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Turbulent drag reduction in one and two dimensions
Physica A 298 (2001) 140–154
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Turbulent drag reduction in one and two dimensions Theo Odijk ∗ Section Theory of Complex Fluids, Faculty of Applied Sciences, Delft University of Technology, Delft, The Netherlands
Abstract Drag reduction is investigated in one- and two-dimensional turbulent )ows that are stationary and homogeneous. In the two-dimensional case, the polymer chains are deformed though advected passively through the Kraichnan cascades within a scaling analysis. The typical rate of shear must then be larger than the time of deformation of a chain. Ultimately, elastic forces compete with Reynolds stresses at an elastic cut-o0 similar to that de1ned in the Tabor–de Gennes scenario in three dimensions. There are several regimes in two dimensions because there are two cascades. In the one-dimensional case, a Burgers-type equation is coupled to a frame-indi0erent equation for the viscoelastic Maxwell stress together with a stochastic force. Dissipation arises within viscoelastic shocks. The (e0ectively longitudinal) speed of the elastic waves is a cut-o0 for the shocks. This elastic cut-o0 shows up in velocity correlations. The c 2001 Elsevier Science B.V. implications for soap 1lm and wire experiments are discussed. All rights reserved. PACS: 47.27Gs; 61.25.Hq; 83.60Yz Keywords: Turbulence; Drag reduction; Polymers; Elastic waves; Elasticity; Rheology
1. Introduction The experimental literature on drag reduction in three-dimensional turbulent )ows is vast and con)icting, and impossible to review here. On the theoretical side, the increase of the viscous sublayer in pipe )ow was put forward as a primary mechanism
Presented at a symposium on the occasion of the 60th birthday of Dick Bedeaux. Correspondence address: P.O. Box 11036, 2301 EA Leiden, Netherlands. Tel.=Fax: 31-71-5145346. E-mail address: [email protected] (T. Odijk). ∗
c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 2 1 5 - 1
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for polymer drag reduction a long time ago [1,2]. The Lumley picture was amended by Ryskin [3] who accounted for the increase in the elongational viscosity by the unravelling of the polymer chains by the turbulent shearing. This amendment has a certain measure of success in explaining the change in the Prandtl resistance law at the expense of adjustable numerical coeJcients. Nevertheless, it stands to reason that the elastic nature of the polymer rheological )uid must play some role in drag reduction, possibly at all scales [4]. Tabor and de Gennes proposed that the passive elastic elongation of chains in the inertial regime might increase the lower scale of the Kolmogorov cascade [5]. This e0ect could thus also lower the turbulent friction. Recently, Sreenivasan and White [7] gave a succinct summary of the experimental situation and assessed the elastic theory [5,6] in terms of the onset of drag reduction and the reduction asymptote connected with chain overlap. A cut-o0 of the inertial range caused by polymer elasticity does not seem unreasonable. On the whole, the evidence for an elasticity induced attenuation of the velocity )uctuations at small scales is also fairly convincing [8–16]. Yet there are also recent strong claims that the role of elasticity in drag reduction is minor or even adverse [17]. These developments have prompted us to reinvestigate viscoelastic models for turbulent drag reduction, though in one and two dimensions because of current experimental activity in these areas and because such an analysis would give additional checks on any ideas concerning the three-dimensional case. A recent study by the author focuses on postulated negative temperatures states in two-dimensional rheological )uids after the turbulent )ow has decayed to a stationary state [18]. The turbulent )ow competes with normal polymeric stresses in coherent vortices. Here, I wish to discuss forced viscoelastic turbulence that is homogeneous and stationary. Turbulent )ow in two-dimensional soap 1lms was 1rst quanti1ed by Couder [19]. He found that there were coherent vortices merging in accord with scaling predictions originally put forward by Batchelor [20]. The coupling of the 1lm )ow to that of the surrounding air [21] is potentially a problem but the linear increase of the vortex size with time is also borne out in experiments at low air pressures [22]. Evidence for the existence of the Kraichnan energy and enstrophy cascades [23] in stationary turbulent 1lms was adduced by Gharib and Derango [24]. The signature of the energy spectrum of the enstrophy cascade found at small scales has been consistent with that found in other 1lm experiments [22,25 –27]. Electromagnetic forcing of a thin electrolyte layer also induces )ows that are closely two-dimensional [28]. The forced energy spectra [29,30] are in agreement with the Kraichnan–Batchelor picture. Turbulent decay [28,31], however, does not follow the original prediction [20] based on energy conservation alone but is consistent with a scaling picture [32] of coalescing coherent vortices of constant amplitude. There is numerical evidence that the preemption of Batchelor decay may be associated with the magnitude of the initial Reynolds number [33]. Experiments on turbulent decay in two-layer strati1ed )uids also show that viscous drag may well play a substantial role even at high initial Reynolds numbers [34]. Numerical simulations of forced two-dimensional turbulence [35 –38] have
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exhibited the spectrum characteristic of the inverse energy cascade although there is evidence [39] that coherent vortices [40] could perturb this at suJciently high Reynolds numbers. Simulations [41,42] also appear to bear out the predicted spectrum for the enstrophy cascade [23]. In decaying turbulence [43], the same cascade is again witnessed but signs of the perturbing in)uence by coherent vortices again crop up. Ultimately, a stationary state is reached solely governed by the dynamics of the vortices [44]. To summarize, on an empirical level there is considerable evidence that the two-cascade scenario for forced two-dimensional turbulence is basically correct but there may be perturbing in)uences owing to intermittency and the presence of coherent structures, especially if the Reynolds number is not high enough. It would thus appear sound to develop an approach to drag reduction in two dimensions based on the scaling analysis by Kraichnan [23]. Moreover, the addition of polymer to soap solutions in 1lm turbulence experiments would not seem to pose serious technical diJculties so that such a theory could be put to the test. A )uid streaming down a long thin wire may develop Rayleigh-like instabilities [45]. Typically, the cross-sectional area of the )uid is not uniform and rather diJcult to analyze theoretically. In a certain regime, the )ow becomes turbulent on longitudinal scales larger than the thickness of the wire [46]. Thus, one can in e0ect speak of experimental one-dimensional turbulence albeit for e0ectively compressible )ow owing to the )uctuations in the cross section of the (incompressible) )uid [46]. The e0ective one-dimensional )uid would thus seem to satisfy a stochastic Burgers equation [47]. Indeed, Kellay and Rouch 1nd by laser Doppler velocimetry that the usual Kolmogorov scaling for the energy spectrum actually holds (upon assuming the turbulence is frozen aL la Taylor) and so a recent version of the randomly driven Burgers equation studied by Chekhlov and Yakhot seems applicable [48,49]; besides the Kolmogorov scaling, there is semiquantitative agreement with regard to scaling of the correlation of the velocity )uctuations as a function of the longitudinal distance along the wire. Higher moments deviate strongly from the classic Kolmogorov cascade owing to the presence of strong shocks [48,49]. A lot of e0ort has been devoted to computing the probability functions of the velocity and its derivatives [50 –58]. Given this state of a0airs, it is obviously of interest to investigate drag reduction in this type of one-dimensional liquid. From an experimental point of view [46], its onset owing to a slight addition of )exible polymer ought to be discernible. In the following, we 1rst study the impact of forced stationary turbulence on passively advected polymer chains in the two-dimensional case. Because there are now two cascades, we have to delineate a variety of regime by comparing time, length and velocity scales. Ultimately, there is a lower cut-o0 determined by balancing the kinetic energy of the )uid against the polymer elastic energy. Then, a more quantitative analysis of elasticity in turbulent )ow is attempted for a one-dimensional “Maxwell–Burgers )uid” that is introduced here. The stationary distribution of viscoelastic shocks is analyzed in terms of a characteristic velocity of the elastic waves. It is argued that, under certain conditions, viscoelastic turbulence may exhibit simple inertial scaling.
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2. Two-dimensional drag reduction A statistical analysis of the equation of motion of a forced two-dimensional Newtonian )uid shows that the energy and the enstrophy could be conserved quantities at high enough Reynolds numbers [20,23]. In the formal inviscid limit (kinematic viscosity = 0), there is an in1nity of local invariants but the dissipation at very small scales in a realistic )uid reduces this to a 1nite number (presumably two). Moreover, it can be argued that the energy is constant at fairly small scales [20,23] so there could be an enstrophy cascade in an inertial regime. Its energy spectrum is given by [23] E 2=3 k −3
[ki ¡ k ¡ kd = (=3 )1=6 ] ;
(1)
in terms of the wave number k. The energy (per unit mass) is injected at a wave number ki and the rate of dissipation of the enstrophy with time is assumed constant. 1 d!2 : (2) 2 dt The square of the vorticity ! is averaged over the ensemble of forced stationary states. At the dissipation scale kd−1 the cascade is cut o0 owing to friction. By contrast, in the second inertial regime where the enstrophy remains constant, the energy cascades upwards to larger scales =
E 2=3 k −5=3
(k ¡ ki ) :
(3)
Here, the rate of dissipation of the energy is assumed not to vary 1 du2 : (4) 2 dt where ˜u is the velocity of the )uid. The enstrophy cascade (Eq. (1)) ought to have logarithmic corrections to scaling [59,60]. Eq. (1) has been doubted in the past [61,62] but recent exact work [63] has corroborated Eqs. (1) and (3) apart from possible corrections due to intermittency. We next wish to study a two-dimensional non-Newtonian )uid which is a dilute suspension of )exible polymer chains in a Newtonian solvent (all quantities relevant to the latter bear the index 0). We investigate the impact of the )ow given by the cascades Eqs. (1) and (3) in the case when the chains may be considered to be advected passively through the )uid. In certain regimes de1ned below, the chains are assumed to have negligible import on the )uid cascade laws. Our line of argumentation will generally follow that of Tabor and de Gennes [5,6], although peculiarities show up that are particular to two-dimensional drag reduction. We 1rst introduce the average rate of shear on the basis of dimensional arguments
=
(k 2 )1=3 (k ¡ ki ) ; 1=3 (k ¿ ki ) :
(5)
Within the enstrophy cascade at smaller scales, this rate is constant. This immediately implies a regime in which the polymer coils are not perturbed at all.
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2.1. Case I The characteristic time of deformation of a )exible chain is given by the Zimm time [64] z ≡
0 R3 : kB T
(6)
(In this section we neglect all numerical coeJcients) The typical dimension of the chain is represented by the radius of gyration R, the viscosity of the solvent at temperature T is 0 (0 = 0 0 where 0 is its density) and kB is Boltzmann’s constant. It is only at high enough rates of shear that we expect the chains to be deformed or start to unravel. This happens when the rate of dissipation is high enough (implying stronger )ows in an experimental set-up). Hence, the chains remain undisturbed whenever z ¡ 1, or in view of Eq. (5) if 1=3 z ¡ 1 :
(7)
As we increase the average rate of )ow, we inevitably reach the next stage. 2.2. Case II We 1rst reach this case when the enstrophy cascade alone interacts with the macromolecules 1=3 −1 z :
(8)
The upper scale r∗ at which the polymers are a0ected is simply the injection scale r∗ ki 1 :
(9)
The chains are supposed to be advected by the turbulent )ow cascading downwards to smaller scales u(r) 1=3 r
(r ¡ r∗ ) :
(10)
Concomitantly, the elongation of the chains is assumed to be a power law in terms of the only two relevant scales r and r∗ r m ∗ : (11) r The exponent m need not be equal to zero despite the invariance of the typical rate of shear in the enstrophy cascade. First, there are corrections to scaling [59,60]. Second, one may argue that Eq. (10) is analogous to expressions for converging )ows [5,6] where advection scaling laws also apply. Equations like Eq. (11) have been criticized [3] in the three-dimensional case on the grounds that a chain does not belong to any one eddy. However, in a blob of turbulent )uid of size r ¿ kd−1 , there are very many chains extended to various elongations and Eq. (11) is the simplest relation conceivable (in the
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absence of a complete dynamic theory of chain elongation). In the three-dimensional case, the velocity 1eld in the inertial regime is given by u ( r)1=3 :
(12)
From their analysis of the onset of drag reduction in a variety of three-dimensional experiments, Sreenivasan and White [7] conclude that the exponent n3 in the elongation Ansatz [5,6] r n3 ∗ (13) r should be close to 2=3 (their analysis is not entirely satisfactory in view of the failure of all the curves in their Fig. 5 to collapse onto one universal curve). The supposition of the chain advection remaining passive must break down on a scale r∗∗ . As the chains become more elongated, their elastic energy increases. At the same time, the Reynolds stress in the )uid decreases as the velocity diminishes at small scales (Eq. (10)). Ultimately, the macromolecules will start to play an active role in the dynamics of the )uid when their elastic energy competes with the )uid kinetic energy u2 (r∗∗ ) G2 (r∗∗ ) :
(14)
For a dilute suspension of concentration c (number of polymer chains per unit volume), we write the elastic modulus as G ckB T :
(15)
For the sake of simplicity the chains are here assumed to be Gaussian (hence the square of in Eq. (14)). It is also convenient to introduce a characteristic velocity of the elastic waves resulting from the polymeric elasticity [4,65 – 68] ve = (G=)1=2 :
(16)
Using Eq. (10), we now rewrite Eq. (14) in a scaled form so as to eliminate the dissipative rate of enstrophy. u(r∗ ) 1=3 ki−1 ; u(r∗∗ ) i
r∗∗ r∗
(17)
u(r∗ ) ;
≡ ve =u(r∗ ) = ve =u(ki−1 ) ;
r∗∗ r∗
! i
(! ≡ 1=(m + 1)) :
(18) (19) (20)
Eq. (20) predicts the onset of elastic perturbation of the )ow by the polymer, an onset that should be discernible in experiment (see Ref. [7] in the three-dimensional case). The lower scale must of course be larger than the Kraichnan dissipation scale:
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r∗∗ kd ¿ 1. We note that we remain in the elliptic regime with respect to the occurrence of elastic waves (u 6 u(r∗ ) 6 ve , see Ref. [4]). 2.3. Case III As we next increase the externally applied rate of )ow, we enter a regime involving both cascades. The rate of shear in the inverse energy cascade equals the reciprocal Zimm time at a scale given by r∗ 1=2 z3=2 :
(21)
Thus, the elongation of the polymer chains is expressed by an assumed scaling law Eq. (13). However, the exponent n2 in two dimensions certainly di0ers from the one in three [5,6] r n2 ∗ (r∗ ¿ r ¿ ki−1 ) : (22) r This expression does not apply at scales below the injection scale ki−1 . Instead, the chains enter the enstrophy cascade so their elongation is now expressed by Eq. (11). The exponents n2 and m are probably not identical in the respective cascades. Therefore, we have (ki r∗ )n2 = (rki ¡ 1) : (23) (ki r)m Again, we introduce a cut-o0 where polymeric elasticity starts to compete with the Reynolds stress (see Eq. (14)). In soap 1lm experiments, the )uid velocity at the injection scale is a known quantity as well as the injection scale itself [22,26]. It is thus expedient to scale Eq. (14) in the following way: r∗∗ ki
! !n2 i (ki r∗ )
:
(24)
The characteristic velocity at injection shows up in the dimensionless form, Eq. (19). 2.4. Case IV If one keeps on increasing the external )ow—implying an increase in the rates of dissipation and —we may attain a new regime. Provided n2 ¿ 2=3, the cut-o0 r∗∗ could occur in the inverse energy cascade according to Eq. (24) (when r∗∗ ki ¿ 1). This scenario is identical to the one pertaining to three dimensions as introduced by Tabor and de Gennes [5,6] but with an exponent n2 instead of n3 so it will not be described here. 3. One-dimensional drag reduction Chekhlov and Yakhot have recently simulated the Burgers equation with a special type of stochastic forcing function [48,49]. The white-in-time force has a correlation
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function depending on the wave number in such a way that the energy spectrum is Kolmogorov-like to within a logarithmic factor @u @2 u @u + u = 2 + f ; @t @x @x
(25)
f(k; t)f(k ; t ) ˙ |k|−1 %(k + k )%(t − t ) :
(26)
This is an apt model for one-dimensional turbulence that is compressible. The simulations explicitly show well-separated shocks in a random background [48]. At high Reynolds numbers, the shocks may be viewed as very thin boundary layers: the rate of dissipation of the kinetic energy of the ensemble of shocks displays inertial scaling independent of the viscosity. This model turns out to have statistical features similar to the quasi-one-dimensional turbulence exhibited by water )owing down a thin wire at high enough rates of )ow [46]. It is thus of some interest to investigate the impact of polymer on this type of one-dimensional )ow, even though there are obvious di0erences between the successive transport mechanisms as we reduce the dimension from three to two to one. Let us 1rst regard the equation of motion perturbed by some general viscoelastic stress arising from the polymer @u @u @ +u = : (27) @t @x @x The stress [u(x; t)] is a functional of the velocity u(x; t). Eq. (27) is here also perturbed by a random force given by Eq. (26). If the one-dimensional interval of length L containing the )uid is long enough, the resulting turbulent )ow at high enough Reynolds number Lkd is homogeneous (kd = dissipation wave number for the viscoelastic )uid). The rate of dissipation of the kinetic energy is then given by 1 du2 @ : (28)
= = u 2 dt @x Next, one may suppose the shocks appearing in the Newtonian case are now modi1ed, but not too dramatically, by the new form of the stress . In particular, the boundary layers in which dissipation occurs remain thin and the shocks are well-separated coherent structures (see Fig. 1). Thus, the velocity may be expressed as u(x; t) us; i (x; t) + uR (x; t) ; (29) i
where the shocks (index s) are well-de1ned entities translating within a random background. Shocks are continually being formed and are continually coalescing but they are also dissipated by friction. Here, the forced turbulence is in a steady state. Upon using arguments going back to Burgers [69] in the Newtonian case, we may suppose that the shocks are e0ectively stationary. The equation of motion of each shock is given by us
@ @us : @x @x
(30)
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Fig. 1. One realization of the velocity u(x; t) at time t as a function of x. The shocks are sawtooth structures perturbed by a random background.
Hence, the rate of dissipation is expressed as @
us; i @x s i 3 uA; i @us; i us;2 i = ; @x s 3r i i
(31)
where the average is over intervals of size 2r centered on the shocks of amplitude uA . In e0ect, the random background )uctuates without energy losses. This is the case for the Newtonian )uid [48,49] so it remains plausible for a viscoelastic )uid under the restrictions stated. We see that Eq. (31) has a form valid for an inertial regime independent of the viscoelasticity. There may be fewer shocks than in the Newtonian case, however, but this has to be analyzed in a concrete model as we now do. A variety of rheological models have been employed to describe non-Newtonian phenomena at high rates of shear and elongation [4,66,70]. A special type of interpolated Maxwell model is introduced here for a one-dimensional viscoelastic )uid that is frame-indi0erent: @u @u @m +u = ; (32) @t @x @x @u @m @m + m = : +u (33) @t @x @x This has one time constant to be discussed below and displays a simple elastic response that is advected with the )ow. In the practical case of a viscoelastic )uid running down a wire [45], would be an e0ective elongational kinematic viscosity. The turbulence arising from the action of a forcing function like Eq. (26) on the highly nonlinear equations (32) and (33) may be termed as Maxwell–Burgers turbulence. Eqs. (30), (32) and (33) yield an expression for a quasi-stationary shock in the turbulent background (m = m [u(x)]) @2 us @us 1 @2 u 3 = 2 − 2s ; us (34) @x @x 3 @x
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with boundary conditions us → −uA us → uA
(x → ∞) ; (x → −∞) :
(35)
Eq. (34) displays the Hadamard-like instability associated with the change of type from elliptic to hyperbolic [4,66] as the velocity increases. The solution to Eq. (34) is )w + (1 − )) arctanh(w) = − s ;
(36) uA2 =ve2 ,
where in terms of the dimensionless variables w ≡ us =uA , s ≡ uA x=(2), and ) ≡ ve ≡ (=)1=2 is a characteristic velocity of the longitudinal elastic waves. There is no solution of Eq. (34) for ) ¿ 1. Hence, the viscoelastic shocks are cut o0 at an upper velocity ve . In order to analyze the width of a shock, we Taylor expand arctanh(w) as w tends to unity. Thus, a characteristic velocity w∗ separating two regimes may be introduced 2 ; (37) w∗ ≡ 1 − 2 exp )−1 and the velocity in the respective regimes is given by w −s w 1 − 2 exp
2|s| )−1
(w 6 w∗ ) ; (w ¿ w∗ ) ; (s ¡ 0; ) ¡ 1) :
(38)
We conclude that the shock width is never greater than that in the Newtonian case () = 0). Thus, the original supposition behind the superposition Eq. (29) is self-consistent. The rate of dissipation is inertia-like (Eq. (31)) but this holds up to a scale re = ve3 = :
(39)
At velocities higher than ve , there is a regime where only elasticity perturbs inertia, if at all. We write the dissipation via dimensional analysis as 1=3 1=3 u = (r ) h r 2=3 2 re 2=3 = (r )1=3 h (r ¿ re ) : (40) r The smallest width of a shock is smaller than or equal to the Kolmogorov scale K = (3 = )1=4 . Accordingly, at high Reynolds numbers we have 2 = 6 (K =re )4=3 and so 2 =1. Therefore, the dissipation rate in this regime is virtually una0ected by elasticity and is hence inertia-like. At this stage, it is well to compare our one-dimensional )uid with the two- and three-dimensional polymer suspensions discussed in the previous section. The relaxation time is not precisely the Zimm time but rather z =(cR3 ) ;
(41)
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at least if the polymer solution is dilute. This is required if the elastic waves are to propagate at the same velocity in all three cases (see Eqs. (6), (15) and (16)) To summarize, in Maxwell–Burgers turbulence driven by a force given by Eq. (26), the cascade is Kolmogorov-like, both for the shocks and the random background at scales larger than the dissipative cut-o0. Viscoelasticity in)uences neither at high Reynolds numbers. Therefore, we may speak of passive advection so one may introduce r∗ again as given by Eq. (21). In addition, we are led to posit a polymer elongation (in a one-dimensional polymeric )uid )owing down a wire and assumed to be well-represented by Eqs. (32) and(33)): r n1 ∗ : (42) r The exponent n1 probably di0ers from n2 and n3 . Again it would be possible to introduce a lower cut-o0 r∗∗ within both Kolmogorov cascades. This cut-o0 is the same as that conjectured by Tabor and the Gennes [5,6] but with n1 instead of n3 . This second e0ect due to elasticity is discussed below. The rate of dissipation within the shocks is written as
=
3 uA; i
3r
i
(uA; i ¡ ve ) :
(43)
This di0ers from that in the Newtonian )uid studied by Chekhlov and Yakhot who cut o0 their shock cascade by an upper velocity depending on the system size L [48,49] Umax ( L)1=3 :
(44)
We disregard a logarithmic factor. Chekhlov and Yakhot present a semiquantitative analysis of the probability functions for the velocity [49] in terms of Umax so we simply follow their argumentation by replacing Umax by ve . If we introduce the probability Pr (u) for the shock amplitude u within some interval 2r, we may write the continuum version of Eq. (43) in terms of the identity
ve Lve : (45) du u3 Pr (u) r ln =L The lower cut-o0 pertains to the weakest shock; according to Eq. (37) the kinematic viscosity remains relevant in spite of the elasticity. The right-hand side of Eq. (45) is computed from the rate of energy dissipation via Eq. (32) with an additional force f added to the momentum balance whose statistical properties are given by Eq. (26). We have shown above that both shocks and random background have statistical properties that are inertia-like. Hence, the end result is the analog of that derived in Ref. [49]. Eq. (45) yields the probability function for the shocks
r (u ¡ ve ) ; u4 0 (u ¿ ve ) :
Pr (u)
(46)
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Of interest are the correlation functions S‘ (r) ≡ (u(x + r) − u(x))‘ ; = (Tu)‘ :
(47)
The probability function for Tu is asymmetric in view of the shock structures given by the boundary equations (35). This asymmetry leads to a peculiar dependence of S‘ on ‘ [49] 0 6 ‘ 6 3: ‘ ¿ 3:
S‘ (r) ( r)‘=3 ;
rve‘−3 ‘−3
(ve ¿ ( r)1=3 ) ;
( r)‘=3
(ve ¡ ( r)1=3 ) :
S‘ (r)
(48)
(49)
The structure is Kolmogorov-like except at small r and for ‘ ¿ 3. Shocks dominate the )uid structure for ‘ ¿ 3. For fairly large negative Tu, the probability Pr (Tu) scales in the same way as Eq. (46). For 0 6 ‘ 6 3, we see only the signature of the Kolmogorov background. In the Newtonian case, Eqs. (46) – (48) (with Umax instead of ve ) agree well with the data from computer simulations [48,49]. Note that for the non-Newtonian )uid, Eq. (49) exhibits two regimes because the elastic velocity is less than the maximum cut-o0 (Eq. (44)). In conclusion, in this section we have discussed two routes to the e0ect of elasticity on one-dimensional turbulence. With the help of a crude rheological model (Eq. (33)), we have argued for the existence of a velocity cut-o0 in the correlation functions (Eqs. (48) and (49)). But Eq. (33) is too primitive to allow for passive advection of the chains leading to their elongation described by Eq. (42). The latter e0ect would implicate elasticity via a lower elastic cut-o0 r∗∗ . Nevertheless, it is possible to reconcile both points of view, at least qualitatively, because, fortunately, in both models there are inertial regimes that are Kolmogorov-like. We could introduce a (complicated) rheological model in Eq. (27) which implicitly accounts for passive advection as expressed by Eq. (42). Shocks then occur within the stationary turbulent )ow caused by the stochastic forcing, but they simply obey an inertial law given by Eq. (31). On the whole, the assumption of chain elongation by passive advection remains self-consistent down to the cut-o0 r∗∗ , provided r∗∗ ¡ re , the latter scale being de1ned by Eq. (39). We do not address the precise elastohydrodynamics at scales less than r∗∗ although an expression like Eq. (34) for the viscoelastic shocks remains plausible at scales greater than r∗∗ . Therefore, the structure functions should still be given by Eqs. (48) and (49) but cut o0 at lower scales by r∗∗ . 4. Concluding remarks The present analysis is entirely based on the implications for turbulent drag arising from elasticity alone. Any change in the elongational viscosity, for instance, is totally
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disregarded. Recent experiments [71] and simulations [72] concerning DNA in elongational )ow show that a simple coil–stretch transition does not exist. There is a host of complicated elongated shapes as a function of the Deborah number. Thus, primitive scaling forms like Eqs. (11), (13), (22) and (42) would seem to be qualitatively useful for the moment. Is it possible to see the elastic cut-o0 (Eq. (24)) in a quasi-two-dimensional experiment? In a recent set-up involving strati1ed layers [73], a )exible polymer did not seem to in)uence the turbulent )ow but the Reynolds number was not exorbitant. Vortex streets in )owing soap 1lms at intermediate Reynolds numbers are markedly perturbed by polymer at rather small concentrations [74]. Flowing soap 1lms have, nevertheless, been shown to display turbulence at quite high Reynolds numbers (see Ref. [24]; Rei 104 at injection). Moreover, if we focus on a long semi)exible polymer like double-stranded DNA, much longer Zimm times are possible. More concretely, let us introduce the following dimensionless numbers: Deborah number De ≡ 1=3 z , coil volume fraction ’ ≡ cR3 , injection Reynolds number Rei ≡ ui =(ki ), and the variable V ≡ ’0 =(z ui2 ). Using Eqs. (6), (15), (16), (19), (21) and (24), we can see whether the polymer would be elongated by both the energy and enstrophy cascades (Case III). The chain relaxation time must be long enough: De ¿ 1 : The elastic cut-o0 must be positioned within the enstrophy cascade: r∗∗ ki ¡ 1 or VDe3n2 ¡ 1 : The elastic cut-o0 must be larger than the Kraichnan dissipation scale: Rei VDe3n2 ¿ 1 : For example, 5 m long DNA exhibiting a weak excluded-volume e0ect would have a Zimm time of 0:1 s within a soap 1lm having a kinematic viscosity of 0:02 cm2 =s. The 1lm could be thick enough for the DNA coil to move quite freely. The solution is dilute (’ = 0:1 say). If we set Rei = 104 , ui = 100 cm=s, = 107 s−3 and n2 n3 2=3, say, we obtain V = 2 × 10−6 and De 20. Therefore, the inequalities above could be easily ful1lled so the elastic cut-o0 should be discernible. In the quasi-one-dimensional case, a 0:1 g=l polyethylene oxide suspension in water would have an elastic speed in the range of cm=s [4]. These velocities can be monitored in wire experiments like those of Kellay and Rouch [46]. Hence, the proposal for the velocity correlations being cut o0 by elasticity can be checked. Acknowledgements I would like to thank Dick Bedeaux for his friendship and loyalty dating back to his practical course in “Quantum Statistical Mechanics” at the Lorentz Institute, University of Leiden in 1975. He was kind enough to think seriously about my proposal for a
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new polyelectrolyte theory in 1976, at a time when I dearly needed such support. In the past 25 years, we have witnessed a lot of ups and downs together.
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www.elsevier.com/locate/physa
Turbulent drag reduction in one and two dimensions Theo Odijk ∗ Section Theory of Complex Fluids, Faculty of Applied Sciences, Delft University of Technology, Delft, The Netherlands
Abstract Drag reduction is investigated in one- and two-dimensional turbulent )ows that are stationary and homogeneous. In the two-dimensional case, the polymer chains are deformed though advected passively through the Kraichnan cascades within a scaling analysis. The typical rate of shear must then be larger than the time of deformation of a chain. Ultimately, elastic forces compete with Reynolds stresses at an elastic cut-o0 similar to that de1ned in the Tabor–de Gennes scenario in three dimensions. There are several regimes in two dimensions because there are two cascades. In the one-dimensional case, a Burgers-type equation is coupled to a frame-indi0erent equation for the viscoelastic Maxwell stress together with a stochastic force. Dissipation arises within viscoelastic shocks. The (e0ectively longitudinal) speed of the elastic waves is a cut-o0 for the shocks. This elastic cut-o0 shows up in velocity correlations. The c 2001 Elsevier Science B.V. implications for soap 1lm and wire experiments are discussed. All rights reserved. PACS: 47.27Gs; 61.25.Hq; 83.60Yz Keywords: Turbulence; Drag reduction; Polymers; Elastic waves; Elasticity; Rheology
1. Introduction The experimental literature on drag reduction in three-dimensional turbulent )ows is vast and con)icting, and impossible to review here. On the theoretical side, the increase of the viscous sublayer in pipe )ow was put forward as a primary mechanism
Presented at a symposium on the occasion of the 60th birthday of Dick Bedeaux. Correspondence address: P.O. Box 11036, 2301 EA Leiden, Netherlands. Tel.=Fax: 31-71-5145346. E-mail address: [email protected] (T. Odijk). ∗
c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 2 1 5 - 1
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for polymer drag reduction a long time ago [1,2]. The Lumley picture was amended by Ryskin [3] who accounted for the increase in the elongational viscosity by the unravelling of the polymer chains by the turbulent shearing. This amendment has a certain measure of success in explaining the change in the Prandtl resistance law at the expense of adjustable numerical coeJcients. Nevertheless, it stands to reason that the elastic nature of the polymer rheological )uid must play some role in drag reduction, possibly at all scales [4]. Tabor and de Gennes proposed that the passive elastic elongation of chains in the inertial regime might increase the lower scale of the Kolmogorov cascade [5]. This e0ect could thus also lower the turbulent friction. Recently, Sreenivasan and White [7] gave a succinct summary of the experimental situation and assessed the elastic theory [5,6] in terms of the onset of drag reduction and the reduction asymptote connected with chain overlap. A cut-o0 of the inertial range caused by polymer elasticity does not seem unreasonable. On the whole, the evidence for an elasticity induced attenuation of the velocity )uctuations at small scales is also fairly convincing [8–16]. Yet there are also recent strong claims that the role of elasticity in drag reduction is minor or even adverse [17]. These developments have prompted us to reinvestigate viscoelastic models for turbulent drag reduction, though in one and two dimensions because of current experimental activity in these areas and because such an analysis would give additional checks on any ideas concerning the three-dimensional case. A recent study by the author focuses on postulated negative temperatures states in two-dimensional rheological )uids after the turbulent )ow has decayed to a stationary state [18]. The turbulent )ow competes with normal polymeric stresses in coherent vortices. Here, I wish to discuss forced viscoelastic turbulence that is homogeneous and stationary. Turbulent )ow in two-dimensional soap 1lms was 1rst quanti1ed by Couder [19]. He found that there were coherent vortices merging in accord with scaling predictions originally put forward by Batchelor [20]. The coupling of the 1lm )ow to that of the surrounding air [21] is potentially a problem but the linear increase of the vortex size with time is also borne out in experiments at low air pressures [22]. Evidence for the existence of the Kraichnan energy and enstrophy cascades [23] in stationary turbulent 1lms was adduced by Gharib and Derango [24]. The signature of the energy spectrum of the enstrophy cascade found at small scales has been consistent with that found in other 1lm experiments [22,25 –27]. Electromagnetic forcing of a thin electrolyte layer also induces )ows that are closely two-dimensional [28]. The forced energy spectra [29,30] are in agreement with the Kraichnan–Batchelor picture. Turbulent decay [28,31], however, does not follow the original prediction [20] based on energy conservation alone but is consistent with a scaling picture [32] of coalescing coherent vortices of constant amplitude. There is numerical evidence that the preemption of Batchelor decay may be associated with the magnitude of the initial Reynolds number [33]. Experiments on turbulent decay in two-layer strati1ed )uids also show that viscous drag may well play a substantial role even at high initial Reynolds numbers [34]. Numerical simulations of forced two-dimensional turbulence [35 –38] have
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exhibited the spectrum characteristic of the inverse energy cascade although there is evidence [39] that coherent vortices [40] could perturb this at suJciently high Reynolds numbers. Simulations [41,42] also appear to bear out the predicted spectrum for the enstrophy cascade [23]. In decaying turbulence [43], the same cascade is again witnessed but signs of the perturbing in)uence by coherent vortices again crop up. Ultimately, a stationary state is reached solely governed by the dynamics of the vortices [44]. To summarize, on an empirical level there is considerable evidence that the two-cascade scenario for forced two-dimensional turbulence is basically correct but there may be perturbing in)uences owing to intermittency and the presence of coherent structures, especially if the Reynolds number is not high enough. It would thus appear sound to develop an approach to drag reduction in two dimensions based on the scaling analysis by Kraichnan [23]. Moreover, the addition of polymer to soap solutions in 1lm turbulence experiments would not seem to pose serious technical diJculties so that such a theory could be put to the test. A )uid streaming down a long thin wire may develop Rayleigh-like instabilities [45]. Typically, the cross-sectional area of the )uid is not uniform and rather diJcult to analyze theoretically. In a certain regime, the )ow becomes turbulent on longitudinal scales larger than the thickness of the wire [46]. Thus, one can in e0ect speak of experimental one-dimensional turbulence albeit for e0ectively compressible )ow owing to the )uctuations in the cross section of the (incompressible) )uid [46]. The e0ective one-dimensional )uid would thus seem to satisfy a stochastic Burgers equation [47]. Indeed, Kellay and Rouch 1nd by laser Doppler velocimetry that the usual Kolmogorov scaling for the energy spectrum actually holds (upon assuming the turbulence is frozen aL la Taylor) and so a recent version of the randomly driven Burgers equation studied by Chekhlov and Yakhot seems applicable [48,49]; besides the Kolmogorov scaling, there is semiquantitative agreement with regard to scaling of the correlation of the velocity )uctuations as a function of the longitudinal distance along the wire. Higher moments deviate strongly from the classic Kolmogorov cascade owing to the presence of strong shocks [48,49]. A lot of e0ort has been devoted to computing the probability functions of the velocity and its derivatives [50 –58]. Given this state of a0airs, it is obviously of interest to investigate drag reduction in this type of one-dimensional liquid. From an experimental point of view [46], its onset owing to a slight addition of )exible polymer ought to be discernible. In the following, we 1rst study the impact of forced stationary turbulence on passively advected polymer chains in the two-dimensional case. Because there are now two cascades, we have to delineate a variety of regime by comparing time, length and velocity scales. Ultimately, there is a lower cut-o0 determined by balancing the kinetic energy of the )uid against the polymer elastic energy. Then, a more quantitative analysis of elasticity in turbulent )ow is attempted for a one-dimensional “Maxwell–Burgers )uid” that is introduced here. The stationary distribution of viscoelastic shocks is analyzed in terms of a characteristic velocity of the elastic waves. It is argued that, under certain conditions, viscoelastic turbulence may exhibit simple inertial scaling.
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2. Two-dimensional drag reduction A statistical analysis of the equation of motion of a forced two-dimensional Newtonian )uid shows that the energy and the enstrophy could be conserved quantities at high enough Reynolds numbers [20,23]. In the formal inviscid limit (kinematic viscosity = 0), there is an in1nity of local invariants but the dissipation at very small scales in a realistic )uid reduces this to a 1nite number (presumably two). Moreover, it can be argued that the energy is constant at fairly small scales [20,23] so there could be an enstrophy cascade in an inertial regime. Its energy spectrum is given by [23] E 2=3 k −3
[ki ¡ k ¡ kd = (=3 )1=6 ] ;
(1)
in terms of the wave number k. The energy (per unit mass) is injected at a wave number ki and the rate of dissipation of the enstrophy with time is assumed constant. 1 d!2 : (2) 2 dt The square of the vorticity ! is averaged over the ensemble of forced stationary states. At the dissipation scale kd−1 the cascade is cut o0 owing to friction. By contrast, in the second inertial regime where the enstrophy remains constant, the energy cascades upwards to larger scales =
E 2=3 k −5=3
(k ¡ ki ) :
(3)
Here, the rate of dissipation of the energy is assumed not to vary 1 du2 : (4) 2 dt where ˜u is the velocity of the )uid. The enstrophy cascade (Eq. (1)) ought to have logarithmic corrections to scaling [59,60]. Eq. (1) has been doubted in the past [61,62] but recent exact work [63] has corroborated Eqs. (1) and (3) apart from possible corrections due to intermittency. We next wish to study a two-dimensional non-Newtonian )uid which is a dilute suspension of )exible polymer chains in a Newtonian solvent (all quantities relevant to the latter bear the index 0). We investigate the impact of the )ow given by the cascades Eqs. (1) and (3) in the case when the chains may be considered to be advected passively through the )uid. In certain regimes de1ned below, the chains are assumed to have negligible import on the )uid cascade laws. Our line of argumentation will generally follow that of Tabor and de Gennes [5,6], although peculiarities show up that are particular to two-dimensional drag reduction. We 1rst introduce the average rate of shear on the basis of dimensional arguments
=
(k 2 )1=3 (k ¡ ki ) ; 1=3 (k ¿ ki ) :
(5)
Within the enstrophy cascade at smaller scales, this rate is constant. This immediately implies a regime in which the polymer coils are not perturbed at all.
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2.1. Case I The characteristic time of deformation of a )exible chain is given by the Zimm time [64] z ≡
0 R3 : kB T
(6)
(In this section we neglect all numerical coeJcients) The typical dimension of the chain is represented by the radius of gyration R, the viscosity of the solvent at temperature T is 0 (0 = 0 0 where 0 is its density) and kB is Boltzmann’s constant. It is only at high enough rates of shear that we expect the chains to be deformed or start to unravel. This happens when the rate of dissipation is high enough (implying stronger )ows in an experimental set-up). Hence, the chains remain undisturbed whenever z ¡ 1, or in view of Eq. (5) if 1=3 z ¡ 1 :
(7)
As we increase the average rate of )ow, we inevitably reach the next stage. 2.2. Case II We 1rst reach this case when the enstrophy cascade alone interacts with the macromolecules 1=3 −1 z :
(8)
The upper scale r∗ at which the polymers are a0ected is simply the injection scale r∗ ki 1 :
(9)
The chains are supposed to be advected by the turbulent )ow cascading downwards to smaller scales u(r) 1=3 r
(r ¡ r∗ ) :
(10)
Concomitantly, the elongation of the chains is assumed to be a power law in terms of the only two relevant scales r and r∗ r m ∗ : (11) r The exponent m need not be equal to zero despite the invariance of the typical rate of shear in the enstrophy cascade. First, there are corrections to scaling [59,60]. Second, one may argue that Eq. (10) is analogous to expressions for converging )ows [5,6] where advection scaling laws also apply. Equations like Eq. (11) have been criticized [3] in the three-dimensional case on the grounds that a chain does not belong to any one eddy. However, in a blob of turbulent )uid of size r ¿ kd−1 , there are very many chains extended to various elongations and Eq. (11) is the simplest relation conceivable (in the
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absence of a complete dynamic theory of chain elongation). In the three-dimensional case, the velocity 1eld in the inertial regime is given by u ( r)1=3 :
(12)
From their analysis of the onset of drag reduction in a variety of three-dimensional experiments, Sreenivasan and White [7] conclude that the exponent n3 in the elongation Ansatz [5,6] r n3 ∗ (13) r should be close to 2=3 (their analysis is not entirely satisfactory in view of the failure of all the curves in their Fig. 5 to collapse onto one universal curve). The supposition of the chain advection remaining passive must break down on a scale r∗∗ . As the chains become more elongated, their elastic energy increases. At the same time, the Reynolds stress in the )uid decreases as the velocity diminishes at small scales (Eq. (10)). Ultimately, the macromolecules will start to play an active role in the dynamics of the )uid when their elastic energy competes with the )uid kinetic energy u2 (r∗∗ ) G2 (r∗∗ ) :
(14)
For a dilute suspension of concentration c (number of polymer chains per unit volume), we write the elastic modulus as G ckB T :
(15)
For the sake of simplicity the chains are here assumed to be Gaussian (hence the square of in Eq. (14)). It is also convenient to introduce a characteristic velocity of the elastic waves resulting from the polymeric elasticity [4,65 – 68] ve = (G=)1=2 :
(16)
Using Eq. (10), we now rewrite Eq. (14) in a scaled form so as to eliminate the dissipative rate of enstrophy. u(r∗ ) 1=3 ki−1 ; u(r∗∗ ) i
r∗∗ r∗
(17)
u(r∗ ) ;
≡ ve =u(r∗ ) = ve =u(ki−1 ) ;
r∗∗ r∗
! i
(! ≡ 1=(m + 1)) :
(18) (19) (20)
Eq. (20) predicts the onset of elastic perturbation of the )ow by the polymer, an onset that should be discernible in experiment (see Ref. [7] in the three-dimensional case). The lower scale must of course be larger than the Kraichnan dissipation scale:
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r∗∗ kd ¿ 1. We note that we remain in the elliptic regime with respect to the occurrence of elastic waves (u 6 u(r∗ ) 6 ve , see Ref. [4]). 2.3. Case III As we next increase the externally applied rate of )ow, we enter a regime involving both cascades. The rate of shear in the inverse energy cascade equals the reciprocal Zimm time at a scale given by r∗ 1=2 z3=2 :
(21)
Thus, the elongation of the polymer chains is expressed by an assumed scaling law Eq. (13). However, the exponent n2 in two dimensions certainly di0ers from the one in three [5,6] r n2 ∗ (r∗ ¿ r ¿ ki−1 ) : (22) r This expression does not apply at scales below the injection scale ki−1 . Instead, the chains enter the enstrophy cascade so their elongation is now expressed by Eq. (11). The exponents n2 and m are probably not identical in the respective cascades. Therefore, we have (ki r∗ )n2 = (rki ¡ 1) : (23) (ki r)m Again, we introduce a cut-o0 where polymeric elasticity starts to compete with the Reynolds stress (see Eq. (14)). In soap 1lm experiments, the )uid velocity at the injection scale is a known quantity as well as the injection scale itself [22,26]. It is thus expedient to scale Eq. (14) in the following way: r∗∗ ki
! !n2 i (ki r∗ )
:
(24)
The characteristic velocity at injection shows up in the dimensionless form, Eq. (19). 2.4. Case IV If one keeps on increasing the external )ow—implying an increase in the rates of dissipation and —we may attain a new regime. Provided n2 ¿ 2=3, the cut-o0 r∗∗ could occur in the inverse energy cascade according to Eq. (24) (when r∗∗ ki ¿ 1). This scenario is identical to the one pertaining to three dimensions as introduced by Tabor and de Gennes [5,6] but with an exponent n2 instead of n3 so it will not be described here. 3. One-dimensional drag reduction Chekhlov and Yakhot have recently simulated the Burgers equation with a special type of stochastic forcing function [48,49]. The white-in-time force has a correlation
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function depending on the wave number in such a way that the energy spectrum is Kolmogorov-like to within a logarithmic factor @u @2 u @u + u = 2 + f ; @t @x @x
(25)
f(k; t)f(k ; t ) ˙ |k|−1 %(k + k )%(t − t ) :
(26)
This is an apt model for one-dimensional turbulence that is compressible. The simulations explicitly show well-separated shocks in a random background [48]. At high Reynolds numbers, the shocks may be viewed as very thin boundary layers: the rate of dissipation of the kinetic energy of the ensemble of shocks displays inertial scaling independent of the viscosity. This model turns out to have statistical features similar to the quasi-one-dimensional turbulence exhibited by water )owing down a thin wire at high enough rates of )ow [46]. It is thus of some interest to investigate the impact of polymer on this type of one-dimensional )ow, even though there are obvious di0erences between the successive transport mechanisms as we reduce the dimension from three to two to one. Let us 1rst regard the equation of motion perturbed by some general viscoelastic stress arising from the polymer @u @u @ +u = : (27) @t @x @x The stress [u(x; t)] is a functional of the velocity u(x; t). Eq. (27) is here also perturbed by a random force given by Eq. (26). If the one-dimensional interval of length L containing the )uid is long enough, the resulting turbulent )ow at high enough Reynolds number Lkd is homogeneous (kd = dissipation wave number for the viscoelastic )uid). The rate of dissipation of the kinetic energy is then given by 1 du2 @ : (28)
= = u 2 dt @x Next, one may suppose the shocks appearing in the Newtonian case are now modi1ed, but not too dramatically, by the new form of the stress . In particular, the boundary layers in which dissipation occurs remain thin and the shocks are well-separated coherent structures (see Fig. 1). Thus, the velocity may be expressed as u(x; t) us; i (x; t) + uR (x; t) ; (29) i
where the shocks (index s) are well-de1ned entities translating within a random background. Shocks are continually being formed and are continually coalescing but they are also dissipated by friction. Here, the forced turbulence is in a steady state. Upon using arguments going back to Burgers [69] in the Newtonian case, we may suppose that the shocks are e0ectively stationary. The equation of motion of each shock is given by us
@ @us : @x @x
(30)
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Fig. 1. One realization of the velocity u(x; t) at time t as a function of x. The shocks are sawtooth structures perturbed by a random background.
Hence, the rate of dissipation is expressed as @
us; i @x s i 3 uA; i @us; i us;2 i = ; @x s 3r i i
(31)
where the average is over intervals of size 2r centered on the shocks of amplitude uA . In e0ect, the random background )uctuates without energy losses. This is the case for the Newtonian )uid [48,49] so it remains plausible for a viscoelastic )uid under the restrictions stated. We see that Eq. (31) has a form valid for an inertial regime independent of the viscoelasticity. There may be fewer shocks than in the Newtonian case, however, but this has to be analyzed in a concrete model as we now do. A variety of rheological models have been employed to describe non-Newtonian phenomena at high rates of shear and elongation [4,66,70]. A special type of interpolated Maxwell model is introduced here for a one-dimensional viscoelastic )uid that is frame-indi0erent: @u @u @m +u = ; (32) @t @x @x @u @m @m + m = : +u (33) @t @x @x This has one time constant to be discussed below and displays a simple elastic response that is advected with the )ow. In the practical case of a viscoelastic )uid running down a wire [45], would be an e0ective elongational kinematic viscosity. The turbulence arising from the action of a forcing function like Eq. (26) on the highly nonlinear equations (32) and (33) may be termed as Maxwell–Burgers turbulence. Eqs. (30), (32) and (33) yield an expression for a quasi-stationary shock in the turbulent background (m = m [u(x)]) @2 us @us 1 @2 u 3 = 2 − 2s ; us (34) @x @x 3 @x
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with boundary conditions us → −uA us → uA
(x → ∞) ; (x → −∞) :
(35)
Eq. (34) displays the Hadamard-like instability associated with the change of type from elliptic to hyperbolic [4,66] as the velocity increases. The solution to Eq. (34) is )w + (1 − )) arctanh(w) = − s ;
(36) uA2 =ve2 ,
where in terms of the dimensionless variables w ≡ us =uA , s ≡ uA x=(2), and ) ≡ ve ≡ (=)1=2 is a characteristic velocity of the longitudinal elastic waves. There is no solution of Eq. (34) for ) ¿ 1. Hence, the viscoelastic shocks are cut o0 at an upper velocity ve . In order to analyze the width of a shock, we Taylor expand arctanh(w) as w tends to unity. Thus, a characteristic velocity w∗ separating two regimes may be introduced 2 ; (37) w∗ ≡ 1 − 2 exp )−1 and the velocity in the respective regimes is given by w −s w 1 − 2 exp
2|s| )−1
(w 6 w∗ ) ; (w ¿ w∗ ) ; (s ¡ 0; ) ¡ 1) :
(38)
We conclude that the shock width is never greater than that in the Newtonian case () = 0). Thus, the original supposition behind the superposition Eq. (29) is self-consistent. The rate of dissipation is inertia-like (Eq. (31)) but this holds up to a scale re = ve3 = :
(39)
At velocities higher than ve , there is a regime where only elasticity perturbs inertia, if at all. We write the dissipation via dimensional analysis as 1=3 1=3 u = (r ) h r 2=3 2 re 2=3 = (r )1=3 h (r ¿ re ) : (40) r The smallest width of a shock is smaller than or equal to the Kolmogorov scale K = (3 = )1=4 . Accordingly, at high Reynolds numbers we have 2 = 6 (K =re )4=3 and so 2 =1. Therefore, the dissipation rate in this regime is virtually una0ected by elasticity and is hence inertia-like. At this stage, it is well to compare our one-dimensional )uid with the two- and three-dimensional polymer suspensions discussed in the previous section. The relaxation time is not precisely the Zimm time but rather z =(cR3 ) ;
(41)
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at least if the polymer solution is dilute. This is required if the elastic waves are to propagate at the same velocity in all three cases (see Eqs. (6), (15) and (16)) To summarize, in Maxwell–Burgers turbulence driven by a force given by Eq. (26), the cascade is Kolmogorov-like, both for the shocks and the random background at scales larger than the dissipative cut-o0. Viscoelasticity in)uences neither at high Reynolds numbers. Therefore, we may speak of passive advection so one may introduce r∗ again as given by Eq. (21). In addition, we are led to posit a polymer elongation (in a one-dimensional polymeric )uid )owing down a wire and assumed to be well-represented by Eqs. (32) and(33)): r n1 ∗ : (42) r The exponent n1 probably di0ers from n2 and n3 . Again it would be possible to introduce a lower cut-o0 r∗∗ within both Kolmogorov cascades. This cut-o0 is the same as that conjectured by Tabor and the Gennes [5,6] but with n1 instead of n3 . This second e0ect due to elasticity is discussed below. The rate of dissipation within the shocks is written as
=
3 uA; i
3r
i
(uA; i ¡ ve ) :
(43)
This di0ers from that in the Newtonian )uid studied by Chekhlov and Yakhot who cut o0 their shock cascade by an upper velocity depending on the system size L [48,49] Umax ( L)1=3 :
(44)
We disregard a logarithmic factor. Chekhlov and Yakhot present a semiquantitative analysis of the probability functions for the velocity [49] in terms of Umax so we simply follow their argumentation by replacing Umax by ve . If we introduce the probability Pr (u) for the shock amplitude u within some interval 2r, we may write the continuum version of Eq. (43) in terms of the identity
ve Lve : (45) du u3 Pr (u) r ln =L The lower cut-o0 pertains to the weakest shock; according to Eq. (37) the kinematic viscosity remains relevant in spite of the elasticity. The right-hand side of Eq. (45) is computed from the rate of energy dissipation via Eq. (32) with an additional force f added to the momentum balance whose statistical properties are given by Eq. (26). We have shown above that both shocks and random background have statistical properties that are inertia-like. Hence, the end result is the analog of that derived in Ref. [49]. Eq. (45) yields the probability function for the shocks
r (u ¡ ve ) ; u4 0 (u ¿ ve ) :
Pr (u)
(46)
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Of interest are the correlation functions S‘ (r) ≡ (u(x + r) − u(x))‘ ; = (Tu)‘ :
(47)
The probability function for Tu is asymmetric in view of the shock structures given by the boundary equations (35). This asymmetry leads to a peculiar dependence of S‘ on ‘ [49] 0 6 ‘ 6 3: ‘ ¿ 3:
S‘ (r) ( r)‘=3 ;
rve‘−3 ‘−3
(ve ¿ ( r)1=3 ) ;
( r)‘=3
(ve ¡ ( r)1=3 ) :
S‘ (r)
(48)
(49)
The structure is Kolmogorov-like except at small r and for ‘ ¿ 3. Shocks dominate the )uid structure for ‘ ¿ 3. For fairly large negative Tu, the probability Pr (Tu) scales in the same way as Eq. (46). For 0 6 ‘ 6 3, we see only the signature of the Kolmogorov background. In the Newtonian case, Eqs. (46) – (48) (with Umax instead of ve ) agree well with the data from computer simulations [48,49]. Note that for the non-Newtonian )uid, Eq. (49) exhibits two regimes because the elastic velocity is less than the maximum cut-o0 (Eq. (44)). In conclusion, in this section we have discussed two routes to the e0ect of elasticity on one-dimensional turbulence. With the help of a crude rheological model (Eq. (33)), we have argued for the existence of a velocity cut-o0 in the correlation functions (Eqs. (48) and (49)). But Eq. (33) is too primitive to allow for passive advection of the chains leading to their elongation described by Eq. (42). The latter e0ect would implicate elasticity via a lower elastic cut-o0 r∗∗ . Nevertheless, it is possible to reconcile both points of view, at least qualitatively, because, fortunately, in both models there are inertial regimes that are Kolmogorov-like. We could introduce a (complicated) rheological model in Eq. (27) which implicitly accounts for passive advection as expressed by Eq. (42). Shocks then occur within the stationary turbulent )ow caused by the stochastic forcing, but they simply obey an inertial law given by Eq. (31). On the whole, the assumption of chain elongation by passive advection remains self-consistent down to the cut-o0 r∗∗ , provided r∗∗ ¡ re , the latter scale being de1ned by Eq. (39). We do not address the precise elastohydrodynamics at scales less than r∗∗ although an expression like Eq. (34) for the viscoelastic shocks remains plausible at scales greater than r∗∗ . Therefore, the structure functions should still be given by Eqs. (48) and (49) but cut o0 at lower scales by r∗∗ . 4. Concluding remarks The present analysis is entirely based on the implications for turbulent drag arising from elasticity alone. Any change in the elongational viscosity, for instance, is totally
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disregarded. Recent experiments [71] and simulations [72] concerning DNA in elongational )ow show that a simple coil–stretch transition does not exist. There is a host of complicated elongated shapes as a function of the Deborah number. Thus, primitive scaling forms like Eqs. (11), (13), (22) and (42) would seem to be qualitatively useful for the moment. Is it possible to see the elastic cut-o0 (Eq. (24)) in a quasi-two-dimensional experiment? In a recent set-up involving strati1ed layers [73], a )exible polymer did not seem to in)uence the turbulent )ow but the Reynolds number was not exorbitant. Vortex streets in )owing soap 1lms at intermediate Reynolds numbers are markedly perturbed by polymer at rather small concentrations [74]. Flowing soap 1lms have, nevertheless, been shown to display turbulence at quite high Reynolds numbers (see Ref. [24]; Rei 104 at injection). Moreover, if we focus on a long semi)exible polymer like double-stranded DNA, much longer Zimm times are possible. More concretely, let us introduce the following dimensionless numbers: Deborah number De ≡ 1=3 z , coil volume fraction ’ ≡ cR3 , injection Reynolds number Rei ≡ ui =(ki ), and the variable V ≡ ’0 =(z ui2 ). Using Eqs. (6), (15), (16), (19), (21) and (24), we can see whether the polymer would be elongated by both the energy and enstrophy cascades (Case III). The chain relaxation time must be long enough: De ¿ 1 : The elastic cut-o0 must be positioned within the enstrophy cascade: r∗∗ ki ¡ 1 or VDe3n2 ¡ 1 : The elastic cut-o0 must be larger than the Kraichnan dissipation scale: Rei VDe3n2 ¿ 1 : For example, 5 m long DNA exhibiting a weak excluded-volume e0ect would have a Zimm time of 0:1 s within a soap 1lm having a kinematic viscosity of 0:02 cm2 =s. The 1lm could be thick enough for the DNA coil to move quite freely. The solution is dilute (’ = 0:1 say). If we set Rei = 104 , ui = 100 cm=s, = 107 s−3 and n2 n3 2=3, say, we obtain V = 2 × 10−6 and De 20. Therefore, the inequalities above could be easily ful1lled so the elastic cut-o0 should be discernible. In the quasi-one-dimensional case, a 0:1 g=l polyethylene oxide suspension in water would have an elastic speed in the range of cm=s [4]. These velocities can be monitored in wire experiments like those of Kellay and Rouch [46]. Hence, the proposal for the velocity correlations being cut o0 by elasticity can be checked. Acknowledgements I would like to thank Dick Bedeaux for his friendship and loyalty dating back to his practical course in “Quantum Statistical Mechanics” at the Lorentz Institute, University of Leiden in 1975. He was kind enough to think seriously about my proposal for a
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new polyelectrolyte theory in 1976, at a time when I dearly needed such support. In the past 25 years, we have witnessed a lot of ups and downs together.
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