Fluid Flow Linearized Open Cavity Model

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Description

Motivation

Figure 1: Open cavity geometry and flow computed on FreeFem.

Figure 2: Open cavity schematic view: actuator on the left and sensor on the right of the cavity.

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of and also (with a more dynamical system view point) in , and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of $193,874$ triangles, corresponding to $n=680,974$ degrees of freedom for the velocity variables along the $x$ and $y$ axis. After linearization around three fixed points for varying Reynolds numbers $Re=\{4000,5250,6000\}$ three dynamical models $\{H_l\}_{l=1}^3$ can be described as a DAE realization of order $n=680,974$ given as (matrices, about 100Mo each, can be provided under request to Charles Poussot-Vassal )

$E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}$

where the input $u(t)$ is the vertical pressure actuator located upstream of the cavity and the output $y(t)$ is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number $Re$. The transfer functions read

$H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}$

In , the IRKA approach (being a realization based $\mathcal H_2$-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way. The approach involves the Loewner framework as in . We then construct the data as follows:

First, the frequency response $H_l(\xi_k)$ of each $l=1,\dots,3$ configurations for frozen complex values $\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N$, with $\omega_i\in\mathbb R_+$, $\xi_0\in\mathbb R_+$ and $k=1,\dots,61$ is computed. Second, ten intermediate parametric configurations between each $l$-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses $H_l(\xi_k)$, leading to $\{\rho_j\}_{j=1}^M\in\mathbb R$, for $j=1,\dots,M=21$ configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

Considered data

File:FluidControl data.png

Figure 3: Open cavity geometry.

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

$\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},$

where $H_{kl}\in\mathbb C$ represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values $\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N$ and parameters $\{\rho_l\}_{l=1}^M\in\mathbb R$.

Origin

Collaboration between ONERA DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

Data

Description

The FluidFlowCavityDataParametric.zip (36ko) repository contains three files:

• The dataONERA_FluidFlowOpenCavity.mat data file, with
  • P : the parametric values (real $1 \times 21$ vector).
  • S : the complex values where the model is evaluated (real $1 \times 123$ vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
  • H : transfer function matrix evaluation at different S,P couples (complex $1 \times 1 \times 123\times 21$ matrix).

• The dataONERA_FluidFlowOpenCavity_withMOR.mat data file, with a parametric ROM obtained with the MOR toolbox using the parametric Loewner method.
  • Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

• The startONERA_FluidFlowOpenCavity.m script file, used to loads and plots the data for illustration and a solution.

Objective

Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property seems very promising.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, Fluid Flow Linearized Open Cavity Model. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

• For illustration of the Loewner-driven parametric model construction involvin this use-case, see:

@article{GoseaHNA:2022,
   author  = {V. Gosea and C. Poussot-Vassal and A.C. Antoulas},
   title   = {Data-driven modeling and control of large-scale dynamical systems in the Loewner framework},
   journal = {Handbook of Numerical Analysis},
   year    = {2022},
   volume  = {23},
   number  = {Numerical Control: Part A},
   pages   = {499-530},
}

• For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
  author    = {C. Poussot-Vassal and  D. Sipp},
  title     = {Parametric reduced order dynamical model construction of a fluid flow control problem},
  booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
  address   = {Grenoble, France},
  month     = {October},
  year      = {2015},
  pages     = {133-138},
}

• For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
  author  = {A. Barbagallo and D. Sipp and P.J. Schmid},
  journal = {Journal of Fluid Mechanics},
  pages   = {1-50},
  title   = {Closed-loop control of an open cavity flow using reduced-order models},
  volume  = {641},
  year    = {2008}
}

Contact

Charles Poussot-Vassal