Fluid Flow Linearized Open Cavity Model

Note: This page has not been verified by our editors.

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 the xx--CrossReference--dft--fig:cavity--xx and xx--CrossReference--dft--fig:cavityScheme--xx.

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 $\{H_j\}_{j=1}^3$ can be described as a DAE realisation of order $n=680,974$ given as (matrices, about 100Mo each, can be provided under request to Charles Poussot-Vassal )

$E_j \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}$

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

$H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}$

In \cite{PoussotLPVS:2015}, 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 Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

Considered data

File:FluidControl data.png

Figure 3: Open cavity geometry.

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where $n\approx 700,000$):

$E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}$

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

$\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},$

where $H_{ij}\in\mathbb C^{n_y\times n_u}$ represents the transfer from $n_u=1$ input signal (upward cavity pressure) to $n_y=1$ measurement output (downward cavity pressure), evaluated at varying complex values $\xi_i\in\mathbb C$ and $p_j\in\mathbb R$, for $j=1,\dots,M=3$. Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

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.

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 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