Tomographic PIV Measurement of Turbulence Energy Budget Equation Terms in a Square Shaped Stirred Flow Mixer

Transcription

1 Tomographic PIV Measurement of Turbulence Energy Budget Equation Terms in a Square Shaped Stirred Flow Mixer Chandra Shekhar1,*, Kazunao Takahashi2, Takuya Matsunaga3, Koichi Nishino2 1: Heat and Fluid Dynamics Laboratory, IHI Corporation, Yokohama, Japan 2: Department of Mechanical Engineering, Yokohama National University, Yokohama, Japan 3: Department of Systems Innovation, The University of Tokyo, Tokyo, Japan * correspondent author: chandraiitk@yahoo.co.in Abstract Tomographic PIV measurements are carried out inside a square cylinder, which is filled with water and agitated by a commercially available, pitched, three-blade, axial flow impeller. The impeller rotates at the constant angular speed of 150 RPM. The Reynolds number based on the tip velocity of the impeller blades and its diameter is equal to The measurements are performed at four different angular locations, inside vertical volume sheets of 7 mm thickness, in order to understand flow behavior inside the whole 120 o angular space between any two of the symmetric impeller blades. The results show that the blade tips generate high-turbulence vortices, which are convected down by the axial flow motion. The rates of the turbulence energy production, its convection, and its transports by the processes of turbulence diffusion and viscous diffusion are also investigated. The results reveal that the turbulence production rate and the rate of energy transport due to the turbulence diffusion process, both are negative inside the tip vortices, as well as underneath the impeller blades; but positive in the regions surrounding the vortcies and above the impeller blades. The rate of energy convection, on the other hand, is found to positive in these regions, with much larger magnitudes. Furthermore, when the turbulent fluid produced at inner sections of the rotating blades is convected down by the bulk-mean flow motion, it forms a highly turbulent, narrow, tail-looking structure, which extends towards the impeller's axis of rotation. 1. Introduction Turbulent flow mixers are widely used in various practical applications, starting from home appliances to medical applications and in industries. In general, mixers can be divided into passive and active mixers. Passive mixers usually consist of a static, rigid unit, which is inserted into a flow to enhance mixing in the wake region. In active mixers, on the other hand, the flow is agitated by power-driven units, such as, impellers, which impart momentum to the flow in order to enhance mixing. In general, impellers used to agitate a flow can be divided into two types: radial-flow type and axialflow type. Radial flow type impellers usually consist of vertical paddles which rotate about a given axis to enhance mixing. These impellers are known to provide a high level of mixing at moderate rotation speeds. One of the widely researched radial flow type mixers consists of Ruston turbine, perhaps due to its simplistic design. Both numerical and experimental techniques are used. Wu and Petterson (1989) used LDV technique to measure turbulence statistics and energy spectra inside a baffled tank agitated by a Ruston turbine. After carrying out energy balance around the impeller, they estimated that about 60% of the energy that the impeller transmitted into the tank is dissipated in the region surrounding the impeller. Costes and Cauderc (1988) also studied similar flows using LDV technique. First, they carried out measurement in an axisymmetric plane lying right in-between two baffle plates, and then in another axisymmetric plane that contains a baffle plate. They found that when the mean velocity components and the velocity fluctuations are non-dimensionalized with the tip velocity of the rotating blades, the resulting distribution profiles become independent of the rotation speed of the impeller, but depend on size of the flow domain. Ranade et al (2001) investigated trailing vortices of a Ruston turbine, both numerically and by PIV experiments. They compared the two results in order to assess the accuracy with which the numerical technique can predict the flow behavior. Schafer et al (1997) found that the turbulence length scale in a baffled mixer is about same as the height of the rotating blades of the Ruston turbine. Sharp and Adrian (2001) studied turbulence structures near tip of the rotating blades by carrying out two-dimensional PIV measurements in vertical planes. They found the turbulence being highly anisotropic at the spatial scales they could resolve. Moreover, they also used their experimental results to assess accuracy of various numerical methods. Ducci and Yianneskis -1-

2 NOMENCLATURE D : Diameter of the impeller; in the present study, it is 90 mm. Re : Reynolds number (defined as ωd2/2γ), t : Time X : Horizontal axis in the rightward direction, with the origin lying on the tip of the rotating shaft. Y : Vertical axis in the upward direction, with the origin lying on the tip of the rotating shaft. Z : Direction perpendicular to the X-Y plane, considering a right-handed Cartesian coordinate system. U : X-component of the instantaneous flow velocity. u : Turbulence fluctuation in the U component of the velocity; u U U. V : Y-component of the instantaneous flow velocity. v : Turbulence fluctuation in the V component of the velocity; v V V. V L : Instantaneous velocity vector; VL V L = (UU + VV + WW)0.5. W : Z-component of the instantaneous flow velocity. w : Turbulence fluctuation in the W component of the velocity; w W W. GREEK ALPHABETS γ : Kinematic viscosity of the working fluid. φ : Phase angle (angular location of the mid-plane of the measured axisymmetric volume sheet). ω : Impeller's rotation speed in clockwise direction, 150 RPM (15.7 rad/s). ABBREVIATIONS PIV : Particle Image Velocimetry LDV : Laser-Doppler Velocimetry RMS : Root mean square. RPM : Revolutions per minute. RSS : Reynolds shear stress; e.g. RSSuv uv etc. TKE : Turbulence kinetic energy (uu + vv + ww)/2. CONVENTIONS {a} : A physical quantity, a, in its dimensional form. a : Mean value of a physical quantity, a. arms : RMS value of a physical quantity, a. (2005) tried to measure velocity fluctuation derivatives of Re = flows directly, by two-point LDA experimental technique, in order to calculate the turbulence dissipation rate. In this process, they also assumed the local turbulence being isotropic. When they non-dimensionalized the obtained dissipation rate with ω3d2, they found that the resulting quantity remains approximately independent of the angular position, whereas the velocity gradients themselves vary greatly. It should be noted that successful direct measurement of turbulence dissipation rate and turbulence shear stress components has been a matter of challenge till date, because it requires knowledge of all the velocity fluctuation gradients, with spatial resolution comparable to the Kolmogorov length scale. Baldi and Yianneskis (2004) could resolve spatial scales as small as 0.1 mm in their PIV study of a mixer where the Reynolds number varied in the range They found that the normalized dissipation rate in the flow stream is in the range When they compared their velocity gradients with previously reported results obtained using indirect methods, some of the reported results matched reasonably well, whereas some results did not. Yoon et al (2001) tried to estimate the dissipation rate by means of a combined experimental and computational approach. They carried out two-dimensional phase-locked PIV measurements in six axisymmatric planes, as well as in an azimuthal plane that -2-

3 approximately touches the tip of a rotating blade. Based on the experimental results, they developed some simplistic theoretical models, which they later used in their Reynolds-averaged Navier-Stokes simulations as boundary conditions. The simulation results, however, quantitatively differed from the original experimental results. Despite continued popularity of Ruston turbines, some researchers investigated axial-flow mixers (including the present study), because axial-flow mixers leave smaller shear stress footprints in the flow, as well they consume less energy compared to radial-flow type impellers. Such mixers utilize pitches blades, which yield high axial discharge rates; but, in general, they induce all the three velocity components in the flow. Sheng et al (1998) carried out two-dimensional PIV measurements in an axisymmetric plane of a round baffled tank, in which the flow was agitated by a four-blade pitched impeller. They compared the obtained mean velocity and some simple turbulence statistics with the numerical simulations that they carried out under similar conditions and after setting the boundary conditions based on the experimental results. They found that the computed mean velocity components match well with the experimental results, whereas the turbulence statistics match only in a qualitative sense. Bakker et al (1996) also used two-dimensional PIV results as boundary conditions in their numerical simulations, and found that the results well predict the flow behavior when it is laminar. The same was true for the mean velocity components of turbulent flows, but the numerical results fell short of predicting turbulence statistics. Khopkar et al (2003) and Ge et al (2014) also took a similar approach with more or less same outcome. All these studies suggest that flows in turbulent flow mixers are fairly complex and cannot be predicted by popular CFD models, such as, Reynolds-averaged Navier-Stokes model or k-ε model. Ranade et al (2002), however, reported reasonably matching results with experiments, both the bulk flow region and in the near-blade region, when they adopted computational snapshot approach (Ranade and Dommeti, 1996) in their numerical simulations. In fact, they went on to suggest that accuracy of the computational results obtained by this computational approach might be adequate for industrial application designing. Khan et al (2006) carried out stereo PIV measurements in axisymmetric planes, and found that the flow away from the blades are approximately isotropic, indicating that a simpler, two-dimensional PIV experiments can be adequate to estimate simple turbulence statistics. They further observed that about 44% of the energy transmitted by the impeller to the flow dissipates in the region surrounding the impeller. Delafosse et al (2011) carried out two-dimensional PIV measurements in vertical, horizontal, and tangential planes of a stirred tank, and found that the flow in the impeller discharge region is significantly anisotropic. They proposed a new expression to estimate the turbulence dissipation rate in such anisotropic flows based on two-dimensional measurement data. They also analyzed effects of the spatial resolution with which the flow field could be resolved in their experiments, and found that estimated turbulence dissipation rate triples when the spatial resolution is doubled. When they estimated the dissipation rate based on the highest spatial resolution they could achieve, the maximum dissipation rate turned out to be as much as 50 times larger than the bulk-mean dissipation rate with the same resolution. Shekhar et al (2012) carried out phase-locked, stereo PIV measurements inside a baffled cylindrical tank with water as the working fluid. They used a commercially available, pitched-blade impeller, called HR-100 (Satake Chemical Equipment Limited, Japan), as the agitator. They obtained the mean velocity, the turbulence kinetic energy, and the Reynolds shear stress components in six axisymmetric planes, which were separated with each-other by the constant angle of 20o, so that the whole 120o angular space between any two of the impeller-blades could be covered. They observed the tip-generate vortex in which the turbulence level were very high. The up-to-down motion in the impeller discharge region convected the tip vortices downward, along with the turbulence generated at the inner sections of the blades. The turbulence level inside the vortices gradually decreased in the process of the convection, whereas the vortex size increased. The vortices formed three backward-bending helical arcs inside the flow domain, similar to those found in wind turbines. A schematic diagram showing these arcs are presented in Fig.1. Note that although these arcs persist until long distances in the wake of wind turbines, these diffused fast in the studied case because the working medium is water whose viscosity is much larger than air. Later, Shekhar et al (2013) carried out phase-locked stereo PIV measurements in three horizontal planes inside the same flow geometry. The measurement planes lied below the impeller, in the discharge region. They combined the obtained results with the previously measured vertical plane results (Shekhar et al, 2012), which enabled them to estimate all the spatial derivatives of the velocity along the 18 radial lines where the vertical and the horizontal measurement planes intersected. From these derivatives, they estimated the production, the convection, and the viscous diffusion terms of the turbulence kinetic energy budget equation along the intersection lines, followed by estimation of the turbulence dissipation term as the residual of the energy budget equation, while simultaneously neglecting -3-

4 150RPM Impeller blades arc -3 arc -2 arc -1 Vortices Helic al v o r path tex Fig. 1 An isometric, instantaneous, schematic view of the tip-generated vortices produced by rotating impeller blades. the pressure and the turbulence transport terms. The results showed that the mean velocity, the turbulence kinetic energy, and the turbulence dissipation rate are largest in the immediate wake of an impeller blade, but they significantly reduce by the time the succeeding blade approaches. This distribution pattern gradually changed with increasing downward distance from the blades, as expected. It is evident from the above literature survey that although researches have taken various approaches to understand flow behavior inside flow mixers, it remains far from being fully understood, primarily because of the difficulties associated with (1) simultaneous measurements of the velocity gradients in all the three directions and (2) flow measurements with spatial resolutions sufficient to resolve spatial scales comparable to the Kolmogorov length scale of the flow (which is sometimes as small as a few micrometers in a practically used mixers). Shekhar et al (2014) tried to measure all the velocity gradients inside a squareshaped flow mixer through phase-locked tomographic PIV experiments. Although they used a flow tank different from in their previous studies, the impeller used to agitate the flow was the same (which is the commercially available HR-100 impeller). They carried out the measurements at four different angular locations and estimated all the nine components of the turbulence shear stress tensor with the spatial resolution that they could achieve in the experiments. The present study is an extension of the same study, and a part of our comprehensive study that aims to understand flow characteristics inside baffled mixers with an axial-flow type impeller as the agitator. It worth mentioning that we used the square cylinder in the present study because it allows the flow field to be visualized from outside without any nonlinear distortion (which is essential for a successful tomographic PIV measurement), while providing a similar order of mixing as in baffled circular flow tanks (Myers et al, 2002). Here, the asymmetry induced by the four corners of the square tank plays a role similar to baffle plates present inside circular flow tanks. Nilpawar et al (2006) experimentally showed that when viscosity of the working fluid in a mixer is large, magnitude of a flow property at a given spatial location varies with the fundamental frequency smaller than the blade-passing frequency of the rotating impeller. However, the two frequencies match in the case where low-viscosity fluid like water is used. It justifies that statistical behavior of our flow at a given spatial location can be understood by means of phase-locked measurements. Prime objective of this part of our tomographic PIV study is to understand energetics of the flow, by analyzing the production, the convection, the turbulence diffusion, and the viscous diffusion terms of the turbulence kinetic energy budget equation, which are the velocity-based terms that can be estimated without calculating instantaneous velocity-fluctuation derivatives, and therefore does not require a spatial resolution comparable to the Kolmogorov scale. We would describe the flow geometry and the experimental methodology, in sections 2 and 3, respectively. In section 4, we would validate accuracy of our tomographic PIV measurements by comparing mean velocity and some basic turbulence statistics that we obtained from stereo PIV measurements carried out under same experimental conditions. Thereafter, in the same section, we would discuss general flow behavior and its energetics by presenting the measured energy budget equation terms. Finally, we would draw conclusions in section

5 (a) (b) Fig. 2 (a) Schematic diagrams of the flow domain, with the spatial dimensions non-dimensionalized with respect to the impeller diameter. It also shows the camera positions. (b) The pitched impeller, HR-100 (manufactured by Satake Chemical Equipment Limited, Japan), which is used in the present study as the agitator. The figure also shows schematic diagrams of the top views of the impeller at different phase angles. 1.1 Non-dimensionalization All physical quantities are presented in non-dimensional forms. The spatial dimensions, the mean velocity and the velocity fluctuation components, and the turbulence kinetic energy are non-dimensionalized with respect to the impeller diameter (D), the blade's tip velocity (ωd/2), and square of the blade's tip velocity (ω2d2/4), respectively. Similarly, the budget equation terms are normalized with respect to ω3d2/8. Note that dimensional form of a physical quantity is represented by putting it inside curly braces, like { }. 2. System details The flow geometry consists of a square-shaped cylindrical tank filled with water, with the rotating HR100 impeller mounted on a thin, round shaft and placed along the central axis of the tank. Schematic diagrams of the flow geometry are shown in Fig.2(a), which also contain the right-handed Cartesian coordinate system used in the present study. The impeller diameter is 90mm, and its isometric view is shown in Fig.2(b). It is driven in the clockwise direction (when viewed from the top), with the angular speed of 150 RPM (15.7 rad/s), by an electric motor. All the measurements are performed at the water temperature of 19oC. The corresponding Reynolds number is In order to minimize bubble formation in the flow, the water is deaerated before filling into the flow tank. It is done by first heating it until 70 oc, and then letting it naturally cool down until 19 oc, while manually stirring it on hourly basis. We carried out the measurements in 7 mm thick vertical volume sheets, at four different angular locations, as shown in the top view of the Fig.2(a). The mid-planes of the volume sheets pass through the central axis of the flow tank, such that any two of the consecutive mid-planes form 30 o angle with eachother. These measurements help us understand the overall flow behavior in the whole 120 o angular space -5-

6 between any two of the impeller blades. We refer the angular locations of these mid-planes as phase angles (φ), where the location φ = 0o plane lies approximately in-between any two of the impeller blades (the exact location lies at the center of the first of the two radial, M4 screw holes that are present on the hub peripheral of the clockwise-rotating HR-100 impeller). In addition, we also carried out stereo PIV measurements at φ = 0o after reducing the thickness of the laser light sheet to 2 mm. 3. Experimental setup The overall experimental setup consists of (1) the flow domain (with the impeller), (2) spherical Nylon particles of average diameter 10 μm, in order to seed the flow, (3) four Ethernet-connected CCD cameras, in order to acquire particle images, (4) a double-pulse Nd:YAG laser with 30 mj/pulse of the maximum energy output, (5) an optical system that converts the round laser beam into a uniform 7 mm thick vertical symmetrical laser light sheet whose mid plane passes through the central axis of the square cylinder (6) a digital time-delay generator, (7) a precise, laser-based, phase-synchronization mechanism that allowed us to acquire the particle images precisely at the desired phase angles, and (8) a computer to synchronize the electronic equipments and to store the acquired particle images. All the experimental apparatus are fixed in space, except the impeller (which rotates at 150 RPM). In order to convert the round laser beam into the 7 mm thick light sheets, the beam is first passed through a self-made beam expander, which increases the beam diameter from 2.5 mm to 12.5 mm. The expanded beam, which has the Gaussian energy distribution across its cross-section, is then passed through a square-shaped thin slit of side-length 7 mm, with the axis of the laser beam aligned along the axis of the square. The slit cuts the low-energy portions of the round beam that exists at the peripheral and yields a parallel, high-energy laser beam of square cross-section. When this square-shaped beam is passed through a cylindrical lens, it results into a diverging vertical light-sheet of the uniform thickness 7 mm, which is finally used to illuminate the flow. The laser-based phase-synchronization system consists of a small, continuous laser, a small mirror that reflects the continuous laser light, and a laser light detector embedded in the laser emitting device. The mirror is fixed at some distance from the emitting device, perpendicularly, so that the laser light is reflected back to the emitting device where it is detected. When a thin metal bar, which is fixed with the rotating shaft on which the impeller is mounted, passed through the space between the laser device and the mirror, it interrupts the light and the laser device transmit an electric signal to the digital delay generator, which recognized it as the input signal. According to its user-controllable settings, the delay generator sends further signals to the Nd:YAG laser and the CCD cameras after some time interval. By manually controlling this time delay, the particle images are captured at precise angular locations of the rotating blades. The four CCD cameras used to obtain the particle images are also schematically shown in Fig.2(a). The cameras are arranged at the four ends of the plus shape (+) and looking at the rotating impeller through a same vertical face of the flow domain, with the normal axis of the plus shape parallel to the Z axis, but at some offset from it, so that view of the tip generated vortex falls in the middle of the acquired particle images. The angle that the line of sights of the viewing cameras make with the normal of the viewing face of the flow domain is about 25o. The Scheimpflug condition, in order to sharply visualize the illuminated volume, is achieved by using tilt-shift lenses and lensbaby. The experimental system is kept the same even in the stereo PIV experiment, except reducing the thickness of the laser light sheet to 2 mm. While analyzing the particle images, we used the images acquired by only the left and the right cameras (the cameras 1 and 2 in Fig.2(a)), while discarding the images acquired by the other two cameras. At each phase angle, a total of pairs of the particle images of size pixels are acquired, which would yield 9000 three-dimensional instantaneous velocity fields after the tomographic PIV analysis. The acquired particle images at a random time instant are shown in Fig.3. Such particle images are analyzed using a self-developed GPU program written in the CUDA programming language, which first reconstructs a three-dimensional brightness intensity field using the iterative MART algorithm (Herman and Lent, 1976), followed by calculation of the three-dimensional cross correlation coefficients after dividing the reconstructed volumes into smaller interrogation volumes. In total, pairs of the full particle images are used to reconstruct 3000 pairs of the intensity field of size voxel3, which are then divided into the interrogation volumes of size voxel3 to obtain 3000 instantaneous velocity fields. -6-

7 Fig. 3 Sample particle images acquired by the four cameras at φ = 0o (from left to right, respectively). Fig. 4 Images of the calibration plate acquired by the four cameras (from left to right, respectively). In the physical space, the size of the interrogation volume translates to mm3. The overlap ratio of two consecutive interrogation volumes is kept equal to 50%, which yields the instantaneous velocity vectors separated by 1 mm. Therefore, from one pair of the reconstructed volume, the three-component velocity field could be obtained in a total of five different vertical planes, which are {Z} = 2 mm, 1 mm, 0, 1 mm, and 2 mm. Our analysis program takes about 18 minutes of time to yield one three-dimensional instantaneous velocity field, on a GeFORCE GTX TITAN GPU (NVIDIA Corporation). The mean velocity field and some simple turbulence statistics, such as, turbulence kinetic energy and Reynolds shear stress components, are calculated based on these 3000 instantaneous velocity fields. Central region of the remaining pairs of the particle images are first cropped to extract the same number of smaller particle images, each of which contains the high turbulence region surrounding the tip vortices. Depending on spatial position of the vortex at a phase angle of measurement, the exact position and size of the extracted particle images differ. All the extracted images are subjected to the process of tomographic reconstruction, followed by the calculation of the cross-correlation coefficient, while keeping the interrogation volume size and the overlap ratio same as those during the analysis of the 3000 full-sized images. The obtained 6000 instantaneous velocity vector fields are used to calculate the energy budget equation terms in the vortex region. Note that a reliable calculation of the budget equation terms in the vortex-containing region requires a larger number of instantaneous velocity fields than in rest of the region, because the turbulence level in the former region is much higher than in the latter. Here, the particle images are cropped merely to reduce the analysis time. The interrogation window size in the stereo PIV analysis is equal to 2 2 mm2, with the same 50% of the overlap ratio. Moreover, since the thickness of the illuminated volume sheet is 2 mm, the obtained results can be reliably compared with the corresponding tomographic PIV results. 3.1 Camera calibration The calibration is performed with a dark carbon plate having white circular spots of diameter equal to 2 mm printed on it. A circular spot of diameter 5 mm is also printed at the center of the plate, for reference purposes. The spatial accuracy of these spots is better than 10 μm. The spots are centered on the equidistant square grids separated by 5 mm. Images of the calibration plate are acquired after moving it to 7 different Z locations, from {Z} = 3 mm to {Z} = +3 mm at the constant interval of 1 mm. The average calibration error is less than 0.5 pixels, which is within the range recommended by Scarano (2013). The calibration images, as acquired by the four cameras at the location Z = 0, are shown in Fig.4. Here, we would like to emphasize on the importance of the plus arrangement of the cameras, which yielded nearly-circular images of the white calibration spots and led to the aforementioned small calibration error. When we tried to place the cameras at the ends of the cross shape ( ), the images of the calibration spots became elliptical, which, in turn, substantially increased the calibration error. -7-

8 Fig. 5 Mean velocity vector field, with the color contours showing the velocity magnitude. The mean velocity is obtained from 3000 instantaneous nominal velocity fields. The thickness of the volume sheet is equal to in the non-dimensionalized coordinate. (a) (b) Fig. 6 (a) Color contours of TKE. The thickness of the volume sheet is equal to in the non-dimensional coordinate. (b) Plane-wise view of the same, where the Z direction is expanded out of proportion in order to visualize the en-volume distributions properly. The TKE is calculated from 3000 instantaneous nominal velocity fields. For the stereo PIV experiment, we used the same set of calibration, but only those which are acquired at the locations {Z} = 1 mm, 0, and +1 mm. 3.2 Validation of the image-analysis program The accuracy of the program is validated using artificial particle images of the PPP (particle per pixel) value The images are generated after considering the three flow velocity components varying as follows: U = U 0 sin ( 4 π X / L1 ); V = U 0 sin ( 3π Y / L 2 ) ; and W = U 0 (sin ( 2 π Z / L3 ) + cos ( 2 π Z / L 3) ) ; where {U0} = 100 mm/s, {L1} = {L3} = 100 mm, and {L2} = 75 mm. Note that these variations do not satisfy the three-dimensional continuity equation, but can be used to correctly verify the accuracy of the analysis program. The U, V, and W components obtained after analyzing the artificial particle images differed with the -8-

9 same calculated using the above equations by less than 5% of U Filtering of spurious velocity vectors Due to some practically unavoidable imperfections in our PIV measurements, the obtained velocity vectors at some spatial points do not reflect the true value of the local flow velocity. We used a statistical approach propose by Thompson (1935) to discard such spurious vectors. In essence, this method recognizes an instantaneous velocity vector at a given point in space as an spurious vector if its velocity fluctuation component exceeds a pre-defined limit. In the present study, we set this limit to three times the standard deviation of the flow velocity at the same spatial point. 4 Results and discussion The three-dimensional mean velocity vector field obtained from the 3000 full-sized instantaneous velocity fields at φ = 0o is plotted in Fig.5. The thickness of the volume sheet is in the non-dimensional coordinate. The same plot also contains color contours of the velocity magnitude in the background. Since a rotating blade has just passed, the plot clearly shows the tip vortex with large velocity magnitudes. Large downward mean flow velocity component is also evident in the X < 0.4 region, as typical with axial-flow type impellers. In the X > 0.6 region, the velocity is small and directed upward. Although not clearly apparent in the plot, the flow also possesses significant radial and peripheral velocity components. Turbulence kinetic energy distribution at the same phase angle is presented in Fig.6(a). Its plane-wise view is also presented, in Fig.6(b), in order to show the inner distributions. Note that the spatial Z direction in the plane-wise view is expanded out of proportion, for clarity purposes. The turbulence kinetic energy plots demonstrate very high level of turbulence inside the tip vortex, along with a second, relatively diffused high turbulence region below the tip vortex. This region corresponds to the tip vortex that was generated by the preceding blade and convected downward by the bulk-mean flow motion. The bulk-mean flow motion also convects the turbulence generated at the inner sections of the blade, which forms a narrow, high turbulence region below the impeller, which resembles a tail and extends towards the impeller's axis of rotation. 4.1 Validation of tomographic PIV results Mean velocity and simple turbulence statistics, as obtained from the tomographic PIV experiment and the stereo PIV experiment, are compared in the Z = 0 plane at φ = 0o. In Fig.7, the mean velocity components W, U, and V are compared. Similarly, in Figs.8 and 9, the turbulence intensity components (namely, wrms, urms, and vrms) and the three Reynolds shear stress components (namely, RSSuv, RSSvw, and RSSwu) are compared, respectively. It is evident from these figures that the tomographic PIV results match reasonably well with the corresponding stereo PIV results, except in the vicinity of the center of the tip vortex where the W magnitude, as obtained from the tomographic PIV experiment, is smaller than the same resulting from the stereo PIV measurement. This discrepancy is likely due to large out-of-plane flow motion in the stereo PIV measurement. Apart from validating the accuracy, the mean velocity and turbulence statistics plots in Figs.7~9 also provide significant insight into the flow behavior itself. Fig.7 shows that the flow velocity remains large even inside the tip vortex generated by the preceding blade, despite being convected downward by the mean flow. Moreover, the figure also confirms that the downward mean velocity component generated by inner sections of the blades is very large. Figure 8, apart from containing the tail-looking high-turbulence feature, shows that all the three turbulence intensity components are comparable to each other, as expected in regions far from a solid surface. Similarly, Fig.9 shows that all the three Reynolds shear stress components are comparable in magnitude, but have different signs in different regions. Furthermore, these components are an order of magnitude smaller than the turbulence kinetic energy (see Figs.6(a) and (b)). It is evident from Figs.8 and 9 that the turbulence activity remains confined below the area swept by the rotating blades, because the bulk-mean flow motion convects all the turbulence generated at the blade-water interface in the downward direction. Since the turbulence gradually diffuse in the convection process, magnitudes of the turbulence statistics gradually reduce with the downward distance. Here, we would like to mention that the presented turbulence statistics that involve the velocity fluctuation component w differ greatly from the same measured inside a baffled tank in one of our previous studies (Shekhar et al, 2012) that used stereo PIV technique. However, since the flow field in both the flow -9-

10 (a) Stereo PIV Tomographic PIV (b) Stereo PIV Tomographic PIV (c) Stereo PIV Tomographic PIV Fig. 7 Comparison of the three mean velocity components obtained from the tomographic PIV experiment with the same as obtained from the stereo PIV experiment, at φ = 0o. The comparison is made in the middle plane (Z = 0) of the measurement volume sheet. domains remain similar (Myers et al, 2002) and the same impeller is used to agitate the flow in both the measurements, the results are expected to remain comparable. The apparent differences are likely caused by some unexpected misalignment of the laser light sheet with the calibration plate in the previous stereo PIV experiment, which leads to large registration error that affects the Z component of the measured velocity

11 (a) Stereo PIV Tomographic PIV (b) Tomographic PIV Stereo PIV (c) Tomographic PIV Stereo PIV Fig. 8 Comparison of the three turbulence intensity components obtained from the tomographic PIV experiment with the same as obtained from the stereo PIV experiment, at φ = 0o. The comparison is made in the middle plane (Z = 0) of the measurement volume sheet. 4.2 Turbulence kinetic energy budget equation terms The production, the convection, the turbulence diffusion, and the viscous diffusion terms of the turbulence kinetic energy budget equation can be given as follows:

12 (a) Stereo PIV Tomographic PIV (b) Tomographic PIV Stereo PIV (c) Tomographic PIV Stereo PIV Fig. 9 Comparison of the three turbulence intensity components obtained from the tomographic PIV experiment with the same as obtained from the stereo PIV experiment, at φ = 0o. The comparison is made in the middle plane (Z = 0) of the measurement volume sheet. Production term: U U U V V V W W W uu uv uw vu vv vw wu wv ww X Y Z X Y Z X Y Z (1)

13 Fig. 10 The number of correct velocity vectors in the middle three planes of the measured volume sheet at at φ = 0o. Here, the depth direction (Z) is expanded out of proportion in order to visualize the plane-wise distributions properly. (a) (b) (c) (d) Fig. 11 (a) The production, (b) the convection, (c) the turbulence diffusion, and (d) the viscous diffusion terms of the turbulence kinetic energy budget equation in the Z = 0 plane, at φ = 0o. Convection term: (TKE ) (TKE ) (TKE ) U V W X Y Z (2)

14 (a) (b) (c) (d) Fig. 12 Rate of turbulence production in the Z = 0 plane at the four different phase angles of the measurements. Turbulence diffusion term: 1 q 2 u q2 v q 2 w X Y Z ( (3) ) where q2 u2 + v2 + w2. Viscous diffusion term: (TKE) (TKE ) (TKE ) Re X Y Z ( ) (4) The partial derivatives in the above terms are discretized by the standard, second-order accurate central difference scheme. Therefore, the budget equation terms could be calculated only in the middle three planes of the measured volume sheets, namely, in the planes {Z} = 1 mm, 0, and 1 mm. At first, the budget equation terms are discussed at φ = 0o. The number (N) of the correct velocity vectors in the middle three planes (after removal of the spurious velocity vectors) are presented in Fig.10, which shows that at least 2000 instantaneous velocity vectors are involved in the calculations of the budget equation terms. Note that the number of the correct vectors has reduced significantly in the vortex core, likely because the local velocity fluctuations are large, which leads to accordingly large number of the spurious velocity vectors. The number remains approximately the same even for the other phase angles. The four budget equation terms at Z = 0 are presented in Figs.11(a)~(d), respectively (the distributions remain almost the same even in the other two Z planes). From Fig.11(a), it is readily apparent that the turbulence production rate is negative in the core of the tip vortex; that is, the flow tends to relaminarize in this region after returning the turbulence energy back to the mean flow. In the nearby region surrounding the core, however, the production rate is positive, but significantly smaller in magnitude. Simultaneously, some positive production can be observed in the tail region also. Figure 11(b) reveals that the core region gains turbulence energy by the process of convection, whereas the surrounding region loses the energy. The figure also shows two layers of negative and positive convection rates in the tail region. Later (from Figs.13(a)), it becomes evident that when a rotating blade cuts

15 (a) (b) (c) (d) Fig. 13 Rate of turbulence energy convection in the Z = 0 plane at the four different phase angles of the measurements. through the measurement region, the energy convection terms right above and below the blades are negative and positive, respectively, indicating that the aforementioned two layers in the tail region is a direct consequence of the movement of the turbulent fluid from the near-blade region by the downward bulk-mean flow motion. Figure 11(c) shows that the energy transport due to the turbulence diffusion process is negative in the core region, whereas positive in the surrounding region. This is as expected, because the turbulence level happens to be the largest in the core region, which would, in turn, induce fluctuations in the relatively lowturbulence surrounding region. A layered tail region can be observed in this plot also. The viscous diffusion term plot in Fig.11(d) shows that it is an order of magnitude smaller than the other terms, implying that its influence on the flow turbulence is negligible The budget equation terms at the four phase angles The production term at the phase angles φ = 30o, 0o, 30o, and 60o in the Z = 0 is plotted in Figs.12(a)~(d) respectively. The plot in Fig.12(a) shows that when a blade passes through the measured volume sheet, the turbulence is produced near its tip, but the flow tends to relaminarize right below the blade. Here, it should be noted that the region below the blade is not in the immediate vicinity of the blade's surface where a strong shear layer forms (as typical with a flow close to a solid boundary) in which the turbulence production rate is likely to be positive and large in magnitude. This wall-bounded shear layer is not resolved in the present study. Figures 12(b) shows presence of the tip generated vortex after the blade has passed measurement region. The negative production rate in the core of the vortex, as well as the positive production rate in the nearby surrounding region and in the tail region, gradually reduces with time and almost vanishes by the time the next vortex cuts through the measurement region (see Figs.12(c) and (d)). The convection term at the four phase angles are plotted in Figs.13(a)~(d), respectively, in the same Z = 0 plane. The plot at φ = 30o reveals that the rate is negative right above the blade, but positive below it. After the blade has passed the measurement region, the core of the vortex gains energy by the convection process, whereas its surrounding region loses the energy, (see Fig.13(b)). The same figure also shows existence of the two-layer tail. The overall distribution pattern remains approximately the same even at the subsequent phase angles, but the magnitudes become smaller, because the turbulence tends to gradually damp down in the wake of the blade. The rate of energy transport due to the turbulence diffusion is plotted in Figs.14(a)~(d), respectively, for

16 (b) (a) (d) (c) Fig. 14 Rate of turbulence energy transport due to the turbulence diffusion in the Z = 0 plane at the four different phase angles of the measurements. the same four phase angles. These plots well resemble the production term plots (see Figs.12(a)~(d)), except in the tail region where the turbulence diffusion term plots clearly show presence of the positive and negative layers (as a direct consequence of the downward convection of the turbulent fluid from the near-blade region). All these budget equation term plots further show that magnitudes of the energy convection rate are much larger than the individual magnitudes of the turbulence production rate and the rate of energy transport due to the turbulence diffusion. Moreover, the sign of the convection term is opposite to the other two terms, suggesting that the process of convection more or less balances out the energy changes caused by the other two processes. 5 Conclusions Tomographic PIV measurements were carried out inside a square water tank with a commercially available pitched, axial-flow type impeller as the agitator, at four different angular locations. The results confirmed the presence of highly turbulent tip-generated vortices, which were convected downward by the bulk-mean flow motion. In this process, the turbulence inside the vortex gradually diffused, which increased its size and decreased the local turbulence level. The results revealed that the flow inside axial-flow mixers are utterly complex and demands special care in numerical simulations. Inside the core of the tip vortex, the turbulence production rate was found to be negative, but positive in the region surrounding the vortex core. Rate of the turbulence energy transport due to the turbulence diffusion process also exhibited a similar distribution pattern and with a comparable magnitude. The energy convection rates, on the other hand, were opposite in sign and significantly larger in magnitudes, which appeared to approximately balance out the turbulence energy changes induced by the production and the turbulence diffusion processes, combined. It was further observed that when an impeller blade cuts through the measurement region, it generates highly turbulent fluid even at its inner sections. The rate of energy production and the rate of energy transport by the turbulence diffusion process, both were found to be positive above the blade, but negative

17 below it. The signs for the energy convection rate, on the other hand, were opposite to the above two. When this near-blade fluid was transported downward by the bulk-mean flow, it formed a narrow, high-turbulence, tail-looking structure that extended towards the impeller's axis of rotation, while keeping the signs of the convection and the turbulence diffusion terms unchanged. It, in turn, constituted layers in the tail region. The results also showed that the energy transport due to the process of viscous diffusion is an order of magnitude smaller than the other three budget equation terms, and therefore it can be neglected. References Bakker A, Myers KJ, Ward RW, Lee CK (1996) The laminar and turbulent flow pattern of a pitched blade turbine. Trans IchemE 74(A): Baldi S, Yianneskis M (2004) On the quantification of energy dissipation in the impeller stream of a stirred vessel from fluctuating velocity gradient measurements. Chemical Engineering Science 59(13): Costes J, Couderc JP (1988) Study by laser Doppler anemometry of the turbulent flow induced by a Rushton turbine in a stirred tank: Influence of the size of the units I. Mean flow and turbulence. Chemical Engineering Science 43(10): Delafosse A, Collignon ML, Crine M, Toye D (2011) Estimation of the turbulent kinetic energy dissipation rate from 2D-PIV measurements in a vessel stirred by an axial Mixel TTP impeller. Chemical Engineering Science 66: Ducci A, Yianneskis M (2005) Direct determination of energy dissipation in stirred vessels with two-point LDA. AIChE Journal 51(8): Ge CY, Wang JJ, Gu XP, Feng LF (2014) CFD simulation and PIV measurement of the flow field generated by modified pitched blade turbine impellers. Chemical Engineering Research and Design 92: Herman GT, Lent A (1976) Iterative reconstruction algorithms. Computers in Biology and Medicine 6: Khan FR, Rielly CD, Brown DAR (2006) Angle-resolved stereo-piv measurements close to a downpumping pitched-blade turbine. Chemical Engineering Science 61(9): Khopkar AR, Aubin J, Xuereb C, Sauze NL, Bertrand J, Ranade VV (2003) Gas-liquid flow generated by a pitched blade turbine: PIV measurements and CFD simulations. Industrial & Engineering Chemistry Research 42(21): Myers KJ, Reeder MF, Fasano JB (2002) Optimize mixing by using the proper baffles. Chemical Engineering Progress 91: Nilpawar AM, Reynolds GK, Salman AD, Hounslow MJ (2006) Surface velocity measurement in a high shear mixer. Chemical Engineering Science 61: NVIDIA Corporation ( Ranade VV, Dommeti S (1996) Computational snapshot of flow generated by axial impellers in baffled stirred vessels. Chemical Engineering Research and Design 74: Ranade VV, Perrard M, Sauze NL, Xuereb C, Bertrand J (2001) Trailing Vortices of Rushton Turbine: PIV Measurements and CFD Simulations with Snapshot Approach. Chemical Engineering Research and Design 79(1):3 12. Ranade VV, Tayalia Y, Krishnan H (2002) CFD predictions of flow near impeller blades in baffled stirred vessels: assessment of computational snapshot approach. Chemical Engineering Communications 189(7): Satake Chemical Equipment Limited, Japan (

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