Fractional Closure Models for Turbulence

• Fractional closure models are nonlocal turbulence models employing power-law operators, such as the fractional Laplacian, derived from kinetic theory to capture long-range interactions.
• They leverage isotropic α-stable Lévy laws and variable-order derivatives to unify inertial-range scaling with wall turbulence behavior under a single framework.
• Calibration with DNS and experiments validates these models by accurately replicating energy spectra, non-Gaussian tails, and two-point correlation dynamics.

Fractional closure models for turbulence introduce nonlocal, power-law–type operators—typically variants of the fractional Laplacian or variable-order fractional derivatives—into the closure of the Reynolds-averaged Navier-Stokes (RANS) equations or large-eddy simulation (LES) frameworks. These models are rigorously derived from kinetic theory or filtered Boltzmann equations, often using Lévy α-stable statistics to reflect the heavy-tailed, non-Gaussian, and long-range correlated nature of turbulent fluctuations. Unlike local eddy-viscosity or diffusive closures, fractional models encode both enhanced turbulent transport across scales and nonlocality inherent to turbulent flows, unifying classical laws and inertial-range phenomenology under a single framework.

1. Theoretical Foundations: From Kinetic Theory to Fractional Closures

Fractional closure models originate with the filtered Boltzmann equation (BTE) or BGK-type kinetic formulations, where the single-particle equilibrium distribution $f^{eq}_\alpha$ is modeled by an isotropic α-stable Lévy law (Epps et al., 2018, Samiee et al., 2019, Akhavan-Safaei et al., 2020). The distinctive characteristic function scaling as $\exp(-|k|^\alpha)$ leads to heavy tails ($0<\alpha<2$), permitting rare, large velocity excursions mimicking turbulent eddies. Incorporating this statistical ansatz into the BTE and passing to macroscopic (ensemble- or spatially-averaged) moments yields an additional nonlocal integral contribution to the stress tensor: $$\rho \frac{D\bar{\mathbf u}}{D t} = -\nabla p + \mu_\alpha\,\nabla^2\bar{\mathbf u} + \rho\,C_\alpha \int_{\mathbb{R}^d} \frac{\bar{\mathbf u}(\mathbf{x}') - \bar{\mathbf u}(\mathbf{x})}{|\mathbf{x}'-\mathbf{x}|^{\alpha+d}}\,d\mathbf{x}'$$ where the final term is the singular-integral representation of the fractional Laplacian, $(−\Delta)^{\alpha/2}\bar{\mathbf u}$ (Epps et al., 2018, Samiee et al., 2019). The coefficient $C_\alpha$ and enhanced viscosity $\mu_\alpha$ are linked to kinetic parameters and moments of the equilibrium distribution.

Variable-order generalizations—where the fractional order $\alpha(x)$ varies in space—are constructed to reflect the spatially inhomogeneous nonlocality in wall-bounded or stratified turbulence (Song et al., 2018, Mehta, 2023). In these, the Reynolds stress closure is replaced by a variable-order Caputo derivative, further discussed below.

2. Mathematical Form and Properties of the Fractional Operators

The core mathematical objects are the fractional Laplacian and variable-order fractional derivatives:

• Spectral definition: $$(−\Delta)^{\alpha} u(x) = \mathcal{F}^{−1}[\,|\xi|^{2\alpha}\,\mathcal{F}u(\xi)\,](x)$$.
• Singular integral (Riesz) form:

$$(−\Delta)^\alpha u(x) = C_{d,\alpha}\,PV\int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+2\alpha}}\;dy,\quad 0<\alpha\leq1$$

with $$C_{d,\alpha}=2^{2\alpha}\Gamma(\alpha+d/2)/(\pi^{d/2}\Gamma(-\alpha))$$ (Samiee et al., 2019, Akhavan-Safaei et al., 2020).

For wall turbulence, variable-order Caputo derivatives are used: $$D_y^{\alpha(y)}U(y) = \frac{1}{\Gamma(1-\alpha(y))}\int_0^y (y-\tau)^{-\alpha(y)}\frac{dU}{d\tau}(\tau)d\tau$$ which allows $\alpha(y)\to1$ close to the wall but decays to smaller values in the bulk (Song et al., 2018). Two-sided models (symmetric Caputo or Riesz-type) are essential for flows where nonlocality aggregates from both boundaries (Mehta, 2023).

Tempered fractional operators introduce exponential (or sharp) cutoffs to the power-law kernel, yielding finite-variance statistics and improved physical behavior in unbounded domains (Samiee et al., 2021, Mehta, 2023). The tempered fractional Laplacian is defined as: $$(\Delta+\lambda)^\alpha u(\mathbf{x}) = C_{d,\alpha}\;PV\int_{\mathbb{R}^d}\frac{u(\mathbf{x})-u(\mathbf{y})}{e^{\lambda|\mathbf{x}-\mathbf{y}|}|\mathbf{x}-\mathbf{y}|^{d+2\alpha}}\,d\mathbf{y}$$

3. Physical Interpretation and Modeling Universality

Fractional closure models directly encode turbulent superdiffusion, backscatter, and nonlocality:

• α = 2 recovers classical Navier-Stokes with local, Gaussian statistics (Epps et al., 2018).
• α = 1 (Cauchy) produces the law-of-the-wall/logarithmic profiles and manifests enhanced wall-driven superdiffusion (Epps et al., 2018).
• α = 2/3 is linked (via Richardson dispersion) to inertial-range scaling and $r^2\sim t^3$ particle pair superdiffusion (Gunzburger et al., 2016, Hannani et al., 3 Aug 2025).
• Variable $\alpha(y)$ in wall turbulence is found to be universal: DNS and experiments demonstrate collapse of $\alpha^*(y^+)$ across wide Reynolds numbers and flow types (channel, Couette, pipe) (Song et al., 2018).

Monotonic decay of $\alpha$ away from walls is interpreted as a diagnosis of increasingly nonlocal, energy-containing eddy interactions—quantifying the effective range/memory of turbulent transport (Song et al., 2018).

4. Calibration, Computational Implementation, and Validation

Model coefficients (fractional order $\alpha$, diffusion prefactors, tempering parameter $\lambda$) are inferred:

• Data-driven fitting from DNS: Optimal $\alpha^{opt}$ is selected to maximize two-point correlation between modeled and DNS SGS forces, with regression slope close to unity; regression or neural network approaches are used for spatially varying parameters (Samiee et al., 2019, Mehta, 2023, Song et al., 2018).
• A priori & a posteriori validation: Model predictions are tested against PDF tails of SGS dissipation, two-point correlations, mean velocity profiles, and energy spectra versus DNS and experiments (including the Princeton superpipe) (Akhavan-Safaei et al., 2020, Song et al., 2018, Samiee et al., 2021).
• Numerical schemes: Rational approximations, eigen-decomposition of Laplacian matrices, and IMEX modular time-stepping facilitate integration of fractional operators with minimal intrusion to legacy codes; error analysis and stability proofs are provided (Gunzburger et al., 2016).

Modeling key aspects:

Model Type	Operator Form	Primary Calibration Variable
Fractional Laplacian (SGS/RANS)	$(-\Delta)^{\alpha}$	$\alpha$ (fractional order)
Variable-order Caputo (Wall)	$D_{y}^{\alpha(y)}$	$\alpha(y)$ (from DNS)
Tempered fractional	$(\Delta+\lambda)^{\alpha}$	$\alpha,\;\lambda$

Tempered and truncated models further allow specifying finite interaction horizons or exponential cutoffs to regularize extreme events and ensure convergence of all moments—physically modeling the transition to dissipative, small scales (Mehta, 2023, Samiee et al., 2021).

5. Extension to Passive Scalars, Synthetic Turbulence, and Stochastic FPDEs

Fractional closures have been extended to model subgrid-scale passive scalar fluxes (Akhavan-Safaei et al., 2020) and to generate stochastic ensembles of velocity fields:

• SGS scalar flux: Closure of filtered Boltzmann transport via a power-law (α-stable) equilibrium yields fractional Laplacian terms in the scalar flux, $$q^R \sim -C_\alpha(-\Delta)^{\alpha/2}\widetilde{\Phi}$$, with parameters inferred by matching two-point correlations and PDFs of scalar dissipation (Akhavan-Safaei et al., 2020).
• Stochastic FPDE models: Linear fractional PDEs driven by Gaussian noise yield velocity fields of Matérn/tuned covariance, with spectral exponent directly linked to the fractional order; wall effects are enforced via mixed Dirichlet/Robin BCs on a vector potential (Keith et al., 2020). This approach captures prescribed two-point/turbulent statistics but neglects intermittency and mean-flow feedback.

6. Impact, Limitations, and Future Directions

Fractional closure models introduce several advances:

• Unified multiscale modeling: By tuning only the order $\alpha$ (and, if present, tempering), these models interpolate between local diffusion, inertial-range, and superdiffusive/memory effects without introducing extraneous empirical constants (Epps et al., 2018, Song et al., 2018, Mehta, 2023).
• Improved statistical fidelity: Fractional models better match non-Gaussian tails in PDFs, long-range spatial and temporal correlations, and energy spectra across ranges, especially at moderate-to-large filter widths (Samiee et al., 2019, Akhavan-Safaei et al., 2020, Samiee et al., 2021).
• Open challenges:
  • Rigorous enforcement of solid-wall boundary conditions for global (Riesz) fractional operators remains unresolved; truncation, blending, or horizon strategies are under development (Hannani et al., 3 Aug 2025, Mehta, 2023).
  • Hybridization with local LES/RANS is an open research direction; reconciling $\alpha$ with explicit filter-widths and grid resolutions requires further paper.
  • Calibration of fractional orders across flow regimes and development of robust, efficient numerical solvers (e.g., preconditioned Lanczos, domain truncation) are active topics (Gunzburger et al., 2016, Hannani et al., 3 Aug 2025).
  • Gaussianity limitations in some stochastic FPDE formulations: fractional linear models underpredict intermittency and cannot capture third- or higher-order velocity-increment statistics (Keith et al., 2020).

7. Summary Table of Major Model Classes and Distinctions

Reference	Closure Operator	Calibrated Parameter(s)	Application Focus
(Epps et al., 2018)	Fractional Laplacian $(−\Delta)^{\alpha/2}$	$\alpha$	RANS momentum, law of the wall
(Song et al., 2018)	Caputo $(D_y^{\alpha(y)})$	$\alpha(y)$ (universal fit)	Channel/pipe/Couette, wall turbulence
(Samiee et al., 2019)	Fractional Laplacian SGS	$\alpha(\mathcal L, Re_\lambda)$	LES SGS closure H.I.T.
(Mehta, 2023)	Variable-order Caputo (one-/two-sided)	$\alpha(y^+)$	RANS wall flows, truncated/tempered
(Samiee et al., 2021)	Tempered Laplacian $(\Delta+\lambda)^\alpha$	$\alpha,\;\lambda$	LES, unbounded domains
(Keith et al., 2020)	Fractional FPDE (stochastic)	$\alpha_1,\alpha_2,L_i(z)$	Synthetic turbulence generation
(Gunzburger et al., 2016)	Fractional Laplacian + Modular Solver	$\alpha$	Richardson, energy spectrum, numerics
(Hannani et al., 3 Aug 2025)	Riesz/Caputo $(−\Delta)^{1/3}$, time-fractional	$\alpha, \beta$	Non-Markovian NSE, spectral DNS

Fractional closure models thus provide a physically-derived, mathematically consistent, and data-validated nonlocal generalization of traditional turbulence closures, offering improved representation of multiscale transport, non-Gaussian statistics, and scaling laws, while presenting new challenges in parameterization, boundary treatment, and computational feasibility.