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Michelson–Sivashinsky equation

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In combustion, Michelson–Sivashinsky equation describes the evolution of a premixed flame front, subjected to the Darrieus–Landau instability, in the small heat release approximation. The equation was derived by Gregory Sivashinsky in 1977, who along the Daniel M. Michelson, presented the numerical solutions of the equation in the same year. Let the planar flame front, in a uitable frame of reference be on the x y {\displaystyle xy} -plane, then the evolution of this planar front is described by the amplitude function u ( x , t ) {\displaystyle u(\mathbf {x} ,t)} (where x = ( x , y ) {\displaystyle \mathbf {x} =(x,y)} ) describing the deviation from the planar shape. The Michelson–Sivashinsky equation, reads as

∂ u ∂ t + 1 2 ( ∇ u ) 2 − ν ∇ 2 u − 1 8 π 2 ∫ | k | e i k ⋅ ( x − x ′ ) u ( x , t ) d k d x ′ = 0 , {\displaystyle {\frac {\partial u}{\partial t}}+{\frac {1}{2}}(\nabla u)^{2}-\nu \nabla ^{2}u-{\frac {1}{8\pi ^{2}}}\int |\mathbf {k} |e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')}u(\mathbf {x} ,t)d\mathbf {k} d\mathbf {x} '=0,}

where ν {\displaystyle \nu } is a constant. Incorporating also the Rayleigh–Taylor instability of the flame, one obtains the Rakib–Sivashinsky equation (named after Z. Rakib and Gregory Sivashinsky),

∂ u ∂ t + 1 2 ( ∇ u ) 2 − ν ∇ 2 u − 1 8 π 2 ∫ | k | e i k ⋅ ( x − x ′ ) u ( x , t ) d k d x ′ + γ ( u − ⟨ u ⟩ ) = 0 , {\displaystyle {\frac {\partial u}{\partial t}}+{\frac {1}{2}}(\nabla u)^{2}-\nu \nabla ^{2}u-{\frac {1}{8\pi ^{2}}}\int |\mathbf {k} |e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')}u(\mathbf {x} ,t)d\mathbf {k} d\mathbf {x} '+\gamma \left(u-\langle u\rangle \right)=0,\quad }

where ⟨ u ⟩ ( t ) {\displaystyle \langle u\rangle (t)} denotes the spatial average of u {\displaystyle u} , which is a time-dependent function and γ {\displaystyle \gamma } is another constant.

N-pole solution

The equations, in the absence of gravity, admits an explicit solution, which is called as the N-pole solution since the equation admits a pole decomposition,as shown by Olivier Thual, Uriel Frisch and Michel Hénon in 1988. Consider the 1d equation

u t + u u x − ν u x x = ∫ − ∞ + ∞ e i k x u ^ ( k , t ) d k , {\displaystyle u_{t}+uu_{x}-\nu u_{xx}=\int _{-\infty }^{+\infty }e^{ikx}{\hat {u}}(k,t)dk,}

where u ^ {\displaystyle {\hat {u}}} is the Fourier transform of u {\displaystyle u} . This has a solution of the form

u ( x , t ) = − 2 ν ∑ n = 1 2 N 1 x − z n ( t ) , d z n d t = − 2 ν ∑ l = 1 , l ≠ n 2 N 1 z n − z l − i s g n ( I m z n ) , {\displaystyle {\begin{aligned}u(x,t)&=-2\nu \sum _{n=1}^{2N}{\frac {1}{x-z_{n}(t)}},\\{\frac {dz_{n}}{dt}}&=-2\nu \sum _{l=1,l\neq n}^{2N}{\frac {1}{z_{n}-z_{l}}}-i\mathrm {sgn} (\mathrm {Im} z_{n}),\end{aligned}}}

where z n ( t ) {\displaystyle z_{n}(t)} (which appear in complex conjugate pairs) are poles in the complex plane. In the case periodic solution with periodicity 2 π {\displaystyle 2\pi } , the it is sufficient to consider poles whose real parts lie between the interval 0 {\displaystyle 0} and 2 π {\displaystyle 2\pi } . In this case, we have

u ( x , t ) = − ν ∑ n = 1 2 π cot ⁡ x − z n ( t ) 2 , d z n d t = − ν ∑ l ≠ n cot ⁡ z n − z l 2 − i s g n ( I m z n ) {\displaystyle {\begin{aligned}u(x,t)&=-\nu \sum _{n=1}^{2\pi }\cot {\frac {x-z_{n}(t)}{2}},\\{\frac {dz_{n}}{dt}}&=-\nu \sum _{l\neq n}\cot {\frac {z_{n}-z_{l}}{2}}-i\mathrm {sgn} (\mathrm {Im} z_{n})\end{aligned}}}

These poles are interesting because in physical space, they correspond to locations of the cusps forming in the flame front.

Dold–Joulin equation

In 1995, John W. Dold and Guy Joulin generalised the Michelson–Sivashinsky equation by introducing the second-order time derivative, which is consistent with the quadratic nature of the dispersion relation for the Darrieus–Landau instability. The Dold–Joulin equation is given by

∂ 2 φ ∂ t 2 + I ( ∂ φ ∂ t − 1 2 ( ∇ φ ) 2 − ν ∇ 2 φ − ν I ( φ ) ) = 0 , {\displaystyle {\frac {\partial ^{2}\varphi }{\partial t^{2}}}+{\mathcal {I}}\left({\frac {\partial \varphi }{\partial t}}-{\frac {1}{2}}(\nabla \varphi )^{2}-\nu \nabla ^{2}\varphi -\nu {\mathcal {I}}(\varphi )\right)=0,}

where I ( e i k ⋅ x ) = | k | e i k ⋅ x {\displaystyle {\mathcal {I}}(e^{i\mathbf {k} \cdot \mathbf {x} })=|\mathbf {k} |e^{i\mathbf {k} \cdot \mathbf {x} }} corresponds to the non-local integral operator.

Joulin–Cambray equation

In 1992, Guy Joulin and Pierre Cambray extended the Michelson–Sivashinsky equation to include higher-order correction terms, following by an earlier incorrect attempt to derive such an equation by Gregory Sivashinsky and Paul Clavin. The Joulin–Cambray equation, in dimensional form, reads as

∂ ϕ ∂ t + S L 2 ( 1 + ϵ 2 ) | ∇ ϕ | 2 + ϵ S L 4 ⟨ | ∇ ϕ | 2 ⟩ = S L ϵ 2 ( 1 + ϵ 2 ) ( ν ∇ 2 ϕ + I ( ϕ , x ) ) . {\displaystyle {\frac {\partial \phi }{\partial t}}+{\frac {S_{L}}{2}}\left(1+{\frac {\epsilon }{2}}\right)|\nabla \phi |^{2}+\epsilon {\frac {S_{L}}{4}}\langle |\nabla \phi |^{2}\rangle ={\frac {S_{L}\epsilon }{2}}\left(1+{\frac {\epsilon }{2}}\right)\left(\nu \nabla ^{2}\phi +I(\phi ,\mathbf {x} )\right).}

See also

References

  1. Sivashinsky, G.I. (1977). "Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations". Acta Astronautica. 4 (11–12): 1177–1206. doi:10.1016/0094-5765(77)90096-0. ISSN 0094-5765.
  2. Michelson, Daniel M., and Gregory I. Sivashinsky. "Nonlinear analysis of hydrodynamic instability in laminar flames—II. Numerical experiments." Acta astronautica 4, no. 11-12 (1977): 1207-1221.
  3. Matalon, Moshe. "Intrinsic flame instabilities in premixed and nonpremixed combustion." Annu. Rev. Fluid Mech. 39 (2007): 163-191.
  4. Rakib, Z., & Sivashinsky, G. I. (1987). Instabilities in upward propagating flames. Combustion science and technology, 54(1-6), 69-84.
  5. ^ Thual, O., U. Frisch, and M. Henon. "Application of pole decomposition to an equation governing the dynamics of wrinkled flame fronts." In Dynamics of curved fronts , pp. 489-498. Academic Press, 1988.
  6. Frisch, Uriel, and Rudolf Morf. "Intermittency in nonlinear dynamics and singularities at complex times." Physical review A 23, no. 5 (1981): 2673.
  7. Joulin, Guy. "Nonlinear hydrodynamic instability of expanding flames: Intrinsic dynamics." Physical Review E 50, no. 3 (1994): 2030.
  8. Matsue, K., & Matalon, M. (2023). Dynamics of hydrodynamically unstable premixed flames in a gravitational field–local and global bifurcation structures. Combustion Theory and Modelling, 27(3), 346-374.
  9. Clavin, Paul, and Geoff Searby. Combustion waves and fronts in flows: flames, shocks, detonations, ablation fronts and explosion of stars. Cambridge University Press, 2016.
  10. Vaynblat, Dimitri, and Moshe Matalon. "Stability of pole solutions for planar propagating flames: I. Exact eigenvalues and eigenfunctions." SIAM Journal on Applied Mathematics 60, no. 2 (2000): 679-702.
  11. Dold, J. W., & Joulin, G. (1995). An evolution equation modeling inversion of tulip flames. Combustion and flame, 100(3), 450-456.
  12. Joulin, G., & Cambray, P. (1992). On a tentative, approximate evolution equation for markedly wrinkled premixed flames. Combustion science and technology, 81(4-6), 243-256.
  13. Sivashinsky, G. I., & Clavin, P. (1987). On the nonlinear theory of hydrodynamic instability in flames. Journal de Physique, 48(2), 193-198.
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