A variational framework for flow optimization using semi-norm constraints

When considering a general system of equations describing the space-time evolution (flow) of one or several variables, the problem of the optimization over a finite period of time of a measure of the state variable at the final time is a problem of great interest in many fields. Methods already exist in order to solve this kind of optimization problem, but sometimes fail when the constraint bounding the state vector at the initial time is not a norm, meaning that some part of the state vector remains unbounded and might cause the optimization procedure to diverge. In order to regularize this problem, we propose a general method which extends the existing optimization framework in a self-consistent manner. We first derive this framework extension, and then apply it to a problem of interest. Our demonstration problem considers the transient stability properties of a one-dimensional (in space) averaged turbulent model with a space- and time-dependent model "turbulent viscosity". We believe this work has a lot of potential applications in the fluid dynamics domain for problems in which we want to control the influence of separate components of the state vector in the optimization process.Comment: 30 page

Localization of flow structures using infinity-norm optimization

International audienceStability theory based on a variational principle and finite-time direct-adjoint optimization commonly relies on the kinetic perturbation energy density E-1(t ) = (1/V-Omega) integral(Omega) e(x, t) d Omega (where e(x, t) = vertical bar u vertical bar(2)/2) as a measure of disturbance size. This type of optimization typically yields optimal perturbations that are global in the fluid domain Omega of volume V-Omega. This paper explores the use of p-norms in determining optimal perturbations for 'energy' growth over prescribed time intervals of length T. For p = 1 the traditional energy-based stability analysis is recovered, while for large p >> 1, localization of the optimal perturbations is observed which identifies confined regions, or 'hotspots', in the domain where significant energy growth can be expected. In addition, the p-norm optimization yields insight into the role and significance of various regions of the flow regarding the overall energy dynamics. As a canonical example, we choose to solve the infinity-norm optimal perturbation problem for the simple case of two-dimensional channel flow. For such a configuration, several solutions branches emerge, each of them identifying a different energy production zone in the flow: either the centre or the walls of the domain. We study several scenarios (involving centre or wall perturbations) leading to localized energy production for different optimization time intervals. Our investigation reveals that even for this simple two-dimensional channel flow, the mechanism for the production of a highly energetic and localized perturbation is not unique in time. We show that wall perturbations are optimal (with respect to the infinity-norm) for relatively short and long times, while the centre perturbations are preferred for very short and intermediate times. The developed p-norm framework is intended to facilitate worst-case analysis of shear flows and to identify localized regions supporting dominant energy growth

Optimal mixing in two-dimensional plane Poiseuille flow at finite Peclet number

International audienceWe consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in a two-dimensional plane Poiseuille flow at finite Reynolds and Péclet numbers, below the linear instability threshold. We use a nonlinear-adjoint-looping approach to identify optimal perturbations leading to maximum time-averaged energy as well as maximum mixing in a freely evolving flow, measured through the minimisation of either the passive scalar variance or the so-called mix-norm, as defined by Mathew, Mezić & Petzold (Physica D, vol. 211, 2005, pp. 23-46). We show that energy optimisation appears to lead to very weak mixing of the scalar field whereas the optimal mixing initial perturbations, despite being less energetic, are able to homogenise the scalar field very effectively. For sufficiently long time horizons, minimising the mix-norm identifies optimal initial perturbations which are very similar to those which minimise scalar variance, demonstrating that minimisation of the mix-norm is an excellent proxy for effective mixing in this finite-Péclet-number bounded flow. By analysing the time evolution from initial perturbations of several optimal mixing solutions, we demonstrate that our optimisation method can identify the dominant underlying mixing mechanism, which appears to be classical Taylor dispersion, i.e. shear-augmented diffusion. The optimal mixing proceeds in three stages. First, the optimal mixing perturbation, energised through transient amplitude growth, transports the scalar field across the channel width. In a second stage, the mean flow shear acts to disperse the scalar distribution leading to enhanced diffusion. In a final third stage, linear relaxation diffusion is observed. We also demonstrate the usefulness of the developed variational framework in a more realistic control case: mixing optimisation by prescribed streamwise velocity boundary conditions

Constraints and evolution in cosmology

We review some old and new results about strict and non strict hyperbolic formulations of the Einstein equations.Comment: To appear in the proceedings of the first Aegean summer school in General Relativity, S. Cotsakis ed. Springer Lecture Notes in Physic

Harmonic Initial-Boundary Evolution in General Relativity

Computational techniques which establish the stability of an evolution-boundary algorithm for a model wave equation with shift are incorporated into a well-posed version of the initial-boundary value problem for gravitational theory in harmonic coordinates. The resulting algorithm is implemented as a 3-dimensional numerical code which we demonstrate to provide stable, convergent Cauchy evolution in gauge wave and shifted gauge wave testbeds. Code performance is compared for Dirichlet, Neumann and Sommerfeld boundary conditions and for boundary conditions which explicitly incorporate constraint preservation. The results are used to assess strategies for obtaining physically realistic boundary data by means of Cauchy-characteristic matching.Comment: 31 pages, 14 figures, submitted to Physical Review

Harmonic coordinate method for simulating generic singularities

This paper presents both a numerical method for general relativity and an application of that method. The method involves the use of harmonic coordinates in a 3+1 code to evolve the Einstein equations with scalar field matter. In such coordinates, the terms in Einstein's equations with the highest number of derivatives take a form similar to that of the wave equation. The application is an exploration of the generic approach to the singularity for this type of matter. The preliminary results indicate that the dynamics as one approaches the singularity is locally the dynamics of the Kasner spacetimes.Comment: 5 pages, 4 figures, Revtex, discussion expanded, references adde

First order hyperbolic formalism for Numerical Relativity

The causal structure of Einstein's evolution equations is considered. We show that in general they can be written as a first order system of balance laws for any choice of slicing or shift. We also show how certain terms in the evolution equations, that can lead to numerical inaccuracies, can be eliminated by using the Hamiltonian constraint. Furthermore, we show that the entire system is hyperbolic when the time coordinate is chosen in an invariant algebraic way, and for any fixed choice of the shift. This is achieved by using the momentum constraints in such as way that no additional space or time derivatives of the equations need to be computed. The slicings that allow hyperbolicity in this formulation belong to a large class, including harmonic, maximal, and many others that have been commonly used in numerical relativity. We provide details of some of the advanced numerical methods that this formulation of the equations allows, and we also discuss certain advantages that a hyperbolic formulation provides when treating boundary conditions.Comment: To appear in Phys. Rev.

Hamiltonian Time Evolution for General Relativity

Hamiltonian time evolution in terms of an explicit parameter time is derived for general relativity, even when the constraints are not satisfied, from the Arnowitt-Deser-Misner-Teitelboim-Ashtekar action in which the slicing density $\alpha(x,t)$ is freely specified while the lapse $N=\alpha g^{1/2}$ is not. The constraint ``algebra'' becomes a well-posed evolution system for the constraints; this system is the twice-contracted Bianchi identity when $R_{ij}=0$. The Hamiltonian constraint is an initial value constraint which determines $g^{1/2}$ and hence $N$, given $\alpha$.Comment: 4 pages, revtex, to appear in Phys. Rev. Let

Yang's gravitational theory

Yang's pure space equations (C.N. Yang, Phys. Rev. Lett. v.33, p.445 (1974)) generalize Einstein's gravitational equations, while coming from gauge theory. We study these equations from a number of vantage points: summarizing the work done previously, comparing them with the Einstein equations and investigating their properties. In particular, the initial value problem is discussed and a number of results are presented for these equations with common energy-momentum tensors.Comment: 28 pages, to appear in Gen. Rel. Gra

Geometrical Hyperbolic Systems for General Relativity and Gauge Theories

The evolution equations of Einstein's theory and of Maxwell's theory---the latter used as a simple model to illustrate the former--- are written in gauge covariant first order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom [the spatial shift vector $\beta^{i}(t,x^{j})$ and the spatial scalar potential $\phi(t,x^{j})$, respectively] are not among the dynamical variables: the gauge and the physical quantities in the evolution equations are effectively decoupled. For example, the gauge quantities could be obtained as functions of $(t,x^{j})$ from subsidiary equations that are not part of the evolution equations. Propagation of certain (``radiative'') dynamical variables along the physical light cone is gauge invariant while the remaining dynamical variables are dragged along the axes orthogonal to the spacelike time slices by the propagating variables. We obtain these results by $(1)$ taking a further time derivative of the equation of motion of the canonical momentum, and $(2)$ adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier $\beta^{i}$) or of the Gauss's law constraint of electromagnetism (Lagrange multiplier $\phi$). General relativity also requires a harmonic time slicing condition or a specific generalization of it that brings in the Hamiltonian constraint when we pass to first order symmetric form. The dynamically propagating gravity fields straightforwardly determine the ``electric'' or ``tidal'' parts of the Riemann tensor.Comment: 24 pages, latex, no figure