Computational Study of Turbulent Laminar Patterns in Couette Flow

Turbulent-laminar patterns near transition are simulated in plane Couette flow using an extension of the minimal-flow-unit methodology. Computational domains are of minimal size in two directions but large in the third. The long direction can be tilted at any prescribed angle to the streamwise direction. Three types of patterned states are found and studied: periodic, localized, and intermittent. These correspond closely to observations in large-aspect-ratio experiments.

Figure 1

Turbulent-laminar pattern at Reynolds number 350. The computational domain (outlined in white, aligned along $x^{\prime }$, $z^{\prime }$) is repeated periodically to tile an extended region. The kinetic energy is visualized in a plane midway between and parallel to the plates moving in the streamwise ($x$) direction. Uniform black corresponds to laminar flow. The sides of the image are 60 times the plate separation $L_y=2$; the pattern wavelength is $20L_y$. Streamwise streaks, with spanwise extent approximately $L_y$, are visible at the edges of the turbulent regions.

Figure 2

Simulation domains. The wall-normal direction $y$ is not seen; $L_y=2$. The gray or colored bars represent streamwise vortex pairs with a spanwise spacing of 4. (The vortices are schematic; these are dynamic features of the actual flow.) (a) MFU domain of size $6\times 4$. (b) Central portion of a domain [on the same scale as (a)] tilted to the streamwise direction. $\alpha ,{\alpha }^{\prime }$, and $\beta ,{\beta }^{\prime }$ are pairs of points identified under periodic boundary conditions in $x^{\prime }$. (c) Full tilted domain with $L_{x^{\prime }}=10$, $L_{z^{\prime }}=120$, and $\theta =24°$. On this scale the MFU domain, shown for comparison, is small.

Figure 3

Space-time evolution of turbulent-laminar patterns in the domain $L_{x^{\prime }}\times L_{z^{\prime }}=10\times 120$, $\theta =24°$. Time evolves upward with changes in $\mathrm{R}\mathrm{e}$ indicated on the right. States seen upon decreasing $\mathrm{R}\mathrm{e}$ from uniform turbulence at $\mathrm{R}\mathrm{e}=420$, through various patterned states, ending in simple Couette flow at $\mathrm{R}\mathrm{e}=290$. Center: Time-averaged kinetic energy $⟨E⟩$ on a space-time grid. The same scale is used for all space-time plots, with $⟨E⟩=0$ in white. Right: kinetic energy plotted over a time window $T=500$ in a turbulent and laminar region. Left: Average spectral components $⟨|{\widehat{w}}_m|⟩$ of spanwise velocity with $m=3$ (solid line) and $m=2$ (dotted line), $m=0$ (long-dashed line), and $m=1$ (short-dashed line).

Figure 4

Simulations at $\mathrm{R}\mathrm{e}=350$, $\mathrm{R}\mathrm{e}=300$, and $\mathrm{R}\mathrm{e}=410$ illustrating three distinct states: periodic, localized, and intermittent. Space-time representation of $⟨E⟩$ is as in Fig. 3. For $\mathrm{R}\mathrm{e}=350$ and $\mathrm{R}\mathrm{e}=300$ the domain length is increased from $L_{z^{\prime }}=50$ to $L_{z^{\prime }}=140$ in increments of 5. The integrated energy profile $\overline{E}(z^{\prime })$ is shown at the final time. For $\mathrm{R}\mathrm{e}=410$ a single long simulation is shown for $L_{z^{\prime }}=40$, accompanied by $m=1$ (solid line) and $m=0$ (dashed line) spectral components.

Figure 5

Turbulent-laminar patterns at minimum ($\theta =15°$) and maximum ($\theta =66°$) angle for which they have been computed at $\mathrm{R}\mathrm{e}=350$. Conventions as in Fig. 1.