Effective treatment of the singular line boundary problem for three-dimensional grids
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Palmer, Grant↗
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On constructing three-dimensional overlapping grids with CMPGRD
Techniques for the construction of three-dimensional composite overlapping grids, using the grid construction program CMPGRD, are described. The overlapping approach can be used to generate grids for regions of complicated geometry. The grids can be constructed to be smooth and free from coordinate singularities. The ability to create smooth grids for complicated regions is an important first step towards the accurate numerical solution of partial differential equations. The creation of grids for surfaces defined by cross-sections such as an airplane wing is described. A method for integrating the patched surfaces generated by a computer aided design (CAD) package with the CMPGRD program is described. Additionally, the creation of grids in regions where surfaces intersect is described.
Henshaw, William D.↗
Addition of the AUTOSPC feature to NASTRAN
A new capability called the AUTOSPC feature has been incorporated into the April 1987 release of NASTRAN. It gives the user the option of automatically applying single-point constraints for the purpose of removing potential grid and scalar point singularities that have not been otherwise already constrained out. Details of this implementation are given. Its usage is described and it is illustrated with an example problem.
Pamidi, P. R.↗
Finite-surface method for the Maxwell equations with corner singularities
The finite-surface method for the two-dimensional Maxwell equations in generalized coordinates is extended to treat perfect conductor boundaries with sharp corners. Known singular forms of the grid and the electromagnetic fields in the neighborhood of each corner are used to obtain accurate approximations to the surface and line integrals appearing in the method. Numerical results are presented for a harmonic plane wave incident on a finite flat plate. Comparisons with exact solutions show good agreement.
Vinokur, Marcel↗
Textbook Multigrid Efficiency for Computational Fluid Dynamics Simulations
Considerable progress over the past thirty years has been made in the development of large-scale computational fluid dynamics (CFD) solvers for the Euler and Navier-Stokes equations. Computations are used routinely to design the cruise shapes of transport aircraft through complex-geometry simulations involving the solution of 25-100 million equations; in this arena the number of wind-tunnel tests for a new design has been substantially reduced. However, simulations of the entire flight envelope of the vehicle, including maximum lift, buffet onset, flutter, and control effectiveness have not been as successful in eliminating the reliance on wind-tunnel testing. These simulations involve unsteady flows with more separation and stronger shock waves than at cruise. The main reasons limiting further inroads of CFD into the design process are: (1) the reliability of turbulence models; and (2) the time and expense of the numerical simulation. Because of the prohibitive resolution requirements of direct simulations at high Reynolds numbers, transition and turbulence modeling is expected to remain an issue for the near term. The focus of this paper addresses the latter problem by attempting to attain optimal efficiencies in solving the governing equations. Typically current CFD codes based on the use of multigrid acceleration techniques and multistage Runge-Kutta time-stepping schemes are able to converge lift and drag values for cruise configurations within approximately 1000 residual evaluations. An optimally convergent method is defined as having textbook multigrid efficiency (TME), meaning the solutions to the governing system of equations are attained in a computational work which is a small (less than 10) multiple of the operation count in the discretized system of equations (residual equations). In this paper, a distributed relaxation approach to achieving TME for Reynolds-averaged Navier-Stokes (RNAS) equations are discussed along with the foundations that form the basis of this approach. Because the governing equations are a set of coupled nonlinear conservation equations with discontinuities (shocks, slip lines, etc.) and singularities (flow- or grid-induced), the difficulties are many. This paper summarizes recent progress towards the attainment of TME in basic CFD simulations.
Brandt, Achi↗
Elliptic grid generation with orthogonality and spacing control on an arbitrary number of boundaries
A procedure for the generation of two and quasi-three-dimensional grids with control of orthogonality and spacing with respect to any and/or all boundaries of the domain is described. The elliptic grid generation equations of Thompson are solved implicitly. Control of the grid behavior is achieved through the introduction of forcing functions terms in the manner of Steger and Sorenson or in a modification of the method of Hilgenstock. The forcing function terms are constructed on the boundaries and propagated into the domain using transfinite Lagrangian bivariate interpolation. An anisotropic transfinite stencil is introduced and is shown to produce excellent grid behavior particularly in the vicinity of corner singularities. Emphasis is placed on the generation of viscous grids and the method is shown to be suited for use in the generation of grids for internal as well as external flow geometries. A FORTRAN program named PISCES has been written to implement the algorithm. Examples of grids for internal and external flows are given that highlight the characteristics and behavior of the algorithm.
White, J. A.↗
Computation at a coordinate singularity
Coordinate singularities are sometimes encountered in computational problems. An important example involves global atmospheric models used for climate and weather prediction. Classical spherical coordinates can be used to parameterize the manifold – that is, generate a grid for the computational spherical shell domain. This particular parameterization offers significant benefits such as orthogonality and exact representation of curvature and connection (Christoffel) coefficients. But it also exhibits two polar singularities and at or near these points typical continuity/integral constraints on dependent fields and their derivatives are generally inadequate and lead to poor model performance and erroneous results. Other parameterizations have been developed that eliminate polar singularities, but problems of weaker singularities and enhanced grid noise compared to spherical coordinates (away from the poles) persist. Here, in this study reparameterization invariance of geometric objects (scalars, vectors and the forms generated by their covariant derivatives) is utilized to generate asymptotic forms for dependent fields of interest valid in the neighborhood of a pole. The central concept is that such objects cannot be altered by the metric structure of a parameterization. The new boundary conditions enforce symmetries that are required for transformations of geometric objects. They are implemented in an implicit polar filter of a structured grid, nonhydrostatic global atmospheric model that is simulating idealized Held–Suarez flows. A series of test simulations using different configurations of the asymptotic boundary conditions are made, along with control simulations that use the default model numerics with no absorber, at three different grid sizes. Typically the test simulations are ~ 20% faster in wall clock time than the control—resulting from a decrease in noise at the poles in all cases. In the control simulations adverse numerical effects from the polar singularity are observed to increase with grid resolution. In contrast, test simulations demonstrate robust polar behavior independent of grid resolution.
97 MATHEMATICS AND COMPUTING↗
Accuracy Analysis for Finite-Volume Discretization Schemes on Irregular Grids
A new computational analysis tool, downscaling test, is introduced and applied for studying the convergence rates of truncation and discretization errors of nite-volume discretization schemes on general irregular (e.g., unstructured) grids. The study shows that the design-order convergence of discretization errors can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all. The downscaling test is a general, efficient, accurate, and practical tool, enabling straightforward extension of verification and validation to general unstructured grid formulations. It also allows separate analysis of the interior, boundaries, and singularities that could be useful even in structured-grid settings. There are several new findings arising from the use of the downscaling test analysis. It is shown that the discretization accuracy of a common node-centered nite-volume scheme, known to be second-order accurate for inviscid equations on triangular grids, degenerates to first order for mixed grids. Alternative node-centered schemes are presented and demonstrated to provide second and third order accuracies on general mixed grids. The local accuracy deterioration at intersections of tangency and in flow/outflow boundaries is demonstrated using the DS tests tailored to examining the local behavior of the boundary conditions. The discretization-error order reduction within inviscid stagnation regions is demonstrated. The accuracy deterioration is local, affecting mainly the velocity components, but applies to any order scheme.
Diskin, Boris↗
Two-way coupled particle-turbulence interaction: Effect of numerics and resolution on fluid and particle statistics
Euler-Lagrange point-particle simulation has emerged as a premier methodology for studying dispersed particle-laden flows. This method's popularity stems from its ability to resolve fine-scale fluid structures while also tracking individual particles at reduced cost using an appropriate particle acceleration model. However, the point-particle model has known convergence issues in that refinement of the fluid grid can lead to changes in the predicted statistics. The reasons for non-convergence are twofold: the point-particle two-way coupling force in the Navier-Stokes equations requires a numerical regularization and, without careful implementation, yields a singular force on the fluid with grid refinement. The second factor that yields grid-dependent statistics is that the point-particle force model in general depends on the undisturbed fluid velocity. When the undisturbed fluid velocity is not robustly modeled in a grid-insensitive way, the calculated force for both particles and fluid will be grid-dependent, contaminating their respective statistics. While the first issue regarding regularizing the point-particle source term has received attention in the literature, the consequences of robustly modeling the undisturbed velocity in the context of grid refinement of turbulence has received little attention. Here, we consider decaying homogeneous isotropic turbulence laden with particles at different Stokes numbers. For a given Stokes number, we systematically refine the grid and demonstrate that explicitly modeling the undisturbed fluid velocity yields relative grid insensitivity for the energy of the particle and fluid phases, as well as acceleration of the particles. We also demonstrate that an appropriately defined dissipation rate is also grid-insensitive when an undisturbed fluid velocity correction is used. In contrast, when the undisturbed fluid velocity is modeled using the conventional approach of interpolating the local fluid velocity to the particle location, we show this procedure yields divergent statistics with grid refinement. In particular, we show that higher-order interpolation of the fluid velocity in two-way coupled problems is worse than lower-order interpolation, in the absence of a correction procedure to estimate the undisturbed fluid velocity. We also examine velocity derivative statistics of the fluid phase and demonstrate that these statistics are not in general convergent even when the undisturbed fluid velocity is explicitly modeled. Collectively, the observations in this paper are used to present a philosophy on the types of questions which are answerable with point-particle methods.
42 ENGINEERING↗
Towards Verification of Unstructured-Grid Solvers
New methodology for verification of finite-volume computational methods using unstructured grids is presented. The discretization order properties are studied in computational windows, easily constructed within a collection of grids or a single grid. Tests are performed within each window and address a combination of problem-, solution-, and discretization/grid-related features affecting discretization error convergence. The windows can be adjusted to isolate particular elements of the computational scheme, such as the interior discretization, the boundary discretization, or singularities. Studies can use traditional grid-refinement computations within a fixed window or downscaling, a recently-introduced technique in which computations are made within windows contracting toward a focal point of interest. Grids within the windows are constrained to be consistently refined, allowing a meaningful assessment of asymptotic error convergence on unstructured grids. Demonstrations of the method are shown, including a comparative accuracy assessment of commonly-used schemes on general mixed grids and the identification of local accuracy deterioration at boundary intersections. Recommendations to enable attainment of design-order discretization errors for large-scale computational simulations are given.
Thomas, James L.↗
Bipolar coordinates for computation of transition duct flows
Numerical simulation techniques for flows in jet-engine transition ducts with changing cross sections are developed and demonstrated. The boundary-conforming grids required are generated using bipolar coordinates, and the singularities arising at the no-slip boundaries are treated as described by de Vahl Davis (1979) and Tsai and Levy (1987). Grids for several typical configurations are employed in computations with the three-dimensional laminar/turbulent viscous-flow solver of Levy et al. (1983), and the results are presented graphically. The scheme used to deal with singularities is shown to be robust, suggesting that the bipolar grids may be applicable to Navier-Stokes computations.
Madabhushi, R. K.↗
Grid Convergence for Turbulent Flows(Invited)
A detailed grid convergence study has been conducted to establish accurate reference solutions corresponding to the one-equation linear eddy-viscosity Spalart-Allmaras turbulence model for two dimensional turbulent flows around the NACA 0012 airfoil and a flat plate. The study involved three widely used codes, CFL3D (NASA), FUN3D (NASA), and TAU (DLR), and families of uniformly refined structured grids that differ in the grid density patterns. Solutions computed by different codes on different grid families appear to converge to the same continuous limit, but exhibit different convergence characteristics. The grid resolution in the vicinity of geometric singularities, such as a sharp trailing edge, is found to be the major factor affecting accuracy and convergence of discrete solutions, more prominent than differences in discretization schemes and/or grid elements. The results reported for these relatively simple turbulent flows demonstrate that CFL3D, FUN3D, and TAU solutions are very accurate on the finest grids used in the study, but even those grids are not sufficient to conclusively establish an asymptotic convergence order.
Diskin, Boris↗
PARC Navier-Stokes code upgrade and validation for high speed aeroheating predictions
Applications of the PARC full Navier-Stokes code for hypersonic flowfield and aeroheating predictions around blunt bodies such as the Aeroassist Flight Experiment (AFE) and Aeroassisted Orbital Transfer Vehicle (AOTV) are evaluated. Two-dimensional/axisymmetric and three-dimensional perfect gas versions of the code were upgraded and tested against benchmark wind tunnel cases of hemisphere-cylinder, three-dimensional AFE forebody, and axisymmetric AFE and AOTV aerobrake/wake flowfields. PARC calculations are in good agreement with experimental data and results of similar computer codes. Difficulties encountered in flowfield and heat transfer predictions due to effects of grid density, boundary conditions such as singular stagnation line axis and artificial dissipation terms are presented together with subsequent improvements made to the code. The experience gained with the perfect gas code is being currently utilized in applications of an equilibrium air real gas PARC version developed at REMTECH.
Liver, Peter A.↗
Characterization of dynamic stall phenomenon using two-dimensional unsteady Navier-Stokes equations
Among the new significant aspects of the present work are: (1) the treatment of the far-field boundary; (2) the use of C-grid topology, with the branch-cut singularity treated analytically; (3) evaluation of the effect of the envelope of prevailing initial states, and finally; (4) the ability to employ streakline/pathline 'visualization' to probe the unsteady features prevailing in vortex-dominated flows. The far-field boundary is placed at infinity, using appropriate grid stretching. This contributes to the accuracy of the solutions, but raised a number of important issues which needed to be resolved; this includes determining the equivalent time-dependent circulation for the pitching airfoil. A secondary counter-clockwise vortex erupts from within the boundary layer and immediately pinches off the energetic leading-edge shear layer which then, through hydrodynamic instability, rolls up into the dynamic stall vortex. The streakline/pathline visualization serves to provide information for insight into the physics of the unsteady separated flow.
Ghia, K. N.↗
Chapter 32 - Power Grid Resilience
The new energy paradigm is altering the current power grid trend from synchronous generator reliant system toward power-electronics-based distributed energy resources (DERs). The resilient operation of modern power grid is highly dependent upon cyber and physical reliable operation of DERs. Power grid resilience broadly refers to the ability of the grid to be robust to eventualities and singularities that may disrupt the continuity of reliable power flow. This could involve resilience related to the physical system but more recently, the focus on cyber resilience has been evolving rapidly especially given the burgeoning growth of distributed generation and power-electronics-based DERs under smart grid and given that such threats to reliable operation do not have to evolve localized to where the physical asset is. Commonly, in power grids dominated by DERS, control and energy management are structured at a multitime scale multilayer fashion: (i) primary control layer at microseconds time scale, (ii) secondary control layer at millisecond time scale, and (iii) tertiary control layer at seconds to minutes time scale. The cyber-related issues that may affect the power grid resilience can be introduced in through primary, secondary, and tertiary control layers. The failure of each of these control layers may impact the reliable and resilient operation of the overall network and cause widespread failures which leads to unintended blackouts. As such, this chapter focuses on cyber-related resilience issues in primary control layer, secondary control layer, tertiary control layer, wide area/utility control, and anomaly detection and resilient communication.
anomaly detection↗
TTHRESH: Tensor Compression for Multidimensional Visual Data
Memory and network bandwidth are decisive bottlenecks when handling high-resolution multidimensional data sets in visualization applications, and they increasingly demand suitable data compression strategies. In this paper, we introduce a novel lossy compression algorithm for multidimensional data over regular grids. It leverages the higher-order singular value decomposition (HOSVD), a generalization of the SVD to three dimensions and higher, together with bit-plane, run-length and arithmetic coding to compress the HOSVD transform coefficients. Our scheme degrades the data particularly smoothly and achieves lower mean squared error than other state-of-the-art algorithms at low-to-medium bit rates, as it is required in data archiving and management for visualization purposes. Further advantages of the proposed algorithm include very fine bit rate selection granularity and the ability to manipulate data at very small cost in the compression domain, for example to reconstruct filtered and/or subsampled versions of all (or selected parts) of the data set.
97 MATHEMATICS AND COMPUTING↗
A novel numerical treatment of the near-wall regions in the k – ω class of RANS models
In this study, we discuss a novel approach to modeling the near-wall region in the class of k – ω to models. The proposed methodology obviates the need for ad hoc boundary conditions of to omega on the wall as typically required in the k – ω model. The primary motivation of this work is to provide a formulation equivalent to the standard k – ω to and k – ω to SST models, but which at the same time overcomes their limitations in the context of their implementation in high-order methods. This is achieved by subtracting the asymptotically known singular behavior of omega at walls. Imposing a grid-dependent value of omega at walls in high-order codes is not straightforward as is demonstrated below and it causes instability as well as accuracy issues. The mathematical formulation of the two novel approaches, termed as the "regularized k – ω model" and the "regularized k – ω OJ SST model", is discussed in detail. A consistency and verification study for these two approaches is performed by proving that the regularized models recover the results of the standard models in various canonical problems, such as turbulent channel and pipe flows, and a systematic investigation of convergence, using both p- (polynomial order) as well as h- (grid) refinement is reported. Furthermore, comparisons highlighting the performance of the proposed methods in more complex configurations such as flow over a backward facing step and the turbulent mixing of fluid streams of different temperatures in a T-junction are also presented.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Simulations of Solar Wind Turbulence
Recently we have restructured our approach to simulating magnetohydrodynamic (MHD) turbulence in the solar wind. Previously, we had defined a 'virtual' heliosphere that contained, for example, a tilted rotating current sheet, microstreams, quasi-two-dimensional fluctuations as well as Alfven waves. In this new version of the code, we use the global, time-stationary, WKB Alfven wave-driven solar wind model developed by Usmanov and described in Usmanov and Goldstein [2003] to define the initial state of the system. Consequently, current sheets, and fast and slow streams are computed self-consistently from an inner, photospheric, boundary. To this steady-state configuration, we add fluctuations close to, but above, the surface where the flow become super-Alfvenic. The time-dependent MHD equations are then solved using a semi-discrete third-order Central Weighted Essentially Non-Oscillatory (CWENO) numerical scheme. The computational domain now includes the entire sphere; the geometrical singularity at the poles is removed using the multiple grid approach described in Usmanov [1996]. Wave packets are introduced at the inner boundary such as to satisfy Faraday's Law [Yeh and Dryer, 1985] and their nonlinear evolution are followed in time.
Goldstein, Melvyn L.↗