Resolvent-based estimation of space–time flow statistics

Abstract

We develop a method to estimate space-time flow statistics from a limited set of known data. While previous work has focused on modeling spatial or temporal statistics independently, space-time statistics carry fundamental information about the physics and coherent motions of the flow and provide a starting point for low-order modeling and flow control efforts. The method is derived using a statistical interpretation of resolvent analysis. The central idea of our approach is to use known data to infer the statistics of the nonlinear terms that constitute a forcing on the linearized Navier-Stokes equations, which in turn imply values for the remaining unknown flow statistics through application of the resolvent operator. Rather than making an a priori rank-1 assumption, our method allows the known input data to select the most relevant portions of the resolvent operator for describing the data, making it well-suited for highrank turbulent flows. We demonstrate the predictive capabilities of the method using two examples: the Ginzburg-Landau equation, which serves as a convenient model for a convectively unstable flow, and a turbulent channel flow at low Reynolds number. * Email address for correspondence: towne@umich.edu power spectral densities (PSDs) using knowledge of the mean flow field and power spectra at a few locations. This is accomplished using a least-squares fit at each frequency between the known power spectra and the leading singular response mode obtained from the resolvent operator (McKeon & Sharma 2010), which is derived from the linearized Navier-Stokes equations. This strategy explicitly assumes that the spectral content at frequencies of interest is dominated by the leading resolvent mode, and the method performs well when the matching points are located in regions where this hypothesis is valid. Specifically, excellent PSD estimates were obtained for the flow over a backward-facing step (Beneddine et al. 2016) and an initially laminar jet (Beneddine et al. 2017).Zare et al. (2017) developed a method that uses arbitrary known entries in the spatial covariance tensor to estimate the remaining unknown entries. Their approach is also based on linearized flow equations and entails solving a convex optimization problem that determines a matrix controlling the structure and statistics of the associated nonlinear forcing terms. The optimization problem is subject to two constraints on the estimated covariance tensor: it must reproduce the known entries and obey a Lyapunov equation that relates the forcing and flow statistics. The constrained optimization problem is computationally demanding and requires a customized algorithm (Zare et al. 2015(Zare et al. , 2017.The objective of the present paper is to build on these previous methods to estimate unknown two-point space-time flow statistics. Both the PSDs (one-point temporal statistics) and spatial covariances (two-point spatial statistics) are subsets of two-point space-time correlations, so our approach represents a generalization of these previous m...

Stochastic Dynamical Modeling of Turbulent Flows

“…For the modified dynamics given by Equation 22, Figure 11(b) shows the premultiplied spatio-temporal energy spectrum as a function of the wall-normal coordinate and temporal frequency in inner (viscous) units, i.e., y + := (1 + y)Re and ω + := ω/Re. This spectrum is computed by integrating ω diag (Tvw(k, ω)T * vw (k, ω)) over k and is concentrated around y + ≈ 15 within a frequency band ω + ∈ (0.01, 1), which is in agreement with the trends observed in DNS-generated energy spectra (82). Improving the accuracy in matching the temporal correlations resulting from DNS may require closer examination of the role of parameter γ or the addition of extra constraints in problem CC-1 and is a subject of ongoing research.…”

“…As described in Section 3.2, the temporal dependence of such statistics is captured by the spectral density matrix Svv(k, ω). This matrix can be used to provide real-time estimates of the flow state (79), and recent efforts have been directed at estimating Svv(k, ω) by either matching individual entries at specified temporal frequencies (80,81,82) or the spectral power (83), trace (Svv(k, ω)). Either way it should be independently considered whether the so-constructed colored-in-time forcing models preserve important aspects of the original linearized NS dynamics.…”

A tale of two airfoils: resolvent-based modelling of an oscillator versus an amplifier from an experimental mean

“…the A10 case, it could be possible to devise new strategies for predicting their statistics, e.g. Towne et al (2019), or modelling their effect on the large-scale structures, e.g. Illingworth et al (2018).…”

“…whereŜff (ω) = E{f (x, ω)f * (x, ω)} is the cross-spectral density of the nonlinear forcing as in Towne et al (2019).Ŝûû(ω) may be low-rank in two scenarios. The first is when the resolvent operator is low-rank, which occurs when there is a linear amplification mechanism.…”

Self-similar hierarchies and attached eddies

“…The weights χ j (λ, c) hold the key to nonlinear closure of the resolvent framework; the work herein suggests that analytical progress to complement data-driven resolvent approaches, e.g., Refs. [20,28,29], may be made. Connections between the resolvent results, the AEM, the self-similar minimal unit and exact coherent solutions, and the mean flow similarity of the MMB are the topic of ongoing work.…”