From oblique-wave forcing to streak reinforcement: A perturbation-based frequency-response framework

TL;DR

This work develops a perturbation-based frequency-response framework that expands fluctuations about the laminar base flow in the forcing amplitude $\epsilon$, linking linear resolvent analysis to nonlinear interactions and unifying non-modal amplification with streak formation and modal instability. At second order, quadratic interactions of unsteady oblique waves generate steady streamwise streaks via the lift-up mechanism, with the streak structure captured by the second output singular function of the streamwise-constant resolvent; higher-order terms can reinforce or attenuate these streaks depending on phase, until a practical critical forcing $\epsilon_{\mathrm{cr}}$ marks the breakdown of the weakly nonlinear regime and the onset of sustained unsteadiness. The authors validate the framework against direct numerical simulations and secondary-stability analyses, showing that the breakdown aligns with modal instability of the distorted base flow and that the dominant streak physics are governed by resolvent modes. Collectively, the framework provides a mechanistically transparent, computationally efficient route to describe subcritical transition by unifying non-modal amplification, streak formation, and modal instability within a Navier–Stokes–based formulation.

Abstract

We develop a perturbation-based frequency-response framework for analyzing amplification mechanisms that are central to subcritical routes to transition in wall-bounded shear flows. By systematically expanding the input-output dynamics of fluctuations about the laminar base flow with respect to forcing amplitude, we establish a rigorous correspondence between linear resolvent analysis and higher-order nonlinear interactions. At second order, quadratic interactions of unsteady oblique waves generate steady streamwise streaks via the lift-up mechanism. We demonstrate that the spatial structure of these streaks is captured by the second output singular function of the streamwise-constant resolvent operator. At higher orders, nonlinear coupling between oblique waves and induced streaks acts as structured forcing of the laminar linearized dynamics, yielding additional streak components whose relative phase governs reinforcement or attenuation of the leading-order streak response. Our analysis identifies a critical forcing amplitude marking the breakdown of the weakly nonlinear regime, beyond which direct numerical simulations exhibit sustained unsteadiness. We show that this breakdown coincides with the onset of secondary instability, revealing that the nonlinear interactions responsible for streak formation also drive the modal growth central to classical transition theory. The resulting framework provides a mechanistically transparent and computationally efficient description of transition that unifies non-modal amplification, streak formation, and modal instability within a single formulation derived directly from the Navier-Stokes equations.

Figures (28)

• Figure 1: Pressure driven flow between two parallel infinitely long plates with parabolic laminar velocity profile.
• Figure 2: For small-amplitude exogenous inputs, $\mathbf{d} = \epsilon \mathbf{d}^{(1)}$, perturbation analysis transforms the NS equations \ref{['eq.NS']} into a conveniently coupled hierarchy of linearized systems around the equilibrium profile $\bar{\mathbf{u}}$. The dynamics of velocity fluctuations $\mathbf{u}^{(n)}$ at $\mathcal{O}(\epsilon^n)$ are driven by: (a) the exogenous input $\mathbf{d}^{(1)}$ for $n = 1$; and (b) the nonlinear modulation $\mathbf{d}^{(n)} \mathrel{\mathop:}= - \sum_{r = 1}^{n - 1} (\mathbf{u}^{(r)} \! \cdot \nabla)\mathbf{u}^{(n - r)}$ of lower-order responses $\mathbf{u}^{(r)}$ for $r = 1, \dots, n - 1$, when $n \geq 2$. (c) Block diagram illustrating the coupling structure, i.e., the propagation of information, up to $\mathcal{O}(\epsilon^4)$; operators $\mathcal{N}$ and $\mathcal{M}$ are defined in \ref{['eq.NM']}.
• Figure 3: (a) Square of the $H_{\infty}$ norm for Poiseuille flow with $Re = 2000$. The black dot marks the pair $(k_x, k_z)$ corresponding to the largest amplification. (b) Dependence of the principal singular value on temporal frequency $\omega$ for streamwise streaks with $(k_x, k_z) = (0, 1.65)$ (solid), oblique waves with $(k_x, k_z) = (0.74, 1.14)$ (dashed), and two-dimensional OS modes with $(k_x, k_z) = (1.17, 0)$ (dotted). While the strongest amplification of streamwise streaks occurs for steady disturbances ($\omega = 0$), oblique waves and OS modes exhibit pronounced amplification for unsteady disturbances with negative $\omega$.
• Figure 4: (a) Spatial wavenumbers that arise in the $\mathcal{O}(\epsilon^2)$ expansion of the response of the NS equations to small-amplitude three-dimensional disturbances. Input wavenumbers are marked by crosses, the $\mathcal{O}(\epsilon)$ response by squares, and nine $\mathcal{O}(\epsilon^2)$ terms by black dots. (b) Small-amplitude harmonic input \ref{['eq.dow']} to the NS equations \ref{['eq.NS']}, with wavenumbers $(\pm k_x, k_z)$ and temporal frequency $\omega = \mp k_x c$, excites $\mathcal{O}(\epsilon)$ oblique waves whose quadratic interactions generate $\mathcal{O}(\epsilon^2)$ steady vortical forcing and thereby streamwise streaks. The operator $\widetilde{\mathcal{M}}$ denotes the Fourier-transformed nonlinear operator $\mathcal{M}$ given in \ref{['eq.NM']}, $\widetilde{\mathcal{M}}(\mathbf{u}_{\mathbf{k}}, \mathbf{u}_{\mathbf{k}'}) \mathrel{\mathop:}= -(\mathbf{u}_{\mathbf{k}} \cdot \nabla_{\mathbf{k}'}) \mathbf{u}_{\mathbf{k}'} - (\mathbf{u}_{\mathbf{k}'} \cdot \nabla_{\mathbf{k}}) \mathbf{u}_{\mathbf{k}}$, where $\nabla_{\mathbf{k}} \mathrel{\mathop:}= [\, \mathrm{i} k_x \; \partial_y \; \mathrm{i} k_z ]^T$. The operators $\mathcal{G}_{(\pm k_x, k_z)}(c)$ and $\mathcal{G}_{(0, 2k_z)}(c)$, associated with the linearized NS equations at $(\pm k_x, k_z, \mp k_x c)$ and $(0, 2 k_z, 0)$, completely characterize the resulting responses.
• Figure 5: (a) Energy amplification as a function of the spanwise wavenumber $k_z$ obtained from the primary linearized (dashed) and perturbation (solid) analyses. The dashed curve shows $\sigma_{\max}^2(\mathcal{H}_{(0,k_z)}(0))$, while the solid curve displays the $\mathcal{O}(\epsilon^2)$ steady streak response induced by oblique waves with $(k_x,c)$ chosen to maximize streak amplification. Both curves are normalized to unit peak value; absolute magnitudes depend on the choice of $\epsilon$. (b,c) Color contours show the most energetic streamwise velocity fluctuations, while black contours indicate streamwise vortices. The spanwise wavelengths are $2\pi/1.65$ in (b) (primary linearized analysis) and $2\pi/2.28$ in (c) (perturbation analysis).

Theorems & Definitions (3)

• Theorem 1
• Remark 1
• Remark 2: Procedure for estimating $\epsilon_{\mathrm{cr}}$