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Train yourself: self-compressing reduced-order models of turbulent flows

Ian Addison-Smith, Igor A. Maia, Benjamin Herrmann, Andre V. G. Cavalieri

TL;DR

The paper addresses the high dimensionality of ROMs for turbulent flows by introducing a data-free self-compression of a controllability-mode ROM (SCROM). It builds a compressed modal basis from the time-series of ROM coefficients via an energy-based eigen-decomposition of a wavenumber-resolved covariance, yielding a reduced set of coefficients whose evolution preserves key first- and second-order statistics relative to DNS. The SCROM produces POD-like spatial structures without relying on DNS data, and its performance is assessed against DNS and the original ROM for plane Couette flow at $Re=500$, showing substantial mode reduction with maintained statistics. This approach broadens the applicability of reduced-order modeling by delivering data-free, computation-efficient turbulence representations suitable for control and high-Re flows, while linking data-driven and equation-based methodologies.

Abstract

Reduced-order models (ROMs) of turbulent flows based on Galerkin projection often require many degrees of freedom to resolve the dynamics of the turbulence, or simulation data to obtain an optimal modal basis. However, obtaining simulation data is computationally expensive, and the amount of data required to obtain a converged modal basis can increase this cost. Using the linearized Navier-Stokes equations, one can achieve spatial modes through the controllability and observability Gramians, which can yield a ROM without prior simulation data. In this work, we propose a self-compression of a ROM based on controllability modes, where the time series of the modal coefficients are leveraged to reduce the dimension of the ROM. In the self-compressed ROM (SCROM), we can maintain accurate first- and second-order statistics with respect to the DNS simulation, but in a further reduced dimension. The SCROM recovers spatial structures equivalent to proper orthogonal decomposition (POD) without relying on any simulation data, recombining spatial modes from linearized equations. This method leads to a novel ROM that can represent turbulence statistics in a data-free approach in a further reduced state space.

Train yourself: self-compressing reduced-order models of turbulent flows

TL;DR

The paper addresses the high dimensionality of ROMs for turbulent flows by introducing a data-free self-compression of a controllability-mode ROM (SCROM). It builds a compressed modal basis from the time-series of ROM coefficients via an energy-based eigen-decomposition of a wavenumber-resolved covariance, yielding a reduced set of coefficients whose evolution preserves key first- and second-order statistics relative to DNS. The SCROM produces POD-like spatial structures without relying on DNS data, and its performance is assessed against DNS and the original ROM for plane Couette flow at , showing substantial mode reduction with maintained statistics. This approach broadens the applicability of reduced-order modeling by delivering data-free, computation-efficient turbulence representations suitable for control and high-Re flows, while linking data-driven and equation-based methodologies.

Abstract

Reduced-order models (ROMs) of turbulent flows based on Galerkin projection often require many degrees of freedom to resolve the dynamics of the turbulence, or simulation data to obtain an optimal modal basis. However, obtaining simulation data is computationally expensive, and the amount of data required to obtain a converged modal basis can increase this cost. Using the linearized Navier-Stokes equations, one can achieve spatial modes through the controllability and observability Gramians, which can yield a ROM without prior simulation data. In this work, we propose a self-compression of a ROM based on controllability modes, where the time series of the modal coefficients are leveraged to reduce the dimension of the ROM. In the self-compressed ROM (SCROM), we can maintain accurate first- and second-order statistics with respect to the DNS simulation, but in a further reduced dimension. The SCROM recovers spatial structures equivalent to proper orthogonal decomposition (POD) without relying on any simulation data, recombining spatial modes from linearized equations. This method leads to a novel ROM that can represent turbulence statistics in a data-free approach in a further reduced state space.

Paper Structure

This paper contains 9 sections, 17 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Reduced-order model formulation
  4. Self-compressing reduced-order model
  5. Numerical implementation
  6. Results
  7. Statistics of the self-compressed form
  8. Interpretation of the self-compressed form
  9. Conclusions

Figures (4)

  • Figure 1: (a) Eigenvalues of the matrix $\mathsfbi{R}$, different thresholds are shown by horizontal lines in descending order $r=0.9,0.925,0.95,0.975,0.99$ which correspond to $n=72,93,126,181,252$ retained modes, respectively. (b) Error of the first and second-order statistics for the same thresholds as (a).
  • Figure 2: (a) First-order statistics and (b) second-order statistics for different thresholds of the SCROM
  • Figure 3: Visualization of the leading modes in the ROM, SCROM and POD bases for wavenumber pairs $(k_x/\alpha,k_z/\beta)=(0,1)$ in (a) and $(k_x/\alpha,k_z/\beta)= (1,1)$ in (b). The positive and negative contours indicate streamwise component, whereas the arrows indicates the wall-normal and spanwise component
  • Figure 4: Number of modes retained in the SCROM basis for each wavenumbers pair $(k_x/\alpha,k_z/\beta)$: (a) threshold of $r = 0.95$; (b) threshold of $r=0.975$.