Using Nonlinear Transient Growth to Construct the Minimal Seed for Shear Flow Turbulence

Linear transient growth analysis is commonly used to suggest the structure of disturbances which are particularly efficient in triggering transition to turbulence in shear flows. We demonstrate that the addition of nonlinearity to the analysis can substantially change the prediction made in pipe flow from simple two-dimensional streamwise rolls to a spanwise and cross-stream localized three-dimensional state. This new nonlinear optimal is demonstrably more efficient in triggering turbulence than the linear optimal indicating that there are better ways to design perturbations to achieve transition.

Figure 1

The evolution of the linear and nonlinear optimals at $\mathrm{Re}=1750$. The blue (upper) line corresponds to the nonlinear optimal for $E_0=2\times {10}^{-5}$ while the red (lower) line is the linear optimal ($E_0\to 0$). The nonlinear result produces more growth and actually reaches its maximum at a slightly earlier time than $T$.

Figure 2

Three snapshots of the linear optimal (top) and five snapshots (middle and bottom) of the 3D optimal for $\mathrm{Re}=1750$ and $E_0=2\times {10}^{-5}$ during its evolution. Labels refer to Fig. 1, arrows indicate cross-sectional velocities, and colors axial velocity beyond the laminar flow (white or light for positive and red or dark for negative: outside shade represents zero). The bar chart shows the ratio of energy in each streamwise Fourier mode of the initial nonlinear optimal (a).

Figure 3

$\mathrm{Re}=1750$, $E_0=2\times {10}^{-5}$. The iterations are seeded with a noisy version of the 2D optimal which converges to the 3D optimal by way of an intermediate “saddle” state (shown).

Figure 4

$\mathrm{Re}=2500$. The green (lowest at $t=50D/U$) line shows the evolution of the 3D optimal when given initial energy ${\mathcal{E}}_c$. Because it is on the laminar-turbulent boundary two nearly identical initial conditions diverge after, in this case, $220D/U$. The blue (middle at $t=50D/U$) line is the evolution of the 2D optimal for the exact initial energy ${\mathcal{E}}_s^{\mathrm{lin}}$ for which the streaks become linearly unstable. The red (upper at $t=50D/U$) line shows the 2D optimal given initial energy ${\mathcal{E}}_c^{\mathrm{lin}}$ and allowed to evolve until it reaches a maximum amplitude whereupon 0.1% by amplitude unstable perturbation is added. Again the laminar-turbulent boundary can be identified.

Figure 5

The effect of the initial energy on the growth of $A{\mathbf{u}}_{\mathrm{lin}}$ (red) and $A{\mathbf{u}}_{3\mathrm{D}}(\mathbf{x},2\times {10}^{-5},1750)$ (blue) at $\mathrm{Re}=1750$. For small $E_0$ the 2D result is the optimal but after $E_{3\mathrm{D}}=1.35\times {10}^{-5}$, the 3D optimal takes over. The vertical dashed line corresponds to ${\mathcal{E}}_c$, with the dotted lines being the relevant error bars. Inset: The dependence of $E_{3\mathrm{D}}$ (blue or lowest), ${\mathcal{E}}_c$ (green or middle), and ${\mathcal{E}}_s^{\mathrm{lin}}$ (red or upper) on $\mathrm{Re}$ (${\mathcal{E}}_c^{\mathrm{lin}}$ is even higher). For $\mathrm{Re}<2000$, error bars on ${\mathcal{E}}_c$ indicate the energy range over which short to extended turbulent episodes are triggered.