Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T02:11:04.783Z Has data issue: false hasContentIssue false
  • English
  • Français

Pressure-driven flows in helical pipes: bounds on flow rate and friction factor

Published online by Cambridge University Press:  06 October 2020

Anuj Kumar Open the ORCID record for Anuj Kumar [Opens in a new window]
Anuj Kumar*
Affiliation:
Department of Applied Mathematics, Baskin School of Engineering, University of California, Santa Cruz, CA95064, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we use the well-known background method to obtain a rigorous lower bound on the volume flow rate through a helical pipe driven by a pressure differential in the limit of large Reynolds number. As a consequence, we also obtain an equivalent upper bound on the friction factor. These bounds are also valid for toroidal and straight pipes as limiting cases. By considering a two-dimensional background flow with varying boundary layer thickness along the circumference of the pipe, we obtain these bounds as a function of the curvature and torsion of the pipe and therefore capture the geometrical aspects of the problem. In this paper, we also present a sufficient criterion for determining which pressure-driven flow and surface-velocity-driven flow problems can be tackled using the background method.

JFM classification

Mathematical Foundations: Variational methods Boundary Layers: Pipe flow boundary layer
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 904 , 10 December 2020 , A5
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192–196.CrossRefGoogle ScholarPubMed
Berger, S. A., Talbot, L. & Yao, L. S. 1983 Flow in curved pipes. Annu. Rev. Fluid Mech. 15, 461–512.CrossRefGoogle Scholar
Busse, F. H. 1969 On Howard's upper bound for heat transport by turbulent convection. J. Fluid Mech. 37 (3), 457–477.CrossRefGoogle Scholar
Busse, F. H. 1970 Bounds for turbulent shear flow. J. Fluid Mech. 41 (1), 219–240.CrossRefGoogle Scholar
Canton, J., Örlü, R. & Schlatter, P. 2017 Characterisation of the steady, laminar incompressible flow in toroidal pipes covering the entire curvature range. Intl J. Heat Fluid Flow 66, 95–107.CrossRefGoogle Scholar
Canton, J., Rinaldi, E., Örlü, R. & Schlatter, P. 2020 Critical point for bifurcation cascades and featureless turbulence. Phys. Rev. Lett. 124, 014501.CrossRefGoogle ScholarPubMed
Canton, J., Schlatter, P. & Örlü, R. 2016 Modal instability of the flow in a toroidal pipe. J. Fluid Mech. 792, 894–909.CrossRefGoogle Scholar
Caulfield, C. P. 2005 Buoyancy flux bounds for surface-driven flow. J. Fluid Mech. 536, 367–376.CrossRefGoogle Scholar
Caulfield, C. P. & Kerswell, R. R. 2001 Maximal mixing rate in turbulent stably stratified Couette flow. Phys. Fluids 13 (4), 894–900.CrossRefGoogle Scholar
Cioncolini, A. & Santini, L. 2006 An experimental investigation regarding the laminar to turbulent flow transition in helically coiled pipes. Exp. Therm. Fluid Sci. 30, 367–380.CrossRefGoogle Scholar
Constantin, P. & Doering, C. R. 1995 Variational bounds on energy dissipation in incompressible flows. II. Channel flow. Phys. Rev. E 51 (4), 3192–3198.CrossRefGoogle ScholarPubMed
Dean, W. R. 1927 Note on the motion of fluid in a curved pipe. Phil. Mag. 4, 208–223.CrossRefGoogle Scholar
Dean, W. R. 1928 The stream-line motion of fluid in a curved pipe. Phil. Mag. 5, 673–695.CrossRefGoogle Scholar
Dennis, S. C. R. 1980 Calculation of the steady flow through a curved tube using a new finite-difference method. J. Fluid Mech. 99, 449–467.CrossRefGoogle Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69 (11), 1648–1651.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49 (5), 4087–4099.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53 (6), 5957–5981.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 2001 On upper bounds for infinite Prandtl number convection with or without rotation. J. Math. Phys. 42 (2), 784–795.CrossRefGoogle Scholar
Eustice, J. 1910 Flow of water in curved pipes. Proc. R. Soc. Lond. A 84, 107–118.Google Scholar
Eustice, J. 1911 Experiments on streamline motion in curved pipes. Proc. R. Soc. Lond. A 85, 119–131.Google Scholar
Fantuzzi, G. 2018 Bounds for Rayleigh–Bénard convection between free-slip boundaries with an imposed heat flux. J. Fluid Mech. 837, R5.CrossRefGoogle Scholar
Fantuzzi, G., Nobili, C. & Wynn, A. 2020 New bounds on the vertical heat transport for Bénard–Marangoni convection at infinite Prandtl number. J. Fluid Mech. 885, R4.CrossRefGoogle Scholar
Fantuzzi, G., Pershin, A. & Wynn, A. 2018 Bounds on heat transfer for Bénard–Marangoni convection at infinite Prandtl number. J. Fluid Mech. 837, 562–596.CrossRefGoogle Scholar
Fantuzzi, G. & Wynn, A. 2015 Construction of an optimal background profile for the Kuramoto–Sivashinsky equation using semidefinite programming. Phys. Lett. A 379 (1-2), 23–32.CrossRefGoogle Scholar
Fantuzzi, G. & Wynn, A. 2016 Optimal bounds with semidefinite programming: an application to stress-driven shear flows. Phys. Rev. E 93 (4), 043308.CrossRefGoogle ScholarPubMed
Gammack, D. & Hydon, P. E. 2001 Flow in pipes with non-uniform curvature and torsion. J. Fluid Mech. 433, 357–382.CrossRefGoogle Scholar
Germano, M. 1982 On the effect of torsion on a helical pipe flow. J. Fluid Mech. 125, 1–8.CrossRefGoogle Scholar
Germano, M. 1989 The dean equations extended to a helical pipe flow. J. Fluid Mech. 203, 289–305.CrossRefGoogle Scholar
Goluskin, D. 2015 Internally heated convection beneath a poor conductor. J. Fluid Mech. 771, 36–56.CrossRefGoogle Scholar
Goluskin, D. & Doering, C. R. 2016 Bounds for convection between rough boundaries. J. Fluid Mech. 804, 370–386.CrossRefGoogle Scholar
Hagstrom, G. & Doering, C. R. 2010 Bounds on heat transport in Bénard–Marangoni convection. Phys. Rev. E 81 (4), 047301.CrossRefGoogle ScholarPubMed
Hagstrom, G. I. & Doering, C. R. 2014 Bounds on surface stress-driven shear flow. J. Nonlinear Sci. 24 (1), 185–199.CrossRefGoogle Scholar
Hopf, E. 1957 Lecture Series of the Symposium on Partial Differential Equations, Berkeley, 1955. University of Kansas.Google Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17 (3), 405–432.CrossRefGoogle Scholar
Hüttl, T. J. & Friedrich, R. 2001 Direct numerical simulation of turbulent flows in curved and helically coiled pipes. Comput. Fluids 30, 591–605.CrossRefGoogle Scholar
Itō, H. 1959 Friction factors for turbulent flow in curved pipes. Trans. ASME J. Basic Engng 81, 123–134.CrossRefGoogle Scholar
Kao, H. C. 1987 Torsion effect on fully developed flow in a helical pipe. J. Fluid Mech. 184, 335–356.CrossRefGoogle Scholar
Kühnen, J., Braunshier, P., Schwegel, M., Kuhlmann, H. C. & Hof, B. 2015 Subcritical versus supercritical transition to turbulence in curved pipes. J. Fluid Mech. 770, R3.CrossRefGoogle Scholar
Liu, S. & Masliyah, J. H. 1993 Axially invariant laminar flow in helical pipes with a finite pitch. J. Fluid Mech. 251, 315–353.CrossRefGoogle Scholar
Marchioro, C. 1994 Remark on the energy dissipation in shear driven turbulence. Physica D 74 (3-4), 395–398.CrossRefGoogle Scholar
McConalogue, D. J. & Srivastava, R. S. 1968 Motion of a fluid in a curved tube. Proc. R. Soc. Lond. A 307, 37–53.Google Scholar
Naphon, P. & Wongwises, S. 2006 A review of flow and heat transfer characteristics in curved tubes. Renewable Sustainable Energy Rev. 10 (5), 463–490.CrossRefGoogle Scholar
Otero, J., Wittenberg, R. W., Worthing, R. A. & Doering, C. R. 2002 Bounds on Rayleigh–Bénard convection with an imposed heat flux. J. Fluid Mech. 473, 191–199.CrossRefGoogle Scholar
Plasting, S. C. & Ierley, G. R. 2005 Infinite-Prandtl-number convection. Part 1. Conservative bounds. J. Fluid Mech. 542, 343–363.CrossRefGoogle Scholar
Plasting, S. C. & Kerswell, R. R. 2003 Improved upper bound on the energy dissipation rate in plane Couette flow: the full solution to Busse's problem and the Constantin–Doering–Hopf problem with one-dimensional background field. J. Fluid Mech. 477, 363–379.CrossRefGoogle Scholar
Plasting, S. C. & Kerswell, R. R. 2005 A friction factor bound for transitional pipe flow. Phys. Fluids 17 (1), 011706.CrossRefGoogle Scholar
Rinaldi, E., Canton, J. & Schlatter, P. 2019 The vanishing of strong turbulent fronts in bent pipes. J. Fluid Mech. 866, 487–502.CrossRefGoogle Scholar
Sreenivasan, K. R. & Strykowski, P. J. 1983 Stabilization effects in flow through helically coiled pipes. Exp. Fluids 1 (1), 31–36.CrossRefGoogle Scholar
Tang, W., Caulfield, C. P. & Young, W. R. 2004 Bounds on dissipation in stress-driven flow. J. Fluid Mech. 510, 333–352.CrossRefGoogle Scholar
Taylor, G. I. 1929 The criterion for turbulence in curved pipes. Proc. R. Soc. Lond. A 124, 243–249.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Tilgner, A. 2017 Bounds on poloidal kinetic energy in plane layer convection. Phys. Rev. Fluids 2 (12), 123502.CrossRefGoogle Scholar
Tilgner, A. 2019 Time evolution equation for advective heat transport as a constraint for optimal bounds in Rayleigh–Bénard convection. Phys. Rev. Fluids 4 (1), 014601.CrossRefGoogle Scholar
Tuttle, E. R. 1990 Laminar flow in twisted pipes. J. Fluid Mech. 219, 545–570.CrossRefGoogle Scholar
Van Dyke, M. 1978 Extended stokes series: laminar flow through a loosely coiled pipe. J. Fluid Mech. 86, 129–145.CrossRefGoogle Scholar
Vashisth, S., Kumar, V. & Nigam, K. D. P. 2008 A review on the potential applications of curved geometries in process industry. Ind. Engng Chem. Res. 47 (10), 3291–3337.CrossRefGoogle Scholar
Vester, A. Kalpakli, Örlü, R. & Alfredsson, P. H. 2016 Turbulent flows in curved pipes: recent advances in experiments and simulations. Appl. Mech. Rev. 68, 050802.Google Scholar
Wang, X. 1997 Time averaged energy dissipation rate for shear driven flows in $\mathbb {R}^n$. Physica D 99 (4), 555–563.CrossRefGoogle Scholar
Webster, D. R. & Humphrey, J. A. C. 1993 Experimental observations of flow instability in a helical coil (Data bank contribution). Trans. ASME J. Fluids Engng 115, 436–443.CrossRefGoogle Scholar
Webster, D. R. & Humphrey, J. A. C. 1997 Traveling wave instability in helical coil flow. Phys. Fluids 9, 407–418.CrossRefGoogle Scholar
Wen, B., Chini, G., Dianati, N. & Doering, C. R. 2013 Computational approaches to aspect-ratio-dependent upper bounds and heat flux in porous medium convection. Phys. Lett. A 377 (41), 2931–2938.CrossRefGoogle Scholar
Wen, B., Chini, G. P., Kerswell, R. R. & Doering, C. R. 2015 Time-stepping approach for solving upper-bound problems: application to two-dimensional Rayleigh–Bénard convection. Phys. Rev. E 92 (4), 043012.CrossRefGoogle ScholarPubMed
White, C. M. 1929 Streamline flow through curved pipes. Proc. R. Soc. Lond. A 123, 645–663.Google Scholar
Whitehead, J. P. & Doering, C. R. 2011 Ultimate state of two-dimensional Rayleigh–Bénard convection between free-slip fixed-temperature boundaries. Phys. Rev. Lett. 106 (24), 244501.CrossRefGoogle ScholarPubMed
Whitehead, J. P. & Wittenberg, R. W. 2014 A rigorous bound on the vertical transport of heat in Rayleigh–Bénard convection at infinite Prandtl number with mixed thermal boundary conditions. J. Math. Phys. 55 (9), 093104.CrossRefGoogle Scholar
Wittenberg, R. W. 2010 Bounds on Rayleigh–Bénard convection with imperfectly conducting plates. J. Fluid Mech. 665, 158–198.CrossRefGoogle Scholar
Yamamoto, K., Akita, T., Ikeuchi, H. & Kita, Y. 1995 Experimental study of the flow in a helical circular tube. Fluid Dyn. Res. 16, 237–249.CrossRefGoogle Scholar
Yamamoto, K., Yanase, S. & Yoshida, T. 1994 Torsion effect on the flow in a helical pipe. Fluid Dyn. Res. 14, 259–273.CrossRefGoogle Scholar