Armin Zare

Colour of turbulence

In this paper, we address the problem of how to account for second-order statistics of turbulent flows using low-complexity stochastic dynamical models based on the linearized Navier–Stokes equations. The complexity is quantified by the number of degrees of freedom in the linearized evolution model that are directly influenced by stochastic excitation sources. For the case where only a subset of velocity correlations are known, we develop a framework to complete unavailable second-order statistics in a way that is consistent with linearization around turbulent mean velocity. In general, white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics. We develop models for coloured-in-time forcing using a maximum entropy formulation together with a regularization that serves as a proxy for rank minimization. We show that coloured-in-time excitation of the Navier–Stokes equations can also be interpreted as a low-rank modification to the generator of the linearized dynamics. Our method provides a data-driven refinement of models that originate from first principles and captures complex dynamics of turbulent flows in a way that is tractable for analysis, optimization and control design.

Low-Complexity Modeling of Partially Available Second-Order Statistics: Theory and an Efficient Matrix Completion Algorithm

State statistics of linear systems satisfy certain structural constraints that arise from the underlying dynamics and the directionality of input disturbances. In the present paper, we study the problem of completing partially known state statistics. Our aim is to develop tools that can be used in the context of control-oriented modeling of large-scale dynamical systems. For the type of applications we have in mind, the dynamical interaction between state variables is known while the directionality and dynamics of input excitation is often uncertain. Thus, the goal of the mathematical problem that we formulate is to identify the dynamics and directionality of input excitation in order to explain and complete observed sample statistics. More specifically, we seek to explain correlation data with the least number of possible input disturbance channels. We formulate this inverse problem as rank minimization, and for its solution, we employ a convex relaxation based on the nuclear norm. The resulting optimization problem is cast as a semidefinite program and can be solved using general-purpose solvers. For problem sizes that these solvers cannot handle, we develop a customized alternating minimization algorithm (AMA). We interpret AMA as a proximal gradient for the dual problem and prove sublinear convergence for the algorithm with fixed step-size. We conclude with an example that illustrates the utility of our modeling and optimization framework and draw contrast between AMA and the commonly used alternating direction method of multipliers (ADMM) algorithm.

Alternating direction optimization algorithms for covariance completion problems

Second-order statistics of nonlinear dynamical systems can be obtained from experiments or numerical simulations. These statistics are relevant in understanding the fundamental physics, e.g., of fluid flows, and are useful for developing low-complexity models. Such models can be used for the purpose of control design and analysis. In many applications, only certain second-order statistics of a limited number of states are available. Thus, it is of interest to complete partially specified covariance matrices in a way that is consistent with the linearized dynamics. The dynamics impose structural constraints on admissible forcing correlations and state statistics. Solutions to such completion problems can be used to obtain stochastically driven linearized models. Herein, we address the covariance completion problem. We introduce an optimization criterion that combines the nuclear norm together with an entropy functional. The two, together, provide a numerically stable and scalable computational approach which is aimed at low complexity structures for stochastic forcing that can account for the observed statistics. We develop customized algorithms based on alternating direction methods that are well-suited for large scale problems.

Completion of partially known turbulent flow statistics

Second-order statistics of turbulent flows can be obtained either experimentally or via high fidelity numerical simulations. The statistics are relevant in understanding fundamental flow physics and for the development of low-complexity models. For example, such models can be used for control design in order to suppress or promote turbulence. Due to experimental or numerical limitations it is often the case that only partial flow statistics are known. In other words, only certain correlations between a limited number of flow field components are available. Thus, it is of interest to complete the statistical signature of the flow field in a way that is consistent with the known dynamics. Our approach to this inverse problem relies on a model governed by stochastically forced linearized Navier-Stokes equations. In this, the statistics of forcing are unknown and sought to explain the given correlations. Identifying suitable stochastic forcing allows us to complete the correlation data of the velocity field. While the system dynamics impose a linear constraint on the admissible correlations, such an inverse problem admits many solutions for the forcing correlations. We use nuclear norm minimization to obtain correlation structures of low complexity. This complexity translates into dimensionality of spatio-temporal filters that can be used to generate the identified forcing statistics.

The use of the r* heuristic in covariance completion problems

We consider a class of structured covariance completion problems which aim to complete partially known sample statistics in a way that is consistent with the underlying linear dynamics. The statistics of stochastic inputs are unknown and sought to explain the given correlations. Such inverse problems admit many solutions for the forcing correlations, but can be interpreted as an optimal low-rank approximation problem for identifying forcing models of low complexity. On the other hand, the quality of completion can be improved by utilizing information regarding the magnitude of unknown entries. We generalize theoretical results regarding the r* norm approximation and demonstrate the performance of this heuristic in completing partially available statistics using stochastically-driven linear models.

Perturbation of system dynamics and the covariance completion problem

We consider the problem of completing partially known sample statistics in a way that is consistent with underlying stochastically driven linear dynamics. Neither the statistics nor the dynamics are precisely known. Thus, our objective is to reconcile the two in a parsimonious manner. To this end, we formulate a convex optimization problem to match available covariance data while minimizing the energy required to adjust the dynamics by a suitable low-rank perturbation. The solution to the optimization problem provides information about critical directions that have maximal effect in bringing model and statistics in agreement.

Low-complexity stochastic modeling of spatially-evolving flows

Low-complexity approximations of the Navier-Stokes (NS) equations are commonly used for analysis and control of turbulent flows. In particular, stochastically-forced linearized models have been successfully employed to capture structural and statistical features observed in experiments and high-fidelity simulations. In this work, we utilize stochastically-forced linearized NS equations and the parabolized stability equations to study the dynamics of flow fluctuations in transitional boundary layers. The parabolized model can be used to efficiently propagate statistics of stochastic disturbances into statistics of velocity fluctuations. Our study provides insight into the interaction of the slowly-varying base flow with streamwise streaks and Tollmien-Schlichting waves. It also offers a systematic, computationally efficient framework for quantifying the influence of stochastic excitation sources (e.g., free-stream turbulence and surface roughness) on velocity fluctuations in weakly non-parallel flows.

The effect of sponge layers on global stability analysis of Blasius boundary layer flow

In this paper, we conduct a parametric study on the influence of sponge layer strength on temporal eigenvalue problems arising from the one-dimensional wave equation and the linearized Navier-Stokes equations. Sponge layers have shown to stabilize eigenmodes and introduce additional spatial growth to eigenfunctions. As the strength of sponge layers increases, temporal eigenvalues are displaced and the spatial growth rates of their associated eigenfunctions are modified. In both wave and linearized Navier-Stokes equations, the linear relationship between temporal damping and spatial growth can be specified as an approximate dispersion relation. It can also be shown that an over strengthened sponge layer can reflect spatially propagating waves. This reflection can lead to a destabilization of the otherwise stable eigenspectrum with alteration of eigenfunction wavelengths. We provide an empirical guideline for determining the desirable sponge layer strength and demonstrate the efficacy of our method in the global stability analysis of the linearized Navier-Stokes equations.

Turbulent drag reduction by streamwise traveling waves

For a turbulent channel flow with zero-net-mass-flux surface actuation in the form of streamwise traveling waves we develop a model-based approach to design control parameters that can reduce skin-friction drag. In contrast to the traditional approach that relies on numerical simulations and experiments, we use turbulence modeling in conjunction with stochastically forced linearized equations to determine the effect of small amplitude traveling waves on the drag. Our simulation-free approach is capable of identifying drag reducing trends in traveling waves with various control parameters. High-fidelity simulations are used to verify quality of our theoretical predictions.