Quasihomogeneous Distributions


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The high concentration spikes occurred at a saddle point of the velocity field based on the large-scale structure of turbulent eddies. Moreover, according to [14] [15] , the occurrence frequency as well as the mean concentration of the high concentration spikes decreases with increasing distance downstream from the source. Although the high concentration spikes and their occurrence mechanism have attracted many researchers, those calculated statistics are not enough to derive a diffusion theory or prediction method regarding the high concentration spikes.

Clarification of the diffusion theory of the high concentration spikes is an important challenge in reducing the damage from pollutant diffusion by means of a quick localization of the emission source. With the aim of calculating the scalar statistics of high concentration spikes, which relate to the diffusion theory, we experimentally demonstrated the concentration and velocity fields of passive scalar dye diffusing from a fixed emission point in a quasi-homogeneous turbulent flow with simultaneous PIV Particle Image Velocimetry and PLIF Planar Laser Induced Fluorescence measurement.

As a simple test case, we employed a water channel flow between two parallel walls. The high concentration spike we focus on in this study occurs intermittently in the turbulent flow, and its unconditional statistical quantities such as mean concentration and concentration fluctuation cannot be calculated by the normal ensemble-average operation. In order to extract the high concentration spikes, the conditional sampling technique [16] was applied to a spatial dye- diffusion image obtained by PLIF measurement.

From the experimental results of the scalar statistics of the extracted spikes, which include spanwise diffusion width, average length scale, mean concentration, and turbulent diffusion coefficient, the diffusion theory of the spikes as well as its difference with that of the concentration plume were discussed. Figure 1 shows a schematic diagram of our main experimental apparatus, which was a closed-circuit water channel. The water flow under constant temperature was driven by an inverter controlled centrifugal pump with the constant volume. A rectifier was installed at the entrance to ensure a steady flow by damping the large vortices.

A fully developed two-dimensional turbulent channel flow was established in the measurement section, which was located - mm downstream from the channel inlet. The flow rate Q was measured by a flow-meter to monitor the constant Reynolds number. In this study, we focus on the statistically modeled quasi-homogeneous turbulence to investigate the fundamental characteristics of the high concentration spikes.

Accordingly, the uniformities of the time-average velocity and turbulent intensity in the turbulent flow are important. Figure 2 shows the spanwise distributions of time-average streamwise velocity U c and turbulent intensity at the channel center obtained by PIV measurement. We can confirm that the velocity field in the measurement area was fully developed and the assumption of no velocity gradient in spanwise direction is.

Figure 1. Experimental apparatus. Figure 2. Spanwise distributions of time-average streamwise velocity and turbulent intensity at channel center obtained by PIV measurement.

Submission history

This means that the flow field we focus on in this study is quasi-homogeneous turbulence in the two-dimensional plane of the streamwise and spanwise directions. In order to measure the spatial velocity field and concentration field at the same time and at the same position, we employed simultaneous PIV and PLIF measurement.

This simultaneous measurement was extensively used to investigate the mixing processes [17] [18] [19]. From the nozzle located at the center of the channel, a solution containing fluorescent dye was introduced into water flow for PLIF measurement. The water flow was laden with tracer particles for PIV measurement. The nozzle had an internal diameter of 1 mm and outside diameter of 2 mm.

Both coordinate systems define x as the streamwise direction, z as the spanwise direction, y as the wall-normal direction, and the nozzle tip as the origin point. The injection speed of the dye solution was.

Quasihomogeneous Distributions, Volume 165

Figure 3. We assumed that the amount volume of released dye solution was so small that any effect on the turbulence would be neglected. We used USS as the tracer particles, which are sufficiently small and can be assumed to faithfully follow the flow dynamically, and also used Rhodamine WT as the fluorescent dye, which exhibits a high Schmidt number of approximately in water. A laser emission device Nd: YAG laser was installed at the side of the channel while cameras to capture the light for measurement of velocity and concentration were installed below the channel.

Alaser sheet with a wavelength of nm was emitted to the height of the channel center after expanding to a plane with a thickness of 1 mm by a cylindrical lens. With laser illumination, scattering light from the tracer particles with a wavelength of nm and fluorescent light from the dye with a wavelength of nm were captured by the CCD cameras Flow Sense 4M , after being separated by a beam-splitter. The concentration of the dye was calculated assuming that the fluorescent intensity was proportional to the local dye concentration.

Simultaneous PIV and PLIF measurements were taken at a sequence of streamwise distances in order to acquire the spatial concentration and velocity distributions at various downstream distances from the nozzle. In this experiment, the photographing rate was 4 Hz and consecutive distributions of velocity and concentration were captured simultaneously.

The total duration of the sampling was thus s, which was significantly longer than the largest time scales in the turbulent flow. In addition, the photographed area was approximately 4. The mixing-length scale in the outer layer of wall turbulence was roughly estimated as 0. For a reference, note also that the streamwise integral length scale obtained by integration of the streamwise two-point correlation coefficient was about 0. To calculate the scalar statistics of the high concentration spikes it is necessary to indicate the position and area of the spikes in the PLIF image describing the diffusing dye.

In this study, a conditional sampling technique was used, which comprised the moving averaging, pattern recognition and binarizing by setting a threshold, as shown in Figure 4. Firstly, spatial data on concentration fluctuation C' x, z were obtained for different x at a fixed z from the raw PLIF images.

Then C' x, z was smoothed by the moving averaging method [21]. After that, by applying pattern recognition to the smoothed data, the corresponding waveform pattern was extracted. When the maximum value of concentration fluctuation in the waveform pattern recognized by the pattern recognition was higher than the threshold of concentration, the waveform pattern was extracted. In this technique, the threshold was the root mean square of C' x, z.

Thereafter, the concentration on the extracted waveform pattern was binarized as either zero or one: the concentration between the position of the largest gradient and the smallest gradient in the waveform pattern was binarized as one, while the other part was binarized as zero. When the above analysis was performed for all z on each PLIF image, the time series of the two-dimensional binarized images was obtained and the spikes were extracted.

This spike extraction method is based on. Figure 4. Procedure of conditional sampling technique to extract high-concentration spikes. For different streamwise locations, areas of the same size were selected based on the recorded PLIF images, and the further spike statistical analysis was done focused on the selected areas. Figure 5 takes the original PLIF images for three locations for example.

Quasihomogeneous Distributions, de O. von Grudzinski - Leia online

In Figure 5 , higher brightness means higher fluorescent intensity, which is proportional to the local dye concentration. Figure 5 shows dye plumes with low and high concentration areas. Figure 6 shows the corresponding binarized images obtained by the conditional sampling technique mentioned above, and the length-scale mark is also shown in the figures.

Figure 5. Images of diffusion and convection of dye emitted from nozzle obtained by PLIF measurement. Figure 6. Binarized concentration images visualizing high concentration spikes obtained by the conditional sampling. In the diffusing plume, the concentration fluctuates in space and there are areas having a high concentration and concentration gradient compared with the surroundings. Moreover, a meandering motion of the diffusing plume is not identified definitely in Figure 5 , which was observed under the turbulent flow background at high Reynolds numbers [22].

The occurrence of the meandering motion depends on the relationship between the Lagrangian time scale of the turbulent field and the diffusion time of the plume, and can be observed when the diffusion time is shorter than the Lagrangian time scale.

Lecture notes

The Lagrangian time scale consists of the Eulerian length scale L E corresponding to the streamwise integral length scale and the streamwise turbulent intensity. The white areas in Figure 6 stand for the high concentration spikes, while the black areas indicate zero or low concentration areas. Therefore, Figure 6 shows the existence of the high concentration spikes in the diffusing plume. Note here that Figure 6 of the binarized images are obtained from Figure 5 of the original raw PLIF images by applying the above-mentioned sampling technique.

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Quasihomogeneous Distributions

Comparing the corresponding original and binarized images, it can be seen that high concentration spikes are distinguished with clear outlines by extraction. Moreover, the high concentration spikes are scattered randomly in Figure 6. From Figure 5 and Figure 6 , it can be seen that a large number of high concentration spikes exist even at the edge of the plume where the concentration is very low.

The reason is that such a low-concentration edge area has a large concentration gradient, so the spikes can be recognized. Actually, through extraction by the conditional sampling technique, the low-concentration edge area of the plume is also taken seriously, which is more reasonable than analyzing only based on the plume concentration. We study solutions to partial differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter transformation group such that all eigenvalues of the infinitesimal matrix are positive.

The infinitesimal matrix may contain a nilpotent part.

1st Edition

In the asymptotic scale of regularly varying functions, we find conditions under which such differential equations have asymptotically homogeneous solutions in the critical case. Unable to display preview. Download preview PDF. Skip to main content.

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