Download - Turbulent pipe flow downstream of a 90° bend

J. Fluid Mech. (2013), vol. 735, R7, doi:10.1017/jfm.2013.534

Turbulent pipe flow downstream of a 90 bendLeo H. O. Hellström1,†, Metodi B. Zlatinov1, Guangjun Cao2

and Alexander J. Smits1,3

1Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA2CARDC, PO Box 211 Mianyang, Sichuan 621000, PR China3Mechanical and Aerospace Engineering, Monash University, VIC 3800, Australia

(Received 7 August 2013; revised 1 October 2013; accepted 3 October 2013;first published online 29 October 2013)

Time-resolved stereoscopic PIV was used to investigate the curvature-inducedstructures downstream of a 90 bend at Reynolds numbers between 20 × 103 and115 × 103. Data were taken at three downstream locations to investigate the evolutionof the structures. Snapshot proper orthogonal decomposition (POD) analysis showsthat the most energetic structure is not the well-known Dean motion but a bimodalsingle cell structure with alternating direction of rotation, called the ‘swirl switching’mode. The strengths of the Dean motion and the swirl-switching structures are similar,indicating that the difference in energy is related to their duration of occurrence, wherethe Dean motion is associated with a comparatively rapid transition between the twostates in the swirl switching mode.

Key words: low-dimensional models, turbulent boundary layers, vortex dynamics

1. Introduction

Turbulent flow through bends can be found in many fluid applications, includingflow through heat exchangers, industrial plant piping systems, and in rivers. Bendstend to introduce secondary motions in the downstream flow that are associated withpressure losses (Hawthorne 1951) and the distortion of the velocity profile, which maylead to scouring and non-uniform heat transfer (Berger, Talbot & Yao 1983). In rivers,for example, this scouring process is believed to cause meandering.

The theoretical work by Dean (1928) was the first to identify such curvature-inducedmotions, consisting of a pair of steady vortices in the streamwise direction, with thevortices located on either side of the symmetry plane (that is, the plane of the bend).These vortices are often called the Dean motions, and their strength is described interms of the Dean number, De, where De = (D/2Rc)

1/2ReD. Here, ReD = DUb/ν isthe Reynolds number, D is the diameter of the pipe, Ub is the area-averaged or bulk

† Email address for correspondence: [email protected]

c© Cambridge University Press 2013 735 R7-1

mailto:[email protected]

L. H. O. Hellström, M. B. Zlatinov, G. Cao and A. J. Smits

velocity, ν is the fluid kinematic viscosity, and Rc is the radius of the bend curvaturemeasured to the pipe centreline.

Most previous work has focused on flows with a small Dean number because ofits importance in, for example, arterial flows (Boiron, DelPlano & Pelissier 2007),as well as other biological applications. Investigations of higher Dean number flows,corresponding to turbulent flow, are not so common. Tunstall & Harvey (1968) werethe first to study the distortion of the velocity profile downstream of a sharp bend(D/Rc →∞) for turbulent pipe flow at Reynolds numbers up to 21.7 × 104, andthis work was extended by Enayet et al. (1982), Anwer, So & Lai (1989), Anwer& So (1993), So & Anwer (1993), and Sudo, Sumida & Hibara (1998) to includethe evolution of the velocity profile and the development of the secondary motionsthrough the bend. The consensus view from these extended studies was that whenthe flow enters the bend the pressure is greatest at the outer wall, where the flowdecelerates while the flow near the inner wall accelerates, thereby shifting the velocityprofile towards the inner wall. At ∼30 downstream of the bend inlet, centrifugalforces begin to dominate, creating two counter-rotating vortices that move fluid in thesymmetry plane towards the outer part of the bend, so that upon exiting the bendthe faster fluid has reached the outer wall. Downstream of the bend, the depressionin the velocity profile, first located at the inner wall, moves towards the outer wall,whilst the secondary flow gradually weakens. Sudo et al. (1998) found that at 10Ddownstream of the bend the flow had recovered its symmetrical profile with anattendant breakdown of the secondary flow, although they indicated that a longerdistance would be needed for full recovery.

The original work by Tunstall & Harvey (1968) noted that instead of a pair ofmore or less steady Dean motions, the flow downstream of the bend was governedby a single streamwise vortex that filled the pipe, and this vortex abruptly changeddirection on a relatively long time scale. This unsteady feature was referred to as‘swirl switching’, and the switching frequency f was estimated to be 65 Hz, whichcorresponded to a spatial extent of ∼12D. They also noted an additional frequencyassociated with the region of separated flow located downstream of the inner corner ofthe bend, which was approximately twice the value of the swirl switching frequency.These frequencies correspond to Strouhal numbers of St = 0.19 and 0.35, respectively,where St = fD/Ub. It was also found that the maximum velocities associated with thesesingle vortex secondary motions was O(0.1Ub), and that they appeared to be triggeredby the oscillations of the separation bubble located at the inner corner of the bend.

Brucker (1998) investigated the flow downstream of a bend using particle imagevelocimetry (PIV) for ReD = 5000 and D/Rc = 1. In this work, the swirl switchingmode was not observed, but a structure similar to the Dean motion was seen, wherethe plane of symmetry was oscillating about the plane of the bend. It was also shownthat the tangential velocity at the bend symmetry plane was governed by two distinctStrouhal numbers, 0.03 and 0.12. It is, however, unclear which flow structures theseStrouhal numbers corresponded to.

Rutten, Schroder & Meinke (2005) conducted an LES study at three differentReynolds numbers over the range 11 × 103 to 27 × 103 for D/Rc = 1 and 3, andidentified structures that were similar to the ones described by Brucker (1998) butwith one of the cells noticeably suppressed in strength. The flow was found tobe governed by two different Strouhal numbers, St = 0.01 and 0.2–0.3. The higherStrouhal number was ascribed to the shear layer instability between the deceleratedflow at the inner wall downstream of the bend, and the outer, faster flow. The lowerStrouhal number was related to an oscillation of the Dean motion, measured by

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Turbulent pipe flow downstream of a 90 bend

monitoring the movement of the outer wall stagnation point. It was further stated thatthe symmetric Dean motion is a time-averaged phenomenon, while the instantaneousstructure is asymmetric and additional vortices may appear at the outer side of thebend. Also, the azimuthal instability was found to occur without the presence ofan upstream separated region, contrary to what was surmised by Tunstall & Harvey(1968). Rutten et al. (2005) further proposed that the transition between the structuresis not as abrupt as described by Tunstall & Harvey (1968) but that it occurs moresmoothly, where there are not only two stable positions of the Dean stagnation point,as suggested by Brucker (1998), but that the stagnation point can be found at anyazimuthal position within ±40 of the symmetry plane.

More recently, Hellstrom et al. (2011b) investigated the flow structure downstreamof a 90 bend using snapshot proper orthogonal decomposition (POD), and found thatthe Dean motions were very weak in the mean and appeared to co-exist with large-scale swirl switching motions, which do not contribute to the mean. The transitionsbetween these modes were not further explored. Kalpakli & Orlu (2013) investigateda similar flow and also used snapshot POD to visualize the structures. Their flowreconstruction did not show any transitions, probably due to insufficient samplingrates.

These various observations are inherently inconclusive, and the objective ofthe current research is to experimentally investigate the secondary flow structuresdownstream of a 90 bend to try and resolve some of these ambiguities. Specifically,we will investigate the nature of the swirl switching mode, and its relationship to theDean motions, using snapshot POD.

2. Experimental setup

The experiment was conducted in a pipe flow facility consisting of a glass pipe witha 40 mm inner diameter and 4.75 mm wall thickness. The facility has a 150D longdevelopment length before entering a 90 bend with D/Rc = 1. Downstream of thebend there is another 40D long straight recovery section. The working fluid is water,and the flow is recirculated through the system using a 1 hp pump. The mean flowwas investigated for Reynolds numbers ranging from 20× 103 to 115× 103, while theturbulence and secondary flow measurements were limited to Reynolds numbers lessthan 25× 103 due to resolution limitations with the PIV system.

The PIV test section consisted of a Plexiglas box surrounding the pipe. The box wasfilled with water to minimize the optical distortion due to refraction through the pipewall. The box was fitted with O-ring seals, which allowed it to be placed at differentlocations upstream or downstream of the bend.

Two different arrangements were employed for acquiring the velocity field. Thefirst setup used stereoscopic PIV implemented with a single camera to investigatethe temporal behaviour and the large-scale structures in the flow. The view from ahigh-speed camera (Redlake MotionXtra HG-LE) was divided into two views andredirected to the laser sheet using optics in such a way so that the laser sheet was seenfrom two directions orthogonal to each other. The 1.5 mm thick laser sheet was usedto illuminate a cross-section of the pipe at three different downstream locations: 5D,12D and 18D. The camera operated in continuous mode at 1200 fps with a resolutionof 1040 × 640 pixels. At a maximum Reynolds number of 25 × 103, this mode ofoperation provides time-resolved velocity field ∼13D long. The resolution of the datais ∼2.15 vectors per mm2, and data near the wall (for 1 − r/R < 0.1, where r is

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measured from the centreline) were discarded because of insufficient resolution causedby optical refraction problems.

The second setup used a planar PIV system to investigate the time-averagedmean velocity profiles and their evolution with increasing downstream location andReynolds number. A dual-frame camera (Megaplus ES4.0) operating at 15 fps with aresolution of 2K × 2K pixels was paired with an Nd:YAG double pulsed laser with200 mJ pulse−1 to illuminate the flow. The 1 mm thick laser sheet was oriented alongthe symmetry plane of the bend. The flow was investigated at positions 5D, 10D,20D and 50D downstream of the bend, together with one reference case located 15Dupstream of the bend.

3. Results and discussion

3.1. Flow fieldThe mean velocity distributions at different downstream locations are shown infigure 1. The profiles, when integrated across the pipe cross-section, agreed withthe bulk velocity as measured using a flowmeter to within 5 %. The procedure wasrepeated for the axisymmetric velocity profiles measured 15D upstream, having asimilar accuracy to within 5 %.

As noted by previous investigators, there is a clear shift of the peak velocitytoward the outer part of the bend for downstream distances up to 20D, but by 50Dthe velocity profile is approximately symmetric. The profiles show a clear Reynolds

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FIGURE 2. Streamwise velocity (Ux/Ub) contours at ReD = 25× 103, at (a) 15D upstream ofthe bend, and (b) 5D downstream of the bend (I marks the inner part of the bend, O the outer

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number dependence in the regions downstream of the inner part of the bend, but inthe locations downstream of the outer part of the bend, where the peak in the velocityprofile appears, the flow is almost Reynolds number independent. Hence the recoveryof the profile appears to be driven by turbulent transport, which is expected to be moresignificant at higher Reynolds numbers. The most upstream locations have a fullervelocity distribution, indicating that a high degree of large-scale mixing takes placewithin the bend. Also, the profiles at 50D show a fuller profile at higher Reynoldsnumbers, as expected from studies of fully developed turbulent pipe flow (Zagarola &Smits 1998).

The cross-sectional mean velocity distributions, obtained using stereoscopic PIV at15D upstream of the bend and 5D downstream of the bend, are compared in figure 2.Again, we see a significant shift of the peak velocity toward the outer wall, where theregion of high velocity is flattened out along the outer wall for about half of the pipecircumference. A similar behaviour was observed by Enayet et al. (1982) and Sudoet al. (1998).

The curvature-induced velocities are said to increase mixing and cause additionalpressure losses (Berger et al. 1983). We see from the turbulence intensity distributionsshown in figure 3 that the distortion of the turbulence intensity is biased towards theouter wall. The radial turbulence component shows an increase along the symmetryline, whereas the tangential turbulence component shows an increase near the upperand lower walls. These trends are consistent with the in-plane behaviour of the meanDean motion, suggesting that the variance captures the unsteady behaviour of thesecondary motions, rather than the background turbulent fluctuations. The axisymmetryof the streamwise turbulence is recovered 18D downstream, where the intensity isreduced to half. The in-plane turbulence intensities, however, continue to show theinfluence of the bend.

Figure 4 shows contour plots of the streamwise component of the clockwisevorticity, Ωx, in a cross-sectional plane 5D downstream of the bend, for two Reynoldsnumbers, 20 × 103 and 25 × 103. The streamwise direction points into the plane. Thevorticity is scaled using Ub and the pipe radius. The Dean motion can be found in

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the mean vorticity in figure 4(a), while figure 4(c) shows the tilted Dean motion,mentioned by Brucker (1998). The mean flow structures are, however, not evidentamong the much higher levels of instantaneous vorticity (figure 4b,d).

3.2. Snapshot POD analysisTunstall & Harvey (1968) estimated the rotation velocity of the curvature-inducedstructures to be O(0.1Ub), and it is therefore proposed that the curvature-inducedvortices may be the most energetic structures downstream of the bend and could beextracted using snapshot POD, as suggested by Meyer, Pedersen & Ozcan (2007)in the context of a jet in cross-flow. The snapshot POD analysis was performedby only considering the in-plane velocity components, in order to eliminate fromthe eigenvalue problem the effect of the distorted streamwise velocity field and anystreamwise structures inherent to pipe flow such as the very large-scale motionsidentified by Hellstrom, Sinha & Smits (2011a) and Sakakibara & Machida (2012).

Figure 5 shows the energy distribution for the 20 most energetic snapshot PODmodes for ReD = 25 × 103 at three different downstream locations. It can be seen thatthe first mode contains ∼8 % of the total energy at 5D downstream of the bend, andthe relative energy of this mode gradually decreases to about a third of its originalvalue at 18D downstream of the bend, reflecting the slow relaxation of the flow.

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FIGURE 4. Contours of the streamwise vorticity, 5D downstream of the bend: (a,b) Re =20 × 103, (c,d) Re = 25 × 103; (a,c) time-averaged vorticity, (b,d) sample instantaneousrealization of the vorticity field.

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Figure 6 shows the first three snapshot POD modes for each downstream location.The first mode at each downstream location shows a single cell structure spanningthe entire cross-section. Figure 6(a,b) shows a clockwise rotation, whereas figure 6(c)indicates an anticlockwise rotation. Since POD extracts modes having an orthogonalbasis, the two counter-rotating motions share the same basis and would therefore be

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FIGURE 6. First three snapshot POD modes for ReD = 25×103: (a,d,g) x= 5D, (b,e,h) x= 12D,(c,f,i) x = 18D; (a–c) mode 1, (d–f ) mode 2, and (g–i) mode 3. Symbols: , inner stagnationpoint; •, outer stagnation point. The streamwise direction points into the plane.

represented by a single mode. This POD mode may correspond to the swirl switchingmotion, although the snapshot POD mode will not reveal any time dependence.

Figure 6(d–f ) displays the second POD mode, and reveals a Dean-like motionsimilar to the structure described by Brucker (1998). The inner stagnation point isazimuthally displaced further from the bend symmetry plane compared to the outerstagnation point, indicating that not only are the structures rotated with respect to thesymmetry plane but one of the cells is simultaneously suppressed.

Figure 6(g) shows the third POD mode at 5D downstream, which is a mirror imageof the second mode, again showing the bimodal nature of the curvature-induced flow.This antisymmetry is not so clear at the 12D and 18D locations, (figure 6h,i), primarilybecause the energy level for the third mode at these locations has decreased and noiseis an issue. It does, however, show the presence of additional vortices next to the Deanmotion, as previously described by Rutten et al. (2005).

A further inspection of figure 6(b,c), showing the single cell structure at 12D and18D downstream of the bend, reveals two counter-rotating structures where the centreof each cell is shifted to opposite sides of the bend symmetry plane. The location ofeach cell centre coincides with the location where the similarly rotating cell centreof the Dean motion was found. This observation, together with the cell suppressionnoted in modes 2 and 3, may indicate that the single cell structure is induced bythe suppression of one of the cells in the Dean motion. While the cause of this cellsuppression is unknown, the amplitude difference of the inner and outer stagnationpoints suggests that the suppression is triggered by an instability at the inner bend.

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This suggestion was first made by Rutten et al. (2005), who noted that the Strouhalnumber related to the instabilities was similar to the ones associated with unstableshear layers.

The suspected unsteadiness in the flow is revealed by inspecting when each of thesnapshot POD modes are active, which is done through the snapshot POD coefficients,α(τ), where τ = tUb/D and t is the time at which each PIV image pair was recorded.The coefficients are determined by projecting each mode onto the ‘random’ velocityfield, estimating the strength of that mode at each snapshot (Sirovich 1987). Figure 7shows the first three snapshot POD coefficients 5D downstream of the bend. A slidingaverage has been performed with a window of τ = 0.1 to reduce the noise due to thebackground turbulence. All modes have both positive and negative values, indicatingthat all modes in figure 6 are bidirectional. The Strouhal numbers associated witheach mode can be found by Fourier transforming the snapshot POD coefficients.Figure 8 shows spectra averaged over five independent data blocks, where |α|2 is thesquared magnitude of the Fourier transformed signal of α(τ). We see that there aretwo characteristic Strouhal numbers, 0.16 and 0.33, in good agreement with the onesreported by Tunstall & Harvey (1968) and Rutten et al. (2005). The lower Strouhalnumber represents the oscillation of the Dean motion while the larger Strouhal numberrepresents the swirl switching mode. There is, however, an interaction between thetwo modes, where modes 1 and 2 share the lower Strouhal number. It is unclear at

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FIGURE 9. Spatiotemporal flow field reconstructed using the snapshot POD modes. (a) Firstmode only, (b) first 2 modes, (c) first 3 modes, (d) first 8 modes. Flow direction is fromleft to right, placing the first time instance at the right and the last time instance at the left.The structures are visualized by isosurfaces of the tangential velocity, with magnitude 0.1Ub,showing the direction of rotation, where red is positive and blue is negative azimuthal velocity.

this point if the reversed Dean motion is reversed due to an interaction between thesemodes or due to the instability originating at the bend. However, the length of thestructure associated with this common Strouhal number is approximately the same asthe distance to the downstream location (6.25D compared to 5D), suggesting that itis certainly possible that the rotation of the Dean motion is due to the interactionbetween the two modes.

Figure 9 shows the reconstruction of the snapshot POD modes, where each temporalslice has been added after one another, creating a 13D long spatiotemporal extrusion(Hellstrom et al. 2011a). The flow is viewed from the inner bend along the bendsymmetry plane, where the flow direction is from left to right. The spatial field isconverted from the temporal domain using a convection velocity given by Ub. Thespatiotemporal conversion implies that the first acquired data are placed in the front(rightmost) of the extrusion. The figure shows isosurfaces of the tangential velocity,with magnitude 0.1Ub, that are used to indicate direction of rotation.

The reconstructed first mode (the swirl switching mode), shown in figure 9(a),reveals its bimodal nature with time. The flow rotates in one direction beforeswitching to a rotation in the opposite direction. Figure 9(b–d) shows the effectsof superimposing an increasing number of modes in the reconstruction. The additionalmodes seem to be related to the transition between the two rotation directions forswirl switching, although the transition appears to occur at different rates. Considerfigure 9(d). At time I, an abrupt transition occurs that spans a distance of 0.25D,while the transition between III and IV is comparatively slower, spanning 2.5D.The transitions appear consistently to pass through a Dean-like motion, indicatedby an inclined plane separating positively and negatively rotating fluid. The Dean-likestructure between II and III consists of a counter-rotating Dean motion, while thetransition at II forms the classical Dean motion.

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4. Conclusion

The secondary motions formed downstream of a 90 bend appear to be governedprincipally by two states, the swirl switching motion and the Dean motion. A PODanalysis suggests that the Dean motion is a transitional state that occurs betweenthe two states of the swirl switching motion, during which one of the cells ofthe Dean motion is being suppressed, resulting in a single swirl. These transitionspersist for time scales corresponding to 0.25D–2.5D. As a result, the Dean motionsare only weakly evident in the mean streamwise vorticity, and not seen at all inthe instantaneous vorticity. The mechanism causing the suppression is unclear butis believed to be related to the shear flow region at the inner corner of the bendwhich is sensitive to upstream perturbations, as suggested by Rutten et al. (2005)and Sakakibara & Machida (2012), and leaves open the possibility of controlling thetransition among secondary flow states.

Acknowledgement

This work was supported under ONR grant N00014-09-1-0263 (Program Manager R.Joslin).

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