Observation of fine structures in the Kuramoto-Sivashinsky turbulence

Chaos, Solitons & Fractals

Vol. 2, No. 1, pp.73-79, 1992

0960-0779/9255.00+ O0 © 1992PergamonPress Ltd

Printed in Great Britain

Observation of Fine Structures in the Kuramoto-Sivashinsky Turbulence H. MIURA and Y. AIZAWA D e p a r t m e n t of Applied Physics, Waseda University, O k u b o 3-4-1, Shinjuku, Tokyo 169, Japan

A b s t r a c t - - T u r b u l e n c e in the KS equation is numerically studied by use of the level-crossing technique. The soliton-like excitation and the more detailed structure e m b e d d e d in the KS turbulence are elucidated by the symbolic coding method and by the return m a p reconstruction. New scaling aspects of the KS turbulence are discovered.

1. I N T R O D U C T I O N

Kuramoto-Sivashinsky (KS) equation

ut + UUx + uxx + Uxxxx = 0

(1)

was derived to describe universal aspects of turbulence in various field of physics; phase turbulence in reaction-diffusion systems [1] and flame turbulence in combustion fluid [2]. The turbulent wave form u(x, t) and its spatial power spectrum are shown in Fig. 1. The KS equation is quite similar to fluid mechanical ones such as the K o r t e w e g - d e Vries equation and the Burgurs equation [3], but the physical processes in the KS equation are essentially different from others. In a phenomenological sense, two dissipative effects (positive and negative) are competing in the KS turbulence. The negative dissipative regime is squeezed towards the higher and the lower Fourier components, and the positive regime appears in the intermediate modes. As a result, some unstable intermediate modes are excited to create a relatively stable solitary excitation with a certain non-linear structure [4]. But the fundamental pattern-like soliton does not propagate with constant velocity, since the dispersive effect (as in the KdV equation) is missing in the KS equation. The velocity of the soliton-like pattern is only fluctuating according to the birth-and-death process of soliton-like excitation. The annihilation of the excitation is mainly induced by the global effect due to the boundary condition or the system size effect. Statistical aspects of the KS turbulence is surmised to be explained by taking account two competing factors; one is the local stabilization of soliton-like patterns, and another is the global destabilization due to boundary effect. This picture is quite different from the soliton turbulence discussed before [5]. The basic idea to attack this new type of soliton turbulence in the KS equation is to apply the level-crossing technique based on the symbolic realization of turbulent wave form. As is shown in Fig. 2 the coding rule of the symbolic sequence is as follows; o(x) -- _+1 for u(x) ~ 0 where the zero-tangential point (3,u = 0 at u(x) = 0) is neglected. The detailed wave form of u(x) is suppressed in this coding, but the merit of this realization is to amplify the fine information embedded in the small size fluctuation of u(x). In this paper we will show new scaling aspects of the KS turbulence, which are concealed in the ordinary analysis based on the Fourier mode expansion of u(x). 73

74

H. MIURAand Y. AIZAWA

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T h e creation and the annihilation process of the soliton-like excitation is especially emphasized in this coding o(x). F u r t h e r m o r e , the pattern r a n d o m i z a t i o n mechanism also can be sharpened. For m o r e precise analysis the crossing-level should be changed, but this paper is limited only to the zero-level crossing sequence o(x) which conserves the spatial s y m m e t r y of the KS equation. The KS turbulence is uniquely controlled by a bifurcation p a r a m e t e r L which stands for the system size. The spatio-temporal chaos is often generated for L > 30. O u r simulations are carried out with periodic conditions for L = 48 and L = 64, in which almost the same qualitative features are obtained. 2.

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ZERO-LEVEL

CROSSING

ANALYSIS

the nearest n e i g h b o r zero-level points by x,, and x,,+] (i.e. u ( x , , ) the spacing ?;,, is defined by ~5,, = [x,,-x,,+~]. The aspect of the KS

75

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turbulence is imprinted in the sequence of the spacing {6.}t, where the sequence itself is a stochastic time dependent process. In what follows, we assume the ergodicity of the KS turbulence and its symbolic realization { 6 . } . namely all the statistical properties of {(~n}t can be obtained from the ensemble spanned for large period T ( t e [0, T]), though the system size L is finite. The auto-correlation function C(r)=- ((6. - ( 6 . ) ) ( 6 . + r - ( 6 . + ~ ) ) ) / ( ( 6 . - ( 6 . ) ) 2) is shown in Fig. 3 where the bracket () means a long time average. Quick decay of C(r)

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76

H. MIURA and Y. AIZAWA

implies that soliton-like excitations are almost mutually independent, and only the short-ranged effect remains between nearest neighboring excitations. It is reasonable in some sense that only the short-ranged coherence survives, since the dominant structure of the KS turbulence is sustained by the relatively stable excitation. However, the shortranged coherence comes out not only from the soliton-like excitation solely, but also from very fine structures embedded in the KS turbulence. To see this next we consider the detailed statistics of nearest neighboring structure. The distribution of the spacing P l ( t ~ ) is shown in Fig. 4, where two dominant peaks are 6,,~x.~ = 6,,(= 7.5) and (~max,2 = 6b(-'= 4.2) • The structure of the soliton-like pattern which was obtained by Toh [4] is well adjustable by these two spacing distances 6,, and 6b. We can approximate the soliton-like pattern u,(y) by a cubic function u,(y) = ay + (b/2)y 2 + (c/3!)y 3, where the variable y stands for the running coordinate with velocity e, i.e. y = x - ?t. Therefore the soliton-like (unstable) pattern u,(y) satisfies the following separatrix equation, --gU,. +

du~ d 3 Us U~. + dr. + dy 3 - 0.

Then the coefficients of the previous cubic function u~(y) are determined a = 1 + g2, b = g(1 + ~2), c = - ( 1 + gz), and the nearest neighbor spacings 6~(>0) and 6t,(>0) satisfies, 6 , , - 6b = 3g and 6~eb = 6. From these two relations the velocity ~ becomes g - - 1 . 1 , and this value well coincides with the numerical estimation obtained by Toh [4] though the values of 6~ and 6b are quite different. The correlation between nearest neighbor spacings 6~ and 6,,+i is estimated by two ways; one is the probability of the nearest sum P2(6,, + 6,,+1) and another is the convolution probability Pt(6~) x P l ( ~ n + l ) . If the nearest neighbor spacing 6,, and 6,+~ do not have any correlation, the numerical distribution of the sum 6,, + 6,,+1, say P2(g3n + 0 n + l ) , should be equal to the convolution P~(6,) x PI(C~,,+I) where P1(6,) is the numerical one given in Fig.

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77

4. The strength of the nearest neighboring correlation can be characterized by the difference A2 ~-- P2(c~n + (~n+l) -- E i ( b n )

X

El((~n+l).

(2)

Figure 5 shows each distribution. A 2 is strongly enhanced at On + 6n+1 = 8.2. The value is very close to 26b. But at the point On + 6n+1-~ 6a + bb and ~--2ba, the correlation is strongly suppressed. F u r t h e r m o r e , the enhancement around 6, + 6,+1 ~ 10.5 reveals that very fine structures play important roles in the KS turbulence. Indeed, numerical simulations show that very complicated fine structures are often sandwiched between two soliton-like excitations when soliton-like patterns are growing up or going to vanish. More detailed structure e m b e d d e d in such small size fluctuations can be extracted by the return map q~ which is shown in Fig. 6, c¢: 6. ~-~ 6.+~

(3)

where the dominant structure due to soliton-like pattern is demonstrated by a thick branch along the diagonal line (6.+1 = 6.). The remarkable point is that fine structures are clearly observed as several branching lines which are nearly perpendicular to the diagonal. Small size structures (6n < 6b) seem to be generated in a quite systematic way. It is still difficult to explain the mechanism to create all these branches but a few typical ones are identified in Figs 6 ( b ) - ( d ) .

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78

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3. SCALING ASPECTS IN ZERO-LEVEL CROSSING As was shown in the return map qo (Fig. 6), the birth-and-death process of a large cluster pattern (such as the soliton-like excitation) seems to obey a dynamical law, and several branching patterns in the map qo suggest the existance of a certain strange attractor. Though numerical data do not show a strict self-similarity, we will try to find out the evidence of the scaling law immersed in the KS turbulence or the KS strange attractor. Firstly, statistical properties of fine structures were surveyed by the Fourier mode analysis of the dyadic sequence a ( x , t), whose averaged power spectrum S(k, ~) is enlarged in Fig. l(b). In the short wave-length regime, an inverse power law was undoubtedly observed for 2 ~< k.

Kuramoto-Sivashinsky turbulence S ( k ; tr) ~- k - ~ , (o: ~- 1.63).

79 (4)

The value of the index a~ is the same even in the case of L = 48. This regime seems to be a dissipative range of the KS turbulence. The important point is that the power spectrum of the wave form S ( k ; u) does not show any clear cascade in this regime, that is to say, the relation of equation (4) is the first observation of the scaling law embedded in the KS turbulence. Secondly, the distribution of the zero-crossing distance P1(6) was precisely checked (see the magnification in Fig. 4), and the power law was discovered in the region for 0.2 ~< 6 ~< 1.6, P1(6) = 6 e, (/3 = 0.79).

(5)

H e r e we neglected small scale region (6~<0.2) since the numerical resolution of zero-crossing points is quite unstable. Both scaling regimes of equations (4) and (5) are well consistent with each other; the former adjustable regime ( k / > 2) is translated as 6 ~< 1.6 by use of 7r/k ~- 6. However, the inter-relation between these two scaling forms is still unknown. In this paper we have only studied two-point correlations of the dyadic sequence a(x). However the essential feature of the KS turbulence should be searched in the higher order coherence as were discussed by Pumir [6] and the dimension of the strange attractor [7] must be explained in terms of the symbolic realization used in the present paper. Detailed structure of the KS strange attractor will be analysed in the forthcoming paper. REFERENCES

1. 2. 3. 4. 5. 6. 7.

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence. Springer, Berlin (1984). G. I. Sivashinsky, Acta Astronautica 4, 1177 (1977). A. Jeffrey and T. Kawahara, Asymptotic Methods in Nonlinear Wave Theory. Pitman, Belmont, CA (1982). S. Toh, J. Phys. Soc. Japan 56,949 (1987). Y. Aizawa et al., Physica D 45,307 (1990). A. Pumir, J. Physique 46,511 (1985). Y. Pmeau et al. J. Stat. Phys. 37, 39 (1984); P. Manneville, in Macroscopic Modelling of Turbulent Flows and Fluid Mixtures, edited by O. Pirennau, Lecture Notes in Physics. Springer, Berlin (1985).