3.1. Single-point velocity statistics

Figure 3 shows the time-averaged velocity fields for four representative forcing strategies: constant frequency (case CF1, 45 Hz), random frequency (case CF6), random spatial on/off actuation (case CS2, $\phi _{\mathrm{on}}=50\,\%$ ) and random power modulation (case CP5). Among these, the constant-frequency operation exhibits the strongest mean circulation, while the other strategies produce negligible mean motion in the central region of the chamber. The measured in-plane mean velocities for baseline case (case CF6) were $\langle \mathrm{U_1} \rangle = -0.0063\,\mathrm{m\,s^{-1}}$ and $\langle \mathrm{U_2} \rangle = -0.0064\,\mathrm{m\,s^{-1}}$ .

Figure 3.

Temporal average velocity field of cases (a) CF1, (b) CF6, (c) CS2 and (d) CP5.

Figure 3 Long description

Panel A: A heat map showing the temporal average velocity field for case CF1. The x-axis is labeled x1 (m) ranging from -0.04 to 0.04 meters. The y-axis is labeled x2 (m) ranging from -0.04 to 0.04 meters. The color scale on the right indicates the magnitude of the velocity, with yellow representing higher values and dark purple representing lower values. The heat map shows a gradient of colors with distinct patterns and contours. Panel B: A heat map showing the temporal average velocity field for case CF6. The axes and color scale are the same as in Panel A. The heat map displays different patterns and contours compared to Panel A. Panel C: A heat map showing the temporal average velocity field for case CS2. The axes and color scale are consistent with the previous panels. The heat map exhibits unique patterns and contours. Panel D: A heat map showing the temporal average velocity field for case CP5. The axes and color scale remain the same. The heat map presents distinct patterns and contours different from the other panels.

Figure 4.

(a) Probability density functions (p.d.f.s) of the velocity fluctuations $u_1$ and $u_2$ for case CF6; power spectral density of (b) $u_1$ and (c) $u_2$ for cases CF1, CF2, CF3 and CF4.

Figure 4 Long description

Panel A: A line graph shows the probability density functions (PDFs) of velocity fluctuations for case CF6. The x-axis represents velocity (u1/u2) and the y-axis represents the probability density function (PDF) on a logarithmic scale. The graph includes two data series, u1 and u2, represented by circles and triangles, respectively. Panel B: A line graph displays the power spectral density (E11) for cases CF1, CF2, CF3, and CF4 at frequencies 11 Hz, 15 Hz, 22 Hz, and 44 Hz. The x-axis represents frequency (f) in Hertz (Hz) on a logarithmic scale, and the y-axis represents the power spectrum (E11) in meters squared per second (m^2/s) on a logarithmic scale. Different colors represent different frequencies: green for 11 Hz, blue for 15 Hz, red for 22 Hz, and black for 44 Hz. Panel C: A line graph shows the power spectral density (E22) for the same cases and frequencies as Panel B. The x-axis represents frequency (f) in Hertz (Hz) on a logarithmic scale, and the y-axis represents the power spectrum (E22) in meters squared per second (m^2/s) on a logarithmic scale. Different colors represent different frequencies: green for 11 Hz, blue for 15 Hz, red for 22 Hz, and black for 44 Hz.

Probability density functions (p.d.f.s) of the velocity fluctuations were constructed using all two-dimensional (2-D) PIV snapshots for each case. As shown in Figure 4, the p.d.f.s of $u_1$ and $u_2$ fluctuations for case CF6 are approximately Gaussian, indicating the absence of directional bias in either component. The skewness $S$ and kurtosis $K$ fall within the ranges $S\in [-0.015,0.023]$ and $K\in [3.1,3.9]$ across all operating modes, consistent with the Gaussian values $(S=0,\,K=3)$ . In contrast, asymmetric single-panel facilities often exhibit strong positive skewness and elevated kurtosis aligned with the jet axis (Pérez-Alvarado et al. Reference Pérez-Alvarado, Mydlarski and Gaskin2016; Variano & Cowen Reference Variano and Cowen2008). The present symmetric two-panel configuration mitigates such directional imprinting, consistent with the observations of Carter et al. (Reference Carter, Petersen, Amili and Coletti2016).

The actuation in the present facility introduces two characteristic frequencies: the speaker driving frequency, which governs the jet ejection process, and the on/off switching frequency associated with the spatiotemporal forcing algorithm. Either of these frequencies could, in principle, imprint a deterministic signature on the measured velocity fluctuations and thereby contaminate the turbulence statistics. Consequently, it is essential to verify that the observed fluctuations are dominated by turbulent dynamics rather than by periodic forcing. Longitudinal power spectral density (PSD) $E_{11}(f)$ and $E_{22}(f)$ of the velocity fluctuations are therefore examined for all actuation cases, with particular attention paid to both the driving frequency and the on/off switching frequency. PSDs were computed using Welch’s method to improve statistical convergence of the PSD estimates. The velocity time series at each spatial location was divided into overlapping segments of length $128$ , with $50\,\%$ overlap between adjacent segments. Each segment was multiplied by a Hanning window prior to Fourier transformation to reduce spectral leakage. The PSD for each segment was computed from the squared magnitude of the discrete Fourier transform and normalised by the sampling frequency and window energy. The final PSD estimate was obtained by ensemble averaging the segment-wise periodograms. Spatially averaged spectra were then calculated by averaging the PSDs over all PIV locations. For the constant-frequency cases (CF1–CF4), the PSDs shown in Figure 4 reveal a modest increase in spectral energy near the speaker driving frequency for 11, 15 and 22 Hz, while the driving frequency for the 44 Hz case is beyond the measurement cut off. Even in these cases, the spectral contribution remains weak relative to the broadband turbulent energy. For all randomised forcing configurations, including random frequency switching and spatiotemporal on/off forcing, no discernible peaks are observed at any particular frequencies. The absence of narrow-band spectral features confirms that the imposed actuation does not introduce persistent temporal oscillations and that the measured velocity fluctuations are dominated by broadband turbulence rather than deterministic forcing effects.

A key quantitative descriptor of turbulence intensity is the root-mean-square (r.m.s.) velocity fluctuation, defined as

(3.3) \begin{equation} u_i' = \sqrt {\overline {u_i^2}}, \qquad i=1,2,3, \end{equation}

where the overbar denotes temporal averaging. For 2-D PIV, the r.m.s. fluctuations $u_1'$ and $u_2'$ are computed directly, and a weighted r.m.s. magnitude is defined as $u' = (u_1' + 2u_2')/3$ based on the statistical symmetry of the forcing about the $x_1$ axis. The field of $u_1'$ and $u_2'$ measured from the baseline case is shown in Figure 5a,b. The values of $\langle u'\rangle$ for all cases are listed in Table 2; they range from $0.40$ to $0.74\,\mathrm{m\,s^{-1}}$ , with the highest value obtained for the baseline random-frequency case (case CF6) and the lowest for the constant-frequency case at 11 Hz (case CF4). These magnitudes are consistent with those reported for other loudspeaker-driven facilities (Goepfert et al. Reference Goepfert, Marié, Chareyron and Lance2010; Hoffman & Eaton Reference Hoffman and Eaton2021; Hwang & Eaton Reference Hwang and Eaton2004) and two-panel air-jet systems (Carter et al. Reference Carter, Petersen, Amili and Coletti2016).

Figure 5.

The (a) $u_1'$ , (b) $u_2'$ and (c) MFF fields for the baseline case (case CF6).

Isotropy is assessed through the ratio of directional r.m.s. fluctuations. For the present facility, the isotropy ratio $\langle u_1'\rangle /\langle u_2'\rangle$ remains below $1.34$ for all operating conditions, significantly more isotropic than the range $1.35$ – $1.75$ reported for the two-panel air-based facility of Carter et al. (Reference Carter, Petersen, Amili and Coletti2016). However, the water-based two-panel facility investigated by Bellani & Variano (Reference Bellani and Variano2014) achieved a closer approach to isotropic conditions than the present configuration. The baseline case exhibits $\langle u_1'\rangle /\langle u_2'\rangle =1.27$ , in agreement with prior measurements in similar configurations (Carter et al. Reference Carter, Petersen, Amili and Coletti2016; Esteban et al. Reference Esteban, Shrimpton and Ganapathisubramani2019). Interestingly, the constant-frequency cases show the closest approach to unity, suggesting enhanced directional balance under mono-frequency forcing.

Homogeneity is quantified using the homogeneity deviation (HD), defined as

(3.4) \begin{equation} \mathrm{HD} = \frac {2\sigma }{u'}, \end{equation}

where $\sigma$ is the spatial standard deviation of $u'$ . With the exception of the constant-frequency cases, all operating modes yield $\mathrm{HD}\approx \leq 0.1$ , indicating good spatial uniformity of the turbulence field. The baseline case attains $\mathrm{HD}=0.06$ , comparable to values reported in other symmetric facilities (Carter et al. Reference Carter, Petersen, Amili and Coletti2016; Esteban et al. Reference Esteban, Shrimpton and Ganapathisubramani2019).

The turbulent kinetic energy (TKE) is calculated as $k = 0.5(u_1'^2 + u_2'^2 + u_3'^2)$ , where the symmetry of the apparatus justifies the approximation $u_3' \approx u_2'$ . The TKE values for all forcing strategies are reported in Table 2. The baseline case attains $ \langle k\rangle =0.83\,\mathrm{m^2 s^{-2}}$ . The relative strength of mean flow compared with turbulence is characterised by the mean-flow factor (MFF),

(3.5) \begin{equation} \mathrm{MFF} = \frac {|\overline {{U}}|}{u'}, \qquad |\overline {{U}}| = \sqrt {\frac {\overline {{U_1}}^2 + 2\,\overline {{U_2}}^2}3}, \end{equation}

where $|\overline {U}|$ is the magnitude of the mean velocity. While the mean velocities are small compared with $u'$ , certain constant-frequency cases exhibit $\langle \mathrm{MFF}\rangle \gt 0.15$ , underscoring the importance of minimising coherent mean motions. The MFF field for the baseline case is shown in Figure 5c and the spatial average $\langle \mathrm{MFF}\rangle =0.06$ , comparable to the value $\langle \mathrm{MFF}\rangle \approx 0.10$ obtained by Carter et al. (Reference Carter, Petersen, Amili and Coletti2016). An alternative metric used to quantify the relative contribution of mean flow is the ratio of mean kinetic energy to turbulent kinetic energy, $\mathrm{M^*} = (\overline {{U_1}}^2 + 2\,\overline {{U_2}}^2)/(2k)$ which, under the present assumptions, is approximately related to $\mathrm{MFF}$ through $M^* \approx \mathrm{MFF}^2$ . Values of $\langle M^*\rangle$ across the investigated cases range from 0.01 to 0.06. Notably, some constant-frequency operations yield $\langle M^*\rangle \gt 0.05$ , exceeding the threshold for negligible mean flow proposed by Variano & Cowen (Reference Variano and Cowen2008). In contrast, all randomised forcing strategies achieve $\langle M^*\rangle \leq 0.05$ , with the baseline case reaching a value of 0.01, thereby confirming the negligible mean flow.

3.2. Multi-point velocity statistics

The large-scale structure of turbulence is characterised using two-point velocity statistics obtained from the PIV velocity fields. The longitudinal and transverse autocorrelation functions at separation $r_j$ in direction $x_j$ are defined as

(3.6) \begin{equation} \rho _{ii}(r_j) = \frac {\left \langle \overline {u_i(x_j+r_j)\,u_i(x_j)} \right \rangle } {\left \langle \overline {u_i^2(x_j)} \right \rangle }. \end{equation}

The integral length scale, representing the characteristic size of the largest energy-containing eddies, is defined as $L_{ij} = \int _{0}^{\infty } \rho _{ii}(r_j)\,\text{d}r_j$ .

In the present study, several of the spatial autocorrelation functions did not reach zero within the available PIV field of view. Furthermore, the correlation functions became increasingly noisy at large spatial separations, making direct numerical integration unreliable (Griffin et al. Reference Griffin, Wei, Bodenschatz and Bewley2019). Several approaches have been proposed for estimating integral length scales from truncated correlation measurements, including extrapolation of the unresolved correlation tail using various functional forms. Among these, exponential fitting has been widely adopted in PIV-based turbulence studies when the correlation function remains positive over the measurement domain (Bewley et al. Reference Bewley, Chang and Bodenschatz2012; de Jong et al. Reference de Jong, Cao, Woodward, Salazar, Collins and Meng2009; Griffin et al. Reference Griffin, Wei, Bodenschatz and Bewley2019; Han et al. Reference Han, Watanabe and Nagata2026; Mori et al. Reference Mori, Watanabe and Nagata2024). For the present data, the measured autocorrelation functions exhibited an approximately exponential decay over the resolved spatial range and the exponential model provided a satisfactory representation of the observed behaviour. Accordingly, the autocorrelation function was approximated as

(3.7) \begin{equation} \rho _{ii}(r_j) = \exp \!\left (-\frac {r_j}{L_{ij}}\right ), \end{equation}

where the integral length scale, $L_{ij}$ , is obtained through a nonlinear least-squares regression of the measured autocorrelation function (Bewley et al. Reference Bewley, Chang and Bodenschatz2012; Griffin et al. Reference Griffin, Wei, Bodenschatz and Bewley2019; Han et al. Reference Han, Watanabe and Nagata2026; Mori et al. Reference Mori, Watanabe and Nagata2024). Under this formulation, the fitted decay constant corresponds directly to the integral length scale, eliminating the need for explicit tail extrapolation and subsequent numerical integration. Assuming approximate axisymmetry of the turbulence, a weighted estimate of the longitudinal integral length scale is computed as $L_L = (L_{11} + 2L_{22})/3$ following the approach of Carter et al. (Reference Carter, Petersen, Amili and Coletti2016).

Figure 6.

Longitudinal and transverse autocorrelation functions for the baseline configuration CF6.

Figure 6 shows the longitudinal and transverse autocorrelation functions for the baseline configuration CF6 (random-frequency actuation at the highest average power). Integral length scales for all configurations are summarised in Table 2. While the actuation method has minimal effect on the integral length scale, slight variations are observed for the cases involving power modulation. The baseline case yields $L_L = 2.7$ cm, smaller than two-panel jet-driven facilities (Bellani & Variano Reference Bellani and Variano2014; Carter et al. Reference Carter, Petersen, Amili and Coletti2016; Esteban et al. Reference Esteban, Shrimpton and Ganapathisubramani2019) but comparable to other loudspeaker-driven turbulence generators (Goepfert et al. Reference Goepfert, Marié, Chareyron and Lance2010; Hwang & Eaton Reference Hwang and Eaton2004). Transverse integral scales $L_N$ are obtained by $L_N = (L_{12}+L_{21}+L_{22}/2)/3$ (Carter et al. Reference Carter, Petersen, Amili and Coletti2016) and the isotropy ratio $L_L/L_N$ ranges from $1.76$ to $1.94$ , close to the isotropic value of $2$ . No systematic trend with actuation method or input power is observed. The Reynolds number based on the longitudinal length scale is calculated as $Re_L = u' L_L/\nu$ with values reported in Table 2.

The second-order structure function,

(3.8) \begin{equation} D_{ii}(r_j) = \left \langle \overline {\left [u_i(x_j+r_j)-u_i(x_j)\right ]^2} \right \rangle , \end{equation}

is used to estimate the dissipation rate. This approach is preferred for PIV data of HIT because direct gradient-based estimates are sensitive to noise and out-of-plane motion (Buxton et al. Reference Buxton, Laizet and Ganapathisubramani2011; Tanaka & Eaton Reference Tanaka and Eaton2007; de Jong et al. Reference de Jong, Cao, Woodward, Salazar, Collins and Meng2009). According to Kolmogorov’s (Reference Kolmogorov1941) theory, the longitudinal structure function in the inertial range follows

(3.9) \begin{equation} D_{ii}(r_i) = C_2(\varepsilon r_i)^{2/3}, \end{equation}

with $C_2 = 2.0$ (Pope Reference Pope2000). Figure 7a presents $D_{11}(r_1)$ for the baseline case, illustrating the $r^{2/3}$ inertial-range scaling and the $r^2$ dissipative scaling.

Figure 7.

(a) $D_{11}(r_1)$ for the baseline configuration CF6, compensated longitudnal structure functions along (b) $x_1$ and (c) $x_2$ (c) direction for the cases CF1, CF6, CS2 and CP5.

Figure 7 Long description

Panel A: A line graph shows the compensated longitudinal structure function D11 as a function of r1 in meters. The y-axis is labeled D11 in square meters per second squared, and the x-axis is labeled r1 in meters. The graph includes three lines: a red line with circles representing D11(r1), a blue line representing r1 squared, and a black line representing r1 to the power of 2/3. Panel B: A line graph shows the compensated longitudinal structure function along the r1 direction for different cases. The y-axis is labeled 1 divided by the square root of D11(r1) divided by C2 to the power of 3/2, and the x-axis is labeled r1 in meters. The graph includes four lines: a red line with circles for CF6, a blue line with triangles for CF1, a green line with squares for CS2, and a black line with triangles for CP5. Panel C: A line graph shows the compensated longitudinal structure function along the r2 direction for different cases. The y-axis is labeled 1 divided by the square root of D22(r2) divided by C2 to the power of 3/2, and the x-axis is labeled r2 in meters. The graph includes four lines: a red line with circles for CF6, a blue line with triangles for CF1, a green line with squares for CS2, and a black line with triangles for CP5.

The dissipation rate is obtained from the compensated form,

(3.10) \begin{equation} \varepsilon = \left (\frac {D_{ii}(r_i)}{C_2}\right )^{3/2} r_i^{-1}. \end{equation}

Compensated longitudinal structure functions for different actuation protocols are shown in Figure 7b,c. A clear plateau is observed in all cases. The plateau level is slightly higher in the $x_1$ direction than in $x_2$ , with a baseline-case discrepancy of approximately $9.5\,\%$ . This inertial-subrange anisotropy is supportive of balancing the energy cascade as observed by Knutsen et al. (Reference Knutsen, Baj, Lawson, Bodenschatz, Dawson and Worth2020). A weighted average of the converged values from both directions is used as the final dissipation estimate. The resulting $\varepsilon$ values are listed in Table 2. Dissipation increases systematically with actuation power ( $P$ ), shown in Figure 8a. Dimensionally, $\varepsilon \sim P$ and the deviation from linearity suggests a moderate increase of actuator efficiency with power. Constant-frequency forcing yields lower dissipation than random-frequency forcing, while spatial on/off forcing slightly reduces dissipation relative to random-frequency operation.

The Kolmogorov scales follow from $\varepsilon$ as

(3.11) \begin{equation} \eta = \left (\frac {\nu ^3}{\varepsilon }\right )^{1/4}, \qquad \tau _\eta = \left (\frac {\nu }{\varepsilon }\right )^{1/2}, \end{equation}

with values reported in Table 2.

The Taylor microscale and associated Reynolds number are computed as

(3.12) \begin{equation} \lambda = \sqrt {\frac {15\nu \langle u'\rangle ^2}{\varepsilon }}, \qquad Re_\lambda = \frac {\langle u'\rangle \lambda }{\nu }. \end{equation}

For all configurations, $\lambda$ ranges from $3.7$ to $5.3$ mm, decreasing with higher power and actuation frequency.

Figure 8.

(a) Dissipation rate ( $\varepsilon$ ) and (b) $Re_\lambda$ as a function of speaker average power.

The variation of $Re_\lambda$ with actuation power, $P$ , follows a power law consistent with the scaling proposed by Hoffman & Eaton (Reference Hoffman and Eaton2021). Figure 8b shows $Re_\lambda \propto P^{0.12}$ , close to the theoretical exponent $1/6$ . Hoffman & Eaton (Reference Hoffman and Eaton2021) observed a larger exponent ( $0.42$ ) in experiment as opposed to theoretical exponent, attributed to a strong increase in actuator efficiency with power in their configuration, which is more moderate in our case.

3.3. Coherent structures

The present study focuses on qualitative analysis, repeatability and energetic contribution of large-scale coherent structures generated by geometry and different forcing strategies within the HIT region of the facility. To investigate these spatially organised motions, POD (Lumley Reference Lumley, Yaglom and Tatarski1967) is employed. POD provides an energy-based statistical framework capable of identifying spatially organised motions that are not necessarily periodic. The method has been widely applied to turbulent flows containing energetic organised motions, including wakes, jets, a mechanically stirred tank and confined turbulent flows (de Lamotte et al. Reference de Lamotte, Delafosse, Calvo and Toye2018; Lacassagne et al. Reference Lacassagne, Simoëns, EL Hajem and Champagne2021; Linsong et al. Reference Linsong, Tong, Shaoyi, Yongqian, Hamid, Osman, Yang, Haotian, Xinle and Maozhao2025; Rovira et al. Reference Rovira, Engvall and Duwig2021; Vitkovicova et al. Reference Vitkovicova, Yokoi and Hyhlik2020; Yin et al. Reference Yin, Fan, Sandberg and Li2019).

POD decomposes the velocity field into orthogonal spatial modes and corresponding temporal coefficients (Lumley Reference Lumley, Yaglom and Tatarski1967; Sirovich Reference Sirovich1987). In the present study, the snapshot POD formulation (Sirovich Reference Sirovich1987) is employed for its computational efficiency on large spatial datasets with relatively few temporal samples. The instantaneous fluctuating velocity field obtained from Reynolds decomposition (3.1) may be expressed as

(3.13) \begin{equation} u_i(\mathbf{x}, t) = \sum _{j=1}^{N} a^{[j]}(t)\,\psi _i^{[j]}(\mathbf{x}), \end{equation}

where $\psi _i^{[j]}(\mathbf{x})$ are orthonormal spatial modes and $a^{[j]}(t)$ are the corresponding temporal coefficients. The eigenvalues $\Lambda ^{[j]}$ associated with each mode quantify the fraction of turbulent kinetic energy captured.

To compute the POD modes, the fluctuating velocity components are assembled into a stacked snapshot matrix $S(\mathbf{x},t)$ , in which each column represents a temporal snapshot, and all spatial points and velocity components are arranged vertically. The correlation matrix and eigenvalue problem are then given by

(3.14) \begin{equation} C = \frac {1}{N} S^{\mathrm{T}} S, \qquad C\,\phi ^{[j]} = \Lambda ^{[j]}\,\psi ^{[j]}, \end{equation}

where $\psi ^{[j]}$ denotes the stacked POD modes. Unless otherwise noted, the term ‘modes’ refers to the combined POD modes that account for all velocity directions simultaneously.

Figure 9.

(a) Modal energy fraction ( $\Lambda ^{[j]}/\sum _{j=0}^{N} \Lambda ^{[j]}$ ) of each mode; (b) cumulative energy fraction ( $\sum _{j=0}^{n}\Lambda ^{[j]}/\sum _{j=0}^{N} \Lambda ^{[j]}$ ).

To examine coherent structures in the present turbulence facility, four representative cases (cases CF6, CS1, CS2 and CP5) were selected based on overall performance and the degree of randomness in the forcing strategy. The modal energy distributions for these cases are shown in Figure 9. The overall shape of the modal energy distribution is relatively insensitive to the forcing strategy.

The most energetic POD modes (modes 1–6) are shown in Figure 10. It should be emphasised that individual POD modes do not represent instantaneous flow realisations; rather, they constitute an orthogonal basis describing the dominant and repeatable patterns of flow variability. The following discussion therefore focuses on the qualitative interpretation of these energetic coherent modes, consistent with approaches commonly adopted in previous POD-based turbulence studies (Arosemena & Solsvik Reference Arosemena and Solsvik2023).

Figure 10.

Normalised nondimensional mode shapes of modes 1–6 for selected four cases (increasing rank: left to right). Normalisation is applied to show the modal shape rather than the relative energy content.

Figure 10 Long description

Panel CF6: A heat map with x1 and x2 axes in centimeters. The color scale ranges from 0.00 to 1.00, with lighter colors indicating higher values. The map shows a gradient pattern with distinct regions of varying intensity. Panel CS1: A heat map with x1 and x2 axes in centimeters. The color scale ranges from 0.01 to 1.00, with lighter colors indicating higher values. The map displays a mix of high and low-intensity regions with noticeable patterns. Panel CS2: A heat map with x1 and x2 axes in centimeters. The color scale ranges from 0.00 to 1.00, with lighter colors indicating higher values. The map exhibits a central concentration of high-intensity values surrounded by lower-intensity regions. Panel CP5: A heat map with x1 and x2 axes in centimeters. The color scale ranges from 0.01 to 1.00, with lighter colors indicating higher values. The map features a diagonal gradient with varying intensity levels.

Across all forcing conditions, the leading energetic modes, particularly modes 1 and 2, exhibit pronounced panel-to-panel transport aligned with the jet forcing direction. In cases CF6, CS1 and CS2, this large-scale transport is accompanied by weaker vortical motions, whereas case CP5 displays comparatively stronger transverse and asymmetric flow organisation. An additional contributing mechanism may arise from exhaust-to-suction exchange between opposing panels, as suggested by the frequent sign reversals observed in the corresponding temporal coefficients, indicating bidirectional transport behaviour. Higher-order energetic modes, such as mode 3 in cases CF6, CS1 and CS2 or mode 2 in case CP5, exhibit transverse or vertical motions that may be associated with secondary instabilities, including shear-layer roll-up, lateral jet deflection or local pressure redistribution. Several modes also contain single or paired vortical structures that likely originate from geometric confinement and interactions among neighbouring actuators separated by approximately one mesh spacing. Although the PIV field of view captures only a portion of the chamber, the persistence and energetic dominance of these modes suggest that the associated organised motions extend over a substantial fraction of the facility.

A clear distinction between forcing strategies emerges from the modal structures. Case CF6 (random frequency, highest constant power, $\phi _{\mathrm{on}}=1$ ) exhibits strong swirling patterns, indicating that continuous high-energy injection promotes persistent coherent recirculation. Random frequency alone is therefore insufficient to eliminate global recirculation induced by confinement and energy remains trapped at large scales. In case CS1 (random frequency, highest constant power, $\phi _{\mathrm{on}}=0.75$ ), large-scale swirling structures persist, but become less symmetric and more distorted. Increased spatial variability in the streamlines indicates that partial randomisation begins to weaken coherence, although enough jets remain active to sustain energetic structures. In case CS2 (random frequency, highest constant power, $\phi _{\mathrm{on}}=0.5$ ), the modes become visibly more fragmented, vortices are smaller and less domain-filling, and no single dominant circulation pattern is observed. Reduced simultaneous forcing prevents the formation of stable large-scale loops and energy injection becomes more locally distributed. Case CP5 (random frequency and random power, $\phi _{\mathrm{on}}=1$ ) exhibits the most irregular and asymmetric modes. Although vortical structures are present, they are intermittent and vary strongly from mode to mode. Random power destroyed the spatial coherence, as nonuniform jet strength prevents synchronisation even when the number of ejections is high. These qualitative behaviours demonstrate that increasing the degrees of freedom in the forcing strategy effectively suppresses persistent large-scale coherent motions.

Although the most energetic POD modes qualitatively reveal the presence of large-scale coherent organisation, no quantitative criterion exists to distinguish facility-induced coherent structures from other turbulent structures. In randomly forced turbulent flows, organised motions are not necessarily confined to a single dominant mode or a small subset of energetic modes. Instead, coherent and less-coherent motions coexist continuously across the turbulent energy spectrum. Consequently, careful interpretation of POD results is required when identifying system-induced coherent structures.

To distinguish these coherent structures quantitatively, the repeatability of the POD modes across statistically independent realisations was examined within the HIT region, whose size is approximately four integral length scales. The present analysis follows the framework introduced by Butcher & Spencer (Reference Butcher and Spencer2019), in which coherent structures are identified through spatial correlation of POD modes obtained from repeated realisations of the same flow condition. For each operating condition, five statistically independent datasets (Sets A–E), each containing 500 image pairs, were acquired under identical forcing conditions. Snapshot POD was then applied independently to each dataset, yielding $N=500$ spatial modes $\psi ^{[j]}$ for each realisation.

To evaluate modal repeatability, spatial cross-correlations were computed between POD modes from different datasets. For a given mode $j$ in Set A, the spatial correlation was evaluated against all $N$ modes in Set B, producing $N$ correlation coefficients. The maximum correlation value was retained as the representative correlation between mode $j$ in Set A and Set B, corresponding to the best spatial match. This procedure was repeated for Set A paired with Sets C, D and E, resulting in four correlation values for each mode in Set A. The average of these values defines the modal repeatability coefficient $R^{[j]}$ . The entire process was subsequently repeated by treating each dataset as the reference realisation, yielding repeatability coefficients for all $2500$ modes across the five datasets. Modes exhibiting consistently high spatial correlation across independent realisations are interpreted as signatures of persistent system-induced coherent organisation associated with the forcing strategy and facility geometry, whereas weakly correlated modes correspond to less repeatable and more spatially distributed turbulent motions.

Figure 11 presents the normalised modal energy $\Lambda ^{[j]}/\sum \Lambda ^{[j]}$ as a function of modal repeatability coefficient $R^{[j]}$ for all five datasets of the four representative forcing conditions. It should be noted that the modal energies shown here are obtained from independent POD analyses of each dataset, unlike in Figure 9, where the modal energies were computed from a combined POD analysis using all five datasets simultaneously. Across all cases, a clear trend is observed: modes containing larger fractions of turbulent kinetic energy exhibit systematically higher repeatability across realisations. This behaviour supports the interpretation that the energetically dominant modes correspond to persistent coherent structures generated by the forcing strategy and geometry.

The first three POD modes, which were identified qualitatively in Figure 10 as the most organised structures, form a distinct cluster with repeatability coefficients typically exceeding $R^{[j]}\approx 0.75$ . In contrast, modes 4–7 occupy a transition region in which some forcing conditions exhibit clear large-scale organisation, while others display increasingly distorted structures. Consistent with these observations, the corresponding repeatability coefficients generally fall within $0.6 \lesssim R^{[j]} \lesssim 0.75$ . This transition region is particularly important because it differentiates forcing protocols with strongly repeatable coherent structures from those dominated by less-organised motions.

To quantitatively separate these regimes, a reference repeatability threshold was introduced. As noted by Butcher & Spencer (Reference Butcher and Spencer2019), the selection of such a threshold is inherently empirical, and depends on the flow configuration and dataset. In the present study, the lower and upper repeatability bounds of the transition region were identified for all forcing conditions, and an energy-weighted average was computed using the modal energy content as the weighting factor. Similar transition behaviour was also observed in a companion study employing the same turbulence facility and measurement methodology (Gautam & Berk Reference Gautam and Berk2026). Considering all cases from both studies, the resulting weighted-average threshold was found to be $R^{[j]}=0.66$ . As shown in Figure 11, this value approximately marks the onset of the energetic, highly repeatable modal population. Accordingly, modes with $R^{[j]}\gt 0.66$ are interpreted as signatures of strongly repeatable facility-induced coherent structures. Importantly, the qualitative conclusions regarding the relative degree of coherent organisation among the forcing strategies remain insensitive to moderate variations of the threshold within the range $0.6 \le R^{[j]} \le 0.75$ .

Table 3.

Percentage of coherent energy to that of total turbulent kinetic energy

Table 3 Long description

A table comparing percentages of coherent energy to total turbulent kinetic energy across different cases. The table has one row and four columns. Column headers are CF6, CS1, CS2, and CP5. Row 1: CF6, 39.77 percent; CS1, 35.25 percent; CS2, 30.15 percent; CP5, 31.25 percent.

Figure 11.

Modal energy fraction ( $\Lambda ^{[j]}/\sum \Lambda ^{[j]}$ ) versus repeatability coefficient ( $R^{[j]}$ ) for cases (a) CF6, (b) CS1, (c) CS2 and (d) CP5. The legends are the same for all four cases and the blue dashed lines indicate the assumed threshold for coherent modes.

Figure 11 Long description

Panel A: A scatter plot showing the relationship between modal energy fraction and repeatability coefficient for case CF6. The x-axis is labeled R^Uj and ranges from 0.0 to 1.0. The y-axis is labeled λ^Uj/Σλ^Uj and ranges from 0.00 to 0.15. Different sets are represented by various symbols: circles for Set A, diamonds for Set B, downward triangles for Set C, squares for Set D, and upward triangles for Set E. A blue dashed line indicates the assumed threshold for coherent modes at R^Uj = 0.66. Panel B: A scatter plot showing the relationship between modal energy fraction and repeatability coefficient for case CS1. The axes and symbols are the same as in Panel A. Panel C: A scatter plot showing the relationship between modal energy fraction and repeatability coefficient for case CS2. The axes and symbols are the same as in Panel A. Panel D: A scatter plot showing the relationship between modal energy fraction and repeatability coefficient for case CP5. The axes and symbols are the same as in Panel A.

The fraction of turbulent kinetic energy associated with the coherent modes is summarised in Table 3. For the baseline random-frequency forcing case CF6, coherent structures account for approximately $39.77\,\%$ of the total turbulent kinetic energy. Introducing additional spatiotemporal randomisation through on/off actuation reduces this fraction to $35.25\,\%$ for case CS1 and further to $30.15\,\%$ for case CS2. A comparable reduction is observed for the random-power forcing case CP5, for which the coherent-energy fraction is $31.25\,\%$ . These quantitative trends are consistent with the qualitative modal structures shown in Figure 10. Overall, the results demonstrate that a substantial fraction of the turbulent kinetic energy may reside within facility-induced coherent structures, even in nominally homogeneous low-mean-flow turbulence facilities. The energetic contribution of these structures therefore provides an additional and physically meaningful descriptor for characterising turbulence-generating facilities and assessing the influence of forcing strategy and geometry.

3.4. Controlled turbulence-rotation

Negligible-mean-flow turbulence facilities are typically designed to generate homogeneous isotropic turbulence under controlled conditions, with comparatively less emphasis on deliberately introducing large-scale circulation or anisotropy within the same apparatus. The present facility provides an additional degree of control by introducing large-scale mean-flow rotation through spatially varying jet forcing, without mechanically rotating the facility or its internal components. Traditional laboratory studies of rotating turbulence commonly employ rotating frames, grids or internal components such as disks and impellers (Baroud et al. Reference Baroud, Plapp, She and Swinney2002; Campagne et al. Reference Campagne, Machicoane, Gallet, Cortet and Moisy2016; Hopfinger et al. Reference Hopfinger, Browand and Gagne1982; Jacquin et al. Reference Jacquin, Leuchter, Cambonxs and Mathieu1990; Lemasquerier et al. Reference Lemasquerier, Favier and Le Bars2021; Morize et al. Reference Morize, Moisy and Rabaud2005; Simand et al. Reference Simand, Chillà and Pinton2000); a comprehensive review is provided by Godeferd & Moisy (Reference Godeferd and Moisy2015). In contrast, the present approach generates mean-flow rotation while the facility remains stationary.

Mean-flow rotation is induced by imposing a linear power gradient across the opposing jet arrays, with the gradient increasing in one direction on one panel and decreasing in the opposite direction on the facing panel. This configuration generates a large-scale circulation with an effective rotation axis normal to the imposed power gradient (i.e. along the $x_3$ direction). The outer jets, relative to the chamber centre, primarily sustain the large-scale circulation, while the inner jets contribute predominantly to turbulence generation through frequency modulation and/or spatiotemporal on/off switching.

Table 4.

Turbulence and rotation parameters for the case CTR

Table 4 Long description

An equation representing turbulence and rotation parameters. The equation starts with the fraction with numerator U prime squared and denominator U bar squared equals 1.12. Next, epsilon equals 4.1 meters squared per second cubed. Following, Re subscript L equals 85. Then, t subscript L equals 0.07 seconds. Finally, Ro equals 0.38.

Figure 12.

(a) Power-gradient forcing scheme used to introduce mean rotation. (b) Time-averaged velocity field for the controlled turbulence-rotation case (CTR).

An example of the imposed power gradient is shown in Figure 12a: on the right panel, the supplied power decreases linearly from $23.54\,\mathrm{W}$ at the top row to $4.66\,\mathrm{W}$ at the bottom row, with the opposite gradient applied on the left panel. This forcing generates a vertical mean velocity gradient, $\partial \overline {\mathrm{U}}_1/\partial x_2$ . Owing to confinement and continuity, a complementary gradient $\partial \overline {\mathrm{U}}_2/\partial x_1 \approx -\partial \overline {\mathrm{U}}_1/\partial x_2$ develops, while the diagonal gradients $\partial \overline {\mathrm{U}}_1/\partial x_1$ and $\partial \overline {\mathrm{U}}_2/\partial x_2$ remain negligible. The resulting time-averaged velocity field, shown in Figure 12b, reveals a strong, large-scale circulation in the measurement plane.

Table 4 summarises the key turbulence and mean-flow rotation parameters for the controlled turbulence-rotation case. Relative to the baseline non-rotating case (case CF6), the imposed power gradient reduces the turbulent kinetic energy $\langle k\rangle$ , r.m.s. velocity $\langle u'\rangle$ and dissipation rate $\varepsilon$ , while increasing the integral time scale $t_L$ and length scale $L_L$ . These changes indicate that the imposed forcing modifies both the turbulent fluctuations and the large-scale flow.

The mean rotation rate is quantified using the out-of-plane component of the mean vorticity,

(3.15) \begin{equation} \overline {\omega _3} = \frac {\partial \overline {\mathrm{U_2}}}{\partial x_1} - \frac {\partial \overline {\mathrm{U_1}}}{\partial x_2}, \qquad \Omega = \frac {1}{2}\sqrt {\left \langle \overline {\omega _3}^2 \right \rangle }, \end{equation}

where $\Omega$ represents a characteristic angular velocity associated with the measured large-scale circulation. Unlike conventional rotating-turbulence experiments, this angular velocity is not imposed through mechanical rotation of the reference frame, tank or grid; rather, it results from the spatially nonuniform jet forcing while the facility remains stationary.

The relative strength of the turbulent fluctuations and the imposed mean-flow rotation is characterised using the Rossby-like number

(3.16) \begin{equation} Ro = \frac {\langle u' \rangle }{2 \Omega L_L}; \end{equation}

here, $Ro$ is used as a measure of the relative importance of turbulent fluctuations and the imposed mean-flow rotation. In contrast to the classical Rossby number, which characterises the ratio of inertial to Coriolis forces in a rotating reference frame, the present definition is based on the angular velocity associated with the measured mean circulation. Using this definition, all HIT cases in the present study yield $Ro\gt 1$ . In particular, the baseline case CF6 gives $Ro=1.33$ , whereas the rotation case CTR yields a substantially lower value of $Ro=0.38$ . Thus, the characteristic mean-flow rotation is dynamically significant relative to the turbulent turnover based on the present characteristic scales.

The ability to impose spatially varying jet power while independently controlling the stochastic characteristics of the actuation provides a useful degree of freedom for studying the interaction between turbulence and mean-flow rotation. In particular, the strength of the large-scale circulation can be varied while retaining stochastic turbulent forcing, allowing the relative influence of the mean flow and turbulent fluctuations to be systematically investigated within a single compact facility. The present measurements, however, do not establish the flow as rotating turbulence in the classical sense. In particular, they do not demonstrate hallmark features associated with rotating turbulence, such as inertial-wave dynamics, rotation-induced changes in the energy spectrum or an inverse transfer of energy. The present results therefore demonstrate the ability of the facility to generate turbulence with a controlled large-scale mean-flow rotation, rather than establishing a canonical rotating-turbulence state.