The significance of coherent flow structures for the turbulent mixing ...

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¢ ü Øî¥¤ï§¦óØíî¨¤¡© âðï¦ ñ"ó (2.4) 2 Particle Image Velocimetry a pixel, either due to the low sensitivity of the sensor or to the noise, the fill ratio becomes important as the complete image information can disappear when the intensity distribution is located between two adjacent light sensitive areas. Due to the statistical evaluation, which will be described in the following section, the displacement can still be calculated as the probability that all particle images within an interrogation area are unpaired is small, but an increase in noise is unavoidable due to unpaired particle images and to the poor performance of the centre of gravity approach or related methods, see section 2.4 and 2.4.1. This is different when the displacement estimation relies on a single image pair as in particle tracking analysis or as forced in [27]. However, under these conditions the measurement accuracy is strongly affected by the CCD characteristics. 2.4 Particle image analysis In order to extract the displacement information from the two single exposed grey-level patterns acquired atìandìpí, both images are usually evaluated by means of statistical evaluation techniques, for two reasons. First, individual particle image pairs cannot be identified sufficiently reliable due to the high seeding concentration. Secondly, the statistical evaluation is less sensitive to noise and discretisation effects as outlined in the previous section. For the evaluation, the whole image is sampled with an appropriate step-size, typically half the linear dimension of the grey-level sampleØîâðïòñ"ó. For each sampling location a two-dimensional grey-level sampleØîâðïôñkóof certain size and shape is extracted from the source image as indicated in figure 2.1, and it is cross-correlated with the corresponding sampleØíîâðïòñkófrom the search recording according to the following equation 1 [52, 105, 109]. ýÿþ¡ £ôþ¡ ¢ The discrete cross-correlation formula produces one cross-correlation valueõöÂöpøîâðïòñ"ó û for each possible displacement between the two samples by calculating the sum of the products of all overlapping pixel intensities so that the output of this calculation can be displayed in the form of a two dimensional cross-correlation plane, see figure 2.11. Due to the symmetry of the cross-correlation function the templateØÌîâðïòñ"ócan be selected in either of the two images, but the sign of the displacement has to be reversed when the template originates from the image atì © recorded ì. The cross-correlation formula is usually properly normalised so that its amplitude is invariant under scale changes in the ofØ amplitude andØíand thus not affected by variations in the particle image number, size and intensity. This allows to quantify the quality of the signal which is of great importance when aligning the experimental components, such as light-sheet and cameras, with respect to each other, see figure 2.12. õ÷ö´öpøÈîâðïòñ"óúù û ü © 28 1 As each single grey-level sample back-projected in the object space and multiplied by the effective light-sheet thickness, given by the intensity distribution of the light-sheet itself, the particle size distribution, the imaging optics and sensitivity of the CCD camera, represent a fluid volume, the cross-correlation can be considered as weighed volume average.
ýÿþ¡ û û ¢ ¢ £ôþ¡ î¥¤ï§¦ó ! û ¢ ¢ £|þ¡ íî¥¤© âðï§¦ ñkó ñ"ó 2.4 Particle image analysis Shift ( ∆x =-2, ∆y = 2) Shift ( ∆x = 0, ∆y = 0) Shift ( ∆x = 2, ∆y = 2) Shift ( ∆x =-1, ∆y =-1) +2 Shift ( ∆x = 1, ∆y =-2) ∆ y 0 -2 -2 0 ∆ x +2 Cross-correlation plane FIGURE 2.11: Example of the formation of the correlation plane by direct cross-correlation: here a pixel template (grey-square) is correlated with a pixel cross-correlation plane. pixel size sample to produce a © ýqþ¡ û û ¢ ¢ £|þ¡ î¨¤ï§¦ó íî¨¤© âðï§¦ í (2.5) © íó" ýqþ¡ ídenotes the mean intensity value of the search window calculated over the portion which û coincides perfectly with the î¥¤ï§¦óand is the corresponding mean intensity for the template. It is obvious that íhas to be calculated for each possible shift to guarantee #%$ that $'& (negative values are usually not considered as intensities are always positive and the mean value is close to zero for moderate image densities). As a direct implementation of this formula in the spatial domain becomes quite time intense, especially for large correlation windows, the computation of the correlation is usually performed in the frequency domain via fast Fourier transformation (FFT) based algorithms. They are fast as symmetry properties are öÂö taken into account but artifacts like aliasing, circular effects and limitations in the size and shape of the correlation or interrogation window may lower the quality of the velocity estimation and restrict the performance of the method for particular applications, see chapter 3. As the cross-correlation approach is based on the idea that the location of the signal-peak yields the information about the displacement of the image pattern, the conditions for the correctness of this assumption need to be analysed in detail. Due to the fact that this is mathematically not tractable when a single realization of a particle image pattern is considered, the pattern is thought of as a single realization of a random process and it is assumed that this particu- öÂöpøÈîâðïòñ"óúù 29
Page 1 and 2: The significance of coherent flow s
Page 3 and 4: Contents 1 Introduction 5 2 Particl
Page 5 and 6: 1 Introduction One of the fascinati
Page 7 and 8: As order can be always observed whe
Page 9 and 10: were not considered in detail in th
Page 11 and 12: varying signal can be transformed i
Page 13 and 14: 2 Particle Image Velocimetry The tr
Page 15 and 16: 2.1 Principles It is of fundamental
Page 17 and 18: ’ ’“ ” ”• Ž Ž Ž Ž
Page 19 and 20: ¹ and 2.2 Generation of appropriat
Page 21 and 22: 2.2 Generation of appropriate trace
Page 23 and 24: 2.3 Registration of the particle im
Page 25 and 26: 2.3 Registration of the particle im
Page 27: ë ë 2.3 Registration of the parti
Page 31 and 32: or = ö D Q ù ù ?xö ù Y[Z]\ & &
Page 33 and 34: ñ † † † † † † † †
Page 35 and 36: ˜ E ˜ T ˜ T 2.4 Particle image
Page 37 and 38: 3 Stereo-scopic Particle Image Velo
Page 39 and 40: › B › B › J ù ž & 3.1 Princ
Page 41 and 42: Æ e¼c’„ïG ù # #dc’e › B
Page 43 and 44: Ð È Ñ ù Ñ ù Ð È Ð È Ñ ù
Page 45 and 46: 3.2 Evaluation of stereo-scopic ima
Page 47 and 48: ú ¦ 3.3 Calibration validation by
Page 49 and 50: 4 Multiplane Stereo Particle Image
Page 51 and 52: £ è ¤ ó 4.2 Four-pulse-laser S
Page 53 and 54: £ ¤ £ £ 4.2 Four-pulse-laser Sy
Page 55 and 56: ( ì ( æ ( ì è ð ( ( ( ( ( ( ì
Page 57 and 58: æ ( | | | | | | | ì æ õ æ õ
Page 59 and 60: 4.5 Simplified recording system 4.5
Page 61 and 62: 4.7 Monochromatic aberrations refle
Page 63 and 64: º¹ n 4.7 Monochromatic aberration
Page 65 and 66: 4.8 Feasibility study experiments i
Page 67 and 68: 4.8 Feasibility study 10 20 30 40 5
Page 69 and 70: é¥î ëÝïñð€òôó 5 Invest
Page 71 and 72: I L § § § : ? ; õ : I I I W
Page 73 and 74: 5.2 Experimental set-up α4 α3 α2
Page 75 and 76: { : { L Ø { 9 D { 9 D T W ð 9
Page 77 and 78: : T ž ž ò 5.3 Statistical pro
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¾ ¾ 5.3.2 Spatial auto- and cross
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½ Ë 5.3 Statistical properties of
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½ Ë 5.3 Statistical properties of
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ô ó ñ ½ ñ ô ó 5.3 Statistica
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æ å ä ã ã ã ½ Ë ä æ å ½
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æ æ å ä À Â ½ 5.3 Statistica
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½ 5.4 Properties of coherent veloc
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» % %QÄ ¾ Ó turbulent velocit
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* Ó , Ó * Á , Â 5.4 Properties
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! 6 Investigation of the xz-plane O
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6.1 Experimental set-up the (virtua
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ß ß 6.2 Statistical properties
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g ˆ g ˆ 6.2 Statistical propertie
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— ¯ 6.2 Statistical properties o
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å å ç æ æ ã ã ã å å è æ
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6.2.3 Spatial cross-correlation fun
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6.2 Statistical properties of the b
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6.2 Statistical properties of the b
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performed at and the second at
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6.3 Spatio-temporal buffer layer st
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M M æ æ æ æ æ æ ã ã ã H M
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V W V 6.4 Properties of coherent ve
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6.4 Properties of coherent velocity
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Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
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d d 6.4 Properties of coherent velo
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'§ 'Î ª ¿ ¦ '§ ')( 7 Investi
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% { s s s s 7.1 Experimental set-up
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‘ — s 7.2 Statistical propertie
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7.2 Statistical properties of the l
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¿ 7.2.2 Spatial cross-correlations
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¿ 7.2 Statistical properties of th
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ô ô 7.3 Spatio-temporal correlati
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7.3 Spatio-temporal correlations wi
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¨ ¨ ¨ 7.3 Spati
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£ 7.4 Spatio-temporal correlations
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Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
¿ 7.5 Properties of coherent veloc
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¿ 8 Summary The first part of this
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aperture, it is shown how to elimin
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¡ ¿ tures keep their spatial orga
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Bibliography [1] ACARLAR MS, SMITH
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[30] HINSCH K (1995) Three-dimensio
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[60] KNEUBÜHL FK, SIGRIST MW (1995
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[93] SMITH CR, METZLER SP (1983) Th
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Z k l l l l l { l R q ‡ ˆ ’
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Ø á a Õ Œ v momentum thickness,
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Index Kolmogorov micro-scale . . .
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Acknowledgement First of all I woul
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Lebenslauf von Christian Joachim K
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