SEARCH · Engineering Papers

Results for “high-order numerics”

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.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 163 records · Page 9

All sources NASA NTRS DOE OSTI

NASA NTRS1992

Direct numerical simulation of laminar breakdown in high-speed, axisymmetric boundary layers

The compressible Navier-Stokes equations are solved using spectral collocation and high-order compact-difference techniques to simulate the laminar breakdown in high-speed, axisymmetric boundary-layer flow. Mach 4.5 flow along a hollow cylinder and Mach 6.8 flow along a sharp cone are considered. Data obtained replicate two previously unexplained phenomena, namely, the appearance of so-called 'rope-like waves' and 'the precursor transition effect', in which transitional flow originates near the critical layer well upstream of the transition location at the wall. The numerical data also reveal that neither of these effects can be explained, even qualitatively, by linear stability theory alone. It is shown that rope-like appearance arises from secondary instability. Certain features of the precursor transition effect also emerge from secondary instability but its nature is revealed to be fundamentally nonlinear.

Pruett, C. D.↗

NASA NTRS1993

Direct calculations of waves in fluid flows using a high-order compact difference scheme

The solution of the unsteady Euler equations by a sixth-order compact difference scheme combined with a fourth-order Runge-Kutta method is investigated. Closed-form expressions for the amplification factors and their corresponding dispersion correlations are obtained by Fourier analysis of the fully discretized, two-dimensional Euler equations, and the numerical dissipation, dispersion, and anisotropic effects are assessed. It is found that the CFL limit for stable calculations is about 0.8. For a CFL number equal to 0.6, the smallest wavelength which is resolved without numerical damping is about 6 to 8 grid nodes. For phase speeds corresponding to acoustic waves, the corresponding time period is resolved by about 200 to 300 time steps. Three numerical examples of waves in compressible flow are included.

Yu, Sheng-Tao↗

NASA NTRS1993

Numerical simulations of a high Mach number jet flow

The present 2D simulations of plane and axisymmetric jets, which are relevant to current efforts to suppress the jet exhaust noise of prospective high-speed civil transports, were conducted by solving full Navier-Stokes equations by means of a high-order finite-difference scheme. The results obtained, which were able to generate the correct mode shape after an adjustment region of about 10 diameters, are in good agreement with linear-theory predictions of the growth of instability waves.

Hayder, M. E.↗

NASA NTRS1993

Uniform high-order spectral methods for one- and two-dimensional Euler equations

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one-dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two-dimensional shock wave and a rotating vortex is simulated.

Cai, Wei↗

NASA NTRS1988

Maximum likelihood identification for large space structures

This paper examines the use of on-orbit identification based on Maximum Likelihood Estimation (MLE) to provide these high-order, high-accuracy control design models for large space structures (LSS's). First, it outlines a general MLE identification algorithm, together with a covariance-analysis procedure to assess algorithm performance in terms of systematic and stochastic errors. Next, it examines various simplifications appropriate for the LSS identification application. Simplified analytical performance results are presented, as are numerical results to support these analyses. Finally, a graphical interpretation of these results is given.

Barrett, Michael F.↗

NASA NTRS1993

Multigrid direct numerical simulation of the whole process of flow transition in 3-D boundary layers

A new technology was developed in this study which provides a successful numerical simulation of the whole process of flow transition in 3-D boundary layers, including linear growth, secondary instability, breakdown, and transition at relatively low CPU cost. Most other spatial numerical simulations require high CPU cost and blow up at the stage of flow breakdown. A fourth-order finite difference scheme on stretched and staggered grids, a fully implicit time marching technique, a semi-coarsening multigrid based on the so-called approximate line-box relaxation, and a buffer domain for the outflow boundary conditions were all used for high-order accuracy, good stability, and fast convergence. A new fine-coarse-fine grid mapping technique was developed to keep the code running after the laminar flow breaks down. The computational results are in good agreement with linear stability theory, secondary instability theory, and some experiments. The cost for a typical case with 162 x 34 x 34 grid is around 2 CRAY-YMP CPU hours for 10 T-S periods.

Liu, Chaoqun↗

NASA NTRS1994

An HP Adaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws

This dissertation addresses various issues for model classes of hyperbolic conservation laws. The basic approach developed in this work employs a new family of adaptive, hp-version, finite element methods based on a special discontinuous Galerkin formulation for hyperbolic problems. The discontinuous Galerkin formulation admits high-order local approximations on domains of quite general geometry, while providing a natural framework for finite element approximations and for theoretical developments. The use of hp-versions of the finite element method makes possible exponentially convergent schemes with very high accuracies in certain cases; the use of adaptive hp-schemes allows h-refinement in regions of low regularity and p-enrichment to deliver high accuracy, while keeping problem sizes manageable and dramatically smaller than many conventional approaches. The use of discontinuous Galerkin methods is uncommon in applications, but the methods rest on a reasonable mathematical basis for low-order cases and has local approximation features that can be exploited to produce very efficient schemes, especially in a parallel, multiprocessor environment. The place of this work is to first and primarily focus on a model class of linear hyperbolic conservation laws for which concrete mathematical results, methodologies, error estimates, convergence criteria, and parallel adaptive strategies can be developed, and to then briefly explore some extensions to more general cases. Next, we provide preliminaries to the study and a review of some aspects of the theory of hyperbolic conservation laws. We also provide a review of relevant literature on this subject and on the numerical analysis of these types of problems.

Bey, Kim S.↗

NASA NTRS1994

High-order centered difference methods with sharp shock resolution

In this paper we consider high-order centered finite difference approximations of hyperbolic conservation laws. We propose different ways of adding artificial viscosity to obtain sharp shock resolution. For the Riemann problem we give simple explicit formulas for obtaining stationary one and two-point shocks. This can be done for any order of accuracy. It is shown that the addition of artificial viscosity is equivalent to ensuring the Lax k-shock condition. We also show numerical experiments that verify the theoretical results.

Gustafsson, Bertil↗

NASA NTRS1994

SCISEAL: A CFD code for analysis of fluid dynamic forces in seals

A viewgraph presentation is made of the objectives, capabilities, and test results of the computer code SCISEAL. Currently, the seal code has: a finite volume, pressure-based integration scheme; colocated variables with strong conservation approach; high-order spatial differencing, up to third-order; up to second-order temporal differencing; a comprehensive set of boundary conditions; a variety of turbulence models and surface roughness treatment; moving grid formulation for arbitrary rotor whirl; rotor dynamic coefficients calculated by the circular whirl and numerical shaker methods; and small perturbation capabilities to handle centered and eccentric seals.

Athavale, Mahesh↗

NASA NTRS1994

Low-dissipation and -dispersion Runge-Kutta schemes for computational acoustics

In this paper, we investigate accurate and efficient time advancing methods for computational acoustics, where non-dissipative and non-dispersive properties are of critical importance. Our analysis pertains to the application of Runge-Kutta methods to high-order finite difference discretization. In many CFD applications multi-stage Runge-Kutta schemes have often been favored for their low storage requirements and relatively large stability limits. For computing acoustic waves, however, the stability consideration alone is not sufficient, since the Runge-Kutta schemes entail both dissipation and dispersion errors. The time step is now limited by the tolerable dissipation and dispersion errors in the computation. In the present paper, it is shown that if the traditional Runge-Kutta schemes are used for time advancing in acoustic problems, time steps greatly smaller than that allowed by the stability limit are necessary. Low-Dissipation and -Dispersion Runge-Kutta (LDDRE) schemes are proposed, based on an optimization that minimizes the dissipation and dispersion errors for wave propagation. Order optimizations of both single-step and two-step alternating schemes are considered. The proposed LDDRK schemes are remarkably more efficient than the classical Runge-Kutta schemes for acoustic computations. Moreover, low storage implementations of the optimized schemes are discussed. Special issues of implementing numerical boundary conditions in the LDDRK schemes are also addressed.

Hu, F. Q.↗

NASA NTRS1995

Unified approach for incompressible flows

A unified approach for solving incompressible flows has been investigated in this study. The numerical CTVD (Centered Total Variation Diminishing) scheme used in this study was successfully developed by Sanders and Li for compressible flows, especially for the high speed. The CTVD scheme possesses better mathematical properties to damp out the spurious oscillations while providing high-order accuracy for high speed flows. It leads us to believe that the CTVD scheme can equally well apply to solve incompressible flows. Because of the mathematical difference between the governing equations for incompressible and compressible flows, the scheme can not directly apply to the incompressible flows. However, if one can modify the continuity equation for incompressible flows by introducing pseudo-compressibility, the governing equations for incompressible flows would have the same mathematical characters as compressible flows. The application of the algorithm to incompressible flows thus becomes feasible. In this study, the governing equations for incompressible flows comprise continuity equation and momentum equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. Thus, the CTVD schemes can be implemented. In addition, the physical and numerical boundary conditions are properly implemented by the characteristic boundary conditions. Accordingly, a CFD code has been developed for this research and is currently under testing. Flow past a circular cylinder was chosen for numerical experiments to determine the accuracy and efficiency of the code. The code has shown some promising results.

Chang, Tyne-Hsien↗

NASA NTRS1994

A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow

High order essentially non-oscillatory (ENO) schemes, originally designed for compressible flow and in general for hyperbolic conservation laws, are applied to incompressible Euler and Navier-Stokes equations with periodic boundary conditions. The projection to divergence-free velocity fields is achieved by fourth-order central differences through fast Fourier transforms (FFT) and a mild high-order filtering. The objective of this work is to assess the resolution of ENO schemes for large scale features of the flow when a coarse grid is used and small scale features of the flow, such as shears and roll-ups, are not fully resolved. It is found that high-order ENO schemes remain stable under such situations and quantities related to large scale features, such as the total circulation around the roll-up region, are adequately resolved.

Weinan, E.↗

NASA NTRS1994

Direct calculations of waves in fluid flows using high-order compact difference scheme

The solution of the unsteady Euler equations by a sixth-order compact difference scheme combined with a fourth-order Runge-Kutta method is investigated. Closed-form expression for the amplification factors and their corresponding dispersion correlations are obtained by Fourier analysis of the fully discretized, two-dimensional Euler equations. The numerical dissipation, dispersion, and anisotropic effects are assessed. It is found that the Courant-Friedrichs-Lewy (CFL) limit for stable calculations is about 0.8. For a CFL number equal to 0.6, the smallest wavelength which is resolved without numerical damping is about six - eight grid nodes. For phase speeds corresponding to acoustic waves, the corresponding time period is resolved by about 200 - 300 time steps. Three numerical examples of waves in compressible flow are included: (1) sound propagation in a duct with linear shear, (2) linear wave growth in a compressible free shear layer, and (3) vortex pairing in a compressible free shear layer perturbed at two frequencies.

Yu, Sheng-Tao↗

NASA NTRS1997

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge-Kutta Discontinuous Galerkin methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, their high-order formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L(sup 2)-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are k-th order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

Cockburn, Bernardo↗

NASA NTRS1997

Runge-Kutta Methods for Linear Ordinary Differential Equations

Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODES) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge-Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge-Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.

Zingg, David W.↗

NASA NTRS1998

Real Gas Computation Using an Energy Relaxation Method and High-Order WENO Schemes

In this paper, we use a recently developed energy relaxation theory by Coquel and Perthame and high order weighted essentially non-oscillatory (WENO) schemes to simulate the Euler equations of real gas. The main idea is an energy decomposition into two parts: one part is associated with a simpler pressure law and the other part (the nonlinear deviation) is convected with the flow. A relaxation process is performed for each time step to ensure that the original pressure law is satisfied. The necessary characteristic decomposition for the high order WENO schemes is performed on the characteristic fields based on the first part. The algorithm only calls for the original pressure law once per grid point per time step, without the need to compute its derivatives or any Riemann solvers. Both one and two dimensional numerical examples are shown to illustrate the effectiveness of this approach.

Montarnal, Philippe↗

NASA NTRS1995

A 3D High-Order Unstructured Finite-Volume Algorithm for Solving Maxwell's Equations

A three-dimensional finite-volume algorithm based on arbitrary basis functions for time-dependent problems on general unstructured grids is developed. The method is applied to the time-domain Maxwell equations. Discrete unknowns are volume integrals or cell averages of the electric and magnetic field variables. Spatial terms are converted to surface integrals using the Gauss curl theorem. Polynomial basis functions are introduced in constructing local representations of the fields and evaluating the volume and surface integrals. Electric and magnetic fields are approximated by linear combinations of these basis functions. Unlike other unstructured formulations used in Computational Fluid Dynamics, the new formulation actually does not reconstruct the field variables at each time step. Instead, the spatial terms are calculated in terms of unknowns by precomputing weights at the beginning of the computation as functions of cell geometry and basis functions to retain efficiency. Since no assumption is made for cell geometry, this new formulation is suitable for arbitrarily defined grids, either smooth or unsmooth. However, to facilitate the volume and surface integrations, arbitrary polyhedral cells with polygonal faces are used in constructing grids. Both centered and upwind schemes are formulated. It is shown that conventional schemes (second order in Cartesian grids) are equivalent to the new schemes using first degree polynomials as the basis functions and the midpoint quadrature for the integrations. In the new formulation, higher orders of accuracy are achieved by using higher degree polynomial basis functions. Furthermore, all the surface and volume integrations are carried out exactly. Several model electromagnetic scattering problems are calculated and compared with analytical solutions. Examples are given for cases based on 0th to 3rd degree polynomial basis functions. In all calculations, a centered scheme is applied in the interior, while an upwind matching scheme is employed at material interfaces and the Engquist-Majda non-reflecting boundary condition is implemented at the numerical outer boundaries. The staggered leapfrog scheme and the Runge-Kutta methods are utilized for the time integration. Excellent agreements are found between the numerical and analytical solutions.

Liu, Yen↗

NASA NTRS2001

Traveling-Wave Tube Amplifier Model to Predict High-Order Modulation Intersymbol Interference

Demands for increased data rates in satellite communications necessitate higher order modulation schemes, larger system bandwidth, and minimum distortion of the modulated signal as it is passed through the traveling wave tube amplifier (TWTA). One type of distortion that the TWTA contributes to is intersymbol interference (ISI), and this becomes particularly disruptive with wide-band, complex modulation schemes. It is suspected that in addition to the dispersion of the TWT, frequency dependent reflections due to mismatches within the TWT are a significant contributor to ISI. To experimentally investigate the effect of these mismatches within the physical TWT on ISI would be prohibitively expensive, as it would require manufacturing numerous amplifiers in addition to the acquisition of the required digital hardware. In an attempt to develop a more accurate model to correlate IS1 with the TWTA and the operational signal, a fully three-dimensional (3D), time-dependent, TWT interaction model has been developed using the electromagnetic particle-in-cell (PIC) code MAFIA (solution of Maxwell's equations by the Finite-Integration-Algorithm). The model includes a user defined slow-wave circuit with a spatially tapered region of loss to implement a sever, and spatially varied geometry (such as helical pitch) to implement a phase velocity taper. The model also includes user defined input/output coupling and an electron beam contained by solenoidal, electrostatic, or periodic permanent magnet (PPM) focusing allowing standard or novel TWTs to be investigated. This model comprehensively takes into account the effects of frequency dependent nonlinear distortions (MAM and AMPM); gain ripple due to frequency dependent reflections at the input/output coupling, severs, and mismatches from dynamic pitch variations; drive induced oscillations; harmonic generation; intermodulation products; and backward waves.

Kory, Carol L.↗