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An intrinsic stabilization scheme for proper orthogonal decomposition based low-dimensional models
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10.1063/1.2723149
/content/aip/journal/pof2/19/5/10.1063/1.2723149
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/5/10.1063/1.2723149

Figures

Image of FIG. 1.
FIG. 1.

Cartesian coordinates of the computational domain for the flow past a square cylinder (not to scale).

Image of FIG. 2.
FIG. 2.

(Color online) Partial view of a typical flow field for flow past a square cylinder at (a) and (b) showing vorticity; (clockwise rotation) and (counterclockwise).

Image of FIG. 3.
FIG. 3.

(Color online) Stream function of typical flow field for the driven cavity flow at (a) and (b); and . Dots mark discernible vortices.

Image of FIG. 4.
FIG. 4.

(Color online) Vorticity of typical flow field for the driven cavity flow at (a) and (b); (clockwise rotation) and (counterclockwise).

Image of FIG. 5.
FIG. 5.

(Color online) Comparison of solutions to the driven cavity flow at : , et al., et al. (a) along vertical center line [plotted ]; (b) along horizontal center line [plotted ].

Image of FIG. 6.
FIG. 6.

(Color online) Comparison of solutions to the driven cavity flow at : , red et al. (a) along vertical center line [plotted ]; (b) along horizontal center line [plotted ].

Image of FIG. 7.
FIG. 7.

(Color online) Comparison of solutions to the driven cavity flow at : , grid, red et al. (a) along vertical center line [plotted ]; (b) along horizontal center line [plotted ].

Image of FIG. 8.
FIG. 8.

(Color online) Mode 1, and unstabilized ODE solution. Figures 8–11 refer to the low-dimensional model for driven cavity flow at . Red dots: exact from POD; black: evolved.

Image of FIG. 9.
FIG. 9.

(Color online) Asymptotic behavior of mode 1, and unstabilized ODE solution. See Fig. 8 for details.

Image of FIG. 10.
FIG. 10.

(Color online) Mode 1, and intrinsically stabilized ODE solution. See Fig. 8 for details.

Image of FIG. 11.
FIG. 11.

(Color online) Asymptotic behavior of mode 1, and intrinsically stabilized ODE solution. See Fig. 8 for details.

Image of FIG. 12.
FIG. 12.

Error, , for mode 1 of the driven cavity flow at .

Image of FIG. 13.
FIG. 13.

(Color online) Mode 1, and unstabilized ODE solution. Figures 13–16 refer to the low-dimensional model for flow past a square cylinder at . Red dots: exact from POD; black: evolved.

Image of FIG. 14.
FIG. 14.

(Color online) Asymptotic behavior of mode 1, and unstabilized ODE solution. See Fig. 13 for details.

Image of FIG. 15.
FIG. 15.

(Color online) Mode 1, and intrinsically stabilized ODE solution. See Fig. 13 for details.

Image of FIG. 16.
FIG. 16.

(Color online) Asymptotic behavior of mode 1, and intrinsically stabilized ODE solution. See Fig. 13 for details.

Image of FIG. 17.
FIG. 17.

(Color online) Time history of envelopes of temporal modes for a four-mode intrinsically stabilized dynamical system of flow past a square cylinder at for 1000 shedding cycles. One shedding cycle is 6.6. (a) Mode 1; (b) mode 2; (c) mode 3; (d) mode 4.

Image of FIG. 18.
FIG. 18.

Square root of eigenvalues, POD at .

Image of FIG. 19.
FIG. 19.

Zoom of square root of eigenvalues, POD at .

Image of FIG. 20.
FIG. 20.

(Color online) Intrinsically stabilized square cylinder wake flow, , mode 19. from POD, from stabilized dynamical system.

Image of FIG. 21.
FIG. 21.

(Color online) Intrinsically stabilized square cylinder wake flow, , mode 20. Red: exact from POD; black: evolved from stabilized dynamical system.

Image of FIG. 22.
FIG. 22.

(Color online) Phase portrait of modes 1 and 2. Four-mode intrinsically stabilized dynamical system of flow past a square cylinder at for 1000 shedding cycles in black; red dots mark one cycle of corresponding temporal modes from POD.

Image of FIG. 23.
FIG. 23.

(Color online) Phase portrait of modes 3 and 4. See Fig. 22 for details.

Image of FIG. 24.
FIG. 24.

(Color online) Mode 1 of 16-mode model for driven cavity flow at . Red dashed: exact from POD; black: evolved. (a) “Raw” coefficients; (b) intrinsically stabilized.

Image of FIG. 25.
FIG. 25.

(Color online) Mode 4 of 16-mode model for driven cavity flow at . Red dashed: exact from POD; black: evolved. (a) “Raw” coefficients; (b) intrinsically stabilized.

Image of FIG. 26.
FIG. 26.

(Color online) Mode 8 of 16-mode model for driven cavity flow at . Red dashed: exact from POD; black: evolved. (a) “Raw” coefficients; (b) intrinsically stabilized.

Image of FIG. 27.
FIG. 27.

(Color online) Mode 16 of 16-mode model for driven cavity flow at . Red dashed: exact from POD; black: evolved. (a) “Raw” coefficients; (b) intrinsically stabilized.

Image of FIG. 28.
FIG. 28.

(Color online) Envelope of first temporal mode for a 16-mode intrinsically stabilized dynamical system of the quasiperiodic driven cavity flow at for 1000 cycles. One cycle .

Image of FIG. 29.
FIG. 29.

(Color online) Envelope of fourth temporal mode. See Fig. 28 for details.

Image of FIG. 30.
FIG. 30.

(Color online) Envelope of temporal mode 8. See Fig. 28 for details.

Image of FIG. 31.
FIG. 31.

(Color online) Envelope of temporal mode 16. See Fig. 28 for details.

Image of FIG. 32.
FIG. 32.

norm of each mode of the parameter continuation of the intrinsically stabilized 16-mode dynamical system derived from the POD of the driven cavity flow at compared with DNS results.

Image of FIG. 33.
FIG. 33.

(Color online) Mode 1 of four-mode model for driven cavity flow at . Red dashed: exact from POD; black: evolved. (a) Perturbed coefficients; (b) intrinsically stabilized.

Image of FIG. 34.
FIG. 34.

(Color online) Mode 2 of four-mode model for driven cavity flow at . Red dashed: exact from POD; black: evolved. (a) Perturbed coefficients; (b) intrinsically stabilized.

Image of FIG. 35.
FIG. 35.

(Color online) Mode 3 of four-mode model for driven cavity flow at . Red dashed: exact from POD; black: evolved. (a) Perturbed coefficients; (b) intrinsically stabilized.

Image of FIG. 36.
FIG. 36.

(Color online) Mode 4 of four-mode model for driven cavity flow at . Red dashed: exact from POD; black: evolved. (a) Perturbed coefficients; (b) intrinsically stabilized.

Image of FIG. 37.
FIG. 37.

(Color online) Evolution of from Lorenz equations. Original, red dashed; perturbed, solid black. (a) Short term (500 steps), . (b) Long term (10000 steps), .

Image of FIG. 38.
FIG. 38.

(Color online) Evolution of from Lorenz equations. Original, red dashed; perturbed, solid black. (a) Short term (500 steps), . (b) Long term (10000 steps), .

Image of FIG. 39.
FIG. 39.

(Color online) Evolution of from Lorenz equations. Original, red dashed; perturbed, solid black. (a) Short term (500 steps), . (b) Long term (10000 steps), .

Image of FIG. 40.
FIG. 40.

(Color online) phase portrait from Lorenz equations. Original, red dashed; perturbed, solid black. (a) Short term (500 steps), vs . (b) Long term (10000 steps), vs .

Image of FIG. 41.
FIG. 41.

(Color online) Evolution of from Lorenz equations. Original, red dashed; corrected, solid black. (a) Short term (500 steps), . (b) Long term (10000 steps), .

Image of FIG. 42.
FIG. 42.

(Color online) Evolution of from Lorenz equations. Original, red dashed; corrected, solid black. (a) Short term (500 steps), . (b) Long term (10000 steps), .

Image of FIG. 43.
FIG. 43.

(Color online) Evolution of from Lorenz equations. Original, red dashed; corrected, solid black. (a) Short term (500 steps), . (b) Long term (10000 steps), .

Image of FIG. 44.
FIG. 44.

(Color online) phase portrait from Lorenz equations. Original, red dashed; corrected, solid black. (a) Short term (500 steps), vs . (b) Long term (10000 steps), vs .

Tables

Generic image for table
Table I.

Constant coefficient, .

Generic image for table
Table II.

Linear coefficient, .

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/content/aip/journal/pof2/19/5/10.1063/1.2723149
2007-05-16
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: An intrinsic stabilization scheme for proper orthogonal decomposition based low-dimensional models
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/5/10.1063/1.2723149
10.1063/1.2723149
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