A Numerical study of Flow through Sigmoid Duct

Prasanta K.Sinha et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 5, Issue 11, (Part - 5) November 2015, pp.91-99
www.ijera.com 91 | P a g e
A Numerical study of Flow through Sigmoid Duct
Prasanta K.Sinha1
, S. Mukhopadhya2
, B.Majumdar3
1
Durgapur Institute of Advanced Technology& Management, West Bengal, India
2
Department of Power Management Institute, National thermal power Corporation, India
3
Jadavpur University, Kolkata, West Bengal, India
Abstract
Curved diffusers are an integral component of the gas turbine engines of high-speed aircraft. These facilitate
effective operation of the combustor by reducing the total pressure loss. The performance characteristics of
these diffusers depend on their geometry and the inlet conditions. In the present investigation the distribution of
mean velocity, static pressure and total pressure are experimentally studied on a S-shape Diffusing Duct of
45°/45° angle of turn with an area ratio of 1.65 aspect ratio 3.95 keeping inlet width 55 mm with centre line
length 460 mm. The experimental results then were numerically validated with the help of Fluent. The velocity
distribution shows that generation of secondary motion in the form of counter rotating vortices within the 1st
half
of the diffuser. The secondary motion changes their sense of rotation after the inflexion plane of the test
diffuser. The maximum values of the mass average static Pressure recovery and total pressure loss are 36% and
13% compared to the predicted results of 39% and 11% respectively, which shows a good agreement between
the experimental and predicted results.
Keywords: S-shape diffuser, k-ε model, Fluent solver, Five-hole probe
Nomenclature
Ar Area ratio
As Aspect ratio
CC Concave or inward wall
Cpr Coefficient of pressure recovery
CV Convex or outward wall
W1 Inlet width of the Diffuser
W1 Inlet width of the Diffuser
L Centerline length of the diffuser
Re Reynold’s number
U Velocity of air
ρ Mass density of fluid
 Angle of turn of the curvature
 Kinematic Viscosity
μ Dynamic Viscosity
ρ Mass density of fluid
ζ Coefficient of pressure loss
I. Introduction
Diffusers are used in many engineering
application to decelerate the flow or to convert the
dynamic pressure into static pressure. Depending on
application, they have been designed in many
different shapes and sizes. The sigmoid diffuser is
one of such design and is an essential component in
many fluid handling systems. sigmoid diffusers are
an integral component of the gas turbine engines of
high-speed aircraft. It facilitates effective operation
of the combustor by reducing the total pressure loss.
The performance characteristics of these diffusers
depend on their geometry and the inlet conditions.
sigmoid diffusers are used in wind tunnels,
compressor crossover, air conditioning and
ventilation ducting systems, plumes, draft tubes, etc.
The use of such diffusers mainly depends either on
the specific design of the machine or on the space
limitation where compactness is desired or both.
The objective of the present study is to
investigate the flow characteristics within a
rectangular cross sectioned sigmoid diffuser.
An exhaustive survey of the reported literatures
on various types of diffusers reveals that the studies
on straight diffusers are available in considerable
numbers in open literatures. Literatures for curved
diffusers, specially, sigmoid diffusers are less in
number and detailed flow measurements
methodologies for these diffusers are very limited in
the open literatures. The earliest work on curved
diffuser was reported by Stanitz [1]. He designed the
diffuser based on potential flow solution for two-
dimensional, inviscid, incompressible and irrotational
flow. The first systematic studies on 2-D curved
subsonic diffusers were carried out by Fox & Kline
[2]. The centerline of the diffuser was taken as
circular with a linearly varying area distribution
normal to the centerline. They established a complete
map of flow over a range of the L/D ratio and at
RESEARCH ARTICLE OPEN ACCESS

different values of and at different values of ∆β. A
qualitative measurement of the mean flow quantities
in a 40° curved diffuser of rectangular cross section
with Ar =1.32 and inlet As =1.5 have been reported
by McMillan [3]. The result clearly showed the
development of strong counter rotating vortices
between two parallel walls, which dominate the flow
and performance characteristics. Seddon [4] has
made extensive experimental investigations to
explain the self-generated swirl within the S-shaped
diffuser of rectangular to circular cross-section
having Ar =1.338. He attempted to improve the
performance of S-shaped diffusing ducts by
introducing fences of 10 different configurations
within the first bend of the diffuser and observed a
significant improvement in the performance and exit
flow distribution. The relation between the shape of
curvature and cross sectional area can be understood,
if their effects are considered separately. In the early
parts of 1980's researchers started working on how to
improve the performance by introducing vortex
generator, fences etc. within the diffusers to change
the magnitude and direction of the generated
secondary motion. Vakili et al. [5] reported the
experimental studies in an S-shaped diffusing duct of
∆β = 30°/30° having circular cross-section and Ar
=1.5. They observed that there is a significant
improvement in the exit flow distribution and
pressure recovery by introducing vortex generator at
the inlet. Yaras [6] experimentally investigated the
flow characteristics of 90° curved diffuser with strong
curvature having Ar =3.42 for different values of inlet
boundary layer thickness and turbulence intensity.
Measurements were taken by the help of seven-hole
pressure probe. He observed that the performance
parameters were almost independent by the variations
in the inlet boundary layer. Reichert and Wendt [7]
experimentally studied the effect of vortex on the
flow field of a diffusing S-duct with ∆ β = 30° /30°
and Ar =1.5. The objective was to reduce flow
distortion and improve total pressure recovery within
the diffuser by using the vortex generation at the
inlet. They concluded that the mechanism responsible
for improved aerodynamic performance is not
boundary layer re-energization from shed axial
vortices but rather the suppression of detrimental
secondary flows by redirecting the flow. Majumder et
al. [8] also studied the performance characteristics of
a 90° / 90° S-shaped diffuser of rectangular cross
section with Ar = 2.0 and inlet As= 6.0 by using
three-hole pressure probe. They observed a detached
flow at the inflexion point and overall pressure
recovery in comparison to a straight diffuser is low.
Sonoda et al. [9] studied the flow characteristics
within an annular S-shaped duct. The effect on the
flow of a downstream passage is also carried out.
They observed that the total pressure loss near the
hub is larger due to the instability of the flow, as
compared with that near the casing, in the case of
curved annular downstream passage. The total
pressure loss near the hub is greatly increased
compared with the straight annular passage. Mullick
and Majumdar [10] studied the performance of a
fully developed subsonic turbulent flow in 22.5° /
22.5° circular cross section S-shaped diffusing duct.
The experiment was carried out with Re = 8.4 x l05
and measurements were taken with the help of a pre-
calibrated five-hole pressure probe. The experimental
results indicated the generation of secondary flow in
the form of a pair of contra rotating vortices in the
first half, which changes its senses of rotation in the
second half and overall static pressure recovery was
40%. Numerical Simulation of flow development
through turbine diffusers was reported by Dominy et
al. [11] they performed the experiment in an S-
shaped annular duct of inlet hub and core diameters
arc 0.286m and 0.421m respectively and Ar =1.5 with
inlet Re = 3.9 x 105
. They used 34 numbers inlet
swirl vanes. The results show that the influence of
wakes and swirl upon the flow has a significant effect
upon the development of flow. The numerical
simulations also give a good matching of the flow
development within this diffuser. Fuji et al [12]
studied the Curved Diffusing Annulus Turbulent
Boundary layer Development and concluded a
simplified but reliable method for boundary layer
calculation with acceptable assumptions for special
flow situations. A numerical and experimental
investigation of turbulent flows occurring in a 180°
bend annular diffuser with an aperture in front of the
bend was reported by Xia et al. [13] .They observed
that the pressure recovery coefficient increases with
increasing blow of mass flow rate and inlet pressure
but remains nearly constant if the inlet pressure is
higher than about 10 bars. The numerical prediction
is compared with the experimental data and excellent
agreement is achieved. Sing et al. [14] conducted
investigation to select the range of the inlet swirl
intensity for the best performance of annular
diffusers with different geometries but having the
same equivalent cone angle. This is analyzed on the
basis of the static pressure recovery and total pressure
loss coefficients. The results show that the parallel
diverging hub and casing annular diffuser produces
the best performance at high-swirl intensities. Sinha
et al. [15] conducted an experiment on 30° curved
annular diffuser. They measured the mean velocity,
static pressure and total pressure along the flow
passage of the diffuser. They observed that the high
velocity fluids shifted and accumulated at the
concave wall of the exit section. They also observed
that with the increase in area ratio pressure recovery
increases upto certain point than with further increase
in area ratio Pressure recovery decreases. Benin et
al.[16], 2007,studeturbulent flows in smooth pipes
using CFD analysis and find loss correlations for

developing incompressible turbulent flow. They used
different k–ε turbulent models and compare the result
with experimental data. Kahrom and Shokrgozar [17]
studied the flow with distributed turbulence intensity
and tested the accuracy by different turbulence
models compared the results with the experimental
data. They used numerical solutions of one equation
Spalart-Almaras Model, two equations high Reynolds
number k-ε Model, two equations Shear Stress
Transport k- Ω Model and anisotropy Reynolds
Stress Models are compared to experimental
measurements and results are discussed. Sinha et al.
[18] conducted an experiment on 40° curved diffuser.
They measured the mean velocity, static pressure and
total pressure along the flow passage of the diffuser.
They observed that the high velocity fluids shifted
and accumulated at the concave wall of the exit
section. They also observed that with the increase in
area ratio pressure recovery increases upto certain
point than with further increase in area ratio Pressure
recovery decreases. Biswas et al. [19] investigated
experimentally a 45°/45° constant area S-duct. They
measured the wall pressure and mean velocity along
the flow passage of the duct. They observed that the
bulk flow shifting from outer wall to the inner wall in
the first half and inner wall to the outer wall in the
second half along the flow passage of curved duct is
very instinct. Sinha et al. [20] investigated
experimentally a 22.5°/22.5° sigmoid diffusing S-
duct. They measured the mean velocity and turbulent
intensity along the flow passage of the duct. They
observed that the velocity distribution shows that
generation of secondary motion in the form of
counter rotating vortices within the 1st half of the
diffuser. The secondary motion changes their sence
of rotation after the inflexion plane of the test
diffuser. Elsafty and Elazm [21] studied improving
air quality in enclosed parking facilities using
ventilation system design with the aid of CFD
simulation. The study draws attention to the effect of
exhaust fan height on the carbon monoxide (CO)
dispersion in enclosed parking facilities. A
mathematical model is presented in a general
computer code that can provide detailed information
on CO concentrations as well as airflow fields
prevailing in three-dimensional enclosed spaces of
any geometrical complexity. Ahmed et al. [22]
numerical predicted the effect on air flow rate in the
presence of heated obstruction within a room and two
notable points are presented; first, higher flow rate is
depending on throw of jet and its effect on the CO2
concentration and temperature distribution in upper
zone more than occupied zone with presence the
obstruction. The second; in low flow rate buoyancy
effect is considerable. Vertical temperature gradient
above the obstruction implies that both fresh air and
CO2 concentration.
From the available literature on curved diffusers
it is apparent that the studies are generally related to
straight or curved diffusers with circular cross-
section. The present experimental investigation aims
for a systematic study on the flow pattern of a
sigmoid diffuser through the measurements of mean
velocity and Turbulence intensity distribution, under
the influence of different geometrical and flow
parameters.
II. Material and Methodology
A test rig for the present investigation has been
constructed at Fluid Mechanics & Machinery
Laboratory of Power Engineering Department
Jadavpur University to investigate the flow
characteristics within a rectangular cross sectioned
sigmoid diffuser. The geometry of the test diffuser is
shown in Fig.1 with co-ordinate system and
measurement locations. The entire set up was
fabricated from mild steel sheet except the test
diffuser.
The test diffuser was designed with increase in
area from inlet to exit and it distributed normal to the
centerline as suggested by Fox and Kline [2]. The test
diffuser was designed based on an area ratio of 1.65
and aspect ratio 3.95 with the centerline length of
.460m. To develop the test diffuser geometry, a
straight diffuser of inlet width 0.06m, center line
length 0.46m and area ratio 1.65 was first drawn. The
first half of the centerline of the straight diffuser was
given a circular shape of radius of curvature 0.576m
and the 2nd
half was then turned in the reverse
direction with same radius of curvature to achieve the
complete contour of the test diffuser. Top and bottom
wall of the test diffuser was kept parallel to ensure
2D geometry of the test diffuser. The test diffuser is
made of transparent plexiglass.
In order to avoid the pressure losses and flow
distortion at the inlet and exit, two constant area
connectors were attached at the inlet and exit of the
test diffuser. A pre-calibrated five-hole pressure
probe was used to obtain detailed flow parameters
like mean velocity and its components, total and
static pressure and secondary motions along the
entire length of the diffuser. Ambient air was used as
working fluid.
For measuring mean velocity and its
components, turbulence intensity and static and total
pressure surveys along the entire cross section of
curved diffuser, the test piece was divided into seven
planes, one at the Inlet section 2W1 upstream of the
test diffuser, three planes, Section A, Section B and
Section C at 18°, 30° and 45° turn along the first half
of the diffusing passages and Section D Section E
and Section F at -30°,-18° and -0° turn along the
second half of the diffusing passages. The number of
measuring stations at each section varied from 5 to 7,
from inlet to exit depending on the width of

corresponding sections as shown in Fig.2.The details
of measured planes are shown in Fig.1 and Fig.2.
Turbulence intensity at seven different sections
of test diffuser has been measured with a single wire
hot wire anemometer. The wire has been place into
the flow field at desire location keeping it normal to
the flow direction.
FIGURE 1.Schematic diagram of the experimental set up
FIGURE 2. Geometry of test diffuser and measuring locations
The pre-calibrated five hole pressure probes was
mounted in a traversing mechanism and the probe
inserted into the flow field through 6 mm slot
provided at the top wall. The probe was placed within
2 mm of bottom wall for the first reading. The probe
was then moved Z-direction up and placed at the
desired location as shown in Fig.1.
Instrumentation for the present study was chosen
such that the experimental errors are minimum and
also to have quick response to the flow parameters.
The pre-calibrated hemispherical tip five-hole
pressure probe used for the present study. The probe
was calibrated and using null technique was used to
measure the flow parameter. All the five sensing
ports of the probe were connected to a variable
inclined multi tube manometer. The readings were
recorded with respect to atmospheric pressure.
The mean velocity and components of mean
velocity distribution have been drawn with the help
of SURFER software
The assessment of errors resulting from the
readings of the present five hole pressure probe was
made as a function of all incidence angles for all flow
characteristics in all the probe sectors and discussed
in details‎[23].
III. Results and discussion
The flow characteristics have been evaluated by
mass average mean velocity, between the curved and
parallel (top and bottom) walls, turbulence intensity,
total pressure and static pressure of the flow at
various cross sections. Measured flow quantities have
been presented in the form of 2-D profiles. All the
velocities and pressures were normalized with respect
to the inlet mass average velocity and inlet dynamic
pressure respectively
III. 1. Mean velocity contour
The normalized mean velocity distribution in the
form of contour plots at at seven measuring sections
of different angle of turn for this diffuser has been
discussed here and is shown in Fig.3.
At section I, Fig (a) velocity distribution is
uniform throughout the section except very close to
the top and bottom walls. This is mainly due to the
corner effect. At section A, 18° turn, Fig.(b) shows
that the high velocity zone is located near to the

convex side. Here the diffusion of flow is seen more
along the concave wall. Diffusion of flow is also seen
to be faster along the concave wall, whereas marginal
diffusion is seen along the convex wall. The figure
also depicts that the high velocity fluid has
accumulated to the top half along the convex wall.
This may be due to the momentum transfer from one
curved wall to other curved wall. At 30° turn, Fig.(c),
depicts an overall diffusion of flow throughout the
entire cross section, though the high velocity fluid
adhere to the convex wall sides as seen in previous
section. However, near to the parallel walls and very
close to the convex side, the nature of the iso-velocity
lines are ending towards the middle. it indicates the
generation of flow instability at these zone. As the
flow moves downward and reaches the inflexion
plane as in Fig.(d) the instability of flow, seen at 30°
turn of 1st bend enhances and is clearly visible along
the convex side. The high velocity fluid occupied
more cross sectional area at this section. The bulk
flow has also moved towards the concave side and as
a result of which the flow has accelerated along this
wall. This is attributed to the transfer of momentum
from convex to concave side. Further downstream, at
30° turn of the 2nd bend Fig.(e) the high velocity
fluid has completely shifted towards the convex side.
The transfer of momentum, seen at inflexion plane,
continues at this section even change in centerline
curvature. The low velocity fluid has accumulated
along the concave side of the 2nd
bend and occupied
the entire depth of the section along this wall. The
bulging out of the contour lines near to the top and
bottom parallel walls further indicates the
accumulation of more low fluid at this zone
compared to the mid plane of the diffuser. This type
of phenomenon has also been reported by Majumder
et al. [8] for 90°/90° diffuser. Further downstream,
at -18° turn of the right bend as per Fig.(f) more low
velocity fluid accumulates along concave side,
occupying more flow area. This indicates the change
in flow direction as the curvature of the centerline
changes after the inflexion plane. The flow along the
convex side has also accelerated a little at this
section. The movement of fluid from concave to
convex continues further downstream Fig.(f). The
high velocity core has occupied more or less entire
exit cross section except at the corners near to the
concave side.
(a) Inlet section (b) Section A (c) Section B (d) Section C

(e) Section E (f) Section F (g) Section D
FIGURE 3. Normalized mean velocity distribution
III.2. Pressure recovery and loss coefficients
The variation of normalized mass averaged static
pressure recovery and total pressure loss coefficients
based on the mass average pressure at different
sections of the test diffuser are shown in Fig.4. It was
calculated based on the static pressure difference
between the succeeding and preceding sections. The
figure shows that the coefficient of pressure recovery
increases steadily up to the Section A in S-shape
diffuser and then the increase takes place in rapidly
up to the Section F. This is mainly due to the
formation of secondary motions The overall mass
average pressure recovery coefficient is nearly 36%
for this diffuser.
The mass average total pressure loss coefficient
increases sharply in the test diffuser up to the Section
A. Then the loss coefficients in curved diffuser
increases in steadily up to Section F. The overall
mean value in mass average pressure loss coefficient
is nearly 13% for this test diffuser.
FIGURE 4. Variation of mass average pressure
recovery and loss coefficients
III.3. Numerical Validation
In the present study a preliminary investigation
was carried out using different turbulence models
available in FLUENT. The main differential
equations used here continuity equation, Momentum
equation and transport equations used in standard k-ε
model.
The law of conservation of mass applied to a
fluid passing through a fixed control volume yields
the following equation of continuity For a Cartesian
coordinate system, where u, v, w represent the x, y, z,
components of the velocity vector, can be written as
0)()()( 











w
z
v
y
u
xt


The Navier-Stokes equations form the basis upon
which the entire science of viscous flow theory has
been developed. If the flow is incompressible and the
coefficient of viscosity  is assumed to be constant,
The Navier-Stokes equations used in simpler form as
Upf
Dt
UD 2
 

In our investigation flow velocity is 40 m/s i.e. the
flow is completely incompressible. Consequently, the
temperature gradient is ignored at the time of analysis
and the detailed discussion of energy equation is not
needed for our study.
The transport equations used in the standard k-ε
model are as follows:
Turbulent kinetic energy:



 





























j
i
ijijr
jk
r
j x
u
ks
x
k
rxDt
kD
3
2
2
Dissipation rate:




 





























j
i
ijijr
jk
r
j x
u
ks
x
k
rxDt
kD
3
2
2
The terms on the right hand side of the Equation
from left to right can be interpreted as the diffusion,
generation and dissipation rates of ε.
Based on the Intensive investigation it was found
that Standard k – ε model of turbulence provides the
best result and results obtained from computational
analysis match both in qualitatively and
quantitatively with the experimental results.
It is to be noted here that the inlet velocity
profiles obtained during experiment are fed as an
inlet condition and normal atmospheric pressure at
outlet as outlet condition during the validation with
FLUENT Some of the validation figures are shown in
Fig.5(a), Fig.5(b), Fig.5(c) and Fig.5(d) respectively
All four figures indicate that the mass averaged
mean velocity contours obtained by computational
and experimental investigations, which shows a
qualitative matching to each other.
The mean velocity distribution at the Section C,
Section D, Section E and Section F are shown in Fig.
5(a), Fig. 5(b), Fig.5(c) and Fig.5 (d) show a
reasonably good agreement of the computational
investigation with the experimental results.
Experimental Computational Experimental Computational
(a) Section C (b) Section D
Experimental Computational Experimental Computational
(C) Section E (d) Section F
FIGURE 5. Comparison of normalized velocity distribution at Section C, Section D, Section E and Section F
through Computational and Experimental investigation

FIGURE 6. Comparison of performance parameters obtained through computational and experimental
investigation
Fig.6 shows the comparison of performance
parameters like coefficient of static pressure recovery
and coefficient of total pressure loss obtained through
experimental and computational investigation. From
the figure it has been observed that coefficient of
pressure recovery Cpr for the computational
investigation was obtained as 39% compared to the
experimental investigation, which obtained as 36%.
Similarly the coefficient of pressure loss is obtained
as 11% in computation investigation compared to the
13% of experimental study. This shows very good
matching of the predicted results with the
experimental one.
These agreements confirm that the CFD code
using Standard k – ε model can predict the flow and
performance characteristics reasonably well for
similar geometries with same boundary conditions
IV. Conclusions
From the present investigation the following
conclusions have been drawn:
1. High velocity fluids shifted and accumulated at
the convex wall of the exit section. .
2. The mass average static pressure recovery and
total pressure loss for the curved test diffuser is
continuous from Inlet section to Section F.
3. Performance parameter like coefficient of mass
average static pressure recovery and coefficient
of mass average total pressure loss are 36% and
13% respectively.
4. A comparison between the experimental and
predicated results for the sigmoid diffuser
show good qualitative agreement between the
two.
5. The coefficient of mass averaged static pressure
recovery and total pressure loss are obtained as
39% and 11% in predicted results and in the
experimental results their values obtained as
36% and 13% respectively, which indicate a
good matching between the experimental and
predicted results
6. Among the different turbulence models within
the fluent solver a standard k-ε model shows the
best results and predicts the flow and
performance characteristics well for S-shape
diffusing ducts with uniform flow at inlet.
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