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NASA NTRS1995

On the Dynamics of Some Discretizations of Convection-Diffusion Equations

Numerical discretizations of differential equations which model physical processes can possess dynamics quite different from that of the equations themselves. Recently the emphasis has been on the the dynamics of numerical discretizations for Ordinary Differential Equations (ODEs). For Partial Differential Equations (PDEs) using a method of lines approach the situation is more complex. First, the spatial discretisation may introduce dynamics not present in the original equations; second, the solution of the resulting system of ODEs is open to the modified dynamics of the ODE solver used. These two effects may interact in a complex manner. In this talk we present some results of our recent work on the dynamics of discretizations of convection-diffusion equations, including those produced using Total Variation Diminishing (TVD) schemes and adaptive grid techniques. A more general overview of the area may be found on our accompanying poster presentation.

Sweby, Peter K.↗

NASA NTRS2015

Numerical Simulations of Shock/Plume Interaction Using Structured Overset Grids

Computational simulations using structured overset grids with the Launch Ascent and Vehicle Aerodynamics (LAVA) solver framework are presented for predicting oblique shock/plume interaction effects to near-field sonic boom signatures. Standard second-order accurate as well as higher-resolution numerical discretizations are utilized and compared in the study. The numerical approach is compared with supersonic wind-tunnel data for three cases. The cases include an empty wind-tunnel at the operating conditions, an isolated shockgenerating diamond wedge within the tunnel, and a nozzle with diamond wedge configuration at five different nozzle pressure ratios. Solution sensitivity to numerical discretization is analyzed. Favorable comparisons between the computational results and experimental data of near-field pressure signatures are obtained. A simple prediction method for plume induced shock deflection is developed and results are compared with the CFD data.

Housman, Jeffrey A.↗

NASA NTRS1995

On the Importance of the Dynamics of Discretizations

It has been realized recently that the discrete maps resulting from numerical discretizations of differential equations can possess asymptotic dynamical behavior quite different from that of the original systems. This is the case not only for systems of Ordinary Differential Equations (ODEs) but in a more complicated manner for Partial Differential Equations (PDEs) used to model complex physics. The impact of the modified dynamics may be mild and even not observed for some numerical methods. For other classes of discretizations the impact may be pronounced, but not always obvious depending on the nonlinear model equations, the time steps, the grid spacings and the initial conditions. Non-convergence or convergence to periodic solutions might be easily recognizable but convergence to incorrect but plausible solutions may not be so obvious - even for discretized parameters within the linearized stability constraint. Based on our past four years of research, we will illustrate some of the pathology of the dynamics of discretizations, its possible impact and the usage of these schemes for model nonlinear ODEs, convection-diffusion equations and grid adaptations.

Sweby, Peter K.↗

NASA NTRS2003

Implementation of Preconditioned Dual-Time Procedures in OVERFLOW

Preconditioning methods have become the method of choice for the solution of flowfields involving the simultaneous presence of low Mach and transonic regions. It is well known that these methods are important for insuring accurate numerical discretization as well as convergence efficiency over various operating conditions such as low Mach number, low Reynolds number and high Strouhal numbers. For unsteady problems, the preconditioning is introduced within a dual-time framework wherein the physical time-derivatives are used to march the unsteady equations and the preconditioned time-derivatives are used for purposes of numerical discretization and iterative solution. In this paper, we describe the implementation of the preconditioned dual-time methodology in the OVERFLOW code. To demonstrate the performance of the method, we employ both simple and practical unsteady flowfields, including vortex propagation in a low Mach number flow, flowfield of an impulsively started plate (Stokes' first problem) arid a cylindrical jet in a low Mach number crossflow with ground effect. All the results demonstrate that the preconditioning algorithm is responsible for improvements to both numerical accuracy and convergence efficiency and, thereby, enables low Mach number unsteady computations to be performed at a fraction of the cost of traditional time-marching methods.

Pandya, Shishir A.↗

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 NTRS1995

Automation based on knowledge modeling theory and its applications in engine diagnostic systems using Space Shuttle Main Engine vibrational data

Humans can perform many complicated tasks without explicit rules. This inherent and advantageous capability becomes a hurdle when a task is to be automated. Modern computers and numerical calculations require explicit rules and discrete numerical values. In order to bridge the gap between human knowledge and automating tools, a knowledge model is proposed. Knowledge modeling techniques are discussed and utilized to automate a labor and time intensive task of detecting anomalous bearing wear patterns in the Space Shuttle Main Engine (SSME) High Pressure Oxygen Turbopump (HPOTP).

Kim, Jonnathan H.↗

NASA NTRS1993

Convection equation modeling: A non-iterative direct matrix solution algorithm for use with SINDA

The determination of the boundary conditions for a component-level analysis, applying discrete finite element and finite difference modeling techniques often requires an analysis of complex coupled phenomenon that cannot be described algebraically. For example, an analysis of the temperature field of a coldplate surface with an integral fluid loop requires a solution to the parabolic heat equation and also requires the boundary conditions that describe the local fluid temperature. However, the local fluid temperature is described by a convection equation that can only be solved with the knowledge of the locally-coupled coldplate temperatures. Generally speaking, it is not computationally efficient, and sometimes, not even possible to perform a direct, coupled phenomenon analysis of the component-level and boundary condition models within a single analysis code. An alternative is to perform a disjoint analysis, but transmit the necessary information between models during the simulation to provide an indirect coupling. For this approach to be effective, the component-level model retains full detail while the boundary condition model is simplified to provide a fast, first-order prediction of the phenomenon in question. Specifically for the present study, the coldplate structure is analyzed with a discrete, numerical model (SINDA) while the fluid loop convection equation is analyzed with a discrete, analytical model (direct matrix solution). This indirect coupling allows a satisfactory prediction of the boundary condition, while not subjugating the overall computational efficiency of the component-level analysis. In the present study a discussion of the complete analysis of the derivation and direct matrix solution algorithm of the convection equation is presented. Discretization is analyzed and discussed to extend of solution accuracy, stability and computation speed. Case studies considering a pulsed and harmonic inlet disturbance to the fluid loop are analyzed to assist in the discussion of numerical dissipation and accuracy. In addition, the issues of code melding or integration with standard class solvers such as SINDA are discussed to advise the user of the potential problems to be encountered.

Schrage, Dean S.↗

NASA NTRS2000

Residual Distribution Schemes for Conservation Laws Via Adaptive Quadrature

This paper considers a family of nonconservative numerical discretizations for conservation laws which retains the correct weak solution behavior in the limit of mesh refinement whenever sufficient order numerical quadrature is used. Our analysis of 2-D discretizations in nonconservative form follows the 1-D analysis of Hou and Le Floch. For a specific family of nonconservative discretizations, it is shown under mild assumptions that the error arising from non-conservation is strictly smaller than the discretization error in the scheme. In the limit of mesh refinement under the same assumptions, solutions are shown to satisfy an entropy inequality. Using results from this analysis, a variant of the "N" (Narrow) residual distribution scheme of van der Weide and Deconinck is developed for first-order systems of conservation laws. The modified form of the N-scheme supplants the usual exact single-state mean-value linearization of flux divergence, typically used for the Euler equations of gasdynamics, by an equivalent integral form on simplex interiors. This integral form is then numerically approximated using an adaptive quadrature procedure. This renders the scheme nonconservative in the sense described earlier so that correct weak solutions are still obtained in the limit of mesh refinement. Consequently, we then show that the modified form of the N-scheme can be easily applied to general (non-simplicial) element shapes and general systems of first-order conservation laws equipped with an entropy inequality where exact mean-value linearization of the flux divergence is not readily obtained, e.g. magnetohydrodynamics, the Euler equations with certain forms of chemistry, etc. Numerical examples of subsonic, transonic and supersonic flows containing discontinuities together with multi-level mesh refinement are provided to verify the analysis.

Barth, Timothy↗

NASA NTRS1997

Some Experiences with High Order Discretization Schemes for Compressible Fluid Flow

It is widely recognized that the structure and complexity of compressible fluid flow places high demands on numerical discretization techniques for the fluid flow equations. Fluid flows arising in external aerodynamics often contain both flow field discontinuities and fluid boundary-layers. Both must be accurately resolved to provide useful information to aerodynamic design and analysis engineers. These accuracy requirements motivated the present author to examine a class of finite-volume techniques on arbitrary triangulated domains based on linear or quadratic reconstruction of integral-averaged data followed by upwind flux function evaluation and small time evolution. More recently, we have considered some new upwind techniques which yield compact discretizations while maintaining higher order accuracy. In the mini-symposium talk we will discuss both of these techniques as well as demonstrate the relative merits of each method by computing a number of aerodynamic flows containing shock waves and boundary-layers.

Barth, Timothy J.↗

NASA NTRS2023

Numerical Investigation of C L,max Prediction on the NASA High-Lift Common Research Model

An assessment of a Hybrid RANS/LES (HRLES) approach for C L,max prediction is presented for the NASA High-Lift Common Research Model (CRM-HL). Both free air and wind tunnel configurations of the CRM-HL are investigated and the results are compared to the QinetiQ wind tunnel experiments and two other numerical approaches: Reynolds Averaged Navier-Stokes (RANS) and Wall-Modeled Large Eddy Simulations (WMLES). For the free-air configuration, HRLES is shown to address some of the known shortcomings with RANS and prevent inboard and outboard flow separation particularly in the region of C L,max and post-stall. To achieve these improvements over RANS, LES-appropriate grids and numerical discretizations are required. It was also found that when applying HRLES to a RANS best practice grid and numerics that the HRLES method significantly underperformed RANS. For the in-tunnel configuration, HRLES showed good agreement with the loads, surface pressure and oil-flow photographs obtained in the experiment. HRLES was able to improve upon the RANS simulations, which showed a sharp loss of lift at the two highest angles-of-attack due to large scale inboard and outboard separation on the wing, by correctly predicting the corner flow separation and showing remarkably close agreement in the flow topologies with the experiment.

TTT↗

NASA NTRS1983

Initialization of mesoscale models - The possible impact of remotely sensed data

Little or no improvement appears to have been achieved in mesoscale numerical prediction. This has been blamed on inherent unpredictability, insufficient spatial resolution, inadequate parameterization of important physical processes, and errors in the numerical discretization of the governing hydrodynamic equations. Attention is presently given to the more fundamental difficulty of forecasting when present conditions are not known with sufficient accuracy, and to the matching of remotely sensed sounding-capability satellite and all-weather Doppler radar data with mesoscale numerical models. This matching is not straightforward; simple and variational forms of four-dimensional assimilation, and Kalman filtering, are suggested approaches but their practical testing remains to be accomplished. The most severe problem faced is that of the temperature and humidity profiles' vertical resolution.

Gal-Chen, T.↗

NASA NTRS1985

Volcanism of the Colorado Plateau - Basin and Range Transition: Implications for Crustal Processes

On the Colorado Plateau-Basin and Range transition in Arizona and New Mexico, basaltic volcanism takes two physiographic forms: (1) Major central composite volcanoes (e.g., San Francisco Peaks) in which basaltic magma differentiates at shallow crustal levels to siliceous and members and (2) large numbers of small cinder cones and a few domes that result from numerous discrete magma batches which apparently had little opportunity for differentiation. This study represents the first look at the petrogenesis of Plateau-bounding volcanic fields, based on rigorous stratigraphic controls, in combination with major and trace-element constraints. The emerging pattern of large volcanic constructs associated with shallow fractional crystallization, and numerous cinder cones associated with limited or no shallow differentiation may provide an analogue for interpreting petrogenetic patterns and/or tectonic settings on Mars.

Condit, C. D.↗

NASA NTRSDate not supplied

A Hybrid RANS-LES Perspective for the High Lift Common Research Model Using LAVA

An assessment of a Hybrid RANS/LES (HRLES) approach for𝐶𝐿,max prediction is presented for the NASA High-Lift Common Research Model (CRM-HL). Both the free air and the wind tunnel configuration of the CRM-HL are investigated and the results are compared to the QinetiQ wind tunnel experiments and to two other numerical approaches: Reynolds Averaged Navier-Stokes (RANS) and Wall-Modeled Large Eddy Simulations (WMLES). For the free-air configuration, HRLES was shown to address some of the known shortcomings in RANS methods and prevent inboard and outboard flow separation particularly in the region of𝐶𝐿,max and post-stall. To achieve these improvements over RANS, LES-appropriate grids and numerical discretizations are required. HRLES predicts a weak pitch break at the highest angle-of-attack due to onset of wing-root corner flow separation whereas the free-air corrected experiment values indicate an occurrence of a much stronger pitch break. The improvements of HRLES over a URANS approach has been objectively shown by computing a set of solutions with the same grid, same numerics and time-step size and comparing the solutions. It was also found that when applying HRLES to a RANS best practice grid and numerics that the HRLES method significantly under performed RANS. For the in tunnel configuration, HRLES showed good agreement with the loads, surface pressure and oil-flow photographs obtained in the experiment. HRLES was able to improve upon the RANS simulations, which showed a sharp loss of lift at the two highest angles-of-attack due to large scale inboard and outboard separation on the wing, by correctly predicting the corner flow separation and showing remarkably close agreement in the flow topologies with the experiment.

TTT↗

NASA NTRS1994

New Developments in the Method of Space-Time Conservation Element and Solution Element-Applications to Two-Dimensional Time-Marching Problems

A new numerical discretization method for solving conservation laws is being developed. This new approach differs substantially in both concept and methodology from the well-established methods, i.e., finite difference, finite volume, finite element, and spectral methods. It is motivated by several important physical/numerical considerations and designed to avoid several key limitations of the above traditional methods. As a result of the above considerations, a set of key principles for the design of numerical schemes was put forth in a previous report. These principles were used to construct several numerical schemes that model a 1-D time-dependent convection-diffusion equation. These schemes were then extended to solve the time-dependent Euler and Navier-Stokes equations of a perfect gas. It was shown that the above schemes compared favorably with the traditional schemes in simplicity, generality, and accuracy. In this report, the 2-D versions of the above schemes, except the Navier-Stokes solver, are constructed using the same set of design principles. Their constructions are simplified greatly by the use of a nontraditional space-time mesh. Its use results in the simplest stencil possible, i.e., a tetrahedron in a 3-D space-time with a vertex at the upper time level and other three at the lower time level. Because of the similarity in their design, each of the present 2-D solvers virtually shares with its 1-D counterpart the same fundamental characteristics. Moreover, it is shown that the present Euler solver is capable of generating highly accurate solutions for a famous 2-D shock reflection problem. Specifically, both the incident and the reflected shocks can be resolved by a single data point without the presence of numerical oscillations near the discontinuity.

Chang, Sin-Chung↗

NASA NTRS1976

Numerical comparison of discrete Kalman filter algorithms - Orbit determination case study

Numerical characteristics of various Kalman filter algorithms are illustrated with a realistic orbit determination study. The case study of this paper highlights the numerical deficiencies of the conventional and stabilized Kalman algorithms. Computational errors associated with these algorithms are found to be so large as to obscure important mismodeling effects and thus cause misleading estimates of filter accuracy. The positive result of this study is that the U-D covariance factorization algorithm has excellent numerical properties and is computationally efficient, having CPU costs that differ negligibly from the conventional Kalman costs. Accuracies of the U-D filter using single precision arithmetic consistently match the double precision reference results. Numerical stability of the U-D filter is further demonstrated by its insensitivity to variations in the a priori statistics.

Bierman, G. J.↗

NASA NTRS1981

Blade loss transient dynamics analysis, volume 2. Task 2: Theoretical and analytical development. Task 3: Experimental verification

The component element method was used to develop a transient dynamic analysis computer program which is essentially based on modal synthesis combined with a central, finite difference, numerical integration scheme. The methodology leads to a modular or building-block technique that is amenable to computer programming. To verify the analytical method, turbine engine transient response analysis (TETRA), was applied to two blade-out test vehicles that had been previously instrumented and tested. Comparison of the time dependent test data with those predicted by TETRA led to recommendations for refinement or extension of the analytical method to improve its accuracy and overcome its shortcomings. The development of working equations, their discretization, numerical solution scheme, the modular concept of engine modelling, the program logical structure and some illustrated results are discussed. The blade-loss test vehicles (rig full engine), the type of measured data, and the engine structural model are described.

Gallardo, V. C.↗

NASA NTRS2019

Effect of Spatial Filtering in Implicit Large-Eddy Simulations of Separated Flows

The relatively high Reynolds number of turbulent flows encountered in various applications puts these problems well beyond the reach of direct numerical simulation (DNS) at present. Meanwhile, lower-fidelity Reynolds-averaged Navier-Stokes (RANS) calculations are known to be not accurate enough in complex problems, such as smooth-body flow separation and other flows involving highly-unsteady phenomena. Hence, given the current infeasibility of DNS and the unsatisfactory performance of RANS, intermediate techniques such as large-eddy simulation (LES) and hybrid RANS-LES, whose fidelity lie between RANS and DNS, have received much attention for application to various problems of practical importance. Modeling of the effect of missing scales on resolved scales, also known as subgrid-scale (SGS) modeling, is an important subject for LES. SGS models can be broadly categorized as explicit or implicit approaches. The explicit approach is based on an SGS model that explicitly appears in the governing equations expressed in the form of so-called “filtered Navier-Stokes equations”, which describe the evolution of the turbulence scales resolved by the LES grid. The effect of the scales unresolved by the grid is represented by the SGS model. The implicit modeling approach, on the other hand, does not employ an explicit model but instead treats the intrinsic dissipation of the numerical discretization scheme as an implicit SGS model. An LES without an explicit SGS model is commonly termed as an implicit LES (ILES). The relative merits of one SGS modeling approach over another is a subject of ongoing debate. We have opted to employ an ILES methodology, based on high-order compact finite-difference and spatial filtering schemes, in our recent investigations of separated flow problems [1, 2]. Further discussion of our preference of ILES over explicit LES is provided in Uzun and Malik [2]. The spatial filtering operation, described in the next section, is treated as an implicit SGS model for the ILES. Some observations made during the course of our recent investigations, which pointed out to excessive numerical dissipation in certain parts of the flowfield, prompted us to take a closer look at the potential effect of the spatial filter on ILES predictions. This technical note is therefore devoted to spatial filter effects in the context of a high Reynolds number, transonic shock-induced separated flow.

Ali Uzun↗

NASA NTRS1981

Multiple Alfven wave reflections excited by Io Origin of the Jovian decametric arcs

The recent Voyager radio astronomy measurements near Jupiter show that the Jovian decametric radiation consists of numerous discrete features called decametric arcs which are observed at all Jovian longitudes. It is generally believed that these arcs are produced by an interaction of Io with the Jovian magnetosphere. In this paper we propose that the large number of decametric arcs is caused by multiple reflections of a standing Alfven wave current system excited by Io. Estimates of the reflection coefficient at the ionosphere and other damping processes show that a large number of reflections can occur, with the Alfven wave current system possibly extending completely around Jupiter. This source geometry can account for a number of otherwise puzzling characteristics of the Jovian decametric radiations,

Gurnett, D. A.↗