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NASA NTRS2012

Effects of Mesh Irregularities on Accuracy of Finite-Volume Discretization Schemes

The effects of mesh irregularities on accuracy of unstructured node-centered finite-volume discretizations are considered. The focus is on an edge-based approach that uses unweighted least-squares gradient reconstruction with a quadratic fit. For inviscid fluxes, the discretization is nominally third order accurate on general triangular meshes. For viscous fluxes, the scheme is an average-least-squares formulation that is nominally second order accurate and contrasted with a common Green-Gauss discretization scheme. Gradient errors, truncation errors, and discretization errors are separately studied according to a previously introduced comprehensive methodology. The methodology considers three classes of grids: isotropic grids in a rectangular geometry, anisotropic grids typical of adapted grids, and anisotropic grids over a curved surface typical of advancing layer grids. The meshes within the classes range from regular to extremely irregular including meshes with random perturbation of nodes. Recommendations are made concerning the discretization schemes that are expected to be least sensitive to mesh irregularities in applications to turbulent flows in complex geometries.

Diskin, Boris↗

DOE OSTI2023

A Numerical Scheme for Wave Turbulence: 3-Wave Kinetic Equations

Here, we introduce a finite volume scheme to solve a special case of isotropic 3-wave kinetic equations. We test our numerical solution against theoretical results concerning the long time behavior of the energy and observe that our solutions verify the energy cascade phenomenon. To our knowledge, this is the first numerical scheme that can capture the long time asymptotic behavior of solutions to those isotropic 3-wave kinetic equations, where the energy cascade can be observed. Our numerical energy cascade rates are in good agreement with previously obtained theoretical results. The finite volume scheme given here relies on a new identity, allowing one to reduce the number of terms needed in the collision operators.

3-wave equation↗

NASA NTRS1990

Computation and control of asymmetric vortex flow around circular cones using Navier-Stokes equations

The unsteady, compressible, thin-layer and full Navier-Stokes equations are used to numerically simulate steady and unsteady asymmetric, supersonic, locally conical flows around a 5-deg semiapex angle circular cone. The main computational scheme is the implicit, upwind, flux-difference splitting, finite-volume scheme. Comparison of asymmetric flow solutions using the thin-layer and full Navier-Stokes equations is presented and discussed. The implicit, upwind, flux-vector splitting, finite-volume scheme has also been used to solve for the unsteady asymmetric flow with vortex shedding. The unsteady-flow solution using the flux-vector splitting scheme perfectly agrees with the previously obtained solution using the flux-difference splitting scheme. Passive control of asymmetric flows has been demonstrated and studied using sharp- and round-edged, thick and thin strakes.

Kandil, Osama A.↗

NASA NTRS2017

Stabilized Finite Elements in FUN3D

A Streamlined Upwind Petrov-Galerkin (SUPG) stabilized finite-element discretization has been implemented as a library into the FUN3D unstructured-grid flow solver. Motivation for the selection of this methodology is given, details of the implementation are provided, and the discretization for the interior scheme is verified for linear and quadratic elements by using the method of manufactured solutions. A methodology is also described for capturing shocks, and simulation results are compared to the finite-volume formulation that is currently the primary method employed for routine engineering applications. The finite-element methodology is demonstrated to be more accurate than the finite-volume technology, particularly on tetrahedral meshes where the solutions obtained using the finite-volume scheme can suffer from adverse effects caused by bias in the grid. Although no effort has been made to date to optimize computational efficiency, the finite-element scheme is competitive with the finite-volume scheme in terms of computer time to reach convergence.

Anderson, W. Kyle↗

NASA NTRS1990

Prediction of steady and unsteady asymmetric vortical flows around cones

Steady and unsteady, supersonic asymmetric vortical flows and their passive control around circular and noncircular cones are considered in this paper. These problems are formulated by using the unsteady, compressible, single and double, thin-layer. Navier-Stokes equations. The equations are solved by using an implicit, upwind, flux-difference splitting, finite-volume scheme, either in a pseudotime stepping or in an accurate-time stepping. An implicit, approximately-factored, central-difference finite-volume scheme has also been used to validate some applications of the upwind scheme. Steady asymmetric vortical flows have been predicted by using random and controlled disturbances for circular and noncircular cones. Unsteady asymmetric vortex-shedding flows have also been predicted, for the first time, using time-accurate solutions, for circular and noncircular cones. Control of flow asymmetry have been demonstrated computationally, for the first time, by inserting a vertical fin the leeward plane of geometric symmetry.

Kandil, Osama A.↗

DOE OSTI2022

A Moving Embedded Boundary Approach for the Compressible Navier-Stokes Equations in a Block-Structured Adaptive Refinement Framework

A computational technique has been developed to perform compressible flow simulations involving moving boundaries using an embedded boundary approach within the block-structured adaptive mesh refinement (SAMR) framework of AMReX [1], [91], [92]. We leverage the SAMR capability to obtain quantitatively accurate results whilst using robust, second-order finite volume schemes. A conservative, unsplit, cut-cell approach is utilized and a ghost-cell approach is developed for computing the flux on the moving, embedded boundary faces. A third-order least-squares formulation has been developed to compute the wall velocity gradients, and was found to significantly improve the performance of the solver in terms of the quantitative comparison of surface quantities such as the skin friction coefficient. Various test cases are performed to validate the method, and compared with analytical, experimental, and other numerical results in literature. Inviscid and viscous test cases are performed that span a wide regime of flow speeds - acoustic (harmonically pulsating sphere), smooth flows (expansion fan created by a receding piston) and flows with shocks (shock-cylinder interaction, shock-wedge interaction, pitching NACA 0012 airfoil and shock-cone interaction). A closed system with moving boundaries - an oscillating piston in a cylinder, showed that the percentage error in mass within the system decreases with refinement, demonstrating that the numerical scheme is conservative with grid refinement, but is not discretely conservative. Viscous test cases involve that of a horizontally moving cylinder at Re = 40, an inline oscillating cylinder at Re = 100, and a transversely oscillating cylinder at Re = 185. The judicious use of adaptive mesh refinement with appropriate refinement criteria to capture the regions of interest leads to well-resolved flow features, and good quantitative comparison is observed with the results available in literature.

adaptive refinement↗

NASA NTRS2010

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↗

NASA NTRSDate not supplied

Finite-Volume Diffusion Schemes for Svard's Eulerian Governing Equations

This paper, contains discussion on the implementation of new diffusion schemes for Svard’s Eulerian flow (EF) governing equations. It contains an alpha-damping type scheme and a new hyperbolic Eulerian flow (HEF) discretization paralleling previous work on hyperbolic Navier-Stokes (HNS). It shows these equations have simplified hyperbolic discretizations due to the simplicity of the new stress tensor and compares these schemes to established Navier-Stokes (NS) discretizations to verify the predictive utilities of such diffusion schemes.

Computational Fluid Dynamics↗

NASA NTRS2024

Finite-Volume Diffusion Schemes for Svard's Eulerian Governing Equations

Computational Fluid Dynamics↗

NASA NTRS1986

Embedded mesh solution of the 2-D Euler equations - Evaluation of interface formulations

Solution of the steady 2-D Euler equations using mesh embedding, or local grid refinement, with a cell-centered finite volume scheme is investigated. Embedded regions which are topologically similar to the global grid are considered. An isoenergetic model for the governing equations is used in Jameson's finite volume multistage scheme with modifications to the boundary conditions and smoothing. A detailed study of the embedding interface flux and smoothing formulations is conducted. Taylor expansion analysis reveals that local second order spatial accuracy is not possible if a conservative interface flux formulation is used. The analysis also gives constraints for local first order accuracy. An energy stability analysis indicates that downwind weighting of interface fluxes causes local instabilities. Analysis shows that conservative interface smoothing formulations must have a locally convective component, but that correct interface formulations allow globally dissipative smoothing. Embedded mesh solutions obtained with this scheme are presented for a transonic airfoil. They show that if embedding interfaces are close to the shocks, then small modifications in the interface location can have large effects on converge and solution accuracy.

Allmaras, S. R.↗

NASA NTRS1993

Order of accuracy of QUICK and related convection-diffusion schemes

This report attempts to correct some misunderstandings that have appeared in the literature concerning the order of accuracy of the QUICK scheme for steady-state convective modeling. Other related convection-diffusion schemes are also considered. The original one-dimensional QUICK scheme written in terms of nodal-point values of the convected variable (with a 1/8-factor multiplying the 'curvature' term) is indeed a third-order representation of the finite volume formulation of the convection operator average across the control volume, written naturally in flux-difference form. An alternative single-point upwind difference scheme (SPUDS) using node values (with a 1/6-factor) is a third-order representation of the finite difference single-point formulation; this can be written in a pseudo-flux difference form. These are both third-order convection schemes; however, the QUICK finite volume convection operator is 33 percent more accurate than the single-point implementation of SPUDS. Another finite volume scheme, writing convective fluxes in terms of cell-average values, requires a 1/6-factor for third-order accuracy. For completeness, one can also write a single-point formulation of the convective derivative in terms of cell averages, and then express this in pseudo-flux difference form; for third-order accuracy, this requires a curvature factor of 5/24. Diffusion operators are also considered in both single-point and finite volume formulations. Finite volume formulations are found to be significantly more accurate. For example, classical second-order central differencing for the second derivative is exactly twice as accurate in a finite volume formulation as it is in single-point.

Leonard, B. P.↗

NASA NTRS1988

Separation-bubble flow solution using Euler/Navier-Stokes zonal approach with downstream compatibility conditions

The two-dimensional flow over a blunt leading-edge plate is simulated on the basis of an Euler/Navier-Stokes zonal scheme. The scheme uses an implicit upwind finite-volume scheme, which is based on the van Leer flux-vector splitting. It is shown that the Euler/Navier-Stokes zonal scheme with downstream boundary-layer compatibility conditions is accurate and efficient.

Liu, C. H.↗

NASA NTRS1998

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Chapter 1 briefly reviews several related topics associated with the symmetrization of systems of conservation laws and quasi-conservation laws: (1) Basic Entropy Symmetrization Theory; (2) Symmetrization and eigenvector scaling; (3) Symmetrization of the compressible Navier-Stokes equations; and (4) Symmetrization of the quasi-conservative form of the magnetohydrodynamic (MHD) equations. Chapter 2 describes one of the best known tools employed in the study of differential equations, the maximum principle: any function f(x) which satisfies the inequality f(double prime)>0 on the interval [a,b] attains its maximum value at one of the endpoints on the interval. Chapter three examines the upwind finite volume schemes for scalar and system conservation laws. The basic tasks in the upwind finite volume approach have already been presented: reconstruction, flux evaluation, and evolution. By far, the most difficult task in this process is the reconstruction step.

Bart, Timothy J.↗

NASA NTRS2001

2-D/Axisymmetric Formulation of Multi-dimensional Upwind Scheme

A multi-dimensional upwind discretization of the two-dimensional/axisymmetric Navier-Stokes equations is detailed for unstructured meshes. The algorithm is an extension of the fluctuation splitting scheme of Sidilkover. Boundary conditions are implemented weakly so that all nodes are updated using the base scheme, and eigen-value limiting is incorporated to suppress expansion shocks. Test cases for Mach numbers ranging from 0.1-17 are considered, with results compared against an unstructured upwind finite volume scheme. The fluctuation splitting inviscid distribution requires fewer operations than the finite volume routine, and is seen to produce less artificial dissipation, leading to generally improved solution accuracy.

Wood, William A.↗

NASA NTRS1989

Unsteady Navier-Stokes computations past oscillating delta wing at high incidence

The unsteady, thin-layer, compressible Navier-Stokes equations, written in the moving frame of reference for the flow relative motion, is solved for the steady and unsteady supersonic flow around a round-edged delta wing. For supersonic flow, local conical flow solution has been obtained from the three-dimensional equations. Pseudotime stepping is used for the steady flow, while time-accurate stepping is used for the unsteady flow. The computational scheme is an implicit approximately-factored finite volume scheme which uses explicit and implicit dissipation terms. The scheme is verified for the steady flow solution. The scheme is then applied to a delta wing undergoing rolling oscillation at a reduced frequency of 1.137 with 15- deg maximum amplitude about a mean angle of attack of 10 deg for a Mach number of 2 and a Reynolds number of 500,000.

Kandil, Osama A.↗

DOE OSTI2021

Applying Time-Parallelization to Turbulent Flows

Parallelization of the temporal domain is explored for the solution of turbulent flows. Multigrid reduction-in-time (MGRIT) is used to advance the large-scale fluid dynamics in time sequentially on the coarsest space-time grid but propagate the information in time parallel on all other levels. The goal of this process is to accurately and efficiently resolve the coarse-scale turbulence structure and use that to drive the fine-scales of the turbulent flow. The extra forcing from nonlinear multigrid facilitates the coupling and interaction between fine and coarse scales, through which the multiscale nonlinear physics is properly captured. Adaptive mesh refinement is employed to finely resolve only the regions with strong gradients, which provides further computational efficiency. The underlying computational fluid dynamics solver is a fourth-order finite-volume scheme with the standard 4-stage Runge-Kutta method. An advanced approach is devised and implemented to enable MGRIT to solve highly turbulent flows successfully. Furthermore, the method is applied to solve a Taylor-Green vortex problem and a doubleshear-layer turbulent mixing flow. Results are promising, validating that MGRIT with the filtering approach has the potential to efficiently solve general turbulent flows.

Computational Fluid Dynamics↗

NASA NTRS1989

A time accurate finite volume high resolution scheme for three dimensional Navier-Stokes equations

A time accurate, three-dimensional, finite volume, high resolution scheme for solving the compressible full Navier-Stokes equations is presented. The present derivation is based on the upwind split formulas, specifically with the application of Roe's (1981) flux difference splitting. A high-order accurate (up to the third order) upwind interpolation formula for the inviscid terms is derived to account for nonuniform meshes. For the viscous terms, discretizations consistent with the finite volume concept are described. A variant of second-order time accurate method is proposed that utilizes identical procedures in both the predictor and corrector steps. Avoiding the definition of midpoint gives a consistent and easy procedure, in the framework of finite volume discretization, for treating viscous transport terms in the curvilinear coordinates. For the boundary cells, a new treatment is introduced that not only avoids the use of 'ghost cells' and the associated problems, but also satisfies the tangency conditions exactly and allows easy definition of viscous transport terms at the first interface next to the boundary cells. Numerical tests of steady and unsteady high speed flows show that the present scheme gives accurate solutions.

Liou, Meng-Sing↗

NASA NTRS2000

Numerical Solutions to the Third CAA Workshop Benchmark Problems

This paper presents numerical solutions to the problems of propagation of sound waves through a transonic nozzle, shock-sound interactions, and automobile noise involving feedback of the third NASA Computational Aeroacoustics (CAA) Workshop on benchmark problems. The numerical algorithm is based on a dual time scheme for temporal discretization and a third-order finite volume scheme for spatial discretization. The aims of this study are to apply a dual time stepping scheme to treat aeroacoustic problems of sound propagation and to validate our CAA solver with the benchmark problems for developing a numerical tool for noise analysis and control.

Loh, Roy H.↗