Characterization of the hairpin vortex solution in plane Couette flow

Quantitative evidence that establishes the existence of the hairpin vortex state (HVS) [T. Itano and S. C. Generalis, Phys. Rev. Lett. 102, 114501 (2009)] in plane Couette flow (PCF) is provided in this work. The evidence presented in this paper shows that the HVS can be obtained via homotopy from a flow with a simple geometrical configuration, namely, the laterally heated flow (LHF). Although the early stages of bifurcations of LHF have been previously investigated, our linear stability analysis reveals that the root in the LHF yields multiple branches via symmetry breaking. These branches connect to the PCF manifold as steady nonlinear amplitude solutions. Moreover, we show that the HVS has a direct bifurcation route to the Rayleigh-Bénard convection.

Figure 1

Geometrical configuration of the fluid flow in a vertical slot between laterally heated boundaries moving in the opposite directions.

Figure 2

Bifurcation sequence in pure LHF $ϵ=1$. The various higher-order states are depicted, as well as the laminar state. The latter is depicted by the line $\tau =-2$. Here, $ϵ=1$ for the pure LHF states and S, T, and Q stand for secondary, tertiary, and quaternary respectively. The number following each letter indicates the branch of each (bifurcating) state.

Figure 3

Bifurcation diagram for $\text{Re}=300$. Several states obtained in LHF show their connections to PCF via the homotopy parameter $ϵ$. Here, the various states are denoted in the same way as in Fig. 2.

Figure 4

Bifurcation diagram obtained at $\text{Re}=350$. Here, the various states are denoted in the same way as in Fig. 2.

Figure 5

The tertiary and quaternary states connected to PCF. The value of $\tau$ of tertiary state is calculated by two sets of truncation levels, ○: $\left(N_x,N_y,N_z\right)=\left(8,8,28\right)$ and ×: (32, 16, 60), in order to explicitly show the numerical convergence, necessary for the (absolute) existence of the state.

Figure 6

The quaternary state Q1 at $\left(ϵ,\text{Re}\right)=\left(0,200\right)$. (a) Contour plot of $\overline{u}\left(y,z\right)$, which varies from $-0.59$ to 0.59. Contour levels from $-0.6$ (white) to 0.6 (black) by 0.1. (b) Contour plot of $u_{+}\left(x,y\right)$, which varies from 0 to 0.54. Contour levels from 0.0 (white) to 0.6 (black) by 0.1.

Figure 7

The tertiary state T2 at $\left(ϵ,\text{Re}\right)=\left(0,200\right)$. (a) Contour plot of $\overline{u}\left(y,z\right)$, which varies from $-0.67$ to $+0.67$. Contour levels from $-0.7$ (white) to 0.7 (black) by 0.1. (b) Contour plot of $u_{+}\left(x,y\right)$, which varies from 0 to 0.59. Contour levels from 0.0 (white) to 0.6 (black) by 0.1.

Figure 8

Bifurcation in thermally stratified PCF. The vertical axis denotes the $L_2$ norm of the vertical component of the velocity field. The bifurcation diagram is obtained in thermally stratified PCF. The secondary flow corresponds to longitudinal roll cells. Here, the Re and Prandtl numbers are kept fixed at 210 and 7, respectively. The Rayleigh number value $\text{Ra}\approx 1403$ is the critical value in RBC for the odd mode.