On the properties of steady states in turbulent axisymmetric flows

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by  R. Monchaux

On the properties of steady states in turbulent axisymmetric flows
∗
R. Monchaux, F. Ravelet, B. Dubrulle, A. Chiffaudel, and F. Daviaud
Service de Physique de l’E´tat Condens´e, DSM, CEA Saclay, CNRS URA 2464, 91191 Gif-sur-Yvette, France
(Dated: February 6, 2008)
6 Weexperimentally studytheproperties of mean and most probablevelocity fieldsin a turbulent
0 von K´arm´an flow. These fields are found to be described by two families of functions, as predicted
0
by a recent statistical mechanics study of 3D axisymmetric flows. We show that these functions
2
depend on the viscosity and on the forcing. Furthermore, when the Reynolds numberis increased,

n weexhibitatendencyforBeltramization oftheflow,i.e. avelocity-vorticityalignment. Thisresult
a provides a first experimental evidence of nonlinearity depletion in non-homogeneous non-isotropic
J turbulentflow.

5
2 PACSnumbers: 47.27.eb,47.32.Ef,05.70.Ln

]
h Introduction Ayetunansweredquestioninstatistical similartaskhasbeenundertakenbyLeprovostetal. [13].
c physics is whether stationary out-of-equilibrium systems In the ideal case —force free, inviscid— they proved the
e
m shareanyresemblancewithclassicalequilibriumsystems. existenceofaninfinitenumberofconservedquantities,in
A good paradigm to explore this question is offered by additiontotheenergyandthehelicity. Theyalsoshowed
-
t turbulent flows. Incompressible flows subject to statis- the existence of an infinite number of equilibrium states,
a
tically stationary forcing generally reach a steady state, depending on the conserved quantities. Among them,
t
s in a statistical sense, independent of the initial condi- a Beltrami state is obtained, when only the energy and
.
t tions. Most of the current turbulence modeling is de- the helicity are conserved. They postulated that even in
a
m voted to the understanding of this state, and the role thepresenceofsmallbutfinite viscosityandforcing[14],
of the velocity fluctuations or velocity smallest scales in this featureremainsvalid,with selectionofsteady states
-
d its construction. A distinguished feature of stationary through boundary conditions and forcing.
n largeReynoldsnumber turbulentflowsis the presenceof The purpose ofthe presentLetter is to performanex-
o
coherent structures, under the shape of vortices in 2D perimental check of these findings. We find that mean
c
[ [1, 2], or vorticity thin tubes in 3D [3]. These structures and most probable velocity fields of axisymmetric flows

correspond to regions where vorticity is locally almost canindeed be characterizedby two families of functions,
1
aligned with velocity —phenomenon referred to as Bel- as predicted in Ref. [13]. These functions depend on the
v
2 tramization [4, 5]. This tendency to velocity-vorticity viscosity and forcing. As the Reynolds number is in-
8 alignment induces depletion of nonlinearities in Navier- creased,theyevolvetowardsthefunctionscorresponding
5 Stokesequations. Interestinglyenough,similardepletion to a Beltrami state, providing a first evidence of non-
1
of nonlinearity is also observed in the inviscid counter- linearity depletion in a non-homogeneous, non-isotropic
0
part of the Navier-Stokes equation —the so-called Euler system.
6
0 equation— resulting in a slowing down of the vorticity Theoretical background and definitions Consider an
/ blow-up with respect to rigorous estimates [6]. A theo- incompressible axisymmetric flow, with velocity compo-
t
a reticalquestionofinterestisthereforewhetherturbulent nents in a cylindrical referential (vr, vθ, vz). Due to in-
m flows have a natural tendency for nonlinearity depletion, compressibility and axisymmetry,only two functions are
- and whether this is a characteristic of the steady states. sufficient to describe the flow, namely the angular mo-
d
The answer to the first issue is ambiguous. On one mentum —σ(r,z) = rvθ— and either, the stream func-
n
o hand, a tendency for Beltramization has indeed been tion Ψ(r,z), such that (vr,vz) = −∇×Ψeθ, or the az-
c observed in some numerical simulations of stationary, imuthal vorticity ωθ(r,z). Using a variational method,
: nearly isotropic turbulence [7, 8]. On the other hand, Leprovost et al. showed that the steady states solution
v
i similar study performedonmore generalflows —suchas of axisymmetric Euler equations, in the force-free case
X
boundary layer [9]— provided little evidence of such a obey:
r property .. A similar conclusion has been reached using ′
a FF
experimentaldata[10],withslightlylessreliabilityowing σ =F(Ψ); ξ− r2 =G(Ψ); with ξ =ωθ/r, (1)
to the difficulty to measure vorticity. As for the second
where F and G are arbitrary functions linked with con-
issue, it has only been partially explored in special ge-
servationlawsofthesystem—Casimirsofσ, generalized
ometries. In2D,equilibriumstatesoftheEulerequations
helicity [13]. For example, the steady state correspond-
havebeenclassifiedthroughstatisticalmechanicsprinci-
ing to conservation of energy and helicity is such that F
plebyRobertandhiscollaborators[11,12]. Theyindeed
islinearandG≡0,resultinginalinearrelationbetween
correspond to Beltrami solution. In 3D axisymmetric
vorticity and velocity (~v =λ~ω). This is a Beltrami flow,
flows—anintermediatesituationbetween2Dand3D—a
given by :
2
the flow of most probable velocities (fmpv)[16], obtained
πR r z by taking, at each point, the most probable value of the
vr = −HmJ1(mR)cos(πH) velocity. Thevr componentisoncemorederivedfromthe
λR r z two other components using continuity equation. Fig-
vθ = J1(m )sin(π ) (2) ure 1 compares the two flows computed from the same
m R H
r z data set: they have the same overall structure, but dif-
vz = J0(m )sin(π )
R H fer through the size of the middle shear layer, which is
where λ and m are free parameters and J0 and J1 are thinner in the case of the fmpv. This is due to the fact
first order Bessel functions. that the chaotic wandering of the shear layer around its
Considering further the thermodynamics of the sys- averagepositionaresignificantlylessprobableasonecon-
tem, Leprovost et al. proved that the equilibrium states siders positions further away from the equatorial plane.
atsomefixedcoarse-grainedscalearesuchthatthemost From a theoretical point of view, the two flows differ in
probable velocity flow (mpvf) is a stationary state of the the sense that the mean velocity is not a solution of the
Euler equation, i.e. satisfies Eq.(1). Finally, they pos- Navier-StokesorEulerequations—becauseofthefluctu-
tulated that these force-free, inviscid results can be ex- ations, which induce a Reynolds stress, especially in the
tended to the stationary states in the presence of both shear layer—, while the mpvf is a stationary solution of
viscosityandforcing,providingthefunctionF andGare the Euler equations. Therefore, one can expect theoriti-
selected through boundary conditions and forcing. cal predictions regarding the structure of the stationary
Experimental setup In order to check the theoretical state to be less accurate in the case of the mean flow, as
predictions, we have worked with a simple “axisymmet- the Reynolds number and the fluctuations increase.
ric”configuration: thevonK´arm´anflowgeneratedbytwo
.9
counter-rotating impellers in a cylindrical vessel. The
cylinder radius and height are respectively R=100 mm
z/R 0.5
and H = 180 mm (distance between inner faces of im-
pellers). The impellers consistof185mmdiameter disks
0 0
fitted with sixteen 20 mm high curved blades. More
details about the experimental setup can be found in −0.5
Ref. [15]. The impellers rotation frequencies are both
set equal to f to get exact counter-rotating regime. We −.9 −1
0 r/R 1 r/R 0
definetwoforcingassociatedwiththeconcave(resp. con- .9
vex) face of the blades going forward, denoted in the se-
quel by direction (−) (resp. (+) ). The working fluid is z/R 0.5
either water or glycerol at different dilution rates. The
resulting accessible Reynolds numbers (Re = 2πfR2ν−1 0 0
with ν the kinematic viscosity)vary from102 to 3×105.
−0.5
In the exact counter-rotating regime, whatever the im-
pellers, the flow is divided into two toric cells separated
−.9
by an azimuthal shear layer. This setup is invariant un- 0 r/R 1 r/R 0
der rotations of π (Rπ) around any radial axis passing FIG.1: VelocityprofileforRe=2.5×105,direction(−): col-
throughthecenterofthecylinder. Thetime-averagedve- ored contour for azimuthal component, arrow representation
locity fields we consider hereafter are Rπ-invariant and for poloidal part. Top: fmpv. Bottom: time-averaged flow.
axisymmetric[15]. Velocitymeasurementsaredonewith
a DANTEC Laser Doppler Velocimetry (LDV) system. We first compute the F function by looking at σ as a
Data processing The LDV data only provide the ax- functionofΨ,andthen, we use this estimate to evaluate
ial and azimuthal velocity components on a 170 points G, by looking at ξ2 = ξ− FrF2′ as a function of Ψ. This
grid covering half a meridian plane, through time series procedureislikelytoinducealotofnoiseintheestimate
of about 200,000 randomly sampled values at each grid of G. To check its robustness, we have tested it on a
point. From this time series, it is straightforward to get Beltrami flow (Eq. 2) with a superimposed level of noise
time-averaged axial and azimuthal velocities. The re- comparable to the level of fluctuations in the flow (see
maining radialcomponent of the mean velocity field can Figure 2). Even in the presence of noise the fits give the
thenbeobtainedusingtheincompressibilityandaxisym- correct shapes: F =λΨ and G≡0.
metry: this procedure has been later validated through After this test, we apply this procedure on the real
direct measurements of the radial velocity with a Parti- data, with a further correction,motivated by the follow-
cles Image Velocimetry (PIV) system on the same flow. ingremark. Figure3displaystwotypicalplotofF : one
We also use the time series to extract at each point his- overthe whole apparatus,andone overa50%portionof
togramsoftheaxialandazimuthalvelocity,andcompute the flow obtained after removingregionsclose to the im-
3
.4 σ 1 ξ / ξ similarly. As Reynolds number increases, the F cubic
2 max
curves collapse on the Beltrami line.
.2 .5
At Re = 2.5×105, it is not even possible to distin-
0 0 guish the F fit from a straight line and this for the two
directions of rotation. In the case of G, there is a differ-
.2 −.5
Ψ Ψ / Ψ ence between the two cases. For the fmpv, the best fit
−.4 −1 max is linear, with a slope smaller than the noise level, this
−.1 .05 0 .05 .1 −1 −.5 0 .5 1
FIG. 2: σ and ξ2 = ξ− FrF2′ versus Ψ for the Beltrami flow itshecosnhsaipsteenotf wGitihs Gver≡y d0i.ffeFreonrtt.heAltlimthee-afivetsrapgreedsecnatsea,
defined by equation (2) with a superimposed white noise of
plateau around zero. The greater the Reynolds number,
amplitude 60%. We have represented the F (resp. G) fit on
theleft (resp. right) figure. the wider this plateau, so the wider the range of Ψ —
or equivalently the volume of consideredflow—where G
pellers and along the vessel. These parts correspond to is very close to zero, i.e., very close to the Beltrami G
locations where the viscosity and the forcing (neglected function.
in Ref. [13]) are principally at work, and larger devia- The difference in the shape of G can be explained as
tions from theoretical predictions can be expected. One follows: accordingtoequation(1),G(Ψ)isthedifference
sees that while data on the whole vessel display signifi- of two terms. One linked to F is necessarily a cubic,
cantscatter,preventingthe outcomeofawell-definedF, while the second, connectedto the azimuthal vorticityis
the restricted data gather onto a cubic-shaped function almost a noise in this area of the flow. In the fmpv case,
fitted by a two parameters cubic: F(Ψ) = p1Ψ+p3Ψ3. the magnitude of the cubic term is small compared to
ThisfitisthenusedtoobtainG. Inthesequel,weobtain the vorticity, and we only see the noise coming from the
F and G through fits over this 50% portion of the flow vorticity. In the time-averaged case, the cubic is much
away from the boundaries. larger, so that it determines the behavior of G.
.5 σ σ .4 .8
F(Ψ) fmpv (−) (a) F(Ψ) fmpv (+) (b)
0
.2 .4
−.5 Ψ Ψ Ψ Ψ
−0.04 0 0.04 −0.04 0 0.04 0 0
.4
FIG. 3: σ vs Ψ for experimental time-averaged flow in direc- F(Ψ) time−av. (−) (c) .4 F(Ψ) time−av. (+) (d)
tion (+) at Re = 2100. Left: for the whole flow. Right: for
r≤0.81, −0.56≤z ≤0.56,correspondingto50%oftheflow
volume. The remaining points are clearly gathering along a
.2
cubic-shaped function F. .2
Results We present results computed for both time-
Ψ Ψ
averagedflowandfmpvforthetwodirectionsofrotation, 0 0
0 0.02 0.04 0 0.05 0.1
at different Reynolds numbers. Figure 4 presents the F
fits obtained in each case. We have included on each FIG.4: Ffit. (a)and(b): fmpvdirection(−)and(+)respec-
graph a straight line tangent to experimental curves for tively,(c) and (d): time-averaged field direction (−) and (+)
respectively. Legendforallfigures: Re=100(thin ···),Re=
Ψ = 0 as a reference to a Beltrami flow. The experi-
150(thin ·−·),Re=2100 (thin −−),Re=9100 (thin—),
mental data are much more scattered in the fmpv case, Re = 2.5×105 (thick —). The thin dotted straight line is
debasing the fit accuracy compared to the mean flow —
a Beltrami. As F is odd, we display it for positive values of
R2 ≃0.94 instead of 0.99. Ψ only. Note that the fit at Re = 9100 for fmpv case in di-
In figure 5 we present the G fits obtained in the same rection (+)could not beobtained from thedataduetolarge
conditions. The best fit of the fmpv G functions is lin- scattering.
earwhereascubic-shapedfunctions arerequiredfor time
averaged G. The plots of G consist of wide noisy bands A second observation is the dependence of F and G
surrounding the fits in both cases. The noise for G is on the forcing. Comparison of Fig. 4-(a,c) and 4-(b,d),
largerthan for F, since it is a resultof a two-stepproce- obtained for two different forcing, shows that the slope
dure including a spatial derivation. of F slightly depends on the forcing (p1 = 5 for (−)
Consideringthe variationwithReynoldsnumberofF, direction, p1 = 4 for (+) direction). For any forcing, G
both the fmpv case and the time-averaged case behave remainsclosetozeroatthemeasurementscale. Tocheck
4
Finally, we have shown that the evolution for increas-
1 G(Ψ) fmpv (−) (a) G(Ψ) fmpv (+) (b)
ing Reynolds number is towards a Beltrami state, with
.5 depletion of nonlinearities. This evolution is more ob-
vious when considering the fmpv, as expected from the
0
thermodynamics analysis developed by Leprovost et al..
−.5 To our knowledge,this is the firstexperimentalevidence
of nonlinearity depletion in a non-homogeneous, non-
−1 Ψ Ψ
isotropicturbulence,whereitis verychallengingtomea-
1 G(Ψ) time−av. (−) (c) G(Ψ) time−av. (+) (d) sure simultaneously all components of velocity and vor-
ticity. Ouruse ofthe functions characterizingthe steady
.5
states enables us to lower the experimental constraints
0 for such a check.
−.5
−1 Ψ Ψ F(Ψ) fmpv F(Ψ) time−av.
.2 .2
−0.05 0 0.05 −0.1 −0.05 0 0.05 0.1
FIG. 5: G fit. (a) and (b): fmpv direction (−) and (+) re-
spectively,candd: time-averagedfielddirection(−)and(+) .1 .1
respectively. Same legend as Fig. 4.
Ψ Ψ
thisdependencefurther,wehaveconductedanadditional 0 0
0 0.025 0.05 0 0.025 0.05
experimentathighReynoldsnumber,withmuchsmaller
impellers. As shown in Figure 6, we obtained a different FIG. 6: F fit for 100mm diameter impellers fitted with
straight blades. Left: fmpv, Right: time-averaged. Re =
shape for F, with a different inflection.
1.9×105 (···), Re=2.6×105 (−−),Re=5×105 (—).
Discussion and perspectives An inviscid force-free
theorydevelopedby Leprovostet al. predicteda charac-
terization of the steady state in axisymmetric turbulent
flows through two functions F and G. In our experi-
ments,wehaveconfirmedthis,andmeasuredthesefunc-
tions for different forcing, and different viscosities, using ∗ Electronic address: [email protected]
two different fields as diagnostic: the fmpv as predicted
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[5] M. Farge et al., Phys.Fluids 15, 2886 (2003).
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[10] J. M. Wallace, J.-L. Balint, and L. Ong, Phys. Fluids A
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This suggeststhat stationaryout-of-equilibriumsystems physics/0505084 to bepublished in PRE.
[14] Considering ν =0 and ν →0 limit is different.
like turbulent flows are universal in a weaker sense than
[15] F. Ravelet, L. Mari´e, A. Chiffaudel, and F. Daviaud,
in ordinary equilibrium systems : they can be described Phys. Rev.Lett. 93, 164501 (2004).
in a universal manner through general functions (like F
[16] A priori, fmpv and mpvf are different. Experimentation
and G) determining the “equation of state”. However, only gives access to fmpv. Nevertheless, both may be
these functions are non-universal since they depend on quite similar if the velocity pdf at each point shows a
the fine details of the system (dissipation and forcing). well defined peak.