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Model reduction for fluids using frequential snapshots
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10.1063/1.3590732
/content/aip/journal/pof2/23/6/10.1063/1.3590732
http://aip.metastore.ingenta.com/content/aip/journal/pof2/23/6/10.1063/1.3590732

Figures

Image of FIG. 1.
FIG. 1.

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

Image of FIG. 2.
FIG. 2.

(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.

Image of FIG. 3.
FIG. 3.

(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.

Image of FIG. 4.
FIG. 4.

(Color online) (a) First 14 HSVs σ j and (d) first 140 POD eigenvalues λ j . (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.

Image of FIG. 5.
FIG. 5.

(Color online) (a) Impulse response of the full system G(t) and of the reduced-order models Gr (t) for (a) BPOD models and (b) POD models.

Image of FIG. 6.
FIG. 6.

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

Image of FIG. 7.
FIG. 7.

(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.

Image of FIG. 8.
FIG. 8.

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

Image of FIG. 9.
FIG. 9.

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

Image of FIG. 10.
FIG. 10.

(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).

Image of FIG. 11.
FIG. 11.

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

Image of FIG. 12.
FIG. 12.

(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.

Image of FIG. 13.
FIG. 13.

(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.

Image of FIG. 14.
FIG. 14.

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

Image of FIG. 15.
FIG. 15.

(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.

Image of FIG. 16.
FIG. 16.

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

Image of FIG. 17.
FIG. 17.

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.

Image of FIG. 18.
FIG. 18.

(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

Generic image for table
Table I.

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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/content/aip/journal/pof2/23/6/10.1063/1.3590732
2011-06-10
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Model reduction for fluids using frequential snapshots
http://aip.metastore.ingenta.com/content/aip/journal/pof2/23/6/10.1063/1.3590732
10.1063/1.3590732
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