^{1}, A. Haas

^{1}, N. Aksel

^{1}, M. C. T. Wilson

^{2}, H. M. Thompson

^{2}and P. H. Gaskell

^{2}

### Abstract

Eddy formation and presence in a plane laminar shear flow configuration consisting of two infinitely long plates orientated parallel to each other is investigated theoretically. The upper plate, which is planar, drives the flow; the lower one has a sinusoidal profile and is fixed. The governing equations are solved via a full finite element formulation for the general case and semianalytically at the Stokes flow limit. The effects of varying geometry (involving changes in the mean plate separation or the amplitude and wavelength of the lower plate) and inertia are explored separately. For Stokes flow and varying geometry, excellent agreement between the two methods of solution is found. Of particular interest with regard to the flow structure is the importance of the clearance that exists between the upper plate and the tops of the corrugations forming the lower one. When the clearance is large, an eddy is only present at sufficiently large amplitudes or small wavelengths. However, as the plate clearance is reduced, a critical value is found, which triggers the formation of an eddy in an otherwise fully attached flow for any finite amplitude and arbitrarily large wavelength. This is a precursor to the primary eddy to be expected in the lid-driven cavity flow, which is formed in the limit of zero clearance between the plates. The influence of the flow driving mechanism is assessed by comparison with corresponding solutions for the case of gravity-driven fluid films flowing over an undulating substrate. When inertia is present, the flow generally becomes asymmetrical. However, it is found that for large mean plate separations the flow local to the lower plate becomes effectively decoupled from the inertia dominated overlying flow if the wavelength of the lower plate is sufficiently small. In such cases the local flow retains its symmetry. A local Reynolds number based on the wavelength is shown to be useful in characterizing these large-gap flows. As the mean plate separation is reduced, the form of the asymmetry caused by inertia changes and becomes strongly dependent on the plate separation. For lower plate wavelengths which do not exhibit a kinematically induced secondary eddy, an inertially induced secondary eddy can be created if the mean plate separation is sufficiently small and the global Reynolds number is sufficiently large.

Professor N. Aksel wishes to thank the EPSRC for the provision of Visiting Fellowship, Grant No. EP/E029183/1, which made this collaboration possible.

I. INTRODUCTION

II. PROBLEM SPECIFICATION AND SOLUTION

A. Flow geometry, field equations, and boundary conditions

B. Methods of solution

1. Finite element formulation

2. Semianalytical approach (Stokes flow)

III. RESULTS AND DISCUSSION

A. Kinematically induced eddies in Stokes flow

B. Inertial effects on kinematically induced eddies: Large gaps

C. Inertial effects on kinematically induced eddies: Small gaps

IV. CONCLUSIONS

### Key Topics

- Eddies
- 132.0
- Stokes flows
- 42.0
- Free surface flows
- 32.0
- Shear flows
- 19.0
- Reynolds stress modeling
- 14.0

## Figures

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

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

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.

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.

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 .

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 .

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 .

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 .

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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.

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.

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 .

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 .

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.

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.

## Tables

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.

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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