13 research outputs found
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
Well-Posed Initial-Boundary Evolution in General Relativity
Maximally dissipative boundary conditions are applied to the initial-boundary
value problem for Einstein's equations in harmonic coordinates to show that it
is well-posed for homogeneous boundary data and for boundary data that is small
in a linearized sense. The method is implemented as a nonlinear evolution code
which satisfies convergence tests in the nonlinear regime and is robustly
stable in the weak field regime. A linearized version has been stably matched
to a characteristic code to compute the gravitational waveform radiated to
infinity.Comment: 5 pages, 6 figures; added another convergence plot to Fig. 2 + minor
change
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.
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
An optimization approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar
This article introduces and reviews recent work using a simple optimization technique for analysing the nonlinear stability of a state in a dynamical system. The technique can be used to identify the most efficient way to disturb a system such that it transits from one stable state to another. The key idea is introduced within the framework of a finite-dimensional set of ordinary differential equations (ODEs) and then illustrated for a very simple system of two ODEs which possesses bistability. Then the transition to turbulence problem in fluid mechanics is used to show how the technique can be formulated for a spatially-extended system described by a set of partial differential equations (the well-known Navier–Stokes equations). Within that context, the optimization technique bridges the gap between (linear) optimal perturbation theory and the (nonlinear) dynamical systems approach to fluid flows. The fact that the technique has now been recently shown to work in this very high dimensional setting augurs well for its utility in other physical systems