Session H23: Flow Control III

Modification of conservation laws for 2D turbulent flows

In two-dimensional decaying turbulence, kinetic energy and enstrophy are transferred to large and small scales, respectively.  Identifying the flow structures that affect these transfers is a critical consideration towards flow modification. We propose and numerically implement two flow modification strategies that selectively and simultaneously change kinetic energy and enstrophy to drive the system towards steady states. Energy modification excites characteristic flow structures associated with smaller wavenumbers than enstrophy modification.   

Information-theoretic control of wall-bounded turbulence

Information theory is applied to devise optimal control strategies for wall-bounded turbulence. The case considered is opposition-flow control for drag reduction in a turbulent channel flow. The design parameters are the wall-normal sensor location and the actuator amplitude of the wall jet. The cost functional depends on the Kullback-Leibler (KL) divergence of the wall stress distribution and the mutual information between the actuator and the wall-shear stress. The working principle of our approach differs significantly from traditional optimization methodologies. In the latter, the cost functional is explicitly constructed from the partial differential equations governing the system, whereas the information-theoretic approach is formulated in terms of the Shannon entropy of the flow state. It is shown that maximum drag reduction is achieved for controllers with maximum mutual information and minimum KL divergence. Our results establish the capabilities of information theory as a new venue for optimal control in wall-bounded turbulence.   

Data-Driven blowing suction control in a turbulent channel flow

We use an adaptive data-driven technique to modulate coherent motions within a turbulent channel flow. A Direct Numerical Simulation (DNS) is used with actuators placed along the channel wall, which interact with the flow field via blowing and suction based control. The actuators' actions are determined by an adaptive deep learning technique, which allows for model-free control despite the high dimensional and nonlinear nature of the turbulent flow field. Upon successfully attaining the specified high-level goal, the data-driven controller's decision-making is analyzed to understand its impact on the flow field.   

Nonlinear Optimal Control and Design Optimization: a comparison of Direct-adjoint-Looping and one-shot methods

In our previous work we have considered  PDE-constrained optimization for steady-state problems using i) Direct-Adjoint-Looping (DAL), and ii) one-shot methods. We now extend such analysis to transient problems. For the DAL method, the forward problem, described by nonlinear PDES  (e.g., unsteady Reynolds-Averaged-Navier-Stokes (URANS) or Kuramoto–Sivashinsky (KS) equation, which can be chaotic) is first solved for the given design/control variables. We then derive the continuous adjoint equations, which are solved backward in time. We discuss the checkpointing method, and some variations of it, to handle the backward solution of adjoint equations  in time. We also discuss the impact of the step-size on the convergence rate. In the one-shot method, we couple the adjoint solver with the forward solver and updates of the control variable and we use an approximate Hessian to speed up the convergence rate. We demonstrate the limitations of each method for various classes of  problems and cost functionals.   

Optimal control of PDEs using physics-informed neural networks (PINNs)

Physics-informed neural networks (PINNs) have recently become a popular method for solving forward and inverse problems governed by partial differential equations (PDEs). By incorporating the residual of the PDE into the loss function of a neural network-based surrogate model for the solution, PINNs can seamlessly blend measurement data with known physical constraints. Here, we extend this framework to optimal control problems, for which the governing PDEs are fully known except for a control variable that minimizes a desired cost objective. We show that by adding the cost objective to the training loss function of the PINN, we can find an optimal solution for the control variable. We validate the performance of our method by comparing it to adjoint-based nonlinear optimal control, which is purely based on the governing PDEs. We evaluate the pros and cons of the two approaches in light of several distributed control examples based on the Laplace, Burgers, Kuramoto-Sivashinski and Navier-Stokes equations.   

Control of Partially Observable Flows with Model-Free Reinforcement Learning

Model-free Reinforcement Learning algorithms have been employed recently to discover flow control strategies, for example for efficient drag reduction by suppressing vortex-shedding in the wake of a circular cylinder. These methods rely on probes located in the flow downstream of the body to achieve full-state observability and control. The present approach considers real-world applicability by restricting sensing to pressure probes mounted on the base of a square bluff body. Surface mounted sensing is shown to restrict observability over the flow and reduce drag reduction performance by 65% compared to probes optimally located downstream of the body. A method integrating memory into the control architecture is proposed to improve drag reduction performance in partially observable systems. Memory is integrated by augmenting the input to the controller with a time series of lagged observations. A power expenditure study shows that the active drag reduction strategies discovered with Reinforcement Learning are extremely power efficient. These results are a first step towards realistic implementation of reinforcement learning for active drag reduction in the type of partially observable systems often found in the real-world.   

Sensor-based Temporal Super-Resolution of non-Time-Resolved Flowfields using Deep Learning

This work introduces a novel method to estimate time-resolved (TR) velocity flow-fields from undersampled Particle Image Velocimetry (PIV) measurements and oversampled pressure sensor data. Feedforward and long short-term memory neural networks are utilized to estimate the temporal evolution of a low-dimensional proper orthogonal decomposition (POD) subspace. Unlike conventional techniques that correlate non-TR sensor and POD states, the proposed method leverages available TR sensor history to encode potentially missing correlations into the estimator. The efficacy of the method is demonstrated for laminar flow simulations of two side-by-side cylinders, which is characterized by highly non-linear wake interactions. The technique is also tested on a high Reynolds number turbulent separated flow over a Gaussian speed-bump validation geometry. Our results show promise in advancing sensor-based estimation in two avenues: (i) Educing unresolved dynamics through the estimated fields; (ii) Real-time sensing of turbulent flows made possible with the use of low-order subspaces and oversampled pressure sensors.   

Machine Learning flow control in the few sensors limit

A comparative assessment of machine learning (ML) methods for closed-loop wake control in configurations with a limited number of sensors is presented. The baseline flow field is a two-dimensional simulation of the Kármán vortex street past a cylinder at moderate Reynolds number (Re=100). The actuation is performed by two jets on the sides of a cylinder. The flow is monitored with several sensor probe arrangements including 5, 11 and 151 velocity signals and one case with lift and drag sensors. Two popular alternatives are evaluated: Deep Reinforcement Learning (DRL) as pioneered by Rabault et al. (2019, JFM) and Genetic Programming control (GPC) for tree-based, linear and gradient-enriched realizations as pursued by the authors. All machine learning control methods successfully stabilize the vortex shedding and effectively reduce drag while using small mass flow rates for the actuation. DRL and GPC have complementary strengths. DRL yields higher drag reductions for large numbers of probes and short training periods. In contrast, GPC performs better for cases with fewer sensors and longer training periods. The results hint at combinations of DRL and GPC for further performance improvements.   

Homogenization-based optimization and design of microstructured membranes: flow past a circular cylindrical shell

A formal framework to characterize and optimize the flow past permeable membranes through homogenization is proposed and applied to the two-dimensional wake flow past a permeable cylindrical shell. An effective stress jump condition is employed to model the presence of the membrane, where the normal and tangential velocities at the membrane are respectively proportional to the so-called filtrability and slip coefficient. The characterization of the steady flow solution for several filtrability and slip values, kept uniform over the membrane, shows that the flow is dominantly influenced by the filtrability and exhibits a recirculation region which moves downstream of the body and eventually disappears, and the suppression of the vortex shedding as long as large values of the filtrability are employed. An inverse procedure to obtain the microscopic geometry is implemented and verified in the case of uniform distributions of filtrability and slip over the membrane. Variations of the filtrability and slip along the membrane are then considered and their distributions are optimized to fulfill a given objective in the context of adjoint-based optimization. The optimal distributions are finally linked back to the microscopic geometry via a homogenization-based inverse procedure.   

Learning Galerkin Reduced Order Model Closures for Feedback Flow Control with Differentiable Programming

Turbulent flow control has numerous applications and building POD-based Galerkin projection reduced order models (GP-ROMs) of the flow and the associated feedback control laws are extremely challenging. However, a key limitation is that the ODEs arising from GP ROMs are highly susceptible to instabilities due to truncation of POD modes and lead to deterioration in accuracy. In this work, we propose a differentiable programming approach that learns stable GP-ROMs with a closure term for truncated modes, by embedding neural networks in the ODEs and integrating it with a feedback controller. We test this approach on the isentropic Navier-Stokes equations for compressible flow over a cavity at a moderate Mach number.  We show that differentiable programming as a paradigm can learn arbitrary terms corresponding to different formulations of truncated mode closures in ODEs,  while obeying its constraints. The results show significantly longer and accurate time horizon predictions and effective control when compared to the classical GP-ROM. Key benefits of the differentiable programming-based approach include superior physics-based learning, low computational costs, and a significant increase in interpretability when compared to purely data-driven vanilla neural networks.