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Hartmann layer
Introduction
Flow of an electrically conductive fluid across a magnetic field
induces an electric current. The resulting Lorentz force may profundly
affect the flow behaviour. Examples of such magneto-fluid dynamics
occur in plasmas, liquid metals, molten salts and electrolytes.
Designing power systems or process equipment in which magneto-fluid
dynamic effects are significant requires understanding of the flow
and its development due to the imposed magnetic field.
In the examined cases, a two-dimensional flow layer is subjected
to such transverse, external magnetic field. Due to low
magnetic Reynolds number, the impact of the induced magnetic field
can be neglected. A simplified arrangement, for which an analytical
approximation exists, offers a suitable benchmark test for
CFD modelling capabilities.
Objectives
In flow that is subjected to a transverse magnetic field, the volumetric Lorentz force
may become a dominant factor shaping the flow field. As it acts in the opposite direction to
the local flow velocity vector, it decelerates the flow motion and increases the associated
pressure drop. Therefore, such magneto-fluid dynamic flow regime represents a
significant challange to stability of CFD solvers.
Studied cases represent a very basic arrangement of the magneto-fluid dynamic
regime, where a conductive fluid layer is subjected to an external magnetic field [1].
Three variations with different electric field boundary conditions are analysed
in order to test the solver stability and its ability to find a numerical, steady-state solution.
Geometry
Height of the layer (2 a) is 0.01 m.
Width of the layer (2 b) is 0.04 m.
Length of the simulation domain is 0.06 m.
As the current density field is solenoidal, its two-dimensionality in
the spanwise (z) direction can only be modelled by extending the
width of the simulation domain (2 b).
The flow is periodic in the streamwise direction. Therefore, the length
of the simulation domain can be selected with an aim to enhance
convergence of the CFD simulation.
Where the electrically conductive bottom and top walls are included in
the CFD model (i.e. in the conjugate cases), the simulation domain also contains the additional
horizontal layers with the thickness (`delta`) of 0.001 m.
Loading
The fluid motion is induced by the prescribed pressure gradient in
the streamwise (x) direction that corresponds to a selected Reynolds number (`Re`) value:
`u_0 = (mu Re)/(rho a) ` and `(dp)/(dx) = - (mu u_0)/(a^2)`
where `u_0` is the velocity scale, `mu` is the fluid's dynamic viscosity and `rho` its density.
The magnetic field is defined by selected Hartmann number (`Ha`) values:
`B = (Ha)/a sqrt(mu/sigma)`
where `mu` is the fluid's dynamic viscosity and `sigma` its electric conductivity.
Four groups of cases with an equal Reynolds number, but a different Hartmann number
are used in the assessment. Their pressure drop (`dp//dx`) and magnetic field flux density (`B`) are:
1) `Re = 2000` & `dp//dx =`-`1.625*10^-3` `"Pa"//"m"`, `Ha = 0.0` & `B = 0.0` `"T"`
2) `Re = 2000` & `dp//dx =`-`1.625*10^-3` `"Pa"//"m"`, `Ha = 2.0` & `B = 0.761*10^-2` `"T"`
3) `Re = 2000` & `dp//dx =`-`1.625*10^-3` `"Pa"//"m"`, `Ha = 5.0` & `B = 1.901*10^-2` `"T"`
4) `Re = 2000` & `dp//dx =`-`1.625*10^-3` `"Pa"//"m"`, `Ha = 10.0` & `B = 3.803*10^-2` `"T"`
Material properties
The material properties of NaK eutectic [2 & 3] are used for the simulated cases:
`Y_(K)` is potassium mass fraction of 77.8 %
`rho` is density of 870 kg/kmol
`mu` is dynamic viscosity of 9.4·10-4 Pa s
`sigma` is electrical conductivity of 2.6·106 S/m
For the conjugate cases, material properties of 316 stainless steel [4]
are selected for the solid layers:
`sigma_s` is electrical conductivity of 1.3·106 S/m,
which yields the wall conductance `c = sigma_s delta//sigma a` of 0.1.
Meshing
In all simulated cases, the numerical grid consists of hexahedral grid
elements:
The uniform spacing of 0.001 m is applied in the streamwise (x) direction.
In the horizontal (z) direction, 80 elements are employed with a bias factor (16)
to refine the grid near both vertical boundaries.
In the vertical (y) direction, 60 elements are used with a bias factor (20)
to refine the grid near both horizontal surfaces.
The utilised numerical grid contains 0.301 mil nodes and 0.288 mil elements.
Numerical grid
For the conjugate cases, 12 elements are employed in the vertical direction
for each solid layer with a bias factor (6) to refine the grid near the interface
with the fluid flow domain.
This increases the number of grid nodes to 0.420 mil and the number of elements to 0.403 mil.
Initial conditions
Steady-state CFD simulations are used for comparison with the experimental data.
Although their initial conditions can be arbitrary, they should enhance stability of
the solution procedure.
For that purpose, it is advised to start with the CFD simulation of forced convection
without the external magnetic field. When the flow field is fully developed, the external magnetic
field is reintroduced.
Boundary conditions
In the streamwise (x) direction, the momentum and electric field
boundary conditions are periodic.
At the bottom and the top surface, no-slip conditions `u = 0.0` `"m"//"s"`
are prescribed. Three cases with different electric field boundary
conditions are investigated:
1) non-conductive boundaries with electric current density `j = 0.0` `"A"//"m"^2`
2) infinitely conductive boundaries with electric potential set to `phi = 0.0` `"V"`
3) boundaries coupled to the solid domain with `j` preserved in the normal direction
At the bottom and the top boundary of the solid domain (case 3),
the non-conductive electric current conditions (`j = 0.0` `"A"//"m"^2`)
are prescribed.
At the vertical, side surfaces, the symmetry or equivalent conditions
can be used for the fluid flow. The electric field requires the non-conductive
boundary conditions (`j = 0.0` `"A"//"m"^2`) to enforce solenoidity
of the electric current across the simulation domain.
Results
Steady-state CFD simulations were conducted with a double precision CFD solver.
The timescale 0.1 - 1.0 s was used to obtain converged results.
Electric current density (`j`) for a conjugate case at Ha = 10
The CFD results are compared with an analytical, 2D approximation of
the streamwise flow velocity [1] across the simulation domain. The velocity
profile in the direction of the imposed magnetic field is expressed as:
`u^ star = hat u [1 - cosh (y^ star Ha)/cosh(Ha)]`
where `u^ star = u//u_0` is the dimensionless flow
speed and `y^ star = y//a` the dimensionless vertical location. In addition,
the characteristic magnitude of velocity is defined as:
`hat u = 1/(Ha)[(c+1)/(c Ha + tanh(Ha))]`
A parabolic velocity profile that corresponds to the forced convection flow in absense of a magnetic field (Ha = 0):
`u^ star = 0.5 [1 - (y^ star)^2]`
is added to comparison. Non-conductive bottom and top boundary
Streamwise velocity (`u^ star`) along vertical centreline for non-conductive bottom and top boundary
For validation purposes, quadratic mean (or RMS) of deviation between the analytical approximation [1] and the CFD
simulation results is calculated for the streamwise velocity (`u^ star`) along the centreline:
Hartmann number
RMS of deviation
0.0
7.05E-4
2.0
2.75E-4
5.0
1.41e-4
10.0
1.27e-4
Infinitely conductive bottom and top boundary
Streamwise velocity (`u^ star`) along vertical centreline for infinitely conductive bottom and top boundary
For validation purposes, quadratic mean (or RMS) of deviation between the analytical approximation [1] and the CFD
simulation results is calculated for the streamwise velocity (`u^ star`) along the centreline:
Hartmann number
RMS of deviation
0.0
7.05E-4
2.0
1.22E-4
5.0
1.41e-4
10.0
1.54e-6
Electrically coupled bottom and top boundary
Streamwise velocity (`u^ star`) along vertical centreline for conjugate cases
For validation purposes, quadratic mean (or RMS) of deviation between the analytical approximation [1] and the CFD
simulation results is calculated for the streamwise velocity (`u^ star`) along the centreline:
U. Müller, L. Bühler, Magnetofluiddynamics in channels and containers, Springer-Verlag, 2001, p. 37.
K. Miyazaki, S. Kotake, N. Yamaoka, S. Inoue & Y. Fujii-E, MHD pressure drop of NaK flow in stainless steel pipe, Nucl. Tech. - Fusion, 4:2P2, pp. 447-452.
O.J. Foust, Sodium - NaK engineering handbook, 1972, p. 11.