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Weak turbulence in periodic flows
Physica 17D (1985) 243-255 North-Holland, Amsterdam
WEAK TURBULENCE IN PERIODIC FLOWS
Gregory I. SIVASHINSKY Depurtmentof Muthemoticul
Sciences, Tel-Aviv
Universi!v, Rumut-Aviv,
Tel-Aviv 69978, Isruel
Received 6 August 1984 Revised manuscript received 22 May 1985
We consider the stability of a two-dimensional plane-parallel flow of viscous liquid in an external force field which is a periodic function of one of the coordinates. At sufficiently high Reynolds numbers the plane-parallel flow becomes unstable and a two-dimensional secondary flow ensues. Near the stability threshold, the secondary flow turns out to be large-scale and chaotically self-fluctuating in time.
1. Introduction
One of the most intriguing problems in hydrodynamics is the spontaneous formation of large-scale turbulent structures in a viscous liquid driven at small scales. In a theoretical approach to this phenomenon, it is convenient to understand the latter as a manifestation of long-wave instability of a system of small-scale eddies maintained by an external force field that obstruct their dissipation. One of the simplest hydrodynamic systems of the type in question is the so-called Kolmogorov flow - the two-dimensional flow of a viscous liquid induced by an unidirectional external force field periodic in one of the coordinates (see, e.g., Arnold and Meshalkin [l]; Obukhov [2]). Fig. 1 shows streamlines and velocity profile of the corresponding laminar flow, which obeys a periodic sine law along the y-axis. The problem of the linear stability of such flows was solved by Meshalkin and Sinai [3] and later, independently, by Green [4]. Fig: 2(a) shows the marginal stability curve corresponding to the solution. In 1979, Dolzhansky et al. [5] reproduced Kolmogorov flow under laboratory conditions in a thin layer of electrolite (x, y-plane) placed in an external force field f = [H, j], where H is the magnetic field intensity, with the z-component a periodic function of y; ,j is the current density between the two electrodes in the liquid layer. The magnitude of j can be varied by varying the electrode voltages and thus producing flows with different Reynolds numbers (fig. 3). As shown by Dolzhansky et al. [5], a more accurate description of the observed liquid flow requires a modification of the Kolmogorov model which,.also allows for external friction (friction on the bottom surface). The corresponding equations of two-dimensional hydrodynamics are: u,+uu,+uu,,=
-px+R-‘V2u-~R-‘u-RR-1siny,
u, + uux + UU.” = -pY + R-‘v2u
- pR-‘u,
(1.1)
11, + l+ = 0.
The units of length, time and velocity in these equations are respectively q-i, 0167-2789/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
vqy-l,
vm1qe2, where Y is
244
G.I. Sivashinsky/
Weak turbulence in periodic flows
Fig. 1. Velocity profile and streamlines of Kolmogorov flow.
the viscosity, q and y the space frequency and intensity of the applied force; R = v-*qe3y is the Reynolds number; p = 2q-*h-* is the coefficient of friction, where h is the depth of the liquid layer. The inclusion of external friction is essential for a correct estimate of the critical Reynolds number R,, which is calculated to be of order lo3 (instead of the fi according to Meshalkin and Sinai [3]), and this agrees completely with experiment. The appropriate marginal stability curve is depicted in fig. 2(b). Experiment shows [5] that, after going through the critical value of the Reynolds number R,, unidirectional flow (fig. 1) becomes unstable and a secondary flow appears, in the form of a regular system of stationary eddies (fig. 4). At R = 2.5R, the steady flow becomes unstable and periodic oscillations set in. As the Reynolds number increases, the oscillation period decreases. For R s 3R, the oscillations have a marked monochromatic nature. When the excitation is strong (R 2 3R,) the oscillation pattern becomes more complicated. The spectrum is enriched by higher harmonics. At R x- 3R, one may expect the onset of irregular oscillations, i.e. turbulence. In a system free of external friction (p = 0), the stability region at R L R, = R, is localized near small wavenumbers, i.e., the secondary flow is expected to be long-wave (see fig. 2(a)). The situation changes once external friction has been introduced (fig 2(b)), since then, near the stability threshold; the secondary flow corresponds to a certain finite wavelength - 2~/k,. In the experiment done by Dolzhansky et al. [5].
Fig. 2. Marginal stability curve (a) for periodic flow without inclusion of external friction; (b) flow with external friction included. R-Reynolds number. k-wavenumber of perturbation.
G.I. Sivashinsky/
Weak turbulence in periodic flows
245
H
%I+
a
Y
c
Fig. 3. Diagram of MHD arrangement for simulation of Kolmogorov flow. (a) View from above; (b) side view; (c) profile of vertical component of magnetic field H generated by magnetic system (b). (1) Magnetoelastic rubber; (2) profile of Lorentz force /= [H, j] acting on electrolite upon passage of direct current j; (3) electrolite; (4) force lines of magnetic field.
2a/k. was of the order of unity, i.e., the longitudinal dimension of the eddies was comparable with the period of the external force. Thus, the effect of friction is in this case quite significant. Nevertheless, to facilitate the theoretical analysis-and, hopefully, without too much detriment to the qualitative aspect of the phenomenon - we shall assume throughout that friction is small. In this sense, the asymptotic picture discussed below is, as it were, intermediate between Kolmogorov’s model, which ignores external friction, and the case studied by Dolzhansky et al., where the effect of friction completely excludes the possibility of large-scale secondary flow. In the small friction case k, is also small, and so the secondary flow may again be of a long-wave nature.? From the mathematical point of view, the situation turns out to be very similar to that in the problem of free convection between nearly insulating boundaries (Chapman and Proctor [6]; Depassier and Spiegel [7]; Gertsberg and Sivashinsky [S]).
2. Nonlinear asymptotic analysis. The case of one large-scale space variable For further analysis, it is convenient function, (VW,+
*j,;(vZ*),-
to transform
from system (1.1) to an equation
!Px(~29)y=R-1v%P-pR-1v2!P-R-1co~y.
for the stream
(2.1)
Set R-l
= (1 - e2)Ri1
(e a 1).
(2.2)
Thus, as small parameter e we take the proximity between the Reynolds number and the stability threshold
TWe note that in this case
R,
is close to R,.
246
G.I. Sivushinsky/
Weuk turbulence
in periodicf70ws
Fig. 4. Streamlines of secondary flow near stability threshold [5].
(R = R,). On the assumption that the coefficient of friction is small, p=
A&4,
(2.3)
we introduce the following scaled space and time variables: (=&X,
7 =
E4t.
(2.4)
The selection of these scalings was governed by the analogy with the above-mentioned convection problem [6, 7, 81. As we shall see below, the approximation based on introduction of a single large-scale coordinate 5 provides a successful description of the formation of regular eddy systems (fig. 4). However, it turns out that the resulting asymptotic model is not capable of capturing the effect of turbulence and will subsequently be improved in section 3. In terms of the new variables, eq. (2.1) becomes
E4(V2~)~+E~~(V2~)~-E~~(v2\k)y = (1 - E2)R,‘V4\k -(l - &2)h&%;l+P (v2\k=Ez*[E+
(2.5)
- (1 - &2)R~1COS_y,
Q.
We shall assume that the function \k is 2n-periodic with respect to y. For further analysis, it is useful to integrate eq. (2.5) over the interval 0 of \k and the identity
5 y
s 27r. Using the periodicity
we can write the resulting integral relationship as follows: ?QP,,dy
= (1 - e2)eR,1~2n!I’~i,,,dy
-(l
- e2)he3R,1/02?Pt,dy.
(2.7)
To solve eq. (2.5), we express the solution as an asymptotic expansion,
!P=-P”+E?P1+&2!P2+&3*3+ -.*.
(2.8)
G.I. Sivushinsky / Weak turbulence in periodic flows
For the zerorh approximation,
247
we obtain from (2.5)
q”.“.”= cmY*
(2.9)
Hence (2.10) The integral relationship (2.7) in the zeroth approximation
yields (2.11)
It is readily seen from (2.10) that this relationship is automatically valid for any A’((, T). For the Jirst approximation, we deduce from (2.Q using (2.10) that R,V . YY.Y.Y = -Aisin
y.
(2.12)
P = -R,Aisiny+A’([,T).
(2.13)
Hence
The integral relationship (2.7) in the first approximation
yields (2.14)
Hence, in view of (2.10) and (2.13), we obtain (R,-2R,‘)A;,,=
0,
or
R,=fi.
(2.15)
Thus, we have determined the critical Reynolds number at which the onset of secondary flow is possible. For the second approximation we obtain from (2.5), using (2.10) and (2.13) R3r2
r
.v.v.v.v
-R,(Ai)‘cosy-Aisiny.
(2.16)
Hence
gr== -Ri(
Ai)‘cos
y - R,Aisin
In the second approximation,
y + A’(t,
7).
(2.17)
the integral relationship (2.17) gives (2.18)
By substituting the previously found solutions ?Po,!Pl, !PZ into this relationship, one can verify that it is automatically valid at R, = fi.
248
G. I. Sivashinsky/
For the third approximation;
Weak turbulence
in periodic flows
we again proceed from (2.5), using (2.10), (2.13) and (2.17), to obtain
R,‘YP&, = [ R2,( Ai)) - 3&
- At - Ai] sin y - 2A$4; cosy.
(2.19)
Hence q3= [R;(A;)3-3R,A;Ct-
R,Ai
- R,A;]
sin y - ~R~A~A~cosY
The integral relationship (2.17) in the third approximation
+ A3(Ey T).
(2.20)
gives
(2.21) Substituting the previously following relationship:
found expressions for !PO,‘PI, qlk2,‘k; into this equality, we arrive at the
+([Ro-(R;/3)(A~)*]A~}~~~+XR6’A~,=0.
(AL) T+ (3Ro/2)AL
(2.22)
This determines the required differential equation for A,(& T). Integrating eq. (2.22) twice with respect to 6, we obtain the following simpler equation: +{[Ro-(R;/3)(A;)2]A;)tfXR~1Ao=0. A: + (3Ro/2)A&,,
(2.23)
Once A,([, T) is known, one can determine the stream function of secondary flow from eq. (2.10). Introducing the elementary resealings A”=(3/fi)@,
(=&I,
7=6&,
A=6cu,
(2.24)
one transforms eq. (2.24) to the following more conveniently analyzed form:
~s+9,,,+[(2-cPF)~,],+a~=0.
(2.25)
It is interesting that this equation is a one-dimensional version of the equation for the evolution of the plane form of convective cells in a nearly insulated liquid layer [6, 7, 81. The dispersion relation for linear stability analysis of the trivial solution is o=2k2-k4-a!
Instability
(@-exp(ws+ik<)).
appears at a < OL,= 1, corresponding
(2.26) to sufficiently low friction (at a fixed Reynolds number
R > R,).
Eq. (2.26) indicates the existence of a wavelength (2?r/k, = 2s) corresponding to maximum growth rate of small harmonic disturbances. It would therefore be natural to expect that 2?r is the characteristic cell size
G. I. Sivashinsky / Weak turbulence in periodic flows
249
1.5F
i-
0.5 cp O-
-0.5-
-1 -
-1.5
t -2 -5
I
I
I
I
0
5
10
15
I< 20
I
I
I
25
30
35
1 40
Fig. 5. Steady periodic solution generated by eq. (2.25) at a = 0.5.
of the periodic structure produced during the evolution of the initial disturbances (Rayleigh principle). However, analysis of the stability of the periodic solutions done for the case (Y= 0, showed that the dispersion relation (2.26) is misleading in this respect [6]. At a = 0, the cell size of the secondary structure is determined, not by the period 2s, but by the entire length of the interval on the l-axis in which the disturbances are evolving. Stable structures with a finite period (fig. 4) are possible only provided 0 -z a < a,, when long-wave instability is excluded (see fig. (2b)). Fig. 5 shows the result of numerical solution of eq. (2.25) at (Y= 0.5. The equation was solved in the interval 0 -C{ < 11~ with periodic boundary conditions. The initial-value condition selected was the antisymmetric perturbation @(O,1) = ({ - lS)exp( -(l - 15)‘/10). With the passage of time, a steady periodic structure developed. Eq. (2.25) may be written in variational form,
(2.27) where the functional
F[@]=
-$
/(
F[@] is defined as 4@;-@;-29;+I~*)d~.
(2.28)
In view of the existence of this functional, there cannot exist solutions that oscillate in time (Pomeau and Zaleski [9]). However, experiment shows that such solution modes are nevertheless possible at sufficiently large Reynolds numbers [2]. In this connection, in the next section, we shall construct a modified evolution equation which is adequate to describe the real situation in a larger interval of Reynolds numbers.
G.I. Sivushinsky/
250
3. Two-dimensional
Weuk turbulence in periodic flows
evolution equation for large-scale flow
In order to capture the turbulence effect, it proved necessary to take slowly-varying disturbances into consideration, not only along the x-axis but also along the y-axis. The two-dimensional equation for the large-scale stream function A0 will obviously describe the evolution of large-scale vorticity 9’ = V2Ao; consequently, it should contain a kinematic term of the type
(V2Ao),-A0,(V2Ao)y+A~(v2Ao)x.
(3.1)
Since XE- 1, E4t - 1 and A0 - 1 comparison of orders of magnitude in (3.1) suggests the following choice of the slow coordinate along the y-axis: Gy=?j.
(3.2)
Eq. (2.5) for the stream function is then generalized to E4(V2~)~+E~~(v2~)~+E4~~(V2~)~-~~(V2~)_”-E4~~(v2~)~ = R,‘(l (v**=
- &2)VV - hRil&4(1 - &*)v** - R,‘(l *Yk;y+E*!P[kEE + 2&3eyk,,
- &*)cosy,
(3.3)
+ A&).
As before, the stream function 9 = 9(E, q,~, y) is assumed to be 2n-periodic in y. Integrating eq. (3.3) over the interval 0 I y I 2a, we get
= RGl&(l - &2) 2n Q,,
J0
+ 2&4!P
(
dy - pR,‘e3(1
- E*)/*~( ‘PtiE+ e4qqq) dy.
Solving the problem in the zeroth, first, second and third approximations,
(3.4)
0
we get
!PO=cosy+AO(.$,~,T), 9l=
-R,Aisiny+A’([,q,7),
**=
-R;(A;)*
R,=ti,
cosy-R,Aisiny+A’(E,y,7),
e3 = [ R3,( Ai)3 - 3R,Az,, - R,Ai
- R,A;]
sin y - 2Rf~AiA:cosy
The integral relationship (3.4) for the third approximation a z,
/
2-
!P;
+ A3(4, y, T).
(3.5)
yields
!I$‘!$[ + ‘P’!J’A+ ‘#$I’;~ + ‘P’!P; + 2’$95
+ 2’P;!$,, + 2!;\1;“1) dy
0 a
+z
-R,’
2n
/
(q$P;,,
+ q;+_;,,
+ -P,f9;Y
+ !l$?l’;)
0
J(0
2n *Arc - !&)
dy - ARo’i*“!P$dy
dy - $- I’“( !$‘!l$!Y+ +;.+V + !P;F$ + et*&) 0
dy (3.6)
G. I. Sivashinsky/
Weuk turbulence in periodic flows
251
Inserting into this equation the previously determined solutions for \k,, \k,, \k;, !PJk;,we obtain the evolution equation for A,(& 1,~) - the stream function for large-scale flow
(3.7)
4. Possibility of turbulent flow For a undirectional large-scale flow parallel to the q-axis, the stream function is A0 = A’((, T) and eq. to eq. (2.22). Let us consider the case of undirectional flow at an angle to the q-axis:
(3.6) transforms
(4.1)
A0 = AO(E + CTj,7). In this case eq. (3.6) takes the form of the following spatially one-dimensional (A,O,),-(~~~/2)(A,O)~p+(3~o/2)A,O,,,,,+([~o-(~~/3)(A~)2]A~)
equation:
PPP
+A%‘A,O,=O,
(4.2)
where p = t + CT). Hence, after integrating twice, we obtain A~-(cR2,/2)(~~)2+(3Ro/2)A~ppp+([Ro-(R~/3)(A~)2]A~}
P-3)
+AR,‘A’=O. P
Resealing as follows: A,=(3&)@,
p=&,
7=6tis,
A=6a,
(4.4
,I3=3c,
we bring eq. (4.3) to the form
(4.5)
Q?+9,,,+[(2-~~)~l]z--~~+u~=o. chaotically
behauing
solutions.
Indeed, put
@=fl-‘@.
(4.6)
We then have, by (4.5), 8,+8,,,+[(2-B-28:)8,],-8:+rre=o.
(4.7)
When /3 3*- 1, eq. (4.7) degenerates to s* + @,,, + 28,, - 8; + a@ = 0,
(4.8)
252
G.I. Sivashinsky/
Weak turbulence in periodicjlows
CYP1
E!l!i#
Fig. 6. Nature of flow as function of parameters
10
20
0.1
-
-
+
0.01
-
+
+
(I and /3. Sign “+ ” indicates presence of turbulence, sign “- ” its absence.
an equation which is known to be capable of generating chaotically self-fluctuating solutions for sufficiently small a [Kuramoto [lo], Sivashinsky [ll]]. This property is obviously also conserved in eq. (4.5) for sufficiently large #I and small (Y. Fig. 6-9 reproduce the results of the corresponding numerical experiments. The boundary and initial-value conditions are identical to those used in solving eq. (2.5). Thus, eq. (3.7) is indeed adequately rich to describe not only the formation of steady periodic structures, but also that of turbulent flows. In reality, of course, the large-scale flow is not necessarily unidirectional. It would be interesting in this connection to conduct numerical experiments with the full two-dimensional equation (3.7). Note that the frequency of the turbulent pulsations is a quantity of the order of c4 - (R - ZQ2, i.e., it increases with increasing Reynolds number. This is in qualitative agreement with observation [2]. When there is no external friction (a = 0) and the aspect ratio of the system is sufficiently large, turbulence may set in immediately at the stability threshold (R = R,). Thus the weak turbulence effect described above appears to be qualitatively similar to that observed by Ahlers in Rayleigh-BCnard convection [ 131.
79
r
Fig. 7. Steady-state periodic solution generated by eq. (4.5) with a = 0.01. fl = 1. Note corresponding to maximal amplification of harmonic disturbances in the linear theory.
that the period considerably
exceeds 2r,
G.I. Sivushinsky/
Weuk turbulencein periodic flows
253
1.6 -
1.7 -
1.6 Qi 1.5 -
1.4 -
1.3 -
1.2 -5
t
I
I
I
I
I
0
5
10
15
20
1
I
I
I
25
30
35
40
Fig. 8. Space configuration of solution (P(j, 7) at five consecutive instants of time (a = 0.1, j3 =
20).
,. 1.7 1.6
1.1L
r
1.7 1.6 1.5 @
1.4 1.3 1.2 1.1
L 600
I
I 620
I
I 640
,
T
I 660
I
I 680
Fig. 9. Temporal evolution of @(5. T) at 5 = 10 and { - 20 (a = 0.1, /3 = 20).
I
G.I. Sivushinsky/
254
5. Hydrodynamics
in periodic flows
Weuk turbulence
of negative viscosity
We shall try to write the equations of the large-scale flow in terms of the average velocity (ii, 6):
To this end, we transform to the scaled coordinates [, n, 7, y in the Navier-Stokes Integrating the resulting equations over the interval 0 I y 5 27r, we get
a
E3- ;T
/2”udy
+ x
0
a zn
a
-Iat2
h
a
a2
& =X
ago /
2s
udy+x
/0
JT
u2dy+a2% ’ /2nuudy+X o
o
277
=--
E3-
277 J
h /
2T O
o
udy+R7
udy+e’%
uudy+ezX ES
a2
87 a
2n /
udy a
2r
/
o pdy
x xE3U,
2T
J
a
o v2dy+ezS
equations (1.1).
(5.2)
a
2a2
J
o 0 dy+s’s
a
2n
/ o pdy
x
udy - R~3v,
o
(5.3)
2n
I
o udy=O.
(5.4)
According to (3.5)
(5.5)
(5.6) Inserting (5.5), (5.6) into the integral relationships (5.2)-(5.4), we have the following equations for the principal term of the asymptotic expansion: -R;Et+pE=O, u,+iiu,+uu,+j,= u,+ij,=o,
(5.7) [(R;V2-
R,)u,],-(3R,/2)~~66E-XRolU,
(5.8) (5.9)
where we have put 1 2n0
297
/
pdy=e’ji.
(5.10)
G.I. Sivashinsky/
Weak turbulence in periodicflows
255
The structure of system (5.7)-(5.9) is clearly similar to that of Prandtl’s boundary layer theory. For small fi*, the coefficient of fit6 is negative, corresponding to the well-known effect of negative viscosity sometimes observed in large-scale flows (Starr [12]). The dissipation of short-wave disturbances is ensured by the term The term R@c in eq. (5.7) is the result of averaging u*; it stems from the perturbation of the u-component interacting with itself. This is precisely the term that ensures the possibility of the unidirectional turbulence described in section 4.
Acknowledgements The author would like to thank Dr. V. Yakhot for stimulating discussions.
This research was supported in part by a contract from the US Department of Energy, no. DE-ACOZ 80ER-10559, while the author was on leave at the Institute of Applied Chemical Physics, the City College of New York.
References
ill V.I. Arnold and L.D. Meshalkin, Uspekhi Mat. Nauk 15 (1960) 247. I21A.M. Obukhov, Russ. Math. Survey 38 (1983) 113. [31 L.D. Meshalkin and YaG. Sinai, J. Appl. Math. Mech. (PMM) 25 (1961) 1700. [41 J.S.A. Green, J. Fluid Mech. 62 (1974) 273. [51 N.F. Bondarenko, M.Z. Gak and F.V. Dolzhansky, Atmospheric and Oceanic Physics 15 (1979) 711. [61 C.J. Chapman and M.R.E. Proctor, J. Fluid Mech. 101 (1980) 759. [71 M.C. Depassier and E.A. Spiegel, Astron. J. 86 (1981) 496. PI V.L. Gertsberg and G.I. Sivashinsky, Prog. Theor. Phys. 66 (1981) 1219. 191 Y. Pomeau and S. Zaleski, J. de Physique 42 (1981) 515. WI Y. Kuramoto, Prog. Theor. Phys. Suppl. 64 (1978) 346. illI G.I. Sivashinsky. Ann. Rev. Fluid Mech. 15 (1983) 179. WI V.P. Starr, Physics of Negative Viscosity Phenomena, (McGraw-Hill, New York, 1968). 1131 G. Ahlers and R.P. Behringer, Prog. Theor. Phys. Suppl. 64 (1978) 186.
WEAK TURBULENCE IN PERIODIC FLOWS
Gregory I. SIVASHINSKY Depurtmentof Muthemoticul
Sciences, Tel-Aviv
Universi!v, Rumut-Aviv,
Tel-Aviv 69978, Isruel
Received 6 August 1984 Revised manuscript received 22 May 1985
We consider the stability of a two-dimensional plane-parallel flow of viscous liquid in an external force field which is a periodic function of one of the coordinates. At sufficiently high Reynolds numbers the plane-parallel flow becomes unstable and a two-dimensional secondary flow ensues. Near the stability threshold, the secondary flow turns out to be large-scale and chaotically self-fluctuating in time.
1. Introduction
One of the most intriguing problems in hydrodynamics is the spontaneous formation of large-scale turbulent structures in a viscous liquid driven at small scales. In a theoretical approach to this phenomenon, it is convenient to understand the latter as a manifestation of long-wave instability of a system of small-scale eddies maintained by an external force field that obstruct their dissipation. One of the simplest hydrodynamic systems of the type in question is the so-called Kolmogorov flow - the two-dimensional flow of a viscous liquid induced by an unidirectional external force field periodic in one of the coordinates (see, e.g., Arnold and Meshalkin [l]; Obukhov [2]). Fig. 1 shows streamlines and velocity profile of the corresponding laminar flow, which obeys a periodic sine law along the y-axis. The problem of the linear stability of such flows was solved by Meshalkin and Sinai [3] and later, independently, by Green [4]. Fig: 2(a) shows the marginal stability curve corresponding to the solution. In 1979, Dolzhansky et al. [5] reproduced Kolmogorov flow under laboratory conditions in a thin layer of electrolite (x, y-plane) placed in an external force field f = [H, j], where H is the magnetic field intensity, with the z-component a periodic function of y; ,j is the current density between the two electrodes in the liquid layer. The magnitude of j can be varied by varying the electrode voltages and thus producing flows with different Reynolds numbers (fig. 3). As shown by Dolzhansky et al. [5], a more accurate description of the observed liquid flow requires a modification of the Kolmogorov model which,.also allows for external friction (friction on the bottom surface). The corresponding equations of two-dimensional hydrodynamics are: u,+uu,+uu,,=
-px+R-‘V2u-~R-‘u-RR-1siny,
u, + uux + UU.” = -pY + R-‘v2u
- pR-‘u,
(1.1)
11, + l+ = 0.
The units of length, time and velocity in these equations are respectively q-i, 0167-2789/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
vqy-l,
vm1qe2, where Y is
244
G.I. Sivashinsky/
Weak turbulence in periodic flows
Fig. 1. Velocity profile and streamlines of Kolmogorov flow.
the viscosity, q and y the space frequency and intensity of the applied force; R = v-*qe3y is the Reynolds number; p = 2q-*h-* is the coefficient of friction, where h is the depth of the liquid layer. The inclusion of external friction is essential for a correct estimate of the critical Reynolds number R,, which is calculated to be of order lo3 (instead of the fi according to Meshalkin and Sinai [3]), and this agrees completely with experiment. The appropriate marginal stability curve is depicted in fig. 2(b). Experiment shows [5] that, after going through the critical value of the Reynolds number R,, unidirectional flow (fig. 1) becomes unstable and a secondary flow appears, in the form of a regular system of stationary eddies (fig. 4). At R = 2.5R, the steady flow becomes unstable and periodic oscillations set in. As the Reynolds number increases, the oscillation period decreases. For R s 3R, the oscillations have a marked monochromatic nature. When the excitation is strong (R 2 3R,) the oscillation pattern becomes more complicated. The spectrum is enriched by higher harmonics. At R x- 3R, one may expect the onset of irregular oscillations, i.e. turbulence. In a system free of external friction (p = 0), the stability region at R L R, = R, is localized near small wavenumbers, i.e., the secondary flow is expected to be long-wave (see fig. 2(a)). The situation changes once external friction has been introduced (fig 2(b)), since then, near the stability threshold; the secondary flow corresponds to a certain finite wavelength - 2~/k,. In the experiment done by Dolzhansky et al. [5].
Fig. 2. Marginal stability curve (a) for periodic flow without inclusion of external friction; (b) flow with external friction included. R-Reynolds number. k-wavenumber of perturbation.
G.I. Sivashinsky/
Weak turbulence in periodic flows
245
H
%I+
a
Y
c
Fig. 3. Diagram of MHD arrangement for simulation of Kolmogorov flow. (a) View from above; (b) side view; (c) profile of vertical component of magnetic field H generated by magnetic system (b). (1) Magnetoelastic rubber; (2) profile of Lorentz force /= [H, j] acting on electrolite upon passage of direct current j; (3) electrolite; (4) force lines of magnetic field.
2a/k. was of the order of unity, i.e., the longitudinal dimension of the eddies was comparable with the period of the external force. Thus, the effect of friction is in this case quite significant. Nevertheless, to facilitate the theoretical analysis-and, hopefully, without too much detriment to the qualitative aspect of the phenomenon - we shall assume throughout that friction is small. In this sense, the asymptotic picture discussed below is, as it were, intermediate between Kolmogorov’s model, which ignores external friction, and the case studied by Dolzhansky et al., where the effect of friction completely excludes the possibility of large-scale secondary flow. In the small friction case k, is also small, and so the secondary flow may again be of a long-wave nature.? From the mathematical point of view, the situation turns out to be very similar to that in the problem of free convection between nearly insulating boundaries (Chapman and Proctor [6]; Depassier and Spiegel [7]; Gertsberg and Sivashinsky [S]).
2. Nonlinear asymptotic analysis. The case of one large-scale space variable For further analysis, it is convenient function, (VW,+
*j,;(vZ*),-
to transform
from system (1.1) to an equation
!Px(~29)y=R-1v%P-pR-1v2!P-R-1co~y.
for the stream
(2.1)
Set R-l
= (1 - e2)Ri1
(e a 1).
(2.2)
Thus, as small parameter e we take the proximity between the Reynolds number and the stability threshold
TWe note that in this case
R,
is close to R,.
246
G.I. Sivushinsky/
Weuk turbulence
in periodicf70ws
Fig. 4. Streamlines of secondary flow near stability threshold [5].
(R = R,). On the assumption that the coefficient of friction is small, p=
A&4,
(2.3)
we introduce the following scaled space and time variables: (=&X,
7 =
E4t.
(2.4)
The selection of these scalings was governed by the analogy with the above-mentioned convection problem [6, 7, 81. As we shall see below, the approximation based on introduction of a single large-scale coordinate 5 provides a successful description of the formation of regular eddy systems (fig. 4). However, it turns out that the resulting asymptotic model is not capable of capturing the effect of turbulence and will subsequently be improved in section 3. In terms of the new variables, eq. (2.1) becomes
E4(V2~)~+E~~(V2~)~-E~~(v2\k)y = (1 - E2)R,‘V4\k -(l - &2)h&%;l+P (v2\k=Ez*[E+
(2.5)
- (1 - &2)R~1COS_y,
Q.
We shall assume that the function \k is 2n-periodic with respect to y. For further analysis, it is useful to integrate eq. (2.5) over the interval 0 of \k and the identity
5 y
s 27r. Using the periodicity
we can write the resulting integral relationship as follows: ?QP,,dy
= (1 - e2)eR,1~2n!I’~i,,,dy
-(l
- e2)he3R,1/02?Pt,dy.
(2.7)
To solve eq. (2.5), we express the solution as an asymptotic expansion,
!P=-P”+E?P1+&2!P2+&3*3+ -.*.
(2.8)
G.I. Sivushinsky / Weak turbulence in periodic flows
For the zerorh approximation,
247
we obtain from (2.5)
q”.“.”= cmY*
(2.9)
Hence (2.10) The integral relationship (2.7) in the zeroth approximation
yields (2.11)
It is readily seen from (2.10) that this relationship is automatically valid for any A’((, T). For the Jirst approximation, we deduce from (2.Q using (2.10) that R,V . YY.Y.Y = -Aisin
y.
(2.12)
P = -R,Aisiny+A’([,T).
(2.13)
Hence
The integral relationship (2.7) in the first approximation
yields (2.14)
Hence, in view of (2.10) and (2.13), we obtain (R,-2R,‘)A;,,=
0,
or
R,=fi.
(2.15)
Thus, we have determined the critical Reynolds number at which the onset of secondary flow is possible. For the second approximation we obtain from (2.5), using (2.10) and (2.13) R3r2
r
.v.v.v.v
-R,(Ai)‘cosy-Aisiny.
(2.16)
Hence
gr== -Ri(
Ai)‘cos
y - R,Aisin
In the second approximation,
y + A’(t,
7).
(2.17)
the integral relationship (2.17) gives (2.18)
By substituting the previously found solutions ?Po,!Pl, !PZ into this relationship, one can verify that it is automatically valid at R, = fi.
248
G. I. Sivashinsky/
For the third approximation;
Weak turbulence
in periodic flows
we again proceed from (2.5), using (2.10), (2.13) and (2.17), to obtain
R,‘YP&, = [ R2,( Ai)) - 3&
- At - Ai] sin y - 2A$4; cosy.
(2.19)
Hence q3= [R;(A;)3-3R,A;Ct-
R,Ai
- R,A;]
sin y - ~R~A~A~cosY
The integral relationship (2.17) in the third approximation
+ A3(Ey T).
(2.20)
gives
(2.21) Substituting the previously following relationship:
found expressions for !PO,‘PI, qlk2,‘k; into this equality, we arrive at the
+([Ro-(R;/3)(A~)*]A~}~~~+XR6’A~,=0.
(AL) T+ (3Ro/2)AL
(2.22)
This determines the required differential equation for A,(& T). Integrating eq. (2.22) twice with respect to 6, we obtain the following simpler equation: +{[Ro-(R;/3)(A;)2]A;)tfXR~1Ao=0. A: + (3Ro/2)A&,,
(2.23)
Once A,([, T) is known, one can determine the stream function of secondary flow from eq. (2.10). Introducing the elementary resealings A”=(3/fi)@,
(=&I,
7=6&,
A=6cu,
(2.24)
one transforms eq. (2.24) to the following more conveniently analyzed form:
~s+9,,,+[(2-cPF)~,],+a~=0.
(2.25)
It is interesting that this equation is a one-dimensional version of the equation for the evolution of the plane form of convective cells in a nearly insulated liquid layer [6, 7, 81. The dispersion relation for linear stability analysis of the trivial solution is o=2k2-k4-a!
Instability
(@-exp(ws+ik<)).
appears at a < OL,= 1, corresponding
(2.26) to sufficiently low friction (at a fixed Reynolds number
R > R,).
Eq. (2.26) indicates the existence of a wavelength (2?r/k, = 2s) corresponding to maximum growth rate of small harmonic disturbances. It would therefore be natural to expect that 2?r is the characteristic cell size
G. I. Sivashinsky / Weak turbulence in periodic flows
249
1.5F
i-
0.5 cp O-
-0.5-
-1 -
-1.5
t -2 -5
I
I
I
I
0
5
10
15
I< 20
I
I
I
25
30
35
1 40
Fig. 5. Steady periodic solution generated by eq. (2.25) at a = 0.5.
of the periodic structure produced during the evolution of the initial disturbances (Rayleigh principle). However, analysis of the stability of the periodic solutions done for the case (Y= 0, showed that the dispersion relation (2.26) is misleading in this respect [6]. At a = 0, the cell size of the secondary structure is determined, not by the period 2s, but by the entire length of the interval on the l-axis in which the disturbances are evolving. Stable structures with a finite period (fig. 4) are possible only provided 0 -z a < a,, when long-wave instability is excluded (see fig. (2b)). Fig. 5 shows the result of numerical solution of eq. (2.25) at (Y= 0.5. The equation was solved in the interval 0 -C{ < 11~ with periodic boundary conditions. The initial-value condition selected was the antisymmetric perturbation @(O,1) = ({ - lS)exp( -(l - 15)‘/10). With the passage of time, a steady periodic structure developed. Eq. (2.25) may be written in variational form,
(2.27) where the functional
F[@]=
-$
/(
F[@] is defined as 4@;-@;-29;+I~*)d~.
(2.28)
In view of the existence of this functional, there cannot exist solutions that oscillate in time (Pomeau and Zaleski [9]). However, experiment shows that such solution modes are nevertheless possible at sufficiently large Reynolds numbers [2]. In this connection, in the next section, we shall construct a modified evolution equation which is adequate to describe the real situation in a larger interval of Reynolds numbers.
G.I. Sivushinsky/
250
3. Two-dimensional
Weuk turbulence in periodic flows
evolution equation for large-scale flow
In order to capture the turbulence effect, it proved necessary to take slowly-varying disturbances into consideration, not only along the x-axis but also along the y-axis. The two-dimensional equation for the large-scale stream function A0 will obviously describe the evolution of large-scale vorticity 9’ = V2Ao; consequently, it should contain a kinematic term of the type
(V2Ao),-A0,(V2Ao)y+A~(v2Ao)x.
(3.1)
Since XE- 1, E4t - 1 and A0 - 1 comparison of orders of magnitude in (3.1) suggests the following choice of the slow coordinate along the y-axis: Gy=?j.
(3.2)
Eq. (2.5) for the stream function is then generalized to E4(V2~)~+E~~(v2~)~+E4~~(V2~)~-~~(V2~)_”-E4~~(v2~)~ = R,‘(l (v**=
- &2)VV - hRil&4(1 - &*)v** - R,‘(l *Yk;y+E*!P[kEE + 2&3eyk,,
- &*)cosy,
(3.3)
+ A&).
As before, the stream function 9 = 9(E, q,~, y) is assumed to be 2n-periodic in y. Integrating eq. (3.3) over the interval 0 I y I 2a, we get
= RGl&(l - &2) 2n Q,,
J0
+ 2&4!P
(
dy - pR,‘e3(1
- E*)/*~( ‘PtiE+ e4qqq) dy.
Solving the problem in the zeroth, first, second and third approximations,
(3.4)
0
we get
!PO=cosy+AO(.$,~,T), 9l=
-R,Aisiny+A’([,q,7),
**=
-R;(A;)*
R,=ti,
cosy-R,Aisiny+A’(E,y,7),
e3 = [ R3,( Ai)3 - 3R,Az,, - R,Ai
- R,A;]
sin y - 2Rf~AiA:cosy
The integral relationship (3.4) for the third approximation a z,
/
2-
!P;
+ A3(4, y, T).
(3.5)
yields
!I$‘!$[ + ‘P’!J’A+ ‘#$I’;~ + ‘P’!P; + 2’$95
+ 2’P;!$,, + 2!;\1;“1) dy
0 a
+z
-R,’
2n
/
(q$P;,,
+ q;+_;,,
+ -P,f9;Y
+ !l$?l’;)
0
J(0
2n *Arc - !&)
dy - ARo’i*“!P$dy
dy - $- I’“( !$‘!l$!Y+ +;.+V + !P;F$ + et*&) 0
dy (3.6)
G. I. Sivashinsky/
Weuk turbulence in periodic flows
251
Inserting into this equation the previously determined solutions for \k,, \k,, \k;, !PJk;,we obtain the evolution equation for A,(& 1,~) - the stream function for large-scale flow
(3.7)
4. Possibility of turbulent flow For a undirectional large-scale flow parallel to the q-axis, the stream function is A0 = A’((, T) and eq. to eq. (2.22). Let us consider the case of undirectional flow at an angle to the q-axis:
(3.6) transforms
(4.1)
A0 = AO(E + CTj,7). In this case eq. (3.6) takes the form of the following spatially one-dimensional (A,O,),-(~~~/2)(A,O)~p+(3~o/2)A,O,,,,,+([~o-(~~/3)(A~)2]A~)
equation:
PPP
+A%‘A,O,=O,
(4.2)
where p = t + CT). Hence, after integrating twice, we obtain A~-(cR2,/2)(~~)2+(3Ro/2)A~ppp+([Ro-(R~/3)(A~)2]A~}
P-3)
+AR,‘A’=O. P
Resealing as follows: A,=(3&)@,
p=&,
7=6tis,
A=6a,
(4.4
,I3=3c,
we bring eq. (4.3) to the form
(4.5)
Q?+9,,,+[(2-~~)~l]z--~~+u~=o. chaotically
behauing
solutions.
Indeed, put
@=fl-‘@.
(4.6)
We then have, by (4.5), 8,+8,,,+[(2-B-28:)8,],-8:+rre=o.
(4.7)
When /3 3*- 1, eq. (4.7) degenerates to s* + @,,, + 28,, - 8; + a@ = 0,
(4.8)
252
G.I. Sivashinsky/
Weak turbulence in periodicjlows
CYP1
E!l!i#
Fig. 6. Nature of flow as function of parameters
10
20
0.1
-
-
+
0.01
-
+
+
(I and /3. Sign “+ ” indicates presence of turbulence, sign “- ” its absence.
an equation which is known to be capable of generating chaotically self-fluctuating solutions for sufficiently small a [Kuramoto [lo], Sivashinsky [ll]]. This property is obviously also conserved in eq. (4.5) for sufficiently large #I and small (Y. Fig. 6-9 reproduce the results of the corresponding numerical experiments. The boundary and initial-value conditions are identical to those used in solving eq. (2.5). Thus, eq. (3.7) is indeed adequately rich to describe not only the formation of steady periodic structures, but also that of turbulent flows. In reality, of course, the large-scale flow is not necessarily unidirectional. It would be interesting in this connection to conduct numerical experiments with the full two-dimensional equation (3.7). Note that the frequency of the turbulent pulsations is a quantity of the order of c4 - (R - ZQ2, i.e., it increases with increasing Reynolds number. This is in qualitative agreement with observation [2]. When there is no external friction (a = 0) and the aspect ratio of the system is sufficiently large, turbulence may set in immediately at the stability threshold (R = R,). Thus the weak turbulence effect described above appears to be qualitatively similar to that observed by Ahlers in Rayleigh-BCnard convection [ 131.
79
r
Fig. 7. Steady-state periodic solution generated by eq. (4.5) with a = 0.01. fl = 1. Note corresponding to maximal amplification of harmonic disturbances in the linear theory.
that the period considerably
exceeds 2r,
G.I. Sivushinsky/
Weuk turbulencein periodic flows
253
1.6 -
1.7 -
1.6 Qi 1.5 -
1.4 -
1.3 -
1.2 -5
t
I
I
I
I
I
0
5
10
15
20
1
I
I
I
25
30
35
40
Fig. 8. Space configuration of solution (P(j, 7) at five consecutive instants of time (a = 0.1, j3 =
20).
,. 1.7 1.6
1.1L
r
1.7 1.6 1.5 @
1.4 1.3 1.2 1.1
L 600
I
I 620
I
I 640
,
T
I 660
I
I 680
Fig. 9. Temporal evolution of @(5. T) at 5 = 10 and { - 20 (a = 0.1, /3 = 20).
I
G.I. Sivushinsky/
254
5. Hydrodynamics
in periodic flows
Weuk turbulence
of negative viscosity
We shall try to write the equations of the large-scale flow in terms of the average velocity (ii, 6):
To this end, we transform to the scaled coordinates [, n, 7, y in the Navier-Stokes Integrating the resulting equations over the interval 0 I y 5 27r, we get
a
E3- ;T
/2”udy
+ x
0
a zn
a
-Iat2
h
a
a2
& =X
ago /
2s
udy+x
/0
JT
u2dy+a2% ’ /2nuudy+X o
o
277
=--
E3-
277 J
h /
2T O
o
udy+R7
udy+e’%
uudy+ezX ES
a2
87 a
2n /
udy a
2r
/
o pdy
x xE3U,
2T
J
a
o v2dy+ezS
equations (1.1).
(5.2)
a
2a2
J
o 0 dy+s’s
a
2n
/ o pdy
x
udy - R~3v,
o
(5.3)
2n
I
o udy=O.
(5.4)
According to (3.5)
(5.5)
(5.6) Inserting (5.5), (5.6) into the integral relationships (5.2)-(5.4), we have the following equations for the principal term of the asymptotic expansion: -R;Et+pE=O, u,+iiu,+uu,+j,= u,+ij,=o,
(5.7) [(R;V2-
R,)u,],-(3R,/2)~~66E-XRolU,
(5.8) (5.9)
where we have put 1 2n0
297
/
pdy=e’ji.
(5.10)
G.I. Sivashinsky/
Weak turbulence in periodicflows
255
The structure of system (5.7)-(5.9) is clearly similar to that of Prandtl’s boundary layer theory. For small fi*, the coefficient of fit6 is negative, corresponding to the well-known effect of negative viscosity sometimes observed in large-scale flows (Starr [12]). The dissipation of short-wave disturbances is ensured by the term The term R@c in eq. (5.7) is the result of averaging u*; it stems from the perturbation of the u-component interacting with itself. This is precisely the term that ensures the possibility of the unidirectional turbulence described in section 4.
Acknowledgements The author would like to thank Dr. V. Yakhot for stimulating discussions.
This research was supported in part by a contract from the US Department of Energy, no. DE-ACOZ 80ER-10559, while the author was on leave at the Institute of Applied Chemical Physics, the City College of New York.
References
ill V.I. Arnold and L.D. Meshalkin, Uspekhi Mat. Nauk 15 (1960) 247. I21A.M. Obukhov, Russ. Math. Survey 38 (1983) 113. [31 L.D. Meshalkin and YaG. Sinai, J. Appl. Math. Mech. (PMM) 25 (1961) 1700. [41 J.S.A. Green, J. Fluid Mech. 62 (1974) 273. [51 N.F. Bondarenko, M.Z. Gak and F.V. Dolzhansky, Atmospheric and Oceanic Physics 15 (1979) 711. [61 C.J. Chapman and M.R.E. Proctor, J. Fluid Mech. 101 (1980) 759. [71 M.C. Depassier and E.A. Spiegel, Astron. J. 86 (1981) 496. PI V.L. Gertsberg and G.I. Sivashinsky, Prog. Theor. Phys. 66 (1981) 1219. 191 Y. Pomeau and S. Zaleski, J. de Physique 42 (1981) 515. WI Y. Kuramoto, Prog. Theor. Phys. Suppl. 64 (1978) 346. illI G.I. Sivashinsky. Ann. Rev. Fluid Mech. 15 (1983) 179. WI V.P. Starr, Physics of Negative Viscosity Phenomena, (McGraw-Hill, New York, 1968). 1131 G. Ahlers and R.P. Behringer, Prog. Theor. Phys. Suppl. 64 (1978) 186.