DNS OF SWIRLING FLOW IN A ROTATING PIPE

Bikash SahooDepartment of Mathematics

National Institute of Technology, RourkelaOdisha, INDIA

bikashsahoo@nitrkl.ac.in

Frode NygårdStatoil

Laberget 114, 4020 Stavanger, NORWAYfron@statoil.com

Helge I. AnderssonDepartment of Energy and Process EngineeringNorwegian University of Science and Technology

Trondheim, NORWAYhelge.i.andersson@ntnu.no

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

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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

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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

04/09) to pursue advanced research work at Norwegian Uni-versity of Science and Technology (NTNU), Norway.

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