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Calculation of the virtual current around an electromagnetic velocity probe using the alternating method of Schwarz
Calculation of the virtual current around an electromagnetic velocity probe using the alternating method of Schwarz X. Z. Zhang* and J. H e m p t *Department of Power Engineering, Nanjing Aeronautical Institute, China -t-Department of Fluid Engineering and Instrumentation, Cranfield University, Cranfield, Bedford MK43 0AL, UK
A semi-analytical method is introduced to solve Laplace's equation in a complex geometry under certain boundary conditions and hence to calculate the virtual current around an electromagnetic flow velocity probe situated near a wall. Contours are given of virtual current potential with insulating and conducting walls with the probe at various distances from the wall.
Keywords: electromagnetic velocity probe; virtual current; Schwarz's alternating method
Introduction Electromagnetic (e.m.) velocity probes and flowmeters are used to measure the flow velocity of electrolytes. Their principle of operation is based on Faraday's law of induction that the movement of a conductor in a magnetic field causes an induced electromotive force in it. The e.m. velocity probe has been used in large spaces, such as rivers and oceans. More recently it has been used in pipe flows. However, not much is known about how a pipe wall affects the output signal of an e.m. velocity probe. This effect will, of course, be greatest when the probe is near to the wall. In the theory of electromagnetic flow measurement, the output signal (i.e. the interelectrode potential U2-U1 induced by flow v in a magnetic field B can be expressed as1
U2 - U 1 = I . W ' v d r
(2)
The virtual current is defined as that current density which would be present (in the stationary fluid) if the unit current were emitted from one electrode and absorbed at the other. The potential of the virtual current obeys Laplace's equation (with boundary conditions) in the space r. The determination of the distribution of virtual current is important in the design and optimization of e.m. velocity probes. In the case considered, a plane wall is assumed
146
Flow Meas. Instrum,, 1994 Volume 5 Number 3
(3)
V2G=0
where G is the potential of virtual current j: i = -
vc
(4)
The boundary conditions are
(1)
where the integration space r is the whole fluid volume and W (called the weight vector) is the cross product of B and the virtual current j:
W= Bx j
to be situated near an e.m. velocity probe of cylindrical form (radius R) with a pair of point electrodes on its closed end (see Figure l(a)). The mathematical problem is to solve
aG c3z z=O
= I - 6(r-a)8(O)/r [ + 8(r- a)a(0- rr)/r
aG ar
= 0 r~R
~G c3z
z=L!
O<_r<_R
z-<0
(5)
= 0 for insulating wall
or
G~_q = 0 for conducting wall
where 8if-a), 8(0) and 8(0-~') are Delta functions, L1 is the distance between the end of the e.m. velocity probe and the pipe wall, and a is the radial position of the electrodes. This kind of problem can be solved by numerical methods, for example by the finite element method. We then have to spend much time defining meshes and there are problems regarding the accuracy of the results, CPU time etc. On the other hand, it is impossible at present to obtain a solution analytically. The purpose of this paper is to make use of a
0955-5986/94/030]46-04
© 1994 Butterworth-Heinemann Ltd
X. Z. Zhang and J. Hemp - Virtual current around an electromagnetic velocity probe method, described for example in Ref. 3. We divide the space ~- into two parts expressing the virtual current potential G as in Figure l(b):
z
all
I G1 (r, O, z) G(r, 0, z) = I G2 (r, 0, z)
Coil
/
[
I pIane end
Insulating
a. ~
O<_r<_R2 R<_r (6)
Here L2 and R2 are chosen to be so large that the virtual current near the probe end is not affected by the boundaries at r = R2 and z = -L2. If both the wall and the probe body are insulating, the boundary conditions for G~ and G~ are
#---' I \ n~-
0_
Probe
OG1 ~=R2 = 0 ,
body
OG1 =0 Oz I~=L~
Or
(a)
(7)
OG1 Oz ~=o
-= { + ~(r- a)~( O- 7r)/r
OG~ Orr ~=R
= O,
O<_r<_R
and Y
Region I
-L2~z<-O
OG~ = 0, 0z I~=L~ G21r---> °°
(
L1 •
x
(8)
OG~ =0 az Iz=-L2
= 0
Because of these boundary conditions and because G1 and G2 satisfy Laplace's equation, it is possible to expand them in an infinite series thus:
L2
Gl(r, 0, z) =
L
L
Am,, J~ [(/~n)l/2r]
(9)
rn= 1,3,5... n--1
_g
cos(mS)
\ Region 2
sinh
[(/J,n)l/2L1]
where
Jm [(/~g)1/2 R2)] =
(b)
Figure 1 Geometry of the electromagnetic probe
cosh [(/~,~,)1/2(z-L1)]
velocity
and
L
G2(r,8,z) = method that lies mid-way between a purely analytical and a purely numerical method. This provides more flexibility when applied to complicated boundary problems, while still retaining something of the analytical approach.
The method used to calculate the virtual current In a previous paper 2 a method was presented to solve Laplace's equation with mixed boundary values. In that work, on part of the boundary there was a Dirichlet condition while on another part there was a Neumann condition. A similar method is applied here to the problem of finding the virtual current potential of an e.m. velocity probe. Here the values of each boundary are not mixed, but the space we are dealing with is complicated. It can, however, be regarded as a combination of two overlapping simple geometries. The method we use is Schwarz's alternating
(10)
0
R(p+1)
B~,ocos p8 - -
rP
p--1,3,5..,
+
~,
~, Bpo Kp([~'q/(L2+L1)]r)
p=l.3,s.., q=l
Kp ([Trq/(L2+L,)]R)
~rq Here Jm(t) and Kp(t) are the ordinary and modified Bessel functions of the first and second kind respectively. ]~(t) and K'o(t) are their derivatives. To find the Amn and Boo in equations (10) and (11), we first introduce a function G~°), which has the form of equation (9) with coefficients A(°~) and satisfies
OG(p) ['- ~(r-a) ~(O)/r az z=o = [ +~(r-a) 8(e-~r)/r
0-
(12)
It is easy to calculate the ,.mnA(°)(m = 1,3,5..., n = 1,2,3...). Second, we introduce G(2~), which has the form of equation (11) and satisfies
Flow Meas. Instrum., 1994 Volume 5 Number 3
147
X. Z. Zhang and J. Hemp - Virtual current around an electromagnetic velocity probe aG~) ar r=R =
t"0
-L2 -
aG~°l L a~lr=R
0-
(13)
Thus we obtain the R(~I (p = 1,3,5 .... q = 0,1,2...). Third, we introduce G~~), which has the form of equation (9) and satisfies ~ p q
r l -~r-a)a(O)/r O<_r<_R = ~ [ + ~(r-a)a(O-~r)/r az ~=o /aG~ ~) I. Oz ]~=o R<-r<~R2
aG~2)
(14)
Thus we obtain the Aimed, and so forth. Generally, for p, = 2,4,6... we have G(I#) =
L
L AmnJm[(I, (/~) n 112r] Zm)
m=1,3,5..,
n=l
As the process goes on, G(?~) and G~ ÷1) approach the required values of G1 and G2 in equation (6) and the solution of equations (3) and (5) is found (to any desired accuracy). Figures 2(a) and (b) show approximate computer contour plots of virtual current potential calculated using the method. The wall and probe body are both insulating. The same method has been used in cases where the wall or the curved surface of the probe body is conducting, or where both are conducting. On a conducting wall, the virtual current potential will be zero. The process of finding the expressions for G~ and G2 in these cases is similar to that discussed above. To avoid repeating long expressions, we give only the calculated results. See Figure 3 for the conducting wall and Figure 4 for the case when the curved surface of the probe body is conducting.
Discussion of the method used
cosh [(/.t,%)~/2(z- L~)] x cosm8 sinh [(/~n,)1/2 L~]
(15)
The above computations show that the method used is a useful one for solving Laplace's equation in a geometry
with
F
A(~)mn
I~m[(~) " ~ r ] l l ~ n ) "~ '1
+ E
.
I .00
L - 2lm [(p,~)'/2a]
[ ~K
~K
s,n%_t ' t, t +t,
Oi/
0.75 0.50
(16)
i;I
k=l
0.25 X
IR~2jm[(l'tn)W2r]K'Kmi}{[~'k/(L2+G)]r} ~ ~ ) ] R
rdr]j 0.00
where IIJm[(/~g)~/2r]lJ is the norm of the Besse[ function R 2 n 1/2 of the first kind (i.e. II/,.[(~)"~r]ll = .ro2Jml(#m) r] rdr). Also,
-0.25 -0.50
R(p+ 1 )
G ~ +1 ) =
-0.75
Btp%÷ l) c o s pO rp
p= 1,3,5...
+
~ Bg%+I, Kp{[Trq/(L2+G)I r} L p=~,3,s...q=, K'p {['n'q/(L2+t,)] R}
cos 0co,( ,z-,, 0
,,,,
I I -I .00 0.00 0,25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
r/R
(a)
G= 0.2 0.4 0.6 0.8
0.00
with Bg%+1)
-1
-
(L2+L1)p
k~_~1 =
A~)j,,, k,1/2
-0.25
. k~1/2
(I.~p)
~
G = 0.2
-0.50 x I L' cosh [(/~g)1/2 (z-L~)] dz sinh [(/a,g)1/2 L1] Jo
(18) -0.75
-I.00
B~%+~) = - -2 L A(14.)l..k11/2p ,~pk~pJ j p [ ( ~ pk'j~112R] .
~"q k= I
[b)
× IL, cos_h Jo
sinh [(/.t~,)1/2 L1]
x cos ( ~ 148
( z - G ) ) dz
(19)
Flow Meas. Instrum., 1994 Volume 5 Number 3
.00 0.25 0.50 0.75 I.00 I.25 1.50 1.75 2.00 r/R
Figure 2 Equispaced virtual potential contours in the xz-plane - insulating wall and probe body. (a) L1/R = 1, a/r = 0.8, R2/R = 6, L2/R = 5. (b) LJR = 0.1, a/r = 0.8, R2/R = 5, LJR = 2
X. Z. Zhang and J. Hemp - Virtual current around an electromagnetic velocity probe
~
~
0.10
0.00 -0.25 e~
-0.50 -0.75 -I .00 0.00
I
I
I
0.25
0.50
0.75
1.00
I
I
1.25
1.50
I
1.75 2.00
fir
Figure 3 Equispaced virtual potential contours in the xz-plane - conducting wall and insulating probe body. L1/R = 0.2, a/r = 0.8, R2/R = 6, L2/R = 4 I .00
0.75
0.50
0.25
=0.1 r-,,
0.00 N
-0.25
-0.50
particular case. In most cases we have met, the process itself converges quickly, 10 to 15 cycles being enough to reach an accuracy of 0.01% in the coefficients. Strictly speaking, we should prove that the method converges and estimate rates of convergence. This is not easy in general, however, and no attempt is made to do this here. In the general theory of Schwarz's method, ~'4 convergence is proved for Dirichlet boundary conditions in two- and three-dimensional geometries and the rate of convergence is shown to be at least as fast as a geometric progression (i.e. error < qn, 0 < q < 1, n = number of iterations). This proof depends on the assumption (valid for Laplace's equation) that the values of the unknown function (G in our case) in a region enclosed by a boundary always lie between the maximum and minimum of the values (of G) specified on the boundary. It would seem that this proof provides justification for convergence only in the case, studied by us, where the wall is insulating and the curved surface of the probe is conducting (Fig. 4). In this case we might as well have calculated G = aG/az rather than G. Then G would have satisfied Laplace's equation and would have been specified on the boundaries. The process (for G) would then have converged and G could have been obtained from (~ by term by term integration with respect to z. No general proof of convergence of Schwarz's alternating method for Neumann (or for mixed) boundary conditions is known to the authors, but the present numerical experience seems to show that convergence is at least sometimes present in these cases. The technique used to calculate virtual current potential will be useful in the modelling and optimization of e.m. velocity probes of the sort considered. It may also be applied to other probe geometries.
-0.75
Acknowledgements -1.00
~
-
0.00
0.25
-
0.50
0.75
1.00
1.25
1.S0
1.75
2.00
r/R
Figure 4 Equispaced virtual potential contours in the xz-plane - insulating wall and conducting probe body. L~/R = 2, a/r = 0.8, R2/R = 5, L2/R = 4 that is too complicated for full analytical treatment. The method is no doubt useful in applications other than e.m. velocity probes. What we obtain from the method is a number of series of known functions multiplied by coefficients which are found numerically by the iteration process. The important advantage of this method over a pure numerical method is that the results remain in some degree analytic and therefore convenient for further calculations. Also, they can be obtained in less computation time. In the above examples, less than five minutes of CPU time is needed to obtain accurate values of the coefficients in one
One of the authors (X. Z. Z.) would like to express his thanks to the Royal Society of the United Kingdom for supporting his visit as a K. C. Wong Fellow to the Department of Fluid Engineering and Instrumentation at Cranfield Institute of Technology (Cranfield University). Thanks are due also to A. Guilbert, R. Krafft and other colleagues at the department for helping with computation, and to Professor M. L. Sanderson for his support.
References 1
Bevir, M. K. Theory of induced voltage electromagnetic flowmeters
J. Fluid. Mech. 43 (1970) 577-590 2
Zhang, X. Z. A method for solving Laplace's equation with mixed boundary conditions in electromagnetic flowmeters J. Phys. D: Appl. Phys. 22 (1989) 573-576 3 Kantororich, L. V. and Krylov, V. I. Approximate Methods of Higher Analysis P. Noordhoff Ltd, Amsterdam (1958) 4 Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol II. Interscience Publishers, New York (1965)
Flow Meas. Instrum., 1994 Volume 5 Number 3
149
A semi-analytical method is introduced to solve Laplace's equation in a complex geometry under certain boundary conditions and hence to calculate the virtual current around an electromagnetic flow velocity probe situated near a wall. Contours are given of virtual current potential with insulating and conducting walls with the probe at various distances from the wall.
Keywords: electromagnetic velocity probe; virtual current; Schwarz's alternating method
Introduction Electromagnetic (e.m.) velocity probes and flowmeters are used to measure the flow velocity of electrolytes. Their principle of operation is based on Faraday's law of induction that the movement of a conductor in a magnetic field causes an induced electromotive force in it. The e.m. velocity probe has been used in large spaces, such as rivers and oceans. More recently it has been used in pipe flows. However, not much is known about how a pipe wall affects the output signal of an e.m. velocity probe. This effect will, of course, be greatest when the probe is near to the wall. In the theory of electromagnetic flow measurement, the output signal (i.e. the interelectrode potential U2-U1 induced by flow v in a magnetic field B can be expressed as1
U2 - U 1 = I . W ' v d r
(2)
The virtual current is defined as that current density which would be present (in the stationary fluid) if the unit current were emitted from one electrode and absorbed at the other. The potential of the virtual current obeys Laplace's equation (with boundary conditions) in the space r. The determination of the distribution of virtual current is important in the design and optimization of e.m. velocity probes. In the case considered, a plane wall is assumed
146
Flow Meas. Instrum,, 1994 Volume 5 Number 3
(3)
V2G=0
where G is the potential of virtual current j: i = -
vc
(4)
The boundary conditions are
(1)
where the integration space r is the whole fluid volume and W (called the weight vector) is the cross product of B and the virtual current j:
W= Bx j
to be situated near an e.m. velocity probe of cylindrical form (radius R) with a pair of point electrodes on its closed end (see Figure l(a)). The mathematical problem is to solve
aG c3z z=O
= I - 6(r-a)8(O)/r [ + 8(r- a)a(0- rr)/r
aG ar
= 0 r~R
~G c3z
z=L!
O<_r<_R
z-<0
(5)
= 0 for insulating wall
or
G~_q = 0 for conducting wall
where 8if-a), 8(0) and 8(0-~') are Delta functions, L1 is the distance between the end of the e.m. velocity probe and the pipe wall, and a is the radial position of the electrodes. This kind of problem can be solved by numerical methods, for example by the finite element method. We then have to spend much time defining meshes and there are problems regarding the accuracy of the results, CPU time etc. On the other hand, it is impossible at present to obtain a solution analytically. The purpose of this paper is to make use of a
0955-5986/94/030]46-04
© 1994 Butterworth-Heinemann Ltd
X. Z. Zhang and J. Hemp - Virtual current around an electromagnetic velocity probe method, described for example in Ref. 3. We divide the space ~- into two parts expressing the virtual current potential G as in Figure l(b):
z
all
I G1 (r, O, z) G(r, 0, z) = I G2 (r, 0, z)
Coil
/
[
I pIane end
Insulating
a. ~
O<_r<_R2 R<_r (6)
Here L2 and R2 are chosen to be so large that the virtual current near the probe end is not affected by the boundaries at r = R2 and z = -L2. If both the wall and the probe body are insulating, the boundary conditions for G~ and G~ are
#---' I \ n~-
0_
Probe
OG1 ~=R2 = 0 ,
body
OG1 =0 Oz I~=L~
Or
(a)
(7)
OG1 Oz ~=o
-= { + ~(r- a)~( O- 7r)/r
OG~ Orr ~=R
= O,
O<_r<_R
and Y
Region I
-L2~z<-O
OG~ = 0, 0z I~=L~ G21r---> °°
(
L1 •
x
(8)
OG~ =0 az Iz=-L2
= 0
Because of these boundary conditions and because G1 and G2 satisfy Laplace's equation, it is possible to expand them in an infinite series thus:
L2
Gl(r, 0, z) =
L
L
Am,, J~ [(/~n)l/2r]
(9)
rn= 1,3,5... n--1
_g
cos(mS)
\ Region 2
sinh
[(/J,n)l/2L1]
where
Jm [(/~g)1/2 R2)] =
(b)
Figure 1 Geometry of the electromagnetic probe
cosh [(/~,~,)1/2(z-L1)]
velocity
and
L
G2(r,8,z) = method that lies mid-way between a purely analytical and a purely numerical method. This provides more flexibility when applied to complicated boundary problems, while still retaining something of the analytical approach.
The method used to calculate the virtual current In a previous paper 2 a method was presented to solve Laplace's equation with mixed boundary values. In that work, on part of the boundary there was a Dirichlet condition while on another part there was a Neumann condition. A similar method is applied here to the problem of finding the virtual current potential of an e.m. velocity probe. Here the values of each boundary are not mixed, but the space we are dealing with is complicated. It can, however, be regarded as a combination of two overlapping simple geometries. The method we use is Schwarz's alternating
(10)
0
R(p+1)
B~,ocos p8 - -
rP
p--1,3,5..,
+
~,
~, Bpo Kp([~'q/(L2+L1)]r)
p=l.3,s.., q=l
Kp ([Trq/(L2+L,)]R)
~rq Here Jm(t) and Kp(t) are the ordinary and modified Bessel functions of the first and second kind respectively. ]~(t) and K'o(t) are their derivatives. To find the Amn and Boo in equations (10) and (11), we first introduce a function G~°), which has the form of equation (9) with coefficients A(°~) and satisfies
OG(p) ['- ~(r-a) ~(O)/r az z=o = [ +~(r-a) 8(e-~r)/r
0-
(12)
It is easy to calculate the ,.mnA(°)(m = 1,3,5..., n = 1,2,3...). Second, we introduce G(2~), which has the form of equation (11) and satisfies
Flow Meas. Instrum., 1994 Volume 5 Number 3
147
X. Z. Zhang and J. Hemp - Virtual current around an electromagnetic velocity probe aG~) ar r=R =
t"0
-L2 -
aG~°l L a~lr=R
0-
(13)
Thus we obtain the R(~I (p = 1,3,5 .... q = 0,1,2...). Third, we introduce G~~), which has the form of equation (9) and satisfies ~ p q
r l -~r-a)a(O)/r O<_r<_R = ~ [ + ~(r-a)a(O-~r)/r az ~=o /aG~ ~) I. Oz ]~=o R<-r<~R2
aG~2)
(14)
Thus we obtain the Aimed, and so forth. Generally, for p, = 2,4,6... we have G(I#) =
L
L AmnJm[(I, (/~) n 112r] Zm)
m=1,3,5..,
n=l
As the process goes on, G(?~) and G~ ÷1) approach the required values of G1 and G2 in equation (6) and the solution of equations (3) and (5) is found (to any desired accuracy). Figures 2(a) and (b) show approximate computer contour plots of virtual current potential calculated using the method. The wall and probe body are both insulating. The same method has been used in cases where the wall or the curved surface of the probe body is conducting, or where both are conducting. On a conducting wall, the virtual current potential will be zero. The process of finding the expressions for G~ and G2 in these cases is similar to that discussed above. To avoid repeating long expressions, we give only the calculated results. See Figure 3 for the conducting wall and Figure 4 for the case when the curved surface of the probe body is conducting.
Discussion of the method used
cosh [(/.t,%)~/2(z- L~)] x cosm8 sinh [(/~n,)1/2 L~]
(15)
The above computations show that the method used is a useful one for solving Laplace's equation in a geometry
with
F
A(~)mn
I~m[(~) " ~ r ] l l ~ n ) "~ '1
+ E
.
I .00
L - 2lm [(p,~)'/2a]
[ ~K
~K
s,n%_t ' t, t +t,
Oi/
0.75 0.50
(16)
i;I
k=l
0.25 X
IR~2jm[(l'tn)W2r]K'Kmi}{[~'k/(L2+G)]r} ~ ~ ) ] R
rdr]j 0.00
where IIJm[(/~g)~/2r]lJ is the norm of the Besse[ function R 2 n 1/2 of the first kind (i.e. II/,.[(~)"~r]ll = .ro2Jml(#m) r] rdr). Also,
-0.25 -0.50
R(p+ 1 )
G ~ +1 ) =
-0.75
Btp%÷ l) c o s pO rp
p= 1,3,5...
+
~ Bg%+I, Kp{[Trq/(L2+G)I r} L p=~,3,s...q=, K'p {['n'q/(L2+t,)] R}
cos 0co,( ,z-,, 0
,,,,
I I -I .00 0.00 0,25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
r/R
(a)
G= 0.2 0.4 0.6 0.8
0.00
with Bg%+1)
-1
-
(L2+L1)p
k~_~1 =
A~)j,,, k,1/2
-0.25
. k~1/2
(I.~p)
~
G = 0.2
-0.50 x I L' cosh [(/~g)1/2 (z-L~)] dz sinh [(/a,g)1/2 L1] Jo
(18) -0.75
-I.00
B~%+~) = - -2 L A(14.)l..k11/2p ,~pk~pJ j p [ ( ~ pk'j~112R] .
~"q k= I
[b)
× IL, cos_h Jo
sinh [(/.t~,)1/2 L1]
x cos ( ~ 148
( z - G ) ) dz
(19)
Flow Meas. Instrum., 1994 Volume 5 Number 3
.00 0.25 0.50 0.75 I.00 I.25 1.50 1.75 2.00 r/R
Figure 2 Equispaced virtual potential contours in the xz-plane - insulating wall and probe body. (a) L1/R = 1, a/r = 0.8, R2/R = 6, L2/R = 5. (b) LJR = 0.1, a/r = 0.8, R2/R = 5, LJR = 2
X. Z. Zhang and J. Hemp - Virtual current around an electromagnetic velocity probe
~
~
0.10
0.00 -0.25 e~
-0.50 -0.75 -I .00 0.00
I
I
I
0.25
0.50
0.75
1.00
I
I
1.25
1.50
I
1.75 2.00
fir
Figure 3 Equispaced virtual potential contours in the xz-plane - conducting wall and insulating probe body. L1/R = 0.2, a/r = 0.8, R2/R = 6, L2/R = 4 I .00
0.75
0.50
0.25
=0.1 r-,,
0.00 N
-0.25
-0.50
particular case. In most cases we have met, the process itself converges quickly, 10 to 15 cycles being enough to reach an accuracy of 0.01% in the coefficients. Strictly speaking, we should prove that the method converges and estimate rates of convergence. This is not easy in general, however, and no attempt is made to do this here. In the general theory of Schwarz's method, ~'4 convergence is proved for Dirichlet boundary conditions in two- and three-dimensional geometries and the rate of convergence is shown to be at least as fast as a geometric progression (i.e. error < qn, 0 < q < 1, n = number of iterations). This proof depends on the assumption (valid for Laplace's equation) that the values of the unknown function (G in our case) in a region enclosed by a boundary always lie between the maximum and minimum of the values (of G) specified on the boundary. It would seem that this proof provides justification for convergence only in the case, studied by us, where the wall is insulating and the curved surface of the probe is conducting (Fig. 4). In this case we might as well have calculated G = aG/az rather than G. Then G would have satisfied Laplace's equation and would have been specified on the boundaries. The process (for G) would then have converged and G could have been obtained from (~ by term by term integration with respect to z. No general proof of convergence of Schwarz's alternating method for Neumann (or for mixed) boundary conditions is known to the authors, but the present numerical experience seems to show that convergence is at least sometimes present in these cases. The technique used to calculate virtual current potential will be useful in the modelling and optimization of e.m. velocity probes of the sort considered. It may also be applied to other probe geometries.
-0.75
Acknowledgements -1.00
~
-
0.00
0.25
-
0.50
0.75
1.00
1.25
1.S0
1.75
2.00
r/R
Figure 4 Equispaced virtual potential contours in the xz-plane - insulating wall and conducting probe body. L~/R = 2, a/r = 0.8, R2/R = 5, L2/R = 4 that is too complicated for full analytical treatment. The method is no doubt useful in applications other than e.m. velocity probes. What we obtain from the method is a number of series of known functions multiplied by coefficients which are found numerically by the iteration process. The important advantage of this method over a pure numerical method is that the results remain in some degree analytic and therefore convenient for further calculations. Also, they can be obtained in less computation time. In the above examples, less than five minutes of CPU time is needed to obtain accurate values of the coefficients in one
One of the authors (X. Z. Z.) would like to express his thanks to the Royal Society of the United Kingdom for supporting his visit as a K. C. Wong Fellow to the Department of Fluid Engineering and Instrumentation at Cranfield Institute of Technology (Cranfield University). Thanks are due also to A. Guilbert, R. Krafft and other colleagues at the department for helping with computation, and to Professor M. L. Sanderson for his support.
References 1
Bevir, M. K. Theory of induced voltage electromagnetic flowmeters
J. Fluid. Mech. 43 (1970) 577-590 2
Zhang, X. Z. A method for solving Laplace's equation with mixed boundary conditions in electromagnetic flowmeters J. Phys. D: Appl. Phys. 22 (1989) 573-576 3 Kantororich, L. V. and Krylov, V. I. Approximate Methods of Higher Analysis P. Noordhoff Ltd, Amsterdam (1958) 4 Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol II. Interscience Publishers, New York (1965)
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