Fully developed isotropic turbulence: Nonperturbative renormalization group formalism and fixed-point solution

Area of Science:

• Fluid Dynamics
• Statistical Physics
• Computational Physics

Background:

• Turbulence, governed by the Navier-Stokes (NS) equation, is a fundamental phenomenon in fluid dynamics.
• Understanding homogeneous and isotropic turbulence, especially with stochastic forcing, is crucial for many scientific and engineering applications.
• Existing theoretical frameworks often rely on approximations that may not fully capture turbulent behavior, necessitating advanced methods.

Purpose of the Study:

• To investigate fully developed homogeneous and isotropic turbulence within the Navier-Stokes equation framework.
• To apply the nonperturbative (functional) renormalization group (NPRG) method to analyze turbulent systems.
• To identify the underlying mechanisms for deviations from dimensional scaling laws and the emergence of intermittency.

Main Methods:

• Utilized the nonperturbative (functional) renormalization group (NPRG) to study the Navier-Stokes equation.
• Employed a symmetry-based approximation to obtain fixed-point solutions for the NPRG flow equations.
• Derived exact flow equations in the large wave-number limit to analyze deviations and scale invariance.

Main Results:

• Obtained a fixed-point solution corresponding to fully developed turbulence in both 2 and 3 dimensions.
• Observed deviations from established dimensional scalings (Kolmogorov in d=3, Kraichnan-Batchelor in d=2) for two-point functions.
• Demonstrated that the NPRG fixed point does not exhibit usual scale invariance, revealing a mechanism for intermittency.

Conclusions:

• The NPRG framework provides a robust method for studying Navier-Stokes turbulence beyond traditional approximations.
• The study identifies a key mechanism for the emergence of intermittency in turbulent flows through the breakdown of scale invariance.
• This work lays a detailed foundation for future NPRG investigations into turbulence, including the determination of intermittency exponents.