PIV measurement of high-Reynolds-number homogeneous and isotropic turbulence in an enclosed flow apparatus with fan agitation

With a given physical space, a chamber that is nearly spherical has the highest degree of symmetry and the largest volume, and thereby is most advantageous for achieving high homogeneity, isotropy and ${{R}_{\lambda}}$. However, a spherical chamber is difficult to construct. Chang et al [10] proposed a truncated icosahedron to approximate a sphere for use as an enclosed flow chamber. It has 20 hexagonal faces and 12 pentagonal faces for mounting hardware. This highly symmetrical shape, ideal for an enclosed chamber, was adopted in our chamber design. In principle, all 32 faces could be utilized for mounting flow actuators [10]. However, the hexagonal faces and pentagonal faces have different distances to the center of the truncated icosahedron. To mount actuators on a sphere without varying the length of the actuators, we decided to mount 20 identical actuators on the 20 hexagonal faces, which have equal distances to the center owing to the built-in symmetry of the icosahedron.

Our truncated icosahedron, or 'soccer-ball', chamber had a 1-meter outer diameter. The distances from the center of the solid to the centroids of the pentagonal and hexagonal faces are approximately 47.0 and 45.8 centimeters, respectively.

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An enclosed HIT facility requires symmetrically distributing a number of actuators about the homogeneous and isotropic interrogation volume along multiple axes of symmetry. Both fan or 'continuous jet' [6, 12, 14] and loudspeaker attached to a nozzle or 'synthetic jet' [3, 9–11] have been used as flow actuators in HIT facilities.

We chose continuous jets over synthetic jets to minimize the gravity effect on particles to be studied. A synthetic jet uses a loudspeaker to draw in and push out air (which can be laden with particles) of a nozzle cavity through an orifice. A continuous jet uses a rotating fan to agitate air. Using multiple, symmetrically placed continuous jets, we can produce high mixing efficiency and turbulent kinetic energy dissipation rate, thereby achieving a high Froude number—the ratio of Kolmogorov-scale turbulence acceleration to gravitational acceleration. A high Froude number allows us to minimize the influence of gravity on particle dynamics.

Another reason fans are preferred over synthetic jets in our design is the need to avoid particle deposition in the corners of the nozzle. In synthetic jets, particles tend to be deposited and trapped around the oscillating diaphragm due to vortex generation near the jet orifice [18, 19]. When studying dynamics of particles of different Stokes numbers, it would be difficult to clean out trapped old particles from the synthetic jets. With fans, there are no flow cavities for particles to be deposited in. Moreover, loudspeakers employed in synthetic jets may generate variable standing-wave acoustic fields inside an enclosed chamber, and these sound fields could undermine the repeatability of the HIT facility.

To minimize disturbances to the flow and heating of the fluid within the chamber, motors can be mounted outside the chamber, and only the axes that drive the fans cross the hexagonal walls.

In an enclosed HIT apparatus, symmetrical flow forcing along more than 6 axes can produce turbulence with no preferred directions in the center [7, 10]. It is expected that turbulence in this region is not only homogeneous, but also isotropic. Furthermore, by virtue of the symmetrically placed multiple actuators, such a region has a nearly zero mean velocity.

In order to quantify the degree of homogeneity and isotropy, we define a homogeneity index $\chi$ and an isotropy index $~\gamma ~$ [3, 10, 11]:

$$\begin{eqnarray}&&\chi =\frac{{{u}^{\prime}}\left(\text{local}\right)}{{{u}^{\prime}}\left(\text{center}\right)},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\gamma =\frac{{{\sigma}_{x}}}{{{\sigma}_{y}}},\end{eqnarray} \tag{ 2 }$$

where ${{u}^{\prime}}=\sqrt{\left(\sigma _{x}^{2}+\sigma _{y}^{2}\right)/2}$) is the turbulence strength, ${{\sigma}_{x}}$ and ${{\sigma}_{y}}$ are respectively the root-mean-square (RMS) velocity fluctuation in the horizontal and vertical directions at any point in the chamber, and the designations 'center' and 'local' refer to value at the origin point and a local measurement point in the chamber, respectively. In ideal homogeneous and isotropic turbulence, $\chi =1$ and $~\gamma =1$. However, in a real apparatus, system errors including imbalance and fluctuation of fans and environmental as well as measurement noise will cause $\chi$ and $\gamma$ to deviate from unity. Thus, $\chi =1\pm 0.1$ and $\gamma =1\pm 0.1$ have been frequently accepted [3, 12, 14]. Notably, Chang et al [10] achieved a homogeneity and isotropy region with $\chi$ and $\gamma$ fluctuations within 5% around 1. Since we would like the HIT apparatus to be useful for DNS validation with high reliability, we adopt the 5% requirement and define the 'homogeneous region' as a region in which the velocity fluctuations in all locations are within 5% of that at the center of the chamber, and the 'isotropic region' as a region where RMS velocity fluctuations in different directions are within 5% of each other. As such, a combination of $\chi =1\pm 0.05$ and $\gamma =1\pm 0.05$ qualifies for the homogeneous and isotropic region in our chamber.

The characteristic Reynolds number in homogeneous and isotropic turbulence is conventionally defined as ${{R}_{\lambda}}~$ in terms of the Taylor microscale:

$$\begin{eqnarray}&&{{R}_{\lambda}}=\frac{{{u}^{\prime}}{{\lambda}_{g}}}{\nu},\end{eqnarray} \tag{ 3 }$$

where ${{u}^{\prime}}$ is the turbulence strength, $\nu$ is the viscosity of the flow, and ${{\lambda}_{g}}$ is the Taylor microscale of turbulence. Finding the ${{R}_{\lambda}}~$ dependence on the physical parameters of the enclosed chamber is important for understanding, generating and controlling turbulence in enclosed HIT chambers. Therefore, in what follows we will develop a relationship between ${{R}_{\lambda}}$ and chamber and actuator parameters.

Following Pope [2], we define large-eddy (or integral) length scale as

$$\begin{eqnarray}&&L\equiv \frac{{{k}^{3/2}}}{\varepsilon}\end{eqnarray} \tag{ 4 }$$

and integral-scale Reynolds number as

$$\begin{eqnarray}&&{{\operatorname{Re}}_{L}}\equiv \frac{{{k}^{1/2}}L}{\nu},\end{eqnarray} \tag{ 5 }$$

where the turbulent kinetic energy $k=\frac{3}{2}{{u}^{\prime2}}$. From equations (3)–(5) it follows that

$$\begin{eqnarray}&&{{R}_{\lambda}}={{\left(\frac{20}{3}{{\operatorname{Re}}_{L}}\right)}^{1/2}}\end{eqnarray} \tag{ 6 }$$

or equivalently,

$$\begin{eqnarray}&&{{R}_{\lambda}}={{\left(\frac{20}{3}\right)}^{1/2}}{{k}^{1/4}}{{\left(\frac{L}{\nu}\right)}^{1/2}}.\end{eqnarray} \tag{ 7 }$$

Based on energy conservation, we assume that the turbulent kinetic energy $k~$ at the center of an enclosed, symmetrically forced multiple-actuator HIT chamber is proportional to the kinetic energy provided from all the actuators or jets:

$$\begin{eqnarray}&&k~\sim ~n\cdot U_{0}^{2},\end{eqnarray} \tag{ 8 }$$

where $n$ is the number of actuators and ${{U}_{0}}$ is the mean flow velocity generated by the actuators. Furthermore, we assume that the integral length scale $L$ is proportional to the chamber radius,

$$\begin{eqnarray}&&L~\sim ~R.\end{eqnarray} \tag{ 9 }$$

This is based on the consideration that in an enclosed turbulence system, the fluid continually recirculates inside a finite space, forming the largest flow structure by impinging at the center and then pushing back to the inlet of actuators (figure 2). Substituting equations (8) and (9) into equation (7) and consolidating coefficients into a facility constant $c$, we obtain the ${{R}_{\lambda}}$ dependence on physical parameters of the chamber as:

$$\begin{eqnarray}&&{{R}_{\lambda}}=c\cdot {{n}^{1/4}}{{\left(\frac{{{U}_{0}}R}{\nu}\right)}^{1/2}}.\end{eqnarray} \tag{ 10 }$$

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This important relationship shows that ${{R}_{\lambda}}$ is proportional to the number of actuators $n$ to the power 1/4, the square root of jet velocity ${{U}_{0}}$ at the actuator exit, the square root of chamber radius $R,$ and the square root of fluid viscosity $\nu$. The jet velocity ${{U}_{0}}$ from an actuator depends on the power input and the energy conversion efficiency of the actuator.

The ${{R}_{\lambda}}~$ relation in equation (10) is applicable to any type of actuators, including fans and synthetic jets (figures 2(a) and (b), respectively).

Next, when rotating fans are used as actuators (figure 2(a)), jet velocity ${{U}_{0}}$ can be further determined by the flow volume per fan rotation $V$, fan rotational speed $\omega$, and fan diameter $d$:

$$\begin{eqnarray}&&{{U}_{0}}=\frac{2\omega \cdot V}{{{\pi}^{2}}\cdot {{d}^{2}}}.\end{eqnarray} \tag{ 11 }$$

Substituting equation (11) into equation (10) and consolidating coefficients into a single coefficient $C\equiv \frac{\sqrt{2}c}{\pi}$, we get an updated ${{R}_{\lambda}}$ relation specifically for fan-driven HIT chambers:

$$\begin{eqnarray}&&{{R}_{\lambda}}=C\cdot \frac{{{n}^{1/4}}}{d}{{\left(\frac{\omega \cdot V\cdot R}{\nu}\right)}^{1/2}}.\end{eqnarray} \tag{ 12 }$$

Equation (12) illustrates the dependence of ${{R}_{\lambda}}$ on physical factors in a fan-driven HIT chamber, and was used in the process of designing our apparatus. To realize a desired $~{{R}_{\lambda}}$, we could select the fan parameters ($V$ and $d$), the motor speed of operation ($\omega$), and number ($n$) of fans and the kinematic viscosity ($\nu$) of the fluid. For any commercial fan, a rated CFM (cubic feet per minute) at a given fan speed ${{f}_{0}}$ is always given by the manufacturer. From the rated CFM, the flow volume per fan rotation $V$ can be obtained from dividing CFM by ${{f}_{0}}$. The coefficient $C$ depends on the flow chamber.

To estimate ${{R}_{\lambda}}$ for our new 'soccer ball' chamber, we used our previous apparatus [12], which was fully characterized, as a reference chamber to eliminate the unknown coefficient C. From equation (12) we can take a ratio of the Reynolds number of the new chamber to be estimated,$~R_{\lambda}^{\text{est}}$, to the Reynolds number of the reference chamber (denoted by 'prime'), ${{R}_{\lambda}}^{\prime}$:

$$\begin{eqnarray}&&\frac{R_{\lambda}^{est}}{R_{\lambda}^{\prime}}=\frac{C}{{{C}^{\prime}}}\cdot \frac{{{d}^{\prime}}}{d}\cdot {{\left(\frac{{{\nu}^{\prime}}\cdot \omega \cdot ~V\cdot R}{\nu \cdot {{\omega}^{\prime}}\cdot {{V}^{\prime}}\cdot {{R}^{\prime}}}\right)}^{1/2}}\cdot {{\left(\frac{n}{{{n}^{\prime}}}\right)}^{1/4}}.\end{eqnarray} \tag{ 13 }$$

Since both chambers use the same fluid—room air, we have $\nu ={{\nu}^{\prime}}$. Furthermore, we assume $C\approx {{C}^{\prime}}$. Then

$$\begin{eqnarray}&&R_{\lambda}^{\text{est}}=R_{\lambda}^{\prime}\cdot \frac{{{d}^{\prime}}}{d}\cdot {{\left(\frac{{{\nu}^{\prime}}\cdot \omega \cdot ~V\cdot R}{\nu \cdot {{\omega}^{\prime}}\cdot {{V}^{\prime}}\cdot {{R}^{\prime}}}\right)}^{1/2}}{{\left(\frac{n}{{{n}^{\prime}}}\right)}^{1/4}}.\end{eqnarray} \tag{ 14 }$$

The reference chamber had ${{n}^{\prime}}=$ 8 fans with ${{d}^{\prime}}=$ 11 cm, a maximum motor speed ${{f}^{\prime}}=3900\text{RPM}$ (fan rotational speed ${{\omega}^{\prime}}=2\pi {{f}^{\prime}}$), an equivalent radius of ${{R}^{\prime}}=$ 28.82 cm, and a maximum $R_{\lambda}^{\prime}$  =  184. The fan had a rated $\text{CFM}=102$ at 3700 RPM. This yields $V=7.8~\times {{10}^{-4}}~{{\text{m}}^{3}}$ per rotation. Plugging these numbers into equation (14) allows us to estimate the Taylor-scale Reynolds number for the new chamber, $R_{\lambda}^{\text{est}}$, under given design parameters.

For our new HIT chamber, our design goal was ${{R}_{\lambda}}\approx ~$ 400. We chose a robust motor (A.O. Smith 91 1/15 HP 115/230 Volt) with a maximum rotation speed of $f=\,3500$ RPM ($\omega =2\pi f$). We adopted a plastic fan of diameter $d$  =16 cm (Dayton 5JLP3) with a rated $\text{CFM}=360$ at 3000 RPM. This yields $V=3.4~\times {{10}^{-3}}~{{\text{m}}^{3}}$ per rotation. The new chamber had a radius $R=0.5~\text{m}$ and $n$  =20 fans, as described in section 2.1. Plugging these values and those of the reference chamber into equation (14), we estimated the maximal ${{R}_{\lambda}}$ in the new 'soccer ball' chamber to be $\,R_{\lambda}^{\text{est}}=415.$

Air is usually preferred as the working fluid in the chamber. To change ${{R}_{\lambda}}~$ via varying the fluid viscosity $\,\nu$, either the working fluid or the operating temperature of the chamber [20] can be adjusted. Both numerical [21] and experimental [22, 23] studies have demonstrated that by choosing hexafluorides ($\text{S}{{\text{F}}_{6}}$, $\text{Se}{{\text{F}}_{6}}$, and $\text{Te}{{\text{F}}_{6}}$), oxyfluorides ($\text{S}{{\text{O}}_{2}}{{F}_{2}}$, $\text{PO}{{\text{F}}_{3}}$) or Freon-12 ($\text{CC}{{\text{l}}_{2}}{{\text{F}}_{2}}$) as fluid in wind tunnels, the same ${{R}_{\lambda}}~$ can be obtained with lower power requirement. However, these gases are generally either toxic to the experimenter or environmentally hazardous, thus requiring a well-sealed chamber constructed with low permeability materials. We chose air as the working fluid without pressurizing the chamber for ease of operation.

Fluid viscosity can be sensitive to the operating temperature [24]. To maintain constant air viscosity during data acquisition, the facility should operate at a constant temperature, for which active cooling can be considered. In absence of active temperature regulation, sufficient time lapses should be allowed between experimental runs to avoid appreciable heating, allowing the chamber to operate near room temperature.

Noninvasive diagnosis of the flow or the particle without disrupting flow homogeneous and isotropic characteristics is important. We introduce optical windows on the opaque chamber walls to allow laser light (a light sheet or a 3D slab) to illuminate and cameras to view the flow contained within (or other probes to receive signals), allowing both planar PIV and hybrid digital holographic PIV [14] to image the central region of the chamber. Four 5'' circular optical windows are added on the chamber. Two windows are mounted 180° across from one another. The remaining two windows are 23° from the 180° windows toward one another. Additionally, two 4''  ×  1'' slit windows for laser light sheet entry were mounted 180° from one another. These windows allow for illumination and visualization of the flow from different orientations, as well as the use of multiple cameras for stereoscopic view or multiple views of regular or holographic imaging. Finally, a 0.2'' diameter hole was placed at the bottom of our facility to connect with a particle injector.

Figure 3 shows our implemented 'soccer ball' HIT chamber. With a 1-meter diameter, the chamber is constructed of 3/8'' aluminum plates for its high stiffness and thermal conductivity. A 16 cm plastic fan was mounted on each of the 20 hexagonal faces, with a large torque driving motor mounted outside of the flow field and connected to the fans, which are inside, via shafts going through the chamber wall. The rotation speeds of each fan are synchronously controlled and continuously variable within a reliable range of 1500–3500 RPM (driving frequency from 30 Hz to 70 Hz) by using a power frequency modulated controller (precision of 0.1 Hz).

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By increasing both the size and degree of symmetry of the flow apparatus as well as choosing larger fans, we expected this new HIT chamber to provide a nearly zero-mean turbulent flow with improved flow homogeneity and isotropy, a larger HIT region, and higher $~{{R}_{\lambda}}$ than our previous cubic HIT chamber [12].

Compared to the apparatus by Chang et al [10], our new apparatus not only allows whole-field (such as PIV) flow measurements compared to single-point LDV measurements, but also offers the advantage of minimal gravity effect on inertial particles. The ${{R}_{\lambda}}~\sim ~{{n}^{1/4}}$ relationship indicates that by choosing $n=20$ instead of 32, our ${{R}_{\lambda}}$ is sacrificed moderately (by 11%). However, this keeps the task of balancing the lengths of the fan-motor assemblies simple by using identical assemblies mounted on a 'sphere', as explained in section 2.1.

Several noninvasive (optical) techniques using flow tracer particles are readily available for measuring turbulence: Laser Doppler Velocimetry (LDV), PIV, and Particle Tracking Velocimetry. LDV is a single-point measurement technique. Unlike in wind tunnels where a large mean velocity advects all tracer particles in one direction, Taylor's hypothesis, which relates temporal to spatial fluctuations in turbulent flows, cannot be applied to near-zero-mean turbulence [25]. Furthermore, some turbulence properties such as spatial covariance of velocity and velocity structure function require performing two-point statistics from velocities measured simultaneously at different locations. Although it is possible to perform two-point velocity measurement using two LDV probes, there is a delay between the velocity signals from the two probes [10, 11]. Moreover, two-point statistics by using two-probe LDV requires measurements to be repeated for many different distances between the two points. On the other hand, PIV is an instantaneous whole-field flow measurement technique providing simultaneous multi-point velocity. Hence, PIV is preferred as our flow-field interrogation technique.

However, PIV presents some special challenges when applied to a nearly zero-mean enclosed chamber. For lack of a mean flow, turbulence velocity has a wide dynamic range, making the selection of $\Delta t$ (exposure time delay) in PIV difficult [26]. Likewise, the selection of camera magnification also faces conflicting requirements: zooming in will improve spatial resolution but reduce field of view; consequently each interrogation cell (wherein PIV correlation is performed) will correspond to a smaller physical size in the flow field, thus limiting the upper bound of the measurable velocity. Zooming out, on the other hand, will improve the field of view but sacrifice spatial resolution, thus limiting the lower bound of the measurable velocity. Additionally, if we zoom out to an inadequate spatial resolution, PIV measurement will underestimate turbulent fluctuations [27]. This makes it difficult to acquire PIV images with a high spatial resolution while capturing a large flow field accurately in a single PIV measurement, leading to the need for multiple measurements at different magnification factors.

In previous PIV measurements that require both high spatial resolution and large field of views, various approaches have been taken to expand the dynamic range of PIV, by either varying the lens magnification of a single PIV system, as in the measurement of both large- and small-scale turbulence statistics of an orifice jet by Lacagnina and Romano [28], or by an array of nine cameras, as in turbulent boundary layer measurement in a wind tunnel by de Silva et al [29]. These approaches are not practical in our application. We instead propose a two-scale PIV technique by combining two independent PIV systems, referred to as 'small-view' and 'large-view', with two distinct magnification factors, and overlapping their imaging fields in the central region of the chamber. The different CCD resolutions of these two PIV systems enable one system to measure small velocities at a high spatial resolution and the other system to capture a large flow field including more flow structures. This two-scale approach greatly improves the dynamic range of measurement.

Furthermore, the two-scale PIV approach offers the capability to cross-validate and enhance turbulence statistics measurement in the zero-mean turbulence. We use the flow statistics (mean velocity and turbulence strength) obtained from small-view PIV to improve the large-view PIV setup by fine-tuning its parameters including particle seeding density, $\Delta t$, and depth of focus. By crosschecking the final measurement results from the small- and large-view PIV for consistency, we gain additional confidence over the accuracy of critical turbulence properties such as dissipation rate and ${{R}_{\lambda}}$.

The proposed two-scale PIV measurement approach is illustrated in figure 4. A laser light sheet illuminates the central region of the 'soccer ball' flow chamber. A PIV camera views the flow field through the optical window, which is enlarged in the figure to show the fields of view proportionally for the large- and small-view PIV image acquisition systems. The blue box and red box denote the physical dimensions of the large-view (48.6 mm  ×  43.3 mm, 16.2 μm/pixel) and small-view (7.1 mm  ×  5.7 mm, 5.56 μm/pixel) PIV images, respectively. Note that both PIV measurement areas are located at the geometrical center of the chamber.

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The two PIV systems share the same laser and optical setup in order to ensure that images are obtained from the same plane. The laser was a Solo-PIV miniLase III, providing a light sheet at a thickness around 1 mm through a spherical and a cylindrical lens. The two PIV image acquisition systems had different hardware, as described below. The small-view PIV system consisted of an IDT SharpVision 1300-DE camera (1280  ×  1024 pixel resolution, 8-bit, double exposures at 10 Hz) attached to an Infinity-USA K Series long-distance microscope for image capturing. The small-view PIV camera and the laser were synchronized by the IDT Provision PIV software and an IDT timing hub. The large-view PIV system used a PCO 4000 CCD camera (3000  ×  2672 pixel resolution, 14 bit, double exposures at 2 Hz) with a Nikon-NIKKOR 200 mm f/4 D Close-Up Lens. The large-view PIV camera and the laser were synchronized by two Stanford Pulse Generators, and PCO CamWare software was used to capture images. The PIV cameras determine the data acquisition frequencies for each setup: 10 image pairs per second for the small-view PIV and 2 image pairs per second for the large-view PIV. The laser double-pulse frequency was adjusted accordingly for the two setups. The distance between the camera lens and the visualization window was fixed at 10 cm. Under this condition, the small-view and large-view PIV systems had magnification factors (physical dimension divided by the image dimension) of 0.83 and 1.8, respectively.

We used aluminum powder particles of $1\mu m$ diameter ($\rho =2.7~\text{g}/\text{c}{{\text{m}}^{3}})$ as tracer particles. These particles were injected into the chamber by a solid particle nozzle mounted on top of the apparatus while the fans were running at given rotating speeds. We chose PIV exposure time delay ($\Delta t$) based on the criteria given in equation (8.88) by Adrian and Westerweel [26]. PIV images from both systems were processed using an in-house modified version of the open-source MatPIV 1.6.1 code [30], with 50% overlap of interrogation cells. Interrogation cell size was determined by changing the size of interrogation cell from 28  ×  28 pixels to 98  ×  98 pixels with 10 pixels increments. It was found that at 48  ×  48 pixels, the small-view PIV image pairs produced the clearest distinguishable peaks during image-pair cross-correlation, and the averaged signal-to-noise ratio (SNR)—the height of the primary peak and the height of the second tallest peak—was 2.47. The large-view PIV image pairs produced the clearest, distinguishable correlation peaks when interrogation cell size was 28  ×  28 pixels, which gave an averaged SNR of 3.91. We subsequently processed PIV images using these pixel sizes. These corresponded to resolved cell sizes of $267\times 267$ $\mu m$ and $454\times 454~\mu m$ in the chamber for the small- and large-view PIV setups, respectively. Furthermore, we determined measurement uncertainties of the two-scale PIV system based on the SNR of PIV image pairs [31]. For the small- and large-view PIV measurements, the uncertainties were $\pm$0.12 pixel and $\pm$0.10 pixels, corresponding to $\pm$4% and $\pm$5% of turbulence RMS velocity fluctuation uncertainty, respectively.

Although our 'soccer ball' chamber is equipped with two visualization windows on opposite sides of the chamber, due to confinement of laboratory space, the small-view and large-view PIV measurements were taken through only one window sequentially under identical flow conditions to provide turbulence statistics.

Before performing turbulence characterization, we checked the statistical repeatability of flow in the HIT chamber. If the turbulence in the apparatus is statistically repeatable under a given operating condition, then temporal averaging can be carried out instead of ensemble averaging to reduce the cost of experiments. To that end, we took 6 independent time series of PIV measurement using the small-view PIV system at the fan speed of 3000 RPM. In each time series, we collected 500 PIV image pairs continuously and analyzed velocity statistics (mean velocity and turbulence strength) at the center of the field of view. These statistics were compared across the 6 individual tests to check for measurement consistency.

During HIT turbulence characterization, we performed PIV measurements under 6 flow conditions inside the chamber: varying fan speed from 1500 RPM to 3500 RPM in an increment of 500 RPM provided 5 flow conditions, and an additional fan speed of 3250 RPM was also included to increase the data density in the higher end of ${{R}_{\lambda}}$. To avoid perturbation of the flow from particle injection, PIV data were only recorded after the particle injector had been closed for at least 60 s.

Under each flow and PIV recording condition, PIV measurements were taken in 10 separate runs with an ample time lapse in between, instead of continuously. This approach helped minimize the temperature rise in the flow chamber from heating due to fan operation. The length of the time window for each data collection run was determined by the CCD camera memory. The small-view PIV camera allowed for a batch of 500 image pairs to be recorded and transferred to the computer each time. Its framing rate was 10 image pairs per second. Therefore, each run lasted 50 s. For the large-view PIV measurements, the camera allowed for a batch of 200 image pairs to be recorded and transferred. Its framing rate was at 2 image pairs per second. Therefore, each run lasted 100 s. We have determined that within a 100 s run, the air temperature inside the chamber did not increase significantly. Under the highest fan speed (3500 RPM), the maximum temperature rise was ${{3}^{o}}C$, corresponding to $1.5\%$ increase in air viscosity. Between runs we allowed sufficient time for the facility to cool down to room temperature. In order to determine if the convergence of turbulence statistics was achieved within one run, we plot the mean velocity and turbulence strength against the image pair recording time, both for small-view and large-view PIV.

The size of homogeneous and isotropic region and the degree of homogeneity and isotropy are two important considerations of our HIT chamber (see section 2.3). The whole-field flow measurement capability of PIV allowed us to examine the spatial distributions of homogeneity index $\chi$ and isotropy index $\gamma$. We employed the large-view PIV measurement (field of view $=48.6~\text{mm}\times 43.3~\text{mm}$) to quantify the homogeneous and isotropic size and degree of homogeneity and isotropy of our new flow facility. To describe the spatial domain in the flow field, we introduced a Cartesian coordinate system with the origin point at the geometrical center of the chamber, which was also the center of the PIV view. The X- and Y-axes were along the horizontal and vertical directions, respectively.

By applying equations (1) and (2) at each PIV interrogation cell, we were able to measure homogeneity index $\chi$ and isotropy index $\gamma$ distribution in the field of view. At each interrogation cell, we calculated $\chi$ and $\gamma$ ensemble-averaged over 10 large-view PIV runs. The data presented in section 4.3 was collected at fan speed of 3000 RPM. We found that up to 3500 RPM of fan speed, the test data of $\chi$ and $\gamma$ displayed no significant differences with respect to fan speed.

A challenge in characterizing turbulence in a flow facility is the estimation of turbulent kinetic energy dissipation rate $\varepsilon$. Options for estimating $\varepsilon$ include the scaling argument method [32], spectral fitting method [3, 33, 34], direct method [35–37], large eddy PIV method [38], and the structure function fit method [39]. We discussed in detail the performance of these different methods for estimating $\varepsilon$ from PIV velocity field measurements in our previous HIT chamber [12]. We concluded that the ${{R}_{\lambda}}$-corrected, second-order longitudinal velocity structure function method was the most appropriate method to estimate the dissipation rate in homogeneous and isotropic turbulent flow [12]. Therefore, we adopted the same method in characterizing turbulence dissipation rate in our 'soccer ball' apparatus.

The longitudinal velocity structure function is defined as [12]

$$\begin{eqnarray}&&{{D}_{{{L}^{N}}}}={{\left[\left(\vec{u}\left(\vec{x}+\vec{r}\right)-\vec{u}\left(\vec{x}\right)\right)\cdot \widehat{r}\right]}^{N}},\end{eqnarray} \tag{ 15 }$$

where $N$ is the order of the structure function, $\vec{r}$ is the separation vector of two fluid elements, $\overset{{}}{\mathop{{\widehat{r}}}}$ is the unit vector which points in the direction of $\vec{r}$, and $\vec{u}~$ is the fluctuating velocity in the $\vec{x}$ direction. When $N=2$, equation (15) becomes the second-order longitudinal velocity structure function ${{D}_{LL}}(r)$.

From Kolmogorov's first and second similarity hypotheses, in the inertial subrange the second-order structure function can be expressed as [2]

$$\begin{eqnarray}&&{{D}_{LL}}\left({{r}_{IR}}\right)={{C}_{2}}\left({{R}_{\lambda}}\right){{\left(\varepsilon {{r}_{IR}}\right)}^{2/3}}.\end{eqnarray} \tag{ 16 }$$

Note that this is valid only in the inertial subrange, i.e. when the separation distance $r={{r}_{IR}}$. Rearranging equation (16), we can express turbulence dissipation rate as

$$\begin{eqnarray}&&\varepsilon ={{\left[\frac{{{D}_{LL}}\left({{r}_{IR}}\right)}{{{C}_{2}}\left({{R}_{\lambda}}\right)}\right]}^{\frac{3}{2}}}\left(\frac{1}{{{r}_{IR}}}\right),\end{eqnarray} \tag{ 17 }$$

Where ${{r}_{IR}}$ is any separation distance in the inertial subrange, where $\varepsilon$ is a constant. In order to determine $~{{r}_{IR}}$, we can plot ${{\left[\frac{{{D}_{LL}}(r)}{{{C}_{2}}\left({{R}_{\lambda}}\right)}\right]}^{\frac{3}{2}}}\left(\frac{1}{r}\right)$ as a function of $r$. When it plateaus (reaches a constant), the constant value is taken as $\varepsilon$. Equation (17) provides foundation for estimating turbulence dissipation rate $\varepsilon$ though the second-order structure function ${{D}_{LL}}\left({{r}_{IR}}\right)$.

From $\varepsilon$, we can calculate ${{R}_{\lambda}}$ through Taylor microscale ${{\lambda}_{g}}$, which is related to dissipation rate $\varepsilon$ as [1, 2]:

$$\begin{eqnarray}&&{{\lambda}_{g}}={{u}^{\prime}}\sqrt{15\nu /\varepsilon}.\end{eqnarray} \tag{ 18 }$$

Plugging equation (18) into equation (3), we obtain

$$\begin{eqnarray}&&{{R}_{\lambda}}={{u}^{\prime2}}\sqrt{15/\nu \varepsilon}.\end{eqnarray} \tag{ 19 }$$

To calculate $\varepsilon$ and ${{R}_{\lambda}}$ using equations (17) and (19) respectively, we need to estimate the constant ${{C}_{2}}\left({{R}_{\lambda}}\right)$, which is dependent on the unknown ${{R}_{\lambda}}$. This implicit equation set can be solved by iteration. We assumed an initial value ${{C}_{2}}\left(\infty \right)=2.12~$[40], which enabled the initial estimates of $\varepsilon$ and ${{R}_{\lambda}}$. Subsequent corrections to the constant ${{C}_{2}}\left({{R}_{\lambda}}\right)$ were made by using figure 6 from Yeung and Zhou [41]. With the new, iterated values of ${{C}_{2}}\left({{R}_{\lambda}}\right)$, new estimates for $\varepsilon$ and ${{R}_{\lambda}}$ were obtained. Final values of $\varepsilon$ and ${{R}_{\lambda}}$ were obtained when the iteration results are within 1% differences between successive iterations.

It should be noted that calculating dissipation rate $\varepsilon$ and ${{R}_{\lambda}}$ through velocity structure function requires instantaneous velocity measurement at two points separated by a distance $r$. Furthermore, the separation distance $r$ also needs to be varied to give ${{D}_{LL}}(r)$, and subsequently ${{D}_{LL}}\left({{r}_{IR}}\right)$ in the inertial subrange. The use of PIV is intrinsically more advantageous than LDV for getting instantaneous multi-point velocity measurements for this purpose.

Figure 5(a) shows the time-averaged velocity components $\left({{U}_{\text{mean}}},~{{V}_{\text{mean}}}\right)$ at the center of field of view from six independent tests. These mean velocity values were all within 0.1 m/s from zero, which is negligible compared to the turbulence strength of 1.33 $\text{m}\,{{\text{s}}^{-1}}$. This reflects the 'nearly zero-mean' characteristics of the HIT chamber. There was no preferred sign for the residual mean velocity. Furthermore, turbulence strength values from different tests, plotted in figure 5(b), were within 5% from each other. We thus conclude that our apparatus has acceptable statistical repeatability.

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Figure 6 plots the mean velocity and turbulence strength at the center of field of view as a function of PIV recording time through one single run, both for small- and large-view PIV measurement. It is evidenced that the convergence of turbulence statistics is achieved by the end of the experimental run (500 and 200 image pairs for the small- and large-view PIV setups, respectively). Moreover, all the turbulence statistical characterizations described later in this paper are calculated based on larger sample sizes, 2000 and 5000 image pairs for small- and large-view PIV, respectively.

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After the statistical repeatability test, we carried out the entire turbulence characterization as described in sections 3.4–3.6. Figure 7 plots the mean flow velocity ${{U}_{\text{mean}}}$ and ${{V}_{\text{mean}}}$ in the horizontal and vertical directions, respectively, from both PIV measurements, for all six fan speeds. The mean velocity was essentially zero at low fan speeds. Deviations from zero increased with increasing fan speeds to a maximum of $0.21~\text{m}\,{{\text{s}}^{-1}}$.

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Figure 8 plots turbulence strength ${{u}^{\prime}}$ from both small- and large-view PIV measurements against fan speed $f$, demonstrating a linear dependence. Separate linear regressions were performed for the small- and large-view data, producing almost identical linear fits. The highest ${{u}^{\prime}}$ was 1.57 m s⁻¹ under the highest fan speed of 3500 RPM. The linear dependence of the measured ${{u}^{\prime}}$ on fan speed provides strong experimental support for equation (8) (i.e. linear relationship of turbulence kinetic energy to individual fan velocity), which was a key assumption for our derivation of the ${{R}_{\lambda}}$ expressions in equations (10) and (12). Recall the argument for equation (8) was: The turbulent kinetic energy $k$ in the center of the HIT chamber should be proportional to the kinetic energy input from all the actuators, $n\cdot U_{0}^{2}$, due to energy conservation. Note that $k=\frac{3}{2}{{u}^{\prime2}}$ and ${{U}_{0}}~\sim ~\omega$, as shown in equation (11). Thus the energy argument $~k~\sim ~U_{0}^{2}$ is equivalent to ${{u}^{\prime}}~\sim \omega$. This linear relation of turbulence strength on fan rotational speed is confirmed by the experimental results shown in figure 8.

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Figure 9 shows a spatial distribution of the homogeneity index $\chi$ ensemble averaged from 10 runs of large-view PIV, with figure 9(a) showing 2D distribution of $\chi$ and figure 9(b) showing $\chi$ value plotted along the horizontal line $\,y=0$. We can see that $\chi \approx 1$ with a small range of fluctuations (0.97–1.02), indicating that the HIT chamber provides a high degree of homogeneity. Using the criterion of $~\chi =1\pm 0.05$, we can see that 'homogeneous region' existed in a region of at least $\pm 24\,\text{mm}$ about the center of the chamber.

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Figure 10 shows the measurement results of isotropy index $~\gamma$ distribution, ensemble-averaged over 10 large-view PIV runs, where figure 10(a) shows 2D distribution of $\gamma$ and figure 10(b) shows $\gamma$ along the horizontal line $y=0$. We can see $\gamma \approx 1$ with a small range of fluctuations (0.95–1.03), which indicates that the HIT chamber provides a high degree of isotropy. Using the criterion of $~\gamma =1\pm 0.05$, we can see that 'isotropic region' existed in a region of $\pm 24\,\text{mm}$ in the center of the chamber.

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Assuming spherical symmetry of the apparatus, we infer that the homogeneous and isotropic region is a sphere with a diameter of $48\,\text{mm}$ at the center of the chamber. This is a significant improvement over our previous cubic chamber ($22~\text{mm}$), both size and quality [42].

In order to estimate turbulence dissipation rate $\varepsilon$ using equation (17), we calculated the second-order longitudinal velocity structure function ${{D}_{LL}}(r)$ using equation (15), where $r$ was varied along the $x$ direction. This was done for all experimental conditions. In order to determine the inertial subrange and thus $\varepsilon$, we plotted the compensated longitudinal second-order structure function ${{\left[{{D}_{LL}}(r)/{{C}_{2}}\left({{R}_{\lambda}}\right)\right]}^{\frac{3}{2}}}\left(\frac{1}{r}\right)$ against separation distance $r$. When it plateaus (reaches a constant), the constant value is taken as $\varepsilon$ according to equation (17). Any ${{r}_{IR}}$ in the range of the plateau (inertial subrange) gives the same result of $\varepsilon$. Figures 11(a) and (b) show these functions under different fan speeds for the small-view and large-view PIV measurements, respectively. All $\varepsilon$ values are marked on the axes on the right of the plots.

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Comparing figures 11(a) and (b), we see that the small-view and large-view PIV measurements yielded similar dissipation rate values. However, the large-view measurement accounted for a much larger separation distance r. To compare the measured dissipation rate as a function of fan speed for both sets of PIV measurement, we generated figure 12, which demonstrates an excellent agreement across all flow conditions from the two PIV views. Power fit was used to find the relation between dissipation rate $\varepsilon$ and fan speed $f$. Exponents of the curve-fitting results were found to be 2.83 (small-view PIV) and 2.54 (large-view PIV). This experimental result is roughly consistent with the scaling argument of dissipation rate in equation (4), where $\varepsilon$ is proportional to the turbulent kinetic energy $k$ to the power 3/2, and thus to ${{u}^{\prime}}$, as well as to $f$, to the power 3. This consistency suggests that our two independently measured dissipation rates from the two-scale PIV system are reasonable and in agreement with each other.

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Based on the relationship between $~{{R}_{\lambda}}~$ and $\varepsilon$ given in equation (19), we calculated ${{R}_{\lambda}}~$ for each flow condition using both PIV views. The results are plotted as ${{R}_{\lambda}}$ versus fan speed in figure 13. Power fit was used to find the experimental relationship between ${{R}_{\lambda ~}}$ and fan speed, showing exponents to be 0.49 (small-view PIV) and 0.51 (large-view PIV). A good agreement between the two PIV measurements was achieved. These experimental results also lend strong support to equation (12), the ${{R}_{\lambda ~}}$ equation we derived theoretically and used for our chamber design. Equation (12) predicted that ${{R}_{\lambda ~}}$ scales with the rotational fan speed $f\,$ to the power of 0.5.

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The experimentally validated square-root dependence of ${{R}_{\lambda}}\sim ~{{f}^{1/2}}$ actually reflects a well-known scaling that ${{R}_{\lambda}}~$ is proportional to the square root of the integral-scale Reynolds number $~\text{R}{{\text{e}}_{L}}$, as shown in equation (6) as well as in Pope [2]. In wind tunnels, $~\text{R}{{\text{e}}_{L}}$ is proportional to the inlet flow speed and thus to the fan (blower) speed of the wind tunnel. Here, we have shown both theoretically and experimentally that a fan-driven enclosed chamber also follows a square-root relationship between ${{R}_{\lambda}}$ and fan speed.

Since the large-view PIV measurement accounted for a larger portion of the homogeneous and isotropic field compared to small-view PIV measurement, and since it agreed rather well with the latter, we used the large-view PIV results as the basis for calculating other turbulence quantities.

The ${{R}_{\lambda}}$ in the 'soccer ball' chamber at the maximal fan speed was calculated to be ${{R}_{\lambda}}=384$, which is within 7.5% difference from the theoretical prediction of $R_{\lambda}^{\text{est}}=415.$ This agreement again supports equation (12)—our theoretical relationship between ${{R}_{\lambda}}$ and physical parameters of a fan-driven HIT chamber. The ${{R}_{\lambda}}$ levels measured in our apparatus match the current DNS capabilities.

Table 1 shows the full turbulence characterization of the 'soccer ball' chamber under all six different flow conditions from large-view PIV measurement. From the mean velocity values, we can see that near-zero-mean flow is achieved across all fan speeds in our chamber. From the Froude number values, we can see that the fan-generated turbulence acceleration at the Kolmogorov scale was 4.4–25.0 times of the acceleration of gravity, which indicates that the gravity effect on inertial particles was insignificant compared to the turbulence effect in our chamber. It is also seen that, as fan speed increased, turbulent kinetic energy dissipation rate increased, while the Taylor microscale and Kolmogorov length/time scales decreased.

Table 1. Full turbulence characterization in the 'soccer ball' HIT chamber.

Fan speed (RPM)	1500	2000	2500	3000	3250	3500
${{\sigma}_{x}}\left(\text{m}/\text{s}\right)$	0.68	0.93	1.16	1.35	1.47	1.56
${{\sigma}_{y}}\left(\text{m}/\text{s}\right)$	0.72	0.95	1.17	1.32	1.51	1.56
${{U}_{\text{mean}}}\left(\text{m}/\text{s}\right)$	0.02	0.02	0.09	0.07	0.06	-0.04
${{V}_{\text{mean}}}\left(\text{m}/\text{s}\right)$	0.01	0.06	0.03	0.02	0.02	-0.02
Turbulence strength, ${{u}^{\prime}}\,=\sqrt{\left(\sigma _{x}^{2}+\sigma _{y}^{2}\right)/2~}\left(\text{m}/\text{s}\right)$	0.70	0.94	1.16	1.33	1.49	1.56
Turbulent kinetic energy,$\,k=\frac{3}{4}\left(\sigma _{x}^{2}+\sigma _{y}^{2}\right)$ ($\left. {{\text{m}}^{2}}/{{\text{s}}^{2}}\right)$	0.73	1.33	2.03	2.67	3.32	3.65
Eddy turnover time, ${{T}_{\text{e}}}={{\lambda}_{g}}/{{u}^{\prime}}\left(\text{ms}\right)$	7.9	4.9	3.8	2.9	2.5	2.5
Dissipation rate, $~\varepsilon ~\left({{\text{m}}^{2}}/{{\text{s}}^{3}}\right)$	3.6	9.2	16.5	27.0	35.9	47.0
Large eddy length scale, $L=\frac{{{k}^{3/2}}}{\varepsilon}\left(\text{m}\right)$	0.18	0.16	0.18	0.17	0.17	0.18
Large Eddy Time Scale, ${{T}_{l}}=\frac{L}{{{u}^{\prime}}}\,\left(\text{s}\right)$	0.24	0.18	0.15	0.13	0.11	0.11
Kolmogorov length scale, $~\eta ={{\left({{\nu}^{3}}/\varepsilon \right)}^{1/4}}\left(\mu \text{m}\right)$	179	141	123	109	101	100
Kolmogorov time scale, ${{\tau}_{k}}={{\left(\nu /\varepsilon \right)}^{1/2}}\left(\text{ms}\right)$	2.0	1.3	1.0	0.8	0.7	0.6
Kolmogorov velocity scale, ${{v}_{k}}={{\left(\nu \varepsilon \right)}^{1/4}}\left(\text{m}/\text{s}\right)$	0.09	0.11	0.13	0.14	0.16	0.16
Taylor Microscale, $\,{{\lambda}_{g}}={{(15\nu {{u}^{\prime2}}/\varepsilon )}^{1/2}}(\text{mm})$	5.5	4.6	4.4	3.9	3.8	3.9
Froude number, $Fr={{v}_{k}}/\left(9.8{{\tau}_{k}}\right)$	4.4	8.9	13.4	19.3	24.2	25.0
Taylor-microscale Reynolds number, ${{R}_{\lambda}}={{u}^{\prime}}{{\lambda}_{g}}/\nu$	246	277	324	334	357	384

The Kolmogorov length scale $\eta$ varied from 179 to 100 $\mu \text{m}$ in the HIT chamber. The PIV measurement interrogation cell sizes ($267\times 267~\mu \text{m}$ for large and $454\times 454~\mu \text{m}$ for small-view) were both smaller than 5$\eta$, thus satisfying the spatial resolution requirement for PIV measurement in turbulence [27].