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Numerical computations of turbine blade aerodynamics;

comparison of LES, SAS, SST, SA and k-ε

Carlos Velez1, Patricia Coronado

2, Husam Al-Kuran

3 and Marcel Ilie

4

University of Central Florida, Orlando, FL, 32826

Numerical investigations of turbine blade are carried out using large-eddy simulation

(LES), Scale Adaptive Simulation (SAS), k-ε with extended wall function, Spalart-Allmaras

(SA), and Shear Stress Transport (SST). The goal of the present studies is to investigate the

turbine blade aerodynamics. The simulations are performed for a Reynolds number, Re =

3.67 x 106, based on the chord, c, of the airfoil and free-stream velocity. The computational

results reveal the dissipative nature, of SAS, associated with the turbulence modeling.

Cp = pressure coefficient

Cl = lift coefficient

Cs = Smagorinsky constant

c = chord of airfoil

Dwall = van Driest wall damping function

T = subgrid scale viscosity

ij = subgrid scale tensor

I. Introduction

HE boundary layer for the flow through a Low-Pressure Turbine (LPT) cascade is transitional in nature and the

transition location is not known a-priori. Furthermore, the separation process is highly unsteady with a wide

variation in the separation location. Both these factors tend to limit the predictive capability of the RANS approach

for this flow. Furthermore, conventional RANS simulations provide information only about the mean flow field, and

only limited insight regarding the dynamics of the unsteady separation process can be gained from these simulations.

Developments in computer technology hardware as well as in advanced numerical algorithms have now made it

possible to perform very large-scale computations of these turbine flow fields. Numerical methodologies based on

the large-eddy simulation (LES) technique have emerged as a viable means of investigating the transitional flow

through a LPT. In LES, the large-scale motion is simulated accurately, and the so-called subgrid-scales (SGS) are

modeled. Recent numerical studies of flow in a LPT used LES in conjunction with upwind-biased schemes.

Fujiwara et al. (2002) investigated the unsteady suction side boundary layer of a highly loaded low-pressure

turbine blade, TL10. Simulations were performed using a low-Reynolds number k-ε model and also compressible

LES with the Smagorinsky SGS model. The numerical computations, using the low Re k-ε model, were assumed to

be two-dimensional and steady, whereas the large-eddy simulations were three-dimensional and unsteady. For LES,

the three-dimensional compressible Navier-Stokes equations were solved by evaluating the convective terms using a

third-order upwind biased scheme and evaluating the viscous terms using a second-order central-difference scheme.

The study concerned the Reynolds number effect on the blade aerodynamics. Reynolds number, based on the axial

chord and exit velocity, varied in the range (0.99 ÷ 1.76) x105. The study showed that LES can predict the boundary

layer separation and reattachment process, and its Re-number dependence, while the 2D steady simulation with a k-ε

model cannot capture these flow phenomena. However, some difference between LES and experimental data were

1 Master Student, Department of Mechanical, Materials & Aerospace Engineering, AIAA Student Member

2 PhD Student, Department of Mechanical, Materials & Aerospace Engineering, AIAA Student Member

3 PhD Student, Department of Mechanical, Materials & Aerospace Engineering, AIAA Student Member

4 Assistant Professor, Department of Mechanical, Materials & Aerospace Engineering, AIAA Member

T

47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit31 July - 03 August 2011, San Diego, California

AIAA 2011-5880

Copyright © 2011 by Marcel Ilie. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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observed at the reattachment point. Raverdy et al. (2003) employed the monotonically integrated large-eddy

simulation (MILES) approach to predict the transition process and its interaction with the wake dynamics for a

subsonic turbine blade configuration. The three-dimensional unsteady filtered Navier-Stokes equations were solved

using the finite-volume solver FLU3M, developed by ONERA. No explicit sub-grid scale model was used.

However, the numerical dissipation of the modified AUSM + (P) upwind scheme used to discretize the Euler fluxes

was assumed to transfer the energy from large scales to the small scales at a rate nearly equivalent to the one

provided by a conventional SGS model. The simulations were carried out for the T106 low-pressure turbine blade at

Re = 160,000 based on the exit isentropic velocity. They investigated the influence of the mesh resolution and the

spanwise extent, and concluded that the mean and turbulent quantities compared well with the available

experimental data. Although the numerical results, obtained using upwind-biased schemes, were in good agreement

with the experimental data, the energy in a substantial portion of the resolvable wave number range was damped out

due to the excessive numerical dissipation of the scheme.

According to Mittal and Moin (1997), LES gives good prediction only when is used in conjunction with energy-

conserving schemes such as spectral schemes or high-order central difference schemes. Mittal et al. (2001)

performed a computational study of a flow through a LPT cascade using LES with a dynamic SGS model. The study

employed a completely non-dissipative, mixed finite-difference–spectral spatial discretization scheme, with a

second-order central difference scheme and a Fourier spectral method. Simulations were carried out at Reynolds

numbers of 10,000 and 25,000 based on inlet velocity and axial chord. The spatial and temporal variations of the

flow through a LPT cascade were investigated by examining the mean streamwise velocity profiles and temporal

variation of the instantaneous streamwise velocity along various locations on the suction surface as well as in the

very near wake of the blade. It was observed that at relatively low Reynolds numbers, about 10,000, the dynamics of

the separation phenomenon on the suction surface is governed by the Karman-vortex type shedding behavior in the

wake, but this behavior was not observed for Re = 25,000. The study also concluded that, with the increase of

Reynolds number, the vertical and streamwise extent of the separation bubble on the suction surface decays. It is

worth to notice that the comparisons were based only on the LES results at two different Reynolds numbers, and no

comparison with the experimental data was performed.

Rizzetta and Visbal (2003) conducted a comprehensive numerical study to investigate the subsonic transitional

flow through a LPT cascade using the implicit large eddy simulation (ILES) technique. The ILES technique is

similar to the monotonically integrated large-eddy simulation (MILES). Unlike MILES, which introduces artificial

viscosity by applying a dissipative scheme to discretize the Navier-Stokes equations, in ILES, the truncation errors

of the higher-order accurate dispersive scheme are used to dissipate turbulent energy at the sub-grid scales which

cannot be supported by the computational grid. These studies were carried out for three different Reynolds numbers

2.5x104, 5.0x10

4 and 10.0x10

4. The study revealed the existence of differences between the computed blade-surface

pressure distribution and experimental data, and these differences were associated with the details of the

experimental configuration which were not accounted in the simulations. One of the main conclusions of this study

was that even two-dimensional simulations might provide useful information for preliminary design and parametric

studies.

In spite of the extensive LES studies, the numerical computations of LPT aerodynamics using LES still suffers

from the high grid refinement for resolving boundary layer. In the present study we employ the LES, SAS, Shear

Stress Transport (SST), k-ε and Spalart-Allmaras (SA) models. The comparison of the five turbulence models is

present in this study.

The structure of the paper is as follows. In Section 2 the computational method and models are introduced with

details regarding the numerical approach and computational domain. Section 3 presents the numerical results of the

aerodynamic and aeroacoustic studies. The conclusions regarding the present study are summarized in Section 4.

II. Computational Method and Models

A. Computational Method

In the present work, a LES, SAS, SST, k-ε and SA approaches are used for simulations. The quasi 3-D models

were performed for Re = 3.67x106, based on free stream velocity U and the chord length of the airfoil. The flow

field is solved using the filtered Navier-Stokes equations along with a standard subgrid scale (SGS) model and van

Driest wall damping.

The boundary conditions were assigned as follows. No slip boundary conditions are used at the airfoil wall. Free

slip boundary conditions are used at the top and bottom walls with opening at the end of the computational domain.

Large Eddy Simulation is a result of space averaging operation applied to Navier-Stokes equations. The filtered

Navier-Stokes equations are given by:

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0

i

i

x

u (1)

jj

i

j

ij

i

ji

j

i

xx

u

xx

puu

xt

u

2

Re

1 (2)

where ij is the subgrid scale (SGS) given by:

jijiij uuuu (3)

and is modeled. In the present work the SGS proposed by Smagorinsky and Lilly20-21

is used. In the present analysis

the value of Smagorinsky constant was set to 0.1. The SGS stresses are related to the strain rate tensor by SGS

viscosity, T:

ijTkkijij S 23

1 (4)

The SGS viscosity T is given by:

SDC wallsT

2)( (5)

where Cs is the Smagorinsky constant, Dwall represents the van Driest wall damping factor , is the filter width and

S represents the magnitude of the large-scale strain-rate tensor.

j

j

j

iij

x

u

x

uS

2

1 (6)

B. Simulation Domain

The Aachen 1-1/2 turbine rig stage is used as the case geometry. The turbine’s characteristic dimensions for the

rotor and stator are listed below in Table 1.

Table 1. Characteristic dimensions of blade geometry

Blade Characteristic Rotor Stator

Aspect Ratio 0.917 0.887

Pitch 41.8 mm 47.6mm

Blade # 41 36

RPM 3500 0

Tip Diameter 600mm 600mm

The models requires the definition of a y+ value in order to calculate the adequate distance of the nearest grid

point from the wall in order to resolve the boundary layer thickness. The y+ value is calculated through equation 7.

(7)

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Where, is the distance of the nearest grid point to the wall, is a reference velocity of the flow, is the

kinematic viscosity, is the reference length, and is a non-dimensional value. The corresponding values are

listed below in Table 2.

Table 2. Reference values for calculation

109.9557 m/s

0.3 m

1.038e-5

1

7.8e-6 m

A structured mesh, as seen in Fig. 1, is created with the corresponding y+ value and a span wise expansion ratio

of 1.309 is used to expand the cell width away from the boundary layer region. The entire mesh consists of 1.203e6

nodes split up into three separate blocks for each blade row. The stator blocks consists of 364k nodes each and the

rotor block consists of 475k nodes. The code allows for multi-staged blocks to interface between rotating and non-

rotating rows via a mixing region located at the interface of the stages.

Figure 1. Meshed turbine domain

The models key boundary conditions and settings are summarized below in Table 3.

Table 3. Model Description

Fluid Type Air (perfect Gas)

Axial , Radial, Tangential Flow velocity 109.95, 0, 0 m/s

Inlet Total Pressure 169,500 Pa

Inlet Total Temperature 900 K

Outlet Static Pressure 90,000 Pa

Turbulence Coefficients (k, ) 5

& 30,000

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III. Results and Discussion

Five different turbulence models were investigated and compared, LES, SAS, k-ε with extended wall function,

Spalart-Allmaras (SA), and Shear Stress Transport (SST), to study the effect on the temperature and pressure fields

of turbine blades. The simulations were carried out using in-house codes as well commercial CFD software

NUMECA, which is turbo machinery oriented software. NUMECA was used to carry out the SA, SST and k-ε

simulations, while an in-house code was used to simulate the LES and SAS models.

t = 0.1s

t = 0.2s

t = 0.3s

t = 0.4s

t = 0.5s

Figure 2. Time-dependent velocity magnitude, LES solution

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Figure 2 presents the time-dependent velocity magnitude from the LES computations at different instants in

time. The analysis of the velocity magnitude reveals the presence of the wake and the flow separation. The flow

accelerates at the upper surface of the airfoil and thus, a high velocity is identified at this location. As the flow

develops the separation line moves upward towards the leading edge of the airfoil. Highly turbulent flow is

identified at the trailing edge in the separation region. From Figure it can be seen that the wake remains highly

turbulent.

Figure 3 presents the wake evolution. The computations were carried out by means of Direct Numerical

Simulation using an Adaptive Mesh Refinement (AMR) technique. The analysis reveals the presence of relatively

large vertical structures in the wake of the blade. The flow separation region is well captured as well and in good

agreement with the LES computations.

t = 0.025s

t = 0.030s

t = 0.035s

t = 0.040s

t = 0.045s

Figure 3. Time evolution of the blade wake (DNS computations)

Figure 4 presents the comparison of the LES and SAS computations, based on the velocity magnitude. Overall,

the LES and SAS computations are in good agreement. However, the study reveals that dissipative nature of SAS.

Thus, the wake is smaller in the SAS computations when compared with the LES ones. The differences are well

captured in the vector field as well, as shown in Figure 5.

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a) LES b) SAS

Figure 4. LES/SAS comparison (magnitude velocity)

a) LES b) SAS

Figure 5. LES/SAS comparison (vector field)

Figures 6 and 7 provide a better insight into the differences between LES and SAS. The point wise comparisons

are conducted in the separation region where the difference between the two models is most significant. This

comparison emphasizes the dissipative nature of the SAS method. Although the computational time was reduced by

17% the SAS lacks of capturing the wake. The wake is an important component especially when the rotor-stator-

rotor configuration is employed.

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Figure 6. Comparison of velocity magnitude

Figure 7. Comparison of pressure

0

50

100

150

200

250

300

350

0 0.001 0.002 0.003 0.004 0.005 0.006

Ve

loci

ty [

m/s

]

t [s]

LES

SAS

-80000

-60000

-40000

-20000

0

20000

40000

60000

0 0.001 0.002 0.003 0.004 0.005 0.006

Pre

ssu

re [

Pa}

t[s]

LES

SAS

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Figure 8 presents the spanwise velocity, from LES computations. The analysis reveals a high velocity region at

the upper surface of the airfoil at the leading edge region. This trend remains valid for almost half of the airfoil

chord. In the second half chord of the airfoil the flow separation takes place and this is illustrated by the low values

of velocity, in this region vortices start to form. Close to the trailing edge the vortex formation become dominant

mechanism and thus enhanced flow separation is observed.

x/c = 0.001

x/c = 0.001

x/c = 0.001

x/c = 0.001

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x/c = 0.001

x/c = 0.001

Figure 8. Spanwise velocity magnitude (LES computations)

Figure 9 presents the static temperature distribution of the blade using k-ε with extended wall function. The

results show an average blade temperature of 890.122 K and an average blade pressure of 135,877 Pa, showing a

higher blade temperature at the leading edge. The static temperature distribution of both the trailing edge

and leading edge blade using k-ε are illustrated in Fig. 10.

The static temperature distribution of the blade using SA is presented in Fig. 11. The results show an average

blade temperature of 884.959 K and an average blade pressure of 136,042 Pa. The average temperature of the blade

is lower than that of the k-ε but still showing a higher blade temperature at the leading edge. The static

temperature distribution of both the trailing edge and leading edge blade using SA are illustrated in Fig.

12.

Figure 13 presents the static temperature distribution of the blade using SST. The results show an average blade

temperature of 885.685 K and an average blade pressure of 136,015 Pa, showing a higher blade temperature at the

leading edge. These values are relatively close to those for SA. The static temperature distribution of both

the trailing edge and leading edge blade using SST are illustrated in Fig. 14.

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Figure 9. Static temperature distribution of blade using k-ε

Figure 10. Static temperature distribution of LE and TE blade using k-ε

Leading Edge (LE)

Trailing Edge (TE)

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Figure 11. Static temperature distribution of blade using SA

Figure 12. Static temperature distribution of LE and TE blade using SA

Leading Edge (LE)

Trailing Edge (TE)

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Figure 13. Static temperature distribution of blade using SST

Figure 14. Static temperature distribution of LE and TE blade using SST

Leading Edge (LE)

Trailing Edge (TE)

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t=0.00011

t=0.00023

t=0.00034

a) SST- EWF b) SA c) k

Figure 15. Turbulence Model Comparison of Static Pressure

Figure 15 presents the comparison of static pressure for the SST, SA and k-ε turbulence models at three different

time steps. There is not a significant change in the results between the models. The main difference to note between

turbulence models was the time that the simulations took to be completed, the fastest one being the k-ε, then the SA

and finally the SST with extended wall function.

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t=0.000116

t=0.000232

t=0.000348

a) SST b) SA c) K-

Figure 16. Turbulence Model Comparison of Static Temperature

Figure 16 presents the comparison of static temperature for the SST, SA and k-ε turbulence models at three

different time steps. The results indicate that both the SA and k-ε turbulence models show a larger heat flux close to

the walls, where as the SST model has a more uniform heat distribution throughout the blade. Additionally, the SST

model shows a large drop in temperature towards the trailing edge of the rotor blade that is not found in the SA and

k-ε results.

IV. Conclusions

Numerical computations, using LES, SAS, SST, k-ε and SA approaches, are conducted to investigate the

aerodynamics of turbine blade. For all the computations we will employ the LES based models since it provides a

better prediction of the flow field. The dissipative nature of the SAS is evident in the results when compared to

turbulence models which model the SGS.

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References 1Papadakis, M., Nizampatnam L., and Hoffmann, K., “Computational Investigation of Blade Vortex Interaction

Noise", AIAA Paper 99-0231, 38th

Aerospace Sciences Meeting and Exhibit, Reno (NV), Jan. 10-13, 2000. 2Singer B.A., Brentner K.S., Lockard D.P. and Lilley G.M. "Simulation of Acoustic Scattering from a Trailing

Edge", AIAA Paper 99-0231, 37th Aerospace Sciences Meeting and Exhibit, Reno (NV), Jan. 12-15, 1999.

3Singer B.A., Lockard D.P., Brentner K.S. "Computational Aeroacoustic Analysis of Slat Trailing- Edge

Flow", AIAA Journal, Vol. 38, No. 9, pp. 1558-1564, September, 2000. 4Abelló, J. C., and George, A.R., "Rotorcraft BVI Noise Reduction by Attitude Modification," paper presented

at the 5th AIAA/CEAS Aeroacoustics Conference, Bellevue, Washington, May, 10-12, 1999 5Abello, J., George, A., Wake Displacement Study of Attitude and Flight Parameter Modifications to Reduce

Rotorcraft Blade-Vortex Interaction (BVI) Noise, AIAA 2003-3174, 9th AIAA/CEAS Aeroacoustics Conference,

12-14 May 2003,Hilton Head, SC 6Becker, S., Kaltenbacher, M., Ali, I., Escobar, M., Hahn, C., Sound Generation by Flow around Simple

Geometries: Simulation and Experiment, 12th AIAA/CEAS Aeroacoustics Conference, 8-10 May 2006, Cambridge 7Felten, F., Lund,T., Numerical Simulation of parallel Airfoil/Vortex Interaction Using a Zonal Hybrid

RANS/LES Method, AIAA2005-5127 8Nagarajan, S., Lele, S., Prediction of Sound Generated by a Pitching Airfoil: A comparison of RANS and LES,

12th AIAA/CEAS Aeroacoustics Conference, 8-10 May 2006, Cambridge, MA 9Wang, M., Moin, P., Computation of Trailing-Edge Flow and Noise Using Large-Eddy Simulation, AIAA

Journal, Vol.38, No.12, 2000 10

Magagnato, E., Sorguven, E., Gabi, M., Far Field Prediction by Large Eddy Simulation and Ffowcs -Willimas

Hawkings Analogy, AIAA Paper 3206-2003, May 2003 11

Lardeau, S., Leschziner, M.A., Unsteady Reynolds–Averaged Navier-Stokes Computations of Transitional

Wake/Blade Interaction, AIAA, Vol.42, No.8, 2004, pp. 1559-1571 12

Bernandini,G., Serafini,J., Gennaretti, M., Aeroelsatic Modeling Effect in Rotor BVI Noise Prediction, 12th

AIAA/CEAS Aeroacoustics Conference, 8-10 May 2006, Cambridge, MA 13

Lyrintzis, A., Xue, Y., Study of Noise Mechanisms of Transonic Blade-Vortex Interactions, AIAA Journal,

Vol.29, No.10, 1990, pp.1562-1572 14

Johnson, W., Calculation of Blade–Vortex Interaction Airloads on Helicopter Rotors, Journal of Aircraft ,

Vol.26, No.5, 1989, pp.470-475 15

Seath, D., Kim, J., Wilson, D., Investigation of Parallel Blade-Vortex Interaction at Low Speed, Journal of

Aircraft , Vol.26, No.4, 1989, pp.328-333 16

Oh, W., S., Kim, J.S., Kwon, O.J., Numerical Simulation of Two-Dimensional Blade-Vortex Interactions

Using Unstructured Adaptive Meshes, AIAA Journal, Vol. 40., No.3, 2002, pp. 474-480

17Strawn, R. C., Ahmad, J., Duque, E. P. N, “Rotorcraft Aeroacoustics Computations with Overset Grid CFD

Methods,” 54th

AHS Annual Forum, Washington DC, May 22-24, 1996. 18

Horner, M., Galbraith, R., Coton, F., Examination of Vortex Deformation during Blade-Vortex Interaction,

AIAA Journal, Vol.34, No.6, 1996, pp.1188-1194 19

Hu, H., Jordan,L., CFD Investigation of Double Swept Blade in BVI Noise Reduction, 9th AIAA/CEAS

Aeroacoustics Conference, 12-14 May 2003,Hilton Head, SC 20

Smagorrinsky, J.S., General circulation experiments with the primitive equation, Monthly Weather Rev.91:99-

164, 1963 21

Lilly, D.K., On the application on the eddy viscosity concept in the inertial sub-range of turbulence, NCAR

Manuscript, pp195-200, 1996 22

M. Germano, U. Piomelli, P. Moin, and W. H. Cabot, A dynamic subgrid-scale eddy viscosity model. Phys.

Fluids A, 3(7):1760–1765, 1991