Osborne Reynolds pipe flow: Direct simulation from laminar through gradual transition to fully developed turbulence Xiaohua Wua, Parviz Moinb,1, Ronald J. Adrianc, and Jon R. Baltzerd a Department of Mechanical and Aerospace Engineering, Royal Military College of Canada, Kingston, ON, Canada K7K 7B4; bCenter for Turbulence Research, Stanford University, Stanford, CA 94305-3035; cSchool for the Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 852876106; and dLos Alamos National Laboratory, Los Alamos, NM 87545
The precise dynamics of breakdown in pipe transition is a centuryold unresolved problem in fluid mechanics. We demonstrate that the abruptness and mysteriousness attributed to the Osborne Reynolds pipe transition can be partially resolved with a spatially developing direct simulation that carries weakly but finitely perturbed laminar inflow through gradual rather than abrupt transition arriving at the fully developed turbulent state. Our results with this approach show during transition the energy norms of such inlet perturbations grow exponentially rather than algebraically with axial distance. When inlet disturbance is located in the core region, helical vortex filaments evolve into large-scale reverse hairpin vortices. The interaction of these reverse hairpins among themselves or with the near-wall flow when they descend to the surface from the core produces small-scale hairpin packets, which leads to breakdown. When inlet disturbance is near the wall, certain quasi-spanwise structure is stretched into a Lambda vortex, and develops into a large-scale hairpin vortex. Small-scale hairpin packets emerge near the tip region of the large-scale hairpin vortex, and subsequently grow into a turbulent spot, which is itself a local concentration of small-scale hairpin vortices. This vortex dynamics is broadly analogous to that in the boundary layer bypass transition and in the secondary instability and breakdown stage of natural transition, suggesting the possibility of a partial unification. Under parabolic base flow the friction factor overshoots Moody’s correlation. Plug base flow requires stronger inlet disturbance for transition. Accuracy of the results is demonstrated by comparing with analytical solutions before breakdown, and with fully developed turbulence measurements after the completion of transition.
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introduced at the laminar pipe inlet will induce gradual transition, eventually leading to a state of fully developed turbulence. Finite amplitude is needed because pipe flow is linearly stable with respect to infinitesimal disturbances. Weak and localized inlet disturbance is preferred whenever possible so as not to destroy the overall characteristics of the base parabolic/plug flow. It would be ideal if there exists an extended streamwise range before breakdown over which statistics of the slightly perturbed flow essentially agree with the fully developed laminar solution. This is analogous to the approach for boundary layer bypass transition in the narrow sense (10). There, one is usually concerned with perturbations less than ∼5% of the freestream velocity because stronger disturbances will immediately produce transition or large patches of turbulence (11), which therefore is not useful for a careful study of the transition process. Ref. 12 simulated strong injection and suction through slots opened on the wall of a short pipe (60× the pipe radius R) in a spatially developing computation; however, no quantitative results were presented. The current method differs from previous approaches in which pipe flow was subjected to strong jet-in-cross-flow–type injection or suction to directly generate pipe turbulence. Confidence in our method can be established by evaluating statistics against analytical solutions in the early laminar region before breakdown, and against established data in the fully developed turbulent region after the completion of transition. The validated DNS can subsequently provide data on the dynamics in the Significance
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The precise dynamics of disturbance energy growth and breakdown in pipe transition is a century-old unresolved problem in fluid mechanics. In this paper, we demonstrate that the mystery attributed to the breakdown process of the Osborne Reynolds pipe transition can be partially resolved with a direct, spatially evolving simulation that carries weakly but finitely perturbed laminar inflow through gradual rather than abrupt transition arriving at the fully developed turbulent state. Some of the previously attributed abruptness and mysteriousness was perhaps due to the inability to study the process accurately with very fine spatial and temporal resolution. The energy norm was found to grow exponentially rather than algebraically. The sensitivity of the transition process to pipe entrance conditions is demonstrated.
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he vast and expanding realm of fluid mechanics research on transition and turbulence can actually be traced back to a single point in history: the publication of Osborne Reynolds’ 1883 pipe flow paper (1) in which the concept of Reynolds number was introduced. Given the historical, fundamental, and applied importance of the problem, it is ironic that the Osborne Reynolds pipe transition remains to this day “abrupt and mysterious” (2, 3). Significant progress has been made during the past decade, mostly concentrating on the detection of traveling wave (4, 5), and on the lifetime and reverse transition (relaminarization) of existing pipe flow turbulence produced by strong jet-in-crossflow type of blowing and suction (6, 7). Refs. 8 and 9 reported insightful relaminarization simulations using the axially periodic boundary condition. We tackle directly the Osborne Reynolds pipe transition problem with spatially developing direct numerical simulation (DNS). The disturbance energy growth rate with respect to axial distance, and how the friction factor and vortex structures develop during pipe transition with the distance, is currently unknown. We anticipate that weakly finite and localized disturbances www.pnas.org/cgi/doi/10.1073/pnas.1509451112
Author contributions: X.W., P.M., and R.J.A. designed research; X.W. performed research; P.M. contributed new reagents/analytic tools; X.W., P.M., R.J.A., and J.R.B. analyzed data; and X.W., P.M., R.J.A., and J.R.B. wrote the paper. Reviewers: B.E., University of Marburg; and T.M., University of Manchester. The authors declare no conflict of interest. Freely available online through the PNAS open access option. 1
To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1509451112/-/DCSupplemental.
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Contributed by Parviz Moin, May 18, 2015 (sent for review March 27, 2015; reviewed by Bruno Eckhardt and Thomas Mullin)
transitional region that is bounded by these two verified ends. The DNS will be done in a laboratory reference frame without the axially periodic boundary condition. The governing equations are the continuity and the Navier– Stokes equations for incompressible flow in cylindrical coordinates. The computer program and the numerical scheme were described by Pierce and Moin (13, 14) in their large-eddy simulation of combustion in a coaxial jet combustor. That configuration is degenerated into a spatially developing circular pipe in this work. Unless otherwise noted, the pipe length is 250 R, and the computational mesh size is 8,192 × 200 × 256 in the axial ðzÞ, radial ðrÞ, and azimuthal ðθÞ directions, respectively. The time step remains a constant Δt = 0.008R=V, where V is the bulk velocity. An overbar denotes ensemble averaging, superscript “+” refers to normalization by friction velocity uτ for velocity and by viscous wall unit ν=uτ for distance; ν is the kinematic viscosity. Poisson’s equation for pressure is solved using Fourier and cosine transform along the azimuthal and axial directions, respectively. The entire set of DNS reported here took four calendar years to complete. Six steps were taken to systematically vary the inlet condition ðz = 0Þ and quantitatively assess the downstream response. In step 1, plug inflow was first prescribed without perturbation. The downstream velocity develops into the expected parabolic profile. Replacing the plug inflow with the exact parabolic inflow simply kept the downstream flow unchanged from the inlet. In step 2, perturbations were introduced at the inlet under the guiding principle that the disturbance magnitude should be finite yet weak and well-controlled. Another guiding principle is that the setup should permit transition to proceed to a fully developed turbulent state downstream. Initially, the inlet base flow was parabolic, Re = 2VR=ν = 5,300, and the inlet perturbations were confined within the tiny circle 0 ≤ r=R < 0.02 by superposing either white noise or isotropic grid turbulence on the base flow, but they disappeared rapidly without exciting any noticeable turbulence. We subsequently imposed instantaneous velocity fields extracted from a separate, axially periodic DNS of the turbulent pipe flow at Re = 5,300 over the same radial range 0 ≤ r=R < 0.02, but no transition was observed. Additional tests by increasing the contaminated area from 0 ≤ r=R < 0.02 up to 0 ≤ r=R < 0.10 failed
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to produce transition. Increasing Re from 5,300 to 8,000 did not produce a full transition either. Changing the parabolic base flow to the plug inflow did not produce transition. Of course, if Re is increased to a very large value, transition will eventually be obtained under these disturbances. The resolution constraint required by high-quality DNS prevents us from increasing Re further. Our finite yet localized weak disturbance principle does not encourage us to further enlarge the inlet perturbation area either. Laminar pipe flow below Re = 8,000 is reluctant to transition under such disturbances. In step 3, inspired by the absolute instability of tangential discontinuity in general shear flows, we made the perturbation area over the inlet plane into a narrow ring instead of a small full circle. Within this ring the exact parabolic velocity profile was replaced by the time-dependent, fully developed turbulent DNS field at Re = 5,300. With the ring positioned at 0.4 ≤ r=R < 0.42, it was found that at Re = 5,300 the flow still returned to laminar after a transient turbulent spot. Successful transition was obtained at Re = 8,000 (case S3R8, Figs. 1A and 2H). At the inlet, the perturbed area is well localized and accounts for merely and the energy norm jj E jj = R R 2 1.6%2 of the 2cross-section, 2 2 ðu ′ + u ′ + u ′ Þrdr=ðR V Þ = 5.8 × 10−5 is less than 1% z,rms θ,rms r,rms 0 of the fully developed turbulent pipe value 9.3 × 10−3, and is less than 0.15% compared with the peak value 3.9 × 10−2 attained ′ are the turbulence induring transition. Here u′z,rms , u′r,rms , uθ,rms tensities. This inlet disturbance is analogous to a turbulent wake arising from a very thin wire ring mounted on the inlet plane in a laboratory. Under this perturbation, the parabolic pipe flow gradually breaks down and develops into a fully developed turbulent state downstream. Switching the inlet ring turbulent fluctuation to a mean wake deficit superimposed with white noise failed to produce transition. In the present simulation, there is no imposed axial pressure gradient. After transition, R+ = uτ R=ν = 258.5, uτ = 0.06462V ; grid resolution is Δz+ = 7.5 and ΔðRθÞ+ = 6.3. Auxiliary simulation on the fully developed turbulent pipe flow at Re = 8,000 (case S3R8T) was performed using the conventional axially periodic boundary condition over a 30-R-long domain with a grid 2,048 × 256 × 512. Fig. 3 compares the turbulent statistics between S3R8T and S3R8. Note the
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Fig. 1. Friction factor f (●) and energy norm 10−2 log10 kEk (◇). Dash-dot-dash: f = 64=Re; dash-dot-dot-dash: Moody’s correlation. (A) Case S3R8; (B) case S4R6; (C) case S5R8; (D) case S6R8. Contoured insets are uz ðr, θ, z = 0, tÞ at one random instant.
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Fig. 2. Vortex structures in pipe transition. Case S5R8: (A) two infant turbulent spots; (B) the predecessors of the two turbulent spots in A; (C) subsequent merging of the two turbulent spots in A; (D) isosurface of scalar φ = 0.05 showing scalar breakdown; (E) zoomed view of D showing modulation of scalar field by hairpin packet; (F) zoomed view of D showing a street of indentions on scalar isosurface; (G) planar view of uz; (H) the breakdown region in case S3R8; (I) from inlet to breakdown in case S4R6.
reliability of the approach of S3R8T for developed turbulent channel/pipe flow has been well established since the work of ref. 15; see also ref. 16. All of the statistics, including even the rate of viscous dissipation «+, agree very well between the two computations, demonstrating the quantitative reliability of the present spatially developing pipe transition simulation. The transition is continuous in the sense that no isolated turbulent spot was observed in this case; see the vortex visualization over the breakdown region 30 ≤ z=R < 45 in Fig. 2H and Movie S1 using isosurfaces of swirling strength λci (17); red color indicates uz > 1.3V. The finite-amplitude disturbances from the ring create a roughly annular turbulent wake that grows downstream radially inward and outward. Transition is triggered by vortex filaments drawn from the disturbance region, intensified by stretching, and moving toward the wall. Between 30 and 40 R the filaments induce wall-ward radial flows; in turn, they create large reverse hairpin vortices that Wu et al.
rapidly induce hairpin packets near the wall. Small-scale activity explodes when the packets grow and interact with other vortex structures. The small-scale activity overcomes the larger scale fluctuations from the initial disturbances as it grows and interacts to fill the pipe with turbulence that approaches the fully developed state by about 75 R. A reverse hairpin vortex as part of a diamond structure was reported in previous turbulent channel simulations of ref. 18; the present reverse hairpins occur in a train among a quiescent environment, are of large scale (several R in length), and are much more distinct. In step 4, with identical inlet condition as in S3R8, Re was first reduced from 8,000 to 5,700, and the flow returned completely laminar. Reducing Re from 8,000 to 6,000 (case S4R6) produced delayed transition compared with S3R8. Thus, for this particular inlet disturbance, the critical Reynolds number is between 5,700 and 6,000. In S4R6 after transition, PNAS Early Edition | 3 of 5
Fig. 3. Comparison between turbulent statistics in the axially periodic turbulent case S3R8T (dotted line) and in the fully developed turbulent region ′+ ′+ 210 ≤ z=R ≤ 240 of case S3R8 (solid line). □, u+z =10; ◇, u′+ z,rms ; ★, uθ,rms ; +, ur,rms ; + △, −u′z u′r ; ○, −10«+ .
R+ = 200.1, uτ = 0.0667V , Δz+ = 6.11, ΔðRθÞ+ = 4.91. An auxiliary simulation of the fully developed turbulent pipe flow at Re = 6,000 was also performed (case S4R6T), and the accuracy was further verified by comparing the turbulent statistics from S4R6 and S4R6T. Friction factor f as a function of z is shown in Fig. 1B. Although data on f variation with Re are widely available (19), we cannot find how f varies with z during transition from the literature. The present data show that before breakdown, f agrees very well with the analytical solution 64=Re; after the completion of transition, f agrees with Moody’s correlation for smooth, fully developed turbulent pipe flow. Thus, the quantitative reliability of the present spatially developing DNS is again demonstrated. The peak in f is related to the change of sign of the near-wall mean radial velocity gradient. In boundary layers it is also related to the sudden increase in entrainment of outer flow into the near-wall region (10). Fig. 1 also presents the development of energy norm jj E jj with z. There is an extended, convincing, exponential growth region before breakdown in S4R6. It is known that infinitesimally small disturbances will not be amplified exponentially in a parabolic pipe flow. The four sets of energy norm jj E jj results in Fig. 1 suggest that localized, weakly finite disturbances are often amplified exponentially rather than algebraically with respect to axial distance in the pipe flow. We also found that shortly downstream of the inlet in S4R6 and S3R8, the mean flow profile within the ring quickly recovers to approximately that of the base flow. The vortex structure development of S4R6 shown in Fig. 2I is similar to that in S3R8 with a sequential formation of helical vortex filaments, large-scale reverse hairpin vortices, followed by smallscale hairpin packets and transition without turbulent spots; see also Movie S2. There are several notable overall agreements between our observations in steps 2–4 and previous dynamical system analysis of pipe transition, and the associated numerical work performed using axially periodic temporal DNS. Ref. 20 defined the object of “edge-of-chaos,” which separates, in state space, the regions of initial conditions where the lifetime statistics exhibit a sensitive dependence on initial conditions capable of resulting in turbulence from those decaying to laminar flow. In this work, we have repeatedly encountered situations in which a very minor change in either Reynolds number or inlet disturbance ring location made a decisive difference regarding whether the downstream flow stays laminar or transitions to turbulence––the flow parameter settings are close to the edge-of-chaos in the state 4 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1509451112
space. We have also found in other situations, as in the tests described in step 2, where the pipe flow resists transition even under significant modifications in either Reynolds number or inlet disturbance; these settings are far away from the edge-of-chaos. Ref. 21 reported temporal DNS in a pipe with a length of 10 R, and a pair of streamwise vortices with modulated tilting along the pipe axis as an initial condition. The inset of figure 4 in ref. 21 shows that the energy norm difference between the bounding (amplification and decay) trajectories of the edge-of-chaos grows exponentially with time, which was attributed to a shear flow instability arising from the temporal development of the initially imposed streamwise vortex pair. The present DNS demonstrates that spatial amplification of weakly finite inlet perturbations along the axial direction of a laminar pipe is exponential. The relation between this exponential growth rate and the absolute instability of tangential discontinuity of the inlet condition needs to be clarified in future research. In steps 3 and 4, the inlet disturbance rings are confined in the core region. At the prescribed two Reynolds numbers 8,000 and 6,000, when the large reverse hairpin from the core impinges on the wall, a small-scale hairpin packet is generated at the surface, leading to turbulence. This is somewhat consistent from the results of ref. 22 at Reynolds number less than 2,300. There, it was observed that a turbulent puff from the core could, but not always, impinge on the wall and induce a secondary turbulent puff. It is conceivable that if the Reynolds number is reduced slightly in step 4 from 6,000, the secondary hairpin packet will form an isolated puff instead of resulting continuous turbulence. In step 5, we address the effect of the disturbance ring location by first repositioning it from 0.4 ≤ r=R < 0.42 to 0.9 ≤ r=R < 0.92. At Re = 8,000, the flow transitions immediately at the inlet. This is not useful because it does not permit us to study how the disturbances are amplified, and how the breakdown process happens. The width of the ring was therefore reduced to 0.9 ≤ r=R < 0.91. This minor modification completely relaminarized the flow. Successful and prolonged transition between 50 ≤ z=R < 150 at Re = 8,000 was observed when the ring was positioned at 0.9 ≤ r=R < 0.915, which covers 2.7% of the inlet cross-section (case S5R8). It was further found that at this setting, reducing Re from 8,000 to 7,000 produced relaminarization. Inspired by the colored band visualization of Osborne Reynolds, a passive scalar φ = 1 was also introduced at the inlet over a small circle 0 ≤ r=R < 0.05 (Fig. 2D). Note φ = 0 over the rest of the inflow plane; the molecular Prandtl number was 1. A refined mesh of 16,384 × 200 × 512 was also used (case S5R8F). During transition, there is an overshoot of f over the turbulent correlation in S3R8, S4R6, and S5R8 for which the base flows are parabolic. Vortex structure development is different from that in the previous two cases when the inlet disturbances were in the core region. Here, as shown in Fig. 2 A and B, structures surviving the initial decay near the inlet have the shape of a Lambda vortex, possibly because their oblique orientation is favored by shear flow amplification via the lift-up mechanism. The Lambda vortex subsequently develops into an elongated large-scale (on the order of R) hairpin vortex; further downstream, a small-scale hairpin packet emerges near the tip region of the large-scale hairpin vortex. A turbulent spot is subsequently formed, which is essentially a local concentration of small-scale hairpin vortices (Fig. 2C). This process is analogous to the vortex development in boundary layer bypass transition in the narrow sense (10), and is also broadly similar to the secondary instability and breakdown stage of boundary layer natural transition (23). This similarity was conjectured and argued for in ref. 24 based on interpretation of ensemble-averaged hot-wire signals; here our DNS provides direct, convincing, time-accurate, 3D evidence (see also Movie S3). Although the breakdown of the scalar field in Fig. 2D may seem abrupt at first glance, careful examination shows that the scalar field has already been modulated by hairpin packets long Wu et al.
transition was observed at Re = 8,000. However, reducing Re from 8,000 to 7,500 relaminarized the flow. With the plug inflow, jj E jj continues to exhibit robust exponential growth rate after a prolonged decay to 75 R, but friction factor f does not show the overshoot found in previous cases with the parabolic base inflow; Fig. 1D. This is analogous to the boundary layer bypass transition in the narrow sense in which the skin friction has no overshoot either (10). The sharp rise of skin friction during transition is closely related to the entrainment of outer fluid into the near-wall region. Vortex dynamics is similar to that in case S5R8 when the inlet disturbance is also near the wall, characterized by the sequential formation of a Lambda vortex, hairpin packet, and turbulent spot, which is a localized hairpin forest. The accuracy of the results is demonstrated. Below Re = 8,000 laminar pipe flow is reluctant to transition under weakly finite and localized inlet disturbances. This is particularly pronounced when the base inflow is of the plug type because downstream the nearwall flow develops under an accelerating core. When the disturbance ring is located in the core region of the pipe inlet, helical vortex filaments evolve into a train of large-scale reverse hairpin vortices. The interaction of these reverse hairpins among themselves or with the near-wall flow produces small-scale hairpin packets, which leads to breakdown. When the inlet disturbance ring is located near the wall, certain quasi-spanwise structure is stretched into a Lambda vortex, and develops into a large-scale hairpin vortex. Further downstream, small-scale hairpin packets emerge near the tip region of the large-scale hairpin vortex, and subsequently grow into a turbulent spot, which is itself a local concentration of small-scale hairpin vortices. This process is broadly analogous to boundary layer bypass transition in the narrow sense, and also to the secondary instability and breakdown in natural transition. With the given inlet disturbances, the energy norm was found to grow exponentially rather than algebraically. Friction factor exhibits overshoot with parabolic base flow. The Osborne Reynolds pipe transition is indeed abrupt with respect to the Reynolds number, but can be rather gradual with respect to the axial distance through a sequence of events. Some of the previously attributed abruptness and mysteriousness was perhaps due to the inability to study the process accurately with very fine spatial and temporal resolution.
1. Reynolds O (1883) An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Philos Trans R Soc A 174:935–982. 2. Mullin T (2011) Experimental studies of transition to turbulence in a pipe. Annu Rev Fluid Mech 43:1–24. 3. Eckhardt B (2011) A critical point for turbulence. Science 333:165–166. 4. Faisst H, Eckhardt B (2003) Traveling waves in pipe flow. Phys Rev Lett 91(22):224502. 5. Hof B, et al. (2004) Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305(5690):1594–1598. 6. Peixinho J, Mullin T (2006) Decay of turbulence in pipe flow. Phys Rev Lett 96(9):094501. 7. Hof B, Westerweel J, Schneider TM, Eckhardt B (2006) Finite lifetime of turbulence in shear flows. Nature 443(7107):59–62. 8. Willis AP, Peixinho J, Kerswell RR, Mullin T (2008) Experimental and theoretical progress in pipe flow transition. Philos Trans A Math Phys Eng Sci 366(1876):2671–2684. 9. Moxey D, Barkley D (2010) Distinct large-scale turbulent-laminar states in transitional pipe flow. Proc Natl Acad Sci USA 107(18):8091–8096. 10. Wu X, Moin P, Hickey J-P (2014) Boundary layer bypass transition. Phys Fluids 26(9):091104. 11. Matsubara M, Alfredsson PH (2001) Disturbance growth in boundary layers subjected to free-stream turbulence. J Fluid Mech 430:149–168. 12. Åsén P, Kreiss G, Rempfer D (2010) Direct numerical simulations of localized disturbances in pipe Poiseuille flow. Comput Fluids 39(6):926–935. 13. Pierce CD, Moin P (2004) Progress variable approach for large-eddy simulation of nonpremixed turbulent combustion. J Fluid Mech 504:73–97. 14. Pierce CD, Moin P (2001) Progress variable approach for large-eddy simulation of turbulent combustion. Mechanical Engineering Department Report TF-80 (Stanford University, Stanford, CA).
15. Kim J, Moin P, Moser R (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177:133–166. 16. Wu X, Moin P (2008) A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J Fluid Mech 608:81–112. 17. Adrian RJ (2007) Hairpin vortex organization in wall turbulence. Phys Fluids 19(4):041301. 18. Moin P, Kim J (1985) The structure of the vorticity field in turbulent channel flow. Part 1. Analysis of instantaneous fields and statistical correlations. J Fluid Mech 155: 441–464. 19. McKeon BJ, Swanson CJ, Zagarola MV, Donnelly RJ, Smits AJ (2004) Friction factors for smooth pipe flow. J Fluid Mech 511:41–44. 20. Skufca JD, Yorke JA, Eckhardt B (2006) Edge of chaos in a parallel shear flow. Phys Rev Lett 96(17):174101. 21. Schneider TM, Eckhardt B, Yorke JA (2007) Turbulence transition and the edge of chaos in pipe flow. Phys Rev Lett 99(3):034502. 22. Avila K, et al. (2011) The onset of turbulence in pipe flow. Science 333(6039):192–196. 23. Sayadi T, Hamman CW, Moin P (2013) Direct numerical simulation of complete H-type and K-type transitions with implications for the dynamics of turbulent boundary layers. J Fluid Mech 724:480–509. 24. Han G, Tumin A, Wygnanski I (2000) Laminar–turbulent transition in Poiseuille pipe flow subjected to periodic perturbation emanating from the wall. Part 2. Late stage of transition. J Fluid Mech 419:1–27. 25. Peixinho J, Mullin T (2007) Finite-amplitude thresholds for transition in pipe flow. J Fluid Mech 582:169–178. 26. Tasaka Y, Schneider TM, Mullin T (2010) Folded edge of turbulence in a pipe. Phys Rev Lett 105(17):174502.
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ACKNOWLEDGMENTS. X.W. acknowledges support from the Natural Science and Engineering Research Council of Canada; P.M. is grateful for support from the US Department of Energy and Air Force Office of Scientific Research; R.J.A. and J.R.B. acknowledge support from the US National Science Foundation.
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before breakdown; Fig. 2 E and F. This subtle effect could not have been detected by Reynolds and other subsequent experimentalists when they used less-detailed, low-resolution observation techniques. What they visualized was likely the late-stage phenomenon associated with the exponential growth rate identified in the present study and in ref. 21. This explains, to a certain extent, the abrupt and mysterious breakdown widely attributed to the Osborne Reynolds transition when in fact the vortex development can be a rather slow and gradual process via a sequence of events as shown in Fig. 2 and Movie S3. Refs. 25 and 26 applied periodic disturbances using a push– pull device through holes in the pipe surface. One interesting aspect of their study is that, compared with the previous similar injection–suction approaches such as that in ref. 6, the perturbation amplitude in ref. 26 was quite weak; only 0.1–0.3% of the mean pipe flow flux was applied through the holes. This feature permits a clear observation of the transition process. Waves of hairpin-like structures were observed, which often break down when they cross the centerline of the pipe. The hairpin waves start from locations close to where the disturbance was applied. They also noted that the process was consistent with the temporal algebraic growth theory. The present vortex dynamics is distinct from that in ref. 26 for several reasons. Reverse hairpin streets in steps 3 and 4 start far downstream from the inlet where the disturbance is introduced; and the reverse hairpins are not deliberately created through jet-in-cross-flow. Instead, they are a result of the gradual downstream amplification of the upstream perturbations. The evolution of Lambda vortex to hairpin packet to turbulent spot discussed in step 5 also occurs naturally (without artificial creation) downstream of the inlet at an irregular and much longer time interval compared with that in ref. 26. Of course, the exponential growth rate in disturbance energy is also different from the algebraic growth in ref. 25. Inlet mass flow rate is held constant in the present study, which guarantees a constant Reynolds number. This was also achieved experimentally in refs. 25 and 26 through a clever design. In step 6, we study the effect of switching the parabolic inflow to a plug inflow. We superimposed on the plug base flow the same inlet disturbance ring as that in S3R8 and S4R6, but transition was not observed; using the same ring as that in S5R8 did not result in transition either. Through further testing, it was found that the inlet ring had to be widened and located closer to the wall to produce downstream transition. In case S6R8, the disturbance ring was widened to 0.8 ≤ r=R < 0.94 and successful downstream
The precise dynamics of breakdown in pipe transition is a centuryold unresolved problem in fluid mechanics. We demonstrate that the abruptness and mysteriousness attributed to the Osborne Reynolds pipe transition can be partially resolved with a spatially developing direct simulation that carries weakly but finitely perturbed laminar inflow through gradual rather than abrupt transition arriving at the fully developed turbulent state. Our results with this approach show during transition the energy norms of such inlet perturbations grow exponentially rather than algebraically with axial distance. When inlet disturbance is located in the core region, helical vortex filaments evolve into large-scale reverse hairpin vortices. The interaction of these reverse hairpins among themselves or with the near-wall flow when they descend to the surface from the core produces small-scale hairpin packets, which leads to breakdown. When inlet disturbance is near the wall, certain quasi-spanwise structure is stretched into a Lambda vortex, and develops into a large-scale hairpin vortex. Small-scale hairpin packets emerge near the tip region of the large-scale hairpin vortex, and subsequently grow into a turbulent spot, which is itself a local concentration of small-scale hairpin vortices. This vortex dynamics is broadly analogous to that in the boundary layer bypass transition and in the secondary instability and breakdown stage of natural transition, suggesting the possibility of a partial unification. Under parabolic base flow the friction factor overshoots Moody’s correlation. Plug base flow requires stronger inlet disturbance for transition. Accuracy of the results is demonstrated by comparing with analytical solutions before breakdown, and with fully developed turbulence measurements after the completion of transition.
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pipe flow transition spatially evolving
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turbulence
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direct numerical simulation
introduced at the laminar pipe inlet will induce gradual transition, eventually leading to a state of fully developed turbulence. Finite amplitude is needed because pipe flow is linearly stable with respect to infinitesimal disturbances. Weak and localized inlet disturbance is preferred whenever possible so as not to destroy the overall characteristics of the base parabolic/plug flow. It would be ideal if there exists an extended streamwise range before breakdown over which statistics of the slightly perturbed flow essentially agree with the fully developed laminar solution. This is analogous to the approach for boundary layer bypass transition in the narrow sense (10). There, one is usually concerned with perturbations less than ∼5% of the freestream velocity because stronger disturbances will immediately produce transition or large patches of turbulence (11), which therefore is not useful for a careful study of the transition process. Ref. 12 simulated strong injection and suction through slots opened on the wall of a short pipe (60× the pipe radius R) in a spatially developing computation; however, no quantitative results were presented. The current method differs from previous approaches in which pipe flow was subjected to strong jet-in-cross-flow–type injection or suction to directly generate pipe turbulence. Confidence in our method can be established by evaluating statistics against analytical solutions in the early laminar region before breakdown, and against established data in the fully developed turbulent region after the completion of transition. The validated DNS can subsequently provide data on the dynamics in the Significance
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The precise dynamics of disturbance energy growth and breakdown in pipe transition is a century-old unresolved problem in fluid mechanics. In this paper, we demonstrate that the mystery attributed to the breakdown process of the Osborne Reynolds pipe transition can be partially resolved with a direct, spatially evolving simulation that carries weakly but finitely perturbed laminar inflow through gradual rather than abrupt transition arriving at the fully developed turbulent state. Some of the previously attributed abruptness and mysteriousness was perhaps due to the inability to study the process accurately with very fine spatial and temporal resolution. The energy norm was found to grow exponentially rather than algebraically. The sensitivity of the transition process to pipe entrance conditions is demonstrated.
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he vast and expanding realm of fluid mechanics research on transition and turbulence can actually be traced back to a single point in history: the publication of Osborne Reynolds’ 1883 pipe flow paper (1) in which the concept of Reynolds number was introduced. Given the historical, fundamental, and applied importance of the problem, it is ironic that the Osborne Reynolds pipe transition remains to this day “abrupt and mysterious” (2, 3). Significant progress has been made during the past decade, mostly concentrating on the detection of traveling wave (4, 5), and on the lifetime and reverse transition (relaminarization) of existing pipe flow turbulence produced by strong jet-in-crossflow type of blowing and suction (6, 7). Refs. 8 and 9 reported insightful relaminarization simulations using the axially periodic boundary condition. We tackle directly the Osborne Reynolds pipe transition problem with spatially developing direct numerical simulation (DNS). The disturbance energy growth rate with respect to axial distance, and how the friction factor and vortex structures develop during pipe transition with the distance, is currently unknown. We anticipate that weakly finite and localized disturbances www.pnas.org/cgi/doi/10.1073/pnas.1509451112
Author contributions: X.W., P.M., and R.J.A. designed research; X.W. performed research; P.M. contributed new reagents/analytic tools; X.W., P.M., R.J.A., and J.R.B. analyzed data; and X.W., P.M., R.J.A., and J.R.B. wrote the paper. Reviewers: B.E., University of Marburg; and T.M., University of Manchester. The authors declare no conflict of interest. Freely available online through the PNAS open access option. 1
To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1509451112/-/DCSupplemental.
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Contributed by Parviz Moin, May 18, 2015 (sent for review March 27, 2015; reviewed by Bruno Eckhardt and Thomas Mullin)
transitional region that is bounded by these two verified ends. The DNS will be done in a laboratory reference frame without the axially periodic boundary condition. The governing equations are the continuity and the Navier– Stokes equations for incompressible flow in cylindrical coordinates. The computer program and the numerical scheme were described by Pierce and Moin (13, 14) in their large-eddy simulation of combustion in a coaxial jet combustor. That configuration is degenerated into a spatially developing circular pipe in this work. Unless otherwise noted, the pipe length is 250 R, and the computational mesh size is 8,192 × 200 × 256 in the axial ðzÞ, radial ðrÞ, and azimuthal ðθÞ directions, respectively. The time step remains a constant Δt = 0.008R=V, where V is the bulk velocity. An overbar denotes ensemble averaging, superscript “+” refers to normalization by friction velocity uτ for velocity and by viscous wall unit ν=uτ for distance; ν is the kinematic viscosity. Poisson’s equation for pressure is solved using Fourier and cosine transform along the azimuthal and axial directions, respectively. The entire set of DNS reported here took four calendar years to complete. Six steps were taken to systematically vary the inlet condition ðz = 0Þ and quantitatively assess the downstream response. In step 1, plug inflow was first prescribed without perturbation. The downstream velocity develops into the expected parabolic profile. Replacing the plug inflow with the exact parabolic inflow simply kept the downstream flow unchanged from the inlet. In step 2, perturbations were introduced at the inlet under the guiding principle that the disturbance magnitude should be finite yet weak and well-controlled. Another guiding principle is that the setup should permit transition to proceed to a fully developed turbulent state downstream. Initially, the inlet base flow was parabolic, Re = 2VR=ν = 5,300, and the inlet perturbations were confined within the tiny circle 0 ≤ r=R < 0.02 by superposing either white noise or isotropic grid turbulence on the base flow, but they disappeared rapidly without exciting any noticeable turbulence. We subsequently imposed instantaneous velocity fields extracted from a separate, axially periodic DNS of the turbulent pipe flow at Re = 5,300 over the same radial range 0 ≤ r=R < 0.02, but no transition was observed. Additional tests by increasing the contaminated area from 0 ≤ r=R < 0.02 up to 0 ≤ r=R < 0.10 failed
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to produce transition. Increasing Re from 5,300 to 8,000 did not produce a full transition either. Changing the parabolic base flow to the plug inflow did not produce transition. Of course, if Re is increased to a very large value, transition will eventually be obtained under these disturbances. The resolution constraint required by high-quality DNS prevents us from increasing Re further. Our finite yet localized weak disturbance principle does not encourage us to further enlarge the inlet perturbation area either. Laminar pipe flow below Re = 8,000 is reluctant to transition under such disturbances. In step 3, inspired by the absolute instability of tangential discontinuity in general shear flows, we made the perturbation area over the inlet plane into a narrow ring instead of a small full circle. Within this ring the exact parabolic velocity profile was replaced by the time-dependent, fully developed turbulent DNS field at Re = 5,300. With the ring positioned at 0.4 ≤ r=R < 0.42, it was found that at Re = 5,300 the flow still returned to laminar after a transient turbulent spot. Successful transition was obtained at Re = 8,000 (case S3R8, Figs. 1A and 2H). At the inlet, the perturbed area is well localized and accounts for merely and the energy norm jj E jj = R R 2 1.6%2 of the 2cross-section, 2 2 ðu ′ + u ′ + u ′ Þrdr=ðR V Þ = 5.8 × 10−5 is less than 1% z,rms θ,rms r,rms 0 of the fully developed turbulent pipe value 9.3 × 10−3, and is less than 0.15% compared with the peak value 3.9 × 10−2 attained ′ are the turbulence induring transition. Here u′z,rms , u′r,rms , uθ,rms tensities. This inlet disturbance is analogous to a turbulent wake arising from a very thin wire ring mounted on the inlet plane in a laboratory. Under this perturbation, the parabolic pipe flow gradually breaks down and develops into a fully developed turbulent state downstream. Switching the inlet ring turbulent fluctuation to a mean wake deficit superimposed with white noise failed to produce transition. In the present simulation, there is no imposed axial pressure gradient. After transition, R+ = uτ R=ν = 258.5, uτ = 0.06462V ; grid resolution is Δz+ = 7.5 and ΔðRθÞ+ = 6.3. Auxiliary simulation on the fully developed turbulent pipe flow at Re = 8,000 (case S3R8T) was performed using the conventional axially periodic boundary condition over a 30-R-long domain with a grid 2,048 × 256 × 512. Fig. 3 compares the turbulent statistics between S3R8T and S3R8. Note the
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Fig. 1. Friction factor f (●) and energy norm 10−2 log10 kEk (◇). Dash-dot-dash: f = 64=Re; dash-dot-dot-dash: Moody’s correlation. (A) Case S3R8; (B) case S4R6; (C) case S5R8; (D) case S6R8. Contoured insets are uz ðr, θ, z = 0, tÞ at one random instant.
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Fig. 2. Vortex structures in pipe transition. Case S5R8: (A) two infant turbulent spots; (B) the predecessors of the two turbulent spots in A; (C) subsequent merging of the two turbulent spots in A; (D) isosurface of scalar φ = 0.05 showing scalar breakdown; (E) zoomed view of D showing modulation of scalar field by hairpin packet; (F) zoomed view of D showing a street of indentions on scalar isosurface; (G) planar view of uz; (H) the breakdown region in case S3R8; (I) from inlet to breakdown in case S4R6.
reliability of the approach of S3R8T for developed turbulent channel/pipe flow has been well established since the work of ref. 15; see also ref. 16. All of the statistics, including even the rate of viscous dissipation «+, agree very well between the two computations, demonstrating the quantitative reliability of the present spatially developing pipe transition simulation. The transition is continuous in the sense that no isolated turbulent spot was observed in this case; see the vortex visualization over the breakdown region 30 ≤ z=R < 45 in Fig. 2H and Movie S1 using isosurfaces of swirling strength λci (17); red color indicates uz > 1.3V. The finite-amplitude disturbances from the ring create a roughly annular turbulent wake that grows downstream radially inward and outward. Transition is triggered by vortex filaments drawn from the disturbance region, intensified by stretching, and moving toward the wall. Between 30 and 40 R the filaments induce wall-ward radial flows; in turn, they create large reverse hairpin vortices that Wu et al.
rapidly induce hairpin packets near the wall. Small-scale activity explodes when the packets grow and interact with other vortex structures. The small-scale activity overcomes the larger scale fluctuations from the initial disturbances as it grows and interacts to fill the pipe with turbulence that approaches the fully developed state by about 75 R. A reverse hairpin vortex as part of a diamond structure was reported in previous turbulent channel simulations of ref. 18; the present reverse hairpins occur in a train among a quiescent environment, are of large scale (several R in length), and are much more distinct. In step 4, with identical inlet condition as in S3R8, Re was first reduced from 8,000 to 5,700, and the flow returned completely laminar. Reducing Re from 8,000 to 6,000 (case S4R6) produced delayed transition compared with S3R8. Thus, for this particular inlet disturbance, the critical Reynolds number is between 5,700 and 6,000. In S4R6 after transition, PNAS Early Edition | 3 of 5
Fig. 3. Comparison between turbulent statistics in the axially periodic turbulent case S3R8T (dotted line) and in the fully developed turbulent region ′+ ′+ 210 ≤ z=R ≤ 240 of case S3R8 (solid line). □, u+z =10; ◇, u′+ z,rms ; ★, uθ,rms ; +, ur,rms ; + △, −u′z u′r ; ○, −10«+ .
R+ = 200.1, uτ = 0.0667V , Δz+ = 6.11, ΔðRθÞ+ = 4.91. An auxiliary simulation of the fully developed turbulent pipe flow at Re = 6,000 was also performed (case S4R6T), and the accuracy was further verified by comparing the turbulent statistics from S4R6 and S4R6T. Friction factor f as a function of z is shown in Fig. 1B. Although data on f variation with Re are widely available (19), we cannot find how f varies with z during transition from the literature. The present data show that before breakdown, f agrees very well with the analytical solution 64=Re; after the completion of transition, f agrees with Moody’s correlation for smooth, fully developed turbulent pipe flow. Thus, the quantitative reliability of the present spatially developing DNS is again demonstrated. The peak in f is related to the change of sign of the near-wall mean radial velocity gradient. In boundary layers it is also related to the sudden increase in entrainment of outer flow into the near-wall region (10). Fig. 1 also presents the development of energy norm jj E jj with z. There is an extended, convincing, exponential growth region before breakdown in S4R6. It is known that infinitesimally small disturbances will not be amplified exponentially in a parabolic pipe flow. The four sets of energy norm jj E jj results in Fig. 1 suggest that localized, weakly finite disturbances are often amplified exponentially rather than algebraically with respect to axial distance in the pipe flow. We also found that shortly downstream of the inlet in S4R6 and S3R8, the mean flow profile within the ring quickly recovers to approximately that of the base flow. The vortex structure development of S4R6 shown in Fig. 2I is similar to that in S3R8 with a sequential formation of helical vortex filaments, large-scale reverse hairpin vortices, followed by smallscale hairpin packets and transition without turbulent spots; see also Movie S2. There are several notable overall agreements between our observations in steps 2–4 and previous dynamical system analysis of pipe transition, and the associated numerical work performed using axially periodic temporal DNS. Ref. 20 defined the object of “edge-of-chaos,” which separates, in state space, the regions of initial conditions where the lifetime statistics exhibit a sensitive dependence on initial conditions capable of resulting in turbulence from those decaying to laminar flow. In this work, we have repeatedly encountered situations in which a very minor change in either Reynolds number or inlet disturbance ring location made a decisive difference regarding whether the downstream flow stays laminar or transitions to turbulence––the flow parameter settings are close to the edge-of-chaos in the state 4 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1509451112
space. We have also found in other situations, as in the tests described in step 2, where the pipe flow resists transition even under significant modifications in either Reynolds number or inlet disturbance; these settings are far away from the edge-of-chaos. Ref. 21 reported temporal DNS in a pipe with a length of 10 R, and a pair of streamwise vortices with modulated tilting along the pipe axis as an initial condition. The inset of figure 4 in ref. 21 shows that the energy norm difference between the bounding (amplification and decay) trajectories of the edge-of-chaos grows exponentially with time, which was attributed to a shear flow instability arising from the temporal development of the initially imposed streamwise vortex pair. The present DNS demonstrates that spatial amplification of weakly finite inlet perturbations along the axial direction of a laminar pipe is exponential. The relation between this exponential growth rate and the absolute instability of tangential discontinuity of the inlet condition needs to be clarified in future research. In steps 3 and 4, the inlet disturbance rings are confined in the core region. At the prescribed two Reynolds numbers 8,000 and 6,000, when the large reverse hairpin from the core impinges on the wall, a small-scale hairpin packet is generated at the surface, leading to turbulence. This is somewhat consistent from the results of ref. 22 at Reynolds number less than 2,300. There, it was observed that a turbulent puff from the core could, but not always, impinge on the wall and induce a secondary turbulent puff. It is conceivable that if the Reynolds number is reduced slightly in step 4 from 6,000, the secondary hairpin packet will form an isolated puff instead of resulting continuous turbulence. In step 5, we address the effect of the disturbance ring location by first repositioning it from 0.4 ≤ r=R < 0.42 to 0.9 ≤ r=R < 0.92. At Re = 8,000, the flow transitions immediately at the inlet. This is not useful because it does not permit us to study how the disturbances are amplified, and how the breakdown process happens. The width of the ring was therefore reduced to 0.9 ≤ r=R < 0.91. This minor modification completely relaminarized the flow. Successful and prolonged transition between 50 ≤ z=R < 150 at Re = 8,000 was observed when the ring was positioned at 0.9 ≤ r=R < 0.915, which covers 2.7% of the inlet cross-section (case S5R8). It was further found that at this setting, reducing Re from 8,000 to 7,000 produced relaminarization. Inspired by the colored band visualization of Osborne Reynolds, a passive scalar φ = 1 was also introduced at the inlet over a small circle 0 ≤ r=R < 0.05 (Fig. 2D). Note φ = 0 over the rest of the inflow plane; the molecular Prandtl number was 1. A refined mesh of 16,384 × 200 × 512 was also used (case S5R8F). During transition, there is an overshoot of f over the turbulent correlation in S3R8, S4R6, and S5R8 for which the base flows are parabolic. Vortex structure development is different from that in the previous two cases when the inlet disturbances were in the core region. Here, as shown in Fig. 2 A and B, structures surviving the initial decay near the inlet have the shape of a Lambda vortex, possibly because their oblique orientation is favored by shear flow amplification via the lift-up mechanism. The Lambda vortex subsequently develops into an elongated large-scale (on the order of R) hairpin vortex; further downstream, a small-scale hairpin packet emerges near the tip region of the large-scale hairpin vortex. A turbulent spot is subsequently formed, which is essentially a local concentration of small-scale hairpin vortices (Fig. 2C). This process is analogous to the vortex development in boundary layer bypass transition in the narrow sense (10), and is also broadly similar to the secondary instability and breakdown stage of boundary layer natural transition (23). This similarity was conjectured and argued for in ref. 24 based on interpretation of ensemble-averaged hot-wire signals; here our DNS provides direct, convincing, time-accurate, 3D evidence (see also Movie S3). Although the breakdown of the scalar field in Fig. 2D may seem abrupt at first glance, careful examination shows that the scalar field has already been modulated by hairpin packets long Wu et al.
transition was observed at Re = 8,000. However, reducing Re from 8,000 to 7,500 relaminarized the flow. With the plug inflow, jj E jj continues to exhibit robust exponential growth rate after a prolonged decay to 75 R, but friction factor f does not show the overshoot found in previous cases with the parabolic base inflow; Fig. 1D. This is analogous to the boundary layer bypass transition in the narrow sense in which the skin friction has no overshoot either (10). The sharp rise of skin friction during transition is closely related to the entrainment of outer fluid into the near-wall region. Vortex dynamics is similar to that in case S5R8 when the inlet disturbance is also near the wall, characterized by the sequential formation of a Lambda vortex, hairpin packet, and turbulent spot, which is a localized hairpin forest. The accuracy of the results is demonstrated. Below Re = 8,000 laminar pipe flow is reluctant to transition under weakly finite and localized inlet disturbances. This is particularly pronounced when the base inflow is of the plug type because downstream the nearwall flow develops under an accelerating core. When the disturbance ring is located in the core region of the pipe inlet, helical vortex filaments evolve into a train of large-scale reverse hairpin vortices. The interaction of these reverse hairpins among themselves or with the near-wall flow produces small-scale hairpin packets, which leads to breakdown. When the inlet disturbance ring is located near the wall, certain quasi-spanwise structure is stretched into a Lambda vortex, and develops into a large-scale hairpin vortex. Further downstream, small-scale hairpin packets emerge near the tip region of the large-scale hairpin vortex, and subsequently grow into a turbulent spot, which is itself a local concentration of small-scale hairpin vortices. This process is broadly analogous to boundary layer bypass transition in the narrow sense, and also to the secondary instability and breakdown in natural transition. With the given inlet disturbances, the energy norm was found to grow exponentially rather than algebraically. Friction factor exhibits overshoot with parabolic base flow. The Osborne Reynolds pipe transition is indeed abrupt with respect to the Reynolds number, but can be rather gradual with respect to the axial distance through a sequence of events. Some of the previously attributed abruptness and mysteriousness was perhaps due to the inability to study the process accurately with very fine spatial and temporal resolution.
1. Reynolds O (1883) An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Philos Trans R Soc A 174:935–982. 2. Mullin T (2011) Experimental studies of transition to turbulence in a pipe. Annu Rev Fluid Mech 43:1–24. 3. Eckhardt B (2011) A critical point for turbulence. Science 333:165–166. 4. Faisst H, Eckhardt B (2003) Traveling waves in pipe flow. Phys Rev Lett 91(22):224502. 5. Hof B, et al. (2004) Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305(5690):1594–1598. 6. Peixinho J, Mullin T (2006) Decay of turbulence in pipe flow. Phys Rev Lett 96(9):094501. 7. Hof B, Westerweel J, Schneider TM, Eckhardt B (2006) Finite lifetime of turbulence in shear flows. Nature 443(7107):59–62. 8. Willis AP, Peixinho J, Kerswell RR, Mullin T (2008) Experimental and theoretical progress in pipe flow transition. Philos Trans A Math Phys Eng Sci 366(1876):2671–2684. 9. Moxey D, Barkley D (2010) Distinct large-scale turbulent-laminar states in transitional pipe flow. Proc Natl Acad Sci USA 107(18):8091–8096. 10. Wu X, Moin P, Hickey J-P (2014) Boundary layer bypass transition. Phys Fluids 26(9):091104. 11. Matsubara M, Alfredsson PH (2001) Disturbance growth in boundary layers subjected to free-stream turbulence. J Fluid Mech 430:149–168. 12. Åsén P, Kreiss G, Rempfer D (2010) Direct numerical simulations of localized disturbances in pipe Poiseuille flow. Comput Fluids 39(6):926–935. 13. Pierce CD, Moin P (2004) Progress variable approach for large-eddy simulation of nonpremixed turbulent combustion. J Fluid Mech 504:73–97. 14. Pierce CD, Moin P (2001) Progress variable approach for large-eddy simulation of turbulent combustion. Mechanical Engineering Department Report TF-80 (Stanford University, Stanford, CA).
15. Kim J, Moin P, Moser R (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177:133–166. 16. Wu X, Moin P (2008) A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J Fluid Mech 608:81–112. 17. Adrian RJ (2007) Hairpin vortex organization in wall turbulence. Phys Fluids 19(4):041301. 18. Moin P, Kim J (1985) The structure of the vorticity field in turbulent channel flow. Part 1. Analysis of instantaneous fields and statistical correlations. J Fluid Mech 155: 441–464. 19. McKeon BJ, Swanson CJ, Zagarola MV, Donnelly RJ, Smits AJ (2004) Friction factors for smooth pipe flow. J Fluid Mech 511:41–44. 20. Skufca JD, Yorke JA, Eckhardt B (2006) Edge of chaos in a parallel shear flow. Phys Rev Lett 96(17):174101. 21. Schneider TM, Eckhardt B, Yorke JA (2007) Turbulence transition and the edge of chaos in pipe flow. Phys Rev Lett 99(3):034502. 22. Avila K, et al. (2011) The onset of turbulence in pipe flow. Science 333(6039):192–196. 23. Sayadi T, Hamman CW, Moin P (2013) Direct numerical simulation of complete H-type and K-type transitions with implications for the dynamics of turbulent boundary layers. J Fluid Mech 724:480–509. 24. Han G, Tumin A, Wygnanski I (2000) Laminar–turbulent transition in Poiseuille pipe flow subjected to periodic perturbation emanating from the wall. Part 2. Late stage of transition. J Fluid Mech 419:1–27. 25. Peixinho J, Mullin T (2007) Finite-amplitude thresholds for transition in pipe flow. J Fluid Mech 582:169–178. 26. Tasaka Y, Schneider TM, Mullin T (2010) Folded edge of turbulence in a pipe. Phys Rev Lett 105(17):174502.
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ACKNOWLEDGMENTS. X.W. acknowledges support from the Natural Science and Engineering Research Council of Canada; P.M. is grateful for support from the US Department of Energy and Air Force Office of Scientific Research; R.J.A. and J.R.B. acknowledge support from the US National Science Foundation.
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before breakdown; Fig. 2 E and F. This subtle effect could not have been detected by Reynolds and other subsequent experimentalists when they used less-detailed, low-resolution observation techniques. What they visualized was likely the late-stage phenomenon associated with the exponential growth rate identified in the present study and in ref. 21. This explains, to a certain extent, the abrupt and mysterious breakdown widely attributed to the Osborne Reynolds transition when in fact the vortex development can be a rather slow and gradual process via a sequence of events as shown in Fig. 2 and Movie S3. Refs. 25 and 26 applied periodic disturbances using a push– pull device through holes in the pipe surface. One interesting aspect of their study is that, compared with the previous similar injection–suction approaches such as that in ref. 6, the perturbation amplitude in ref. 26 was quite weak; only 0.1–0.3% of the mean pipe flow flux was applied through the holes. This feature permits a clear observation of the transition process. Waves of hairpin-like structures were observed, which often break down when they cross the centerline of the pipe. The hairpin waves start from locations close to where the disturbance was applied. They also noted that the process was consistent with the temporal algebraic growth theory. The present vortex dynamics is distinct from that in ref. 26 for several reasons. Reverse hairpin streets in steps 3 and 4 start far downstream from the inlet where the disturbance is introduced; and the reverse hairpins are not deliberately created through jet-in-cross-flow. Instead, they are a result of the gradual downstream amplification of the upstream perturbations. The evolution of Lambda vortex to hairpin packet to turbulent spot discussed in step 5 also occurs naturally (without artificial creation) downstream of the inlet at an irregular and much longer time interval compared with that in ref. 26. Of course, the exponential growth rate in disturbance energy is also different from the algebraic growth in ref. 25. Inlet mass flow rate is held constant in the present study, which guarantees a constant Reynolds number. This was also achieved experimentally in refs. 25 and 26 through a clever design. In step 6, we study the effect of switching the parabolic inflow to a plug inflow. We superimposed on the plug base flow the same inlet disturbance ring as that in S3R8 and S4R6, but transition was not observed; using the same ring as that in S5R8 did not result in transition either. Through further testing, it was found that the inlet ring had to be widened and located closer to the wall to produce downstream transition. In case S6R8, the disturbance ring was widened to 0.8 ≤ r=R < 0.94 and successful downstream
