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Masson, C.; Saabas, H. J.; Baliga, B. R.
doi: 10.1002/fld.1650180102pmid: N/A
The formulation of a control‐volume‐based finite element method (CVFEM) for axisymmetric, two‐dimensional, incompressible fluid flow and heat transfer in irregular‐shaped domains is presented. The calculation domain is discretized into torus‐shaped elements and control volumes. In a longitudinal cross‐sectional plane, these elements are three‐node triangles, and the control volumes are polygons obtained by joining the centroids of the three‐node triangles to the mid‐points of the sides. Two different interpolation schemes are proposed for the scalar‐dependent variables in the advection terms: a flow‐oriented upwind function, and a mass‐weighted upwind function that guarantees that the discretized advection terms contribute positively to the coefficients in the discretized equations. In the discretization of diffusion transport terms, the dependent variables are interpolated linearly. An iterative sequential variable adjustment algorithm is used to solve the discretized equations for the velocity components, pressure and other scalar‐dependent variables of interest. The capabilities of the proposed CVFEM are demonstrated by its application to four different example problems. The numerical solutions are compared with the results of independent numerical and experimental investigations. These comparisons are quite encouraging.
Koobus, Bruno; Lallemand, Marie‐Hélène; Dervieux, Alain
doi: 10.1002/fld.1650180103pmid: N/A
We are interested in solving second‐order PDEs with multigrid and unstructured meshes. The multigrid strategy we present here is adapted from the generalized finite volume agglomeration multigrid algorithm we have developed recently for the solution of the Euler equations. We now focus on Poisson's equation. A strategy is defined by introducing a correction factor for the diffusive terms, and some illustrating results are given.
Winterscheidt, Daniel; Surana, Karan S.
doi: 10.1002/fld.1650180104pmid: N/A
A p‐version least squares finite element formulation for non‐linear problems is applied to the problem of steady, two‐dimensional, incompressible fluid flow. The Navier‐Stokes equations are cast as a set of first‐order equations involving viscous stresses as auxiliary variables. Both the primary and auxiliary variables are interpolated using equal‐order C0 continuity, p‐version hierarchical approximation functions. The least squares functional (or error functional) is constructed using the system of coupled first‐order non‐linear partial differential equations without linearization, approximations or assumptions. The minimization of this least squares error functional results in finding a solution vector {δ} for which the partial derivative of the error functional (integrated sum of squares of the errors resulting from individual equations for the entire discretization) with respect to the nodal degrees of freedom {δ} becomes zero. This is accomplished by using Newton's method with a line search. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.
doi: 10.1002/fld.1650180105pmid: N/A
We develop simulation tools for the non‐stationary incompressible 2D Navier‐‐Stokes equations. The most important components of the finite element code are: the fractional step ϑ‐scheme, which is of second‐order accuracy and strongly A‐stable, for the time discretization; a fixed point defect correction method with adaptive step length control for the non‐linear problems (stationary Navier‐Stokes equations); a modified upwind discretization of higher‐order accuracy for the convective terms. Finally, the resulting nonsymmetric linear subproblems are treated by a special multigrid algorithm which is adapted to the quadrilateral non‐conforming discretely divergence‐free finite elements. For the graphical postprocess we use a fully non‐stationary and interactive particle‐tracing method. With extensive test calculations we show that our method is a candidate for a ‘black box’ solver.
doi: 10.1002/fld.1650180106pmid: N/A
A numerical scheme is presented for the solution of the Euler equations of compressible flow of a gas in a single spatial co‐ordinate. This includes flow in a duct of variable cross‐section as well as flow with slab, cylindrical or spherical symmetry and can prove useful when testing codes for the two‐dimensional equations governing compressible flow of a gas. The resulting scheme requires an average of the flow variables across the interface between cells and for computational efficiency this average is chosen to be the arithmetic mean, which is in contrast to the usual ‘square root’ averages found in this type of scheme. The scheme is applied with success to five problems with either slab or cylindrical symmetry and a comparison is made in the cylindrical case with results from a two‐dimensional problem with no sources.
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