Reduced order models for thermally coupled low Mach flows

Ricardo Reyes,Ramon Codina,Sergio Idelsohn,Joan Baiges

doi:10.1186/s40323-018-0122-7

Abstract

In this paper we present a collection of techniques used to formulate a projection-based reduced order model (ROM) for zero Mach limit thermally coupled Navier–Stokes equations. The formulation derives from a standard proper orthogonal decomposition (POD) model reduction, and includes modifications to improve the drawbacks caused by the inherent non-linearity of the used Navier–Stokes equations: a hyper-ROM technique based on mesh coarsening; an implicit ROM subscales formulation based on a variational multi-scale (VMS) framework; and a Petrov–Galerkin projection necessary in the case of non-symmetric terms. At the end of the article, we test the proposed ROM formulation using 2D and 3D versions of the same example: a differentially heated cavity.

Highlights

The main purpose of this paper is to develop a model reduction formulation suitable for thermally coupled flows, by expanding the techniques in projection-based model reduction developed for several applications on fluid dynamics—mostly for incompressible Navier–Stokes equations—to the zero Mach limit Navier–Stokes equations developed in [1, 2].Following the model reduction developments in [3] we choose a proper orthogonal decomposition (POD) model reduction approach
For that purpose we have described a set of tools that allow us to tackle the main problems that arise: lack of stability, added computational cost due to the non-linearities and non-optimal projection over the reduced space
To solve the lack of stability, we have proposed a variational multi-scale (VMS)-finite element (FE)-reduced order model (ROM) formulation with the following characteristics:

Summary

Introduction

The main purpose of this paper is to develop a model reduction formulation suitable for thermally coupled flows, by expanding the techniques in projection-based model reduction developed for several applications on fluid dynamics—mostly for incompressible Navier–Stokes equations—to the zero Mach limit Navier–Stokes equations developed in [1, 2]. The implementation of this technique is done straightforwardly by writing the discrete approximation (Eq 31) in function of the new coarser mesh. Two dimensional case In the 2D problem, we use 2 uniform structured meshes composed of quadrilateral elements: one with 10,000 elements and a mesh size h = 0.01, used for the solving the FOM and the ROM; and one with 2500 elements and a mesh size h = 0.02, for testing the hyper-ROM formulation. The reduction in the computational time for the 3D problem is larger than the one achieved for the 2D problem, using a similar amount of basis vectors

Conclusions

Methods