Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-22T00:01:04.489Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulation of liquid-metal MHD flows in rectangular ducts

Published online by Cambridge University Press:  26 April 2006

A. Sterl
A. Sterl
Affiliation:
Association KfK-EURATOM, Kernforschungszentrum Karlsruhe, Institut für Reaktorbauelemente, Postfach 3640, D-7500 Karlsruhe 1, FRG Max-Planck-Institut für Meterologie, Bundesstrasse 55, D-2000 Hamburg 13, FRG.
Get access Rights & Permissions [Opens in a new window]

Abstract

To design self-cooled liquid metal blankets for fusion reactors, one must know about the behaviour of MHD flows at high Hartmann numbers. In this work, finite difference codes are used to investigate the influence of Hartmann number M, interaction parameter N, wall conductance ratio c, and changing magnetic field, respectively, on the flow.

As liquid-metal MHD flows are characterized by thin boundary layers, resolution of these layers is the limiting issue. Hartmann numbers up to 103 are reached in the two-dimensional case of fully developed flow, while in three-dimensional flows the limit is 102. However, the calculations reveal the main features of MHD flows at large M. They are governed by electric currents induced in the fluid. Knowing the paths of these currents makes it possible to predict the flow structure.

Results are shown for two-dimensional flows in a square duct at different Hartmann numbers and wall conductivities. While the Hartmann number governs the thickness of the boundary layers, the wall conductivities are responsible for the pressure losses and the structure of the flows. The most distinct feature is the side layers where the velocities can exceed those at the centre by orders of magnitude.

The three-dimensional results are also for a square duct. The main interest here is to investigate the redistribution of the fluid in a region where the magnetic field changes. Large axial currents are induced leading to the ‘M-shaped’ velocity profiles characteristic of MHD flow. So-called Flow Channel Inserts (FCI), of great interest in blanket design, are investigated. They serve to decouple the load carrying wall from the currents in the fluid. The calculations show that the FCI is indeed a suitable measure to reduce the pressure losses in the blanket.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 216 , July 1990 , pp. 161 - 191
Copyright
© 1990 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

Aitov, T. N., Kalyntik, A. I. & Tananaev, A. V. 1983 Numerical analysis of three-dimensional MHD-flow in channels with abrupt change of cross section. Magnetohydrodynamics 19, 223–229.Google Scholar
Aitov, T. N., Kalyntik, A. I. & Tananaev, A. V. 1984 Numerical study of three-dimensional MHD flow in a duct of complex configuration using stokes approximation. Magneto-hydrodynamics 20, 288–293.Google Scholar
Alty, C. J. N. 1971 Magnetohydrodynamic duct flow in a uniform transverse magnetic field of arbitrary orientation. J. Fluid Mech. 48, 429–461.Google Scholar
Anderson, D. A., Tannehill, J. C. & Pletcher, R. H. 1984 Computational Fluid Mechanics and Heat Transfer. McGraw-Hill.
Briley, W. R. & McDonald, H. 1980 On the structure and use of linearized block implicit schemes. J. Comp. Phys. 34, 54–73.Google Scholar
Carré, F., Proust, E., Remolouv, J., Racaboy, A. & Tilliette, Z. 1984 Fusion Reactor Blanket—Comparative Evaluation Study. Rapport EMT/SERMA/BP/84/No. 584 T, 3591–21–000, January 17.
Chang, C. C. & Lundgren, Th. S. 1961 Duct flow in magnetohydrodynamics. Z. angew. Math. Phys. 12, 100–114.Google Scholar
Douglas, J. & Gunn, J. E. 1964 A general formulation of alternating direction methods. Part 1. Parabolic and hyperpolic problems. Num. Maths 6, 428–453.Google Scholar
Holroyd, R. J. & Mitchell, J. T. D. 1984 Liquid lithium as a coolant for tokamak fusion reactors. Nucl. Engng Design/Fusion 1, 17–38.Google Scholar
Holroyd, R. J. & Walker, J. S. 1978 A theoretical study of the effects of wall conductivity, non-uniform magnetic fields and variable-area ducts on liquid-metal flows at high Hartmann number. J. Fluid Mech. 84, 471–495.Google Scholar
Hua, T. Q.et al. 1988 Three dimensional MHD flows in rectangular ducts of liquid-metal-cooled blankets. Fusion Tech. 14, 1389–1398.Google Scholar
Hua, T. Q. & Picologlou, B. F. 1989 Heat transfer in rectangular first wall coolant channels of liquid-metal cooled blankets. Fusion Tech. 15, 1174.Google Scholar
Hunt, J. C. R. 1965 Magnetohydrodynamic flow in rectangular ducts. J. Fluid Mech. 21, 577–590.Google Scholar
Hunt, J. C. R. 1969 A uniqueness theorem for magnetohydrodynamic duct flows. Proc. Camb. Phil. Soc. 65, 319–327.Google Scholar
Hunt, J. C. R. & Holroyd, R. J. 1977 Applications of laboratory and theoretical MHD duct flow studies in fusion reactor technology. UKAEA-Rep. CLM-R 169. (The results are also found in Holroyd & Walker 1978.)Google Scholar
Kalis, K. E. & Tsinober, A. B. 1973 Numerical analysis of three dimensional MHD flow problems. Magnetohydrodynamics 2, 35–40.Google Scholar
Khan, S. 1987 Three-dimensional MHD entry flow in a duct with a laterally moving wall. Arab. J. Sci. Engng 12, 405–412.Google Scholar
Khan, S. & Davidson, J. N. 1979 Magnetohydrodynamic coolant flows in fusion reactor blankets. Ann. Nucl. Energy 6, 499–509.Google Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier—Stokes-Equations. J. Comp. Phys. 59, 308–322.Google Scholar
Madarame, M. & Hagiwara, T. 1988 Computer code for analysing liquid metal MHD flow in rectangular ducts under strong transverse magnetic field. First Intl Symp. on Fusion Nucl. Tech. Tokyo 1988. To appear in Fusion Engng Design.Google Scholar
Malang, S., Arheidt, K., Barleon, L., Borgstedt, H. U., Casal, V., Fischer, U., Link, W., Reimann, J. & Reust, K. 1988 Self-cooled liquid-metal blanket concept. Fusion Tech. 14, 1343–1356.Google Scholar
Moffatt, H. K. 1967 On the suppression of turbulence by a uniform magnetic field. J. Fluid Mech. 28, 571–592.Google Scholar
Morozova, V. I., Nagornyi, M. M. & Elkin, A. J. 1983 Motion of a viscous, electrically conducting liquid through distributing and converging collectors in a transverse magnetic field. Magnetohydrodynamics 19, 332–336.Google Scholar
Picologlou, B. F. 1985 Magnetohydrodynamic considerations for the design of self-cooled liquid-metal fusion reactor blankets. Fusion Tech. 8, 276–282.Google Scholar
Ramos, J. I. & Winowich, N. S. 1986 Magnetohydrodynamic channel flow study. Phys. Fluid 29, S. 992–997.Google Scholar
Reed, C. B., Picologlou, B. F., Hua, T. Q. & Walker, J. S. 1987 ALEX results — A comparison of measurements from a round and a rectangular duct with 3-D code predictions. Proc. 12th Symp. on Fusion Engineering, pp. 1267.
Schumann, U. & Sweet, R. A. 1988 Fast Fourier transforms for direct solution of Poisson's equation with staggered boundary conditions. J. Comp. Phys. 75, 123–137.Google Scholar
Sezgin, M. 1987 Magnetohydrodynamic flow in a regular duct. Intl. J. Num. Meth. Fluids 7, 697–718.Google Scholar
Smith, D. L.et al. 1984 Blanket Comparison and Selection Study — Final Report, vol.1–3, ANL/FPP-84–1, Sept. 1984.
Sommeria, J. & Moreau, R. 1982 Why, how and when MHD turbulence becomes two-dimensional. J. Fluid Mech. 118, 507–518.Google Scholar
Sterl, A. 1989 Numerische Simulation magnetohydrodynamischer Flüssig-Metall-Strömungen in rechteckigen Rohren bei grosen Harmannzahlen. Kernforschungszentrum Karlsruhe, KfK-4504.
Swarztrauber, P. & Sweet, R. 1975 Efficient FORTRAN subprograms for the solution of elliptic partial differential equations. NCAR Tech. Note NCAR-LTN/IA-109.Google Scholar
Talmage, G. & Walker, J. S. 1987 Three-dimensional laminar MHD-flow in ducts with thin metal walls and strong magnetic fields. 5th Beer-Sheva Seminar on MHD and Turbulence, Tel-Aviv.
Temperley, D. J. 1984 Reconciliation of some discrepancies in results obtained for magnetohydrodynamic flow in a rectangular duct under an uniform transverse magnetic field at high Hartmann number. Arch. Mech. 35, (5–6), 673–686.Google Scholar
Thompson, J. F. 1985 A survey of dynamically-adaptive grids in the numerical solution of partial differential equations. Appl. Maths 1, 3–27.Google Scholar
Tillack, M. S. 1988 Examination of MHD fluid flow in geometric elements of a fusion reactor blanket using the core flow approach. First Intl Symp. on Fusion Nucl. Techn. Tokyo 1988. To appear in Fusion Engng Design.
Walker, J. S. 1981 Magnetohydrodynamic flows in rectangular ducts with thin conducting walls. Part I: Constant area and variable area ducts with strong uniform magnetic fields. J. Méc. 20, 79–112.Google Scholar
Walker, J. S. 1986a Laminar duct flows in strong magnetic fields. In Liquid-Metal Flows and Magnetohydrodynamics, vol. 2 (ed. H. Branover, P. S. Lykoudis & M. Moon), pp. 3–16. American Institute of Aeronautics and Astronautics, New York.
Walker, J. S. 1986b Liquid-metal MHD flow in a duct whose cross section changes from a rectangle to a trapezoid, with applications in fusion blanket design. Argonne Nat. Lab. Rep. ANL/FPP/TM-206.Google Scholar
Walker, J. S. 1986c Liquid-metal MHD flow in a thin conducting pipe near the end of a uniform magnetic field. J. Fluid Mech. 167, S. 199–217. Argonne Nat. Lab. Rep. ANL/FPP/TM-205.Google Scholar
Walker, J. S. 1986d Liquid-metal flow in a rectangular duct with a non-uniform magnetic field. Argonne Nat. Lab. Rep. ANL/FPP/TM-207.Google Scholar
Walker, J. S. 1986e Liquid-metal flow through a thin-walled elbow in a plane perpendicular to a uniform magnetic field. Argonne Nat. Lab. Rep. ANL/FPP/TM-204, Intl J. Engng Sci. 24, 1741–54.Google Scholar
Walker, J. S. & Ludford, G. S. S. 1974a MHD flow in conducting circular expansions with strong transverse magnetic fields. Intl J. Engng Sci. 12, 193–204.Google Scholar
Walker, J. S. & Ludford, G. S. S. 1974b MHD flow in insulated circular expansions with strong transverse magnetic fields. Intl J. Engng Sci. 12, 1045–1061.Google Scholar
Walker, J. S. & Ludford, G. S. S. 1975 MHD flow in circular expansions with thin conducting walls. Intl J. Engng Sci. 13, 261–269.Google Scholar
Winowich, N. S. & Hughes, W. F. 1982 A finite element analysis of two-dimensional MHD-flows. In Liquid Metal Flows and MHD (ed. M. Branover, P. S. Lykoudis & A. Yakhot). 3rd Intl Sem. in the MHD and Turbulence Series, Beer Sheva, 1981, Am. Inst. of Aeronautics and Astronautics, 1982.
Wu, S. T. 1973 Unsteady MHD duct flow by the finite element method. Intl J. Num. Math. Engng. 6, S.3–10.Google Scholar
Yagawa, G. & Masuda, M. 1982 Finite element analysis of magnetohydrodynamics and its application to lithium blanket design of a fusion reactor. Nucl. Engng Design 71, 121–136.Google Scholar
Zierep, J. 1982 Grundzüge der Strömungslehre, 2nd edn. Karlsruhe.