Applicability of Frozen-Viscosity Models of Unsteady Wall Shear Stress

The validity of assumed frozen-viscosity conditions underpinning an important class of theoretical models of unsteady wall shear stress in transient flows in pipes and channels is assessed using detailed computational fluid dynamics (CFD) simulations. The need for approximate one-dimensional $(1\mathrm{D})\{x,t\}$ models of the wall stress is unavoidable in analyses of transient flows in extensive pipe networks because it would be economically impracticable to use higher order methods of analysis. However, the bases of the various models have never been established rigorously. It is shown herein that a commonly used approach developed by the first authors is flawed in the case of smooth-wall flows although it is more plausible for rough-wall flows. The assessment process is undertaken for a particular, but important, unsteady flow case, namely, a uniform acceleration from an initially steady turbulent flow. First, detailed predictions from a validated CFD method are used to derive baseline solutions with which predictions based on approximate models can be compared. Then, alternative solutions are obtained using various prescribed frozen-viscosity distributions. Differences between these solutions and the baseline solutions are used to determine which frozen-viscosity distributions are the most promising starting points for developing $1\mathrm{D}\{x,t\}$ models of unsteady components of wall shear stress. It is shown that no frozen-viscosity distribution performs well for large times after the commencement of an acceleration. However, even the simplest approximation (laminar) performs well for short durations—which is when the greatest amplitudes of the unsteady components occur.