COMPARATIVE COMPUTATIONAL STUDY OF TURBULENTFLOW IN A 90◦ PIPE ELBOW

R. Röhrig, S. Jakirlić and C. Tropea

Institute of Fluid Mechanics and AerodynamicsTechnical University of Darmstadt

Alarich-Weiss-Str. 10, 64287 Darmstadt, Germany

r.roehrig@sla.tu-darmstadt.de

AbstractTurbulent flow through a 90◦ pipe elbow in a

range of moderately high Reynolds numbers between14000 and 34000 is studied computationally usingwall-resolved Large Eddy Simulation (LES) as wellas various RANS (Reynolds Averaged Navier-Stokes)models aiming at a comparative assessment to illus-trate benefits and drawbacks of different computa-tional approaches for the considered case. Accord-ingly, characteristic secondary motions resembling theaxially oriented counter-rotating Dean vortices, seeBradshaw (1987), are further investigated and theirpredictability using LES is assessed.

1 IntroductionFully turbulent flow through curved conduits and

channels have been studied intensively in the pastby applying theoretical and experimental approaches.Berger et al. (1983) and the earlier publication bySmith (1976) provide a broad range of theoretical in-vestigations regarding curved pipe flow. However,with the scenario of curved flows being considerablymore complex and resource-demanding than flowsthrough straight domains, only recently, in the pastfew decades numerical investigations have been addedto the range of scientific tools assessing curved pipeflows, see e.g. Rütten et al. (2005). With increasingcomputing capacities, the applicable methods to pre-dict curved pipe flows are now shifting from simpler,steady-state approaches to more elaborate, temporallyresolving techniques.

The present study is thus dedicated to the investi-gation of turbulent flow through 90◦ pipe bends usingstate of the art numerical models of different complex-ity, accuracy and computational cost. Prior to simulat-ing the 90◦ pipe elbow configuration, fully-developedflow through a straight pipe at different Reynolds num-bers is computed in order to validate the employedcomputational procedures and to approve the adoptedmesh properties. The results for the case of a 90◦

elbow are subsequently evaluated regarding mean ve-locity and Reynolds stress fields and are compared to

Figure 1: (Left) Schematic overview of the THAI-26 setup:a Helium stratification (shaded) inside a contain-ment is dissolved by an entraining air jet develop-ing from a 90◦ elbow. (Right) Instantaneous ve-locity magnitude for a flow through an elbow pipeas obtained by the present LES.

experimentally obtained data from Kalpakli and Örlü(2013). The RANS models applied in the frameworkof this study include both the basic low Reynolds num-ber k-�-model, see Launder and Sharma (1974), aswell as a near-wall second moment closure model asproposed by Jakirlić and Hanjalić (2002). In the fol-lowing these two RANS approaches are referred toas Eddy Viscosity Model (EVM) and Reynolds StressModel (RSM), respectively.

The simulations performed within this frameworkessentially represent preliminary investigations con-cerning specific inlet conditions for the erosion pro-cess of a Helium stratification by an air jet withinthe containment of a nuclear reactor as motivated bythe THAI-26 program of the Reactor Safety ResearchProject 150 1455, see Fig. 1. The eventual goalis to demonstrate LES’ uprising capabilities for han-dling relevant engineering applications with signifi-cant streamline curvature such as flows through bends,manifolds or helically coiled pipes, as they appear in

e.g. oil pipelines, automotive air intakes or heat ex-changers, respectively.

2 Numerical MethodologyThe equations governing the mass and momen-

tum balance for a flow in a bent pipe are given bythe Navier-Stokes equations for incompressible New-tonian fluids as shown in Eq. (1) and (2), respectively.

∂Ui∂xi

= 0 (1)

∂Ui∂t

+ Uj∂Ui∂xj

= − 1ρ

∂p

∂xi+ ν

∂2Ui∂xj∂xj

(2)

In order to reduce the computational effort in thesense of LES, i.e. to explicitly resolve the larger scaleswhile modelling the fairly universal smaller ones bymeans of approximative assumptions, the above equa-tions are spatially filtered yielding

∂Ũi∂xi

= 0 (3)

∂Ũi∂t

+∂(ŨjŨi)

∂xj= −1

ρ

∂p̃

∂xi+ 2ν

∂

∂xj(S̃ij)−

∂τij∂xj

.

(4)In the above expressions the tilde-operator denotes

a three-dimensional filter operation in space with a fil-ter width of ∆. In Eq. (4), S̃ij is the filtered strainrate tensor, whereas τij = ŨiUj − ŨiŨj representsthe part of the non-resolved (i.e. subgrid) motions inthe governing equation for the resolved momentum,which needs to be modelled in order to close the equa-tion. The common standard nomenclature for velocity,pressure, density and kinematic viscosity being Ui, p,ρ and ν, respectively, is employed.

In this study, the well established approach accord-ing to Boussinesq is utilized, which assumes linear-ity between the subgrid-scale (SGS) tensor τij and themean strain rate tensor S̃ij according to

τij =1

3δijτkk − 2νtS̃ij , (5)

with δij being Kronecker’s delta. In Eq. (5), the artifi-cial SGS viscosity νt is approximated according to theassumptions stated by Smagorinsky (1963) as

νt = (CS∆)2|S̃ij | , (6)

where the model parameter CS is dynamically eval-uated as a function of the smallest resolved scales asfirst proposed by Germano et al. (1991). However, inorder to allow for backscatter, i.e. energy transfer fromsubgrid to resolved scales, the modification accordingto Lilly (1991) is employed throughout this study, thusyielding

νt = CD∆2|S̃ij | . (7)

In Eq. (7), the dynamically determined coefficient CDcan locally become negative (backscattering). We re-fer to Fröhlich (2006) and Lilly (1991) regarding fur-ther details for the different dynamical Smagorinskymodels as well as specific equations for CD.

To account for statistically steady, turbulent inletconditions, a periodic inlet section is used which hasa driving pressure gradient imposed to the momentumbalance in order to maintain a constant mass flow rate.The viscous sublayer of the pipe’s near-wall region isresolved, thus essentially resulting in a wall-resolvedLES. The dimensionless grid spacings in azimuthaland streamwise direction are chosen to be in the or-der of ∆+Θ ≈ 15 and ∆+z ≈ 30, respectively, while thefirst near-wall cell centers are located at y+ ≈ 1.

3 Numerical ValidationIn order to validate the employed numerical pro-

cedures, fully developed turbulent pipe flows at var-ious bulk Reynolds numbers are considered prior tosimulating the case of a 90◦ bend, see Fig. 2. Thediameter-based bulk Reynolds number is hereby de-fined as Reb = UbD/ν, where Ub, D = 2R and ν arethe pipe bulk velocity, the pipe diameter and the fluidkinematic viscosity, respectively. The computationalsetup is hence validated for pipe flow in the Reynoldsnumber range from Reb = 11700 to Reb = 34000.

Figure 2: Pseudocolors of the instantaneous streamwise ve-locity for fully developed pipe flows at Reb =12500 (left) and Reb = 32000 (right) obtainedfrom wall-resolved LES.

Fig. 3 shows the comparison of LES, RSM, EVMand DNS pipe flow results for the non-dimensionalstreamwise velocity U+ = 〈Uz〉/uτ as function ofthe normalised wall distance r+ = uτ (R − r)/ν atReb = 11700. Here uτ is the wall friction velocityand r the radial coordinate in a cylindrical coordinatesystem, while 〈.〉 denotes the temporal averaging overa sufficiently long interval. The DNS data are takenfrom El Khoury et al. (2013) and refer to a frictionReynolds number Reτ = uτ (D/2)/ν of Reτ = 360.The RANS results are obtained using the near-wallReynolds stress model according to Jakirlić and Han-jalić (2002). Without any exceptions, all three ap-proaches yield satisfying results with respect to meanvelocity prediction.

0

5

10

15

20

25

30

35

1 10 100 1000

U+

r+

U+ = r

+

U+ = 1/κ ln(r

+) + B

Equilibrium LawsLES, Re

τ = 360

RSM, Reτ = 360

EVM, Reτ = 360

DNS, Reτ = 360

Figure 3: Non-dimensional streamwise velocity for a fullydeveloped, turbulent pipe flow at a bulk Reynoldsnumber of Reb = 11700 (Reτ = 360). Resultsfrom LES are compared against RSM, EVM, DNSand the universal equilibrium laws of the wall, as-suming κ = 0.41 and B = 5.1; DNS data takenfrom El Khoury et al. (2013).

The statistics of the mean velocity fluctuations cor-responding to the afore-mentioned turbulent pipe flowat ReD = 11700 are provided in Fig. 4 for LESand RSM. Due to the nature of the eddy viscosity ap-proach, no specific turbulence intensities can be pro-vided for the EVM, which results in an isotropic-likesolution: 〈u2z〉 = 〈u2Θ〉 = 〈u2r〉 = 2k/3. For LESand RSM it is observed that all fluctuation componentsmatch fairly well with respect to the DNS results.

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300

ui/u

τ

r+

LES

RSMuz/uτ DNS

uΘ

/uτ DNS

ur/uτ DNS

Figure 4: Non-dimensional turbulent intensities in stream-wise (z), azimuthal (Θ) and wall-normal (r) direc-tions, complementing Fig. 3. DNS data taken fromEl Khoury et al. (2013).

In analogy to the above case of Reb = 11700 pipeflow, the applied procedures of LES, RSM and EVMare also validated for the upper limit of the presentlyconsidered Reynolds number range. Hence, Fig. 5

shows the non-dimensional streamwise velocity for afully developed, turbulent pipe flow at Reb = 34000(Reτ = 910). The graph shows that the LES can re-turn the mean flow velocity in a fairly good agreementwith the DNS data, whereas both RANS approachesresult in a slight underprediction within the logarith-mic region. It should be mentioned that although thecomputations are performed for a slightly lower valueof the friction velocity than the DNS of Reτ = 1000,the comparison can be regarded a proper validationsince no major differences in the velocity profile areto be expected resulting from a mismatch of this orderin the considered Reynolds number range, see also ElKhoury et al. (2013).

0

5

10

15

20

25

1 10 100 1000

U+

r+

U+ = r

+

U+ = 1/κ ln(r

+) + B

Equilibrium LawsLES, Re

τ = 910

RSM, Reτ = 910

EVM, Reτ = 910

DNS, Reτ = 1000

Figure 5: Non-dimensional streamwise velocity for a fullydeveloped, turbulent pipe flow at a bulk Reynoldsnumber of Reb = 34000 (Reτ = 910). Resultsfrom LES are compared against RSM, EVM, DNSand the universal equilibrium laws of the wall, as-suming κ = 0.41 and B = 5.1; DNS data takenfrom El Khoury et al. (2013).

The statistics of the mean velocity fluctuations cor-responding to the afore-mentioned turbulent pipe flowat ReD = 34000 are provided in Fig. 6. It is ob-served that the turbulent intensities in azimuthal andwall-normal direction show a fairly decent match withrespect to the DNS results, while the streamwise fluc-tuations for both RSM and LES deviate from the directnumerical simulation regarding wall-peak and core re-gion, respectively. However, the overall statistics andmean flow fields show reasonable agreement with thereference DNS results.

The objective regarding resolution requirements ofthe performed Large Eddy Simulations is to resolve aminimum of 80% of the total turbulent kinetic energy,i.e. keeping the part of the modelled subgrid-scalesbelow 20%, see e.g. Pope (2011). To evaluate whetherthese requirements are fulfilled, the above pipe flowsimulations are compared against DNS turbulent ki-netic energy spectra from El Khoury et al. (2013)yielding a satisfying fraction of about 85% totally re-

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300

ui/u

τ

r+

LES

RSMuz/uτ DNS

uΘ

/uτ DNS

ur/uτ DNS

Figure 6: Non-dimensional turbulent intensities in stream-wise (z), azimuthal (Θ) and wall-normal (r) direc-tions, complementing Fig. 5. DNS data taken fromEl Khoury et al. (2013).

solved scales. The ratio ∆/η between Kolmogorovscales η and mean LES filter width ∆ is located in theorder of ∆/η ≈ 6 in near-wall regions while it dropsdown to ∆/η ≈ 2 in the pipe core region.

4 Pipe Bend ConfigurationAfter preliminary validation by computing the

fully developed pipe flow, the methods of Large EddySimulation as well as the RANS approaches using aReynolds Stress and an Eddy Viscosity Model are sub-sequently applied to the case of a 90◦ pipe bend. Re-sults of the different approaches are compared to eachother and their engineering applicability is assessed re-garding computational effort, accuracy and usability.

The investigated flow configuration represents apipe of diameter D which is turned around π/2 with adistance between the center line and the pivot of Rc =1.58D. With the curvature parameter κ = D/(2Rc),a non-dimensional group according to

De = Reb√κ =

UbD

ν

√D

2Rc(8)

is built which is referred to as the Dean number De.The Dean number can be used as an indication forwhether a flow develops secondary motions past abend, the so-called Dean vortices. With the investi-gated bulk Reynolds numbers of Reb = 14000 and34000 and the according Dean numbers of De =7900 and 19000, secondary vortices are expected tobe found for both cases.

5 ResultsTo start with the qualitative behavior of the flow

through a bent pipe, Fig. 7 exemplarily illustrates thenormalised mean velocity magnitude inside the elbow

at Reb = 34000 as obtained by the present LES. Al-though it is difficult to judge from the shown cross-section view, it is interesting to notice that the flowdoes not separate from the inner wall of the bendwithin the symmetry plane of the setup, despite a sub-stantial momentum deficit in this region.

Figure 7: Normalised magnitude of the mean velocityacross the symmetry plane of the pipe (color con-tours) and a few corresponding iso-lines. Resultsobtained by the present LES at Reb = 34000.

In accordance with the above illustration, Fig. 8shows various mean streamwise velocity profiles atdifferent positions along the pipe elbow as they arecomputed using LES. The local acceleration of thefluid flow along the inner wall associated with the ve-locity magnitude increase at the outer wall behind thebend is clearly visible.

Figure 8: Mean velocity profile evolution in streamwise di-rection at various downstream positions comple-menting Fig. 7. The shown profiles are true toscale and thus of quantitative nature.

Quantitative comparison of simulation results forthis configuration are specifically shown in Fig. 9where the mean streamwise velocity is plotted at a cer-tain location behind the pipe elbow. It is observed thatthe LES yields notably good results regarding the over-all velocity profile development. Although the resultsobtained by the RSM deviate noticeably from the ex-perimental data on the inner side of the elbow, the pre-diction of the velocity gradient at the wall agrees wellwith the LES results. The standard k-�-model, how-ever, fails entirely with respect to the mean velocity inthe inner region of the bend and thus yields a wrongprediction of the wall shear stress.

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-1 -0.5 0 0.5 1

U/U

b

r/R

LESRSMEVMExp

Figure 9: Comparison of the non-dimensional mean stream-wise velocity evaluated in the domain’s symmetryplane at a position of 0.67D behind the pipe elbowfor Reb = 34000. The position of (r/R = −1)refers to the outer wall of the bend, while (r/R =1) is the inner one.

Fig. 10 shows an exemplary comparison of contourplots for the time-averaged streamwise velocity acrossthe pipe’s cross-section at a position of 0.67D behindthe pipe bend, coinciding with the plane evaluated inFig. 9. Illustrated are the experimentally obtained re-sults using PIV (left) as performed by Kalpakli andÖrlü (2013) and the numerical results of the presentLES. The comparison is supposed to underline thecapability of LES to qualitatively capture the three-dimensional behaviour of the flow behind the bendsince, unfortunately, no relevant quantitative data setsare available.

In analogy to Fig. 10, the qualitative behavior ofthe mean in-plane velocity field is illustrated in Fig.11 as a comparison between PIV and LES results. TheLarge Eddy Simulation predicts the in-plane velocitiessimilar as in the case of the streamwise velocity.

In order to assess the predicted velocity deficiencyoccurring on the inner wall of the elbow using theReynolds Stress Model, the computed turbulence in-tensity components are compared against the resultsobtained by LES, see Fig. 12. The graph reveals

Figure 10: (Left) Mean streamwise velocity contours behindthe elbow obtained from PIV by Kalpakli andÖrlü (2013). (Right) Corresponding numericalresults reproduced by the present LES at ReD =34000.

Figure 11: (Left) Mean in-plane velocity contours at 0.67Dbehind the elbow obtained from PIV by Kalpakliand Örlü (2013). (Right) Corresponding numer-ical results reproduced by the present LES atReD = 34000.

that the overall slopes of the Reynolds stress intensi-ties are captured correctly, however exhibiting a sub-stantial magnitude underprediction. This observationis consistent with the velocity deficiency in the meanstreamwise profile since the underprediction of turbu-lence leads to a lower degree of turbulent mixing, thusresulting in a delayed recovery of the flow.

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

ui/u

τ

r/R

uz/uτ LES

uΘ

/uτ LES

ur/uτ LES

uz/uτ RSM

uΘ

/uτ RSM

ur/uτ RSM

Figure 12: Non-dimensional turbulent intensities in stream-wise (z), azimuthal (Θ) and wall-normal direc-tions (r) in the inner section of the bend, comple-menting Fig. 9.

Figure 13: Instantaneous in-plane velocity field snapshotsshowing counter-clockwise Dean cells. (Left)Vortex as obtained from PIV by Kalpakli et al.(2012). (Right) Vortex captured by the presentLES. Small scale fluctuations have been sup-presses in both cases using a spatial low pass fil-tering operation.

The analysis of the instantaneous flow field showsthat in-plane oscillations, the so-called Dean vortices,can be observed using LES. These vortices rotate withalternating clockwise and counter-clockwise directionin the region past the pipe elbow. Such an exemplarycounter-clockwise Dean cell is illustrated in Fig. 13 incomparison to relevant experimental observations.

6 ConclusionsThe present study embodies a comparative assess-

ment of how the employed computational methods ofLES and RANS can handle turbulent flows with sig-nificant streamline curvature and potential separationregions. It reveals the superiority of LES over the ap-plied RANS approaches, however at the cost of sig-nificant increase in computational effort. It is shownthat the mean velocity field of the flow through a pipeelbow can be predicted accurately and that occurringsecondary vortices are precisely captured.

As addressed above, a future goal is to performa highly transient, multi-species LES of the erosionprocess of a Helium stratification by an entraining jetusing the inlet conditions obtained within this frame-work. Hence, this study additionally yields a databaseof turbulent inlet conditions for this specific case.

The computations of the pipe elbow configurationat Reb = 14000 and Reb = 24000 are currently inprogress and will be presented at the conference.

AcknowledgmentsThe present project was funded by the Federal

Ministry for Economic Affairs and Energy (BMWi,project no. 1501463) on basis of a decision by theGerman Bundestag.

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