From G - Equation to Michelson - Sivashinsky Equation in Turbulent Premixed Combustion Modelling

It is well known that the Michelson-Sivashinky equation describes hydrodynamic instabilities in turbulent premixed combustion. Here a formulation of the flame front propagation based on the G-equation and on stochastic fluctuations imposed to the average flame position is considered to derive the Michelson-Sivashinky equation from a modelling point of view. The same approach was shown to reproduce the G-equation along the motion of the mean flame position, when the stochastic fluctuations are removed, as well ast the Zimont & Lipatnikov model, when a plane front is assumed. The new results here presented support this promising approach as a novel and general stochastic formulation for modelling turbulent premixed combustion

Self-similar stochastic models with stationary increments for symmetric space-time fractional diffusion

An approach to develop stochastic models for studying anomalous diffusion is proposed. In particular, in this approach the stochastic particle trajectory is based on the fractional Brownian motion but, for any realization, it is multiplied by an independent random variable properly distributed. The resulting probability density function for particle displacement can be represented by an integral formula of subordination type and, in the single-point case, it emerges to be equal to the solution of the spatially symmetric space-time fractional diffusion equation. Due to the fractional Brownian motion, this class of stochastic processes is self-similar with stationary increments in nature and uniquely defined by the mean and the auto-covariance structure analogously to the Gaussian processes. Special cases are the time-fractional diffusion, the space-fractional diffusion and the classical Gaussian diffusion

Front Curvature Evolution and Hydrodynamics Instabilities

It is known that hydrodynamic instabilities in turbulent premixed combustion are described by the Michelson-Sivashinsky (MS) equation. A model of the flame front propagation based on the G-equation and on stochastic fluctuations imposed to the mean flame position is considered. By comparing the governing equation of this model and the MS equation, an equation is derived for the front curvature computed in the mean flame position. The evolution in time of the curvature emerges to be driven by the inverse of the dispersion relation and by the nonlinear term of the MS equation.PhD grant “La Caixa 2014

The role of the environment in front propagation

In this work we study the role of a complex environment in the propagation of a front with curvature-dependent speed. The motion of the front is split into a drifting part and a fluctuating part. The drifting part is obtained by using the level set method, and the fluctuating part by a probability density function that gives a comprehensive statistical description of the complexity of the environment. In particular, the environment is assumed to be a diffusive environment characterized by the Erdélyi–Kober fractional diffusion. The evolution of the front is then analysed with a Polynomial Chaos surrogate model in order to perform Sensitivity Analysis on the parameters characterizing the diffusion and Uncertainty Quantification procedures on the modeled interface. Sparse techniques for Polynomial Chaos allowed a limited size for the simulation databases.PhD Grant "La Caixa 2014

Restoring property of the Michelson-Sivashinsky equation

In this paper we propose a derivation of the Michelson-Sivashinsky (MS) equation that is based on front propagation only, in opposition to the classical derivation based also on the flow field. Hence, the characteristics of the flow field are here reflected into the characteristics of the fluctuations of the front positions. As a consequence of the presence of the nonlocal term in the MS equation, the probability distribution of the fluctuations of the front positions results to be a quasi-probability distribution, i.e., a density function with negative values. We discuss that the appearance of these negative values, and so the failure of the pure diffusive approach that we adopted, is mainly due to a restoring property that is inherent to the phenomenology of the MS equation. We suggest to use these negative values to model local extinction and counter-gradient phenomena.Basque Government trough BERC 2014-2017 Spanish Ministry of Economy and Competitiveness MINECO trough Severo Ochoa SEV-2013-0323 "La Caixa" Foundation trough PhD grant "La Caixa 2014

Quasi-probability Approach for Modelling Local Extinction and Counter-gradient in Turbulent Premixed Combustion

In opposition to standard probability distributions, quasi-probability distributions can have negative values which highlight nonclassical properties of the corresponding system. In quantum mechanics, such negative values allow for the description of the superposition of two quantum states. Here, we propose the same approach to model local extinction and counter-gradient in turbulent premixed combustion. In particular, the negative values of a quasi-probability correspond to the local reversibility of the progress variable, which means that a burned volume turns to be unburned and then the local extinction together with the counter-gradient interpretation follows. We derive the Michelson-Sivashinsky equation as the average of random fronts following the G-equation, and their fluctuations in position emerge to be distributed according to a quasi-probability distribution displaying the occurrence of local extinction and counter-gradient. The paper is an attempt to provide novel methods able to lead to new theoretical insights in combustion science.PhD grant "La Caixa 2014

Darrieus-Landau instabilities in the framework of the G-equation

We consider a model formulation of the flame front propagation in turbulent premixed combustion based on stochastic fluctuations imposed to the mean flame position. In particular, the mean flame motion is described by a G-equation, while the fluctuations are described according to a probability density function which characterizes the underlying stochastic motion of the front. The proposed approach reproduces as special cases the G-equation along the motion of the mean flame position, when the stochastic fluctuations are removed, and the Zimont & Lipatnikov model, when a Gaussian density for fluctuations is used together with the assumption of a plane front. The potentiality of the approach is here investigated further focusing on the Darrieus-Landau (hydrodynamic) instabilities. In particular, this model formulation is set to lead to the Michelson-Sivashinsky equation. Furthermore, a formula that connects the consumption speed and the front curvature is established.PhD Grant "La Caixa 2014

Wiener-Hopf Integral Equations in Mean First-passage Time Problems for Continuous-time Random Walks

We study the mean first-passage time (MFPT) for asymmetric continuous time random walks in continuous space characterised by finite mean waiting times and jump amplitudes with both finite average and finite variance. We derive an inhomogeneous Wiener-Hopf integral equation that allows the exact estimation of the MFPT, which depends on the whole distribution of the jump amplitudes, but on the average of the waiting times only. Thus, our findings hold for general non-Markovian processes, since Markovianity emerges solely with an exponential distribution of the waiting times. Through the paradigmatic case study of a general class of asymmetric distributions of the jump-amplitudes that is exponential towards the boundary and arbitrary in the opposite direction, we show, that only the average of the jump amplitudes in the opposite direction of the boundary contributes to the MFPT. Moreover, we determine a length-scale, which depends only of the distribution of jumps in the direction of the boundary, such that for initial positions close to the boundary the MFPT depends on the specific whole distribution of jump amplitudes, in opposition to the appearing universality for initial positions far away from the boundary.PRE2018-08442

Exact calculation of the mean first-passage time of continuous-time random walks by nonhomogeneous Wiener-Hopf integral equations

We study the mean first-passage time (MFPT) for asymmetric continuous-time random walks in continuous-space characterised by waiting-times with finite mean and by jump-sizes with both finite mean and finite variance. In the asymptotic limit, this well-controlled process is governed by an advection-diffusion equation and the MFPT results to be finite when the advecting velocity is in the direction of the boundary. We derive a nonhomogeneous Wiener–Hopf integral equation that allows for the exact calculation of the MFPT by avoiding asymptotic limits and it emerges to depend on the whole distribution of the jump-sizes and on the mean-value only of the waiting-times, thus it holds for general non-Markovian random walks. Through the case study of a quite general family of asymmetric distributions of the jump-sizes that is exponential towards the boundary and arbitrary in the opposite direction, we show that the MFPT is indeed independent of the jump-sizes distribution in the opposite direction to the boundary. Moreover, we show also that there exists a length-scale, which depends only on the features of the distribution of jumps in the direction of the boundary, such that for starting points near the boundary the MFPT depends on the specific whole distribution of jump-sizes, in opposition to the universality emerging for starting points far-away from the boundary.PRE2018-084427