E XPERIMENTAL SET - UP

A schematic overview of the experimental setup is shown in Figure 3.1.

Figure 3.1 Schematic overview of the experimental setup. A rectangular container filled with water is placed on a rotating table. A sharp-edged wall is inserted in the flow field parallel to the rotation axis. A moving cylinder is used to create a

dipolar vortex. The flow field evolution is recorded by a digital camera that co-rotates with the tank.

A rectangular cuboid water tank with dimensions 100 x 150 x 40 is placed on a rotating table and filled with tap water. Before the experiment can start the fluid is brought into solid body rotation at an angular velocity of . This rotation speed value is chosen high enough to ensure the general validity of the assumptions made for the Taylor-Proudman theorem, yet slow enough for the experiment to be practically feasible and safe.

Prior to the experiment the fluid is allows to reach solid-body rotation during at least 60 min. Once the fluid is solid body rotation the centrifugal force will result in a parabolic free surface. The effects of a position dependent water height [17] should be corrected for. In order to have a fluid of uniform depth,

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a parabolic bottom is used such that the depth of the fluid layer is the same everywhere. A dipolar vortex is conveniently created by dragging an open, thin-walled cylinder horizontally along a straight line through the fluid while simultaneously lifting it slowly out of the fluid. During this process the flow is unsteady and the Taylor-Proudman theorem does not apply. As the typical velocity of the cylinder is 10 cm/s and the diameter of the cylinder is 6 cm the Reynolds number associated with the cylinder lifting process is . This implies that the wake behind the cylinder is turbulent. After the cylinder is lifted out the inverse energy cascade [11] causes the small vortex structures to form larger vortex structures and the result is a quasi-2D dipolar vortex structure. This dipole is characterized by a typical length scale associated with its radius R = 10 - 15 cm and a typical propagation speed U = 1 - 2 cm/s.

In the experiments the collision of a dipole with two different geometries is studied. For the experiments with a sharp edge a flat aluminum plate with a thickness of 2 mm is used. Since this thickness is much smaller than the typical length scale associated with the dipole the edge of this plate is considered to be

“sharp”. For the collision with an opening in a wall the fact that the ends of the wall are not sharp is important [7]. Therefore cylinders with a diameter of 3 cm are placed at the edge of the walls. A photo of the used “opening in a wall” geometry is shown below.

Figure 3.2 Plan-view photograph of the gap between two walls (white) surrounded by water that is partially dyed (yellow).

Cylinders are attached to the walls such that the edged have a larger radius of curvature

During the experiment the fluid flow is recorded by a digital camera. The camera co-rotates with the water tank. A way to visualize the fluid flow is to add dye that acts as a visual tracer in the fluid by virtue of contrast between fluid with different colors or transparency. By adding dye near the geometry and in the cylinder before it is lifted out the tank the flow can be recorded by a color camera. It turns out that a large portion of the dye in the cylinder is entrained by the dipole and the evolution of the dipole can thus be monitored. The dye visualizations do not give any quantitative data for comparison with the numerical simulations, neither does it provide flow information for areas where there is no visible contrast. For a quantitative and entire flow field measurement Particle Image Velocimetry (PIV) measurements are done. These measurements are carried out by tracking suspended particles that are

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passively advected by the flow using a 1600 x 1200 gray-scale camera. The camera records at a frame rate of 30 frames per second. Subsequent frames show details of the flow field by deformation of the particle distribution between these frames. A length calibration is necessary to correct for optical defects and gain information of the magnification factor for the images (i.e. mm/pixel). To do this a sheet with a grid is placed in the tank, which is filled with water at the depth of the light sheet that illuminates the particles. The camera takes a snapshot of a frame and this image is then analyzed by PIVMAP 1.2 software. Together with the known grid spacing a reshape mapping and a magnification factor are calculated. After carrying out this process once, all frames can be reshaped. The individual frames from this process are fed to the software program PIVview 3.5, together with magnification factor and the frame rate. The software is able to calculate the flow field by correlating small sections of an image with the corresponding section of the next image. More information on the PIV technique can be found in [33]. In practice the obtained flow field data is noisy. Therefore, outliers in the vorticity field are averaged out by applying a Gaussian filter. Due to the presence of relative weak convection cells in the flow field [15] a non-zero vorticity field is observed outside the primary and secondary vortices. The associated circulation is filtered out by applying thresholding. This post-processing influences the flow field locally, but the extracted data (vortex locations and strengths) concern more global flow field properties that appeared to be independent on the chosen filter parameters. Therefore, the quantitative data from PIV appears well determined despite of the errors in the raw flow field data. An actual error analysis of these quantities is not presented in this thesis.

Due to the dimensions of the rotating tank, the used fluid and the tank’s angular velocity the experiment is limited to only a finite range of experimental parameters. This section gives typical values of important quantities that have an effect on what can be investigated experimentally.

The experimental setup and the dipolar vortex produced in the experiment are both characterized by some typical values that give an estimate of dimensionless numbers discussed in the theory section. The typical values are:

(3.1) From these we obtain the following typical values for the dimensionless numbers:

(3.2)

In the theory both E and Ro were assumed to be small enough for the Taylor-Proudman theorem to be applicable (i.e. E << 1 and Ro << 1). Although E << 1 in the experiments, since ( ) it cannot be concluded that the Taylor-Proudman theorem will apply in the rotating tank experiment and therefore further investigation is needed. The Reynolds number Re >> 1. This indicates the relative small effect of the viscous forces on the flow field evolution in the interior region of the flow (i.e at some distance from any solid boundaries).

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