Structure function of passive scalars in two-dimensional turbulence.

Area of Science:

• Fluid Dynamics
• Turbulence
• Statistical Mechanics

Background:

• Passive scalar advection in turbulent flows is crucial for understanding mixing and transport phenomena.
• Characterizing the statistical properties of passive scalars, such as their structure functions, is key to turbulence theory.
• Previous studies have focused on 3D turbulence, with 2D cases presenting unique challenges and characteristics.

Purpose of the Study:

• To investigate the structure function of a passively advected scalar in two-dimensional (2D) turbulent flow.
• To derive a relationship between the passive scalar's graph fractal dimension and its scaling exponent.
• To analyze the influence of flow properties and driving parameters on scalar dispersion in 2D turbulence.

Main Methods:

• Analysis of the fractal dimension (delta1g) of the passive scalar graph.
• Derivation of relations involving the scalar structure function (D1(theta)r) and the velocity structure function (D2(r)).
• Application of mean-field approximations and models for the energy spectrum in different inertial subranges.

Main Results:

• A novel relation is established between the fractal dimension of the passive scalar graph and its scaling exponent in 2D turbulent flow.
• The 2D scalar structure function exhibits dependence on an additional parameter related to scalar driving, unlike in 3D.
• Specific scaling behaviors for the passive scalar graph are found in the enstrophy inertial subrange (delta1g<2 for intermediate/large Prandtl numbers) and energy inertial subrange (delta1g=5/3).

Conclusions:

• The study provides a theoretical framework for understanding passive scalar behavior in 2D turbulence.
• The derived relationships and observed scaling exponents offer new perspectives on scalar dispersion mechanisms.
• The findings are relevant for interpreting experimental observations of scalar dispersion in nonuniversal 2D flows.