Orr mechanism in transition of parallel shear flow

The Orr mechanism is revisited to understand its precise role in the transition of plane Couette flow. By considering homogeneous shear flow and plane Couette flow, it is identified that the Orr mechanism induces a lift-up effect which significantly amplifies spanwise velocity. An optimal perturbation analysis for an individual velocity component reveals that the amplification of spanwise velocity is most active at the streamwise length comparable to the given spanwise length of the perturbation. The relevance of this mechanism to transition is subsequently examined in plane Couette flow. To this end, a set of initial conditions, which combines the optimal perturbation for spanwise velocity with the one for all the velocity components, is considered by varying their amplitudes. Two representative transition scenarios are found: oblique and streak transitions. In the former, the spanwise velocity perturbation amplified with the Orr mechanism initiates both streak amplification and breakdown, whereas in the latter, its role is limited only to the streak breakdown at the late stage of transition. As such, the oblique transition offers a route to turbulence energetically more efficient than the streak transition, at least for the cases examined in the present paper. Finally, the oblique transition is found to share many physical similarities with the transition by the minimal seed.

Figure 1

Contours of (a) $G_{max}$, (b) $G_{u,max}$, (c) $G_{v,max}$, and (d) $G_{w,max}$ in the ${\kappa }_1\text{−}{\kappa }_3$ plane for $\text{Re}=100$.

Figure 2

Variation of (a) $G_{max}$, (b) $G_{u,max}$, (c) $G_{v,max}$, and (d) $G_{w,max}$ with ${\kappa }_1$ for ${\kappa }_3={10}^{-2}$ and $\text{Re}=100$.

Figure 3

The optimal initial condition maximizing the energy of spanwise velocity perturbation in the (a) $x\text{−}y$ plane and (b) the $y\text{−}z$ plane (${\kappa }_1={\kappa }_3=0.0033$, $\text{Re}=100$). In (a), the contours denote $w$, and the vectors represent $u$ and $v$. In (b), the contours denote $u$, and the vectors represent $v$ and $w$. Here, the energy of the initial condition is normalized to unity.

Figure 4

The time dependence of (a) ${\left|\right|\mathbf{u}\left(t\right)\left|\right|}^2$, (b) ${\left|\right|u\left(t\right)\left|\right|}^2$, (c) ${\left|\right|v\left(t\right)\left|\right|}^2$, and (d) ${\left|\right|w\left(t\right)\left|\right|}^2$ for ${\kappa }_1={\kappa }_3=0.0033$ ($\text{Re}=100$). Here, the initial condition is given by the optimal perturbation maximizing the energy of spanwise velocity.

Figure 5

The contours of (a) $G_{u,max}$ and (b) $G_{w,max}$ in terms of ${\kappa }_1$ and ${\kappa }_3$ at $\text{Re}=400$.

Figure 6

Temporal evolution of (a) ${\left|\right|u\left(t\right)\left|\right|}^2$ from the optimal initial condition for $G_{u,max}$ (${\kappa }_1=0$, ${\kappa }_3=1.26$), and (b) ${\left|\right|w\left(t\right)\left|\right|}^2$ from the one for $G_{w,max}$ (${\kappa }_1=0.63$, ${\kappa }_3=1.26$) at $\text{Re}=400$.

Figure 7

Contours of spanwise velocity at (a) $t=0$, (b) $t=4$, (c) $t=8$, (d) $t=12$, (e) $t=16$, (f) $t=20$, and at $z=0$ plane of the small-amplitude optimal spanwise velocity perturbation with ${\kappa }_1=0.63$, ${\kappa }_3=1.26$, $\text{Re}=400$. Here, the energy of initial condition is normalized to unity.

Figure 8

Contours of lifetime $T$ as a function of ${\lambda }_1$ and ${\lambda }_2$ at $\text{Re}=400$ in the log-log coordinates. Here, ${\lambda }_1$ and ${\lambda }_2$ are varied from $3\times {10}^{-5}$ to ${10}^{-1}$. The red dashed lines represent the initial disturbance energy, $E_0(={\lambda }_1^2+{\lambda }_2^2)={10}^{-8}$, ${10}^{-6}$, ${10}^{-4}$, ${10}^{-2}$.

Figure 9

The time evolution of (a) $E_{u_1}$ (red), $E_{u_2}$ (blue), $E_{u_3}$ (black), (b) $E_{u_1}^{(m,n)}$, (c) $E_{u_2}^{(m,n)}$, and (d) $E_{u_3}^{(m,n)}$ for the initial condition given by Eq. (6) for ${\lambda }_1=0.001$ and ${\lambda }_2=0.007$ ($\text{Re}=400$). In (b)–(d), red solid, $(m,n)=(0,1)$; red dashed, (0,2); blue solid, (1,0); blue dashed, (1,1); green solid, (1,2); green dashed, (2,0); black solid, (2,1); black dashed, (2,2).

Figure 10

Contours of streamwise velocity fluctuation on the plane of $y\approx 0.3$ at (a) $t=0$, (b) $t=20$, (c) $t=40$, (d) $t=60$, (e) $t=80$, and (f) $t=100$. Here, ${\lambda }_1=0.001$, ${\lambda }_2=0.007$, and $\text{Re}=400$.

Figure 11

Time traces of the energetics terms for the streak development and breakdown (${\lambda }_1=0.001$ and ${\lambda }_2=0.007$): (a) $N_{u,y}$ (red), $N_{u,z}$ (blue), $L$ (black); (b) $N_{v,y}$ (red), $N_{v,z}$ (blue), $P_v$ (black); (c) $N_{w,y}$ (red), $N_{w,z}$ (blue), $P_w$ (black); and (d) $T_y$ (red), $T_z$ (blue).

Figure 12

Time traces of (a) $E_{u_1}$ (red), $E_{u_2}$ (blue), $E_{u_3}$ (black), (b) $E_{u_1}^{(m,n)}$, (c) $E_{u_2}^{(m,n)}$, and (d) $E_{u_3}^{(m,n)}$ for the initial condition given by Eq. (6) for ${\lambda }_1=0.07$ and ${\lambda }_2=0.005$ ($\text{Re}=400$). In (b)–(d): red solid, $(m,n)=(0,1)$; red dashed, (0,2); blue solid, (1,0); blue dashed, (1,1); green solid, (1,2); green dashed, (2,0); black solid, (2,1); and black dashed, (2,2).

Figure 13

Contours of streamwise velocity fluctuation on the plane of $y\approx 0.3$ at (a) $t=0$, (b) $t=20$, (c) $t=120$, and (d) $t=150$. Here, ${\lambda }_1=0.07$, ${\lambda }_2=0.005$, and $\text{Re}=400$.

Figure 14

Time traces of (a) $N_{u,y}$ (red), $N_{u,z}$ (blue), $L$ (black), and (b) $T_y$ (red line), $T_z$ (blue) with ${\lambda }_1=0.07$, ${\lambda }_2=0.005$ at $\text{Re}=400$.

Figure 15

A schematic diagram of how the Orr mechanism initiates the oblique transition.

Figure 16

A schematic diagram of how the Orr mechanism initiates the streak transition.

Figure 17

Contours of the wall-normal velocity fluctuation at (a) $t=0$, (b) $t=2$, (c) $t=4$, and (d) $t=6$, and at the $z=0$ plane of the small-amplitude random initial condition with $\text{Re}=400$.

Figure 18

Contours of the streamwise velocity fluctuation at (a) $t=0$, (b) $t=50$, (c) $t=90$, and (d) $t=100$, and at the $y\approx 0.3$ plane of the small-amplitude random initial condition with $\text{Re}=400$.

Figure 19

The time evolution of $E_{u_1}$ (red), $E_{u_2}$ (blue), and $E_{u_3}$ (black) with the random initial condition at $\text{Re}=400$.

Figure 20

The optimal initial condition maximizing the energy of spanwise velocity perturbation in the (a) $x\text{−}y$ plane and (b) the $y\text{−}z$ plane for plane Couette flow. (${\kappa }_1=0.63$, ${\kappa }_3=1.25$, $\text{Re}=400$). In (a), the contours denote $w$, and the vectors represent $u$ and $v$. In (b), the contours denote $u$, and the vectors represent $v$ and $w$. Here, the energy of the initial condition is normalized to unity.