Bartosz Protas

Drag force in the open-loop control of the cylinder wake in the laminar regime

In this paper we are interested in identifying the physical mechanisms that accompany mean drag modifications in the cylinder wake flow subject to rotary control. We consider simple control laws where the obstacle rotates harmonically with frequencies varying from half to more than five natural frequencies. In our investigation we analyze the results of the numerical simulations at Re=150. All the simulations were performed using the vortex method, which in the paper is outlined and benchmarked. We confirm the earlier findings concerning mean drag reduction at higher forcing frequencies and show that for the considered values of Re this control technique is energetically inefficient. The main result is that changes of the mean drag are achieved by modifying the Reynolds stresses and the related mean flow correction. The controlled flows are carefully characterized in terms of these fields. Drag reduction is related to elongation of the recirculation bubble. It is argued that mean drag reduction is associa...

Optimal rotary control of the cylinder wake in the laminar regime

In this paper we develop the Optimal Control Approach to the rotary control of the cylinder wake. We minimize the functional which represents the sum of the work needed to resist the drag force and the work needed to control the flow, where the rotation rate φ(t) is the control variable. Sensitivity of the functional to control is determined using the adjoint equations. We solve them in the “vorticity” form, which is a novel approach and leads to computational advantages. Simulations performed at Re=75 and Re=150 reveal systematic decrease of the total power and drag achieved using a very small amount of control effort. We investigate the effect of the optimization horizon on the performance of the algorithm and the correlation of the optimal controls with the changes of the flow pattern. The algorithm was also applied to the control of the subcritical flow at Re=40, however, no drag reduction was achieved in this case. Based on this, limits of the performance of the algorithm are discussed.

Linear feedback stabilization of laminar vortex shedding based on a point vortex model

In this paper we use the Foppl point vortex system as a reduced-order model for stabilization of the steady symmetric solution in an unstable laminar wake. The downstream location of the Foppl vortices is chosen so as to produce the same recirculation length as in the actual flow at a given Reynolds number. When the cylinder rotation is used as flow actuation, the linearized Foppl system is shown to be stabilizable, but not controllable. With centerline velocity measurements as the system output, the linearized Foppl model is also shown to be fully observable. The Linear-Quadratic-Gaussian (LQG) control design is performed based on the linearized Foppl system which has only four degrees of freedom. Computational results show that thus designed LQG compensator stabilizes the stationary solution of the nonlinear Foppl system. When applied to an actual cylinder wake at Re=75, the LQG compensator stabilizes the downstream region of the flow. Possibilities and limitations of flow control strategies based on po...

Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows

We propose a variational approach to the identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for proper orthogonal decomposition (POD) models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at Re=500 (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at Re=300000 (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark proposed recently by Osth et al. (J. Fluid Mech., vol. 747, 2014, pp. 518–544). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.

Accurate Characterization of Ion Transport Properties in Binary Symmetric Electrolytes Using In Situ NMR Imaging and Inverse Modeling

We used NMR imaging (MRI) combined with data analysis based on inverse modeling of the mass transport problem to determine ionic diffusion coefficients and transference numbers in electrolyte solutions of interest for Li-ion batteries. Sensitivity analyses have shown that accurate estimates of these parameters (as a function of concentration) are critical to the reliability of the predictions provided by models of porous electrodes. The inverse modeling (IM) solution was generated with an extension of the Planck-Nernst model for the transport of ionic species in electrolyte solutions. Concentration-dependent diffusion coefficients and transference numbers were derived using concentration profiles obtained from in situ (19)F MRI measurements. Material properties were reconstructed under minimal assumptions using methods of variational optimization to minimize the least-squares deviation between experimental and simulated concentration values with uncertainty of the reconstructions quantified using a Monte Carlo analysis. The diffusion coefficients obtained by pulsed field gradient NMR (PFG-NMR) fall within the 95% confidence bounds for the diffusion coefficient values obtained by the MRI+IM method. The MRI+IM method also yields the concentration dependence of the Li(+) transference number in agreement with trends obtained by electrochemical methods for similar systems and with predictions of theoretical models for concentrated electrolyte solutions, in marked contrast to the salt concentration dependence of transport numbers determined from PFG-NMR data.

On geometrical alignment properties of two-dimensional forced turbulence

In the present paper we study some geometrical properties of the small scales in 2D numerical turbulence. We analyze the alignment of the vorticity gradient with respect to the eigenvectors of the rate of strain tensor, a phenomenon related to the dynamics of the enstrophy cascade. Numerical simulations with different resolutions and dissipation models are used to show non-trivial dependence of the alignment on both the magnitude of the vorticity gradient and the Reynolds number. These findings are shown to be dynamical in origin and imply organization of the small scales in the flow. ©1999 Elsevier Science B.V. All rights reserved.

Vortex dynamics models in flow control problems

In this article we review the state of the art in the field of control of vortex dynamics. We focus on problems governed by two-dimensional incompressible Euler equations in domains both with and without boundaries. Following a comprehensive review of earlier approaches, we discuss how methods of modern control and optimization theory can be employed to solve control problems for vortex systems. In addition, we address the companion problem of the state estimation for vortex systems. While most of the discussion concerns point vortex systems, in the second part of the article we also introduce a novel approach to the control of Euler flows involving finite-area vorticity distributions. The article concludes with what, in the authors opinion, represent promising new research directions.

Controlling the dual cascade of two-dimensional turbulence

The Kraichnan–Leith–Batchelor (KLB) theory of statistically stationary forced homogeneous isotropic two-dimensional turbulence predicts the existence of two inertial ranges: an energy inertial range with an energy spectrum scaling of k −5/3 , and an enstrophy inertial range with an energy spectrum scaling of k −3 . However, unlike the analogous Kolmogorov theory for three-dimensional turbulence, the scaling of the enstrophy range in the two-dimensional turbulence seems to be Reynolds-number-dependent: numerical simulations have shown that as Reynolds number tends to infinity, the enstrophy range of the energy spectrum converges to the KLB prediction, i.e. E ~ k −3 . The present paper uses a novel optimal control approach to find a forcing that does produce the KLB scaling of the energy spectrum in a moderate Reynolds number flow. We show that the time–space structure of the forcing can significantly alter the scaling of the energy spectrum over inertial ranges. A careful analysis of the optimal forcing suggests that it is unlikely to be realized in nature, or by a simple numerical model.

Adjoint-based optimization of PDEs in moving domains

In this investigation we address the problem of adjoint-based optimization of PDE systems in moving domains. As an example we consider the one-dimensional heat equation with prescribed boundary temperatures and heat fluxes. We discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional. In the first approach we derive the adjoint system after mapping the problem to a fixed domain, whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus. We show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that, while the gradient information contained in both systems is the same, the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically. This approach is then used to solve a moving boundary optimization problem for our model system.

Optimal reconstruction of material properties in complex multiphysics phenomena

We develop an optimization-based approach to the problem of reconstructing temperature-dependent material properties in complex thermo-fluid systems described by the equations for the conservation of mass, momentum and energy. Our goal is to estimate the temperature dependence of the viscosity coefficient in the momentum equation based on some noisy temperature measurements, where the temperature is governed by a separate energy equation. We show that an elegant and computationally efficient solution of this inverse problem is obtained by formulating it as a PDE-constrained optimization problem which can be solved with a gradient-based descent method. A key element of the proposed approach, the cost functional gradients are characterized by mathematical structure quite different than in typical problems of PDE-constrained optimization and are expressed in terms of integrals defined over the level sets of the temperature field. Advanced techniques of integration on manifolds are required to evaluate numerically such gradients, and we systematically compare three different methods. As a model system we consider a two-dimensional unsteady flow in a lid-driven cavity with heat transfer, and present a number of computational tests to validate our approach and illustrate its performance.