## Publications

... in pursuit of knowledge.

# INVITED TALKS

May 12, 2023

## Fractional Calculus and Turbulence

Mehta, P. P. (2023). Nonlocal turbulence modeling by developing a fractional stress-strain relationship for Reynolds-averaged Navier-Stokes Equations. Workshop on Fractional Calculus, Special Functions and Applications, Sapeniza University of Rome, Italy.

Organised by: Prof. Luisa Beghin, Prof. Alessandro De Gregorio, Prof. Francesco Mainardi, Prof. Costantino Ricciuti, Prof. Enrico Scala

Sapeniza University of Rome

April 13, 2022

## Fractional Calculus and Turbulence

Isaac Newton Institute, UK

Mehta, P. P. (2022). Addressing anamolous diffusion of turbulent flows using fractional derivatives. Isaac Newton Institute of Mathematical Sciences, Cambridge, UK

Turbulence exhibits long range interactions and has multiple scales. Resolving all scales with fine grids implies non-locality is addressed by subsequent resolution of local dynamics. However, if only spatially or temporally averaged fields are solved for computational feasibility of real world applications, then addressing non-locality explicitly becomes important due to absence of information of interactions between two points. In this work, the author demonstrates the use of variable order fractional derivatives for wall bounded turbulent flows as a closure model which addresses the duality of local and non-local characteristics.

April 13, 2022

## Fractional Calculus and Turbulence

Mehta, P. P. (2022). Non-local Turbulence modeling using Fractional Derivatives. IMEC, Belgium

IMEC, Lueven, Belgium

Turbulence exhibits long range interactions and has multiple scales. Resolving all scales with fine grids implies non-locality is addressed by subsequent resolution of local dynamics. However, if only spatially or temporally averaged fields are

solved for computational feasibility of real world applications, then addressing non-locality explicitly becomes important due to absence of information of intermediate scale, thus a closure problem arises in Reynolds- averaged Navier-Stokes Equations (RANS), which solve directly for mean quantities.. To address the closure problem in RANS, an eddy-viscosity hypothesis has been a popular choice. In this talk, the author remarks that a classical eddy-viscosity hypothesis is an elaborate form of non-Fickian diffusion equation. Further, we construct a stress-strain relationship using variable-order Caputo fractional derivative, such a hypothesis leads to a fractional diffusion term in RANS. It is to be noted that both non-Fickian and fractional diffusion address anomalous diffusion processes, thereby bridging the two literatures. Nonlocality at a given point is an amalgamation of all effects, thus a two-sided fractional stress- strain relationship gives physically consistent results. We have identified distinct regions, akin to log-law of velocity and law of wake in the fractional order computed. It is also shown computationally that the fractional order curves asymptotes at the infinite Reynolds number limit. The success of the two-sided model led to an investigation with tempered fractional derivatives, by introducing an exponential term and Heaviside function. Its equivalence is studied computationally and a length-scale, called the horizon of nonlocal interactions is defined.

Acknowledgement: This work was carried out at Brown University, USA under the supervision of Prof. George Karniadakis (Brown University). The author credits Prof. Karniadakis for proposing the use of fractional calculus for turbulence modeling.

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June 10, 2022

## Fractional Calculus and Turbulence

Imperial College, London

Mehta, P. P. (2022) Non-local Turbulence modeling using Fractional Derivatives. Dept. of Aeronautics, Imperial College London, UK

Turbulence exhibits long range interactions and has multiple scales. Resolving all scales with fine grids implies non-locality is addressed by subsequent resolution of local dynamics. However, if only spatially or temporally averaged fields are solved for computational feasibility of real world applications, then addressing non-locality explicitly becomes important due to absence of information of interactions between two points. In this work, the author demonstrates the use of variable order fractional derivatives for wall bounded turbulent flows as a closure model which addresses the duality of local and non-local characteristics.

June 23, 2022

## Fractional Calculus and Turbulence

IIT Madras, India

Mehta, P. P. (2022). Non-local Turbulence modeling using Fractional Derivatives. Dept. of Engineering Design, IIT Madras, India

Turbulence exhibits long range interactions and has multiple scales. Resolving all scales with fine grids implies non-locality is addressed by subsequent resolution of local dynamics. However, if only spatially or temporally averaged fields are solved for computational feasibility of real world applications, then addressing non-locality explicitly becomes important due to absence of information of interactions between two points. In this work, the author demonstrates the use of variable order fractional derivatives for wall bounded turbulent flows as a closure model which addresses the duality of local and non-local characteristics.

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# CONFERENCES

June 23, 2022

Mehta, P. P. and Rozza, G (2023) Generalized Jacobi poly-fractonomials for nonlocal spectral methods. 2nd IACM Mechanistic Machine Learning and Digital Engineering for Computational Science and Engineering (MMLDE-CSET 2023)

## Numerical method for Nonlocal Operators

MMLDE-CSET 23, Texas, US

Jacobi poly-fractonomials are the eigen-functions of the fractional Sturm-Liouville problem (Zayernouri and Karniadakis, 2013; J. Comp. Phy.), and can be regarded as a generalization of Jacobi Polynomials. When used as trial functions in variational form, they lead to spectrally accurate methods for fractional operators. We extend the notion of Jacobi poly-fractonomials to Generalized Jacobi poly-fractonomials, which leads to development of spectrally accurate methods in variational form for nonlocal differential equations, which are a further generalization of fractional differential equations.

March 04, 2021

## Fractional Calculus and Turbulence

SIAM CSE 21

Mehta, P.P., Karniadakis, G. (2021). Fractional modelling of wall-bounded turbulence. SIAM Conference on Computational Science and Engineering (CSE21), Texas, USA

Turbulence exhibits long range interactions and has multiple scales. Resolving all scales with fine grids implies non-locality is addressed by subsequent resolution of local dynamics. However, if only spatially or temporally averaged fields are solved for computational feasibility of real world applications, then addressing non-locality becomes important due to absence of information of interactions between two points. In this work, we demonstrate the use of variable order fractional gradients for wall bounded turbulent flows to address the duality of local and non-local characteristics. Tempering assures that the second moment of fractional operators exists as real world applications have finite jump lengths. It is found that the presence of the wall itself implies tempering with a sharp cutoff, which mathematically can be shown with the use of Heaviside functions. As a result of tempering either with exponential or Heaviside function we discover the existence of a minimum horizon of non-local interactions in turbulent flows. Furthermore, we show the equivalence between these two definitions. Also, we generalize the tempered fractional gradients definitions to employ any arbitrary tempering distribution such that the second moment is finite. Since this is generalisable to possibly other non-local operators, the Dirac function transforms any arbitrary non-local operator to address local dynamics, if limited to describe only non-local dynamics.

June 23, 2022

## Tempered Fractional Calculus and Turbulence

ICTAM 2020+1, Italy

Mehta, P. P., Karniadakis, G. (2021). Discovering a universal scaling law using Tempered Fractional two sided derivative for Turbulent flow. 25th International congress on applied and theoretical mechanics. Milan, Italy.

A fractional derivative is used to model non-local effects in turbulence. However, as the second moment does not exist, tempering the Levy distribution leads to finite moments, and thus, a new tempered fractional derivative free from all assumptions is formulated for non-local modelling of turbulent flows. The total shear stress is modelled here as opposed to commonly modelled unclosed terms in turbulence modelling; also, working in wall units does not introduce any coefficient for representing the shear stresses. We demonstrate this new concept for turbulent Couette flow, where the error found in total shear stress is very low compared to results of direct numerical simulations.

Sept. 26-29, 2021

## Fractional Calculus and Turbulence

MMLDT

Mehta, P.P., Karniadakis, G. (2021). Non-local modeling of wall-bounded turbulence using fractional gradients. Mechanistic Machine Learning and Digital Twins for Computational Science, Engineering & Technology conference. San Diego, USA. Link: https://mmldt.eng.ucsd.edu/home

Nov. 23, 2021

## Fractional Calculus and Turbulence

APD DFD 2021

Pranjivan Mehta, P., Karniadakis, G. and Bravo, L., 2021. Fractional Reynolds-averaged Navier Stokes equations (f-RANS) for modeling of separated boundary layers. In APS Division of Fluid Dynamics Meeting Abstracts (pp. T09-003).

The duality of local and non-local regions observed in turbulent flows makes defining mathematically rigorously operators a cumbersome task with literature focused on either local or non-local modeling. Recently, we showed that a variable-order fractional operator can not only model both the regimes but also seamlessly transitions from local to non-local regimes. The effect of pressure gradients further complicates the problem, especially in separated flows, which has been a long standing problem in the turbulence modelling community. Thus, in this work we investigate the role of pressure coupled with Reynolds number effects for adverse pressure gradient boundary layers, including modelling of the recirculation bubble. Upon the formulation of one- and two-sided models using Caputo fractional derivatives, we found that only the two-sided model leads to physical solutions. This is not a surprise as non-locality at a given point is an aggregate effect of all directions and the two-sided model addresses this very fact. The predictive nature of this formulation presents a model free of ad-hoc tuning coefficients thereby providing the basis for a robust engineering tool.

Nov. 24, 2020

## Fractional Calculus and Turbulence

APS DFD 2020

Pranjivan Mehta, P., Zaki, T., Meneveau, C. and Karniadakis, G., 2020. Fractional Reynolds-averaged Navier Stokesequations (f-RANS) for modeling of transitional and turbulent boundary layers. Bulletin of the American Physical Society, 65.

Reynolds averaged Navier-Stokes (RANS) equations often invoke a local model for the Reynolds stresses, while in reality the correlations between these stresses and the strain rate are nonlocal. In this work, we propose to model the Reynolds stress in terms of the mean velocity using fractional gradients, which are nonlocal operators. We demonstrate mathematically that a single model structure is valid for all regimes of the flow. Also, when non-dimensionalized in wall units, there are no additional coefficient to model. Results are presented for modeling statistics from direct numerical simulations of bypass transition from JHTDB, and for analytical expressions of the total shear stress from the literature. The model can match the mean velocity profile in the transitional and fully turbulent regimes. The results demonstrate the mathematical expressivity of the fractional gradient, where a non-local physics are properly captured.

Nov. 23, 2019

Pranjivan Mehta, P., Song, F., Pange, G., Meneveau, C. and Karniadakis, G., 2019. Fractional physical-inform neural networks (fPINNs) for turbulent flows. Bulletin of the American Physical Society, 64.

## Fractional Calculus and Turbulence

APS DFD 2019

We employ fractional operators in conjunction with physics-informed neural networks (PINNs) to discover new governing equations for modeling and simulating the Reynolds stresses in the Reynolds Averaged Navier-Stokes equations (RANS) for wall-bounded turbulent flows at high Reynolds number. In particular, we develop a simple one-dimensional model for fully-developed wall-turbulence that involves a fractional operator with fractional order that varies with the distance from the wall. We use available DNS data bases to infer the function that describes the fractional order, which has an integer value at the wall and decays monotonically to an asymptotic value at the centerline. We show that this function is universal upon re-scaling and hence it can be used to predict the mean velocity profile at all Reynolds numbers. We also extend the fractional RANS for fully-developed turbulent channel flow to a turbulent boundary layer and infer the fractional order in the wake region.

*This work is supported by the DARPA-AIRA grant HR00111990025.

# RESEARCH PAPERS

31 Oct., 2023

Mehta, P.P., 2023. Fractional and tempered fractional models of Reynolds-averaged Navier-Stokes equations for Turbulent flows. Journal of Turbulence, DOI: 10.1080/14685248.2023.2274100

## Fractional Calculus and Turbulence

J. Turbulence

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 applied to channel, couette and pipe flow. 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. An expression for the fractional order is also found, thereby solving the closure for the considered cases.

This study is further extended to tempered fractional calculus, where tempering ensures finite jump lengths, this is an important remark for unbounded flows. Within the context of this paper, we limit ourselves to wall bounded flow. 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, as truncating the domain has computational advantages.

For the above investigations, we carefully designed algorithms, notably, the pointwise version of fractional physics-informed neural network to find the fractional order as an inverse problem.

December, 2019

## Fractional Calculus and Turbulence

Frac. Calculus and Applied Analysis

Mehta, P.P., Pang, G., Song, F. and Karniadakis, G.E., 2019. Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network. Fractional calculus and applied analysis, 22(6), pp.1675-1688.

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%.

8 May, 2021

Mehta, P.P., 2021. Fractional models of Reynolds-averaged Navier-Stokes equations for Turbulent flows. arXiv preprint arXiv:2105.03646.

## Fractional Calculus and Turbulence

preprint

Its is a well known fact that Turbulence exhibits non-locality, however, modeling has largely received local treatment following the work of Prandl over mixing-length model. Thus, in this article we report our findings by formulating a non-local closure model for Reynolds-averaged Navier-Stokes (RANS) equation using Fractional Calculus. Two model formulations are studied, namely one- and two-sided for Channel, Pipe and Couette flow, where the results shown have less 1% error. The motivation of two-sided model lies in recognising the fact that non-locality at a given spatial location is an aggregate of all directions. Furthermore, scaling laws and asymptotic relationship for Couette, Channel and Pipe flow is reported. It is to be noted that modeling in wall units, no additional coefficient appears, thus there models could be applied to complex flows with ease.

# POSTER PRESENTATION

April 13, 2022

## Fractional Calculus and Turbulence

Mehta, P. P. (2022). Addressing anamolous diffusion of turbulent flows using fractional derivatives. Isaac Newton Institute of Mathematical Sciences, Cambridge, UK

Isaac Newton Institute, UK

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# THESIS

October, 2016

## Uncertainty Quantification and Turbulence

The University of Manchester

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Mehta, P. P. (2016). Uncertainty Quantification in CFD Analyses using Generalised Polynomial Chaos Method. The University of Manchester

Uncertainty quantification was conducted using non–intrusive form of Generalised Polynomial Chaos methods for square inline tube bundles. The input was treated as uncertain. Uncertainties were considered in Reynolds number of the flow and distance between two tubes of square inline tube bundles, P/D. The range were considered as 250≤ Re≤ 1000 and 1.5≤ P/D≤ 5. The desired gPC output was to determine the lift, drag and skin friction coefficient for the input range. At first the gPC method were validated for a univariate and a multi–variate case, cosine function and spring body problem respectively.