Hierarchical Model Reduction for Incompressible Flows in Cylindrical Domains: The Axisymmetric Case

51/2016

Hierarchical Model Reduction for Incompressible Flows in Cylindrical Domains: The Axisymmetric Case

Sunday 4th December 2016

Guzzetti, S.; Perotto, S.; Veneziani, A.

Hierarchical Model (HiMod) Reduction provides an efficient way to solve Partial Differential Equations in domains with a geometrically dominant direction, like slabs or pipes. The associated solution is regarded as the combination of mainstream dynamics driven by the geometry and trans- verse components. The latter are generally of secondary importance so to be described by few degrees of freedom of a spectral approximation in- troduced at the top of a finite element discretization of the mainstream. Thus, the 3D nature of the problem is broken into a basically 1D descrip- tion added by transverse details. The versatility of this approach is that the accuracy of the method can be adaptively refined when needed, by judi- ciously selecting the number of transverse modes - as opposed to purely 1D models popular in computational hemodynamics and gasdynamics. After having investigated the basic features of the method in slab-like domains - where the Cartesian tensor product framework facilitates the practical implementation, in this paper we consider cylindrical pipes with polar co- ordinates. The selection of a different coordinate system rises several issues in particular for the most appropriate selection of the modal basis func- tions. Having computational hemodynamics as reference application, we address here the HiMod approximation of Advection-Diffusion-Reaction as well as Incompressible Navier-Stokes equations in axisymmetric domains.