Formulation of a Galerkin spectral element-fourier method for three-dimensional incompressible flows in cylindrical geometries
By Hugh Maurice Blackburn, Spencer J. Sherwin. 2004
271 citations recorded by the source corpus.
A primitive-variable formulation for simulation of time-dependent incompressible flows in cylindrical coordinates is developed. Spectral elements are used to discretise the meridional semi-plane, coupled with Fourier expansions in azimuth. Unlike previous formulations where special distributions of nodal points have been used in the radial direction, the current work adopts standard Gauss-Lobatto-Legendre nodal-based expansions in both the radial and axial directions. Using a Galerkin projection of the symmetrised cylindrical Navier-Stokes equations, all geometric singularities are removed as a consequence of either the Fourier-mode dependence of axial boundary conditions or the shape of the weight function applied in the Galerkin projection. This observation implies that in a numerical implementation, geometrically singular terms can be naively treated by explicitly zeroing their contributions on the axis in integral expressions without recourse to special treatments such as l'Hopital's rule. Exponential convergence of the method both in the meridional semi-plane and in azimuth is demonstrated through application to a three-dimensional analytical solution of the Navier-Stokes equations in which flow crosses the axis.
Subfields: Computational Mechanics
View Paper: https://www.semanticscholar.org/paper/b8fcada2f591251942b9ebc8f46f68b5855e7046
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In Computational Mechanics
Citations in the corpus
- Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book (2012, methodology): “To obtain a better reference value, we have therefore computed the solution using the spectral element code Semtex (Blackburn, 2011; Blackburn and Sherwin, 2004) with up to 80× 80 10th order elements heavily refined in the area in the vicinity of the minimum of the stream function.”
- Direct numerical simulation of turbulent flow over a backward-facing step (2011, methodology): “2.1 and 2.2 was simulated using a spectral element method code Semtex (Blackburn & Sherwin 2004), which discretizes the solution in x and z using a two-dimensional spectral/hp element method (SEM), and Fourier transform in y.”