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Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network

  • Pavan Pranjivan Mehta , Guofei Pang EMAIL logo , Fangying Song and George Em Karniadakis
Published/Copyright: December 31, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 22 Issue 6

Abstract

The first fractional model for Reynolds stresses in wall-bounded turbulent flows was proposed by Wen Chen [2]. Here, we extend this formulation by allowing the fractional order α(y) of the model to vary with the distance from the wall (y) for turbulent Couette flow. Using available direct numerical simulation (DNS) data, we formulate an inverse problem for α(y) and design a physics-informed neural network (PINN) to obtain the fractional order. Surprisingly, we found a universal scaling law for α(y+), where y+ is the non-dimensional distance from the wall in wall units. Therefore, we obtain a variable-order fractional model that can be used at any Reynolds number to predict the mean velocity profile and Reynolds stresses with accuracy better than 1%.

MSC 2010: Primary 76F40; 65N21; Secondary 68T20
Key Words and Phrases: turbulence; Reynolds-averaged Navier-Stokes (RANS) equations; fractional calculus; physics-informed neural networks (PINNS); machine learning

This paper is dedicated to the memory of the late Professor Wen Chen

Acknowledgements

The work was partially supported by the MURI/ARO at Brown University (W911NF-15-1-0562) and DARPA-AIRA (HR00111990025).

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Received: 2019-08-15
Published Online: 2019-12-31
Published in Print: 2019-12-18

© 2019 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA special issue – In memory of late professor Wen Chen (FCAA–Volume 22–6–2019)
  4. Survey Paper
  5. State-of-art survey of fractional order modeling and estimation methods for lithium-ion batteries
  6. An investigation on continuous time random walk model for bedload transport
  7. Tutorial Survey
  8. Porous functions
  9. Research Paper
  10. A time-space Hausdorff derivative model for anomalous transport in porous media
  11. High-order algorithms for riesz derivative and their applications (IV)
  12. Mass-conserving tempered fractional diffusion in a bounded interval
  13. Dispersion analysis for wave equations with fractional Laplacian loss operators
  14. Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application
  15. Some further results of the laplace transform for variable–order fractional difference equations
  16. Robust stability analysis of LTI systems with fractional degree generalized frequency variables
  17. Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network
https://doi.org/10.1515/fca-2019-0086
Keywords for this article
turbulence; Reynolds-averaged Navier-Stokes (RANS) equations; fractional calculus; physics-informed neural networks (PINNS); machine learning

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA special issue – In memory of late professor Wen Chen (FCAA–Volume 22–6–2019)
  4. Survey Paper
  5. State-of-art survey of fractional order modeling and estimation methods for lithium-ion batteries
  6. An investigation on continuous time random walk model for bedload transport
  7. Tutorial Survey
  8. Porous functions
  9. Research Paper
  10. A time-space Hausdorff derivative model for anomalous transport in porous media
  11. High-order algorithms for riesz derivative and their applications (IV)
  12. Mass-conserving tempered fractional diffusion in a bounded interval
  13. Dispersion analysis for wave equations with fractional Laplacian loss operators
  14. Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application
  15. Some further results of the laplace transform for variable–order fractional difference equations
  16. Robust stability analysis of LTI systems with fractional degree generalized frequency variables
  17. Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network
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