Energy cascades in active-grid-generated turbulent flows

The energy cascade and diverse turbulence properties of active-grid-generated turbulence were studied in a wind tunnel via hot-wire anemometry. To this end, two active grid protocols were considered. The first protocol is the standard triple-random mode, where the grid motors are driven with random rotation rates and directions, which are changed randomly in time. This protocol has been extensively used due to its capacity to produce higher values of ${\text{Re}}_{\lambda }$ than its passive counterpart, with good statistical homogeneity and isotropy. The second protocol was a static or open grid mode, where all grid blades were completely open, yielding the minimum blockage attainable with our grid. Center-line streamwise profiles were measured for both protocols and several inlet velocities. It was found that the turbulent flow generated with the triple-random protocol evolved in the streamwise direction consistently with an energy dissipation scaling of the form $\varepsilon =C_{\varepsilon }{u}^{\prime 3}/L$, with $C_{\varepsilon }$ being a constant, $L$ the longitudinal integral length scale, and $u^{\prime }$ the rms of the longitudinal velocity fluctuations. Conversely, for the open-static grid mode, the energy dissipation followed a nonequilibrium turbulence scaling, namely, $C_{\varepsilon }\sim {\text{Re}}_G/{\text{Re}}_L$, where ${\text{Re}}_G$ is a global Reynolds number based on the inlet conditions of the flow and ${\text{Re}}_L$ is based on the local properties of the flow downstream the grid. Furthermore, this open-static grid mode scaling exhibits important differences with other grids, as the downstream location of the peak of turbulence intensity is a function of the inlet velocity; a remarkable observation that would allow one to study the underlying principles of the transition between equilibrium and nonequilibrium scalings, which are yet to be understood. It was also found that a rather simple theoretical model can predict the value of $C_{\varepsilon }$ based on the number density of zero crossings of the longitudinal velocity fluctuations. This theory is valid for both active grid operating protocols (and therefore two different energy cascades).

Figure 1

Picture of the active grid used on the present work.

Figure 2

Power spectral densities of velocity fluctuations obtained at $x=300$ cm for different values of $U_{\infty }$ for the open (a) and the active (b) modes. The black dashed line is a $-5/3$ power law.

Figure 3

Ratios between the fluctuating velocity vector $\overrightarrow{U^{\prime }}=\left(u,v,w\right)$ components for the open (left) and the active (right) modes. Measurements were taken at $U_{\infty }=11.5$ m/s for the first mode and at $U_{\infty }=4.9$ m/s for the latter modes.

Figure 4

(a) Estimation of $\varepsilon$ via the dissipation spectra; the black dashed line corresponds to the modeled frequencies. (b) Typical autocorrelation functions $R_{uu}$ obtained for both operating modes. (c) Example of the method applied to estimate $L$ for the active mode.

Figure 5

Streamwise evolution of ${\text{Re}}_{\lambda }$ (a), turbulence intensity (b), and integral length scale $L$ (c) for all results obtained for the active mode (AG).

Figure 6

(a) Plot of $C_{\varepsilon }$ vs ${\text{Re}}_{\lambda }$. (b) $L/\lambda$ vs ${\text{Re}}_{\lambda }$. Results correspond to all data sets obtained for the active mode.

Figure 7

Streamwise evolution of ${\text{Re}}_{\lambda }$ (a), turbulence intensity (b), and integral length scale $L$ (c) for all results obtained for the open mode. (d) Downstream position of the turbulence intensity maximum $x_{\max }$ vs $U_{\infty }$.

Figure 8

(a) Plot of $C_{\varepsilon }$ vs ${\text{Re}}_G/{\text{Re}}_L$. (b) $L/\lambda$ vs ${\text{Re}}_{\lambda }$. Results correspond to all data sets obtained for the open mode.

Figure 9

$\overline{l}{n}_s$ vs $L/{\eta }_c$ (a). The dashed lines are a $2/3$ power law. $C{s}^{\prime }$ vs $L/{\eta }_c$ (b). Blue lines correspond to the open mode and the red ones to the active one.

Figure 10

(a) Comparison of the values $C_{\varepsilon }^{\mathrm{model}}$ obtained using equation (2) (star symbols) and the experimental ones $C_{\varepsilon }^{\exp }=\varepsilon L/u^{\prime 3}$ (filled symbols). Value of $A$ (b), $C_s^{\prime }$ (c) and the intermittency constant $C$ (d) vs ${\text{Re}}_{\lambda }$ for all data sets.