Flow around a bent duct theory

8. FLOW AROUND A BEND IN A DUCT
AF15 Flow Around a Bend Apparatus
89

Introduction
The engineer is frequently presented with problems of flow contained within tubes
and ducts. Such flows may be classified as internal flows to distinguish them from
flows over bodies such as aerofoils, called external flows. It is sometimes necessary
to shape a duct in such a way that particular requirements are met. For example, it
may be necessary to change the shape of cross-section from square to rectangular with
a small loss of total pressure, or it may be required to form a bend in such a way that
the distribution of velocity at the exit is as nearly uniform as it can be made.
Due to the presence of boundary layers along the duct walls, the fluid mechanics of
such flows are sometimes extremely complicated. Separation may be produced where
the pressure rises in the direction of flow, as illustrated in Figure 8.1(a).
Figure 8.1(a) Schematic Representation of Separating and Reattaching Flow
in a Duct
This shows a duct of increasing cross-sectional area in which the flow decelerates
with an accompanying rise of pressure. Separation of flow from one wall is shown,
followed by a region of severe turbulence in which there is mixing between the main
flow and the region of recirculating flow (often called the separation bubble). The
turbulent mixing leads to loss of total pressure, the size of this loss depending on the
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extent of the separation. It should be emphasised that the flow shown in the figure is
schematic only.
The separation line is rarely steady. The size of the separated zone often fluctuates
violently, and in some cases the separation is intermittent. Separation might occur
over more than one surface and would not normally take place uniformly over one
side as shown for illustrative purposes in the diagram. A further complication arises
from secondary flow which is again due to boundary layer effects.
Figure 8.1(b) Formation of Secondary Flow in a Bend of a Duct
Figure 8.1(b) shows one example of the formation of a secondary flow in a gently-
curving duct of rectangular cross-section. The curvature of the flow is accompanied
by a pressure gradient which rises across the section from the inner to the outer wall.
The pressure gradient extends over the whole section, so that the boundary layers on
the upper and lower walls are subjected to the same pressure gradient as the main
flow. But because the streaming velocity in the boundary layer is less than in the
main part of the flow, the curvature of the streamlines in the boundary layer is more
severe, as indicated. This gives rise to a net inward-directed flow adjacent to the
upper and lower walls, which sets up a secondary flow in the form of a double
rotation, superimposed on the main stream. The motion emerging from the curve in
the duct is therefore a pair of contra-rotating spirals, the strength of which depends on
the amount of curvature and on the thickness of the boundary layer.
91

Simple Theory of Flow in a Bend
Figure 8.2 Assumed Velocity Distribution in a Bend
In this experiment we investigate the flow around a 90° bend in a duct of rectangular
section, using pressure tappings along the walls to establish the pressure distributions.
Figure 8.2 indicates flow approaching a bend with a uniform velocity U. Within the
bend we shall assume a free vortex distribution of velocity, given by
u
C
r
=
(8-1)
where u is the streaming velocity at radius r from the centre of curvature of the bend.
Separation and secondary flow will be neglected. The constant C may be found by
applying the equation of continuity as follows:
( )Q Ub r r b u
r
dr
r
= − =
∫2 1
1
2
(8-2)
where b is the width of the section of the duct. Substituting for u from Equation (8-1)
and performing the integration leads to the result
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C U
r r
r r
=
−2 1
2 1ln ( / )
(8-3)
so the velocity distribution is, in dimensionless form,
u
U
r r
r r r
=
−2 1
2 1ln ( )
(8-4)
The corresponding pressure distribution may be found by assuming that Bernoulli’s
equation may be applied between the upstream section and a section within the bend
as follows:
p U po + = +1
2
2 1
2
2
ρ ρu
(8-5)
where po is the static pressure upstream and p is the pressure at radius r in the bend. It
is convenient to express p in the form of a dimensionless pressure coefficient cp where
c
p p
U
p
o
=
−
1
2
2
ρ
(8-6)
From Equation (8-5) this may be written
c
u
U
p = −1
2
2
(8-7)
which may be evaluated for any radius r by substituting the appropriate value of u/U
obtained from Equation (8-4). A comparison with measured values of cp may be
made as shown in Table 8.2.
Description of Apparatus
Figure 8.3 shows the dimensions of the bend and the positions of the pressure
tappings. There is a reference pressure tapping 0 on the side face near the entry, and
three sets of tappings; one set of 10 along the outer curved wall, one set of 10 along
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the inner curved wall and a set of 9 along a radius of the bend. Air from the
contraction section is blown along the duct and is exhausted to atmosphere.
Figure 8.3 Dimensions of the Bend and the Positions of Pressure Tappings
Experimental Procedure and Results
The pressure tappings along the outer wall, the reference tapping 0 and the pressure
tapping in the airbox are all connected to the manometer. The air speed is adjusted to
a value slightly below the maximum, as indicated by the airbox pressure, and the
pressures are recorded. (The setting of air speed slightly below the maximum is to
ensure that the same setting may be repeated in later tests). The tappings on the inner
wall are then connected in place of the ones on the outer wall. The airbox pressure is
adjusted to the previous value and a further set of readings are recorded. Finally the
procedure is repeated with the third set of pressure tappings. In Table 8.1 the
pressures p are recorded relative to an atmospheric datum and the pressure
coefficients cp are calculated from Equation (8-6).
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Airbox pressure = P = 630 N/m2
Reference tapping pressure = po = 80 N/m2
Velocity pressure of uniform flow along duct P − po =
1
2
2
ρU = 550 N/m2
Tapping Outer Wall Inner Wall Radial Section
No. p (N/m
2
) cp p (N/m
2
) cp p (N/m
2
) cp
1
2
3
4
5
6
7
8
9
10
90
145
205
325
330
320
240
85
35
0
0.02
0.12
0.23
0.45
0.45
0.44
0.29
0.01
–0.08
–0.15
70
40
–240
–415
–440
–395
–255
–25
10
0
–0.02
–0.07
–0.58
–0.90
–0.95
–0.86
–0.61
–0.19
–0.13
–0.15
–265
–150
–40
50
115
170
215
260
295
–0.63
–0.42
–0.22
–0.05
0.06
0.16
0.25
0.33
0.39
Table 8.1 Measured Pressures and Pressure Coefficients
From Figure 8.3 the inner and outer surfaces of the bend have radii
r1 = 50 mm
r2 = 100 mm
From Equation (8-4) the velocity distribution across the section according to the free
vortex assumption is therefore
u
U r
= =
50
2
721
ln( )
.
r
where r is expressed in mm. In Table 8.2 we calculate this ratio and the
corresponding value of cp from Equation (8-7) for a number of values of r.
95

r
(mm)
u
U
cp
50
55
60
65
70
75
80
85
90
95
100
1.443
1.312
1.202
1.110
1.030
0.962
0.902
0.849
0.801
0.759
0.721
–1.081
–0.720
–0.445
–0.232
–0.062
0.075
0.187
0.280
0.358
0.423
0.480
Table 8.2 Calculated Pressure Coefficients
Figure 8.4 Distribution of Pressure Coefficient cp over Walls
Figure 8.4 shows the distribution of measured pressure coefficient over the curved
walls and compares the measured and calculated values across the radial section. It
may be seen that the pressure across the inlet section is nearly uniform. As the flow
approaches the bend, the pressure on the inner wall falls rapidly and on the outer wall
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rises rapidly to values which remain substantially constant round most of the curve.
This indicates that the curvature of the flow is also likely to be reasonably constant.
The distribution of cp over the radial section follows the calculated curve quite
closely, indicating that the assumption of a free vortex velocity distribution made in
Equation (8-1), together with the assumption that Bernoulli’s equation applies to the
flow, give a fairly accurate distribution of the pressure field. The measured pressure
distribution varies rather less steeply than calculated, indicating a vortex strength C
somewhat less than that given by Equation (8-3).
Downstream of the bend, the wall pressures readjust until at the duct exit the pressure
is constant across the section. It is, however, a little lower than the reference pressure
at the inlet, and this difference represents a pressure loss round the bend. It is
convenient to express this loss ∆p in terms of the velocity pressure ½ρU2
in the
uniform approaching flow by the expression
K
p
U
=
∆
1
2
2ρ
(8-8)
where K is the dimensionless loss coefficient. In this case we find, from the change
in cp from the inlet to the outlet sections, the value:
K = 0.15
(8-9)
Conclusion
The distribution of pressure over the curved walls of a 90° bend of rectangular section
has been established by pressure plotting. The pressure coefficient is negative and
almost constant round the inner wall, and positive and almost constant round the outer
wall. Across the 45° cross-section the pressure distribution may be predicted with
reasonable accuracy by assuming free-vortex velocity distribution over the section.
The value of loss coefficient K is 0.15 for this bend.
97

Flow around a bent duct theory

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Flow around a bent duct theory