1. Influence of Laminar-Turbulent Transition on 3D Flow Pattern in Subsonic Turbine Cascade
Sergiy Yershov*, Anton Derevyanko*, Viktor Yakovlev
Institute for Mechanical Engineering Problems of National Academy of Sciences of Ukraine, Kharkiv, Ukraine (* formerly)
Maria Gryzun
National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine

2. Overview Introduction Mathematical model of the flow
The RANS equations
The k-ω SST turbulence model
The Production Term Modification model of transition
Numerical technique
Preliminary simulations of transitional flows
Zero-pressure gradient flat plate flows (ERCOFTAC test cases)
The subsonic compressor cascade (Zierke & Deutsch)
Subsonic turbine cascade flow (VKI-Genoa cascade)
Comparison with experimental results for midspan flow
Influence of transition on secondary flow pattern and cascade losses
Conclusions

3. Introduction
Turbomachinery flows at high Reynolds numbers are typically turbulent and characterized by high levels of turbulent fluctuations in both the boundary layer and the main flow core.
Nevertheless, at the leading edges, where the local Reynolds number is small, the regions of the laminar boundary layer and, consequently, the laminar-turbulent transition are usually present.
In the case of non-separated laminar boundary layer, skin friction and energy losses are significantly less when compared with turbulent boundary layer. Therefore, increasing length of laminar boundary layer region can improve efficiency of cascade flow.
On the other side, laminar boundary layer is more susceptible to separation. Thus, the losses may increase substantially if separation occurs.
Numerical simulation of transitional flows, determination of transition onset (TO), and estimation of energy losses enable the control of transitional flows and consequently the improvement of turbomachinery cascade efficiency.

4. Mathematical model of the flow
The RANS equations in a local curvilinear rotated coordinate system:

5. Turbulence modeling The k-ω SST turbulence model*:
*Menter, F. R., Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, Vol. 32, No 8, 1994

6. Turbulence modeling
The low-Reynolds modification* and the realizability constraint**:*
* D.C.Wilcox, Simulation of Transition with a Two-Equation Turbulence Model, AIAA Journal, Vol. 32, No 2, 1994
** X.Zheng, C.Liu, F.Liu, C.-I.Yang, Turbulent Transition Simulation Using The k-ω Model, Intern. J. Numer. Meth. Engng., Vol. 42, 1998

7. Transition modeling The Production Term Modification (PTM) model*:
* R.B.Langtry and S.A.Sjolander, Prediction of Transition for Attached and Separated Shear Layers in Turbomachinery, AIAA Paper 

8. Transition modeling The Production Term Modification (PTM) model*:
* R.B.Langtry and S.A.Sjolander, Prediction of Transition for Attached and Separated Shear Layers in Turbomachinery, AIAA Paper 

9. Transition modeling The Production Term Modification (PTM) model*:
influence of free stream turbulence intensity
* R.B.Langtry and S.A.Sjolander, Prediction of Transition for Attached and Separated Shear Layers in Turbomachinery, AIAA Paper 

10. Transition modeling The Production Term Modification (PTM) model*:
influence of adverse pressure gradient
* R.B.Langtry and S.A.Sjolander, Prediction of Transition for Attached and Separated Shear Layers in Turbomachinery, AIAA Paper 

11. Numerical approach
Implicit 2nd-order accurate scheme
Newton’s iterations
ENO reconstruction
Exact Riemann solver
Local time stepping
Simplified multigrid approach
Structural hexahedral mesh
In-house CFD code F (pre-processor, mesh generator, solver, post-processor)
ParaView visualisation

12. Methodology of the calculations
In the transition zone, flow is subjected to rapid changes within very short distances => we need good mesh resolution along streamwise and wall-normal directions.
y+≈1
at least 30 cells across the boundary layer
at least 150 cells along blade surface in streamwise direction
cell sizes in wall-normal direction must change smoothly
This all corresponds to meshing guidelines of Menter et al.*
These requirements lead to a few million cells per one blade channel
*F.R.Menter, P.E.Smirnov, T.Liu and R.Avancha, A One-Equation Local Correlation-Based Transition Model, J. Flow, Turbulence and Combustion, 2015, Vol. 95, No 4

13. Preliminary verification
The ERCOFTAC test cases for zero-pressure gradient flat plate flow*
Test case
Velocity, m/s
Tu, %
T3A
5,4
3,0
T3B
9,4
6,0
T3A-
19,8
0,9
♦♦♦ experiment
2D mesh: 80x120 = 9600 cells
HR k-ωSST sets “TO” almost at the LE
LR k-ωSST determines TO downstream, overestimates TKE upstream TO and underestimates TKE downstream TO
LR-PTM k-ωSST determines TO slightly downstream and underestimates TKE downstream TO
LAM – laminar (NS) HR – fully turbulent (RANS & HR k-ωSST) LR – transitional (RANS & LR k-ωSST) LR-PTM – transitional (RANS & LR-PTM k-ωSST)
*J.Coupland, Flat Plate Transitional Boundary Layers, ERCOFTAC Classic Collection database, 1990, available at

14. Preliminary verification
The ERCOFTAC test cases for zero-pressure gradient flat plate flow*
Test case
Velocity, m/s
Tu, %
T3A
5,4
3,0
T3B
9,4
6,0
T3A-
19,8
0,9
♦♦♦ experiment
2D mesh: 80x120 = 9600 cells
HR k-ωSST sets “TO” almost at the LE
LR k-ωSST determines TO downstream, overestimates TKE upstream TO and underestimate TKE downstream TO
LR-PTM k-ωSST determines TO properly and overestimates TKE upstream TO
LAM – laminar (NS) HR – fully turbulent (RANS & HR k-ωSST) LR – transitional (RANS & LR k-ωSST) LR-PTM – transitional (RANS & LR-PTM k-ωSST)
*J.Coupland, Flat Plate Transitional Boundary Layers, ERCOFTAC Classic Collection database, 1990, available at

15. Preliminary verification
The ERCOFTAC test cases for zero-pressure gradient flat plate flow*
Test case
Velocity, m/s
Tu, %
T3A
5,4
3,0
T3B
9,4
6,0
T3A-
19,8
0,9
♦♦♦ experiment
2D mesh: 80x120 = 9600 cells
HR k-ωSST sets “TO” almost at the LE
LR k-ωSST does not predict TO
LR-PTM k-ωSST determines TO close to experimental data
LAM – laminar (NS) HR – fully turbulent (RANS & HR k-ωSST) LR – transitional (RANS & LR k-ωSST) LR-PTM – transitional (RANS & LR-PTM k-ωSST)
LR-PTM k-ωSST works much better than LR k-ωSST for considered flow cases
*J.Coupland, Flat Plate Transitional Boundary Layers, ERCOFTAC Classic Collection database, 1990, available at

16. Preliminary verification
The subsonic compressor cascade*
3D mesh: 120x96x180 ≈ 2·106 cells
fully turbulent
transitional
*W.C.Zierke and S.Deutsch, The Measurement of Boundary Layers on a Compressor Blade in Cascade, NASA CR , 1989

17. 3D Flow in Turbine Cascade
The subsonic turbine cascade VKI-Genoa*
* M.Ubaldi, P.Zunino, U.Campora, and A.Ghiglione, Detailed Velocity and Turbulence Measurements of the Profile Boundary Layer in a Large Scale Turbine Cascade, ASME Paper 96-GT-42, 1996

18. 3D Flow in Turbine Cascade
H-type mesh with quasi-orthogonal cells in boundary layer zones, with refinement near walls and blade edges
128x128x256 ~ 4.2·106 cells
y+ ~ 1
30 cells across the BL
168 cells along the blade sides

19. 3D Flow in Turbine Cascade
Normalized isentropic velocity at the blade surface in the midspan section
The present numerical results are in a satisfactory agreement with the experimental data.
M.Ubaldi, P.Zunino, U.Campora, and A.Ghiglione, Detailed Velocity and Turbulence Measurements of the Profile Boundary Layer in a Large Scale Turbine Cascade, ASME Paper 96-GT-42, 1996

20. 3D Flow in Turbine Cascade
Mach number contours in the midspan section (HR k-ωSST)
fully turbulent
In the fully turbulent flow case the boundary layer at the suction side is thicker, comparing with the transitional flow case.

21. 3D Flow in Turbine Cascade
Mach number contours in the midspan section (LR-PTM k-ωSST)
transitional
In the fully turbulent flow case the boundary layer at the suction side is thicker, comparing with the transitional flow case.

22. 3D Flow in Turbine Cascade
Friction velocity at the blade suction surface in the midspan section
All computations predict a sharp increase in the friction velocity and a short transition zone; whereas, the experiment shows a more extended transition region at the suction side.
M.Ubaldi, P.Zunino, U.Campora, and A.Ghiglione, Detailed Velocity and Turbulence Measurements of the Profile Boundary Layer in a Large Scale Turbine Cascade, ASME Paper 96-GT-42, 1996
R.B.Langtry, A Correlation-Based Transition Model Using Local Variables for Unstructured Parallelized CFD Codes, Ph.D Dissertation, University of Stuttgart, Germany, 2006
P.Malan, K.Suluksna, and E.Juntasaro, Calibrating the γ-Reθ Transition Model for Commercial CFD, AIAA Paper

23. 3D Flow in Turbine Cascade
TKE contours in the midspan section (HR k-ωSST)
fully turbulent –
‘transition onset’
The high-Reynolds model predicts the growth of the TKE near the leading edge at the suction side and somewhat downstream of the leading edge at the pressure side. The boundary layer is turbulent at the both blade sides.

24. 3D Flow in Turbine Cascade
TKE contours in the midspan section (LR-PTM k-ωSST) –
transition onset
transitional
In the transitional flow case the transition onset is near the cascade throat at the suction side, and in the trailing edge separation zone at the pressure side. Similar to experimental data, the boundary layer is laminar along the whole pressure side.

25. 3D Flow in Turbine Cascade
TKE distribution at the distance of pitch from the blade suction side in the midspan section
fully turbulent
transitional
The fully turbulent flow case is characterized by a quick but continuous growth of the TKE somewhat downstream of the leading edge; whereas, in the transitional flow case the TKE increases abruptly at the transition onset.

26. 3D Flow in Turbine Cascade
TKE contours at 1% blade height distance from the endwall surface
(HR k-ωSST)
The flow starts to turbulize in the horseshoe vortex, but the turbulization level between the vortex branches is higher in the fully turbulent flow case comparing with the case of transitional flow.
fully turbulent

27. 3D Flow in Turbine Cascade
TKE contours at 1% blade height distance from the endwall surface
(LR-PTM k-ωSST)
This means that in the transitional flow case the boundary layer in this region is less filled comparing with the fully turbulent flow case and therefore it is more susceptible to cross-flow.
transitional

28. 3D Flow in Turbine Cascade
Limiting streamlines at the endwall surface (HR k-ωSST)
In the fully turbulent flow case the cross-flow between two branches of the horseshoe vortex appears to be less intensive; whereas, the cross-flow upstream of the trailing edge is more intensive, comparing with the transitional flow case.
fully turbulent
Visualized using open software

29. 3D Flow in Turbine Cascade
Limiting streamlines at the endwall surface (LR-PTM k-ωSST)
In the fully turbulent flow case the cross-flow between two branches of the horseshoe vortex appears to be less intensive; whereas, the cross-flow upstream of the trailing edge is more intensive, comparing with the transitional flow case.
transitional
Visualized using open software

30. 3D Flow in Turbine Cascade
The entropy function contours at the crosswise section at the distance of 20 percent of the axial chord downstream of the trailing edges
(HR k-ωSST)
1 – passage vortex
2 – near-endwall discrete vortex
3 – corner vortex
4 – suction-side branch of horseshoe vortex
fully turbulent
In the transitional flow case the passage and near-wall vortices are slightly closer to the endwall comparing with the fully turbulent flow case.

31. 3D Flow in Turbine Cascade
The entropy function contours at the crosswise section at the distance of 20 percent of the axial chord downstream of the trailing edges
(LR-PTM k-ωSST)
1 – passage vortex
2 – near-endwall discrete vortex
3 – corner vortex
4 – suction-side branch of horseshoe vortex
transitional
In the transitional flow case the passage and near-wall vortices are slightly closer to the endwall comparing with the fully turbulent flow case.

32. 3D Flow in Turbine Cascade
The secondary flows patterns (HR k-ωSST )
1 – passage vortex
2 – near-endwall discrete vortex
3 – corner vortex
4 – suction-side branch of horseshoe vortex
5 – pressure-side branch of horseshoe vortex 2 1 1 2 5 4 3
fully turbulent
Visualized using open software

33. 3D Flow in Turbine Cascade
The secondary flows patterns (LR-PTM k-ωSST)
1 – passage vortex
2 – near-endwall discrete vortex
3 – corner vortex
4 – suction-side branch of horseshoe vortex
5 – pressure-side branch of horseshoe vortex 2 1 1 2 5 4 3
transitional
Visualized using open software

34. 3D Flow in Turbine Cascade
The kinetic energy losses along blade span
Mid-span losses
Total losses
The distance downstream the trailing edges; [the axial chord]
40%
100%
Fully turbulent flow
0.078
0.081
0.108
0.123
Transitional flow
0.070
0.073
0.093
0.109
fully turbulent
transitional
In the transitional flow case the total kinetic energy losses are less by more than 1 percentage point, and the mid-span kinetic energy losses are less by more than 0.5 percentage points comparing with the fully turbulent flow case

35. Conclusion & Outlook
We have implemented the PTM transition model in the in-house CFD solver.
A verification of the model demonstrates
an acceptable agreement between computational and experimental data for the skin friction coefficient distribution;
a physically plausible behavior of the transitional boundary layer;
influence of the transition on the secondary flow pattern and kinetic energy losses in the turbine cascade.
The focus of the further research is on the mechanism of the transition effect on the secondary flows to identify various ways of reducing the kinetic energy losses in the turbine cascades

36. Thank you for your attention!
This work was partially performed under the support of the Szewalski Institute of Fluid-Flow Machinery of the Polish Academy of Sciences (Gdansk, Poland), contract No 16/7/13/PS. The authors would like to thank Prof. Piotr Lampart for useful discussions.

37. 3D Flow in Turbine Cascade
TKE contours at 0.1% cascade pitch distance from the suction surface
(HR k-ωSST)
upper endwall
fully turbulent
lower endwall
inlet blade exit
The growth of the TKE at the suction side in the fully turbulent flow case is about the same along blade span except zones of secondary flow vortices.

38. 3D Flow in Turbine Cascade
TKE contours at 0.1% cascade pitch distance from the suction surface
(LR-PTM k-ωSST)
upper endwall
transitional
lower endwall
inlet blade exit
In the transitional flow case the growth of the TKE starts much downstream and correlates with the pressure minimum locations along blade span.

39. 3D Flow in Turbine Cascade
The limiting streamlines in the trailing edge/endwall corner region
(HR k-ωSST)
TE suction side
In the fully turbulent flow case the cross-flow near its origin at the suction surface is less intensive, but downstream, closer to the trailing edge, it is more intensive, comparing with the transitional flow case.
endwall
fully turbulent
Visualized using open software

40. 3D Flow in Turbine Cascade
The limiting streamlines in the trailing edge/endwall corner region
(LR-PTM k-ωSST)
TE suction side
As a result, in the both cases, the zone of secondary flows at the suction surface has approximately the same spanwise extension, however the spanwise extension of near-wall vortex is smaller in the transitional flow case.
endwall
transitional
Visualized using open software