2.1 Apparatus

Experiments are conducted in the DeFrees Hydraulics Laboratory at Cornell University in a 1.000 m tall tank with a 0.800 by 0.800 m horizontal cross-sectional area. Turbulence is generated by RASJAs suspended above the facility. A 1.27 cm thick glass plate is mounted into the bottom of the facility to provide a stable rigid bed. The top of the glass is located 8 cm above the base of the tank for adequate optical access.

The coordinate system is shown in figure 1, with $z=0$ at the top of the bed increasing upwards to $H$ , the height of the jet orifice plane relative to the bed. At the lateral centre of the facility, $x=0$ and $y=0$ , each orthogonal to the side walls and following the right-hand rule. Velocity components $U$ , $V$ and $W$ follow the $x$ -, $y$ - and $z$ -directions, respectively.

Figure 1. Schematic drawing of the turbulence facility with the $8\times 8$ RASJA.

Figure 2. Photo of tank with the $8\times 8$ RASJA.

2.2 Measurement techniques

ADV measurements are made using a Nortek Vectrino with ‘plus’ firmware to record three components of velocity at a single point location. All results shown herein are located 12.0 cm above the bed at the lateral centre of the tank, as shown in figure 2. The instrument is mounted vertically, with the $z$ -axis of the ADV aligned with the $z$ -axis of the tank, and the instrument $x$ - and $y$ -axes orthogonal to the tank walls. Measurements are recorded at a sampling frequency of 100 Hz with a sampling volume length of 7.0 mm and transmit length of 1.8 mm. To ensure convergence of turbulence statistics, data records are at least 30 min long. An adaptive Gaussian (AGW) filter (Cowen & Monismith Reference Cowen and Monismith1997) is applied to eliminate spurious measurements due to instrument noise and/or instantaneously low seeding density. Less than 1 % of the data are eliminated through this method. Filtered data points are linearly interpolated to compute frequency spectra as they require a continuous temporal record.

Arkema Group ORGASOL (R) 2002 ES 3 Nat 3 Polyamide 12 nylon particles are used to seed the flow for accurate ADV measurements. These particles feature an average batch diameter $D_{p}=29.4~\unicode[STIX]{x03BC}\text{m}$ , with 5 % less than 20  $\unicode[STIX]{x03BC}\text{m}$ and 8 % greater than 40  $\unicode[STIX]{x03BC}\text{m}$ . With a specific gravity of 1.03, they are effectively neutrally buoyant. For all flow cases considered, the Stokes number $St=(\unicode[STIX]{x1D70F}_{R}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D707}})$ in which $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D707}}$ represents the Kolmogorov time scale (details presented in § 5.5) and $\unicode[STIX]{x1D70F}_{R}=((S)D_{p}^{2}/18\unicode[STIX]{x1D708})$ is a relaxation time scale, we find values of $St$ consistently less than 0.001. With $St\ll 1$ , we conclude that the particles follow the flow as passive tracers.

PIV is used to record spatio-temporal data in the lateral centre of the tank, in the $x{-}z$ plane, at the bottom boundary. Measurements are collected using an Imperx Bobcat IGV-2020 camera with 2056 by 2060 pixel resolution and either a Nikkor 50 mm lens or Nikon 60 mm lens, each with $f/2.8$ . Illumination is provided by a Coherent Innova 90 Argon Ion laser operated at approximately 3 W power with wavelengths of 488 and 514.5 nm in multi-line mode. The beam, which is 1.4 mm in diameter, passes through a 2.5 mm mechanical shutter (NM technologies LS200) and is scanned through the planar field-of-view (FOV) with a mirror (Cambridge Technologies 6M8505X-V) attached to a galvanometer (Cambridge Technologies 6860). Each scan is completed within 5 ms, and the time, $\unicode[STIX]{x0394}t$ , between the scans is 8 ms.

The shutter, mirror and camera are synchronized using a National Instruments analogue output card (PCI-6711) controlled by MATLAB. The same nylon ORGASOL particles are also used for PIV measurements, though PIV and ADV data are not collected simultaneously. Velocity fields are acquired at a sampling frequency $f_{s}$ of 1 Hz to ensure uncorrelated samples. Results reported herein are for tests of 30 min in duration, which was found to be a sufficient period of time for statistics to converge to the same levels as if tests were run over a 24 h period (Variano & Cowen Reference Variano and Cowen2008).

Image analysis is performed in MATLAB using a sub-pixel cross-correlation peak locating PIV algorithm developed by Liao & Cowen (Reference Liao and Cowen2005) and based on Cowen & Monismith (Reference Cowen and Monismith1997). This accurately determines particle displacements between image pairs separated by a time $\unicode[STIX]{x0394}t$ , with a spectral continuous subwindow shifting method to improve subpixel particle displacements. In order to reduce error caused by tracer particles that are highly sheared due to the energy of the turbulent flow, we artificially expand the illuminated tracer particles by convolving the images with a $4\times 4$ Gaussian kernel using MATLAB’s imfilter function, as is described in Variano (Reference Variano2007).

Before analysing the images, we perform pre-processing to remove background noise introduced by uniform ambient light and reflections off of the glass bed. As in Cowen & Monismith (Reference Cowen and Monismith1997), we look temporally across all pixels in the first image of every pair and compute the minimum light intensity to compile a single background image. We repeat this process to compute a background image for all of the second images. These background images are subtracted from all of the raw images.

An initial interrogation is completed with $64\times 64$ pixel subwindows with 50 % overlap in order to determine a first estimate for particle displacements across the FOV. There are 6 iterations for the algorithm to converge upon pixel displacements in each subwindow. The resulting converged vectors from the initial pass are smoothed according to the median of valid vectors within a $3\times 3$ array of neighbouring subwindows. The results of the median filter smoothing of valid particle displacements are then used to guide a more refined grid with $32\times 32$ pixel subwindow interrogation of the images, again with 50 % overlap, to obtain the final particle displacements. The resulting spatial resolution for the experiments presented herein is 1.76 mm from subwindow centre-to-centre with an approximately $20\times 20~\text{cm}$ FOV, depending on the precise location of the camera.

Several post-processing filters are applied to reduce spurious velocity vectors from the measurements. First, unconverged vectors are removed. An AGW filter is then applied to remove the uniformly distributed spurious vectors that lie outside the statistical bounds of the assumed Gaussian distributed turbulence measurements. This is performed across all time at given subwindow heights above the bed. Finally, a $5\times 5$ local median filter, with a threshold determined by the user (Westerweel Reference Westerweel1994; Cowen & Monismith Reference Cowen and Monismith1997) is applied to remove spurious data within an instantaneous image that lie within the statistical bounds of the assumed Gaussian distributed turbulence measurements. Between 80 and 97 % of data are declared valid, with regions of high shear contributing to fewer valid vectors. Removed data are not replaced through interpolation, as interpolation can significantly alter resulting statistical analyses.

Velocity data are Reynolds decomposed such that $U(x,y,z,t)=\langle U(x,y,z)\rangle +u(x,y,z,t)$ and likewise for $V$ and $W$ . The angle brackets denote a temporal average and lower case letters represent fluctuations. According to ADV data, the flow is radially symmetric about the $z$ -axis (see Johnson (Reference Johnson2016) for details), thus we only report statistics along the $x{-}z$ plane. Lateral variations across the 20 cm FOV are sufficiently small to allow us to invoke horizontal homogeneity. We use an overbar to indicate averages that include time and space (i.e. the horizontal average of $\langle U\rangle$ is  $\overline{U}$ ).

We use the bootstrap method (Efron & Tibshirani Reference Efron and Tibshirani1993) to construct 95 % confidence intervals of the turbulence statistics to compute uncertainty bounds. With our assumptions of convergence and horizontal homogeneity, the bootstrap analyses presented typically utilize between 1800 and 225 000 data points, resampled 1000 times, with replacement, to generate ordered random samples. The 95 % confidence interval is determined by the 97.5 percentile and 2.5 percentile statistic.

3.1 Mean and fluctuating velocities

As expected, temporally averaged flow fields show near-zero mean velocity profiles for both RASJAs. Of the 15 experimental cases considered with the $8\times 8$ RASJA, we find typical mean horizontal velocities with magnitudes of approximately $0.15~\text{cm}~\text{s}^{-1}$ . The vertical forcing of the jets induces a weak mean decaying downward flow in the centre of the tank, with return flows at the walls. Typical values of these downward velocities are approximately $0.47~\text{cm}~\text{s}^{-1}$ in the mixed region of the facility (i.e. the region below which the jet wakes have merged and above which the flow is affected by the presence of the boundary). These velocities are low relative to the velocity fluctuations, as is presented in § 3.2.

We define the bed-normal r.m.s. velocity as $u^{\prime }=\sqrt{\overline{u^{2}}}$ (and likewise for $v^{\prime }$ and $w^{\prime }$ ) to consider the strength of the turbulent velocity fluctuations. Figure 3 explores the influence of $T_{on}$ (left) and $\unicode[STIX]{x1D6F7}_{on}$ (right) on $u^{\prime }$ and $w^{\prime }$ for the $8\times 8$ RASJA. While $\unicode[STIX]{x1D6F7}_{on}$ has a relatively negligible impact on $u^{\prime }$ and $w^{\prime }$ , we observe a strong dependence upon $T_{on}$ . As subsequent statistical analyses also show a greater dependence upon $T_{on}$ rather than $\unicode[STIX]{x1D6F7}_{on}$ , we present the remaining statistical analyses controlled only by varying $T_{on}$ for a selected $\unicode[STIX]{x1D6F7}_{on}$ of 6.25 %.

Figure 3. Dependence of $u^{\prime }$ and $w^{\prime }$ on $T_{on}=4$  s ( $u^{\prime }=-$ , $w^{\prime }=-\cdot$ ), 6 s ( $u^{\prime }=--$ , $w^{\prime }=\cdot \cdot \cdot$ ), 8 s ( $u^{\prime }=-$ (grey), $w^{\prime }=-\cdot$ (grey)) for $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ (a) and $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ ( $u^{\prime }=-$ , $w^{\prime }=-\cdot$ ), 7.7 % ( $u^{\prime }=--$ , $w^{\prime }=\cdot \cdot \cdot$ ), 9.1 % ( $u^{\prime }=-$ (grey), $w^{\prime }=-\cdot$ (grey)), 10.5 % ( $u^{\prime }=--$ (grey), $w^{\prime }=\cdot \cdot \cdot$ (grey)), 12.5 % ( $u^{\prime }=-$ (grey), $w^{\prime }=-\cdot$ (grey)) for $T_{on}=4$  s (b). $8\times 8$ RASJA.

Given the relative independence of the turbulent velocity fluctuations on $\unicode[STIX]{x1D6F7}_{on}$ observed with the $8\times 8$ RASJA, we select a single $\unicode[STIX]{x1D6F7}_{on}$ of 3.1 % with which to perform 5 tests on the $16\times 16$ RASJA with varying $T_{on}$ . Results from these experiments are summarized in figure 4. Similar trends are observed between the $8\times 8$ and $16\times 16$ RASJAs, with a clear relationship between the r.m.s. velocities and $T_{on}$ . Because the r.m.s. velocities are similar in magnitude in the $8\times 8$ RASJA trial for $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $T_{on}=4$  s and the $16\times 16$ RASJA trial for $\unicode[STIX]{x1D6F7}_{on}=3.1\,\%$ , $T_{on}=0.8$  s, we use these two cases for several sample plots in the remainder of this article to draw comparisons.

Figure 4. Dependence of $u^{\prime }$ and $w^{\prime }$ on $T_{on}=0.8$  s ( $u^{\prime }=-$ , $w^{\prime }=-\cdot$ ), 1.0 s ( $u^{\prime }=--$ , $w^{\prime }=\cdot \cdot \cdot$ ), 1.2 s ( $u^{\prime }=-$ (grey), $w^{\prime }=-\cdot$ (grey)), 1.4 s ( $u^{\prime }=--$ (grey), $w^{\prime }=\cdot \cdot \cdot$ (grey)) 1.6 s ( $u^{\prime }=-$ (light grey), $w^{\prime }=-\cdot$ (light grey)). $\unicode[STIX]{x1D6F7}_{on}=3.1\,\%$ , $16\times 16$ RASJA.

3.2 Turbulent kinetic energy and secondary flows

We compute turbulent kinetic energy, $k$ , from PIV data as $k=1/2(2u^{\prime 2}+w^{\prime 2})$ by invoking radial symmetry. As expected from the r.m.s. velocity results, $k$ increases with $T_{on}$ in the turbulence facility using either the $8\times 8$ RASJA or the $16\times 16$ RASJA, as shown in figures 5 and 6, respectively.

Figure 5. Turbulent kinetic energy profiles for $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $T_{on}=4$  s ( $--$ ), 6 s ( $-\cdot$ ), 8 s ( $\cdot \cdot \cdot$ ).  $8\times 8$ RASJA.

Figure 6. Turbulent kinetic energy profiles for $\unicode[STIX]{x1D6F7}_{on}=3.1\,\%$ , $T_{on}=0.8$  s ( $--$ ), 1.0 s ( $-\cdot$ ), 1.2 s ( $\cdot \cdot \cdot$ ); 1.4 s ( $--$ grey), 1.6 s ( $-\cdot$ grey). $16\times 16$ RAJSA.

As in the literature review of Variano et al. (Reference Variano, Bodenschatz and Cowen2004), we consider $M_{1}$ and $M_{3}$ , the ratios of mean velocity to r.m.s. velocity in the bed-parallel and bed-normal directions, respectively, to evaluate the strength of secondary flows. Clearly, if averaged spatially over the entire facility, both $M_{1}$ and $M_{3}$ tend to zero, as any flow must be balanced by a return flow in another location. Because of this, we consider temporally averaged values of $M_{1}$ and $M_{3}$ across the entire FOV before computing $M_{1}=(\langle U\rangle /\langle u^{\prime }\rangle )$ and $M_{3}=(\langle W\rangle /\langle w^{\prime }\rangle )$ to ensure that averaging along the $x$ -axis accurately represents typical values across the entire width of the FOV. Additionally, we consider a relative mean flow strength $M^{\ast }$ , defined as the ratio of the mean kinetic energy ( $\langle U\rangle ^{2}+(1/2)\langle W\rangle ^{2}$ ) to the turbulent kinetic energy.

Across all cases with both the $8\times 8$ RASJA and $16\times 16$ RASJA, we find $M^{\ast }$ consistently less than 3 % throughout the FOV. We find typical values of $M_{1}$ of approximately 4 %, and typical values of $M_{3}$ around 7 %. Values of $M^{\ast }$ are on average 0.8 % for the $8\times 8$ RASJA and 1 % with the $16\times 16$ RASJA. $M_{1}$ , $M_{3}$ , and $M^{\ast }$ are not correlated with $T_{on}$ . Additional details are presented in Johnson (Reference Johnson2016). As secondary flows were found to be negligible by Variano & Cowen (Reference Variano and Cowen2008) for flows in which $M_{1}$ and $M^{\ast }$ do not exceed 5 %, both RASJAs perform adequately to study turbulence with negligible recirculations or secondary flows.

3.3 Integral length scale

Using PIV data, we compute the integral length scale, ${\mathcal{L}}$ , as the characteristic length scale of the largest eddies of our turbulent flow. At every height above the bed, we compute the spatial longitudinal autocorrelation function

(3.1) $$\begin{eqnarray}a_{11,1}(r)=\frac{\left\langle u(x_{c}-{\displaystyle \frac{r}{2}})u(x_{c}+{\displaystyle \frac{r}{2}})\right\rangle }{\left(\left\langle u(x_{c}-{\displaystyle \frac{r}{2}})^{2}\right\rangle \left\langle u(x_{c}+{\displaystyle \frac{r}{2}})^{2}\right\rangle \right)^{1/2}}\end{eqnarray}$$

such that $r$ is the spatial separation along the horizontal axis, as presented in Variano & Cowen (Reference Variano and Cowen2008). Similarly, the transverse autocorrelation $a_{33,1}(r)$ is computed as a function of $w$ . For sufficiently large measurement regions, $a(r)$ converges to zero with increasing $r$ , and the integral length scale can be computed directly as ${\mathcal{L}}_{L}=\int a_{11,1}(r)\,\text{d}r$ at every height in the FOV. However, our FOV is too narrow to consistently capture this convergence, so we fit an exponential curve $a_{L}(r)=e^{-r/{\mathcal{L}}_{L}}$ to the longitudinal autocorrelation data, as shown in figure 7, to determine the fit parameter ${\mathcal{L}}_{L}$ in $a_{L}(r)$ that best matches $a_{11,1}(r)$ . This modelled curve fit consistently shows coefficient of determination $R^{2}$ values of 0.99 between $a_{L}(r)$ and $a_{11,1}(r)$ , demonstrating an excellent match to the autocorrelation data.

Figure 7. Exponential curve fit $(-\cdot )$ to autocorrelation function $a_{11,1}(\ast )$ . $T_{on}=4$  s,  $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $z=13.02$  cm,  $8\times 8$ RASJA.

By assuming isotropy to address the transverse autocorrelation, we invoke the relationship from Pope (Reference Pope2000),

(3.2) $$\begin{eqnarray}a_{33,1}(r)=a_{11,1}(r)+\frac{1}{2}r\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}a_{11,1}(r),\end{eqnarray}$$

which is modelled as $a_{T}(r)=e^{-r/{\mathcal{L}}_{t}}(1-r/2{\mathcal{L}}_{t})$ according to the exponential fit for $a_{L}(r)$ . Using this model, we solve for ${\mathcal{L}}_{t}$ , as ${\mathcal{L}}_{T}=(1/2){\mathcal{L}}_{t}$ from assuming isotropy (Pope Reference Pope2000). Although this is an appropriate assumption in the mixed region of the flow (to be discussed further in § 4), this assumption is violated near the bed, in particular below the point where $u^{\prime }>w^{\prime }$ due to the kinematic boundary condition. The modelled curve fit has $R^{2}$ values of 0.99. Sample profiles of the longitudinal and transverse integral length scales are shown in figure 8.

Figure 8. Profiles of ${\mathcal{L}}_{L}(--)$ and ${\mathcal{L}}_{T}(-\cdot )$ with 95 % confidence intervals for $T_{on}=4$  s,  $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $8\times 8$ RASJA.

Although the integral length scale in experimental turbulence facilities is often thought to scale strictly with the grid spacing or other geometric constraints, as discussed in Hopfinger & Toly (Reference Hopfinger and Toly1976), it has been shown that the integral length scale can be controlled by varying the parameters of an active grid in wind tunnel experiments (Makita Reference Makita1991; Mydlarski & Warhaft Reference Mydlarski and Warhaft1996). Additionally, the integral length scale is strongly dependent upon distance from the grid in both GST (Thompson & Turner Reference Thompson and Turner1975) and moving-bed experiments (Thomas & Hancock Reference Thomas and Hancock1977).

We consider the integral length scale of the turbulent flow to be equivalent to the longitudinal integral length scale in the mixed region of the tank (to be discussed in the following section) where the energetics and structure of the turbulence are independent of $z$ . We find that we are able to control ${\mathcal{L}}$ by varying $T_{on}$ , as observed in figure 9. Longer $T_{on}$ means the jets penetrate deeper into the flow, injecting more energy, enhancing the degree of turbulent stirring and increasing the integral length scale. Therefore changing $T_{on}$ is equivalent to changing the geometry within a facility, and is a suitable manner in which to vary ${\mathcal{L}}$ using a single RASJA.

Figure 9. Profiles of ${\mathcal{L}}_{L}$ with 95 % confidence intervals for $T_{on}=4$  s $(--)$ , 6 s $(-\cdot )$ , 8 s $(\cdot \cdot \cdot )$ , $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $8\times 8$ RASJA.

For comparison to other experimental and theoretical data, we also compute the integral length scale via the scaling determined relationship ${\mathcal{L}}^{\ast }=k^{3/2}/\unicode[STIX]{x1D716}$ , where $\unicode[STIX]{x1D716}$ is the dissipation rate of turbulent kinetic energy, to be discussed in depth in § 5. As this is not a direct measurement of the integral length scale, it results in values that are within the right order-of-magnitude only of the exponential fit method. Results are presented in § 4.

Because the variations in $T_{on}$ in the $16\times 16$ RASJA are very small compared to those in the $8\times 8$ RASJA, incremented by 0.2 s rather than 2 s, we do not observe discernible differences in the resulting integral length scale measured by the exponential fit to the autocorrelation function. Although ${\mathcal{L}}_{L}$ does not strictly increase with $T_{on}$ , a trend of increasing ${\mathcal{L}}^{\ast }$ is observed with $T_{on}$ , and the incremental changes suggest that changing $T_{on}$ should still result in changing integral length scale as long as the interval in $T_{on}$ is sufficiently large. Due to the different geometry and energetics of the flow between the two facilities, however, we do observe noticeably smaller ${\mathcal{L}}$ in the $16\times 16$ RASJA as compared to the $8\times 8$ RASJA. Resulting values of ${\mathcal{L}}$ , ${\mathcal{L}}_{T}$ , and ${\mathcal{L}}^{\ast }$ will be presented in the following section.

3.4 Integral time scale

As in Peters (Reference Peters1999), we estimate the integral time scale as

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{int}=\frac{{\mathcal{L}}}{\sqrt{k}}\end{eqnarray}$$

or the ratio of the integral length scale to the r.m.s. turbulent velocity scale. For the 15 cases considered with the $8\times 8$ RASJA, we find an average $\unicode[STIX]{x1D70F}_{int}$ of 1.17 s, which is close to our PIV sampling frequency, $F_{s}$ of 1 Hz that was selected to achieve uncorrelated samples. There is no distinct trend between the sunbathing parameters and $\unicode[STIX]{x1D70F}_{int}$ . With the $16\times 16$ RASJA, $\unicode[STIX]{x1D70F}_{int}$ is reduced to 0.67 s.

4.1 Mixed region

4.1.1 The $8\times 8$ RASJA

Profiles of $u^{\prime }$ , $w^{\prime }$ and $k$ show a fairly straightforward structure of the turbulence facility with the $8\times 8$ RASJA. In figures 3 and 5, we observe $u^{\prime }$ and $k$ to be constant with $z$ for much of the upper one half to two thirds of the FOV. If we also consider profiles of the integral length scale, we observe ${\mathcal{L}}_{L}$ also approximately constant for $z>1.5{\mathcal{L}}_{L}$ . This is the lower bound of our mixed region. As $z$ increases above this height, $u^{\prime }$ continues to be relatively independent of $z$ , and so it appears that in experiments with the $8\times 8$ RASJA, the mixed region extends to the top of the FOV. We define metrics with subscript $m$ as the mean of those values in the mixed region; for example, $u_{m}^{\prime }$ is the mean of the horizontal r.m.s. velocity in the mixed region. A sample of resulting statistics is summarized in tables 1 and 2.

Table 1. Turbulent (r.m.s.) velocities and turbulent kinetic energy. All values shown are the mean value of the statistic in the mixed region from PIV data. $8\times 8$ RASJA.

In the mixed region, we expect the flow to be nearly isotropic, as it is fully mixed and unaffected by the boundary below. The ratio $w^{\prime }/u^{\prime }$ provides a measure of isotropy. Though we do not achieve unity, we find a ratio $w^{\prime }/u^{\prime }=1.29$ ; this is consistent with isotropy ratios observed in other facilities with forcing from only one side of the tank. Variano & Cowen (Reference Variano and Cowen2008) found an isotropy ratio of 1.27, and several GST experiments found ratios of 1.1–1.3 (Hopfinger & Toly Reference Hopfinger and Toly1976) and 1.4 (McDougall Reference McDougall1979). Due to the forcing along the vertical axis in particular, one would expect to observe $w^{\prime }/u^{\prime }>1$ away from boundaries.

Table 2. Integral length scale results. All values shown are the mean value of the statistic in the mixed region from PIV data. $8\times 8$ RASJA.

We can also look to the profiles of integral length scale to evaluate the assumption of isotropy used in measuring the transverse integral length scale. Our assumption of isotropy at both small and energy-containing scales ought to lead us to the relationship ${\mathcal{L}}_{L}/{\mathcal{L}}_{T}=2$ (Pope Reference Pope2000). However, the data produce a relationship of approximately ${\mathcal{L}}_{L}/{\mathcal{L}}_{T}=1.29$ in the mixed region across the 15 cases considered with the $8\times 8$ RASJA, which is consistent with the ratio of 1.19 reported in Variano & Cowen (Reference Variano and Cowen2008).

4.1.2 The $16\times 16$ RASJA

When using the $16\times 16$ RASJA, the flow regions are organized slightly differently. Recall that the $16\times 16$ RASJA is non-dimensionally ‘deeper’, with $H/J=13$ , compared to $H/J=7.1$ with the $8\times 8$ RASJA. With the $16\times 16$ RASJA, the lower bound of the mixed region is again observed at $z=1.5{\mathcal{L}}_{L}$ , where $u^{\prime }$ and ${\mathcal{L}}_{L}$ are relatively independent of $z$ . Interestingly, although $u^{\prime }$ is constant in this range, $k$ decreases with distance from the jets. We will explore this further in the coming section. Above the mixed region, there is a decay in $u^{\prime }$ , $w^{\prime }$ and $k$ with decreasing $z$ (or increased distance from the jets). This is noticeably different between the two RASJAs, as shown when comparing figures 5 and 6. As mixing occurs a distance of $6J$ from the jet orifice plane, we are confident that jets are fully merged prior to reaching the top of the FOV, and that likely the significantly greater $H/J$ parameter generates this different energetic structure.

Resulting values of r.m.s. velocities, turbulent kinetic energy, and integral length scale computed in the mixed region are presented in tables 3 and 4. We observe similar behaviour regarding isotropy ratios, with $w^{\prime }/u^{\prime }=1.26$ and ${\mathcal{L}}_{L}/{\mathcal{L}}_{T}=1.20$ across all 5 cases with the $16\times 16$ RASJA.

Table 3. Turbulent (r.m.s.) velocities and turbulent kinetic energy. All values shown are the mean value of the statistic in the mixed region from PIV data. $16\times 16$ RASJA.

Table 4. Integral length scale results. $16\times 16$ RASJA.

We are interested in exploring the rate of decay observed with the $16\times 16$ RASJA and in characterizing the turbulence in the mixed region directly above the boundary layer. Recalling the decay relationship proposed by Hopfinger & Toly (Reference Hopfinger and Toly1976) suggesting that turbulence facilities with turbulence generated by oscillating grids should show decay of $u^{\prime }$ and $w^{\prime }$ away from the source, we draw comparisons between the results obtained in our two facilities to determine the relevance of the concept of decay away from the RASJAs.

As shown in figures 10 and 11, turbulent decay is not always an accurate representation of the observations in a facility with an oscillating grid or random jet array, and we are excited to be able to generate such different flow patterns with similarly designed facilities. For experiments with the $8\times 8$ RASJA, with $H/J=7$ .1, both $u^{\prime }$ and $w^{\prime }$ are nearly independent of $z$ as discussed previously, aligning well with curves from Hunt & Graham (Reference Hunt and Graham1978). However, with the $16\times 16$ RASJA, with $H/J=13$ , there is a much clearer region of decay for both $u^{\prime }$ and $w^{\prime }$ that closely aligns with the Hopfinger & Toly (Reference Hopfinger and Toly1976) relationship for $z/H>0.15$ . Although figures 10 and 11 only show one profile from each RASJA, all curves collapse well within their respective facility. However, we do not see collapse between the facilities, in part because of the type of non-dimensionalization performed in this figure. As in the convention of Brumley & Jirka (Reference Brumley and Jirka1987), the profiles are normalized according to the value of $u^{\prime }$ at which $z/H=0.24$ , which is not in the mixed region with the $16\times 16$ RASJA. We also note that because the two RASJA experiments shown have similar values of $k$ , $u^{\prime }$ and $w^{\prime }$ near the bed, the flow structure of the facility appears more closely linked to the proximity of the turbulence source to the bed, or rather, the ratio $H/J$ , than to the turbulence levels themselves.

Figure 10. Comparison of experimental $u^{\prime }$ data. $T_{on}=4$  s,  $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $8\times 8$ RASJA $(-\cdot )$ , $T_{on}=0.8$  s,  $\unicode[STIX]{x1D6F7}_{on}=3.1\,\%$ , $16\times 16$ RASJA ( $--$ ) with Brumley & Jirka (Reference Brumley and Jirka1987) GST data (▫), Hopfinger & Toly (Reference Hopfinger and Toly1976,  $-$ grey), and Hunt & Graham (Reference Hunt and Graham1978) ( $\cdot \cdot \cdot$ grey). Vertical axis normalized by jet height $H=71$  cm and $H=65$  cm for $8\times 8$ and $16\times 16$ RASJAs, respectively; horizontal axis normalized to average 1 at $z/H=24\,\%$ , as in Brumley & Jirka (Reference Brumley and Jirka1987).

Figure 11. Comparison of experimental $w^{\prime }$ data. See previous figure for legend.

4.2 Bed-influenced region

Beneath the mixed region, there is a clear intercomponent energy transfer as bed-normal velocity fluctuations begin to decay and bed-parallel velocity fluctuations rise as a result of interactions with the solid boundary, describing a kinematic effect. We term the flow between the mixed layer and bed the boundary layer, as the flow patterns we observe are a direct result of the flow encountering a solid interface.

Considering profiles of $u^{\prime }$ and $w^{\prime }$ with the $8\times 8$ RASJA, there is a gradual increase in $u^{\prime }$ beneath the mixed layer, or beneath approximately $z=1.5{\mathcal{L}}$ . This continues until approximately $z/{\mathcal{L}}=0.5$ , at which point $u^{\prime }=w^{\prime }$ . This region is the transition between the mixed region and the source region. Interestingly, we do not observe this transition when using the $16\times 16$ RASJA, but instead observe a sharper transition where $u^{\prime }$ is constant then rapidly increases beneath the point at which $u^{\prime }=w^{\prime }$ .

Beneath the crossing point at which point $u^{\prime }=w^{\prime }$ , we observe a significant decrease in bed-normal velocity fluctuations as bed-parallel fluctuations rise due to the kinematic boundary condition and inability of bed-normal motions to penetrate the bed. The anisotropy between $u^{\prime }$ and $w^{\prime }$ increases until roughly $z/{\mathcal{L}}=0.15$ , at which point we observe a peak in both $u^{\prime }$ and $k$ . The region for which $0.15<z/{\mathcal{L}}<0.5$ is the source region, where the presence of the bed induces significant intercomponent energy transfer and dynamic turbulent splats (Perot & Moin Reference Perot and Moin1995a ). We also recall that whereas ${\mathcal{L}}_{T}$ decreases towards the bed in the boundary layer, there is a notable increase in ${\mathcal{L}}_{L}$ due to the significant restructuring of energy from homogeneous isotropic motions to strongly energetic bed-parallel motions throughout the boundary layer.

Below $z/{\mathcal{L}}=0.15$ , $w^{\prime }$ continues to diminish to zero, while $u^{\prime }$ , $k$ , and dissipation, which we will consider in the following section, rapidly decay to zero due to the no-slip boundary condition. In physical space, this region encompasses a thickness of the order of 1 cm, and so clearly this is not a true viscous sublayer. We term the region beneath $z/{\mathcal{L}}=0.15$ as the buffer region, and it extends as high as the greatest bed-parallel energy that arises from the kinematic boundary condition. This region is comparable to the viscous region described in Hunt & Graham (Reference Hunt and Graham1978), though in our experiments it extends beyond a true viscous region. As we will show in the following section, the bottom-most layer within this region is our viscous sublayer, in which we see the final decay to zero for the first-order flow statistics considered.

Looking again at figures 10 and 11, we can compare prior theory and experimental data in the boundary layer. We find the expected decay in GST theory neglects the enhancement of $u^{\prime }$ in the kinematic region and the viscous decay of $u^{\prime }$ and $w^{\prime }$ at the bed that we observe with the RASJAs. Instead, our data from both experiments are better aligned with the experimental GST results of Brumley & Jirka (Reference Brumley and Jirka1987) and theory of Hunt & Graham (Reference Hunt and Graham1978) in the measurement region considered, even though the Brumley & Jirka (Reference Brumley and Jirka1987) experiments consider the mean shear free turbulent boundary layer at a free surface rather than a flat plate. A free surface differs from a flat plate boundary in that there is free slip and a deformable free surface, leading to potentially weaker intercomponent energy transfer near the surface and minimal viscous decay at the surface itself. Even with these differences, the Brumley & Jirka (Reference Brumley and Jirka1987) experiments show a significant transfer from surface-normal to surface-parallel energy, with $u^{\prime }/u_{24}^{\prime }$ values around 1.4, compared to 0.8 predicted by the combined Hopfinger & Toly (Reference Hopfinger and Toly1976) and Hunt & Graham (Reference Hunt and Graham1978) theories.

It has been suggested that turbulence at a free surface can generate conditions in which $\unicode[STIX]{x2202}w^{\prime }/\unicode[STIX]{x2202}z=0$ at a surfactant-contaminated surface (Shen, Yue & Triantafyllou Reference Shen, Yue and Triantafyllou2004; Khakpour, Shen & Yue Reference Khakpour, Shen and Yue2011), and that such a contaminated surface may model a solid boundary due to reduced slip at the boundary (Herlina & Wissink Reference Herlina and Wissink2016). However, we do not observe this behaviour in our experimental turbulence facility. At the resolution of our PIV experiments (0.176 cm), it appears that both $u^{\prime }$ and $w^{\prime }$ approach zero at non-zero vertical gradients. Indeed, it is likely extremely difficult to fully prevent surfactants from contaminating a free surface, even in a controlled laboratory environment (Variano & Cowen Reference Variano and Cowen2008), which perhaps contributes to the significant increase in $u^{\prime }$ in the Brumley & Jirka (Reference Brumley and Jirka1987) experiments that agree so well with our results with the $8\times 8$ RASJA in the boundary layer, despite the mobility of the free surface that differs critically from a solid boundary that necessitates significant intercomponent energy transfer.

5.1 Dissipation

5.1.1 Direct method

To accurately compute dissipation, following Cowen & Monismith (Reference Cowen and Monismith1997), we use the direct formulation $\unicode[STIX]{x1D716}\equiv 2\unicode[STIX]{x1D708}\langle S_{ij}S_{ij}\rangle$ , with $S_{ij}\equiv (1/2)(\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j})+(\unicode[STIX]{x2202}u_{j}/\unicode[STIX]{x2202}x_{i})$ . This method is advantageous in that there are no empirical constants or isotropy requirements. In the boundary layer under investigation, we know the flow is anisotropic near the bed, as is evident when considering the intercomponent transfer from $w^{\prime }$ to $u^{\prime }$ presented in figures 3 and 4. Although the flow is more isotropic in the mixed region, we still observe weak anisotropy, due to the vertical forcing of the facility. Thus, we cannot assume that horizontal and vertical statistics are equivalent, which is a key requirement in the alternate methods of computing $\unicode[STIX]{x1D716}$ , explored further in Johnson (Reference Johnson2016).

Because we can only directly measure $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ , $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z$ , $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}x$ , and $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ from PIV data, we invoke continuity, as in Doron et al. (Reference Doron, Bertuccioli, Katz and Osborn2000), such that

(5.2) $$\begin{eqnarray}\overline{\left(\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}y}\right)^{2}}=\overline{\left(-\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}-\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}\right)^{2}}=\overline{\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}\right)^{2}}+2\overline{\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}\right)}.\end{eqnarray}$$

Radial symmetry allows for substitutions $\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x=\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ , $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y=\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ , $\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}z=\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z$ , and $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}y=\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}x$ , which produces a two-dimensional radially symmetric direct formulation

(5.3) $$\begin{eqnarray}\unicode[STIX]{x1D716}=2\unicode[STIX]{x1D708}\left[4\overline{\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}\right)^{2}}+\overline{\left(\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}x}\right)^{2}}+2\overline{\left(\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}\right)^{2}}+2\overline{\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}\right)}+2\overline{\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}x}\right)}\right].\end{eqnarray}$$

The cross-terms, $\overline{((\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x)(\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z))}$ and $\overline{((\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z)(\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}x))}$ , only account for approximately 3 % of the total dissipation in the direct method, and diagonal elements of $S_{ij}$ dominate. This relationship holds in both the isotropic mixed region and anisotropic boundary layer.

Spatial resolution is of critical importance in using the direct method, as noise is amplified if the resolution is refined beyond the smallest length scales, and PIV interrogation regions that are too large average the turbulent length scales causing an underestimation of the dissipation rate. Given our spatial resolution of $9\unicode[STIX]{x1D702}$ , where $\unicode[STIX]{x1D702}$ represents the Kolmogorov length scale (to be explored further in § 5.5), an integration of the universal spectrum proposed by Pao (Reference Pao1965) suggests that computing the spatial derivatives directly from our PIV data is sufficient for capturing 92 % of the total dissipation with the direct method (Cowen & Monismith Reference Cowen and Monismith1997). Thus, resulting values are scaled up by a factor of 1.09 to account for this slight under-resolution.

Typical resulting profiles of dissipation computed directly are shown in figure 12, revealing that dissipation is essentially constant as a function of height except very near the bed in the buffer region, beneath approximately $z/{\mathcal{L}}=0.15$ , where the magnitude of dissipation increases dramatically due to the no-slip boundary condition. In figure 12, we see reasonable collapse of three selected $\unicode[STIX]{x1D716}$ profiles, when normalized by their values in the mixed region. Hunt (Reference Hunt1984) predicts $\unicode[STIX]{x1D716}\propto z^{-1}$ as $z/{\mathcal{L}}$ approaches zero, and indeed it appears our data fall between exponential values of $-3/4$ and $-1$ across all experimental cases.

Figure 12. Normalized dissipation profiles with $\unicode[STIX]{x1D716}\propto z^{-3/4}$ ( $-$ ),  $\unicode[STIX]{x1D716}\propto z^{-1}$ ( $-$ grey). $T_{on}=4$  s $(--)$ , 6 s $(-\cdot )$ , 8 s $(\cdot \cdot \cdot )$ , $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $8\times 8$ RASJA.

As a means of validating our calculation of $\unicode[STIX]{x1D716}$ , we consider the energy spectrum as computed from bed-parallel PIV data. Specifically, we compute the dissipation spectrum, as in Pao (Reference Pao1965), Cowen & Monismith (Reference Cowen and Monismith1997) and Pope (Reference Pope2000), among others. A sample dissipation spectrum computed in the mixed region is shown in figure 13. The dissipation spectrum is normalized by $u_{\unicode[STIX]{x1D702}}^{3}$ , where the Kolmogorov velocity scale $u_{\unicode[STIX]{x1D702}}=(\unicode[STIX]{x1D708}\unicode[STIX]{x1D716})^{1/4}$ is computed in the mixed region. We observe strong agreement between the modelled dissipation spectrum and the present experimental data, noting high wavenumber noise for $\unicode[STIX]{x1D705}\unicode[STIX]{x1D702}>0.3$ .

By integrating the dissipation spectrum, we can also compute the cumulative dissipation, as shown in figure 13. Although the spectrum from the data does not extend as far as $\unicode[STIX]{x1D705}\unicode[STIX]{x1D702}\approx 1.6$ , where approximately 100 % of the dissipation is captured, we see agreement of 93 % between our data and the modelled curve for $\unicode[STIX]{x1D705}\unicode[STIX]{x1D702}<0.3$ when normalizing the cumulative dissipation by the value of $\unicode[STIX]{x1D716}_{m}$ computed via the direct method.

Figure 13. Normalized dissipation spectrum ( $-$ as in Pope (Reference Pope2000);  $-\cdot$ present data) and cumulative dissipation ( $\cdot \cdot \cdot$ as in Pope (Reference Pope2000);  $--$ present data) normalized by $\unicode[STIX]{x1D716}_{m}$ as computed by the direct method. Present experimental data shown for $z=11.8$  cm,  $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $T_{on}=4$  s,  $8\times 8$ RASJA.

5.1.2 Eulerian frequency spectra

Whereas several methods are available from which to compute dissipation rates from spatial PIV data, it is valuable to further develop methods of determining dissipation from point measurements such as ADV. Such instruments are typically more readily deployable in field experiments or in laboratory set-ups with limited visual access. Without a mean flow, we are unable to convert temporal records to spatial records via Taylor’s frozen turbulence hypothesis, and so we instead turn to Eulerian frequency spectra (Tennekes Reference Tennekes1975).

Figure 14. Frequency spectrum of vertical velocity from 100 Hz ADV measurement at $z=12$  cm, 200 ensemble averages. $T_{on}=4$  s,  $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $8\times 8$ RASJA.

Temporal frequency spectra, an example of which is shown in figure 14, are computed from 100 Hz ADV records taken in the mixed region at a point $z=12$  cm above the bed for experiments using the $8\times 8$ RASJA only. Although the spectra are only plotted up to the Nyquist frequency, all frequency spectra shown are normalized such that the integral of the spectra over $\unicode[STIX]{x1D714}\in (-\infty ,\infty )$ is equal to the variance of the velocity signal. The $S_{ww}$ spectra are least affected by noise in the configuration used, due to the geometric considerations of the ADV, so we are only considering the bed-normal velocity spectra in our analysis.

Through a simple scaling of $2\unicode[STIX]{x03C0}$ , temporal frequency spectra are transformed into Eulerian frequency spectra; a sample compensated Eulerian frequency spectrum is shown in figure 15. By identifying the plateau of the compensated spectrum, we utilize the relationships introduced by Tennekes (Reference Tennekes1975)

(5.4) $$\begin{eqnarray}E\left(\unicode[STIX]{x1D714}\right)=B_{o}\unicode[STIX]{x1D716}^{2/3}(\sqrt{2k})^{2/3}\unicode[STIX]{x1D714}^{-5/3}\end{eqnarray}$$

and further developed by Kit, Fernando & Brown (Reference Kit, Fernando and Brown1995) as

(5.5) $$\begin{eqnarray}E\left(\unicode[STIX]{x1D714}\right)=B_{1}\unicode[STIX]{x1D716}^{2/3}w^{\prime 2/3}\unicode[STIX]{x1D714}^{-5/3}\end{eqnarray}$$

to deduce the empirical constants $B_{0}$ and $B_{1}$ that relate dissipation to the Eulerian frequency spectra.

Figure 15. Eulerian frequency spectrum of vertical velocity from 100 Hz ADV measurement at $z=12$  cm, 80 ensemble averages. $T_{on}=4$  s,  $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $8\times 8$ RASJA.

Few experimental studies have been conducted to determine the empirical fit coefficients. In GST studies, De Silva & Fernando (Reference De Silva and Fernando1994) found a value of $B_{1}=8$ . In these experiments r.m.s. velocities were estimated by the Hopfinger & Toly (Reference Hopfinger and Toly1976) equations and scaling arguments rather than measured directly in the facility. Later GST experiments by Kit et al. (Reference Kit, Fernando and Brown1995) used a two-component fibre optic laser Doppler velocimeter and reported a value of coefficient $B_{1}=0.7$ for vertical velocity records after using isotropic strain rate relationships to determine the dissipation rate from the vertical velocity records. Variano & Cowen (Reference Variano and Cowen2008) reported $B_{0}=0.23$ and $B_{1}=0.35$ for a vertical velocity time series, with dissipation rates computed from the second-order longitudinal structure function and r.m.s. velocities computed from PIV measurements and turbulent spectra computed from ADV data.

To compute current estimates of $B_{0}$ and $B_{1}$ , our dissipation estimate comes from the corrected direct method using PIV data, whereas $E(\unicode[STIX]{x1D714})$ , $w^{\prime }$ and $k$ come directly from ADV measurements. When averaged across all 15 trials, we find resulting values of $B_{0}=0.14$ with a 95 % confidence interval of [0.13, 0.14] and $B_{1}=0.19$ with a 95 % confidence interval of [0.18, 0.20]. These values satisfy the Tennekes (Reference Tennekes1975) and Kit et al. (Reference Kit, Fernando and Brown1995) models, respectively, for vertical velocity measurements.

Table 5. Resulting values of dissipation, calculated directly, and coefficients for Eulerian frequency spectra estimates of dissipation. Values of dissipation are averaged in the mixed region. $8\times 8$ RASJA.

Table 6. Resulting values of dissipation, calculated directly, averaged in the mixed region. $16\times 16$ RASJA.

5.2 Turbulent transport

Because the spatial resolution of the analysed PIV data is sufficient to directly compute spatial derivatives for higher-order statistical analysis such as dissipation, we can also directly compute the remaining terms in the turbulent kinetic energy balance such as turbulent transport, $T=-(1/2)(\unicode[STIX]{x2202}\langle u_{j}u_{j}u_{i}\rangle /\unicode[STIX]{x2202}x_{i})$ . We invoke radial symmetry to simplify our equation to

(5.6) $$\begin{eqnarray}T=-\frac{1}{2}\left[4\frac{\unicode[STIX]{x2202}\left\langle uuu\right\rangle }{\unicode[STIX]{x2202}x}+2\frac{\unicode[STIX]{x2202}\left\langle wwu\right\rangle }{\unicode[STIX]{x2202}x}+2\frac{\unicode[STIX]{x2202}\left\langle uuw\right\rangle }{\unicode[STIX]{x2202}z}+\frac{\unicode[STIX]{x2202}\left\langle www\right\rangle }{\unicode[STIX]{x2202}z}\right].\end{eqnarray}$$

In doing so, we find negligible contributions from the $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x$ terms, as is expected from horizontal homogeneity. The triple correlation is an inherently noisy calculation, with relatively wide fluctuations throughout the FOV. The $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z$ terms fluctuate greatly, ranging across magnitudes of $-\unicode[STIX]{x1D716}$ to $2\unicode[STIX]{x1D716}$ , and so a local median smoothing filter is applied to produce the final profile of $T$ shown in figure 16. This is done via MATLAB’s medfilt1 function, a one-dimensional median filter, along 6 point-long segments.

5.3 Production

Similarly, we compute production, $P=-\langle u_{i}u_{j}\rangle \unicode[STIX]{x2202}\langle U_{i}\rangle /\unicode[STIX]{x2202}x_{j}$ , which expands to

(5.7) $$\begin{eqnarray}P=-\left[4\langle uu\rangle \frac{\unicode[STIX]{x2202}\langle U\rangle }{\unicode[STIX]{x2202}x}+2\langle uw\rangle \frac{\unicode[STIX]{x2202}\langle U\rangle }{\unicode[STIX]{x2202}z}+2\langle uw\rangle \frac{\unicode[STIX]{x2202}\langle W\rangle }{\unicode[STIX]{x2202}x}+\langle ww\rangle \frac{\unicode[STIX]{x2202}\langle W\rangle }{\unicode[STIX]{x2202}z}\right],\end{eqnarray}$$

when invoking radial symmetry. Whereas $\langle uu\rangle$ and $\langle ww\rangle$ have significant magnitudes, as expected from our analysis of r.m.s. velocities previously, $\langle uw\rangle$ is essentially zero, with flow being equally likely to move in any radial direction. In considering the mean velocity gradients, we observe negligible contributions from $\unicode[STIX]{x2202}\langle U\rangle /\unicode[STIX]{x2202}x$ and $\unicode[STIX]{x2202}\langle W\rangle /\unicode[STIX]{x2202}x$ due to horizontal homogeneity. There are, however, weak contributions from the vertical gradients of both horizontal and vertical mean velocities due to the kinematic effect of the bed.

When combining the products of the mean velocity gradients and $\langle u_{i}u_{j}\rangle$ terms, the summation ultimately results in low levels of production. For all trials with the $8\times 8$ RASJA, $P$ is approximately zero at the bed. For the trials explored with $M^{\ast }<1\,\%$ , production remains negligible near the bed and in the mixed region, though not all 15 trials show such negligibly small values of $P$ throughout the entire boundary layer. While more than half of these trials show near-zero magnitudes of production, some trials exhibit weak production in the source layer ranging from 2 to 6 $\text{cm}^{2}\,\text{s}^{-3}$ . It appears that non-zero production results from non-zero $\unicode[STIX]{x2202}\langle U\rangle /\unicode[STIX]{x2202}x$ in particular, and additional details are explored in Johnson (Reference Johnson2016). Even in cases with non-negligible values of $P$ , the values of production obtained remain small relative to dissipation. Whereas production and dissipation are typically in balance in the near-wall region in boundary layers resulting from mean flows, the magnitude of production is at most only half as great as the magnitude of dissipation and is typically negligible in boundary layers absent mean flows. Because the computations of $P$ are quite noisy about $P=0$ , a local median smoothing filter along 4 point segments is applied to the final profile shown in figure 16.

Figure 16. Dissipation ( $--$ ), turbulent transport ( $-\cdot$ ), production ( $\cdot \cdot \cdot$ ), estimated pressure diffusion ( $-$ ), non-dimensionalized by $\unicode[STIX]{x1D716}_{m}$ from direct method. $T_{on}$ = 4 s, $\unicode[STIX]{x1D6F7}_{on}=6.25\,\%$ , $8\times 8$ RASJA.

5.4 Pressure diffusion

At present, we do not make in situ pressure measurements in the turbulence facility, nor do we have measurements with sufficient temporal resolution to allow for the direct computation of pressure gradients from PIV data using methods such as those shown by Dabiri et al. (Reference Dabiri, Bose, Gemmeli, Colin and Costello2014). However, by considering the turbulent kinetic energy balance and computing pressure diffusion as the residual, it is apparent that pressure plays a significant role at the bed to balance non-zero dissipation, since all other terms approach zero at the bed. In particular, the bed-normal component $p_{z}=-(1/\unicode[STIX]{x1D70C}_{0})(\unicode[STIX]{x2202}\overline{w^{\prime }p^{\prime }}/\unicode[STIX]{x2202}z)$ would likely be of greatest significance, and it is something that we will revisit in future experiments to evaluate its role in the energy balance.

Figure 16 shows typical relative contributions of production and dissipation (and the remaining terms in the energy balance) for the case of near-zero production. The relative magnitude of the pressure term is significant within the buffer layer. Due to this increase, we hypothesize that significant pressure fluctuations in the boundary layer contribute to the increase in $k$ observed in figures 5 and 6 in the absence of an active source of turbulent kinetic energy near the stationary solid bed.

5.5 Turbulence metrics

Having completed analysis of many higher-order turbulence statistics, we can compute traditional metrics of the flow such as Kolmogorov scales, Taylor scales, Reynolds numbers and others. The quantities included are all computed in the mixed region of the flow. Using the estimate for integral length scale found via the exponential fit to the longitudinal autocorrelation function and the dissipation rate calculated directly, we compute the Kolmogorov time scale, $\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D716})^{1/2}$ , which ranges from 0.023 to 0.034 s with the $8\times 8$ RASJA. Values of the Kolmogorov length scale, $\unicode[STIX]{x1D702}\equiv (\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D716})^{1/4}$ , range from 0.016 to 0.019 cm within the $8\times 8$ RASJA. Within the $16\times 16$ RASJA, values of $\unicode[STIX]{x1D70F}$ range from 0.017 to 0.025 s, and values of $\unicode[STIX]{x1D702}$ range from 0.013 to 0.016 cm. It appears that both $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D702}$ decrease with increasing $T_{on}$ , as shown by the sample statistics in tables 7 and 8, consistent with expectations given that turbulent kinetic energy increases with $T_{on}$ .

Table 7. Kolmogorov and Taylor scales, $8\times 8$ RASJA.

Table 8. Kolmogorov and Taylor scales, $16\times 16$ RASJA.

We also compute the Taylor microscale, $\unicode[STIX]{x1D706}_{g}=\sqrt{10}\unicode[STIX]{x1D702}^{2/3}{\mathcal{L}}^{1/3}$ , which gives us an intermediate length scale of the turbulence. In the $8\times 8$ RASJA, $\unicode[STIX]{x1D706}_{g}$ ranges from 0.40 to 0.45 cm across all 15 cases. In the $16\times 16$ RASJA, $\unicode[STIX]{x1D706}_{g}$ ranges from 0.29 to 0.33 cm, with no clear dependence of $T_{on}$ on $\unicode[STIX]{x1D706}_{g}$ . The Reynolds number based on the Taylor microscale, $Re_{\unicode[STIX]{x1D706}}=((2/3)k)\sqrt{15/\unicode[STIX]{x1D708}\unicode[STIX]{x1D716}}$ , in which we use the average $k$ from the mixed region of the flow, provides a traditional metric of grid turbulence. Across all 15 trials with the $8\times 8$ RASJA, our values range from 277 to 378, as shown in table 9, consistent with results in Variano & Cowen (Reference Variano and Cowen2008). Using the $16\times 16$ RASJA, $Re_{\unicode[STIX]{x1D706}}$ ranges from 197 to 262, shown in table 10.

We compute a grid Reynolds number, $Re_{G}=2(\sqrt{(2/3)k}{\mathcal{L}}/\unicode[STIX]{x1D708})$ , for comparison to prior experiments in GSTs, and $Re_{L}=Re^{\ast }=k^{2}/\unicode[STIX]{x1D716}\unicode[STIX]{x1D708}$ for comparison to moving-bed experiments. These Reynolds numbers show highly turbulent flow, with values greater than much of the prior literature in this field, such as Brumley & Jirka (Reference Brumley and Jirka1987) with $Re_{\unicode[STIX]{x1D706}}=74$ , $Re_{L}=390$ , Uzkan & Reynolds (Reference Uzkan and Reynolds1967) with $Re_{L}=90$ , Thomas & Hancock (Reference Thomas and Hancock1977) with $Re_{L}=2000$ , and Perot & Moin (Reference Perot and Moin1995a ) with $Re_{L}$ ranging from 6.2 to 374.

From the Reynolds number, we can approximate a viscous sublayer thickness, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}\approx 2{\mathcal{L}}_{L}Re_{L}^{-1/2}$ , from scaling arguments as in Brumley & Jirka (Reference Brumley and Jirka1987), Calmet & Magnaudet (Reference Calmet and Magnaudet2003) and Variano & Cowen (Reference Variano and Cowen2008). For all 15 of the cases considered with the $8\times 8$ RASJA, we consistently obtain a value of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}\approx$ 0.24 $\pm$ 0.01 cm. This is very close to the value of 0.26 cm determined by Variano & Cowen (Reference Variano and Cowen2008). It includes one data point given the spatial resolution of 0.176 cm used in our PIV measurements. Thus, this prevents us from measuring velocity gradients within the viscous sublayer. With the $16\times 16$ RASJA, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}\approx$ 0.11 $\pm$ 0.01 cm, which is too thin to measure with our current experimental set-up. Upon determining $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ , we can also consider the relationship ${w^{\prime }}^{2}=\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D716}^{2/3}(z+\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})^{2/3}$ as presented in Hunt (Reference Hunt1984), Calmet & Magnaudet (Reference Calmet and Magnaudet2003) and Variano & Cowen (Reference Variano and Cowen2008). Given the substantial increase in $\unicode[STIX]{x1D716}$ as $z$ approaches zero, we use $\unicode[STIX]{x1D716}_{m}$ and find the relationship to hold for approximately $10\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}<z<0.7{\mathcal{L}}$ , with $\unicode[STIX]{x1D6FD}=1.75\pm 0.23$ across the 15 cases with the $8\times 8$ RASJA. Although we find a wide range of $\unicode[STIX]{x1D6FD}$ , this is near the theoretical prediction of $\unicode[STIX]{x1D6FD}\approx 1.8$ (Hunt Reference Hunt1984) and experimental results $\unicode[STIX]{x1D6FD}\approx$ 2.0 (Calmet & Magnaudet Reference Calmet and Magnaudet2003) and $\unicode[STIX]{x1D6FD}\approx 1.5$ (Variano & Cowen Reference Variano and Cowen2008).

The Péclet number, in general, defines the ratio of advective transport to diffusive transport – large Péclet number indicates that advection dominates while small indicates that diffusion dominates. In this set-up we use a simple scaling analysis to define a Péclet number. We represent the time scale of the advective processes as the integral length scale of the turbulence divided by the mean velocity or $\unicode[STIX]{x1D70F}_{adv}={\mathcal{L}}/\overline{U}$ . We represent the time scale of the turbulent diffusive processes as $\unicode[STIX]{x1D70F}_{diff}={\mathcal{L}}^{2}/\unicode[STIX]{x1D705}_{t}$ where $\unicode[STIX]{x1D705}_{t}=\sqrt{k}{\mathcal{L}}$ and hence $\unicode[STIX]{x1D70F}_{diff}={\mathcal{L}}/\sqrt{k}$ . Thus, in our facility, we use Péclet $\text{number}=Pe=\overline{U}/\sqrt{k}$ , which is the inverse of the turbulence intensity, as a measure of the relative importance of turbulent stirring to advection by the mean flow. Values of less than 0.1 confirm the highly diffusive nature of the facility in which turbulent transport overwhelms advective transport. The aforementioned statistics are summarized in tables 9 and 10.

Table 9. Reynolds numbers and Péclet number, $8\times 8$ RASJA.

Table 10. Reynolds numbers and Péclet number, $16\times 16$ RASJA.