# Elasticity of flame fronts and numerical experiments

U. Frisch, Z. She, Olivier Thual

► To cite this version:

U. Frisch, Z. She, Olivier Thual. Elasticity of flame fronts and numerical experiments. Numerical Simulation of Combustion Phenomena, Roland Glowinski, Bernard Larrouturou, Roger Temam, May 1985, Sophia-Antipolis, France. pp.234-236, 10.1007/BFb0008663. hal-04740433

HAL Id: hal-04740433

https://hal.science/hal-04740433v1

Submitted on 18 Oct 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
U. Frisch, CNRS, Observatoire de Nice, B.P. 139, 06003 Nice Cedex, France
Z.S. She, Observatoire de Meudon, 92195 Meudon Principal Cedex, France
and O. Thual, CNRM, 42 avenue Coriolis, 31057 Toulouse Cedex, France

We have studied the Kuramoto-Sivashinsky equation

$$ \partial_t u + u \partial_x u + \partial_x^2 u + v \partial_x^4 u = 0 \quad (1) $$

with $2\pi$-periodic boundary conditions. The controlling parameter $v$ plays here a role analogous to the viscosity in the Navier-Stokes equation. When $v$ is small compared to unity the number of unstable modes in the linearized problem $(0(v^{-1/2}))$ is large and chaotic solutions are usually expected. On the other hand eq. (1) admits steady solutions of periodicity $2\pi/N$ with only one excited wavenumber in the unstable band (provide $vN^2$ is close to one). Writing $v = (1-\eta)/N^2$, we have found that such cellular solutions can be stable for $\eta_1 < n < \eta_2$ where $\eta_1 \approx 0.3$ and $\eta_2 \approx 0.4$. This result was obtained by a multi-scale (homogenization) analysis of the Kuramoto-Sivashinsky equation linearized around the cellular solution. The details of this analysis are reported in Ref. 2. The essence is now given. Perturbations of wavelength large compared to $2\pi/N$ are governed by an effective transport equation. This equation involves two phenomena: elasticity (which has its root in the Galilean invariance of the problem) and (eddy) diffusivity. Thus, the cellular solutions can be visco-elastic. The window of stability $\eta_1 < n < \eta_2$ is characterized by simultaneous positivity of the elasticity and diffusivity coefficients. When a cellular solution is subject to a strong perturbation, dislocations may appear in the visco-elastic structure. Chaotic behaviour is then observed. In many instances this chaos will disappear after some (fairly long) time $2^{-3}$. It has even been suggested that the chaos is metastable and will always disappear$^4$.

The theoretical predictions have been checked by a number of numerical experiments which are reported in Ref. 2. The numerical method used for this was a variant of the spectral method. The only novel aspect was in the time-stepping procedure where we had to face a difficulty more likely to arise in flame fronts than in other hydrodynamic problems as we now explain. The Kuramoto-Sivashinsky equation has a fourth power of the wavenumber in the linear damping term. At the high k end of the simulation the modes are linearly very quickly damped; hence they adiabatically adjust
to whatever input is provided by the nonlinear term which varies on a considerably slower time scale since it involves interaction of modes with lower k's. In Haken's terminology the high k modes are "slaved". A simple linear model that displays this phenomenon is

$$dq/dt = - \alpha q + f(t) \quad (2)$$

Suppose that $\alpha > 0$ and $\alpha T \gg 1$ where T is the typical time scale of variation of $f(t)$. Then, to leading order in $(\alpha T)^{-1}$, we get the slaved solution

$$q(t) = f(t)/\alpha \quad (3)$$

We wish to find a time-stepping scheme for (2) which

i) reduces to the leap-frog scheme for $\alpha = 0$ (this scheme, when appropriately stabilized, is very convenient for spectral calculations; other explicit second order schemes can also be dealt with),

ii) is second order in time,

iii) is unconditionally stable when $f = 0$,

iv) reproduces slavery to leading order in $(\alpha T)^{-1}$.

Consider the following standard schemes

1. Leap-Frog-Crank-Nicolson

$$q_{n+1} = \frac{(1-\alpha\delta t)q_{n-1} + 2\delta t f_n}{1 + \alpha\delta t}, \quad q_n = q(t_n), \quad f_n = f(t_n) \quad (4)$$

2. Exponential Leap Frog

$$q_{n+1} = e^{-2\alpha\delta t} q_{n-1} + \frac{2\delta t e^{-\alpha\delta t} f_n}{\alpha} \quad (5)$$

Both satisfy i) through iii) but not iv). For large $\alpha\delta t$ scheme 2 gives exponentially small q's instead of algebraically small ones. Scheme 1 will essentially make $q_n$ reverse its sign at each time step. If $f_n$ is kept constant, eventually $q_n$ goes to the correct value $f/\alpha$ but this takes a time $O(\alpha\delta t^2)$ instead of $O(\alpha^{-1})$.

A satisfactory scheme is

3. "Slaved Frog"

$$q_{n+1} = e^{-2\alpha\delta t} q_{n-1} + \frac{1 - e^{-2\alpha\delta t}}{\alpha} f_n \quad (6)$$
This is obtained from the exact relation

$$q(t + \delta t) = e^{-2\alpha\delta t} q(t - \delta t) + \int_{t-\delta t}^{t+\delta t} e^{-\alpha(t + \delta t - s)} f(s) ds \quad , \quad (7)$$

by pulling out $f(s)$ at the middle time $t = t_n$.

Implementation of the Slaved Frog in a spectral simulation is straightforward. For example, for the KS equation one interprets $q$ to be the Fourier amplitude of mode $k$, $\alpha$ is $k^2 - \sqrt{k^4}$ and $f$ is the Fourier transform of $-a_x(u^2/2)$. The Slaved Frog may also be useful for certain stiff problems such as convection at low Prandtl numbers. After this work was completed, we learned that C. Basdevant, N. Corfield and P.-L. Sulem have been using similar schemes in problems other than combustion.

REFERENCES.

1. Sivashinsky, G.I.: 1983, Ann. Rev. Fluid Mech. 15, 179.

2. Frisch, U., She, Z.S. and Thual, O.: 1984, "On the Elastic Behaviour of Turbulence": A Case Study of the Kuramoto-Sivashinsky Model", preprint Observatoire de Nice.

3. Manneville, P.: 1984, private communication.

4. Shraiman, B.: 1985, "Order, Disorder and Phase Turbulence", preprint, James Franck Institute, Chicago.