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Fractional and tempered fractional models for Reynolds-averaged Navier-Stokes equations

Pavan Pranjivan Mehta

Abstract

Turbulence is a non-local phenomenon and has multiple-scales. Non-locality can be addressed either implicitly or explicitly. Implicitly, by subsequent resolution of all spatio-temporal scales. However, if directly solved for the temporal or spatially averaged fields, a closure problem arises on account of missing information between two points. To solve the closure problem in Reynolds-averaged Navier-Stokes equations (RANS), an eddy-viscosity hypotheses has been a popular modelling choice, where it follows either a linear or non-linear stress-strain relationship. Here, a non-constant diffusivity is introduced. Such a non-constant diffusivity is also characteristic of non-Fickian diffusion equation addressing anomalous diffusion process. An alternative approach, is a fractional derivative based diffusion equations. Thus, in the paper, we formulate a fractional stress-strain relationship using variable-order Caputo fractional derivative. This provides new opportunities for future modelling effort. We pedagogically study of our model construction, starting from one-sided model and followed by two-sided model. Non-locality at a point is the amalgamation of all the effects, thus we find the two-sided model is physically consistent. Further, our construction can also addresses viscous effects, which is a local process. Thus, our fractional model addresses the amalgamation of local and non-local process. We also show its validity at infinite Reynolds number limit. This study is further extended to tempered fractional calculus, where tempering ensures finite jump lengths, this is an important remark for unbounded flows. Two tempered definitions are introduced with a smooth and sharp cutoff, by the exponential term and Heaviside function, respectively and we also define the horizon of non-local interactions. We further study the equivalence between the two definitions.

Fractional and tempered fractional models for Reynolds-averaged Navier-Stokes equations

Abstract

Turbulence is a non-local phenomenon and has multiple-scales. Non-locality can be addressed either implicitly or explicitly. Implicitly, by subsequent resolution of all spatio-temporal scales. However, if directly solved for the temporal or spatially averaged fields, a closure problem arises on account of missing information between two points. To solve the closure problem in Reynolds-averaged Navier-Stokes equations (RANS), an eddy-viscosity hypotheses has been a popular modelling choice, where it follows either a linear or non-linear stress-strain relationship. Here, a non-constant diffusivity is introduced. Such a non-constant diffusivity is also characteristic of non-Fickian diffusion equation addressing anomalous diffusion process. An alternative approach, is a fractional derivative based diffusion equations. Thus, in the paper, we formulate a fractional stress-strain relationship using variable-order Caputo fractional derivative. This provides new opportunities for future modelling effort. We pedagogically study of our model construction, starting from one-sided model and followed by two-sided model. Non-locality at a point is the amalgamation of all the effects, thus we find the two-sided model is physically consistent. Further, our construction can also addresses viscous effects, which is a local process. Thus, our fractional model addresses the amalgamation of local and non-local process. We also show its validity at infinite Reynolds number limit. This study is further extended to tempered fractional calculus, where tempering ensures finite jump lengths, this is an important remark for unbounded flows. Two tempered definitions are introduced with a smooth and sharp cutoff, by the exponential term and Heaviside function, respectively and we also define the horizon of non-local interactions. We further study the equivalence between the two definitions.
Paper Structure (45 sections, 2 theorems, 128 equations, 11 figures, 2 tables, 2 algorithms)

This paper contains 45 sections, 2 theorems, 128 equations, 11 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary : Fractional Calculus
  3. Grünwald-Letnikov Definition
  4. Riemann-Liouville Definition
  5. Caputo Definition
  6. Left and right sided fractional derivative
  7. Fractional Reynolds-averaged Navier-Stokes equations (f-RANS)
  8. One- and two-sided f-RANS models
  9. Application to couette, channel and pipe flows
  10. Turbulent couette flow
  11. Turbulent channel flow
  12. Turbulent pipe flow
  13. Numerical Scheme: Physics-informed Neural Network
  14. Inverse problem: computing fractional order (pointwise)
  15. Results : fractional order of one-sided f-RANS model applied to turbulent couette, channel and pipe flow
  16. ...and 30 more sections

Key Result

Corollary 7.1

For $y \rightarrow 0$, within the viscous sub-layer, the fractional order asymptotes to 1. Consider (eq:anal_couette) written for couette flow. We suppress, $(.)^+$ to avoid any confusion with the signs in power. We consider each term individually. The first term in (eq:anal_couette) tends to 1. For Since (eq:anal_channel) and (eq:anal_pipe) for channel and pipe, respectively, has a similar struct

Figures (11)

  • Figure 1: A schematic representation for the inverse modeling, here the spatial location ($y^+$) is the input of the feed forward neural network, while the fractional order ($\alpha$) is the output, which is used to computed the loss function which comprises of f-RANS model using velocity from DNS databases
  • Figure 2: Fractional order of one-sided f-RANS model for channel, couette and pipe flow. Here the x-axis of the plot: channel and couette: $y^+/Re_\tau$, pipe: $(1-r)^+/R^+$. It is observed as the Reynolds / Karman number increases the fractional order lowers corresponding to higher turbulence intensity. The channel and pipe show an artifact due to symmetry, where a fractional order if unity is not a physical solution at the center-line, merely a numerical solution of our constructed model
  • Figure 3: Fractional order of one-sided f-RANS model for channel, couette and pipe flow. Here the x-axis of the plot: channel and couette: $y^+$, pipe: $(1-r)^+$. Remarkably, the fractional order for couette flow shows universality. The channel and pipe shows an artifact due to symmetry, exposing the limitation of the one-sided model. For all three cases, the the fractional order is universal/overlaps in the viscous sub-layer and to some extent the buffer layer.
  • Figure 4: Fractional order of two-sided f-RANS model for channel, couette and pipe flow. Here the x-axis of the plot: channel and couette: $y^+/Re_\tau$, pipe: $(1-r)^+/R^+$. It is observed as the Reynolds / Karman number increases the fractional order lowers corresponding to higher turbulence intensity. The artifact due symmetry of one-sided model is no longer present in this case, as the non-locality is considered in a physical manner.
  • Figure 5: Comparison of total shear stress obtained using one- and two-sided f-RANS model with DNS databases for couette, channel and pipe flow. Here the x-axis of the plot: channel and couette: $y^+/Re_\tau$, pipe: $(1-r)^+/R^+$. The error in either model is less than 1%
  • ...and 6 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Corollary 7.1
  • Remark 6
  • Corollary 7.2
  • Remark 7
  • Remark 8
  • ...and 3 more