Tracers take the tube
Three-dimensional (3D) time-periodic mixing of laminar flows is present in many physical processes in nature and industry. Food industry, polymer industry, geological and biological fields as well as micro-fluidic flows are among many applications that one can name for 3D chaotic advection. In spite of its universality, mixing problems are mostly treated individually and lack of general views, rules and regulations is evident and sensible.
Aim of this project is a numerical as well as an experimental study of fundamental characteristics of 3D viscous mixing under inertial perturbations. The starting point is a numerical investigation of the flow topology for a selected number of problems in cylindrical cavities. Some of the most remarkable 3D phenomena are made predictable and topologically explainable. Moreover this study has been extended to experimentally feasible configurations which make the laboratory validation of these trends possible.
Time-periodic flows in a square cylindrical domain are considered. The flow is generated via translation of the top or bottom wall of the cylinder under specific forcing protocols. Each forcing protocol has a particular configuration consisting of n steps. The n-step flow (n = 2, 3 and 4) is approximated as a piece-wise linear combination of n separate steady flows. For applying numerically all of the forcing protocols to the flow we have used the outcome of an in-house code for the steady state flow field. The Navier-Stokes solver, which utilizes the spectral method, enables us to make sophisticated numerical simulations including effects due to inertial perturbations highly accurate.
Two key elements of the flow are the periodic-1 points and the invariant surfaces. Periodic points of each time-periodic map are those that will return to their initial position after one complete period. They may have an elliptic, a parabolic or a hyperbolic character. The invariant surfaces or constants of the motion of this flow are topologically equivalent to spheres. The passive tracer stays on the surface of one of them according to its initial position. The two-dimensional (2D) pattern formed on the surface, visualized by displaying the Poincaré map (see the figure below), can vary from a combination of some elliptic islands embedded in chaotic areas to an entire chaotic sea.
By adding inertial disturbances to the flow and solving the problem for a certain Reynolds number, despite the 2D character of the system in the absence of inertia, we gain a complex 3D bifurcation to chaos. For very small Reynolds number the invariant surfaces expected to survive as adiabatic shells, meaning that they keep their topological shape to a high extent and instead of being a well-defined rigid surface, obtain a certain thickness. This indicates the presence of small motion of the passive tracer perpendicular to the surface, in general just showing a perturbed version of the same topology. But what will happen here for increasing inertial perturbations is the following: new 3D coherent structures form which consist of two adiabatic shells and one connecting tube in between of them.
We perform the experiment in exactly the same way as the numerical simulation was done. This engages keeping the correct coordinates, the same boundary conditions, an equivalent forcing protocol and of course the exact analogy of the cylindrical domain.
The (8cm x 8cm) cylinder is placed in a tank (30cm x 40cm x 60cm) filled with silicon oil. This is partly visible in the figure below. The gravity effects on the tracer particles can safely be neglected in this experiment. There is an opening (4cm) in the side of the cylinder which makes the access into the cylinder possible (to add small tracer particles or coloured dye). The moving wall is controlled by stepper motors in two directions which are able to be positioned as accurate as 1mm. A needle is also designed for positioning purposes which is controlled by a manual micrometer in the z-direction and by a stepper motor in the y-direction.
Dye visualization method has been used in order to verify the analytical and numerical prediction for the position as well as the characteristics of the periodic-1 points.
Two periodic points on the periodic-1 line of one of the forcing protocols is chosen, one of them supposed to be hyperbolic and the other one elliptic. Both the location and the behaviour of the points are in quite satisfactory agrements with our numerical simulations.