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Variable-Order Fractional Models for Wall-Bounded Turbulent Flows.

Fangying Song1, George Em Karniadakis2

  • 1College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China.

Entropy (Basel, Switzerland)
|July 2, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces a novel fractional calculus approach to model turbulent flow velocity profiles across all regions. The new model reveals universal properties of turbulent diffusion and nonlocality, improving upon existing methods.

Keywords:
fractional PINNfractional conservations lawsphysics-informed learningturbulent flowsvariable fractional model

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Area of Science:

  • Fluid Dynamics
  • Turbulence Modeling
  • Fractional Calculus

Background:

  • Wall-bounded turbulent flow modeling remains a significant challenge in physics.
  • Current models like the log law only capture intermediate flow regions.
  • Progress in accurately describing the entire velocity profile has been slow.

Purpose of the Study:

  • To develop a new model for the entire mean velocity profile in wall-bounded turbulent flows.
  • To utilize fractional calculus for representing Reynolds stresses with a variable-order fractional derivative.
  • To investigate the universality of this fractional order across different flow types and Reynolds numbers.

Main Methods:

  • Application of fractional calculus to model Reynolds stresses as a non-local, variable-order fractional derivative.
  • Training the variable-order function using direct numerical simulation (DNS) databases.
  • Validation against independent DNS data and experimental measurements (e.g., Princeton superpipe).
  • Development of a two-sided fractional derivative model for total shear stress.
  • Implementation of finite difference methods and fractional physics-informed neural networks (fPINNs) for solving inverse and forward problems.

Main Results:

  • A universal variable fractional order was identified for channel, Couette, and pipe flows across various Reynolds numbers.
  • The model captures the continuous change in turbulent diffusion rate and the nonlocality of turbulent interactions.
  • The two-sided fractional derivative model provides smooth profiles across the entire domain and improves upon one-sided models.
  • Fractional physics-informed neural networks (fPINNs) offer efficient solutions for both inverse and forward problems.

Conclusions:

  • Fractional calculus provides a powerful framework for modeling the entire velocity profile of wall-bounded turbulent flows.
  • The identified universal fractional order highlights fundamental aspects of turbulence nonlocality and diffusion.
  • The proposed models, especially with fPINNs, offer significant improvements in accuracy and computational efficiency for turbulence research.