Dynamic state of low-Reynolds-number turbulent channel flow

We numerically study the dynamic state of a low-Reynolds-number turbulent channel flow from the viewpoints of symbolic dynamics and nonlinear forecasting. A low-dimensionally (high-dimensionally) chaotic state of the streamwise velocity fluctuations emerges at a viscous sublayer (logarithmic layer). The possible presence of the chaotic states is clearly identified by orbital instability-based nonlinear forecasting and ordinal partition transition network entropy in combination with the surrogate data method.

Figure 1

(A) Temporal evolution of streamwise velocity $u$. (B) Instantaneous streamwise velocity field $u$ on $x\text{−}z$ plane. (a) $y^{+}=4$ and (b) $y^{+}=35$.

Figure 2

Frequency distributions of the OPTN entropy $S_t$ for the original and surrogate data. (a) $y^{+}=4$ and (b) $y^{+}=35$. Here, the sampling interval ${\tau }^{+}=1.08$.

Figure 3

(a) Temporal evolutions of the original streamwise velocity $u_o$ and predicted streamwise velocity $u_p$ at $y^{+}=4$, together with (b) power spectrum and (c) probability density distribution. Here, the sampling interval ${\tau }^{+}=1.08$.

Figure 4

(a) Variation in the correlation coefficient $C$ between $u_o$ and $u_p$ at $y^{+}=4$ and 35 as a function of the predicable time $t_p^{+}$ for the OIFM. (b) Variation in the predicable time $t_{p,c}^{+}$ in terms of $y^{+}$ for the OIFM and RC. Here, the sampling interval ${\tau }^{+}=1.08$.

Figure 5

Variation in the correlation coefficient $C$ between $u_o$ and $u_p$ as a function of the sampling interval ${\tau }^{+}$ at different $y^{+}$.

Figure 6

Variation in the OPTN entropy $S_t$ for (a) $u_o$ and (b) $u_p$ as a function of the sampling interval ${\tau }^{+}$ at different vertical $y^{+}$ locations.