Experimental estimation of turbulence modification by inertial particles at moderate ${\mathrm{Re}}_{\lambda }$

We advance a method to estimate the carrier-flow dissipation ${\varepsilon }_p$ in the presence of inertial sub-Kolmogorov particles at moderate ${\text{Re}}_{\lambda }$. Its foundations rely on the unladen flow dissipation calculation using the Rice theorem, and the density of zero crossings of the longitudinal velocity fluctuation $u^{\prime }\left(x\right)$ coming from a laser Doppler anemometry device. Our experimental results provide strong evidence regarding the non-negligible effect that sub-Kolmogorov particles have on the carrier-flow energy cascade at ${\varphi }_v=O\left({10}^{-5}\right)$, and ${\text{Re}}_{\lambda }\in [200–600]$.

Figure 1

(a) Longitudinal energy density spectra $F_{p11}$. The darker the color, the larger is the mean velocity $U_{\infty }$. (b) Normalized zero crossing density $n_s\overline{l}$ against the normalized filter size $L/{\eta }_C$. The thicker the line, the larger is the liquid fraction ${\varphi }_v$ (see Table 1).

Figure 2

Several parameters plotted against the unladen ${\text{Re}}_{\lambda }$. In all figures, the larger the marker size, the larger is the concentration, and the darker its color, the larger is the ${\text{Re}}_{\lambda }$ (see Table 1). (b) and (c) cover the data sets from (a) with the condition $A<40,{\varphi }_v\in [1.2–4.4]\times {10}^{-5}$. (a) $A={\eta }_{pC}/\eta$. Our values of $A$ are in good agreement with previously reported unladen flow values of (MV 2008: Mazellier and Vassilicos) [14, 15, 29], i.e., $A=O\left(10\right)\approx 7.8+9.1{log}_{10}{\text{Re}}_{\lambda }$ [14]. (b) Stokes number modification $\text{St}/{\text{St}}_0={({\varepsilon }_p/\varepsilon )}^{1/2}$. In the inset, $\left(\mathrm{\Delta }\right)$ represents the results of Dejoan and Monchaux [5] for $V_T/u=1$ (in their notation $V_T$ is the particle terminal velocity), and $(\square )$ illustrate the results of Bosse et al. [3]. (c) Ratio of integral length scales $L_p/L$, a factor close to 2 with respect to the unladen flow [15] is due to the method [25] to estimate $L_p$. (d) Ratio of Kolmogorov length scales; ${\eta }_P/\eta ={(\varepsilon /{\varepsilon }_p)}^{1/4}$.

Figure 3

Several parameters plotted against the unladen ${\text{Re}}_{\lambda }$ for the data found in Fig. 2. In the figures, the larger the marker size, the larger is the concentration, and the darker its color, the larger is the ${\text{Re}}_{\lambda }$ (see Table 1). (a) $C_{\varepsilon }^p={\varepsilon }_p{L}_p/u_p^3$. The inset shows $C_{\varepsilon }^p$ vs ${\varphi }_v$. (b) $C_{\varepsilon }^p$ vs ${\text{Re}}_M^{1/2}/{\text{Re}}_{\lambda }$ as in Ref. [34].