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Data-Driven POD-Galerkin Reduced Order Model for Turbulent Flows

Saddam Hijazi, Giovanni Stabile, Andrea Mola, Gianluigi Rozza

arXiv:1907.09909v3math.NAcs.CE

TL;DR

The paper addresses the cost of full-order simulation and the stability challenges of projection-based ROMs for turbulent flows. It introduces Mixed-ROM, combining POD-Galerkin projection for velocity and pressure with data-driven eddy-viscosity reduction, and validates it on steady and unsteady cases with accurate fields and substantial speedups. The approach still leaves long-time unsteady stabilization and broader multiphysics extensions as future work.

  • Problem

    Full-order simulations are costly for real-time and multi-query applications, while projection-based ROMs for turbulent flows face energy-stability issues and prior models often omit pressure or turbulence fields.

  • Method

    Mixed-ROM uses POD-Galerkin projection for velocity and pressure and radial-basis-function interpolation to compute reduced eddy-viscosity coefficients.

  • Results

    The Mixed-ROM accurately reconstructs steady and unsteady turbulent-flow fields, including lift-coefficient histories, with speedups of around 1000 for steady flow and 10 for unsteady flow.

  • Takeaways & Limitations

    The mixed projection/data-driven formulation provides accurate turbulent-flow predictions at reduced computational cost in the tested finite-volume settings.

  • Takeaways & Limitations

    The work identifies a need for stabilization techniques for long-time integration of unsteady flows and for multiphysics problems.

Abstract

from arXiv · show

In this work we present a Reduced Order Model which is specifically designed to deal with turbulent flows in a finite volume setting. The method used to build the reduced order model is based on the idea of merging/combining projection-based techniques with data-driven reduction strategies. In particular, the work presents a mixed strategy that exploits a data-driven reduction method to approximate the eddy viscosity solution manifold and a classical POD-Galerkin projection approach for the velocity and the pressure fields, respectively. The newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with Reynolds up to Re=O(10^5).

1. Introduction

Classical full-order simulations can be too costly for real-time and multi-query applications, motivating reduced-order models. This paper combines projection-based reduction for velocity and pressure with data-driven approximation of turbulent eddy viscosity.

  • Full-order simulations can be computationally prohibitive for real-time control and multi-query tasks, motivating reduced-order methods.ROMs reduce computational time while targeting repeated or time-sensitive computations.
  • Projection-based ROMs are established for laminar flows but face energy-stability issues in turbulent flows because POD favors large energetic scales over turbulent small scales.The paper frames turbulent-flow stabilization as an unresolved challenge for standard projection methods.
  • Existing stabilization and turbulence-ROM strategies include closure models, LES at the full-order level, H1-based POD, minimum-residual formulations, DMD, and constrained Galerkin methods.The introduction surveys several alternatives for improving dissipative behavior or long-time stability.
  • Prior reduced models often omitted pressure and turbulence-related fields, whereas this work targets velocity, pressure, and turbulent viscosity together.Including eddy viscosity supports consistency with the full-order turbulence formulation, while pressure matters for outputs such as surface forces.
  • The proposed mixed approach uses traditional projection methods for reduced velocity and pressure degrees of freedom and radial-basis-function interpolation for eddy-viscosity coefficients.The data-driven component approximates the eddy-viscosity solution manifold without requiring the additional turbulence variables k, ϵ, or ω.

2. The full order model (FOM)

The full-order model discretizes parametrized incompressible Navier–Stokes equations with a two-dimensional finite volume method. High-Reynolds-number turbulence is modeled with RANS and an eddy-viscosity closure based on Reynolds averaging.

  • The mathematical problem: parametrized Navier-Stokes equations: The parametrized incompressible Navier–Stokes problem is posed on a domain with inlet, outlet, and wall boundaries, with velocity, pressure, viscosity, and initial data specified.The parameter vector controls the inlet velocity function, while the time interval and initial velocity define the evolution problem.
  • The finite volume discretization: The finite volume discretization integrates the governing equations over polygonal control volumes in two-dimensional tessellations, whose degrees of freedom define the FOM dimension.Cell-centered and face-centered quantities are connected through interpolation and face-flux calculations.
  • The finite volume discretization: Pressure is obtained from a Poisson equation and used with the momentum and continuity equations to update the mass flux through each control-volume face.The face flux combines pressure-gradient and momentum contributions through the discretized operators.
  • The finite volume discretization: Velocity unknowns are evaluated at face centers through interpolation from cell-center values, while non-orthogonal meshes require a correction to the diffusion discretization.The discretization supports central, upwind, second-order upwind, and blended differencing schemes.
  • Turbulence modeling: Because resolving all scales is infeasible at high Reynolds numbers, the model uses RANS rather than DNS or LES to represent turbulence.RANS evolves mean fields after Reynolds decomposition and time averaging; LES is identified as an alternative not used here.
  • RANS equations: The RANS closure represents the Reynolds stress tensor with the Boussinesq eddy-viscosity assumption, where νt may be obtained from additional turbulence transport equations.The paper mentions Spalart–Allmaras, k–ϵ, and SST k–ω models as examples of eddy-viscosity closures.

3. The reduced order model (ROM)

The Mixed-ROM combines POD-Galerkin projection for velocity and pressure with data-driven approximation of the eddy-viscosity field. It uses POD modes for the solution fields and RBF interpolation for eddy-viscosity coefficients, while supremizer stabilization closes the reduced velocity-pressure system.

  • POD-Galerkin projection: The ROM represents velocity and pressure with POD expansions whose coefficients depend on time and parameters.POD compresses snapshots into orthogonal modes, and Galerkin projection produces the reduced governing system.
  • POD-Galerkin projection: Supremizer enrichment supplies additional velocity modes so the continuity equation can be projected onto the pressure space.This makes the velocity modes no longer all divergence free and introduces the reduced mass and continuity matrices.
  • Mixed-ROM for turbulent flows: The turbulence treatment reduces only the eddy viscosity rather than the additional turbulence variables, avoiding turbulence-model-specific reduced transport equations.A POD expansion represents eddy viscosity, while the alternative would project equations for variables such as k, ϵ, or ω and increase reduced unknowns.
  • Mixed-ROM for turbulent flows: RBF interpolation computes the scalar coefficients of the eddy-viscosity POD expansion during the online stage.The coefficient vector g is therefore obtained through a data-driven POD-I approach using radial basis functions.
  • Mixed-ROM for turbulent flows: For unsteady cases, the time-based RBF formulation cannot extrapolate in time, motivating interpolation based on reduced velocity coefficients instead.The proposed reformulation changes the independent variable from time to combinations of reduced velocity coefficient vectors.

4. Numerical results

The Mixed-ROM was evaluated on steady and unsteady turbulent-flow benchmarks, including different turbulence models and parameterized inlet conditions. Across these tests, it reproduced full-order fields and lift-coefficient behavior more accurately than the P-ROM in the reported comparisons.

  • Steady case: Using 10 modes for each field, the Mixed-ROM qualitatively approximated the k-ε full-order velocity and pressure fields, unlike the P-ROM.The comparison used velocity, pressure, supremizer, and eddy-viscosity modes for both reduced models.
  • Steady case: The SST k-ω tests again showed good qualitative agreement between Mixed-ROM and FOM fields, whereas the P-ROM results did not match as closely.The comparison included velocity, pressure, and eddy-viscosity fields at U = 7.0886 m/s.
  • Steady case: The Mixed-ROM maintained close pressure-field values to the FOM across both k-ε and SST k-ω turbulence models.The result addresses the model’s ability to reconstruct pressure while accounting for turbulence-model variation.
  • Unsteady case: For unsteady cylinder flow, the Mixed-ROM was assessed through time-dependent velocity, pressure, lift, peak, and period errors using parameterized inlet-velocity snapshots.The study varied inlet velocity and treated time as a PDE parameter, with snapshots collected after the flow reached its final regime.
  • Unsteady case: The Mixed-ROM lift-coefficient error reached values as low as 3%, while the P-ROM errors remained above 16% relative to FOM values.The comparison used L2 relative lift-coefficient errors across online modal truncation orders.
  • Unsteady case: With 12 velocity modes and 10 modes for the other fields, the Mixed-ROM achieved 1.9654% lift error, 2.0672% maximum peak error, and 0.2% period error.The reported average periods were 0.3641 s for FOM and 0.3642 s for Mixed-ROM.
  • Unsteady case: The Mixed-ROM remained sensitive to the turbulence model used in the FOM solver, despite solving no additional turbulent-quantity PDEs at reduced order.The observation comes from extrapolated-in-time lift-coefficient comparisons for two turbulence models.

5. Conclusions and Outlook

The Mixed-ROM combines data-driven eddy-viscosity reduction with projection-based treatment of turbulent-flow fields and performs accurately for steady and unsteady flows at Reynolds numbers of order 10^5. It reconstructs lift histories with speedups of 10 in the unsteady case and around 1000 in the steady case, while long-time and multiphysics stabilization remain future-work needs.

  • Conclusions and Outlook: The Mixed-ROM combines projection-based reduction with a non-intrusive, RBF-interpolated reduced eddy-viscosity field for turbulent flows.The interpolation uses either time-parameter inputs or velocity projection coefficients and their time derivatives.
  • Conclusions and Outlook: The method accurately reconstructs fluid-dynamics fields in steady and unsteady cases with Reynolds numbers on the order of 10^5.
  • Conclusions and Outlook: The Mixed-ROM reconstructs the lift-coefficient time history around a circular cylinder and provides satisfactory extrapolation in time.The lift history mainly depends on local flow features around the cylinder.
  • Conclusions and Outlook: SU = 10 in the unsteady case and around SU = 1000 in the steady case.These speedups are reported alongside accurate predictions and acceptable computational cost.
  • Conclusions and Outlook: Future work includes improving eddy-viscosity approximation, exploring DMD for unsteady extrapolation, and developing stabilization for long-time and multiphysics problems.

Appendix A. List of abbreviations and symbols

The appendix lists the paper’s abbreviations and symbols for finite-volume reduced-order modeling, including field dimensions, reduced variables, operators, tensors, and interpolation quantities.

  • Appendix A. List of abbreviations and symbols: Mixed-ROM denotes the mixed projection/data-driven reduced order model developed in this work.
  • Appendix A. List of abbreviations and symbols: FEM, FVM, POD, RANS, RBF, RB, and ROM denote finite element methods, finite volume methods, Proper Orthogonal Decomposition, Reynolds Averaged Navier–Stokes, radial basis functions, reduced basis, and reduced order model.
  • Appendix A. List of abbreviations and symbols: The symbol list includes reduced velocity and pressure unknown vectors, correlation and turbulent tensors, and offline-calculated viscous and pressure-force matrices.
  • Appendix A. List of abbreviations and symbols: The interpolation notation includes observed outputs formed from viscosity-mode projection coefficients and a generic parameter-time vector.
  • Appendix A. List of abbreviations and symbols: Nh_p, Nh_u, and Nh_νt denote the full-order numbers of pressure, velocity, and eddy-viscosity unknowns, respectively.

Appendix B. Lift and drag forces offline/online computations

The appendix develops an offline/online procedure for computing viscous and pressure forces on a domain boundary from reduced velocity and pressure representations, avoiding full-order mesh integration online.

  • Appendix B. Lift and drag forces offline/online computations: The force-computation section introduces offline and online calculations for surface forces acting on a portion of the domain called BΩf.
  • Appendix B. Lift and drag forces offline/online computations: The total force on BΩf is defined as the sum of viscous and pressure contributions through an integral.
  • Appendix B. Lift and drag forces offline/online computations: The reduced force computation avoids resorting to the full-order mesh, preserving a reduced approach for problems such as flow past a circular cylinder.
  • Appendix B. Lift and drag forces offline/online computations: Inserting the reduced approximation into the force integral produces the expression used to separate offline and online computations.
  • Appendix B. Lift and drag forces offline/online computations: Offline quantities are defined from reduced velocity and pressure basis contributions and stored for later online use.
  • Appendix B. Lift and drag forces offline/online computations: For a new time-parameter vector z˚, the online stage computes the forces using the stored reduced quantities.
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