A statistical state dynamics approach to wall turbulence

Abstract

One contribution of 14 to a theme issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number' . This paper reviews results obtained using statistical state dynamics (SSD) that demonstrate the benefits of adopting this perspective for understanding turbulence in wall-bounded shear flows. The SSD approach used in this work employs a second-order closure that retains only the interaction between the streamwise mean flow and the streamwise mean perturbation covariance. This closure restricts nonlinearity in the SSD to that explicitly retained in the streamwise constant mean flow together with nonlinear interactions between the mean flow and the perturbation covariance. This dynamical restriction, in which explicit perturbation-perturbation nonlinearity is removed from the perturbation equation, results in a simplified dynamics referred to as the restricted nonlinear (RNL) dynamics. RNL systems, in which a finite ensemble of realizations of the perturbation equation share the same mean flow, provide tractable approximations to the SSD, which is equivalent to an infinite ensemble RNL system. This infinite ensemble system, referred to as the stochastic structural stability theory system, introduces new analysis tools for studying turbulence. RNL systems provide computationally efficient means to approximate the SSD and produce self-sustaining turbulence exhibiting qualitative features similar to those observed in direct numerical simulations despite greatly simplified dynamics. The results presented show that RNL turbulence can be supported by as few as a single streamwise varying component interacting with the streamwise constant mean flow and that judicious selection of this truncated support or 'bandlimiting' can be used to improve quantitative accuracy of RNL turbulence. These results suggest that the SSD

Restricted nonlinear model for high- and low-drag events in plane channel flow

“…Farrell & Ioannou (2012) used the RNL system with a stochastic forcing (see also the overview in Farrell et al (2017)). Their results confirmed that this RNL system supports a realistic self-sustaining process (SSP).…”

Adjoint sensitivity analysis of chaotic systems using cumulant truncation

“…The original d nonlinear equations F from (1) are recovered by differentiating (13) with respect to the vector P. Associated with the original system (1) are an infinite hierarchy of cumulant equations,…”

“…Fortunately, heterogeneous flows that are dominated by the interaction of eddies with a mean shear are amenable to relatively simple closures, because the evolution of third-order cumulants, describing eddy-eddy interactions, can sometimes be neglected [16]. Statistical state dynamics, or direct statistical simulation [46,1] has therefore been applied with success in simulations of planetary jets [37,47,14,7] and wall-bounded shear-flow [17,13]. Whilst strongly nonlinear systems, such as the model for Rayleigh-Bénard convection given by the Lorenz equations [31], require a more sophisticated treatment that accounts for the role of cumulants beyond second order, direct statistical simulation can nevertheless produce accurate predictions [2].…”

Prospectus: towards the development of high-fidelity models of wall turbulence at large Reynolds number

“…An emerging theme underlying a number of recent model constructions (such as RNL [42] and the resolvent model framework [11]) is that linear mechanisms play a greater role in the overall dynamics than perhaps previously imagined, and especially so with increasing Reynolds number. The contribution by Cossu & Hwang [12] summarizes their recent results, revealing elements that simultaneously incorporate a number of the physical and conceptual features studied by other authors in this theme issue.…”

“…[41]). In their contribution, Farrell et al [42] describe how the RNL ideas can be employed within a statistical state dynamics (SSD) framework that respectively represents the mean velocity and Reynolds stresses by the first and second cumulant of ensemble averages. These ensemble averages are constructed using a Leray projection of the Navier-Stokes equations and by employing temporal white noise forcing to generate the members of the ensemble.…”