SUMMARYIn this paper, the locally conservative Galerkin (LCG) method (Numer. Heat Transfer B Fundam. 2004; 46:357-370; Int. J. Numer. Methods Eng. 2007) has been extended to solve the incompressible NavierStokes equations. A new correction term is also incorporated to make the formulation to give identical results to that of the continuous Galerkin (CG) method. In addition to ensuring element-by-element conservation, the method also allows solution of the governing equations over individual elements, independent of the neighbouring elements. This is achieved within the CG framework by breaking the domain into elemental sub-domains. Although this allows discontinuous trial function field, we have carried out the formulation using the continuous trial function space as the basis. Thus, the changes in the existing CFD codes are kept to a minimum. The edge fluxes, establishing the continuity between neighbouring elements, are calculated via a post-processing step during the time-stepping operation. Therefore, the employed formulation needs to be carried out using either a time-stepping or an equivalent iterative scheme that allows post-processing of fluxes. The time-stepping algorithm employed in this paper is based on the characteristic-based split (CBS) scheme. Both steady-and unsteady-state examples presented show that the element-by-element formulation employed is accurate and robust.
SUMMARYAn element-wise locally conservative Galerkin (LCG) method is employed to solve the conservation equations of diffusion and convection-diffusion. This approach allows the system of simultaneous equations to be solved over each element. Thus, the traditional assembly of elemental contributions into a global matrix system is avoided. This simplifies the calculation procedure over the standard global (continuous) Galerkin method, in addition to explicitly establishing element-wise flux conservation. In the LCG method, elements are treated as sub-domains with weakly imposed Neumann boundary conditions. The LCG method obtains a continuous and unique nodal solution from the surrounding element contributions via averaging. It is also shown in this paper that the proposed LCG method is identical to the standard global Galerkin (GG) method, at both steady and unsteady states, for an inside node. Thus, the method, has all the advantages of the standard GG method while explicitly conserving fluxes over each element.Several problems of diffusion and convection-diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the LCG method. Both linear and quadratic elements are used in the calculations. For convection-dominated problems, Petrov-Galerkin weighting and highorder characteristic-based temporal schemes have been implemented into the LCG formulation.
Purpose -To improve inviscid compressible flow solution. Design/methodology/approach -A local element-size calculation procedure in the streamline direction and a local variable smoothing approach are employed to improve inviscid compressible flow solution. The characteristic based split approach is used as basic solution procedure to demonstrate the employed improvements. Findings -Results show that employing the element size in the streamline direction improves the solution accuracy in the transonic flow region. The nodal variable smoothing is very effective below a Mach number of 0.85 and produces results without any spatial oscillations. Originality/value -This paper fills the gap by suggesting novel procedures to study Mach number range between zero and supersonic flow.
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