On an analytical explanation of the phenomena observed in accelerated turbulent pipe flow

Abstract

This research presents a new theory that explains analytically the behaviour of any fully developed incompressible turbulent pipe flow, steady or unsteady. We propose the name of theory of underlying laminar flow (TULF), because its main consequence is the description of any turbulent pipe flow as the sum of two components: the underlying laminar flow (ULF) and the purely turbulent component (PTC). We use the framework of the TULF to explain analytically most of the important and interesting phenomena reported in He & Jackson (J. Fluid Mech., vol. 408, 2000, pp. 1–38). To do so, we develop a simple model for the pressure gradient and Reynolds shear stress that could be applied to the linearly accelerated pipe flow described by He & Jackson (2000). The following features of the unsteady flow are explained: the deformation undergone by the mean velocity profiles during the transient, the velocity overshoot observed in the more rapid excursions, the dual deformation of mean velocity profiles when overshoots are present, the laminarisation effects described during acceleration, the rapid decrease of the Reynolds shear stress upon approaching the wall that brings forth the laminar sublayer, and some other minor effects. A new field is defined to characterise the degree of turbulence within the flow, directly calculable from the theory itself. Arguably, this new field would describe the degree of turbulence in a pipe flow more accurately than the familiar turbulence intensity parameter. Finally, a paradox is found in the deformation of the unsteady mean velocity profiles with respect to equal-Reynolds-number steady profiles, which is duly explained. The research also predicts the occurrence of mean velocity undershoots if the flow is decreased rapidly enough.

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

“…One can check by zooming into figure 2 how each pair SPi-ULFi is coincident about the wall, including the curve's slope at . In a previous paper (García García & Fariñas Alvariño 2019 c ), we showed that the near-wall viscous sublayer observed in turbulent Hagen–Poiseuille flow is a domain where the S-ULF prevails and the S-PTC is virtually zero. The field is negligible about the wall in any S-flow and wall-related quantities, such as or , are exclusively defined by the S-ULF.…”

“…The turbulence is seen as an agent that detaches the flow from its best possible configuration, which is the ULF. A turbulent Hagen–Poiseuille flow verifies ; otherwise put, the viscous sublayer is a retrodiction of the TULF, see García García & Fariñas Alvariño (2019 c ) and .…”

“…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 , and is defined as the ratio of the mean velocity to the ULF (see García García & Fariñas Alvariño 2019 c , 2020, 2021), with in a S-flow, and in regions where the flow is mean laminar; that is, in regions where . We shall see later that is very useful in determining where a turbulent flow is laminarised, and the degree of laminarisation it presents.…”

“…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.…”

About Inverse Laplace Transform of a Dynamic Viscosity Function

“…Among such solutions, there are solutions for accelerated [5][6][7][8][9], decelerated [10], reverse [11], pulsating [12,13], and other types of flows [14][15][16][17][18][19]. Relatively recently, new solutions have been developed in which a variable viscosity has been introduced during the duration of unsteady pipe flow [20,21], and with the help of newly derived methods, the previously known laminar models can be extended to turbulent flows [22,23].…”