Fractional Conservational Laws For Fluid Flow From First Principles

Turbulence is a non-local and non-Markovian process, exhibiting both memory (in time) and non-locality (in space). Fluid flows are further complicated by presence of viscous action, which is a local process. As result, three regimes can be identified, purely local (due to viscosity alone), purely non-local (due to turbulence alone) and the regime in-between which is an amalgamation of the two. Resolving all the scales of motions addresses this phenomenon. However, due to computational consideration, if one directly solves for time-averaged or spatial filtered fields, then addressing non-locality explicitly becomes important on account of missing interactions due to unresolved fields or scales. This leads to a closure problem. Over the past five decades, there have been many models proposed to solve the closure problem. The models proposed in the literature are empirical and do not account for non-locality, in-spite – turbulence being a non-local phenomenon a well known fact. As a result, over five decades of turbulence modeling research has not yielded in any general model applicable to all flow settings. Furthermore, there is very limited to none modeling capability for predicting transition and separation often observed in most practical application. In order to account for non-locality, we will employ fractional calculus, which is the generalization of classical calculus, where the order an is arbitrary number defined in R + . Paradoxically, these are non-local operators for non-integer orders, while local for integer-orders. Thus, well suited to address the duality of local and non-local processes. As a result, the author has introduced the “fractional stress-strain” hypothesis. The hypothesis is now validated not only for canonical flows, such as channel and Couette but also for transitional and separated boundary layers. In order to avoid building empirical models, the author shall derive - “Fractional Navier-Stokes equations” (FNS) from first principles, which required first developing the “Fractional Vector Calculus”. To the best of our knowledge, a fractional conservational law was not previously derived rigorously. In order to solve for these new type of fractional differential equations, the author has introduced a novel basis function, namely, “Jacobi Convolution series” (JCS). By projecting fractional operators onto spaces constructed by JCS, we are able to construct a spectral method, which achieves higher-order accuracy. Additionally, the author has also developed numerical methods within the modern paradigm of Scientific Machine Learning, notably, • Extension of fractional physics informed neural networks (fPINNs) to compute the fractional order as an inverse problem. The novelty of the proposed algorithm, it does not need boundary conditions, since the fractional order for complex settings is rather unknown. • Secondly, the author has extended fPINNs to tempered fractional operators, and proposed an algorithm to study the equivalence between the two definitions. • A hybrid method is proposed in the context of general fractional operators, since automatic differentiation is not applicable for such operators. The resulting algorithm leverages both techniques of both spectral methods, and physics-informed neural network, there by an accurate, and efficient method.