3.1 Decay of turbulence

Physical mechanisms responsible for the decay of turbulence have been under debate for decades (Reference de Karman and Howarthde Karman & Howarth 1938; Reference Batchelor and TownsendBatchelor & Townsend 1947; Reference FrenkielFrenkiel 1948). In turbulence-generating facilities, regions of the flow may experience a reduction in the strength of turbulence due to either increasing distance from the source (spatial decay) or unsteadiness in the driving mechanism (temporal decay). The variety of extant turbulence facilities provides an opportunity to investigate the dynamics of decay for different methods of forcing. While decay can be seen in integral scales, energy spectra, dissipation rate and many other flow metrics, we explore decay through the lens of turbulent kinetic energy. As the turbulence decays, the turbulent kinetic energy decreases, and other turbulence statistics respond accordingly.

In wind and water tunnels or flumes with upstream passive grid-generated turbulence (hereafter referred to as WWTs), spatial decay can be studied directly. Temporal decay can subsequently be determined by invoking Taylor's frozen turbulence hypothesis, given the mean velocity of the flow. In WWTs, $k$ has been observed to decay as a power law with respect to time (or space) with a stage-dependent exponent of decay, $n$. This can be summarized by (3.1), in which $x$ is a position downstream of virtual origin $x_0$ (Reference de Karman and Howarthde Karman & Howarth 1938; Reference KolmogorovKolmogorov 1941; Reference SaffmanSaffman 1967; Reference Comte-Bellot and CorrsinComte-Bellot & Corrsin 1971; Reference MakitaMakita 1991):

(3.1)\begin{equation} k \sim (x-x_0)^{{-}n} .\end{equation}

In the first stage, the ‘near-field’ region, turbulence develops from the interaction of wakes from the passive or active grid. This region extends approximately $30\mathcal {L}_L\unicode{x2013}50\mathcal {L}_L$ beyond the grid (Reference Krogstad and DavidsonKrogstad & Davidson 2012). The second stage, the ‘far-field’ region, starts where HIT is developed. The value of $n$ in both of these regions is highly dependent on the initial conditions and the initial Reynolds number (e.g. $n \approx [1.1\unicode{x2013}1.4]$ in the far field) (Reference Valente and VassilicosValente & Vassilicos 2012; Reference Thormann and MeneveauThormann & Meneveau 2014). The final stage of decay, sometimes referred to as the ‘tired turbulence’ region (Reference BatchelorBatchelor 1953), where the dynamics of the flow is mainly affected by viscous rather than inertial forces, can be observed in sufficiently long tunnels assuming infinite boundaries. A general decay law with a single value of $n$ can be applied to a broad range of studies for this region (Reference Batchelor and TownsendBatchelor & Townsend 1948; Reference Comte-Bellot and CorrsinComte-Bellot & Corrsin 1966; Reference Bennett and CorrsinBennett & Corrsin 1978; Reference GeorgeGeorge 1992; Reference Touil, Bertoglio and ShaoTouil, Bertoglio & Shao 2002; Reference Biferale, Boffetta, Celani, Lanotte, Toschi and VergassolaBiferale et al. 2003). The decay of the turbulence can be characterized as Saffman turbulence where $n\sim 3/2$ (Reference SaffmanSaffman 1967; Reference Skrbek and StalpSkrbek & Stalp 2000; Reference Krogstad and DavidsonKrogstad & Davidson 2010). However, before reaching this stage, $\mathcal {L}_L$ increases as $k$ decays, leading to the confinement of large-scale motions by the smallest dimension of the domain. Consequently, these motions are of the same order of magnitude as the characteristic length scale and the turbulence decays at a higher rate (Reference Skrbek and StalpSkrbek & Stalp 2000).

Direct measurements of the velocity in zero-mean-flow HIT facilities enable the study of decay in different directions, which is beneficial when exploring how isotropy may vary in relation to decay. Both temporal and spatial decay can be observed directly from single-point, planar or volumetric velocity data. In facilities with planar forcing, spatial decay can be calculated as a function of distance from the source (e.g. Reference Variano and CowenVariano & Cowen 2008; Reference Khorsandi, Gaskin and MydlarskiKhorsandi et al. 2013; Reference Pérez-Alvarado, Mydlarski and GaskinPérez-Alvarado et al. 2016). Temporal decay can be measured directly by halting the forcing (i.e. turning off the actuators) (Reference Hwang and EatonHwang & Eaton 2004; Reference Goepfert, Marié, Chareyron and LanceGoepfert et al. 2010; Reference Esteban, Shrimpton and GanapathisubramaniEsteban et al. 2019). We can therefore compare various stages of decay between WWTs and zero-mean-flow HIT facilities.

In RJA facilities, the ‘jet-merging’ region is analogous to the near-field region of WWTs. In this region, energy from the mean flow of the individual jets transfers to the turbulent kinetic energy of the bulk HIT flow (Reference Variano and CowenVariano & Cowen 2008; Reference Johnson and CowenJohnson & Cowen 2018). While the near-field region in WWTs is characterized by the exponential decay of $k$, we hypothesize that contrasting behaviour occurs in RJA-based facilities due to turbulence production from the time-varying jets; however, turbulence statistics have not explicitly been measured in this region of RJA-based or other zero-mean-flow HIT facilities. Thus, to estimate the length of the near-field region (or jet-merging region) in these facilities, we used the distance between the actuators and the reported HIT region. When normalizing this distance with respect to $\mathcal {L}_L$, we observed that the near-field region is much smaller in RJA-based facilities than in WWTs, indicating that HIT forms more rapidly in RJA facilities due to the stochastic forcing. According to our analysis, beyond a distance from the array of $z_J/S>6$, which is approximately equivalent to $6\mathcal {L}_L\unicode{x2013}10\mathcal {L}_L$ (where $\mathcal {L}_L$ is measured within the homogeneous and isotropic region), the zero-mean-flow HIT region is formed. In this far-field region, $k$ decays with a power law in the forcing direction. The value of $n$ reported for RJA facilities (e.g. Reference Variano and CowenVariano & Cowen 2008) was higher than those reported for WWTs. Reference Variano and CowenVariano & Cowen (2008) reported $n$ to be approximately 2.3, which is closer to the values for the final stage of decay observed in WWTs. Similarly, in GST facilities, this value was reported to be approximately 2 (Reference Hopfinger and TolyHopfinger & Toly 1976).

Reference Hwang and EatonHwang & Eaton (2004) measured the temporal decay of the fluctuating velocities after turning off the actuators in a loudspeaker-driven facility with a spherical symmetric forcing system. The power-law decay exponent of the turbulent kinetic energy in this facility was $n = 1.86$, which is higher than the value estimated for the far field in WWT (Reference Valente and VassilicosValente & Vassilicos 2012). During the decay, Reference Hwang and EatonHwang & Eaton (2004) found that the turbulence statistics were modified, yet remained isotropic. In a spherical symmetric forcing facility with no boundary, Reference Goepfert, Marié, Chareyron and LanceGoepfert et al. (2010) also studied temporal decay and found $n$ to be approximately equal to 1.6. While $n$ is somewhat facility-dependent, it is clear that the decay is faster in these two facilities than in the far-field region of WWT experiments; we hypothesize that this is also the case for zero-mean-flow HIT facilities in general.

Unlike Reference Hwang and EatonHwang & Eaton (2004) and Reference Goepfert, Marié, Chareyron and LanceGoepfert et al. (2010), who found a single value for $n$ to characterize the temporal decay of $k$, Reference Esteban, Shrimpton and GanapathisubramaniEsteban et al. (2019) categorized multiple phases of decay based on different values of $n$. They found that immediately after turning off the jets, $k$ decayed at a high rate, aligning well with the near-field region in WWT studies. Following this initial stage, the decay rate decreased and decelerated close to the range reported in the far-field region of WWTs. Reference Esteban, Shrimpton and GanapathisubramaniEsteban et al. (2019) continued the measurements further in time and found that the decay rate of $k$ increased while the integral length scale remained constant. However, in the first two stages of decay, they observed a decrease in $k$ accompanied by an increase in the integral length scale. They stated that the final phase of the observed temporal decay of the turbulence corresponds to the integral length scale being constrained by the facility size (i.e. indicating the saturation phase; Reference Skrbek and StalpSkrbek & Stalp 2000).

The decay rate in the final stage found by Reference Esteban, Shrimpton and GanapathisubramaniEsteban et al. (2019) is similar to the rates found by Reference Hwang and EatonHwang & Eaton (2004) and Reference Goepfert, Marié, Chareyron and LanceGoepfert et al. (2010), and it is higher than the decay rate in the far-field stage of WWT studies. Reference Esteban, Shrimpton and GanapathisubramaniEsteban et al. (2019) hypothesized that since the integral length scale in Reference Hwang and EatonHwang & Eaton (2004) was comparable to the size of the facility, their observed decay rate corresponded to the final saturation phase. By contrast, the lower decay rate reported by Reference Goepfert, Marié, Chareyron and LanceGoepfert et al. (2010) could be due to the absence of boundaries in their facility, thus precluding the presence of high velocity gradients (and concomitant energy dissipation) that develop close to a boundary. Analytical studies, such as Reference Skrbek and StalpSkrbek & Stalp (2000), suggest that in the region where $\mathcal {L}_L$ cannot grow further, due to confinement, $k$ diminishes at an exponential decay rate of $n=2$, which is similar to the values found in the aforementioned experimental studies (e.g. Reference Hwang and EatonHwang & Eaton 2004; Reference Variano and CowenVariano & Cowen 2008; Reference Esteban, Shrimpton and GanapathisubramaniEsteban et al. 2019). This is another indication that facility size strongly affects the integral length scale of the flow. As the Taylor-scale Reynolds number increases, $\mathcal {L}_L$ changes. However, the magnitude of $\mathcal {L}_L$ remains constrained by the size of the facility, as we discuss further in § 3.3.

Temporal decay can be directly measured in a zero-mean-flow facility and compared with decay in WWTs. However, if we consider comparing different stages of the spatial decay in WWTs and the zero-mean-flow HIT facilities, the spatial decay of turbulence does not have a counterpart in the far-field region of WWTs in terms of the decay exponent (as shown in figure 6). In addition to the absence of mean flow in these facilities (and thus the smaller distance needed to achieve HIT), we hypothesize that this is due to their different generation mechanisms and resulting flow dynamics. In WWTs, the very presence of mean flow drives production of shear and turbulence, and thus a longer distance is unavoidably necessary for the complete development of HIT. However, this is not the case in zero-mean-flow HIT facilities. According to the values of $n$ presented for the spatial decay in HIT facilities (Reference Esteban, Shrimpton and GanapathisubramaniEsteban et al. 2019), turbulent length scales are likely to be affected by the confinement of the facility before the far-field region develops. In most of the HIT studies presented here, the magnitude of the integral length scale is comparable to the size of the facility, consistent with the saturation phase. We propose that this is the reason faster decay rates (i.e. higher values of $n$) were observed as compared with in the far-field section of WWTs.

Figure 6. Regions of decay in (a) RJA facility versus (b) WWTs. Distances are not to scale between (a) and (b) to enhance visualization.

3.2 Forcing geometry and physical aspects of facilities

The spatial distribution of forcing elements affects the interaction of input momentum from individual actuators and can ultimately impact the resultant characteristics of the zero-mean-flow HIT region. Actuators may be directionally aligned in a closely spaced planar array (as in RJAs) or spaced widely, with generated flows oblique to one another (as in a spherical arrangement of loudspeakers). The forcing distribution may be categorized as symmetric or asymmetric. Symmetric forcing systems generate turbulence from multiple directions, resulting in a core volume of turbulence. Asymmetric forcing, with momentum input from a single direction, has been used to study mean-shear-free boundary layer dynamics or scalar transport at planar interfaces. Understanding the impact of the forcing distribution is crucial for optimizing the design of turbulence facilities and tailoring the resulting flow.

In asymmetrically forced facilities, where the source of input momentum is orthogonal to a single plane, it is reasonably assumed that the flow is radially symmetric in the plane perpendicular to the forcing direction at some distance downstream from the actuators (Reference Variano and CowenVariano & Cowen 2008; Reference Johnson and CowenJohnson & Cowen 2018). In this type of forcing system, it is important to identify the location where the flow transitions from a region dominated by individual actuator activity to HIT with negligible mean flow (recall §§ 2.3 and 3.1). Several studies have explored the location of this transition (Reference Variano and CowenVariano & Cowen 2008; Reference Masuk, Salibindla, Tan and NiMasuk et al. 2019); however, further systematic investigation is needed.

In symmetric facilities, the HIT region is expected to be located at the centre of the facility. However, the size and shape of this region differ based on the type of actuator, the size of the facility, the forcing distribution and the forcing strength. For instance, spherical forcing systems (shown in figure 7b) generate spherical regions of HIT, while a cylindrical forcing system generates a long narrow turbulent region (the HIT region in the cylindrical facility of Reference Hoffman and EatonHoffman & Eaton (2021) (figure 7c) had a high aspect ratio, with a height of $13\mathcal {L}_L$ and diameter of approximately $0.25 \mathcal {L}_L$).

Figure 7. Different types of symmetrically forced facilities. (a) Planar symmetric facility of Reference Bellani and VarianoBellani & Variano (2014), (b) spherical symmetric facility of Reference Chang, Bewley and BodenschatzChang et al. (2012) and (c) cylindrical symmetric facility of Reference Hoffman and EatonHoffman & Eaton (2021). 2-D, two-dimensional.

In systems with spherical forcing, achieving negligible mean flow may not require stochastic forcing. This is evident in the studies of Reference Krawczynski, Renou, Danaila and DemoulinKrawczynski et al. (2006, Reference Krawczynski, Renou and Danaila2010), where turbulence produced with constant input forcing from all directions was not homogeneous, but a small region with negligible mean flow was identified. Interestingly, in the study of Reference Chang, Bewley and BodenschatzChang et al. (2012), regions with higher values of $k$ were associated with lower mean flow. We believe the resultant turbulence characteristics in these facilities are due to the interaction of input momentum from multiple opposing directions. This results in total reduction of mean flow and generation of turbulence with $\varOmega$ close to 1.

Symmetric forcing may also be achieved using planar forcing with multiple actuator arrays, such as two or more facing RJAs, an example of which is shown in figure 7(a) (Reference Bellani and VarianoBellani & Variano 2014; Reference Carter, Petersen, Amili and ColettiCarter et al. 2016; Reference Bang and PujaraBang & Pujara 2023). This forcing arrangement produces a cuboid-shaped zero-mean-flow HIT region that is widest in the plane perpendicular to the input forcing direction and narrowest in the direction of the input forcing (i.e. the depth of the HIT region). Multi-planar forcing systems can produce a larger HIT region compared with alternative symmetric forcing geometries. In these facilities, the depth of the HIT region, which is influenced by the decay of turbulence and the interaction of momentum-driven flows, is comparable to the size of the integral length scale of the turbulence (Reference Bellani and VarianoBellani & Variano 2014). The height and width of the HIT region are constrained by the decay of the fluctuating velocities due to the zero-energy condition at the boundaries (Reference Bellani and VarianoBellani & Variano 2014; Reference Johnson and CowenJohnson & Cowen 2018). We note that in several facilities, especially RJA-, fan- and loudspeaker-driven facilities, the homogeneous region has been shown to encompass multiple integral length scales, which can be useful for applications such as particle–turbulence interactions.

As previously discussed, flow behaviour and turbulence characteristics are strongly influenced by boundaries. This extends to all types of facilities and all boundaries (e.g. free surfaces, sediment beds, solid walls). Consequently, the influence of the boundaries on the flow statistics has been the focus of several theoretical, experimental and numerical studies (e.g. Reference Thomas and HancockThomas & Hancock 1977; Reference Hunt and GrahamHunt & Graham 1978; Reference Perot and MoinPerot & Moin 1995). Many of the known laboratory-based zero-mean-shear boundary layer studies are performed in planar forcing facilities. For boundary planes perpendicular to the input forcing direction, turbulence metrics remain unaffected by the presence of boundaries at a distance greater than $1.5 \mathcal {L}_L$ from the boundary (Reference Delbos, Weitbrecht, Bleninger, Grand, Chassaing, Lincot, Kerrec and JirkaDelbos et al. 2009; Reference Johnson and CowenJohnson & Cowen 2018). In addition, the effect of boundaries parallel to the input forcing extends $2\mathcal {L}_L$ into the facility (Reference Variano and CowenVariano & Cowen 2008; Reference Bellani and VarianoBellani & Variano 2014). The type of boundary is also a crucial factor. Reference Johnson and CowenJohnson & Cowen (2020) reveal significant differences in the boundary layer dynamics of turbulence metrics, such as fluctuating velocities and the integral length scale, depending on the type of the boundary (e.g. solid impermeable bed, flat sediment bed and rippled sediment bed).

In laboratory facilities, a distinct flow pattern may be established based on the momentum generated by the actuators and the characteristics of the boundaries, such as their position and type. The generated flow may develop into a large mean circulation or enhance standing waves with a specific frequency (i.e. loudspeakers; Reference Sabban and van HoutSabban & van Hout 2011). One approach to reducing the likelihood of persistent mean circulations is to introduce spatial variability in the forcing pattern to interrupt the flow. Another technique is to induce temporal variability in the forcing mechanism, as discussed in § 1.1.3. For instance, in RJA facilities, the flow is constantly disturbed by jet flows generated by the pumps changing between on and off states, making it unlikely for large mean circulations to persist. While stochastic forcing generally reduces mean flows, a recent study with a planar impeller array found a peak in the temporal energy spectra (Reference Lawson and GanapathisubramaniLawson & Ganapathisubramani 2022), which was not observed in RJA facilities (Reference Johnson and CowenJohnson & Cowen 2018), indicating that the rotational signature of impeller-type actuators may be challenging to eradicate in the resulting flow. It should be mentioned that even in RJA facilities with $M^*<5\,\%$, tank-scale toroidal flows are often observed as bulk mean recirculations. Even though their contribution to the turbulence dynamics is negligible, these flows exist – both locally, to maintain conservation of mass immediately surrounding the actuators, and at the tank scale. Similarly, in loudspeaker-driven tanks, the frequency of the loudspeakers must be refined to avoid the generation of standing waves or recirculations of a certain mode in an enclosed facility. According to Reference Goepfert, Marié, Chareyron and LanceGoepfert et al. (2010), in facilities without boundaries, since the condition on mass flux is lifted, the forcing algorithm does not require variability to achieve negligible mean flow turbulence.

There are multiple approaches to change the turbulent flow characteristics once a facility is constructed. One is to add a mesh or perforated plate in front of the actuators (Reference Hwang and EatonHwang & Eaton 2004; Reference Bellani, Byron, Collignon, Meyer and VarianoBellani et al. 2012; Reference Carter, Petersen, Amili and ColettiCarter et al. 2016; Reference Pujara, Clos, Ayres, Variano and Karp-BossPujara et al. 2021). Insertion of a mesh introduces smaller turbulence length scales by breaking up eddies into smaller sizes, altering the actuator-dominated flows and their interactions with the same input energy. For example, in RJA facilities with the same jet voltage, $T_{{on}}$ and $T_{{off}}$, using a mesh immediately downstream of the pumps resulted in lower turbulent kinetic energy (Reference Carter, Petersen, Amili and ColettiCarter et al. 2016). Other modifications based on actuator type can be incorporated to further change the flow characteristics. In fan facilities, for example, changing the number of blades and/or pitch angle (recall § 2.1.1) can change flow metrics such as the turbulent Reynolds number and the integral length scale of the resultant HIT. We hypothesize that a higher number of blades leads to a reduction in $\mathcal {L}_L$ because the time for one blade's generated flow to be disturbed by the adjacent blade is shorter; therefore, the generated eddies are smaller in size, and the turbulent kinetic energy is lower.

3.3 Synthesis of forcing geometry and turbulence length scales across zero-mean-flow facilities

Previously, we discussed how physical aspects of a facility (forcing distribution, facility size, actuator characteristics, etc.) influence characteristics of the turbulence produced. In this section, we quantitatively investigate the extent of this influence. The importance of this topic is especially apparent in studies that employ the same flow-generation mechanism but with different overall size or forcing element spacing (e.g. Reference Bellani and VarianoBellani & Variano 2014; Reference Carter, Petersen, Amili and ColettiCarter et al. 2016; Reference Esteban, Shrimpton and GanapathisubramaniEsteban et al. 2019). We use available data from many zero-mean-flow turbulence facilities (see the Appendix, table 6) to establish the relative importance of these variables.

Table 6. Properties of turbulent flow generated in zero-mean-flow HIT facilities. Superscripts as explained in table 2.

Length scales inherent to a facility (i.e. overall facility size, actuator size and actuator spacing) can affect the resulting turbulent length scales. As discussed in § 3.1, it is likely that $\mathcal {L}_L$ is constrained by the size of the facility. Figure 8(a) reveals that larger facilities (as measured by $D/2$, the half-width of the smallest facility dimension) tend to have larger turbulent length scales. This is especially apparent when considering $\mathcal {L}_L$, more so than $\mathcal {L}$ (which is determined via scaling arguments when direct measurements of $\mathcal {L}_L$ are not available). Therefore, we focus our subsequent analysis on changes in $\mathcal {L}_L$ only; we note this precludes many facilities from the subsequent analysis due to a lack of data on these parameters, but we include as many facilities as possible to identify trends.

Figure 8. Absolute (dimensional) variation of (a) large eddy length scale $\mathcal {L}$ and integral length scale $\mathcal {L}_L$ with facility half-width and (b) integral length scale with outlet jet diameter. The following markers represent facility type: loudspeakers ($\blacksquare$, red); RJAs (*, green); fans ($\bullet$, blue); and rotating elements ($\blacklozenge$, pink). The filled symbols represent $\mathcal {L}_L$ and the open symbols represent $\mathcal {L}$. See the Appendix (table 6) for full listing of sources from which data points were generated.

While $\mathcal {L}_L$ and $D/2$ are positively correlated, deviations from this correlation indicate the influence of other factors on the turbulent length scales. Additionally, some studies have reported different values of $\mathcal {L}_L$ in facilities with the same dimensions (e.g. Reference Variano and CowenVariano & Cowen 2008; Reference Johnson and CowenJohnson & Cowen 2018), or observed changes in the value of $\mathcal {L}_L$ with varying input driving parameters (e.g. Reference Carter, Petersen, Amili and ColettiCarter et al. 2016; Reference Johnson and CowenJohnson & Cowen 2018); this further supports the idea that other factors besides facility size can impact $\mathcal {L}_L$. Another physical length scale that can directly impact $\mathcal {L}_L$ is the outlet jet diameter, $d_J$. Parameter $d_J$ is only relevant to RJAs and some of the loudspeaker-driven facilities with orifice plates. As is apparent in figure 8(b), there is a strong positive association between jet diameter and $\mathcal {L}_L$.

The equivalent actuator spacing, $S_e$, is a proposed counterpart to the mesh spacing in grid-generated turbulence (Reference Hopfinger and TolyHopfinger & Toly 1976; Reference Kurian and FranssonKurian & Fransson 2009) and to actuator spacing in multi-jet array facilities (Reference Yin, Zhang and LinYin, Zhang & Lin 2007; Reference Berk, Gomit and GanapathisubramaniBerk, Gomit & Ganapathisubramani 2016). Parameter $S_e$ describes a characteristic distance between active actuators, taking into account the stochasticity of spatial forcing in some facilities. It is equal to $S$ in (non-stochastically forced) fan- and loudspeaker-driven facilities (see figure 9a), because all actuators are operating concurrently. In RJA facilities, since not all jets are activated simultaneously, $S_e$ will be larger than the jet-to-jet spacing $S$ (recall § 2.3). To define $S_e$ in RJA facilities, we evenly distribute the number of jets that are, on average, operating at the same time across the overall area occupied by the original footprint of the jet array. The average number of jets operating simultaneously can be found by multiplying the total number of actuators, $N$, in an RJA facility by $\phi _{{on}}$. These hypothetical ‘active’ actuators are then distributed across the original array footprint so as to maximize their spacing and ensure forcing across the entire facility, as shown in figure 10. We note that as $\phi _{{on}}$ decreases, $S_e$ approaches facility size $D$, whereas as $\phi _{{on}}$ increases, $S_e$ approaches geometric jet spacing $S$.

Figure 9. Schematic diagrams of facilities with (a) symmetric vertex-mounted actuators and (b) planar actuator arrays indicating actuator spacing, equivalent actuator spacing and facility half-width.

Figure 10. Visualizations of jet spacing $S$ (left) and equivalent actuator spacing $S_e$ (right) for an $8 \times 8$ array of jets in an RJA facility with $\phi _{on}=14\,\%$.

Figure 11(a) indicates that the integral length scale weakly increases with $S_e$ (assuming $S_e = S$ for facilities with spatially uniform forcing). However, when we non-dimensionalize these two variables with respect to $D/2$ (figure 11b), this relationship disappears. This can be explained by the observation that, in general, as $D/2$ increases, $S_e$ also increases. Non-dimensionalizing $\mathcal {L}_L$ with respect to $D/2$ removes its effect, resulting in no significant trends between $S_e$ and $\mathcal {L}_L$. However, it is worth noting that in WWTs, there appears to be a positive correlation between the mesh spacing in grids and $\mathcal {L}_L$. Moreover, when non-dimensionalizing $\mathcal {L}_L$ with mesh spacing, the $\mathcal {L}_L$ values for different mesh spacings tend to collapse into a single curve (Reference Kurian and FranssonKurian & Fransson 2009).

Figure 11. Relationship between $\mathcal {L}_L$ and equivalent jet spacing $S_e$ in (a) dimensional and (b) non-dimensional form. See figure 8(a) for legend.

Another relevant scale is the ratio of the actuator diameter to equivalent actuator spacing, $d_J/S_e$, representing the length (or area, if squared) occupied by an outlet between two actuators. We note that $d_J/S_e$ is the inverse of the more commonly used non-dimensional jet spacing in many synthetic and non-synthetic jet studies (Reference Greco, Ianiro, Astarita and CardoneGreco et al. 2013; Reference San and ChenSan & Chen 2014), which is known to influence characteristics of the generated turbulent flow. As shown in figure 12(a), a positive correlation exists between $d_J/S_e$ and $\mathcal {L}_L$, indicating the possibility for higher values of $\mathcal {L}_L$ when the jet diameter occupies a larger proportion of the distance between actuators. It should be noted that we cannot imply that by simply increasing the number of active actuators (which would also decrease $S_e$), $Re_{\lambda }$ will necessarily increase. Previous work with RJAs has explored the effect of increasing the mean percentage of active jets, and found an optimum of 12.5 % (Reference Variano and CowenVariano & Cowen 2008), indicating that the trend shown here is more intricate than solely changing $d_J/S_e$.

Figure 12. Normalized integral length scale ${\mathcal {L}_L}/{D/2}$ versus (a) $d_J/S_e$ (ratio of jet diameter to equivalent spacing) in RJA and loudspeaker-driven facilities and (b) outlet jet Reynolds number (lower axis) and fan rotational speed (Reference Ravi, Peltier and PetersenRavi et al. (2013), upper axis). Datapoints from Reference Ravi, Peltier and PetersenRavi et al. (2013) indicated with +. See figure 8(a) for legend of remaining markers.

The outlet Reynolds number of the actuators, $Re_J$, is a crucial component of the interactions between the input momentum fluxes, and subsequently the generated HIT. Since $d_J$ was shown to influence $\mathcal {L}_L$ (see figure 8b), it is plausible to assume that $Re_J$ also affects $\mathcal {L}_L$. However, we find that the normalized integral length scale $\mathcal {L}_L$ does not vary with $Re_J$ (figure 12b). This is consistent with findings from fan facilities, where ${\mathcal {L}_L}/{D/2}$ remains unchanged as fan rotational speed increases, as shown in figure 12(b) for a study by Reference Ravi, Peltier and PetersenRavi et al. (2013).

While $Re_J$ does not directly affect $\mathcal {L}_L$ relative to facility size, it still plays an important role in other aspects of flow generation. For example, consider the ratio between the strength of the actuator outlet flux and the facility dimensions. If the flows generated by the actuators dissipate before interacting with each other, there will be no opportunity for HIT to form. This can occur if the actuators are not strong enough compared with the size of the facility. Conversely, if the actuators are spaced too closely, or if their outlet flow and input energy are too strong in relation to the available volume for HIT generation, the momentum may become amplified in particular directions, leading to higher levels of anisotropy (Reference Chang, Bewley and BodenschatzChang et al. 2012; Reference Carter, Petersen, Amili and ColettiCarter et al. 2016; Reference Esteban, Shrimpton and GanapathisubramaniEsteban et al. 2019). From here we can discern a general governing principle for the design of turbulence-generating facilities: the size of the facility must be selected based on the ability of the actuator to impart sufficient momentum over a distance, which is a function of the actuator strength (Reference Dou, Pecenak, Cao, Woodward, Liang and MengDou et al. 2016; Reference Hoffman and EatonHoffman & Eaton 2021).

The ratio between $Re_J$ and the available volume for turbulence generation can also affect the generation of total mean flow, especially when a negligible mean flow is desired. Small chambers often exhibit a smaller overall mean velocity compared with larger ones (Reference Hwang and EatonHwang & Eaton 2004; Reference Hoffman and EatonHoffman & Eaton 2021). Additionally, $Re_{\lambda }$ is typically lower in smaller facilities than in larger ones. Figure 13(a) shows that as the size of the facility increases, turbulent flow with higher $Re_{\lambda }$ is generated. This may be due to the fact that achieving a high $Re_{\lambda }$ requires high input axial momentum. We note that figure 13(a) includes only the maximum values of $Re_{\lambda }$ reported for each facility. In small facilities, the limited space may impede momentum decay, resulting in a very high mean flow. Therefore, to maintain a negligible mean flow state, the input energy in the facility should be adjusted based on its size. For example, Reference Rusello and CowenRusello & Cowen (2015) (recall § 2.3) observed high values of mean flow strength due to high input energy relative to the size of the facility. Therefore, to produce zero-mean-flow HIT with desired characteristics, we speculate that higher values of $Re_J$ are needed in larger facilities. This theory is supported by figure 13(b), which shows a positive relationship between $Re_J$ and $Re_{\lambda }$. We note that for fan facilities, $Re_J$ is calculated using the blade length or radius of the fan (whichever is available) as the length scale and $f D_{fan}$ as the velocity scale.

Figure 13. The reported maximum $Re_{\lambda }$ of generated turbulence with (a) facility length scale $D/2$ and (b) outlet Reynolds number for RJA and fan facilities. See figure 8(a) for legend.

As a final thought, we note that some facility designs may be more appropriate for certain investigative questions. For example, studies of decay may benefit from the use of actuators whose energy injection can be halted instantaneously. However, if the goal is to increase the overall energy injected into the flow, rotating elements may be more appropriate (recalling the previous discussion on how facility size may create a ceiling for $Re_\lambda$, and how injecting more energy may serve to increase mean flow up to the point where the flow cannot be assumed to have zero mean velocity). Those interested in Lagrangian turbulence or particle transport may wish to prioritize a design that maximizes the size of the homogeneous and isotropic region, due to the importance of the history of the flow. Similar questions may also require greater tunability of the Kolmogorov length and time scales; however, we did not observe any consistent functional dependence of $\eta$ or $\tau _\eta$ on the design parameters considered here. In all cases, facility designers should keep in mind the following major design parameters: (1) facility size and geometry, (2) actuator type, (3) actuator spacing/arrangement, (4) actuator energy level (e.g. $Re_J$ or $f$) and (5) spatiotemporal variability in forcing. While there is no guaranteed formula for designing a turbulence facility with specific flow characteristics, the analysis provided within this section can provide a starting point.