Characterization of Turbulent Properties in the EPA Baffled Flask for Dispersion Effectiveness Testing

Abstract

The baffled flask test (BFT) has been proposed by United States Environmental Protection Agency to be adopted as the official standard protocol for testing dispersant effectiveness. The mixing energy in the baffled flask is investigated in this paper. Particle image velocimetry (PIV) was used to measure the water velocity in the flask placed at an orbital shaker that was rotated at seven rotation speeds: 100, 125, 150, 160, 170, 200, and 250 rpm. Two dimensional velocity fields in large and small vertical cross sections of the flask for each rotation speed were obtained. The one-dimensional (1D) energy spectra indicates the existence of inertial subrange. The estimated average energy dissipation rates were in the range 7.65×10⁻³ to 4 W/kg for rotation speeds of Ω=100–250 rpm, of which it is larger than the one estimated by prior studies using single-point velocity measurement techniques for Ω=100 and 200 rpm. Factors such as instruments used, velocity components measured, and different analysis methods could contribute to the discrepancies in the results. The Kolmogorov scale estimated in this study for all seven rotation speeds approached the size of oil droplets observed at sea, which is 50–400 μm. The average energy dissipation rate, ε and Kolmogorov microscale, η, in the flasks were correlated to the rotation speed, and it was found that $\overline{\epsilon }=9.0\times {10}^{-5}$ Exp (0.043Ω) with R² = 0.97 and $\overline{\eta }=1,463$ Exp (−0.015Ω) with R² = 0.98.

Introduction

Oil spills, because of their potential large sizes, often cause both short-term and long-term adverse effects on the environment and wildlife. Once a spill event occur, various countermeasures are considered including the use of chemical dispersants to cause the oil to spread in the water column (labelled oil dispersion), which would prevent the slick from reaching the shorelines where it could cause the most damage (Fingas 2011; NRC 2003, 2005). Oil dispersant is a mixture of surfactants and solvents to help an oil slick break into small droplets in a process known as dispersion (Kaku et al. 2006a, b, 2010). The term dispersion used here is from the oil literature and is different from the spreading of chemicals attributable to the spatial variation of velocity. The mixing energy imposed by waves, especially breaking waves will enhance the dispersion process (Delvigne 1993).

Field and laboratory experiments have been conducted to study the dispersant effectiveness (DE) under different turbulence intensity of the sea state. Because of the complex environment in the sea, field studies often introduce experimental uncertainties in the measurements and are also very difficult and expensive to implement. Alternatively, the standard laboratory flask test has been widely used in oil spill research to study dispersant effectiveness on oil (Fingas 2000, 1991). To better understand the representative of these flask tests to various sea states, it is necessary to fully understand the flow dynamics and turbulence structures in the flask tests.

Prior investigations on this issue were conducted by Kaku et al. (2006a, b). They used a hot wire anemometer (HWA) to characterize mixing dynamics in the swirling flask (SF) and the baffled flask (BF). They concluded that the turbulence in the baffled flask closely resembles the turbulence occurring in the top few centimeters of a breaking wave, hence, the baffled flask is more representative of mixing at sea attributable to breaking waves than the swirling flask (Kaku et al. 2006a, b). Based on their studies, the baffled flask test was proposed to be adopted as the official standard protocol replacing the swirling flask test by the U.S. Environmental Protection Agency (EPA) (Venosa and Holder 2013). However, the instrument they used for the measurement of the velocity, HWA (adopts single-point velocity measurement technique), can only measure time series of point velocities and cannot provide a whole field (or a cross section) spatial velocity variations in the flasks. This method gives the temporal spectrum and needs to rely on Taylor’s frozen turbulence hypothesis (Hinze 1975) to convert temporal series into spatial data. The hypothesis states that turbulence is advected with space without change. This hypothesis is achievable in unidirectional flows in which the turbulence velocity is less than 10 to 20% of the advection velocity. However, in a rotating fluid, there might be difficulty in using the hypothesis; also, this method could also limit the estimation approaches of energy dissipation. For this reason, in this paper, we are using the PIV to measure turbulence structures in the BF.

The particle image velocimetry (PIV) has been widely used for the measurement of spatial phenomena in flows, which uses an optical imaging technique to obtain simultaneous measurement of two components of the velocity at many points (normally more than thousands of points) in a flow field. For example, Liu et al. (1991) used a high resolution PIV to measure the turbulent velocity field for fully developed flow in an enclosed channel and the statistical properties of the velocity were in good agreement with laser-Doppler velocimeter (LDV) measurements and with direct numerical simulations. Hyun et al. (2003) evaluated the performance of PIV in the open-channel flow and concluded that PIV is fast reaching a stage in which it can be applied with a level of confidence similar to LDV. Cheng et al. (1997) was the first to use PIV technique to study flow related to water treatment processes in a stirred tank. Since then, studies of mixing energy in stirred tank or reactors using PIV data have been well tested and evaluated (e.g., Delafosse et al. 2011; Kilander and Rasmuson 2005; Micheletti et al. 2004; Sheng et al. 2000).

In this study, the particle-image velocimetry (PIV) was used for the measurements of velocity fields (velocities at thousands of points within a chosen cross section) in the baffled flask used for dispersant effectiveness testing. Most of the stirred vessels investigated in previous studies are a cylindrical or square tank (has an amount of more than 1 L liquids) with an impeller inside. This is the first time to use PIV techniques to evaluate the flow dynamics and turbulent structures in a laboratory flask with a volume of 200 mL. Seven rotation speeds were measured during the experiments, which were Ω = 100, 125, 150, 160, 170, 200, and 250 rpm. The measured spatial velocity data were then used to evaluate the overall energy structures in the baffled flask.

Evaluation of Energy Dissipation

The baffled flask has four baffles in it, resulting in an irregular geometry in the flask. Such geometry forms an over-and-under motion of water flow which could represent more characteristic of the type of mixing that occurs from breaking waves at sea (Kaku et al. 2006a, b; Nelkin 1994). To study the turbulent structure in the flask, there are several characteristic parameters of interest in the turbulent mixing process. The energy dissipation rate, ε, can be used as an appropriate scaling parameter to characterize the intensity of the mixing energy (Delvigne and Sweeney 1988). The integral length based on the velocity field characterizes the size of eddies containing most of the turbulence mixing energy (Tennekes and Lumley 1972). The Kolmogorov microscale provides an estimate for the smallest eddy that can exist in the turbulent flow.

For calculation of these characteristic parameters from PIV data, a turbulent velocity field must first be obtained. Considering a two-dimensional (2D) velocity field obtained from PIV, the velocity at each location can be expressed as

$$u_i(x,z)=\overline{U}(x,z)+u_i^{\prime }(x,z)i=1,2,3,\dots ,N$$ (1)

where the index i represents different realizations (replicates from PIV measurements); $\overline{U}$ = average of N realizations at each location (m/s); $u_i^{\prime }$ = turbulent component of velocity (m/s); and x, z = coordinates of each point in the measured cross section (m).

The internal dynamics of turbulence transfer energy from large scale to small scale. This energy transfer proceeds at a rate dictated by the energy of the large eddies (of order u²) and their time scale (of order l/u), where l is characteristic length scale and u is velocity (Tennekes and Lumley 1972). Thus, the evaluation of energy dissipation rate ε could be estimated using the dimensional argument analysis (Cheng et al. 1997; Kresta and Wood 1993; Tennekes and Lumley 1972), which is based on the relationship between dissipation rate and integral length scale. The energy dissipation rate can be written as

$${\epsilon }_j=A\frac{(u_{rmsj}^{\prime }{)}^3}{{\tau }_l}$$ (2)

where $u_{rmsj}^{\prime }$ = root mean square of turbulent component of velocity at each measuring location j (m/s); A = constant of order unity; and τ_l = integral length scale of turbulence (m) which can be estimated as (Tennekes and Lumley 1972)

$${\tau }_l={\int }_0^{\infty }{R}_{l}dl$$ (3)

where R_l = spatial autocorrelation function as a function of distance class l. The one-dimensional (1D) autocorrelation function is obtained as

$$R_l=\frac{\overline{u_k^{\prime }(x,z)u_k^{\prime }(x+\tau ,z)}}{(u_k^{\prime }{)}^2}\text{horizontal direction}k=x,z$$ (4a)

$$R_l=\frac{\overline{u_k^{\prime }(x,z)u_k^{\prime }(x,z+\tau )}}{(u_k^{\prime }{)}^2}\text{vertical direction}k=x,z$$ (4b)

where τ = distance lag. In the calculation of Eq. (3), the point of first zero crossing is used to replace the upper limit of the integration. The Kolmogorov microscale is estimated based on dimensional arguments (Tennekes and Lumley 1972) as

$$\eta =(\frac{v^3}{\epsilon }{)}^{1∕4}$$ (5)

The velocity gradient is estimated as

$$G=(\frac{{}^{\epsilon }}{{}_v}{)}^{1∕2}$$ (6)

Experiment Setup

A particle image velocimetry (PIV) system was employed to measure the velocity field in the baffled flask at different rotation speeds. A 200 mL baffled trypsinizing flask was used during the experiments. The flask contained 120 mL of tap water and was mounted in the center of a transparent cubic container which was filled with tap water to eliminate optical distortion. The cubic container with the flask inside was then mounted on the center of an orbital shaker (MaxQ 2000, Thermo Scientific, New Jersey). The shaking speed can be varied from approximately 40–400 rpm. An orbital diameter, d₀ = 20 mm was traced by the shaker. Seven rotation speeds were tested during the experiments: 100, 125, 150, 160, 170, 200, and 250 rpm. All the experiments were conducted under room temperature (approximately 20°C).

The Dantec PIV system was employed consisting of a Dantec Dynamics HiSense PIV/PLIF CCD camera (Model No. C4742-53-12NRB, Dantec Dynamics A/S, Denmark). A double pulsed Nd: YAG laser with 120 mJ output energy (New Wave Research model Solo 120 15 Hz, Fremont, California) was used to produce a 5-mm-thick light sheet, which illuminated a thin vertical sheet passing through the center of the baffled flask. Silver-coated hollow borosilicate glass spherical particles with a density of 1.4 g/cm³ and median diameter of 10 μm were seeded in the flask to identify the velocity field at each cycle. The seeding density is maintained as 1525 particles/interrogation cell (Sheng et al. 2000) during all the experiments. The schematic of the PIV system is shown in Fig. 1. Measurements were taken when the container (with the baffled flask in the center) reached its position furthest to the left side as shown in Fig. 1(a). The axial and radial velocity components were measured.

Fig. 1.

Schematic of PIV experimental setup to measure the water velocity in the baffled flask (BF): (a) position of the flask on the shaker; (b) illustration how the PIV is used to measure the water velocity in vertical planes within the BF

The camera we used provided a resolution of 1264 × 1008 pixels per frame, with a compact Nikon lens of AF Micro-Nikkor 60 mm f/2.8D for extreme close-up/macro shooting. Although the camera could be positioned at different distances from the flask, to have the best spatial resolution within the capacity of the camera and lenses, data reported in this paper were obtained with the camera approximately 25 cm (the minimum distance that the lens required) from the plane of the laser sheet. This provided a viewing area of 37.5 × 29.9 mm. The images were processed using the Dantec’s FlowManager software provided by Dantec Dynamics A/S. Adaptive cross-correlation technique was employed. To maintain high measurement accuracy, a compromise between spatial resolution and velocity dynamic range had to be made. Based on the suggestions from Dantec Dynamics A/S (FlowManager), to maximize the number of vectors in the PIV vector map, a 32×32 pixel interrogation areas (approximately 0.95 mm width) with a 75% overlap were chosen in this study, which gave 0.24 mm step size. Although the overlapping of the interrogation areas does not indicate an increase in the fundamental spatial resolution, it may provide some inherent correlation among the adjacent vectors (Cheng et al. 1997). A total of 19,625 (157 × 125 in the x, z plane) vectors were generated within each frame.

The time between pulses is a critical data acquisition parameter which is generally selected to suit the velocity of the flow field and the size of the interrogation area so that there is low noise in the particle correlation function (FlowManager). For a given measurement scenario, the higher the velocity the shorter the time between pulses. The recommended maximum displacement is ± 1/4 of the length of the side of the interrogation cell (FlowManager). For our experiments, the time between pulses were chosen from 100 to 900 μs for the seven rotation speeds (250 – 100 rpm) evaluated to obtain suitable particle displacement for each flow field.

Two vertical planes were measured for each rotation speed. Fig. 2 shows the positions and dimensions of the large cross section (A_large) and the small cross section (A_small). Images in Figs. 2(b and c) were captured by a second lens (Nikon 28 mm f/2.8 D AF Nikkor lens) to show a full view of the flask cross sections, in which the full large cross section has the dimension of approximately 76 × 44 mm, and the small cross section of approximately 57 × 43 mm. Both cross sections are larger than the PIV field of view (37.5 × 29.9 mm) from the experimental lens (Nikon 60 mm f/2.8D AF Micro-Nikkor lens). To obtain a full picture of the velocity field in the flask, the A_large was divided into 6 PIV fileds [Fig. 2(b)], and A_small was divided into 4 PIV fields [Fig. 2(c)]. Each PIV field of view was measured independently. This gives a total 10 individual experiments for each rotation speed. For each PIV field of view, 200 PIV realizations were obtained, which indicates that 200 instantaneous velocity fields were generated for each PIV view at each rotation speed.

Fig. 2.

Specifications of the large cross section and the small cross section of the velocity field: (a) plane view; (b and c) vertical views of the large and small cross section, respectively; a second lens (Nikon 28 mm f/2.8 D AF Nikkor lens) was used to capture a full view of the flask cross sections and images were shown in (b) and (c); The boxes in (b) and (c) indicate the actual PIV field of view for the two cross sections using the experimental lens (Nikon lens of AF Micro-Nikkor 60 mm f/2.8 D)

The PIV camera was moved to capture each of the different fields of view outlined in Fig. 2 and the camera position was calculated based on the dimension shown in [Figs. 2(b and c)] with proper overlapping distance between views. After the experiments, images of PIV fields of view were printed out and critical points, such as flask boundaries, water surface, and baffles, were identified. In conjunction with the critical points and the design position of the camera, the PIV fields of view were stitched together to present the whole flow fields in the flask. For the large cross section [Fig. 2(b)], the overlap vectors were 70 (up) and 64 (bottom) between left and middle views, and 84 (up) and 83 (bottom) between middle and right views, respectively. The overlap vectors between up and bottom views are 88. The stitching locations were chosen along the baffles. For the small cross section [Fig. 2(c)], the overlap vectors were 88 (up) and 86 (bottom) between left and right views, and seven between up and bottom views, respectively. A full view of the left PIV fields (with five vectors cut off along the boundaries to eliminate possible errors at the edges) was remained, whereas the rest was from the right PIV fields for small cross sections. For both cross sections, a full view of bottom PIV fields (also with five vectors cut off along the boundaries) was kept intact whereas the rest were from the top fields. All the analysis presented in the next section used the stitched data.

Results

Velocity Fields in the Baffled Flask

The average flow fields in the baffled flask are plotted in Fig. 3 for rotation speeds of Ω = 100, 150, 200, and 250 rpm. As stated in section “Experiment Setup,” the large and small cross sections are divided into small PIV fields of view (6 fields for A_large, 4 fields for A_small as shown in Fig. 2), stitched data of these small PIV fields were used for results analysis. Although velocity fields of these PIV fields of view were measured independently, only small data discrepancies were observed in the boundaries between views during the integration as shown in Fig. 3. Integrated velocity fields can generally present the whole-field velocity fields in the flask.

Fig. 3.

Average velocity fields in the baffled flask at rotation speed of (a) Ω=100 rpm (A_large); (b) Ω=100 rpm (A_small); (c) Ω=150 rpm (A_large); (d) Ω=150 rpm (A_small); (e) Ω=200 rpm (A_large); (f) Ω=200 rpm (A_small); (g) Ω=250 rpm (A_large); (h) Ω=250 rpm (A_small); A_large and A_small represent large and small cross section, respectively; the water surface was found and masked for each rotation speed based on the images captured during the experiments

The vector map obtained from PIV is a square field with 157 × 125 points in the x, z plane for each small PIV field of view. However, the map of each PIV field contains vectors outside the domain of interest (fluid inside the flask) [Figs. 2(b and c)]. These vectors (vectors above the water surface and beyond the flask boundaries) were found and masked based on the images captured from the experiments for each rotation speed. To avoid any possible measurement affects by the water surface, especially for high rotation speed that the splash of water could affect the PIV recordings (movement of the seeding particles within the splash regions may not follow or represent the real motion of water flow and also the splash of water may trap air bubbles in the water), the vector mask of water surface was measured a few millimeters (10–15 vectors) below the real surface. The same methods were also employed for the three wall boundaries, and vectors (inside the flask) approximately one mm (four to five vectors) away from the wall was removed to mask the wall boundaries. Only vectors inside the mask covering the fluid in the flask were used for result analysis. Because the vectors were extremely dense, only every 10 vectors were plotted in Fig. 3.

Velocities from individual realizations are plotted in Appendix S1, whereas average velocities are shown in Fig. 3. The average velocities were computed by taking the average more than 200 realizations for both radial and axial water speeds, $\overline{\mathrm{U}}$ _i (i = x, z in Fig. 3) at each interrogation cell. The contours are the magnitude of the average velocities of the two components, $\overline{\mathrm{U}}=\sqrt{{\overline{\mathrm{U}}}_x^2+{\overline{\mathrm{U}}}_z^2}$. For large cross section [Figs. 3(a, c, e, and g)], the zone of high speed velocity is essentially observed near the water surface, and moving downward as rotation speed increases. A zone of low speed occupies the lower portion of the flask at Ω ≤ 150 rpm. With the increase of the rotation speed, the water is more agitated throughout the flask. For Ω ≥ 200 rpm, the high speeds occur from the water surface down to the bottom and walls of the flask. The flow in the flask is towards up left side of the wall because the measuring position is the furthest left side that the shaker moved to [Fig. 1(a)]. The right and middle portion of the water surface decrease and the water flow are going more upwards on the left portion as the rotation speed increases also indicate that more intense turbulence occurs in the flask.

For small cross section [Figs. 3(b, d, f, and h)], the high speed zone occurs on the left and middle portion from surface to bottom at Ω ≤ 200 rpm. The water flowing directions in the small cross section are similar to that in the large cross section [Figs. 3(a, c, e, and g)], which are towards up left side of the flask wall, and going more upwards as Ω increases. The average velocities in A_small appear to be larger than that in A_large for all rotation speeds. The relative differences of average velocities in the large and small cross sections are in the range of approximately 20–50% (mean velocity data are presented in Appendix S2) from large to small Ω. The simple explanation is that, as shown in Figs. 2(b and c), A_small is approximately 30% smaller than A_large. Based on the mass conservations, ${\overline{\mathrm{U}}}_{large}{A}_{large}={\overline{\mathrm{U}}}_{small}{A}_{small}$, the mean velocity in the small cross section should be approximately 30% larger than that in the large cross section. Also, there exist dead zones with very small velocities (e.g., the bottom-left corner of the flask as shown in Figs. 3(a, c, e, and g) in the large cross section, which could also cause smaller average velocity in the large cross section. At Ω = 250 rpm, the water surface is very close to the wall, and because of that, the wall and surface may have more effects on the PIV measurements. Some high speed values are observed close to the left wall. Because of the boundary effects (water surface and wall) and the extreme rotation speed, the results shown in Fig. 3(h) (Ω = 250 rpm in small cross section) may not present the real flow speed in the small cross section. The water were also highly aerated (more air bubbles are trapped in the water which could give extra reflection lights and may cause optical distortion) causing obstructions of the optical paths. Because of the effects of the wall and surface, measurements at Ω = 250 rpm for small cross sections may not be eligible to present the water flow in the flask.

Appendix S1 shows the flow field of the instantaneous velocity and the turbulent component (one of the 200 realizations). The contour of the instantaneous velocity field is close to that of the average velocity, but exhibit patchlike structures. The flow directions appear to be more random than the average velocity. High speed values are observed along the baffles for each rotation speed in the large cross section. The turbulent component also shows high speed values along the baffles in the large cross section. This may confirm the early assumption that it is because of the baffles that the average velocity in the small cross section is larger than that in the large one. In small cross section, relatively high speed values of turbulent component are observed in the bottom-center portion of the flask. The flow pattern is more like a vortex flow occurring in a small area very close to the center of the flask.

Fig. 4 shows the average velocity and its standard deviation at each realization. This average velocity [Fig. 4(a)] was calculated by averaging the magnitude of the two velocity components at each interrogation cell for each realization. The average velocities [Fig. 4(a)] are not deviate far from one another over the 200 realizations for each rotation speed, demonstrating the representative samples of these 200 replicates. The standard deviation of these 200 average velocities at each rotation speed is within the range of 3.12–5.53% of the mean speed values (this mean was calculated by averaging the 200 average velocities shown in Fig. 4(a) for each rotation speed). The deviations slightly increase as Ω increases, indicating that high speed flows introduce more uncertainties in the sampling. The standard deviation as shown in Fig. 4(b) is subject to the calculated velocity magnitude at each interrogation cell to the average for each realization. Similar to average velocities in Fig. 4(a), the evolution of standard deviation over the 200 realizations [Fig. 4(b)] shows the increases of standard deviation with the increase of Ω. This indicates that for small rotation speed, more uniform flow occur in the flask, but with the increase of rotation speed, the turbulent behavior is more intense, speed values are more random as shown in the instantaneous velocity field (Appendix S1, left panel).

Fig. 4.

Average velocity $\overline{U}$ and its standard deviation as a function of the PIV realizations for the seven rotation speeds

Energy Spectra Analysis

A quantitative means to detect the presence of turbulence is through evaluation of the Fourier spectrum, which represents the kinetic energy content at various scales. Based on Kolmogorov’s second similarity hypothesis, the spectrum has the following property (De Jong et al. 2009; Frisch 1995; Kolmogorov 1941):

$$E(k)\propto k^{-5∕3}$$ (7)

where k = wave number (/m). Eq. (7) characterizes the spectral slope in the inertial subrange (Kilander and Rasmuson 2005; Kresta and Wood 1993).

Fig. 5 contain plots of the energy spectra as a function of the wave number for the large cross section based on the magnitude of the velocity U (speed $U=\sqrt{U_x^2+U_z^2}$) [Figs. 5(a, c, e, and g)] and radial U_x and axial U_z velocities [Figs. 5(b, d, f, and h)]. The energy spectra of the small cross section are very similar to that of the large cross section and are not reported here for brevity. The calculations of 1D Fourier transform of the spatial data are obtained from velocity values on horizontal lines (in the x-direction of the cross section). The spectra are averaged along y according to an homogeneity hypothesis. Then averaging the spectral amplitudes on the 200 realizations is computed and presented in Fig. 5. The theoretical −(5/3) slope based on Kolmogorov theory (Kolmogorov 1941) is also plotted in the figures. At low wave number (large length scale), the spectrum scales in a different way and the energy decreases slower than −(5/3). The fluctuation energy is produced at the large eddies which also absorb some energy directly from the mean flow. This region is also called the energy-producing or energy-containing range (Kaimal and Finnigan 1994; Tennekes and Lumley 1972). The energy exchange between the mean flow and the turbulence is governed by the dynamics of the large eddies which contribute most to the turbulent production. Then the large eddies break down to smaller and smaller eddies and the energy cascade down the spectrum to high wave number region. For all spectra obtained from PIV measurements as shown in Fig. 5, one could tell that there exists a region with −(5/3) slope which demonstrate the existence of the inertial subrange and the presence of turbulent flow.

Fig. 5.

Average energy spectra over 200 realizations as a function of wavenumber in the large cross section (A_large) for rotation speed of: (a) Ω=100 rpm (U); (b) Ω=100 rpm (U_x and U_z); (c) Ω=150 rpm (U); (d) Ω=150 rpm (U_x and U_z); (e) Ω=200 rpm (U); (f) Ω=200 rpm (U_x and U_z); (g) Ω=250 rpm (U); (h) Ω=250 rpm (U_x and U_z); spectra were calculated horizontally (in the x-direction of the cross section) based on the spatial data of turbulent velocities in the baffle flasks; U represents the velocity magnitude ($U=\sqrt{U_x^2+U_z^2}$ ), and U_x, U_z represent the radial and axial velocity, respectively; the −5/3 slope is also reported; the spectra seem to match the −5/3 slope closely

The energy spectra of the radial U_x and axial U_z velocities [Figs. 5(b, d, f, and h)] are not exactly the same, but they are comparable. The relative differences of log(E) between radial and axial velocities are in the range of 3.2–6.9% for all rotation speeds. However, the energy from U_z is larger than that from U_x by more than 300% for Ω = 100–200 rpm. The energy spectra of radial and axial velocities at Ω = 250 rpm [Fig. 5(h)] are very close, which may be an indication of local isotropy. Kolmogorov’s hypothesis of local isotropy states that at sufficiently high Reynolds numbers, the small-scale turbulent motions are statistically isotropic (Kolmogorov 1941). In addition, the measured spatial resolution in the current study was 0.95 mm width per interrogation cell, which may not fully resolve the turbulent fields conducted in the current study to show the characteristics of isotropic flow. As discussed for all the rotation speeds, the scale of the energy spectrum is changed at a small wavelength λ of approximately 1 mm (the wavenumber κ ≈ 1,000 cycles/m). This may be caused by the fundamental spatial resolution of the PIV measurement (approximately 0.95 mm width of the interrogation cell) that the turbulent behavior in the smaller scale is not properly captured because of the limitations of the instrument in the current study, which gives a white noise in the small wavelength of the spectra.

Fig. 6 shows the average energy spectra on one realization for Ω=100 rpm [Figs. 6(a and b)] and Ω = 200 rpm [Figs. 6(c and d)] for both velocity speed and two velocity components. Compare to the spectra in Figs. 5(a and b) and Figs. 5(e and f), the spectra of one realization are noisier because of the more random behavior of instantaneous velocity at one realization. The energy spectra at small scale wavelength λ < 1 mm (the wavenumber κ > 1,000 cycles/m) appear to be more like noises. This may well confirm the early statement that the change of turbulent structure scale observed in the average spectra over 200 realizations is attributable to the fundamental spatial resolution of the PIV measurements adopted in the current study.

Fig. 6.

Average energy spectra of one realization as a function of wavenumber in the large cross section (A_large) for rotation speed of (a) Ω=100 rpm (U); (b) Ω=100 rpm (U_x and U_z); (c) Ω=200 rpm (U); (d) Ω=200 rpm (U_x and U_z); spectra were calculated horizontally (in the x-direction of the cross section) based on the spatial data of turbulent velocities in the baffled flasks; U represents the velocity magnitude ($U=\sqrt{U_x^2+U_z^2}$ ), and U_x, U_z represent the radial and axial velocity, respectfully

Fig. 7 shows the 1D average energy spectra of 200 realizations calculated from velocities on the vertical lines (in the z-direction of the cross section) for Ω = 100 rpm [Figs. 7(a and b)] and Ω = 200 rpm [Figs. 7(c and d)]. Comparing with the average spectra in Figs. 5(a and b) and Figs. 5(e and f) (calculated from the velocities on the horizontal lines), difference can be observed in the spectra. The energy-producing region (low wave number) is smoother (less fluctuated) than the spectra calculated from the horizontal lines. This is because the rotational behavior of the large eddies in the horizontal direction [the flask is mounted on the rotational shaker as shown in Fig. 1(a)]. The overall energy spectra are smaller than that on the horizontal lines. Similar to the one on the horizontal lines, the energy from U_z is larger than that from U_x, which is reasonable for rotation speeds analyzed.

Fig. 7.

Average energy spectra over 200 realizations as a function of wavenumber in the large cross section (A_large) for rotation speed of: (a) Ω=100 rpm (U); (b) Ω=100 rpm (U_x and U_z); (c) Ω=200 rpm (U); (d) Ω=200 rpm (U_x and U_z); spectra were calculated vertically (in the z-direction of the cross section) based on the spatial data of turbulent velocities in the baffle flasks; U represents the velocity magnitude ($U=\sqrt{U_x^2+U_z^2}$ ), and U_x, U_z represent the radial and axial velocity, respectively

Energy Dissipation

The energy dissipation rate, ε, was estimated using Eq. (2) with the coefficient A set equal to 1.0 (Kresta and Wood 1993). To obtain integral length scale, the autocorrelation coefficient was calculated based on 1D spatial turbulent velocity for each realization. Fig. 8 shows the normalized autocorrelation functions in all direction of the large cross section for Ω = 100, 150, 200, and 250 rpm. Similar results were obtained for all other cases (small cross section and other rotation speeds). The autocorrelation function rapidly descends to zero within 5–35 mm distance lag, indicating that the autocorrelation function is not periodical rather presents chaotic (or random) behavior. For small rotation speed of Ω = 100 rpm, the radial velocity $U_x^{\prime }$ has very close trend in horizontal and vertical lines, but with more fluctuation in the vertical line. The same behavior is also applied for the axial velocity $U_z^{\prime }$. A shorter spatial lag is needed for the autocorrelation function of $U_z^{\prime }$ to reach zero than that of $U_x^{\prime }$, indicating a more random behavior of axial flows in the flask. With increase of rotation speeds, the similar trend in horizontal and vertical lines as observed for Ω = 100 rpm is not clear, but the autocorrelation function that is first reach zero is the one of $U_z^{\prime }$ in the horizontal line for all the cases present. The distance lag for autocorrelation functions in all directions (four lines in Fig. 8) to reach zero decreases with the increase of rotation speeds, demonstrating a more random flow as mixing intensity increases in the flask. The integral length scale was evaluated for each realization by integrating from length zero to the length of first zero crossing of the autocorrelation function. The trapezoidal rule was used for the numerical integration of Eq. (3). Although differences were observed in the autocorrelation functions at different directions (Fig. 8), the computed integral length scales are close, with a difference of approximately 10–50% in all directions. The average integral length scale was obtained by taking the average of integral length scales in the two directions for all 200 realizations.

Fig. 8.

Autocorrelation functions calculated based on the turbulent components of the velocities in the large cross section for rotation speeds of 100, 150, 200, and 250 rpm; integration of Eq. (3) occurs until the first zero crossing on the distance lag axis

Energy dissipation rate was estimated based on Eq. (2) for both radial (U_x) and axial velocities (U_z). The root mean square $u_{rms}^{\prime }$ of turbulent component of radial and axial velocities were computed based on the speed values of 200 PIV realizations at each interrogation cell. Then the energy dissipation rate was calculated for both radial and axial velocities at each interrogation cell based on the corresponding root mean square velocity $u_{rmsri}^{\prime }$ and the average integral length scale. The average energy dissipation rate $\overline{\epsilon }$ is the average of the two velocity components (radial and axial).

Fig. 9 shows the contour plots of the average energy dissipation rate in the large [Figs. 9(a, c, e, and g)] and small cross section [Figs. 9(b, d, f, and h)]. In the large cross section, for Ω = 100 rpm, the zone with high ε is distributed along the baffles. At Ω = 150 rpm, high ε zone shows on the top left portion close to the surface which is expected because that is the high velocity zone located (Fig. 3). At Ω ≥ 200 rpm, high ε is distributed downwards and throughout the cross section, especially on the left and right portion. For small cross section, similar to the findings from velocity fields of the turbulent components (Appendix S1), relatively high ε is observed at the area close to the center of the flask. Future studies of the energy distribution on the horizontal plane (horizontal cross sections at different elevations) may provide more insights on this. The distributions are similar at Ω ≤ 150 rpm with slight increase of ε on the water surface as Ω increases. At Ω ≥ 170 rpm (plot of 170 rpm is not shown in Fig. 9), as in the large cross section, high ε is also distributed throughout the cross section, demonstrating that the turbulence is intense under such high rotation speed. At the extreme case of Ω = 250 rpm in the small cross section, the water is basically accumulated close to the left wall of the cross section, and very high ε is generated in that region. However, as discussed in section “Velocity Fields in the Baffled Flask,” because the high intensity of the turbulence and the water is so close to the wall and water surface, also with the air bubbles trapped in the water, the quality of the PIV measurements for this extreme case is in question. Therefore, results of Ω=250 rpm in the small cross section will not be presented in the paragraphs that follow. Further investigations are needed for this case and new technology may be needed.

Fig. 9.

Contour plots of average energy dissipation rate ($\stackrel{\rightharpoonup }{\epsilon }$) in the baffled flask for rotation speed of (a) Ω=100 rpm (A_large); (b) Ω=100 rpm (A_small); (c) Ω=150 rpm (A_large); (d) Ω=150 rpm (A_small); (e) Ω=200 rpm (A_large); (f) Ω=200 rpm (A_small); (g) Ω=250 rpm (A_large); (h) Ω=250 rpm (A_small); A_large and A_small represent large and small cross section, respectively; legends represent values of ε (W/kg)

The average energy dissipation rate was obtained for each rotation speed by taking the ε values of all interrogation cells in the cross section and plotted in Fig. 10 as a function of rotation speed. The average εε in the baffled flask increase with the increases of rotation speed. Considering all the points in Fig. 10 for both large and small cross section, the average energy dissipation rate appears to increase exponentially with the rotation speed

$$\overline{\epsilon }=9.0\mathrm{x}{10}^{-5}Exp(0.043\Omega ),\text{with}{R}^2=0.97$$ (8)

Fig. 10.

Variation of energy dissipation rate ε as a function of the rotation speed for the large and small cross sections

The average Kolmogorov scale $\overline{\eta }$ and velocity gradient was calculated based on Eqs. (5) and (6) using the energy dissipation rate for each velocity components (radial and axial) at each PIV interrogation cell, and then taking the average for each rotation speed. Fig. 11 shows the results of the average Kolmogorov microscale $\overline{\eta }$ in both large and small cross sections. $\overline{\eta }$ decreases from 296 to 32 μm with the rotation speed increases from 100 to 250 rpm. Similar to the energy dissipation rate (Fig. 10), a fitting line is generated for $\overline{\eta }$ using the data from both large and small cross sections

$$\overline{\eta }=1,463\text{Exp}(-0.015\Omega ),{\text{with R}}^2=0.98$$ (9)

Fig. 11.

Variation of Kolmogorov microscale η as a function of the rotation speed for the large and small cross sections

Appendix S2 shows the overall average parameters estimated in this study. Results for both large and small cross sections are shown in the tables. The average of the two cross sections is considered as the final results in the baffled flask, except Ω = 250 rpm that only results in the large cross section are presented. The mean velocities presented in Appendix S2 were the average magnitude of the average speed values of radial (${\overline{\mathrm{U}}}_x$) and axial velocity (${\overline{\mathrm{U}}}_z$) components over 200 realizations at each PIV interrogation cell. The mean velocities increase from 0.017 to 0.14 m/s with the rotation speed increases from 100 to 250 rpm. The RMS velocity $u_{rms}^{\prime }$ also increases with the increase of Ω. The integral length scales at Ω ≤ 160 rpm has the same magnitude with 0.63–0.69 mm for Ω = 100–150 rpm and 0.91 mm for Ω = 160 rpm. Then the integral length scales increase an order of magnitude at Ω ≥ 170 rpm to 1.19–1.27 mm. The average energy dissipation rates are in the range of 7.65 × 10⁻³ − 4 W/kg for the rotation speed investigated in the current study (Ω = 100–250 rpm). The Kolmogorov scales estimated in this study for all seven rotation speeds (32–296 μm) are well in the range of the sizes of oil droplets observed at sea, which is 50–400 μm (Delvigne and Sweeney 1988). The velocity gradient increases from 26.3 to 1,472.6/s based on the measurements obtained in this study. Kaku et al. (2006a, b) suggested that the energy dissipation rate ε is more presentative of the turbulent intensity because ε could be stipulated based on the law of conservation of energy, whereas there is no law that requires the conservation of velocity or its gradient.

Discussion

Rather than single-point velocity measurement techniques (e.g., HWA), the PIV system can provide a 2D or three-dimensional (3D) instantaneous velocity fields depending on the system chosen. However, the limitations of using the PIV system should be recognized and considered during the experiments. There are many sources can introduce errors in the PIV measurements such as CCD camera noise, tracer characteristics, light scattering, and velocity gradient (Tanaka and Eaton 2007). As discussed previously, air bubbles are also found for some of the high speed experiments in both cross sections. The effects of air bubbles and other error sources on the results presented in this study require further investigations in the near future.

The velocity value from PIV measurements is not speed value at a single point like the one measured by HWA rather than an average velocity of each interrogation cell (FlowManager). Bounded by the measuring techniques and capacity of the instruments (e.g., camera and laser), compromise has to be made between spatial resolutions and accuracy of the measuring data as discussed previously in the paper. Such compromise could also limit the methods used for analyzing the energy distribution and turbulent structure within the vessel (e.g., energy dissipation rate estimation). The most accurate estimation method of ε the is direct measurement of strain rate tensors, which is the definition of dissipation rate from fundamental turbulence theory (Cheng et al. 1997; Tennekes and Lumley 1972)

$$\epsilon =2v\overline{\overline{S_{mn}{S}_{mn}}}=\frac{v}{2}\overline{\overline{(\frac{\partial u_m^{\prime }}{\partial x_n}+\frac{\partial u_n^{\prime }}{\partial x_m}{)}^2}}$$ (10)

where double overbar = ensemble average; S_mn = turbulence rate of strain; ν = kinematic viscosity; $u_m^{\prime }$ and $u_n^{\prime }=m\text{th}$ and nth component of the fluctuating velocity; and x_m and x_m = coordinates. However, this method requires 3D flow field measurements, and the spatial resolution has to be smaller than Kolmogorov length scale. So far, only three-dimensional direct numerical simulation (DNS) is able to provide a full-field estimation of the dissipation rate by this method (Sheng et al. 2000). The estimation method from turbulence energy spectrum can use one-dimensional spectra when homogeneous, isotropic turbulence is assumed. However, to fully resolve E(k) in Eq. (7), again, the velocity measurements must be sampled at a rather high wave numbers to allow an application of this method.

In the current study, the most widely used energy dissipation estimation method—dimensional argument analysis—was chosen to compute the ε. This method has been proved to provide rather accurate results from rotational flow in stirred tank experiments (e.g., Cheng et al. 1997; Costes and Couderc 1988; Kresta and Wood 1993; Sheng et al. 2000). In theory, this method applies only to turbulence that is homogeneous, isotropic, and in spectral equilibrium, but in practice, it is rather difficult to meet these conditions (Sheng et al. 2000). The fine-scale structure of most actual nonisotropic turbulent flows is locally nearly isotropic; therefore, many features of isotropic turbulence may thus be applied to phenomena in actual turbulence that are determined primarily by the fine-scale structure (Hinze 1955; Luo and Svendsen 1996). It was reported that the dimensional argument analysis could produce reliable average energy dissipation but may be incorrect when local ε values and distribution are estimated (Baldi and Yianneskis 2003). Further studies using finer scale in the measurements may provide more insights on the turbulent flow characteristics and in the application of different estimation techniques in the future.

Table 1 shows the comparison of the average turbulent parameters between the current study and the one from Kaku et al. (2006a, b) for Ω = 100, 200 rpm. Most of the turbulent parameters obtained from the current study are in the same order of magnitude as the ones obtained from Kaku et al. (2006a, b). The mean velocities from Kaku et al. are four to six times greater than the ones from the current study, whereas the energy dissipation rate is smaller than the ones from this study by 6–24%. Fig. 12 shows a comparison of the average radial velocities for Ω = 200 rpm at locations comparable to those in Kaku et al. between the two studies. The overall magnitude of the velocities obtained from the current study [Figs. 12(a and b)] is apparently smaller than that from Kaku et al. (2006a, b) by approximately 40%. However, the flow patterns are similar for the portion under water surface (the top right corner), especially velocities in the small cross section that a patchlike flow structure can be observed in the middle portion of the flask from both studies [Figs. 12(b and c)].

Table 1.

Comparison of Average Turbulent Parameters (Data from Kadu et al. 2006a, b)

Rotation speed (rpm)	Data source	Mean velocity uave(m/s)	RMS velocity U′rms (m/s)	Average energy dissipation rate $\overline{\epsilon }$ (W/kg)	Kolmogorov microscale $\overline{\eta }$ (μm)	Average velocity gradient $\overline{\mathit{G}}$ (/s)
100	Results of the current study	1.74×10−2	6.28×10−3	7.65×10−3	287.5	26.3
	Kaku et al. (2006a, b)	6.31×10−2	3.68×10−3	4.78×10−4	276.7	17.2
200	Results of the current study	8.81×10−2	5.62×10−2	6.70×10−1	63.8	446.7
	Kaku et al. (2006a, b)	5.33×10−1	4.95×10−2	1.63×10−1	57.0	349.9

Fig. 12.

Comparison of the average radial velocities in the baffled flask for Ω=200 rpm at locations comparable to those in Kaku et al. (2006a, b): (a) ${\overline{U}}_{\mathrm{x}}$ in the large cross section (the current study); (b) ${\overline{U}}_{\mathrm{x}}$ in the small cross section (the current study); (c) contours obtained from Kaku et al. (2006b)

Several factors may have contributed to the difference between the current study and Kaku et al. (2006a, b)

1. One factor is the window or support of the measurement. The HWA used in Kaku et al. is a single point velocity measuring instrument (the sensing area is 3 mm long and 25 μm in diameter), whereas PIV measurements are the average velocity in square cells with approximately 0.95 mm width.

2. The velocity components obtained and used for analysis in Kaku et al. were radial and azimuthal velocities, whereas those conducted in this research were radial and axial velocities.

3. Kaku et al. (2006a, b) used temporal data and therefore had to invoke the Taylor’s frozen turbulence hypothesis (that turbulence is advected in space without alteration of its properties) to compute turbulence properties. Such an assumption was not needed in this study as the PIV provides spatial data. It is thus possible that the frozen turbulence hypothesis was not valid at all locations and/or at all times in Kaku et al. (2006a, b). For spatial data, the autocorrelation and spectrums can be evaluated directly from the measurements without resorting to Taylor’s hypothesis. Thus, this allows one to capture anisotropic turbulence, at least at the flask scale.

4. The equation of the energy dissipation rate in this study differs from that of Kaku et al. (2006a, b). The traditional expression of dimensional argument for ε is based on the integral length scale as the one used in this paper [Eq. (2)], whereas that used in Kaku et al. relies on using a characteristic time scale derived from the characteristic spatial scale. Although, it was reported that consistent results can be obtained from both methods, results with slight differences could occur (Kresta and Wood 1993).

Bounded by available techniques, measurements from different types of instruments have their advantages and limitations in analyzing properties of turbulent flows. In theory, if fully resolved (high spatial resolution) three-component velocity measurements in a 3D volume can be obtained in a turbulent flow. Accurate results of turbulent properties would be obtained. However, such criteria normally cannot be met in practice. Correction methods have been proposed based on data with relatively low spatial resolution (e.g., De Jong et al. 2009; Sheng et al. 2000; Tanaka and Eaton 2007), but the accuracy and application of these methods is still need further investigation. In summary, although differences (Table 1 and Fig. 12) have been observed between results obtained from the current study and the one from Kaku et al. (2006a, b), they are not far from one another. Different instruments, velocity components measured, and consequently different analysis methods are all contribute to the discrepancy in the results. Additional studies are needed to further validate the results obtained from this study, especially three-component velocity measurements would help for further investigations in the future.

Conclusion

Measurements of the water velocity in the baffled flask were obtained using particle image velocimetry (PIV) for seven rotation speeds of 100, 125, 150, 160, 170, 200, and 250 rpm. Two vertical cross sections were measured to obtain the velocity profiles in the flask: the one with the largest diameter of the flask and the one with the smallest diameter. Noises were observed in spectrum at smaller scale than the fundamental spatial resolution of the PIV measurements. The energy dissipation rate was estimated using dimensional argument for both cross sections, and the overall dissipation rates were the average of the results from the two cross sections. An exponential model was created for both average energy dissipation rate and the Kolmogorov microscale as a function of rotation speed with R² ≥ 0.97. The estimated average energy dissipation rates are in the range of 7.65 × 10⁻³ – 4 W/kg for rotation speeds of Ω = 100–250 rpm, of which it is larger than the one estimated by Kaku et al. (2006a, b) for Ω = 100 and 200 rpm. Different instruments, velocity components measured, and consequently different analysis methods all contribute to the discrepancy in the results of our studies and Kaku et al.’s. The Kolmogorov scale estimated in this study for all seven rotation speeds approached the size of oil droplets observed at sea, which is 50–400 μm (Delvigne and Sweeney 1988).