Cellular Structures in Instabilities pp 137-155 | Cite as

# Modelisation and simulation of convection in extended geometry

## Abstract

Models of convection such as the one of $2 give an interesting starting point for a theoretical analysis devoted to the elucidation of concepts which seem important for an understanding of weak turbulence at large aspect ratios.In particular one can study in nearly realistic conditions the problems of pattern selection and defect motion using the standard amplitude equation formalism. As a matter of fact one can easily recover from (3a,b) the modified amplitude equations derived by Siggia and Zippelius for nearly straight rolls in 3-D convection with stress free BC. However it should be kept in mind that, due to the approximations made in order to have models sufficiently simple to remain tractable, one should not expect more than semi-quantitative results at best.

Another and perhaps more promising use of these models is for numerical simulations in domains of lateral extent incomparably larger than those accessible to 3-D simulations. This study is at its very beginning. The above example of a turbulent transient observed at low Prandtl number (Pr=1.6) shows what one can obtain from such simulations. First it seems that they are able to reproduce the often quoted laboratory observation of alternating agitated and calm periods. A careful examination of the evolution allows to interpret this phenomenon in terms of “dynamical frustration“ linked to the compatibility between the large scale drift flow which tends to advect the rolls and the overall curvature of the current structure which generates the vertical vorticity. Calm periods then correspond to the slow distortion of the structure by the flow and agitated periods to the nucleation and motion of dislocations responding to distortions getting too large.Whether this feed-back mechanism is responsible for the occurrence of weak turbulence at large aspect ratios is not yet known but this is a very tempting conjecture in view of our preliminary results.

## Keywords

Large Aspect Ratio Boundary Condition Amplitude Equation Weak Turbulence Vertical Vorticity## Preview

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## References

- [1]D.Ruelle and F.Takens: Comm.math.Phys. 20 (1971)167, see also: a) J.P.Eckmann: Rev.Mod.Phys. 53(1981)643, b) Workshop Common Trends in Particule and Condensed Matter Physics Les Houches (March 1983) Physics Reports, to appear.CrossRefGoogle Scholar
- [2]P.Manneville: “the transition to turbulence in “wide systems”, in ref.lb above.Google Scholar
- [3]M.C.Cross: Phys.Rev. A25 (1982)1065.Google Scholar
- [3]aE.D.Siggia and A.Zippelius: Phys.Rev.A24 (1981)1036.Google Scholar
- [3]bY.Pomeau, S.Zaleski and P.Manneville: Phys.Rev.A27 (1983)2710.Google Scholar
- [3]cP.Manneville and Y.Pomeau: Phil.mag.A48 (1983)607.Google Scholar
- [4]G.Ahlers and R.P.Behringer: Phys.Rev.Lett. 40 (1978)712 andCrossRefGoogle Scholar
- [4]aJ.Maurer and A.Libchaber: J.Physique Lettres41 (1980)L–515.Google Scholar
- [5] a)C.Normand, Y.Pomeau, M.G.Velarde, Rev.Mod.Phys. 49 (1977)581CrossRefGoogle Scholar
- [5]b)F.H.Busse: Rep.Prog.Phys.41 (1973)1929.CrossRefGoogle Scholar
- [6]a1)A.C.Newell and J.Whitehead: J.Fluid Mech. 38 (1979)279Google Scholar
- [6]a1a)L.A.Segel: J.Fluid Mech. 38 (1969)203Google Scholar
- [6]a2)J.Wesfreid,Y.Pomeau,M.Dubois,C.Normand and P.Berge: J.Physique 39 (1978)725.Google Scholar
- [6]b)M.C.Cross and A.C.Newell: “Convection patterns in large aspect ratio systems“ Physica D (1983)Google Scholar
- [7]E.Siggia and A.Zippelius: Phys.Rev.Lett. 47 (1981)835 and Phys.Fluids 26(1983)2905.CrossRefGoogle Scholar
- [8]see for example: J.B.McLaughlin and S.A.Orszag: J.Fluid Mech. 122 (1982)123.Google Scholar
- [9]J.Swift and P.Hohenberg: Phys.Rev. A15 (1977)319 (Appendix A) and C.Normand: Z.Angew.Math.Phys. 32(1981)81.Google Scholar
- [10]P.Manneville: J.Physique 44 (1983)759.Google Scholar
- [11]a)H.S.Greenside,W.M.Coughran,Jr and N.L.Schryer: Phys.Rev.Lett. 49 (1982)726, H.S.Greenside and W.M.Coughran,Jr: “Non-linear Pattern Formation near the Threshold of Rayleigh-Benard Convection“ Phys.Rev.A(1933).CrossRefGoogle Scholar
- [11]b)P.Manneville: J.Physique 44 (1983)563.Google Scholar
- [12]P.Manneville: J.Physique Lettres 44 (1963)L–903.Google Scholar
- [13]See for example: P.Glansdorff and I.Prigogine: Structure. Stabilite et Fluctuations Chap.ll, Sect.10 (Masson Ed., Paris,1971).Google Scholar
- [14]P.Manneville: Phys.Lett. 95A (1983)463.Google Scholar
- [15]R.D.Richtmyer and K.W.Morton: Difference Methods for Initial Value Problems (Wiley,New-York,1967).Google Scholar
- [16]F.S.Acton: Numerical Methods that Work (Harper and Row, New-York,1970).Google Scholar