Non Dimensional Numbers

The experimental conditions for each run of the experiment are characterized in terms of an ejection Reynolds number and an ejection Rossby number. These two non dimensional numbers are defined as

Re0 =

u0d

ν , (3.21)

Ro0= u0

Ωd, (3.22)

where u0 is the velocity of the jet at the source, ν is the kinematic viscosity

of the jet liquid. Moreover a local Rossby number dependent on the height above the source is used. This local Rossby number is based on the azimuthal velocity and it is defined by

Rol=

vθ

Ωl, (3.23)

wherevθ is the maximum circumferentially averaged azimuthal velocity and l

is the radius at which it occurs.

Following Hunt & Kaye (2001) a source parameter, or Richardson Number, (Γ) for the plumes at the source is defined as

Γ= 5Q2 0B0 4αM5/2 0 , (3.24)

whereB0,M0andQ0are Buoyancy, Momentum and Volume flux at the source.

3.4

Particle Image Velocimetry

PIV is a non-intrusive, laser-based, optical measurement technique. Figure 3.2 shows a schematic diagram of a typical experimental PIV arrangement. As illustrated in the figure a basic PIV measuring system consists of a CCD (charge couple device) camera to record images of the flow, a laser device to illuminate the flow, optics to create a light sheet and PIV tracer particles for seeding the flow. Figure 3.2 illustrates how images of the seeded flow are

recorded by means of the CCD camera. The underlying rationale of the PIV measurement technique requires that the seeding particles faithfully follow the flow. Information regarding the flow-field velocities can then be obtained by filming the the particle motion in the illuminated plane of the light sheet and a subsequent analysis of these recordings.

Figure 3.2: Schematic diagram of a PIV experimental arrangement (Raffel

et al., 2007).

There are two techniques that can be used to record images, namely single frame/multi exposure technique and the multi frame/single exposure tech- nique. Figure 3.3 illustrates the way that images get recorded when these two techniques are used. Both techniques rely on information of the particle velocities being available for successive instants, t, in time separated by short intervals ∆t.

Figure 3.3: (a) Single frame with single exposure of light to the camera lens, (b) Single frame with multi exposures of light to the camera lens (Raffelet al., 2007).

In modern PIV, the data analysis is conducted using dedicated computer pack- ages and the terminology Digital Particle Image Velocimetry (DPIV) is now common. For DPIV the pixel array representing an image is divided into smaller areas referred to as ‘interrogation areas’.

Figure 3.4: Schematic diagram of analysing a PIV image pair (Raffel et al., 2007).

As illustrated in figure 3.4 image pairs at timet and t′

=t+∆t are taken into consideration. Each image is then subdivided into a number of small interro- gation regions. The interrogation regions have to be small enough such that it can be assumed that all particles within each interrogation move homoge- neously as a whole. These interrogation areas are the key factor that define the desired density of the vector field obtained after processing the images. For the data analysis one interrogation area from the first image (image taken

at timet) is selected and the respective interrogation area of the second image (image taken at time t′_{) is shifted with respect to the first. The cross cor-} relation of the image densities at each pixel is calculated. Then the peak of the correlation map shows the degree of match between the two interrogation areas and hence can be used to estimate the particle displacement. Most of the software packages use fast Fourier transform to calculate the cross correlation.

In order to find the velocity vectors, the particle displacement has to be quan- tified. This is achieved by means of an image calibration. Image calibration provides a conversion factor that relates the distance in terms of pixels to its corresponding real-world distance in units of length. For the calibration pro- cess a calibration image is used. Figure 3.5 displays a typical calibration image which, in this particular case, consists of rows of black circles of constant radius on a white background. The radius of the circles and the distance between their centres represent known reference lengths. Once the real distance of par- ticle displacement is known a velocity vector for the considered interrogation area is determined. A velocity vector map over the whole image area is ob- tained by repeating the above explained procedure for each interrogation area over the image pair. In order to increase the spatial resolution, a technique called ‘overlapping of interrogation areas’ can be used. Figure 3.6 shows, three interrogation windows and, as it can be seen, the interrogation window 2 has a 50% overlap with the interrogation window 1. Also the interrogation window 3 has a 50% overlap with the interrogation window 2. Since one velocity vec- tor is assigned for one interrogation window, the overlapping of interrogation windows increases the spatial resolution of the vector field. Another technique of improving the results of PIV is offsetting the interrogation windows. Figure 3.7 shows two interrogation windows in an image pair. The cross correlation is calculated by comparing the interrogation window 1 in image 1 shown in the figure with the interrogation window 1 in image 2. As it can be observed in the figure the position of the interrogation window in image 2 has a slight offset when compared with the position of the interrogation area in image 1. The offset has to be carefully chosen to improve the results obtain from PIV images - for details refer to Westerweel et al. (1997).

Figure 3.5: A typical PIV calibration image (Qu´enot et al., 2001).

Figure 3.7: Interrogation windows in an image pair with respective interroga- tion window in second image has an offset.