The Conjecture of a General Law of the Wall for Classical Turbulence Models, Implying a Structural Limitation

Abstract

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 which are built from the internal quantities of the model and are 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. We only use channels and boundary layers, but have seen no evidence that pipe flow is different. The conjectured General Law of the Wall (GLW), in contrast with the classical law for mean velocity \(U\) only, states that every quantity Q in the model (e.g., dissipation, stresses) is the product of four quantities: powers of the friction velocity \({u}_{\tau }\) and y which satisfy dimensional analysis; a constant C characteristic of the model; and a function f of the wall distance y in wall units, which closely approaches 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, and it rigidly constrains the y-dependence of Q outside the wall region: in particular, all the stresses are on plateaus. 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 \(\kappa\). 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. A Structural Limitation any Classical Model would suffer from then arises because the results of experiments (not shown here) and Direct Numerical Simulations contradict the GLW, already for some of the Reynolds stresses in simple flows and all the way to the wall (the conflict between the GLW as predicted by Classical turbulence theory, on which the models are based, and measurements was discussed by Townsend as early as Townsend in J Fluid Mech 11:97–120, 1961). 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 implicitly in the literature, it extends all the way to the wall, and it has theoretical interest. It creates a danger for the developing “data-driven” efforts based on Machine Learning in turbulence modelling, which generally involve all six Reynolds stresses and possibly other quantities such as budget terms.