The steady, inviscid flow through and around a screen inclined at a uniform angle to the incoming flow was investigated. For a screen placed in an infinite flow field, an asymptotic analysis for small resistance coefficients was performed, and the effects of inclination were determined. The velocity at first order in the asymptotic expansion was nonuniform along the screen. This nonuniformity caused the wake behind the screen to contain distributed vorticity at second order. These effects therefore occurred at one order lower than for normal screens

Durbin, P. A.

Muramoto, K. K.

English

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NASA Technical Memorandum 86979
A	 k
N	 el
Flow Through Very porous
Inclined Screens
(/AS1-T M-( 697:)	 FLCk THFCUGH VERY FORCUS	 Ne5-257E1
11CLIM) SCHINS IN1SJ)	 10 E EC 102/MF 101
*	 C SC L 2CD
Unclas
G3/34 21137
Kenneth K. Muramoto and Paul A. Durbin
Lewis Research :.inter
Cleveland, Ohio
It.
It
Prepared for the
18th Fluid Dynamics and Plasmadynamics and Lasers Conference
sponsored by the American Institute of Aeronautics and Astronautics
Cincinnati, Ohio, July 16-18, 1985
^^	 1
A^z ^q
I
i
I	 ;
teady, inviscid flow through and around a
clined at a uniform angle to the incoming
investigated. For a screen placed in an
flow field, an asymptotic analysis for
small resistance coefficients was performed, and
the effects of inclination were determined. The
velocity at first order in the asymptotic expansion
was nonuniform along the screen. This nonuniform-
ity caused the wake behind the screen to contain
distributed vorticity at second order. These
effects therefore occurred at one order lower than
for normal screens.
Introduction
A screen is a device that can be used to modify
the velocity distribution of a fluid flow in a
controlled manner. When fluid passes through a
screen, the static pressure of the flow is reduced
and the oncoming streamlines are deflected toward
normal to the screen. These properties have
p.	 ted researchers to treat a screen as a surface
of c, :ontinuity in the flow.
The first analysis of the flow around a very
porous screen was carried out by Taylor (1963).
He modeled the screen as a sheet of uniformly dis-
tributed sources. Although only the flow field
outside the wake can be determined wit ►. his model,
Taylor's formula for the drag coefficient agrees
with experiment for values of the resistance coef-
ficient k up to approximately 4.
This distributed source analysis was later
extended by Koo and James (1973). In their mathe-
matical model, a wake is introduced by making
certain assumptions about the streamline pattern
behind the screen. Along with numerical solutions
to their model, Koo and James also presented some
closed-form results for both normal and inclined
screens. However, their analytical results are
solutionc not to their model but to an approximate
form of their model. Although Koo and James'
solutions are at best formally accurate to the same
degree as Taylor's, viz, 0(k), their expression for
the drag coefficient agrees with experimental val-
ues for k as large as 10.
A formal solution to the governing equations
for steady and oscillatory flows through a normal
screen has recently been given by Durbin and
Muramoto (1985). They carded out an asymptotic
expansion accurate to 0(k ) and found that at
this order the effects of tangential drag and wake
vorticity have entered into the solution. A Pad6
approximant to their drag formula also agrees with
experiment for k as large as 10. In this paper,
the analysis of Durbin and Muramoto is extended to
consider the steady flow through a screen inclined
to the oncoming flow. Only the special case of a
finite screen in an infinite flow field is
considered.
iolla:ing Durbin and Muramoto (1985). nondi-
mensional variables will be used. Thus, the fluid
density, the projected screen half-height, and the
far-upstream velocity are assumed to be unity. For
steady, inviscid. incompressible flow, the conti-
nuity and momentum equations are
au +
 av0
ax	 ay
ax	 ay	 Ti
u av +v av^ - aP
ax	 ay	 ay
These governing equations must be solved, subject
to the following jump and boundary conditions. The
pressure drop across the screen [P] is generally
expressed by a dimensionless resistance coefficient
k (Taylor, 1963), defined by
[P] . 2 kU 2
	(2)
where U is the velocity normal to the screen.
Since mass flow through the screen must be
conserved,
[U] . 0
	 (3)
Because the screen deflects the streamlines, there
is a jump in the tangential velocity [V]. given by
[V] . BV,
	
(4)
where B is the tangential resistance coefficient
and V+ is the tangential velocity on the upstream
side of the screen. If we consider a screen placed
in a uniform flow field, the upstream boundary con-
dition is u - 1 and v • 0 as x • - . Note that
lower case u and v refer to x,y components,
while upper case refers to normal and tangential
components.
Asymptotic Analysis
Let the location of the screen be given by the
equation s - x(y). Here we will take
x. ,gn(y)ay
	 -1 S y 5 1	 (5)
and thus depending on the sign of a taken, both
forward-tilting (z > 0) and backward-tilting (a < 0)
screens ca , 
	 considered. According to equa-
tion (5), the unit vectors normal and tangent to
the screen are, respectively,
T
i - sgn(y)aj
1+a2
L	 ai + s9n(y)j
l+a
FLOW THROUGH VERY POROUS INCLINED SCREENS
Kenneth K. Muramoto and Paul A. Durbin
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
Abstract	 Governing Equations
n
LnN
L,
7
t	 r
r s l
We seek a solution to the problem by expanding
the dependent variables in powers of k, that is
u. uO + ku I + k 2u 2 + 	.
V . v O + kv I + k 
2 
v 2 +	 (7)
P . PO + kP 1 + k 2P 2 +	 .
Similarly, we will assume that B can be written
in the form
	
B . kB I + k 2 a, + k 38 3 + . . .
	 (8)
By inspection, the zeroth-order solution is simply
u0- 1, v0 . 0. PO . 0	 (9)
Proceeding to the first-order problem, a formal
solution must be obtained to the following
equations:
au	 av
a l + ay . 0
	 (10a)
x 
a p I 	 aul
ax - ax	
(lOb)
with [U1] - 0. [ V I] - 81V	 [P1]	 112 ua.
Integration of equation (Q0b) yields
	
u 1 . -P I + f l ( y )	 (11)
where f 1 (y) is an arbitrary function of y,
except for a jump across the screen. Now since
P1 satisfies Laplace's equation, we may introduce
the harmonic function QI conjugate to P1.
Therefore, by the Cauchy-Riemann equations and the
boundary conditions to the problem
y r . Q I
	(12)
To determine the unknown quantities f 11 (y) and
[Q 1], the jump conditions on U and V at the
screen will be used. Substituting equations (11)
and (12) into (3) and (4) gives the following pair
of algebraic equations:
[ f 1 (Y)] nx + [Q I ] ny - [PI]nx
(13)
[ f I (Y)] t x + [Q 1 ]ty . [P 1 ]t x + 81V0
where nx n i, n  . n j, t x . t • i, and
ty - t • 3. Substituting [PI] . nJ12 and VO . tx
and after using equation (6) for n and t
gives the solution to the system of equations (13):
1 + 2a2B
aB l
 sgn(Y)
i*a
Since the jumps [P I] and [Q I ] are now
known, the sectionally analytic function P1 + iQl
can be determined according to the Plemelj formulas
(Roos, 1969):
I	 [PI]	 I
P I * iQI . T7 
-^ 1— z dz + ,	 T z d 
1 —^rZ - ' 1 1
	I 	 aa
	L 	 +	 J4w1(1 + a )
+aB1 In (z2 - 2az + 1 + a2/ (16)2 w(1 + a 2
	 \	 z
where the branch cuts of the logarithms are along
the screen. Hence, by equations (11) and (12),
u 
	
ivl
 • -	
r
In z + a
s +	 ]
	
4wi(1 + a )	 Lz
- a^In Cz2-2az+ 1 + a2
/
	
2w(l + a )	 z
- C(a)f(y)
	
(17)
1 + 2a2B
C(a) . 2(2
	
(18)
and
0, x<ay
f (Y) -	 0, x > ay. IYJ > 1	 (19)
	
1,	 x > ay, J Y J < 1
Note here that the velocity given by equation (17)
has logarithmic singularities at the edges of the
screen z . 0, a + i. As in the case of a normal
screen, vortici + y is absent inside the wake at this
order because f is constant there:	 orticity is
confined to a sheet at Jyl . 1. On the other
hand, an inclined screen generates a first-order
velocity whose normal component is not constant
along the screen.
To avoid a nonuniformity in the asymptotic
expansion at the edge of the wake, we will follow
the method used by Durbin and Muramoto (1985) and
replace f(y) by f[y - sgn(y)s(x)] with s .
ksl + k 2 s2 *	 . . so that the wake boundary
lies on y - 1 + s(x). This transformation does
not affect the first-order solution given by
equations (17) to (19).
Moving on to next order, the following second-
order equations may be derived:
2u2
ax + ay
	
. 
0	 (20a)
	
a p t 	 avl	 where
	
- +y	 ax	 (10^ )
^.J
to
• n
2
+1
a P 
2	
3u2	ds1 
of	 au1	
au1
- ax ax + C(a) T;-- liy I + u  ax + v l ay
(20b)
3P  3v2
	
av1
	
3v1
	
Ty- ` ax + u l ax + vI ay	
(20c)
To eliminate the singularity at Jyj . 1 associated
with the au 1 /ay term in equation (20b), we are
required to £ake vl(x,l) . dsl/dx. From equation
(17), the first-order wake boundary is
Y
wake ` 1 + ks1(x)
. 1 + k - —^ (x - a)ln	 x - a
2
8.(1 + a )	 [4+(x-a)]
2
• a In 4
—°	 - 4 Itan-1 (L-- +tan-1 (! )]^
 )
+ a1—^ (x - a) tan-1
 ( x 2 a) -a tan -1 (a )
2.(1 + a )
)2
- 2x tan 1 	
+
(')
 + In 4	 x+
1+x2
- ln(4 + a2 )]	 (21)
If we let H 2 - P 2 + 112(ui + vi), the second-
order momentum equations (20) can be written in the
form
aH2	 au 
- ax	 ax	
(22a)
-	
-
aN2	 av2 +
	 of
C(a)u	 (22b)
ay	 ax	 1 ay
Equation (22a) can be integrated immediately to
yield
	
u2 . -H2 + f2 ( y )	 (23)
where f 2 (y) has a jump across the screen. From
equation (22) it is evident that H2 satisfies
Laplace's equation everywhere except for a jump
across the vortex sheets of
	
(H 2) . C(a)u 1 ( x .l y l - 1) sgn (Y)	 (24)
where the angular brackets denote a jump across the
sheets and ul(x,JyJ . 1) stands for the average
of the velocities above and below lyl . 1.	 It
follows from equation (17) that
(H 2 ) . -C(a) [.7 C( a) + P 1 ( x,l) I sgn (y)	 (25)
In a manner analogous to what was done at first
order, we introduce the function R2, the harmonic
conjugate to H 2 . By the Cauchy-Riemann equations,
one can easily show that
	
V2 . R2
	(26)
The jump of R 2
 across the vortex sheets can be
determined after removing the nonuniformity appearing
in the equations at third order. Thus, to remove the
nonuniformity, it is necessary to take
ds 2	dsl	 av1
	
V2 ` T + ul ax - s l ay	 (27)
Using equation (26) in (27), the jump of R2
across the upper vortex sheet is just
ds1
(R 2 ) - (u 1)xa
- C(a)vI(x,l)
- C(a)Q I (x.1)	 (28)
since none of the other terms on the right side of
equation (27) have jumps. A similar expression can
be deriveo for the jump across the lower sheet.
The jumps of H2 and R2
 across the screen
can also be found in a straigh t forward fashion.
The result for H2 is
[ " 2] s [P2J + 2 [u 1 ] + 2 [v1]
U 0 1 + u lsLu l ) + vls[v1i
` u ls nx + vls n x nY + u ls B l tx + v1s81txtY
1 + a281  
1—
^)
uls +	 a s9n(Y)vls
(2g)
where u ls and v1s denote the average of the
velocities in front of and behind the screen. By
equation (17), uls and v 	 are functions of
y. Inserting equations (23}s and (26) into the two
jump conditions (3) and (4) provides two algebraic
equations for the unknowns 1f2(y)J and [R2]:
	
[f2(Y)]nx + [R 2 ]ny ` [H2]nx	
^ (30)
L f 2 (Y)]t x + L R 2 ] ty . [H 2 ]t , + " ' I+ + YO
Hence,
81-1
[f2(Y)] - 2C(a)u ls +
(21+
— r
 a sgn(y)vIs
a
a 2 B 2 + 2B2
	
+	 (31)
2(1 + a )
3
QO7 Mt.
[ R2 ] - s n
	 [Bl[auls + sgn(y)vls]l+a
C - \1 + 2a261
l k ( a 2g 1 [66 1 (l _ a 2 ) - 81 - 1
D I 
+
1	 a	 4(1 * a2)
+ I a (Bi + 262
)1
	 (32)
Because u 55 and v 1	are functions of y, so is
f 2 . Since l the downstream profile is given by u.
	 1
- [fl]k - ([f ] - (HZ)x.)k 2 - .	 , the wake at sec-
ond order contains distributed vorticity. This
should be contrasted with the normal screen result,
where there is no vorticity in the wake until third
order in the asymptotic expansion (Durbin and
Muramoto, 1985).
By applying the Plemelj formulas, one can in
principle determine a sectionally analytic function
H2
 + iR2 with u s (25), (28), (29), and (32)
and then find ^P	 Because a Pads appioximant
to a two-term asymptotic expansion proviues a very
good representation of the drag coefficient over a
large range of k (Durbin and Muramoto, 1985), we
will not be concerned here with calculating these
higher order contributions. The total drag and
lift coefficients based on the projected area of
an inc li-d screen are, respectively,
C  . f ([P]cos a + U[V] sin a) ds
- Cos a	 1 ^k[P 1 ] + ^ 2[P 2 ] +	 .^
x Nrl + a 2 dy + sin s
x	 JkUOLV13 + k2 ( UO [ V2 ] + U 1 [ V 1 ]) + . ..}
x Nrl + a dy	 (33)
C U . 2 J 	 J-[P]	 ))) sin a + U[V] cus a ds^
1
1
- -2 sin a i I k[Pl] + 
k2[P 2 ] +
	+ 2a 2B2	aB1(1 + 2a281) 
In	
4	
l
* 7	 ^1 + z1	 a	 2.(1 + a 2 )	 a
	
a28
1	 1
(1 + 2B ) + 1 r
_	 \^ _ tan lal
.(1 + a2)
2a 2B 1 (1 + B 1 + a 2B 1 ) + 1
2.(1 + a2l)
x \- - 2 tan -1
a)) 
k 2	 (35)
l+a J
I k
(a[1 - 2B 1 (1 - 3a 2 ) + 6B2 ( 1 - a2)]
+	 40 + a2)
+ 2aB 2 + 6 1 (1 + , a 2 - 2a281) In
	
4
1 + a2	 2.(1 + a2)	 `1 + a7)
all - 6 - 28 1+	 1	
(*
	
tan-la
w(1 + a2)
a[1 + 2B 1 (a 2
- B 1 - a2B1)]
_	 2*(l	
a2)^
2 tan- 
la/J 
k 2	 (36)
The formul s reduce to CD - K - k 212 and CC .
(81 In 2)k /x when a • 0.
If we assume that the deflection coefficient De
(Taylor and Batchelor, 1949) is independent of the
angle of incidence and that the following formula
can be used,
x1 + a2 dy + 2 cos a
x •1 1 S kU O[ V I] + k2 ( UO [ V 2] + UI[ V I ]) + . . .^
0	
1
x	 1 + a 2 dy	 (34)
in which a - tan-l a. It should be pointed out
that the CL defined in equation (34) is the
lift coefficient that would be measured for a
half-screen located next to an infinite wall at
y - 0. Substituting the appropriate quantities
into equations (33) and (34)ields U2 following
expressions accurate to U(k 4:
-1/2
Da . (1 + k)	 (37)
then since B . 1 - De, for small k
Bl^
	
B2._8	 (38)
These values of B1 and B 2 have been used in
computing all of the curves in figures 1 to 3.
In figure 1, the second-order downstream
velocity in the wake is displayed for both forward-
tilting and backviard-tilting screens. Actually,
u2(x . -,y) has logarithmic singularities at
y . 0 and Jyj . 1. The maximum velocity in the
4
.I
wake occurred at the centerline for a > 0 and at
the edge	 y . 1 for a < 0. Thus, the shapes
of these asymptotic profiles are in qual'tative
agreement with those measured by Koo and James
(1973). A direct comparison cannot be made, how-
ever, since the experimental data are for screens
partially spanning the width of a channel. Note
that the analytical solution of Koo and James
yields a uniform velocity profile in the wake that
is not a good approximation to the actual profile.
Their numerical results are in qualitative agree-
ment with figure 1.
In figure 2, Pade approximants formed from the
two-term expansion for CD are plotted versus
k for a series.of values of a. This figure shows
that CD . CD(k) is a nonsymmetric function of
a; the maximum drag occurred not at a . 0 but at
i . -0.11. In contrast, the drag coefficient pre-
dicted by Koo and James' analytical solution is
independent of the sign of a. Finally, the lift
coefficient is shown in figure 3 for various values
of a. One observes that CL . CL(k) first
• ncreases for 0.2 ,Z a z -1 and then decreases
for -1 > a > -
 -. For the values of B1 and Bp
given, the maximum lift and the point of zero lift
occurred at a - -0.9 and a . 0.23, respectively.
Conclusion;
The inviscid flow through a very porous screen
inclined 'Lo the oncoming flow has been investigated
An asymptotic analysis for small resistance coef-
ficients showed thrt when the screen is submerged
in an infinite flow field, the first-order velocity
is nonuniform along the screen. This nonuniformity
gives rise to the presence of vorti:ity within the
wake at second order. The present method also
applies to axisymmetric screens or to free surface
jets flowing through infinite screens. In those
cases, Green's formulas for axisymmetric flow would
be used in place of the Plemelj formulas.
References
Durbin, P.A. and Muramoto, K.K., "Flow Through Very
Porous Screens," NASA TP-2436, 1985.
Koo, J.-K, and James, D.F., "Fluid Flow Around and
Through a Screen," Journal of Fluid Mechanics,
Vol. 60, Part 3, 1973, pp.
	 -
Roos, B.W., Analytic Functions and Distributions
in Physics and Engineering, John Wil ey
 t.`>o- ns, New
TorK, 19D9.
Taylor, G.I., "Air Resistance of a Flat Plate of
Very Porous Material," The Science Pa ers of G.I.
Taal^loor^ , vol. 3, edited yG.K.Batchelor,Cambridge
University Press, 1944, pp. 383-386.
Taylor, G.I. and Batchelor, G.K., "The Effect of
Wire Gauze on Small Disturbances in a Uniform
Stream," Quarterly Journal of Mechanics and Applied
Mathematics, Vol.art 1, 1949, pp. 1-29.
(a) Forward-tilting screens.
` v iI
1.0
.8
.6
Y
.4
.2
0
1.0
.8r
.6
Y
4
.2
0
-1
-J3
/ -3.732
.1	 0	 .1	 .2	 .3	 .4	 .5	 6
U2(+-, y)
(b) Backward-tilting screens.
Fig. 1. - Second-order downstream velocity within wake
with B 1 • 112 and B 2 - -318.
Ip
1
.8
.6
CD
.4
.2
2
.1	 .2	 .4	 .6	 .8	 1	 2	 4
k
Fig. 2. - Drag coefficient as a function of resistance coefficient
for various values of a with B 1 . 112 and B 2 - -318. Pad.
approximant formed from eq. M).
I
64
2
1
C L .8
.6
.4
.2
^f
.4	 .6	 .8	 1	 2	 4
k
Fig. 3. - Lift coefficient as a function of resistance
coefficient for backward-tilting screens with 13 1
-112 and 82- -318. Pad6 approximant formed from
eq. (36).
C^. 1
asr i
1. Report No. 2. Government Acoeasion No. 3. Recipient's Catalog No.
NASA TM-86979
4. Title and Subtitle 5. Report Date
Flow Through Very Porous Inclined Screens
d. Performing Org aniza tion code
505-31-04
7. Author(s) S. Performing Organization Report No.
Kenneth K. Muramoto and Paul A. Durbin E-2517
10. Work Unit No.
g. Performing Organization Name and Address
11. Contract or Grant No.
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio
	
44135 13. Type of Report and Period Gowned
Technical Memorandum12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration	 14. Sponsoring Agency Code
Washington,	 D.C.	 20546
15. Supplementary Notes
Prepa red for the 18th Fluid Dynamics & Plasmadynarrics & Lasers Conference
sponsored by the American Institute of Aeronautics and Astronautics, Cincinnati,
Ohio,	 July	 16-18,	 1935.
16. Abstract
The steady,	 inviscid flow through and around a screen inclined at a uniform angle
to the incoming flow was investigated. 	 For a screen placed in an infinite flow
field, an asymptotic analysis for small resistance coefficients was performed,
and the effects of inclination were determined.	 The velocity at first order in
the asymptotic expansion was nonuniform along the screen. 	 This nonuniformity
caused the wake behind the screen to contain distributed vorticity at second
order-.	 These effects therefore occurred at one order lower than for normal
screens.
17. Key Words (Suggested by Author(s)) 18. Distribution Statement
Screen Unclassified - unlimited
Drag coefficient STAR Category 34
Plemelj formula
19. Secur i ty Clarsll. (of this report) 20. Security Clasall. (of this paps) 21. No. of pages 22. Prtee'
Unclassified Unclassified
. For sale by the National Technical Information Semite. Springfield. Virginia 22161 	
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Flow through very porous inclined screens

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