations. The matrix preconditioner has also been embedded into a Newton algorithm for solving the nonlinear Euler and Navier-Stokes equations governing compressible flow. The remainder of the talk will show numerous examples in CFD to demonstrate the efficiency and robustness of the techniques.

Barth, Timothy J.↗

NASA NTRS1983

Compressive behavior of unidirectional fibrous composites

The longitudinal compressive behavior of unidirectional fiber composites was investigated by using the Illinois Institute of Technology Research Institute (IITRI) test method with thick and thin test specimens. The test data obtained are interpreted by means of stress/strain curves from back-to-back strain gages, examination of fracture surfaces by scanning electron microscope, and predictive equations for distinct failure modes including fiber compression failure. Euler buckling, delamination, and flexure. The results show that longitudinal compressive fracture is induced by a combination of delamination, flexure, and fiber tier breaks. No distinct fracture surface characteristics can be associated with unique failure modes. An equation is described that can be used to extract the longitudinal compressive strength from the longitudinal tensile and flexural strengths of the same composite system.

Sinclair, J. H.↗

NASA NTRS1982

Compression behavior of unidirectional fibrous composite

The longitudinal compression behavior of unidirectional fiber composites is investigated using a modified Celanese test method with thick and thin test specimens. The test data obtained are interpreted using the stress/strain curves from back-to-back strain gages, examination of fracture surfaces by scanning electron microscope, and predictive equations for distinct failure modes including fiber compression failure, Euler buckling, delamination, and flexure. The results show that the longitudinal compression fracture is induced by a combination of delamination, flexure, and fiber tier breaks. No distinct fracture surface characteristics can be associated with unique failure modes. An equation is described which can be used to extract the longitudinal compression strength knowing the longitudinal tensile and flexural strengths of the same composite system.

Sinclair, J. H.↗

NASA NTRS1984

Transonic viscous-inviscid interaction using Euler and inverse boundary-layer equations

The paper is concerned with the use of a zonal method for the computation of transonic viscous-inviscid interacting flow about airfoils. The inviscid portion of the flow is treated by using an Euler equation solution method, while an inverse integral compressible turbulent boundary-layer solution method is used for the viscous portion of the flow. The matching of the viscous and inviscid solutions is discussed, and some numerical results as well as comparisons with experimental data are presented.

Whitfield, D. L.↗