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Turbulent mixing of oil droplets in a round water jet

Turbulent mixing of oil droplets in a round water jet

diameter

diameter of a 20 µm droplet is 4.7 µm which is smaller than the pixel size, thus peak-locking isexpected. The depth of field is, acccording to equation 3.3, equal to 2.7 mm so larger than thethickness of the light sheet.As the edges of the CCD are not illuminated, the active pixel region is decreased to 1024 inthe horizontal direction by 512 in the vertical direction. This decreases the read-out time of thecamera and increases the number of images that can be stored in memory. In this configuration,a frequency of 4 Hz can be obtained and 200 double images can be stored in memory. Theexperiment has to wait while the software (CamWare) writes data from memory to harddisk. Inorder to get the statistics of more than 200 double images, a number of equal experiments has tobe performed in sequence. Before the measurements are performed, a background greyvalue imageis recorded. The background is subtracted from every measured greyvalue image. To decrease thebackground noise, multiple background images can be recorded and averaged before subtractingfrom the measurements.Regarding the size of the image field (6.3 cm x 3.15 cm) and the distance to the lens (60 cm), theangle of the incident light beams on the camera, scattered by the oil droplets, varies between 87and 93 degrees. According to figure 3.5 this angle dependency of the scattering will be visible.The camera position is adjustable in every degree of freedom in order to point it straight at thelight sheet. The z-coordinate measured from the jet outlet is read from a millimeter grid attachedto the measurement section.3.3.4 DisplacementTo calculate the displacement vectors of the double images the software of DaVis, version 6.2,is used. It reads in the greyvalue images as stored by CamWare and applies a standard cyclicFFT-based algorithm to obtain the cross-correlation function [LaVision, 2001]. A multi-passalgorithm is applied, starting at an interrogation area of 64 x 64 pixels and ending at 32 x 32with 50% overlap. It is possible within the software to try other algorithms to calculate thecorrelation function. Regarding the size of an interrogation area, the edges of 16 pixels width arenot calculated correctly. Therefore, they will be removed when the data are analysed. Outliersare defined as vectors having a component with a difference with the median of its neighbourslarger than two times the rms of its neighbours.3.4 Mean velocity and turbulent statisticsIt is possible that the camera is slightly misaligned or the jet is not issued from the tube perfectlyvertically. To correct for a rotation of the jet in the image plane, the mean velocity data are fit to aGaussian curve on every height. If the middle of this curve, x0, is shifted as a function of the height,the camera is misaligned. The shift is calculated by fitting x0 to a linear function. Thereafter, therotational angle compensating this shift is applied to the velocity field and coordinate system.The mean velocity field is calculated by averaging over the velocity calculations of all N images.The fluctuations in the mean velocity, as a measure of the turbulence, are calculated according to:u rms = √ 1 NN∑(u − u) 2 =1√u 2 − u 2 (3.6)in which the Reynolds condition uu = ūū is used (see section 2.1.1, N ≫ 1) From the mean axialvelocity, the mean centreline velocity, u s , and jet width, l, are obtained by fitting the mean velocityto a Gaussian curve on every axial position. For calculating the best fit, only the measurementsinside the jet are used (-1 < η < 1), as the outer regions are disturbed by the coflow. The fitfunction has the following form:( ) 2u = rp(3)e − x−rp(1)rp(2)(3.7)26

The regression parameter rp(1) is the position of the jet axis (x0), rp(2) the width of the jet (l)and rp(3) the centreline velocity (u s ). Subsequently, the spreading rate, α, and virtual origin, z0,can be calculated by imposing the self-similarity conditions 2.16 and 2.17 to the jet width andcentreline velocity and fit a Gaussian curve to the velocity data of all positions (z, x) in the imagefield:( )u =r(4)2x−r(1)z − r(2) e−r(3) z−r(2)(3.8)The expression r(4)/(z − r(2)), is the centreline velocity for the jet with imposed self-similarity.The regression parameter r(2) is equal to the virtual origin, z0, while r(3) determines the spreadingrate by r(3) = 1/α 2 .The turbulent transport of momentum, the Reynolds shear stress, is calculated according to:u ′ v ′ = 1 NN∑(u − u)(v − v) = uv − ū¯v (3.9)1Finally, the uncertainty, due to the finite number of samples, in the mean velocity measurementsis determined as follows:σ u = u rms√ (3.10)NThe velocity measurements are assumed to be independent, so the time interval between twomeasurements (250 ms) is larger than the time scale of the flow. The theoretical profiles (seesection 2.1.3) will be compared to the experimental profiles. Besides, the boundary-layer equations(2.7) will be verified by substituting the self-similar profiles, as a function of the dimensionlessvariable (x − x0)/(z − z0).3.5 Mean concentration and turbulent statisticsThe first way to estimate the concentration of oil is to take the intensity of the scattered light asa measure for the concentration of droplets present in the flow. This is allowed if the droplets aremonodisperse and their images do not overlap. Besides, it is necessary that the greyvalues of therecorded images are proportional to the amount of scattered light. By averaging the greyvaluesover the area of N by N pixels surrounding a pixel, the single droplet images are smeared out andan estimation of the concentration is acquired on every pixel position. The chosen number of pixels(N) of the averaging area, depends on the concentration. The chance of finding no droplets inthis area, when in fact the concentration is non-zero, should be minimal as it gives an unphysicalcontribution to the rms-values of the concentration field. On the other hand, enough informationof the spatial development must be preserved. Therefore, N is chosen equal to the size of theinterrogation area of the PIV-measurements. An average of 15 droplets is observed in each interrogationarea and it gives the highest possible spatial resolution for the combined velocity andconcentration measurements.The estimation is not optimal because the light intensity over the light sheet is not homogeneous.Light is absorbed by the droplets, scattering is angle dependent (forward scattering is higher thanbackward scattering) and the light intensity of the light sheet itself is not distributed homogeneously:it has vertically a more or less Gaussian shape whereas it decreases from left to right dueto divergence of the light sheet. To compensate for these effects, a reference intensity measurementis used to scale the greyvalue images. The reference intensity is obtained by adding oil dropletsto the coflow and record the intensity of the scattered light. However, the result will still not beoptimal: the droplet distribution during the reference measurement differs from the distributionduring the measurements. As a consequence, the local absorption in the light sheet changes: the27

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