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Prediction of 3D Turbulence Induced Secondary Flows
in Rotating Square Ducts
Suabsakul Gururatana1, Vejapong Juttijudata2, Ekachai Juntasaro3 & Varangrat Juntasaro1
1Department of Mechanical Engineering, Faculty of Engineering, Kasetsart University, Bangkhen,
Bangkok 10900, Thailand, Email: fengvrj@ku.ac.th
2Department of Aerospace Engineering, Faculty of Engineering, Kasetsart University, Bangkhen,
Bangkok 10900, Thailand, Email: fengvej@ku.ac.th
3School of Mechanical Engineering, Institute of Engineering, Suranaree University of Technology,
Nakhon Ratchasima 30000, Thailand, Email: junta@sut.ac.th
Keywords: Cooling turbine blades, Reynolds-averaged Navier-Stokes (RANS), Rotating square duct
Abstract: The cooling turbine blades are one of the challenging problems at present. This is because the
characteristics of the flow field in the internal passage are affected by the combined effects of the secondary
flows occurred in the passage and the rotation of the blades. These secondary flows have the great effect on the
characteristics of the primary flow and the cooling rate. The accurate modeling of the flow in the near wall
regions is therefore necessary. The Reynolds-averaged Navier-Stokes (RANS) turbulence models are more
suitable than the large-eddy simulation (LES) in this case especially for the design optimization. There are
currently three popular near-wall turbulence models: the enhanced wall function, the non-equilibrium wall
function and the k-ω SST model. This paper aims to find the most suitable linear base model for the explicit
algebraic Reynolds stress models (EARSM) and the non-linear turbulence models in predicting the combined
effects of the rotating and secondary flows especially near the wall. From the results, it can be concluded that the
k- ε turbulence model with the enhanced or non-equilibrium wall function is more suitable to be further
developed for the EARSM or non-linear turbulence models.
In the aircraft industry, one of the major concerns
is the gas turbine engine. The increase in the efficiency
of the engine causes the temperature of the gas raisen.
Because the turbine blades are limited to the material
manufacturing, the cooling turbine blades are necessary
for heat reduction. The important flow characteristic in
the cooling turbine blades is the combined effects of the
rotating and secondary flows. The rotating square duct
is used as a test case to represent this combined effect
in the current work.
In the case of the rotating square ducts, the
structure of the secondary flows is modified by the
effect of the Coriolis force due to the rotation. The
cross-sectional area that is perpendicular to the
steamwise direction is constituted of four counter-
rotating vortices instead of eight vortices as found in
the non-rotating square duct. Recently, Martensson et al.
(2005) has discussed that the cross-stream flows
become more asymmetric and the vortex near the top
walls (pressure side) increases as the rotating number
increases. The magnitude of the secondary flows grows
approximately linearly with the rotating number.
The modeling of the turbulent flows in rotating
square ducts is still not satisfactory even for the
advanced Reynolds-averaged Navier-Stokes (RANS)
turbulence models like Reynolds stress models (RSM),
explicit algebraic Reynolds stress models (EARSM)
and non-linear turbulence models. Pettersson Reif and
Andersson (2003) has pointed out that the Reynolds
stress models cannot predict the turbulence quantities
in rotating ducts correctly. Belhoucine et al. (2004) has
proposed the new EARSM model for the rotating
square ducts. However, the model still fails to predict
the Reynolds stresses and the velocity profiles
accurately especially in the near wall regions.
At present, the turbulence models that have been
proved to be good at predicting the near wall regions
are the k- ε model with the enhanced wall function
(Kader 1993), the k- ε model with the non-equilibrium
wall function (Kim and Choudhury 1995) and the k- ω
SST model (Menter 1994). This paper aims to find the
linear model that has higher accuracy than the other
linear models in predicting the combined effects of
rotating and secondary flows to be used as the based
model for the non-linear turbulence models. The
chosen test case for the evaluation is the turbulent
flows through a rotating straight square duct at
Reynolds number = 48,000 and rotating numbers =
0.0133, 0.0266 and 0.12 (Pallares and Davidson 2000).
The results of the enhanced wall function, the non-
equilibrium wall function and the k- ω SST model are
compared with the LES data of Pallares and Davidson
(2000) and also with the result of the EARSM model of
Belhoucine et al. (2004) in predicting the combined
effects of the rotating and secondary flows. The
comparison is made where the data are available for the
mean streamwise velocity along the width of the duct
from the suction side to the pressure side (z/h=0.5) at
Ro = 0.0133 and 0.12, and along the corner bisector at
Ro = 0.0266 in Figs. 1 to 3, respectively.
It can be seen in Fig. 1 that the velocity profile
near the pressure side (y/h=1) using the EARSM model
is closer to the LES data than the other models as
expected. This is because there are the large secondary
flows on this side and it is well known that the linear
models cannot predict the secondary flows. However,
the linear models can predict the velocity profile better
than the EARSM model in the near wall region on the
suction side (y/h=0) where there are the small
secondary flows. Amongst the near-wall linear models,
the k- ε model with the enhanced wall function shows
the better results than the k-ω SST model.
The predicted streamwise velocity profiles by the
EARSM and the linear models on the pressure side and
the suction side at the higher rotating number (Ro =
0.12) are similar to those at the lower rotating number
as can be seen in Fig. 2. The non-equilibrium and the
standard wall functions are better than other linear
models at the core of the duct in this case for the higher
rotating number.
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8 1
y/h
U/u
LES, ( Pallares and Davidson 2000)
Standard k-epsilon
Enhanced wall function k-eps ilon
Non-equilibrium wall func tion k-epsilon
k-omega SST
EARSM, (Belhoucine et al. 2004)
τ
Fig. 1 Mean streamwise velocity at z/h=0.5 for
Ro=0.0133
0
5
10
15
20
0 0.2 0.4 0.6 0.8 1
y/h
U/u
LES, (Pallares and Davidson 2000)
Standard k-epsilon
Enhanced wall function k-epsi lon
Non-equilibrium wall function k-epsilon
k-omega SST
EARSM, (Belhoucine et al. 2004)
τ
Fig. 2 Mean streamwise velocity at z/h=0.5 for
Ro=0.12
In Fig. 3, it is clearly seen that the EARSM
model cannot predict the streamwise velocity in
the near wall region along the corner bisector of
the duct. The non-equilibrium wall function, the
enhanced wall function and the k-ω SST models
perform much better in this case.
0
3
6
9
12
15
18
21
24
27
0 0.1 0.2 0.3 0.4 0.5
y/h
U/u
LES, ( Pallares and Davidson 2000)
Standard k-epsilon
Enhanced wall function k-epsilon
Non-equilibrium wall f unction k-epsilon
k-omega SST
EARSM, (Belhoucine et al. 2004)
τ
Fig. 3 Mean streamwise velocity along the corner
bisector at Ro = 0.0266
The performance of the popular near-wall
turbulence models to predict the combined effects of
the rotating and secondary flows is assessed in this
paper and compared with the result of EARSM model
and the LES data for the rotating square ducts at three
rotating numbers. The near-wall turbulence models
even in the linear form perform better than the EARSM
model in some cases especially in the near wall region.
It is therefore important to have the suitable linear
model as the base for the non-linear or EARSM models.
It is found that the k-ε model with the enhanced or
non-equilibrium wall function performs better than the
k- ω SST model.
Belhoucine, L., Deville, M., Elazehari, A. R., and
Bensalah, M. O., 2004, Explicit Algebraic
Reynolds stress Model of Incompressible
Turbulent Flow in Rotating Square Duct,
Computers & Fluids, 33, pp.179-199.
Kader, B., 1993, Temperature and Concentration
Profiles in Fully Turbulent Boundary Layers, Int. J.
Heat Mass Transfer, 24, pp.1541-1544.
Kim, S. E. and Choudhury, D., 1995, A Near-Wall
Treatment Using Wall Functions Sensitized to
Pressure Gradient, In ASME FED, 217.
Martensson, G. E., Brethouwer, G., and Johansson, A.
V., 2005, Direct Numerical Simulatiom of Rotating
Turbulent Duct Flow, The Fourth International
Symposium on Turbulence and Shear Flow
Phenomena (TSFP4), June 27-29 , Virginia,
U.S.A. , pp.911-916.
Menter, F. R., 1994, Two-Equation Eddy-Viscosity
Turbulence Models for Engineering Applications,
AIAA Journal, 32, pp.1598-1605.
Pallares, J., and Davidson, L., 2000, Large-eddy
Simulations of Turbulent Flow in a Rotating
Square Duct, Phy. Fluids, 12, pp.2878-2894.
Pettersson Reif, B. A., and Andersson, H. I., 2003,
Turbulent Flow in a Rotatind Duct : A Modelling
Study, J. of Turbulence, 4.
