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Linear stability analysis and numerical simulation of miscible two-layer channel flow
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10.1063/1.3116285
/content/aip/journal/pof2/21/4/10.1063/1.3116285
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/4/10.1063/1.3116285

Figures

Image of FIG. 1.
FIG. 1.

Schematic of the “three-layer” base state flow whose linear stability is analyzed. Here, layers of fluids 1 and 2, corresponding to the more and less viscous fluids, respectively, are separated by mixed regions of thickness, . The interfaces between these regions and the region occupied by the less viscous fluid are located at , respectively.

Image of FIG. 2.
FIG. 2.

The effect of increasing the order of Chebyshev polynomials, , on the variation of the growth rate, , with . The parameters are , , , , and .

Image of FIG. 3.
FIG. 3.

Isocontours of and in the complex wavenumber plane for and are shown in (a) and (b), and (c) and (d), respectively. The value of at the saddle point is and for and , respectively. The rest of the parameter values are , , , and .

Image of FIG. 4.
FIG. 4.

Contour deformation procedure following the Briggs method (Ref. 54) with showing the coalescence of two spatial branches initially present in the upper and lower halves of the complex plane. Panels (a), (c), and (e) show the effect of contour deformation in the complex plane, while (b), (d), and (f) show the analogous effect in the complex plane. The rest of the parameters are the same as in Fig. 3. The inset in panel (e) shows details of the branch point in the complex -plane.

Image of FIG. 5.
FIG. 5.

Stability diagram showing the regions of convective and absolute instability in space. (a) Effect of Sc with ; (b) effect of with . In both panels, .

Image of FIG. 6.
FIG. 6.

The effect of on the dispersion curves, vs , for and , shown in (a) and (b), respectively. The rest of the parameters are , , and .

Image of FIG. 7.
FIG. 7.

Variations of the different contributions to the rate of change in the disturbance kinetic energy with Re with and , and and , shown in (a) and (b), respectively. The rest of the parameter values are and .

Image of FIG. 8.
FIG. 8.

(a) Comparison of the temporal evolution of the vertical velocity perturbation from the linear stability analysis with that obtained from the numerical simulation with 256 points in the and directions for . The direct numerical simulation result with 361 points in the and directions is virtually indistinguishable from the one shown in this figure. Comparison of the contour of the vertical velocity perturbation from the linear stability analysis (dashed line) with that from numerical simulations (solid line) at (b) and (c) . The rest of the parameters are , , , , and .

Image of FIG. 9.
FIG. 9.

Spatiotemporal evolution of the contours of for (a) and (b) ; solutions are shown at (dotted lines), (dashed-dotted line), and (solid line). For clarity of presentation, the dashed-dotted and solid lines in panels (a) and (b) are displaced by 0.01 and 0.03, respectively, from their original position in the vertical axis. The rest of the parameter values are , , , and . Enlarged views of the contour near the channel inlet for and , shown in (c) and (d), respectively. (e) and (f) show the dispersion curve for and the contour of in the complex -plane. The estimates of the disturbance wavelength shown panels (c) and (d) correspond approximately to those of the most dangerous temporal mode and the mode associated with the pinch point singularity at the saddle point in panels (e) and (f), respectively.

Image of FIG. 10.
FIG. 10.

Spatiotemporal evolution of the concentration contours for , , , , and .

Image of FIG. 11.
FIG. 11.

Spatiotemporal evolution of the vorticity contours. The parameter values remain unchanged from Fig. 10.

Image of FIG. 12.
FIG. 12.

Spatiotemporal evolution of the concentration contours for . The rest of the parameter values remain unchanged from Fig. 10.

Image of FIG. 13.
FIG. 13.

Concentration contours obtained in the presence and absence of inlet forcing shown in (a) and (b), and (c) and (d), respectively at . The results depicted in (a) and (c) were generated with , while those in (b) and (d) were obtained with . The rest of the parameter values remain unchanged from Fig. 10.

Image of FIG. 14.
FIG. 14.

Spatiotemporal evolution of the concentration contours for . The rest of the parameter values remain unchanged from Fig. 12.

Image of FIG. 15.
FIG. 15.

Spatiotemporal evolution of the concentration contours for . The rest of the parameter values remain unchanged from Fig. 12.

Image of FIG. 16.
FIG. 16.

Spatiotemporal evolution of the concentration contours for . The rest of the parameter values remain unchanged from Fig. 12.

Image of FIG. 17.
FIG. 17.

Spatiotemporal evolution of the vorticity contours for the same parameter values as in Fig. 16.

Image of FIG. 18.
FIG. 18.

Spatiotemporal evolution of the concentration contours for , , and .

Image of FIG. 19.
FIG. 19.

(a) Mass fraction of the displaced fluid and (b) temporal evolution of the position of the leading front separating the two fluids, , obtained using different mesh densities for the same parameters as in Fig. 18. The dashed line of constant slope has been included in (a) to demonstrate the constant displacement rate in the flow regime dominated by the unperturbed penetration of a finger of the less viscous fluid into the more viscous one.

Image of FIG. 20.
FIG. 20.

Spatiotemporal evolution of the concentration contours for , , and .

Image of FIG. 21.
FIG. 21.

Spatiotemporal evolution of the concentration contours for . The rest of the parameter values remain unchanged from Fig. 20.

Image of FIG. 22.
FIG. 22.

Spatiotemporal evolution of the concentration contours for . The rest of the parameter values remain unchanged from Fig. 20.

Image of FIG. 23.
FIG. 23.

Spatiotemporal evolution of the concentration contours for , , and .

Image of FIG. 24.
FIG. 24.

Spatiotemporal evolution of the concentration contours for . The rest of the parameter values remain unchanged from Fig. 23.

Image of FIG. 25.
FIG. 25.

Spatiotemporal evolution of the concentration contours for . The rest of the parameter values remain unchanged from Fig. 23.

Image of FIG. 26.
FIG. 26.

Spatiotemporal evolution of the concentration contours for , , and .

Image of FIG. 27.
FIG. 27.

Spatiotemporal evolution of the concentration contours for . The rest of the parameter values remain unchanged from Fig. 26.

Image of FIG. 28.
FIG. 28.

Effect of Re, Sc, and on the mass fraction of the displaced fluid 1 and the temporal evolution of the position of the leading front separating the two fluids, shown in (a),(c), (e), and (b), (d), (f), respectively. The rest of the parameter values are and in (a) and (b), and in (c) and (d), and and in (e) and (f).

Tables

Generic image for table
Table I.

Variation of the critical layer location with for and . Two different values of are considered, and , and the width of the mixed layer is .

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/content/aip/journal/pof2/21/4/10.1063/1.3116285
2009-04-15
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Linear stability analysis and numerical simulation of miscible two-layer channel flow
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/4/10.1063/1.3116285
10.1063/1.3116285
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