Osborne Reynolds Pipe Flow: Direct Simulation from Laminar Through Gradual Transition to Fully Developed Turbulence

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

aDepartment 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 85287- 6106; and dLos Alamos National Laboratory, Los Alamos, NM 87545

Contributed by Parviz Moin, May 18, 2015 (sent for review March 27, 2015; reviewed by Bruno Eckhardt and Thomas Mullin) The precise dynamics of breakdown in pipe transition is a century- introduced at the laminar pipe inlet will induce gradual transi- old unresolved problem in fluid mechanics. We demonstrate that tion, eventually leading to a state of fully developed turbulence. the abruptness and mysteriousness attributed to the Osborne Rey- Finite amplitude is needed because pipe flow is linearly stable nolds pipe transition can be partially resolved with a spatially de- with respect to infinitesimal disturbances. Weak and localized veloping direct simulation that carries weakly but finitely perturbed inlet disturbance is preferred whenever possible so as not to laminar inflow through gradual rather than abrupt transition arriv- destroy the overall characteristics of the base parabolic/plug ing at the fully developed turbulent state. Our results with this flow. It would be ideal if there exists an extended streamwise approach show during transition the energy norms of such inlet range before breakdown over which statistics of the slightly perturbations grow exponentially rather than algebraically with ax- perturbed flow essentially agree with the fully developed laminar ial distance. When inlet disturbance is located in the core region, solution. This is analogous to the approach for boundary layer helical vortex filaments evolve into large-scale reverse hairpin vor- bypass transition in the narrow sense (10). There, one is usually tices. The interaction of these reverse hairpins among themselves or concerned with perturbations less than ∼5% of the freestream with the near-wall flow when they descend to the surface from the velocity because stronger disturbances will immediately produce core produces small-scale hairpin packets, which leads to break- transition or large patches of turbulence (11), which therefore is down. When inlet disturbance is near the wall, certain quasi-span- not useful for a careful study of the transition process. Ref. 12 wise structure is stretched into a Lambda vortex, and develops into simulated strong injection and suction through slots opened on a large-scale hairpin vortex. Small-scale hairpin packets emerge the wall of a short pipe (60× the pipe radius R) in a spatially near the tip region of the large-scale hairpin vortex, and subse- developing computation; however, no quantitative results were quently grow into a turbulent spot, which is itself a local concen- presented. The current method differs from previous approaches tration of small-scale hairpin vortices. This vortex dynamics is in which pipe flow was subjected to strong jet-in-cross-flow–type broadly analogous to that in the boundary layer bypass transition injection or suction to directly generate pipe turbulence. Confi- and in the secondary instability and breakdown stage of natural dence in our method can be established by evaluating statistics transition, suggesting the possibility of a partial unification. Under ’ against analytical solutions in the early laminar region before parabolic base flow the friction factor overshoots Moody s corre- breakdown, and against established data in the fully developed lation. Plug base flow requires stronger inlet disturbance for tran- turbulent region after the completion of transition. The validated sition. Accuracy of the results is demonstrated by comparing with DNS can subsequently provide data on the dynamics in the analytical solutions before breakdown, and with fully developed turbulence measurements after the completion of transition. Significance pipe flow | transition | turbulence | direct numerical simulation | spatially evolving 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 he vast and expanding realm of fluid mechanics research on the mystery attributed to the breakdown process of the transition and turbulence can actually be traced back to a T Osborne Reynolds pipe transition can be partially resolved with single point in history: the publication of Osborne Reynolds’ a direct, spatially evolving simulation that carries weakly but 1883 pipe flow paper (1) in which the concept of Reynolds finitely perturbed laminar inflow through gradual rather than number was introduced. Given the historical, fundamental, and abrupt transition arriving at the fully developed turbulent applied importance of the problem, it is ironic that the Osborne state. Some of the previously attributed abruptness and mys- Reynolds pipe transition remains to this day “abrupt and mys- ” teriousness was perhaps due to the inability to study the terious (2, 3). process accurately with very fine spatial and temporal resolu- Significant progress has been made during the past decade, tion. The energy norm was found to grow exponentially rather mostly concentrating on the detection of traveling wave (4, 5), than algebraically. The sensitivity of the transition process to and on the lifetime and reverse transition (relaminarization) of pipe entrance conditions is demonstrated. existing pipe flow turbulence produced by strong jet-in-cross- flow type of blowing and suction (6, 7). Refs. 8 and 9 reported Author contributions: X.W., P.M., and R.J.A. designed research; X.W. performed research; insightful relaminarization simulations using the axially periodic P.M. contributed new reagents/analytic tools; X.W., P.M., R.J.A., and J.R.B. analyzed data; boundary condition. and X.W., P.M., R.J.A., and J.R.B. wrote the paper. We tackle directly the Osborne Reynolds pipe transition Reviewers: B.E., University of Marburg; and T.M., University of Manchester. problem with spatially developing direct numerical simulation The authors declare no conflict of interest. (DNS). The disturbance energy growth rate with respect to axial Freely available online through the PNAS open access option. distance, and how the friction factor and vortex structures develop 1To whom correspondence should be addressed. Email: [email protected]. during pipe transition with the distance, is currently unknown. This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. We anticipate that weakly finite and localized disturbances 1073/pnas.1509451112/-/DCSupplemental.

7920–7924 | PNAS | June 30, 2015 | vol. 112 | no. 26 www.pnas.org/cgi/doi/10.1073/pnas.1509451112 Downloaded by guest on September 25, 2021 transitional region that is bounded by these two verified ends. to produce transition. Increasing Re from 5,300 to 8,000 did not The DNS will be done in a laboratory reference frame without produce a full transition either. Changing the parabolic base flow the axially periodic boundary condition. to the plug inflow did not produce transition. Of course, if Re is The governing equations are the continuity and the Navier– increased to a very large value, transition will eventually be Stokes equations for incompressible flow in cylindrical coordi- obtained under these disturbances. The resolution constraint re- nates. The computer program and the numerical scheme were quired by high-quality DNS prevents us from increasing Re fur- described by Pierce and Moin (13, 14) in their large-eddy simu- ther. Our finite yet localized weak disturbance principle does not lation of combustion in a coaxial jet combustor. That configura- encourage us to further enlarge the inlet perturbation area either. tion is degenerated into a spatially developing circular pipe in this Laminar pipe flow below Re = 8,000 is reluctant to transition work. Unless otherwise noted, the pipe length is 250 R, and the under such disturbances. computational mesh size is 8,192 × 200 × 256 in the axial ðzÞ, In step 3, inspired by the absolute instability of tangential radial ðrÞ, and azimuthal ðθÞ directions, respectively. The time step discontinuity in general shear flows, we made the perturbation remains a constant Δt = 0.008R=V,whereV is the bulk velocity. An area over the inlet plane into a narrow ring instead of a small full overbar denotes ensemble averaging, superscript “+” refers to circle. Within this ring the exact parabolic velocity profile was normalization by friction velocity uτ for velocity and by viscous replaced by the time-dependent, fully developed turbulent DNS wall unit ν=uτ for distance; ν is the kinematic viscosity. Poisson’s field at Re = 5,300. With the ring positioned at 0.4 ≤ r=R < equation for pressure is solved using Fourier and cosine transform 0.42,itwasfoundthatatRe= 5,300 the flow still returned to along the azimuthal and axial directions, respectively. The entire set laminar after a transient turbulent spot. Successful transition of DNS reported here took four calendar years to complete. was obtained at Re = 8,000(case S3R8, Figs. 1A and 2H). At the Six steps were taken to systematically vary the inlet con- inlet, the perturbed area is well localized and accounts for dition ðz = 0Þ and quantitatively assess the downstream response. merelyR 1.6% of the cross-section, and the energy norm jj E jj = Rðu′2 + u′2 + u′2 Þrdr=ðR2V 2Þ = × −5 In step 1, plug inflow was first prescribed without perturbation. 0 z,rms θ,rms r,rms 5.8 10 is less than 1% The downstream velocity develops into the expected parabolic of the fully developed turbulent pipe value 9.3 × 10−3, and is less profile. Replacing the plug inflow with the exact parabolic inflow than 0.15% compared with the peak value 3.9 × 10−2 attained simply kept the downstream flow unchanged from the inlet. In during transition. Here u′z,rms , u′r,rms, uθ′,rms are the turbulence in- step 2, perturbations were introduced at the inlet under the tensities. This inlet disturbance is analogous to a turbulent wake guiding principle that the disturbance magnitude should be finite arising from a very thin wire ring mounted on the inlet plane in a yet weak and well-controlled. Another guiding principle is that laboratory. Under this perturbation, the parabolic pipe flow the setup should permit transition to proceed to a fully developed gradually breaks down and develops into a fully developed tur- turbulent state downstream. Initially, the inlet base flow was bulent state downstream. Switching the inlet ring turbulent parabolic, Re = 2VR=ν = 5,300, and the inlet perturbations were fluctuation to a mean wake deficit superimposed with white confined within the tiny circle 0 ≤ r=R < 0.02 by superposing ei- noisefailedtoproducetransition. In the present simulation, ther white noise or isotropic grid turbulence on the base flow, but there is no imposed axial pressure gradient. After transition, + + they disappeared rapidly without exciting any noticeable turbu- R = uτR=ν = 258.5, uτ = 0.06462V; grid resolution is Δz = 7.5 + lence. We subsequently imposed instantaneous velocity fields and ΔðRθÞ = 6.3. Auxiliary simulation on the fully developed extracted from a separate, axially periodic DNS of the turbulent turbulent pipe flow at Re = 8,000 (case S3R8T) was performed pipe flow at Re = 5,300 over the same radial range 0 ≤ r=R < 0.02, using the conventional axially periodic boundary condition over a but no transition was observed. Additional tests by increasing the 30-R-long domain with a grid 2,048 × 256 × 512. Fig. 3 com- contaminated area from 0 ≤ r=R < 0.02 up to 0 ≤ r=R < 0.10 failed pares the turbulent statistics between S3R8T and S3R8. Note the

AB ENGINEERING 0.04 0.04

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● −2 k k ◇ = = ’ Fig. 1. Friction factor f ( ) and energy norm 10 log10 E ( ). 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 uzðr, θ, z = 0, tÞ at one random instant.

Wu et al. PNAS | June 30, 2015 | vol. 112 | no. 26 | 7921 Downloaded by guest on September 25, 2021 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 rapidly induce hairpin packets near the wall. Small-scale activity channel/pipe flow has been well established since the work of ref. explodes when the packets grow and interact with other vortex 15; see also ref. 16. All of the statistics, including even the rate of structures. The small-scale activity overcomes the larger scale viscous dissipation «+, agree very well between the two com- fluctuations from the initial disturbances as it grows and interacts putations, demonstrating the quantitative reliability of the pre- to fill the pipe with turbulence that approaches the fully developed sent spatially developing pipe transition simulation. The transition state by about 75 R. A reverse hairpin vortex as part of a diamond is continuous in the sense that no isolated turbulent spot was structure was reported in previous turbulent channel simulations of observed in this case; see the vortex visualization over the break- ref. 18; the present reverse hairpins occur in a train among a down region 30 ≤ z=R < 45 in Fig. 2H and Movie S1 using iso- quiescent environment, are of large scale (several R in length), and surfaces of swirling strength λci (17); red color indicates uz > 1.3V. are much more distinct. The finite-amplitude disturbances from the ring create a roughly In step 4, with identical inlet condition as in S3R8, Re was annular turbulent wake that grows downstream radially inward and first reduced from 8,000 to 5,700, and the flow returned outward. Transition is triggered by vortex filaments drawn from the completely laminar. Reducing Re from 8,000 to 6,000 (case disturbance region, intensified by stretching, and moving toward S4R6)produceddelayedtransitioncomparedwithS3R8. the wall. Between 30 and 40 R the filaments induce wall-ward Thus, for this particular inlet disturbance, the critical Reynolds radial flows; in turn, they create large reverse hairpin vortices that number is between 5,700 and 6,000. In S4R6 after transition,

7922 | www.pnas.org/cgi/doi/10.1073/pnas.1509451112 Wu et al. Downloaded by guest on September 25, 2021 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,anda 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 Fig. 3. Comparison between turbulent statistics in the axially periodic turbulent case S3R8T (dotted line) and in the fully developed turbulent region the wall, a small-scale hairpin packet is generated at the surface, + 210 ≤ z=R ≤ 240 of case S3R8 (solid line). □, u =10; ◇, u′+ ; ★, u′+ ; +, u′+ ; leading to turbulence. This is somewhat consistent from the re- + z z,rms θ,rms r,rms △ − ′ ′ ○ − «+ , uzur ; , 10 . sults of ref. 22 at Reynolds number less than 2,300. There, it was observed that a turbulent puff from the core could, but not al- ways, impinge on the wall and induce a secondary turbulent puff. + + + R = 200.1, uτ = 0.0667V, Δz = 6.11, ΔðRθÞ = 4.91. An auxil- It is conceivable that if the Reynolds number is reduced slightly iary simulation of the fully developed turbulent pipe flow at in step 4 from 6,000, the secondary hairpin packet will form an Re = 6,000 was also performed (case S4R6T), and the accuracy isolated puff instead of resulting continuous turbulence. was further verified by comparing the turbulent statistics from In step 5, we address the effect of the disturbance ring location S4R6 and S4R6T. Friction factor f as a function of z is shown in by first repositioning it from 0.4 ≤ r=R < 0.42 to 0.9 ≤ r=R < 0.92. Fig. 1B. Although data on f variation with Re are widely available At Re = 8,000, the flow transitions immediately at the inlet. This (19), we cannot find how f varies with z during transition from the is not useful because it does not permit us to study how the literature. The present data show that before breakdown, f agrees disturbances are amplified, and how the breakdown process very well with the analytical solution 64=Re; after the completion happens. The width of the ring was therefore reduced to of transition, f agrees with Moody’s correlation for smooth, fully 0.9 ≤ r=R < 0.91. This minor modification completely relami- developed turbulent pipe flow. Thus, the quantitative reliability of narized the flow. Successful and prolonged transition between the present spatially developing DNS is again demonstrated. The 50 ≤ z=R < 150 at Re = 8,000 was observed when the ring was peak in f is related to the change of sign of the near-wall mean positioned at 0.9 ≤ r=R < 0.915, which covers 2.7% of the inlet radial velocity gradient. In boundary layers it is also related to the cross-section (case S5R8). It was further found that at this set- sudden increase in entrainment of outer flow into the near-wall ting, reducing Re from 8,000 to 7,000 produced relaminarization. region (10). Fig. 1 also presents the development of energy norm Inspired by the colored band visualization of Osborne Rey- jj E jj with z. There is an extended, convincing, exponential growth nolds, a passive scalar φ = 1 was also introduced at the inlet over region before breakdown in S4R6. It is known that infinitesimally a small circle 0 ≤ r=R < 0.05 (Fig. 2D). Note φ = 0 over the rest of ENGINEERING small disturbances will not be amplified exponentially in a para- the inflow plane; the molecular Prandtl number was 1. A refined bolic pipe flow. The four sets of energy norm jj E jj results in Fig. 1 mesh of 16,384 × 200 × 512 was also used (case S5R8F). During suggest that localized, weakly finite disturbances are often am- transition, there is an overshoot of f over the turbulent corre- plified exponentially rather than algebraically with respect to axial lation in S3R8, S4R6, and S5R8 for which the base flows are distance in the pipe flow. We also found that shortly downstream parabolic. Vortex structure development is different from that in of the inlet in S4R6 and S3R8, the mean flow profile within the the previous two cases when the inlet disturbances were in the ring quickly recovers to approximately that of the base flow. The core region. Here, as shown in Fig. 2 A and B, structures sur- vortex structure development of S4R6 shown in Fig. 2I is similar to viving the initial decay near the inlet have the shape of a Lambda that in S3R8 with a sequential formation of helical vortex fil- vortex, possibly because their oblique orientation is favored by aments, large-scale reverse hairpin vortices, followed by small- shear flow amplification via the lift-up mechanism. The Lambda scale hairpin packets and transition without turbulent spots; see vortex subsequently develops into an elongated large-scale (on also Movie S2. the order of R) hairpin vortex; further downstream, a small-scale There are several notable overall agreements between our hairpin packet emerges near the tip region of the large-scale observations in steps 2–4 and previous dynamical system analysis hairpin vortex. A turbulent spot is subsequently formed, which is of pipe transition, and the associated numerical work performed essentially a local concentration of small-scale hairpin vortices using axially periodic temporal DNS. Ref. 20 defined the object (Fig. 2C). This process is analogous to the vortex development in of “edge-of-chaos,” which separates, in state space, the regions boundary layer bypass transition in the narrow sense (10), and is of initial conditions where the lifetime statistics exhibit a sensi- also broadly similar to the secondary instability and breakdown tive dependence on initial conditions capable of resulting in stage of boundary layer natural transition (23). This similarity turbulence from those decaying to laminar flow. In this work, we was conjectured and argued for in ref. 24 based on interpretation have repeatedly encountered situations in which a very minor of ensemble-averaged hot-wire signals; here our DNS provides change in either Reynolds number or inlet disturbance ring lo- direct, convincing, time-accurate, 3D evidence (see also Movie cation made a decisive difference regarding whether the down- S3). Although the breakdown of the scalar field in Fig. 2D may stream flow stays laminar or transitions to turbulence––the flow seem abrupt at first glance, careful examination shows that the parameter settings are close to the edge-of-chaos in the state scalar field has already been modulated by hairpin packets long

Wu et al. PNAS | June 30, 2015 | vol. 112 | no. 26 | 7923 Downloaded by guest on September 25, 2021 before breakdown; Fig. 2 E and F. This subtle effect could not transition was observed at Re = 8,000. However, reducing Re from have been detected by Reynolds and other subsequent experi- 8,000 to 7,500 relaminarized the flow. With the plug inflow, jj E jj mentalists when they used less-detailed, low-resolution obser- continues to exhibit robust exponential growth rate after a pro- vation techniques. What they visualized was likely the late-stage longed decay to 75 R, but friction factor f does not show the phenomenon associated with the exponential growth rate iden- overshoot found in previous cases with the parabolic base inflow; tified in the present study and in ref. 21. This explains, to a Fig. 1D. This is analogous to the boundary layer bypass transition certain extent, the abrupt and mysterious breakdown widely at- in the narrow sense in which the skin friction has no overshoot tributed to the Osborne Reynolds transition when in fact the either (10). The sharp rise of skin friction during transition is vortex development can be a rather slow and gradual process via closely related to the entrainment of outer fluid into the near-wall a sequence of events as shown in Fig. 2 and Movie S3. region. Vortex dynamics is similar to that in case S5R8 when the Refs. 25 and 26 applied periodic disturbances using a push– inlet disturbance is also near the wall, characterized by the se- pull device through holes in the pipe surface. One interesting quential formation of a Lambda vortex, hairpin packet, and tur- aspect of their study is that, compared with the previous similar bulent spot, which is a localized hairpin forest. injection–suction approaches such as that in ref. 6, the pertur- The accuracy of the results is demonstrated. Below Re = 8,000 – bation amplitude in ref. 26 was quite weak; only 0.1 0.3% of the laminar pipe flow is reluctant to transition under weakly finite and mean pipe flow flux was applied through the holes. This feature localized inlet disturbances. This is particularly pronounced when permits a clear observation of the transition process. Waves of the base inflow is of the plug type because downstream the near- hairpin-like structures were observed, which often break down wall flow develops under an accelerating core. When the distur- when they cross the centerline of the pipe. The hairpin waves bance ring is located in the core region of the pipe inlet, helical start from locations close to where the disturbance was applied. vortex filaments evolve into a train of large-scale reverse hairpin They also noted that the process was consistent with the tem- vortices. The interaction of these reverse hairpins among them- poral algebraic growth theory. The present vortex dynamics is selves or with the near-wall flow produces small-scale hairpin distinct from that in ref. 26 for several reasons. Reverse hairpin packets, which leads to breakdown. When the inlet disturbance ring streets in steps 3 and 4 start far downstream from the inlet where is located near the wall, certain quasi-spanwise structure is stretched the disturbance is introduced; and the reverse hairpins are not into a Lambda vortex, and develops into a large-scale hairpin vor- deliberately created through jet-in-cross-flow. Instead, they are a tex. Further downstream, small-scale hairpin packets emerge near result of the gradual downstream amplification of the upstream the tip region of the large-scale hairpin vortex, and subsequently perturbations. The evolution of Lambda vortex to hairpin packet grow into a turbulent spot, which is itself a local concentration of to turbulent spot discussed in step 5 also occurs naturally small-scale hairpin vortices. This process is broadly analogous to (without artificial creation) downstream of the inlet at an ir- boundary layer bypass transition in the narrow sense, and also to the regular and much longer time interval compared with that in ref. secondary instability and breakdown in natural transition. With the 26. Of course, the exponential growth rate in disturbance energy is also different from the algebraic growth in ref. 25. Inlet mass given inlet disturbances, the energy norm was found to grow flow rate is held constant in the present study, which guarantees exponentially rather than algebraically. Friction factor exhibits a constant Reynolds number. This was also achieved experi- overshoot with parabolic base flow. The Osborne Reynolds pipe mentally in refs. 25 and 26 through a clever design. transition is indeed abrupt with respect to the Reynolds number, but In step 6, we study the effect of switching the parabolic inflow to can be rather gradual with respect to the axial distance through a a plug inflow. We superimposed on the plug base flow the same sequence of events. Some of the previously attributed abruptness inlet disturbance ring as that in S3R8 and S4R6, but transition was and mysteriousness was perhaps due to the inability to study the not observed; using the same ring as that in S5R8 did not result in process accurately with very fine spatial and temporal resolution. transition either. Through further testing, it was found that the ACKNOWLEDGMENTS. X.W. acknowledges support from the Natural Science inlet ring had to be widened and located closer to the wall to and Engineering Research Council of Canada; P.M. is grateful for support from produce downstream transition. In case S6R8, the disturbance the US Department of Energy and Air Force Office of Scientific Research; R.J.A. ring was widened to 0.8 ≤ r=R < 0.94 and successful downstream and J.R.B. acknowledge support from the US National Science Foundation.

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