Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-20T17:51:08.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Turbulent secondary flows in channels with no-slip and shear-free boundaries

Published online by Cambridge University Press:  23 April 2021

Nikolay Nikitin Open the ORCID record for Nikolay Nikitin [Opens in a new window]
Nikolay Nikitin*
Affiliation:
Institute of Mechanics, Lomonosov Moscow State University, 1 Michurinsky prospect, 119899Moscow, Russia
*
†Email address for correspondence: nvnikitin@mail.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

A simple principle is formulated, which makes it possible to explain and, in some cases, to predict the shape of secondary flows of Prandtl's second kind that arise in turbulent flows in straight pipes of non-circular cross-section. The motion along the no-slip wall is directed in the direction of reduction of pressure. Therefore, the direction of the secondary motion along the perimeter of the pipe cross-section can be determined by the location of the mean wall pressure local extrema. Velocity fluctuations along a curved wall cause the mean wall pressure change, the greater the change, the greater the curvature of the wall. Thus, the location of local wall pressure extrema is consistent with the wall curvature. In a number of cases, an approximate picture of secondary flows can be predicted a priori using only symmetry considerations and a wall curvature analysis. The conditions for the formation of a secondary flow change on a free surface where, in addition to pressure, Reynolds stresses act on the fluid particles, and the result of their interaction is unknown in advance. A turbulent flow in a rectangular channel is considered, in which one of the walls (upper) is shear-free. Application of the formulated principle dictates the presence of secondary flows in the upper corners, with a downward motion along the solid sidewalls. This contradicts the popular point of view about the shape of secondary flows in open channels, according to which the motion along the sidewalls is directed upward towards the corner and then towards the centre of the channel along the free surface. The paper presents the direct numerical simulation (DNS) results showing that along with the large-scale secondary flow, which generally agrees with the traditional picture, there is a smaller-scale single-vortex secondary flow with the opposite direction of rotation in the vicinity of each upper corner. The presence of a no-slip wall is an important element of the described mechanism of secondary flow formation. In this regard, the question logically arises about the possibility of the occurrence of secondary flows in the corners between two free boundaries. A series of DNS of turbulent flows in a square duct with two solid and two free adjacent walls have been carried out. It was found that the Reynolds stress forces in the vicinity of the corner between free boundaries are potential, are neutralized by the pressure gradient, and therefore do not create a secondary flow.

JFM classification

Turbulent Flows: Pipe flow Turbulent Flows: Turbulence simulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 917 , 25 June 2021 , A24
Check for updates
Copyright
© The Author(s), 2021. 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

Albayrak, I. & Lemmin, U. 2011 Secondary currents and corresponding surface velocity patterns in a turbulent open-channel flow over a rough bed. J. Hydraul. Engng 137 (11), 1318–1334.CrossRefGoogle Scholar
Broglia, R., Pascarelli, A. & Piomelli, U. 2003 Large-Eddy simulations of ducts with a free surface. J. Fluid Mech. 484, 223–253.CrossRefGoogle Scholar
Brundrett, E. & Baines, W.D. 1964 The production and diffusion of vorticity in duct flow. J. Fluid Mech. 19 (3), 375–394.CrossRefGoogle Scholar
Dai, Y.-J. & Xu, C.-X. 2019 Wall pressure and secondary-flow origination in a square duct. Phys. Fluids 31, 085104.Google Scholar
Demuren, A.O. & Rodi, W. 1984 Calculation of turbulence-driven secondary motion in non-circular ducts. J. Fluid Mech. 140, 189–222.CrossRefGoogle Scholar
Eichelbrenner, E.A. & Preston, J.H. 1971 On the role of secondary flow in turbulent boundary layers in corners (and salients). J. Méc. 10, 91–112.Google Scholar
Einstein, H.A. & Li, H. 1958 Secondary currents in straight channels. Trans. Am. Geophys. Union 39 (6), 1085–1088.CrossRefGoogle Scholar
Gavrilakis, S. 1992 Numerical simulation of low-Reynolds-number turbulent flow through a straight square duct. J. Fluid Mech. 244, 101–129.CrossRefGoogle Scholar
Gessner, F.B. 1973 The origin of secondary flow in turbulent flow along a corner. J. Fluid Mech. 58, 1–25.CrossRefGoogle Scholar
Gessner, F.B. & Jones, J.B. 1965 On some aspects of fully-developed turbulent flow in rectangular channels. J. Fluid Mech. 23 (4), 689–713.CrossRefGoogle Scholar
Grega, L.M., Wey, T., Leighton, R.I. & Neves, J.C. 1995 Turbulent mixed-boundary flow in a corner formed by a solid wall and a free-surface. J. Fluid Mech. 294, 17–46.CrossRefGoogle Scholar
Hinze, J. 1967 Secondary currents in wall turbulence. Phys. Fluids (Suppl.) 10, S122–S125.CrossRefGoogle Scholar
Hinze, J.O. 1973 Experimental investigation of secondary currents in the turbulent flow through a straight conduit. Appl. Sci. Res. 28, 453–465.CrossRefGoogle Scholar
Hoagland, L.C. 1960 Fully developed turbulent flow in straight rectangular ducts: secondary flow, its cause and effect on the primary flow. PhD thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology.Google Scholar
Huser, A. & Biringen, S. 1993 Direct numerical simulation of turbulent flow in a square duct. J. Fluid Mech. 257, 65–95.CrossRefGoogle Scholar
Jing, S., Yang, W. & Chen, Y. 2019 Smooth open channel with increasing aspect ratio: influence on secondary flow. Water 11, 1872.CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R.D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.CrossRefGoogle Scholar
Kornilov, V.I. 2017 Three-dimensional turbulent near-wall flows in streamwise corners: current state and questions. Prog. Aerosp. Sci. 94, 46–81.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_\tau =5200$. J. Fluid Mech. 774, 395–415.CrossRefGoogle Scholar
Maslen, S.H. 1958 Transverse velocities in fully-developed flows. Q. Appl. Maths 16, 173–175.CrossRefGoogle Scholar
Moinuddin, K.A.M., Joubert, P.N. & Chong, M.S. 2004 Experimental investigation of turbulence-driven secondary motion over a streamwise external corner. J. Fluid Mech. 511, 1–23.CrossRefGoogle Scholar
Moissis, R. 1957 Secondary flow in rectangular ducts. Master's thesis, Massachusetts Institute of Technology.Google Scholar
Nagata, K., Hunt, J.C.R., Sakai, Y. & Wong, H. 2011 Distorted turbulence and secondary flow near right-angled plates. J Fluid Mech. 668, 446–479.CrossRefGoogle Scholar
Nagata, K., Wong, H., Hunt, J.C.R., Sajjadi, S.G. & Davidson, P.A. 2006 Weak mean flows induced by anisotropic turbulence impinging onto planar and undulating surfaces. J. Fluid Mech. 556, 329–360.CrossRefGoogle Scholar
Nezu, I. & Nakagawa, H. 1993 Turbulence in Open-Channel Flows, Monograph of IAHR. Balkema.Google Scholar
Nezu, I. & Rodi, W. 1985 Experimental study on secondary currents in open channels flow. In Proceedings of 21st Congress of IAHR, Vol. 2, Melbourne, Australia, pp. 19–23. Institution of Engineers.Google Scholar
Nikitin, N.V. 1997 Numerical simulation of turbulent flows in a pipe of square cross section. Dokl. Phys. 42 (3), 158–162.Google Scholar
Nikitin, N. 2006 a Third-order-accurate semi-implicit Runge–Kutta scheme for incompressible Navier–Stokes equations. Intl J. Numer. Meth. Fluids 51 (2), 221–233.CrossRefGoogle Scholar
Nikitin, N. 2006 b Finite-difference method for incompressible Navier–Stokes equations in arbitrary orthogonal curvilinear coordinates. J. Comput. Phys. 217, 759–781.CrossRefGoogle Scholar
Nikitin, N.V. 2006 c Direct simulation of turbulent flow in eccentric pipes. Comput. Maths Math. Phys. 46 (3), 489–504.CrossRefGoogle Scholar
Nikitin, N.V., Pimanov, V.O. & Popelenskaya, N.V. 2019 Mechanism of formation of Prandtl's secondary flows of the second kind. Dokl. Phys. 64 (2), 61–65.CrossRefGoogle Scholar
Nikitin, N., Wang, H. & Chernyshenko, S. 2009 Turbulent flow and heat transfer in eccentric annulus. J. Fluid Mech. 638, 95–116.CrossRefGoogle Scholar
Nikitin, N. & Yakhot, A. 2005 Direct numerical simulation of turbulent flow in elliptical ducts. J. Fluid Mech. 532, 141–164.CrossRefGoogle Scholar
Pan, Y. & Banerjee, S. 1995 Numerical investigation of free-surface turbulence in open-channel flows. Phys. Fluids 7, 1649–1664.CrossRefGoogle Scholar
Perkins, H.J. 1970 The formation of streamwise vorticity in turbulent flow. J. Fluid Mech. 44, 721–740.CrossRefGoogle Scholar
Prandtl, L. 1926 Über die ausgebildete Turbulenz [Turbulent Flow]. Verh. 2nd Intl Kong. NACA Tech. Memo 62, 2nd Intl Kong. für Tech. Mech., Zürich, p. 435.Google Scholar
Prandtl, L. 1952 Essentials of Fluid Dynamics. Hafner.Google Scholar
Speziale, C.G. 1982 On turbulent secondary flows in pipes of noncircular cross-section. Intl J. Engng Sci. 20 (7), 863–872.CrossRefGoogle Scholar
Stoesser, T., McSherry, R. & Fraga, B. 2015 Secondary currents and turbulence over a non-uniformly roughened open-channel bed. Water 7 (9), 4896–4913.CrossRefGoogle Scholar
Tominaga, A., Nezu, I., Ezaki, K. & Nakagawa, H. 1989 Threedimensional turbulent structure in straight open-channel flows. J. Hydraul. Res. 27, 149–173.CrossRefGoogle Scholar
Vidal, A., Vinuesa, R., Schlatter, P. & Nagib, H.M. 2017 Impact of corner geometry on the secondary flow in turbulent ducts. In Proceedings of the 10th International Symposium on Turbulence and Shear Flow Phenomena, TSFP-10, Chicago, USA.Google Scholar
Vidal, A., Vinuesa, R., Schlatter, P. & Nagib, H.M. 2018 Turbulent rectangular ducts with minimum secondary flow. Intl J. Heat Fluid Flow 72, 317–328.CrossRefGoogle Scholar
Voronova, T.V. & Nikitin, N.V. 2006 Direct numerical simulation of the turbulent flow in an elliptical pipe. Comput. Maths Math. Phys. 46, 1378–1386.CrossRefGoogle Scholar
Voronova, T.V. & Nikitin, N.V. 2007 Results of direct numerical simulation of turbulent flow in a pipe of elliptical cross-section. Fluid Dyn. 42 (2), 201–211.CrossRefGoogle Scholar
Xu, H. & Pollard, A. 2001 Large eddy simulation of turbulent flow in a square annular duct. Phys. Fluids 13 (11), 3321–3337.CrossRefGoogle Scholar
Yang, S.-Q., Tan, S.K. & Wang, X.-K. 2012 Mechanism of secondary currents in open channel flows. J. Geophys. Res. 117, F04014.Google Scholar