1. Sergey YERSHOV, Andrey RUSANOV
XVII International Symposium on Air Breathing Engines, September 6, 2005, Munich, Germany
AERODYNAMIC IMPROVEMENT OF TURBOJET ENGINE FLOWPATH USING 3D VISCOUS FLOW COMPUTATION
Dear Ladies and Gentlemen,
I would like to present the paper “AERODYNAMIC IMPROVEMENT OF TURBOJET ENGINE FLOWPATH USING 3D VISCOUS FLOW COMPUTATION”. My co-author, Andrey Rusanov and me represent two institutions: Institute of Mechanical Engineering Problems of Ukrainian Academy of Sciences and company FlowER Ltd.
Sergey YERSHOV, Andrey RUSANOV

2. CONTENTS Introduction Optimisation problem statement
In my speech after brief introduction I will describe the statement of the optimisation problem and the technique of the 3D flow simulation.
Than I will tell about results of numerical optimisation of LP turbine stage. At the end the conclusions will be presented.
Introduction
Optimisation problem statement
3D flow simulation
LP turbine stage optimisation
Conclusions

3. INTRODUCTION 3D flow solver Direct problem, verification computations
1D, 2D, Q3D flow
Optimisation techniques
At present the methods of the 3D viscous flow simulation are widely used within design process, generally for verification purposes.

4. INTRODUCTION 3D flow solver Direct problem, verification computations
1D, 2D, Q3D flow
Optimisation techniques
At the same time the turbine design process is usually performed using the simpler 1D, 2D or Q3D techniques.

5. INTRODUCTION 3D flow solver Direct problem, verification computations
1D, 2D, Q3D flow
Optimisation techniques
It leads up to the inevitable contradiction between the results obtained using the simpler but less reliable methods and the results computed with the more sophisticated and more accurate ones. The contradiction can be removed only if 3D flow simulation and optimisation methods are used simultaneously.

6. INTRODUCTION 3D flow solver Direct problem, verification computations
1D, 2D, Q3D flow
Optimisation techniques
3D flow solver
Optimisation techniques
In the present paper we solve such problem.

7. We consider the statement of a classical optimisation problem:
PROBLEM STATEMENT
We consider the statement of a classical optimisation problem:
Find
with constraints
where f — objective function r — vector of operating conditions parameters g — vector of design parameters
Our aim is to find minimum (or maximum) of objective function f with the constraints on operating condition parameters and on geometrical design parameters.

8. OBJECTIVE FUNCTION AND OPERATING CONDITIONS PARAMETERS
We use the rotor wheel torque, stage efficiency or stage kinetic energy losses as an objective functions.
Objective function
torque
efficiency
KE losses
f :
Operating condition parameters
mass flow rate
stage reaction
absolute exit angle
r :

9. OBJECTIVE FUNCTION AND OPERATING CONDITIONS PARAMETERS
We consider the mass flow rate, the stage reaction and absolute exit angle as operating condition parameters.
Objective function
torque
efficiency
KE losses
f :
Operating condition parameters
mass flow rate
stage reaction
absolute exit angle
r :

10. OBJECTIVE FUNCTION AND OPERATING CONDITIONS PARAMETERS
The objective function and operating condition parameters are determined by the computations of 3D viscous flow through turbomachinery stages using solver FlowER.
Objective function
torque
efficiency
KE losses
f :
Operating condition parameters
mass flow rate
stage reaction
absolute exit angle
r :

11. OBJECTIVE FUNCTION AND OPERATING CONDITIONS PARAMETERS
3D flow solver
FlowER
Objective function
torque
efficiency
KE losses
f :
Operating condition parameters
mass flow rate
stage reaction
absolute exit angle
r :

12. DESIGN (GEOMETRICAL) PARAMETERS
Varied design geometrical parameters for blading are the following:
Stagger angle at root
Twist angle
Blade number
Straight lean
Straight sweep
Compound lean
Compound sweep
Compound twist
In the nearest future we plan to use additionally parameters that define the blade section shape.

13. CONSTRAINTS
on geometrical parameters: , (including “engineering sense” of turbine flowpath)
For each geometrical parameter the constraint is imposed as a tolerance range. An arbitrary combination of geometrical parameters satisfying tolerance range can make no engineering sense. To exclude such cases special geometrical constraints are additionally used.

14. CONSTRAINTS
on geometrical parameters: , (including “engineering sense” of turbine flowpath)
on operating condition parameters: (stage reaction, mass flow rate, exit absolute angle)
Invariability of flow condition is ensured by the constraint on the turbine stage mass-flow-rate. Besides, it is possible to impose additional operative condition constraints such as stage reaction and exit absolute angle.

15. Exterior penalty functions
CONSTRAINTS
on geometrical parameters: , (including “engineering sense” of turbine flowpath)
on operating condition parameters: (stage reaction, mass flow rate, exit absolute angle) 
Exterior penalty functions
All of the constraints are of inequality type and they are implemented as exterior penalty functions.

16. The objective function is generally non-smooth
OPTIMISATION METHODS
The objective function is generally non-smooth 
Search methods
For considered problem the objective function is generally non-smooth. Therefore, it is desirable to use the search methods, since they do not require the calculation of derivatives of objective function.

17.  OPTIMISATION METHODS The objective function is generally non-smooth
Search methods
Now we use the following optimisation methods:
Direct Search Method of Hooke & Jeeves
Deformed Polyhedron Method of Nelder & Mead
Direct Pattern Search Method of Torczon
Genetic Algorithm

18. 3D FLOW SIMULATION
We simulate the 3D viscous flow through turbine stages with unsteady RANS equations that are written in general form for curvilinear body-fitted co-ordinate system.

19. TWO-EQUATION TURBULENCE MODEL
c4=0, c5=0 – Wilcox model
c4=0, c5=1 – BSL Menter
c4=1, c5=1 – SST Menter
We close the governing equations with the two-equation k-ω family of turbulence models that includes, in particular, the Menter’s SST model.

20. 2nd ORDER ACCURATE ENO-SCHEME
explicit step, finite volume, Riemann problem
piece-wise parabolic reconstruction
ENO- limiter: selection of smooth stencil
We integrate the RANS equations with the second-order accurate high-resolution ENO-scheme. The scheme is constructed on the basis of the Godunov’s one and its explicit step uses the piece-wise parabolic reconstruction with the ENO-limiter and the exact Riemann solver.

21. 2nd ORDER ACCURATE ENO-SCHEME
ADI implicit step
scalar Thomas algorithm
matrix Thomas algorithm
The implicit step of the scheme uses the ADI technique with the matrix or scalar Thomas algorithm. The acceleration of convergence is reached with simplified multi-grid approach and local time stepping.

22. Optimus – FlowER INTERACTION
The developed problem statement and numerical approach are implemented as the code Optimus. The shell of the code is developed using the Java-2 programming language. The optimiser itself and some other modules are written using the programming language Fortran-95.
Optimus (java2+fortran95)
FlowER database updating
optimisation method
geometry updating
optimisation step
convergence check
objective function analysis
Optimus database writing
constraints accounting
FlowER database reading

23. Optimus – FlowER INTERACTION
FlowER (fortran95)
geometry check (control.exe)
grid construction
coors.exe
coord.exe
metrics.exe
initial flowfield (vinfi.exe)
flow computations (flow.exe)
data processing and result writing (results.exe)
The code FlowER, which is widely used in Eastern Europe, is written using the programming language Fortran-95.
For optimisation purposes we use only several modules of the code.

24. Optimus – FlowER INTERACTION
FlowER (fortran95)
geometry check (control.exe)
grid construction
coors.exe
coord.exe
metrics.exe
initial flowfield (vinfi.exe)
flow computations (flow.exe)
data processing and result writing (results.exe)
Optimus (java2+fortran95)
FlowER database updating
optimisation method
geometry updating
optimisation step
convergence check
objective function analysis
Optimus database writing
constraints accounting
FlowER database reading
End
Start
The interaction of the codes FlowER and Optimus provides its reliable operation.

25. LP TURBINE STAGE OPTIMISATION
The objective of the current investigation is to increase the LPT efficiency
Shown here is meridional view of the turbine flow path that is under investigation. The turbine consists of high-pressure stage (HPT), low-pressure stage (LPT), transitional diffuser with struts of different shape and power turbine stage. The objective of the current investigation is to increase the low-pressure turbine stage efficiency.

26. LP TURBINE STAGE OPTIMISATION
Intensive air-cooling through orifices at meridional and blade surfaces
HPT
ΔG = GIH – GEL ≈ 0.25 GIH
The HPT and LPT stages are film-cooled as it is shown in this outline. The orifices for cooling air injection are located at the hub and tip endwalls as well as at the blade surfaces, in particular at the trailing edges. The mass flow rate of cooling air is significant, it amounts to approximately 25 % of inlet mass flow rate.

27. TRANSITIONAL DIFFUSER
The geometry of transitional diffuser and the computational grid for it are shown in the figure.
struts
struts with fairing

28. COMPUTATIONAL GRID y+≈12
The coarse computational grid of 145’000 cells is used for turbine stage flow computations within the optimisation. In this case a gasdynamic computation took approximately 1 hour on PC Athlon XP It enabled more than 100 computations per week at automatic operation.

29. COMPUTATIONAL GRID y+≈12
It is necessary to note that the computational grid for verification computations before and after optimisation process was more than 1 million cells per stage. The condition for y+ was satisfied on verification computations.

30. COMPUTATIONAL GRID y+≈12
The total number of cells for a whole turbine throughflow computations was between 3.5 millions and 4.5 millions approximately.

31. TEMPERATURE CONTOURS AT TANGENTIAL SURFACES OF HPT
root
mid
tip
Shown in the figure are the temperature contours for root, mid and tip sections of HPT stator blade. Film cooling generates cooled zones at endwalls and blade surfaces. The cooled flow is clearly seen to locate near injection regions. The film cooling influences significantly both stage efficiency and mass flow rate, therefore it can not be ignored during optimisation process.

32. OPTIMISATION OF LPT STAGE
Objective function is stage efficiency calculated on turbine wheel torque M
We have considered the stage efficiency, determined from the rotor wheel torque as an objective function for the LP turbine optimisation. Although the criterion is rather questionable for stages with intensive film cooling, it is acceptable for comparative analysis of turbine flowpaths.

33. OPTIMISATION OF LPT STAGE
Objective function is stage efficiency calculated on turbine wheel torque M
Varied geometrical parameters are
stagger root angles of stator and rotor blades
twist angles of stator and rotor blades
stator straight lean
The varied geometrical parameters for LPT optimisation were the following, stagger root angles and twist angles of stator and rotor blades, stator straight lean.

34. OPTIMISATION OF LPT STAGE
Objective function is stage efficiency calculated on turbine wheel torque M
Varied geometrical parameters are
stagger root angles of stator and rotor blades
twist angles of stator and rotor blades
stator straight lean
Constraint: constancy of mass flow rate ±0.4 kg/s
The operating condition constraint has been imposed on the mass flow rate with some tolerance for LPT optimisation.

35. OPTIMISATION OF LPT STAGE
Objective function is stage efficiency calculated on turbine wheel torque M
Varied geometrical parameters are
stagger root angles of stator and rotor blades
twist angles of stator and rotor blades
stator straight lean
Constraint: constancy of mass flow rate ±0.4 kg/s
Nelder – Mead deformed polyhedron method
We have used the deformed polyhedron method of Nelder and Mead for the LPT optimisation.

36. GEOMETRICAL PARAMETERS OF INITIAL AND MODIFIED LPT STAGE
Shown in the Table are varied geometrical parameters of initial and modified LPT stage. The stagger and twist angles are clearly seen to be changed significantly. The stagger angle of stator decreases near the root and increases at the tip. The rotor stagger angle slightly increases through the full blade length.

37. INITIAL AND MODIFIED LP TURBINE STAGE
The blading geometry for initial and modified turbine stage is demonstrated in the figure. It is seen that the geometry changes lead to increase of twist of stator and rotor.

38. TURBINE STAGE REACTION
blade length
reaction I M
root reaction is increased, negative reaction is eliminated
tip reaction is approximately unchanged
Due to blade geometry changing, the rearrangement of flow pattern occurs. The decrease of the root stagger angle of stator results in the stage reaction increasing near the root and the negative reaction being eliminated there. At the same time, since the tip stagger angle of stator slightly increased, the tip reaction was nearly constant and it is very important in view of a radial gap leakage.

39. NEAR ROOT FLOW IN LPT ROTOR WHEEL
initial
modified
flow acceleration
separation
As a result of optimisation, a flow deceleration region at the suction side root sections near the leading edge, as well as the flow separation evoked by it, have been eliminated completely. The flow that was not accelerating near root section for initial design becomes pronounced accelerating for modified one.

40. FLOW IN TRANSITIONAL DIFFUSER
Initial LPT design
Modified LPT design
Mach number contours at 75 % section from root
Mach number contours at cross-flow section behind struts
Shown in the figure are Mach number contours for tangential and cross flow sections of transitional diffuser. The separation zones near transitional diffuser struts for turbomachine of initial design are clearly seen to be substantially smaller than those of improved design. However, the separations contribute insignificantly to the turbomachine total kinetic energy losses through smallness of the transitional diffuser heat drop.

41. PERFORMANCES OF HPT AND LPT STAGES (3D throughflow computations)
The performances obtained in three-dimensional throughflow computations of HPT and LPT for initial and modified turbines are given in the Table. It is seen that mass flow rate is approximately the same for initial and modified flowpaths. It provided the invariability of flow conditions since the turbine pressure drop was constant at the computations.

42. PERFORMANCES OF HPT AND LPT STAGES (3D throughflow computations)
The rotor wheel torque significantly increases for LPT stage, whereas it remains approximately the same for HPT stage.

43. PERFORMANCES OF HPT AND LPT STAGES (3D throughflow computations)
The LPT stage power increases significantly, whereas it remains approximately the same for HPT stage. The total power of HPT and LPT stages is increased more than by one point one percent.

44. PERFORMANCES OF HPT AND LPT STAGES (3D throughflow computations)
The LPT efficiency gain amounts to about two point two percent. Here it is very important that the HPT efficiency just as its power is not changed significantly.

45. CONCLUSIONS
The main conclusions of the investigation are the following.
The aerodynamic optimisation of the spatial shape of blading is performed for the film-cooled LPT.
Computations of the flow through the turbomachine verify favourable effect of flowpath modernisation.
The LPT stage reaction is increased for root sections.
Separation bubble near suction surface of LPT rotor wheel is eliminated.
The efficiency of the LPT stage is increased more than by percent.
The total power of the two stage turbomachine is increased more than by 1 percent.

46. ACKNOWLEDGEMENTS
The work has been performed under contract with the State Enterprise «Zorya–Mashproyekt», Nikolayev, Ukraine. The authors wish sincerely to thank designers and engineers of the enterprise for many useful discussions, namely
chief designer B.V.Isakov,
chief engineer V.Ye.Spitsyn,
engineers M.A.Sharovskiy and A.A.Usatenko.

47. Thank you for your attention