Three-dimensional eddy structure in a cylindrical container

We consider Stokes flow in a cylindrical container of circular section induced by the uniform translatory motion of one of the endwalls. This flow field is of interest because it is possible to get reliable analytical descriptions of important three-dimensional structures such as the primary and corner eddies. It is shown, using a result of TranCong and Blake, that separable solutions exist which can be combined to yield vector eigenfunctions that satisfy the sidewall boundary conditions provided the eigenvalues satisfy the transcendental equation 2kJ. The eigenstructure in the complex plane is somewhat unusual because the eigenvalues form two distinct sequenes: a real-sequence and a complex both of which need to be used to satisfy the top and bottom boundary conditions. The complex eigenvlue with the smallest real part, approximately 2.568+1.123i, determines the spacing and decay of the intensity of the primary eddies in deep containers. The above vector eigenfunctions are first combined, using a least-squares procedure, to determine the flow field in a container of infinite height. It is found, as in the corresponding two-dimensional case, that there is an infinite number of almost equally spaced counter-rotating primary eddies spaced about 2.8 container radii apart. For containers of finite height the number of primary eddies depends on the height of the container; computations show that for container heights of 1 and 2 (based on the radius) there is a single primary eddy while there are two and four respectively for heights of 4 and 10. More interestingly, the corner eddies in the plane of symmetry, rather than being made up of closed streamlines, consist of streamlines that connect the fwo foci at opposite corners in the plane of symmetry. Detailed three-dimensional streamline plots show that away from this plane the flow is almost entirely azimuthal in the corner, a result that would not be evident from two-dimensional results. Other interesting three-dimensional features of the eddy structure are also found. Finally, the growth and merger of the corner eddy as the container height h is increased beyong 3 is studied carefully. Among a number of interesting features, it is found that at some stage streamlines cease to flow into the focus in the centreplane and start, rather, to stream out, resulting on opposite sides of the eddy in the symmetry plane, with local flow from one focus to the other, takes place before the eddy is fully developed. Full merger then takes place when following which the merged eddy gives way to the second primary eddy. We find the corner eddy structure to be quite complicated during the merger process; three-dimensional streamline plots show intricate and rather beautiful patterns in the flow fiel