1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Compressible proper orthogonal decomposition/Galerkin reduced-order model of self-sustained oscillations in a cavity
Rent:
Rent this article for
USD
10.1063/1.2998448
/content/aip/journal/pof2/20/11/10.1063/1.2998448
http://aip.metastore.ingenta.com/content/aip/journal/pof2/20/11/10.1063/1.2998448

Figures

Image of FIG. 1.
FIG. 1.

DNS results. (a) Positive (—) and negative (gray line) isocontours of instantaneous vorticity during one period of oscillation. Time trace (b), and spectrum (c) of fluctuating pressure on the cavity floor (, ).

Image of FIG. 2.
FIG. 2.

POD eigenvalues as function of the mode number with (a) or without (b) the mean subtracted. (○–○) 74 snapshot basis; (○○) 37 snapshot basis.

Image of FIG. 3.
FIG. 3.

Visualizations of the POD modes 1 (a), 2 (b), 3 (c), 4 (d), 5 (e), 6 (f) for the isentropic model. For each mode, vorticity modes (top) and dilatation modes (bottom).

Image of FIG. 4.
FIG. 4.

POD/Galerkin reconstruction with the isentropic model with the mean substracted. Temporal evolutions of fluctuating velocities at four sensors located in the shear layer. (a) Influence of the number of modes: 2 ; 4 (– – –); 8 (—); 12 . (b) Influence of the number of snapshots: 37 (—); 74 ; 180 (– ⋅ – ⋅). DNS results (•••).

Image of FIG. 5.
FIG. 5.

POD/Galerkin reconstruction with the isentropic model without the mean substracted. Temporal evolutions of fluctuating velocities at four sensors located in the shear layer. (a) Influence of the number of modes: 2 ; 4 (– – –); 8 (—); 12 . (b) Influence of the number of snapshots: 37 (—); 74 ; 180 (– ⋅ – ⋅). DNS results (•••).

Image of FIG. 6.
FIG. 6.

POD/Galerkin reconstruction with the full model with the mean substracted. Temporal evolutions of fluctuating velocities at four sensors located in the shear layer. (a) Influence of the number of modes: 2 ; 4 (– – –); 8 (—); 12 . (b) Influence of the number of snapshots: 37 (—); 74 ; 180 (– ⋅ – ⋅). DNS results (•••).

Image of FIG. 7.
FIG. 7.

Erroneous limit cycles. time trace at the sensor , during iterations: isentropic model (a) and full model (b). DNS results (•••).

Image of FIG. 8.
FIG. 8.

Trajectories of the POD/Galerkin model for coefficients 2–5, during iterations: isentropic model (a) and full model (b). DNS results (•••).

Image of FIG. 9.
FIG. 9.

Stabilization of the POD/Galerkin reconstruction (with the mean substracted) by the addition of a viscosity term. (a) Influence of on the isentropic model: (—); ; (– – –). (b) Influence of on the full model: (—); ; (– – –). DNS results (•••).

Image of FIG. 10.
FIG. 10.

Stabilization of the POD/Galerkin reconstruction (with the mean substracted). (a) Influence of for the use of a Sobolev norm in the isentropic model: (—); ; (– – –). (b) Influence of the parameter for the use of a penalty term in the isentropic model: (—); ; (– – –). DNS results (•••).

Image of FIG. 11.
FIG. 11.

Calibration of the POD/Galerkin reconstruction. (a) Influence of the calibration parameter in the isentropic model with mean substracted: ; (—); (– – –). (b) Calibration with of the isentropic model with mean included ; of the full model with mean substracted (—); of the full model with mean included (– – –). DNS results (•••).

Image of FIG. 12.
FIG. 12.

(a) Influence of the calibration parameter on the reconstruction error, measured by the ratio of the mean value of the maxima of the signals for four sensors located in the shear layer over the mean of the same maxima from DNS. Pressure field after iterations from the DNS (b) and reconstructed from the calibrated POD/Galerkin model (c) , with the same color scale.

Image of FIG. 13.
FIG. 13.

POD/Galerkin reconstruction with the mean included . Time traces of coefficients , , and for the (a) isentropic and (b) full model: calibrated model (– – –), calibrated model with a penalty term for the mean (—), and exact coefficients from DNS (●).

Image of FIG. 14.
FIG. 14.

(a) POD/Galerkin reconstruction for the extended DNS domain with the isentropic model (mean removed) without any stabilization (—). DNS results (•••). (b) Reconstructed pressure field after .

Image of FIG. 15.
FIG. 15.

Trajectories of the POD/Galerkin model (, , mean substracted) for coefficients 2–5, during iterations: from top to bottom, constant viscosity term for the isentropic model; constant viscosity term for the full model; use of a Sobolev norm with for the isentropic model; calibration of the isentropic model with ; calibration of the isentropic model with ; snapshots from a DNS with an extended outlet zone for the isentropic model with no other stabilization. DNS results (•••).

Image of FIG. 16.
FIG. 16.

(a) Sketch of the computational domain, and (b) instantaneous fluctuating pressure field in the median plane (levels between and 500 Pa).

Image of FIG. 17.
FIG. 17.

Instantaneous traces of the pressure fluctuations in the acoustic radiation at (, , 0).

Image of FIG. 18.
FIG. 18.

(a) Power spectral density at (, , 0), and (b) bispectrum of the fluctuating pressure near the upstream corner of the cavity, at (0, , 0).

Image of FIG. 19.
FIG. 19.

Ten snapshots of the vorticity in the midplane during five successive cycles. The visualized instants (a)–(j) (top) correspond to the dashed lines on the pressure time trace at (, , 0) (bottom).

Image of FIG. 20.
FIG. 20.

POD eigenvalues as function of the mode number with snapshots (—) (mean included); (○○) modes retained; modes retained. The vertical lines indicate the number of modes necessary to capture 90% , 99% , and 99.9% (– – –) of the fluctuating energy. (a) Logarithmic scale and (b) linear scale.

Image of FIG. 21.
FIG. 21.

Spatial modes 1–6 of the 3D cavity flow from top to bottom. The u-component of the velocity is represented in the first column, the v-component in the second column, and the w-component in the third column.

Image of FIG. 22.
FIG. 22.

Spatial modes 7–12 of the 3D cavity flow.

Image of FIG. 23.
FIG. 23.

Spatial modes 13–18 of the 3D cavity flow.

Image of FIG. 24.
FIG. 24.

3D POD/Galerkin reconstruction with the isentropic model with the mean removed; 30 modes are retained from a 110 snapshot basis. Histories of the first three POD coefficients for the noncalibrated (a) and the calibrated (b) models .

Image of FIG. 25.
FIG. 25.

3D POD/Galerkin reconstruction with the isentropic model with the mean removed; 50 modes are retained from a 328 snapshot basis. Histories of the first three POD coefficients for the noncalibrated (a) and the calibrated (b) models .

Image of FIG. 26.
FIG. 26.

Long-time behavior of the first four coefficients of the 3D POD/Galerkin reconstruction. Obtained with the isentropic model (mean removed), calibrated with . 30 modes are retained from a 110 snapshot basis.

Image of FIG. 27.
FIG. 27.

Long-time behavior of the first four coefficients of the 3D POD/Galerkin reconstruction. Obtained with the isentropic model (mean removed), calibrated with . 50 modes are retained from a 328 snapshot basis.

Image of FIG. 28.
FIG. 28.

3D POD/Galerkin reconstruction with the isentropic model with the mean included (, , ). Histories of the coefficients, (representing the mean field), and .

Image of FIG. 29.
FIG. 29.

Long-time behavior of the first four coefficients of the 3D POD/Galerkin reconstruction. Obtained with the isentropic model with the mean included (, , ).

Image of FIG. 30.
FIG. 30.

3D POD/Galerkin reconstruction with the isentropic model with the mean included (, , ). Histories of the coefficients, (representing the mean field), and with a penalty term imposed for at the upper boundary, to prevent a drift of the mean flow.

Image of FIG. 31.
FIG. 31.

Long-time behavior of the first four coefficients of the 3D POD/Galerkin reconstruction. Obtained with the isentropic model with the mean included (, , ). A penalty term is imposed for at the upper boundary, to prevent a drift of the mean flow

Image of FIG. 32.
FIG. 32.

Same legend as in Fig. 31 for iterations.

Image of FIG. 33.
FIG. 33.

Temporal evolutions of reconstructed velocity for four sensors located in the shear layer (isentropic model, , , mean included, calibration , penalty term for the mean).

Tables

Generic image for table
Table I.

Energy of the ten first POD modes expressed as a percentage of the total fluctuating energy (modes 1–10 with the mean subtracted and modes 2–11 with the mean included) for 37 and 74 snapshot bases.

Loading

Article metrics loading...

/content/aip/journal/pof2/20/11/10.1063/1.2998448
2008-11-25
2014-04-20
Loading

Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Compressible proper orthogonal decomposition/Galerkin reduced-order model of self-sustained oscillations in a cavity
http://aip.metastore.ingenta.com/content/aip/journal/pof2/20/11/10.1063/1.2998448
10.1063/1.2998448
SEARCH_EXPAND_ITEM