2 lectures on turbulence with Philippe Spalart at Centrale Nantes on 15-16 June 2023

June 15, 2023 at 3:00 pm: An Old-Fashioned Framework for Machine Learning in Turbulence Modelling

The objective is to provide clear and well-motivated guidance to Machine Learning (ML) teams, founded on our experience in empirical turbulence modeling. Guidance is also needed for modeling outside ML. ML is not yet successful in turbulence modeling, and many papers have produced unusable proposals either due to errors in math or physics, or to severe overfitting. We believe that “Turbulence Culture” (TC) takes years to learn and is difficult to convey especially considering the modern lack of time for careful study; important facts which are self-evident after a career in turbulence research and modeling and extensive reading are easy to miss. In addition, many of them are not absolute facts, a consequence of the gaps in our understanding of turbulence and the weak connection of models to first principles. Some of the mathematical facts are rigorous, but the physical aspects often are not. Turbulence models are surprisingly arbitrary. Disagreement between experts confuses the new entrants. In addition, several key properties of the models are ascertained through non-trivial analytical properties of the differential equations, which puts them out of reach of purely data-driven ML-type approaches. The best example is the crucial behavior of the model at the edge of the turbulent region (ETR). The knowledge we wish to put out here may be divided into “Mission” and “Requirements,” each combining physics and mathematics. Clear lists of “Hard” and “Soft” constraints are presented. A concrete example of how DNS data could be used, possibly allied with ML, is first carried through and illustrates the large number of decisions needed. Our focus is on creating effective products which will empower CFD, rather than on publications.

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June 16, 2023 at 10:00 am: A Conjecture of a General Law of the Wall for Classical Turbulence Models

We call classical a transport model in which each governing equation comprises a production term proportional to velocity gradients, and terms such as diffusion and dissipation built from the internal quantities of the model and local. They may depend on the wall-normal coordinate y. We consider the layer along a wall in which the total shear stress is uniform, and y is much smaller than the thickness of the full wall layer. The Generalized Law of the Wall (GLW) states that every quantity Q in the model (e.g., dissipation, stresses) is the product of four quantities: powers of the friction velocity and y which satisfy dimensional analysis; a constant C of the model; and a function f of the wall distance y in wall units, which equals 1 outside the viscous and buffer layers. This is independent of any flow Reynolds number such as the friction Reynolds number in a channel, once it is large enough. In the widely accepted velocity law of the wall, the shear rate dU/dy satisfies such a law with C the inverse of the Karman constant. Both variables in the k-epsilon model also do. We cannot prove the GLW property as a theorem, but we provide extensive arguments to the effect that any Classical equation set allows it, and many numerical results support it. The Structural Limitation then arises because the results of experiments and Direct Numerical Simulations contradict the GLW, already for some of the Reynolds stresses in simple flows and all the way to the wall. This implies that no modification of a model that remains within the classical type can make it agree closely with this key body of results. This has been tolerated for decades, but the GLW is stated here more precisely than it has been implied in the literature, it has theoretical interest, and it creates a danger for the developing “data-driven” efforts in turbulence modelling, which generally involve all six Reynolds stresses and possibly other quantities such as budget terms.