0
Research Papers: Fundamental Issues and Canonical Flows

A Functional Relationship for Modeling Laminar to Turbulent Flow Transitions

[+] Author and Article Information
George Papadopoulos

Fellow ASME
Innoveering, LLC,
100 Remington Boulevard,
Ronkonkoma, NY 11779
e-mail: George.Papadopoulos@innoveering.net

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 30, 2015; final manuscript received March 12, 2017; published online June 20, 2017. Assoc. Editor: D. Keith Walters.

J. Fluids Eng 139(9), 091202 (Jun 20, 2017) (10 pages) Paper No: FE-15-1606; doi: 10.1115/1.4036594 History: Received August 30, 2015; Revised March 12, 2017

A dimensional analysis which is based on the scaling of the two-dimensional Navier–Stokes equations is presented for correlating bulk flow characteristics arising from a variety of initial conditions. The analysis yields a functional relationship between the characteristic variable of the flow region and the Reynolds number for each of the two independent flow regimes, laminar and turbulent. A linear relationship is realized for the laminar regime, while a nonlinear relationship is realized for the turbulent regime. Both relationships incorporate mass-flow profile characteristics to capture the effects of initial conditions (mean flow and turbulence) on the variation of the characteristic variable. The union of these two independent relationships is formed leveraging the concept of flow intermittency to yield a generic functional relationship that incorporates transitional flow effects and fully encompasses solutions spanning the laminar to turbulent flow regimes. Empirical models to several common flows are formed to demonstrate the engineering potential of the proposed functional relationship.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Li, X. , and Djilali, N. , 1995, “ On the Scaling of Separation Bubbles,” JSME Int. J., 38(4), pp. 541–548. [CrossRef]
Bejan, A. , 1984, Convection Heat Transfer, Wiley, New York.
Papadopoulos, G. , and Pitts, W. M. , 1998, “ Scaling the Near-Field Centerline Mixing Behavior of Axisymmetric Turbulent Jets,” AIAA J., 36(9), pp. 1635–1642. [CrossRef]
Papadopoulos, G. , and Pitts, W. , 1999, “ A Generic Centerline Velocity Decay Curve for Initially Turbulent Axisymmetric Jets,” ASME J. Fluids Eng., 121(1), pp. 80–85. [CrossRef]
George, W. , 1989, “ Self-Preservation of Turbulent Flows and Its Relation to Initial Conditions and Coherent Structures,” Advances in Turbulence, W. George and R. Arndt , eds., Springer, New York.
Kays, W. M. , 1950, “ Loss Coefficients for Abrupt Changes in Flow Cross Section With Low Reynolds Number Flow in Single and Multiple-Tube Systems,” Proceedings of the ASME Heat Transfer Division Spring Meeting, Washington, DC, Apr. 12–14, pp. 1067–1074.
Sautet, J. C. , and Stepowski, D. , 1995, “ Dynamic Behavior of Variable-Density, Turbulent Jets in Their Near Development Fields,” Phys. Fluids, 7(11), pp. 2796–2806. [CrossRef]
Patel, V. , 1974, “ A Simple Integral Method for the Calculation of Thick Axisymmetric Turbulent Boundary Layers,” Aeronaut. Q., 25(1), pp. 47–58. [CrossRef]
Shah, R. , and London, A. , 1978, Laminar Forced Convection in Ducts, Supplements to Advances in Heat Transfer, Academic Press, New York.
Papadopoulos, G. , and Durst, F. , 1997, “ Influence of Tripping Level and Development Length on Pipe Flow Discharge Characteristics,” AIAA Paper No. AIAA97-0552.
Durst, F. , and Ünsal, B. , 2006, “ Forced Laminar-to-Turbulent Transition of Pipe Flows,” J. Fluid Mech., 560, pp. 449–464. [CrossRef]
Nishi, M. , Ünsal, B. , Durst, F. , and Biswas, G. , 2008, “ Laminar-to-Turbulent Transition of Pipe Flows Through Puffs and Slugs,” J. Fluid Mech., 614, pp. 425–446. [CrossRef]
White, F. M. , 1991, Viscous Fluid Flow, 2nd ed., McGraw-Hill, New York.
Klebanoff, P. S. , 1955, “ Characteristics of Turbulence in Boundary Layer With Zero Pressure Gradient,” National Bureau of Standards, Washington, DC, Technical Report No. NACA-TR-1247. https://ntrs.nasa.gov/search.jsp?R=19930092249
Corrsin, S. , and Kistler, A. L. , 1955, “ Freestream Boundaries of Turbulent Flows,” Johns Hopkins University, Baltimore, MD, Technical Report No. NACA-TR-1244. https://ntrs.nasa.gov/search.jsp?R=19930092246
Avila, K. , Moxey, D. , de Lozar, A. , Avila, M. , Barkley, D. , and Hof, B. , 2011, “ The Onset of Turbulence in Pipe Flow,” Science, 333(6039), pp. 192–196. [CrossRef] [PubMed]
Patel, V. , and Head, M. , 1969, “ Some Observations on Skin Friction and Velocity Profiles in Fully Developed Pipe and Channel Flows,” J. Fluid Mech., 38(1), pp. 181–201. [CrossRef]
Nikuradse, J. , 1933, “ Gesetzmäßigkeiten der Turbulenten Strömung in Glatten Rohren (Nachtrag),” Forschungsarb. Ingenieurwes., 4(1), p. 44.
Papadopoulos, G. , Lekakis, I. , and Durst, F. , 1997, “ Reynolds Number Asymptotic Covariance for Turbulent Pipe Flow Past a Sudden Expansion,” ASME Paper No. FEDSM97-3323.
den Toonder, J. M. J. , and Nieuwstadt, F. T. M. , 1997, “ Reynolds Number Effects in a Turbulent Pipe Flow for Low to Moderate Re,” Phys. Fluids, 9(3398), pp. 3398–3409. [CrossRef]
Pinho, F. , and Whitelaw, J. , 1990, “ Flow of Non-Newtonian Fluids in a Pipe,” J. Non-Newtonian Fluid Mech., 34(2), pp. 129–144. [CrossRef]
Rhodes, D. G. , and New, A. P. , 2000, “ Preston Tube Measurements in Low Reynolds Number Turbulent Pipe Flow,” J. Hydraul. Eng., 126(6), pp. 407–415. [CrossRef]
Clauser, F. H. , 1956, “ The Turbulent Boundary Layer,” Adv. Appl. Mech., 4, pp. 1–51.
Bandyopadhyay, P. R. , 1992, “ Reynolds Number Dependence of the Freestream Turbulence Effects on Turbulent Boundary Layers,” AIAA J., 30(7), pp. 1910–1912. [CrossRef]
Bandyopadhyay, P. R. , 1987, “ Rough-Wall Turbulent Boundary Layer in the Transition Regime,” J. Fluid Mech., 180, pp. 231–266. [CrossRef]
Blair, M. F. , and Werle, M. J. , 1980, “ The Influence of Free-Stream Turbulent on Boundary Layer Development,” United Technologies Research Center, East Hartford, CT, Report No. R80-914388-12.
Gillis, J. C. , 1980, “ Turbulent Boundary Layer on a Convex, Curved Surface,” Ph.D. thesis, Stanford University, Stanford, CA.
Anders, J. B. , 1989, “ Boundary Layer Manipulators at High Reynolds Numbers,” Structure of Turbulent and Drag Reduction, Springer-Verlag, New York, pp. 475–482.
Hasan, M. , and Hussain, A. , 1982, “ The Self-Excited Axisymmetric Jet,” J. Fluid Mech., 115, pp. 59–89. [CrossRef]
Russ, S. , and Strykowski, P. , 1993, “ Turbulent Structure and Entrainment in Heated Jets: The Effect of Initial Conditions,” Phys. Fluids A, 5(12), p. 3216. [CrossRef]
Boersma, B. J. , Brethouwer, G. , and Nieuwstadt, F. T. M. , 1998, “ A Numerical Investigation on the Effect of the Inflow Conditions on the Self-Similar Region of a Round Jet,” Phys. Fluids, 10(4), pp. 899–909. [CrossRef]
Kuhlman, J. , 1987, “ Variation of Entrainment in Annular Jets,” AIAA J., 25(3), pp. 373–379. [CrossRef]
Hussein, H. , Capp, S. , and George, W. , 1994, “ Velocity Measurements in a High-Reynolds-Number, Momentum-Conserving, Axisymmetric, Turbulent Jet,” J. Fluid Mech., 258, pp. 31–75. [CrossRef]
Lane, J. C. , and Loehrke, R. I. , 1980, “ Leading Edge Separation From a Blunt Plate at Low Reynolds Number,” ASME J. Fluids Eng., 102(4), pp. 494–496. [CrossRef]
Ota, T. , Asano, Y. , and Okawa, J. I. , 1981, “ Reattachment Length and Transition of the Separated Flow Over Blunt Flat Plates,” Bull. JSME, 24(192), pp. 941–947. [CrossRef]
Djilali, N. , Gartshore, I. S. , and Salcudean, M. , 1991, “ Turbulent Flow Around a Bluff Rectangular Plate—Part 2: Numerical Predictions,” ASME J. Fluids Eng., 113(1), pp. 60–67. [CrossRef]
Tafti, D. K. , and Vanka, S. P. , 1991, “ A Numerical Study of Flow Separation and Reattachment on a Blunt Plate,” Phys. Fluids A, 3(1749), pp. 1749–1759. [CrossRef]
Back, L. H. , and Roschke, E. J. , 1972, “ Shear-Layer Flow Regimes and Wave Instabilities and Reattachment Lengths Downstream of an Abrupt Circular Channel Expansion,” ASME Appl. Mech., 39(3), pp. 677–681. [CrossRef]
Iribarne, A. , Frantisak, F. , Hummel, R. , and Smith, J. , 1972, “ An Experimental Study of Instabilities and Other Flow Properties of a Laminar Pipe Jet,” AIChE J., 18(4), pp. 689–698. [CrossRef]
Macagno, E. , and Hung, T. K. , 1967, “ Computational and Experimental Study of a Captive Annular Eddy,” J. Fluid Mech., 28(1), pp. 43–64. [CrossRef]
Pak, B. , Cho, Y. I. , and Choi, S. U. S. , 1990, “ Separation and Reattachment of Non-Newtonian Fluid Flows in a Sudden Expansion Pipe,” J. Non-Newtonian Fluid Mech., 37(2–3), pp. 175–199. [CrossRef]
Latornell, D. J. , and Pollard, A. , 1986, “ Some Observations on the Evolution of Shear Layer Instabilities in Laminar Flow Through Axisymmetric Sudden Expansions,” Phys. Fluids, 29(9), pp. 2828–2835. [CrossRef]
Drikakis, D. , and Papadopoulos, G. , 1996, “ Experimental and Numerical Investigation of Laminar-to-Transitional Pipe Flow Past a Sudden-Expansion,” Proceedings of the ASME Fluids Engineering Division Summer Meeting, San Diego, CA, July 7–11, pp. 679–684.
Khezzar, L. , Whitelaw, J. H. , and Yianneskis, M. , 1985, “ An Experimental Study of Round Sudden Expansion Flows,” Fifth Symposium on Turbulent Shear Flows, Ithaca, NY, Aug. 7–9, pp. 5.25–5.30.
Lekakis, I. , Durst, F. , and Sender, J. , 1994, “ LDA Measurements in the Near-Wall Region of an Axisymmetric Sudden Expansion,” Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, pp. 11–14.
Devenport, W. J. , and Sutton, E. P. , 1993, “ An Experimental Study of Two Flows Through an Axisymmetric Sudden Expansion,” Exp. Fluids, 14(6), pp. 423–432. [CrossRef]
Stieglmeier, M. , Tropea, C. , Weiser, N. , and Nitsche, W. , 1992, “ Experimental Investigation of the Flow Through Axisymmetric Expansions,” ASME J. Fluids Eng., 111, pp. 464–471. [CrossRef]
Tinney, C. E. , Glauser, M. N. , Eaton, E. L. , and Taylor, J. A. , 2006, “ Low-Dimensional Azimuthal Characteristics of Suddenly Expanding Axisymmetric Flows,” J. Fluid Mech., 567, pp. 141–155. [CrossRef]
So, R. M. C. , 1987, “ Inlet Centerline Turbulence Effects on Reattachment Length in Axisymmetric Sudden-Expansion Flows,” Exp. Fluids, 5(6), pp. 424–426. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Velocity-distribution coefficients and coefficient of nonuniformity as functions of radial extent of plug flow (laminar regime)

Grahic Jump Location
Fig. 2

Velocity-distribution coefficients for fully developed turbulent pipe flow

Grahic Jump Location
Fig. 3

Normalized bulk velocity versus Reynolds number for fully developed pipe flow

Grahic Jump Location
Fig. 4

Reynolds number effect on Clauser’s shape parameter

Grahic Jump Location
Fig. 5

Centerline velocity decay data for a variety of axisymmetric jets

Grahic Jump Location
Fig. 6

Application of coefficient of profile uniformity to jet decay data for laminar axisymmetric jets issuing from short pipes

Grahic Jump Location
Fig. 7

Jet decay data plotted using the effective diameter term along with fitted distributions (dashed lines) based on proposed relationship

Grahic Jump Location
Fig. 8

Variation of the separation bubble length with Reynolds number for the flow over a blunt rectangular plate

Grahic Jump Location
Fig. 9

Reattachment length versus Reynolds number for axisymmetric sudden expansion flow: low Re range

Grahic Jump Location
Fig. 10

Reattachment length versus Reynolds number for axisymmetric sudden expansion flow: moderate to high Re range

Grahic Jump Location
Fig. 11

Reattachment length normalized by the effective step height for axisymmetric sudden expansion flow: low Re range

Grahic Jump Location
Fig. 12

Reattachment length data fitted using the functional relationship of Eq. (29)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In