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

Pagnini, G.

Trucchia, A.

English

BCAM's Institutional Repository Data

JOINT MEETINGTHE GERMAN AND ITALIAN SECTIONSOF THE COMBUSTION INSTITUTE SORRENTO, ITALY –  2018QUASI-PROBABILITY APPROACH FORMODELLING LOCAL EXTINCTION AND COUNTER-GRADIENT IN TURBULENT PREMIXED COMBUSTIONG. Pagnini*^, A. Trucchia*¨gpagnini@bcamath.org*BCAM- Basque Center for Applied Mathematics,Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country - Spain^Ikerbasque, Calle de María Díaz de Haro 3, E-48013 Bilbao, Basque Country - Spain¨Department of Mathematics, University of the Basque Country UPV/EHU,Apartado 644, E-48080 Bilbao, Basque Country - SpainAbstractIn opposition to standard probability distributions, quasi-probability distributionscan have negative values which highlight nonclassical properties of thecorresponding system. In quantum mechanics, such negative values allow for thedescription of the superposition of two quantum states. Here, we propose the sameapproach to model local extinction and counter-gradient in turbulent premixedcombustion. In particular, the negative values of a quasi-probability correspond tothe local reversibility of the progress variable, which means that a burned volumeturns to be unburned and then the local extinction together with the counter-gradient interpretation follows. We derive the Michelson-Sivashinsky equation asthe average of random fronts following the G-equation, and their fluctuations inposition emerge to be distributed according to a quasi-probability distributiondisplaying the occurrence of local extinction and counter-gradient. The paper is anattempt to provide novel methods able to lead to new theoretical insights incombustion science. IntroductionTurbulent premixed combustion requires a set of governing equations and a richphenomenology follows. The set of governing equation includes: mass andmomentum conservation, equation of state for gases, energy and speciesconservation. The nonlinearity of the problems requires closures and modelling.Self-extinction and counter-gradient are phenomena occurring in premixedcombustion that require non-standard modelling approach or ad hoc modellingwhen standard approaches are adopted. Here we proceed with a research programpresented in the last years at the Meetings of the Italian Section of the CombustionInstitute [1,2,3]. The aim is to provide novel methods able to lead to newtheoretical insights in combustion science. In particular, in the following we brieflyreport the derivation of the Michelson-Sivashinsky equation as the average ofJOINT MEETINGTHE GERMAN AND ITALIAN SECTIONSOF THE COMBUSTION INSTITUTE SORRENTO, ITALY –  2018random fronts following the G-equation [2,3], and we discuss that, since theirfluctuations in position emerge to be distributed according to a quasi-probabilitydistribution, then local extinction and counter-gradient are displayed by negativevalues of the quasi-probability emerging from such modelling approach. In the nextsection the main equations are derived, discussion and conclusion are reported in afurther and ending section. Statistical derivation of the Michelson-Sivashinsky equationThis section is based on [2,3] and it is here included for highlighting the role of theemerging quasi-probability into the proposed modelling approach.Let the scalar function G ( x ,t ) , x∈ℝn , be a level surface that represents thefront which splits the considered domain into burned and unburned sub-domains.Let xc  be a point on the level surface G=c at the instant t 0 . The levelsurface propagates with a consumption speed given by the laminar burning velocitysL  in the normal direction n relative to the mixture element and its evolution isdescribed by the following Hamilton-Jacobi equation where the flow velocity fieldis u∂G∂ t+u ⋅∇G=sL‖∇G‖  ,   n=−∇G‖∇G‖   . (1)Let the front motion be described by the random process Xcω ( x̂ , t ) where ωlabels any independent realization and the mean value of Xcω  be denoted by⟨Xω ( x̂ , t ) ⟩= x̂ (t ) , then if Pc ( xc ; t| x̂ )  is the corresponding PDF, with initialcondition Pc ( xc ;t 0|x̂ )=δ ( x− x0 ) , the mean flame position is given by theintegral⟨ xc ⟩=∫ℝnxc Pc ( xc ; t|x̂ )d xc= x̂ (t ) . (2)Introducing Ǧ ( x̂ ,t ) ,with Ǧ ( x̂ ,t 0 )=Ǧ ( x0 , t 0 )=c , as the implicit formulationof the mean flame position x̂ , the ensemble averaging of (1) gives [5]∂Ǧ∂ t+û ⋅∇ Ǧ=ŝL n ⋅∇ Ǧ . (3)Since the G-equation can be derived on the basis of considerations aboutsymmetries, there is a unique model for the RHS term of equation (3) providing arelation between the laminar burning velocity sL  and the turbulent burningJOINT MEETINGTHE GERMAN AND ITALIAN SECTIONSOF THE COMBUSTION INSTITUTE SORRENTO, ITALY –  2018velocity sT  [5], i.e.ŝL n=sT ň   ,    ň=−∇ Ǧ‖∇ Ǧ‖s . (4)Finally, combining equation (3) and (4), the G-equation that describes the surfacemotion along the mean flame position results to be∂Ǧ∂ t+û ⋅∇ Ǧ=sT‖∇ Ǧ‖ . (5)The one-dimensional Michelson-Sivashinsky equation reads [6]∂g∂ t=∂2g∂ x2−( ∂ g∂ x )2−Dx1 g , (10)where Dx1 is the fractional derivative of order 1 in the Riesz-Feller sense withFourier symbol −|ξ| , which differs from the classical first derivative, and it isrelated to the Hilbert transform by the formula Dx1 g= 1πddx ∫−∞+∞ g ( x ' )(x '− x )d x ' . (11)Let us introduce the field g (x ,t )  in analogy with [7], i.e., g (x,t )=∫Ǧ ( x̂ , t )Pc (x− x̂ , t ) d x̂ . (12)It is well-known that the dispersion relation of equation (10) is e−ξ2 t+|ξ|t . (13)This suggests the following fractional differential equation for the PDF of thefluctuations of the front position:∂PC∂ t=ΔPC+(−Δ )1/2PC ,    PC ( x ,0 )=δ ( x ) . (14)JOINT MEETINGTHE GERMAN AND ITALIAN SECTIONSOF THE COMBUSTION INSTITUTE SORRENTO, ITALY –  2018Actually, the dispersion relation (13) is the Fourier transform of the Green functionof (14). It is here highlighted that the PDF of fluctuations which solves (14)emerges to be a quasi-probability distribution showing negative values that requireshigh care, see Figure.         Figure: Quasi-probability distribution solution of (14).The evolution equation of the function g (x ,t ) results to be: ∂g∂ t=∂2g∂ x2−Dx1 g+∫ sT ( x̂ , t ) PC (x− x̂ , t ) d x̂ . (15)Comparing (10) and (15) we have −(∂ g∂ x )2=∫ sT ( x̂ t )PC (x− x̂ , t ) d x̂≡ω , (16) that in Fourier domain reads  ~ω=~sT~Pc=~sT exp (−ξ 2t+|ξ|t ) , (17)and the turbulent burning velocity turns out to be sT ( x ,t )=12 π∫ exp (− i ξ x ) exp (ξ2t−|ξ|t )~ω (ξ , t ) d ξ . (18)JOINT MEETINGTHE GERMAN AND ITALIAN SECTIONSOF THE COMBUSTION INSTITUTE SORRENTO, ITALY –  2018Discussion and conclusionIn the previous section we showed that when the Michelson-Sivashinsky equationis derived as the ensemble average of random fronts governed by the G-equation,then the spatial fluctuations of the positions of the random fronts are distributedaccordingly to a quasi-probability. Quasi-probability distributions display negativevalues, see Figure, that require high care in their interpretation. In particular, byusing Fourier inverse transformation, the quasi-probability that solves (14) can bewritten in the following integral form                       Pc ( x ,t )=12 π∫ exp (− i ξ x ) exp (−ξ2 t+|ξ|t )d ξ ,                        (19)such that it results to be close to the representation of the Wigner quasi-distributionfor quantum optics. The negative values highlight where statistically the fraction of burned mixture isreplaced by unburned mixture. In fact, if we integrate (12) in space we have that incorrespondence of the negative values of the quasi-probability there is a reductionof the mass amount. The propagation of the front is slowed. For this reason, wepropose that these negative values can be interpreted as due to local extinction andcounter-gradient phenomena. Local reversibility of the value of the progressvariable occurs because of the entering of fresh mixture into a volume just nowfully burned. This effect can be ascribed first to the local extinction, that stops thepropagation of the combustion, and then to the so-called counter-gradient, which isgenerated by the density difference between reactants and products, that pushesback the front of the burned mixture. In formulae , th is in terpre ta t ion can be s ta ted as fo l lows. LetC=|C|eiQ=CR+iCI be the progress variable with real and imaginary part.Then it holds CR2=|C|2−CI2 , and if the g (x ,t ) defined in (12) correspondsto the real part, i.e. g=CR , then we have                                                     g=|C|+F (CI ) ,                                               (20)where                                    |C|=∫PC>0 Ǧ ( x̂ , t )Pc (x− x̂ , t ) d x̂ ,                            (21)                             F (CI )=∫PC<0 Ǧ ( x̂ , t )Pc ( x− x̂ , t ) d x̂<0.                         (22)The imaginary part results to be       CI=ei (Q−π /2)∫PC>0 Ǧ ( x̂ , t ) Pc (x− x̂ , t ) d x̂+i∫Ǧ ( x̂ , t )Pc ( x− x̂ , t ) d x̂ .      (23)JOINT MEETINGTHE GERMAN AND ITALIAN SECTIONSOF THE COMBUSTION INSTITUTE SORRENTO, ITALY –  2018Acknowledgments This research is supported by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and CompetitivenessMINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323and through project MTM2016-76016-R ”MIP”. AT is supported by the PhD grant”La Caixa 2014”.References[1] Pagnini, G., Akkermans, R.A.D., Buchmann, N., Mentrelli, A., “Reaction-diffusion equation and G-equation approaches reconciled in turbulentpremixed combustion”, Proceedings/Extended Abstract Book (6 pages) ofthe XXXVIII Meeting of the Italian Section of the Combustion Institute,Lecce, Italy, September 20–23, 2015. ISBN 978-88-88104-25-6. Paper doi:10.4405/38proci2015.I4[2] Pagnini, G., “From G-equation to Michelson–Sivashinsky equation inturbulent premixed combustion modelling”, in M. Commodo, W. Prins, F.Scala, A. Tregrossi (Eds.): Proceedings/Extended Abstract Book (6 pages) ofthe XXXIX Meeting of the Italian Section of the Combustion Institute,Naples, Italy, July 4–6, 2016. ISBN 9788888104171. Paper doi:10.4405/39proci2016.II3 [3]  Pagnini G., Trucchia A., Front curvature evolution and hydrodynamicsinstabilities, in F. Scala, M. Valorani, M. Commodo, A. Tregrossi, H.G. Im,P. Glarborg (Eds) Proceedings/Extended Abstract Book (6 pages) of the XLMeeting of the Italian Section of the Combustion Institute, Rome, Italy, June7–9, 2017. ISBN 9788888104188.[4] Pagnini, G., Trucchia, A., “Darrieus-Landau instabilities in the framework ofthe G-equation”, Digital proceedings of the 8th European CombustionMeeting, Dubrovnik, Croatia, April 18-21, 2017.  ISBN 978-953-59504-1-7 ,Paper M_371, pages 1678-1683[5] Oberlack, M.,Wenzel, H., Peters, N., “On symmetries and averaging of theG-equation for premixed combustion”, Combust. Theor. Model. 5:363-383(2001).[6] Creta, F, Matalon, M, “Propagation of wrinkled turblent flames in thecontext of hydrodynamic theory”, J. Fluid Mech. 680:225-264 (2011).[7] Pagnini, G., Bonomi, E., “Lagrangian formulation of turbulent premixedcombustion”, Phys. Rev. Lett. 107:044503 (2011).

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

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

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