The Myth of URANS

In the 1990s, RANS modeling practitioner using finer grids and higher-order models began to observe cases for which simulations never reached a steady state. Instead, they observed the emergence of structures which were, at least qualitatively, similar to the turbulent structures observed in experiments (e.g. Johansson et. al, Int. J. Num. Meth. Fld. 1993; Durbin, AIAA J. 1995). These studies raise numerous questions. Fundementally, it is not clear why some RANS simulations result in unsteady solutions, and other do not. And the results of these studies have a mixed record of quantitiatve accuracy when compared to experiments. Spalart (Int. J. Heat Fld. Flow, 2000) present systematic critiques of what they refer to as the unsteady RANS (URANS) approach, notably, the lack of a theoretical justification. The current work addresses this lacuna with a theoretical model for both the partition between the resolved and modeled contributions to the kinetic energy, as well as for the bulk parameters of the turbulence. The principle predictions of the theory are that, in the absence of forcing, URANS tends to relax to a steady solution, and, depending on how the URANS is initialized, strongly nonphysical evolution can occur.