General variational model reduction applied to incompressible viscous flows

Abstract

In this paper, a method is introduced that allows calculation of an approximate proper orthogonal decomposition (POD) without the need to perform a simulation of the full dynamical system. Our approach is based on an application of the density matrix renormalization group (DMRG) to nonlinear dynamical systems, but has no explicit restriction on the spatial dimension of the model system. The method is not restricted to fluid dynamics. The applicability is exemplified on the incompressible Navier–Stokes equation in two spatial dimensions. Merging of two equal-signed vortices with periodic boundary conditions is considered for low Reynolds numbers Re≤800 using a spectral method. We compare the accuracy of a reduced model, obtained by our method, with that of a reduced model obtained by standard POD. To this end, error functionals for the reductions are evaluated. It is observed that the proposed method is able to find a reduced system that yields comparable or even superior accuracy with respect to standard POD method results.

A short remark on Chien’s variational principle of maximum power losses for viscous fluids

“…The variational theory for viscous flow has been caught much attention, and various variational formulations were established for some special cases (Bogner, 2008;Chen et al, 2010;Ecer, 1980;He, 1998;Petrov, 2015;Fei et al, 2013;Jia et al, 2014;Li and Liu, 2013;Tao and Chen, 2013). As early as 1868, Helmholtz proposed a minimum principle for viscous fluids neglecting kinetic effect (Finlayson, 1972), and it was once proved that there exists no variational representation for Navier-Stokes equations (Finlayson, 1972).…”