Content uploaded by Jacques Magnaudet
Author content
All content in this area was uploaded by Jacques Magnaudet on Dec 20, 2016
Content may be subject to copyright.
Turbulence-induced secondary motion in a buoyancy-driven flow
in a circular pipe
Yannick Hallez and Jacques Magnaudet
INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Université de Toulouse,
Allée Camille Soula, F-31400 Toulouse, France and CNRS, IMFT, F-31400 Toulouse, France
共Received 13 June 2009; accepted 17 July 2009; published online 25 August 2009兲
We analyze the results of a direct numerical simulation of the turbulent buoyancy-driven flow that
sets in after two miscible fluids of slightly different densities have been initially superimposed in an
unstable configuration in an inclined circular pipe closed at both ends. In the central region located
midway between the end walls, where the flow is fully developed, the resulting mean flow is found
to exhibit nonzero secondary velocity components in the tube cross section. We present a detailed
analysis of the generation mechanism of this secondary flow which turns out to be due to the
combined effect of the lateral wall and the shear-induced anisotropy between the transverse
components of the turbulent velocity fluctuations. © 2009 American Institute of Physics.
关DOI: 10.1063/1.3213246兴
Since Prandtl’s group pioneering investigations,1high-
Reynolds-number incompressible flows in curved channels
or in straight noncircular ducts are known to involve second-
ary mean motions. Centrifugal or other nonconservative
body forces perpendicular to the main motion tend to skew
the primary spanwise vorticity in developing flows, leading
to transverse motions whose magnitude can be up to 20%–
30% of the streamwise velocity. Secondary flows typically
one order of magnitude smaller are also known to exist in
fully-developed turbulent flows in straight rectangular ducts.2
These weak secondary motions, known as Prandtl’s second-
ary flows of second kind, are due to the inhomogeneity of the
transverse Reynolds stresses near the corners of the duct and
result in a pair of counter-rotating streamwise vortices near
each corner. They have been the subject of several detailed
investigations, as they induce significant transport of mo-
mentum and heat within the duct cross section and also be-
cause their prediction represents a serious challenge for one-
point turbulence models.3During a joint experimental/
numerical investigation of buoyancy-induced turbulent
mixing in an inclined tube 共see Refs. 4and 5for more detail
on the experiments兲, we observed the existence of a second-
ary mean motion whose topology and sustaining mechanism
differ from those of the secondary flows reported so far,
since it arises in a straight circular pipe in presence of a
fully-developed primary flow. It is the purpose of this letter
to describe the structure of this, apparently undocumented,
secondary flow and elucidate its generating mechanism.
The physical configuration we consider is as follows. A
circular pipe of diameter dand length Lclosed at both ends
共x=⫾L/2兲is filled with two miscible 共i.e., with zero inter-
facial tension兲low-viscosity fluids of different densities 共
2
⬎
1兲in such a way that each fluid initially occupies half of
the pipe length, the two fluids being separated by a lock at
x=0. The pipe is tilted at an angle ⌰from the vertical in
such a way that the arrangement is unstable, owing to buoy-
ancy. After the lock is removed at time t= 0, the light 共heavy兲
fluid tends to flow along the upper 共lower兲part of the cross
section toward the upper 共lower兲end wall, forming a stably
stratified turbulent countercurrent flow where turbulent mix-
ing takes place. As far as the fronts of both currents remain
far from the end walls, this relative motion results in a
turbulent shear flow which, in the region 兩x兩ⰆL/2 on which
we focus in this letter, is statistically independent of both x
and t.
The results to be discussed below were obtained by solv-
ing the three-dimensional, time-dependent Navier–Stokes
equations for a variable-density incompressible flow 共with no
explicit reference to the Boussinesq approximation兲, together
with the density equation, assuming molecular diffusivity to
be negligibly small. The computations were carried out in a
176dlong circular pipe of diameter d, using a cylindrical
grid with 32⫻64⫻2816 grid points in the radial, azimuthal,
and streamwise directions, respectively. Details regarding the
computational technique and the validation of the code may
be found elsewhere6and are not repeated here. In the ex-
ample discussed below, we select a tilt angle ⌰=15°, an
Atwood number At=共
2−
1兲/共
2+
1兲=10−2 and a Reynolds
number Re=Vtd/
=790 with Vt=共At gd兲1/2,gdenoting
gravity and
being the kinematic viscosity which we assume
to be identical in both fluids. The 共x,t兲averaging procedure
was performed throughout a 9d-long window centered at x
=0, starting at time TI共At g/d兲1/2=177 and continued until
the final time of the computation, TF共At g/d兲1/2=554, so that
the averaged quantities 共hereinafter denoted with angle
brackets兲result from averaging over 180 000 time steps and
384 grid cells.
Let the x-axis be along the streamwise ascending direc-
tion, the z-axis be such that the 共x,z兲plane is vertical with
z⬎0 in the ascending direction, and the y-axis be horizontal
and such that the Cartesian 共x,y,z兲coordinate system is
right-handed with y=z=0 on the tube centerline. The aver-
aged velocities in the vertical diametrical plane y=0 are
shown in Fig. 1. The streamwise velocity 具U典exhibits the
expected S-shape with an almost constant shear except in the
PHYSICS OF FLUIDS 21, 081704 共2009兲
1070-6631/2009/21共8兲/081704/4/$25.00 © 2009 American Institute of Physics21, 081704-1
Downloaded 25 Aug 2009 to 193.48.203.101. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
vicinity of the wall. The puzzling feature of the flow is the
crosswise velocity 具W典which is seen to be nonzero even
though it is typically only 3%–5% of 具U典. This secondary
velocity is negative 共positive兲in the upper 共lower兲part of the
vertical midplane, indicating that the fluid converges toward
the center of the pipe for y⬇0. The complete spatial struc-
ture of the secondary velocity field is shown in Fig. 2, to-
gether with the so-called swirling strength ci which is com-
monly used as a criterion allowing the identification of
vortical structures in three-dimensional flows 共ci is defined
as the imaginary part of the conjugate eigenvalues of the
velocity gradient tensor7兲. This plot confirms the presence of
four stationary streamwise vortices within the pipe cross sec-
tion. Note that the secondary flow revealed by Fig. 2only
exhibits approximate left-right and top-bottom symmetries,
owing to the limited sampling time allowed by the computa-
tional run.
Obviously the nonzero secondary mean flow is directly
related to the existence of a nonzero mean component of the
streamwise vorticity ⍀共we omit the brackets for simplicity兲
whose magnitude 共not shown here兲is observed to be
O关共At g/d兲1/2兴, even though the mean secondary velocities
themselves are only a few percents of the primary velocity
具U典. We shall come back to this point later.
To elucidate the origin of the secondary mean flow, we
need to consider the balance equation for ⍀. Since the mean
flow is both stationary and homogeneous in the streamwise
direction, this equation reduces to
具V典
y⍀+具W典
z⍀=B+R+
共
yy
2+
zz
2兲⍀,共1兲
where
B=具共
y
zp−
z
yp兲/
2典,
共2兲
R=
yz
2共具v⬘2典−具w⬘2典兲 −共
yy
2−
zz
2兲具v⬘w⬘典,
p,
,v⬘, and w⬘standing for the pressure, density, and tur-
bulent velocity fluctuations along the y- and z-axes, respec-
tively. The first and last terms in Eq. 共1兲represent the trans-
port of ⍀by the mean secondary flow and its diffusion by
viscous effects, respectively, so that both of them remain
zero if the mean flow is parallel. Therefore the generation of
⍀is due to term Bresulting from baroclinic effects and to
term Rinduced by variations of the secondary Reynolds
stresses within the cross section.
The numerical data were used to evaluate the latter two
terms throughout the pipe cross section. Figure 3indicates
that both terms are concentrated near the wall and exhibit a
four-lobe structure which is antisymmetric with respect to
the horizontal and vertical axes. The baroclinic torque Bis
typically five times smaller than the turbulent term R.
Hence, while buoyancy effects drive the main flow, they are
not at the root of the secondary flow. The turbulent term R
appears to be the key of the generation of the streamwise
vorticity, suggesting that the present secondary flow is of
Prandtl’s second type. To elucidate the origin of the second-
ary flow, it is thus necessary to understand the spatial struc-
FIG. 1. Averaged velocity profiles in the vertical midplane. Bold line:
streamwise velocity; thin line: crosswise velocity. Lengths are normalized
by the pipe diameter dand velocities are normalized by Vt=共At gd兲1/2.
−
0.5
−
0.4
−
0.3
−
0.2
−
0.1 0 0.1 0.2 0.3 0.4 0.5
−
0.5
−
0.4
−
0.3
−
0.2
−
0.1
0
0.1
0.2
0.3
0.4
0
.5
0.15
0.2
0.25
0.3
0.35
0.4
FIG. 2. Mean secondary flow and streamwise vortices identified by the
magnitude of the swirling strength 共ci criterion兲. The normalization is simi-
lar to that of Fig. 1.
−
0
.4 −
0
.2
0 0
.2
0
.4
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−
0
.4 −
0
.2
0 0
.2
0
.4
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
FIG. 3. Map of the turbulent term R共left兲and baroclinic term B共right兲in
Eq. 共1兲, both normalized by 共Vt/d兲2. The contour levels of Rrange from
⫺0.342 to 0.342 with a step of 0.049, while those of Brange from ⫺0.085
to 0.085 with a step of 0.015. Solid 共dashed兲lines denote positive 共negative兲
values. The zero level has been removed for clarity.
081704-2 Y. Hallez and J. Magnaudet Phys. Fluids 21, 081704 共2009兲
Downloaded 25 Aug 2009 to 193.48.203.101. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
ture of the secondary Reynolds stresses. The spatial distribu-
tion of Rand Brevealed by Fig. 3clearly suggests that the
lateral wall plays a key role in the mechanism, even though
no geometrical anisotropy exists here, in contrast with the
case of a rectangular duct. The other crucial ingredient,
which is specific to the present situation, is the existence of
the nonaxisymmetric mean shear
z具U典in the flow. This sa-
lient characteristic of the primary flow, which results directly
from the existence of the light ascending and heavy descend-
ing currents, does not exist in a usual Poiseuille flow, allow-
ing 具U典to be almost constant in the core in the latter case.
Several classical studies of the turbulent structure in homo-
geneous shear flows have revealed that, in presence of a
constant and nonzero
z具U典, the three main Reynolds stresses
are such that 具u⬘2典⬎具v⬘2典⬎具w⬘2典.8–10 This is because the
turbulent fluctuation, which is produced only along the
x-direction, is then unevenly redistributed along the other
two directions, owing to the anisotropic transport and
stretching induced by the shear. In particular Refs. 9and 10
found the ratio 具v⬘2典1/2/具w⬘2典1/2to range from 1.15 to 1.5,
depending on the time ratio Sⴱ=Tt
z具U典,Ttdenoting the
large-eddy turnover time. In presence of a stable stratifica-
tion characterized by the Richardson number Ri
=共g具
z
典/
兲共
z具U典兲−2,具v⬘2典1/2/具w⬘2典1/2has been found to in-
crease from about 1.28 for Ri=0 to about 1.44 for Ri=1,
which reflects the gradual inhibition of vertical motions as Ri
increases.11 Present results indicate 具v⬘2典1/2/具w⬘2典1/2=1.3
with Ri=0.07 near the diametrical midplane z=0, Ri being
based on the effective gravity gsin ⌰in the z-direction.
Hence the anisotropy level determined in the present compu-
tations is consistent with known results and indicates that the
main cause of the difference in magnitude between the span-
wise fluctuation v⬘and the crosswise fluctuation w⬘in the
core of the flow is the shear rather than the stratification.
Within a large part of the pipe cross section, the mean
shear is almost constant 共see Fig. 1兲so that the flow is close
to a homogeneous turbulent shear flow. Hence the anisotropy
level determined above does not vary much and the cross-
correlation 具v⬘w⬘典is also almost zero since the y-axis is a
principal axis of the time-averaged flow. This allows us to
conclude that the whole term Ris negligibly small outside
the peripheral region directly influenced by the wall. Since
the main gradients lie along the radial direction in the near-
wall region, the analysis of this region is more easily
achieved in cylindrical coordinates. Defining the polar coor-
dinate system 共r,
兲so that 共r,
,x兲is right-handed and intro-
ducing the associated radial and azimuthal velocity
fluctuations 共vr,v
兲, it may be shown that the generation
term becomes R=共1/r2兲
r
2关r共具vr
2典−具v
2典兲兴−关
rr
2−共1/r2兲
2
+共3/r兲
r兴具vrv
典. In a constant-density turbulent flow in a cir-
cular pipe, no average quantity depends on
and vrand v
are uncorrelated, so that the above expression of Rmakes it
clear that no secondary flow can exist in this case. Consid-
ering that the thickness
␦
共
兲of the near-wall boundary layer
is much less than the pipe radius implies that, in the vicinity
of the wall, the radial derivative of any Reynolds stress in-
volved in Ris proportional to 1/
␦
共
兲and thus dominates
over terms with a 1/rprefactor. Hence, although the relative
magnitude of 具vrv
典and 具vr
2典−具v
2典is unknown, one has at
leading order in both contributions R⬇共1/r兲
r
2共具vr
2典−具v
2典兲
−
rr
2具vrv
典.
Within the viscous sublayer adjacent to the wall 共which
given the modest Reynolds number extends approximatively
up to r/d=0.4 here兲,v
共vr兲varies linearly 共quadratically兲
with the distance d/2−r, owing to the no-slip condition and
the divergence-free constraint. Therefore we may write 具vr
2典
−具v
2典⬇−
␣
共
兲共d/2−r兲2, with
␣
共
兲⬎0, by which we con-
clude that the radial derivative
r共具vr
2典−具v
2典兲 is positive close
to the wall, whatever the azimuthal location
. Since the
mean velocity 具U典reaches its maximum at some radial loca-
tion r=rM共
兲and returns to zero for r=d/2, a negative mean
shear exists in the range rM共
兲⬍r⬍d/2. This negative shear
is maximum in the vertical midplane 共see Fig. 1兲and is zero
in the diametrical midplane z=0 where the mean velocity is
almost zero whatever the spanwise location. Hence, owing to
the generic shear-induced anisotropy of the Reynolds
stresses discussed above, we expect 具v
2典to be larger near the
top and bottom of the cross section, where it corresponds to
the spanwise fluctuation, than near the midplane z=0 where
it corresponds to the crosswise fluctuation. This behavior,
which is confirmed by Fig. 4共left兲, implies
␣
⬎0共⬍0兲in
the first and third 共second and fourth兲quadrants, so that the
same conclusion holds for the whole term 共1/r兲
r
2共具vr
2典
−具v
2典兲.
Let us now consider the contribution to Rof the cross-
correlation 具vrv
典which, according to the near-wall behavior
of vrand v
, evolves like

共
兲共d/2−r兲3within the viscous
sublayer. To determine the sign of

, it is useful to note
the geometrical property 具vrv
典=−1
2共具v⬘2典−具w⬘2典兲sin 2
+具v⬘w⬘典cos 2
. We may then take advantage of the left-right
and top-bottom symmetries of all second-order moments
共keeping in mind that in a homogeneous turbulent shear flow
with a mean shear
z具U典the y-axis is a principal direction兲to
conclude that 具vrv
典is zero for both y=0 and z= 0. Hence we
expect 具vrv
典to reach its maximum strength along the two
diagonals
=⫾
/4, where 具vrv
典=⫿共具v⬘2典−具w⬘2典兲/2.
These various conclusions are confirmed by Fig. 4共right兲.As
we saw above, the mean shear results in a positive difference
具v⬘2典−具w⬘2典throughout the core of the pipe, so that

is
negative 共positive兲along the first 共second兲diagonal. As the
sign of the second derivative
rr具vrv
典follows that of

,we
conclude that the contribution −
rr具vrv
典to Ris positive
共negative兲within the first and third 共second and fourth兲quad-
rants.
Coming back to Eq. 共1兲it turns out that, whatever their
relative magnitude, both leading terms in Rhave the same
four-lobe structure and the same sign in each quadrant, i.e.,
Ris ⬎0共⬍0兲within the first and third 共second and fourth兲
quadrants, in agreement with the results displayed in Fig. 3
共left兲.
The final point to be addressed is that of the scaling of
the secondary flow obtained through the above mechanism.
The first estimate we need is that of R. In the present flow,
the wall shear stress is responsible for the turbulent velocity
fluctuations in the region where Ris nonzero. The mean
axial velocity goes from zero at the wall to its maximum
081704-3 Turbulence-induced secondary motion Phys. Fluids 21, 081704 共2009兲
Downloaded 25 Aug 2009 to 193.48.203.101. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
O共Vt兲value within the boundary layer of thickness
␦
=O共Re−1/2d兲, so that the wall shear stress scales like
O共Re1/2
Vt/d兲. Thus the magnitude uof the velocity fluc-
tuations obeys
u2=Re1/2
Vt/d, implying u2/Vt
2=Re−1/2
关note, however, that within the viscous sublayer 具vrv
典is
smaller than 具v
2典by O共Re−1/2兲, owing to its cubic variation
with the distance to the wall as compared to the quadratic
variation of 具v
2典兴. It is then straightforward to determine the
magnitude of Rby noting that the radial variation of the
turbulent stresses arises within the viscous sublayer of thick-
ness O共
␦
兲, whereas their azimuthal variation arises over one-
fourth of the pipe perimeter, i.e., over an O共d兲distance. Ob-
viously this azimuthal variation 共and the correlation between
vrand v
兲also depends on the dimensionless shear rate Sⴱ
through some function F共Sⴱ兲that vanishes for Sⴱ=0. Here
the turnover time of the large-scale turbulent motions is gov-
erned by the mean shear, so that Sⴱ=O共1兲. The aforemen-
tioned results on homogeneous sheared turbulence9–11 indi-
cate that the Reynolds stress anisotropy does not vary much
with Sⴱin this range, so that here Fmay simply be regarded
as a nonzero constant. All this implies that Ris of
O关共Vt/d兲2兴, as indicated by Fig. 3共left兲. Very close to the
wall, Ris balanced by the viscous diffusion of the stream-
wise vorticity 关last term in Eq. 共1兲兴. This diffusion arises
within the boundary layer, so that the diffusion term is of
O共
⍀/
␦
2兲, which implies ⍀=O共Vt/d兲. Finally, the second-
ary velocities reach their typical magnitude VSnear r=d/2
−
␦
, so that the transport term in Eq. 共1兲is of O关VSVt/共
␦
d兲兴
and this term balances Rat this radial location, thus imply-
ing VS=O共Vt
␦
/d兲=O共Re−1/2Vt兲. These conclusions are con-
sistent with the features displayed by the computational re-
sults. In particular we recover the already noticed feature that
⍀is of the order of Vt/d, even though the mean secondary
velocities are only a few percents of Vt共here Re=790, so that
Re−1/2⬇0.036兲.
The above qualitative analysis allows the spatial struc-
ture and sign of the source term Rin Eq. 共1兲and the scaling
of the resulting streamwise vorticity and secondary velocities
to be correctly predicted. This supports the idea that the sec-
ondary flow observed in our computations and recently con-
firmed in experiments5is a special case of Prandtl’s second-
ary flow of second kind which results from the combined
effect of the mean shear and the lateral wall on the turbu-
lence structure.
This research has been funded by the ANR through
Grant No. ANR-07-BLAN-0181. We are grateful to F. Moisy
for stimulating discussions on the generation mechanism of
the secondary flow.
1L. Prandtl, “Über die ausgebildete turbulenz,” Verfahren diese Zweite In-
ternationale Kongress für Technische Mechanik, Zürich 共1926兲关“Turbu-
lent flow,” NACA Technical Memo 435 共1927兲,p.62兴.
2P. Bradshaw, “Turbulent secondary flows,” Annu. Rev. Fluid Mech. 19,
53 共1987兲.
3A. O. Demuren and W. Rodi, “Calculation of turbulence-driven secondary
motion in non-circular ducts,” J. Fluid Mech. 140, 189 共1984兲.
4T. Séon, J. P. Hulin, D. Salin, B. Perrin, and E. J. Hinch, “Buoyant mixing
of miscible fluids in tilted tubes,” Phys. Fluids 16, L103 共2004兲.
5J. Znaien, “Étude locale du mélange de deux fluides dans la géométrie
confinée d’un tube incliné,” Ph.D. thesis, Université Paris 11, 2009.
6Y. Hallez and J. Magnaudet, “Effects of channel geometry on buoyancy-
driven mixing,” Phys. Fluids 20, 053306 共2008兲.
7J. Zhou, S. Balachandar, and T. M. Kendall, “Mechanisms for generating
coherent packets of hairpin vortices in channel flow,” J. Fluid Mech. 387,
353 共1999兲.
8V. G. Harris, J. A. H. Graham, and S. Corrsin, “Further experiments in
nearly homogeneous turbulent shear flow,” J. Fluid Mech. 81,657共1977兲.
9S. Tavoularis and S. Corrsin, “Experiments in nearly homogenous turbu-
lent shear flow with a uniform mean temperature gradient. Part 1,” J. Fluid
Mech. 104,311共1981兲.
10S. Tavoularis and U. Karnik, “Further experiments on the evolution of
turbulent stresses and scales in uniformly sheared turbulence,” J. Fluid
Mech. 204, 457 共1989兲.
11H.-J. Kaltenbach, T. Gerz, and U. Schumann, “Large-eddy simulation of
homogeneous turbulence and diffusion in stably stratified shear flow,” J.
Fluid Mech. 280,1共1994兲.
FIG. 4. 共Color online兲Map of the secondary anisotropy 具vr
2典−具v
2典共left兲and cross correlation 具vrv
典共right兲, both normalized by Vt
2.
081704-4 Y. Hallez and J. Magnaudet Phys. Fluids 21, 081704 共2009兲
Downloaded 25 Aug 2009 to 193.48.203.101. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
