Stability analysis of transitional flows based on disturbance magnitude

Abstract

We propose a novel stability criterion for incompressible shear flows by combining input-output analysis and the small-gain theorem. The criterion yields an explicit threshold on the magnitude of velocity perturbations about a given base flow that guarantees stability. If this threshold is crossed--either due to nonmodal growth, exponential growth, or a bypass transition scenario--our analysis predicts a loss of stability that may lead to transition to turbulence. We consider three approximated models for nonlinearity: unstructured, structured with non-repeated blocks, and structured with repeated blocks. We show that the imposed threshold obtained by these three methods complies with a hierarchical relationship, where the unstructured case is the most conservative, imposing the lowest bound on disturbance magnitude. We apply this approach to three canonical and well-studied base flows: Couette, plane Poiseuille, and Blasius. For these three base flows, we compare our results with experiments, direct numerical simulation results, nonmodal nonlinear stability results, and linear stability theory (LST). In the limit of infinitesimally small perturbation magnitude, our stability criterion for the unstructured case recovers the results of LST. For finite perturbations, the structured cases that account for nonlinear interactions provided stability thresholds that are consistent with experimental observations and simulation results of transition at both subcritical and post-critical Reynolds numbers for the considered base flows in our study. In particular, we utilize our stability criterion to demonstrate that Couette flow can become unstable and transition can be triggered at different Reynolds numbers, which is consistent with past experimental observations.

Figures (16)

• Figure 1: Interconnection loop block diagram for the nonlinear interactions. Adapted from liu2021.
• Figure 2: Illustration of uncertainty matrices with relevant structures: (a) uncertainty structure that is based on the feedback interconnection as defined in Eq. \ref{['eq:2.25']}, (b) repeated block structure as defined in Eq. \ref{['eq:2.26']}, (c) non-repeated block structure as defined in Eq. \ref{['eq:2.27']}. Visualizations of the uncertainty matrices are shown for (a) ${\textbf{U}}_\Xi \in \mathbf{\Delta}_{\mathbf{u}}$, (b) ${\textbf{U}}_\Xi \in \mathbf{\Delta}_{r}$, (c) ${\textbf{U}}_\Xi \in \mathbf{\Delta}_{nr}$.
• Figure 3: Contour plots in logarithmic scale of (from left to right): $\norm{\mathscr{H}_{\nabla}}_{\infty}^{-1}$, $\norm{\mathscr{H}_{\nabla}}_{\mu_{\mathbf{\Delta}_{nr}}}^{-1}$, and $\norm{\mathscr{H}_{\nabla}}_{\mu_{\mathbf{\Delta}_{r}}}^{-1}$ at (a) $\Rey=358$ and (b) $\Rey=8000$ for Couette flow. The most dominant mode is marked by an X (cyan color indicates that the mode's $(k_x,k_z)$ value is not denoted in the table), whereas the other modes from Table \ref{['tbl:1']} are marked by colored circles (magenta - DLR mode, black - TS mode, red - SPS mode, blue - DHR mode). Results are showed for (a) $Re=358$ and (b) $Re=8000$.
• Figure 4: Evolution curves of imposed thresholds due to flow structures associated with pre-selected modes of interest denoted in Table \ref{['tbl:1']} (magenta solid - DLR mode, black dotted - TS mode, red dashed - SPS mode, blue dashed-dotted - DHR mode) for Couette base flow as a function of the Reynolds number in terms of (a) $\norm{\mathscr{H}_{\nabla}}_{\infty}^{-1}$, (b) $\norm{\mathscr{H}_{\nabla}}_{\mu_{\mathbf{\Delta}_{nr}}}^{-1}$ and (c) $\norm{\mathscr{H}_{\nabla}}_{\mu_{\mathbf{\Delta}_{r}}}^{-1}$.
• Figure 5: Contour plots in logarithmic scale of (from left to right): $\norm{\mathscr{H}_{\nabla}}_{\infty}^{-1}$, $\norm{\mathscr{H}_{\nabla}}_{\mu_{\mathbf{\Delta}_{nr}}}^{-1}$, and $\norm{\mathscr{H}_{\nabla}}_{\mu_{\mathbf{\Delta}_{r}}}^{-1}$ at (a) $\Rey=690$, (b) $\Rey=5700$, and (c) $\Rey=6000$ for plane Poiseuille flow. The most dominant mode is marked by an X (cyan color indicates that the mode's $(k_x,k_z)$ value is not denoted in the table), whereas the other modes from Table \ref{['tbl:1']} are marked by colored circles (magenta - DLR mode, black - TS mode, red - SPS mode). Results are shown for (a) $Re=690$, (b) $Re=5700$, (c) $Re=6000$.

Theorems & Definitions (1)

• Theorem 1