Download Lorentz force velocimetry based on time-of-flight

PHYSICS OF FLUIDS 22, 125101 共2010兲
Lorentz force velocimetry based on time-of-flight measurements
Axelle Viré,1,a兲 Bernard Knaepen,1 and André Thess2
1
Faculté des Sciences, Université Libre de Bruxelles, Bd. du Triomphe CP231, B-1050 Brussels, Belgium
Institute of Thermodynamics and Fluid Mechanics, Ilmenau University of Technology, P.O. Box
100565, 98684 Ilmenau, Germany
2
共Received 28 June 2010; accepted 20 October 2010; published online 10 December 2010兲
Lorentz force velocimetry 共LFV兲 is a contactless technique for the measurement of liquid metal
flowrates. It consists of measuring the force acting upon a magnetic system and arising from the
interaction between an external magnetic field and the flow of an electrically conducting fluid. In
this study, a new design is proposed so as to make the measurement independent of the fluid’s
electrical conductivity. It is made of one or two coils placed around a circular pipe. The forces
produced on each coil are recorded in time as the liquid metal flows through the pipe. It is
highlighted that the auto- or cross-correlation of these forces can be used to determine the flowrate.
The reliability of the flowmeter is first investigated with a synthetic velocity profile associated with
a single vortex ring, which is convected at a constant speed. This configuration is similar to the
movement of a solid rod and enables a simple analysis of the flowmeter. Then, the flowmeter is
applied to a realistic three-dimensional turbulent flow. In both cases, the influence of the coil radii,
coil separation, and sign of the coil-carrying currents is systematically assessed. The study is
entirely numerical and uses a second-order finite volume method. Two sets of simulations are
performed. First, the equations of motion are solved without accounting for the effect of the
magnetic field on the flow 共kinematic simulations兲. Second, the Lorentz force is explicitly added to
the momentum balance 共dynamic simulations兲, and the influence of the external magnetic field on
the flow is then quantified. © 2010 American Institute of Physics. 关doi:10.1063/1.3517294兴
I. INTRODUCTION
A key feature of electromagnetism is that a force is generated when an electrically conducting material moves
through a magnetic field. If the material is in a fluid state,
this principle can be used to determine its flowrate—a technique that is generally referred to as electromagnetic flow
measurement. The origin of this technique can be traced back
to the work of Faraday,1 who attempted to determine the
flowrate of the river Thames by measuring the potential difference between two electrodes located across the river.
Faraday’s experiments have been followed by many others,
based on the same principle; first in oceanography and later
in the framework of liquid metals.2 At temperatures below
200 ° C, flowrates can be measured through Faraday-type inductive flowmeters, as reviewed in Ref. 3. By contrast, measurements in metallurgical flows of liquid metals at high
temperatures cannot be carried out using conventional inductive flowmeters since electrodes, which are indispensable to
apply Faraday’s principle, cannot be inserted in the flow. The
present work is devoted to a noncontact electromagnetic flow
measurement technique called Lorentz force velocimetry
共LFV兲.4,5 It is based on measuring the force acting upon a
magnetic system that interacts with the flow of an electrically
conducting fluid. More precisely, the goal of the present
work is to demonstrate numerically the feasibility of a particular improvement, which makes this technique essentially
independent of the electrical conductivity of the fluid.
In LFV, the measured force signal depends both on the
a兲
Electronic mail: [email protected].
1070-6631/2010/22共12兲/125101/15/$30.00
velocity of the fluid and on its electrical conductivity. In
practice, the electrical conductivity is rarely known exactly
in a given metallurgical application since it depends on the
temperature and on the composition of the alloy, and both
can vary significantly in time. It is thus a challenge to develop a Lorentz force flowmeter 共LFF兲, whose measurement
would be independent of the unknown and poorly controlled
electrical conductivity. In the present work, we numerically
analyze a novel version of the LFV technique based on temporal correlations, taken from force measurements at several
locations. The technique will be referred to as time-of-flight
LFV.
This paper is organized as follows. The general principle
of the new measurement technique and its differences with
the previously developed flowmeters are explained in Sec. II.
The governing equations and their discretization in the numerical method are presented in Sec. III. Section IV shows
the characteristics of the magnetic field distributions used for
the present analysis. The results are then divided into two
parts. First, the technique is validated on a vortex ring moving at a constant velocity. The results of this validation are
described in Sec. V. Since the ring is not distorted during its
advection and its velocity is known a priori, this configuration can be solved semianalytically, so it is referred to as a
synthetic case. Second, the measurement principle is applied
to a numerically simulated three-dimensional turbulent flow,
which is advected according to the Navier–Stokes equations
and is referred to as a realistic case 共Sec. VI兲. In both cases,
the measurement is performed using either one or two coils,
and the influence of various coil parameters is investigated
22, 125101-1
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125101-2
Phys. Fluids 22, 125101 共2010兲
Viré, Knaepen, and Thess
turn, create an induced magnetic field b, referred to as the
secondary magnetic field. In this work, we assume that the
magnetic diffusion time is much smaller than the time scale
of large eddies. Therefore, the secondary magnetic field becomes negligible with respect to the primary magnetic field,
namely, 兩b兩 Ⰶ 兩B兩. The coupling between the fluid motion and
the primary magnetic field is then only one way: the applied
magnetic field affects the flow through the presence of an
electromagnetic force but the flow does not alter significantly
the magnetic field. This is referred to as the quasistatic
approximation7 and implies that the magnetic Reynolds number is small. In this framework, the eddy currents are described by a simplified Ohm’s law, which is expressed by
B
R
y
z
rm
F
u
J
j
F r
x
L
(a)
B
u
y
z
Δ
F1
rm
j J1
x
F2
J2
j = ␴共− ⵱␾ + u ⫻ B兲.
F2r
F1r
(b)
FIG. 1. 共Color online兲 Principle of Lorentz force velocimetry. One 共a兲 or
two 共b兲 current-carrying coils produce the primary magnetic field B, which
interacts with the flow u of an electrically conducting fluid and induces eddy
currents j. The force acting upon the coil共s兲 共denoted by a superscript r兲 is
equal in magnitude 共but opposite in direction兲 to the Lorentz force F in 共a兲
and the Lorentz forces F1 and F2 in 共b兲.
systematically. Finally, a comparison is made between the
results obtained from kinematic simulations 共where the Lorentz force is neglected in the equations of motion兲 and dynamic simulations 共which also take into account the Lorentz
force in the right-hand side of the momentum balance兲 to
assess the effects of the magnetic field on the flow dynamics.
We finally summarize our work and indicate some questions
that should be investigated in future work.
II. BASIC PRINCIPLE
A LFF measures the integrated or bulk Lorentz force
resulting from the interaction between a liquid metal in motion and an applied magnetic field. The magnetic field can be
generated in two ways, namely, by permanent magnets or
electromagnets consisting of current-carrying coils. In the
present paper, we shall consider magnetic systems that consist in current-carrying coils placed around a circular pipe.
Moreover, the resulting magnetic field is constant in time.
For Nc coils placed around the pipe, the so-called primary magnetic field B is given by the Biot–Savart law,6
Nc
B共r兲 =
Nc
兺 B␣共r兲 = ␣兺=1
␣=1
␮ 0J ␣
4␲
冖
C␣
dl⬘ ⫻ 共r − r⬘兲
,
兩r − r⬘兩3
共1兲
where J␣ is the magnitude of the primary electrical current
circulating in the ␣th coil, ␮0 = 4␲10−7 H / m is the vacuum
permeability, dl⬘ is a length element of the coil of contour
C␣, r⬘ is the position of the coil element, and r denotes the
location where the magnetic field is evaluated. The magnetic
field lines are sketched in Fig. 1, for one and two coils
wrapped around a pipe of length L. Since the magnetic field
interacts with the flow velocity u, eddy currents, also called
secondary currents, are induced in the liquid metal. These, in
共2兲
This assumes that the electrical field is the gradient of the
electrical potential ␾. The electrical conductivity of the fluid
is denoted by ␴. According to the conservation of electric
charge and the hypothesis of quasineutrality, eddy currents
are divergence-free. Hence, the electrical potential satisfies
the Poisson equation, ⵜ2␾ = ⵜ · 共u ⫻ B兲, which has to be
solved in order to determine j. The total Lorentz force density acting on the flow is then
FL共r兲 = j共r兲 ⫻ B共r兲,
共3兲
which dissipates energy. For the single-coil flowmeter,
shown in Fig. 1共a兲 and treated in Ref. 4, the Lorentz force
density peaks upstream and downstream from the coil, where
the eddy currents are maximum 共as shown by loops located
aside from the coil兲. For two coils with same-sign currents,
the total Lorentz force exhibits four maxima if the coil separation ⌬ is large enough so that the magnetic fields produced
by each coil do not interfere. Conversely, when the magnetic
fields overlap, the Lorentz force is maximum in two regions
only, where j is maximum, as indicated in Fig. 1共b兲.
While the magnetic field produced by the coils exerts a
force on the flow, the converse is also true: the induced magnetic field b interacts with the primary electrical current J␣
and induces on the ␣th coil a reaction force given by
F␣r =
1
2␲rc
冖
J␣共r⬘兲 ⫻ b共r⬘兲dl⬘ ,
共4兲
C␣
where r⬘ is again the position of a coil element dl⬘ and rc is
the coil radius. By virtue of the reciprocity principle,5 the
reaction force is equal to the opposite of the integrated Lorentz force due to the ␣th coil, namely,
F␣r = − F␣ = −
1
V
冕
j共r兲 ⫻ B␣共r兲dV,
共5兲
where V is the volume of the pipe.6 This implies that the
Lorentz force due to the ␣th coil can, in fact, be obtained by
measuring the reaction force on the coil itself. Previous
works4,5 have looked at the time-averaged values of F␣r to
determine the mean flowrate assuming the fluid conductivity
is known. In metallurgy, which is the main field of application of LFV, the electrical conductivity of the melt is, however, often unknown or fluctuates in time. This is due to the
fact that ␴ is a function of the temperature and of the com-
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125101-3
position of the alloy, and both may undergo fluctuations that
are difficult to measure in situ. It is therefore desirable to
develop Lorentz force flowmeters that operate independently
of the electrical conductivity of the fluid.
To remove the dependence on the fluid conductivity, we
shall investigate a variant of the LFF based on time-of-flight
measurements and focus on the time-dependence of the Lorentz forces. The general principle is the following. Since
metallurgical flows are turbulent, the force acting upon a coil
is not constant in time. Its temporal behavior reflects turbulent eddies that are swept across the region on which the
Lorentz force acts. Since the turbulent eddies experience
little changes when passing two nearby coils, the signals of
the two coils should be strongly correlated. The time evolutions of F in Fig. 1共a兲, or F1 and F2 in Fig. 1共b兲, are recorded. The signals are then auto- or cross-correlated in time
to provide information about the time of convection of the
flow. The flowmeter can be calibrated by measuring the correlation time for known flowrates in order to estimate a calibration curve relating flowrates and correlation times. During
the measurements, the flowrate is then determined through
the evaluation of the correlation time and the knowledge of
the calibration curve.
Although this approach is novel in the framework of
LFV, the success of time-of-flight measurements in fluid dynamics has already been demonstrated.8 For example, in turbulent thermal convection, temperature probes have been
used successfully to determine the magnitude of the largescale circulation 共or “mean wind”兲 in cryogenic helium,
where direct velocity measurements are difficult.9,10 If the
probes are placed in the direction parallel to the wind, the
mean wind should pass the second probe a short time after it
has passed the first one. Hence, this time interval is determined through time correlations of the temperature signals
provided by each probe. This correlation technique is also
the basis of the widely used ultrasonic Doppler
velocimetry.11 In that case, the sensors are ultrasonic beams,
whose waves modulate with the flow velocity. By knowing
the distance between the ultrasonic beams, the crosscorrelation of the modulated signals at each beam yields the
mean velocity of the flow.
In our application, the novelty stands in using the coils
as sensors. The forces F␣ acting on the coil共s兲 are nondimensionalized by removing their mean 具F␣典 and by dividing
them by their root-mean-square F␣⬘ , i.e.,
F̃␣ =
Phys. Fluids 22, 125101 共2010兲
Lorentz force velocimetry based on time-of-flight measurements
F␣ − 具F␣典
F␣⬘
.
共6兲
The cross-correlation between two forces F̃1 and F̃2 is defined by
C共T兲 = 具F̃1共t兲F̃2共t + T兲典,
共7兲
in which T is the time shift between F̃1 and F̃2 and the
angular brackets stand for time averaging. By examining the
maximum of the curve C共T兲, one can obtain a time scale T p
for the crossing of a turbulent structure through the flowmeter 共see details below兲. The idea is to obtain a measurement
that is independent of the fluid properties. The present flow-
meter, however, requires unsteadiness of the flow because it
is sensitive to force fluctuations. Thus, it cannot be applied to
laminar flows. This contrasts with the previously developed
LFF.4,5
In the present time-of-flight technique, two quantities are
of particular interest to evaluate the quality of the measurement. First, the magnitude of the correlation peak indicates
the reliability of the measuring device. The higher the magnitude of the correlation peak, the smaller the random error
that inherently affects the measurement. Second, the product
between the correlation time and the actual flowrate should
be as constant as possible for different geometrical parameters of the pipe and flowmeter. In that case, the calibration
curve would remain rather unchanged when the pipe radius
varies, for example, which is advantageous. By defining the
bulk velocity as Ub = 1 / V兰udV 共where V is the pipe volume
and u is the flow velocity兲, the choice of an arbitrary distance
d yields a characteristic time Tb = d / Ub associated with the
flowrate. The calibration factor can then be defined as the
ratio between the correlation time T p and this characteristic
time Tb so that
T p U bT p
.
=
Tb
d
共8兲
The knowledge of the distance d and the correlation time T p,
in turn, gives a velocity ULFF = d / T p. The present study compares the correlation time with the characteristic time of the
mean flow through the calibration factor T p / Tb. This is
equivalent to the comparison between ULFF and Ub. Although the distance d can be chosen arbitrarily, it is convenient for a single-coil flowmeter to define the reference
length d as the distance between the regions of maximum
Lorentz force, located upstream and downstream from the
coil because these regions contribute mostly to the fluctuations of the reaction force acting upon the coil. Using a
double-coil flowmeter, the reference length is chosen as the
distance between the planes of the coils. The motivation for
this choice is the following. If the coils are well separated,
namely, the magnetic field produced by the coils do not overlap 共ideal flowmeter兲, and the flow moves at a constant velocity Ub without distortions 共like a solid bar兲, the time variations of the force acting on the upstream coil are identical to
those of the force acting on the downstream coil, and the
time shift between these forces equals the time-of-flight of
the flow over the distance between the coil planes. Furthermore, this time is unambiguously given by the time T p obtained from the cross-correlation of the forces. Therefore, the
calibration factor T p / Tb equals unity 共with d chosen as the
distance ⌬ between the coil planes兲. In practice, the calibration factor differs from unity for mainly two reasons. First,
the magnetic fields produced by each coil interfere with each
other. Second, the correlation time T p essentially measures
the advection velocity of turbulent eddies. Previous
works12,13 showed that the advection velocity of an eddy
approximately equals the value of the mean velocity profile
at the location of the vortex, except in the near-wall region.
Therefore, the measurement provided by the present flowmeter would further differ from the exact flowrate in two ways.
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125101-4
Phys. Fluids 22, 125101 共2010兲
Viré, Knaepen, and Thess
First, it might differ from the value of the mean velocity
profile at the radial location corresponding to the region of
sensitivity of the flowmeter 共i.e., where the Lorentz force is
maximum兲. Second, this value of the mean velocity does not
equal the flowrate, except at one radial location. For all these
reasons, the calibration factor, as previously defined, is expected to differ from unity. Attention will, however, be put
on its invariance with the parameters of the magnetic system.
The questions addressed here are summarized as follows.
共i兲
共ii兲
共iii兲
共iv兲
Does the measured velocity relate to the actual mean
flowrate in a reliable and systematic way? The degree
of reliability is determined by the amplitude of the
correlation peak. It will be quantified for different coil
radii. When two coils are used, its dependence on the
coil separations and the sign of the coil currents will
be further investigated.
How does the ratio between the actual flowrate and
ULFF vary with the geometrical parameters of the
coils, namely, their radius rm and their separation ⌬?
This ratio is defined as the calibration factor.
What is the region of maximum sensitivity 共i.e., maximum Lorentz force兲 of the flowmeter?
How strong is the effect of the magnetic field on the
flow? This issue will be investigated through comparisons between kinematic and dynamic results. In the
kinematic simulations, it is assumed that the Lorentz
force does not affect the flow dynamics. In the dynamic simulations, the force is added to the right-hand
side of the momentum balance and may alter the flow
evolution.
III. GOVERNING EQUATIONS AND NUMERICAL
METHOD
For the single-coil flowmeter shown in Fig. 1共a兲, our
time-of-flight measurement approach is based on the autocorrelation of the integrated Lorentz force given by
F共t兲 =
1
V
=
␴
V
冕
冕
FL共x,t兲dV
关− ⵱␾共x,t兲 + u共x,t兲 ⫻ B共x兲兴 ⫻ B共x兲dV.
共9兲
For a double-coil device shown in Fig. 1共b兲, crosscorrelations are performed between the Lorentz forces due to
each coil,
F1共t兲 =
␴
V
冕
关− ⵱␾共x,t兲 + u共x,t兲 ⫻ Btot共x兲兴 ⫻ B1共x兲dV
共10兲
and
F2共t兲 =
␴
V
冕
关− ⵱␾共x,t兲 + u共x,t兲 ⫻ Btot共x兲兴 ⫻ B2共x兲dV,
共11兲
where Btot = B1 + B2. In both cases, the Lorentz force density
should thus be recorded as time evolves. Since the Lorentz
forces depend on the electrical potential and the velocity
field, these quantities are needed at each time step and are
computed using two different approaches.
In the kinematic simulations, we assume that the flow is
unaffected by the Lorentz force. The incompressible flow
dynamics are then governed by the Navier–Stokes equations,
as in classical hydrodynamics. Two types of velocity profiles
will be considered in this paper: either synthetic velocity
fields, if the velocity is prescribed analytically 共Sec. V兲, or
realistic velocity fields, when a three-dimensional turbulent
flow evolves according to the Navier–Stokes equations 共Sec.
VI兲.
The alternative approach to the simplified kinematic
simulations is to account for the Lorentz force in the equations of motion. This approach will be referred to as dynamic
simulations. Following the quasistatic approximation, the
Lorentz force is added to the momentum balance as an additional source term. The magnetohydrodynamics equations
used for the dynamical simulations become
⳵u
FL
+ 共u · ⵱兲u = − ⵱p + ␯ⵜ2u +
,
⳵t
␳
共12a兲
⵱ · u = 0,
共12b兲
ⵜ2␾ = ⵱ · 共u ⫻ B兲.
共12c兲
Here, p is the kinematic pressure 共equal to the ordinary pressure divided by ␳兲, and ␯ and ␳ are the kinematic viscosity
and density of the fluid, respectively. These equations of motion are discretized spatially using an unstructured finite volume method based on a collocated formulation. The method
is analogous to that used in the previous studies,5,14 and it is
thus not detailed here.
Simulations have shown that using a single coil, the
force correlation is sensitive to the pipe length for L ⱗ 17R.
This limit was found by varying systematically the length of
the pipe in the range of 10R ⱕ L ⱕ 20R. The pipe length is
thus L = 20R for the synthetic cases using the single-coil
flowmeter. By contrast, runs on the synthetic velocity profile
and a double-coil flowmeter showed that the results obtained
with a pipe of length L = 10R were identical to those obtained
with L = 20R. This is because the peak of interest in the correlation of the double-coil flowmeter is less affected by the
limited length of the pipe than that of the single-coil device.
For the double-coil flowmeter, the length of the pipe is thus
set at L = 10R. The simulation meshes are the following. The
pipe is discretized with 65 points in the streamwise direction.
The mesh resolution in the radial and azimuthal directions
varies with the azimuthal angle since the mesh is unstructured. The pipe wall is discretized with 64 points in the azimuthal direction. The number of points along the pipe diameter is approximately 49, depending on the azimuthal
position. In the synthetic case 共Sec. V兲, the resolution is set
to 455 points in the streamwise direction and 255 points in
the radial direction because the simulations are much less
costly than in the realistic case.
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125101-5
Phys. Fluids 22, 125101 共2010兲
Lorentz force velocimetry based on time-of-flight measurements
TABLE I. Summary of the coil geometrical parameters for the synthetic and realistic cases.
Case
Velocity
Coils
Sign of 共J1 , J2兲
Range of rm
Range of ⌬
No. of 共rm , ⌬兲
1.05⬍ rm / R ⬍ 3.05
1.05⬍ rm / R ⬍ 3.05
1.05⬍ rm / R ⬍ 3.05
1.01⬍ rm / R ⬍ 2.885
1.1⬍ rm / R ⬍ 2.66
1.1⬍ rm / R ⬍ 2.66
¯
0.5⬍ ⌬ / R ⬍ 5
0.5⬍ ⌬ / R ⬍ 5
¯
0.6⬍ ⌬ / R ⬍ 4
0.6⬍ ⌬ / R ⬍ 4
40
875
875
13
144
144
A1
Synthetic
1
¯
A2S
A2O
T1
T2S
T2O
Synthetic
Synthetic
Realistic
Realistic
Realistic
2
2
1
2
2
Same
Opposite
¯
Same
Opposite
IV. SPECIFICATION OF THE PRIMARY MAGNETIC
FIELDS
The first step of our analysis consists of the specification
of the primary magnetic field produced by the coil共s兲. The
inputs for the analysis of the single-coil flowmeter are its
radius rm and the electrical current J. The input parameters
for the double-coil LFF are the coil radius rm 共both coils
being assumed to be equal in size兲, the coil separation ⌬, and
the currents J1 and J2 flowing through each coil. We limit
our attention to the particular cases J1 = J2, corresponding to
same-sign currents, as shown in Fig. 1共b兲, and J1 = −J2 corresponding to opposite-sign currents 共not shown in Fig. 1兲.
Note that the latter configuration is referred to as a cusp-type
magnetic field in the crystal growth community.15 The range
of variations of these parameters is presented in Table I for
the synthetic and realistic cases.
The magnetic field produced by an infinitely thin and
circular coil can be obtained using Eq. 共1兲. Here, this expression is implemented in the solver by discretizing the coil in
1000 segments.
A. Single-coil flowmeter
For the single-coil flowmeter, shown in Fig. 1共a兲, the coil
plane coincides with the pipe midplane, i.e., xm = L / 2. It is
characterized by a single dimensionless parameter, namely,
the ratio between the radius of the coil and that of the pipe.
An illustration of the magnetic field contours is given by Fig.
2 for a coil radius twice larger than that of the pipe 共rm
= 2R兲. The magnitude of the magnetic field is illustrated
through the coloring, from intense 共black兲 to weak 共white兲
intensities. At the location of the coil plane, the magnetic
field is purely axial, while its radial component increases
upstream and downstream from the coil. The total magnetic
field is maximum at the position of the coil plane and close
to the wall. Finally, the magnetic field is symmetric with
respect to the coil plane.
B. Double-coil flowmeter
With two coils, two additional parameters need to be
specified along with rm: the coil separation ⌬ and the sign of
the coil currents 共Table I兲. The coil radii are assumed identical for the two coils and the coils are located symmetrically
with respect to the streamwise midplane x = L / 2, which is
thus a plane of symmetry for the total magnetic field.
The total magnetic field contours are illustrated in Fig. 3
for coil radii and separation equal to twice the pipe radius.
Again, the coloring indicates the magnitude of the magnetic
field, from intense 共black兲 to weak 共white兲 intensities. Far
from the coils, the orientation of the magnetic field lines
depends on the sign of the currents flowing through the coils,
while their pattern do not. Rather, the pattern of the magnetic
field lines highly depends on the current signs between the
coils. For same-sign currents 关see Fig. 3共a兲兴, the magnetic
field is mainly axial between the coils. In their vicinity, the
intensity of the magnetic field is further distributed quite
equally in the pipe, although it slightly increases close to the
(a)
(b)
FIG. 2. Contours of the magnetic field for a single-coil flowmeter and a coil
radius twice larger than that of the pipe 共rm = 2R兲. The coil is represented by
circles: the bullet corresponds to an outward electrical current, while the
cross represents an inward current.
FIG. 3. 共Color online兲 Contours of the magnetic field for a double-coil
flowmeter with coil radii and separation equal to twice the pipe radius
共rm = ⌬ = 2R兲: same-sign currents 共a兲 and opposite-sign currents 共b兲. The coil
is represented by circles: the bullet corresponds to an outward electrical
current, while the cross represents an inward current.
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Phys. Fluids 22, 125101 共2010兲
Viré, Knaepen, and Thess
pipe axis and near the walls at the location of the coil planes.
Conversely, the magnetic field is mainly radial between the
coils with opposite-sign currents 关see Fig. 3共b兲兴. In addition,
it is very intense close to the pipe walls around the coil
planes.
V. RESULTS FOR SYNTHETIC VELOCITY FIELDS
共13兲
where 共xv , rv兲 are the axial and radial positions of the vortex.
The intensity ⌫ of the vortex tube is chosen as
⌫⬅
冖
u␣dl = 0.01 m2/s,
共14兲
c
where u␣ is the tangential velocity in a cross section of the
ring and c is a closed contour surrounding the cross section.
The velocity field associated with such a circular vortex ring
is analogous to the magnetic field produced by the coil, if the
product ␮0J is replaced by the intensity ⌫. For this particular
case, the Biot–Savart law can be expressed analytically as
follows.6,16 The formulation is derived in a cylindrical frame
of reference 共x , r , ␪兲 for the streamwise, radial, and azimuthal
coordinates, respectively. Since the geometry is axisymmetric and the velocity field is solenoidal, the velocity field can
be expressed in terms of a stream function ␺, such that
u共r兲 = ⵱ ⫻
and
冉 冊
␺
e␪
r
冋
4rrv
.
共x − xv兲2 + 共r + rv兲2 + a2
共15兲
共17兲
The vortex core radius a is introduced in the definition of mv
to avoid undefined velocities for x → xv and r → −rv.17 In the
simulations, we choose a = 0.01R. The velocity components
are then given by
ux共r兲 =
In this section, we investigate the feasibility of time-offlight LFF by using an analytically prescribed velocity field,
whose advancement velocity is known a priori. A constant
value of the convective velocity is thus imposed and the
given field evolves through the pipe without being distorted.
This configuration is similar to the movement of a solid rod.
Moreover, convective and bulk velocities are equal because
the convective velocity is prescribed to a constant value. The
case of a single vortex ring presents several advantages.
First, it is a velocity perturbation that enables a simple analysis of the flowmeter, while satisfying the divergence-free
constraint. Hence, it is appropriate to interpret the shape of
the integrated forces and their correlations. Second, the velocity field associated with a given vorticity distribution ␻
can be derived analytically. The vortex ring is assumed infinitely thin, so the component of the vorticity normal to the
plane 共x , r兲 is expressed in terms of Dirac’s functions ␦,
namely,
␻ = ⌫␦共x − xv兲␦共r − rv兲e␪ ,
mv =
冋
册
mv共r + rv兲 − 2r
⌫ rv冑mv
E共mv兲 ,
3/2 2rK共mv兲 +
␲ 共4rrv兲
1 − mv
共18a兲
ur共r兲 =
冋
册
2 − mv
⌫ rv冑mv
E共mv兲 − 2K共mv兲 .
3/2 共x − xv兲
␲ 共4rrv兲
1 − mv
共18b兲
In order to implement these expressions in the numerical
method, the elliptic integrals are computed using fourthorder polynomial approximations in terms of mv, as described by Hastings.18 These approximations are evaluated at
each mesh node, to which is associated a value of mv. Since
K共mv兲 → ⬁ and E共mv兲 → 1 close to the vortex core 共when
mv → 1兲, the values K = 1000 and E = 1 are imposed at the
nodes for which mv = 1. Finally, the velocity field has been
derived for a vortex ring placed in an unbounded domain.
Therefore, it does not satisfy the no-slip condition at the wall
of the pipe. If the radius of the vortex ring is smaller than
82% of the radius of the pipe, the deviation of its synthetic
velocity field from the no-slip boundary condition is smaller
than 10% of the velocity at the vortex core. However, this
drawback does not affect the general conclusions of this
study because the no-slip boundary condition is violated by
the translating synthetic velocity field anyway. Such a convection of the vortex ring is necessary to avoid its distortion,
and hence, consider the synthetic velocity field as the translation of a frozen velocity field.
The vortex ring is advected throughout the pipe, namely,
0 ⱕ xv ⱕ L. The vortex radius takes a fixed value in the range
of 0.4R ⱕ rv ⱕ 0.95R and different cases are considered for
each value of rv. For a given vortex radius, the vortex ring is
first placed at the pipe center, and the subsequent velocity
field is computed. It is illustrated by Fig. 4 for rv / R = 0.9. The
velocity field is then translated to the inlet of the pipe for the
initial iteration. At every time step ⌬t = tn+1 − tn, the vortex
ring is shifted, of one mesh spacing ⌬x = L / Nx, in the streamwise direction x 共Nx being the number of mesh nodes along
x兲 such that
u共x + ⌬x,r, ␪,t + ⌬t兲 = u共x,r, ␪,t兲.
共19兲
The constant convective velocity is then
册
⌫
冑rrv 2 − mv K共mv兲 − 2 E共mv兲 ,
␺共r兲 =
冑m v
冑m v
2␲
Uc =
共16兲
where e is the unit vector, K共m兲 and E共m兲 are the complete
elliptic integrals of the first and second kind, respectively,
and
⌬x
.
⌬t
共20兲
The simulations are run for Nx iterations, which is the number of iterations needed for the vortex ring to pass once
through the whole pipe. At each time step, the Lorentz force
is computed as follows. Since the vortex has no swirl
共u␪ = 0兲, Eq. 共12c兲 simplifies to
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125101-7
Phys. Fluids 22, 125101 共2010兲
Lorentz force velocimetry based on time-of-flight measurements
−3
6
8
x 10
ux
ur
5
6
4
Fx
3
r m /R = 1.15
r m /R = 2.9
T c /τ c
4
2
2
0
1
−2
0
0
−1
0
(a)
5
10
15
x/R
x 10
0.6
0.8
1
FIG. 5. Time evolution of the integrated Lorentz force acting on a single
vortex ring, which moves across a single-coil LFF 共case A1兲. The vortex
ring is located at rv = 0.9R, and the radius of the coil equals either
rm = 1.15R or rm = 2.9R. Time is nondimensionalized by the time of convection through the pipe, ␶c = L / Uc, where Uc is the convective velocity of the
vortex ring and L is the pipe length. The time elapsed between the braking
peaks is denoted by Tc, as illustrated for rm = 1.15R.
ux
ur
8
0.4
t/τc
−3
10
0.2
20
6
4
A. Single-coil flowmeter
2
The integrated force F̃x is recorded in time as the vortex
moves through the pipe. The result is shown in Fig. 5 for two
different radii of the coil. Time is nondimensionalized by the
time of convection ␶c. Moreover, the streamwise coordinate
of the vortex ring is given by xv共t兲 = Uct. Thus,
0
−2
−4
0
(b)
5
10
15
20
x/R
FIG. 4. Streamwise distribution of the velocity field associated to a vortex
ring, whose radial position is rv = 0.9R: r = 0 共a兲 and r = 0.5R 共b兲.
ⵜ2␾ = B · ␻ − u · 共⵱ ⫻ B兲 = 0
共21兲
because of the axisymmetry of the pipe and the magnetic
field. Since ␾ = 0 is solution of the Poisson equation, the
problem is greatly simplified and can be solved analytically.
The Lorentz force due to the ␣th coil is then given by
F L␣ = j ⫻ B ␣ = − j ␪B ␣re x + j ␪B ␣xe r
= ␴共uxBr − urBx兲共− B␣rex + B␣xer兲,
共22兲
denoting Bx = 兺␣B␣x and Br = 兺␣B␣r the total streamwise and
radial components of the magnetic field, respectively. In this
work, we choose to measure the streamwise component of
the force since it can easily be measured through the axial
displacement of the coil. By contrast, the measurement of the
radial force would require the determination of the coil dilatation. The integral of the axial force produced by the ␣th
coil is expressed as
F␣x共t兲 =
−␴
V
冕
B␣r关ux共x,r, ␪,t兲Br − ur共x,r, ␪,t兲Bx兴dV.
共23兲
It is nondimensionalized by removing its mean and dividing
by its root-mean-square 关see Eq. 共6兲兴. Finally, the time of
convection through the pipe is defined as
␶c =
L
.
Uc
共24兲
xv共t兲 t
= .
L
␶c
共25兲
As a consequence, Fig. 5 can also be interpreted as the evolution of the integrated force F̃x with the streamwise position
of the vortex ring xv共t兲. As shown, the force is symmetric
with respect to the streamwise midplane x = L / 2, corresponding to t = ␶c / 2 关see Eq. 共25兲兴.
When the vortex ring passes through the region of influence of the flowmeter, it experiences a braking force upstream from the coil, followed by an accelerating force when
it is located in the immediate vicinity of the coil. By symmetry, the force decreases as the vortex ring leaves the vicinity of the coil and eventually reverses to brake the flow.
Note that the effect of the force on the vortex is speculative
here because the vortex is forced to move at a constant
speed. The shape of F̃x can be explained from both Eq. 共23兲
and the contours of the magnetic field 共Fig. 2兲. If the vortex
velocity was purely streamwise, the streamwise magnetic
field would play no role in the force because urBx would be
zero. In that case, the induced current j␪, and hence the
streamwise force F̃x, would be maximum where the radial
magnetic field peaks, namely, upstream and downstream
from the coil only. Moreover, in both regions, the Lorentz
force would act against the flow, following the right-hand
rule. Here, the vortex possesses a radial velocity, which complicates the interpretation. However, two regions of negative
force are still observed upstream and downstream from the
coil plane. In addition, the integrated streamwise force presents a third peak, which is positive and coincides with the
coil plane. This positive peak is solely due to the component
urBx of the induced current since the radial magnetic field
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125101-8
Phys. Fluids 22, 125101 共2010兲
Viré, Knaepen, and Thess
8
2
6
1.8
T >0
T <0
4
Fx
r v /R = 0.9
r v /R = 0.7
r v /R = 0.5
1.6
Tp /Tc
2
1.4
0
1.2
-2
-0.5
0
0.5
1
1
1
1.5
t/τc
(a)
2
2.5
3
2.5
3
r m /R
T p /τ c
1
1.5
(a)
0.2
r m /R = 1.15
r m /R = 2.9
0.15
r v /R = 0.9
r v /R = 0.7
r v /R = 0.5
0.5
CM 2
C
CM 2
0.1
0
0.05
−0.5
−1
(b)
−0.5
0
0.5
0
1
1
T /τc
FIG. 6. 共a兲 Illustration of the shifting procedure for the autocorrelation. 共b兲
Autocorrelation of the integrated Lorentz force acting on a single vortex
ring, which moves across a single-coil LFF 共case A1兲. The vortex ring is
located at rv = 0.9R, and the radius of the coil equals either rm = 1.15R or
rm = 2.9R. Time is nondimensionalized by the time of convection through the
pipe, ␶c = L / Uc, where Uc is the convective velocity of the vortex ring and L
is the pipe length. The time shift T p corresponds to the secondary positive
peak, whose amplitude is C M2, as illustrated for rm = 1.15R.
cancels out at the location of the coil plane. Finally, the integral of F̃x over time equals zero 共not shown兲.
Following the shape of the integrated force, the regions
of braking can be used as sensors for the measurement. The
time elapsed between the two maximum braking forces is
directly measurable from Fig. 5 and is denoted by Tc. The
idea is to assess whether this time can be determined, without ambiguity, by autocorrelating F̃x. The autocorrelation is
obtained as follows. The original signal is zero-padded for
t ⬍ 0 and t ⬎ ␶c, so that its length is doubled to 2␶c. The
resulting signal is multiplied by its copy, which is shifted in
time in the range −␶c ⬍ T ⱕ ␶c 关see Fig. 6共a兲兴. The corresponding autocorrelation function is shown in Fig. 6共b兲 for
two values of the coil radius. This function is maximum for
zero shift, which is expected since the signal is perfectly
correlated with itself when T = 0. Following the shape of the
streamwise integrated force F̃x, the time-of-flight between
the braking regions is given by the time shift T p such that the
downstream and upstream braking forces coincide. This corresponds to a secondary positive peak in the autocorrelation.
The amplitude of this peak is denoted by C M2.
Defining T p as the measured time needed for the vortex
to travel the distance d between the maximum braking
forces, and Tc is the exact time-of-flight, the calibration factor T p / Tc determines the ratio of Uc = d / Tc to ULFF = d / T p.
Along with the magnitude of the correlation peak, it is shown
(b)
1.5
2
r m /R
FIG. 7. Case of a single vortex ring, located at rv / R = 0.9 and moving across
a single-coil LFF 共case A1兲: calibration factor T p / Tc, defined as the ratio
between the measured and exact time elapsed between the maximum braking forces 共a兲. Amplitude of the secondary peak in the autocorrelation of the
integrated Lorentz force 共b兲.
in Fig. 7 as a function of the coil radius and for three different radial positions of the vortex ring. The slight irregularities observed in the curve of the calibration factor 关see Fig.
7共a兲兴 are due to the limited grid resolution. The calibration
factor decreases as the coil moves away from the pipe, for
rm ⱗ 1.5. At larger values of the coil radius, the calibration
factor is almost independent of the radius of the coil. Moreover, its value is rather insensitive to the radial position of
the vortex. Both these characteristics are particularly attractive. The magnitude C M2 of the peak is also an important
quantity since it indicates the reliability of the measurement.
Figure 7共b兲 shows that as for the calibration factor, the magnitude of the secondary peak is quite invariant to the position
of the coil, for rm ⲏ 1.5. However, it increases as the vortex
moves toward the wall of the pipe. This can be related to the
fact that the Lorentz force is larger near the walls than in the
core of the pipe 共not shown兲.
In conclusion, we note that a vortex ring is subjected to
two distinct braking Lorentz forces when it passes through a
single-coil LFF. The time elapsed between these forces can
be measured by autocorrelating the time evolution of the
bulk force, generated by the coil, and acting globally in the
streamwise direction. The detection of the secondary positive
peak of the correlation is reliable 共i.e., large amplitude of the
peak兲 provided that the coil is placed sufficiently far from the
pipe. Typically, the coil radius should be 50% larger than that
of the pipe 共rm ⲏ 1.5R兲 to have constant values of the calibration factor and of the magnitude of the secondary peak in
the correlation.
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125101-9
Phys. Fluids 22, 125101 共2010兲
Lorentz force velocimetry based on time-of-flight measurements
4
T p /τ c
F1x
F2x
3
1
2
Fx
0.5
1
CM
C
0
0
−1
−0.5
−2
0
0.2
0.4
0.6
0.8
1
t/τc
(a)
−1
−0.5
4
F1x
F2x
3
0.5
1
0.5
1
1
0.5
2
Fx
0
T /τc
(a)
1
C
0
0
−1
−2
0
−0.5
0.2
(b)
0.4
0.6
0.8
1
t/τc
FIG. 8. 共Color online兲 Time evolution of the integrated forces for a single
vortex ring moving across a double-coil LFF: 共a兲 same-sign currents flowing
through the coils 共case A2S兲 and 共b兲 opposite-sign currents 共case A2O兲. The
vortex is located at rv / R = 0.9. The coil radii and separation are equal to
twice and four times the pipe radius, respectively. Times are nondimensionalized by the time of convection through the pipe, ␶c = L / Uc, where Uc is the
prescribed convective velocity of the vortex.
B. Double-coil flowmeter
In this section, the vortex ring moves through a doublecoil LFF 共cases A2S and A2O in Table I兲. The corresponding
forces F̃1x and F̃2x produced by each coil are presented in
Fig. 8: 共a兲 for same-sign currents 共case A2S兲 and 共b兲 for
opposite-sign currents 共case A2O兲. The figures are for coil
radii and separation equal to twice and four times the pipe
radius, respectively. Moreover, the circles correspond to the
streamwise position of the coils. With same-sign currents,
each force is similar to that produced by a single-coil LFF,
namely, it exhibits a large positive peak 共accelerating force兲
between two negative peaks 共braking forces兲. However, as
opposed to the single-coil case, the force produced by a coil
is not symmetric with respect to the coil plane. By contrast,
the pipe midplane x = L / 2 is a plane of symmetry for the sum
of the forces produced by each coil. With opposite-sign currents, the forces have similar shapes. However, as the coil
separation decreases, the negative peaks located between the
coils increase in magnitude and may overwhelm the accelerating peak. Further, the negative peaks get closer to each
other and, eventually, coincide. This is not observed with
same-sign currents and is explained by the large radial component of the magnetic field between the coils 关see Fig.
3共b兲兴.
With two coils, we choose to cross-correlate F̃1x and F̃2x
−1
(b)
−0.5
0
T /τc
FIG. 9. Cross-correlation of the integrated forces produced by each coil
when a single vortex ring moves across a double-coil LFF: 共a兲 same-sign
currents 共case A2S兲 and 共b兲 opposite-sign currents 共case A2O兲. The vortex is
located at rv / R = 0.9. The radius of the coil and their separation are twice and
four times larger than the pipe radius, respectively. Times are nondimensionalized by the time of convection through the pipe, ␶c = L / Uc, where Uc is the
prescribed convective velocity of the vortex.
关see Fig. 9 for same-sign 共a兲 and opposite-sign 共b兲 currents兴.
Following Fig. 8, the correlation is expected to be maximum
for time shifts such that the acceleration peaks of both forces
coincide. This differs from the single-coil LFF, which measures the time shift between braking regions. With two coils,
the measured time-of-flight is then given by the primary
positive peak of the correlation. As shown, it can be clearly
identified for both current signs, and its magnitude is close to
unity.
According to Fig. 8, the locations of the maximum accelerating force acting upon the vortex ring almost coincide
with the position of the coil planes. As a result, the time-offlight measured by a double-coil LFF in the synthetic case is
expected to be close to the time Tc needed for the vortex to
travel the distance ⌬ separating the coil planes. Figures 10
and 11 共same-sign currents and opposite-sign currents, respectively兲 summarize the calibration factor T p / Tc and the
maximum correlation C M as a function of the coil parameters. For both signs of the currents, T p / Tc and C M tend to
unity when the coil separation ⌬ increases and the coil radius
rm decreases. With same-sign currents, the measurement is
reliable for almost the entire range of 共rm , ⌬兲 because C M is
close to unity everywhere. Moreover, the calibration factor
varies by less than 5% for half of the domain considered.
Interestingly, the difference in the ratio T p / Tc is also within
5% for the same range of parameters with opposite-sign cur-
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125101-10
Phys. Fluids 22, 125101 共2010兲
Viré, Knaepen, and Thess
1.5
1.5
2
1
1.5
0.8
Tp
T c 0.6
Tp
Tc
1
1
0.4
1
0.5
0.2
0
3
0
3
r m /R
2
1
0.85
0.95
1
2
5
4
3
0.5
Δ/R
(a)
1.5
1.05
1.2
r m /R
2
1
1
(a)
3
2
5
4
0.5
Δ/R
0.88
0.98
0.9
2.5
r m /R
2
0.6
0.4
0.96
r m /R
0.5
2.5
0.92
0.94
2
0.8
0.94
0.98
1.5
0.9
1.5
0.7
0.96
0.98
1
(b)
2
3
1
4
Δ/R
(b)
2
3
4
Δ/R
FIG. 10. 共Color online兲 Single vortex ring moving across a double-coil LFF
with same-sign currents flowing through the coils 共case A2S兲: ratio of the
measured to the exact time-of-flight over the distance ⌬ between the coils
共a兲; contours of the peak amplitude CM in the cross-correlation between the
integrated Lorentz forces due to each coil 共b兲. The vortex is located at
rv / R = 0.9.
FIG. 11. 共Color online兲 Single vortex ring moving across a double-coil LFF
with opposite-sign currents flowing through the coils 共case A2O兲: ratio of
the measured to the exact time-of-flight over the distance ⌬ between the
coils 共a兲; contours of the peak amplitude CM in the cross-correlation between the integrated Lorentz forces due to each coil 共b兲. The vortex is
located at rv / R = 0.9.
rents 共case A2O兲 关see Fig. 11共a兲兴. However, in the latter, T p
overestimates Tc, while it is underestimated with same-sign
currents 共case A2S兲. This is caused by the interference between the coils, which brings the positive peaks of the forces
closer to the pipe center with same-sign currents and away
from it with opposite-sign currents. Since the correlation
gives the time shift between these peaks, T p ⬍ Tc when currents have same signs 共because the distance between the
peaks becomes smaller than ⌬兲, and the opposite is true for
opposite-sign currents. For clarity, the calibration factor is
clipped at T p / Tc = 2 when the coil separation is small and the
currents have opposite signs.
The study of the synthetic case provided a simplified
analysis of the time-of-flight LFV technique, and therefore,
enabled to highlight the differences between single- and
double-coil flowmeters. A single-coil LFF measures the time
elapsed between the regions of maximum braking forces occurring on both sides of the coil, whereas a double-coil LFF
measures the time-of-flight between the accelerating force
due to each coil. The time-of-flight measured by a doublecoil LFF is close to the time needed for the vortex to travel
the distance ⌬ separating the coils because the location of the
maximum acceleration force acting upon the vortex ring almost coincides with the position of the coil planes. In particular, the coils and maximum bulk Lorentz forces are almost perfectly aligned in the streamwise direction when the
interference between the coils is weak. The results have
shown that a double-coil flowmeter with same-sign currents
flowing through the coils is particularly reliable in measuring
the convective velocity of a vortex ring because the integrated force produced by each coil exhibits one main peak,
which is well separated from that of the other coil. The crosscorrelation between the forces can then effectively detect the
time spacing between the signals. With opposite-sign currents, the interference between the coils is strong at small
coil separations and further increases with the coil radius.
Reliable measurements can, however, be made provided that
the coil separation is larger than twice the pipe radius.
VI. RESULTS FOR REALISTIC VELOCITY FIELDS
Section IV demonstrated the feasibility of a time-offlight LFF to measure to convective velocity of a synthetic
velocity profile. The range of parameters, which leads to reliable and unambiguous measurements, was also determined.
In this section, a similar analysis is done for a realistic threedimensional turbulent flow. Together with investigating the
reliability of the measurement, the effect of the magnetic
field on the flow will be assessed.
The flow is initialized with the following turbulentlike
velocity profile on which random perturbations are superposed 共see Ref. 5 for details兲. Furthermore, the flow is driven
by a constant pressure gradient such that the bulk Reynolds
number fluctuates around Reb = 2UbR / ␯ ⬇ 3600, where
Ub = 1 / V兰udV = 2 / R2兰R0 具ux典rdr is the bulk velocity, V is the
pipe volume, and 具ux典 is the mean streamwise velocity profile
in the radial direction. The mean value of the bulk velocity is
fixed through
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125101-11
U2b =
Phys. Fluids 22, 125101 共2010兲
Lorentz force velocimetry based on time-of-flight measurements
冉 冊
4R ⳵ p
,
␳f ⳵x
1
共26兲
r m /R = 1.15
r m /R = 2.9
0.8
0.6
where f is the friction factor taken from Ref. 19.
As opposed to the synthetic case, in a turbulent realistic
flow, the convective velocity of each vortex differs from both
the mean velocity of the flow and its flowrate.12 Since the
purpose of a LFF is often to measure the flowrate, the reference time used in the calibration factor is based on the bulk
velocity Ub and is denoted by Tb = d / Ub, instead of
Tc = d / Uc used in the synthetic case. The choice of d depends
on the type of flowmeter. It can either be the distance between the braking regions in a single-coil LFF, or the distance between the coil planes in a double-coil LFF. Times are
nondimensionalized by the crossing time of the mean flow
through the pipe, ␶b = L / Ub.
Two series of runs are performed. First, the Lorentz
force is neglected in the right-hand side of Eq. 共12a兲, the
so-called kinematic simulations. In that case, the flowmeter
does not affect the flow, so several coil parameters can be
simulated at the same time using the same flow. Second,
dynamic simulations are performed, where the Lorentz force
is added to the momentum balance. Because its effects depend on the coil parameters, an individual simulation has to
be run for each couple of coil radius and separation. Here,
the difference with kinematic results will be assessed for few
values of the coil radius, while keeping the coil separation
equal to twice the pipe radius. The results shown here are for
a fully developed turbulent regime and the statistics are recorded over time intervals Ta ⬇ 8.2␶b, for kinematic simulations, and Ta ⬇ 100␶b, for dynamic cases. For the kinematic
simulations, longer time intervals 共Ta ⬇ 40␶b兲 were also considered for three values of the coil radius, namely,
rm / R = 1.4; 2 ; 2.6, while keeping the coil separation fixed at
⌬ / R = 2. Notwithstanding the increase of the time interval,
the time shift T p of the correlation peak was practically unchanged 共it differed by a maximum of 1%兲. When
Ta ⬇ 40␶b, the magnitude of the correlation peak in the kinematic simulations increases by 3% and 7% with same- and
opposite-sign currents, respectively, compared to the values
obtained with Ta ⬇ 8.2␶b.
0.4
C
0.2
0
−0.2
0
4
6
T /τc
FIG. 12. 共Color online兲 Autocorrelation of the integrated Lorentz force acting on a turbulent flow, across a single-coil LFF 共case T1兲. The radius of the
coil is equal to either rm = 1.15R or rm = 2.9R. Time is nondimensionalized by
the crossing time of the mean flow through the pipe, ␶b = L / Ub, where Ub is
the bulk velocity and L is the pipe length.
one coil. Because of this ambiguity, the results obtained from
the correlations are not further presented in the case of a
single-coil flowmeter.
Interesting observations can, however, be made on the
shape of the Lorentz force in the radial direction. Defining
the dimensionless radial distance from the wall as
r+ = 共R − r兲u␶ / ␯ and the friction velocity as u␶ = Ub冑 f / 8,20 Fig.
13 shows the radial distribution of the Lorentz force averaged over 共x , ␪兲 and nondimensionalized by its extremum
value for clarity, namely,
1
u x /u x M
u r /u x M
r m /R = 1.15
r m /R = 2.9
0.8
0.6
∗
F Lx
0.4
0.2
r F+
(a)
0
0
20
40
60
r+
80
100
120
20
A. Single-coil flowmeter
Due to the large number of turbulent eddies in the flow,
the force autocorrelation exhibits several harmonic peaks,
which give information on two different times: that needed
for a given eddy to travel between two regions of maximum
force and the time between the crossing through one region
of two neighboring vortices 共see Fig. 12 for two coil radii兲.
In that case, it is uncertain whether the first secondary peak
in the autocorrelation indeed corresponds to the time-offlight of a perturbation between the two regions of maximum
braking forces. This time might rather be given by the next
peaks. The correlations can be smoothed out by dividing the
time interval into samples and averaging the correlations obtained for each sample. However, this procedure does not
eliminate the peaks due to the neighboring vortices crossing
2
18
r F+
16
14
12
(b)
1.5
2
2.5
r m /R
FIG. 13. Turbulent flow moving across a single-coil flowmeter 共case T1兲.
ⴱ
典, averaged along
共a兲 Radial distribution of the streamwise Lorentz force 具FLx
共x , ␪兲, and nondimensionalized by its extremum value. 共b兲 Radial position of
ⴱ
典 as a function of the coil radius rm. The radial
the maximum value of 具FLx
coordinate is expressed in wall units, i.e., r+ = 共R − r兲u␶ / ␯, where R is the
pipe radius and u␶ is the friction velocity.
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125101-12
Phys. Fluids 22, 125101 共2010兲
Viré, Knaepen, and Thess
3
0.8
2
0.6
0.8
0.6
0.4
1
Fx
C
C
0
0.4
−1
0.2
0
1
T /τ b
0
−1
−0.2
−2
−3
165
F1x
F2x
165.5
−0.4
166
166.5
167
167.5
168
−5
t/τb
(a)
0
5
T /τb
(a)
0.8
3
0.8
0.6
2
C
0.6
0.4
1
Fx
C
0
F1x
F2x
−2
165.5
166
(b)
1
166.5
167
167.5
−0.2
−0.4
168
(b)
t/τb
FIG. 14. 共Color online兲 Time evolution of the integrated forces produced by
each coil when a turbulent flow moves across a double-coil LFF: 共a兲 samesign currents in the coils 共case T2S兲 and 共b兲 opposite-sign currents 共case
T2O兲. The coil radius and separation are equal to twice the pipe radius.
Times are nondimensionalized by the crossing time of the mean flow
through the pipe, ␶b = L / Ub, where Ub is the bulk velocity.
ⴱ
具FLx
典
0
T /τ b
0
−1
−3
165
0.4
−1
0.2
具FLx典
=
,
具FLx典 M
共27兲
where 具FLx典 M denotes the extremum value of 具FLx典. It is
compared to the average root-mean-square of the velocity
perturbations, denoted by 具u⬘典, where u⬘ = u − 具u典. Remarkably, the radial location rF+ of the maximum force is close to
that of the maximum of 具ux⬘典. By contrast, 具ur⬘典 and similarly
具u␪典 are maximum at larger radial positions 共typically around
r+ ⬇ 50兲. As the coil radius increases, the profile of the Lorentz force broadens, and, in turn, the location of its maximum slightly moves toward the pipe center 关see Fig. 13共b兲兴.
The figure also shows that the location of the maximum
force flattens for coil radii larger than twice the pipe radius.
This section highlighted the ambiguity in using a singlecoil LFF to determine the flowrate of turbulent flows. Nevertheless, the sensitivity of the flowmeter was analyzed
through the radial distribution of the Lorentz force. In particular, we have seen that the force produced by the coil is
highly correlated with the turbulent structures that contribute
to the streamwise velocity fluctuations.
B. Double-coil flowmeter
This subsection investigates the reliability of the flowrate measurement with a double-coil LFF for several coil
parameters. To fix ideas, the integrated forces produced by
each coil are shown by Fig. 14: 共a兲 for same-sign currents
−5
0
5
T /τb
FIG. 15. Cross-correlations of the integrated forces produced by each coil
when a turbulent flow moves across a double-coil LFF: 共a兲 same-sign currents in the coils 共case T2S兲 and 共b兲 opposite-sign currents 共case T2O兲. The
coil radius and separation are equal to twice the pipe radius. Times are
nondimensionalized by the crossing time of the mean flow through the pipe,
␶b = L / Ub, where Ub is the bulk velocity.
flowing through the coils and 共b兲 for opposite-sign currents.
For clarity, the time history is limited to three crossing times
of the mean flow through the pipe. Since the flow is turbulent, the forces exhibit strong fluctuations in time. As opposed to the case of the synthetic velocity field, in which the
vortex ring was convected without being deformed, here vortices are distorted, some are dissipated and others are created. Therefore, the flow passing through the downstream
coil differs locally from that which has passed through the
upstream coil. However, since the flow is statistically stationary and homogeneous in the streamwise direction, the conclusions drawn for the synthetic velocity may help in interpreting the present case. In particular, following the results
obtained in the synthetic case, the cross-correlation of the
forces should allow to estimate the time elapsed between the
accelerating forces occurring in the vicinity of each coil. The
cross-correlation are illustrated in Fig. 15 for same-sign 共a兲
and opposite-sign 共b兲 currents. In both cases, they exhibit a
sharp positive peak, whose magnitude CM indicates the reliability of the measurement. The associated time T p is compared to the characteristic time Tb of the mean flow.
Figure 16 shows the systematic variations of the calibration factors T p / Tb and magnitudes C M of the correlation peak
for several coil radii rm and coil separation ⌬, for same-sign
currents flowing through the coils. Two properties of the ratio T p / Tb deserve particular attention in Fig. 16共a兲. First, the
ratio is close to unity for all values of the coil radii and
separations considered. In fact, the ratios are slightly smaller
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125101-13
1.1
1
2.66
2
1.05
0.8
Tp
T b 0.6
1
0.9
0.95
0.4
1
2
0.9
0.2
0
2.5
Phys. Fluids 22, 125101 共2010兲
Lorentz force velocimetry based on time-of-flight measurements
0.85
r m /R
Fx
0.95
0.8
0.8
0.85
0.85
0.8
0.75
0.75
2
0.95
1
r m /R 1.5
0.7
0.6
0.9
2
1
3
4
0.7
1.1
1
Δ/R
2
0
0.65
3
0.55
−1
4
Δ/R
−2
FIG. 16. 共Color online兲 Turbulent flow across a double-coil LFF with samesign currents in the coils 共case T2S兲: ratio of the measured to the exact
time-of-flight over the distance ⌬ between the coils 共left兲; contours of the
peak amplitude CM in the cross-correlation between the integrated Lorentz
force due to each coil 共right兲.
280
F1x
F2x
281
282
283
284
t/τb
(a)
1
0.8
than unity, except at very low 共rm , ⌬兲, which implies that the
measured velocity slightly overestimates the real flowrate. In
other words, the velocity ULFF computed from the correlation
should be multiplied by a factor smaller than unity to get Ub.
Second, their ratio depends only weakly on the geometrical
parameters of the coil. This means that the calibration factor
of a time-of-flight LFF is rather insensitive to the geometric
parameters of the magnetic system provided that same-sign
currents flow through the coils. Another particular feature of
the double-coil LFF with same-sign currents is that the magnitude C M of the correlation peak is large and almost independent of the coil radii for coil separations typically smaller
than twice the pipe radius. This makes the measurement particularly reliable. As the coil separation increases, the forces
produced by both coils decorrelate owing to the flow modifications, through production and dissipation of turbulent eddies.
The results obtained with opposite-sign currents are
shown in Fig. 17. The main characteristics of T p / Tb, observed with same-sign currents, are recovered for large coil
separations only. If the coil separation is larger than twice the
pipe radius, the calibration factor T p / Tb is close to unity and
is independent of the coil parameters. However, for small
coil separations, it sharply decreases. These results agree
with the observations made on the synthetic case. It was
shown that at small coil separations, the two braking forces
located, respectively, downstream from the first coil and upstream from the second coil almost coincide and their magnitude is comparable to that of the accelerating peak in the
force. Therefore, the cross-correlation of the forces yields the
time-of-flight between these negative peaks, instead of being
C
0.6
C
1
1.05
0.8
Tp
T b 0.6
0.95
0.9
0.2
0.85
2
r m /R
0.65
0.7
0.75
0.65
0.8
0.5
0.7
0.75
2
r m /R 1.5
0.8
1
2
Δ/R
3
4
0.7
0
T /τ b
1
0
−0.2
−100
(b)
−50
0
50
100
T /τb
FIG. 18. 共Color online兲 Dynamic simulation of a turbulent flow across a
double-coil LFF with same-sign currents in the coils 共case T2S兲. Time evolution of the integrated forces produced by each coil 共a兲 and their crosscorrelation 共b兲. The coil radius and separation are equal to twice the pipe
radius, i.e., rm = ⌬ = 2R. Times are nondimensionalized by the time of mean
flow through the pipe ␶b = L / Ub, where Ub is the bulk velocity.
sensitive to the accelerating peaks, and the corresponding
time shift is nearly zero. With opposite-sign currents, coil
separations typically smaller than twice the pipe radius
should be avoided, except if the coil radii are small
共rm ⬍ 1.5R, for example兲. Another difference with same-sign
currents stands in the magnitude CM of the correlation peak
关see Fig. 17共b兲兴. With opposite-sign currents, CM highly depends on the coil radii and separation and it decreases when
these parameters increase.
The following discussion examines whether the external
magnetic field influences the flow. In magnetohydrodynamics, the effects of the magnetic field are commonly assessed
using the interaction parameter N, which represents the ratio
between electromagnetic and inertial forces. It is defined
here as
2
R
2␴Bmax
,
␳Ub
共28兲
0.7
1
0.4
0
2.5
2.66
0.4
−1
0.4
0.2
N=
1.1
0.8
0.6
1.1
0.75
0.9 0.85 0.8
1
2
0.55
3
4
Δ/R
FIG. 17. 共Color online兲 Turbulent flow across a double-coil LFF with
opposite-sign currents in the coils 共case T2O兲: ratio of the measured to the
exact time-of-flight over the distance ⌬ between the coils 共left兲; contours of
the peak amplitude CM in the cross-correlation between the integrated Lorentz force due to each coil 共right兲.
where Bmax is the maximum amplitude of the primary magnetic field. So far, the Lorentz force was neglected in the
right-hand side of the momentum balance, which is a valid
approximation for small values of N only. The effect of the
magnetic field on the flow is now evaluated through dynamic
simulations, in which the Lorentz force is added to the righthand side of the momentum balance. The time evolutions of
the forces produced by each coil and their cross-correlation
are illustrated in Fig. 18 for same-sign currents, with the coil
radii and separation equal to twice the pipe radius. In that
case, the interaction parameter equals N = 0.31. As for the
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125101-14
Phys. Fluids 22, 125101 共2010兲
Viré, Knaepen, and Thess
TABLE II. Ratios between dynamic and kinematic results for a double-coil LFF with same-sign 共case T2S兲 and
opposite-sign 共case T2O兲 currents in the coils.
Case
rm / R
⌬/R
N
共Fx兲 M / F f
共Reb兲dyn / 共Reb兲kin
共CF兲dyn / 共CF兲kin
共C M 兲dyn / 共C M 兲kin
T2S
1.4
2
1.03
0.13
0.88
0.99
0.98
T2S
T2S
2
2.6
2
2
0.31
0.21
0.04
0.02
0.97
1
1.01
1.01
0.98
0.99
T2O
1.4
2
0.83
0.23
0.79
0.93
1.01
T2O
2
2
0.11
0.05
0.95
0.98
0.99
T2O
2.6
2
0.03
0.01
1
0.99
0.99
kinematic results, only a sample of the time history is shown
for clarity. The forces produced by each coil are slightly
smoothed out, in comparison with the kinematic results. The
cross-correlation of the forces further exhibits several secondary peaks, although a sharp primary peak can still be
identified without ambiguity. Details of the dynamic results
are given in Table II for three values of the coil radius rm and
the coil separation fixed to twice the pipe radius. The results
are compared to the kinematic ones, obtained over a time
interval Ta ⬇ 40␶b. Together with the evaluation of the interaction parameter, the extremum magnitude of the integrated
force 共Fx兲 M is compared to the driving force per unit volume,
given by
Ff =
␳ fU2b
.
4R
共29兲
Table II shows that the interaction parameters are larger with
same-sign currents than with opposite-sign currents. The opposite is observed for the ratio between Lorentz and driving
forces. This is because the interaction parameter is based on
a single local value of the magnetic field, whereas the bulk
Lorentz force depends on the distribution of the magnetic
field in the whole pipe. Therefore, the ratio between Lorentz
and driving forces is more appropriate than the interaction
parameter to assess the influence of the flowmeter on the
flow. For both signs of the current in the coils, the dynamic
bulk Reynolds number decreases with respect to its kinematic value when the ratio between Lorentz and driving
forces increases. This of course means that the Lorentz force
slows down the mean flow. Their ratio tends to unity as the
coils radius increases. This is expected since the magnitude
of the magnetic field diminishes as the coils move away from
the pipe, which, in turn, decreases the interaction between
the magnetic field and the flow. The ratios between the dynamic and kinematic values of the calibration factor 共denoted
CF兲 and the magnitude CM of the correlation peak are presented in Table II. With both signs of the currents, the magnitude of the correlation peak is almost unaffected by the
magnetic field since it differs by 2% from its kinematic values. This, in turn, lies within the incertitude interval associated with the measurements. Similar differences are observed
on the calibration factor, except for the smaller coil radius
and opposite-sign currents, which presents a decrease of 7%
in the dynamic calibration factor. This case also corresponds
to the largest ratio between Lorentz and driving forces. It
can, however, be concluded that the kinematic and dynamic
results are substantially in agreement, and therefore, that the
reliability of the measurement and the calibration factor are
independent of the interaction parameter. This, in turn,
means that the measurement does not depend on the electrical conductivity of the fluid.
VII. CONCLUSION
This work demonstrated numerically the feasibility of a
Lorentz force flowmeter based on time-of-flight measurements. The proposed technique was first validated on a synthetic case, consisting of a vortex ring moving at a constant
speed. In that case, the vortex ring experiences two forces
having antagonistic effects as it moves through the coil. For
a single-coil flowmeter, an accelerating force coincides with
the coil plane and two braking forces are located upstream
and downstream from the coil. The force autocorrelation was
successful in measuring, without ambiguity, the time-offlight of the vortex between the regions of maximum braking. For a double-coil flowmeter, the measurement is slightly
different since the cross-correlation of the forces produced
by each coil yields the time between the acceleration regions,
located close to each coil. If same-sign currents flow through
the coils, the ratio between the measured and the exact timeof-flight 共defined as the calibration factor兲 is almost independent of the coil parameters and the correlation peak is particularly high. With opposite-sign currents, coil separations
smaller than the pipe radius should be avoided, as well as
large coil radii 共the limit depends on the coil separation兲.
The flowmeters were then applied to a three-dimensional
turbulent flow. In that case, different times-of-flight are measured by the correlations: a time-of-flight between two distinct regions of maximum force, as in the synthetic case, and
a crossing time of two neighboring vortices through a single
region of maximum force. A single-coil flowmeter cannot
distinguish between these various times because they all correspond to secondary peaks in the autocorrelation of the
force. Reliable measurements are thus impossible. Conversely, the cross-correlations provided by a double-coil
flowmeter do exhibit a sharp primary peak, associated with
the time-of-flight between the accelerating forces produced
by each coil. Remarkably, the calibration factor is almost
independent of the coil parameters, although small coil separations should be avoided when opposite-sign currents are
used, as for the synthetic velocity field. The main difference
with the synthetic case is the decorrelation of the forces as
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125101-15
Lorentz force velocimetry based on time-of-flight measurements
the coil separation increases. This is expected since the turbulent flow is modified by the production and dissipation of
turbulent eddies as it evolves through the pipe. This affects
the reliability of the measurement as the coil separation increases. In a realistic turbulent flow, the coil parameters
should thus compromise between reliability of the measurement and invariance of the calibration factor.
The sensitivity of the single-coil flowmeter was analyzed
through the radial distribution of the Lorentz force averaged
in the axial and azimuthal directions. It was shown that the
region of maximum Lorentz force broadens toward the core
of the pipe when the coil radius increases. Moreover, the
location of the maximum force is almost constant for coil
radii larger than twice the pipe radius. Remarkably, the radial
distribution of the Lorentz force is highly correlated with that
of the streamwise velocity fluctuations.
Finally, the influence of the magnetic field on the flow
was assessed by accounting for the Lorentz force in the momentum balance, in the particular case of a coil separation
equal to twice the pipe radius. Kinematic and dynamic results were shown to differ by a maximum of 2% in terms of
calibration factors and magnitudes of the correlation peak.
The difference in the calibration factor reached 7% in one
case, for which the bulk Lorentz force exceeded 20% of the
driving force. Therefore, the influence of the flowmeter on
the flow does not alter the reliability of the measurement and
only slightly modifies the calibration factor. This, in turn,
demonstrates that the measurement provided by the time-offlight flowmeter is independent of the electrical conductivity
of the fluid.
Future work will be devoted to experimentally testing
our numerical predictions. Moreover, it would be interesting
to study more sophisticated magnetic systems and more
components of the Lorentz force than just the axial one.
ACKNOWLEDGMENTS
A.V. was supported by the Fonds pour la Recherche dans
l’Industrie et dans l’Agriculture 共FRIA-Belgium兲. This work,
conducted as part of the award 共Modeling and simulation of
turbulent conductive flows in the limit of low magnetic Reynolds number兲 made under the European Heads of Research
Councils and European Science Foundation EURYI 共European Young Investigator兲 Awards scheme, was supported by
funds from the Participating Organisations of EURYI and the
EC Sixth Framework Programme. The content of the publi-
Phys. Fluids 22, 125101 共2010兲
cation is the sole responsibility of the authors and it does not
necessarily represent the views of the Commission or its services. The support of FRS-FNRS Belgium is also gratefully
acknowledged. A.T. is grateful to the Deutsche Forschungsgemeinschaft for the support of the present work in the
framework of the Research Training Group “Lorentz force
velocimetry and Lorentz force eddy current testing” at the
Ilmenau University of Technology.
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