The wall-normal extent of the large-scale structures modelled by the linearized Navier-Stokes equations subject to stochastic forcing is directly compared to Direct Numerical Simulation (DNS) data. A turbulent channel flow at a friction Reynolds number of Re τ = 2000 is considered. We use the two-dimensional (2-D) linear coherence spectrum (LCS) to perform the comparison over a wide range of energy-carrying streamwise and spanwise length scales. The study of the 2-D LCS from DNS indicates the presence of large-scale structures that are coherent over large wall-normal distances and that are self-similar. We find that, with the addition of an eddy viscosity profile, these features of the largescale structures are captured by the linearized equations, except in the region close to the wall. To further study this coherence, a coherence-based estimation technique, spectral linear stochastic estimation (SLSE), is used to build linear estimators from the linearized Navier-Stokes equations. The estimator uses the instantaneous streamwise velocity field or the 2-D streamwise energy spectrum at one wall-normal location (obtained from DNS), to predict the same quantity at a different wall-normal location. We find that the addition of an eddy viscosity profile significantly improves the estimation.

Abstract

An understanding of how individuals shape and impact the evolution of society is vastly limited due to the unavailability of large-scale reliable datasets that can simultaneously capture information regarding individual movements and social interactions. We believe that the popular Indian film industry, “Bollywood”, can provide a social network apt for such a study. Bollywood provides massive amounts of real, unbiased data that spans more than 100 years, and hence this network has been used as a model for the present paper. The nodes which maintain a moderate degree or widely cooperate with the other nodes of the network tend to be more fit (measured as the success of the node in the industry) in comparison to the other nodes. The analysis carried forth in the current work, using a conjoined framework of complex network theory and random matrix theory, aims to quantify the elements that determine the fitness of an individual node and the factors that contribute to the robustness of a network. The authors of this paper believe that the method of study used in the current paper can be extended to study various other industries and organizations.

We use the linearized Navier-Stokes equations to study the large-scale flow structures in unstably stratified turbulent channel flows. The impulse response of the linear operator at bulk Richardson numbers from Ri b = 0.001 to Ri b = 1.0 are considered, corresponding to the increasing influence of buoyancy relative to shear. We compare the streamwiseconstant flow structures predicted by the linear model to the quasistreamwise rolls that emerge with increasing Ri b in direct numerical simulations (DNSs) [e.g., Pirozzoli et al., J. Fluid Mech. 821, 482 (2017)]. The linearized Navier-Stokes equations augmented with eddy-viscosity and eddy-diffusivity capture the emergence of the quasistreamwise rolls well. With increasing Ri b , the temperature fluctuations of the streamwise-constant flow structures transition from having a peak in intensity at the channel centerline to having two peaks, one at each wall, consistent with DNS.