Two-loop calculation of the turbulent Prandtl number

Area of Science:

• Fluid Dynamics
• Turbulence Theory

Background:

• The turbulent Prandtl number is crucial for understanding heat and mass transfer in turbulent flows.
• Previous calculations often relied on approximations that limited accuracy.

Purpose of the Study:

• To calculate the turbulent Prandtl number using a more advanced theoretical approach.
• To explain the discrepancy between theoretical predictions and experimental results in turbulence.

Main Methods:

• Utilized the two-loop approximation within the epsilon expansion of stochastic turbulence theory.
• Analyzed the behavior of divergences in two-dimensional systems.

Main Results:

• The two-loop correction to the turbulent Prandtl number was found to be surprisingly small.
• This small correction reconciles theoretical predictions with experimental data, unlike other turbulence quantities.
• Identified mutual cancellation of divergences in two dimensions as the key factor.

Conclusions:

• The two-loop approximation provides a more accurate prediction for the turbulent Prandtl number.
• The unique behavior in two dimensions is critical for understanding turbulence theory results.
• This study offers insights into the limitations and successes of current turbulence models.