Interactions of velocity structures between large and small scales in microelectrokinetic turbulence

Recent investigations on electrokinetic (EK) flows have indicated turbulentlike flow can be realized by applying strong and high frequency ac electric field to flows with high-conductivity-gradient interface, even though under low bulk flow Reynolds number. Relative to conventionally hydrodynamic turbulence in a high Reynolds number, the ac EK turbulent flow exhibits high randomness with stronger intermittency [Wang et al., Phys. Rev. E 93, 013106 (2016)]. The abnormally high intermittency could be attributed to the ascending probability density function of velocity gradients (dominated by small scale velocity structure function) far from the equilibrium state. By evaluating the intermittency of the ac EK turbulent flow with hierarchical structures, we astonishingly find the intermittency factor of hierarchical structures in SL94 law, i.e., $\beta$ factor, which is commonly believed to be between 0 and 1, exhibits larger value than 1 in the ac EK turbulent flow. This result indicates the different hierarchical relations of flow structures in ac EK turbulence flow from that in the conventional hydrodynamic turbulence, as further illustrated by the probability density function of velocity structures among different spatial scales.

Figure 1

Influence of $\beta$ on the relation of $P_l$ and $P_{l_0}$. (a) $0<\beta \le 1$, (b) $\beta >1$.

Figure 2

Schematic of the relations between $P_{l_0}$ and $P_l$ at $l^{*}=0.01$. Here, $C=2$, $\beta =3/2$, $\gamma =2/3$. (a) $P_{l_0}$ has Gaussian distributions with ${\alpha }_{l_0}=1$, and (b) $P_{l_0}$ has exponential decaying with $A=1$.

Figure 3

Schematic of the relations between $P_{l_0}$ and $P_l$ at $l^{*}=0.1$. Here, $C=2$, $\beta =3/2$, $\gamma =2/3$. (a) $P_{l_0}$ has Gaussian distributions, and (b) $P_{l_0}$ has exponential decaying.

Figure 4

$P_{l,2}$ calculated at different $\beta$ and $\gamma$, when $C=2$. (a) $\beta =3/2$, $\gamma =2/3$; (b) $\beta =9/8$, $\gamma =5/12$.

Figure 5

Schematic of the microchip and time series of normalized velocity fluctuations $u^{\prime }/\sqrt{\langle u^{\prime }2\rangle }$. (a) Schematic of the microchannel for ac EK turbulence. Two streams have a conductivity ratio of 5000:1. The length ($l_c$), height ($h$) and initial width ($w_c$) of microchannel are 5 mm, 240 μm and 130 μm respectively. (b) Time series of normalized velocity fluctuations $u^{\prime }/\sqrt{\langle u^{\prime }2\rangle }$ at different $x^{*}$, where $u^{\prime }=u-\langle u\rangle$.

Figure 6

Scaling behavior of velocity structure function. (a) $S_p$ ($l$) vs $l$ at $x^{*}=0.77$, where ${\xi }_p$ are 0.68, 1.01 and 1.62 for $p=2,3$, and 6, respectively; (b) $S_2\left(l\right)$ vs $l$ at four different $x^{*}$ positions, where ${\xi }_2$ are 0.68, 0.69, 0.65, and 0.62 as $x^{*}$ increases from 0.77 to 3.84.

Figure 7

$P_l$ of different $l$ at $x^{*}=0.77$. The length increment $l_1$, $l_2$ and $l_3$ are plotted in Fig. 2.

Figure 8

$H_{p+1,m+1}\left(l\right)$ vs $H_{p,m}\left(l\right)$ for β-test. The slope of the curve is equal to $\beta$. (a), (b) at $x^{*}=0.77$ and (c), (d) at $x^{*}=1.54$. (a), (c) different $p$ at $m=1$, (b), (d) different $m$ at $p=5$.

Figure 9

Proportion of $\beta \left(p,m\right)$ calculated from Eq. (12) for all $p$ and $m$. The red dashed line indicates ${\beta }_{\mathrm{avg}}$ at (a) $x^{*}=0.77$ and (b) $x^{*}=1.54$ respectively. The former has ${\beta }_{\mathrm{avg}}=1.51$ and latter has ${\beta }_{\mathrm{avg}}=1.13$.