A Mixed-Metric Two-Field Framework for Turbulence: Emergent Stress Anisotropy and Wall Asymptotics from a Single Scalar
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| Formato: | Preprint |
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2026
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| author | Sanchis-Agudo, Marcial Vinuesa, Ricardo |
| author_facet | Sanchis-Agudo, Marcial Vinuesa, Ricardo |
| contents | In our previous work~\cite{SanchisAgudoVinuesa2025PRL}, we argued that viscous dissipation in turbulence can be understood as the macroscopic imprint of microscopic path uncertainty, and showed that a kernel variance field $s(y)$ constrained by a balance condition yields both the Kolmogorov scales and the logarithmic law of the wall from a single stochastic principle. In the present work we promote $s$ to a dynamical field $s(\bm{x},t)$ with units of kinematic viscosity and develop a two-field framework in which the velocity $\ve$ and an \emph{intermittency} (or stochastic diffusivity) field $s$ evolve in a coupled way. The effective viscosity is $ν_{\mathrm{eff}}=ν_0+s$, but the stress tensor is generalized to include a non-linear closure driven by the commutator of strain and rotation, $[\bm{S}, \bmΩ]$, capturing emergent anisotropy. The evolution of $s$ is defined as a mixed-metric gradient flow: a Wasserstein-2 gradient flow for morphology, $\Div(s\grad s)$, combined with a local $L^2$ gradient flow driven by an objective coupling term $q$. The coupling is decomposed as $q=q_{\mathrm{prod}}-q_{\mathrm{relax}}$, where production is driven by a vortex-stretching invariant, $\mathcal{I} = \|\bm{S}\boldsymbolω\|^2$. This choice ensures that production vanishes identically in strictly two-dimensional flows. We show that, under standard assumptions of constant stress, high Reynolds number and overlap-layer scale invariance, the only scale-invariant overlap-layer solution of the mixed-metric equation is $s(y)\propto y$, which recovers the logarithmic velocity profile. Thus the same mixed-metric equation organizes both wall-resolved and wall-modeled asymptotics within a single, energetically constrained framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Mixed-Metric Two-Field Framework for Turbulence: Emergent Stress Anisotropy and Wall Asymptotics from a Single Scalar Sanchis-Agudo, Marcial Vinuesa, Ricardo Fluid Dynamics In our previous work~\cite{SanchisAgudoVinuesa2025PRL}, we argued that viscous dissipation in turbulence can be understood as the macroscopic imprint of microscopic path uncertainty, and showed that a kernel variance field $s(y)$ constrained by a balance condition yields both the Kolmogorov scales and the logarithmic law of the wall from a single stochastic principle. In the present work we promote $s$ to a dynamical field $s(\bm{x},t)$ with units of kinematic viscosity and develop a two-field framework in which the velocity $\ve$ and an \emph{intermittency} (or stochastic diffusivity) field $s$ evolve in a coupled way. The effective viscosity is $ν_{\mathrm{eff}}=ν_0+s$, but the stress tensor is generalized to include a non-linear closure driven by the commutator of strain and rotation, $[\bm{S}, \bmΩ]$, capturing emergent anisotropy. The evolution of $s$ is defined as a mixed-metric gradient flow: a Wasserstein-2 gradient flow for morphology, $\Div(s\grad s)$, combined with a local $L^2$ gradient flow driven by an objective coupling term $q$. The coupling is decomposed as $q=q_{\mathrm{prod}}-q_{\mathrm{relax}}$, where production is driven by a vortex-stretching invariant, $\mathcal{I} = \|\bm{S}\boldsymbolω\|^2$. This choice ensures that production vanishes identically in strictly two-dimensional flows. We show that, under standard assumptions of constant stress, high Reynolds number and overlap-layer scale invariance, the only scale-invariant overlap-layer solution of the mixed-metric equation is $s(y)\propto y$, which recovers the logarithmic velocity profile. Thus the same mixed-metric equation organizes both wall-resolved and wall-modeled asymptotics within a single, energetically constrained framework. |
| title | A Mixed-Metric Two-Field Framework for Turbulence: Emergent Stress Anisotropy and Wall Asymptotics from a Single Scalar |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2601.03314 |