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Vorticity field in a cascade model of turbulence
Fluid Dynamics North-Holland
Research
3 (1988) 327-330
327
Vorticity field in a cascade model of turbulence Tohru
NAKANO
Department
of Physics, Chuo University, Bunkyo-ku,
Tokyo 112, Japan
Abstract. We propose a model which interprets the behavior of the spontaneous structure in fully developed turbulence simultaneously. The model is justified model of turbulence.
singularity and the intermittent in the framework of a cascade
1. Introduction There have been a great deal of interest in the dynamics of vortex tubes [I]. The tube is represented as a space curve, and the Biot-Savart law is then used to generate the velocity field. The velocity field convects, deforms and stretches the tube. The local characteristic size such as a radius of the tube, a radius of the curvature of the tube and a separation distance between the tubes becomes finer in a certain region. If a fluid is inviscid, the characteristic size appears to approach zero in a finite time, and the numerical calculation blows up, although it has not been proved yet mathematically in three dimensions. In a viscous fluid the vortex tubes are reconnected in the closest approach region and become entangled. Chorin [2] studied turbulence starting with a few vortex elements and found that the active region gets localized in space in due time. It is quite remarkable if turbulence is connected to the fundamental features of the vortex system, but it seems to be a very hard task. A usual statistical theory of turbulence is situated on the other side. It regards turbulence as a structureless entity and ignores its organized structure. However, it has been well-observed [3] that a turbulent flow has a vortex-like coherent structure. The present paper aims to figure out the vertical structure in turbulence from a statistical point of view.
2. Dynamical scaling relation Large scale fluctuations, externally supplied to turbulence, are transferred to higher wavenumber by the action of the vortex stretching. We pursue the way of propagation in wavenumber space. Divide the wavenumber space logarithmically into bands with a band width b (of order of two), so that the n th band contains the wavenumbers such as b” < kL < b”+‘, where L is a size of the largest eddy. We found [4] that w,(t), the vorticity fluctuation in the n th band, would obey the dynamical scaling relation ~,~(t) = bZnf(bZ”(t-
t,)),
(1)
t,, - tn_, = Ab-‘“.
Here A is a positive constant. approaches 2’,, reaches a peak function shows a remarkable number becomes infinite, and diverge. Thus we have two 0169-5983/88/$2.00
(2)
The characteristic of the scaling function f is that it grows as t at t = t,, and then enters the decay region. This dynamical scaling feature that at t = t, = t, + A/(b’ - 1) the characteristic wavethe generalized enstropies L’,,(t) = ((~Pu/&x~)~) are expected to time domains separated by t = 1,. For t -c rm the fluctuations
0 1988, The Japan
Society
of Fluid
Mechanics
T. Nakrrno
328
/ Cascade model of turbulence
propagate from low to high wavenumbers, but the characteristic wavenumber is still finite. This domain is not affected essentially by a molecular viscosity. It should be noted that the energy enter the decay turbulence conservation law does not hold there. For t > t, the fluctuations region under a molecular viscosity, where the energy conservation is guaranteed, and the power-law decaying scaling function shows the Kolmogorov scaling. The scenario constructed from the above consideration fits the numerical simulation of the Taylor-Green flow by Brachet et al. [S] quite well. 3. Determination of the scaling function Let us discuss the scaling function in more detail. function and a value of z for a cascade model equation band,
Recently we calculated (41 the scaling in the n th for x,~, velocity fluctuations
E,, = h”(x,~~,,~, + C.~,f~~x,,). (3) %,/at = (c,, - E,,+ I J/-x,, with Here the first term in E,, represents a straining effect by an (n - 1)th eddy, and C is a positive constant. There we set t,,))* _%I,,( t) = h ‘~(=.,9j”&“:li(~_ (4) We have and calculate series { f,, } and {z,, } for an initial function f. with A as a parameter. A <<. A
Z, A
1.Q _
1.0 -
", 5
0 0
n
0.9 -
0
d bb
0 n
0.5 -
n
0.8 -
n n
0
0 0
n a
0.7 -
0
0
0
_1 0
0.6 0
1
g 0.4
h
8 0.8
18
1.2
Fig. 1. The fixed value zi as a function constant C.
Fig. 3. The normalized scaling function power law decay curve is followed.
0
' 1.6
'
C
of the coupling
f*(s)
0 0
Fig. 2. The critical
vs. .s in the region
0.4
0.8
1.2
value A, as a function
s i 7,). (a) C= 0.30 and (b) c‘=l.O.
1.6 ')C
of C
For .r > s,, a
T. Nakano / Cascade model of turbulence
329
found that there is a critical value A, on A. For A close to A, the series approach fixed values f* and z* and eventually diverge from them after many iterations. In fig. 1 we have plotted the fixed value of z as a function of C; it increases toward unity with decreasing C. It means that a relatively large strain rate yields a more intermittent solution. Note that z = 2/3 corresponds to the Kolmogorov scaling at all orders of structure functions, while z = 1 does to the extremely intermittent scaling. In fig. 2 the critical value of A is shown. On the other hand, the fixed scaling function f* is depicted in fig. 3a for C = 0.30 and in fig. 3b for C = 1.0 (the horizontal scale is s = b’“(t - t,)). It is to be noted that those scaling functions are limited to the time domain t -c t,. Beyond t, the power-law decaying function f(s) - sey with y = 1 - 2/3z is followed.
4. Structure
function
Based on the above scaling formula, function defined through (x:)
- (T)~/~(
L/r)‘”
the exponent
with
r = Lb-”
yP associated
with the pth order structure
(5)
has been calculated [6,7]. The exponent shows a crossover; for p pc yP = (z - 2/3)( p -p,). It yields a good agreement with the recent data about the structure functions by Anselmet et al. [8], provided z = 0.84 is chosen.
5. Spontaneous
singularity
The scaling relation was also applied [9] to the problem of spontaneous singularity [lo] in the Taylor-Green flow, where the width of vortex line or sheet or something else is expected to become zero in a finite time. The scaling formula indicates that the characteristic size of the how the generalized enstropy Q,(t) vortex structure decreases as (t, - t)“’ . We investigated diverge at t = t,. The results show that (a) the singular structure is excited in a sheet-like way, and (b) for p = 1 z = 0.83 agrees with MOF, while for p 2 2 the helicity cascade, if any, yields the stronger singularity than the energy cascade, implying that the helicity cascade might be important there.
6. Summary Summarize our results. There are two time regions in the obtained dynamical scaling function. (1)~For t -c t, the fluctuations supplied to the largest eddies are transferred to smaller scales in a coherent way, corresponding to the intermittent scaling. The scaling function is determined ‘kinematically from a cascade equation and does not satisfy the energy conservation law at this level. The molecular viscosity does not affect the propagation essentially. (2) For 1 > tr the energy conservation, i.e. a constant energy flow in wavenumber space, is guaranteed, and the system is in a quasi-equilibrium state under a molecular viscosity, showing the Kolmogorov scaling. The whole process composed of the above two stages is repeated randomly as suggested by a theory of chaos [ll], though our model does not take into account it explicity.
330
References [l] [2] [3] [4] [5] [6] [7] [X] [9] [lo] [ll]
A. Leonard. Ann. Rev. Fluid Mech. 17 (1985) 523. A.J. Chorin. Commun. Pure Appl. Math. 34 (1981) X53. B.J. Cantwell. Ann. Rev. Fluid Mech. 13 (1981) 457. T. Nakano. preprint (1987). M.E. Brachet. D. Meiron, S.A. Orzag. B. Nickel. R. Morf and U. Friach. J. Fluid Mech. 30 (1983) 411 T. Nakano and M. Nelkm. Phya. Rev. A 31 (1985) 1980. T. Nakano. Prog. Theor. Phys. 75 (1986) 1295. F. Anselmet. Y. Gagne. E.J. Hopfinger and R.A. Antonia. J. Fluid Mech. 140 (1984) 63. T. Nakano, Frog. Theor. Phys. 73 (1985) 629. R.H. Morf. S.A. Orszag and U. Friach. Phys. Rev. Lett. 44 (1980) 572. E.N. Lorenz. J. Atmos. Sci. 20 (1963) 130.
Research
3 (1988) 327-330
327
Vorticity field in a cascade model of turbulence Tohru
NAKANO
Department
of Physics, Chuo University, Bunkyo-ku,
Tokyo 112, Japan
Abstract. We propose a model which interprets the behavior of the spontaneous structure in fully developed turbulence simultaneously. The model is justified model of turbulence.
singularity and the intermittent in the framework of a cascade
1. Introduction There have been a great deal of interest in the dynamics of vortex tubes [I]. The tube is represented as a space curve, and the Biot-Savart law is then used to generate the velocity field. The velocity field convects, deforms and stretches the tube. The local characteristic size such as a radius of the tube, a radius of the curvature of the tube and a separation distance between the tubes becomes finer in a certain region. If a fluid is inviscid, the characteristic size appears to approach zero in a finite time, and the numerical calculation blows up, although it has not been proved yet mathematically in three dimensions. In a viscous fluid the vortex tubes are reconnected in the closest approach region and become entangled. Chorin [2] studied turbulence starting with a few vortex elements and found that the active region gets localized in space in due time. It is quite remarkable if turbulence is connected to the fundamental features of the vortex system, but it seems to be a very hard task. A usual statistical theory of turbulence is situated on the other side. It regards turbulence as a structureless entity and ignores its organized structure. However, it has been well-observed [3] that a turbulent flow has a vortex-like coherent structure. The present paper aims to figure out the vertical structure in turbulence from a statistical point of view.
2. Dynamical scaling relation Large scale fluctuations, externally supplied to turbulence, are transferred to higher wavenumber by the action of the vortex stretching. We pursue the way of propagation in wavenumber space. Divide the wavenumber space logarithmically into bands with a band width b (of order of two), so that the n th band contains the wavenumbers such as b” < kL < b”+‘, where L is a size of the largest eddy. We found [4] that w,(t), the vorticity fluctuation in the n th band, would obey the dynamical scaling relation ~,~(t) = bZnf(bZ”(t-
t,)),
(1)
t,, - tn_, = Ab-‘“.
Here A is a positive constant. approaches 2’,, reaches a peak function shows a remarkable number becomes infinite, and diverge. Thus we have two 0169-5983/88/$2.00
(2)
The characteristic of the scaling function f is that it grows as t at t = t,, and then enters the decay region. This dynamical scaling feature that at t = t, = t, + A/(b’ - 1) the characteristic wavethe generalized enstropies L’,,(t) = ((~Pu/&x~)~) are expected to time domains separated by t = 1,. For t -c rm the fluctuations
0 1988, The Japan
Society
of Fluid
Mechanics
T. Nakrrno
328
/ Cascade model of turbulence
propagate from low to high wavenumbers, but the characteristic wavenumber is still finite. This domain is not affected essentially by a molecular viscosity. It should be noted that the energy enter the decay turbulence conservation law does not hold there. For t > t, the fluctuations region under a molecular viscosity, where the energy conservation is guaranteed, and the power-law decaying scaling function shows the Kolmogorov scaling. The scenario constructed from the above consideration fits the numerical simulation of the Taylor-Green flow by Brachet et al. [S] quite well. 3. Determination of the scaling function Let us discuss the scaling function in more detail. function and a value of z for a cascade model equation band,
Recently we calculated (41 the scaling in the n th for x,~, velocity fluctuations
E,, = h”(x,~~,,~, + C.~,f~~x,,). (3) %,/at = (c,, - E,,+ I J/-x,, with Here the first term in E,, represents a straining effect by an (n - 1)th eddy, and C is a positive constant. There we set t,,))* _%I,,( t) = h ‘~(=.,9j”&“:li(~_ (4) We have and calculate series { f,, } and {z,, } for an initial function f. with A as a parameter. A <<. A
Z, A
1.Q _
1.0 -
", 5
0 0
n
0.9 -
0
d bb
0 n
0.5 -
n
0.8 -
n n
0
0 0
n a
0.7 -
0
0
0
_1 0
0.6 0
1
g 0.4
h
8 0.8
18
1.2
Fig. 1. The fixed value zi as a function constant C.
Fig. 3. The normalized scaling function power law decay curve is followed.
0
' 1.6
'
C
of the coupling
f*(s)
0 0
Fig. 2. The critical
vs. .s in the region
0.4
0.8
1.2
value A, as a function
s i 7,). (a) C= 0.30 and (b) c‘=l.O.
1.6 ')C
of C
For .r > s,, a
T. Nakano / Cascade model of turbulence
329
found that there is a critical value A, on A. For A close to A, the series approach fixed values f* and z* and eventually diverge from them after many iterations. In fig. 1 we have plotted the fixed value of z as a function of C; it increases toward unity with decreasing C. It means that a relatively large strain rate yields a more intermittent solution. Note that z = 2/3 corresponds to the Kolmogorov scaling at all orders of structure functions, while z = 1 does to the extremely intermittent scaling. In fig. 2 the critical value of A is shown. On the other hand, the fixed scaling function f* is depicted in fig. 3a for C = 0.30 and in fig. 3b for C = 1.0 (the horizontal scale is s = b’“(t - t,)). It is to be noted that those scaling functions are limited to the time domain t -c t,. Beyond t, the power-law decaying function f(s) - sey with y = 1 - 2/3z is followed.
4. Structure
function
Based on the above scaling formula, function defined through (x:)
- (T)~/~(
L/r)‘”
the exponent
with
r = Lb-”
yP associated
with the pth order structure
(5)
has been calculated [6,7]. The exponent shows a crossover; for p
5. Spontaneous
singularity
The scaling relation was also applied [9] to the problem of spontaneous singularity [lo] in the Taylor-Green flow, where the width of vortex line or sheet or something else is expected to become zero in a finite time. The scaling formula indicates that the characteristic size of the how the generalized enstropy Q,(t) vortex structure decreases as (t, - t)“’ . We investigated diverge at t = t,. The results show that (a) the singular structure is excited in a sheet-like way, and (b) for p = 1 z = 0.83 agrees with MOF, while for p 2 2 the helicity cascade, if any, yields the stronger singularity than the energy cascade, implying that the helicity cascade might be important there.
6. Summary Summarize our results. There are two time regions in the obtained dynamical scaling function. (1)~For t -c t, the fluctuations supplied to the largest eddies are transferred to smaller scales in a coherent way, corresponding to the intermittent scaling. The scaling function is determined ‘kinematically from a cascade equation and does not satisfy the energy conservation law at this level. The molecular viscosity does not affect the propagation essentially. (2) For 1 > tr the energy conservation, i.e. a constant energy flow in wavenumber space, is guaranteed, and the system is in a quasi-equilibrium state under a molecular viscosity, showing the Kolmogorov scaling. The whole process composed of the above two stages is repeated randomly as suggested by a theory of chaos [ll], though our model does not take into account it explicity.
330
References [l] [2] [3] [4] [5] [6] [7] [X] [9] [lo] [ll]
A. Leonard. Ann. Rev. Fluid Mech. 17 (1985) 523. A.J. Chorin. Commun. Pure Appl. Math. 34 (1981) X53. B.J. Cantwell. Ann. Rev. Fluid Mech. 13 (1981) 457. T. Nakano. preprint (1987). M.E. Brachet. D. Meiron, S.A. Orzag. B. Nickel. R. Morf and U. Friach. J. Fluid Mech. 30 (1983) 411 T. Nakano and M. Nelkm. Phya. Rev. A 31 (1985) 1980. T. Nakano. Prog. Theor. Phys. 75 (1986) 1295. F. Anselmet. Y. Gagne. E.J. Hopfinger and R.A. Antonia. J. Fluid Mech. 140 (1984) 63. T. Nakano, Frog. Theor. Phys. 73 (1985) 629. R.H. Morf. S.A. Orszag and U. Friach. Phys. Rev. Lett. 44 (1980) 572. E.N. Lorenz. J. Atmos. Sci. 20 (1963) 130.