Chapter 12
FLOW CONTROL
Flow through a plane gauze, or screen, is accompanied by a pressure
drop and, if the flow is not normal to the screen, by a flow deflection
toward the normal, much like the refraction of light when moving
from an optically less dense into an optically more dense medium.
Screens are usually woven wire, but may be cloth or may be
perforated plates or have other geometries. There are two effects to
be considered. One is attenuation of turbulence existing upstream
and the other is generation of new turbulence to be studied for its
own sake or for its effect on other phenomena, such as transition,
surface friction, or heat transfer. That is, make a non-uniform flow
uniform or vice versa. I propose not to become involved with turbu-
lence for its own sake, as this subject is very difficult and is covered
in monographs by Hinze, Batchelor, Townsend, Monin and Yaglom,
and elsewhere. (
Comment on curious identity of sizes of woven-wire
screens available from different manufacturers, as if they bought from
each other.
)
(
Look up the reasons why each author was interested in screens
to show wide applicability. Note that early wind tunnels had no con-
tractions; see paper by Prandtl. Mention Wright brothers, van der
Hegge Zijnen.
)
The earliest competent study of the behavior of screens is by
577
578
CHAPTER 12.
FLOW CONTROL
Figure 12.1: Flow through a screen. (Figure and caption
added by K. Coles.)
TAYLOR and BATCHELOR (1949).
Assume that the resistance of the screen depends only on the
component of velocity normal to its plane. If
p
2
−
p
1
is the pressure
drop, the loss coefficient for flow normal to the screen (
θ
= 0) will be
defined as
(consider including solidity; what happens? Note
that overall velocity decreases if there is a deflection; com-
pare to shock wave.) (Put solidity
s
in denominator.)
(Need
FIGURE X.)
1
C
n
=
p
1
−
p
2
1
2
ρQ
2
1
(12.1)
where
Q
,
u
,
v
are velocities in screen coordinates;
u
is the component
normal to the screen and is necessarily conserved; i.e.,
u
2
=
u
1
. The
component parallel to the screen is not conserved, being reduced by
the drag of the screen elements. Here a second loss coefficient can be
defined as
C
t
=
F
1
2
ρu
1
v
1
(12.2)
1
A sketch found in ms that may be the one cited is included here as Figure 12.1.
579
where
T
2
is the force per unit area in the plane of the screen and
the subscript
t
refers to the tangential component.
(Notation is a
problem. Can the Taylor and Batchelor argument be put in
vector form?)
Note that both coefficients are designed to be of order
unity, although they can also be expected to depend on solidity as
well as on Reynolds number, Mach number, and geometrical details.
The relationships
u
1
=
Q
1
cos
φ
1
u
2
=
Q
2
cos
φ
2
v
1
=
Q
1
sin
φ
1
(12.3)
v
2
=
Q
2
sin
φ
2
and the tangential momentum equation
T
=
ρu
1
(
v
1
−
v
2
)
(12.4)
allow equation (12.2) to be put in the form
C
t
= 2
(
1
−
cos
φ
1
sin
φ
1
sin
φ
2
cos
φ
2
)
.
(12.5)
This coefficient
C
t
is called
F
θ
/θ
by Taylor and Batchelor. The anal-
ogy with optics can be made explicit by writing
n
=
sin
φ
1
sin
φ
2
(12.6)
in which case, to first order in
θ
(
give also exact expression
),
C
t
= 2
(
n
−
1
n
)
(12.7)
(
Cite experiments on
φ
2
vs
φ
1
, especially Schubauer, Spangenberg,
and Klebanoff. Mention experimental setup. Mention fit
F
θ
/θ
= 2
−
2
.
2
/
(1 +
k
θ
)
1
2
and T & B’s figure 5. See JAS.
)
2
T
may be same as F in 12.2, i.e., a transcription error from ms.
580
CHAPTER 12.
FLOW CONTROL
The proper geometric parameters for analyzing screen prefor-
mance are the solidity
s
and the index of refraction
n
. For a square
wire-mesh screen, the solidity is defined by the sketch;
s
=
blocked area
total area
=
2
dD
−
d
2
D
2
= 2
d
D
−
d
2
D
2
(12.8)
The resistance coefficient
C
n
can be expected to increase with
increasing solidity, as in sketch
A
.
There is a weak Reynolds-number effect, as indicated in sketch
B
, which can be expected to look like the drag coefficient of a cylin-
der. Finally, there is a Mach number effect, as shown in sketch
C
(if
density changes are appreciable, take them into account).
(Combine to reproduce figure 5 of Taylor and Batchelor. Can
solidity effect in
A
be estimated by adding up cylinders? See Wieghardt.)
(Structural strength is a factor; work out some details. Differ-
ent companies sell the same screens. Better to put screen in low-
velocity region, for sake of lower loads and lower losses. Keep Re
based on stream velocity and wire diameter below shedding frequency.)
(In handout, note better scheme used by Dryden and Schubauer
for determining angle; Simmons and Cowdrey were clumsy.)
To determine the effect of the screen on a small disturbance in
the oncoming flow, linearize the problem. Suppose that the upstream
flow is two-dimensional, with a perturbation that depends only on
y
;
say
(now
U
, not
Q
; comment)
U
1
=
U
0
+
u
1
cos
κy .
(12.9)
This flow is assumed to be normal to the screen, and
u
1
is now
the amplitude of the perturbation. If viscosity is neglected, except
perhaps in the close vicinity of the screen, the vorticity is constant
on streamlines;
Dζ
Dy
= 0
.
(12.10)
In the two-dimensional flow, define a stream function
ψ
, with
ζ
=
−∇
2
ψ .
(12.11)
581
To first order, therefore,
∂
∇
2
ψ/∂x
= 0, and
∇
2
ψ
=
f
(
y
)
.
(12.12)
Note that this analysis is essentially for a two-dimensional
screen; in three dimensions the vortex-stretching terms would ap-
pear and also there would be no stream function. But see pp. 11–12
of Taylor and Batchelor for a counter-argument. Far upstream,
ζ
=
−
∂U
1
∂y
=
κu
1
sin
κy
(12.13)
and in general, for the perturbation,
∇
2
ψ
1
=
−
κu
1
sin
κy .
(12.14)
The solution, easily obtained by separation of variables, is of the form
ψ
(
x, y
) =
Ce
±
κx
sin
cos
κy .
(12.15)
To this must be added the particular solution. For an anti-symmetric
flow, with
ψ
(
x,
0) = 0 and with the upstream boundary condition
(12.9),
ψ
1
=
(
u
1
κ
+
Ae
κx
)
sin
κy .
(12.16)
A similar argument for the downstream region gives
ψ
2
=
(
u
2
κ
+
B e
−
κx
)
sin
κy
(12.17)
where
U
2
=
U
0
+
u
2
cos
κy .
(12.18)
Three conditions are needed to determine
A
,
B
, and
u
2
/u
1
.
First, the component
u
=
∂ψ/∂y
must be continous. At the screen
x
= 0, therefore,
u
1
+
κA
=
u
2
+
κB
=
u
s
,
say
.
(12.19)
The meaning of the quantity
u
s
(for screen) is indicated in
the sketch. Where the stream velocity is higher, more resistance will
582
CHAPTER 12.
FLOW CONTROL
be encountered, and the stream will diverge and reach the screen
at an angle
(how does linearization prevent appearance of
sin 2
κy
, etc.?)
. The amplitude
u
1
will decrease to
u
3
; see below.
Since
n
= sin
φ
1
/
sin
φ
2
=
v
1
/v
2
, the
v
-components across the screen
are related by
v
2
=
−
∂ψ
2
∂x
=
v
1
n
=
−
1
n
∂ψ
1
∂x
(12.20)
from which
A
+
nB
= 0
.
(12.21)
The final condition is obtained from Bernoulli’s equation. To
first order, with
U
1
=
U
2
=
U
, the total pressures differ by the loss
at the screen;
p
1
+
ρ
2
(
U
2
+ 2
U u
1
cos
κy
)
−
p
2
−
ρ
2
(
U
2
+ 2
U u
2
cos
κy
)
=
=
C
n
ρ
2
(
U
2
+ 2
U u
s
cos
κ y
)
.
(12.22)
This becomes, after cancelling terms of order unity,
u
1
−
u
2
=
C
n
u
s
.
(12.23)
When
A
,
B
, and
u
s
are eliminated from equations (12.19),
(12.21), and (12.23), the result is a formula for attenuation;
u
2
u
1
=
1 +
n
−
C
n
1 +
n
+
nC
n
.
(12.24)
(Note that this result is independent of
κ
and that the numera-
tor may vanish. There is no practical upper limit on
C
n
. If
n
= 3
/
2
,
then
C
n
= 5
/
2
removes the upstream perturbation completely. Note
that
v
-perturbations are attenuated by a factor
1
/n
. This might be
neater in a vector notation.)
The velocity
u
s
can be expressed in two ways:
u
s
u
1
=
1 +
n
1 +
n
+
nC
n
(12.25)
u
s
u
2
=
1 +
n
1 +
n
−
C
n
(12.26)
583
from which it is obvious that
u
1
> u
s
> u
2
.
(12.27)
In all of this, the screen is assumed to have no structure, so
that the effect on the turbulence spectrum is not treated. For a screen
with high resistance, there is a high price in power required. The
method has been used in diffusers (
comment on effect on bound-
ary layer or on separated region
) and is worth the cost if a good
downstream flow is essential. The same kind of analysis can be used
for turning vanes. Compare S-duct in Lockheed 1011, Boeing 727.
Reprise of Taylor and Batchelor; see 20 April, 22 April.
Glauert et al., R & M 1469, 1932
Flachsbart, IV Lief., 112, 1932
Ower and Warden, R & M 1559, 1934
Collar, R & M 1867, 1939
Eckert and Pfluger, Lufo. 18, 142, 1941
Czarnecki, WR L-342, 1942
Simmons et al., ARC, 1943
Taylor, ARC R & M 2236, 1944
Simmons and Cowdrey, R & M 2276, 1945
Adler, NACA, 1946
Dryden and Schubauer, JAS 14, 221, 1947
Kovasznay, Cambr. Phil. 44, 58, 1948
Dryden and Schubauer, appendix, 1949
Hoerner, AFTR, 1950
Schubauer et al., TN2001, 1950
Baines and Peterson, ASME 73, 467, 1951
Weighardt, AQ 4, 186, 1953
MacDougall, thesis, 1953
Annand, JRAS 57, 141, 1953
Grootenhuis, PIME 168, 837, 1954
Gedeon and Grele, RM, 1954
Dannenberg et al., TN 3094, 1954
Tong, Stanford, 1956
Tong and London, ASME 79, 1558, 1957
Davis, thesis, 1957
584
CHAPTER 12.
FLOW CONTROL
Cornell, ASME 80, 791, 1958
Morgan, JRAS 63, 474, 1959
Morgan, JRAS 64, 359, 1960
London et al., ASME 82, 199, 1960
Siegel et al., TN D2924, 1965
Morgan, Austr. A347, 1966
Pinker and Herbert, JMES 9, 11, 1967
Blockley, thesis, 1968
Luxenberg and Wiskind, CWRV, 1969
Reynolds, JMES 11, 290, 1969
Rose, JFM 44, 767, 1970
Valensi and Rebont, AGARD, 27, 1970
Castro, JFM 46, 599, 1971
Carrothers and Baines, ASME 97, 116, 1975
Bernardi et al., ASME 98, 762, 1976
Graham, JFM 73, 565, 1976
Richards and Norton, JFM 73, 165, 1976
Murota, WES, 105, 1976
Durbin and Muramoto, NASA, 1985
Shaped Screens
Taylor, ZAMM 15, 91, 1935.
Okaya and Hasegawa, JJP 14, 1, 1941.
Stevens, ARC, 1942.
Taylor and Davies, R & M 2237, 1944.
Squire and Hogg, RAE, 1944.
Taylor and Batchelor, QJMAM 2, 1, 1949
Townsend, QJMAM 4, 308, 1951.
Bonnerville and Harper, thesis, 1951.
Lockard, thesis, 1951.
Owen and Zienkiewicz, JFM 2, 521, 1957.
Elder, JFM 5, 355, 1959
.
Brighton, thesis, 1960.
Mc Carthy, JFM 19, 1964.
Livesey and Turner, IJMES 6, 371, 1964.
Davis, Austr 191, 1964
.
Livesey and Turner, JFM 20, 201, 1964.
585
Vickery, NPL 1143, 1965.
Rose, JFM 25, 97, 1966.
Cockrell and Lee, JRAS 70, 724, 1966.
Livesey, JRAS 70, 1966.
Kotansky, AIAA 4, 1490, 1966
.
Uberoi and Wallace, PF 10, 1216, 1967.
Lau and Baines, JFM 33, 721, 1968.
Mobbs, JFM 33, 227, 1968.
Reynolds, JMES 11, 290, 1969
.
Turner, JFM 36, 367, 1969.
Cockrell and Lee, AGARD, 13, 1970.
Maull, AGARD, 16, 1970
.
Hannemann, thesis, 1970.
Durgin, MIT, 1970.
Champagne et al., JFM 41, 81, 1970.
Kachhara, thesis, 1973.
Sajben et al., AIAA, 1973.
Koo and James, JFM 60, 513, 1973.
Livesey and Laws, AIAA 11, 184, 1973.
Livesey and Laws, JFM 59, 737, 1973
.
Livesey and Laws, CES 29, 306, 1974.
Laws and Livesey, ARFM 10, 247, 1978
.
Tan-atichat, thesis, 1980.
Scheiman, NASA, 1981.
A gauze, or screen, or grid, is a high-drag device that is nor-
mally used to redistribute the flow in a channel. Other uses include
prevention of separation in diffusers, generation of turbulence, and
reduction of turbulence, depending on the properties of the screen.
(See Taylor and Batchelor for turbulence reduction.)
The basic problem considered by ELDER (
ref
) is modification
of flow in a straight channel by a single shaped screen located in the
vicinity of
x
= 0.
The coordinates are (
x, y
), and the corresponding velocities
are (
U, V
). The flow is uniform but rotational far upstream and far
downstream, and the effect of the screen is to introduce a discontinu-
ity in vorticity at
x
= 0. The screen has no structure and is treated
586
CHAPTER 12.
FLOW CONTROL
Figure 12.2: Flow in a channel modified by a single
shaped screen. (Figure and caption added by K. Coles.)
like an actuator sheet.
Suppose that the flow is rotational but steady, incompressible,
inviscid, and two-dimensional. Start with the flow in the sketch.
3
A
subscript 1 or 2 denotes upstream conditions or downstream con-
ditions, respectively. Denote by a superscript 0 the one-dimensional
flow that coincides with the initial or final state far from the screen;
typical variables are
ψ
0
,
U
0
=
∂ψ
0
/∂y
,
ζ
0
=
−
∂U
0
/∂y
.
In the two-dimensional flow, the continuity equation is satisfied
if
~
U
=
∇
ψ
×∇
z
(12.28)
3
Two sketches found in ms that appear related to this discussion, the first of
which may be the one cited, are included here as Figure 12.2.
587
where
ψ
is a stream function. Taking the curl yields
ζ
=
−∇
2
ψ .
(12.29)
The conditions already stated also imply
Dζ
0
Dt
=
~
U
0
·
grad
ζ
0
.
(12.30)
For the flow in the sketch,
Dζ
0
Dt
=
U
0
d
ζ
0
d
x
= (
ζ
0
2
−
ζ
0
1
)
δ
(
x
)
.
(12.31)
However, the velocity cannot be continuous, because
∂U
0
/∂y
is dif-
ferent for the upstream and downstream regions. It is necessary to
add another flow near the screen. If the basic flow carries the vortic-
ity, the proper composition is
∇
2
ψ
=
∇
2
ψ
0
+
∇
2
ψ
′
(12.32)
where the perturbation
ψ
′
is irrotational;
∇
2
ψ
0
=
−
ζ
0
,
(12.33)
∇
2
ψ
′
= 0
.
(12.34)
The assumptions are:
1. The jump in vorticity is carried by the basic flow
ψ
0
.
2. The condition that
U
is constant through the screen is carried
by the combined flow.
3. A jump in
V
, to implement the jump in
ζ
, is carried by the per-
turbation flow. In particular, both
ψ
0
and
ψ
′
are discontinuous
at the screen, but the sum is continuous.
The solution of the equation
∇
2
ψ
′
= 0 in rectangular coordi-
nates is easily obtained by separation of variables. The solution can
be written in dimensionless form for the upstream region
ψ
′
1
L
U
=
∑
1
mπ
P
m
e
mπ x/L
sin
mπ
y
L
(
x <
0)
(12.35)
588
CHAPTER 12.
FLOW CONTROL
and for the downstream region,
ψ
′
2
L
U
=
∑
1
mπ
Q
m
e
−
mπ x/L
sin
mπ
y
L
(
x >
0)
(12.36)
where
L
is the channel width and
U
is the mean velocity over the
cross section,
U
=
1
L
∫
U
d
y .
(12.37)
It follows from the geometry that this velocity is the same far from
the screen in both directions.
The flow represented by
ψ
′
vanishes at
x
=
±∞
. The form
automatically satisfies the requirement that the flow follow the two
walls, since
ψ
′
= 0 at
y
= 0 and at
y
=
L
. Note that there is no net
flow, so that the condition
U
1
=
U
2
must be satisfied by the primary
flow.
The velocity components associated with
ψ
′
1
are
U
′
1
U
=
∑
P
m
e
mπx/L
cos
mπ
y
L
,
(12.38)
V
′
1
U
=
−
∑
P
m
e
mπx/L
sin
mπ
y
L
,
(12.39)
and for
ψ
′
2
are
U
′
2
U
=
∑
Q
m
e
mπx/L
cos
mπ
y
L
,
(12.40)
V
′
2
U
=
∑
Q
m
e
−
mπx/L
sin
mπ
y
L
.
(12.41)
The screen properties are referred to screen coordinates, as
shown in the sketch;
The velocity normal to the screen is
u
1
=
U
1
cos
θ
−
V
1
sin
θ ,
(12.42)
u
2
=
U
2
cos
θ
−
V
2
sin
θ .
(12.43)
Since
u
1
=
u
2
, it follows that
(
U
1
−
U
2
) = (
V
1
−
V
2
) tan
θ .
(12.44)
589
The velocity parallel to the screen is
v
1
=
U
1
sin
θ
+
V
1
cos
θ ,
(12.45)
v
2
=
U
2
sin
θ
+
V
2
cos
θ ,
(12.46)
from which
v
1
−
v
2
= (
U
1
−
U
2
) sin
θ
+ (
V
1
−
V
2
) cos
θ
(12.47)
or, in view of equation (12.44),
v
1
−
v
2
=
V
1
−
V
2
cos
θ
.
(12.48)
Elder notices that the combination (his BUT)
G
=
(
v
1
−
v
2
v
1
)
U
1
sin
θ
cos
θ
(12.49)
can be developed by using equation (12.45) to eliminate sin
θ
;
G
=
(
v
1
−
v
2
)
v
1
(
v
1
−
V
1
cos
θ
)
cos
θ
=
(
v
1
−
v
2
)
cos
θ
−
v
1
−
v
2
)
v
1
V
1
(12.50)
=
(
V
1
−
V
2
)
cos
2
θ
−
(
v
1
−
v
2
)
v
1
V
1
where the last step requires equation (12.48). When terms in
V
1
and
V
2
are collected and the identity 1
/
cos
2
θ
= 1 + tan
2
θ
is used, this
becomes
G
=
V
1
[
1
−
(
v
1
−
v
2
)
v
1
+ tan
2
θ
]
−
V
2
(
1 + tan
2
θ
)
(12.51)
or finally, if tan
2
θ
is neglected on the right-hand side,
U
1
is replaced
by
U
, and the original form (12.49) is restored,
(
v
1
−
v
2
)
v
1
U
tan
θ
=
[
1
−
(
v
1
−
v
2
)
v
1
]
V
1
−
V
2
.
(12.52)
590
CHAPTER 12.
FLOW CONTROL
In terms of the index of refraction, with the approximation
sin
φ
= tan
φ
and the condition
u
1
=
u
2
,
tan
φ
1
=
v
1
u
1
= sin
φ
1
,
tan
φ
2
=
v
2
u
1
= sin
φ
2
(12.53)
and therefore
v
1
=
nv
2
. Equation (12.52) becomes
(
n
−
1
n
)
U
tan
θ
=
V
1
n
−
V
2
.
(12.54)
This expression defines the jump in
V
at the screen when the screen
properties
n
and
θ
are specified. The approximations include neglect-
ing tan
2
θ
compared to unity, replacing tan
φ
by sin
φ
, and replacing
U
1
by
U
.
When the right-hand side of equation (12.54) is rewritten using
equations (12.39) and (12.41) with
x
= 0, the result is
(
n
−
1
n
)
tan
θ
=
∑
(
−
P
m
n
−
Q
m
)
sin
mπ
y
L
.
(12.55)
This relation provides the coefficients in a Fourier series for tan
θ
if
the coefficients
P
m
and
Q
m
are known. (
Define
n
for a honey-
comb.
)
It remains to express the jump in pressure in the same way.
The momentum equation can be written
−
1
ρ
∇
p
=
∇
~u
·
~u
2
+ (
curl ~u
)
×
~u .
(12.56)
The
y
-component of this equation is
−
1
ρ
∂p
∂y
=
∂
∂y
(
U
2
+
V
2
2
)
+
ζU .
(12.57)
Far upstream and downstream of the screen (
close to screen?
)
−
1
ρ
∂p
1
∂y
=
∂
∂y
U
2
1
2
+
ζ
1
U
1
(12.58)
−
1
ρ
∂p
2
∂y
=
∂
∂y
U
2
2
2
+
ζ
2
U
2
(12.59)
591
and therefore
1
ρ
∂
∂y
(
p
1
−
p
2
) =
∂
∂y
(
U
2
2
−
U
2
1
)
2
+
ζ
2
U
2
−
ζ
1
U
1
.
(12.60)
This condition should be applied at the screen, and the approxima-
tion is made that streamline displacements are small. If the second
term is discarded, on the ground that
U
1
=
U
2
=
U
approximately,
then
1
ρ
∂
∂y
(
p
1
−
p
2
) =
ζ
2
U
2
−
ζ
1
U
1
.
(12.61)
By definition,
p
1
−
p
2
=
1
2
ρu
2
C
n
(12.62)
where
u
=
u
1
=
u
2
. Moreover, from (12.42),
u
=
U
cos
θ
approxi-
mately. Substitution gives
1
2
∂
∂y
U
2
cos
2
θC
n
=
U
(
−
d
U
0
2
d
y
+
d
U
0
1
d
y
)
(12.63)
where it is also assumed on the right that
U
1
=
U
2
=
U
. One inte-
gration gives
1
2
U
2
cos
2
θC
n
=
U
(
U
0
1
−
U
0
2
)
+
C
(12.64)
where
C
is a constant of integration. Put
U
=
U
(1 +
)
(12.65)
to obtain
1
2
U
(1 +
) cos
2
θC
n
=
U
(
U
0
1
−
U
0
2
)
+
C
U
(1
−
)
.
(12.66)
A second integration from
y
= 0 to
y
=
L
, with cos
2
θ
treated as
constant, and with
∫
L
0
d
y
= 0
(12.67)
592
CHAPTER 12.
FLOW CONTROL
by virtue of equation (?)
4
, gives
C
=
U
2
2
C
n
.
(12.68)
Equation (12.64) can therefore be written, to leading order, and with
cos
2
θ
taken as unity,
(
U
−
U
)
C
n
2
=
U
0
1
−
U
0
2
.
(12.69)
The stage is now set for the solution of equations (?)
5
and
(12.69) above. Continuity of the streamwise velocity at the screen
requires (Elder’s Eq. 2.5)
U
0
1
+
U
∑
P
m
cos
mπ
y
L
=
U
0
2
+
U
∑
P
m
cos
mπ
y
L
.
(12.70)
The difference in the
V
-component across the screen is
V
′
1
−
V
′
2
=
−
U
∑
(
P
m
+
Q
m
) sin
mπ
y
L
.
(12.71)
(
There is some missing algebra here.
) After some algebra,
there is obtained
(
U
0
2
U
−
1
)
(
n
+ 1 +
n
C
n
2
)
n
C
n
2
−
(
U
0
1
U
−
1
)
(
n
+ 1
−
C
n
2
)
n
C
n
2
(12.72)
=
−
∑
α
m
cos
mπ
y
L
= 0
(12.73)
where
α
m
=
P
m
n
+
Q
m
.
(12.74)
Equation (12.73) has to be compared to equation 12.55, which
can be written
(
n
−
1
n
)
U
tan
θ
=
−
∑
α
m
sin
mπ
y
L
.
(12.75)
4
Equation number not recorded
5
Equation number not recorded
593
Equations (12.73) and (12.75) define two functions expressed as Fourier
series. The series have the same coefficients, but one is in terms of
sin
nπ y/L
and the other is in terms of cos
nπ y/L
. There is a theo-
rem due to Hardy that applies to this situation. Given
g
(
θ
) =
∑
h
m
sin
mθ
(12.76)
g
∗
(
θ
) =
∑
h
m
cos
mθ
(12.77)
valid in the interval 0
< θ < π
, it follows that
g
∗
=
H
(
g
)
, g
=
H
∗
(
g
∗
)
(12.78)
where
H
(
g
) =
1
2
π
∫
π
0
[
g
(
θ
+
t
)
−
g
(
θ
−
t
)] cot
t
2
d
t .
(12.79)
(
Look up the theorem and describe the symmetry. Did this theorem
drive Elder’s analysis, or was it discovered in time to save the analy-
sis? Ask him?
) The theorem connects the screen angle to the veloci-
ties upstream and downstream. The analysis should reduce for
θ
= 0
and
m
= 1 to the Taylor-Batchelor formula.
(There is an error in Elder’s application of his analysis; see
Livesey and Laws and others.)