Proceedings of the 5th IASME / WSEAS International Conference on Fluid Mechanics and Aerodynamics, Athens, Greece, August 25-27, 2007
238
Numerical Investigation of the Role of the Inlet Swirl Velocity
Profile on Decay of Swirl in Pipe Flow
A. C. BENIM
Department of Mechanical and Process Engineering
Duesseldorf University of Applied Sciences
Josef-Gockeln-Str. 9, D-40474 Duesseldorf
GERMANY
[email protected]
F. GUL
Department of Mechanical and Process Engineering
Duesseldorf University of Applied Sciences
Josef-Gockeln-Str. 9, D-40474 Duesseldorf
GERMANY
[email protected]
E. PASQUALOTTO
Development Engineer, Pelton Turbines
Andritz VA Tech Hydro AG
Hardstrasse 319, CH-8023 Zuerich
SWITZERLAND
[email protected]
Abstract: - The influence of the inlet swirl velocity profile shape on the decay of swirl in steady-state, incompressible,
laminar pipe flows is investigated by means of computational fluid dynamics. The investigation is carried out for a
Reynolds number of 2000, which marks the upper limit of the laminar flow behavior for unidirectional pipe flow.
Computations are performed for two different mean swirl numbers, namely, for 0.2 and 0.5. As the integral parameter
for describing the swirl intensity, the swirl number is used, in contrast to other work relying on the circulation, instead.
It is observed that decay of swirl number along the pipe length occurs at different rates for different shapes of the inlet
swirl velocity profile, although the inlet value of the swirl number is kept constant when changing profile shapes.
Although this result is obtained for the laminar flow, which is not as important as the turbulent one for engineering
applications, it implies the possibility of a similar dependency for turbulent flows, and, thus, calls for the further
investigations of the phenomena for turbulent flow.
Key-Words: - Swirling Pipe Flow, Decay of Swirl, CFD.
Nomenclature
a
C
D
n
r
R
Re
S
u
radial extension of solid body rotation
core
empirical parameter
pipe diameter
exponent in empirical correlation
radial coordinate
pipe radius
Reynolds number (Re=UD/ ν)
swirl number
axial velocity component
U
w
Wmax
x
constant inlet axial velocity
swirl velocity component
maximum swirl velocity at inlet
axial coordinate
Greek symbols
κ
empirical parameter
ν
kinematic viscosity
Subscripts
0
inlet value
Proceedings of the 5th IASME / WSEAS International Conference on Fluid Mechanics and Aerodynamics, Athens, Greece, August 25-27, 2007
1 Introduction
Information on the swirl decay in pipe flow is important
in a wide range of engineering applications, such as heat
transfer enhancement techniques by means of swirl [1],
phänomenological models for internal combustion
engines [2] and fluidic vortex valves [3]. Thus,
investigation of swirling flow development in pipes has
attracted attention over several decades [4]. Since
turbulent flows are much more frequently encountered in
technical applications than laminar ones, the main focus
of investigation has been the turbulent swirling pipe
flows. However, one of the major problems in dealing
with turbulent swirling flows is that they are hardly
amenable to computational analysis. Although
Computational Fluid Dynamics (CFD) based modelling
approaches became quite mature in modelling many
flows or practical relevance, the computational
modelling of turbulent swirling flows continues to
remain to be a great challenge. This will be discussed in
more detail below. Thus, although computational
investigations have continuously been performed [5] the
studies on swirl decay have mainly been experimental
[6]. For application purposes, it is important to
determine the decay rate of swirl intensity along the
tube, and express this by convenient correlation
functions. Such correlations functions based on
experimental studies were already proposed in the
literature [7].
In the experimental investigations, many authors
investigated the dependency of the swirl decay rate on
parameters such as the inlet swirl intensity and the
Reynolds number [7]. However, the influence of the
form of the inlet swirl velocity profile was never
investigated. The swirl intensity was identified by a
suitable integral parameter, such as the swirl number, or,
the circulation. However, such an integral parameter
does not uniquely determine the radial swirl velocity
profile shape, as an infinite number of different profiles
can satisfy a given value of the integral parameter. The
present note aims to investigate the role of the inlet swirl
velocity shape on the swirl decay.
Although the turbulent flow is much more relevant
for engineering applications than the laminar one, the
main computations, addressing the effect of the inlet
swirl velocity profile shape, are carried out for laminar
flow (at a rather high Reynolds number of 2000). The
reason for this has been the poor performance of the
RANS (Reynolds Averaged Navier-Stokes equations)
turbulence models observed in the initial phase of the
present study. The poor prediction was observed, not
only for turbulent viscosity based models (TVM), but
also for the more sophisticated Reynolds Stress Model
(RSM) [8]. This practice has been in agreement with our
previous experience. In turbulent swirling flows, it is
known that certain Reynolds stress components are
strongly modified due to the action of flow curvature and
239
pressure gradient. Consequently, the standard TVM
perform unsatisfactorily. This phenomenon can
principally be accounted for within an RSM closure.
However, although RSM is observed to perform
satisfactorily in some applications [9], it is known to
perform poorly in some others [10]. Our previous
experience implies that the omission of the transience of
coherent flow structures within the framework of a
RANS formulation may be the cause of this inconsistent
performance, as these structures may be of different
importance from flow to flow. Thus, consistently reliable
predictions for turbulent swirling flows, for varying flow
parameters, seem to be expected only within a transient,
three-dimensional modelling strategy, which, at the same
time, properly accounts for the non-isotropic turbulence
structure induced by swirl. This framework is provided
by modeling procedures such as Large Eddy Simulations
(LES) and Detached Eddy Simulations (DES) [11-13].
The application of such sophisticated procedures for the
present purposes would go far beyond the scope of this
present preliminary study. Thus, it has been decided to
analyze the problem, first, for the laminar flow. The
analysis is based on the commercial CFD package Fluent
6.2 [14], which employs the finite volume method in
conjunction with an unstructured grid definition.
Since no modeled equations are used in the laminar
flow, the predictions may be assumed to approach the
exact solution, for the given boundary conditions and
geometry, provided that the assumption of flow
steadiness and symmetry are valid and numerical errors
are made sufficiently small. The present findings may be
useful for the further investigations as an indication of
the trends to be expected for the turbulent flows.
2 Overview of Swirl Decay Correlations
In [7] an overview of experimental swirl decay studies
was provided. As the quantity to measure the swirl
decay, some authors preferred to use the swirl number:
R
2
∫ u w r dr
S= 0
R
R ∫ u 2 r dr
0
(1)
Existing correlations generally assume an exponential
decay of swirl by an expression of type
  x n 
S = C exp − κ  
  D  
(2)
where the parameters C and κ are mainly functions of
the inlet swirl number and the Reynolds number.
Proceedings of the 5th IASME / WSEAS International Conference on Fluid Mechanics and Aerodynamics, Athens, Greece, August 25-27, 2007
3 Preliminary Study for Turbulent Flow
Originally, it was intended to perform the analysis for
turbulent flows, using RANS turbulence models. Before
starting the main investigation, a validation study was
performed using the experiments of Steenbergen [15] for
. 5
turbulent swirling flow at Re=3 10 as data base. In this
section, the result of this study shall briefly be discussed.
At the pipe inlet, measured profiles [15] of the axial
and tangential velocities are prescribed as boundary
conditions (Fig. 1). For turbulence quantities, for RSM,
the six measured normal stress components are directly
prescribed at the inlet (Fig. 1). For TVM, the inlet profile
of turbulent kinetic energy is also based on the measured
normal Reynolds stresses. For the turbulent length scale,
which is needed to derive the boundary conditions for
the turbulence dissipation rate, a macro length scale
equal to 30% of pipe diameter is first assumed. This was
also varied later within a sensitivity study. The wallfunctions approach [16] is used for near-wall flow,
+
assuring that quite optimal y wall values between 40100 result, by accordingly adjusted grids.
240
Fig. 2 shows the measured [15] and presently predicted
radial profiles of the axial and swirl velocity components
at an axial cross section (x/D=7.2). Predictions are based
the RNG k-ε model [17] and the RSM [8]. One can
observe that the agreement of the predictions with the
measurements is quite poor, especially for the swirl
velocity component. It is surprising, and, disappointing
at the same time, that the RSM does not perform better
than the turbulent viscosity based RNG k-ε model. In the
computations, second order discretization schemes were
used, and grid independency studies were performed.
Thus a high enough numerical accuracy can be assumed.
Sensitivity studies were also performed for the assumed
inlet distribution of the macro length scale. Thus, one
may conclude that the discrepancy between the
predictions and measurements are mainly due to
turbulence modeling. A discussion of the inferior
behavior of RANS turbulence models was given above.
Fig. 2. Measured [15] and predicted profiles of axial and
. 5
swirl velocity at x/D=7.2 (Re=3 10 ).
4 Modeling for Laminar Flow
Fig. 1. Measured [15] inlet profiles;
(a) velocity components, (b) Reynolds stresses.
The performance of the RANS turbulence models has
been found not to be sufficient for the present purpose
(Fig. 1). Thus, it was decided to perform the main
investigation for the laminar flow, first, since the
application of the three-dimensional transient procedures
such as LES and DES for the intended parametric study
would go far beyond the envisaged scope of the present
study. Thus, the laminar flow is analyzed at the present
stage, expecting that the findings would have qualitative
implications on the behavior of turbulent flows.
The computations are performed for Re = 2000 ,
which practically represents the upper limit of the
laminar regime for the pipe flow. One can argue that the
additional swirl momentum may trigger an earlier onset
Proceedings of the 5th IASME / WSEAS International Conference on Fluid Mechanics and Aerodynamics, Athens, Greece, August 25-27, 2007
of turbulence compared to the unidirectional pipe flow.
Nevertheless, the fact that the axisymmetric, steady-state
(laminar) computations converge to smooth solutions
indicates that the real flow under the given boundary
conditions remains rather laminar.
Incompressible, steady-state, laminar flow of a
constant viscosity Newtonian fluid in a circular pipe is
considered. The Navier-Stokes and continuity equations
are solved for two-dimensional axisymmetric swirling
flow. For treating the velocity-pressure coupling, the
SIMPLEC procedure [18] is applied. The rectangular
solution domain, which is defined to be 100 pipe
diameters long is bounded by the symmetry, wall, inlet
and outlet boundaries. At the inlet boundary, velocity
components are prescribed (constant axial velocity, zero
radial velocity, a given radial variation of the swirl
velocity). At the outlet boundary, zero-gradient
conditions are prescribed for all variables except the
pressure. For the pressure, a radial profile is prescribed,
which results form prescribing a zero gauge pressure at
the centerline and assuming “radial equilibrium”. In the
numerical discretization, a second order upwind scheme
[19] is used for the convective terms of all variables. The
computational grid is a structured one, consisting of
rectangular finite volumes, and results from a detailed
grid independency study performed in the initial phase
of the analysis. The final grid has 600 cells in the axial
and 50 cells in the radial directions. In the radial
direction, an equidistant spacing is used. In the axial
direction, the grid lines are concentrated in the inlet
boundary and expand towards the outlet boundary by
quite mild geometric expansion factors.
4.1 Velocity Profiles at the Inlet
value U is assumed. Based on these inlet velocity
profiles, the inlet swirl number can be expressed as
S0 =
Wmax
U
 a 1  a 3 
 −   
 R 2  R  
Wmax
r
a
for
r≤a
the maximum swirl velocity Wmax is adjusted in such a
way that the inlet swirl number (4) remains constant.
It is known that the so-called “vortex breakdown”
[20,21] occurs for a swirl number above a critical value,
which means a flow reversal in the central region. In
[22] critical swirl numbers as functions of swirl velocity
profiles were theoretically investigated, where values
around 0.5-0.6 were found to represent a lower bound,
for the investigated profile shapes. It is known that at
such high swirl numbers, unsteady phenomena such as
vortex core precession may occur, which would no more
be compatible with the present steady-state,
axisymmetric modelling approach. In the present
analysis, two values of the constant inlet swirl number is
investigated, namely S0 = 0.2 , and S0 = 0.5 . Fig. 3
shows the five inlet swirl velocity profiles (3), for five
different vortex core radii, and all indicating the constant
swirl number of S0 = 0.5 .
(3a)
and a perfect free vortex outside
w0 =
Wmax a
r
for
a≤r≤R
(3b)
where the parameter a denotes the vortex core radial
extension. For the axial velocity at the inlet, a constant
(4)
The free parameters of the swirl velocity profile (4) are
the core radius a and the maximum swirl velocity
Wmax . For a fixed inlet swirl number (4), five different
inlet swirl velocity profiles are investigated. The five
profiles are obtained by locating the core radius a at
five different positions along the pipe radius, namely at
0.1 R , 0.3 R , 0.5 R , 0.7 R , 0.9 R . For each profile,
The integral parameters such as the circulation, or swirl
number do not uniquely prescribe the shape of the swirl
velocity profile. To investigate the isolated influence of
the inlet swirl velocity profile shape, computations are
performed using different inlet swirl velocity profile
shapes, which imply the same inlet swirl number.
For clarity, a quite simplified swirl velocity profile
shape at the inlet is assumed, which consists of a perfect
solid body rotation in the core
w0 =
241
Fig. 3. Inlet swirl velocity profiles ( S0 = 0.5 ).
Proceedings of the 5th IASME / WSEAS International Conference on Fluid Mechanics and Aerodynamics, Athens, Greece, August 25-27, 2007
242
5 Results
Fig. 4 shows the change of the radial position of the
maximum swirl velocity ( rmax ) with the axial distance
for five different inlet profiles, for S0 = 0.2 , and
S0 = 0.5 . The parameter rmax converges to a value at
about x= 60D for S0 = 0.2 , and at about x=70D for
S0 = 0.5 case. It is interesting to note that rmax
converges quite exactly to the same value for the both
swirl numbers, which turns out to be rmax = 0.42 R .
Axial variations of the swirl number predicted for the
five inlet velocity profiles, for S0 = 0.2 , and S0 = 0.5
are shown in Fig. 5 and Fig. 6. In both cases, an
influence of the shape of the inlet swirl profile on swirl
decay is observed, as the swirl numbers obtained for
different inlet profiles steadily diverge with increasing
axial distance. In both cases, the lowest values are
obtained by the curve corresponding to the inlet swirl
profile with a = 0.9 R , whereas the highest values are
provided by the curve corresponding to the inlet swirl
profile with a = 0.1 R . The curves corresponding to the
intermediate values of a are distributed rather smoothly
Fig. 5. Axial variations of swirl number for S0 = 0.2 .
Fig. 6. Axial variations of swirl number for S0 = 0.5 .
Fig. 4. Axial variation of radial position of maximum
swirl velocity; (a) S0 = 0.2 , (b) S0 = 0.5 .
within the area bounded by these curves. Thus, one
conclude that the swirl decays more strongly as the
relative inlet vortex core radius ( a / R ) increases. The
local maximum percentage deviations of the swirl
number are computed depending on the velocity profile
shape according to
− S a =0.9 R
S
% ∆S = 100 x a =0.1R
S0 / 2
(5)
Axial variations of the percentage deviations of the swirl
number for both cases are shown in Figure 7. One can
see that the axial variation of the percentage deviation in
local swirl number is quite similar for S0 = 0.2 and
S0 = 0.5 , qualitatively and quantitatively.
Figure 7. Axial variations of percentage deviation
of local swirl number depending on inlet swirl velocity
profile shape (Eq.(5)), for S0 = 0.2 and S0 = 0.5 .
Proceedings of the 5th IASME / WSEAS International Conference on Fluid Mechanics and Aerodynamics, Athens, Greece, August 25-27, 2007
The maximum deviation can be observed to be as high
as 20% for both cases. The predicted deviations are not
necessarily small, and give an impression of the error
which can result by a straightforward application of
existing swirl decay correlations, without knowing the
details of the swirl velocity profile shape. It indicates
that the accuracy of the swirl decay correlations may be
increased, when they are additionally sensitised to the
swirl velocity profile shape.
6 Conclusion
Influence of inlet swirl velocity profile on swirl decay in
steady-state, incompressible pipe flows is investigated
by CFD. Since results by RANS turbulence models were
unsatisfactory, the analysis is carried out for laminar
flow, for Re=2000, at the present stage. The swirl
number is used to indicate the swirl level at a pipe cross
section. Results show that swirl decay is a function of
the shape of the inlet swirl velocity profile. Present
results obtained for laminar flow can be seen to imply a
possibly similar behavior for turbulent flows. Thus, one
can expect that the accuracy of the swirl decay
correlations, normally based on the swirl number, may
be increased, if they are additionally sensitized to the
shape of the inlet swirl velocity profile. Dependency of
swirl decay on inlet swirl velocity profile in turbulent
pipe flows will be investigated in the future work using
more sophisticated modeling strategies such as LES.
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