Front Propagation in Random Media

This PhD thesis deals with the problem of the propagation of fronts under random circumstances. A statistical model to represent the motion of fronts when are evolving in a media characterized by microscopical randomness is discussed and expanded, in order to cope with three distinct applications: wild-land fire simulation, turbulent premixed combustion, biofilm modeling. In the studied formalism, the position of the average front is computed by making use of a sharp-front evolution method, such as the level set method. The microscopical spread of particles which takes place around the average front is given by the probability density function linked to the underlying diffusive process, that is supposedly known in advance. The adopted statistical front propagation framework allowed a deeper understanding of any studied field of application. The application of this model introduced eventually parameters whose impact on the physical observables of the front spread have been studied with Uncertainty Quantification and Sensitivity Analysis tools. In particular, metamodels for the front propagation system have been constructed in a non intrusive way, by making use of generalized Polynomial Chaos expansions and Gaussian Processes.The Thesis received funding from Basque Government through the BERC 2014-2017 program. It was also funded by the Spanish Ministry of Economy and Competitiveness MINECO via the BCAM Severo Ochoa SEV-2013-0323 accreditation. The PhD is fundend by La Caixa Foundation through the PhD grant “La Caixa 2014”. Funding from “Programma Operativo Nazionale Ricerca e Innovazione” (PONRI 2014-2020) , “Innotavive PhDs with Industrial Characterization” is kindly acknowledged for a research visit at the department of Mathematics and Applications “Renato Caccioppoli” of University “Federico II” of Naples

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

Surrogate based Global Sensitivity Analysis of ADM1-based Anaerobic Digestion Model

In order to calibrate the model parameters, Sensitivity Analysis routines are mandatory to rank the parameters by their relevance and fix to nominal values the least influential factors. Despite the high number of works based on ADM1, very few are related to sensitivity analysis. In this study Global Sensitivity Analysis (GSA) and Uncertainty Quantification (UQ) for an ADM1-based Anaerobic Digestion Model have been performed. The modified version of ADM-based model selected in this study was presented by Esposito and co-authors in 2013. Unlike the first version of ADM1, focused on sewage sludge degradation, the model of Esposito is focused on organic fraction of municipal solid waste digestion. It his recalled that in many applications the hydrolysis is considered the bottleneck of the overall anaerobic digestion process when the input substrate is constituted of complex organic matter. In Esposito's model a surfaced based kinetic approach for the disintegration of complex organic matter is introduced. This approach allows to better model the disintegration step taking into account the effect of particle size distribution on the digestion process. This model needs thus GSA and UQ to pave the way for further improvements and reach a deep understanding of the main processes and leading input factors. Due to the large number of parameters to be analyzed a first preliminary screening analysis, with the Morris' Method, has been conducted. Since two quantities of interest (QoI) have been considered, the initial screening has been performed twice, obtaining two set of parameters containing the most influential factors in determining the value of each QoI. A surrogate of ADM1 model has been defined making use of the two defined quantities of interest. The output results from the surrogate model have been analyzed with Sobol’ indices for the quantitative GSA. Finally, uncertainty quantification has been performed. By adopting kernel smoothing techniques, the Probability Density Functions of each quantity of interest have been defined

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

RandomFront 2.3: a physical parameterisation of fire spotting for operational fire spread models-implementation in WRF-SFIRE and response analysis with LSFire+

Fire spotting is often responsible for dangerous flare-ups in wildfires and causes secondary ignitions isolated from the primary fire zone, which lead to perilous situations. The main aim of the present research is to provide a versatile probabilistic model for fire spotting that is suitable for implementation as a post-processing scheme at each time step in any of the existing operational large-scale wildfire propagation models, without calling for any major changes in the original framework. In particular, a complete physical parameterisation of fire spotting is presented and the corresponding updated model RandomFront 2.3 is implemented in a coupled fire?atmosphere model: WRF-SFIRE. A test case is simulated and discussed. Moreover, the results from different simulations with a simple model based on the level set method, namely LSFire+, highlight the response of the parameterisation to varying fire intensities, wind conditions and different firebrand radii. The contribution of the firebrands to increasing the fire perimeter varies according to different concurrent conditions, and the simulations show results in agreement with the physical processes. Among the many rigorous approaches available in the literature to model firebrand transport and distribution, the approach presented here proves to be simple yet versatile for application to operational large-scale fire spread models.This research was supported by the Basque Government through the BERC 2014–2017 and BERC 2018–2021 programs. It was also funded by the Spanish Ministry of Economy and Competitiveness MINECO via the BCAM Severo Ochoa SEV-2013-0323 and SEV-2017-0718 accreditations, the MTM2013-40824-P “ASGAL” and MTM2016-76016-R “MIP” projects, and the 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

Wildland fire propagation modeling: fire-spotting parametrisation and energy balance

Present research concerns the physical background of a wild-fire propagation model based on the split of the front motion into two parts - drifting and fluctuating. The drifting part is solved by the level set method and the fluctuating part describes turbulence and fire-spotting. These phenomena have a random nature and can be modeled as a stochastic process with the appropriate probability density function. Thus, wildland fire propagation results to be described by a nonlinear partial differential equation (PDE) of the reaction-diffusion type. A numerical study of the effects of the atmospheric stability on wildfire propagation is performed through its effects on fire-spotting. Moreover, it is shown that the solution of the PDE as an indicator function allows to construct the energy balance equation in terms of the temperature.PhD Grant "La Caixa 2014

Wildland fire propagation modelling

Wildfire propagation modelling is a challenging problem due to its complex multi-scale multi-physics nature. This process can be described by a reaction- diffusion equation based on the energy balance principle. Alternative technique is the so-called level-set method (LSM), used in wildfire modelling as well as in many other fields. In the present study a methodology for fire propagation modelling that reconciles these approaches is proposed. This methodology is distinguishable and significant from both academical and industrial point of view because of the inclusion of the ran- dom effects by preserving the existing algorithms and direct implementation as a post-processing numerical routine. The random behaviour of the fire front is caused, for example, by the turbulence and the fire-spotting phenomenon. A probability density function (PDF) is employed in order to describe the random process. In earlier studies it has been shown that new independent ignitions can increase the rate of spread (ROS) of fire and therefore should be carefully studied. In this respect, a physical parametrization of the fire-spotting distribution was proposed. Special attention in the present study is paid to the atmospheric stability conditions. The parametrization proposed in previous works is completed by the multiple fire-spotting modelling. Afterwards special attention is paid to the study of uniqueness of the PDF and consistency with the energy balance equation. Numerical results and discussions complete the study.PhD grant ”La Caixa 2014