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NASA NTRS2010

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Viscous Fluxes

Discretization of the viscous terms in current finite-volume unstructured-grid schemes are compared using node-centered and cell-centered approaches in two dimensions. Accuracy and complexity are studied for four nominally second-order accurate schemes: a node-centered scheme and three cell-centered schemes - a node-averaging scheme and two schemes with nearest-neighbor and adaptive compact stencils for least-square face gradient reconstruction. The grids considered range from structured (regular) grids to irregular grids composed of arbitrary mixtures of triangles and quadrilaterals, including random perturbations of the grid points to bring out the worst possible behavior of the solution. Two classes of tests are considered. The first class of tests involves smooth manufactured solutions on both isotropic and highly anisotropic grids with discontinuous metrics, typical of those encountered in grid adaptation. The second class concerns solutions and grids varying strongly anisotropically over a curved body, typical of those encountered in high-Reynolds number turbulent flow simulations. Tests from the first class indicate the face least-square methods, the node-averaging method without clipping, and the node-centered method demonstrate second-order convergence of discretization errors with very similar accuracies per degree of freedom. The tests of the second class are more discriminating. The node-centered scheme is always second order with an accuracy and complexity in linearization comparable to the best of the cell-centered schemes. In comparison, the cell-centered node-averaging schemes may degenerate on mixed grids, have a higher complexity in linearization, and can fail to converge to the exact solution when clipping of the node-averaged values is used. The cell-centered schemes using least-square face gradient reconstruction have more compact stencils with a complexity similar to that of the node-centered scheme. For simulations on highly anisotropic curved grids, the least-square methods have to be amended either by introducing a local mapping based on a distance function commonly available in practical schemes or modifying the scheme stencil to reflect the direction of strong coupling. The major conclusion is that accuracies of the node centered and the best cell-centered schemes are comparable at equivalent number of degrees of freedom.

Diskin, Boris↗

NASA NTRS1979

Recent progress in finite-volume calculations for wing-fuselage combinations

Progress in the application of finite-volume methods to the calculation of transonic potential flows past general wing-body combinations is reviewed. Two different methods of generating boundary-conforming grids are investigated, and the results compared to provide an estimate of solution sensitivity to grid geometry. Both conservative and quasi-conservative difference schemes are used in one of the coordinate systems. Results show that the error introduced by the quasi-conservative formulation seems to be small, although a one-dimensional analysis suggests that schemes of this type do not necessarily produce mass-conserving shocks. Comparison of calculated results with experimental data for realistic fuselage geometries clearly shows the importance of modelling the effect of fuselage geometry upon the wing pressure distribution.

Caughey, D. A.↗

NASA NTRS1981

Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes

A new combination of a finite volume discretization in conjunction with carefully designed dissipative terms of third order, and a Runge Kutta time stepping scheme, is shown to yield an effective method for solving the Euler equations in arbitrary geometric domains. The method has been used to determine the steady transonic flow past an airfoil using an O mesh. Convergence to a steady state is accelerated by the use of a variable time step determined by the local Courant member, and the introduction of a forcing term proportional to the difference between the local total enthalpy and its free stream value.

Jameson, A.↗

NASA NTRS1990

Using finite volume methods for aeroacoustics

This paper is concerned with the application of unsteady finite volume methods to the numerical calculation of aeroacoustic problems. Some discussion is made of how the acoustic analogy of Lighthill has led to a separation between the calculation of acoustic source, or flow, and the subsequent sound field. It is pointed out that for transonic flow this separation is not necessarily useful. The example problem of an impulsively started cylinder is given for speed of Mach .1 and .5. In the Mach .5 case a strong shock is followed as it develops. A further example of the cylinder stopping impulsively is given as well. MacCormack's explicit predictor-corrector method is used for all the examples.

Brentner, Kenneth S.↗

NASA NTRSDate not supplied

Edge-Based Viscous Method for Mixed-Element Node-Centered Finite-Volume Solvers

A novel, efficient, edge-based viscous (EBV) discretization method has been recently developed, implemented in a practical, unstructured-grid, node-centered, finite-volume flow solver, and applied to viscous-kernel computations that include evaluations of meanflow viscous fluxes, turbulence-model and chemistry-model diffusion terms, and the corresponding Jacobian contributions. Initially, the EBV method had been implemented for tetrahedral grids and demonstrated multifold acceleration of all viscous-kernel computations. This paper presents an extension of the EBV method for mixed-element grids. In addition to the primal edges of a given mixed-element grid, virtual edges are introduced to connect cell nodes that are not connected by a primal edge. The EBV method uses an efficient loop over all (primal and virtual) edges and features a compact discretization stencil based on the nearest neighbors. This study verifies the EBV method and assesses its efficiency on mixed-element grids by comparing the EBV solution accuracy and iterative convergence with those of well-established solutions obtained using a cell-based viscous (CBV) discretization method. The EBV solver’s memory footprint is optimized and often smaller than the memory footprint of the CBV solver. A multifold speedup is demonstrated for all viscous-kernel computations resulting in significant reduction of the time to solutions for several benchmark mixed-element-grid computations, including simulations of a flow around NASA’s juncture-flow model and a hypersonic, chemically reacting flow around a blunt body.

CFD↗

NASA NTRSDate not supplied

Numerical Analysis of Cavitation-Induced Two-Phase Fluid Transients in Cryogenic Feedlines Using Finite Volume Procedure

This article presents a numerical investigation of the cavitation induced fluid transient in cryogenic feedline due to sudden closure of the valve A one dimensional numerical model based on finite volume procedure consisting of nodes and branches using Generalized Fluid System Simulation Program ( has been employed for the prediction of pressure variations The unsteady friction formulation accurately determines wall shear stress, and a homogeneous mixture model is used for two phase flow modeling Numerical results obtained with the present formulation are compared with experimental results and Method of Characteristic ( based results It is observed that low residual in flow variable calculation improves the solution providing numerically stable results GFSSP is able to reproduce the fluid transient peak due to vapor collapse Subsequently, it can retain the shape of pressure wave for two phase fluid transients, but it is dampening faster compared to experiments Further investigation is required with set of experiments to validate the present numerical approach for different experimental data.

Fluid Transient↗

NASA NTRS1977

Parallel computation of unsteady, three-dimensional, chemically reacting, nonequilibrium flow using a time-split finite-volume method on the Illiac IV

A description is presented of the split finite-volume method which is a viable numerical procedure for performing with the aid of a modern special purpose vector computer numerical simulation studies of complicated flow fields, including chemical reactions, about geometrically complex bodies. Such numerical studies are needed for the development of atmospheric entry vehicles such as the space shuttle. The equations which are approximated are quite general and can be used in studies of combustion, pollution, and other chemically reacting flow phenomena, where convective transport effects dominate the influence of radiative, viscous, and other transport mechanisms. The shock perturbed flow about a shuttle orbiter flying at a large angle of attack during atmospheric entry is illustrated. The method uses a time splitting of the convection differencing operator to achieve efficient data management.

Reinhardt, W. A.↗

NASA NTRS1982

A vectorized, finite-volume, adaptive-grid algorithm for Navier-Stokes calculations

An adaptive grid, finite-volume method has been used to solve the Navier-Stokes equations for complete (forebody and afterbody) flowfields around blunt bodies. The code, which is applicable for axisymmetric or two-dimensional flows, allows the mesh to adjust during the computation to provide a closer spacing of mesh points in regions of high gradients, thus minimizing the number of required computational points. The solution technique is explicit, utilizing a maximum time-step advancement at each grid point to accelerate convergence to the steady state. The code has been fully vectorized for efficient solution on the CYBER 203 computer. A very flexible rezoning routine is used to concentrate mesh points anywhere in the field, either by a user-defined weighting function or by allowing high gradient regions to adjust the grid. The grid adjustment routine is implicit in nature and represents a very small portion of the total computational cost. Currently, the code runs in approximately 0.000016 seconds per grid point per iteration.

Gnoffo, P. A.↗

NASA NTRS1993

A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems

The finite-volume approach is presently used to obtain a 2D, high-order accurate and basically nonoscillatory shock-capture method whose high-order spatial accuracy is obtained by means of a piecewise polynomial approximation of the solution from cell averages. Attention is given to a high-order spatial operator that is able to both retain high-order accuracy in smooth regions and avoid the oscillations that are associated with interpolations across steep gradients. The operator is extended to hyperbolic systems of equations and curvilinear meshes.

Casper, Jay↗

NASA NTRS1993

On positivity preserving finite volume schemes for compressible Euler equations

Positivity preserving property of first and higher order finite volume schemes for one and two dimensional compressible Euler equations of gas dynamics is considered. A general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on exact or approximate Riemann solver and the reconstruction are both positivity preserving. Appropriate limitation to achieve high order positivity preserving reconstruction is described.

Perthame, Benoit↗

NASA NTRS2009

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Viscous Fluxes - Part 1

Discretization of the viscous terms in current finite-volume unstructured-grid schemes are compared using node-centered and cell-centered approaches in two dimensions. Accuracy and efficiency are studied for six nominally second-order accurate schemes: a node-centered scheme, cell-centered node-averaging schemes with and without clipping, and cell-centered schemes with unweighted, weighted, and approximately mapped least-square face gradient reconstruction. The grids considered range from structured (regular) grids to irregular grids composed of arbitrary mixtures of triangles and quadrilaterals, including random perturbations of the grid points to bring out the worst possible behavior of the solution. Two classes of tests are considered. The first class of tests involves smooth manufactured solutions on both isotropic and highly anisotropic grids with discontinuous metrics, typical of those encountered in grid adaptation. The second class concerns solutions and grids varying strongly anisotropically over a curved body, typical of those encountered in high-Reynolds number turbulent flow simulations. Results from the first class indicate the face least-square methods, the node-averaging method without clipping, and the node-centered method demonstrate second-order convergence of discretization errors with very similar accuracies per degree of freedom. The second class of tests are more discriminating. The node-centered scheme is always second order with an accuracy and complexity in linearization comparable to the best of the cell-centered schemes. In comparison, the cell-centered node-averaging schemes are less accurate, have a higher complexity in linearization, and can fail to converge to the exact solution when clipping of the node-averaged values is used. The cell-centered schemes using least-square face gradient reconstruction have more compact stencils with a complexity similar to the complexity of the node-centered scheme. For simulations on highly anisotropic curved grids, the least-square methods have to be amended either by introducing a local mapping of the surface anisotropy or modifying the scheme stencil to reflect the direction of strong coupling.

Diskin, Boris↗

NASA NTRS2007

A Time-Accurate Upwind Unstructured Finite Volume Method for Compressible Flow with Cure of Pathological Behaviors

A time-accurate, upwind, finite volume method for computing compressible flows on unstructured grids is presented. The method is second order accurate in space and time and yields high resolution in the presence of discontinuities. For efficiency, the Roe approximate Riemann solver with an entropy correction is employed. In the basic Euler/Navier-Stokes scheme, many concepts of high order upwind schemes are adopted: the surface flux integrals are carefully treated, a Cauchy-Kowalewski time-stepping scheme is used in the time-marching stage, and a multidimensional limiter is applied in the reconstruction stage. However even with these up-to-date improvements, the basic upwind scheme is still plagued by the so-called "pathological behaviors," e.g., the carbuncle phenomenon, the expansion shock, etc. A solution to these limitations is presented which uses a very simple dissipation model while still preserving second order accuracy. This scheme is referred to as the enhanced time-accurate upwind (ETAU) scheme in this paper. The unstructured grid capability renders flexibility for use in complex geometry; and the present ETAU Euler/Navier-Stokes scheme is capable of handling a broad spectrum of flow regimes from high supersonic to subsonic at very low Mach number, appropriate for both CFD (computational fluid dynamics) and CAA (computational aeroacoustics). Numerous examples are included to demonstrate the robustness of the methods.

Loh, Ching Y.↗

NASA NTRS2012

Implementation of Finite Volume based Navier Stokes Algorithm Within General Purpose Flow Network Code

This paper describes a finite volume based numerical algorithm that allows multi-dimensional computation of fluid flow within a system level network flow analysis. There are several thermo-fluid engineering problems where higher fidelity solutions are needed that are not within the capacity of system level codes. The proposed algorithm will allow NASA's Generalized Fluid System Simulation Program (GFSSP) to perform multi-dimensional flow calculation within the framework of GFSSP s typical system level flow network consisting of fluid nodes and branches. The paper presents several classical two-dimensional fluid dynamics problems that have been solved by GFSSP's multi-dimensional flow solver. The numerical solutions are compared with the analytical and benchmark solution of Poiseulle, Couette and flow in a driven cavity.

Schallhorn, Paul↗

NASA NTRS1976

Finite-volume solution of the Euler equations for steady three-dimensional transonic flow

A numerical finite-volume method is proposed for solving the Euler equations of a steady three-dimensional flow. The subsonic regions embedded in steady supersonic flow about aircraft are computed by introducing a time-dependent coordinate system that maps the fuselage and wing into one coordinate surface. The suitability of the method to a delta-winged aircraft is demonstrated, and its applicability to rectangular-winged aircraft and problems in internal transonic flow is pointed out.

Rizzi, A.↗

NASA NTRS1989

Compact finite volume methods for the diffusion equation

An approach to treating initial-boundary value problems by finite volume methods is described, in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, it is possible to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl = 0 and curl grad = 0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and alternating direction implicit (ADI) methods. The treatment of general curvilinear coordinates is shown to result from a specialization of these general results.

Rose, Milton E.↗

NASA NTRS1990

Finite-volume application of high order ENO schemes to multi-dimensional boundary-value problems

The finite volume approach in developing multi-dimensional, high-order accurate essentially non-oscillatory (ENO) schemes is considered. In particular, a two dimensional extension is proposed for the Euler equation of gas dynamics. This requires a spatial reconstruction operator that attains formal high order of accuracy in two dimensions by taking account of cross gradients. Given a set of cell averages in two spatial variables, polynomial interpolation of a two dimensional primitive function is employed in order to extract high-order pointwise values on cell interfaces. These points are appropriately chosen so that correspondingly high-order flux integrals are obtained through each interface by quadrature, at each point having calculated a flux contribution in an upwind fashion. The solution-in-the-small of Riemann's initial value problem (IVP) that is required for this pointwise flux computation is achieved using Roe's approximate Riemann solver. Issues to be considered in this two dimensional extension include the implementation of boundary conditions and application to general curvilinear coordinates. Results of numerical experiments are presented for qualitative and quantitative examination. These results contain the first successful application of ENO schemes to boundary value problems with solid walls.

Casper, Jay↗

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↗

NASA NTRS2013

Implementation of Implicit Adaptive Mesh Refinement in an Unstructured Finite-Volume Flow Solver

This paper explores the implementation of adaptive mesh refinement in an unstructured, finite-volume solver. Unsteady and steady problems are considered. The effect on the recovery of high-order numerics is explored and the results are favorable. Important to this work is the ability to provide a path for efficient, implicit time advancement. A method using a simple refinement sensor based on undivided differences is discussed and applied to a practical problem: a shock-shock interaction on a hypersonic, inviscid double-wedge. Cases are compared to uniform grids without the use of adapted meshes in order to assess error and computational expense. Discussion of difficulties, advances, and future work prepare this method for additional research. The potential for this method in more complicated flows is described.

Schwing, Alan M.↗