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

Pagnini, G.

English

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 XXXIX Meeting of the Italian Section of the Combustion Institute  From G-Equation to Michelson-Sivashinsky Equation in Turbulent Premixed Combustion Modelling  G. PAGNINI gpagnini@bcamath.org BCAM – Basque Center for Applied Mathematics Ikerbasque – Basque Foundation for Sciences Alameda Mazarredo 14, 48009 Bilbao, Basque Country - Spain   Abstract 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.  Introduction The research presented at the XXXVIII Meeting of the Italian Section of the Combustion Institute [1] is here continued. Further results on a novel promising formulation for modelling turbulent premixed combustion [1,2], see also [3], are derived. In particular, it is here reminded that such approach starts from a G-equation that describes the mean flame position, and when stochastic fluctuations are introduced, a reaction-diffusion equation which describes the effective burned fraction follows under the assumption that the probability density function (PDF) of the random process underlying the random front motion is known [1,2,3].         This formulation shows that two different approaches to model turbulent premixed combustion, and furthermore considered alternatives to each other, i.e. the G-equation and the reaction-diffusion equation, are indeed complementary and they can be reconciled. When stochastic fluctuations are removed, such formulation reduces to the G-equation along the motion of the mean flame position, and when a plane front is assumed, the Zimont & Lipatnikov model [4,5] is recovered. Considered these results other efforts are porsued to develop further this novel and general stochastic formulation for modelling turbulent premixed combustion.  Here this approach is used to derive the Michelson-Sivashinky equation [6,7,8]  that describes hydrodynamic instabilities in turbulent premixed combustion.  XXXIX Meeting of the Italian Section of the Combustion Institute  Model Formulation The following presentation of the model formulation is based on [1]. Let the scalar function 𝐺(𝑥, 𝑡), 𝑥 ∈ ℝ𝑛, be a level surface that represents the front which devides burned and unburned domains and the front be denoted by 𝛤(𝑡), 𝑡 ≥ 0. Let 𝑥𝑐be a point on the level surface G=c at the instant 𝑡0, such that the corresponding front is  𝛤0 = 𝛤(𝑡0) = {𝑥 = 𝑥0 ∈ 𝑆|𝐺(𝑥0, 𝑡0) = 𝑐}, where 𝑆 ⊆ ℝ𝑛. The level surface propagates with a consumption speed  given by the laminar burning velocity 𝑠𝐿in the normal direction n relative to the mixture element and its evolution is described by the following Hamilton-Jacobi equation where the flow velocity field is u      𝜕𝐺𝜕𝑡+ 𝑢 ⋅ 𝛻𝐺 = 𝑠𝐿‖𝛻𝐺‖ ,   𝑛 = −𝛻𝐺‖𝛻𝐺‖  .                        (1)  Let the front motion be described by the random process 𝑋𝑐𝜔(?̂?, 𝑡)where 𝜔 labels any independent realization, such that the random contour is           𝛤𝜔(𝑡) = {𝑥 = 𝑋𝑐𝜔(𝑡) ∈ 𝑆|𝐺𝜔(𝑋𝑐𝜔, 𝑡) = 𝑐}.                          (2)  Let the mean value of 𝑋𝑐𝜔be denoted by ⟨𝑋𝜔(?̂?, 𝑡)⟩ = ?̂?(𝑡), then if  𝑃(𝑥𝑐; 𝑡|?̂?) is the corresponding PDF, with initial condition  𝑃(𝑥𝑐; 𝑡0|?̂?) = 𝛿(𝑥 − 𝑥0), the mean flame position is given by the integral                                ⟨𝑥𝑐⟩ = ∫ 𝑥𝑐𝑃(𝑥𝑐; 𝑡|?̂?)𝑑𝑥𝑐 = ?̂?(𝑡)ℝ𝑛 .                               (3)  Introducing ?̌?(?̂?, 𝑡), with ?̌?(?̂?, 𝑡0) = ?̌?(𝑥0, 𝑡0) = 𝑐, as the implicit formulation of the mean flame position ?̂?, the ensemble averaging of (1) gives [9]                                  𝜕?̌?𝜕𝑡+ ?̂? ⋅ 𝛻?̌? = −𝑠𝐿?̂? ⋅ 𝛻?̌?.                                        (4)  Since the G-equation can be derived on the basis of considerations about  symmetries, there is a unique model for the RHS term of equation (4) providing a relation between the laminar burning velocity 𝑠𝐿and the turbulent burning velocity 𝑠𝑇[9], i.e.                      𝑠𝐿?̂? = 𝑠𝑇?̌?  ,        ?̌? = −𝛻?̌?∥𝛻?̌?∥.                                      (5)  Finally, combining equation (4) and model (5), the G-equation that describes the surface motion along the mean flame position results to be                     𝜕?̌?𝜕𝑡+ ?̂? ⋅ 𝛻?̌? = 𝑠𝑇‖𝛻?̌?‖.                                (6)  Note in (5) that the normal vector of the mean flame front, i.e. ?̌?, is different from  XXXIX Meeting of the Italian Section of the Combustion Institute  the mean of the normal vectors to the random flame front, i.e. ?̂?. In general, the mean of the random level surface ⟨𝐺𝜔⟩is different from the level surface ?̌?depicted by the mean position of the flame [10].  Applying properties of the Dirac 𝛿-function, it follows                       𝐺𝜔(𝑋𝑐𝜔, 𝑡) = ∫ 𝐺(𝑥, 𝑡)𝛿(𝑥 − 𝑋𝑐𝜔(?̂?ℝ𝑛, 𝑡))𝑑𝑥,                     (7)  as well as a formula including the stochastic fluctuations around the front depicted by the mean flame position, i.e.                             𝜙𝜔(𝑥, 𝑡) = ∫ ?̌?ℝ𝑛(?̂?, 𝑡)𝛿(𝑥 − 𝑋𝑐𝜔(?̂?, 𝑡))𝑑?̂?.                  (8) Given the level surface ?̌?, the inner domain Ω̌(𝑡)enclosed by the front contour ?̌?(𝑡) = {𝑥 ∈ 𝜕Ω̌(𝑡)}can be understood as the effective volume occupied by the burned fraction. Then the following indicator function is introduced                            𝐼Ω̌(𝑡) = {1, 𝑥 ∈ Ω̌(𝑡),0, 𝑥 ∉ Ω̌(𝑡).                                            (9)  In analogy with (8), the random indicator associated to the surface which enclose the volume of the burned fraction is given by the following formula          𝐼Ω̌𝜔(𝑥, 𝑡) = ∫ 𝐼Ω̌(?̂?, 𝑡)𝛿(𝑥 − 𝑋𝑐𝜔(?̂?, 𝑡))𝑑?̂?ℝ𝑛= ∫ 𝛿(𝑥 − 𝑋𝑐𝜔(?̂?, 𝑡))𝑑?̂?Ω̌.          (10)  Finally, ensemble averaging of (10) gives the effective fraction of the burned mass          ⟨𝐼Ω̌𝜔(𝑥, 𝑡)⟩ = ∫ ⟨𝛿(𝑥 − 𝑋𝑐𝜔(?̂?, 𝑡))⟩𝑑?̂?Ω̌= ∫ 𝑃(𝑥; 𝑡|?̂?)𝑑?̂?Ω̌(𝑡)= 𝑉(𝑥, 𝑡).          (11)                         Applying the Reynolds transport theorem to formula (11), a reaction-diffusion equation follows [3]:                   𝜕𝑉𝜕𝑡= ∫𝜕𝑃𝜕𝑡𝑑?̂? + ∫Ω̌(𝑡)𝛻𝑥 ⋅ [𝑠𝑇?̌?𝑃(𝑥; 𝑡|?̂?)]𝑑?̂?Ω̌(𝑡) .                      (12)  Equation (12) reduces to a Hamilton-Jacobi equation when no diffusion is assumed [1,2,3], and, when a plane front is assumed, it reduces to the same equation derived by Zimont & Lipatnikov [4] and studied in [5], see [1,2,3]. This suggests that equation (12) can be considered as the natural extension of Zimont & Lipatnikov model to the case with non null mean curvature.  Derivation of the Michelson-Sivashinky equation Let −(−𝛻2)𝑠, 𝑠 ∈ (0,1)be the fractional Laplacian defined by its Fourier symbol −|𝑘|2𝑠. When 𝑠 = 1the classical Laplacian is recovered. It is here reminded that the  XXXIX Meeting of the Italian Section of the Combustion Institute  Lévy stable densities are the Green functions of the space-fractional diffusion equation                                                𝜕𝑃𝜕𝑡= −(−𝛻2)𝑠𝑃,                                               (13)  and in particular the Green function corresponds to the Gaussian density when 𝑠 = 1and to the Lorentzian density when𝑠 = 1 2⁄ . The Gaussian density is associated to classical diffusion and the Lorentzian density can be associated to a lightly damped linear oscillator.  Consider now equation (12), by setting                                           𝜕𝑃𝜕𝑡= 𝛻2𝑃 − [−(−𝛻2)1 2⁄ 𝑃],                                   (14)  where the second term on the RHS, because of the sign minus and 𝑠 = 1 2⁄ , may be understood as a counter-damping effect of an harmonic oscillator, and by setting                                                𝑠𝑇 = −(∫Ω̌(𝑡) 𝛻?̂?𝑃(𝑥;𝑡|𝑥)𝑑𝑥)2∫ 𝛻?̂?⋅[?̌?𝑃(𝑥;𝑡|𝑥)]𝑑𝑥Ω̌(𝑡),                               (15)  the multi-dimensional Michelson-Sivashinky equation is obtained. In fact, in the one-dimensional case it holds                                            𝜕𝑉𝜕𝑡=𝜕2𝑉𝜕𝑥2− (𝜕𝑉𝜕𝑥)2 − 𝐷𝑥1𝑉,                                      (16)  which is the Michelson-Sivashinky equation [6,7,8], where𝐷𝑥1is the fractional derivative of order 1 in the Riesz-Feller sense with Fourier symbol is−|𝑘|, which differs from the classical first derivative, and is related to the Hilbert transform by the formula                                         𝐷𝑥1𝑉 =1π𝑑𝑑𝑥∫+∞−∞𝑉(𝑥′)(𝑥′−𝑥)𝑑𝑥′.                               (17)  The solution𝑉(𝑥, 𝑡)to the Michelson-Sivashinky equation (16), or to its multi-dimensional version, can be obtained computing the integral (11) where the kernel function𝑃(𝑥; 𝑡|?̂?)is the Green function of (14) and the domain of integration Ω̌(𝑡)is obtained by the indicator function (9) solving the G-equation (6) with 𝑠𝑇defined in (15). This procedure constitutes a practical scheme to compute numerically the solution to the Michelson-Sivashinky equation which is alternative to the pole decomposition method [11].  Summary and Outlook In the present extended abstract the evolution equation of reaction-diffusion type is  XXXIX Meeting of the Italian Section of the Combustion Institute  briefly derived for an observable that can be understood as the effective burned fraction. When stochastic fluctuations are removed, such equation reduces to the G-equation along the motion of the mean flame position, which suggests that approaches based on reaction-diffusion equations and G-equation are indeed complementary and they can be reconciled. Moreover, when a plane front is assumed, the Zimont & Lipatnikov model is recovered.  This promising approach has been adopted here to derive the Michelson-Sivashinky equation from a modelling point of view. The random process underlying the front motion involves the classical diffusion and a second effect that may be understood as a counter-dumping effect when compared with a lightly damped linear oscillator whose intensity of oscillations follows the Lorentzian function. This second effect can be linked to the mechanism that, a given pressure gradient, accelerates the light products more than the heavier reactants and generates counter-gradient diffusion and turbulent energy production [12]. This physical interpretation will be the investigated in the future.    Acknowledgements This research is supported by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.  References  [1]  Pagnini, G., Akkermans, R.A.D., Buchmann, N., Mentrelli, A., “Reaction-diffusion equation and G-equation approaches reconciled in turbulent premixed combustion”, Proceedings/Extended Abstract Book (6 pages) of the 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., Akkermans, R.A.D., Mentrelli, A., Buchmann, N., “Reacting fronts in random environments”, Proceedings of the 14th International Conference on Non-Linear Analysis, Non-Linear Systems and Chaos (NOLASC ’15), Rome, Italy, November 7–9, 2015. pp. 20–25. ISSN: 1790-5117; ISBN: 978-1-61804-345-0. [3]  Pagnini, G., Bonomi, E., “Lagrangian formulation of turbulent premixed combustion”, Phys. Rev. Lett. 107:044503 (2011). [4]  Zimont, V.L., Lipatnikov, A.N., “A numerical model of premixed turbulent combustion of gases, Chem. Phys. Reports 14:993-1025 (1995). [5]  Zimont, V.L., “Gas premixed combustion at high turbulence. Turbulent flame closure combustion model”, Exp. Therm. Fluid Sci. 21:179-186 (2000).  XXXIX Meeting of the Italian Section of the Combustion Institute  [6]  Sivashinsky, G.I., “Nonlinear analysis of hydrodynamic instability in laminar flames. I. Derivation of basic equations”, Acta Astron. 4:1177-1206 (1977). [7]  Michelson, D.M., Sivashinsky, G.I., “Nonlinear analysis of hydrodynamic instability in laminar flames. II. Numerical experiments”, Acta Astron. 4:1207-1221 (1977). [8]  Creta, F., Matalon, M., “Propagation of wrinkled turblent flames in the context of hydrodynamic theory”, J. Fluid Mech. 680:225-264 (2011). [9]  Oberlack, M., Wenzel, H., Peters, N., “On symmetries and averaging of the G-equation for premixed combustion”, Combust. Theor. Model. 5:363-383 (2001). [10]  Sabelnikov, V.A., Lipatnikov, A.N., “Rigorous derivation of an unclosed mean G-equation for statistically 1D premixed turbulent flames”, Int. J. Spray Dyn.  2:301-324 (2010). [11]  Thual, O., Frisch, U., Hénon, M., “Application of pole decomposition to an equation governing the dynamics of wrinkled flame fronts”, J. Phys. 46:1485-1494 (1985). [12]  Pope, S., “Turbulent premixed flames”, Ann. Rev. Fluid Mech. 19:237-270 (1987).  doi: 10.4405/39proci2016.II3 

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

Pagnini G.

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

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