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Updated: Aug 8, 2026
11:03
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Generalized flows, intrinsic stochasticity, and turbulent transport
1Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
Summary
This study explores passive scalar transport in turbulent flows, revealing how different regularizations affect outcomes. It establishes principles of uniqueness and self-similarity in various compressibility regimes.
Area of Science:
- Fluid Dynamics
- Turbulence Theory
- Statistical Mechanics
Background:
- Passive scalar transport in turbulent velocity fields presents challenges due to non-uniqueness of solutions.
- Generalized flows arise from probability distributions of solutions to associated ODEs.
- Regularization techniques are crucial for analyzing ill-posed problems in fluid dynamics.
Purpose of the Study:
- To investigate passive scalar transport in turbulent velocity fields using two natural regularization methods.
- To examine phenomena arising from non-spatially regular velocity fields, specifically white-in-time random fields.
- To analyze the impact of compressibility on the behavior of passive scalars.
Main Methods:
- Regularization via molecular diffusion and velocity field smoothing.
- Analysis of white-in-time random velocity fields.
- Isolation of three compressibility regimes and examination of scaling behaviors.
Main Results:
- Two distinct scaling behaviors for passive scalar structure functions emerge in the intermediate compressibility regime, dependent on the Prandtl number.
- In weakly and strongly compressible regimes, regularizations yield identical generalized flows.
- The 'one force, one solution' principle is established for scalar fields and their differences in specific regimes.
- Existence and uniqueness of invariant measures are proven under suitable forcing.
- Incomplete self-similarity is demonstrated.
Conclusions:
- The choice of regularization impacts passive scalar transport scaling in intermediate compressibility.
- Fundamental principles of uniqueness and invariant measures hold in specific turbulent flow regimes.
- The study provides insights into the complex dynamics of passive scalars in turbulent environments.
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