Geometric solution of turbulent mixing

Area of Science:

• Fluid Dynamics
• Statistical Physics
• Turbulence Theory

Background:

• Passive scalar transport in turbulence is crucial for understanding mixing processes.
• Previous models often rely on scaling laws, which may not capture complex geometric structures.
• High Reynolds number turbulence presents significant challenges for analytical and numerical studies.

Purpose of the Study:

• To derive an analytic solution for passive scalar density in decaying homogeneous turbulence.
• To describe the geometric structure of scalar distribution under specific turbulence conditions.
• To investigate the impact of finite diffusivity and forcing on the derived structure.

Main Methods:

• Utilizing the Euler ensemble for velocity statistics, derived from Navier-Stokes loop equations.
• Formulating the scalar advection-diffusion problem as a closed linear loop equation.
• Solving the loop equation analytically to obtain the scalar density.

Main Results:

• An analytic solution for passive scalar density was derived.
• The solution exhibits a unique structure of expanding concentric shells with piecewise parabolic profiles.
• This shell structure is supported at discrete radii and influenced by Euler totients, differing from conventional scaling.
• Finite diffusivity or forcing smooths discontinuities but preserves the overall geometry.

Conclusions:

• The study provides a novel geometric description of scalar transport in decaying turbulence.
• The findings are potentially applicable to astrophysical and quantum fluid regimes with weak dissipation.
• The statistical signature of the shell structure is observable through volume-averaged scalar density.