Existence of positive skewness of velocity gradient in early transition

The present study uses direct numerical simulations to calculate two different transitional flows from laminar flow to turbulence, that is, the two-scale wake flow and the two-dimensional Reyleigh-Taylor unstable flow, respectively. Both results show that the skewness of the longitudinal velocity gradient, $S_k$, can become positive in the early transition stage, which is beyond our expectation since the turbulent equilibrium state always implies negative values of $S_k$. These phenomena are explained analytically by considering only two dominant Fourier modes with harmonic relations. It is illustrated that the sign of $S_k$ is not only affected by the amplitudes of the perturbation velocities, but also related to their phases. We expect that the present results will be helpful for understanding the formation of turbulent equilibrium state and for constructing a new measurable criterion of the transition onset.

Figure 1

Sketch of the computational domain of the two-scale wake. The membrane is highlighted. (a) Global view; (b) zoomed view near inlet.

Figure 2

Streamwise evolution of $\mathrm{\Delta }/\eta$ at different positions in the two-scale wake, respectively. (a) The spanwise positions, denoted as 1, 2, and 3, respectively. The dashed line indicates the location of $x_2/L=0.25$; (b) streamwise evolution of $\mathrm{\Delta }/\eta$.

Figure 3

Snapshots of $U_1$ at different $(x_2,x_3)$ planes in the two-scale wake.

Figure 4

A snapshot of $U_1$ at the $x_2/L=0.25$ plane in the two-scale wake.

Figure 5

A snapshot of $P$ at the $x_2/L=0.25$ plane in the two-scale wake.

Figure 6

Streamwise evolution of (a) turbulence intensity $I$ and (b) Taylor-scale Reynolds number ${\mathrm{Re}}_{\lambda }$ in the two-scale wake.

Figure 7

Streamwise evolution of $S_k$ in the two-scale wake.

Figure 8

Spanwise flow structures in the $L\times L$ cell at different $x_1$ locations in early transition of the two-scale wake, respectively. Arrows indicate the spanwise velocities, while contours are streamwise velocities. (a) $x_1/L=1$; (b) $x_1/L=2$; (c) $x_1/L=3$.

Figure 9

Sketch of the main structures in the $L\times L$ cell of streamwise vortices in early transition of the two-scale wake. Arrows indicate the flow direction in a streamwise slice.

Figure 10

The evolution of $a$, $b$, $c$, and $b/c$ along the streamwise direction in the two-scale wake. The red vertical line indicates the location behind the membrane $x_1/L=0.05$. The blue horizontal line corresponds to $b/c=2.765$, a critical value for the sign of $S_k$ according to our model.

Figure 11

Streamwise evolution of (a) kinetic energy and (b) ${(\partial u_1/\partial x_1)}^2$ at several wave numbers in the two-scale wake. The wave number in normalized by $2\pi /L$.

Figure 12

Evolution of the amplitude of the fundamental ($k=4.672$) and harmonic ($k=9.344$) perturbations in the two-dimensional RT flow.

Figure 13

Contours of the density at $t=18$, 19 and 20 for the DNS result of the two-dimensional RT flow, respectively.

Figure 14

Contours of the skewness $S_k(x,t)$ in the two-dimensional RT flow.

Figure 15

Eigenprofiles of the fundamental and second-order harmonic perturbations for different $t$ in the two-dimensional RT flow. The curves for $t=14$ and 15 are shifted for plotting convenience.

Figure 16

Comparison of normalized skewness by its maximum value, $S_k/S_{k,max}$, between the theoretical prediction and the simulation results.

Figure 17

Evolution of the the skewness values of $\partial u_1/\partial x_1$, $\partial u_2/\partial x_2$, and $\partial u_3/\partial x_3$ in the two-scale wake, respectively.