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Nonlinear input-output analysis of transitional shear flows using small-signal finite-gain $\mathcal{L}_p$ stability

Zhengyang Wei, Chang Liu

TL;DR

This work develops a nonlinear input-output analysis framework for transitional shear flows using the Small-Signal Finite-Gain ($L_p$) stability theorem. It combines Linear Matrix Inequalities (LMI) and Sum-of-Squares (SOS) optimization to certify a quadratic Lyapunov function for a nine-mode Moehlis shear-flow model, yielding explicit permissible forcing amplitudes and finite nonlinear $L_p$ gains across $p\in[1,\infty]$. The nonlinear gains exceed linear predictions by several orders of magnitude, and the predicted bounds align with long-time simulations, with the accuracy improving for larger $p$. The results highlight the amplitude-dependent nature of transition to turbulence and provide a conservative but practical framework for predicting and constraining external disturbances in transitional flows, with future work aimed at reducing conservatism and extending to full Navier–Stokes dynamics and time-varying conditions.

Abstract

This SSFG Lp stability theorem can predict permissible forcing amplitudes below which a finite nonlinear input-output gain can be maintained. Our analysis employs Linear Matrix Inequalities (LMI) and Sum-of-Squares (SOS) as the primary tools to search for a quadratic Lyapunov function of an unforced nonlinear system. The resulting Lyapunov function can certify the SSFG Lp stability of a nonlinear input-output system. We demonstrate the applicability of the SSFG Lp stability theorem using a nine-mode shear flow model with a random body force. The predicted nonlinear input-output Lp gain is consistent with numerical simulations; the Lp norm of the output from numerical simulations remains bounded by the theoretical prediction from SSFG Lp stability theorem, with the gap between simulated and theoretical bounds narrowing as $p \rightarrow \infty$. The input-output gain obtained from the nonlinear SSFG Lp stability theorem is higher than the linear Lp gain. Both nonlinear Lp gain and linear Lp gain are valid for each $p\in [1,\infty]$, and such generalizability leads to much higher upper bounds on input-output gain than those predicted by linear L2 gain. The SSFG Lp stability theorem requires the input forcing to be smaller than a permissible forcing amplitude to maintain finite input-output gain, which is an inherently nonlinear behavior that cannot be predicted by linear input-output analysis. We also identify such permissible forcing amplitude using numerical simulations and bisection search, where below such forcing amplitude the output norm at any time will be lower than a given threshold value. The permissible forcing amplitude identified from the SSFG Lp stability theorem is conservative but also consistent with that obtained by numerical simulations and bisection search.

Nonlinear input-output analysis of transitional shear flows using small-signal finite-gain $\mathcal{L}_p$ stability

TL;DR

This work develops a nonlinear input-output analysis framework for transitional shear flows using the Small-Signal Finite-Gain () stability theorem. It combines Linear Matrix Inequalities (LMI) and Sum-of-Squares (SOS) optimization to certify a quadratic Lyapunov function for a nine-mode Moehlis shear-flow model, yielding explicit permissible forcing amplitudes and finite nonlinear gains across . The nonlinear gains exceed linear predictions by several orders of magnitude, and the predicted bounds align with long-time simulations, with the accuracy improving for larger . The results highlight the amplitude-dependent nature of transition to turbulence and provide a conservative but practical framework for predicting and constraining external disturbances in transitional flows, with future work aimed at reducing conservatism and extending to full Navier–Stokes dynamics and time-varying conditions.

Abstract

This SSFG Lp stability theorem can predict permissible forcing amplitudes below which a finite nonlinear input-output gain can be maintained. Our analysis employs Linear Matrix Inequalities (LMI) and Sum-of-Squares (SOS) as the primary tools to search for a quadratic Lyapunov function of an unforced nonlinear system. The resulting Lyapunov function can certify the SSFG Lp stability of a nonlinear input-output system. We demonstrate the applicability of the SSFG Lp stability theorem using a nine-mode shear flow model with a random body force. The predicted nonlinear input-output Lp gain is consistent with numerical simulations; the Lp norm of the output from numerical simulations remains bounded by the theoretical prediction from SSFG Lp stability theorem, with the gap between simulated and theoretical bounds narrowing as . The input-output gain obtained from the nonlinear SSFG Lp stability theorem is higher than the linear Lp gain. Both nonlinear Lp gain and linear Lp gain are valid for each , and such generalizability leads to much higher upper bounds on input-output gain than those predicted by linear L2 gain. The SSFG Lp stability theorem requires the input forcing to be smaller than a permissible forcing amplitude to maintain finite input-output gain, which is an inherently nonlinear behavior that cannot be predicted by linear input-output analysis. We also identify such permissible forcing amplitude using numerical simulations and bisection search, where below such forcing amplitude the output norm at any time will be lower than a given threshold value. The permissible forcing amplitude identified from the SSFG Lp stability theorem is conservative but also consistent with that obtained by numerical simulations and bisection search.

Paper Structure

This paper contains 12 sections, 6 theorems, 57 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Nonlinear SSFG $\mathcal{L}_p$ stability theorem
  4. Sum-of-squares (SOS) optimization
  5. Nonlinear input-output stability analysis
  6. Comparison of nonlinear SSFG $\mathcal{L}_p$ stability analysis with numerical simulations
  7. Comparison with linear finite-gain $\mathcal{L}_p$ stability and linear $\mathcal{L}_2$ stability
  8. Permissible forcing amplitude compared with bisection
  9. Conclusions and future work
  10. The nine-mode shear flow model
  11. Componentwise upper bound of quadratic nonlinearity in \ref{['eq:local_upper_bound_upsilon_m']}
  12. Influences of LMI result by the $\varepsilon$

Key Result

Theorem 1

Khalil2002 Consider the nonlinear input-output system in eq:simp_sys_1-eq:simp_sys_2 and take $\delta>0$ and $r_u>0$ such that $\{\|\boldsymbol{a}\| \leq \delta\} \subset D$ and $\left\{\|\boldsymbol{f}\| \leq r_u\right\} \subset D_u$. Suppose that Then, for each $\boldsymbol{a}_0$ with $\|\boldsymbol{a}_0\| \leq \delta \sqrt{c_1 / c_2}$, the system eq:simp_sys_1-eq:simp_sys_2 is small-signal fin

Figures (12)

  • Figure 1: Illustration of sinusoidal shear flow following moehlis2004lowMoehlis2005.
  • Figure 2: (a) $\text{log}_{10}[\gamma(\text{Re},\delta)]$, (b) $\text{log}_{10}[\gamma_{\text{SOS}}(\text{Re},\delta)]$, (c) $\text{log}_{10}[f_{\text{LMI}}(\text{Re},\delta)]$, and (d) $\text{log}_{10}[f_{\text{SOS}}(\text{Re},\delta)]$. The white region indicates parameter regimes where the LMI or SOS is infeasible.
  • Figure 3: (a) The norm of output $\|\boldsymbol{y}_\tau(t)\|$ from simulations at Re = 200, 1000, 1900, and $\delta = 10^{-6}$. (b) $\|\boldsymbol{y}_\tau(t)\|$ at Re = 200, and $\delta = 10^{-6}, 10^{-4},$ and $5\times10^{-3}$. All simulations are driven by initial disturbance amplitude as $\|\boldsymbol{a}_0\|=\delta_f$ based on \ref{['eq:delta_f_LMI']} and random forcing amplitude as $\|\boldsymbol{f}(t)\|=f_{\text{LMI}}$ based on \ref{['eq:forcing_upper_bound']}. In both panels, horizontal lines are the theoretical upper bound $\xi$ of the $\mathcal{L}_\infty$ norm of the output as defined in equation \ref{['eq:thupper_output']}.
  • Figure 4: (a) The norm of output $\|\boldsymbol{y}_\tau(t)\|$ from simulations at Re = 200, 1000, 1900, $\delta = 10^{-6}$ driven by initial condition ($\boldsymbol{a}_0\neq\boldsymbol{0}$ with $\|\boldsymbol{a}_0\|=\delta_f$ based on \ref{['eq:delta_f_LMI']}) without input forcing ($\boldsymbol{f}=\boldsymbol{0}$). (b) The norm of output $\|\boldsymbol{y}_\tau(t)\|$ from simulations at Re = 200, 1000, 1900, $\delta = 10^{-6}$ driven by input forcing ($\boldsymbol{f}\neq \boldsymbol{0}$ and $\|\boldsymbol{f}(t)\|=f_{\text{LMI}}$ based on \ref{['eq:forcing_upper_bound']}) without initial condition ($\boldsymbol{a}_0=0$).
  • Figure 5: The left-hand side (LHS) of inequality \ref{['eq:gamma_upper_bound']} ($\left\|\boldsymbol{y}_\tau\right\|_{\mathcal{L}_p}$) and the right-hand side (RHS) of inequality \ref{['eq:gamma_upper_bound']} ($\gamma\left\|\boldsymbol{f}_\tau\right\|_{\mathcal{L}_p}+\beta$) versus $p$ at Re = 200 and $\delta$ = $10^{-6}$. The gap between the LHS and RHS converges to a constant value as $p$ increases. The horizontal lines correspond to the LHS ($\left\|\boldsymbol{y}_{\tau}\right\|_{\mathcal{L}_\infty}$) and RHS ($\xi=\gamma\left\|\boldsymbol{f}_{\tau}\right\|_{\mathcal{L}_\infty} + \beta_\infty$ in \ref{['eq:thupper_output']}) of inequality \ref{['eq:gamma_upper_bound']} at $p$ = $\infty$.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • Theorem 5
  • Lemma 6
  • proof