On the general analytic solution for unsteady turbulent incompressible flow between parallel plates

Abstract

This research reports the mean-velocity field associated with a general plane-parallel flow, regardless of its being laminar, turbulent, steady-state, unsteady, or transient. The Reynolds-averaged Navier–Stokes equation is posed, with proper initial and boundary conditions, and the solution is developed in an easy-to-follow fashion. The time-dependent general analytic solution is obtained, which is the sum of three components: (i) the transient decay of the initial mean velocity, (ii) the unsteady mean velocity directly created by the mean pressure gradient, and (iii) the unsteady mean velocity originated from the flow's Reynolds shear-stress gradient. Each one of these components has a different evolution in time, resulting in an asynchronism among them that yields deformed mean-velocity profiles. The formalism is applied to study the transient flow created when an initial steady-state flow is accelerated or decelerated up to a final steady state. A rapidly accelerated version of this problem was already studied in a DNS by He and Seddighi [“Turbulence in transient channel flow,” J. Fluid Mech. 715, 60–102 (2013)]. We show that DNS results concerning mean velocity are qualitatively reproduced by our approach, including surprising features like the flattening of transient mean-velocity profiles, the middle mean-velocity overshoot, global laminarization, and the enhancement of the viscous sublayer related to the notion of hyperlaminarity already introduced by the authors. The decelerated case presents sharpening of mean-velocity profiles, middle mean-velocity undershoots, global turbulentization, and the destruction of the viscous sublayer, and poses predictions that await experimental confirmation. 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

“…Therefore, near the wall the subtracting term is lower in absolute value (the PTC is generally negative), and that yields a locally higher mean velocity, which is the observed phenomenon. The flattening of U-profiles is abundantly reported in our previous papers (García García & Fariñas Alvariño 2019b,c, 2020, 2021, and occurs every time the U-ULF takes precedence over the U-PTC, or otherwise put, any time the turbulence index increases. The mathematical details can be found in those references.…”

“…(iv) 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. (v) Under its own dynamics, the ULF tends to follow the behaviour of the MPG, for Π(τ ) is the source of u L (τ, α) (García García & Fariñas Alvariño 2019b,c, 2020, 2021. The u L (τ, α) reacts rather quickly to changes in Π(τ ), although it does not trail the MPG evolution instantaneously.…”

“…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,…”

“…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.…”

“…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.…”