Bifurcation structure of unstable periodic orbits in plane Couette flow with the Smagorinsky model

To obtain insights into dynamics of developed plane Couette turbulence, this paper considers the bifurcation structure of unstable periodic orbits (UPOs) in the large-eddy-simulation (LES) system with the Smagorinsky-type eddy viscosity model. Treating the Smagorinsky constant as a bifurcation parameter, we detect the bifurcation points connecting two known UPOs which were separately discovered in the Navier-Stokes system [G. Kawahara and S. Kida, J. Fluid Mech. 449, 291 (2001)]. At the moderately high Reynolds number, the LES UPO of the present study possesses the spanwise vortices, which seem to be caused by the streak instability and appear in the central region of the channel. We note that to our knowledge a spanwise vortex has not been reported through the UPO analysis of wall flows. The stretched spanwise vortices of the LES UPO enhance transfer of the streamwise turbulent momentum as in developed near-wall turbulence.

Figure 1

Recalculation of Kawahara and Kida [9] and newly found UPOs using the number of grid points $(N_1,N_2,N_3)=(48,65,48)$. (a) Projection of the UPOs at $\text{Re}=400$. The horizontal and vertical axes represent the energy injection $I$ and the energy dissipation $D$, respectively. The red and blue lines indicate the solution orbits of the vigorous and gentle UPOs [9], respectively. Furthermore, the green line means that of the active UPO, which is newly examined. All the orbits turn clockwise. The gray scale shows the probability density distribution of the NS turbulence. The two yellow dots on the green line shows the phases of panels in Fig. 6. (b) Mean velocity profile and (c) RMS velocity profiles at $\text{Re}=400$: the active UPO (lines), the NS turbulence (symbols).

Figure 2

Statistical properties of LES and NS turbulence at $\text{Re}=2000$ in our numerical computation. (a) Mean velocity profile. The linear laminar profile $U_{\mathrm{Lam}}^{+}$ has been subtracted. (b) RMS velocity profiles. The lines represent the NS turbulence for $h^{+}=121$, while the crosses and open circles denote the LES turbulence at $C_S=0.1$ using the number of grid points $(N_1,N_2,N_3)=(24,37,24),(h^{+}=122)$ and $(50,65,50),(h^{+}=121)$, respectively. In panel (b), the blue, orange, and green colors indicate the RMS velocities ${\widehat{u}}_1^{+}, {\widehat{u}}_2^{+}$, and ${\widehat{u}}_3^{+}$, respectively.

Figure 3

Bifurcation diagram arising from the NS vigorous and gentle UPOs: (a) $\text{Re}=400$, (b) $C_S=0.1$. The solid and open circles represent the vigorous and gentle UPOs in the NS system ($C_S=0$) [9], respectively. The blue and green lines represent the branches of the LES gentle and active UPOs arising through the saddle-node bifurcation. Additionally, the red line indicates the branch stemming from the NS vigorous UPO, while the orange line means the upper branch bifurcating from the branch of the LES vigorous UPO at the turning point. The solid triangles, solid squares, solid diamonds, and open triangles indicate the LES UPOs at $\text{Re}=400$ and $C_S=0.1$, which we call $V, V_1, V_2$, and $G$, respectively. The crosses on the branch of the LES active UPO in (a) represent the bifurcation points at $C_S=0.5040$ and 0.3657, which we refer to as $B_1$ and $B_2$. To avoid needless confusion we omit numerous turning points and branches.

Figure 4

Bifurcation diagram of the LES gentle and active UPOs at $\text{Re}=2000$. The colored lines, from dark-blue to dark-red, represent the sets of the number of grid points from $(N_1,N_2,N_3)=(24,33,24)$ to (50,65,50), and so the filter widths.

Figure 5

Snapshots of the LES active UPO of the period $T=93.72$ at $\text{Re}=2000$ and $\langle \mathrm{\Delta }\rangle C_S=0.05$ ($C_S=0.5797,(N_1,N_2,N_3)=(36,51,36),h^{+}=136$): (a) $t=0$, (b) $t=T/4$, and (c) $t=T/2$. The yellow object denotes the isosurface of the streamwise velocity $u_1=0.15(u^{+}=12.5)$, while the red and blue objects represent the vortical structure visualized by the isosurfaces of the second invariant of velocity gradient tensor $Q=0.05$ with the positive and negative streamwise vorticity.

Figure 6

Snapshots of the NS active UPO of the period $T=62.13$ at $\text{Re}=400(h^{+}=35.0)$: (a) $t=0$, (b) $t=T/5$, and (c) $t=T/2$. The definition of isosurfaces is the same as Fig. 5, but $u_1=0.2$ and $Q=0.2$. See also Fig. 1.

Figure 7

Projection of the LES active UPO at $\text{Re}=2000$ and $\langle \mathrm{\Delta }\rangle C_S=0.01658(C_S=0.2598,(N_1,N_2,N_3)=(50,65,50),h^{+}=105)$. The horizontal and vertical axes mean the energy injection $I$ and the cross-flow energy $E_{2\mathrm{D}}$. The red line indicates the solution orbit of the LES active UPO. The gray scale shows the probability density distribution of the LES turbulence for $h^{+}=110$. The yellow dots represent the phases of panels (a) to (i) in Fig. 9. The orbit turns counterclockwise.

Figure 8

(a) Mean velocity and (b) RMS velocity at $\text{Re}=2000$. The lines mean the LES active UPO at $\text{Re}=2000$ and $\langle \mathrm{\Delta }\rangle C_S=0.01658[C_S=0.2598,(N_1,N_2,N_3)=(50,65,50),h^{+}=105]$, while the open circles represent the LES turbulence $(h^{+}=110)$ at the same parameters. Additionally, the crosses signify the NS turbulence in our numerical computation, already shown in Fig. 2. In panel (a), the gray line indicates the NS turbulence for $h^{+}=550$ of the large domain [40], whereas in panel (b), the definition of colors is the same as Fig. 2.

Figure 9

Snapshots of the LES active UPO of the period $T=79.25$ at $\text{Re}=2000$ and $\langle \mathrm{\Delta }\rangle C_S=0.01658[C_S=0.2598,(N_1,N_2,N_3)=(50,65,50),h^{+}=105]$. Time elapses from (a) to (i) by $T/20$. The definition of isosurfaces is the same as Fig. 5, but $u_1=0.2(u^{+}=21.6)$ and $Q=0.1$.

Figure 10

Development of spanwise vortices. The yellow objects are the isosurfaces of the streamwise velocity $u_1=0.3$, while the blue to red objects denote $Q=0.1$ within the range of $-0.75\le {\omega }_3\le 0.75$. Time elapses from (a) to (c) by $T/20(3T/20\le t\le 5T/20)$ and from (d) to (f) by $T/40(55T/200\le t\le 65T/200)$. The black circles are used for emphasis on the spanwise vortices.

Figure 11

Reynolds shear stress $-u_1^{\prime }{u}_2^{\prime }$ of the LES active UPO. The blue to red objects indicate the isosurface $|u_1^{\prime }{u}_2^{\prime }|/\left({\widehat{u}}_1{\widehat{u}}_2\right)=1.75$ where the color means the value of the Reynolds shear stress $-u_1^{\prime }{u}_2^{\prime }$ from 0 to 0.014. The green surfaces indicate $Q=0.1$. The time series of (a) to (c) and of (d) to (f) are the same as Fig. 10.