We interpret measurements of the Reynolds number dependence of the torque in
Taylor-Couette flow by Lewis and Swinney [Phys. Rev. E 59, 5457 (1999)] and of
the pressure drop in pipe flow by Smits and Zagarola, [Phys. Fluids 10, 1045
(1998)] within the scaling theory of Grossmann and Lohse [J. Fluid Mech. 407,
27 (2000)], developed in the context of thermal convection. The main idea is to
split the energy dissipation into contributions from a boundary layer and the
turbulent bulk. This ansatz can account for the observed scaling in both cases
if it is assumed that the internal wind velocity $U_w$ introduced through the
rotational or pressure forcing is related to the the external (imposed)
velocity U, by $U_w/U \propto Re^\xi$ with xi = -0.051 and xi = -0.041 for the
Taylor-Couette (U inner cylinder velocity) and pipe flow (U mean flow velocity)
case, respectively. In contrast to the Rayleigh-Benard case the scaling
exponents cannot (yet) be derived from the dynamical equations.Comment: 7 pages, 4 ps figures with 4 program files included in the source.
  European Physical Journal B, accepte

Eckhardt, Bruno

Grossmann, Siegfried

Lohse, Detlef

English

arXiv

We interpret measurements of the Reynolds number dependence
of the torque in Taylor-Couette flow by 
Lewis and Swinney [Phys. Rev. E 59, 5457 (1999)] and 
of the pressure drop in pipe flow
by Smits and Zagarola [Phys. Fluids 10, 1045 (1998)]
within the scaling theory of Grossmann and Lohse
[J. Fluid Mech. 407, 27 (2000)], developed in the context of 
thermal convection. The main idea is to split the energy dissipation
into contributions from a boundary layer and the turbulent bulk. 
This ansatz can account for the observed scaling in both cases
if it is assumed that the internal wind velocity Uw introduced
through the rotational or pressure forcing
is related to the 
external (imposed)
velocity U, by $U_{\rm w}/U \propto Re^\xi$ with 
$\xi = -0.051$ and 
$\xi = -0.041$ for the Taylor-Couette (U inner cylinder velocity) and 
pipe flow (U mean flow velocity) case, respectively.
In contrast to the Rayleigh-Bénard case the
scaling exponents cannot (yet) be derived from the dynamical equations

B. Eckhardt

S. Grossmann

D. Lohse

EDP Sciences OAI-PMH repository (1.2.0)

Scaling of global momentum transport in Taylor-Couette  and pipe flow

Crossref

Scaling of global momentum transport in Taylor-Couette and pipe flow

A misprint appeared in the volume number on page 541, within the heading inserted by the production
department:
 
Eur. Phys. J. B 8, 541-544 (2000)   should read   Eur. Phys. J. B 18, 541-544 (2000)

We interpret measurements of the Reynolds number dependence of the torque in Taylor-Couette flow by Lewis and Swinney [Phys. Rev. E 59, 5457 (1999)] and of the pressure drop in pipe flow by Smits and Zagarola [Phys. Fluids 10, 1045 (1998)] within the scaling theory of Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)], developed in the context of thermal convection. The main idea is to split the energy dissipation into contributions from a boundary layer and the turbulent bulk. This ansatz can account for the observed scaling in both cases if it is assumed that the internal wind velocity introduced through the rotational or pressure forcing is related to the external (imposed) velocity U, by with and for the Taylor-Couette (U inner cylinder velocity) and pipe flow (U mean flow velocity) case, respectively. In contrast to the Rayleigh-Bénard case the scaling exponents cannot (yet) be derived from the dynamical equations

Eckhardt, B.

Grossmann, S.

University of Twente Research Information

Eur. Phys. J. B 18, 541–544 (2000) THE EUROPEANPHYSICAL JOURNAL Bc©EDP SciencesSocietà Italiana di FisicaSpringer-Verlag 2000This article was initially published with the erroneous reference “Eur. Phys. J. B 8, 541-544”.This web edition lists the correct bibliographic reference.Scaling of global momentum transport in Taylor-Couetteand pipe flowB. Eckhardt1, S. Grossmann1, and D. Lohse2,a1 Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, 35032 Marburg, Germany2 Department of Applied Physics and J.M. Burgers Centre for Fluid Dynamics, University of Twente,7500 AE Enschede, The NetherlandsReceived 9 September 2000Abstract. We interpret measurements of the Reynolds number dependence of the torque in Taylor-Couetteflow by Lewis and Swinney [Phys. Rev. E 59, 5457 (1999)] and of the pressure drop in pipe flow by Smitsand Zagarola [Phys. Fluids 10, 1045 (1998)] within the scaling theory of Grossmann and Lohse [J. FluidMech. 407, 27 (2000)], developed in the context of thermal convection. The main idea is to split the energydissipation into contributions from a boundary layer and the turbulent bulk. This ansatz can account forthe observed scaling in both cases if it is assumed that the internal wind velocity Uw introduced throughthe rotational or pressure forcing is related to the external (imposed) velocity U , by Uw/U ∝ Reξ withξ = −0.051 and ξ = −0.041 for the Taylor-Couette (U inner cylinder velocity) and pipe flow (U mean flowvelocity) case, respectively. In contrast to the Rayleigh-Bénard case the scaling exponents cannot (yet) bederived from the dynamical equations.PACS. 47.27.-i Turbulent flows, convection, and heat transfer1 IntroductionThe relation between global flow properties and drivingforces is interesting from a fundamental point of viewand for upscaling from laboratory experiments to appli-cations. Examples are the change in mean flow througha pipe as a function of pressure drop, the dependenceof the heat transport as a function of temperature dif-ference (Rayleigh-Benard (RB) flow), and the increase intorque required to maintain a certain rotation speed in aTaylor-Couette (TC) system. In all three systems the ef-fects of the boundary layers are of prime importance butthe way in which they are dealt with differs. For pipe flowthe boundary effects are usually discussed in terms of thePrandtl-van Karman theory [1] which assumes a logarith-mic law for the profile1. For the connection between meanflow and pressure drop it predicts an implicit logarithmicrelationship, the so called skin friction law [1], in reason-able agreement with the experimental data.On the other hand, Rayleigh-Bénard (RB) convectionhas mainly been discussed in terms of algebraic relations:In the last decade the power law relation Nu ∼ Ra2/7 be-tween the Nusselt number Nu and the Rayleigh numberRa was thought to be an appropriate description of theexperimental data [3,4]. Recently, it has turned out thata e-mail: lohse@tn.utwente.nl1 Barenblatt and coworkers however also succeeded to de-scribe the data in terms of power laws [2].the dependences are more involved [5–7]. On the theoret-ical side, the analysis by Grossmann and Lohse [8] of thedifferent dominant dissipation mechanisms leads to a de-tailed phase diagram for Rayleigh-Bénard convection thatis in good agreement with the latest [6,7] and older exper-iments. In this theory the relations between Nu and Raare again algebraic.For the Taylor-Couette system with rotating innercylinder and resting outer one a description of the rela-tion between the dimensionless torqueG and the Reynoldsnumber Re both in terms of a skin friction law and a purepower law has been tried [9,10]. The dimensionless torqueG = T/ρν2L and the Reynolds number Re = Ωa(b−a)/νare defined with T the torque, ρ the fluid density and ν itskinematic viscosity, L the length of the cylinders, Ω theangular rotation rate, and b and a the radii of the outerand inner cylinder, respectively. Lewis and Swinney’s anal-ysis [9] of their experimental data clearly shows that a purepower law G ∼ Reα with α = 5/3 as suggested in refer-ences [11,12] does not describe the data. A description interms of the skin friction law is in better agreement withthe data. However, it still is not fully satisfactory, either,as the systematic drifts in their Figure 1 show.It is our aim here to adopt Grossmann and Lohse’sRayleigh-Bénard theory to Taylor-Couette and pipe flow.In contrast to the RB case, it is presently not possi-ble to derive the scaling exponents fully from dynamicalequations. Instead, there will be one exponent that has542 The European Physical Journal B0 200 400 600 800Re1/2+ξ/2050100150G/Re3/2+5ξ/2(a)104105106Re−0.02−0.010.000.010.02relative error(b)Fig. 1. (a) Compensated plot G/Re3/2+5ξ/2 vs. Re1/2+ξ/2 with ξ = −0.051. The points are Lewis and Swinney’s data [9], theline the fit (8). Re varies in the range 104 through 9 × 105. (b) Relative error (G − Gfit)/G of the suggested combination ofpower laws (8) with ξ = −0.051 (open boxes) and relative error (f − ffit)/f of the friction law fit (11) (filled circles).to be and can be fitted consistently to data for bothsystems.We begin in Section 2 with a discussion of the theoryand the comparison to Lewis and Swinney’s [9] experimen-tal data for TC flow. In Section 3 we will adopt it to pipeflow for which precise high Re data on the pressure dropwere obtained by Smits’s group [13]. In Section 4 we com-pare the quality of the data fit of this theory with thatof the standard skin friction law theory. Section 5 givesconclusions.2 Taylor-Couette flowThe basic idea [8] behind the analysis of the thermal con-vection experiments is the splitting of the energy dissipa-tion ε into contributions from the boundary layer (BL)and the bulk,ε = εBL + εbulk. (1)For the energy dissipation ε in TC flow one strictly has [9]ε =ν2GΩ2π(b2 − a2) · (2)This can be derived as usual by considering the energybalance in a Navier-Stokes flow. In analogy to reference [8](see there for an extensive discussion) εbulk is estimated asεbulk ∼U3wb− a · (3)Here, Uw is the typical velocity difference between theturbulent and the laminar (linear) profile. It is a mea-sure for the turbulent activity induced by the rotationof the inner cylinder and it defines a Reynolds numberRew = Uw(b − a)/ν. In the laminar case Uw = Rew = 0,and therefore εbulk = 0. Obviously, Uw must not be con-fused with the velocity U = 2πaΩ of the inner cylinder orthe corresponding Reynolds number Re = Ωa(b − a)/ν.The Reynolds number Re (or U) is imposed on the flowwhereas Rew (or Uw) is the response of the system. Thesituation can be compared with RB convection where theRayleigh number is imposed on the cell whereas the re-sponse of the system is the large scale wind of turbulence,which again defines a wind Reynolds number Rew. There-fore, in analogy, also here we call Uw the wind velocity.It is this wind velocity which leads to the formation ofa boundary layer of thickness λu. As in reference [8] weassume the BL to be of Blasius type [1],λu ∼ (b− a)/√Rew. (4)For very small Rew the BL will of course not diverge butsaturate at a scale λu ∼ (b−a) which introduces differentscaling relations [14], but in the present work we are notinterested in this very low Reynolds number regime.With its thickness λu as the relevant length scale weestimate the energy dissipation in the BL as [8]εBL ∼ νU2wλ2uλub− a · (5)Putting equations (2,5) together one obtainsGRe = c1Re5/2w + c2Re3w, (6)where c1 and c2 are two unknown constants. The first termis the BL contribution, the second one the bulk contribu-tion.The central question now is: How does Rew depend onRe? We do not know, but for large enough Re it seemsreasonable to assume a power law dependence,RewRe∼ UwU∼ Reξ. (7)B. Eckhardt et al.: Title Suppressed Due to Excessive Length 5430 1000 2000 3000 4000 5000Re1/2+ξ/205101520p’/Re3/2+5ξ/2(a)104105106107108Re−0.03−0.02−0.010.000.010.020.03relative error(b)Fig. 2. (a) Compensated plot p′/Re3/2+5ξ/2 vs. Re1/2+ξ/2 with ξ = −0.041. The points are data from reference [13], the linethe fit (10). Re here varies between 3.16 × 104 and 3.53 × 107. (b) Relative error (p′ − p′fit)/p′ of the suggested combination ofpower laws (10) with ξ = −0.041 (open boxes) and relative error (f − ffit)/f of the friction law fit (11) (filled circles).Therefore,G = c1Re3/2+5ξ/2 + c2Re2+3ξ. (8)We perform a nonlinear fit of Lewis and Swinney’s data [9]to equation (8), obtaining ξ = −0.051, c1 = 10.5, andc2 = 0.196. The best way to check the quality of the fit(8) is to plot G/Re3/2+5ξ/2 vs. Re1/2+ξ/2 so that accordingto equation (8) a straight line should result. This indeedis the case, as shown in Figure 1a. The quality of the fitis underlined by the relative error shown in Figure 1b.3 Pipe flowConsider now pressure driven flow through a pipe of radiusR. The pressure gradient and the energy dissipation ε pervolume are related byε =∆pρux (9)where ux =: U is the average of the x-velocity over thecross section of the pipe, which defines the Reynolds num-ber Re = 2RU/ν. As for the TC flow we define the windvelocity Uw as the maximal difference between the tur-bulent mean velocity profile and the laminar (parabolic)one. This maximum will occur close to the walls. Uw againdefines a wind Reynolds number Rew = 2RUw/ν.We split the energy dissipation in a boundary layerpart and a bulk part as in equation (1) and estimate bothcontributions as above, equations (3,5), with b−a replacedby R, and the thickness of the Blasius boundary layer be-ing λu ∼ R/√Rew. With this the equation for the dimen-sionless pressure drop p′ = ∆pR3/(ρν2) becomesp′ = c′1Re3/2+5ξ/2 + c′2Re2+3ξ, (10)where c′1 and c′2 are two unknown constants and ξ thepower law exponent of equation (7). Equation (10) is theanalog of equation (8) in the TC case.In order to test equation (10) we consider the highprecision pressure drop data of Smits and Zagarola [13].A nonlinear fit then results in ξ = −0.041, c′1 = 0.226,and c′2 = 0.00373. In Figure 2a we show p′/Re3/2+5ξ/2 vs.Re1/2+ξ/2. If equation (10) holds, a straight line shouldresult which is the case. Again the quality of the fit isdemonstrated with the relative errors in Figure 2b.4 The skin friction lawFinally we would like to compare our description of thedata with the standard skin friction law [1]. Define thefriction coefficient f = G/Re2 for TC flow and f = p′/Re2for pipe flow. One then has, in both cases [1,9,10],1√f= c′′1 lg(Re√f) + c′′2 (11)with two flow dependent constants c′′1 and c′′2 which canbe connected to the von Karman constant [9]. Employingequation (11) one can indeed fit both data sets reasonably.We do not show the fits as they have already been shownelsewhere (see Fig. 4a of Ref. [9] for the TC case), but wepresent the relative error (f −ffit)/f in Figures 1a and 2band compare it with the relative errors of the combinedpower laws equations (8) and (10), respectively. For bothTC and pipe flow the relative error of the friction law isroughly of the same order as that of the fits (8) and (10),respectively, maybe somewhat larger.544 The European Physical Journal B5 ConclusionsThe preceding analysis shows that the splitting of thedissipation into a bulk and a boundary layer contribu-tion as used in the Rayleigh-Bénard theory can also beused to describe the Taylor-Couette flow [9] and the pipeflow [13] data. However, in contrast to the RB case, onlyone global balance equation is available, the one for the en-ergy dissipation. Therefore, it is not possible to derive theasymptotic scaling exponents for both the wind Reynoldsnumber and the dimensionless torque (dimensionless pres-sure drop) in the TC case (pipe case). Instead, one scal-ing exponent (ξ) must be fitted to the data. The ratioUw/U ∼ Reξ scales similarly in both cases, ξ = −0.051and ξ = −0.041 for TC and pipe flow, respectively. Therelative error in both cases is less than one percent andwithin this precision the data indicate really different ex-ponents. The origin of this difference is unclear.For shear flow, the strict upper bound for energy dis-sipation is cε = εL/U3 ≤ 0.01087 for Re → ∞ [15,16].All the Couette experiments clearly lie below this bound,and even show a trend towards a scaling that is slowerthan U3. When assuming the Kolmogorov length scaleas smallest length scale on which dissipation contributes,Nicodemus et al. [17] could numerically show that aroundRe = 105−106 one has cε ∼ Re−0.08. Kerswell assumes thesame cutoff and a certain background flow profile and findscε ∼ Re−1/7 for Re → ∞ [18]. Interestingly enough, theexponents 3ξ = −0.12 (TC) and 3ξ = −0.15 (pipe) thatwe find from the experimental data are very close to thatvalue. However, the scaling Uw/U ∼ Reξ with negative ξimplies that the wind does not increase as rapidly as theexternal velocity with Re and that according to our defini-tion of Uw the relative difference between the laminar andthe mean turbulent velocity profiles vanishes. Given thesmallness of ξ the Reynolds numbers at which this couldbecome significant are not experimentally accessible. Butthe situation remains unsatisfactory and it clearly wouldbe highly desirable to calculate this exponent more rigor-ously from the Navier-Stokes equations and to understandthe relation to the mean flow profile better.The authors thank Ch. Doering and K. R. Sreenivasan forvery helpful discussions and H. Swinney, G. Lewis, A.J. Smits,and M.V. Zagarola for supplying us with their experimentaldata. The work is part of the research program of the Sticht-ing voor Fundamenteel Onderzoek der Materie (FOM), whichis financially supported by the Nederlandse Organisatie voorWetenschappelijk Onderzoek (NWO). This research was alsosupported by the German-Israeli Foundation (GIF), by the Eu-ropean Union (EU) under contract HPRN-CT-2000-00162, andin part by the National Science Foundation (NSF) under GrantNo. PHY94-07194. We thank the members of the Institute forTheoretical Physics in Santa Barabara for their hospitality.References1. L.D. Landau, E.M. Lifshitz, Fluid Mechanics (PergamonPress, Oxford, 1987).2. G.J. Barenblatt, J. Fluid Mech. 248, 513 (1993); G.J.Barenblatt, A. Chorin, V.M. Prostokishin, Appl. Mech.Rev. 50, 413 (1997).3. B. Castaing et al., J. Fluid Mech. 204, 1 (1989).4. E.D. Siggia, Annu. Rev. Fluid Mech. 26, 137 (1994).5. X. Chavanne et al., Phys. Rev. Lett. 79, 3648 (1997).6. J. Niemela, L. Skrebek, K.R. Sreenivasan, R. Donelly, Na-ture 404, 837 (2000).7. X. Xu, K.M.S. Bajaj, G. Ahlers, Phys. Rev. Lett. 84, 4357(2000).8. S. Grossmann, D. Lohse, J. Fluid. Mech. 407, 27 (2000).9. G.S. Lewis, H.L. Swinney, Phys. Rev. E 59, 5457 (1999).10. D.P. Lathrop, J. Fineberg, H.S. Swinney, Phys. Rev. A 46,6390 (1992).11. A. Barcilon, J. Brindley, J. Fluid Mech. 143, 429 (1984).12. G.P. King, Y. Li, H.L. Swinney, P.S. Marcus, J. Fluid.Mech 141, 365 (1984).13. A.J. Smits, M.V. Zagarola, Phys. Fluids 10, 1045 (1998).14. S. Grossmann, D. Lohse (preprint, 2000).15. F.H. Busse, Adv. Appl. Mech. 18, 77 (1978).16. R. Nicodemus, S. Grossmann, M. Holthaus, Phys. Rev.Lett. 79, 4170 (1997); Phys. Rev. E 56, 6774 (1997); J.Fluid Mech. 363, 281 (1998); J. Fluid Mech. 363, 301(1998).17. R. Nicodemus, S. Grossmann, M. Holthaus, Eur. Phys. J.B 10, 385 (1999).18. R.R. Kerswell (preprint, 2000).

Scaling of global momentum transport in Taylor-Couette and pipe flow

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