A self-sustaining process theory for uniform momentum zones and internal shear layers in high Reynolds number shear flows

Many exact coherent states (ECS) arising in wall-bounded shear flows have an asymptotic structure at extreme Reynolds number Re in which the effective Reynolds number governing the streak and roll dynamics is O(1). Consequently, these viscous ECS are not suitable candidates for quasi-coherent structures away from the wall that necessarily are inviscid in the mean. Specifically, viscous ECS cannot account for the singular nature of the inertial domain, where the flow self-organizes into uniform momentum zones (UMZs) separated by internal shear layers and the instantaneous streamwise velocity develops a staircase-like profile. In this investigation, a large-Re asymptotic analysis is performed to explore the potential for a three-dimensional, short streamwise- and spanwise-wavelength instability of the embedded shear layers to sustain a spatially-distributed array of much larger-scale, effectively inviscid streamwise roll motions. In contrast to other self-sustaining process theories, the rolls are sufficiently strong to differentially homogenize the background shear flow, thereby providing a mechanistic explanation for the formation and maintenance of UMZs and interlaced shear layers that respects the leading-order balance structure of the mean dynamics

Gradient Information and Regularization for Gene Expression Programming to Develop Data-Driven Physics Closure Models

Learning accurate numerical constants when developing algebraic models is a known challenge for evolutionary algorithms, such as Gene Expression Programming (GEP). This paper introduces the concept of adaptive symbols to the GEP framework by Weatheritt and Sandberg (2016) to develop advanced physics closure models. Adaptive symbols utilize gradient information to learn locally optimal numerical constants during model training, for which we investigate two types of nonlinear optimization algorithms. The second contribution of this work is implementing two regularization techniques to incentivize the development of implementable and interpretable closure models. We apply $L_2$ regularization to ensure small magnitude numerical constants and devise a novel complexity metric that supports the development of low complexity models via custom symbol complexities and multi-objective optimization. This extended framework is employed to four use cases, namely rediscovering Sutherland's viscosity law, developing laminar flame speed combustion models and training two types of fluid dynamics turbulence models. The model prediction accuracy and the convergence speed of training are improved significantly across all of the more and less complex use cases, respectively. The two regularization methods are essential for developing implementable closure models and we demonstrate that the developed turbulence models substantially improve simulations over state-of-the-art models

Mean equation based scaling analysis of fully-developed turbulent channel flow with uniform heat generation

Multi-scale analysis of the mean equation for passive scalar transport is used to investigate the asymp- totic scaling structure of fully developed turbulent channel flow subjected to uniform heat generation. Unlike previous studies of channel flow heat transport with fixed surface temperature or constant inward surface flux, the present flow has a constant outward wall flux that accommodates for the volumetrically uniform heat generation. This configuration has distinct analytical advantages relative to precisely eluci- dating the underlying self-similar structure admitted by the mean transport equation. The present anal- yses are advanced using direct numerical simulations (Pirozzoli et al., 2016) that cover friction Reynolds numbers up to d þ 1⁄4 4088 and Prandtl numbers ranging from Pr 1⁄4 0:2–1:0. The leading balances of terms in the mean equation are determined empirically and then analytically described. Consistent with its asymptotic universality, the logarithmic mean temperature profile is shown analytically to arise as a sim- ilarity solution to the mean scalar equation, with this solution emerging (as d þ ! 1) on an interior domain where molecular diffusion effects are negligible. In addition to clarifying the Reynolds and Prandtl number influences on the von Kármán constant for temperature, k h , the present theory also pro- vides a couple of self-consistent ways to estimate, k h . As with previous empirical observations, the pre- sent analytical predictions for k h indicate values that are larger than found for the mean velocity von Kármán constant. The potential origin of this is briefly discussed