Transition Flow Dynamics

• Transition flow is the dynamic pathway through which systems evolve between qualitatively distinct regimes, such as laminar and turbulent flows.
• It is characterized by localized structures (e.g., puffs, slugs, bands) whose behavior is modeled using critical thresholds, scaling laws, and excitable-medium frameworks.
• Beyond fluid dynamics, transition flow also describes learned mappings in generative modeling, facilitating efficient state interpolation and advanced image synthesis.

Transition flow denotes the flow state or dynamical evolution through which a system passes between qualitatively distinct regimes. In fluid mechanics, the term most commonly refers to finite-amplitude, often localized dynamics that connect laminar and turbulent states, such as puffs, slugs, oblique bands, laminar gaps, or time-periodic post-bifurcation motions. In other settings it denotes regime-changing flows produced by pulsation, vibration, interfacial breakup, topological rearrangements, or imposed parameter jumps. A distinct modern usage appears in generative modeling, where “transition flow” is defined as a learned mapping from an intermediate state at time $t$ to a later state at time $r$ (Manneville, 2017, Xu et al., 2017, Ma, 16 Mar 2026).

1. Canonical meaning in wall-bounded shear flows

In wall-bounded shear flows, transition flow is classically associated with a globally subcritical scenario in which laminar and turbulent domains coexist over a finite Reynolds-number interval. Manneville’s review characterizes this interval by an upper threshold $R_t$, above which turbulence is uniform, and a lower threshold $R_g$, below which all turbulence eventually decays. For $R_g<R<R_t$, the statistically steady state consists of laminar and turbulent coexistence, typically in the form of oblique bands. Near $R_g$, the decay of these bands is described in terms of directed percolation, with turbulent fraction $F_t \sim (R-R_g)^\beta$, spatial correlation length $\xi \sim |R-R_g|^{-\nu_\perp}$, and temporal correlation $\tau \sim |R-R_g|^{-\nu_\parallel}$, using the two-dimensional directed-percolation exponents $\beta \approx 0.58$, $r$0, and $r$1 (Manneville, 2017).

Barkley’s reduced description places this phenomenology in a two-variable excitable-medium framework with turbulence intensity $r$2 and mean shear $r$3. In the continuous model, the laminar state is excitable for $r$4, which yields localized puff-like pulses, while for $r$5 the system becomes bistable and supports expanding slugs. The discrete extension embeds a chaotic repeller and reproduces finite puff lifetimes, puff splitting, sustained spatiotemporal intermittency, and a directed-percolation-like onset of sustained turbulence at $r$6, with order-parameter scaling $r$7 and $r$8 (Barkley, 2011).

The dynamical-state inventory has been extended further by Frishman and Grafke, who introduced the antipuff and the gap-edge as counterparts to the puff and decay edge state. In their Barkley-model landscape, laminar gaps inside homogeneous turbulence are interpreted as antipuffs that nucleate and decay through the gap-edge, producing a symmetric picture in which localized turbulent states and localized laminar states each organize part of the transitional dynamics (Frishman et al., 2021).

A complementary unification was proposed by Tao and Xiong through a locally defined Reynolds number,

$r$9

which is interpreted as the maximum local energy-input rate relative to local dissipation. Using $R_t$0, they identified three transition stages common to linearly stable shear flows: equilibrium localized turbulence, temporally persistent turbulence, and uniform turbulence. In plane-Poiseuille flow, these occur at approximately $R_t$1, $R_t$2, and $R_t$3, and the corresponding threshold bands collapse closely for pipe, channel, and Couette flows (Tao et al., 2017). A related comparison of fluid-element-based criteria showed that Hanks’s $R_t$4, Tao’s $R_t$5, and $R_t$6 reduce to the same algebraic forms in Hagen–Poiseuille and plane–Poiseuille flows, while $R_t$7 and $R_t$8 remain well defined in plane Couette flow (Tao, 2023).

2. Thresholds and time-scale competition in pulsating pipe flow

In pulsating pipe flow, transition flow is controlled not only by Reynolds number but also by the interaction between turbulence dynamics and the pulsation period. The relevant dimensionless groups are the Womersley number

$R_t$9

the mean Reynolds number $R_g$0, the oscillatory Reynolds number $R_g$1, and the instantaneous Reynolds number

$R_g$2

For moderate pulsation amplitudes, the first instability remains subcritical and produces localized turbulent puffs analogous to those in steady pipe flow (Xu et al., 2017).

The onset was mapped through puff survival rather than immediate breakdown. Using the instantaneous escape rate $R_g$3, with steady-flow puff lifetime

$R_g$4

the survival probability over one cycle is

$R_g$5

and the threshold $R_g$6 is defined by $R_g$7 (Xu et al., 2017).

This construction yields three regimes. For $R_g$8, the flow is quasi-steady over each cycle, turbulence decays during the slow phase, and transition is significantly delayed. In the $R_g$9 limit,

$R_g<R<R_t$0

because the minimum instantaneous Reynolds number must stay above the steady critical value. For $R_g<R<R_t$1 and $R_g<R<R_t$2, this gives $R_g<R<R_t$3. For $R_g<R<R_t$4, turbulence cannot adjust to the instantaneous $R_g<R<R_t$5, puff lifetimes revert to their steady-flow values at $R_g<R<R_t$6, and the threshold becomes essentially unaffected by pulsation, $R_g<R<R_t$7, depending on the survival criterion. Between these limits, roughly $R_g<R<R_t$8, the threshold drops steeply from the quasi-steady asymptote toward the steady-pipe value (Xu et al., 2017).

A different threshold proposal, due to Trinh, uses the normalized viscous-layer thickness $R_g<R<R_t$9. In that construction, the asymptotic value $R_g$0 is equated with the normalized pipe radius $R_g$1, leading to

$R_g$2

for Newtonian pipe flow (Trinh, 2010). This is a source-specific criterion rather than a consensus one, but it is representative of attempts to express transition in terms of near-wall elemental dynamics.

3. Geometry, perturbation structure, and constitutive effects

Outside the straight-pipe setting, transition flow is strongly shaped by geometry, perturbation form, curvature, interfacial physics, and constitutive response. The studies below show that the transitional state is often not fixed by a single global Reynolds number, but by a coupled structure of local instability, nonlinear interaction, and finite-amplitude pathway selection (Nguyen et al., 2018, Biau et al., 2010, Razzak et al., 2019, Masuda et al., 29 Aug 2025, Yatou, 2010).

System	Transition sequence or threshold	Dominant mechanism
Sudden-expansion pipe	$R_g$3; LS, US2, US1; hysteresis	Symmetry-broken recirculation and convective shear instability
Square duct	$R_g$4 optimals fail nonlinearly; $R_g$5 sub-optimals trigger	Quadratic generation of streamwise-mean distortion and edge dynamics
Wide-gap Taylor–Couette	ATVF for $R_g$6; transitional non-axisymmetry for $R_g$7	Outer-wall separation-region viscous-layer thickening
Liquid–liquid TCP flow	SF $R_g$8 DSF $R_g$9 transitional droplet flow $F_t \sim (R-R_g)^\beta$0 UBF $F_t \sim (R-R_g)^\beta$1 SBF	Interplay of $F_t \sim (R-R_g)^\beta$2, $F_t \sim (R-R_g)^\beta$3, and droplet inertia $F_t \sim (R-R_g)^\beta$4
Wavy viscoelastic channel	Convective $F_t \sim (R-R_g)^\beta$5 transition $F_t \sim (R-R_g)^\beta$6 elastic regimes	Competition between convective and elastic forcing

In the sudden-expansion pipe with expansion ratio $F_t \sim (R-R_g)^\beta$7, the minimum inlet-vortex amplitude required for sustained unsteady flow obeys the power law $F_t \sim (R-R_g)^\beta$8, with least-squares fit $F_t \sim (R-R_g)^\beta$9, $\xi \sim |R-R_g|^{-\nu_\perp}$0. A weak inlet vortex first generates a weak unsteady pattern in the recirculation bubble, and a new instability, denoted US2, emerges from the high-shear region produced by symmetry-broken recirculation. The disturbance peak convects downstream according to $\xi \sim |R-R_g|^{-\nu_\perp}$1 with $\xi \sim |R-R_g|^{-\nu_\perp}$2. The same system exhibits hysteresis: at $\xi \sim |R-R_g|^{-\nu_\perp}$3, one reported case gives $\xi \sim |R-R_g|^{-\nu_\perp}$4, $\xi \sim |R-R_g|^{-\nu_\perp}$5, $\xi \sim |R-R_g|^{-\nu_\perp}$6, and $\xi \sim |R-R_g|^{-\nu_\perp}$7 (Nguyen et al., 2018).

In square-duct flow, linear optimal perturbations take the form of longitudinal vortices, and at $\xi \sim |R-R_g|^{-\nu_\perp}$8 the global optimum at $\xi \sim |R-R_g|^{-\nu_\perp}$9 reaches $\tau \sim |R-R_g|^{-\nu_\parallel}$0 at $\tau \sim |R-R_g|^{-\nu_\parallel}$1. Yet these optimals decay in nonlinear DNS even for initial energy up to $\tau \sim |R-R_g|^{-\nu_\parallel}$2. By contrast, streamwise-modulated sub-optimal disturbances at $\tau \sim |R-R_g|^{-\nu_\parallel}$3 and $\tau \sim |R-R_g|^{-\nu_\parallel}$4 trigger transition for $\tau \sim |R-R_g|^{-\nu_\parallel}$5 of order $\tau \sim |R-R_g|^{-\nu_\parallel}$6, because quadratic interactions rapidly generate a streamwise-homogeneous distorted mean mode that becomes unstable. At the laminar–turbulent boundary the edge state is a self-sustained orbit with alternating pairs of large-scale vortices, and the reported threshold is $\tau \sim |R-R_g|^{-\nu_\parallel}$7 with minimal perturbation energy $\tau \sim |R-R_g|^{-\nu_\parallel}$8 for $\tau \sim |R-R_g|^{-\nu_\parallel}$9 (Biau et al., 2010).

In wide-gap Taylor–Couette flow at radius ratio $\beta \approx 0.58$0, the axisymmetric Taylor-vortex flow occupies $\beta \approx 0.58$1, a transitional non-axisymmetric regime appears for $\beta \approx 0.58$2, and wavy-vortex flow emerges beyond $\beta \approx 0.58$3. The proposed source of non-axisymmetric disturbance is a sudden increase in viscous-layer thickness in the outer-wall separation region. This localized thickening produces periodic secondary flow, raises the natural wavelength from $\beta \approx 0.58$4 to $\beta \approx 0.58$5 in the transition window, and reduces the dimensionless torque by as much as $\beta \approx 0.58$6 relative to axisymmetric Taylor-vortex flow (Razzak et al., 2019).

In liquid–liquid Taylor–Couette–Poiseuille flow with axial through-flow, the experimentally observed cascade is stratified flow, disturbed stratified flow, transitional droplet flow, unstable banded flow, and stable banded flow. The two principal transition lines in the $\beta \approx 0.58$7–$\beta \approx 0.58$8 plane obey power laws $\beta \approx 0.58$9 and $r$00. For TR I, $r$01 remains in the range $r$02 across glycerol concentration, while for TR II, $r$03 decreases from $r$04 to $r$05 with increasing viscosity. Stable banded flow is linked to loss of droplet inertia, and the UBF $r$06 SBF transition coincides with a critical $r$07 in the high-viscosity case (Masuda et al., 29 Aug 2025).

In a two-dimensional wavy-walled viscoelastic channel governed by the FENE-P model, Yatou identified three steady regimes—convective, transition, and elastic—in the $r$08–$r$09 plane. The first transition occurs when streamwise elastic forcing becomes comparable to convection. During this transition a separation vortex disappears, a jet induced by viscoelasticity moves into the bulk, viscous wall friction drops abruptly, and elastic wall friction rises sharply. Near the first transition boundary the stress scales as $r$10, and the critical line obeys $r$11 (Yatou, 2010).

4. Coherent structures, front dynamics, and reduced diagnostics

Transition flow is often diagnosed through coherent-structure analysis rather than solely through bulk statistics. In hypersonic axisymmetrical compression-ramp flow at Mach $r$12, the transition scenario was decomposed using Spectral Proper Orthogonal Decomposition and resolvent analysis about the mean flow. The fluctuation field satisfies

$r$13

and the SPOD and resolvent modes were compared across the attached boundary layer, mixing layer, and reattachment region. The observed transition starts with the linear amplification of oblique first modes over a broad frequency band, followed by nonlinear interaction of oblique modes into streamwise streaks, and then linear amplification and breakdown of those streaks near reattachment. At $r$14, the leading SPOD mode captures approximately $r$15 of the energy, while the corresponding resolvent and SPOD structures align with $r$16. The paper explicitly concludes that early nonlinear interaction is essential to the transition process (Lugrin et al., 2020).

A very different but equally structured example appears in quasi-two-dimensional bounded granular heap flow after a step change in feed rate. There, the transition between steady states is modeled through local mass conservation,

$r$17

and a linear flux–slope law,

$r$18

In deviation variables, the problem reduces to a diffusion equation,

$r$19

together with a moving boundary at the wedge front. The front position obeys

$r$20

the front speed decays as $r$21, and a second, slower transient follows once the front reaches the endwall. The transition flow is therefore represented as a diffusion-controlled moving-boundary problem rather than an instability cascade (Xiao et al., 2017).

These two cases use different mathematics—one spectral and non-normal, the other diffusive and kinematic—but both treat the transition flow as a finite-time organization of localized structures: oblique modes and streaks in the hypersonic boundary-layer interaction, and a propagating wedge in the granular heap (Lugrin et al., 2020, Xiao et al., 2017).

5. Pattern-forming, topological, and oscillatory transition flows

Transition flow also denotes regime change in systems where the relevant order parameter is not turbulence intensity alone. In thermal vibrational convection, direct numerical simulation in a square box revealed three regimes: periodic circulation, columnar, and columnar-broken. With dimensionless vibration amplitude $r$22, angular frequency $r$23, and fixed $r$24, the critical control parameter for the first transition is the vibrational Rayleigh number

$r$25

The onset of the columnar regime is reported at $r$26. Below this threshold, the flow is an oscillatory circulation; above it, thermal plumes merge into nearly stationary columns and mid-height velocity fluctuations decrease; at still higher forcing the columns break and the flow reorganizes into a large-scale circulation with a sudden increase in fluctuations (Guo et al., 2023).

In quasi-geostrophic channel flow, the phrase “transition flow” refers to time-periodic solutions born when a zonal jet loses stability through a Hopf bifurcation. The normal form on the center manifold is

$r$27

and the real part of the cubic coefficient,

$r$28

determines whether the transition is continuous (Type I, $r$29) or catastrophic (Type II, $r$30). Numerical evaluation for $r$31 and $r$32 yielded $r$33, typically $r$34, suggesting that catastrophic transition is preferred in the parameter region examined (Dijkstra et al., 2015).

In two-dimensional passive nematics, flow coupling changes the topology of the defect-mediated transition itself. Without flow alignment, the transition is consistent with the Berezinskii–Kosterlitz–Thouless scenario, with the standard defect-pair free-energy balance leading to $r$35. With hydrodynamics and $r$36, the defect-creation threshold is approximately $r$37; without flow coupling it is approximately $r$38. For strain-rate-aligning nematics with $r$39, bend–splay walls appear already at $r$40, the defect-creation threshold drops to $r$41, and once defects are created they remain unbound across the entire fluctuation range in both forward and backward protocols. In this sense, flow alignment suppresses the reversible binding–unbinding mechanism characteristic of BKT (Chattopadhyay et al., 9 Oct 2025).

In confluent epithelial monolayers, the flow generated by a T1 transition is a localized saddle. After rotating and averaging many events, Jain et al. obtained an ensemble-averaged velocity field with

$r$42

near the four-cell vertex, together with quadrupolar vorticity around the rearranging cells. The corresponding tissue fluidization is quantified through the mean separation $r$43 of initially neighboring cells, which satisfies

$r$44

with $r$45 cell-diameters per T1 for the chosen parameters, and the same linear law remains nearly unchanged under variations in deformability, rotational noise, adhesion, and active fraction (Jain et al., 2024).

6. Transition flow as a learned map in generative modeling

A conceptually distinct use of the term appears in diffusion- and flow-based generative modeling. In “Transition Flow Matching,” the transition flow is defined on a standard flow-matching coupling $r$46, $r$47, with interpolation

$r$48

Given a later time $r$49, the conditional transition state is

$r$50

and the transition flow is the conditional expectation

$r$51

By definition, $r$52, and the associated average velocity is

$r$53

This makes transition flow a global transport quantity rather than a local velocity field (Ma, 16 Mar 2026).

Training uses a neural approximation $r$54 and a conditional loss based on the computable linear-interpolation target $r$55. The total derivative is obtained by a Jacobian–vector product,

$r$56

and the standard Transition Flow Matching objective is

$r$57

When $r$58, the formulation reduces to mean-velocity flow, since

$r$59

The method supports one-step sampling $r$60 and few-step sampling on arbitrary time grids (Ma, 16 Mar 2026).

The empirical results reported in the paper place this abstract notion of transition flow in direct algorithmic use. On CIFAR-10 with a $r$61M-parameter UNet, the reported FIDs are $r$62 for $r$63, $r$64 for $r$65, $r$66 for $r$67, and $r$68 for $r$69. On latent ImageNet $r$70 with a $r$71M-parameter transformer, the one-step FID is $r$72 and the two-step FID is $r$73 (Ma, 16 Mar 2026).

Taken together, these literatures suggest that transition flow is not a single mechanism but a class of structured intermediate dynamics. A plausible implication is that the most useful descriptions are rarely purely global: they are localized in space, scale, or state space, and they emphasize puffs, bands, streaks, wedges, columns, defect walls, or explicit maps $r$74 as the natural objects that organize the transition.