T. J. Barth, Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes. AIAA Paper 91-0721, Jan. 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....5.1 Discretization of Viscous Terms On tetrahedral meshes, the viscous terms for the full Navier Stokes equations are discretized using a finite element Galerkin approximation. This discretization results in a nearest neighbor stencil, and can be represented using an edge based data structure [2, 27]. Furthermore, the resulting edge coefficients are symmetric and thus only 6 additional coefficients per edge are required for the viscous terms. For hexahedral meshes, the discretization of the full Navier Stokes terms invariably leads to a 27 point stencil involving points which are not ....

T. J. Barth. Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes. AIAA Paper 91-0721, January 1991.

....Of course, Delaunay triangulation has its faults. For example, it doesn t directly control the maximum angle and thus may still produce nearly degenerate triangles this is a particular problem for highly stretched meshes used in aerodynamics, and has prompted the use of a MinMax triangulation[3]. This issue hasn t come up during the testing for this thesis, so I have not followed this possibility. Another problem with Delaunay triangulation is that an M matrix (and accompanying maximum minimum properties) is not guaranteed for elliptic PDE s other than Laplace s equation, particularly ....

T. Barth, Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes, technical report AIAA 91-0721, Reno, 1991.

.... algorithms using either full matrix or matrix free implementations of Krylov methods are described in references [2, 4, 10, 11, 14, 16] The storage requirement can also be reduced by using a quasi Newton method in which the exact Jacobian matrix is replaced by a lower order approximation [1, 3, 5, 7, 12, 17, 21]. Although the number of Newton iterations to convergence may increase, the cost per GMRES iteration is reduced due to the improved conditioning of the matrix and the reduced operation count. Venkatakrishnan and Mavriplis [21] found this quasi Newton strategy to be competitive with explicit ....

Barth, T. J., "Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes," Tech. Rep. AIAA--91--0721, AIAA, 1991. 29th Aerospace Sciences Meeting.

....Of course, Delaunay triangulation has its faults. For example, it doesn t directly control the maximum angle and thus may still produce nearly degenerate triangles this is a particular problem for highly stretched meshes used in aerodynamics, and has prompted the use of a MinMax triangulation[3]. This issue hasn t come up during the testing for this thesis, so I have not followed this possibility. Another problem with Delaunay triangulation is that an M matrix (and accompanying maximum minimum properties) is not guaranteed for elliptic PDE s other than Laplace s equation, particularly ....

T. Barth, Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes, technical report AIAA 91-0721, Reno, 1991.

.... finite element solutions of Euler and NavierStokes flows, on unstructured grids composed by triangles and tetrahedra, are most costeffective when the residuals are computed employing an edge based data structure [10] 11] These data structures were introduced within the finite volume context [12] [13], and involve the derivation of special edge operators for the underlying discrete partial differential equations. The first experiments [14] with implicit solution schemes employing edge based finite volume formulations on 2D unstructured grids were promising. Such implicit finite volume ....

T. J. Barth, Numerical aspects of computing viscous high reynolds number flow on unstructured meshes, AIAA Paper 91-0721, 1991.

....was introduced in [15] Unfortunately, this method does not apply to highly stretched meshes. It usually results in a poor reconnection in the region where the nodes of the mesh are not regularly distributed. In order to overcome this difficulty, an edge swapping technique may be employed [16, 17]. The Delaunay reconnection of a set of four nodes results in two triangles where the minimum angle is maximized (Fig.5.a) In lieu of preserving this connectivity it is possible to swap the edges by minimizing the maximum angle of the two triangles (Fig.5.b) This technique has proved very ....

T. J. Barth. Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes. AIAA Paper 91-0721, 1991.

....h = b 3 , y g h = c 1 1 c 0 , c 2 0 c 3 , x g h = c 4 , y g h = x y , g b 0 1 c 1 1 b 1 0 c 0 0 b 0 c 1 1 = b 1 c 0 0 = 110 tivity and shape of the mesh, statements about positivity of the stencil may be proven. This is shown in [7] for finite volume solutions of Laplace s equation upon triangular tetrahedral meshes (in two and three dimensions) using a vertex based scheme with a special covolume upon which conservation is maintained. This procedure is shown to be equivalent to a Galerkin formulation using linear ....

....each face is perpendicular to the vector joining the centroids that share the face, so that the cotangent is everywhere zero. Example meshes would be a triangular mesh of all equilateral triangles or a quadrilateral mesh that is either unstretched, or stretched along the coordinate axes only. In [7] it is shown that in twodimensions, a Delaunay mesh guarantees positivity when using a linear Galerkin, finite element formulation, although for the reconstruction scheme here, a Delaunay mesh does not guarantee it. A deeper analysis of the positivity criteria for the scheme here, upon a general, ....

T.J. Barth. Numerical Aspects of Computing Viscous High Reynolds Number Flows on Unstructured Meshes. AIAA Paper AIAA-91-0721, 1991.

....j i Gamma i ru j j Delta ij( V Delta n ij ) u x ) i Deltax 2 6 (u xxx ) i Deltax Deltay 6 (u xxy ) i Deltay 2 6 (u xyy ) i Gamma 1 4 i Deltax 2 (u xxx ) i Deltax Deltay(u xxy ) i Deltay 2 (u xyy ) i j O(h 4 ) 32 Selon T. Barth [BAR91], il peut #tre aussi avantageux de conserver la partie centr#e Galerkin du sch#ma, et de ne reconstruire la variable u que pour la partie dissipative du sch#ma. La reconstruction pour la partie dissipative du AEux OE D donne une dissipation du quatri#me ordre: OE D = D 4 M u i O(h 5 ) o# D ....

T. BARTH. Numerical aspects of computing viscous high Reynolds number AEows on unstructured meshes. AIAA-91-0721, 1991. 29th Aerospace Sciences Meeting, Reno (nevada).

....to general elements. While the underlying Navier Stokes methodologies have advanced rapidly in recent years, they are still less mature than their more established structured grid counterparts. Legitimate questions still remain regarding the solution accuracy of unstructured NavierStokes schemes [3,4], and the user community does not yet have a sufficient experience base from which to derive full confidence. Thus, there is a strong need for more fundamental analyses and systematic application studies which address the key issues of solution accuracy, robustness, and efficiency on a range of ....

Barth, T. J.: "Numerical Aspects of Computing Viscous High Reynolds Number Flows on Unstructured Meshes", AIAA Paper 91-0721, January 1991.

.... local adaptation that are comparable to those obtained through global refinement remains an area where further work is required [6] Although work remains to fully realize their potential, much progress has been reported in computing viscous flows on unstructured grids (See for example [7] [8], 9] 10] The purpose of this study is to present computational results obtained with a particular unstructured grid method [11] 12]that has been applied to several flows over multielement airfoils. Comparisons between computational results and experimental data are Fluid Mechanics ....

.... to undeflected position) Computational Method The computational method used in this study is a node based, implicit, unstructured, upwind flow solver described in reference [11] In this code, the discretization of the convective and viscous terms is handled similarly to the method of reference [8]. The inviscid fluxes are obtained using Roe s approximate Riemann solver [13] the viscous terms are evaluated with a Galerkin type approximation that results in a central difference formulation for these terms. Two different turbulence models are presently utilized in the code. These include ....

Barth, T. J., "Numerical Aspects of Computing Viscous High Reynolds Number Flows on Unstructured Meshes," AIAA 91--0721, 1991.

....in two dimensions are illustrated in Figures 1 and 2. The definition of the median dual as the control volume in a cell vertex scheme comes about by showing the equivalence with a Galerkin finite element method employing piecewise linear basis functions while computing the gradient of a function [107, 9]. An alternative definition of a control volume for a cell vertex scheme is a Voronoi cell which is composed of perpendicular bisectors drawn between pairs of grid points. The Voronoi cell of a site (grid point) is defined as that region of space containing points which are closer to the site than ....

T. J. Barth, Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes. AIAA Paper 91-0721, Jan. 1991.

....the ideas of Hlavacek et al. 21] to estimate nodal derivatives and hence second derivatives. 3 Mesh Movement Redistribution in 2 3D The idea that it is important for the the shape of the elements to reflect local solution behaviour, particularly for highly directional flow problems, is well known [15, 9, 27]. One of the significant steps in realising this understanding was the Moving Finite Element method of Keith Miller, see Baines [6] which continuously moves the mesh for transient problems. Some of the meshes shown by Baines are highly distorted. A similar approach, but rather simpler, was ....

....who applied a simple local iterative procedure based on quantities such as pressure gradients to produce stretched meshes for highly directional Euler equations flow problems. A key part of their algorithm is a simple Laplacian smoothing approach that has also been used by many others, e.g. Barth [9, 10]. A slightly different approach still is employed by Tourigny and Baines [42] who investigate the construction of locally optimal piecewise polynomial fits to data and produce meshes which vary from smooth to skewed, depending on the solution. The idea is further extended by Tourigny and Hulseman ....

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T.J.Barth, Numerical Aspects of Computing Viscous High Reynolds Flows on Unstructured Meshes. AIAA Paper 91-0721, 29th Aerospace Sciences Meeting, January 7-10 1991, Reno Nevada.

....code. As discussed in reference [11] the poor performance of the central difference formulation is attributed to the scalar artificial dissipation formulas commonly used to damp odd even oscillations and to provide non linear stability. For upwind calculations on unstructured grids, Barth [12] has described methodology for utilizing Roe s approximate Riemann solver [10] for the inviscid flux computations and a Galerkin formulation for the viscous terms. In this work, a sparse matrix solver is used in conjunction with a Runge Kutta time stepping algorithm for updating the solution at ....

....is utilized for updating the solution. At each time step, the linear system of equations is approximately solved with a red black type relaxation procedure. This method circumvents the need to assemble large matrices and, therefore, significantly reduces the required memory. As in reference [12], the Baldwin Barth turbulence model is used for computing flows at high Reynolds number. In addition, a recently developed turbulence model due to Spalart and Allmaras [14] is also utilized and comparisons between solutions obtained with each model are shown. Results are shown for subsonic flow ....

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Barth, T. J., "Numerical Aspects of Computing Viscous High Reynolds Number Flows on Unstructured Meshes," AIAA 91--0721, 1991.

....on the order of one, thus resulting in some negative weight factors. While it can be demonstrated that the computed weight factors produce an exact linear reconstruction, those with negative values violate the principle of positivity, with a detrimental impact on stability during convergence [16,17]. It is, thus, necessary to clip the weight factors between 0 and 2, thereby losing some of the exactness of the linear reconstruction, but ensuring a more stable scheme. Viscous Fluxes The viscous fluxes G(Q) are approximated at the cell face centroids by linear reconstruction which provides a ....

....solving 3 D viscous problems with the present cell centered scheme has shown that maximum Courant, Friedrichs, Lewy (CFL) number is limited to approximately 25. This limitation is a consequence of violating the principle of positivity in weighting factors, as noted an earlier section and in Refs. [16, 17]. The inherent stability limitation can be improved by scaling the CFL number according to the deviation of cell aspect ratio from the ideal value of an isotropic tetrahedron. This enables the dominate flow field to evolve quickly with the higher CFL numbers, while restricting the more ....

Barth, T. J.: "Numerical Aspects of Computing Viscous High Reynolds Number Flows on Unstructured Meshes", AIAA Paper

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T. J. Barth, Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes. AIAA Paper 91-0721, Jan. 1991.

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T. J. Barth, Numerical aspects of computing viscous high Reynolds number flows on unstructured meshes. AIAA Paper 91-0721, Jan. 1991.

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Barth, T.J.,"Numerical Aspects of Computing Viscous High Reynolds Number Flows on Unstructured Meshes", AIAA paper 91-0721, January, 1991.

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Barth, T. J.: Numerical Aspects of Computing Viscous High Reynolds Number Flows on Unstructured Meshes. AIAA 91-0721.

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Barth, T. J., "Numerical Aspects of Computing Viscous High Reynolds Number Flows on Unstructured Meshes," AIAA 91-0721,

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Barth, T. J., "Numerical Aspects of Computing Viscous High Reynolds Number Flows on Unstructured Meshes," AIAA Paper 91--0721, 1991.

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