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Eddy genesis and manipulation in plane laminar shear flow
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Image of FIG. 1.
FIG. 1.

Schematic of the defining geometry for plane laminar shear flow between a flat moving upper and a fixed corrugated lower plate.

Image of FIG. 2.
FIG. 2.

Sketch of the four measurables used to quantify resulting flow behavior. The solid streamline and open circle mark the separatrix and eddy center positions under general flow conditions, while the dashed streamline and cross give their corresponding positions for Stokes flow. The quantities and therefore refer to the difference in the position of the eddy center and the eddy depth between the general and Stokes flow cases.

Image of FIG. 3.
FIG. 3.

Calculations of the height, , of the eddy center above the bottom of the topography as a function of the gap under Stokes flow conditions. The discrete points give corresponding semianalytical results with .

Image of FIG. 4.
FIG. 4.

Velocity profiles along the vertical centerline. The curves illustrate the effect of doubling the mean gap (film thickness) on the velocity profiles in the case of rigid plate laminar shear (free-surface film) flow. Lengths are scaled by . The thick solid line indicates the lower plate surface and the dotted lines indicate the positions of the upper boundary when and .

Image of FIG. 5.
FIG. 5.

Comparison of the shape and position of the separatrix for different gaps and two different lower plate geometries. In (a) and in (b) . Lengths are scaled by and the dotted line in each plot shows the position of the upper boundary when . Graph (c) gives the variation in eddy depth with the clearance.

Image of FIG. 6.
FIG. 6.

Map of parameter space showing critical combinations of the waviness, , and clearance, , at which the first and subsequent eddies appear. The row of streamline plots all have , and values of are equal to 0.4, 1.5, 3, 4.5, and 6. Streamline plot corresponds to and . For free surface film flow, the corresponding critical combinations of the waviness, , and Nusselt film thickness, , at which the first and the second eddy appear are indicated by the gray-shaded lines.

Image of FIG. 7.
FIG. 7.

Critical values of waviness, , at which a new eddy appears. The solid circles represent critical values extracted from Fig. 6 at . For comparison, the open circles indicate corresponding free-surface film results for a Nusselt film thickness . The solid lines are linear fits to the data.

Image of FIG. 8.
FIG. 8.

Critical curves of Fig. 6 replotted using the amplitude, , of the lower plate as the length scale. The streamline plots show the flow structure for the parameter values at the indicated locations.

Image of FIG. 9.
FIG. 9.

Effect of increasing inertia on the horizontal shift of the eddy center, (lower graph) and corresponding change in eddy depth, (upper graph), for and . The overlaid plots from A to F show the separatrix and eddy center position at the values of Re indicated by the corresponding black circles on the graphs. In plots B–F, the separatrix is shown as a solid line and the eddy center as an open circle. The Stokes flow separatrix and eddy center position from A are included in each case as a dashed line and “×” symbol, respectively.

Image of FIG. 10.
FIG. 10.

Effect of on the horizontal shift of the eddy center, (lower graph) and on eddy depth, (upper graph) for large gaps and . The streamline plot insets illustrate the flow structure for parameter values corresponding to the points shown as black circles. The open circles mark the transition from a kinematically to inertially induced eddy.

Image of FIG. 11.
FIG. 11.

Effect of increasing inertia on eddy shape for and , with the waviness of the lower plate . In , in , in , and in .

Image of FIG. 12.
FIG. 12.

Effect of inertia on the horizontal shift of the eddy center, , for and a number of different gaps. The solid curve, which begins near the center of the plot, corresponds to an inertially induced secondary eddy.

Image of FIG. 13.
FIG. 13.

Streamline plots illustrating the generation and development of an inertially induced secondary eddy as Re increases: (a) , (b) , (c) , (d) , (e) , and (f) . The geometry is given by and , or, equivalently, and .

Image of FIG. 14.
FIG. 14.

Effect of Reynolds number on the critical gap for the appearance of an eddy at long wavelengths. Here , corresponding to . The streamline plots correspond to the conditions at the indicated black circles.


Generic image for table
Table I.

Position and intensity ratios for the sequence of eddies shown in Fig. 6(e) . The ratios between the third and fourth eddies are not given as the fourth eddy is not fully formed in this case.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Eddy genesis and manipulation in plane laminar shear flow