In this study, flow phenomena associated with inflectional and boundary-layer instabilities, as well as a mixed instability mode in quasi-two-dimensional magnetohydrodynamic flows in a rectangular duct are accessed with the help of a parametrical model, where the basic velocity profile with near-wall jets and associated points of inflection are produced by imposing an external flow-opposing force. By varying this force, various instability modes and transition scenarios are reproduced. First, linear stability analysis is performed and then nonlinear effects are studied using direct numerical simulation for Hartmann numbers 100 and 200 and Reynolds numbers from 1800 to 5000. A special attention is paid to the location of the inflection point with respect to the duct wall. A more complex flow dynamics, including various vortex-wall and vortex-vortex interactions, and even negative turbulence production are observed and analyzed as the inflection point approaches the wall. The analysis of obtained results and their comparison with relevant data for magnetohydrodynamic duct flows gain an insight into what is typically called “jet instability,” including linear and nonlinear mechanisms.

1.
R.
Moreau
,
Magnetohydrodynamics
(
Kluwer
,
The Netherlands
,
1990
).
2.
L.
Bühler
, “
Liquid metal magnetohydrodynamics for fusion blankets
,” in
Magnetohydrodynamics: Historical Evolution and Trends
, edited by
S.
Molokov
,
R.
Moreau
, and
H. K.
Moffat
(
Springer
,
New York
,
2007
), pp.
171
194
.
3.
L.
Bühler
, “
Instabilities in quasi-two-dimensional magnetohydrodynamic flows
,”
J. Fluid Mech.
326
,
125
(
1996
).
4.
Yu. B.
Kolesnikov
, “
Two-dimensional turbulent flow in a channel with inhomogeneous electrical conductivity of the walls
,”
Magnetohydrodynamics
8
,
308
(
1972
).
5.
U.
Burr
,
L.
Barleon
,
U.
Müller
, and
A.
Tsinober
, “
Turbulent transport of momentum and heat in magnetohydrodynamic rectangular duct flow with strong sidewall jets
,”
J. Fluid Mech.
406
,
247
(
2000
).
6.
L.
Bühler
and
S.
Horanyi
, “
Measurements of time-dependent liquid-metal magnetohydrodynamic flows in a flat rectangular duct
,”
Fusion Eng. Des.
84
,
518
(
2009
).
7.
C. B.
Reed
and
B. F.
Picologlou
, “
Sidewall flow instabilities in liquid metal MHD flow under blanket relevant conditions
,”
Fusion Tech.
15
,
705
(
1989
).
8.
M.
Kinet
,
B.
Knaepen
, and
S.
Molokov
, “
Instabilities and transition in magnetohydrodynamic flows in ducts with electrically conducting walls
,”
Phys. Rev. Lett.
103
,
154501
(
2009
).
9.
K.
Fajimura
, “Stability of MHD Flow Through a Square Duct,” UCLA-FNT-023,
1989
.
10.
A. L.
Ting
,
J. S.
Walker
,
T. J.
Moon
,
C. B.
Reed
, and
B. F.
Picologlou
, “
Linear stability analysis for high-velocity boundary layers in liquid-metal magnetohydrodynamic flows
,”
Int. J. Eng. Sci.
29
,
939
(
1991
).
11.
J.
Priede
,
S.
Aleksandrova
, and
S.
Molokov
, “
Linear stability of Hunt's flow
,”
J. Fluid Mech.
649
,
115
(
2010
).
12.
J. C. R.
Hunt
, “
Magnetohydrodynamic flow in rectangular ducts
,”
J. Fluid Mech.
21
,
577
(
1965
).
13.
Yu. M.
Gel'fgat
,
V. S.
Dorofeev
, and
E. V.
Shcherbinin
, “
Experimental investigation of the velocity structure of an MHD flow in a rectangular channel with two conducting walls
,”
Magnetohydrodynamics
7
,
26
(
1971
).
14.
I. A.
Platnieks
and
Ya. Zh.
Freiberg
, “
Turbulence and some problems in the stability of flows with M-shaped velocity profiles
,”
Magnetohydrodynamics
8
,
164
(
1972
).
15.
J.
Sommeria
and
R.
Moreau
, “
Why, how and when MHD turbulence becomes two-dimensional
,”
J. Fluid Mech.
118
,
507
(
1982
).
16.
Yu. B.
Kolesnikov
and
A. B.
Tsinober
, “
Magnetohydrodynamic flow in the region of a jump in the conductivity at the wall
,”
Magnetohydrodynamics
8
,
61
(
1972
).
17.
S.
Smolentsev
and
R.
Moreau
, “
Modeling quasi-two-dimensional turbulence in MHD duct flows
,” in
Proceedings of the 2006 Summer Program
, edited by
P.
Moin
and
N. N.
Mansour
(
CTR, Stanford University
,
2006
), p.
419
.
18.
S.
Smolentsev
and
R.
Moreau
, “
Modeling of quasi-two-dimensional magnetohydrodynamic turbulence
,” in
Proceedings of the 7th International PAMIR Conference on Fundamental and Applied MHD
, edited by
A.
Alemany
,
J.-P.
Chopart
, and
J.
Freibergs
(
CNRS
,
France
,
2008
), p.
857
.
19.
D.
Krasnov
,
O.
Zikanov
,
M.
Rossi
, and
T.
Boeck
, “
Optimal linear growth in magnetohydrodynamic duct flow
,”
J. Fluid Mech.
653
,
273
(
2010
).
20.
A.
Pothérat
,
J.
Sommeria
, and
R.
Moreau
, “
Numerical simulations of an effective two-dimensional model for flows with a transverse magnetic field
,”
J. Fluid Mech.
534
,
115
(
2005
).
21.
R.
Klein
and
A.
Pothérat
, “
Appearance of three dimensionality in wall-bounded MHD flows
,”
Phys. Rev. Lett.
104
,
034502
(
2010
).
22.
A.
Thess
and
O.
Zikanov
, “
Transition from two-dimensional to three-dimensional MHD turbulence
,”
J. Fluid Mech.
579
,
383
(
2007
).
23.
R.
Moreau
,
S.
Smolentsev
, and
S.
Cuevas
, “
MHD flow in an insulating rectangular duct under a non-uniform magnetic field
,”
PMC Physics B
3
(
2010
).
24.
S. Yu.
Smolentsev
, “
Averaged model in MHD duct flow calculations
,”
Magnetohydrodynamics
33
,
42
(
1997
).
25.
S.
Cuevas
,
S.
Smolentsev
, and
M.
Abdou
, “
On the flow past a magnetic obstacle
,”
J. Fluid Mech.
553
,
227
(
2006
).
26.
I. V.
Lavrent'ev
,
S. Yu.
Molokov
,
S. I.
Sidorenkov
, and
A. R.
Shishko
, “
Stokes flow in a rectangular magnetohydrodynamic channel with nonconducting walls within a nonuniform magnetic field at large Hartmann numbers
,”
Magnetohydrodynamics
26
,
325
(
1990
).
27.
T. N.
Aitov
,
A. B.
Ivanov
, and
A. V.
Tananaev
, “
Flow of liquid metal in a chute in a coplanar magnetic field
,”
Magnetohydrodynamics
23
,
78
(
1987
).
28.
K.
Messadek
and
R.
Moreau
, “
An experimental investigation of MHD quasi-two-dimensional turbulent shear flows
,”
J. Fluid Mech.
456
,
137
(
2002
).
29.
S.
Smolentsev
,
N.
Morley
, and
M.
Abdou
, “
Code development for analysis of MHD pressure drop reduction in a liquid metal blanket using insulation technique based on a fully developed flow model
,”
Fusion Eng. Des.
73
,
83
(
2005
).
30.
W.
Huang
and
D. M.
Sloan
, “
The pseudospectral method for solving differential eigenvalue problems
,”
J. Comput. Phys.
111
,
399
(
1994
).
31.
S. A.
Orszag
, “
Accurate solution of the Orr-Sommerfeld stability equation
,”
J. Fluid Mech.
50
,
689
(
1971
).
32.
R. C.
Lock
, “
The stability of the flow of an electrically conducting fluid between parallel planes under a transverse magnetic field
,”
Proc. R. Soc. London, Ser. A
233
,
105
(
1955
).
33.
A.
Pothérat
, “
Quasi-two-dimensional perturbations in duct flows under transverse magnetic field
,”
Phys. Fluids
19
,
074104
(
2007
).
34.
A.
Tsinober
,
An Informal Introduction to Turbulence
(
Kluwer
,
The Netherlands
,
2001
).
35.
J. C.
Tannehill
,
D. A.
Anderson
, and
R. H.
Pletcher
,
Computational Fluid Mechanics and Heat Transfer
(
Taylor & Francis
,
London
,
1997
).
36.
A.
Arakawa
, “
Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I
,”
J. Comput. Phys.
1
,
119
(
1966
).
37.
P.
Moresco
and
T.
Alboussière
, “
Experimental study of the instability of the Hartmann layer
,”
J. Fluid Mech.
504
,
167
(
2004
).
38.
T. L.
Doligalski
,
C. R.
Smith
, and
J. D. A.
Walker
, “
Vortex interactions with walls
,”
Annu. Rev. Fluid Mech.
26
,
573
(
1994
).
39.
S.
Smolentsev
and
R.
Moreau
, “
One-equation model for quasi-two-dimensional turbulent magnetohydrodynamic flows
,”
Phys. Fluids
19
,
078101
(
2007
).
You do not currently have access to this content.