Talks | Publications

Abstract : Turbulence is a non-local and multi-scale phenomenon. Resolving all scales implies non-locality is addressed implicitly. However, if spatially or temporally averaged fields are considered for computational feasibility, then addressing non-locality explicitly becomes important as a result of missing information of all scales.

Our previous work involved constructing fractional closure model [1,2] for Reynolds-averaged Navier–Stokes equations, which are temporally averaged. In [2] ”fractional stress-strain hypothesis” was introduced using a a variable-order (spatially-dependent) Caputo fractional derivative. Indeed, it addresses the amalgamation of local and non-local effects [1, 2]. The results of two-sided model [2] were very encouraging, where, we find a power-law behaviour of fractional order akin to logarithmic regime for velocity and also a law of wake. Further, we investigated tempered fractional definitions in [2]. Since turbulence is a decay process, it led to define a ”horizon of non-local interactions” in [2].

Despite the success, one overwhelming question remains, ”how do we derive a fractional conservation law from first principles?” Thus, in this talk, I shall introduce the recently developed control volume approach in [3] to derive fractional conservation law from first principles. The first step was to extend the fractional vector calculus developed in [4] to two-sided operators [3]. Subsequently, derive the fractional analogue of Reynolds transport theorem [3]. By virtue of this theorem, we derive the ”fractional continuity” and ”fractional Cauchy equations”, which follows conservation of mass and momentum, respectively [3]. The stress tensor of fractional Cauchy equation is treated with a fractional stress-strain relationship (which was developed in [2]) to get to the final form of ”fractional Navier–Stokes equations”.

Biography: Pavan Pranjivan Mehta is a PhD student within the mathematics area of SISSA, Italy. He hold’s two Master’s degree, namely, Thermal and Fluid Engineering from University of Manchester, UK and Applied Math from Brown University, USA; with an undergrad in Aeronautical Engineering. He has held research positions in France, USA, UK and India; with an internship at Airbus Group and visiting researcher at Newton Institute, Cambridge for two scientific programs, namely, turbulence and fractional differential equations. Pavan's organisational activities for fractional calculus, includes mini-symposiums at prestigious conferences and seminar series, such as JINX seminar’s at Newton Institute and weekly seminars at SISSA. His current research interests are non-local turbulence modeling and numerical methods for fractional PDE.

Bibliography

[1] 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), 1675–1688.

[2] Pranjivan Mehta, P. (2023).Fractional and tempered fractional models for Reynolds- averaged Navier–Stokes equations. Journal of Turbulence, 24(11-12), 507–553.

[3] Pranjivan Mehta, P. (2024). Fractional vector calculus and fractional Navier-Stokes equations. (in preparation).

[4] Tarasov, V. E. (2008). Fractional vector calculus and fractional Maxwell’s equations. Annals of Physics, 323(11), 2756-2778

Abstract : 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

Abstract : 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.

Watch Lecture : https://www.newton.ac.uk/seminar/35713/ 

Abstract  : 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

Abstract  : 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

Download  : https://mathlab.sissa.it/sites/default/files/Fractional_Calculus_Seminars_Book_of_Abstracts_2024.pdf  

Abstract  : Recently, general fractional calculus was introduced by Kochubei (2011) and Luchko (2021) as a further generalisation of fractional calculus, where the derivative and integral operator admits arbitrary kernel. Such a formalism will have many applications in physics and engineering, since the kernel is no longer restricted. We first extend the work of Al-Refai and Luchko (2023) on finite interval to arbitrary orders. Followed by, developing an efficient Petrov-Galerkin scheme by introducing Jacobi convolution series as basis functions. A notable property of this basis function, the general fractional derivative of Jacobi convolution series is a shifted Jacobi polynomial. Thus, with a suitable test function it results in diagonal stiffness matrix, hence, the efficiency in implementation. Furthermore, our method is constructed for any arbitrary kernel including that of fractional operator, since, its a special case of general fractional operator

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 nonlinear 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 problem 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.

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%.

Abstract  : General fractional calculus is a further generalisation of fractional calculus which admits an arbitrary kernel [1, 2, 3]. In the context of numerics for such differential equations, we propose, general fractional physics-informed neural networks (GF-PINNs). In our method, we employ, separation of variable technique by introducing Jacobi convolution polynomial [3], whilst taking advantage of neural network’s ability as a non-linear solver. Furthermore, we remark automatic differentiation can now be carried out for integer-order operators, thereby mitigating the need of a splitting scheme. The resulting scheme is both accurate and computationally efficient, since we do not have to compute discrete convolution.

Download : https://casml.cc/wp-content/uploads/2024/12/PavanPranjivan_Paper_GFPINNS.pdf  

Abstract  : 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.

Abstract  : Fractional calculus is the generalization of integer-order calculus, where the order is defined in R+. Paradoxically, for non-integer order’s, these operators are non-local. As a result they are best suited to model non-local processes, such as turbulence [1, 2]. However, one can ask, whether all non-local processes can be modeled using fractional operators? The answer, is clearly, no. Thus, one seeks further generalization of fractional operators.

In order to generalize the fractional operator, Sonine [3] recognized a key property in the Abel integral formula, that the convolution of kernel’s of derivative and integral type operators is unity, thereby proposed a condition for any kernels, which is now refereed as Sonine condition. Following Sonine condition, Kochubei [4] introduced a set of conditions for the kernels. As a result, any differential and integral operator defined with these kernel shall satisfy the fundamental theorems of calculus, thereby the birth of ”general fractional calculus”. However, the conditions imposed on these kernels by Kochubei were in Laplace domain, which is inconvenient to use for practical purposes. Thus, Luchko [5] generalized the Sonine condition and termed it as modified Sonine condition. Recently, the idea’s of Luchko were extended on finite interval for arbitrary orders [6]. Thus, the mathematical theory of general fractional calculus is now complete [5, 6]. Indeed, differential and integral operator defined by kernels satisfying the modified Sonine condition, shall satisfy the fundamental theorems of calculus [5, 6].

On the numerical side, the computation of convolution-type operators, especially with singular kernel poses a challenge to construct an efficient and accurate (higher-order) scheme. Recently, we introduced the ”Jacobi convolution series” (JCS), as basis functions [6]. As a result, we obtain a higher-order Petrov-Galerkin scheme with a diagonal stiffness matrix. Furthermore, we extended our numerical approach within the paradigm of physics-informed neural networks.

In this talk, we shall first review the theory of general fractional calculus, followed by our numerical work on spectral methods and general fractional physics-informed neural networks (GFPINNs).

References: 

[1] P. P. Mehta, G. Pang, F. Song, G. Karniadakis, Discovering a universal variable-order fractional model for turbulent couette flow using a physics-informed neural network, Fractional Calculus and Applied Analysis 22 (2019) 1675–1688

[2] P. Pranjivan Mehta, Fractional and tempered fractional models for reynolds-averaged navier–stokes equations, Journal of Turbulence 24 (2023) 507–553

[3] N. Sonine, Sur la g´en´eralisation d’une formule d’abel (1884)

[4] A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equations and Operator Theory 71 (2011) 583–600.

[5] Y. Luchko, General fractional integrals and derivatives with the sonine kernels, Mathematics 9(2021). doi:10.3390/math9060594.

[6] P. P. Mehta, and G. Rozza. ”Jacobi convolution series for Petrov-Galerkin scheme and general fractional calculus of arbitrary order over finite interval.” arXiv preprint arXiv:2411.08080v2 (2024)

Abstract  : Fractional calculus is a generalization of classical (integer-order) derivatives, where the order can be arbitrary. Paradoxically, these are nonlocal operators, addressing anomalous diffusion processes. Although fractional derivatives have found applications in real world phenomena, such as turbulence [1], it begs the question for physics and engineers, can a power-law kernel describe for all physical processes? Indeed, the answer is, No; and one needs to seek further generalisation. One such mathematical foundation is laid down by Kochubei [2] and Luchko [3] for arbitrary kernels following Sonine. In our recent work [4], we extended the theory for general fractional operators to finite intervals for arbitrary orders. Furthermore, we introduced a novel basis function, namely, the Jacobi convolution polynomial. The notable property of such a basis function, the general fractional derivative of this basis function, is a shifted Jacobi polynomial. Thus, allowing the construction of an efficient Petrov-Galerkin scheme where the stiffness matrix is diagonal and our error analysis shows convergence rate is spectral. In this talk, we will introduce the operators, followed by our results for constructing efficient spectral methods for these new types of operators, which will potentially have many applications in physics and engineering. Needless to mention, since fractional operators are a special case of general fractional operators, our methods are valid for fractional cases too. 

References: 

[1] Pavan Pranjivan Mehta. Fractional and tempered fractional models for reynolds averaged navier–stokes equations. Journal of Turbulence, 24(11-12):507–553, 2023 

[2] Anatoly N Kochubei. General fractional calculus, evolution equations, and renewal processes. Integral Equations and Operator Theory, 71(4):583–600, 2011. 

[3] Yuri Luchko. General fractional integrals and derivatives of arbitrary order. Symmetry, 13(5):755, 2021. 

[4] Pavan Pranjivan Mehta and Gianluigi Rozza. Jacobi convolution polynomial for petrov-galerkin scheme and general fractional calculus of arbitrary order over finite interval. arXiv:2411.08080, 2024

Abstract  : Turbulence is a non-local and multi-scale phenomenon. Resolving all scales implies non-locality is addressed implicitly. However, if spatially or temporally averaged fields are considered for computational feasibility, then addressing non-locality explicitly becomes important as a result of missing information of all scales. Thus it is natural to introduce a fractional or a non-local operator. However, an overwhelming question remains, ”how do we derive a fractional conservation law from first principles?” Thus, in this talk, I shall introduce the recently developed control volume approach in [3] to derive fractional conservation law from first principles. Subsequently, derive the fractional analogue of Reynolds transport theorem [3]. By virtue of this theorem, we derive the ”fractional continuity” and ”fractional Cauchy equations”, which follows conservation of mass and momentum, respectively [3]. The stress tensor of fractional Cauchy equation is treated with a fractional stress-strain relationship (which was developed in [2]) to get to the final form of ”fractional Navier–Stokes equations”.

Bibliography:

[1] 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), 1675–1688.

[2] Pranjivan Mehta, P. (2023).Fractional and tempered fractional models for Reynolds- averaged Navier–Stokes equations. Journal of Turbulence, 24(11-12), 507–553.

[3] Pranjivan Mehta, P. (2024). Fractional vector calculus and fractional Navier-Stokes equations. (in preparation). 

Abstract  : Turbulence is a non-local and nonlinear phenomenon. As a closure model to Reynolds-averaged Navier–Stokes equations, we formulate a fractional stress-strain relationship [2] using a a variable-order (spatially-dependent) Caputo fractional derivative, which is a non-local operator. For wall bounded flows, near the wall (within the viscous sub-layer) the flow is purely local, while the outer region is governed by a non-local process. Indeed, our model addresses the amalgamation of local and non-local effects [1, 2]. For the two-sided model constructed in [2] we find scaling relationship for the fractional order akin to velocity scaling for the considered cases. Notably, a power-law behaviour of fractional order akin to logarithmic regime for velocity. By virtue of this power-law the fractional order asymptotes at infinite Reynolds number limit thereby justifying its validity. Further, a law of wake has been observed for the fractional order. Indeed, the two-sided model constructed in [2] demonstrates its physical consistency. The success of two-sided model motivated to invoke tempered fractional calculus in [2]. Tempering ensures finite jump lengths, an important remark for unbounded flows. Since turbulence is a decay process, it led to define a ”horizon of non-local interactions” in [2]. We pedagogically studied with two tempering distributions resulting a smooth and sharp cutoff, by an exponential and Heaviside function respectively for wall bounded flows. Its equivalence is studied computationally. For numerical investigation, we designed algorithms in [2], notably, the pointwise fractional physics-informed neural network to find the fractional order as an inverse problem. 

REFERENCES 

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

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

Abstract  : 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

Abstract  : 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.

Abstract  : 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

Abstract  : 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.

Abstract  : 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.

Abstract  : 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

Abstract : 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.