Bifurcations of a plane parallel flow with Kolmogorov forcing

We study the primary bifurcations of a two-dimensional Kolmogorov flow in a channel subject to boundary conditions chosen to mimic a parallel flow, i.e., periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. The control parameter is the Reynolds number based on the friction coefficient, denoted as $\text{Rh}$. We find that as we increase $\text{Rh}$ the laminar steady flow can display different types of bifurcation depending on the forcing wave number of the base flow. This is in contrast with the case of doubly periodic boundary conditions for which the primary bifurcation is stationary. In the present case, both stationary and Hopf bifurcations are observed. In addition, we discover a type of bifurcation with both the oscillation frequency and the amplitude of the growing mode being zero at the threshold, that we call a stationary drift bifurcation. A reduced four-mode model captures the scalings that are obtained from the numerical simulations. As we increase $\text{Rh}$ further we observe a secondary instability which excites the largest mode in the domain. The saturated amplitude of the largest mode is found to scale as a $3/2$ power law of the distance to the threshold which is also explained using a low-dimensional model.

Figure 1

Sketch of the domain under study. The red line represents the spatial form of the Kolmogorov forcing. Note that $f_0{K}_{f}cos\left(K_{f}y\right)$ profile corresponds to the force that acts on $u$, the $x$-component of the velocity field.

Figure 2

(a) Time series of the growing mode ${\widehat{\psi }}_{3,1}$ for different values of $\text{Rh}$ close to the threshold. (b) The standard deviation $\sigma \left({\widehat{\psi }}_{3,1}\right)$ and the oscillation frequency ${\omega }_f$ of the mode ${\widehat{\psi }}_{3,1}$ as a function of the distance to the threshold. The solution corresponds to a traveling wave that moves in the negative $x$-direction.

Figure 3

Snapshots of the $x$-component (a, c) and $y$-components of the velocity fields (b, d) are shown at two different time instances for the case $\text{Rh}\approx 0.91$. The solution corresponds to a modulated traveling wave that moves in the negative $x$-direction.

Figure 4

The standard deviation of the largest scale mode ${\widehat{\psi }}_{0,1}$ as a function of the distance to the threshold $\text{Rh}-{\text{Rh}}_2^c$. The dashed lines indicate the scalings ${(\text{Rh}-{\text{Rh}}_2^c)}^{3/2}$ and ${(\text{Rh}-{\text{Rh}}_2^c)}^{1/2}$ for comparison.

Figure 5

All the interacting triads in the reduced four-mode model. In panel (b) the mode ${\widehat{\psi }}_{\mathbf{r}}={\widehat{\psi }}_{0,2}$ is repeated twice as it is excited by the modes ${\widehat{\psi }}_{\mathbf{p}}^{*}={\widehat{\psi }}_{-3,1}$, ${\widehat{\psi }}_{\mathbf{q}}^{*}={\widehat{\psi }}_{3,3}$ and is also responsible for the oscillation of the mode ${\widehat{\psi }}_{3,1}$. The red arrow indicates the base flow mode ${\widehat{\psi }}_{0,4}$, which becomes unstable at the first threshold ${\text{Rh}}_1^c$.

Figure 6

All the interacting triads of the reduced eight-mode model. The red arrows indicate the modes that become unstable at the second threshold ${\text{Rh}}_2^c$, and the blue arrow indicates the large-scale mode ${\widehat{\psi }}_{0,1}$.