Experiments and low-order modelling of intermittent transitions between clockwise and anticlockwise spinning thermoacoustic modes in annular combustors

Annular combustion chambers of gas turbines and aircraft engines are subject to unstable azimuthal thermoacoustic modes leading to high amplitude acoustic waves propagating in the azimuthal direction. For certain operating conditions, the propagating direction of the wave switches randomly. The strong turbulent noise prevailing in gas turbine combustors is a source of random excitation for the thermoacoustic modes and can be the cause of these switching events. A low-order model is proposed to describe qualitatively this property of the dynamics of thermoacoustic azimuthal modes. This model is based on the acoustic wave equation with a destabilizing thermoacoustic source term to account for the flame’s response and a stochastic term to account for the turbulent combustion noise. Slow-flow averaging is applied to describe the modal dynamics on times scales that are slower than the acoustic pulsation. Under certain conditions, the model reduces formally to a Fokker-Planck equation describing a stochastic diffusion process in a potential landscape with two symmetric wells: One well corresponds to a mode propagating in the clockwise direction, the other well corresponds to a mode propagating in the anticlockwise direction. When the level of turbulent noise is sufficient, the stochastic force makes the mode jump from one well to the other at random times, reproducing the phenomenon of direction switching. Experiments were conducted on a laboratory scale annular combustor featuring 12 hydrogen-methan flames. System identification techniques were used to fit the model on the experimental data, allowing to extract the potential shape and the intensity of the stochastic excitation. The statistical predictions obtained from the Fokker–Planck equation on the mode’s behaviour and the direction switching time are in good agreement with the experiments.     

Detailed unsteady dynamics of flame-flow interactions during combustion instability and its transition scenario for lean-premixed low-swirl hydrogen turbulent flames

This experimental study elucidates the unsteady dynamics of flame-flow interactions during unique thermoacoustic instability (TI) and the transition mechanism from stable combustion to TI for lean-premixed hydrogen turbulent jet flames in a low-swirl combustor (LSC), where a swirler assembly consists of an unswirled central region (CR) and an annular swirler region (SR). Simultaneous 200-kHz pressure fluctuation p’ measurements and 10-kHz OH* chemiluminescence imaging, as well as 40-kHz stereoscopic particle image velocimetry (SPIV) and two-dimensional PIV measurements for steady-state and transient data acquisitions, respectively, were conducted. The SPIV was performed in multiple planes to explore three-dimensional velocity fields. During TI, periodic flashback was possibly caused by significant axial velocity oscillations, resulting in the local mixture velocity falling below the turbulent flame speed. The large-scale vortex ring generated by the velocity oscillations caused axisymmetric radial velocity V_r oscillations with switching signs during the TI period. Similar to a typical low-swirl flow, the positive V_r away from the combustor axis created diverging flow, whereas unlike the typical flowfield, the negative V_r toward the combustor axis generated converging flow while flattening the axial velocity distributions, which was the signature phenomenon for this TI. Using the transient data and dynamic mode decomposition, variations in delay times between the mixture injection and its convection to a region with positive local Rayleigh indices were investigated. During stable combustion, the mixture jet from the SR predominantly induced thermoacoustic coupling (TC). As the combustion transitioned into the TI, the mixture jet from the CR began to induce TC and, eventually, achieved predominance in inducing TC during fully evolved TI. The transition from the SR jet- into CR jet-dominant TI arising from the dynamic flame-flow interactions resulted from the inherent physical characteristics of hydrogen flames, thereby yielding the larger p’ amplitude compared to typical TIs.     

Short-time correlations of many-body systems described by nonlinear Fokker–Planck equations and Vlasov–Fokker–Planck equations

Analytical expressions for short-time correlation functions, diffusion coefficients, mean square displacement, and second order statistics of many-body systems are derived using a mean field approach in terms of nonlinear Fokker–Planck equations and Vlasov–Fokker–Planck equations. The results are illustrated for the Desai–Zwanzig model, the nonlinear diffusion equation related to the Tsallis statistics, and a Vlasov–Fokker–Planck equation describing bunch particles in particle accelerator storage rings.     

Effect of flame response asymmetries on the modal patterns and collective states of a can-annular lean-premixed combustion system

We experimentally study the effect of rotational asymmetries in the flame response distribution on the thermoacoustic oscillations of four turbulent lean-premixed combustors coupled in a ring network. The asymmetries are created via different combinations of high-swirl (HS) and low-swirl (LS) nozzles. By analyzing the inter-combustor acoustic interactions in terms of discrete thermoacoustic modes, we find a variety of modal patterns: (i) global alternating push–pull modes emerge for most pair-wise asymmetric nozzle combinations, (ii) 2-can push–pull modes emerge for an alternating 2-fold symmetric nozzle combination, and (iii) strong mode localization and global push–push modes emerge when the HS nozzles outnumber the LS nozzles. Using a complex systems framework, we reinterpret these modal patterns as collective states, such as a weak breathing chimera, a weak anti-phase chimera, and in-phase/anti-phase synchronization. This study shows that changing the flame response distribution of a multi-combustor system, via changes in the nozzle swirl distribution, can induce a variety of modal patterns and collective states. This sets the stage for the potential use of rotational asymmetries in the passive control of thermoacoustic modes in can-annular combustion systems.     

Stochastic feedback,nonlinear families of Markov processes,and nonlinear Fokker–Planck equations

We discuss two fundamental aspects of Fokker–Planck equations that are nonlinear with respect to probability densities. First, we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions. Second, we demonstrate that nonlinear Fokker–Planck equations can be interpreted as linear Fokker–Planck equations describing nonlinear families of Markov diffusion processes. We exploit this finding in order to derive complete hierarchies of probability densities from nonlinear Fokker–Planck equations.     

Phase synchronization and collective interaction of multiple flamelets in a laboratory scale spray combustor

In this paper, we investigate the coupled behvior of the acoustic field in the confinement and the unsteady flame dynamics in a laboratory scale spray combustor. We study this interaction during the intermittency route to thermoacoustic instability when the location of the flame is varied inside the combustor. As the flame location is changed, the synchronization properties of the coupled acoustic pressure and heat release rate signals change from desynchronized aperiodicity (combustion noise) to phase synchronized periodicity (thermoacoustic instability) through intermittent phase synchronization (intermittency). We also characterize the collective interaction between the multiple flamelets anchored at the flame holder and the acoustic field in the system, during different dynamical states observed in the combustor operation. When the signals are desynchronized, we notice that the flamelets exhibit a steady combustion without the exhibition of a prominent feedback with the acoustic field. In a state of intermittent phase synchronization, we observe the existence of a short-term coupling between the heat release rate and the acoustic field. We notice that the onset of collective synchronization in the oscillations of multiple flamelets and the acoustic field leads to the simultaneous emergence of periodicity in the global dynamics of the system. This collective periodicity in both the subsystems causes enhancement of oscillations during epochs of amplitude growth in the intermittency signal. On the contrary, the weakening of the coupling induces suppression of periodic oscillations during epochs of amplitude decay in the intermittency signal. During phase synchronization, we notice a sustained synchronized movement of all flamelets with the periodicity of the acoustic cycle in the system.     

Similarity solutions of the Fokker–Planck equation with time-dependent coefficients

In this work, we consider the solvability of the Fokker–Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker–Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulting ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker–Planck equations are presented, and their properties analyzed.     

A Fokker–Planck approach to a moment closure for mixing in variable-density turbulence

ABSTRACT

We develop a theory for the cascade mixing terms in a moment closure approach to binary active scalar mixing in variable-density turbulence. To address the variable-density complications we apply, as a principle and constraint, the conservation of the probability density function (PDF) through a Fokker–Planck equation with bounded sample space whose attractor is the beta PDF with skewness. Mixing is related to a single-point PDF as a realisability principle to provide mathematically rigorous expressions for the small scale statistics in terms of largescale moments. The problem of the unknown small-scale mixing is replaced with the determination of the drift and diffusion terms of a Fokker–Planck equation in a beta-PDF-convergent stochastic process. We find that realisability of a beta-convergent process requires the mixing time-scale ratio, taken as a constant in passive scalar mixing, to be a function of the mean mass fraction, mean fluid density, the Atwood number, the density-volume correlation and moments of the density field. We develop and compare the new model with direct numerical simulations data of non-stationary homogeneous variable-density turbulence.     

Covariant formulation of non-equilibrium statistical thermodynamics

The Fokker Planck equation is considered as the master equation of macroscopic fluctuation theories. The transformation properties of this equation and quantities related to it under general coordinate transformations in phase space are studied. It is argued that all relations expressing physical properties should be manifestly covariant, i.e. independent of the special system of coordinates used. The covariance of the Langevin-equations and the Fokker Planck equation is demonstrated. The diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space. Covariant drift vectors associated to the Langevin- and the Fokker Planck equation are found. It is shown that special coordinates exist in which the covariant drift vector of the Fokker Planck equation and the usual non-covariant drift vector are equal.The physical property of detailed balance and the equivalent potential conditions are given in covariant form. Finally, the covariant formulation is used to study how macroscopic forces couple to a system in a non-equilibrium steady state. A general fluctuation-dissipation theorem for the linear response to such forces is obtained.     

Variability and Self-Organization in Nanosystems

In supersaturated media nanoparticles grow with fluctuating rate. On a surface of growing nanoparticles a mechanism of synchronization of molecular fluctuations operates increasing their scale up to macroscopic one. Thereof the kinetic equation for states distribution function of nanoparticles approaches the Fokker–Planck equation that is experimentally established on an example of nanoparticles of different nature. In a series of suspensions and aerosols, where the ordered aggregates of nanoparticles are formed, aggregation is multistage process, and at each stage the change of a size distribution function of aggregates is also described by the Fokker–Planck equation. Multistage aggregation is completed by appearance of the particles in system with multilevel hierarchical texture having tendency to accept polyhedral shape. Aforementioned features of nanosystems can be treated as development of variability and ability of systems to self-organization.     

Numerical approximation of the Vlasov–Maxwell–Fokker–Planck system in two dimensions

A numerical method is developed for solving the Vlasov–Maxwell–Fokker–Planck system in two spatial dimensions. This system of equations is a model for a collisional plasma in the presence of a self consistent electromagnetic field. The numerical procedure is a type of deterministic particle method and is an extension to include the full electromagnetic field of the approximation method of Wollman and Ozizmir [S. Wollman, E. Ozizmir, Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in two dimensions, J. Comput. Phys. 228 (2009) 6629–6669]. In addition, the long time asymptotic behavior of solutions is studied. It is determined that the solution to the Vlasov–Maxwell–Fokker–Planck system converges to the same steady state solution as that for the Vlasov–Poisson–Fokker–Planck system.     

Route to chaos for combustion instability in ducted laminar premixed flames

Complex thermoacoustic oscillations are observed experimentally in a simple laboratory combustor that burns lean premixed fuel-air mixture, as a result of nonlinear interaction between the acoustic field and the combustion processes. The application of nonlinear time series analysis, particularly techniques based on phase space reconstruction from acquired pressure data, reveals rich dynamical behavior and the existence of several complex states. A route to chaos for thermoacoustic instability is established experimentally for the first time. We show that, as the location of the heat source is gradually varied, self-excited periodic thermoacoustic oscillations undergo transition to chaos via the Ruelle-Takens scenario.     

A review of Vlasov–Fokker–Planck numerical modeling of inertial confinement fusion plasma

The interaction of intense lasers with solid matter generates a hot plasma state that is well described by the Vlasov–Fokker–Planck equation. Accurate and efficient modeling of the physics in these scenarios is highly pertinent, because it relates to experimental campaigns to produce energy by inertial confinement fusion on facilities such as the National Ignition Facility. Calculations involving the Vlasov–Fokker–Planck equation are computationally intensive, but are crucial to proper understanding of a wide variety of physical effects and instabilities in inertial fusion plasmas. In this topical review, we will introduce the background physics related to Vlasov–Fokker–Planck simulation, and then proceed to describe results from numerical simulation of inertial fusion plasma in a pedagogical manner by discussing some key numerical algorithm developments that enabled the research to take place. A qualitative comparison of the techniques is also given.     

Dynamical equations of quantum information and Gaussian channel

A subdynamic based kinetic equation (SKE) for quantum information density (QID) is presented and using this is shown that the Liouville equation, Master equation and Fokker–Planck equation for QID all share the same formalism as the density operator. This allows one to directly use QID for studying quantum communication and to construct a quantum Gaussian channel. The channel is described by a quantum Fokker–Planck equation, which permits harmonic oscillator encoded information to transmit quantum signals with quantum parallelism. The quantum dynamical mutual information for this channel is also calculated.     

Photoluminescence decay of silicon nanocrystals and Lévy stable distributions

We incorporate the tools of Lévy processes and distributions to describe the photoluminescence of silicon nanocrystals. The method relies on two novel features: first we use exact forms of one-sided Lévy distributions to get an excellent reproduction of experimental data. Then we show that the dynamics leading to photoluminescence decay can be modelled in terms of fractional Fokker–Planck equation.     

Solution of Fokker Planck equations with and without manifest detailed balance

Previous work on Fokker Planck equations with manifest detailed balance is generalized to include also the case without manifest detailed balance. The two cases are unified by exhibiting a general time reversal transformation with respect to which any Fokker Planck equation satisfies detailed balance, provided its steady state distribution exists. We also introduce a new method for solving some Fokker Planck equations with nonvanishing steady state drift by analytic continuation of the solution of a hermitian eigenvalue problem.     

Solution of Fokker Planck equations with and without manifest detailed balance

Previous work on Fokker Planck equations with manifest detailed balance is generalized to include also the case without manifest detailed balance. The two cases are unified by exhibiting a general time reversal transformation with respect to which any Fokker Planck equation satisfies detailed balance, provided its steady state distribution exists. We also introduce a new method for solving some Fokker Planck equations with nonvanishing steady state drift by analytic continuation of the solution of a hermitian eigenvalue problem.     

Diffusion front capturing schemes for a class of Fokker–Planck equations: Application to the relativistic heat equation

In this research work we introduce and analyze an explicit conservative finite difference scheme to approximate the solution of initial-boundary value problems for a class of limited diffusion Fokker–Planck equations under homogeneous Neumann boundary conditions. We show stability and positivity preserving property under a Courant–Friedrichs–Lewy parabolic time step restriction. We focus on the relativistic heat equation as a model problem of the mentioned limited diffusion Fokker–Planck equations. We analyze its dynamics and observe the presence of a singular flux and an implicit combination of nonlinear effects that include anisotropic diffusion and hyperbolic transport. We present numerical approximations of the solution of the relativistic heat equation for a set of examples in one and two dimensions including continuous initial data that develops jump discontinuities in finite time. We perform the numerical experiments through a class of explicit high order accurate conservative and stable numerical schemes and a semi-implicit nonlinear Crank–Nicolson type scheme.