Extended Self-Similarity in Turbulent Systems: an Analytically Soluble Example

4 Pages·0.11 MB·English

by  Daniel Segel

Physical Review Letters Version of February 5, 2008
Extended Self-Similarity in Turbulent Systems: an Analytically Soluble Example
Daniel Segel∗, Victor L’vov† and Itamar Procaccia‡
Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
A partial resolution of these difficulties has been re-
Inturbulentflowsthen’thorderstructurefunctionsSn(R) cently offered [7] by changing the way that the experi-
scale like Rζn when R is in the ”inertial range”. Extended
mentaldataispresented. Insteadofplottinglog-logplots
Self-Similarity refers to the substantial increase in the range
of S (R) vs. R, it was argued that plotting log-logplots
of power law behaviour of Sn(R) when they are plotted as a n
6 functionofS2(R)orS3(R). InthisLetterwedemonstratethis ofSn(R) vs. S2(R)or S3(R) revealsmuchlongerscaling
9 phenomenonanalytically inthecontextofthe“multiscaling” ranges that can be used to fit accurate exponents. This
9 turbulentadvectionof apassivescalar. Thismodelgivesrise method wasdubbed ExtendedSelf Similarity. While the
1 toaseriesofdifferentialequationsforthestructurefunctions effectiveness of this method has been demonstrated [8],
n Sn(R)whichcanbesolvedandshowntoexhibitextendedself the reason for its success remains unclear. Therefore its
a similarity. The phenomenon is understood by comparing the reliability is not guaranteed. The aim ofthis Letter is to
J equations for Sn(R) to those for Sn(S2). address this method theoretically for Kraichan’s model
3
of a passive scalar advected by a very rapidly varying
PACS numbers47.27.Gs, 47.27.Jv, 05.40.+j
velocity field [9]. In this model multiscaling is known
1
v tooccur[10,11,13],andwehavesufficentanalyticunder-
2 The fundamental theory of turbulence concerns itself standingofthemechanismof(multi)scalingtoassesswhy
0 Extended Self Similarity could hold. One should stress
withtheuniversalityofthesmallscalestructureofturbu-
0
that in the case of normal scaling Extended Self Simi-
lent flows and their characterizationby universalscaling
1
larity can be rationalized rather straightforwardly [14].
0 laws [1]. Since the days of Kolmogorov [2] one expects
6 the structure functions S (R) to scale with R as long as To our knowledge no analytic derivation of this effect is
n
9 R is in the inertial range available in a case exhibiting multiscaling.
/ In this model the equation for the passive scalar con-
n
y Sn(R)∼Rζn, for η ≪R≪L . (1) centration T(x) is given by
d
o- sHpeercetivηelay.nd LFoarrea thgievevnisctouursbualenndt ofiuetledr lue(nrg,htt)s trhee- ∂∂Tt +(u·∇)T =κ∇2T , (2)
a
h n’th order structure function is defined as Sn(R) = where the velocity field u is taken as Gaussian random
c h[u(r+R,t)−u(r,t)]niwheretheangularbracketsh...i
and with a δ-function correlationin time. The structure
: denote an average over r and t. Komogorov’s 1941 pic-
v
i ture of turbulence (K41) predicted that the exponents functions S2n(R)are knowto obey differentialequations
X [9,11]
of the structure functions of the velocity field satisfy
r ζ = n/3. Further research along the same lines sug-
a gnestedthatthestructurefunctionsofpassivescalarsden- R1−d ∂ Rd−1h(R) ∂ S2n(R)=J2n(R) . (3)
∂R ∂R
sities also have the same exponents [3,4]. On the other
hand experiments [5,6] have indicated deviations from Here h(R) = H(R/L)ζh for R < L where H is a dimen-
this scaling and even the possibility of multiscaling, i.e. tional constant and L is some characteristic outer scale
a non-linear dependence of ζ on n. These findings raise ofthedrivingvelocityfield. TheScalingexponentζ isa
n h
fundamental questions as to our understanding of the chosenparameterinthismodel,takinganydesiredvalue
universality of the small scale structure of turbulence. in the interval (0,2). The function J2n(R) is known ex-
Unfortunately, the experimental measurement of the actlyforRintheinertialrange[11],butforourpurposes
scaling exponents in turbulent flows is hampered by the weneedtoknowitalsointhenearviscousrange. Aform
fact that the range of scales that exhibit scaling be- thatisappropriateinbothrangeshasbeensuggestedby
haviour is rather small for Reynolds numbers Re acces- Kraichnan [10]:
sible to date. Even though the viscous scale η decreases
like Re−3/4, the scaling behaviour (1) is expected to set J2n(R)=2nκS2n(R)[∇2S2(R)−∇2S2(0)] . (4)
in only at about 10η, and cease at about L/100. Thus S2(R)
even at Re∼107 the inertial range is not broad enough
Thisformhasbeenrecentlysupportedbynumericalsim-
to allow accurate measurements of ζ , particularly for
n
ulations[12]. Inthe inertialrangethe lasttwoequations
larger values of n.
1
have a multiscaling solution for S2n(R) with the expo- with a leading order scaling solution ξ(y) = B2ny−ζ2.
nents The constant is found from Eq.(11) and the final result
for ζ2 <1 is
ζ2n = 12 (3−ζ2)2+12nζ2+ζ2−3 . (5)
Our first stephpin studying the issue of Exitended Self- D2n(y)=A2nyζ2n(cid:20)1+ (2ζ2n+33nC−22ζ2)yζ2(cid:21) . (12)
SimilarityusingEqs. (3,4)istonon-dimensionalizethem.
Rescale lengths by rd ≡ L(κ/H)1/ζh and the scalar At this point we can assess the efficacy of Extended Self
density by Θ = rd ǫ/κ. The dimensionless functions, Similarity for the case ζh > 1. If all functions D2n were
D2n(y) are: p an exact power law in argument D2 we would have
D2n(y)≡S2n(yrd)/Θ2n, y ≡R/rd . (6) ESS 2(yζ2 +C2) ζ2n/ζ2
D (y)= , (13)
They satisfy the equations 2n (cid:20) 3ζ2 (cid:21)
d2D2n 2+ζη dD2n 2nD2n d2D2 or in the same order
+ = 1− . (7)
Thesduyc2cessofeyxtenddeydselfsDim2yilζaηrit(cid:20)yforthd2isym(cid:21)odelcan D2EnSS(y)=A2nyζ2n(cid:18)1+ ζζ22nyCζ22(cid:19) . (14)
be assessed from Fig.1 which compares plots of D2n vs. Theratior1 betweenthecoefficientsofy−ζ2 inEqs. (12)
y withplotsofD2n vs. D2. Thedataisbasedonthe nu- and (14) measures the effectiveness of this presentation.
mericalsolutionofEq.(7)usingtheIMSL/IDL’sAdams-
Using Eq. (5) one finds
GearODE-solver. The efficacyofExtendedSelfSimilar-
ity speaks for itself. 3nζ2 nζ2
r1 = = . (15)
Next we exhibit the phenomenon analytically by find- ζ2n(2ζ2n+3−2ζ2) 2nζ2−ζ2n
ing the first order dissipative correction to the inertial
range scaling of the structure functions. In the inertial Ifr1 =1wewouldhaveperfectExtendedSelfSimilarity;
range the exponent of D2(y) is ζ2 = 2−ζη [9]. We can r1 = 0 is what we would get if we just plotted S2n as a
find the first correction to the power law, going into the function of r. In Fig. 2(a) we can see a graph of r1 as
dissipation range, by writing D2(y) = 2yζ2/3ζ2 +ǫ(y). a function of ζ2 for n = 2,n = 4 and n = 6. We can
see that r is always in the interval [1/2,1]: from Eq. (5)
We then find the following ODE for ǫ(y):
limn→∞r1 = 12, and from Eqs. (5),(15) limζ2→0r1 = 1.
d2ǫ 2+ζ dǫ 4(1−ζ ) The convergence to 1/2 is rather slow and not uniform
η η
+ + =0 . (8)
d2y y dy 3y2ζη in ζ2.
The mainpointofthe ExtendedSelf Similarityis that
There are two possibilities for the leading order solution
S2 (which is a function of R) is a “better” variable than
of this equation. If all the terms are of the same order
the distance R itself when it comes to scalingbehaviour.
the solution is ǫ1 = −2y2−2ζη/3(1+ζ2). If the first two Thus,tocompleteouranalysiswewanttowritetheequa-
terms are larger we have ǫ2 = B2 + ǫ1, where now ǫ1 tionsforS2n asODE’swithrespecttoargumentS2 with
appearsasthe nextordercorrectionafter aconstantB2. the inertialrangescaling plus a remainderterm. Chang-
Onecanseethatthedominantsolutionforsmallζh isǫ1, ing variables and writing the S2n as a function of S2,
with a transition to ǫ2 at ζh = 1. Thus the near-inertial we expect to find that the remainder term, evaluated
solution for D2 is in terms of the inertial range estimates, is smaller than
2yζ2 ζ2yζ2−2 when using r as a variable. To this aim we introduce a
D2(y)= 3ζ2 1− 1+ζ2 for ζη <1, (9) new set of function F2n(x) of argument x=D2(y):
(cid:18) (cid:19)
D2(y)= 23yζζ22 1+ yCζ22 for ζη >1 , (10) F2n(x)≡D2n(y), x≡D2(y) . (16)
(cid:18) (cid:19)
Then Eqs. (7) gives for F2n(x) the following ODE
where C2 is a constant. Note that for the case ζh = 2/3
the ODE for D2(y) can be solved exactly in terms of d2F2n + 2 dF2n − nF2n(x)
elementaryoperationsandtheresultsareconsistentwith d2x 2+[f(x)]2−ζ2 dx x
(cid:20) (cid:21)
this solution. 2
df (4−ζ2) df
Next we repeat this procedure for D2n(y). We substi- × + =0 , (17)
tute D2n(y) = A2nyζ2n[1+ξ(y)] in Eq.(7). For the case "(cid:18)dx(cid:19) f(x) dx#
ζ >1 we find
h
where f(x) is the inverse function of S2(y): f(x) = y if
d2ξ 2dξ 3nC2ζ2 x = S2(y). Now let us introduce a third set of dimen-
dy2 + ydy(2ζ2n+4−ζ2)+ y2+ζ2 =0 , (11) sionless structure functions, G2n(z) such that
2
G2n(x)≡D2n(y), x≡2yζ2/3ζ2 . (18) Now comparing Eqs. (15) and (23) one sees that this
relation is true. Extended Self Similarity as represented
Note that the functions F2n andG2n are identical in the by r1 very near unity indeed translates into very small
inertialrange,since oneis afunction ofS2 andthe other r2, that is into a small remainder term. In Fig. 2(b)
isafunctionofitspowerlawintheinertialrange. Indeed,
we can see r2 as a function of ζ2 for n = 2,n = 4 and
it will be seen below that they satisfy the same ODE in
n = 6. As n → ∞ the ratio becomes poor, r2 → −1
the inertial range with a power solution xζ2n/ζ2. Using but the convergenceis slow as we cansee from Fig. 2(b)
(7) one derives the following ODE for G2n(x): and again not uniform in ζ2, since one can verify that
d2G2n 3 dG2n nG2n(x) limζ2→0r2 =0.
dx2 + ζ2x dx − g(x) The situation for ζ2 > 1 is much messier algebraically
(cid:20) (cid:21) althoughjustthesameprocedurecanbefollowedinprin-
3nG2n(x) dg d2g ciple. WehavealreadygiveninEq. (10)thenear-inertial
+ g(x)x2/ζ2 (cid:20)(ζ2−1)dx +ζ2xdx2(cid:21)=0 (19) form for D2(y) for ζ2 > 1; we now substitute this and
with the function g(x) satisfying g(x=D2(y))=D2(y). D2n(y)=A2nyζ2n[1+ξ(y)] [consideredas a definition of
ξ(y)] into Eq. (7) to find the equation
Considerfirsttheζ2 <1case,andusethenear-inertial
approximationforS2(y),Eq. (10). Invertingthisrelation d2ξ dξ
we find the relevant approximation for f in terms of x, dy2 +(2ζ2n+4−ζ2)ydy
i.e.
3ζ2x 1/ζ2 2C2 + nζ2(1+(ζ22ζ+2)1()2−ζ2)yζ2−4 =0 . (25)
f(x)= 1− . (20)
2 3ζ2x
(cid:18) (cid:19) (cid:18) 2 (cid:19) This equation has an obvious power law solution which
Using this and introducing constants C20n S2n = leads to
iCn20enrxtiζa2ln/vζe2rsiinonthoefrEeqm.a(i1n7d)eristerm we find that the near- D2n(y)=A2nyζ2n 1+ nζ2(1+2ζ2) yζ2−2 (26)
(cid:20) (ζ2+1)(1+2ζ2n) (cid:21)
d2F2n 3 dF2n nF2n(x)
d2x + ζ2x dx − x (21) (for ζ2 >1), which gives
(cid:20) (cid:21)
+ 2Cζ22C320n(ζ2n−nζ2)xζ2n/ζ2−3 =0 . r1 = ζn2ζn2((11++22ζζ22n)) . (27)
Doing the same to Eq. (19) we have InFig. 3wecanseer1 asafunctionofζ2 forn=2,n=4
and n=6. For very large n we can see that
D2G2n 3 dG2n nG2n(x)
+ − (22)
d2x ζ2x(cid:20) dx x (cid:21) lim r1 = 1+2ζ2 ∈[1,5] , (28)
+ 2C2C20nnxζ2n/ζ2−3 =0 . n→∞ 6 2 6
ζ3 but the convergence is slow as we can see from Fig. 3
2
and again not uniform in ζ2, since one can verify that
Indeed, we see that the first lines of these equations ,
which are the inertial range contributions, are the same. limζ2→0r1 =1. Again one can verify that the efficacy of
ESSis reflected in a smaller remainder termin the ODE
Also the second lines are of the same order in y but the
rteartmiosr)2isbetween the coefficients in these lines (reminder ffoorr Sw2hnicuhsihnegreSt2oaosra2v=ar1ia−blre1−,1a.s expressed by a ratio r2
Thesuccessofthemethodinthiscasecannotbeglibly
ζ2n−nζ2 extendedtoanyturbulentsystem. Weseefromtheanal-
r2 = nζ2 . (23) ysisthatthe improvementinthe scalingbehaviourofSn
when presented as a function of S2 is not a fundamen-
Since the Ho¨lder inequalities imply that ζ2n ≤ nζ2, we talresultbutastatementonthesimilarbehaviourofthe
have|r2|≤1;thismeansthatindeedtheremainderterm structurefunctionsgoingintothedissipationrange. This
that “spoils” the inertial range scaling to first order is is measured by the value of the parameter r1 of Eq.(15).
smaller in the new representation. One can see, by solv-
We can imagine in principle - although no examples are
ing Eq. (17) in terms of the coefficient of the remainder known - a system for which the dissipative scale for S
n
term, using F2n(x) to express D2n(y) back again and issmallerthanthatforS2. Insuchasystemthe method
comparing with Eq. (12), that one should have the rela-
will not improve the scaling plots. In this passive scalar
tion modelthe dissipativescalesofallthe structurefunctions
1 are of the same order, as has been shown in [13]. This
r2 =1− r1 . (24) is one of the deep reasons for the success of the method
3
in the present case. It is tempting to conjecture that
the success of the method in Navier-Stokes turbulence
is an indication that the dissipative scales of S are ei-
n
ther of the same order or become largerwith n. The full
understanding of the method in that case must await a
better analytic understanding of the structure functions
and their functional form in the near dissipative regime.
∗ Segel’s e-mail: [email protected].
† L’vov’v e-mail: [email protected]
‡ Procaccia’s e-mail: [email protected].
[1] For a recent text see: Uriel Frisch. Turbulence: The
LegacyofA.N.Kolmogorov.CambridgeUniversityPress,
Cambridge, 1995.
[2] A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 30, 229
(1941)
[3] A.M.Obukhov,Izv.Akad.Nauk.SSSRser.,Geogr.Geofiz.
13, 58 (1949).
[4] S.Corrsin, J. Appl.Phys. 22, 469 (1949)
[5] F. Anselmet, Y. Gagne, E.J. Hopfinger, and R.A. Anto-
nia, J. Fluid Mech. 140, 63 (1984)
[6] C.MeneveauandK.R.Sreenivasan,Nucl.Phys.B,Proc.
Suppl.2, 49 (1987)
[7] R.Benzi,S.Ciliberto,R.Tripiccione,C.Baudet,F.Mas-
saioli and S.Succi, Phys. Rev.E 48, R29 (1993)..
[8] R.Benzi,S.Ciliberto,C.Baudet,G.RuizChavarriaand
R.Tripiccione, Europhs. Lett., 24, 275 (1993).
[9] R.H.Kraichnan, Phys.Fluids, 11, 945 (1968)
[10] R.H.Kraichnan, Phys.Rev. Lett., 72 1016 (1994).
[11] A.L.Fairhall,O.Gat,V.S.L’vovandI.Procaccia,Phys.
Rev.E, submitted.
[12] R.H.Kraichnan, privatecommunication, 1995.
[13] V.S. L’vov and I.Procaccia, “Fusion Rules in Turbulent
Systems with Flux Equilibrium”, Phys. Rev. Lett, sub-
mitted.
[14] V.S. L’vov and I. Procaccia, Phys. Rev. E 49, 4044
(1994).
Figure Legends:
Fig.1: The dimensionless structre functions D2n(y)
plotted in log-log plots as functions of y (solid line) and
S2(y) (dashed-dotted line) respectively. In dashed line
the exact scaling law is shown. In panels (a) and (b)
ζ = 1/2 and n = 2 and n = 6 respectively. Panels (c)
h
and (d) show results for ζ =3/2 and the same value so
h
of n. One sees an improvement of the scaling law in at
least 54 orders of magnitude ???.
Fig.2: panel (a): The ratio r1 of Eq.(15) as a function
ofζ2 forn=2(solidline),n=4(dashedline)andn=6
(dotted-dashed line). panel (b): The ratio r2 of Eq.(25)
for the same values of n.
Fig.3 The ratio r1 of Eq.(15) for ζ2 >1, and the same
values of n as in Fig.2.
4