Numerical simulation of a liquid-metal flow in a poorly conducting pipe subjected to a
strong fringing magnetic field
X. Albets-Chico, H. Radhakrishnan, S. Kassinos, and B. Knaepen
Citation: Physics of Fluids 23, 047101 (2011); doi: 10.1063/1.3570686
View online: http://dx.doi.org/10.1063/1.3570686
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PHYSICS OF FLUIDS 23, 047101 共2011兲
Numerical simulation of a liquid-metal flow in a poorly conducting pipe
subjected to a strong fringing magnetic field
X. Albets-Chico,1,a兲 H. Radhakrishnan,1 S. Kassinos,1 and B. Knaepen2
1
Department of Mechanical and Manufacturing Engineering, Computational Science Laboratory–UCYCompSci, University of Cyprus, 75 Kallipoleos, Nicosia 1678, Cyprus
2
Physique statistique et des plasmas, Faculté des Sciences, Université Libre de Bruxelles, Campus de la
Plaine–CP231, Boulevard du Triomphe, 1050 Bruxelles, Belgium
共Received 17 February 2010; accepted 25 January 2011; published online 26 April 2011兲
Using high resolution numerical simulations, we study the flow of a liquid metal in a pipe subjected
to an intense decreasing magnetic field 共fringing magnetic field兲. The chosen flow parameters are
such that our study is directly relevant for the design of fusion breeder blankets. Our objectives are
to provide a detailed description of the numerical method and of the results for benchmarking
purposes but also to assess the efficiency of the so-called “core flow approximation” that models
liquid-metal flows under the influence of intense magnetic fields. Our results are in excellent
agreement with available experimental measurements. As far as the pressure drop is concerned, they
also match perfectly the predictions of the core flow approximation. On the other hand, the velocity
profiles obtained in our numerical simulations show a significant departure from this approximation
beyond the inflection point of the magnetic field’s profile. By plotting the momentum budget of the
MHD equations, we provide evidence that this discrepancy can be attributed to the role of inertia
that is neglected in the core flow approximation. We also consider a case with vanishing outlet
magnetic field and we briefly illustrate the transition to turbulence arising in the outlet region of the
pipe. © 2011 American Institute of Physics. 关doi:10.1063/1.3570686兴
I. INTRODUCTION
Liquid-metal flows, governed by magnetohydrodynamics 共MHD兲, are commonly encountered in many industrial
applications. Examples include the casting of steel, aluminum reduction, or the fabrication of glass and semiconductors. MHD flows are also very important in the design of
nuclear fusion reactors. Indeed, in such devices, a high flux
of neutrons is created by the nuclear reactions and it is envisaged to stop these neutrons in breeder blankets 共acting as
coolant and breeder material兲 that consist of liquid lithium
flows. These blankets often have circular cross-section inlets
and outlets where the flow enters or leaves a region of space
where the magnetic field used for the confinement of the
fusion plasma is present. This intense and strongly varying
magnetic field induces a strong pressure drop and redistribution of the flow. The accurate computation of the pressure
drop is critical in the design and development of the selfcooled blankets. Its magnitude depends on the magnetic field
intensity, Reynolds number of the flow, and electrical conductivity of the blanket walls.1
Pioneering work in understanding liquid-metal MHD
flows was performed in Refs. 2 and 3. In the case of the
homogeneous, laminar, pipe flow 共constant magnetic field兲,
analytical solutions for insulating walls4 and conducting
walls5,6 have been obtained. Unfortunately, these solutions
take the form of infinite series of modified Bessel functions
and are very difficult to evaluate for intense magnetic fields,
a兲
Author to whom correspondence should be addressed. Also at Physique
Statistique et des Plasmas, Université Libre de Bruxelles. Electronic mail:
xalbets@ucy.ac.cy.
1070-6631/2011/23共4兲/047101/11/$30.00
certainly when the intensity of the Lorentz forces compared
to viscous forces, as measured in terms of the Hartmann
number Ha, is high 共see below for a precise definition of Ha兲.
In that case, extremely thin boundary layers, called Hartmann layers, develop in the vicinity of walls perpendicular to
the magnetic field and the flow exhibits sharp gradients in
that region; in the rest of the pipe, the flow mostly adopts a
uniform core in the direction of the magnetic field. At high
values of Ha, the governing equations can however be simplified and approximate analytical solutions can be obtained
through asymptotic methods that assume an inviscid and inertialess core surrounded by the thin exponential Hartmann
layers. In the case of the homogeneous, laminar, MHD pipe
flow, this framework was successfully applied in Refs. 7–9.
Recently, this geometry has been reinvestigated using numerical simulations and the presence of overspeed regions in
the side-layers 共region where the wall is parallel to the magnetic field兲 has been demonstrated for certain values of the
wall conductivity.10
The study of the MHD pipe flow in a nonuniform magnetic field is noticeably more difficult. Here we focus our
attention on the flow in a weakly conducting pipe that exists
a region of space embedded in a strong magnetic field 共fringing magnetic field兲. This configuration has been analyzed
analytically through an asymptotic calculation in Ref. 11 and
the major characteristics of the flow could be described 共see
also Ref. 12兲. In particular, the authors showed how the fully
developed flow present in the upstream part of the pipe was
redistributed in the fringing region to form two high speed
jets in the side-layers and a low velocity core; this redistribution was also shown to induce a very important additional
23, 047101-1
© 2011 American Institute of Physics
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Albets-Chico et al.
pressure drop. This era also saw the birth of a powerful modeling technique of liquid-metal flows under the influence of
strong magnetic fields.13,14 In this method, referred to as the
core flow approximation, one assumes that within the core
flow, inertia and viscous effects are negligible. This implies
that momentum conservation reduces to a linear equation
expressing the balance between the pressure gradient and the
Lorentz force. Upon integration of the three-dimensional
共3D兲 problem along the magnetic field lines, the problem
then reduces to a two-dimensional 共2D兲 system for three unknown functions, thereby greatly reducing the computational
cost.15 In the case of the MHD pipe flow in a fringing magnetic field, the core flow approximation has been implemented in various numerical codes16–23 and compared to the
asymptotic analysis of Ref. 11 or experimental studies of this
flow configuration.16,24–26
Although the core flow approximation is a powerful tool
for investigating numerically MHD flows at high Hartmann
numbers in relatively complex geometries, it does not come
without limitations. First, for flows in which inertia plays a
significant role, it cannot provide an accurate description;
second, a finite 共and strong兲 magnetic field must be applied
everywhere in the computational domain to fulfill its assumptions. Regarding the role of inertia, its magnitude depends on the particular flow considered but in general it is
more important for cases where the magnetic field is less
intense and the wall conductivity of the duct assembly is low
共the braking effect of the Lorentz force is then globally small
since the current loops close mostly within the fluid兲. Several
authors have assessed this issue and compared the accuracy
of the core flow approximation against more complete numerical solutions 共see for example Refs. 22, 27, and 28兲 but
given the extremely high computational requirements, these
studies have been limited to cases at moderate Hartmann
numbers. As a consequence, the range of parameters for
which the core flow approximation offers enough precision is
still an open question. For instance, for the geometry and
parameters we consider, the authors in Refs. 16 and 19 concluded that the agreement between the core flow approximation and experimental data was excellent. This conclusion
was reached by examining data for the pressure over the
whole pipe and the velocity profiles up to a short distance
downstream of the magnetic field’s inflection point. For even
lower values of the Hartmann number 共Ha⬇ 300兲 and for a
similar test case, the authors of Ref. 29 also concluded that
the core flow approximation and a full solution of the 3D
MHD equations agree well in terms of core velocity. Based
on these findings and order of magnitude estimates, the core
flow approximation has been used in several engineering
studies of the MHD pipe flow in a fringing magnetic field
with values of the Hartmann number in the outlet region as
low as 20 or 200 共see, e.g., Ref. 22兲. However, as we will
show later in this article, this approach is not valid for subtle
quantities like velocity profiles at specific downstream positions 共the later were not explored in previous studies兲. Finally, we recall that the second limitation of the core flow
approximation implies that it cannot address the transition of
flows from a high magnetic field region to another in which
no magnetic field is present since it requires high Hartmann
Phys. Fluids 23, 047101 共2011兲
numbers everywhere in the domain. In the case of fringing
fields, it cannot predict velocity profiles as the flow moves
away from the magnets and in particular it cannot capture the
possible transition to turbulence occurring downstream. Following this discussion, one easily understands the practical
difficulty in applying the core flow approximation in the case
of fringing magnetic fields: to satisfy its assumptions, a sufficiently high outlet magnetic field has to be applied; however, the lowest possible value has to be chosen to obtain the
correct physical behavior of the flow away from the magnets.
As reckoned by several authors,19,28,29 high resolution numerical solution of the 3D MHD equations are then indispensable to fully validate the core flow approximation for a
given geometry and flow parameters.
The objective of this paper is to study the case of the
MHD pipe flow with poorly conducting wall in the presence
of a fringing magnetic field, by numerically computing full
solutions of the quasistatic MHD equations. Given the
progress in numerical algorithms and the access to higher
computational resources, full solutions are now available for
flow regimes at high Hartmann numbers that are relevant for
fusion applications. Our test case is based on previously performed experiments16,30,31 and is a benchmark problem of
the International Energy Agency 共IEA兲 Implementing Agreement on Nuclear Technology of Fusion Reactors 共Annex 01,
Tritium Breeding Blanket, Radiation Shielding and Tritium
Processing Systems of Fusion Reactors兲.32 This case was
briefly considered in Refs. 33 and 34 to assess the performance of a novel discretization of the Lorentz force in the
finite volume framework. Here, we extend the results in several ways. First we consider a much longer computational
domain downstream of the fringing field region. This allows
us to examine the evolution of the jets in the weak magnetic
field region and demonstrate the transition to turbulence of
the flow that was not documented in any previous studies.
Also, we make a careful comparison of our results with the
core flow approximation and explicitly evaluate both the
roles of inertia and of the nonzero outlet magnetic field necessary for this approximation. Even though full solutions of
the MHD equations at high Hartmann numbers become more
easily obtainable, they still require very large resources for
cases such as the one considered in this paper. For this reason, we limit our attention to the parameters of the benchmark case and we do not perform a systematic parametric
study for the geometry considered. We also deliberately only
describe qualitatively the transition to turbulence occurring
at the outlet of the pipe since the simulations could not be
run long enough to gather reliable statistics such as turbulence intensities etc. Indeed, the time step required by the
intense magnetic field is two orders of magnitude smaller
than the turbulent hydrodynamic one.
The paper is organized as follows. In Sec. II, we define
the test case considered and recall the governing equations,
along with the proper boundary conditions. Sec. III is devoted to the numerical aspects; in particular, we describe the
numerical algorithm 共based on a finite volume discretization兲
and the grid used. In Sec. IV, we present the computational
results that include the pressure gradient, velocity profiles
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Phys. Fluids 23, 047101 共2011兲
Direct numerical simulation of a liquid-metal flow
tw = 3.27 ⫻ 10−3 m,
outlet
y
x
z
M
d
fiel
etic
n
ag
(x)
= By
where U is the average flow velocity, tw is the pipe’s wall
thickness and ␴w is the wall’s electrical conductivity.
To obtain nondimensional equations, we rescale the variables as follows: u → Uu, B → B01y, ⵜ → R−1ⵜ, t → 共R / U兲t,
j → ␴UB0j, ␾ → RUB0␾, and p → ␳U2 p. Here B0 is the intensity of the magnetic field at the inlet section of the pipe. The
momentum balance and Ohm’s law then read,
D=2R
x=15 R
inlet
30
R
x=-15 R
FIG. 1. Geometry of the duct and magnetic field. The dimensions are reported in multiples of the duct radius.
and momentum budget. Finally, conclusions are drawn in
Sec. V.
II. PROBLEM DESCRIPTION
冉
A. Geometry and governing equations
The geometry considered 共see Fig. 1兲 is based on experiments conducted at Argonne’s Liquid Metal Experiment
共ALEX兲 facility.16,30 It consists of a straight circular duct in
which a liquid-metal flow exits from a region subject to an
intense magnetic field 共the exact form of the magnetic field is
given in Sec. II B兲. As stressed above, this type of flow is
very important in the design of nuclear fusion blankets. The
length unit is chosen as the pipe radius R and the whole
computational domain spans 30R. The axis of coordinates
are placed at the geometrical center of the pipe and the x
direction is aligned with its axis.
For most laboratory flows involving liquid metals, the
so-called quasistatic approximation of the MHD equations35
holds. The momentum balance can then be written as:
⳵u
+ u · ⵜu = − ⵜ共p/␳兲 + ␯ⵜ2u + 共1/␳兲j ⫻ B,
⳵t
共1兲
j = ␴共− ⵜ␾ + u ⫻ B兲,
共2兲
ⵜ · u = 0,
共3兲
ⵜ · j = 0.
共4兲
In the above equations, u , B , j , p , ␾ , ␳ , ␯ , ␴ denote respectively the velocity field, the imposed magnetic field, the electric current, the pressure, the electric potential, the density of
the fluid, its viscosity and its electric conductivity. The extra
term appearing last in the RHS of Eq. 共1兲 is traditionally
referred to as the Lorentz force. The electric potential can be
obtained by solving the Poisson equation, ⵜ2␾ = ⵜ · 共u ⫻ B兲
resulting from Eqs. 共2兲 and 共4兲.
The ALEX experimental flow conditions can be reproduced by adopting the parameters defined in the IEA liquid
breeder blanket subtask benchmark problem definition referred to in the Introduction:
␴ = 2.80 ⫻ 106 S/m,
U = 0.07 m/s,
␳ = 865 kg/m3,
␯ = 9.5 ⫻ 10−7 m2/s,
冊
1 2
⳵u
+ u · ⵜu = − ⵜp +
ⵜ u + Nj ⫻ 1y ,
Re
⳵t
共5兲
j = − ⵜ␾ + u ⫻ 1 y .
共6兲
The two nondimensional numbers appearing in Eq. 共5兲 are
the 共bulk兲 Reynolds number Re and the interaction parameter
N,
Re =
R = 0.0541 m,
␴w = 1.39 ⫻ 106 S/m.
UR
,
␯
N=
␴B20R
.
␳U
共7兲
From Eq. 共5兲, we clearly see that inertia will be negligible
compared to the Lorentz force when N is large and that viscous contributions will also be small for high values of the
Hartmann number,
Ha = 冑N Re = BR
冑
␴
.
␳␯
共8兲
In that case, the flow evolution is governed by the balance,
ⵜp = Nj ⫻ 1y ,
共9兲
which together with appropriate boundary conditions to take
into account the Hartmann layers, constitutes the basis of the
core flow approximation.15 In the case of a spatially varying
magnetic field, N is a function of space, based on the local
strength and direction of the magnetic field.
Finally, we note that based on the benchmark parameters, the bulk velocity Reynolds number is,
共10兲
Re = 3986.
B. Imposed magnetic field
In the ALEX facility, the magnetic field is produced by
an iron core electromagnet capable of generating a highly
uniform 2.08 T field in a region that spans 0.76 m wide
⫻ 1.83 m long. At the outlet of the electromagnet, the magnetic field decays rapidly and its y-component can be approximated by the following fit function:
Bexp
y 共x兲 = Binlet
1 − tanh关␥共x/R + d兲兴
.
2
共11兲
We note that the above formula allows a nearly perfect fit of
the available experimental data. Binlet is the intensity of the
magnetic field inside the magnet 共where it is uniform兲, ␥ is a
decreasing factor and d is a coordinate dependent translation
parameter relative to the inflection point of the magnetic
field. Formula 共11兲 has already been used in several other
studies of liquid-metal flows exiting a region of intense magnetic field19,33,34 and is also part of the IEA liquid breeder
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Phys. Fluids 23, 047101 共2011兲
Albets-Chico et al.
blanket subtask benchmark problem definition. Unfortunately, since it is not available and to allow a fair comparison
with the core flow approximation, Bx is set to zero so that the
magnetic field has a nonzero curl. In order to match the
experimental magnetic field we have,
Binlet = 2.08 T,
␥ = 0.45,
d = − 0.33.
共12兲
The above magnetic field allows a direct comparison
with the experimental results.16 However, we are also interested in the comparison of the full solution data with results
obtained through the core flow approximation. For that reason, we also consider a second magnetic field given by,
Bcfa
y 共x兲 =
再
Bexp
y 共x兲
for x ⬍ 3.6,
0.05Binlet for x ⱖ 3.6.
冎
共13兲
冑
␴
= 6569,
␳␯
N = Ha2/Re = 10824.
⳵n␾ = ⵜ␶ · 共cⵜ␶␾兲
共at the walls兲,
共14兲
共15兲
In the outlet region, we have Ha→ 0 or Ha= 330 respectively
for the cases FSBexp or FSBcfa.
C. Boundary conditions
At the walls, the velocity satisfies the no-slip condition
uwall = 0. At the entrance of the pipe 共inlet兲, the velocity field
has to be specified. The profile is chosen as the solution of a
laminar pipe flow subject to a homogeneous magnetic field
and obtained through an independent numerical simulation
共see Ref. 10 for details兲. At the outlet of the pipe, the
convective 共or nonreflective兲 boundary condition for the
velocity is adopted: 共⳵u / ⳵t兲 + Uconv共⳵u / ⳵n兲 = 0. Uconv
= 共1 / S兲兰outletu · ndS is the bulk velocity computed at the outlet cross section S and n is the outward normal unit vector.
The boundary conditions for the Poisson equation determining the electric potential are more intricate. First, we assume that the walls are thin compared to the pipe’s radius. In
that case, we can use the so-called thin wall approximation,36
共16兲
where ⵜ␶ stands for the component of the ⵜ-operator tangential to the wall. c is the so-called wall-conductance ratio defined as c = 共␴wdw / ␴R兲, ␴w and dw being, respectively, the
wall conductivity and thickness 共here c = 0.027兲. It is easily
seen that in the limit of perfectly conducting 共insulating兲
walls, we recover the familiar Dirichlet 共Neumann兲 condition. At the inlet of the pipe, we impose that no electric
currents are entering the domain. Combining Eq. 共2兲 and u 储 n
at the inlet, we get:
共⳵ ␾/⳵ n兲兩inlet = 0.
Indeed, one of the main assumptions of the core flow approximation is that the Hartmann number is very high and
therefore the magnetic field’s intensity cannot decrease toward zero for large values of x. The core flow approximation
results presented later in this paper 共L. Bühler, private communication兲, are obtained using the same outlet residual field
共0.05Binlet兲. We note however that for x ⬍ 3.6, they are not
computed with the magnetic field based on the fitting function but from interpolated values of the experimental magnetic field 共this results in a difference of maximum 1%兲.
In the far downstream region, the flow will essentially be
a hydrodynamic flow when Bexp
y is used while it will remain
strongly influenced by the magnetic field when Bcfa
y is used.
Numerical results obtained with either fields are referred to
in the sequel as cases FSBexp and FSBcfa.
Based on bulk flow quantities and given the imposed
magnetic field, we have the following values for the Hartmann number and interaction parameter in the inlet region:
Ha = BR
which assumes that currents discharge tangentially in the
wall 共i.e., they have no normal component兲. This condition
reads:
共17兲
At the outlet, we also impose that no electric current leaves
the domain.37 The general boundary condition is thus,
共⳵␾ / ⳵n兲 兩outlet = 共u ⫻ B兲 · n. However, for the cases we consider, this relation reduces again to,
共⳵ ␾/⳵ n兲兩outlet = 0.
共18兲
Indeed, as described above, we either consider that: Boutlet
⬇ 0 or that Boutlet is sufficiently large so that any turbulence
produced in the fringing region is damped before reaching
the outlet 共as a consequence, u 储 n in that region兲.
III. NUMERICAL ASPECTS
A. Numerical Algorithm
Our computations are performed using the CDP code
developed at the Center For Turbulence Research 共Stanford/
NASA Ames兲. The details of the code have been described
extensively in Refs. 38–41 and benchmarked in a variety of
hydrodynamic complex flows. For this study, we have
complemented this code with a module to compute the Lorentz force and include it in the momentum balance. The
implementation and integration of this module in CDP has
been validated for several MHD flows and in various geometries including the MHD channel flow,42 the MHD duct flow
共Hunt’s case兲43 and the MHD pipe flow.10,44,45 Below, we
provide an extra validation of the code in the present test
case by comparing our results with those obtained by another
independent finite volume solver.33,34
CDP uses a collocated finite volume discretization of the
incompressible Navier–Stokes equations in a node-based formulation. A typical grid element is illustrated in Fig. 2. The
label C corresponds to the location of the centroid of the
element in the original volume-based grid. In the dual mesh,
the node-based control volumes are centered around each of
the vertices 共nodes兲 of this original mesh. In the figure, P
represents such a node of the dual mesh. The variables
共ui , ji , p , ␾兲 are stored at these nodes and the 共independent兲
face normal velocity U f are stored at the centroid of the dual
volume’s faces. The discrete equations of motion are solved
using the following fractional step method:
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Direct numerical simulation of a liquid-metal flow
5
n + 1 / 2, through the Poisson system defined by the incompressibility condition,
6
3
共1/⌬t兲 兺 u쐓i,f ni,f A f = 兺 共⳵ j pn+1/2兲 f n j,f A f .
4
f
C
1
P
E
(ui, ji , p, φ)
In Eq. 共25兲, the face normal flux of the pressure gradient is
computed through:
共⳵ j pn+1/2兲 f n j,f =
2
f’ F
Uf
FIG. 2. 共Color online兲 Illustration of the collocated mesh. The shaded area
represents a face belonging to the dual mesh.
ûi,P − uni,P
V P + 兺 Un+1/2
un+1/2
Af
f
i,f
⌬t
f
f
共19兲
共20兲
쐓
un+1
i,P − ui,P
= − 共⳵i p兲n+1/2
.
P
⌬t
共21兲
In the above equations, V P is the volume of the control volume surrounding point P and the sums are extended to all
faces f delimiting this control volume. The face areas are
denoted A f and their unit normal vectors are written ni,f .
Variables have the subscripts P or f depending on whether
they are evaluated at the center of the control volume or the
face. For any quantity ␨, the interpolation from nodal to face
values is done through the following expression:
␨ P + ␨nbr
,
2
共27兲
Un+1
= u쐓i,f ni,f − ⌬t共⳵n p兲n+1/2
.
f
f
共28兲
In Eq. 共27兲, the node value of the pressure gradient is computed by making use of the discrete Gauss theorem which
expresses the derivative of a quantity ␨ with a summationby-part 共SBP兲 operator,41 to provide corrections for skewed
or stretched elements:
f
u쐓i,P − ûi,P
= 共⳵i p兲n−1/2
,
P
⌬t
共␨兲 f =
共26兲
쐓
n+1/2
,
un+1
i,P = ui,P − ⌬t共⳵i p兲 P
共 ⳵ i␨ 兲 PV P = 兺 兺
= − 共⳵i p兲n−1/2
V P + N关jnP ⫻ 1y兴i
P
1
共⳵ jun+1/2
兲 f n j,f A f
i
Re
n+1/2
pnbr
− pn+1/2
P
,
储x j,nbr − x j,P储
where xi,P are the coordinates of P 共an identical expression is
兲 f n j,f in 共19兲 as a semi-implicit
also used to evaluate 共⳵ jun+1/2
i
contribution兲. Once the pressure is known, the nodal and face
normal velocities are updated using:
nbr
f
+兺
共25兲
f
共22兲
where the subscript nbr denotes the node located on the other
side of f, with respect to P. We further have,
3
1
= Unf − Un−1
,
Un+1/2
f
2
2 f
共23兲
1
= 共ûi,f + uni,f 兲.
un+1/2
i,f
2
共24兲
In other words, the face normal velocity is advanced in time
using the Adams–Bashforth scheme, while the Crank–
Nicholson advancement scheme is used for the nodal velocity. Equation 共21兲 is used to compute the pressure at step
f⬘
␨E + ␨F + ␨C
ni,f ⬘A f ⬘ .
3
共29兲
E, F, and C are respectively the center of the edge between P
and nbr, the center of the considered control volume face and
the center of the control volume. In this scheme, each face is
thus decomposed into several subfaces f ⬘ and the subface
value is obtained by averaging over its circumcenter values.
For example, in the case of the cartesian dual-mesh cell illustrated in Fig. 4, ␨E = 共␨ P + ␨nbr兲 / 2, ␨F = 共␨ P + ␨nbr + ␨1 + ␨2兲 / 4,
and ␨C = 共␨ P + ␨nbr + ␨1 + ␨2 + ␨3 + ␨4 + ␨5 + ␨6兲 / 8.
As seen in Eq. 共19兲, the MHD term 共Lorentz force兲 is
treated explicitly as an additional force term. At high Hartmann numbers, this severely limits the time-step to ensure
numerical stability. For example, at Ha= 7000, the time-step
size is two orders of magnitude smaller than the one needed
for the turbulent hydrodynamic flow at the same bulk Reynolds number.
Also, from the spatial discretization point of vue, the
accurate computation of the Lorentz force is the most challenging aspect in the full solution calculations of MHD flows
at moderate and high Hartmann numbers. As seen from
Ohm’s Law 关Eq. 共2兲兴, the current density depends on the
difference between ⵜ␾ and u ⫻ B. Any error in this difference is then multiplied by the interaction parameter 共when
the equations are written in nondimensional form兲 for computing the Lorentz force 关see Eq. 共5兲兴. This may generate an
error in the momentum balance that can in some cases be
larger than the other terms present by several orders of magnitude 共depending on the Ha and the boundary conditions for
the electric potential兲, resulting in heavily inaccurate predictions of the flow behavior. An extensive description of the
different ways to discretize the Lorentz force is given in Ref.
34 共see also Ref. 46 for a recent new proposal兲. In particular,
the authors of Ref. 34 advocate the use of a so-called “con-
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Phys. Fluids 23, 047101 共2011兲
Albets-Chico et al.
O(Ha−1 cos θ)
Hartmann layers
1
θ
θ
n
0.5
O(Ha−1/3 )
u
B
O(Ha−2/3 )
y/R
r
0
Core
-0.5
y
Roberts layers
z
-1
-1
x
sistent and conservative” discretization that guarantees that
no global spurious Lorentz force is added to the momentum
balance because of interpolation errors. While this is an essential and desired feature when the solid walls are electrically insulating 共in that case all the currents loop in the fluid
domain and the net induced Lorentz force vanishes兲, we have
observed in our case that this discretization leads to a more
unstable numerical solution. This was apparently not observed in the work of Ref. 34, but we note that their analysis
was conducted on a cell-centered spatial grid while our code
共CDP兲 uses a node-based grid. For this reason, we have
adopted a conventional discretization of ohm’s law that is
adequate for electrically conducting walls when the flow
phenomenology is dominated by the extremely large Lorentz
force braking:
1
兺 兺 ␾nf ⬘A f ⬘n f ⬘ + unP ⫻ 1y .
V P faces f
0
0.5
1
z/R
FIG. 3. Cross section of a circular duct and flow subregions at high Ha
numbers.
jnP = −
-0.5
共30兲
⬘
In Eq. 共30兲, ␾ f ⬘ is the electric potential value at the subface
f ⬘ 共Fig. 2兲 and it is obtained by discretizing the Poisson
equation ⵜ2␾ = ⵜ · 共u ⫻ 1y兲 in exactly the same way as the
pressure Poisson problem.
B. Numerical grid
In MHD flows, the boundary layers may become extremely thin at moderate and high Ha numbers. For example,
for an infinite circular duct 共see Fig. 3兲, the thickness of the
Hartmann layer is inversely proportional to the Ha number,
i.e., ␦Ha = O共Ha−1兲, and the dimensions of the Roberts 共side兲
layers47 scale as ␦y ⫻ ␦z = O共Ha−1/3兲 ⫻ O共Ha−2/3兲. At high
Hartmann numbers, these layers might be significantly thinner than traditional hydrodynamic viscous layers.
Here we use a mesh of the type represented in Fig. 4. It
is structured and made of quadrilateral elements in the boun-
FIG. 4. Cross-section of the mesh used.
drary layer, while in the core it is unstructured and made of
triangular elements. The streamwise direction 共x兲 grid is
structured, with 750 elements.
The circumference 共␪ 兲 has been meshed with 156 elements at the pipe’s wall, i.e., at r = R. The boundary layer
region is meshed with 19 points in the radial direction. The
first point is located at a distance 5 ⫻ 10−5R from the wall.
The grid spacing is then progressively increased by a stretching factor of 1.424 between two consecutive points until the
core part of the grid is reached. A posteriori, we determined
to have a minimum of 7 points in the radial direction to
capture the Hartmann layer, even at the inlet of pipe where
Ha is the largest. In the core, the grid along the radius 共r兲 is
unstructured and approximately composed of 50 elements.
Overall, the mesh sizes are also fine enough to simulate a
turbulent hydrodynamic pipe flow at Reynolds number Re
⬇ 4000 or Re␶ ⬇ 500. Indeed, in terms of wall units, we have
+
+
+
= 10.4, ⌬rmax
⬃ 10.2, and ⌬rmin
= 0.013,
⌬x+ = 10.4, ⌬␪max
which are similar to the values used in Refs. 48–50 共except
+
which has to be significantly lower here
of course for ⌬rmin
to accommodate for Hartmann layers兲.
As an initial test of the numerical algorithm and the
mesh quality, we have compared the prediction of the code
and those of the core flow approximation for a straight pipe
subjected to a homogeneous magnetic field with Re= 3986,
Ha= 6569, and c = 0.027 共the core flow approximation is very
accurate for this fully developed flow兲. For these parameters,
the bulk velocity and the centerline velocity agree within
0.3% and 0.02%, respectively. In the next section, our computations are benchmarked against experimental results, the
core flow approximation and earlier computations by Refs.
33 for the flow described in Sec. II A.
IV. RESULTS
A. Pressure gradients
The first diagnostics we consider 共Figs. 5 and 6兲 are the
pressure gradients, measured along the pipe’s wall at, y
= 0 , z = R and y = R , z = 0. Overall, the agreement between
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Phys. Fluids 23, 047101 共2011兲
Direct numerical simulation of a liquid-metal flow
0.05
0.02
P (x0R)−P (xR0)
Rσ U b B 2
∂p/∂x(x,0,R)
σUb B 2
0.025
0.015
FSBcfa
0.01
FSBcfa
0.005
Exp. [16]
Core flow approx.
0
Ni et al. [32]
−0.005
−15
−10
−5
FSBexp
FSBcfa
0.04
Exp. [16]
0.03
Core flow approx.
Ni et al. [32]
0.02
0.01
0
0
X/R
5
10
15
FIG. 5. Streamwise pressure gradient along the axis y = 0 , z = R 共the cases
DNSBexp and DNSBcfa cannot be distinguished兲.
our full solution and the one obtained in Ref. 33, the core
flow approximation and experimental results is excellent 共unfortunately, experimental results are not available along the
axis y = R , z = 0兲. We only note a slight difference between
our full solution and the one obtained in Ref. 33 for the inlet
of the pipe. We could not explain this difference but observe
that our results are in complete agreement with the core flow
approximation. Comparing the cases FSBexp and FSBcfa we
conclude that the nonzero Boutlet used here has no influence
on the pressure drop 共the curves cannot be distinguished兲.
The reason is of course that the upstream pressure drop is
two orders of magnitude larger than the downstream pressure
drop. From the comparison of case FSBcfa and core flow approximation results, we also note that inertial effects are negligible as far as the pressure drop is concerned.
Exactly the same conclusions can be drawn from Fig. 7
where the transverse pressure difference is plotted. However,
we note a mismatch of about 15% of the maximum transverse pressure difference between the full solutions and the
0.04
−0.01
−15
−10
−5
0
X/R
5
10
15
20
FIG. 7. Transverse pressure difference along the pipe 共the cases DNSBexp
and DNSBcfa cannot be distinguished兲.
core flow approximation when compared to the experiments.
A similar difference was reported in Ref. 51 where the authors show that by using a 3D analytical model of the experimental magnetic field for the simulations, a better agreement is obtained.
B. Velocity profiles and momentum budget
In Fig. 8 we plot the streamwise velocity profile along
the z-axis at x / R ⬇ 0 共i.e., at the inflexion point of the magnetic field’s profile兲. Overall, the agreement between our full
solutions, the core flow approximation and experimental results is very good with slight deviations in the near wall
region 共this diagnostic is not reported in Ref. 33兲. We note a
slight change in the derivative of the full solution profile
around z / R = ⫾ 0.9. As the flow is perfectly laminar until
much further downstream in all cases, this observation cannot be related to an instability occurring in the boundary
FSBexp
FSBcfa
FSBexp
0.03
Exp. [16]
FSBcfa
Core flow approx.
0.02
Core flow approx.
Ni et al. [32]
0.01
0
−0.01
−15
−10
−5
0
X/R
5
10
15
FIG. 6. Streamwise pressure gradient along the axis y = R , z = 0 共the cases
DNSBexp and DNSBcfa cannot be distinguished兲.
FIG. 8. Normalized mean streamwise velocity as a function of z in the mid
plane 共y = 0兲 at downstream position x / R ⬇ 0 共the cases DNSBexp and
DNSBcfa cannot be distinguished兲.
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Phys. Fluids 23, 047101 共2011兲
Albets-Chico et al.
1.2
30
1
Momentum budget
U(x,0,0)
Ub
0.6
0.4
0.2
0
FSBexp
FSBcfa
Exp. [16]
−0.2
Core flow approx.
−0.4
Ni et al. [32]
−0.6
−20
Centerline (y=0,z=0)
20
0.8
10
0
-10
−ux ∂x ux
−uz ∂z ux
-20
−10
0
X/R
10
−∂x p
20
Njz By
-30
FIG. 9. Normalized streamwise velocity as a function of x along the centerline 共y = 0 , z = 0兲.
4
6
8
x/R
10
12
14
FIG. 11. Streamwise momentum budget along the centerline 共y = 0 , z = 0兲.
layer. Moreover, since the grid used changes from unstructured to structured elements in that region, this slight change
in the derivative could have a local numerical origin. Finally,
we also observe that at this streamwise location, the effect of
the downstream magnetic field on this profile is negligible
since the results for the cases FSBexp and FSBcfa cannot be
distinguished.
A better insight into the roles of inertia and the presence
of a downstream magnetic field is obtained by considering
streamwise velocity profiles along the direction of the pipe.
Two such profiles are presented in Figs. 9 and 10, respectively, for the centerline 共y = 0 , z = 0兲 and the line 共y = 0 , z
= 0.9R兲. Unfortunately, the full solution of Ref. 33 and the
experimental data are only available up to x / R ⬇ 2. Up to this
streamwise location, all full solutions, the core flow approximation and the experimental results are in excellent agreement. Further downstream, a significant difference between
the profiles of our full solution 共case FSBcfa兲 and the core
flow approximation is observed in both figures as the recovery to the fully developed MHD flow corresponding to the
nonzero outlet field occurs faster in the framework of the
core flow approximation. Indeed, for the centerline profile,
both computational methods start to diverge around x / R ⬇ 3
while this occurs around x / R ⬇ 0 for the wall profile 共y
= 0 , z = 0.9R兲. In order to trace the origin of these discrepancies, we plot in Figs. 11 and 12 for case FSBcfa, the contributions of the pressure gradient, the Lorentz force and inertia
to the streamwise momentum budget. As recalled in the introduction, the core flow approximation assumes that the
pressure gradient and the Lorentz force balance each other
perfectly. For the centerline of the pipe, we observe in the
simulations that this holds very accurately up to x / R ⬇ 4.
However, Fig. 11 shows that for 4 ⱗ x / R ⱗ 11, the inertial
contribution −ux⳵xux is certainly nonnegligible when compared to the pressure gradient and the Lorentz force. On the
line 共y = 0 , z = 0.9R兲, we also observe a very significant contribution of inertia for −2 ⱗ x / R ⱗ 10 共Fig. 12兲 共at x / R ⬇ 3.5,
the flow undergoes a very sharp variation in space related to
the discontinuity in the derivative of the magnetic field Bcfa
y 兲.
300
7
FSBexp
U(x,0,0.9R)
Ub
5
200
FSBcfa
Core flow approx.
4
Ni et al. [32]
3
2
1
0
−20
Line (y=0, z=0.9R)
Exp. [16]
Momentum budget
6
100
0
-100
−ux ∂x ux
−uz ∂z ux
−∂x p
Njz By
-200
−10
0
X/R
10
20
FIG. 10. Normalized streamwise velocity as a function of x along the line
共y = 0 , z = 0.9R兲.
-300
-2
0
2
4
6
x/R
8
10
12
14
FIG. 12. Streamwise momentum budget along the line 共y = 0 , z = 0.9R兲.
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1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0
z/R
0
z/R
0
z/R
0
z/R
Direct numerical simulation of a liquid-metal flow
z/R
047101-9
-0.5
-0.5
-0.5
-0.5
-0.5
-1
0
2
4
6
-1
0
4
6
-1
0
2
4
6
-1
0
4
6
-1
0
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0
0
z/R
0
z/R
0
z/R
u/U b x/R=5
1
-0.5
-0.5
-0.5
-0.5
-0.5
-1
0
2
4
6
-1
0
2
4
6
-1
0
2
4
6
-1
2
4
6
u/U b x/R=10
z/R
u/U b x/R=0
2
z/R
u/U b x/R=-10
2
u/U b x/R=-5
u/U b x/R=-5
0
0
0
2
4
6
-1
0
2
4
6
FIG. 13. Normalized mean streamwise velocity as a function of z in the midplane 共y = 0兲 at several downstream positions for the cases DNSBcfa 共top兲 and
DNSBexp 共bottom兲.
Interestingly, for 4 ⱗ x / R ⱗ 8, the Lorentz force is mostly
compensated by the inertial term −ux⳵xux. The fact that the
pressure gradient is not exactly balanced by the Lorentz
force, even for an outlet field corresponding to Ha= 330 共case
FSBcfa兲, is certainly an important element giving rise to the
differences in velocity profiles 共although other factors like
the other simplifications of the core flow approximation or
the slight difference in magnetic fields used 共see Sec. II B兲
could also play a role兲.
Returning to Figs. 9 and 10, we also observe the impact
of the nonzero cut-off magnetic field Boutlet downstream of
x / R ⬇ 3.5 as the curves corresponding to FSBexp and FSBcfa
start to diverge. As expected, the velocity profiles recover
their inlet value when Boutlet ⫽ 0 共at high Hartmann numbers
the fully developed MHD flow has a flat core and we must
conserve inlet massflow兲, while this is not observed when
Boutlet = 0 共see below兲. We also conclude that the influence of
the nonzero outlet field does not propagate significantly upstream of the point from which it is applied.
This drastic difference in velocity profiles is further illustrated in Fig. 13 where the streamwise velocity along the
z-axis is plotted at several downstream locations. Up to
x / R ⬇ 0 the cases FSBexp and FSBcfa provide nearly identical
velocity profiles. Further downstream, the flow recovers the
fully developed MHD profile as the side-wall jets are rapidly
damped by the outlet magnetic field in the FSBcfa case. In the
FSBexp case, the flow is governed by plain hydrodynamics
after the exit of the magnet. We observe that the jets are
significantly stronger than in the FSBcfa case and also that
they persist much further downstream. In Fig. 14 two contour plots of the streamwise velocity profile are shown to
highlight the different behaviors at the outlet of the magnetic
field. In the FSBexp the jets become unstable and turbulence
develops. At very far downstream positions 共not computed
here兲, the flow ultimately would adopt the turbulent pipe
flow profile since the Reynolds number is quite large
共Re= 3986兲. Of course, the core flow approximation cannot
capture the transition to turbulence since its domain of validity does not extend outside the strong magnetic field region.
V. CONCLUDING REMARKS
In this paper we have described some numerical simulations of a liquid-metal flow that exits a region where an
intense magnetic field is present 共flow in a fringing magnetic
field兲. The chosen parameters are based on the IEA liquid
breeder blanket subtask benchmark problem definition that
correspond to experiments conducted at Argonne’s Liquid
Metal Experiment facility.
From the numerical point of view, the excellent agreement with experimental results demonstrate that full solutions on moderately sized clusters are now obtainable for this
complex problem, even for parameters relevant to fusion
blankets. For future comparison, we have provided complete
descriptions of the numerical algorithm and mesh used.
Our analysis also focuses on the comparison between
full solution data and the core flow approximation. As far as
the pressure drop is concerned, the results confirm that the
core flow approximation constitutes a very valuable engineering tool for the configuration studied since this design
parameter can be accurately predicted at a reduced computational cost. In a configuration with electrically insulating
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Phys. Fluids 23, 047101 共2011兲
Albets-Chico et al.
FIG. 14. 共Color online兲 Coutour plots of the dimensionless streamwise velocity 共ux / Ub兲 in the 共y = 0兲 plane for the cases DNSBcfa 共top兲 and DNSBexp 共bottom兲.
boundaries, the situation could be different since the net
braking Lorentz force is much smaller and inertia is expected
to play a more significant role.
As far as the velocity profiles are concerned, full solution computations and the core flow approximation are in
excellent agreement up to approximately the streamwise location of the magnetic field’s inflection point. Beyond that
point, two different flow behaviors are observed depending
on the outlet magnetic field. If, as required by the core flow
approximation, a nonzero outlet field is imposed, the velocity
profiles computed through this method converge to the fully
developed MHD profiles faster than observed with a full solution computation. By examining the momentum budget,
we have provided evidence that this discrepancy is at least in
part due to inertia, even though the Hartmann number and
interaction parameters corresponding to the nonzero outlet
field used here are still quite large 共Ha= 330 and N = 27, respectively兲. As expected, when the outlet magnetic field is
set to zero, the velocity profiles are completely different. The
side layer jets produced in the fringing magnetic field are not
destroyed and persist throughout the rest of the computational domain. Beyond x / R ⬇ 10 and for the flow parameters
chosen, a transition to turbulence is observed resulting from
the high shear present at the sides of the jets. In order to
better study this transition, a much longer computational domain and time evolution are required, both being beyond the
scope of this work. In this regard, the fact that full solution
computation allows a zero outlet magnetic field provides a
significant advantage over the core flow approximation. Indeed, to properly predict local phenomena such as heat transfer and corrosion effects, a precise knowledge of velocity
profiles and turbulence intensities is required.
ACKNOWLEDGMENTS
We are particularly grateful to Leo Bühler for providing
the core flow approximation computational results used in
this work. We also acknowledge fruitful discussion with
Sergei Molokov, Stijn Vantieghem, Axelle Viré and Evgeny
Votyakov. This work has been performed under the UCYCompSci project, the EURYI 共European Young Investigator兲
scheme and with financial support from a Marie Curie Transfer of Knowledge 共TOK-DEV兲 grant 共Contract No. MTKDCT-2004-014199兲 and a Center of Excellence grant from the
Norwegian Research Council to the Center of Biomedical
Computing.
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