A segregated reduced order model of a pressure-based solver for turbulent compressible flows

TL;DR

This paper develops a segregated reduced-order model for turbulent compressible flows discretized with a finite-volume method. The approach couples POD-Galerkin projections for pressure, velocity, and energy with a data-driven neural network that predicts the eddy-viscosity field mu_t, enabling a turbulence-model–independent ROM within a SIMPLE-based framework. Offline training builds three snapshot-based bases (pressure, velocity, energy) and an eddy-viscosity basis, while online evaluation solves a reduced-order SIMPLE loop with relaxation, using mu_t predicted from parametric inputs. Two parametrized benchmarks (physical and geometrical) demonstrate that the ROM reproduces full-order solutions with substantially fewer degrees of freedom, offering substantial computational savings for high-Reynolds/high-Mach compressible flows and parametric studies.

Abstract

This article provides a reduced-order modelling framework for turbulent compressible flows discretized by the use of finite volume approaches. The basic idea behind this work is the construction of a reduced-order model capable of providing closely accurate solutions with respect to the high fidelity flow fields. Full-order solutions are often obtained through the use of segregated solvers (solution variables are solved one after another), employing slightly modified conservation laws so that they can be decoupled and then solved one at a time. Classical reduction architectures, on the contrary, rely on the Galerkin projection of a complete Navier-Stokes system to be projected all at once, causing a mild discrepancy with the high order solutions. This article relies on segregated reduced-order algorithms for the resolution of turbulent and compressible flows in the context of physical and geometrical parameters. At the full-order level turbulence is modeled using an eddy viscosity approach. Since there is a variety of different turbulence models for the approximation of this supplementary viscosity, one of the aims of this work is to provide a reduced-order model which is independent on this selection. This goal is reached by the application of hybrid methods where Navier-Stokes equations are projected in a standard way while the viscosity field is approximated by the use of data-driven interpolation methods or by the evaluation of a properly trained neural network. By exploiting the aforementioned expedients it is possible to predict accurate solutions with respect to the full-order problems characterized by high Reynolds numbers and elevated Mach numbers.

Figures (16)

• Figure 1: Relation between two neighbor cells of the tessellation $\mathcal{T}$ for a certain variable $v$.
• Figure 2: Scheme of the snapshots selection for $\Delta = 2$: black dots are discarded intermediate solutions, blue dots are saved intermediate solutions while the red dot represents the final solution.
• Figure 3: Schematic perspective of a fully connected neural network composed by an input layer, two hidden layers and an output layer, linking parameters $\pi_i$ and reduced velocity coefficients $b_i$ to reduced eddy viscosity coefficients $m_i$, being $N_{\pi}$ the number of parameters possibly existing in the problem.
• Figure 4: Cumulative eigenvalues trends.
• Figure 5: Loss function decay for both train and test sets.