Tailor-Designed Models for the Turbulent Velocity Gradient through Normalizing Flow.

Area of Science:

• Fluid Dynamics
• Turbulence Modeling
• Computational Physics

Background:

• Small-scale turbulence is complex, often modeled using stochastic equations.
• The accuracy of traditional models relies heavily on assumptions for pressure and viscous forces.
• Velocity gradients offer a promising basis for low-dimensional turbulence models.

Purpose of the Study:

• To develop a data-driven machine learning model for velocity gradients in turbulence.
• To capture the statistical properties of turbulence directly from data.
• To create a dynamical system that replicates turbulence statistics and time correlations.

Main Methods:

• Utilized normalizing flows to learn the probability density function (PDF) of velocity gradients from direct numerical simulations (DNS).
• Constructed a deterministic dynamical system based on the learned PDF.
• Optimized model time correlations using gauge terms to match DNS data.

Main Results:

• The machine learning model successfully learned the velocity gradient PDF.
• The resulting dynamical system exhibited the desired steady-state PDF.
• Model-generated time series closely resembled DNS data in statistical properties.

Conclusions:

• A data-driven approach using machine learning provides a powerful alternative for turbulence modeling.
• Normalizing flows are effective for learning complex probability distributions in fluid dynamics.
• The developed model demonstrates high fidelity in capturing turbulent flow statistics and dynamics.