From bypass transition to flow control and data-driven turbulence modeling: An input-output viewpoint

1. INTRODUCTION

• Non-normality couples decaying modes, making wall-bounded laminar flows sensitive to modeling imperfections such as roughness, thermal fluctuations, and incoming-stream irregularities.This sensitivity contributes to discrepancies between conventional stability predictions and observations in wall-bounded shear flows (Schmid 2007).Source: §1 · p. 3
• Input-output analysis maps specified disturbances to selected flow quantities, complementing transient-growth and resolvent analyses while exposing responses to modeling imperfections.Applications include disturbance amplification and turbulent-flow structure, as well as transition mechanisms in viscoelastic fluids and noise generation by turbulent jets (Hoda et al. 2008, 2009, Jovanovi´c & Kumar 2011; Jeun et al. 2016).Source: §1 · p. 3

2. INPUT-OUTPUT VIEWPOINT: BEYOND RESOLVENT ANALYSIS

• The review applies the framework to channel flow and outlines questions involving complex-fluid transition, flow control, and data-driven turbulence modeling.It also describes periodic base-flow modifications for controlling turbulence onset or identifying an oscillation period for turbulent drag reduction (Moarref & Jovanovi´c 2010, 2012).Source: §2 · p. 4, §1 · p. 4

2.1. From evolution model to input-output representation

• A state-space model represents linearized fluctuations using a state, an input map, and an output map; channel-flow examples use wall-parallel wavenumbers and the Orr-Sommerfeld/Squire operator.For Newtonian channel flow, the state can be wall-normal velocity and vorticity, with other velocity and vorticity components recovered through kinematic relations (Kim et al. 1987; Jovanovi´c & Bamieh 2005).Source: §2.1 · pp. 4–5

2.1.1. Resolvent, transfer function, impulse and frequency response operators.

• The transfer function maps inputs to outputs, and restricting the forcing region or selecting outputs can reveal mechanisms obscured by the standard resolvent.For turbulent jets, restricting inputs to the turbulent region and choosing far-field pressure as output identified acoustic-source wavepackets consistent with experiments and revealed noise-generation mechanisms (Jeun et al. 2016).Source: §2.1.1 · pp. 5–6

2.2. Amplification of deterministic inputs

• For deterministic harmonic inputs, the largest singular value of the frequency-response operator quantifies worst-case amplification across spatial profiles and frequency.The H∞ norm also gives the induced gain from finite-energy inputs to outputs and quantifies robustness to modeling imperfections.Source: §2.2 · p. 7

2.3. Amplification of stochastic inputs

• Persistent stochastic forcing sustains fluctuations after natural responses decay, while control-theoretic tools can use the linearized model to analyze amplification without costly stochastic simulations.In the Re = 2000 Poiseuille-flow example, averages over 20 realizations reach a steady-state limit in the absence of modal instability.Source: §2.3 · p. 7

FREQUENCY RESPONSE OPERATOR

• The H∞ norm is the peak of the maximum singular value of the frequency response and quantifies robustness to modeling imperfections.Imperfections with norm 1/√G_k can trigger instability of A_k + B_kΓ_kC_k.Source: FREQUENCY RESPONSE OPERATOR · p. 8
• For 20 realizations of stochastic forcing to the linearized NS equations in Poiseuille flow at Re = 2000, the variance averaged over all simulations is shown by a thick black line.The wavenumbers are k = (0, 1.78), (1, 1), and (1, 0).Source: FREQUENCY RESPONSE OPERATOR · p. 9
• Integrating the spectral density over frequency yields the steady-state covariance, whose kernel determines stationary two-point correlations of velocity fluctuations.The inverse Fourier transform of the two-point spectral density gives the two-point correlation tensor in the streamwise and spanwise directions.Source: Time-invariant systems · pp. 9–10
• For white-in-time forcing, the differential Lyapunov equation evolves state covariance and provides fluctuation energy without costly stochastic simulations.The Reynolds-Orr equation’s terms, including viscous dissipation, base-shear exchange, and body-force work, can also be computed from that covariance solution.Source: Time-invariant systems · p. 10, Lyapunov equation · p. 10
• For linearly unstable flows, the steady-state covariance may be unbounded or nonexistent, although its time-dependent solution can still be computed.In the absence of modal instability, the steady-state covariance is well-defined.Source: Lyapunov equation · p. 10
• Colored-in-time inputs allow sign-indefinite cross terms in the covariance equation, extending the white-input formulation.The white-input Lyapunov equation uses a non-negative covariance contribution on its right-hand side.Source: Lyapunov equation · p. 10

LYAPUNOV EQUATION: TWO-POINT CORRELATIONS OF LINEAR SYSTEMS

• The Lyapunov equation propagates two-point correlations of white stochastic input into colored state statistics.The derivation presented is for finite-dimensional discrete-time linear time-invariant systems.Source: LYAPUNOV EQUATION: TWO-POINT… · p. 11
• For white-in-time input, the discrete-time Lyapunov equation propagates the state covariance; its steady-state limit exists only for stable systems.When stable, it converges to the algebraic Lyapunov equation for stationary covariance.Source: LYAPUNOV EQUATION: TWO-POINT… · p. 11
• For colored inputs, cross terms involving input-state correlations modify the covariance evolution and yield a Lyapunov-like equation.For stable systems, that equation converges asymptotically to its algebraic form.Source: LYAPUNOV EQUATION: TWO-POINT… · p. 11
• A colored-in-time input can be represented as a white input passed through a subsystem, equivalently modifying the dynamics through state feedback.The resulting subsystem maps white input to colored input while preserving the stationary state covariance.Source: LYAPUNOV EQUATION: TWO-POINT… · pp. 11–12
• For time-periodic systems, a harmonic Lyapunov equation computes the Fourier-series coefficients of the periodic state covariance under white-in-time forcing.The covariance is periodic with the base-flow period.Source: LYAPUNOV EQUATION: TWO-POINT… · p. 12, Time-periodic systems · p. 12

3. UNCOVERING MECHANISMS IN WALL-BOUNDED SHEAR FLOWS

• The input-output review identifies streaks, oblique waves, and Orr-Sommerfeld modes as resonances, and describes a viscoelastic analogue of inertial lift-up that may trigger elastic turbulence.It also surveys analysis in turbulent Newtonian channel and pipe flows and periodic modifications for transition control or identifying an optimal oscillation period.Source: §3 · p. 12

3.1. Bypass transition in channel flows of Newtonian fluids

• At Re = 2000, streamwise-constant structures with k_z = 1.78 are much more energetic than oblique waves with k_x = k_z = 1 and Orr-Sommerfeld modes with k_x = 1, k_z = 0.The comparison is for Poiseuille flow.Source: §3.1 · p. 12

POWER SPECTRAL DENSITY AND ENERGY AMPLIFICATION

• For white-in-time, spatially uncorrelated forcing, H2 energy amplification gives steady-state variance and impulse-response L2 norm, whereas H∞ captures peak gain over profiles and frequency.The H2 norm can be obtained from the algebraic Lyapunov equation without integrating over frequency.Source: POWER SPECTRAL DENSITY AND ENERGY… · p. 13, Comparison of H2 and H∞norms · p. 13
• Streamwise velocity contains most energy at small kx and O(1) kz, while classical stability structures play a marginal role in stochastic-disturbance amplification.Oblique modes resonate in the spanwise-velocity response but are less amplified than streamwise-elongated structures.Source: Comparison of H2 and H∞norms · pp. 13–14

3.1.1. Streamwise constant model: lift-up mechanism.

• At kx = 0, wall-normal and spanwise forcing drive O(Re) velocity fluctuations, then vortex tilting produces O(Re2) wall-normal vorticity and streamwise velocity fluctuations.The vortex-tilting coupling is the source of amplification; without it, the self-adjoint Orr-Sommerfeld and Squire dynamics are viscously dissipative.Source: §3.1.1 · p. 15

3.2. Early stages of transition to elastic turbulence: viscoelastic lift-up

• In inertialess Oldroyd-B flow, base-shear polymer-stress stretching lets wall-normal and spanwise forcing generate streamwise velocity proportional to We; other responses are We-independent.This lift-up mechanism parallels Newtonian vortex tilting, but the underlying physical mechanism differs.Source: §3.2 · p. 16

3.3. Turbulent channel and pipe flows of Newtonian fluids

• Earlier turbulent-flow studies found that linear amplification of eddies interacting with mean shear induces streamwise streaks even without walls, and removing vortex tilting yielded near-wall turbulence decay.Source: §3.3 · p. 16

• Boxes denote system components, circles sum signals, and arrows show signal flow; the thick lines mark vortex tilting in Newtonian flow and polymer stretching in viscoelastic flow.Source: BLOCK DIAGRAMS: A TOOL FOR REVEALING… · p. 17
• For Newtonian flows, gains from d2 and d3 to u scale as Re2; in inertialess Oldroyd-B flows they scale as We, while other singular values are We-independent.The Newtonian diagram also reports gains proportional to Re for the other specified nonzero mappings and zero gain from d1 to v and w.Source: Figure 6 · p. 17, BLOCK DIAGRAMS: A TOOL FOR REVEALING… · p. 17
• Eddy-viscosity-enhanced linearizations and resolvent analyses have been used to qualitatively capture turbulent-flow features and characterize energetic structures by convection speeds and wavelengths.Source: BLOCK DIAGRAMS: A TOOL FOR REVEALING… · p. 18
• At Re = 547 and kx = 0, stochastic forcing produces channel-wide and near-wall streaks, with the channel-wide structures forming the most energetic response.The two peaks in the premultiplied energy spectrum determine the spanwise length scales of the most energetic responses.Source: BLOCK DIAGRAMS: A TOOL FOR REVEALING… · p. 18

4. CONTROL OF TRANSITIONAL AND TURBULENT FLOWS

• Periodic actuation changes the base flow to introduce sensor-less feedback, and the review presents a simulation-free framework for assessing control effects in transitional and turbulent flows.The framework combines receptivity analysis, control-oriented turbulence modeling, and control-theory techniques.Source: §4 · p. 19, Turbulence modeling · p. 24

4.1. Controlling the onset of turbulence by streamwise traveling waves

• Traveling-wave blowing and suction create a spatially periodic base flow, and perturbation analysis makes small-amplitude assessments of fluctuation amplification comparable in complexity to uncontrolled-flow computations.For sufficiently small α, the amplification ratio is 1 + α²gk(ωx, c, Re) + O(α⁴).Source: §4.1 · p. 20, §4.1.2 · p. 21

4.1.1. Net efficiency of modified base flow.

• For turbulent uncontrolled flow, sufficiently small-amplitude upstream and downstream waves can improve net efficiency, although blowing and suction reduce it relative to laminar uncontrolled flow.At fixed pressure gradient, upstream waves reduce the modified laminar base-flow skin-friction coefficient, whereas downstream waves increase it.Source: §4.1.1 · p. 20

4.1.2. Dynamics of velocity fluctuations.

• Downstream traveling waves with (ωx = 2, c = 5) reduce energy amplification across all θ and kz, and DNS confirms their effectiveness for controlling transition.Upstream waves with (ωx = 0.5, c = −2) promote amplification over a broad range and can promote transition even when uncontrolled flow stays laminar.Source: §4.1.2 · p. 21, DNS verification · p. 21

4.2. Turbulent drag reduction by spanwise wall oscillations

• A stochastically forced eddy-viscosity-enhanced linearization estimates how oscillations alter turbulent viscosity, using fluctuation statistics to update the turbulence model.Spatial correlations of white-in-time forcing are selected to match the uncontrolled flow's two-dimensional energy spectra, guided by DNS data.Source: Turbulence modeling · p. 23
• For constant-bulk-flux turbulent channel flow at friction Reynolds number Re = 186, input-output analysis predicts optimal period T+ = 102.5, closely matching DNS T+ ≈100 reported at Re = 200.At T+ = 102.5, oscillations suppress energy at large streamwise wavelengths and increase amplification at small streamwise wavelengths. They also minimize turbulent viscosity near the buffer/log-law interface.Source: Turbulence modeling · p. 23

5. PHYSICS-AWARE DATA-DRIVEN TURBULENCE MODELING

• Merging control theory with convex optimization identifies stochastic forcing spectra, modeling neglected nonlinear interactions through a low-rank perturbation to the linearized dynamics.The resulting stochastic model is presented as a physics-aware alternative to machine-learning tools that do not incorporate physical constraints.Source: §5 · pp. 24–25

5.1. Completion of partially available channel flow statistics

• White-in-time forcing cannot recover the desired statistics from turbulent-mean linearization, motivating colored-in-time inputs for covariance completion.The review notes that white-in-time forcing is often too narrow to capture observed turbulent-flow statistics.Source: §5 · pp. 24–25, §5.1 · p. 25
• The completion problem enforces consistency with turbulent-mean linearized dynamics, exactly matches available covariance entries, and uses a singular-value penalty to restrict input rank.A logarithmic barrier ensures the state covariance remains positive definite.Source: §5.1 · p. 26
• The unique optimum reproduces available DNS covariance entries and completes unavailable second-order statistics with a low-complexity stochastic model.The resulting model refines linearized Navier–Stokes predictions while preserving relevant flow physics.Source: §5.1 · p. 26
• Figure 12 compares DNS covariances with the completion: available one-point correlations match exactly, while the recovered two-point correlations are high quality.The comparison is for turbulent channel flow at Re = 186 and k = (2.5, 7), with γ = 300.Source: §5.1 · p. 26, Alternative formulations · p. 27
• Compared with standard resolvent analysis, the componentwise input-output approach considerably improves matching of turbulent-channel spectra and co-spectra.The review also describes alternative formulations, including an optimal-control trade-off between control energy and the number of feedback couplings.Source: §5.1 · p. 26, Alternative formulations · p. 27

SUMMARY POINTS

• Componentwise input-output analysis identifies streaks, oblique waves, and Orr–Sommerfeld modes as forcing-to-velocity resonances, and reveals a viscoelastic analogue of lift-up.The viscoelastic mechanism arises when base shear stretches polymer-stress fluctuations and can amplify them even without inertia.Source: SUMMARY POINTS · p. 27
• The review presents input-output modeling as a data-driven refinement of physics-based models that supports statistical consistency and tractable analysis, optimization, and control.It identifies control theory and convex optimization as tools for addressing shortcomings of physics-agnostic machine learning.Source: SUMMARY POINTS · p. 28

1. Complex fluids and complex flows.

• Input-output analysis may clarify how surface roughness and free-stream disturbances contribute to physical mechanisms in complex flows.The review also identifies modeling uncertainties from chemical reactions and gas-surface interactions in hypersonic flows as an application area.Source: §1 · p. 28
• The tools require O(n^3) computations for evolution models with n degrees of freedom.Such computations are described as routine for canonical flows.Source: §1 · p. 28
• Spatially discretized models in complex geometries induce significant computational overhead.This contrasts with the stated routine computational cost for canonical flows.Source: §1 · p. 28

2. Computational complexity.

• Relating high flow sensitivity to nonlinearity remains a challenge for capturing later disturbance development, identifying transition routes, and designing nonlinear Navier–Stokes control.Source: §2 · p. 28

3. Nonlinear interactions.

• Machine-learning methods must respect physical constraints and generalize to flow regimes absent from available data.Source: §3 · p. 28
• The review also calls for convergence, performance, and robustness guarantees comparable to model-based flow-control approaches.Source: §3 · p. 28

4. Data-driven techniques.

• Noisy measurements and optimal sensor and actuator placement remain challenges for high-Reynolds-number feedback control.Source: §4 · p. 28
• Efficient optimal and robust controller computation, along with structured and distributed control synthesis, remains necessary.Source: §4 · p. 28
• Data-driven reinforcement learning strategies still require progress on convergence and sample complexity.Source: §4 · p. 28

Complex fluids and complex flows

• Global input-output analysis of a hypersonic compression-ramp interaction linked upstream counter-rotating vortices to reattachment streaks with a preferred spanwise scale (Dwivedi et al. 2019).The discussion points to roughness, free-stream disturbances, and modeling uncertainties from chemical reactions and gas-surface interactions as areas where the analysis may clarify physical mechanisms.Source: §6 · p. 29

Computational complexity

• For models with n degrees of freedom, the reviewed tools require O(n^3) computations, while iterative or randomized methods and sparse-system algorithms may help scale analyses to complex geometries.Efficient iterative Lyapunov methods are noted for systems with few inputs and outputs and sparse dynamic matrices.Source: §6 · p. 29

Nonlinear interactions

• The interplay between high flow sensitivity and nonlinearity remains an open challenge for capturing later transition stages and designing effective control strategies.The discussion describes extensions of input-output analysis to nonlinear Navier–Stokes equations using dissipation inequalities and harmonic balance.Source: §6 · p. 29

Data-driven techniques

• Data-driven flow models and control still need physical constraints, generalization beyond observed regimes, and convergence, performance, and robustness guarantees comparable to model-based approaches.DNS data supports reduced-order modeling, while the discussion also identifies sensor and actuator placement, controller computation, and reinforcement-learning sample complexity as challenges for high-Reynolds-number feedback control.Source: §6 · p. 30

• The author reports no known affiliations, memberships, funding, or financial holdings that might affect the review’s objectivity.Source: DISCLOSURE STATEMENT · p. 30