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VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF AN AXIAL HYDRAULIC TURBINE
A. F. Tiago
Mechanical Engineering Department Instituto Superior Técnico
Universidade Técnica de Lisboa Av. Rovisco Pais, 1, 1049-001 Lisboa
e-mail: [email protected] Abstract
The paper reports a numerical study of the flow through a 0.5 m diameter axial hydraulic turbine
rotor. The two-dimensional flow around the blade cascade was modelled using a panel method.
Pressure distribution, lift and drag coefficients results are presented for incidence angle values
close to the design condition, assuming inviscid and viscous flow. The latter, includes the
boundary-layer calculation according to a weak viscous-inviscid interaction formulation. Results
are presented for several rotor-blade sections located at different radial positions.
The three-dimensional viscous flow was computed using the FLUENT code. The Spalart-
Allmaras, the standard k-ε and the k-ω (proposed by Wilcox) turbulence models were used. Non-
structured and structured meshes were tested, considering a maximum of about 2x106 elements.
A mesh dimension dependence study was performed. Pressure, velocity and angular momentum
distributions at the inlet and outlet sections and on the blade surface, obtained with the different
turbulence models, are presented and compared with the results from three-dimensional inviscid
simulations. Pressure distributions on the blade surface were also compared with two-
dimensional inviscid (design condition) and viscous results.
Keywords: rotor, axial turbine, FLUENT, panel method
Nomenclature A area c chord
DC drag coefficient
LC lift coefficient
PC pressure coefficient
0C total pressure coefficient
D outer diameter DH hydraulic diameter Din inner diameter H static head I turbulence intensity k angular momentum
gn number of meshes
P static pressure
relP0 relative total pressure Q mass flow volume r radius Re Reynolds number U∞ external velocity ( )φU uncertainty
aV axial velocity
rV radial velocity
refV reference velocity
Vθ tangential velocity
θW relative tangential
velocity υ kinematic viscosity ρ density μ dynamic viscosity
MΔ estimated error φ numerical solution
exactφ exact solution Γ swirl number β radial component angle η efficiency
aζ loss coefficient α angle of attack
Introduction The paper reports the numerical study performed to validate the tubular-type hydraulic turbine
rotor design method described in [1], considering as test case a turbine designed to be used in a
small hydroelectric power plant.
Two methods were used: a two-dimensional cascade panel method coupled with a weak
viscous-inviscid boundary-layer calculation[2]; and a three-dimensional incompressible viscous
flow calculation, solving the Reynolds averaged Navier-Stokes (RANS) equations, using the
FLUENT code (6.2.16 version) [3], with meshes generated with the GAMBIT 2.2 code [4].
Design Method and Geometry Description The turbine blades are contained between two approximately cylindrical surfaces with a
common axis, which coincides with the turbine’s rotation axis. The outer diameter is D = 0.5 m
and the rotor diameter ratio is Din/D = 0.428.
The rotor blades were designed using the streamline curvature throughflow method for the
axis-symmetric calculation and a first-order panel method for the calculation of the flow between
the blades [5].
The cylindrical section radius was assumed to be the meridian flow streamlines average radius
at the rotor inlet and outlet sections. The rotor’s inlet section tangential velocity component is Vθ =
K/r, in which K is the inlet section angular momentum, constant along the radius. In the outlet
section, a zero tangential velocity component was assumed and, consequently, the angular
momentum is zero.
For the design condition, a nominal flow rate Q=3.362 m3, a static head H=75.6 m and a
rotation speed of 2500 r.p.m were imposed [5]. Air at ambient pressure and temperatures was
assumed in the performed simulations. The Reynolds number is approximately 4.6x106.
Two-dimensional Method (Panel Method) In the two-dimensional study, the panel method was used with a boundary-layer computation
based on the weak viscous-inviscid interaction theory. This inviscid flow solution is based on the
Laplace equation using sources and vortices superficial distributions. The airfoil is discretized
using flat panels with constant superficial sources distributions in each panel [6]. The normal
velocity component on the panel is equal to zero for the purely potential flow, in a first calculation.
However, considering the boundary-layer effect, this is equal to the wall-transpiration velocity [2].
As for the viscous flow solution, the influence of the wake is not taken into account and, as a
consequence, only the laminar region and the turbulent boundary-layer are considered [2].
The inviscid and viscous solutions compatibility is obtained by an iterative calculation using the
“weak interaction” theory.
The twelve profiles used in the rotor blade geometry definition were discretized using 121
panels on the pressure and suction sides, obeying a cosine function distribution. The Reynolds
number υcU∞=Re , the chord-to-pitch ratio, the cascade stagger angle and the attack angle are
the code entry variables. Except for the attack angle, only design values were simulated.
Three-dimensional Method In this study, the flow solution was obtained using the FLUENT 6.2 code [3]. This code solves
the Navier-Stokes equations, for the mean time-dependent values (Reynolds equations).
The turbulence is modelled by different turbulence viscosity models: the standard k-ε model [7]
(used for high Reynolds numbers), the k-ω model proposed by Wilcox [8] and the
Spalart-Allmaras model [9].
The segregated solver was used to solve the flow equations (continuity and momentum). This
solver uses smaller computational resources, and is more suited, as referred in [3], for
simulations using a rotating coordinates system.
Several algorithms can be used to solve the velocity-pressure coupling. The SIMPLEC [10]
algorithm was selected because it is more consistent than the SIMPLE algorithm [3].
The FLUENT code solves the linearized form of the discretized transport equations, using a
Gauss-Siedel point-by-point solution algorithm, together with an algebraic multiple mesh method.
The equations’ diffusive terms are discretized using the second-order central-differences scheme
[3]. The convective terms are interpolated by the third-order QUICK [11] scheme. Among the
different schemes that can be used to obtain the pressure on the control volume faces, the
second-order interpolation scheme was selected [3]. Simple precision variables were used in the
performed simulations.
Relative pressures were used in the computations to minimize the round off error effect. The
reference pressure was defined at the inlet section, with a standard value of 101325 Pa. The air
density and viscosity were assumed constant, and their values at ambient temperature (300K),
are: ρ = 1.225 kg/m3 e μ = 1.789 x 10-5 Pa.s.
In the simulations, five different types of boundary conditions were used: fluid, velocity inlet,
outflow, periodic surfaces and solid surfaces.
Due to the hub rotation, it is necessary to define the rotating coordinates system. To do this, in
the fluid’s boundary condition, a 2500 r.p.m rotation velocity is imposed. At the inlet boundary the
velocity components for the absolute cylindrical referential system are imposed. This boundary
condition is identified in the code FLUENT as velocity inlet. At the inlet the radial velocity
component is null, the axial velocity component is constant, 96.20=AQ m/s and the tangential
velocity component is rV 491.2=θ m/s. The tangential velocity and the axial velocity components
are obtained from the angular momentum and the design flow rate, respectively. The inlet section
turbulence values are obtained from the turbulence intensity, I , and the hydraulic diameter,
286.0=−== inH DDPAD m. For the turbulence intensity, a value of 4% was used, as
suggested in [3].
On the outlet section, the free flow condition was used. This condition, identified as outflow in
the code, corresponds to a variables zero degree extrapolating, from the interior of the domain to
the boundary, followed by a mass flow rate correction.
On the solid surfaces the impermeability condition was imposed for all the simulations.
However, tanking into account the rotation of the domain, it was necessary to define the rotation
of the solid boundaries. On the outer boundary, on the hub inlet and outlet region a zero rotational
velocity was defined. On the central region of the hub and on the pressure and suction faces of
the blade a 2500 r.p.m. velocity was imposed. On the periodic boundaries, the rotational
periodicity was imposed.
Mesh Generation In order to do the three-dimensional simulation, structured hexahedral and non-structured
tetrahedral meshes were built.
The computational domain for the flow between blades was obtained from the boundary lines of
the meridian flow. The radial coordinate ranges from 0.250 to 0.107 m and the axial coordinate
ranges from 0.670 to 1.670 m. The inlet boundary is 1.5 chords upstream of the leading edge and
the outlet boundary is 3 chords downstream of the trailing-edge. The blades shape was defined
from the twelve project sections divided into suction and pressure sides.
For the three turbulence models tested, different mesh thicknesses near the walls were used,
due to different the near-wall treatments. For the k-ε standard model the wall functions are
applied, so the first element height must be included in the logarithmic region, i.e. 30030 << +y .
For the k-ω and Spalart-Allmaras models, the meshes should be thin enough to solve laminar
sub-layer, 5<+y .
To facilitate the mesh generation, the tip clearance was ignored, modifying the load on this
blade region.
For the k-ε standard model four geometrically similar structured hexahedral meshes were
generated, with 129168, 319158, 831138 and 2152516 elements, respectively. Also for this
model, four non-structured tetrahedral meshes were generated, with 148318, 334002, 797531
and 1936342 elements. For the Spalart-Allmaras and k-ω models four geometrically similar
structured hexahedral meshes were generated, with 127188, 299782, 765824 and 2033040
elements, respectively.
Numerical Error Estimation and Solution Convergence
There are three types of error that influences the numerical solution of a computational fluid
dynamics problem: round-off error, iterative error and discretization error [13, 14]. If the round-off
and the iterative errors are negligible compared to the discretization error the numerical
uncertainty is only due to this last term.
The numerical uncertainty estimation allows us to determine an uncertainty band that includes
the exact solution:
( ) ( )φφφφφ UU exact +≤≤− . (1)
The method used in this study, proposed by Eça [14, 15], estimates the error, MΔ , as the
maximum difference between the available results:
( ) gijM nji ≤≤−=Δ ,1max φφ . (2)
This method is appropriate to complex flows due to the almost inevitability of scattered results.
When monotonic convergence is not observed the uncertainty is estimated by [14]:
( ) MU Δ= 3φ . (3)
The numerical errors for the three turbulence models were compared for some computed
integral parameters. The considered variables were: the swirl number, Γ , and the radial
component angle average values, β , at the inlet and outlet; the efficiency, η ; and the loss
coefficient, aζ .
For the k-ω model four structured meshes were used. However, a convergence solution was
only obtained with two of them, the 127188 and 765824 element meshes, respectively. In those
cases the error estimation is quite limited due to the use of only two mesh solutions.
Table 1 presents the swirl number, Γ , numerical uncertainty for the simulations made. The
uncertainties for the other parameters are presented in [12].
Γinlet Γoutlet
Models Ø1 U1 |U1/Ø1| Ø1 U1 |U1/Ø1|
k-ε Structured 0,63196 0,000334 0,000528 0,0551952 0,05073 0,91908 k-ε Non-Structured 0,63221 0,000925 0,001464 0,0517542 0,03280 0,63366 Spalart-Allmaras 0,63083 0,0015323 0,002429 0,039447 0,00524 0,13271
Project 0,63199 0
Tab. 1 – Discretization error numerical uncertainty for the three models used.
The designt value, for this variable is within the uncertainty band of the obtained solutions
11 U±φ , except for the outlet swirl number. For the Spalart-Allmaras model, the uncertainty
assumes relatively lower values in relation to the k-ε model.
Results For the two-dimensional method, the drag coefficient is obtained by the Squire & Young
approximations [2], cUcU
LCL∞∞
Γ−=−=
2
21 2ρ
.
Figure 1 shows the friction coefficient evolution as function of the drag coefficient for the
sections where separation doesn’t occur. In most cases, the design attack angle is on the low
range of drag coefficient, named laminar bucket, however, for the sections with radius equal to
0.1945 m and 0.2065 m, nothing could be concluded because the computation couldn’t continue
for the respectively design attack angles.
CD
CL
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-0.2
0
0.2
0.4
0.6
0.8
1
R=0.148 F. RealR=0.165 F. RealR=0.1805 F. Real
CD
CL
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-0.2-0.1
00.10.20.30.40.50.60.70.80.9
1
R=0.1945 F. RealR=0.2065 F. RealR=0.219 F. Real
CD
CL
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-0.2-0.1
00.10.20.30.40.50.60.70.80.9
1
R=0.230 F. RealR=0.2405 F. RealR=0.250 F. Real
(a) (b) (c)
Fig. 1 – Evolution of the friction coefficient as function of the drag coefficient on the sections where separation doesn’t
occur.
α
CL
-4 -3 -2 -1 0-0.5
0
0.5
1
1.5
2
2.5
R=0.107 F. PerfeitoR=0.118 F. PerfeitoR=0.1295 F. Perfeito
α
CL
-4 -3 -2 -1 0-0.5
0
0.5
1
1.5
2
2.5
R=0.1805 F. PerfeitoR=0.1805 F. Real
α
CL
-4 -3 -2 -1 0-0.5
0
0.5
1
1.5
2
2.5
R=0.250 F. PerfeitoR=0.250 F. Real
(a) (b) (c)
Fig. 2 – Evolution of lift coefficient on the range of α ’s used in some sections
Figure 2 presents the evolution of the lift coefficient in the range of α used for some sections,
for ideal and real fluids cases without flow separation. The evolution of the lift coefficient for the
other sections is presented in [12].
The evolution of the lift coefficient with α is approximately linear in the attack angles range
considered. A linear evolution of the lift coefficient with α is also well verified on real flow for the
range of low attack angles, but with an lower rate than the ideal fluid model.
The drag coefficients become higher in the proximity of the hub. This increase is caused by the
decrease of the chord on those sections and the constant angular momentum imposed along the
radius.
For the three-dimensional method, the presented results were computed with the finest meshes
for which the obtained results were considered enough accurate: The non-structured mesh with
1936342 elements and structured mesh with 2152516 elements for the k-ε model; the structured
mesh with 2033040 elements for the Spalart-Allmaras model; and the structured mesh with
765824 elements for the k-ω model. This last mesh presents a lower refinement when compared
with the others; it corresponds to the finest mesh for which the desired convergence was
obtained.
D*
Va*
0 0.25 0.5 0.75 10.990
0.995
1.000
1.005
1.010
K-Epsilon Não EstruturadaK-Epsilon EstruturadaK-OmegaSpalart-Allmaras
D*
Va*
0 0.25 0.5 0.75 10.90
0.92
0.94
0.96
0.98
1.00
1.02
1.04
K-Epsilon Não EstruturadaK-Epsilon EstruturadaK-OmegaSpalart-Allmaras
(a) (b)
Fig. 3 – Axial velocity axis-symmetric averages: (a) inlet section; (b) outlet section.
The axial velocity component axis-symmetric averages, *aV , on the inlet section are present in
figure 3a). The values show an increase close to the solid boundaries (hub and casing). This
increase is more evident on the k-ω and Spalart-Allmaras models than on k-ε model, because in
the first ones the laminar sub-layer is solved and in the last one the wall functions are applied.
This velocity increase is caused by the boundary condition imposed at the inlet – uniform axial
velocity – to guarantee the desired mass flow.
The axial velocity axis-symmetric averages profiles at the outlet section (figure 3b)) present
shape closer to the flow inside a duct with the viscous effects near the walls visible.
The angular momentum on the outlet section shows small differences between the turbulence
models. Near the hub, its value is zero or slightly negative and near the casing it presents a small
region with positive values; this means that not all the energy was extracted from the flow.
Figure 5 presents the pressure coefficient distribution on the twelve project sections. The
pressure coefficient is calculated by 20
21
ref
relp
V
PPC
ρ
−= , where
222
222 ⎟⎟⎠
⎞⎜⎜⎝
⎛ ++⎟⎟
⎠
⎞⎜⎜⎝
⎛ ++⎟⎟
⎠
⎞⎜⎜⎝
⎛ += rOrIOIaOaI
refVVWWVV
V θθ and each referred variable represents the
circumferential average for the respective radius coordinate.
(a) (b)
(c) (d)
Fig. 4 – Angular momentum distribution on the outlet section: (a) non-structured k-ε model; (b) structured k-ε model; (c) k-ω model; (d) Spalart-Almaras model.
The distribution on the hub is the one that differs most for the three turbulence models used,
the largest difference occurs between the leading edge and the suction peak. In the intermediate
sections a small difference is verified only in the leading edge area [12]. For the casing section,
the distributions are significantly different and vary according to the turbulence models near wall
treatment.
Between R = 0.107 m and R = 0.219 m the pressure coefficient presents a load exchange
close to the leading edge and in those cases the stagnation point is positioned on the suction side
of the blade [12]. The peak suction is predicted between 50% and 75% of the chord, and moves
towards the trailing edge direction as the radius diminishes.
X/C
Cp
0 0.25 0.5 0.75 1-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5R=0.107 Não Estruturada k-epsilonR=0.107 Estruturada k-epsilonR=0.107 k-omegaR=0.107 Spalart-Allmaras
X/C
Cp
0 0.25 0.5 0.75 1-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5R=0.1805 Não Estruturada k-epsilonR=0.180.5 Estruturada k-epsilonR=0.1805 k-omegaR=0.1805 Spalart-Allmaras
X/C
Cp
0 0.25 0.5 0.75 1-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5 R=0.250 Não Estruturada k-epsilonR=0.250 Estruturada k-epsilonR=0.250 k-omegaR=0.250 Spalart-Allmaras
(a) (b) (c)
Fig. 5 – Pressure coefficient distribution in three sections for the used models.
For the three turbulence models used, the k-ω model was the one that presented the biggest
difficulty to obtain convergence. The residual instability required more computation time and, in
some cases, it was not possible to get convergence. Compared to the k-ω model, the k-ε model
does not need such thin meshes near the solid boundaries, so it requires less calculation time.
However, for details on the walls, this model may not be as efficient. Compared to the last two
models, less computational resources are required for the Spalart-Allmaras model and, at the
same time, this model can solve the laminar sub-layer.
For the tested cases, the Spalart-Allmaras model was the one that presented more stability on
the calculation, allowing us to conclude that it has better performance for these types of studies.
So, considering the flow characteristics, it is reasonable to consider that the Spalart-Allmaras
model produces good quality results, and so, only this model results will be compared with the
two-dimensional method ones.
The total pressure coefficient, 2
00
21
refVPP
Cρ
∞−= , for the Spalart-Allmaras model at the inlet and
outlet sections, respectively, are present in figure 6. At the inlet section, the effect of the boundary
layer is visible. The non-uniformity of the pressure fields at the outlet section central region
suggests that the distance of this section to the trailing edge should be increased.
(a) (b)
Fig. 6 –Total pressure coefficient distribution on Spalart-Allmaras model: (a) inlet section; (b) outlet section.
Two-dimensional and Three-dimensional Methods Comparison For the comparison of the two models used, the pressure coefficient distribution on the same
blade profiles shown on the previous section is presented in figure 7.
The project values were determined by the panel method with ideal fluid [5], as expected they
coincide with the results obtained by the two-dimensional study for ideal fluid which uses the
same method.
X/C
Cp
0 0.25 0.5 0.75 1-1
-0.5
0
0.5
1
1.5
2
2.5 R=0.107 Spalart-AllmarasR=0.107 F. PerfeitoR=0.107 ProjectoR=0.107 Invíscido
X/C
Cp
0 0.25 0.5 0.75 1-1
-0.5
0
0.5
1
1.5
2
2.5R=0.1805 Spalart-AllmarasR=0.1805 F. PerfeitoR=0.1805 F. RealR=0.1805 ProjectoR=0.1805 Invíscido
X/C
Cp
0 0.25 0.5 0.75 1-1
-0.5
0
0.5
1
1.5
2
2.5R=0.250 Spalart-AllmarasR=0.250 F. PerfeitoR=0.250 F. RealR=0.250 ProjectoR=0.250 Invíscido
(a) (b) (c)
Fig. 7 – Pressure coefficient distribution for some profiles using the Spalart-Allmaras and the bi-dimensional models.
The differences between the inviscid [5] and the viscous simulations are not significant, except
for the sections coincident with the hub and the casing. These differences are due to the viscous
effects at those solid boundaries. The load exchange on the leading edge is more visible for the
viscous simulation.
There is a good agreement between three-dimensional and two-dimensional flow simulations in
the central zone of the blade. These results diverge in the hub and in the casing adjacent zones.
Conclusions From the three turbulence models used in the three-dimensional method, k-ω model was the
one which presented more difficulties in converging. The residuals’ instability required more
calculation time. For this model, in some cases, the solution convergence was not possible to
achieve.
The k-ε model using the standard wall function, comparatively to k-ω model, does not need
very refined meshes near the walls, so less computation time is required. However, the quality of
the solution near the walls may be less accurate.
Compared to the last two models, less computational resources are required for the
Spalart-Allmaras model. This model, as the k-ω model, can also solve the laminar sub-layer. For
the cases tested, the Spalart-Allmaras model was the one that presented more stability on the
calculation, allowing us to conclude that it has better performance for these types of studies.
The differences between the inviscid [5] and viscous simulations are not significant, except for
the sections coincident with the hub and the casing.
The design values were determined by the panel method considering ideal fluid [5], and as
expected they coincide with the two-dimensional study results obtained for ideal fluid which uses
the same method.
For a significant part of the flow in the middle of the blade span, the three-dimensional method
may be approached to the two-dimensional, which on a first stage of the design may be a very
useful tool, since it has a much lower demanding level.
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