Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-23T02:57:23.326Z Has data issue: false hasContentIssue false

Direct numerical simulation of quasi-two-dimensional MHD turbulent shear flows

Published online by Cambridge University Press:  01 April 2021

Long Chen
Affiliation:
School of Engineering Science, University of Chinese Academy of Sciences, Beijing101408, PR China
Alban Pothérat
Affiliation:
Fluid and Complex Systems Research Centre, Coventry University, CoventryCV15FB, UK
Ming-Jiu Ni*
Affiliation:
School of Engineering Science, University of Chinese Academy of Sciences, Beijing101408, PR China
René Moreau
Affiliation:
Laboratoire SIMAP, Groupe EPM, Université de Grenoble, BP 75, 38402Saint Martin d'Hères, France
*
Email address for correspondence: mjni@ucas.ac.cn

Abstract

High-resolution direct numerical simulations are performed to study the turbulent shear flow of liquid metal in a cylindrical container. The flow is driven by an azimuthal Lorentz force induced by the interaction between the radial electric currents injected through electrodes placed at the bottom wall and a magnetic field imposed in the axial direction. All physical parameters, are aligned with the experiment by Messadek & Moreau (J. Fluid Mech. vol. 456, 2002, pp. 137–159). The simulations recover the variations of angular momentum, velocity profiles, boundary layer thickness and turbulent spectra found experimentally to a very good precision. They further reveal a transition to small scale turbulence in the wall side layer when the Reynolds number based on Hartmann layer thickness $R$ exceeds 121, and a separation of this layer for $R \geq 145.2$. Ekman recirculations significantly influence these quantities and determine global dissipation. This phenomenology well captured by the 2-D PSM model (Pothérat, Sommeria & Moreau, J. Fluid Mech. vol. 424, 2000, pp. 75–100) until small-scale turbulence appears and incurs significant extra dissipation only captured by 3-D simulations. Secondly, we recover the theoretical law for the cutoff scale separating large quasi-two-dimensional (Q2-D) scales from small three-dimensional ones (Sommeria & Moreau, J. Fluid Mech. vol. 118, 1982, pp. 507–518), and thus establish its validity in sheared magnetohydrodynamics (MHD) turbulence. We further find that three-componentality and three-dimensionality appear concurrently and that both the frequency corresponding to the Q2-D cutoff scale and the mean energy associated with he axial component of velocity scale with the true interaction parameter $N_t$, respectively, as $0.063 N_t^{0.37}$ and $0.126N_t^{-0.92}$.

Type
JFM Papers
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

Alboussière, T., Uspenski, V. & Moreau, R. 1999 Quasi-two-dimensional MHD turbulent shear layers. Exp. Therm. Fluid Sci. 20, 1924.CrossRefGoogle Scholar
Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Phys. Rep. 767–769, 1101.CrossRefGoogle Scholar
Baker, N.T., Pothérat, A. & Davoust, L. 2015 Dimensionality, secondary flows and helicity in low-Rm MHD vortices. J. Fluid Mech. 779, 325350.CrossRefGoogle Scholar
Baker, N.T, Pothérat, A., Davoust, L. & Debray, F. 2018 Inverse and direct energy cascades in three-dimensional magnetohydrodynamic turbulence at low magnetic Reynolds number. Phys. Rev. Lett. 120 (22), 224502.CrossRefGoogle ScholarPubMed
Davidson, P.A. 1997 The role of angular momentum in the magnetic damping of turbulence. J. Fluid Mech. 336, 123150.CrossRefGoogle Scholar
Davidson, P.A. & Pothérat, A. 2002 A note on Bödewadt–Hartmann layers. Eur. J. Mech. B/Fluids 21 (5), 545559.CrossRefGoogle Scholar
Eckert, S., Gerbeth, G., Witke, W. & Langenbrunner, H. 2001 MHD turbulence measurements in a sodium channel flow exposed to a transverse magnetic field. Intl J. Heat Fluid Flow 22 (3), 358364.CrossRefGoogle Scholar
Klein, R. & Pothérat, A. 2010 Appearance of three dimensionality in wall-bounded MHD flows. Phys. Rev. Lett. 104 (3), 034502.CrossRefGoogle ScholarPubMed
Kljukin, A.A. & Kolesnikov, Y.B. 1989 Liquid Metal Magnetohydrodynamics, vol. 10. Kluwer.Google Scholar
Kobayashi, H. 2006 Large eddy simulation of magnetohydrodynamic turbulent channel flows with local subgrid-scale model based on coherent structures. Phys. Fluids 18 (4), 045107.CrossRefGoogle Scholar
Kobayashi, H. 2008 Large eddy simulation of magnetohydrodynamic turbulent duct flows. Phys. Fluids 20 (1), 015102.CrossRefGoogle Scholar
Kolesnikov, Y.B. & Tsinober, A.B. 1974 Experimental investigation of two-dimensional turbulence behind a grid. Fluid Dyn. 9 (4), 621624.CrossRefGoogle Scholar
Messadek, K. & Moreau, R. 2002 An experimental investigation of MHD quasi-two-dimensional turbulent shear flows. J. Fluid Mech. 456, 137159.CrossRefGoogle Scholar
Moffatt, H.K. 1967 On the suppression of turbulence by a uniform magnetic field. J. Fluid Mech. 28 (3), 571592.CrossRefGoogle Scholar
Moresco, P. & Alboussière, T. 2003 Weakly nonlinear stability of Hartmann boundary layers. Eur. J. Mech. B/Fluids 22 (4), 345353.CrossRefGoogle Scholar
Moresco, P. & Alboussiere, T. 2004 Experimental study of the instability of the Hartmann layer. J. Fluid Mech. 504, 167181.CrossRefGoogle Scholar
Ni, M.-J., Munipalli, R., Huang, P., Morley, N.B. & Abdou, M.A. 2007 A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part 2. On an arbitrary collocated mesh. J. Comput. Phys. 227 (1), 205228.CrossRefGoogle Scholar
Pothérat, A. & Dymkou, V. 2010 Direct numerical simulations of low-Rm MHD turbulence based on the least dissipative modes. J. Fluid Mech. 655, 174197.CrossRefGoogle Scholar
Pothérat, A. & Klein, R. 2014 Why, how and when MHD turbulence at low Rm becomes three-dimensional. J. Fluid Mech. 761, 168205.CrossRefGoogle Scholar
Pothérat, A. & Kornet, K. 2015 The decay of wall-bounded MHD turbulence between walls, at low Rm. J. Fluid Mech. 683, 605636.CrossRefGoogle Scholar
Pothérat, A. & Schweitzer, J.-P. 2011 A shallow water model for magnetohydrodynamic flows with turbulent Hartmann layers. Phys. Fluids 23 (5), 055108.CrossRefGoogle Scholar
Pothérat, A., Sommeria, J. & Moreau, R. 2000 An effective two-dimensional model for MHD flows with transverse magnetic field. J. Fluid Mech. 424, 75100.CrossRefGoogle Scholar
Pothérat, A., Sommeria, J. & Moreau, R. 2005 Numerical simulations of an effective two-dimensional model for flows with a transverse magnetic field. J. Fluid Mech. 534, 115143.CrossRefGoogle Scholar
Roberts, P.H. 1967 An Introduction to Magnetohydrodynamics, vol. 6. Longmans.Google Scholar
Schumann, U. 1976 Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field. J. Fluid Mech. 74 (1), 3158.CrossRefGoogle Scholar
Sommeria, J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139168.CrossRefGoogle Scholar
Sommeria, J. & Moreau, R. 1982 Why, how, and when, MHD turbulence becomes two-dimensional. J. Fluid Mech. 118, 507518.CrossRefGoogle Scholar
Stelzer, Z., Cébron, D., Miralles, S., Vantieghem, S., Noir, J., Scarfe, P. & Jackson, A. 2015 a Experimental and numerical study of electrically driven magnetohydrodynamic flow in a modified cylindrical annulus. I. Base flow. Phys. Fluids 27 (7), 077101.CrossRefGoogle Scholar
Stelzer, Z., Miralles, S., Cébron, D., Noir, J., Vantieghem, S. & Jackson, A. 2015 b Experimental and numerical study of electrically driven magnetohydrodynamic flow in a modified cylindrical annulus. II. Instabilities. Phys. Fluids 27 (8), 084108.CrossRefGoogle Scholar
Tabeling, P. & Chabrerie, J.P. 1981 Magnetohydrodynamic Taylor vortex flow under a transverse pressure gradient. Phys. Fluids 24 (3), 406412.CrossRefGoogle Scholar
Zhao, Y. & Zikanov, O. 2012 Instabilities and turbulence in magnetohydrodynamic flow in a toroidal duct prior to transition in Hartmann layers. J. Fluid Mech. 692, 288316.CrossRefGoogle Scholar