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NUMERICAL ANALYSIS OF THE HEAT TRANSFER IN A
TRAILING EDGE COOLING DUCT IN STATIONARY AND
ROTATING CONDITIONS
A. Andreini - C. Bianchini - B. Facchini
Department of Energy Engineering “Sergio Stecco”
Via Santa Marta, 3 - 50139 Firenze, Italy
Email: [email protected]
ABSTRACT
The accurate prediction of the heat transfer in the internal channel of a turbine blade represents
a key factor for the evaluation of the entire cooling system performance. This objective requires
to assess the available numerical and physical models against reliable experimental results to
determine their strength and weakness as well as the field of application of the adopted models.
This paper presents a numerical analysis aimed at estimating the heat transfer coefficient over
the heated endwall of a trailing edge internal cooling system. The coolant is blown from a radial
inlet and redirected towards the axial outlet by means of a wedge shaped converging duct with
7 pedestals.
The domain is tested under both stationary and rotating conditions at Re of 20000 based on
inlet hydraulic diameter. The effect of rotation is evaluated for rotational numbers up to 0.275
at three different angular velocities. Two different blowing conditions were investigated namely
the open and the closed tip condition, to assess the effect of different incident angle on the
pedestal leading edge. A constant heat flux is imposed on the heated wall while the pedestals
and the other solid surfaces are modeled as adiabatic.
Computations were performed using commercial and open source unstructured finite volume
codes to benchmark the accuracy for steady state analysis in the stationary and rotating frame
of reference.
The domain consists of a 6 million cells multiblock-structured mesh with fine near wall clus-
tering allowing the integration of turbulence quantities up to the wall. Turbulence is modeled
adopting the k − ω SST RANS model available in both codes.
Results are compared with experimental measurements for the stationary case in terms of heat
transfer coefficient maps over the endwall and spanwise profiles inside the pedestal region.
1
NOMENCLATURED pedestal leading edge diameter [mm]Dh inlet hydraulic diameter [mm]H1 initial duct height [mm]H2 final duct height [mm]K1 linear loss coefficient [kg m−3s−1]K2 quadratic loss coefficient [kg m−4]L pedestal lenght [mm]k turbulent kinetic energy [m2/s2]p static pressure [Pa]Py interpedestal pitch [mm]
Q̇ wall heat flux [W/m2]Re Reynolds number Re = ρ · Ub ·Dh/µ [−]Ro Rotational number ωDh/Ub [−]T static temperature [K]U velocity magnitude [m/s]Ub bulk velocity [m/s]v y velocity component [m/s]x axial coordinate [mm]y quasi-radial coordinate [mm]z quasi-tangential coordinate [mm]Greeks
ρ density [kg/m3]µ dynamic viscosity [kg/ms]ω rotational speed [rad/s]δ Kronecker delta [−]Acronyms
HTC heat transfer coefficient Q̇/(Tw − Tref ) [W/m2K]PIV particle image velocimetry
RANS Reynolds Averaged Navier-Stokes
Subscripts
app apparent
body volumetric forces
i, j generic vector component
L0 entrance region
L1 TE wedge shaped region
meanX corresponding to the mean x line
meanY corresponding to the mean y line
w wall
INTRODUCTION
Today advances in aero-engine design are mainly driven by the requirement of higher thrust/weight
ratios, reduction of fuel consumption and pollutant emissions. To achieve these challenging goals,
manufacturers have generally increased the operating temperature, i.e. the turbine inlet temperature,
that in turn has prompted them to use efficient cooling systems and thermal barrier coatings for the
components that are subjected to high thermal loads.
Among these engine components, the trailing edge is one of the most critical part of gas turbine
2
blades or vanes, due to quite strict aerodynamic, thermal and structural requirements, therefore a very
efficient cooling system is necessary to keep metal temperature below critical values. Internal cooling
schemes aiming at augmenting the heat transfer from the solid parts to the cooling fluid, often relays
on sets of turbulators useful to promote high level of turbulence. Such turbulators, generally named
pedestals, are circular or oblong pin fins that serve also as structural devices. In brief, coolant system
design must adequately trade off increased heat transfer rates and minimal pressure loss.
The heat transfer onto a pinned surface has been quite extensively studied with numerical methods,
especially regarding staggered and in-line configuration of variously shaped pins (square, circular, ob-
long and diamond shaped). After the early work of Donahoo et al. [2001] and Hamilton et al. [2002],
in which the effect of spacing was assessed on a constant section duct in terms of heat dissipation
and required pumping power using a steady solver with RANS modeling and wall functions, Saha
and Acharya [2004] performed an extensive unsteady analysis with the k-ε turbulence model, using a
LES simulation for validation purposes, on a square pin array. Recently Delibra et al. [2009] studied,
with both a wall resolved 4 equation eddy viscosity model and a LES computation, the heat transfer
and flow field around a periodic array composed of 8 rows of eight circular pins with partially suc-
cessful results in terms of heat transfer predictions. D’Agaro and Comini [2008] included the effect
of heat conduction inside the pin by means of a conjugate simulation of a bi-periodic pinned flat plate
in a staggered configuration. Only few attempts however could be found investigating pin arrays per-
formances considering the effect of mixed axial-radial inflow conditions [Kulasekharan and Prasad,
2008] and variable cross section ducts [Di Carmine et al., 2008] and more recently Bianchini et al.
[2010b].
The effect of rotation on the flow and the heat transfer at the trailing edge has been studied com-
putationally in the past for both radially oriented [Prakash and Zerkle, 1992] and U-bended ducts
[Iacovides et al., 2001], while no significant literature was found for wedge shaped duct with mixed
axial-radial flow under rotation. The same can be stated concerning turbulence promotors like en-
larged pedestals: typical turbolators investigated under the effect of rotation are in fact mostly ribs
[Schuler et al., 2010, Prakash and Zerkle, 1995] or dimples [Tafti and Elyyan, 2010].
This paper aims at studying the heat transfer performance of a wedge-shaped converging duct with
seven pedestal aligned with the outflow under rotating conditions. Present study takes into account
the axial redirection of the inlet radial flow and the effect of the tip mass flow rate as well. For this
purpose a closed tip condition was considered together with a 5 holes tip discharging at ambient pres-
sure too. Numerical calculations were performed fixing the inlet velocity profile to obtain a Reynolds
of 20000 in the feeding channel. Rotational speeds investigated range from 0 to 25[rads−1] obtaining
a maximum rotational number of 0.275.
Each numerical calculation was reproduced twice using different CFD codes: the commercial code
CFX R© v. 11 and the open source code OpenFOAM R©. Such comparison permitted to asses the predic-
tion capabilities in terms of agreement with available experiments under stationary conditions and to
evaluate the different implementations of the rotational effects.
INVESTIGATED TEST CASES
The investigated geometries accurately reproduce the experimental rig described in Bianchini
et al. [2010a]. It basically consists of a main channel receiving air in the radial direction and turning
the flow towards the trailing edge in which seven pedestals were inserted in the vicinity of the trailing
edge. The feeding plenum was excluded from the computational domain being available adequate
boundary conditions at the inlet channel.
The duct used to confine the air flow around the pedestals is composed by 2 parts: L0 and L1 as
depicted in Fig.1(a).
The entrance region L0 consists of a smooth constant height duct and of a redirecting element that
3
(a) Domain overview (b) Wedge shaped duct (c) Tip configurations
Figure 1: Computational domain definition
produce a reduction of the passage area towards the tip outlet channel and bends the air stream in the
trailing edge direction. The inlet hydraulic diameter Dh = 58.18 [mm] is defined at the hub section.
Region L1 is a converging duct with angle α = 10 [deg] and initial height H1 = 33 [mm], as pointed
out in Fig.1(b).
The seven pedestals are equally spaced inside the L1 region with a span-wise pitch of Py =75 [mm]. The diameter of each pedestal end is D = 12 [mm] while its length L = 84 [mm]In the experimental analysis only one side was heated and measured that is the flat part of the geometry
where optical access was easier, conventionally taken as the pressure side.
Two different tip geometries were tested: the first corresponding to a smooth plate completely
closing the outflow the second one composed of 5 parallel cylindrical holes allowing some coolant
discharge as depicted in Fig.1(c).
The axis of rotation is oriented along the Z axis (outward pointing in Fig.1(a)) and is placed on a
line corresponding to the middle of the inlet surface at distance of 198 [mm]. The trailing edge outlet
normal vector lays on plane orthogonal to the rotation axis. This choice was driven by a simpler
experimental set-up and the consideration that real engine first stage blades are often designed with
very high curvature resulting in high outlet blade angle.
NUMERICAL TOOLS AND METHODOLOGY
Solvers specification
Numerical simulations were conducted using both the commercial solver Ansys R© CFX R© 11.0 and
an in-house developed CFD solver based on the open-source toolbox for continuum mechanics called
OpenFOAM R©. The extended release of OpenFOAM R© version 1.5 was used in this work. Steady-
state assumption and RANS turbulence modeling were exploited in order to reduce the cost of the
simulation up to an affordable level.
Concerning CFX R© simulations: energy was solved in terms of total enthalpy for the stationary
case and of rothalpy for the rotating case since Navier-Stokes equations were solved in the rotating
4
frame of reference. Convective terms were discretized following the High Resolution scheme that is
a bounded second order upwind scheme.
The in-house code differs from the previous solver mainly for the treatment of pressure momen-
tum coupling: CFX R© uses a coupled algorithm while the OpenFOAM R© C++ library, implements finite
volume discretization aimed at solving pressure and momentum in a segregated way [Weller et al.,
1998]. The compressible Navier-Stokes Equations are solved using a SIMPLE-like (Semi-Implicit
Method for Pressure-Linked Equations) algorithm with a convective diffusive equation for the pres-
sure correction to impose mass continuity considering density variation [Andreini et al., 2007]. Con-
vective schemes uses a second order upwind interpolation scheme based on the NVA (Normalized
Variable Approach) known in literature as Self Filtered Centered Difference [Jasak, 1996] blended
with a deferred approach with a first order upwind scheme. The rotating simulations were performed
in the rotating frame of reference solving for relative velocity and rothalpy as well.
Turbulence was modeled in both codes by means of the k − ω SST model with an hybrid near
wall treatment in CFX R© and a purely Low Reynolds implementation in OpenFOAM R©.
Momentum source terms
Apparent forces
Since Navier-Stokes equations were solved in a non-inertial frame of reference, as described in
Eq.1 where the unknown U is the relative velocity, additional source terms should be included to take
into account apparent forces.
∂(ρUi)
∂t+∂(ρUiUj)
∂xj
+∂p
∂xi
−∂
∂xj
(
µ
(∂Ui
∂xj
+∂Uj
∂xi
)
+2
3µδij
∂Uj
∂xj
)
= Fapp,i + Fbody,i (1)
In case of steady rotation the source terms take the form:
−→F app =
−→F Coriolis +
−→F Centrifugal = −2 ρ−→ω ∧
−→U − ρ−→ω ∧ (−→ω ∧ −→r ) (2)
These external contribution were computed explicitly in the in-house code.
Porous media
Due to the quite low pressure drop and the high aspect ratio of the duct, as soon as rotational
speed exceeds critical values of around 9[rad/s] air is ingested from the trailing edge outlet section.
Such recirculation starts at the low radii because of the stream-line migration towards the tip due to
the centrifugal effects. In order to avoid such entrainment, a 20 [mm] thick porous media was inserted
at the trailing edge discharge section. The higher aerodynamic resistance permits a pressurization of
the channel ensuring a more uniform outflow split along the radius. For the open-tip the same kind
of material guarantee an equivalent pressure drop at the tip outlet with the aim of not completely alter
the tip to trailing edge air split.
The behaviour of this foam was simulated inserting additional body forces in the momentum
equation neglecting the effect of porosity. For the in-house code such additional external forces were
modeled as isotropic imposing frictional stresses aligned with velocity as described in Eq.3:
−→F body =
−→F linear +
−→F quadratic = −K1
−→U −K2 ‖
−→U ‖−→U = −(K1 +K2 ‖
−→U ‖)
︸ ︷︷ ︸
diagonal coefficient
−→U . (3)
Such choice makes it easier to discretize these source terms implicitly, the diagonal coefficient is
reported in Eq.3, resulting in an increased stability.
For CFX instead, a directional loss model was imposed to enhance the flow alignment with the
trailing edge direction. The linear and quadratic coefficients are hence vectors, see Eq.4, whose
5
normal losses were fixed 105 higher than the parallel one resulting in additional counter pressure in
case of misaligned flow as reported in Eq.4. In the case of trailing edge foam for example−→K1 =
[K1x; 105 ·K1x; 105 ·K1x].
Fbody,i = Flinear,i + Fquadratic,i = −K1,iUi −K2,i‖−→U ‖ Ui. (4)
However since the flow itself is already almost aligned with the outlet normal direction without
the porous media as well, no meaningful difference in the two treatments was pointed out.
The effective capacity of the porous media to avoid the entrainment depends obviously on the
numerical values chosen for the constantsK1 andK2. The higher those values the higher the rotational
speed at which ingestion occur. At the same time increasing the pressure drop across the material
would affect the mass flow distribution through the interpedestal ducts. The sizing of such material
was thus performed numerically to match the incipient ingestion condition at the highest rotational
speed investigated. Such preliminary analysis performed with CFX under isothermal conditions found
out the optimal values of K1 = 1000 [kg m−3 s−1] and K2 = 50 [kg m−4].
COMPUTATIONAL SETTINGS
Computational grid
The mesh is a multiblock structured mesh composed by 6.2 · 106 hexahedral cells. It is the final
step of a mesh independence analysis aimed at verifying the stabilization of the recirculation bubble
downstream the sharp corner at the lowest radius and on the first pedestal. Correct near-wall integra-
tion is guaranteed by y+ < 1 and 15-20 elements inside the thermal boundary layer on all viscous
surfaces. An overview of the element size and distribution around and on the pedestal is given in
Fig.2.
(a) Grid distribution near pedestal leading edge (b) Grid spacing on the pedestal
Figure 2: Computational grid details
Boundary conditions
The boundary conditions applied followed a classical scheme for incompressible and LowMach
number flow simulations. A constant velocity is imposed at the inlet surface while ambient static
pressure (101325 Pa) is maintained at the outlet. Inlet velocity profile was extracted from the exper-
imental data obtained by Armellini et al. [2010] on an equivalent test rig. Two measurement lines
were set up along the symmetry planes of the inlet section. The time averaged velocity recorded at
these locations, see Fig.3, were used to perform a 2D interpolation to reconstruct normal velocity for
every inlet boundary face following Eq.5.
vinlet(x/D, z/D) = vmeanX(z/D) · vmeanZ(x/D)/vmeanZ(10). (5)
6
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Figure 3: Experimental inlet velocity profiles from Armellini et al. [2010]
The resulting Reynolds number is 20000 based on inlet hydraulic diameter, corresponding to a bulk
velocity of approximately 5.2[ms−1]. On viscous walls, no slip and adiabatic conditions apply except
for the heated surface on which a constant heat flux (Q̇ = 1500[W/m2]) is guaranteed by a Neumann
boundary condition for temperature.
In order to reproduce the same rotational conditions of real engines the dimensionless parameter
to monitor is the rotation number:
Ro =CoriolisForces
InertialForces=ωDh
Ub
. (6)
Typical values of investigated Ro for internal cooling ducts range between 0.1-0.5 as reported in
Acharya [2010] and Han et al. [2000]. In this case three different rotational speed are investigated,
namely 9, 18 and 25 [rad/s], corresponding to Ro = 0.1, 0.2 and 0.275 respectively.
RESULTS AND DISCUSSIONS
Stationary Case
Before proceeding with the analysis of the rotational effects, the cooling device was simulated
under stationary conditions. Since experimental results were available for such conditions without
the porous media at the outlets [Bianchini et al., 2010a], a preliminary run was performed with no
additional pressure drop in order to validate the predicting capabilities against detailed experiments.
Figures 4 and 5 represent the heat transfer coefficient distribution on the pressure side for the open
and closed tip conditions respectively.
Both conditions show a low heat transfer zone inside the tip cavity, and similar structures in the
first 7 inter pedestal channels and in L0 region. In particular it is possible to note the heat transfer
enhancement on the pedestal leading edge due to the horse-shoe vortex that is convected both in the
pressure and suction side of the pedestal. This effect is maximum for the second pedestal and decrease
towards the higher radii. The imprint on the suction side is limited by the development of a separation
bubble caused by the high angle of attack.
Heat transfer is then increased inside the passage by migration of air from the above pressure
side towards the suction side. The first inter pedestal channel shows a much lower heat transfer level
compared to the other channels due to the big recirculation developing as a consequence of the sharp
corner at L1 inlet.
7
(a) EXP (b) CFX (c) OF
Figure 4: Heat transfer coefficient maps for open tip condition
(a) EXP (b) CFX (c) OF
Figure 5: Heat transfer coefficient maps for closed tip condition
As it is possible to observe comparing the different maps, CFX R© predicts higher heat transfer
level compared to both OpenFOAM R© and experiments, especially it is strongly overpredicting the
effect of the horse shoe vortex. Even though experiments were not directly available for the internal
flow field, the agreement between the two CFD codes is almost complete limiting imprecisions to the
recirculation bubbles in the tip zone and in the first channel. Thus the cause for partial disagreements
8
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Figure 6: Heat transfer coefficient profiles inside L1 - Open tip condition
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Figure 7: Heat transfer coefficient profiles inside L1 - Closed tip condition
in terms of heat transfer should be addressed to the different near wall modeling.
To better quantify the agreement between the codes and the experiments, profiles of heat transfer
coefficient inside L1 region are reported in Figs.6 and 7. Reference x is taken at L1 entrance, x/D =
-2 corresponds to the pedestal leading edge, and X/D = -6 at mid channel position. It is possible to
note how both numerical predictions overestimate leading edge peaks of heat transfer even though the
relative intensity of such peaks is quite well captured. Inside the channels, Figs. and , the agreement
between computations and experiments increases although CFD still maintains higher values close
to the pedestal and lower values close to the mid-channel line. Comparing CFX and OF results, the
former predict much higher peaks at the pedestal corner, however are pointed out by CFX while it
is in a better agreement with the experiments concerning the far from the pedestal zones. At mid
pedestal position OF shows a definitely better agreement with the experiments especially for the open
tip condition.
Stationary analysis was repeated inserting the porous media near the outlet to avoid entrainment
in case of rotation. Apart from a reduction in the dimension of the recirculation bubble in the first
channel no particular dependency is noted for this condition. As a reference, maps of heat transfer
coefficient are reported in Fig.8 for the open tip condition.
9
(a) CFX (b) OF
Figure 8: Heat transfer coefficient maps for open tip condition with porous media
Rotating Cases
The effect of rotation was taken into account studying the behaviour at three different angular
velocities: 9, 18, 25[rad s−1]. As shown in Figs.10-11, increasing rotational speed leads to higher heat
transfer level in the central part of L0 but also to a larger low heat transfer zone near the redirection
device equally for both configurations. For L1 zone the distribution of heat transfer coefficient is
affected as well. The higher radial velocity acquired by the air increases the angle of attack on
the pedestals reducing the heat transfer augmentation along the pressure side and incrementing the
dimension of the separation bubble on the suction side. The suction side part of the horse shoe vortex
is thus bended towards the pressure side. The recirculation bubble in the first duct increases in size,
reaching an almost incipient entrainment from the outlet as this was the chosen criteria to dimension
loss coefficients. Furthermore the augmented heat transfer coefficient zone on the leading edge of the
pedestals is expanded inside L0 region.
The centrifugal effect also increases the heat transfer in the tip region especially on the side facing
the trailing edge and moves the center of the recirculation bubble on the redirecting device towards
lower radii outside the tip zone.
Fig.13 reports the mass of air blown in each channel for CFX computations under isothermal
conditions, in terms of air split. Since the mass flow is made dimensionless with respect to the inlet
mass flow rate, for the open tip the air splits for the 8 interpedestal ducts does not sums up to 1 due to
the tip outflow. With such reduction indeed the air split really represent the amount of air discharged
10
(a) 0 rad/s (b) 9 rad/s (c) 18 rad/s (d) 25 rad/s
Figure 9: HTC maps for open tip condition at various angular velocities - CFX
(a) 0 rad/s (b) 9 rad/s (c) 18 rad/s (d) 25 rad/s
Figure 10: HTC maps for open tip condition at various angular velocities - OpenFOAM
by the channel and comparisons between open and closed tip can be done.
Higher rotational speed promotes a non uniform distribution of mass flowing in various channels
increasing the air blowing at higher radii and reducing those close to the hub. This effect is maximal
for the open tip case due to the increase of the air blown at the tip. Central channel are less affected by
the rotation (i.e.air split for the fifth channel is basically constant). For rotational speed of 9 rad/s−1
no significant change is seen compared to stationary simulation.
11
(a) 0 rad/s (b) 9 rad/s (c) 18 rad/s (d) 25 rad/s
Figure 11: HTC maps for closed tip condition at various angular velocities - CFX
(a) 0 rad/s (b) 9 rad/s (c) 18 rad/s (d) 25 rad/s
Figure 12: HTC maps for closed tip condition at various angular velocities - OpenFOAM
CONCLUSIONS
In the present work, a numerical investigation was performed on the heat transfer capabilities of
an aero-engine blade cooling device consisting of a wedge shaped duct with 7 pedestals to drive the
flow towards the trailing edge. To replicate the cooling system of the blade in a realistic way, the
effect of axial redirection and tip mass flow rate were considered. Reference conditions were fixed at
Reynolds 20000 and two tips discharge section were investigated.
Analysis includes the effects of rotation up to Ro = 0.275 and the dimensioning of a porous media
at the outlets to avoid entrainment in case of rotation. Results are presented in terms of heat transfer
coefficient distribution on the heated wall and are compared with available measurements for the
stationary case.
Steady-state RANS calculations have been performed using two CFD codes, OpenFOAM R© and
12
Figure 13: Air split for the various inter pedestal channels
CFX R© 11.0, in order to allow a direct benchmark of the predicting capabilities for the heat transfer
under rotating conditions. Turbulence was modeled by means of the k − ω SST turbulence model,
mesh near wall clustering allowed a direct integration of turbulence up to the wall.
Comparison of numerical and experimental heat transfer results for the stationary case pointed
out that the CFD predictions are able to correctly reproduce the heat transfer coefficient distribution
on the heated wall. Overestimation of HTC in the proximity of the stagnation point on the pedestal
leading edge was instead reported.
The effect of the tip condition is in general limited to the final part of the main channel and the tip
region: the open tip configuration exhibits a higher dependency on rotational speed.
The effect of rotation is intense especially in the L0 zone including tip region, in the high radii
ducts and for the recirculation bubble in the near hub duct. Better tip cooling is observed at higher
rotational speed, while L0 showed an increasing background value but it pointed out a wider low heat
transfer region close to the redirect device. The heat transfer in the L1 is also affected by rotation,
13
with a reduction in the effect of the horse shoe vortex on the pedestal pressure side but increasing that
one on the suction side. Thus even though averaged heat transfer is definitely increased, certain area
might result critical also at high rotational speeds.
ACKNOWLEDGEMENTS
The authors would like to acknowledge Dr. L. Bonanni and PhD. C. Carcasci for sharing the ex-
perimental results, L. Andrei for the support given in the numerical simulations and PhD. A. Armellini
and PhD. L. Casarsa for providing the boundary conditions. The reported work was performed within
the Italian research project PRIN, the funding of the Italian ministry for education, university and
research (MIUR) is gratefully acknowledged. The other partners involved in the project, namely Uni-
versity of Udine, Genova, Bergamo and Cagliari, should be thanked for sharing the geometry set
up.
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