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A Geometric Foundation for the Universal Laws of Turbulence

Marcial Sanchis-Agudo, Ricardo Vinuesa

TL;DR

The paper reframes turbulence as an ensemble of stochastic trajectories governed by a Schrödinger Bridge variational principle, tying viscous dissipation to microscopic path uncertainty. It derives a diffusion-based macroscopic momentum balance with $D = u + s(x)$ and identifies the SB current velocity with the mean flow, enabling a physically grounded derivation of key scaling laws. The first consequence yields the Kolmogorov diffusion horizon with $ au_ ext{η} = ( u/ ext{ε} )^{1/2} $ and $ η \,= \, ( u^3/ ext{ε} )^{1/4} $, while the second shows the universal law of the wall $ U^+(y^+) = rac{1}{oldsymbol{ ext{kappa}}} ext{ln} y^+ + B $, arising from a linearly growing diffusivity $ s(y) \,\propto\, y $ and stationary-state constraints. Finite-$Re$ corrections are interpreted as geometry- and boundary-condition–driven terms $ O(y/oldsymbol{ extdelta}) $. The framework provides a coherent, physically motivated closure unifying bulk and wall scaling and offers a principled explanation for observed deviations from universality.

Abstract

We propose a theoretical framework where the dissipative structures of turbulence emerge from microscopic path uncertainty. By modeling fluid parcels as stochastic tracers governed by the Schrödinger Bridge (SB) variational principle, we demonstrate that the Navier--Stokes viscous term is a natural linear, second-order macroscopic operator consistent with isotropic microscopic diffusion. We derive two foundational pillars of turbulence from this single principle. First, we show that the Kolmogorov scale $η\sim (ν^3/ε)^{1/4}$ is not merely a dimensional necessity but a geometric diffusion horizon: it is the scale at which the kinetic energy of a fractal trajectory, scaling as $k \sim ν/τ$, balances the macroscopic dissipation rate. Second, we show that the universal law of the wall is the stationary solution to this stochastic process under no-slip constraints. The logarithmic mean profile arises from the scale invariance of the turbulent diffusivity, while finite-Reynolds-number corrections emerge as controlled asymptotic expansions of the stochastic variance. This framework offers a physically grounded derivation of turbulent scaling laws that complements and extends purely phenomenological dimensional analysis.

A Geometric Foundation for the Universal Laws of Turbulence

TL;DR

The paper reframes turbulence as an ensemble of stochastic trajectories governed by a Schrödinger Bridge variational principle, tying viscous dissipation to microscopic path uncertainty. It derives a diffusion-based macroscopic momentum balance with and identifies the SB current velocity with the mean flow, enabling a physically grounded derivation of key scaling laws. The first consequence yields the Kolmogorov diffusion horizon with and , while the second shows the universal law of the wall , arising from a linearly growing diffusivity and stationary-state constraints. Finite- corrections are interpreted as geometry- and boundary-condition–driven terms . The framework provides a coherent, physically motivated closure unifying bulk and wall scaling and offers a principled explanation for observed deviations from universality.

Abstract

We propose a theoretical framework where the dissipative structures of turbulence emerge from microscopic path uncertainty. By modeling fluid parcels as stochastic tracers governed by the Schrödinger Bridge (SB) variational principle, we demonstrate that the Navier--Stokes viscous term is a natural linear, second-order macroscopic operator consistent with isotropic microscopic diffusion. We derive two foundational pillars of turbulence from this single principle. First, we show that the Kolmogorov scale is not merely a dimensional necessity but a geometric diffusion horizon: it is the scale at which the kinetic energy of a fractal trajectory, scaling as , balances the macroscopic dissipation rate. Second, we show that the universal law of the wall is the stationary solution to this stochastic process under no-slip constraints. The logarithmic mean profile arises from the scale invariance of the turbulent diffusivity, while finite-Reynolds-number corrections emerge as controlled asymptotic expansions of the stochastic variance. This framework offers a physically grounded derivation of turbulent scaling laws that complements and extends purely phenomenological dimensional analysis.

Paper Structure

This paper contains 13 sections, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Variational Principle: Viscosity from Uncertainty
  3. Microscopic Stochastic Model
  4. Schrödinger Bridge and macroscopic dissipation
  5. First Consequence: The Kolmogorov Scale
  6. Energy Balance of a Fractal Path
  7. Scale Selection and the Kolmogorov Time
  8. The Diffusion-Length Law
  9. Second Consequence: The Universal Law of the Wall
  10. Scale Invariance and the Logarithmic Law
  11. Asymptotic Corrections
  12. Theoretical Interpretation of the Universality Debate
  13. Conclusions

Figures (2)

  • Figure 1: The geometric origin of viscosity. (a) In an ideal fluid, parcels follow geodesics (paths of least action). (b) In a turbulent fluid, microscopic path uncertainty forces a stochastic trajectory. The macroscopic slowness relative to the geodesic time defines the effective viscosity, and the entropic cost of this deviation is the dissipation.
  • Figure 2: The Law of the Wall as a geometric fixed point. The stochastic diffusivity (red paths) scales linearly with wall distance, $s(y)\propto y$, due to scale invariance. The mean velocity profile (blue dashed line) that satisfies the stationarity condition for this expanding variance is necessarily logarithmic, $U \propto \ln y$. The faded reflection illustrates the Dirichlet image kernel enforcing the no-slip condition.