^{1,2}, D. Sipp

^{2}, J.-C. Robinet

^{1}and A. Barbagallo

^{2,3}

### Abstract

This paper deals with model reduction of high-order linear systems. An alternative method to approximate proper orthogonal decomposition (POD) and balanced truncation is exposed in this paper within the framework of the incompressible Navier-Stokes equations. The method of snapshots used to obtain low-rank approximations of the system controllability and observability Gramians is carried out in the frequency domain. Model reduction is thus performed using flow states that are long-time harmonic responses of the flow to given forcings, we call them frequential snapshots. In contrast with the recent works using time-stepping approach, restricted to stable systems, this one can always be computed for systems without marginal modes while it reduces to the same procedure for stable systems. We show that this method is efficient to perform POD and balanced proper orthogonal decomposition reduced-order models in both globally stable and unstable flows through two numerical examples: the flow over a backward-facing step and the flow over a square cavity. The first one is a globally stable flow exhibiting strong transient growths as a typical noise amplifier system while the second is a globally unstable flow representative of an oscillator system. In both cases, it is shown that the frequency-based snapshot method yields reduced-order models that efficiently capture the input-output behavior of the system. In particular, regarding the unstable cavity flow, our resulting unstable reduced-order models possess the same unstable global modes and stable transfer functions as those of the full system.

The authors gratefully acknowledge the French “Délégation Générale pour l’Armement” (DGA) for financial support.

I. INTRODUCTION

II. MODEL REDUCTION METHODOLOGY

A. Problem formulation

1. Governing equations

2. Model reduction phenomenology

B. Balanced model reduction

1. Controllability and observability Gramians

2. Introduction of the frequential snapshots

3. Computation of the balanced basis

C. POD model reduction

D. Discussion

1. Practical considerations

2. Computation of the snapshots

3. Fall-off of the snapshots norm

E. Numerical methods

III. GLOBALLY STABLE CASE: THE ROUNDED BACKWARD-FACING STEP FLOW

A. Flow configuration

B. Frequential snapshots

C. Reduced-order models

D. Impulse response and transfer function

IV. GLOBALLY UNSTABLE CASE: THE SQUARE CAVITY FLOW

A. Flow configuration

B. Frequential snapshots

C. Reduced-order models

1. Comparison of the unstable subspaces

2. Comparison of the stable input-output dynamics

3. Assessment

V. CONCLUSION

### Key Topics

- Flow instabilities
- 37.0
- Subspaces
- 32.0
- Eigenvalues
- 27.0
- Rheology and fluid dynamics
- 23.0
- Navier Stokes equations
- 20.0

## Figures

(Color online) Streamlines of the base flow at *Re* = 600. The actuator and sensor locations are also depicted.

(Color online) Streamlines of the base flow at *Re* = 600. The actuator and sensor locations are also depicted.

(Color online) The left plot (a) shows the energy of the direct snapshots as a function of . (b), (c), (d), (e) represent the real parts of the longitudinal velocity of the direct snapshots associated with the frequencies ω* _{i} * = 0.2, 0.72, 2, and 3, respectively.

(Color online) The left plot (a) shows the energy of the direct snapshots as a function of . (b), (c), (d), (e) represent the real parts of the longitudinal velocity of the direct snapshots associated with the frequencies ω* _{i} * = 0.2, 0.72, 2, and 3, respectively.

(Color online) The left plot (a) shows the energy of the adjoint snapshots as a function of ω. Note that, owing to the definition (10a) of the snapshots, the relevant quantity to be measured is . Figures (b), (c), (d), and (e) represent the real parts of the longitudinal velocity of the adjoint snapshots associated with the frequencies ω = 0.2, 0.64, 2, and 3, respectively.

(Color online) The left plot (a) shows the energy of the adjoint snapshots as a function of ω. Note that, owing to the definition (10a) of the snapshots, the relevant quantity to be measured is . Figures (b), (c), (d), and (e) represent the real parts of the longitudinal velocity of the adjoint snapshots associated with the frequencies ω = 0.2, 0.64, 2, and 3, respectively.

(Color online) (a) First 14 HSVs σ* _{j} * and (d) first 140 POD eigenvalues λ

*. (b) and (c) stand for the streamwise velocity component of the first and third BPOD modes. Analogously, (e) and (f) stand for the first and third POD modes.*

_{j}(Color online) (a) First 14 HSVs σ* _{j} * and (d) first 140 POD eigenvalues λ

*. (b) and (c) stand for the streamwise velocity component of the first and third BPOD modes. Analogously, (e) and (f) stand for the first and third POD modes.*

_{j}(Color online) (a) Impulse response of the full system G(*t*) and of the reduced-order models *G _{r} *(

*t*) for (a) BPOD models and (b) POD models.

(Color online) (a) Impulse response of the full system G(*t*) and of the reduced-order models *G _{r} *(

*t*) for (a) BPOD models and (b) POD models.

(Color online) Transfer function of the full system and of the reduced-order models for (a) BPOD models and (b) POD models.

(Color online) Transfer function of the full system and of the reduced-order models for (a) BPOD models and (b) POD models.

(Color online) Relative norm of the error as a function of the size *r* of the reduced-order models for (a) the BPOD and (b) the POD modes. Note that the upper and lower bounds on the error, computed by Eq. (28), have been reported in (a) by solid lines.

(Color online) Relative norm of the error as a function of the size *r* of the reduced-order models for (a) the BPOD and (b) the POD modes. Note that the upper and lower bounds on the error, computed by Eq. (28), have been reported in (a) by solid lines.

(Color online) Streamlines of the base flow at *Re* = 7500. The actuator and sensor locations are also depicted.

(Color online) Streamlines of the base flow at *Re* = 7500. The actuator and sensor locations are also depicted.

(Color online) Part of the global eigenspectrum of the square cavity flow at *Re* = 7500 (taken from Barbagallo *et al.* (Ref. 15).

(Color online) Part of the global eigenspectrum of the square cavity flow at *Re* = 7500 (taken from Barbagallo *et al.* (Ref. 15).

(Color online) Energy of the direct and adjoint snapshots as a function of ω. The lines indicate the frequencies of nearby global modes *E* _{2}, *E* _{–1}, *E* _{–3}, *E* _{–2}, and *E* _{0} (ordered from left to right).

(Color online) Energy of the direct and adjoint snapshots as a function of ω. The lines indicate the frequencies of nearby global modes *E* _{2}, *E* _{–1}, *E* _{–3}, *E* _{–2}, and *E* _{0} (ordered from left to right).

(Color online) (a) First 40 HSVs and (b) first 200 POD eigenvalues .

(Color online) (a) First 40 HSVs and (b) first 200 POD eigenvalues .

(Color online) Relative error as a function of the size *r* of the ROMs for the BPOD models. The upper and lower bounds on the error, computed by Eq. (28), are also displayed by the upper and lower solid lines.

(Color online) Relative error as a function of the size *r* of the ROMs for the BPOD models. The upper and lower bounds on the error, computed by Eq. (28), are also displayed by the upper and lower solid lines.

(Color online) Number of unstable modes for (a) the BPOD models and (b) the POD models. Both ROMs exhibits eight unstable modes from the dashed lines standing for *r* = 15 and *r* = 82 respectively.

(Color online) Number of unstable modes for (a) the BPOD models and (b) the POD models. Both ROMs exhibits eight unstable modes from the dashed lines standing for *r* = 15 and *r* = 82 respectively.

(Color online) Unstable eigenspectrum of the full system versus those of (a) BPOD and (b) POD models.

(Color online) Unstable eigenspectrum of the full system versus those of (a) BPOD and (b) POD models.

(Color online) Real part of the longitudinal velocity of the most unstable eigenvector *E* _{–3}. (a) Solution obtained with the full system with a shift and invert Arnoldi algorithm. (b) Solution obtained with a reduced-order model built with 150 POD modes.

(Color online) Real part of the longitudinal velocity of the most unstable eigenvector *E* _{–3}. (a) Solution obtained with the full system with a shift and invert Arnoldi algorithm. (b) Solution obtained with a reduced-order model built with 150 POD modes.

(Color online) Transfer function of the full system stable part compared to those of (a) BPOD and (b) POD models.

(Color online) Transfer function of the full system stable part compared to those of (a) BPOD and (b) POD models.

Relative norm of the error e_{∞s} as a function of the size *r* of the ROMs for (a) the BPOD and (b) the POD models. The limits from which the ROMs exhibit eight unstable global modes are depicted by dashed lines standing for *r* = 15 and *r* = 82 respectively.

Relative norm of the error e_{∞s} as a function of the size *r* of the ROMs for (a) the BPOD and (b) the POD models. The limits from which the ROMs exhibit eight unstable global modes are depicted by dashed lines standing for *r* = 15 and *r* = 82 respectively.

(Color online) (a) Part of the eigenspectrum of POD models of size 120, 140, and 200 superimposed onto that of the full system. A closer view is depicted for the model of size 200 on (b).

(Color online) (a) Part of the eigenspectrum of POD models of size 120, 140, and 200 superimposed onto that of the full system. A closer view is depicted for the model of size 200 on (b).

## Tables

Growth rate α and pulsation ω of the unstable modes labeled *E* _{–} _{3}, *E* _{–} _{2}, *E* _{–} _{1}, and *E* _{0} of the full system and those of several ROMs. The size column stands for the size *r* of the ROMs and is equal to *n* _{1} for the full system.

Growth rate α and pulsation ω of the unstable modes labeled *E* _{–} _{3}, *E* _{–} _{2}, *E* _{–} _{1}, and *E* _{0} of the full system and those of several ROMs. The size column stands for the size *r* of the ROMs and is equal to *n* _{1} for the full system.

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