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Resolvent analysis identifies the most responsive forcings and most receptive states of a dynamical system, in an input--output sense, based on its governing equations. Interest in the method has continued to grow during the past decade due to its potential to reveal structures in turbulent flows, to guide sensor/actuator placement, and for flow control applications. However, resolvent analysis requires access to high-fidelity numerical solvers to produce the linearized dynamics operator. In this work, we develop a purely data-driven algorithm to perform resolvent analysis to obtain the leading forcing and response modes, without recourse to the governing equations, but instead based on snapshots of the transient evolution of linearly stable flows. The formulation of our method follows from two established facts: $1)$ dynamic mode decomposition can approximate eigenvalues and eigenvectors of the underlying operator governing the evolution of a system from measurement data, and $2)$ a projection of the resolvent operator onto an invariant subspace can be built from this learned eigendecomposition. We demonstrate the method on numerical data of the linearized complex Ginzburg--Landau equation and of three-dimensional transitional channel flow, and discuss data requirements. The ability to perform resolvent analysis in a completely equation-free and adjoint-free manner will play a significant role in lowering the barrier of entry to resolvent research and applications.
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The conceptual picture underlying resolvent analysis(RA) is that the nonlinear term in the Navier-Stokes(NS) equations provides an intrinsic forcing to the linear dynamics, a description inspired by control theory. The inverse of the linear operator, defined as the resolvent, is interpreted as a transfer function between the forcing and the velocity response. This inversion obscures the physical interpretation of the governing equations and is prohibitive to analytical manipulation, and for large systems leads to significant computational cost and memory requirements. In this work we suggest an alternative, inverse free, definition of the resolvent basis based on an extension of the Courant-Fischer-Weyl min-max principle in which resolvent modes are defined as stationary points of a constrained variational problem. This leads to a straightforward approach to approximate the resolvent (response) modes of complex flows as expansions in any basis. The proposed method avoids matrix
An analysis of the statistics of the non-linear terms in resolvent analysis is performed in this work for turbulent Couette flow at low Reynolds number. Data from a direct numerical simulation of a minimal flow unit, at Reynolds number 400, is post-processed using Fourier analysis in both time and space, leading to the covariance matrix of the velocity. From the same data, we computed the non-linear terms of the Navier-Stokes equations (treated as forcing in the present formulation), which allowed us to compute the covariance matrix of the forcing for this case. The two covariances are related exactly by the resolvent operator; based on this, we explore the recovery of the velocity statistics from the statistics of the forcing as a function of the components of the forcing term. This is carried out for the dominant structures in this flow, which participate in the self-sustaining cycle of turbulence: (i) streamwise vortices and streaks, and (ii) spanwise coherent fluctuations of spanwise velocity. The present results show a dominance by four of the non-linear terms for the prediction of the full statistics of streamwise vortices and streaks; a single term is seen to be dominant for spanwise motions. A relevant feature observed in these cases is that forcing terms have significant coherence in space; moreover, different forcing components are also coherent between them. This leads to constructive and destructive interferences that greatly modify the flow response, and should thus be accounted for in modelling work.
Using various techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz and geometric singular perturbation theory, we prove the existence of a heteroclinic loop between a turbulent and a laminar steady state and establish a cascade of bifurcations of various traveling waves mediating the transition to turbulence. The most complicated behaviour can be found in an intermediate Reynolds number regime, where the traveling waves exhibit arbitrarily long periodic-like dynamics indicating the onset of chaos. Our analysis provides a systematic mathematical approach to identifying the transition to spatio-temporal turbulent structures that may also be applicable to other models arising in fluid dynamics.
Advanced nuclear reactors often exhibit complex thermal-fluid phenomena during transients. To accurately capture such phenomena, a coarse-mesh three-dimensional (3-D) modeling capability is desired for modern nuclear-system code. In the coarse-mesh 3-D modeling of advanced-reactor transients that involve flow and heat transfer, accurately predicting the turbulent viscosity is a challenging task that requires an accurate and computationally efficient model to capture the unresolved fine-scale turbulence. In this paper, we propose a data-driven coarse-mesh turbulence model based on local flow features for the transient analysis of thermal mixing and stratification in a sodium-cooled fast reactor. The model has a coarse-mesh setup to ensure computational efficiency, while it is trained by fine-mesh computational fluid dynamics (CFD) data to ensure accuracy. A novel neural network architecture, combining a densely connected convolutional network and a long-short-term-memory network, is developed that can efficiently learn from the spatial-temporal CFD transient simulation results. The neural network model was trained and optimized on a loss-of-flow transient and demonstrated high accuracy in predicting the turbulent viscosity field during the whole transient. The trained models generalization capability was also investigated on two other transients with different inlet conditions. The study demonstrates the potential of applying the proposed data-driven approach to support the coarse-mesh multi-dimensional modeling of advanced reactors.
The use of persistently exciting data has recently been popularized in the context of data-driven analysis and control. Such data have been used to assess system theoretic properties and to construct control laws, without using a system model. Persistency of excitation is a strong condition that also allows unique identification of the underlying dynamical system from the data within a given model class. In this paper, we develop a new framework in order to work with data that are not necessarily persistently exciting. Within this framework, we investigate necessary and sufficient conditions on the informativity of data for several data-driven analysis and control problems. For certain analysis and design problems, our results reveal that persistency of excitation is not necessary. In fact, in these cases data-driven analysis/control is possible while the combination of (unique) system identification and model-based control is not. For certain other control problems, our results justify the use of persistently exciting data as data-driven control is possible only with data that are informative for system identification.
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