Mean Flow Formulations in Fluid Dynamics and Geometry

Updated 7 November 2025

Mean flow formulations are mathematical techniques that compute averaged velocity fields in systems characterized by fluctuations.
They employ Eulerian, Lagrangian, and geometric averaging methods to address turbulence closure and capture nonlocal nonlinear interactions.
Their applications span turbulence theory, geometric evolution, generative modeling, and nonlinear wave analysis, offering theoretical and computational insights.

Mean flow formulations encompass a spectrum of mathematical frameworks and techniques aimed at describing the averaged or macroscopic structure of fields—most prominently velocity fields—in systems where microscopic, fluctuating, or wave-like processes are present. These arise extensively in fluid mechanics, geometric analysis, turbulence theory, and generative modeling, with each discipline deploying rigorous mathematical tools to define, analyze, and compute the "mean flow" or "mean velocity" under the constraints of the system dynamics, ensemble averaging, and nonlinear interactions.

1. Conceptual Foundations of Mean Flow Formulation

The mean flow is not a monolithic concept but refers to an averaged field extracted from systems exhibiting fluctuations, oscillations, or underlying stochasticity. In classical fluid mechanics and geometric evolution, the mean flow can be constructed as:

Eulerian mean (Reynolds average): Time or ensemble average at a fixed spatial location, central in turbulence theory.
Lagrangian mean (GLM theory): Average over particle-following trajectories, critical for wave–mean interactions in geophysical contexts.
Geometric mean flow: Defined intrinsically using differential geometric constructs, such as the mean of a set of flow maps (diffeomorphisms) on a manifold (Gilbert et al., 2016, Gilbert et al., 7 May 2024).
Average velocity in generative modeling: Net displacement between states, providing a foundation for rapid sampling in neural ODE flows (Geng et al., 19 May 2025).

The precise definition, existence, and properties of the mean flow depend on the physical context (e.g., compressibility, domain geometry), the averaging procedure (statistical, phase, or coordinate-free), and the intended analytical or computational application.

2. Statistical Mechanics, Closure, and Turbulent Mean Velocity Equations

In turbulence theory, mean velocity formulations address the closure problem induced by nonlinear (quadratic) terms in the Navier-Stokes equations. Statistical mechanics-based approaches start from the N-particle dynamics, employ projection-operator methods, and yield exact evolution equations for conserved densities. For turbulent flows, the mean velocity equation derived via the Zwanzig-Mori formalism takes the form:

$\rho \left( \frac{\partial \mathbf{u}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = - \nabla P + \nabla \cdot \mathbf{R}$

with $\mathbf{R}$ a nonlocal, nonlinear convolution involving high-order time correlation functions (Piest, 2013). This formulation provides explicit closure at third and fourth order via multilinear mode coupling theory:

The friction force includes terms of up to sixth order in velocity gradients, implying fundamentally nonlocal dissipation.
Turbulent dissipation is systematically expanded and expressed as convolution integrals, distinguishing it from empirical (e.g., eddy viscosity) closures.
Numerical evaluation demonstrates that non-linear dissipation terms materialize as mean flow corrections and flow reversal beyond certain Reynolds number thresholds (Piest, 2016).

This statistically grounded approach clarifies how the mean flow departs from the laminar predictions and how universal closure can, in principle, be achieved via analytic correlation functions.

3. Geometric and Lagrangian Mean Flow in Fluid Dynamics

Lagrangian mean theories—rooted in the Generalised Lagrangian Mean (GLM) framework—provide essential tools for mass, momentum, and wave action transport in fluctuating fluid systems (Gilbert et al., 7 May 2024, Gilbert et al., 2016). Key geometric advancements rectify foundational issues:

Intrinsic geometric generalisation: Lagrangian averaging is formulated using pull-back operations on tensor fields (momentum one-form, buoyancy), replacing ill-defined coordinate averages. The mean is the diffeomorphic representative within the group of volume-preserving maps ( $\mathrm{SDiff}(M)$ ) (Gilbert et al., 2016).
Multiple definitions of mean flow: Mean flow maps can be chosen via extended GLM (Euclidean average), optimal transportation (volume preservation), geodesic minimization, or the glm framework (generator with zero mean and divergence-free) (Gilbert et al., 2016).
Mean velocity distinguishes itself from Lagrangian mean velocity: The geometric mean velocity, defined as the velocity of the mean flow map, is generally not equal to the pull-back mean of velocities—this subtlety ensures compatibility with conservation laws and correct treatment of divergence in incompressible fluids.

Explicit leading-order expansions in small-amplitude regimes quantify the mean flow and pseudomomentum, providing the foundations for Kelvin's circulation and wave-action conservation in general geometries. Geometric GLM eliminates the "divergence effect," guaranteeing physically sensible mean flows for incompressible fluids and facilitating application on manifolds such as the sphere.

4. Mean Flow Analysis in Periodic and Statistically Steady Flows

Resolvent analysis and input-output frameworks are deployed to interrogate the linear response properties of statistically steady or oscillatory flows (Leclercq et al., 2022):

Mean resolvent operator ($\mathsfbi{R}_0$): Defined via ensemble averaging over the unsteady base flow, $\mathsfbi{R}_0$ provides the unique linear time-invariant (LTI) approximation of the input–output dynamics for a broad class of flows—periodic, quasiperiodic, chaotic, or stochastic.
Poles and Floquet/Koopman connection: The poles of $\mathsfbi{R}_0$ are the Floquet exponents (for periodic flows), precisely capturing marginally stable modes at harmonic frequencies.
Validation and approximation: In regimes of weak base flow unsteadiness, the classical mean-flow resolvent approximates $\mathsfbi{R}_0$ up to second order, justifying its use in flow control and system identification. Numerical tests in canonical flows support this transferability.

This operator generalizes mean-flow analysis well beyond classical assumptions, enabling rigorous control-oriented modeling even in the absence of steady invariant states.

5. Mean Flow Formulation in Geometric Evolution Equations

Geometric flows, notably mean curvature flows (MCF) and their Lagrangian variants, employ the mean curvature vector as the generator of the flow:

$\partial_t x = \vec{H}(x)$

Developments in the analysis of MCF and related flows:

Singularity formation and regularity: The structure theory for mean convex flows and flow with surgery for two-convex hypersurfaces characterize the topology and partial regularity of evolving surfaces (Haslhofer, 2014). Results such as the preservation of the noncollapsing $\alpha$ -Andrews condition and canonical neighborhood theorems enable a full description of singular set geometries and facilitate topological simplification of manifolds under flow.
Lagrangian mean curvature flow (LMCF): For Lagrangian submanifolds in $\mathbb{C}^2$ , the zero Maslov class constraint and uniformly bounded mean curvature yield rigidity. Notably, the tangent flow at singularities is unique, indicating independence of the blow-up limit from the chosen rescaling sequence (Ghosh, 29 Mar 2025).
Modified mean curvature flows: Variants, such as conformalized MCF, remove anisotropic singularities and stabilize numerical schemes for surface evolution, ensuring non-singular convergence to canonical geometries (e.g., spherical maps for genus-zero surfaces) (Kazhdan et al., 2012).

These geometric mean flow formulations support the regularization, structure theory, and computational tractability of flow-induced surface and manifold evolution problems.

6. Mean Flow in Nonlinear Wave Equations and Generative Modeling

In contexts where the macroscopic mean arises from underlying wave fields or data-generative mechanisms:

High-order nonlinear Schrödinger (NLS) equations: Mean flow terms are explicitly computed as nonlocal convolution integrals of wave envelope derivatives (spatial or temporal Hilbert transforms), essential for accurately reconstructing Lagrangian particle paths, pollutant dispersion, and wave–current interactions across depth regimes (Gomel et al., 2023).
One-step generative modeling: The MeanFlow approach defines the average velocity as the time-integrated net displacement between two points on a trajectory, distinguished from instantaneous velocity. The rigorous MeanFlow Identity relates average and instantaneous velocities, enabling principled neural network training for single-pass generation at state-of-the-art speeds and quality (Geng et al., 19 May 2025).

These mean flow formulations allow physically or algorithmically correct macroscopic predictions from highly nonlinear or data-driven microscopic dynamics.

7. Practical, Computational, and Theoretical Implications

Mean flow formulations underpin rigorous theoretical analysis, provide closure in turbulence modeling, guarantee geometric consistency in complex domains, and yield robust computational algorithms across disciplines. Their implementation depends on:

The choice of averaging (Eulerian, Lagrangian, geometric, phase, ensemble), often constrained by the underlying physics or application.
Resolution of closure and regularity problems via high-order expansions, mode coupling theory, or geometric surgery.
Nonlocal and nonlinear mean flow terms introducing new qualitative and quantitative behaviors beyond leading-order dynamics.

Future research directions involve systematic extensions to compressible and magnetohydrodynamic fluids, quantification of mean flow effects in machine-learned models, and deeper integration with topological data analysis and manifold learning.

Summary Table: Mean Flow Formulations Across Domains

Domain	Mean Flow Formulation	Key Properties and Challenges
Turbulence/statistical	Statistical mechanics mean velocity (Piest, 2013)	Systematic closure, nonlocal convolution
Geometric analysis	Lagrangian/geometric mean (GLM, glm) (Gilbert et al., 2016, Gilbert et al., 7 May 2024)	Intrinsic, divergence-free, frame-invariant
Control/input-output	Mean resolvent operator (Leclercq et al., 2022)	Optimal LTI fit, Floquet poles, generalizable
Wave transport/NLS	Nonlocal mean flow (convolution/Hilbert) (Gomel et al., 2023)	Depth-robust, accurate Lagrangian paths
Generative modeling	Average velocity/MeanFlow Identity (Geng et al., 19 May 2025)	Coarse-grained training, fast sampling
Geometric flows	MCF, LMCF, modified MCF (Haslhofer, 2014, Kazhdan et al., 2012, Ghosh, 29 Mar 2025)	Regularity, singularity structure, uniqueness

Mean flow formulations are foundational in modern applied mathematics, providing deep connections between microstructure, macrodynamics, geometric invariants, and computational modeling in nonlinear, fluctuating, and geometric systems.