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Weakly dispersive hydraulic flows in a contraction: Parametric solutions and linear stability
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10.1063/1.2728938
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1 School of Mathematical Sciences, Monash University, Clayton, Victoria 3168, Australia
Phys. Fluids 19, 056601 (2007)
/content/aip/journal/pof2/19/5/10.1063/1.2728938
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## Figures

FIG. 1.

(Color online) Numerical solution to (3) for both the (i) original solution region defined over with and (ii) reflected solution region defined over with , subject to , , .

FIG. 2.

(Color online) Top panel: Using , a sketch of homoclinic orbits of the KdV equation with center at and saddle point at corresponding to far-field upstream and downstream amplitudes. Bottom panel: sketch of corresponding potential where and showing the center and saddle points.

FIG. 3.

(Color online) Top panel: Numerical solution to (3) over original region defined over , where , , , before applying exponential decay (solid line), after the application of exponential decay (dashed), and the point of the application of the exponential decay denoted by asterisk. Bottom panel: Semilog plot of residue solution , in the original region defined over , to (3), , , , and . The point of application of the exponential decay on the numerical solution is denoted by an asterisk.

FIG. 4.

(Color online) Plot of vs using and . This describes the locus of dispersive hydraulic solutions for (3). Corresponding absolute value of relative error (bottom panels) between our computed and that in GS, to verify the (a) narrow forcing limit and (b) wide forcing limit, both using and .

FIG. 5.

Numerical solution of (3) after solution refinement with and using (a) , (b) , (c) , (d) , (e) , (f) , (g) , (h) , and (i) . Solutions (a)–(i) have each been displaced such that the upstream level corresponds to zero.

FIG. 6.

Plots of eigenvalue spectra, vs , formed by combining the eigenvalue plots using (a) , and (b) , . is the eigenvalue with the largest real part, which can be obtained after solving (20). The smaller box embedded in the bottom panel is a zoomed plot of the spectra for .

FIG. 7.

Plots of the and vs using of the two leading eigenvalues, obtained after solving (20), ranked according to the size of the real components. [(a) and (b)] and [(c) and (d)] .

FIG. 8.

(Color online) Plots of and : real and imaginary components of the eigenfunction associated with , the eigenvalue with the largest real part which can be obtained after solving (20). [(a) and (b)] , ; [(c) and (d)] , ; [(e) and (f)] , ; [(g) and (h)] , ; (i) , . The solutions all have mean value zero but have been displaced for clarity.

FIG. 9.

(Color online) Logarithmic scale plot of perturbation momentum vs time over the combined region , using and for (a) , and (b) , . The dotted lines show the exponential growth due to . Normalized numerical solutions to (17) with and for (c) , and (d) , .

FIG. 10.

(Color online) Logarithmic scale plot of perturbation momentum vs time over combined region , using and for (a) , and (b) , . The dotted lines show the exponential growth due to . Normalized numerical solutions to (17) with and for (c) , and (d) , .

FIG. 11.

(Color online) A depiction of the parameter space for solutions of (1) for . The dashed lines and text denote asymptotic regimes for the solutions of (1) for the trivial initial condition. The solid lines and shading depict allowable boundaries for the dispersive hydraulic solutions considered here.

FIG. 12.

(Color online) Numerical solution of (3) after solution refinement with , along with topographic perturbations using , , and .

/content/aip/journal/pof2/19/5/10.1063/1.2728938
2007-05-10
2014-04-17

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