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DNS OF SWIRLING FLOW IN A ROTATING PIPE
Bikash SahooDepartment of Mathematics
National Institute of Technology, RourkelaOdisha, INDIA
[email protected]
Frode NygårdStatoil
Laberget 114, 4020 Stavanger, [email protected]
Helge I. AnderssonDepartment of Energy and Process
EngineeringNorwegian University of Science and Technology
Trondheim, [email protected]
ABSTRACTNumerical simulation is carried out to study the
com-
bined effects of rotation and induced swirl on the
fullydeveloped turbulent pipe flow at Reynolds number Reb =4900,
based on the bulk velocity and the pipe diameter. Theswirl is
induced by a body force in the tangential momen-tum equation, which
produces a tangential velocity field inthe near wall region.
Results from a single swirl along withfive different strengths of
the rotation are considered. Theeffects of rotation and swirl on
turbulent structures are in-vestigated in detail. Also
instantaneous axial velocity fluc-tuations are provided to
visualize the effects on the turbulentstructures.
IntroductionTurbulent flows of liquids or gases through long
straight pipes occur in a variety of different industrial
ap-plications. Such flows have received considerable atten-tion
throughout the years and are fairly well understood to-day,
although some uncertainties still prevails at very highReynolds
numbers; see e.g. Hultmark et al. (2012). Undercertain
circumstances, however, the streamlines are helicalrather than
straight lines and the mean flow becomes two-componential rather
than one-componential. This happensif a swirling motion arises or
if the pipe is subjected to ax-ial rotation. Swirl may result from
a swirl generator or anupstream elbow, whereas axially rotating
pipes are foundin turbo machinery cooling systems. In both cases, a
cir-cumferential component Uθ of the mean velocity vector co-exists
with the axial mean velocity component Uz. The pres-ence of a
circumferential mean velocity component tends toorient the coherent
near-wall structures with the local meanflow direction. Besides the
tilting of the near-wall struc-tures, the structures may be
strengthened or weakened in atwo-componential mean flow.
Fully developed turbulent flow in axially rotating pipeshas been
studied experimentally by Murakami & Kikuyama(1980) and Imao et
al. (1996) and by means of large-eddysimulations (LES) by Eggels
& Nieuwstadt (1993) and di-rect numerical simulations (DNS) by
Eggels et al. (1994)
and Orlandi & Fatica (1997). It is observed that
rotationresults in drag reduction. Recent DNS studies of
swirlingpipe flow by Nygard & Andersson (2010) showed the
sameinfluence of the induced swirl on the axial mean velocity
asaxial rotation. However, the presence of swirl turned outto have
less clear-cut effects on the turbulence field. In thecases with
stronger swirl, even drag reduction was reportedwhereas weak swirl
gave rise to excess drag.
Swirl and axial rotation both give rise to helical stream-lines
and it is therefore not unexpected that similarities be-tween these
two circumstances can be found. The aim of thepresent study is to
examine how an originally swirling pipeflow reacts to axial
rotation. For a given swirl number, fivedifferent rotation rates (N
= −1,−0.5,0,0.5,1) with bothsenses of rotation will be studied. To
this end, the fullNavier-Stokes equations are solved in
three-dimensionalspace and in time on a computational mesh
sufficiently fineto resolve the energetic large-eddy
structures.
Governing equationsThe governing equations are solved in
cylindrical co-
ordinates θ , r, and z. For practical reasons, the variablesqθ =
ruθ , qr = rur and qz = uz are introduced. Here uθ ,ur, and uz are
velocity components in the respective coor-dinate directions. All
variables in the governing equationsare non-dimensionalized with
the centerline velocity of thePoiseuille profile, Up and the pipe
radius, R.
To simplify, the non-dimensionalized total pressure,ptotal is
divided in to three parts as follows:
ptotal = P̂(θ)+ P̄(z)+ p(θ ,r,z, t). (1)
The first part, P̂(θ), is the artificial transverse pressure
com-ponent. In order to introduce a swirl in the pipe flow, the
az-imuthal pressure gradient, dP̂/dθ , is introduced. The sec-ond
part of Eq. (1) is the mean axial pressure, P̄(z) and fi-nally, p(θ
,r,z, t) represents the remaining part of the totalpressure. The
Navier-Stokes equations in terms of the newvariables, in a
reference frame rotating with the pipe wall
1
-
around the z-axis are
∂ qθ∂ t
+∂qθ/rqr
∂ r+
1r2
∂q2θ∂θ
+∂qθ qz
∂ z+
qθr
qrr+Nqr
=− ∂ p∂θ
− dP̂dθ
+1
Reb
[ ∂∂ r
r∂qθ/r
∂ r− qθ
r2+
1r2
∂ 2qθ∂θ 2
+∂ 2qθ∂ z2
+2r2
∂qr∂θ
](2)
∂qr∂ t
+∂∂ r
q2rr+
∂∂θ
qθ qrr2
+∂qrqz
∂ z−
q2θr2
−Nqθ
=−r ∂ p∂ r
+1
Reb
[ ∂∂ r
r∂ qr/r
∂ r− qr
r2+
1r2
∂ 2qr∂θ 2
+∂ 2qr∂ z2
− 2r2
∂qθ∂θ
](3)
∂qz∂ t
+1r
∂qrqz∂ r
+1r2
∂qθ qz∂θ
+∂q2z∂ z
=−∂ p∂ z
− dPdz
+1
Reb
[1r
∂∂ r
r∂qz∂ r
+1r2
∂ 2qz∂θ 2
+∂ 2qz∂ z2
] (4)
where N = 2ΩR/Up is the non-dimensional rotational num-ber.
The bulk Reynolds number Reb is maintained at 4900by enforcing a
constant bulk velocity Ub in the axial di-rection, which in turn is
sustained by the average pressuregradient dP/dz, found in the axial
momentum equation (4).The imposed non-dimensionalized azimuthal
pressure gra-dient, dP̂/dθ is designed to be constant and non-zero
forr > 0.9R and zero otherwise, as indicated by the shadedarea
in Fig. 1.
R
0.9R
Figure 1. Cross-section of the circular pipe. The shadedannular
region indicates the region where the azimuthalpressure gradient is
imposed.
Numerical Method and Grid ConfigurationThe discretization of the
momentum equations is gen-
eralized as in Orlandi & Fatica (1997) and Nygard &
An-dersson (2010)
(1−αl∆tAiθ )(1−αl∆tAir)(1−αl∆tAiz)∆q̂i= ∆t[γlHni +ρlH
n−1i −αlΨi p
n]+∆t[2αl(Aiθ +Air +Aiz)qni ](5)
where i = 1,2,3 and represent the θ , r, and z directions.∆q̂i =
q̂i − qni and q̂ is an intermediate velocity field. qni is
the velocity field at the old time step, n. Hi contains
thediscretized convective terms. Ψi pn and Aiθ ,Air,Aiz repre-sent
the discretized pressure gradients and the discretizedsecond-order
derivatives, respectively. αl ,γl and ρl are thecoefficients from
the time advancement scheme. The ap-proximate factorization
technique is adopted to reduce theterm in front of ∆q̂i to
tri-diagonal matrices. The discretiza-tions are forward in time and
central in space. Here q̂ isnon-divergence free and is found by the
use of a third-orderhybrid Runge-Kutta/Crank-Nicolson method. An
explicitthird-order low storage Runge-Kutta method is used for
thenonlinear terms and the linear terms are solved by an im-plicit
Crank-Nicolson scheme. The method is second-orderand third-order
accurate in time for nonlinear and linearterms, respectively.
The DNS code is based on staggered grid. Accord-ingly the
computational domain splits up into cells with thevelocities
calculated at the cell faces and the pressure cal-culated at the
cell centers. All the cells are enclosed by sixsides (see Fig. 2)
except for the cells adjacent to the cen-terline. The grid is
described by N1×N2×N3 where N1,N2 and N3 represent the number of
grid points in θ , r andz directions, respectively. The grid is
uniform along the az-imuthal and axial directions and is
non-uniform along theradial direction. Since, the fine grid
requires much largerCPU-time and storage requirements, the present
DNS sim-ulation with induced swirl and pipe rotation has been
carriedout for 65×97×65 grid points with Lz = 10D.
z
×
r
×q p, !
qr ×qz
a)
qz×
b)
q × q r
×
p,
Figure 2. Computational cell: (a) next to the centerlineand (b)
all elsewhere.
1 Results and DiscussionsResults from five cases with different
values of rota-
tion number N and swirl strength d p̂/dθ = 0.0250 are com-puted
and are compared with the results from the DNS sim-ulations by
Nygard & Andersson (2008), without rotation,with d p̂/dθ =
0.0250 and grid size of 64× 96× 64 (seeTable 1). In Fig. 3, mean
axial velocity profiles are plottedfor the current simulations. The
profile corresponding toN = 0 has evidently moved towards the
laminar Poiseuilleprofile. Fig. 4 shows the simultaneous effects of
swirl androtation on the mean azimuthal velocity. It is clear that
the
2
-
amplitude of the normalized mean circumferential velocityis
maximum for high value of opposite (N =−1.0) rotation.
Table 1. Simulation results with N = 0 and dP̂dθ = 0.0250
Present Nygard & Andersson (2008)Rec =UcD/ν 7316 7320Reτ =
uτ D/ν 324 327
Uc/uτ 22.35 22.39Ub/uτ 14.95 14.98Uc/Ub 1.45 1.49
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
y/R
Uz/U
p
N = −1.0 & dP/dθ = 0.0250N = −0.5 & dP/dθ = 0.0250N =
0.0 & dP/dθ = 0.0250N = 0.5 & dP/dθ = 0.0250N = 1.0 &
dP/dθ = 0.0250
Figure 3. Mean axial velocity profiles.
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
y/R
Uθ/U
p
N = −1.0 & dP/dθ = 0.0250N = −0.5 & dP/dθ = 0.0250N =
0.0 & dP/dθ = 0.0250N = 0.5 & dP/dθ = 0.0250N = 1.0 &
dP/dθ = 0.0250
Figure 4. Mean circumferental velocity pro-files.
Sweeps and ejections are the primary sources of theReynolds
shear stress ⟨u′ru′z⟩, shown in Fig. 5 and are re-sponsible for the
production of axial normal stress. Theonly non-negligible stress in
a non-rotating pipe is ⟨u′ru′z⟩.When the pipe rotates, this stress
is reduced and the othertwo stresses ⟨u′ru′θ ⟩ and ⟨u
′θ u
′z⟩ increase. Fig. 5 shows
the profile for ⟨u′ru′z⟩ attains highest peak correspondingto N
= 1 and dP̂/dθ = 0.0250, and gets decreased as therotation number
tends to zero. The magnitude of the rz-component corresponding to
the opposite (N < 0) rotationalways remain less than that of
corresponding to N > 0.This Reynolds shear stress component is a
measure of theturbulent drag. It is evident that maximum drag
reductioncorresponds to N = −0.5 and dP̂/dθ = 0.0250. Fig. 6shows
in the core region of the flow, the behaviour of theflow of ⟨u′ru′θ
⟩ is almost linear. The ⟨u
′θ u
′z⟩-profiles are
shown in Fig. 7 and also here, pronounced effects in
thenear-wall region can be observed. The increase of the⟨u′ru′θ ⟩
and ⟨u
′θ u
′z⟩ components in the near-wall region is
due to the large magnitude of the mean velocity gradientdUθ/dr.
Surprisingly, in presence of swirl, ⟨u′θ u
′z⟩ attains
highest peak value for no-rotation (N = 0), as is clear fromFig.
7.
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
y/R
<u
r’u
z’>
/Up2
N = −1.0 & dP/dθ = 0.0250N = −0.5 & dP/dθ = 0.0250N =
0.0 & dP/dθ = 0.0250N = 0.5 & dP/dθ = 0.0250N = 1.0 &
dP/dθ = 0.0250
Figure 5. Reynolds shear stress profile
for⟨u′ru′z⟩-component.
VisualizationControlling the coherent structures seems important
in
taming the turbulence. In accordance with this, visual-ization
plots of instantaneous axial velocity fluctuations inθ − z plane
are given in Figs. 8 to 12 for all the five cases,considered in
this work. In these figures red and blue colorsvisualize positive
and negative fluctuations respectively. Itis clear that the
presence of swirl creates a velocity field thattilts the streaks.
On the other hand as the rotation numberchanges sign from negative
to positive, the distance betweenthe streaks increases. A reduction
(increase) in the length ofstreaky structures due to negative
(positive) fluctuations isobserved as the rotation number, N
changes sign from neg-ative to positive (compare Figs. 9 and
11).
3
-
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
1.5
2
2.5
y/R
<u
r’u
θ’>
/Up2
N = −1.0 & dP/dθ = 0.0250N = −0.5 & dP/dθ = 0.0250N =
0.0 & dP/dθ = 0.0250N = 0.5 & dP/dθ = 0.0250N = 1.0 &
dP/dθ = 0.0250
Figure 6. Reynolds shear stress profile for⟨u′ru′θ
⟩-component.
0 0.2 0.4 0.6 0.8 1−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
y/R
<u
θ’u
z’>
/Up2
N = −1.0 & dP/dθ = 0.0250N = −0.5 & dP/dθ = 0.0250N =
0.0 & dP/dθ = 0.0250N = 0.5 & dP/dθ = 0.0250N = 1.0 &
dP/dθ = 0.0250
Figure 7. Reynolds shear stress profile for⟨u′θ u
′z⟩-component.
Figure 8. Visualisation of axial velocity fluctuationsu′z/Up, at
y/R = 0.1 for N =−1.0 and dP̂/dθ = 0.0250.
ConclusionsThe work is devoted to the numerical simulation of
a
turbulent pipe flow with rotation and swirl induced by near-wall
body force. The flow is three-dimensional. The nu-merical method is
tested for the non-rotating case (N = 0)by comparing the results
with Nygard & Andersson (2008).In presence of swirl, maximum
drag reduction is achievedfor N = −0.5. It has been taken in to
consideration that
Figure 9. Visualisation of axial velocity fluctuationsu′z/Up, at
y/R = 0.1 for N =−0.5 and dP̂/dθ = 0.0250.
Figure 10. Visualisation of axial velocity fluctuationsu′z/Up,
at y/R = 0.1 for N = 0 and dP̂/dθ = 0.0250.
Figure 11. Visualisation of axial velocity fluctuationsu′z/Up,
at y/R = 0.1 for N = 0.5 and dP̂/dθ = 0.0250.
Figure 12. Visualisation of axial velocity fluctuationsu′z/Up,
at y/R = 0.1 for N = 1.0 and dP̂/dθ = 0.0250.
the simulation has been done with a relative coarse grid,which
have an uncertain influence on the present results.Therefore,
simulations with improved grid resolutions willbe carried out in
future work.
AcknowledgementsOne of the authors (BS) is thankful to the
Department
of Science & Technology, Government of India for award-ing
the prestigious BOYSCAST Fellowship (SR/BY/M-
4
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04/09) to pursue advanced research work at Norwegian Uni-versity
of Science and Technology (NTNU), Norway.
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