- Email: [email protected]
Local singular instabilities in multifractal turbulence: structure functions, mixing blobs, and wrinkled flames
16 May 1994 PHYSICS
LETTERS
A
PhysicsLettersA 188 (1994) 151-156
EISEVIEK
Local singular instabilities in multifractal turbulence: structure functions, mixing blobs, and wrinkled flames A. Bershadskii P.O. Box 39953, Ramat Aviv 61398, Tel Aviv, Israel Received 18 November 1993; revised manuscript received 3 1 January 1994; accepted for publication 2 February 1994 Communicated by A.R. Bishop
AtbSblWt
It is argued that the multifractality of higher order structure functions, active local mixing in troposphere and ocean, and the fractal structure of wrinkled flames arise as a result of local singular instabilities (collapses). The modified Landau approach (with poles) is used for studying supercritical instabilities in these cases. The arguments are supported by comparison with experimental data from laboratory and geophysical experiments of different authors.
1.
IntrodRctlon
The internal intermittency of turbulent flow leads to the existence of subregions with self-generation of turbulence. In the subregions quasi-laminar motion becomes unstable and goes to chaos by one or another scenario. The properties of this chaos depend on the type of scenario. This phenomenon is one of the sources of multifractal behavior of turbulence [ 1,2 1. The well-known Landau approach [ 31 consist of the description of turbulence near the critical Reynolds number with taking into account the most unstable mode with complex frequency w = w1 + f&l, where 01 >> “1. Fluctuations of the velocity field are decomposed as 24= A(t)g(x,
t),
When 01 > 0, representation (2) is valid only for small values of t. To analyze the nonlinear stability problem Landau proposed the following amplitude equation, (3) under the assumption that the average over time intervals smaller than cr; ’ and larger than w; ’ was taken. The simplest nonlinear situation, for small [AI, gives us
with nontrivial result for the asymptotic amplitude (t-+m)
(1) (Ai2--t -o,/cQ.
with amplitude A(t) N exp(fa,t
- iwlt).
(2)
(5)
Here aI > 0 and a2 < 0 (supercritical regime). We investigate the local instabilities in stochastic media and rewrite the amplitude equation as
0375-9601/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SsDIO375-9601(94)00210-G
152
A. Bershadskii/PhysicsLettersA
d.4 dt
=
ISS(1994) 151-156
I + f(f),
[
1
P,(I)
where [ ] denotes the Landau expansion in powers of A and A’ (the asterisk denotes complex conjugation), f(t) is the external stochastic force from the turbulent environment. Let us assume that the correlation radius of “external” (turbulent) forces is smaller than typical times of the dynamical problem, then (in the first approximation) f (t ) and f * ( t ) are Gaussian random functions with mean value equal to zero, (f (t)f’(t))
= 20*fQt-
t’).
(7)
Define a probability density for Z = iA]* as pr = (s&41* - I)).
(8)
Fig. 1. Sketch of PO0(I) for regular supercritical Hopf bifurcation.
After averaging over times of the order of 0;’ and allowing Eq. (4) we obtain the following Fokker-Planck equation for PI,
af+(1) at
= ++,z
+ 2a*&
P,(I)
t
+ a*z*)Pt(z)] .
zz (
>
The stationary solution has the form P,
a
exp
CrlZ +
(
+*z*
202
)
O(Z),
(10)
where 8 (I) is the Heaviside function. The maxima of Pm (I) correspond to the local stable states while the minima correspond to unstable ones (see Refs. [ 4,5] ). Fig. 1 demonstrates the schematic dependence of Pm on I. The original state, Z = 0, is statistically unstable whereas the second state, Ii = -al /a~, is stable (cf. with dynamical result (5 ) ) . Similarly, one can obtain the Fokker-Planck equation and its stationary solution for the general case (3).
2. SiignIsu instabilities (collapses) It is known that the three-dimensional Euler equation has solutions which blow up within a finite time (see, for example, Refs. [ 6- 12 ] ) . This collapse can be described as a singular case of supercritical instability. Let us use the singular expansion in the amplitude equation (6) (and (3) ), instead of the regular one.
I
Fig. 2. Sketch of PO0(I) for supercritical singular bifurcation (collapse PO0(0) = 0). The simplest nontrivial pole of order two, dz
dt=
case is the expansion
cOi)a,zn+ a+r Z
tl=l
with the
(11)
12'
Indeed, in the vicinity of Z = 0, we obtain (cf. with the regular case ( 10) 1 P,
a exp
-&(pZ-’
8(Z).
+ fyZ_*)
(
(12)
>
of Pm on Z for the supercritical situation (fi < 0 and y > 0). In this case Fig. 2 shows the dependence
P, (0) = 0, which corresponds
to the collapse of the
A. Bersh&kii/PhysicsLettersA
original solution. In the regular supercritical situation, Pm (0)# 0,and the function P, (I)has a local minimum in the point Z = 0, which corresponds to an instability of the original solution (I = 0). This solution coexists with the stable secondary one (Ii ) in the regular supercritical case (Fig. 1, continuous line), while in the singular supercritical situation we have the secondary solution only (since Pm (0)= 0) . This solution (Zi = -y/B) is stable because Pm (I)has its local maximum at Z = II (see condition ( 13)). If the poles of order more than 2 are taken into account, more than one equilibrium state emerges (aa = 0 because of Pm (I)divergency).
3. Higher order structure functions and local collapses The equilibrium states, defined above, are different from the Kolmogorov one [ 13 1. The equilibrium states are determined by the condition that the right hand side of (3) or (11) equals zero. Here the term d(A]*/ dt plays the role of the mean input of the energy of fluctuations (E), but in the Kolmogorov approach it is a governing parameter. In the vicinity of equilibrium II, E leads to zero, which means that the Kolmogorov approach does not work here. There are two parameters, p and y, which can be candidates for the role of the governing parameter in this case. The condition for statistical stability of the equilibrium state Ii is d2P b4 CC--<<. d12 Y3
(13)
As one can see from (13), statistical stability of the equilibrium state Ii does not depend on the sign of the parameter /3 and is strongly dependent on the parameter y. That is, the parameter y controls the statistical stability of equilibrium Ii. For this reason, it is reasonable to assume that y is the governing parameter for this equilibrium state. In the Kolmogorov approach, dimensional considerations lead to the estimate [ 3,131 Au o;
$/3,.1/3
,
(14)
where Au = u (X + r ) - u (x ), whereas in the vicinity of equilibrium Ii the same considerations lead to Au oc y’/‘rl/’
(15)
188(1994) 151-156
.
I
I
4
0
153
I
8
I
I
12
I
16
20
P
Fig. 3. Scaling exponent &, of structure functions: (x ) jet (ReA = 852 [ 141) and (0) large wind tunnel in Modane (ReA = 2720 [15]).
(the dimension of the parameter y is obtained from Eq. (11): y CC[L]6[Z-]-7). Numerical experiments show that the collapses are localized in space [ 7- 111. Thus scaling ( 15)) unlike the Kolmogorov one ( 14)) can be observed only in the local subregions. One can separate the subregions with the help of higher order structure functions. Indeed, the structure functions of order p are ((Au,“)
= $- 2
(A~u)~,
(16)
i=l
where Ai corresponds to the subregion with number i (and scale r), while N is the total number of these r-subregions. If in the subregions with relatively large values of Au scaling ( 15) takes place, then for large p ( (Au)~) 0: rd+p’7,
(17)
where d is the dimension of the space (or of the experimental signal) and N CCred. For usual experimental signals d = 1 (see below and Fig. 3). By introducing the exponent cp via the relation [ I] ( (Au)~) 0: r6, and using relation ( 17), it is straightforward the following asymptotic representation, Cp=d
+ $J.
(18) to obtain
(19)
The values of Cp,obtained from two laboratory experiments [ 14,151 (at rather large Reynolds numbers), are shown in Fig. 3, where the straight lines are drawn for comparison with ( 19 ) . In the localized Fokker-Planck-Landau approach the inequality ri >> rf is used, where ri is the charao teristic time “inside” of the “singular” subregion and
154
A. Bershaakkii/Physics LettersA 188 (1994) 151-156
rr is the characteristic time (correlation radius) of environment. According to the singular approach the time scale associated with the “singular” subregions at space scale r is ri cc y-“7r”17. The typical time of the external (environment) forcing at the same scale can be obtained using the Kolmogorov theory as ‘I3 r 2/3. The required condition, ri >> rr, will 7fcc(E)be satisfied only if t > Q, where qs = (.~)-~/~y~/~. Therefore the space scale Q can be used as smallscale boundary of the scaling interval in the “singular” approach, instead the Kohnogorov scale rl = (e)- u4y314 * t , travel
time
(see)
4. Mixing blobs in troposphere and ocean The conservation laws prohibit the collapses of vorticity in strictly 2D-turbulence. This leads to an interesting phenomenon in real quasi-2D-turbulence (as is known, strictly 2D-turbulence is unstable to 3Ddisturbances). The 3D-instability can be realized as a regular topological instability (see, for example, Ref. [ 181) and as a singular instability (the local collapse) because the prohibitive ZD-conservation laws are invalid for 3Ddisturbances. These local 3D-instabilities lead to the appearance of blobs characterized by active mixing If there are conditions for numerous appearances of such blobs, then the phenomenon controls large-scale diffusion of passive scalar. Then, in full analogy with the Kolmogorov approach [ 13 I, it follows from dimensional considerations that the diffusivity, K,, has the following form, 1 d12 K* = --
6 dt
o(
y3/7p7,
(20)
where 1 is the characteristic scale of a cloud of the passive scalar (in the Kolmogorov case K, cc E”314/3 [ 131). Expression (20) is different from both the case of 3D-turbulence (Richardson-Kolmogorov law) and from the case of purely 2D-turbulence. From (20), we obtain 1 rx t716.
(21)
Relations (20) and (2 1) are in good agreement with the results of the experiments on diffusion of the passive scalar in the troposphere [ 191 and the ocean [ 201, as can be seen from Figs. 4 and 5.
Fig. 4. Observationsof widths (horizontal standard deviation) of diffusing tracer as a function of downwind travel time. Different symbols correspond to the results of different authors and dashed lines correspond to the Richardson-Kolmorov law (2 O(t3j2) and to the Taylor law (I CCt). The Figure is adapted from Ref. [ 191.
/ IO3
I
I
IO5 ,cm
e
I
IO7
Fig. 5. Diffusivity versus the scale 1 in tbe ocean., adapted from Ref. [20].
5. Fractal dimensions of a turbulent (wrinkled) flame front In his pioneer paper [ 2 1 ] Landau showed the instability of a plane flame front for typical conditions. He supposed that this instability leads to turbulence of the front and an “eroded” area of combustion appears instead of a very narrow (plane) layer. This “eroded” surface can be described, in modem terms, as a fractal one [ 1,121, and the transition can be described as a singular instability of the original smooth solution within a finite time.
155
A. Bershaa!&i /Physics Letters A 188 (1994) 151-156
We start considering the expansion of a flame front which is spherical symmetric, in the mean, and whose cross section passes through its geometric center (again in the mean) with an effective radius R (t ) . Approximating the boundary of this cross section by a broken line consisting of sections of length r the number N of such sections, N 21 (r/R)-Dz,
(22)
where D = (D2 + 1) is the fractal dimension of the flame front. Now let us find the effective increase, d.r, of the area of the flame front cross section during the time dt
1.0 0.1
1
10
100
U’lU,
d.r N r(Audt)N,
(23)
where Au is the velocity of the section r in the direction normal to it. It follows from (22) and (23) that ds
x
N r’-hAuRD2
(24)
and using the relation AuNrC
(25)
we obtain ds z-’
r-&+c’@.
(26)
Since the rate of increase of the area of the flame front cross section should not depend on r (used for its approximation) it follows from (26 ) that 1- 4 + r = 0. Thus 9
= 1+e
(27)
(see also Ref. [ 121). For the Kolmogorov equilibrium from (14) and (27) we obtain DZ = $, while for Z1equilibrium from ( 15) and (27) 4 = $. The measured fractal dimensions (Dz ) by Mantzaras et al., Marayama and Takeno, and Nort and Santavicca are plotted versus U’/ULin Fig. 6 (adapted from Ref. [22] ). Here U’/UL is the turbulent intensity to the laminar flame speed, and straight lines are drawn for comparison with Kolmogorov’s 9 = $ (with intermittency correction [ 11) and with Z1equilibrium 4 = 9. One can see that for developed turbulence (large values u’/ UL ) the fractal dimension
Fig. 6. Fractal dimensions of turbulent flame fronts versus U’/UL.Solid symbols:Mantzaraset al.; open triangles:North and Santavicca;open circles: Murayama and Takeno. The figure is adapted from Ref. [22]. of the flame fronts is 9 N 1.37 (i.e. rather close to Kolmogorov’s one), while for a transitional type of turbulence (relative small values U’/UL)the fractal dimension is rather close to the II-equilibrium dimension, 4 = $. The value 4 = ! has been connected with singularities of the “front” boundary. Since scalar interfaces in fully developed turbulence have (generally) another nature, this value of 4 is not relevant to them. Perhaps this is possible for scalar surfaces in transitional flows.
Acknowledgement
The author is grateful to J.F. Musy for sending data of a laboratory experiment (Ref. [ 15]), and L. Filyand, A. Mikishev, K.R. Sreenivasan, and A. Tsinober for comments and encouragement. This work was supported in part by the Wolfson Foundation as well as by the Israeli Ministry of Absorbtion. References
[ 1 ] K_R.Sreenivasan,Annu. Rev. Fluid Mech. 23 ( 1991) 539.
A. Be&ad&ii and A. Tsinober, Phys. Rev. E 48 (1993) 282. [3] L.D. Landau and E.M. Lifshitz, Fluid mechanics (Pergamon, Oxford, 1959). [2 ]
156
A. Bersha&ii/PhysicsLettersA
l&3(1994) 151-156
[4] H. Haken, Ann. New York Acad. Sci. 316 (1979) 357.
[14] F. Anselmet, Y. Gagne, E.F. Hopfmger and R.A.
[ 51 G. Nikolis and J.W. Turner, Annu. New York Acad. Sci. 316 (1979) 251. [6] R. Morf, S.A. Orszag and U. Frisch, Phys. Rev. Lett. 44 (1980) 572. [7] A.J. Chorin, Commun. Math. Phys. 83 (1982) 853. [8] A. Pumir and E.D. Siggia, Phys. Fluids A 2 (1990) 220. [9] E.D. Siggia and A. Pumir, Phys. Rev. Lett. 55 (1985 1749. [lo] A.FukuyuandT.Arai,Fluid.Dyn.Res. 7 (1991) 229. [ 111 R.M. Kerr, Phys. Fluids A 5 ( 1993) 1725. [ 121 A. Bershadskii, A. Galper and A. Tsinober, Phys. Lett. A 171 (1993) 29. [ 131 A.S. Monin and A.M. Yaglom, Statistical fluid mechanics, Vol. 2 (MIT Press, Cambridge, 1975 ).
Antonia, J. Fluid Mech. 140 (1984) 63. [ 15] J.F. Muzi, E. Bacry and A. Arneodo, Phys. Rev. Lett. 67 (1991) 3515. [ 161 G. Stolovitzky and K.R. Sreenivasan, Phys. Rev. E 48 (1993) R33. [ 171 R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli and S. Succi, Universita di Roma, report no. ROM 2F/92/54. [ 181 A. Bet-shad&ii, E. Kit and A. Tsinober, Proc. R. Sot. A441 (1993) 147. [ 191 F.A. Gifford, Sixth Symposium on Turbulence and diffusion, Boston (1983), p. 300. [ 201 A. Ocubo, Deep-sea Res. 18 (1971) 789. [21] L.D. Landau, Acta Physicohim. USSR 19 (1944) 77. [22] J. Mantzaras, Combust. Sci. Tech. 86 (1992) 135.
LETTERS
A
PhysicsLettersA 188 (1994) 151-156
EISEVIEK
Local singular instabilities in multifractal turbulence: structure functions, mixing blobs, and wrinkled flames A. Bershadskii P.O. Box 39953, Ramat Aviv 61398, Tel Aviv, Israel Received 18 November 1993; revised manuscript received 3 1 January 1994; accepted for publication 2 February 1994 Communicated by A.R. Bishop
AtbSblWt
It is argued that the multifractality of higher order structure functions, active local mixing in troposphere and ocean, and the fractal structure of wrinkled flames arise as a result of local singular instabilities (collapses). The modified Landau approach (with poles) is used for studying supercritical instabilities in these cases. The arguments are supported by comparison with experimental data from laboratory and geophysical experiments of different authors.
1.
IntrodRctlon
The internal intermittency of turbulent flow leads to the existence of subregions with self-generation of turbulence. In the subregions quasi-laminar motion becomes unstable and goes to chaos by one or another scenario. The properties of this chaos depend on the type of scenario. This phenomenon is one of the sources of multifractal behavior of turbulence [ 1,2 1. The well-known Landau approach [ 31 consist of the description of turbulence near the critical Reynolds number with taking into account the most unstable mode with complex frequency w = w1 + f&l, where 01 >> “1. Fluctuations of the velocity field are decomposed as 24= A(t)g(x,
t),
When 01 > 0, representation (2) is valid only for small values of t. To analyze the nonlinear stability problem Landau proposed the following amplitude equation, (3) under the assumption that the average over time intervals smaller than cr; ’ and larger than w; ’ was taken. The simplest nonlinear situation, for small [AI, gives us
with nontrivial result for the asymptotic amplitude (t-+m)
(1) (Ai2--t -o,/cQ.
with amplitude A(t) N exp(fa,t
- iwlt).
(2)
(5)
Here aI > 0 and a2 < 0 (supercritical regime). We investigate the local instabilities in stochastic media and rewrite the amplitude equation as
0375-9601/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SsDIO375-9601(94)00210-G
152
A. Bershadskii/PhysicsLettersA
d.4 dt
=
ISS(1994) 151-156
I + f(f),
[
1
P,(I)
where [ ] denotes the Landau expansion in powers of A and A’ (the asterisk denotes complex conjugation), f(t) is the external stochastic force from the turbulent environment. Let us assume that the correlation radius of “external” (turbulent) forces is smaller than typical times of the dynamical problem, then (in the first approximation) f (t ) and f * ( t ) are Gaussian random functions with mean value equal to zero, (f (t)f’(t))
= 20*fQt-
t’).
(7)
Define a probability density for Z = iA]* as pr = (s&41* - I)).
(8)
Fig. 1. Sketch of PO0(I) for regular supercritical Hopf bifurcation.
After averaging over times of the order of 0;’ and allowing Eq. (4) we obtain the following Fokker-Planck equation for PI,
af+(1) at
= ++,z
+ 2a*&
P,(I)
t
+ a*z*)Pt(z)] .
zz (
>
The stationary solution has the form P,
a
exp
CrlZ +
(
+*z*
202
)
O(Z),
(10)
where 8 (I) is the Heaviside function. The maxima of Pm (I) correspond to the local stable states while the minima correspond to unstable ones (see Refs. [ 4,5] ). Fig. 1 demonstrates the schematic dependence of Pm on I. The original state, Z = 0, is statistically unstable whereas the second state, Ii = -al /a~, is stable (cf. with dynamical result (5 ) ) . Similarly, one can obtain the Fokker-Planck equation and its stationary solution for the general case (3).
2. SiignIsu instabilities (collapses) It is known that the three-dimensional Euler equation has solutions which blow up within a finite time (see, for example, Refs. [ 6- 12 ] ) . This collapse can be described as a singular case of supercritical instability. Let us use the singular expansion in the amplitude equation (6) (and (3) ), instead of the regular one.
I
Fig. 2. Sketch of PO0(I) for supercritical singular bifurcation (collapse PO0(0) = 0). The simplest nontrivial pole of order two, dz
dt=
case is the expansion
cOi)a,zn+ a+r Z
tl=l
with the
(11)
12'
Indeed, in the vicinity of Z = 0, we obtain (cf. with the regular case ( 10) 1 P,
a exp
-&(pZ-’
8(Z).
+ fyZ_*)
(
(12)
>
of Pm on Z for the supercritical situation (fi < 0 and y > 0). In this case Fig. 2 shows the dependence
P, (0) = 0, which corresponds
to the collapse of the
A. Bersh&kii/PhysicsLettersA
original solution. In the regular supercritical situation, Pm (0)# 0,and the function P, (I)has a local minimum in the point Z = 0, which corresponds to an instability of the original solution (I = 0). This solution coexists with the stable secondary one (Ii ) in the regular supercritical case (Fig. 1, continuous line), while in the singular supercritical situation we have the secondary solution only (since Pm (0)= 0) . This solution (Zi = -y/B) is stable because Pm (I)has its local maximum at Z = II (see condition ( 13)). If the poles of order more than 2 are taken into account, more than one equilibrium state emerges (aa = 0 because of Pm (I)divergency).
3. Higher order structure functions and local collapses The equilibrium states, defined above, are different from the Kolmogorov one [ 13 1. The equilibrium states are determined by the condition that the right hand side of (3) or (11) equals zero. Here the term d(A]*/ dt plays the role of the mean input of the energy of fluctuations (E), but in the Kolmogorov approach it is a governing parameter. In the vicinity of equilibrium II, E leads to zero, which means that the Kolmogorov approach does not work here. There are two parameters, p and y, which can be candidates for the role of the governing parameter in this case. The condition for statistical stability of the equilibrium state Ii is d2P b4 CC--<<. d12 Y3
(13)
As one can see from (13), statistical stability of the equilibrium state Ii does not depend on the sign of the parameter /3 and is strongly dependent on the parameter y. That is, the parameter y controls the statistical stability of equilibrium Ii. For this reason, it is reasonable to assume that y is the governing parameter for this equilibrium state. In the Kolmogorov approach, dimensional considerations lead to the estimate [ 3,131 Au o;
$/3,.1/3
,
(14)
where Au = u (X + r ) - u (x ), whereas in the vicinity of equilibrium Ii the same considerations lead to Au oc y’/‘rl/’
(15)
188(1994) 151-156
.
I
I
4
0
153
I
8
I
I
12
I
16
20
P
Fig. 3. Scaling exponent &, of structure functions: (x ) jet (ReA = 852 [ 141) and (0) large wind tunnel in Modane (ReA = 2720 [15]).
(the dimension of the parameter y is obtained from Eq. (11): y CC[L]6[Z-]-7). Numerical experiments show that the collapses are localized in space [ 7- 111. Thus scaling ( 15)) unlike the Kolmogorov one ( 14)) can be observed only in the local subregions. One can separate the subregions with the help of higher order structure functions. Indeed, the structure functions of order p are ((Au,“)
= $- 2
(A~u)~,
(16)
i=l
where Ai corresponds to the subregion with number i (and scale r), while N is the total number of these r-subregions. If in the subregions with relatively large values of Au scaling ( 15) takes place, then for large p ( (Au)~) 0: rd+p’7,
(17)
where d is the dimension of the space (or of the experimental signal) and N CCred. For usual experimental signals d = 1 (see below and Fig. 3). By introducing the exponent cp via the relation [ I] ( (Au)~) 0: r6, and using relation ( 17), it is straightforward the following asymptotic representation, Cp=d
+ $J.
(18) to obtain
(19)
The values of Cp,obtained from two laboratory experiments [ 14,151 (at rather large Reynolds numbers), are shown in Fig. 3, where the straight lines are drawn for comparison with ( 19 ) . In the localized Fokker-Planck-Landau approach the inequality ri >> rf is used, where ri is the charao teristic time “inside” of the “singular” subregion and
154
A. Bershaakkii/Physics LettersA 188 (1994) 151-156
rr is the characteristic time (correlation radius) of environment. According to the singular approach the time scale associated with the “singular” subregions at space scale r is ri cc y-“7r”17. The typical time of the external (environment) forcing at the same scale can be obtained using the Kolmogorov theory as ‘I3 r 2/3. The required condition, ri >> rr, will 7fcc(E)be satisfied only if t > Q, where qs = (.~)-~/~y~/~. Therefore the space scale Q can be used as smallscale boundary of the scaling interval in the “singular” approach, instead the Kohnogorov scale rl = (e)- u4y314 * t , travel
time
(see)
4. Mixing blobs in troposphere and ocean The conservation laws prohibit the collapses of vorticity in strictly 2D-turbulence. This leads to an interesting phenomenon in real quasi-2D-turbulence (as is known, strictly 2D-turbulence is unstable to 3Ddisturbances). The 3D-instability can be realized as a regular topological instability (see, for example, Ref. [ 181) and as a singular instability (the local collapse) because the prohibitive ZD-conservation laws are invalid for 3Ddisturbances. These local 3D-instabilities lead to the appearance of blobs characterized by active mixing If there are conditions for numerous appearances of such blobs, then the phenomenon controls large-scale diffusion of passive scalar. Then, in full analogy with the Kolmogorov approach [ 13 I, it follows from dimensional considerations that the diffusivity, K,, has the following form, 1 d12 K* = --
6 dt
o(
y3/7p7,
(20)
where 1 is the characteristic scale of a cloud of the passive scalar (in the Kolmogorov case K, cc E”314/3 [ 131). Expression (20) is different from both the case of 3D-turbulence (Richardson-Kolmogorov law) and from the case of purely 2D-turbulence. From (20), we obtain 1 rx t716.
(21)
Relations (20) and (2 1) are in good agreement with the results of the experiments on diffusion of the passive scalar in the troposphere [ 191 and the ocean [ 201, as can be seen from Figs. 4 and 5.
Fig. 4. Observationsof widths (horizontal standard deviation) of diffusing tracer as a function of downwind travel time. Different symbols correspond to the results of different authors and dashed lines correspond to the Richardson-Kolmorov law (2 O(t3j2) and to the Taylor law (I CCt). The Figure is adapted from Ref. [ 191.
/ IO3
I
I
IO5 ,cm
e
I
IO7
Fig. 5. Diffusivity versus the scale 1 in tbe ocean., adapted from Ref. [20].
5. Fractal dimensions of a turbulent (wrinkled) flame front In his pioneer paper [ 2 1 ] Landau showed the instability of a plane flame front for typical conditions. He supposed that this instability leads to turbulence of the front and an “eroded” area of combustion appears instead of a very narrow (plane) layer. This “eroded” surface can be described, in modem terms, as a fractal one [ 1,121, and the transition can be described as a singular instability of the original smooth solution within a finite time.
155
A. Bershaa!&i /Physics Letters A 188 (1994) 151-156
We start considering the expansion of a flame front which is spherical symmetric, in the mean, and whose cross section passes through its geometric center (again in the mean) with an effective radius R (t ) . Approximating the boundary of this cross section by a broken line consisting of sections of length r the number N of such sections, N 21 (r/R)-Dz,
(22)
where D = (D2 + 1) is the fractal dimension of the flame front. Now let us find the effective increase, d.r, of the area of the flame front cross section during the time dt
1.0 0.1
1
10
100
U’lU,
d.r N r(Audt)N,
(23)
where Au is the velocity of the section r in the direction normal to it. It follows from (22) and (23) that ds
x
N r’-hAuRD2
(24)
and using the relation AuNrC
(25)
we obtain ds z-’
r-&+c’@.
(26)
Since the rate of increase of the area of the flame front cross section should not depend on r (used for its approximation) it follows from (26 ) that 1- 4 + r = 0. Thus 9
= 1+e
(27)
(see also Ref. [ 121). For the Kolmogorov equilibrium from (14) and (27) we obtain DZ = $, while for Z1equilibrium from ( 15) and (27) 4 = $. The measured fractal dimensions (Dz ) by Mantzaras et al., Marayama and Takeno, and Nort and Santavicca are plotted versus U’/ULin Fig. 6 (adapted from Ref. [22] ). Here U’/UL is the turbulent intensity to the laminar flame speed, and straight lines are drawn for comparison with Kolmogorov’s 9 = $ (with intermittency correction [ 11) and with Z1equilibrium 4 = 9. One can see that for developed turbulence (large values u’/ UL ) the fractal dimension
Fig. 6. Fractal dimensions of turbulent flame fronts versus U’/UL.Solid symbols:Mantzaraset al.; open triangles:North and Santavicca;open circles: Murayama and Takeno. The figure is adapted from Ref. [22]. of the flame fronts is 9 N 1.37 (i.e. rather close to Kolmogorov’s one), while for a transitional type of turbulence (relative small values U’/UL)the fractal dimension is rather close to the II-equilibrium dimension, 4 = $. The value 4 = ! has been connected with singularities of the “front” boundary. Since scalar interfaces in fully developed turbulence have (generally) another nature, this value of 4 is not relevant to them. Perhaps this is possible for scalar surfaces in transitional flows.
Acknowledgement
The author is grateful to J.F. Musy for sending data of a laboratory experiment (Ref. [ 15]), and L. Filyand, A. Mikishev, K.R. Sreenivasan, and A. Tsinober for comments and encouragement. This work was supported in part by the Wolfson Foundation as well as by the Israeli Ministry of Absorbtion. References
[ 1 ] K_R.Sreenivasan,Annu. Rev. Fluid Mech. 23 ( 1991) 539.
A. Be&ad&ii and A. Tsinober, Phys. Rev. E 48 (1993) 282. [3] L.D. Landau and E.M. Lifshitz, Fluid mechanics (Pergamon, Oxford, 1959). [2 ]
156
A. Bersha&ii/PhysicsLettersA
l&3(1994) 151-156
[4] H. Haken, Ann. New York Acad. Sci. 316 (1979) 357.
[14] F. Anselmet, Y. Gagne, E.F. Hopfmger and R.A.
[ 51 G. Nikolis and J.W. Turner, Annu. New York Acad. Sci. 316 (1979) 251. [6] R. Morf, S.A. Orszag and U. Frisch, Phys. Rev. Lett. 44 (1980) 572. [7] A.J. Chorin, Commun. Math. Phys. 83 (1982) 853. [8] A. Pumir and E.D. Siggia, Phys. Fluids A 2 (1990) 220. [9] E.D. Siggia and A. Pumir, Phys. Rev. Lett. 55 (1985 1749. [lo] A.FukuyuandT.Arai,Fluid.Dyn.Res. 7 (1991) 229. [ 111 R.M. Kerr, Phys. Fluids A 5 ( 1993) 1725. [ 121 A. Bershadskii, A. Galper and A. Tsinober, Phys. Lett. A 171 (1993) 29. [ 131 A.S. Monin and A.M. Yaglom, Statistical fluid mechanics, Vol. 2 (MIT Press, Cambridge, 1975 ).
Antonia, J. Fluid Mech. 140 (1984) 63. [ 15] J.F. Muzi, E. Bacry and A. Arneodo, Phys. Rev. Lett. 67 (1991) 3515. [ 161 G. Stolovitzky and K.R. Sreenivasan, Phys. Rev. E 48 (1993) R33. [ 171 R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli and S. Succi, Universita di Roma, report no. ROM 2F/92/54. [ 181 A. Bet-shad&ii, E. Kit and A. Tsinober, Proc. R. Sot. A441 (1993) 147. [ 191 F.A. Gifford, Sixth Symposium on Turbulence and diffusion, Boston (1983), p. 300. [ 201 A. Ocubo, Deep-sea Res. 18 (1971) 789. [21] L.D. Landau, Acta Physicohim. USSR 19 (1944) 77. [22] J. Mantzaras, Combust. Sci. Tech. 86 (1992) 135.