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Evolving eddy structures in oscillatory Stokes flows in domains with sharp corners

Published online by Cambridge University Press:  09 March 2006

M. BRANICKI  and
H. K. MOFFATT
M. BRANICKI
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, [email protected], [email protected]
H. K. MOFFATT
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, [email protected], [email protected]
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Abstract

Stokes flow of a viscous fluid in a cylindrical container driven by time-periodic forcing, either at the boundary or through oscillation of the cylinder about an axis parallel to its generators, is considered. The behaviour is governed by a dimensionless frequency parameter $\eta$ and by the geometry of the cylinder cross-section. Various cross-sections (square, rhombus, and sector of a circle) are first treated by either finite-difference or analytic techniques and typical transitions of streamline topology during flow reversals are identified. Attention is then focused on the asymptotic behaviour near any sharp corner on the boundary. For small $\eta$, a regular perturbation expansion reveals the manner in which local flow reversal proceeds during each half-cycle of the flow. The behaviour depends on the corner angle, and different regimes are identified for both types of forcing. For example, for an oscillating square domain, eddies grow symmetrically from each corner and participate in the subsequent flow reversal in the interior. For large $\eta$, the corner eddies merge into Stokes-type boundary layers which drive the interior flow-reversal process. In general, the local corner analysis provides a key to an understanding of the global flow evolution.

JFM classification

Mathematical Foundations: Topological fluid dynamics Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 551 , 25 March 2006 , pp. 63 - 92
Copyright
© 2006 Cambridge University Press

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Branicki and Moffatt supplementary movie

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 1.  Time-periodic flow reversal inside an oscillating square cylinder for η = 100. The animation, which corresponds to Figure 1 in the paper, visualises a sequence of instantaneous streamline patterns in a non-inertial frame of reference oscillating with the cylinder. Topological changes in the streamline pattern remain qualitatively the same throughout the low-η regime but the time-scale of the process decreases with decreasing η.

Download Branicki and Moffatt supplementary movie(Video)
Video 20.2 MB

Branicki and Moffatt supplementary movie

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 1.  Time-periodic flow reversal inside an oscillating square cylinder for η = 100. The animation, which corresponds to Figure 1 in the paper, visualises a sequence of instantaneous streamline patterns in a non-inertial frame of reference oscillating with the cylinder. Topological changes in the streamline pattern remain qualitatively the same throughout the low-η regime but the time-scale of the process decreases with decreasing η.

Download Branicki and Moffatt supplementary movie(Video)
Video 5.7 MB

Branicki and Moffatt supplementary movie

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 2.  Time-periodic flow reversal inside an oscillating square cylinder in the high-η regime; η = 5000 (compare this with Figure 14 in the paper). The animation is played four times slower than Movie 1. Note that in contrast to the flow reversal in the low-η regime (Movie 1) the corner eddies blend into Stokes layers through the emergence of wall eddies which develop near the boundary. The flow reversal in the interior is realised through a series of global bifurcations involving these wall eddies rather than the primary corner eddies.

Download Branicki and Moffatt supplementary movie(Video)
Video 70.6 MB

Branicki and Moffatt supplementary movie

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 2.  Time-periodic flow reversal inside an oscillating square cylinder in the high-η regime; η = 5000 (compare this with Figure 14 in the paper). The animation is played four times slower than Movie 1. Note that in contrast to the flow reversal in the low-η regime (Movie 1) the corner eddies blend into Stokes layers through the emergence of wall eddies which develop near the boundary. The flow reversal in the interior is realised through a series of global bifurcations involving these wall eddies rather than the primary corner eddies.

Download Branicki and Moffatt supplementary movie(Video)
Video 24.9 MB

Branicki and Moffatt supplementary movie

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 3.  Time-periodic flow reversal inside a square cavity driven by an oscillating lid (top) for η = 100. The animation visualises a sequence of instantaneous streamline patterns in an inertial frame fixed at the centre of the cavity. The time-scale of the reversal decreases with decreasing η but the sequence of topological changes in the streamline pattern remains the same.

Download Branicki and Moffatt supplementary movie(Video)
Video 15.3 MB

Branicki and Moffatt supplementary movie

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 3.  Time-periodic flow reversal inside a square cavity driven by an oscillating lid (top) for η = 100. The animation visualises a sequence of instantaneous streamline patterns in an inertial frame fixed at the centre of the cavity. The time-scale of the reversal decreases with decreasing η but the sequence of topological changes in the streamline pattern remains the same.

Download Branicki and Moffatt supplementary movie(Video)
Video 5.7 MB

Branicki and Moffatt supplementary movie

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 4.  Time-periodic flow reversal inside an oscillating cylinder with rhomboidal cross-section for η = 100. The animation, which corresponds to Figure 2 in the paper, visualises a sequence of instantaneous streamline patterns in a non-inertial frame of reference which oscillates with the cylinder. Note that the structure in the acute-angle corners is more complicated than in the case of the right-angle corners (applies only to the oscillating-cylinder problem). The local structure of the corresponding corner flow is shown in Movie 8.

Download Branicki and Moffatt supplementary movie(Video)
Video 8.5 MB

Branicki and Moffatt supplementary movie

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 4.  Time-periodic flow reversal inside an oscillating cylinder with rhomboidal cross-section for η = 100. The animation, which corresponds to Figure 2 in the paper, visualises a sequence of instantaneous streamline patterns in a non-inertial frame of reference which oscillates with the cylinder. Note that the structure in the acute-angle corners is more complicated than in the case of the right-angle corners (applies only to the oscillating-cylinder problem). The local structure of the corresponding corner flow is shown in Movie 8.

Download Branicki and Moffatt supplementary movie(Video)
Video 3.2 MB

Branicki and Moffatt supplementary movie

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 5.  Evolution of the infinite eddy structure, characteristic of Regime I (oscillating-lid) and Regime III (oscillating-cylinder), near a 60 deg corner for η = 0.1 (compare with Figure 6 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign. (The full evolution is time-periodic but the eddy structure is almost stationary for most of the period.)

Download Branicki and Moffatt supplementary movie(Video)
Video 8.7 MB

Branicki and Moffatt supplementary movie

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 5.  Evolution of the infinite eddy structure, characteristic of Regime I (oscillating-lid) and Regime III (oscillating-cylinder), near a 60 deg corner for η = 0.1 (compare with Figure 6 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign. (The full evolution is time-periodic but the eddy structure is almost stationary for most of the period.)

Download Branicki and Moffatt supplementary movie(Video)
Video 4.1 MB

Branicki and Moffatt supplementary movie

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 6.  Emergence of a single eddy during the flow reversal in a 160 deg corner (Regime II in the oscillating-lid problem; Regime IV in the oscillating-cylinder problem) for η = 0.1 . There are no infinite eddy structures in the flow at any stage (compare with Figure 7 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign.

Download Branicki and Moffatt supplementary movie(Video)
Video 6 MB

Branicki and Moffatt supplementary movie

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 6.  Emergence of a single eddy during the flow reversal in a 160 deg corner (Regime II in the oscillating-lid problem; Regime IV in the oscillating-cylinder problem) for η = 0.1 . There are no infinite eddy structures in the flow at any stage (compare with Figure 7 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign.

Download Branicki and Moffatt supplementary movie(Video)
Video 3.3 MB

Branicki and Moffatt supplementary movie

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 7.  Flow evolution near a corner of reflex angle 270 deg (Regime II in the oscillating-lid problem; Regime IV in the oscillating-cylinder problem) for η = 0.1 ; this does not involve any eddy during the reversal process (compare with Figure 8 in the paper). As before, this animation shows the flow evolution within a very short time interval when the forcing changes sign.

Download Branicki and Moffatt supplementary movie(Video)
Video 8.3 MB

Branicki and Moffatt supplementary movie

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 7.  Flow evolution near a corner of reflex angle 270 deg (Regime II in the oscillating-lid problem; Regime IV in the oscillating-cylinder problem) for η = 0.1 ; this does not involve any eddy during the reversal process (compare with Figure 8 in the paper). As before, this animation shows the flow evolution within a very short time interval when the forcing changes sign.

Download Branicki and Moffatt supplementary movie(Video)
Video 4.2 MB

Branicki and Moffatt supplementary movie

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 8.  Flow reversal in an oscillating cylinder near a sharp corner representative of Regime II; 2α = 60 deg, η = 0.1, non-inertial frame of reference was adopted which oscillates with the cylinder (compare with Figure 11 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign. Note the emergence of pairs of side wall eddies and their subsequent merger followed by a reversed process closer to the corner. Such pairs of side wall eddies are also involved in the flow reversal in corner angles in Regime I but, in contrast to the situation in Regime II, they are only finite in number.

Download Branicki and Moffatt supplementary movie(Video)
Video 16.4 MB

Branicki and Moffatt supplementary movie

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 8.  Flow reversal in an oscillating cylinder near a sharp corner representative of Regime II; 2α = 60 deg, η = 0.1, non-inertial frame of reference was adopted which oscillates with the cylinder (compare with Figure 11 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign. Note the emergence of pairs of side wall eddies and their subsequent merger followed by a reversed process closer to the corner. Such pairs of side wall eddies are also involved in the flow reversal in corner angles in Regime I but, in contrast to the situation in Regime II, they are only finite in number.

Download Branicki and Moffatt supplementary movie(Video)
Video 7.5 MB