We consider the question of fundamental limitations on the performance of eddy-viscosity closure models for turbulent flows, focusing on the Leith model for 2D Large-Eddy Simulation. Optimal eddy viscosities depending on the magnitude of the vorticit
y gradient are determined subject to minimum assumptions by solving PDE-constrained optimization problems defined such that the corresponding optimal Large-Eddy Simulation best matches the Direct Numerical Simulation. The main finding is that with a fixed cutoff wavenumber $k_c$, the performance of the Large-Eddy Simulation systematically improves as the regularization in the solution of the optimization problem is reduced and this is achieved with the optimal eddy viscosities exhibiting increasingly irregular behavior with rapid oscillations. Since the optimal eddy viscosities do not converge to a well-defined limit as the regularization vanishes, we conclude that the problem of finding an optimal eddy viscosity is not in fact well posed.
The goal of this study is to analyze the fine structure of nonlinear modal interactions in different 1D Burgers and 3D Navier-Stokes flows. This analysis is focused on preferential alignments characterizing the phases of Fourier modes participating i
n triadic interactions, which are key to determining the nature of energy fluxes between different scales. We develop novel diagnostic tools designed to probe the level of coherence among triadic interactions realizing different flow scenarios. We consider extreme 1D viscous Burgers flows and 3D Navier-Stokes flows which are complemented by singularity-forming inviscid Burgers flows as well as viscous Burgers flows and Navier-Stokes flows corresponding to generic turbulent and simple unimodal initial data, such as the Taylor-Green vortex. The main finding is that while the extreme viscous Burgers and Navier-Stokes flows reveal the same relative level of enstrophy amplification by nonlinear effects, this behaviour is realized via modal interactions with vastly different levels of coherence. In the viscous Burgers flows the flux-carrying triads have phase values which saturate the nonlinearity thereby maximizing the energy flux towards small scales. On the other hand, in 3D Navier-Stokes flows with the extreme initial data the energy flux to small scales is realized by a very small subset of helical triads. The second main finding concerns the role of initial coherence. Comparison of the flows resulting from the extreme and generic initial conditions shows striking similarities between these two types of flows, for the 1D viscous Burgers equation as well as the 3D Navier-Stokes equation.
We consider the problem of parameterizing Newman-type models of Li-ion batteries focusing on quantifying the inherent uncertainty of this process and its dependence on the discharge rate. In order to rule out genuine experimental error and instead is
olate the intrinsic uncertainty of model fitting, we concentrate on an idealized setting where synthetic measurements in the form of voltage curves are manufactured using the full, and most accurate, Newman model with parameter values considered true, whereas parameterization is performed using simplifi
We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous resul
ts known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyze equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite-aspect-ratio limit of the stability results known for Kirchhoffs ellipse (Love 1893; Mitchell & Rossi 2008) and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite-aspect-ratio limits of the analogous solutions known for vortex patches.
This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system --- as long as this quantity remains fi
nite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy $mathcal{E}_0$ are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time $T$. Such problems are solved computationally using a large-scale adjoint-based gradient approach derived in the continuous setting. By solving these problems for a broad range of values of $mathcal{E}_0$ and $T$, we demonstrate that the maximum growth of enstrophy is in fact finite and scales in proportion to $mathcal{E}_0^{3/2}$ as $mathcal{E}_0$ becomes large. Thus, in such worst-case scenario the enstrophy still remains bounded for all times and there is no evidence for formation of singularity in finite time. We also analyze properties of the Navier-Stokes flows leading to the extreme enstrophy values and show that this behavior is realized by a series of vortex reconnection events.
We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the Riemann-Hilbert
problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density is real-valued and also vanishes at all sheet endpoints such that the induced velocity field is well-defined; then, the circulation density is determined by evaluating a certain integral formula. As an illustration of this approach, we construct a family of rotating equilibria involving different numbers of straight vortex sheets rotating about a common center of rotation and with endpoints at the vertices of a regular polygon. This equilibrium generalizes the well-known solution involving single rotating vortex sheet. With the geometry of the configuration specified analytically, the corresponding circulation densities are obtained in terms of a integral expression which in some cases lends itself to an explicit evaluation. It is argued that as the number of sheets in the equilibrium configuration increases to infinity, the equilibrium converges in a certain distributional sense to a hollow vortex bounded by a constant-intensity vortex sheet, which is also a known equilibrium solution of the two-dimensional Euler equations.
The shallow water equations (SWE) are a widely used model for the propagation of surface waves on the oceans. We consider the problem of optimally determining the initial conditions for the one-dimensional SWE in an unbounded domain from a small set
of observations of the sea surface height. In the linear case we prove a theorem that gives sufficient conditions for convergence to the true initial conditions. At least two observation points must be used and at least one pair of observation points must be spaced more closely than half the effective minimum wavelength of the energy spectrum of the initial conditions. This result also applies to the linear wave equation. Our analysis is confirmed by numerical experiments for both the linear and nonlinear SWE data assimilation problems. These results show that convergence rates improve with increasing numbers of observation points and that at least three observation points are required for the practically useful results. Better results are obtained for the nonlinear equations provided more than two observation points are used. This paper is a first step in understanding the conditions for observability of the SWE for small numbers of observation points in more physically realistic settings.
This paper concerns feedback stabilization of point vortex equilibria above an inclined thin plate and a three-plate configuration known as the Kasper Wing in the presence of an oncoming uniform flow. The flow is assumed to be potential and is modele
d by the 2D incompressible Euler equations. Actuation has the form of blowing and suction localized on the main plate and is represented in terms of a sink-source singularity, whereas measurement of pressure across the plate serves as system output. We focus on point-vortex equilibria forming a one-parameter family with locus approaching the trailing edge of the main plate and show that these equilibria are either unstable or neutrally stable. Using methods of linear control theory we find that the system dynamics linearised around these equilibria are both controllable and observable for almost all actuator and sensor locations. The design of the feedback control is based on the Linear-Quadratic-Gaussian (LQG) compensator. Computational results demonstrate the effectiveness of this control and the key finding is that Kasper Wing configurations are in general more controllable than their single plate counterparts and also exhibit larger basins of attraction under LQG feedback control. The feedback control is then applied to systems with additional perturbations added to the flow in the form of random fluctuations of the angle of attack and a vorticity shedding mechanism. Another important observation is that, in the presence of these additional perturbations, the control remains robust, provided the system does not deviate too far from its original state. Furthermore, introducing a vorticity shedding mechanism tends to enhance the effectiveness of the control. Physical interpretation is provided for the results of the controllability and observability analysis as well as the response of the feedback control to different perturbations.
In this investigation we revisit the concept of effective free surfaces arising in the solution of the time-averaged fluid dynamics equations in the presence of free boundaries. This work is motivated by applications of the optimization and optimal c
ontrol theory to problems involving free surfaces, where the time-dependent formulations lead to many technical difficulties which are however alleviated when steady governing equations are used instead. By introducing a number of precisely stated assumptions, we develop and validate an approach in which the interface between the different phases, understood in the time-averaged sense, is sharp. In the proposed formulation the terms representing the fluctuations of the free boundaries and of the hydrodynamic quantities appear as boundary conditions on the effective surface and require suitable closure models. As a simple model problem we consider impingement of free-falling droplets onto a fluid in a pool with a free surface, and a simple algebraic closure model is proposed for this system. The resulting averaged equations are of the free-boundary type and an efficient computational approach based on shape optimization formulation is developed for their solution. The computed effective surfaces exhibit consistent dependence on the problem parameters and compare favorably with the results obtained when the data from the actual time-dependent problem is used in lieu of the closure model.
