On the influence of Reynolds shear stress upon the velocity patterns generated in turbulent starting pipe flow

Abstract

Experimental evidence yields a wide variety of starting turbulent pipe flows, usually with greatly deformed mean velocity profiles. Some experimental tests attain profiles with maximum values in the near-wall region, while some others report well-defined local concavities within globally convex mean velocity profiles. To the authors’ knowledge, no available explanation yet exists for such a kind of behavior. This theoretical research studies the specific effects that certain patterns of Reynolds shear stress cause on the mean velocity field obtained in unsteady turbulent flows. A simple model of transient Reynolds shear stress is devised, and this model is used to calculate the mean velocity field of starting turbulent flow. Those mean flow patterns are related to the actual velocity profiles reported in the experimental literature on starting flow. We learn how the different Reynolds shear stress patterns contribute to create the resulting mean velocity patterns and which particular configuration of Reynolds shear stress is responsible for each one of the reported experimental velocity profiles. We explain why, how, where, and when each particular deformation in the unsteady mean velocity profile is caused, and the relationship between each type of deformation and the Reynolds shear stress that brings it forth. We issue some predictions identifying types of flow that, to our knowledge, have not yet been reported in the experimental literature, and we offer clues for those experimental researchers willing to discover the predicted flow patterns. This research continues the exposition of the theory of underlying laminar flow initiated in previous papers.

On the analytic explanation of experiments where turbulence vanishes in pipe flow

“…When data is scarce, or simply not available, one can always presume, ad hoc , some functions (inputs) and check whether the resulting mean-velocity field (the output) would be accurate enough to reproduce the actual flow. We have already performed such a type of exercise in our previous papers (García García & Fariñas Alvariño 2019 a , b , c , 2020, 2021), which we call ‘theoretical experiments’, and the results were satisfactory and provided true explanations for the observed facts, also allowing us to issue predictions that still await confirmation. Regardless of the procedure, the set of particular functions , , , and chosen for any given application is called the ‘model’.…”

“…To end this section, the TULF brings forth a very intuitive field to assess, just at a glance, how turbulent a mean flow is in a given region. This scalar field is called the turbulence index ℑ(α), and is defined as the ratio of the mean velocity to the ULF (see García García & Fariñas Alvariño 2019c, 2020, 2021,…”

“…The zero theorem is only an approximation if the flow is not fully developed or slightly unsteady. With constant, the equation is a mathematical necessity, regardless of the flow being unsteady or stationary. Changes of pressure are transmitted (almost) instantaneously all over the flow, for it is considered incompressible; however, changes in MPG and WSS would take some time to propagate within the flow, for inertia plays a role in U-flows. Under its own dynamics, the ULF tends to follow the behaviour of the MPG, for is the source of (García García & Fariñas Alvariño 2019 b , c , 2020, 2021). The reacts rather quickly to changes in , although it does not trail the MPG evolution instantaneously. Under its own dynamics, the PTC tends to follow the behaviour of the RSSRG, for is the source of (García García & Fariñas Alvariño 2019 b , c , 2020, 2021). In general, reacts more slowly than . Close to the wall, in the viscous sublayer, the S-PTC is negligible and can be safely ignored (García García & Fariñas Alvariño (2019 c ) and ).…”

“…A list of all acronyms used can be found in Appendix A. The present article is the fifth in a series (García García & Fariñas Alvariño 2019 b , c , 2020, 2021) that intends to explain a number of experimental results reported in the literature, which to date remain unexplained.…”

“…In order to fully understand the coming explanations, a certain knowledge of the TULF scope and methods would be necessary. Therefore, we shall begin this work with some basic notions, albeit a comprehensive account can be found in our previous publications (García García & Fariñas Alvariño 2019 b , c , 2020, 2021). The TULF is a theory heavily based upon mathematical analysis, and most of the concepts that will be introduced, however strange they might appear at first sight, arise out of mathematical necessity.…”