Michelson–Sivashinsky equation explained
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,[1] who along the Daniel M. Michelson, presented the numerical solutions of the equation in the same year.[2] Let the planar flame front, in a uitable frame of reference be on the
-plane, then the evolution of this planar front is described by the amplitude function
(where
) describing the deviation from the planar shape. The Michelson–Sivashinsky equation, reads as
[3]
+
(\nablau)2-\nu\nabla2u-
\int|k|eik ⋅ (x-x')u(x,t)dkdx'=0,
where
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),
[4]
+
(\nablau)2-\nu\nabla2u-
\int|k|eik ⋅ (x-x')u(x,t)dkdx'+\gamma\left(u-\langleu\rangle\right)=0,
where
denotes the spatial average of
, which is a time-dependent function and
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.[5] [6] [7] [8] Consider the 1d equation
ut+uux-\nuuxx=
eikx\hatu(k,t)dk,
where
is the Fourier transform of
. This has a solution of the form
[9] \begin{align}u(x,t)&=-2\nu
,\
&=-2\nu
-isgn(Imzn),
\end{align}
where
(which appear in complex conjugate pairs) are poles in the complex plane. In the case periodic solution with periodicity
, the it is sufficient to consider poles whose real parts lie between the interval
and
. In this case, we have
\begin{align}
u(x,t)&=-\nu
,\\
&=-\nu\suml ≠ \cot
-isgn(Imzn)
\end{align}
These poles are interesting because in physical space, they correspond to locations of the cusps forming in the flame front.[10]
Dold–Joulin equation
In 1995,[11] 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
| \partial2\varphi |
| \partialt2 |
+lI\left(
-
(\nabla\varphi)2-\nu\nabla2\varphi-\nulI(\varphi)\right)=0,
where
corresponds to the non-local integral operator.
Joulin–Cambray equation
In 1992,[12] 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.[13] The Joulin–Cambray equation, in dimensional form, reads as
+
\right)|\nabla\phi|2+\epsilon
\langle|\nabla\phi|2\rangle=
\right)\left(\nu\nabla2\phi+I(\phi,x)\right).
See also
Notes and References
- Sivashinsky. G.I.. Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations. Acta Astronautica. 4. 11–12. 1977. 1177–1206. 0094-5765. 10.1016/0094-5765(77)90096-0.
- 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.
- Matalon, Moshe. "Intrinsic flame instabilities in premixed and nonpremixed combustion." Annu. Rev. Fluid Mech. 39 (2007): 163-191.
- Rakib, Z., & Sivashinsky, G. I. (1987). Instabilities in upward propagating flames. Combustion science and technology, 54(1-6), 69-84.
- 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.
- Frisch, Uriel, and Rudolf Morf. "Intermittency in nonlinear dynamics and singularities at complex times." Physical review A 23, no. 5 (1981): 2673.
- Joulin, Guy. "Nonlinear hydrodynamic instability of expanding flames: Intrinsic dynamics." Physical Review E 50, no. 3 (1994): 2030.
- 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.
- Clavin, Paul, and Geoff Searby. Combustion waves and fronts in flows: flames, shocks, detonations, ablation fronts and explosion of stars. Cambridge University Press, 2016.
- 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.
- Dold, J. W., & Joulin, G. (1995). An evolution equation modeling inversion of tulip flames. Combustion and flame, 100(3), 450-456.
- 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.
- Sivashinsky, G. I., & Clavin, P. (1987). On the nonlinear theory of hydrodynamic instability in flames. Journal de Physique, 48(2), 193-198.
