A unified approach for symmetries in plane parallel turbulent shear flows

Abstract

A new theoretical approach for turbulent flows based on Lie-group analysis is presented. It unifies a large set of ‘solutions’ for the mean velocity of stationary parallel turbulent shear flows. These results are not solutions in the classical sense but instead are defined by the maximum number of possible symmetries, only restricted by the flow geometry and other external constraints. The approach is derived from the Reynolds-averaged Navier–Stokes equations, the fluctuation equations, and the velocity product equations, which are the dyad product of the velocity fluctuations with the equations for the velocity fluctuations. The results include the logarithmic law of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couette flow and in the centre of a rotating channel flow, and a new exponential mean velocity profile not previously reported that is found in the mid-wake region of high Reynolds number flat-plate boundary layers. The algebraic scaling law is confirmed in both the centre and the near-wall regions in both experimental and DNS data of turbulent channel flows. In the case of the logarithmic law of the wall, the scaling with the distance from the wall arises as a result of the analysis and has not been assumed in the derivation. All solutions are consistent with the similarity of the velocity product equations to arbitrary order. A method to derive the mean velocity profiles directly from the two-point correlation equations is shown.

New perspective in statistical modeling of wall-bounded turbulence

“…40 yields a log-law; when α = 0 and β = 0, it yields a power low; when α = 0, β = 1, it yields an exponential solution. These are solutions obtained by Oberlack [21]. However, more general cases involves αβ = 0; in this case, direct integration of equation (40) yields a generalized solution…”

“…According to the Lie group theory, Oberlack [20,21] made a "maximum symmetry" assumption to determine a unique transformation of all variables for parallel flows. This transformation maintains four kinds of invariant scaling for U + versus y + , namely, "linear-linear" scaling identified in the sub-layer; "linear-log" and "log-log" scaling corresponding to log-law and power law, respectively; and a newly predicted "log-linear" scaling in the wake.…”

“…The mean momentum equation (21) can be normalized by dividing ρ w u 2 τ . Inspecting the magnitude and keeping the leading terms, we obtain an approximate balance equation (also confirmed by DNS)…”

A LES-Langevin model for turbulence

“…The characteristics of more specific type of models like eddy-viscosity models or similarity models have been studied intensively by many authors ([23, 27,22] From this review, it is clear that our originality is to seek for a systematic derivation of the stochastic effect, rooted in the Navier-Stokes dynamics. By this mean, we hope both to avoid uncontrolled, empirical modeling, and respect of all the symmetries of the original Navier-Stokes equation (a constraint not always easy to respect, see [29]). Our model may also easily be generalized to other more complicated systems, like e.g.…”

DNS and Wavelet Analysis of a Turbulent Channel Flow Rotating about the Streamwise Direction

“…The flow has several common features with the classical rotating channel flow rotating about the spanwise direction [6] but also has some different characteristics. The induction of a mean velocity in x 3 -direction [10] is the most obvious difference compared to the classical case. This cross flow can be deduced by investigating the mean momentum equation and the Reynolds stress transport equation.…”