Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation

By Maciej Balajewicz, Earl H. Dowell, Bernd R. Noack. Journal of Fluid Mechanics, 2013

187 citations recorded by the source corpus.

Abstract We generalize the POD-based Galerkin method for post-transient flow data by incorporating Navier–Stokes equation constraints. In this method, the derived Galerkin expansion minimizes the residual like POD, but with the power balance equation for the resolved turbulent kinetic energy as an additional optimization constraint. Thus, the projection of the Navier–Stokes equation on to the expansion modes yields a Galerkin system that respects the power balance on the attractor. The resulting dynamical system requires no stabilizing eddy-viscosity term – contrary to other POD models of high-Reynolds-number flows. The proposed Galerkin method is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer. Generalizations for more Navier–Stokes constraints, e.g. Reynolds equations, can be achieved in straightforward variation of the presented results.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/aff2065a22422d1f256ae17620c601dad330e478

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistical and Nonlinear Physics

Citations in the corpus

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body (2013, background): “An approach of completely different nature is suggested by Balajewicz et al. (2013).”
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues (2021, unknown): “The last modeling approach that we discuss in this section is that proposed by Balajewicz et al. 259 Although this approach does not add a ROM closure model, it does leverage mathematical arguments to model the effect of the truncated modes.”
Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data (2016, methodology): “Compared to alternating methods, the POD conducts one iteration (returning with m = 1, since 23 it corresponds to the time variable), and hence requires either an explicit elimination of boundary conditions 24 [3], or nonlinear corrections [41].”
Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment (2014, methodology): “These methods [36, 37, 38, 23, 41], motivated by the obs ervation that higher order modes can give rise to nonphysical instabilities in the ROM system , are often physics-based and minimally intrusive to the ROM.”
Constrained sparse Galerkin regression (2016, methodology): “However, a major drawback of regression-based methods is the possible loss of existing symmetries in the governing equations which may otherwise be included in the physics-based Galerkin projection methods described previously (Balajewicz et al., 2013; Carlberg et al., 2015).”