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

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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. Rmerely 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.
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