Sensitivity Analysis in 3D Turbine CFD1120314/FULLTEXT01.pdf · 2017-07-06 · Sensitivity Analysis...

Sensitivity Analysis in 3DTurbine CFD

Simon Kern

Master Thesis at KTH Mekanik

in cooperation withRolls-Royce Deutschland & Co KG

Supervisor KTH: Dr. Stefan Wallin

Supervisors Rolls-Royce: Dr. Torsten WolfDr. Marcus Meyer

Hereby, I declare truthfully that this work is written independently, all resources are statedcorrectly and completely and that everything that is taken both modified and unmodified fromother sources is indicated.

Berlin, July 5, 2017

AbstractA better understanding of turbine performance and its sensitivity to variations in the inletboundary conditions is crucial in the quest of further improving the efficiency of aero engines.Within the research efforts to reach this goal, a high-pressure turbine test rig has been designedby Rolls-Royce Deutschland in cooperation with the Deutsches Zentrum für Luft- und Raumfahrt(DLR), the German Aerospace Center. The scope of the test rig is high-precision measurement ofaerodynamic efficiency including the effects of film cooling and secondary air flows as well as theimprovement of numerical prediction tools, especially 3D Computational Fluid Dynamics (CFD).

A sensitivity analysis of the test rig based on detailed 3D CFD computations was carried outwith the aim to quantify the influence of inlet boundary condition variations occurring in the testrig on the outlet capacity of the first stage nozzle guide vane (NGV) and the turbine efficiency.The analysis considered variations of the cooling and rimseal leakage mass flow rates as well asfluctuations in the inlet distributions of total temperature and pressure. The influence of anincreased rotor tip clearance was also studied.

This thesis covers the creation, calibration and validation of the steady state 3D CFD modelof the full turbine domain. All relevant geometrical details of the blades, walls and the rimsealcavities are included with the exception of the film cooling holes that are replaced by a volumesource term based cooling strip model to reduce the computational cost of the analysis. Thehigh-fidelity CFD computation is run only on a sample of parameter combinations spread overthe entire input parameter space determined using the optimal latin hypercube technique. Thesubsequent sensitivity analysis is based on a Kriging response surface model fit to the sampledata. The results are discussed with regard to the planned experimental campaign on the test rigand general conclusions concerning the impacts of the studied parameters on turbine performanceare deduced.

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AcknowledgementsI thank my supervisors Dr. Torsten Wolf and Dr. Marcus Meyer for their continuous supportand valuable advice during the entire course of my thesis work. I am grateful to Prof. Dr. StefanWallin from the Royal Institute of Technology for giving me the opportunity to complete mydegree with a thesis in cooperation with Rolls-Royce Deutschland. I thank Dr.-Ing. RolandWilhelm for the possibility to do my research in his team at Rolls-Royce Deutschland. The helpI received from Sebastian Brehmer and Jan Lührmann in all questions relating the simulationsusing HYDRA cannot be valued enough. Last but not least, I thank all my colleagues who havegiven me both technical and moral support throughout my time in Dahlewitz.

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Contents

Abstract I

Acknowledgements III

List of Figures VII

List of Tables XI

Nomenclature XIII

1 Introduction 1

2 Fluid Dynamics Theory 32.1 Axial Turbine Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Similarity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Secondary Air System and Cooling . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Steady State RANS Equations . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Modelling Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.4 Modelling Film Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Turbine Performance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Numerical Methods and Setup 213.1 Computational Fluid Dynamics Theory . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 The Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.2 Discretisation of the Governing Equations . . . . . . . . . . . . . . . . . . 213.1.3 The Numerical Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.4 Finite Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.5 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.6 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.1 CAD Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Local Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.4 Boundary Layer Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 HYDRA Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.1 Pre-Processing and Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . 29

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3.3.2 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Reference case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Sensitivity Quantification 374.1 Theory of Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.2 Local versus Global Methods . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.3 Surrogate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.4 Sampling Strategy: Design of Experiments (DoE) . . . . . . . . . . . . . . 394.1.5 Quantification of Main Effects . . . . . . . . . . . . . . . . . . . . . . . . 414.1.6 Interaction Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Variation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.1 Inlet Total Temperature and Pressure Distributions . . . . . . . . . . . . 424.2.2 Cooling Flows and Secondary Air System . . . . . . . . . . . . . . . . . . 454.2.3 Rotor Tip Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Model Calibration and Validation 495.1 Cooling Strip Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.1 Single Row Computations on the NGV1 . . . . . . . . . . . . . . . . . . . 495.1.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Boundary Layer Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2.1 Boundary Layer Flow without Film Cooling . . . . . . . . . . . . . . . . . 575.2.2 Boundary Layer Flow with Film Cooling . . . . . . . . . . . . . . . . . . . 585.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3 Mesh Independence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Results and Discussion 676.1 Results of the Kriging Response Surface Based Analysis . . . . . . . . . . . . . . 67

6.1.1 Kriging Cross-validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.1.2 Capacity Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 686.1.3 Efficiency Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2 Rotor Tip Clearance Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.3 Implications for the Rig Measurements . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Concluding Remarks 837.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Bibliography 87

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List of Figures

2.1 Schematic cut through the test rig indicating the extent of the computationalflow domain along the x-axis as well as the coordinate origin and machine axis ofrotation (dash-dotted line) for reference. The figure shows the two turbine stagesincluding the limits of the individual computational zones. The rimseal inlets ofthe secondary air system are shown. Their denomination refers to the zone inwhich they are introduced. The inflow of the heated and pressurized air is fromthe left. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Midspan cut of the first turbine stage including the direction of rotation. For theNGV, the locations of the leading edge (LE), the trailing edge (TE) as well as thepressure and suction sides of the blade (PS and SS respectively) are shown. Theoriginal geometry has been warped. . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Time history of the axial velocity component U1(t) at the centerline of a turbulentjet. From [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Self similar turbulent boundary layer profile showing the upper limit of the viscoussublayer (−−) as well as the lower limit for the logarithmic region (· · ·). The thincurves show the approximations used in the respective regions. For the bufferlayer no such relation exists. Reproduced from [27]. . . . . . . . . . . . . . . . . 8

2.5 Modern high-pressure turbine rotor internal cooling system combining high-pressure (black) and low-pressure (grey) cooling feeds. From [4, p. 185]. . . . . . 9

2.6 Schematic representation of a typical two-stage turbine expansion.Real process (−) and ideal process (· · ·). . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Dual control volumes in HYDRA. The shaded area around the node is the controlvolume on which the conservation equations are solved. From [6]. . . . . . . . . . 22

3.2 NGV1 blade suction side surface comparing the fully featured model (left) to thecooling strip model used in this work (right). The flow is from left to right. . . . 25

3.3 Sketch of the turbine geometry from the inlet (left) to the NGV2 exit (right)showing geometrical details of the casing wall that were removed (· · ·) or kept(−). On the hub wall the geometry of the seal leakage cavities is shown. . . . . . 26

3.4 Midspan through the ROT1 mesh at the leading edge. . . . . . . . . . . . . . . . 263.5 Comparison of absolute Mach number distributions on a midspan cut in the NGV1

wake region close to the trailing edge overlaid with the mesh with and withoutmesh volume refinement. The flow is from left to right. The Mach number issubsonic in the entire domain and increases from blue to red. . . . . . . . . . . . 27

3.6 y+-distribution for the final mesh of the full HPT. . . . . . . . . . . . . . . . . . 283.7 Selected cuts through the final mesh of the full HPT. . . . . . . . . . . . . . . . . 293.8 Typical convergence history diagrams for a full HPT computation (left) and

preminiary studies on the NGV1 (right). . . . . . . . . . . . . . . . . . . . . . . . 303.9 Convergence history of the NGV1 outlet capacity and efficiency monitors. The

overbar signifies an average over the last 200 iterations. . . . . . . . . . . . . . . 31

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3.10 Meridional view of the computational domain (without exit duct) showing inlets(green arrows), cooling strips (light blue arrows) and wall speed (red is stationary,dark blue is rotating). The flow is from left to right. . . . . . . . . . . . . . . . . 32

3.11 Schematic of the cooling geometry of the first turbine stage. . . . . . . . . . . . . 33

4.1 Schematic of the basic principle of sensitivity analysis to obtain a more completepicture of the variations of the turbine performance by running the CFD modelmultiple times in the input parameter space. The statistical analysis of thecomputed points then allows for inferences in between with varying degrees ofuncertainty (shaded areas). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Comparison of different sampling techniques for ni = 2 input variables. For RSand OLHS the number of sampling points was set to N = 40. . . . . . . . . . . . 41

4.3 Total temperature (top) and total pressure (bottom) rake measurements taken at4 equally spaced circumferential locations in the inlet plenum. Both quantities aremeasured in the same axial plane, the temperature and pressure rakes are shiftedby 45◦ relative to each other. Averages (shown as thick lines) are area weightedover the annulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Inlet boundary conditions for total temperature. . . . . . . . . . . . . . . . . . . 454.5 Measurements of flow capacity through all 34 NGV1 blades for different pressure

differences relative to atmospheric pressure (Pa = 99.4 kPa). Blade capacitynormalized with value at P/Pa = 1.60 corresponding to the reduced case inletconditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.6 Cut through the ROT2 domain in y-normal direction at the location of the tipgap. The two tested tip clearances are superimposed: Small (design) tip clearance(−) and large tip clearance (−−). . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.1 Adiabatic wall temperature distributions (in Kelvin) on the NGV1. . . . . . . . . 505.2 Contours of total temperature (in Kelvin) just before the trailing edge slot lip.

The flow is into the plane, suction side to the left, pressure side to the right of thesurface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 Comparison on the cooling effectiveness on the NGV1 suction side between thecooling strip model (CSM) and the fully-featured computation (FF). All profilesare an average over the range 40-50% span. . . . . . . . . . . . . . . . . . . . . . 52

5.4 Comparison of the span-averaged total temperature distributions over the NGV1pitch between trailing edge and outlet in the NGV1 domain for the cooling stripmodel (CSM) and fully-featured (FF) computations. . . . . . . . . . . . . . . . . 53

5.5 Capacity change relative to the value computed with lc = 1.15 mm versus pene-tration depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.6 Quasi 2D subdomain for boundary layer study. . . . . . . . . . . . . . . . . . . . 555.7 Cut through boundary layer meshes. All meshes are shown at the same scale. . . 565.8 y+-distributions on the NGV1 surface without film cooling. The distance to the

LE is measured positive on the SS and negative on the PS. The curve in the centercorresponds to meshes 1-3 that have virtually identical y+-distributions apartfrom a discrepancy at 15 mm from the leading edge on the SS. . . . . . . . . . . 57

5.9 Results with cooling strip model turned off. Locations A and B (vertical dottedlines) for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.10 Results with cooling strip model turned on. Locations A and B (vertical dottedlines) for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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5.11 Mid-span cut of the domain showing the approximate locations A and B as wellas the locations of the cooling strips (black). . . . . . . . . . . . . . . . . . . . . . 60

5.12 Boundary layer velocity profiles at locations A and B on the suction side of NGV1for the 5 considered meshes (symbols) including the velocity profile for mesh 5 atthe same location without film cooling (NC). The dotted line indicates the filmcooling penetration depth lc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.13 Boundary layer profiles for the compressible turbulent kinetic energy kc at locationsA and B on the suction side of NGV1 for the 5 considered meshes (symbols)including the profile for mesh 5 at the same location without film cooling (NC).The dotted line indicates the film cooling penetration depth lc. . . . . . . . . . . 63

5.14 Change in computed efficiencies for the coarser and finer meshes. Changes givenin percent relative to respective baseline value. . . . . . . . . . . . . . . . . . . . 65

6.1 Cross-validation plots for each output variable including the corresponding R2-value. The percentage delta-values are computed relative to the reference configu-ration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2 Main effects diagrams for the NGV1 outlet capacity C. The plot on the left showsthe main effects of the cooling (COOL) as well as the rimseal leakage mass flowrates (RSL) respectively for each row. The plot on the right shows the main effectsof the total temperature and pressure inlet profiles. Relative to the baseline, thecooling flows are symmetrically varied by 20%, the rimseal flows by 50%. Thetotal temperature and pressure variations are summarized in Tab. 4.1.The bar charts present the magnitude of the main effects for each input parametergroup ordered by importance corresponding to the diagrams above and includethe errorbar indicating the uncertainty in the Kriging fit. The bar colours (lightgrey and dark grey) correspond to positive and negative absolute main effectsrespectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Input parameter interactions on NGV1 outlet capacity over the full parameterranges. The capacity increases from blue to red. . . . . . . . . . . . . . . . . . . 71

6.4 Main effect diagrams for enthalpy based efficiency. The plots on the left showthe main effects of the cooling (COOL) as well as the rimseal leakage mass flowrates (RSL) for each row. The plot on the right shows the main effects of the totaltemperature and pressure inlet profiles. Relative to the baseline, the cooling flowsare symmetrically varied by 20%, the rimseal flows by 50%. The total temperatureand pressure variations are summarized in Tab. 4.1.The bar charts present the magnitude of the main effects for each input parametergroup ordered by importance corresponding to the diagrams above and includethe errorbar indicating the uncertainty in the Kriging fit. The bar colours (lightgrey and dark grey) correspond to positive and negative absolute main effectsrespectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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6.5 Main effect diagrams for the entropy based efficiency. The plot on the left showsthe main effects of the cooling (COOL) as well as the rimseal leakages mass flowrates (RSL) for each row. The plot on the right shows the main effects of the totaltemperature and pressure inlet profiles. Relative to the baseline, the cooling flowsare symmetrically varied by 20%, the rimseal flows by 50%. The total temperatureand pressure variations are summarized in Tab. 4.1.The bar charts present the magnitude of the main effects for each input parametergroup ordered by importance corresponding to the diagrams above and includethe errorbar indicating the uncertainty in the Kriging fit. The bar colours (lightgrey and dark grey) correspond to positive and negative absolute main effectsrespectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.6 Co-influence of the of the midspan total pressure variation and the NGV1 coolingmass flow rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.7 Co-influence of the of the total pressure variation and total temperature variationat midspan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.8 Variations of the performance parameters with the rotor tip clearance. . . . . . . 79

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List of Tables

3.1 Reference boundary conditions for reduced power case. All mass flows are givenas a percentage of W40, the expected main inlet mass flow from a 2D throughflowsolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Reference boundary conditions for cooling flows (excluding TE slots). All massflows are given as a percentage of W40, the expected main inlet mass flow from a2D throughflow solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Final total temperature and pressure ranges for the inlet variations at hub (HUB),midspan (CEN) and casing (CAS). . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.1 Mesh parameters for the boundary layer study. . . . . . . . . . . . . . . . . . . . 565.2 Full HPT mesh summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Global refinement study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1 Summary of the main effects on the on the capacity and efficiencies in percentof the reference value for each input parameter relative to an absolute flow ratechange of 1% W40. Negligible main effects (absolute values below 0.01% per 1%W40) are not shown and indicated with −−. The reference mass flow W40 is theexpected main inlet mass flow from a 2D throughflow solution. . . . . . . . . . . 80

6.2 Summary of the influence of tip clearance (TC) increase on the capacity andefficiencies extrapolated to a standard variation of 1 mm. The value for theenthalpy and power based efficiency is the average of the two values. . . . . . . . 80

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Nomenclature

Latin characters

Symbol Unit DescriptionA [m2] AreaC [kg m2√K/(N s)] Flow capacitycp [J/(kg K)] Specific heat capacity at constant pressurecv [J/(kg K)] Specific heat capacity at constant volumeE [J/kg] Specific internal energyf [N/kg] Specific volume ForceH [J] Enthalpyk [W/(m K)] Thermal conductivityk [m2/s2] Turbulent kinetic energykc [kg/(m s2)] Compressible turbulent kinetic energyL, l m Lengthm [kg/s] Mass flowni, nj - Normal vectorPτ [W/m3] Entropy productionP, p [N/m2] Thermodynamic pressureq [N/m2] Dynamic pressureqi [W/m2] Heat flux vectorR [J/(kg K)] Specific gas constantr [m] RadiusS [m2] Surfaces [J/K] EntropyT [K] Temperaturet [s] TimeU [m/s] Velocityui [m/s] Velocity vectoruτ [m/s] Friction velocityV [m3] VolumeW [W] Powerx, y, z [m] Orthonormal coordinate directionsxi [m] Coordinate vector

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Greek charactersSymbol Unit Descriptionγ - Ratio of specific heatsδij - Kronecker’s deltaε [W/kg] Turbulent dissipation rateη - Efficiencyϕ - Placeholder for a flow variableΦ - Placeholder for T or PΣ - Sumµ [Ns/m2] Dynamic viscosityλ [Ns/m2] Bulk viscosityρ [kg/m3] Densityθ [rad] or [◦] AngleΘij [N/m2] Stress tensorτij [N/m2] Viscous stress tensorω [1/s] Turbulence frequencyψ - Source Term

Subscripts

Index Descriptiontot Total (stagnation) conditionsstat Static conditionsref Reference conditionsa Ambient conditions, local speed of soundc Cooling, Coolant, Compressibleu Upperl Lowert TurbulentH Enthalpy basedP Power basedT Total condition baseds, ss Entropy based, isentropiccrit Criticalloss Lossopt Optimaltip Rotor Tip(a)w (Adiabatic) Wall∞ Free stream conditionsv Volumem Momentume Energyn Normal directionΣ Sum

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AbbreviationsAbbreviation DescriptionCFD Computational Fluid DynamicsCAD Computer Aided DesignDoE Design of ExperimentsDNS Direct Numerical SimulationLES Large Eddy SimulationNS Navier-StokesRANS Reynlds-averaged Navier-StokesDRSM Direct Reynolds Stress Model(E)ARSM (Explicit) Algebraic Reynolds Stress ModelSST Shear Stress TransportNGV Nozzle Guide VaneROT Rotor BladeHPT High Pressure TurbineSAS Secondary Air SystemCS(M) Cooling Strip (Model)FF Fully FeaturedSTG (Turbine) StageFMG Full Multigrid schemePS Pressure SideSS Suction SideLE Leading EdgeTE Trailing EdgeNC No Cooling1D,2D,3D One, Two, Three DimensionalQ2D Quasi-2DBCS Background Cell SizeSCS Surface Cell SizeRMS Root Mean Squaredrpm Revolutions per minuteGPE Gaussian Process Emulator

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1 Introduction

Gas turbines, although relatively new in the developments of energy conversion, have become apillar of worldwide electricity and heat generation and, not least, for propulsion in a wide rangeof aero engine variations. The efficiency of gas turbines has considerably increased since their firstintroduction more than 100 years ago. At the same time, the spread of the technology to evergrowing sectors makes gas turbines responsible for a large proportion of global fuel consumptionand emissions.

Until the year 2050 the European Commission expects a more than 6-fold increase of globaltraffic to 16 billion passengers per year compared to 2011, of which a considerable proportionwill be transported with one of the expected 25 million flights within Europe [9]. To avoid acomparable increase in fuel consumption and pollution, governments and industry are combiningtheir efforts to further improve aero engines.Milestone goals for the reduction of pollutant emissions are set by organizations such as theInternational Air Transport Association (IATA) and the European Environment Agency (EEA)and enforced by increasingly stringent legislation. The aim is a 20% reduction of green housegases by 2020 (compared to 1990) [10] and a 75% decrease by 2050 [9].

The approaches to improve aero engines are manifold, including increasing the fan bypass ratio,higher turbine inlet temperatures through more resistant blade materials, lean-burn combustorsand an optimisation of the cooling and secondary air system in the turbine [18]. Within theseresearch efforts, a high-pressure turbine test rig has been designed by Rolls-Royce Deutschland incooperation with the Deutsches Zentrum für Luft- und Raumfahrt (DLR), the German AerospaceCenter. The scope of this test rig is high-precision measurement of aerodynamic efficiencyincluding the effects of film cooling and secondary air flows as well as the improvement ofnumerical prediction tools, especially 3D Computational Fluid Dynamics (CFD).

In typical simulations of turbine flows, predetermined and fixed boundary conditions representinga real operating point are applied to a deterministic computer model of the aero engine to computethe flow field. As the CFD results are compared to experiments, the choice of the appropriateboundary conditions becomes increasingly difficult: The physical boundary conditions may notbe available with sufficient accuracy or cannot be directly introduced into the computationalmodel and need simplification (e.g. temporal or spatial averaging). The singular output of theCFD computation is therefore a snapshot giving only limited information about the machinebehaviour in the real range of boundary conditions even for a single operating point.Sensitivity analysis is a technique routinely used in risk analysis and high stake decisions toassess the impact of input parameter variability on the output of a model. In this work, themethod of sensitivity analysis is applied to a 3D turbine CFD model to quantify the influence onthe turbine performance of boundary condition variations as they occur in the test rig.

1

1 Introduction

Thesis StructureThis thesis consists of seven chapters.

Introduction: The introduction describes the motivation for the research in which the topic ofthis thesis is inscribed.

Fluid Dynamics Theory: This section presents the test rig as well as the physical and mathemat-ical background for the subsequent numerical analysis. Emphasis is put on the distinctivefeatures of the test rig and their modelling, namely the secondary air system and cooling.The cooling strip model replacing the individual film cooling holes on the blades of the firstturbine stage is introduced. Finally, the performance parameters capacity and efficiencyare introduced.

Numerical Methods and Setup: After a brief summary of the theory of computational fluiddynamics, the different steps in the mesh generation process are detailed and the solverconfiguration described. The last section presents the boundary conditions as well as thereference values on which the sensitivity analysis is based.

Sensitivity Analysis: This section describes the methodology behind a sensitivity analysis andpresents the approach used in this work. Subsequently, the value ranges of the parametersconsidered in the analysis are determined based on preliminary test data from the rig.

Model Calibration and Validation: Before the sensitivity analysis can be carried out, the refer-ence model must be validated, especially due to the cooling strip model. The first sectioncovers the calibration of the cooling strip model to numerical data from a fully-featuredcomputation including the cooling holes on the first turbine stage. Subsequently, theinteraction between the boundary layer mesh and the film cooling flow using the coolingstrip model is analysed, motivating the meshing requirements for the full model.

Results and Discussion: In this section, the results of the sensitivity analysis are presented anddiscussed in view of their implications for the experimental campaign.

Concluding Remarks: The final section summarizes the result of the thesis and provides sugges-tions for future work complementing the analysis performed in this work.

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2 Fluid Dynamics Theory

2.1 Axial Turbine Fluid Dynamics

An axial turbine is a turbomachine designed to extract power from hot gas entering from thecombustion chamber by expanding it to a lower pressure. The turbine in the test rig consistsof an inlet section followed by two consecutive pairs (turbine stages, STG) of stationary statorvanes (Nozzle Guide Vanes, NGV) and moving rotor blades (ROT) leading to an outlet duct,schematically shown in Fig. 2.1. The wall at a lower radius is termed hub, while the outer wall isthe casing. The rotor-to-stator ratio is 2:1 so that 34 stator blades are followed by 68 rotor blades,rotating in counter-clockwise direction when looking along the machine axis in flow direction.

z

xy

STG1 STG2DUCTNGV1 ROT1 NGV2 ROT2

ROT1seal

NGV2seal

ROT2US seal

ROT2DS seal

Computational Domain

Location of optional

swirl generator

Figure 2.1: Schematic cut through the test rig indicating the extent of the computational flowdomain along the x-axis as well as the coordinate origin and machine axis of rotation (dash-dotted line) for reference. The figure shows the two turbine stages including the limits of theindividual computational zones. The rimseal inlets of the secondary air system are shown. Theirdenomination refers to the zone in which they are introduced. The inflow of the heated andpressurized air is from the left.

Although a Cartesian coordinate system is used in the figures for simplicity, a cylindricalcoordinate system (x, r, θ) around the machine axis (x-axis) is used internally in the CFD code.The velocity components in each coordinate direction are denoted U = (ux, ur, uθ). Figure 2.2is a cut through the blades of the first stage at midspan between hub and casing showing, forthe NGV1, the location of the stagnation point at the leading edge as well as the definition ofpressure side (PS) and suction side (SS) used throughout this work.

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LE

TE

SS

PS

Figure 2.2: Midspan cut of the first turbine stage including the direction of rotation. For theNGV, the locations of the leading edge (LE), the trailing edge (TE) as well as the pressure andsuction sides of the blade (PS and SS respectively) are shown. The original geometry has beenwarped.

2.1.1 Similarity Theory

The flow through an axial turbine is governed by several variables that need to be considered,including both flow parameters and material properties. In the following section, fundamentalflow properties and important characteristic ratios used to classify engineering flows are presented.Using similarity theory, these variables are combined to form non-dimensional ratios that reducethe total number of variables and incorporate important features of the flow. Two flows, forwhich these non-dimensional parameters are the same, exhibit similar behaviour, i.e., they aredynamically similar.

Modern aero engine high-pressure turbines operate at transonic and even locally supersonicspeeds and the compressibility of the hot gasses flowing through the turbomachine needs to beaccounted for in the analysis. The pertinent non-dimensional ratio for the study of compressibleflows is the Mach number (Ma), the ratio between local flow velocity and local speed of sound:

Ma = U

a= U√

γRT, (2.1)

where U is the local fluid velocity, a is the local speed of sound, γ is the ratio of specific heatscp/cv, R is the specific gas constant and T is the static temperature of the fluid.

The Mach number is a crucial parameter since, depending on its value, the flow exhibits verydifferent physical behaviour that translates into severe changes in the mathematical structure ofthe governing equations. In fact, while the flow and all the variables are continuous throughoutthe fluid domain for subsonic flow (Ma < 1 everywhere), Mach numbers beyond unity lead to theemergence of shocks, i.e., abrupt changes in the flow variables that are modelled as discontinuities

4


and require special treatment in the numerical solution. Physically speaking, the speed of soundis the speed at which information is propagated through the flow in the form of pressure anddensity fluctuations. If the flow speed exceeds the speed of sound, no information can propagateupstream. Changes in flow situations, such as sharp corners and steps, cannot be smooth butare violent in the form of shocks and expansion fans with large and rapid variations in the localfluid properties. The consequence are separation bubbles and complex shock boundary layerinteractions. A typical threshold below which compressibility effects are considered negligibleis Ma = 0.3 [11]. The appearance of Mach numbers close to unity in the flow considered inthis work prohibit the assumption of incompressibility and requires the solution of the energyequation along with the momentum and continuity equations.

A second central non-dimensional ratio in a flow is the Reynolds number (Re) that representsthe ratio of inertial to viscous forces in the flow

Re = ρUL

µ, (2.2)

where ρ is the fluid density, U is the characteristic flow velocity, L is the characteristic length ofthe turbomachine, usually the blade chord length, and µ is the dynamic viscosity of the fluid.

The Reynolds number is a measure of the dominance of inertial forces over viscous forces inthe flow and is of crucial importance for the general structure of the flow and also for localphenomena, especially boundary layers on solid surfaces. Due to the high speeds in modernaero engine turbine applications, the Reynolds number at the operating point is very high(above 2 · 106 for the NGV1 and around 8 · 105 for the NGV2). Although the main flow isdominated by inertial forces and the boundary layers are very thin, friction must be considered.Close to the solid boundaries in the flow field, the fluid velocity decreases to the speed of theboundary to satisfy the no slip condition, resulting in locally very low Reynolds numbers. In fact,internal flows, as they occur in turbomachines, are heavily influenced by the boundary layerson the blades as well as on hub and casing walls that quickly merge to dominate the flow field [32].

2.1.2 TurbulenceFluid flows can in fact be divided into two different types exhibiting very different characteristicsof engineering importance. At low Reynolds numbers, i.e., viscous effects play a sufficientlyimportant role, the flow tends to be laminar, organized in distinct layers with very little cross-stream mixing. Small disturbances are damped by viscous effects. As the Reynolds numberincreases above a critical threshold, Recrit, and inertial forces start to dominate the flow physics,the laminar flow structure becomes unstable and progressively breaks down into turbulence. Theappearance of random, three-dimensional velocity fluctuations around the mean dramaticallyincrease the mixing rates inside the fluid. The change from one flow regime to the other is calledtransition.

In the first turbine stage, the Reynolds number is sufficiently high for the changes betweendifferent operating conditions to have a negligible effect on the overall flow behaviour that isfully turbulent and independent of the Reynolds number. This is not necessarily true for thesecond stage although the blade dimensions and flow velocities are similar: The temperature anddensity of the air are much lower and the Reynolds number may thus decrease sufficiently to

5


Figure 2.3: Time history of the axial velocity component U1(t) at the centerline of a turbulentjet. From [27].

lead to transitional flow, altering the boundary layer structure entirely.In this work, only small variations of the boundary conditions around a reference case with fullyturbulent flow are investigated so that transitional flow does not occur and the Reynolds numberdependence is negligible.

Turbulent flows are highly unsteady and inherently three-dimensional. They are typicallyassociated with large mean velocities and unpredictable variations around the mean value,illustrated in Fig. 2.3. Characteristic for turbulent flows is the presence of vortical structures oreddies in a large range of sizes. While the largest vortices are of the scale of the flow problem(i.e., several cm in this case) the smallest eddies are of the order of microns. For a completedescription of turbulent flows, all these scales are of importance within the concept of the energycascade first proposed by Richardson then elaborated by Kolmogorov amongst others [27]. Thelargest scales extract energy from the mean flow that breed subsequently smaller eddies. Theseeddies, in turn, transfer the energy to progressively smaller scales within the inertial range untilthe smallest scale is reached, the so-called Kolmogorov microscale. It is only here, where theeddies are smallest and the flow gradients sufficiently large that inertial and viscous forces reachan equilibrium, that the energy contained in the eddies is finally converted into internal energyof the fluid [27].

Due to the large range of length scales appearing in turbulent flows, their computational analysisis very difficult and several different approaches have been devised varying in accuracy andrequired computational time. The main approaches are summarized below listed in order ofdecreasing computational cost.

Direct Numerical Simulation (DNS) : In the DNS approach, as the name suggests, the govern-ing equations are directly solved for all scales of motion without any approximation [11].Due to the large range of lengthscales present in turbulent flows, as well as the shorttimescales of the chaotic fluctuations, that need to be fully resolved for a DNS, the compu-tational costs are currently prohibitive for engineering flows thus restricting this method tofundamental research applications at low Reynolds numbers.

Large Eddy Simulation (LES) : In Large Eddy Simulations, the large scale flow including large

6


eddies in the flow are directly computed while the small scales of motion are modelled.This approach can be seen as a compromise between DNS methods and RANS methods(see below).

Reynolds-averaged Navier-Stokes (RANS) : In this approach, all turbulent motion is modelledand only the mean velocity field is directly computed. In unsteady RANS (URANS) theintrinsic unsteadiness in the mean flow is computed (i.e., the corresponding source termsare introduced in the equations); if they are removed, the mean flow field is steady (steadystate RANS).The information such a simulation provides about the flow is obviously far less accuratethan the results of DNS or even LES computations but is often sufficient for engineeringpurposes since predictions of many quantitative properties of flows, such as average forcesand the degree of mixing, are possible with an acceptable degree of accuracy. The effectof turbulence on the mean flow is accounted for using turbulence models of which a widevariety have been developed with varying degrees of precision and complexity.

In this work, all computations are performed using the steady state RANS method due torestrictions in computational resources and the relatively large amount of simulations that needto be performed. Therefore, only this method will be considered in more detail in section 2.2.3describing the mathematical models.

2.1.3 Boundary LayersAt solid walls, the fluid moves with the speed of the wall. This boundary condition, also calledno-slip condition, leads to the emergence of boundary layers between the wall and the free steamflow in which the fluid velocity experiences very large spatial gradients in wall-normal direction.Especially in flows at high Reynolds numbers, these boundary layers are extremely thin andthe velocity gradients high. Since the wall shear stress, and thus the losses due to skin friction,are directly proportional to the velocity gradient at the wall, it is essential that this value isaccurately computed. The wall shear stress τw is given by

τw = µ∂u

∂y

∣∣∣∣y=0

, (2.3)

where µ is the dynamic viscosity of the fluid. The difficulty in predicting the wall shear stress isdue to the fact that both the velocity gradient that changes rapidly close to the wall must beresolved and an accurate value for the viscosity is required.

When analysing boundary layers, the appropriate parameter is the non-dimensional wall distancey+ and velocity u+ given by

y+ = ρuτµ

y and u+ = u

uτ, (2.4)

where y is the wall distance, u is the local velocity and uτ is the friction velocity defined as

uτ =√τwρ

. (2.5)

When the near wall velocity profiles of fully attached, fully developed turbulent flows are plottedin the (u+, y+)-plane, they collapse into a single profile that exhibits two distinct regions as

7


100 101 102 1030

10

20

30

y+

u+

Figure 2.4: Self similar turbulent boundary layer profile showing the upper limit of the viscoussublayer (−−) as well as the lower limit for the logarithmic region (· · ·). The thin curves showthe approximations used in the respective regions. For the buffer layer no such relation exists.Reproduced from [27].

shown in Fig. 2.4. The region close to the wall (0 < y+ < 5) is called viscous sublayer in whichthe local Reynolds number is close to unity and viscous effects dominate, leading to a linearrelationship between velocity and wall distance. Further from the wall, for 30 < y+ < 300 . . . 500,lies the logarithmic region in which, as the name suggests, the velocity varies logarithmicallywith y+. The region between the logarithmic region and the viscous sublayer is called bufferlayer for which no analytical relationship between velocity and wall distance exists. In the wakeregion beyond the logarithmic region, the profiles are dominated by outer pressure gradients andthe universal similarity breaks down.

When attempting to compute the entire boundary layer numerically, the y+-value must be below1.0 on all surfaces to guarantee a reliable evaluation of the wall shear stress (models with thisrequirement are termed low Reynolds number models). This is a prohibitive requirement formany flows since it leads to very high cell counts. Therefore, turbulence models are often used inconjunction with wall-functions based on the existence of the logarithmic region that bridge theviscous sublayer and buffer layer. These high Reynolds number models heavily reduce the cellcount by allowing the first cell to be placed well inside the logarithmic layer (y+ ≈ 30) where thelocal Reynolds number is already large.

The structure of the boundary layer must be kept in mind for the meshing process since it iscrucial for the accuracy of the simulation with low Reynolds number models to ensure y+ < 5(ideally y+ < 1.0) in order to fully resolve the viscous sublayer and y+ > 25 . . . 30 for highReynolds number models to be sure to be inside the logarithmic region [3, pp. 115–116].

2.1.4 Secondary Air System and CoolingThe aim of film cooling in aero engine turbines is to protect the blade surface from the hightemperature main gas flow coming from the combustion chamber by bleeding a thin layer ofcoolant between the hot gases and the exposed surfaces. This injection is usually achievedthrough individual small holes in the blade surface connected to an inner plenum (or core) insidethe blade that is fed from the compressor. Although film cooling is usually employed in tandem

8


with other cooling techniques such as convection cooling and impingement cooling in the bladecore [19], we will focus only on film cooling since heat transfer is not considered and only filmcooling has an effect on the adiabatic flow field.The secondary air system (SAS) is a collective term for the flow passages around the main gaspath designed to distribute the cooling air to the leakages through rimseals and gaps betweenrotating and stationary parts and the blade cooling inlets.

Traditionally, efficiency improvements for turbines have been achieved mainly through incrementaladvances in technologies pertaining to the main gas path. In recent years, as optimum efficiencyis gradually approached, the secondary air system is increasingly moving into the focus ofdevelopment [36].In the quest of a better understanding of the effects of the various secondary inflows, the testrig studied in this work is equipped with a representative secondary air system and coolingcomprising of streamwise ejection at trailing edge of the stator rows (TE slots) as well as realisticfilm cooling on the blades of both rows of the first turbine stage. The rimseal leakage flowsand the connected cavities as well as the trailing edge slots are included in the simulation asfully-featured components with relatively little modelling.

Figure 2.5: Modern high-pressure turbinerotor internal cooling system combininghigh-pressure (black) and low-pressure(grey) cooling feeds. From [4, p. 185].

Fig. 2.5 shows the intricate geometrical details ofthe individual film cooling holes for a typical tur-bine rotor, similar to the one used in the test rig,that present a considerable challenge for numericalsimulations. Although fully featured computationsare within reach [15], they are still very costly bothin terms of pre- and post-processing and computerhours. Furthermore, their application is not feasiblefor studies involving several calculations. Therefore,numerous models that avoid the complete discreti-sation of the cooling holes have been devised withvarying degrees of simplification.Relatively high-fidelity models are based on the vol-ume source terms locally introduced in the cells ad-jacent to the actual cooling hole locations withoutrequiring them to be meshed. The model parametersare usually evaluated using exterior correlations fromexperimental data, such as in the model proposed byAndreini et al. [2]. Other approaches use 2D patchesto model the film cooling like a conventional inlet,such as a model by Burdet et al. [5]. If additional sim-plifications are applied, the influence of an entire rowof cooling holes can be approximated by uniformlyintroducing a volumetric source term along the bladespan at a certain distance from the wall [17].

An approach similar to the one described in [17] is adopted in this work. The individual coolingrows are replaced by cooling strips at the same location, where volumetric source terms areintroduced into the cells adjacent to the wall in wall-normal direction up to a predetermined

9


penetration depth. The cooling strip model is described in more detail in section 2.2.4.

Although the details of the vortical structures due to localized injection are lost and hence themixing processes are not necessarily reproduced correctly, this cooling strip model was foundto yield good results in terms of simulating the influence of film cooling on the integral turbineperformance parameters that are of interest in this work. Since more of the physics is modelled,certain parameters, especially the penetration depth, need to be calibrated. This process isdescribed more in detail in section 5.1. The cooling mass flow entering the flow domain as filmcooling is considerable (a total of over 10% relative to the main inlet mass flow rate) and hasa strong influence on the turbine performance as a whole. The impact on the boundary layerson the cooled blades is greater still. Therefore, the boundary layer mesh must be constructedappropriately, especially considering that turbulence models and associated wall functions aretypically not calibrated to film cooling data. A detailed study of the interaction between thecooling strip model and the boundary layer is carried out and presented in section 5.2.

2.2 Mathematical Modelling

All fluid mechanical analysis is based on basic conservation laws of mass, momentum and energyapplied to the working medium, i.e. air in this case. These laws can be equivalently expressedeither in differential form, applied to a single point in space, or in integral form pertaining toan extended volume of fluid. The following only gives a rough overview of the derivation, formore details see, e.g., [20]. The resulting set of coupled partial differential equations is called theNavier-Stokes equations and cannot be solved analytically thus requiring a numerical approach.The modelling steps to reach the Reynolds-averaged Navier-Stokes (RANS) equations that aresubsequently solved using the Finite Volume Method are outlined below. Since time-dependentsimulations of complex flows incur very high computational costs that quickly become prohibitivewhen attempting parametric sensitivity studies, this work is confined to steady state simulations.

2.2.1 Conservation Equations

Conservation of Mass

The most basic conservation equation states that the total mass of fluid in a given controlvolume V can change only through changes in density, the flux of mass over the control volumeboundaries ∂V and mass sources inside the control volume. Mathematically the steady, three-dimensional equation for the conservation of mass, also known as continuity equation, is givenby ∫

∂V

ρuini dS =∫V

ψv dV , (2.6)

where ρ is the fluid density, ui are the three velocity vector components, ni are the componentsof the normal vector on the control volume surface ∂V and ψv is a volumetric source term. TheEinstein convention is used, implying summation over the range of any index appearing twice ina given term.

Applying Gauss’ divergence theorem to transform all surface integrals into equivalent volumeintegrals and considering that the control volume V is arbitrary, leads to the differential form of

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the continuity equation:∂(ρui)∂xi

= ψv . (2.7)

The continuity equation (2.7) is typically given without source term. In this work, the sourceterm is introduced since the film cooling for the first turbine stage is modelled by introducingvolumetric source terms to reduce the model size. More details are given in sections 2.1.4 and3.4.

Conservation of Momentum

The second conservation law states that the total momentum of a volume of fluid in the absence ofexternal forces is conserved. Applied to a fixed control volume, the rate of change of momentumis balanced by the flux of momentum across the control volume boundaries and the externalforces acting on the control volume:∫

∂V

ρuiujnj dS =∫∂V

Θijnj dS +∫V

ρfi dV +∫V

ψm dV , (2.8)

where Θij is the stress tensor representing the surface forces due to pressure and viscous stresses, fiare the components of the volume forces acting on the control volume such as gravity, centrifugaland Coriolis forces (when the equations are applied in a rotating frame of reference) and ψm isthe momentum introduced by the volumetric source ψv. Analogously to the continuity equation,the general differential form of the momentum equation can be readily derived:

∂(ρuiuj)∂xj

= ∂Θij

∂xj+ ρfi + ψm . (2.9)

In order to be able to solve the equations above, the stress tensor must be expressed using thevariables for which a conservation equation exists. For many typical engineering fluids such asair, the simplest model, a linear relation between strain rate and velocity gradient, has proven tobe very accurate and is used in this work. For these so-called Newtonian fluids, the stress tensorcan be written:

Θij = τij −(p− λ∂uk

∂xk

)δij , (2.10)

withτij = µ

(∂ui∂xj

+ ∂uj∂xi

)(2.11)

the viscous stress tensor, µ is the dynamic viscosity, δij is Kronecker’s delta and λ is the bulkviscosity which is related to the dynamic viscosity by invoking Stokes’ hypothesis:

λ = −23µ . (2.12)

The dynamic viscosity µ is a fluid property that exhibits a certain temperature dependency butvery little pressure dependency. It is modelled by Sutherland’s law:

µ = 1.461 · 10−6 T 3/2

T + 110.3 , (2.13)

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where T is temperature.

The air flow in the turbine is essentially horizontal so that gravitational effects are negligible incomparison to other forces. Introducing equations (2.10) and (2.12) in (2.9) and omitting thevolume force contribution leads to the fully conservative form of the momentum equation:

∂(ρuiuj)∂xj

= − ∂p

∂xi+ ∂

∂xj

(τij −

23µ

∂uk∂xk

δij

)+ ψm . (2.14)

With ρ, p and ui there are 5 independent variables in the system of equations given by continuityand momentum equations so that the system is not closed in this form. For low speed flows, theerror made by neglecting compressibility is tolerable (typically for Ma < 0.3). In the turbineconsidered in this work, very high fluid velocities of over 300 m/s (or Mach numbers close tounity) are reached so that this approximation is no longer valid. In order to close the equationsystem, another conservation equation is required: the energy equation. This equation for thetotal energy of the system, in conjunction with an equation of state relating density, pressure andinternal energy, closes the system of equations known as compressible Navier-Stokes equations.

Conservation of Energy

The conservation of energy follows from the first law of thermodynamics stating that the energyof a fluid volume is conserved, i.e., can change only through heat fluxes across its boundaries orwork expended on or by its surroundings. Omitting body forces, it follows:∫

∂V

ρEuini dS =∫∂V

Θijujni dS +∫∂V

qini dS +∫V

ψe dV , (2.15)

where E is the total internal energy of the fluid, qi are the components of the heat flux vectorand ψe is the energy source associated with the volumetric source ψv.

The heat flux by thermal conduction is governed by Fick’s law:

qi = −k ∂T∂xi

, (2.16)

where T is temperature and k is the thermal conductivity given by

k = µγ

Pr , (2.17)

where γ is the ratio of specific heats and Pr is the Prandtl number that is relatively constant atPr = 0.72 for air at the considered conditions [24, p. 14].

Applying Gauss’ divergence theorem to equation (2.15) and combining it with equations (2.10)and (2.16) leads to:

∂(ρEui)∂xi

= −∂(pui)∂xi

+ ∂

∂xi

(τijuj −

23µ

∂uj∂xj

ui

)− ∂

∂xi

(k∂T

∂xi

)+ ψe . (2.18)

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Equation of State

To close the system, a thermodynamic equation of state is necessary to link pressure, densityand temperature. The air is modelled as a thermally perfect gas (ideal gas and thus the idealgas law) with properties that are a function of temperature. The ideal gas equation still holds:

p = ρRT = (γ − 1)ρ(E + 1

2uiui)

, (2.19)

where R = 286.96 J/(kgK) is the gas constant for air and γ (and therefore the specific heats) iscomputed from curve fits to account for its temperature dependency [7].

2.2.2 Steady State RANS Equations

In the cases studied in this work, the flow is turbulent and computing the time-dependent flowfield would require the resolution of all scales of turbulence in both space and time which isfar too costly for the study of a full turbine. Therefore, the steady state Reynolds-averagedNavier-Stokes (RANS) equations are considered, that represent the time-averaged versions of theequations of motion in which all turbulent and unsteady motion is modelled. The averaging isbased on the concept of Reynolds decomposition of the instantaneous variables into an averagepart (denoted with an overbar) and an unsteady fluctuation (denoted prime) around the average.For a general dependent variable φ in a statistically steady flow it follows:

φ(t) = φ+ φ′(t) with φ = lim∆t→∞

1∆t

∫ t0+∆t

t0φ(t) dt , (2.20)

where t is time, t0 is an arbitrary reference instant and ∆t is the averaging period. In practice,this period only needs to be much larger than the time scale of the considered flow. The flow ina turbine is inherently unsteady due to the periodic rotor passage but is nevertheless assumedsteady in this work to reduce the computational costs.

In many cases with compressible flow, density-weighted Favre averaging is used instead ofReynolds averaging. In this work, Reynolds averaging is employed nevertheless since the densitychanges, compared to reacting or high Mach number flows, are manageable.

Introducing Reynolds decomposition for the dependent variables into the exact equations ofmotion, averaging over time and omitting the time derivative yields the steady state RANSequations. These equations are very similar to their exact counterparts since all linear termsreappear analogously after time averaging while only the quadratic terms yield additional termsstemming from the correlation of the fluctuating parts:

uiφ = (ui + u′i)(φ+ φ′) = uiφ+ u′iφ′ (2.21)

The non-linearity of the convective terms leads to the emergence of correlations of fluctuatingterms such as ρu′iu′j , called the Reynolds stresses, that represent the transfer of momentum dueto turbulent fluctuations that cannot be expressed simply in terms of the other flow variables.The equations thus contain more unknowns than equations, leading to the notorious closureproblem of turbulence. It is here that turbulence models step in to obtain a closure for theequations of motion.

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2.2.3 Modelling Turbulence

Turbulence and its influence on the mean flow is a complex topic that has been the object ofextensive research in the last decades leading to a large number of different approaches to itsmodelling.

The Reynolds stresses can be seen as a flow quantity that, like kinetic energy, is convectedand diffused in the flow. This interpretation leads to the derivation of transport equationsfor the Reynolds stresses from the Navier-Stokes equations that are solved together with themean flow equations and an additional equation for the turbulent dissipation rate ε (althoughother alternatives such as the turbulence frequency ω exist). This class of turbulence models,termed Differential Reynolds Stress Models (DRSM), relies on relatively little modelling andhas the advantage of capturing turbulence anisotropy effects at the cost of having to solve sevenadditional transport equations. If further modelling assumptions are made, the Reynolds stresstransport equations can be reduced to an (implicit or explicit) algebraic equation solved inconjunction with other transport equations, typically ε or ω as well as the turbulent kineticenergy, k, defined as

k = 12u′iu′i . (2.22)

These models are called Algebraic Reynolds Stress Models (ARSM) or Explicit ARSM (EARSM)[34, pp. 75].The available implementations of this type of model do not support wall functions and thusrequire the complete resolution of the boundary layers (y+ < 1.0). The resulting extremely highcell count precludes the use of the EARSM approach for this study.

The Reynolds stresses can also be interpreted as an additional "turbulent" viscosity leading tothe eddy viscosity models in analogy to Eq. (2.11) (also known as Boussinesq hypothesis). TheReynolds stresses are thus given by

ρu′iu′j = µt

(∂ui∂xj

+ ∂uj∂xi

)− 2

3ρδijk (2.23)

where µt is the eddy viscosity and k is the turbulent kinetic energy.

The definition in Eq. (2.23) does not present a closure in itself since the new unknown µt stillrequires modelling. The form of the equations indicates that this approach assumes isotropicturbulence (the velocity fluctuations ui are treated equally) which is a simplification that isacceptable in many flows. The obvious advantage of these models is their straight forwardimplementation since the turbulent stresses can be simply introduced by adding the eddyviscosity to the laminar viscosity in the transport equations.A series of closures have been put forward to compute the eddy viscosity involving more or lesscomputational effort. The following summarizes the main types of turbulence models of thiskind.

The simplest and cheapest models in terms of computational cost use an algebraic equation toobtain µt (algebraic or zero equation models) relying on a length and velocity scale. Since thelengthscale needs to be adjusted from case to case, algebraic models, despite having merits forpredicting attached turbulent flows near walls, are unreliable for separated flow [35, pp. 53].

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More advanced models use differential equations to compute the turbulence quantities. Althoughone equation models exist and have been used successfully (notably the Spalart-Allmaras model(1992) developed specifically for aerospace applications), two equation models are the preferredchoice in engineering applications today since they are complete (i.e., relying only on localquantities requiring no a priori knowledge of the flow) and hence more general and versatile [35,p. 107].

The two equation models, also called k-ε or k-ω models, typically use a transport equation for kand either ε or ω to obtain the velocity and length scales.The standard k-ε model first developed by Launder and Spalding (1974) performs well for highReynolds numbers and thin attached boundary layers and has been used extensively in bothresearch and industry thus creating an extensive validation database. These models have beenfound to have problems predicting spreading rates of turbulent jets and dealing with adversepressure gradients [34, p. 74].An alternative model, the k-ω model, proposed by Wilcox (1988) shows several advantages overthe k-ε model including the prediction of flows with adverse pressure gradient and the fact thatit can be easily integrated all the way to the wall avoiding the use of damping functions. Knownissues include an unrealistic dependency on free-stream turbulence levels [3, p. 58].Menter’s Shear Stress Transport (SST) k-ω model (1993) is a two-equation model that combinesthe best of both models above. The standard (Wilcox) k-ω model is used close to the walls in theviscous sublayer and the logarithmic region while the k-ε model in an appropriate formulation isused for the outer boundary layer and free shear flows. This way, the model can be used as a lowReynolds number model close to the wall without modification but does not exhibit the typicaldependency on free stream turbulence of the standard k-ω models. Furthermore, the modelincludes a limiter for the turbulence anisotropy that avoids the overprediction of turbulenceproduction in regions with strong shearing [35, pp. 124].

The flow through a turbine exhibits both attached regions (on the blades as well as hub andcasing walls) as well as regions with free shear flows (wakes behind blades and mixing regionsclose to secondary inlets) so that an accurate prediciton of the flow requires a turbulence modelcapable of dealing with both flow configurations. Menter’s SST k-ω model presents the besttrade-off between accuracy and cost and is therefore used throughout this thesis. The exactequations used can be found in the solver user guide [7, pp. 85] based on the seminal paper byMenter [23].

Although Menter’s SST k-ω model can be integrated to the wall without special damping terms,the low Reynolds number formulation is highly demanding with regard to mesh resolution closeto the wall (see section 2.1.3). Since it would be too costly to ensure y+ < 1.0 on all surfaces,wall functions are used that blend into the near wall solution depending on y+ so that the effectbecomes negligible as the mesh quality improves (y+ → 0) [7, p. 91]. While the wall functionsare expected to yield trustworthy results on most surfaces, the adequacy of models for the bladesurfaces with film cooling is not obvious and is considered in more detail in section 5.2.

2.2.4 Modelling Film CoolingAn approach similar to the one described in [17] is adopted in this work. The individual coolinghole rows are replaced by cooling strips at the same location where volumetric source terms areintroduced into the cells adjacent to the wall in direction of the cooling jet up to a predetermined

15


penetration depth. Although the details of the vortical structures due to localized injection arelost and hence the mixing processes are not necessarily reproduced correctly, this cooling stripmodel was found to yield good results in terms of simulating the influence of film cooling on theintegral turbine performance parameters that are of interest in this work.

The model idealizes the cooling holes as strips in the spanwise direction but retains most of theparameters crucial to film cooling. For each individual cooling row, the injection angles (α, β)relative to the blade surface as well as the total area Ac of the bores are extracted directly from theCAD model. The mass flow rate mc was estimated from preliminary design predictions validatedby fully-featured CFD computations from which also the coolant density ρc was evaluated. Thecoolant exit velocity vc is directly computed as

vc = mc

ρcAc. (2.24)

The turbulence parameters for the coolant can unfortunately not be set manually and areautomatically set to the standard values of k = 1 m2/s2 and ω = 5555 s−1. The coolanttemperature was set to 300 K.

The last model parameter, the coolant penetration depth lc controlling how far into the mainflow the volume source terms are introduced, is the only purely empirical parameter of the modeland must be calibrated. This process is described more in detail in section 5.1.

2.3 Turbine Performance Parameters

In order to analyse the performance of a high-pressure turbine, additional non-dimensionalparameters are used that relate the fluid properties to the operation parameters of engineeringimportance such as mass flow rate, pressure ratio or power output. Within the scope of this work,only two performance parameters are studied in detail, the capacity and the turbine efficiency,although several other parameters play an important role in turbine design. The following sectionintroduces the performance parameters.

2.3.1 Capacity

A crucial relationship for turbomachinery is the non-dimensional mass flow rate, also known asflow capacity or simply capacity, is given by:

m√cpTtot

AnPtot= γ√

γ − 1 ·Ma ·(

1 + γ − 12 ·Ma2

)− 12

(γ+1γ−1

), (2.25)

where m is the mass flow rate, Ttot and Ptot are the total temperature and pressure respectivelyand An is the flow area normal to the flow.

The important characteristic of Equation (2.25) is that the RHS is a function of only the Machnumber and fluid properties (γ) and can be used to relate the flow properties at different pointswithin a turbomachine. In real turbomachines, the flow is non-uniform and the application ofEq. (2.25) is not straightforward since some method (averaging) has to be chosen to computethe stagnation properties. Although the absolute values of the capacity is dependent on this

16


choices, the trends are captured by all consistent definitions [16].

To simplify the computation of the capacity, the mass flow weighted area average of the totaltemperature and pressure at the outlet of the NGV1 domain is used. A small error in comparisonto the true throat capacity is expected but this offset is not crucial since this study focuses onvariations instead of absolute values. In this work, following common practice in industry if theworking fluid and machine dimensions are known, the capacity is simplified by removing cp, Anand γ.

This leads to the definition of the NGV1 outlet capacity used throughout this work:

C = m√Ttot

Ptot

∣∣∣∣∣NGV1 outlet

. (2.26)

To lighten the text, the terms NGV1 outlet capacity and capacity will be used synonymously inthis work unless stated otherwise.

For a given fluid and stagnation (total) conditions, the mass flow per unit area (m/An) is afunction of Mach number only. Under subsonic conditions (Ma < 1), its value increases withMach number as the velocity increases faster than the density decreases. For supersonic flows onthe other hand, the reverse is true. Between these two regimes (Ma = 1) the mass flow per unitarea reaches a maximum and the flow is choked. This means that once the flow in a turbinereaches sonic conditions in the section of minimum flow area (throat), and the mass flow cannotbe increased further without changing the inlet stagnation conditions.

The throat capacity is a crucial parameter in turbine design since it defines the maximum flowthrough the turbomachine and sets the operating point of the turbine. The component matchingwithin a turbomachine is based on the throat capacity of the first turbine nozzle (NGV1) wherefull choking occurs first.Modern NGVs are designed with complicated geometries including profiled endwalls, trailingedge slots and strong radial variation of the blade contour and exhibit complex flow fields withsupersonic pockets, coolant injection and leakages as well as 3D flow effects. The nozzle capacityhas been found to be greatly influenced by geometrical shape of the flow passage and bladeloading [1] and the losses incurred by the boundary layers growing along the blade [12]. Theeffect of film cooling has also recently been investigated showing that for a typical turbine a 10%increase of throat flow through film cooling (comparable to the flows in the test rig) noticeablyincreases the capacity by approximately 1.7% [28]. As a consequence, since 1D computationsroutinely lead to errors in the order of 10%, the use of 3D analysis tools is widely recommendedand the boundary layers need to be considered with care in terms of the induced losses, blockageand film cooling interaction in order to yield trustworthy results for sensitivity of the NGVcapacity.

2.3.2 Efficiency

The turbine efficiency is probably the most ubiquitous performance parameter used in turbinedesign. In general, turbine efficiency in the context of fluid dynamics relates the real expansionprocess through the machine to the corresponding ideal, lossless thermodynamic process, i.e.,

17


considering only losses related to the flow. Specifically, this means that mechanical losses in thebearings, shaft losses etc. are not considered. Even with this restriction, there are still severaldifferent definitions used in literature. In this work we will focus on the enthalpy, power andentropy based efficiencies, which are the definitions included in the CFD code HYDRA.

The most straightforward definition of efficiency is based on the total enthalpy change1 ∆Hacross the turbine stage that corresponds to the work output.

h

s

p01

p02

p03

1

2s

3ss

2

3s

3

Figure 2.6: Schematic representation of a typicaltwo-stage turbine expansion.Real process (−) and ideal process (· · ·).

Considering the schematic h-s diagram inFig. 2.6 showing a typical expansion process,the entropy based efficiency for the entire tur-bine can be written as

ηH = H1 −H3H1 −H3ss

, (2.27)

where the nominator represents the real en-thalpy change (work extracted) and the de-nominator represents the maximum enthalpychange possible corresponding to an isentropicprocess. The single stage efficiency is analo-gous.

The important point here is that only the initial and final states are of importance, not the exactpath between them. Translating this concept to CFD means that the mass flow weighted areaaveraged enthalpies are computed at inlet and exit of the considered zones taking into accountall secondary inlets (film cooling, trailing edge slots and rimseals). The reference ideal work foreach inlet is computed using the simplifying assumptions of a uniform total state on all inletsand outlets and using the isentropic relations.

HYDRA also computes a power based efficiency using the same methodology as the enthalpybased efficiency with the difference that the extracted shaft work (W , work of the local pressureand shear stress on all moving walls) is used to quantify the real enthalpy change. Since HYDRA isa fully conservative code, mass, momentum and energy are conserved in a converged solution andthe work output must equal the enthalpy difference over the considered domain. A comparison ofthe power versus the enthalpy based efficiencies is therefore another indicator for the convergenceof the solution.

A second approach to turbine efficiency is based on the fact that all loss mechanisms, regardlessof their origin, lead to an increase in entropy allowing for a unified, description of losses on alocal level [8]. Although this insight is of little practical use to the experimenter since entropycannot be directly measured in a lab, it is very useful within the CFD framework where entropycreation can be readily calculated.

The second law of thermodynamics states that the entropy s of a system can change onlythough production inside the volume as well as heat and mass fluxes across the boundaries. The

1A capital H is used to denote the total enthalpy instead of a subscript "tot" to lighten the formulas.

18


corresponding entropy transport equation for steady, adiabatic flow reads:∫∂V

ms dS =∫V

PτTdV +

∫V

k

T

(∂T

∂xi

)2dV , (2.28)

where the terms on the RHS are the entropy production due to shear stress work (τij) andthermal gradients (∂T/∂xi). Both of these volume source terms are positive.

Considering again the h-s diagram in Fig. 2.6, the difference between maximum available work(H1 −H3ss) and the extracted work (H1 −H3) is equivalent to the integral of the losses due toentropy production over the entire considered fluid domain. It follows:

Wloss = H3 −H3ss =∫ 3

1Tds . (2.29)

The power lost (used to increase entropy) is computed by multiplying the production terms inEq. (2.28) by the local temperature and integrating over all control volumes in the fluid domain.The entropy based efficiency is then defined as

ηs = W

W + Wloss, (2.30)

where the shaft power W is defined as for the power based efficiency.

The power and entropy based efficiencies are not expected to be equivalent even in a fullyconverged solution since they represent two different interpretations of efficiency. In fact, theentropy based approach corresponds to the division of the complete expansion process into afinite number of small steps corresponding to the discrete version of a polytropic expansion.The important difference is that the local isentropic efficiency is computed at cell level and istherefore not exactly constant over the expansion.

19

3 Numerical Methods and Setup

3.1 Computational Fluid Dynamics TheoryThe process of performing a CFD study consists of several steps leading from the definition ofthe objective, the mathematical and numerical setup, the computation of the solution to thepost-processing and assessment of the results. In the following section, the general process isdescribed in more detail before the setup of the study at hand is introduced. This section islargely based on the excellent summary given in [11].

3.1.1 The Mathematical Model

The starting point of any numerical study of fluid flow is the choice of a mathematical model,a set of partial differential equations governing the flow as well as the appropriate boundaryconditions. The immense complexity of fluid flow, especially in application of engineering interest,always requires some degree of simplification of the governing equations. Section 2.2 describesthe modelling approximations applied in this work.

3.1.2 Discretisation of the Governing Equations

Once the mathematical model is chosen, the resulting differential equations need to be discretised,i.e., an appropriate discrete approximation method for the continuous differential operators mustbe chosen in order to build a system of algebraic equations for the variables of interest on discretepoints inside the computational domain. The most common methods are Finite Differences,Finite Element and Finite Volume methods. The Finite Differences method is simple and efficienton structured grids but laborious when applied to unstructured grids. Furthermore, the methodis not conservative by construction so that this issue must be addressed separately. The FiniteElement method is more flexible and uses weighting for every element that is multiplied withthe governing equations before integration. The most common approach in computational fluiddynamics is the Finite Volume method. Since the solver used in this work is based on the FiniteVolume method, this approach is described in more detail.

In the Finite Volume Method the domain is divided into control volumes of arbitrary shape.The scheme can be cell centered with the unknown located in the center of the control volumeof node centered. The HYDRA code employs the latter option and, as shown in Fig. 3.1, asecond set of control volumes is constructed around the computational node where the unknownis located by connecting the centers of mass of the primary volumes. The governing equationsin integral form are solved individually for each control volume of the second set. Analogouslyto the Finite Element Method, the Finite Volume method can be applied to any geometry andis automatically conservative if the fluxes across the common boundary between two adjacentcells are computed the same way. On the other hand, since the Finite Volume Method requiresinterpolation, differentiation and integration at every node, the derivation of approximations ofmore than second order are difficult to derive.

21


Figure 3.1: Dual control volumes in HYDRA. The shaded area around the node is the controlvolume on which the conservation equations are solved. From [6].

3.1.3 The Numerical GridThe choice of the numerical grid or mesh is closely linked to the discretisation method. The meshdefines how the computational domain is divided into control volumes on which the governingequations are to be solved. The meshing methods in CFD are classified in two major groups,structured grids and unstructured grids that are briefly discussed.

Structured mesh : In a structured mesh, the grid points are located at the intersection ofcoordinate lines. Adjacent coordinate lines cannot intersect so that all interior grid pointshave a fixed number of neighbouring cells. For most engineering applications with complexgeometries it is difficult or impossible to find a single mapping of the grid points onto amatrix using one index for each coordinate direction. Therefore, the computational domainis divided into a number of blocks that are meshed individually and then appropriatelylinked (block-structured meshing). The structure of the resulting matrices can be usedfor very efficient solution techniques for the equation system. Despite the computationaladvantages the generation of high-quality structured grids is often very difficult and time-consuming and sometimes even impossible for very complex geometries. Furthermore, thestructured meshing process has only limited flexibility concerning mesh refinement andoften leads to unnecessarily fine grids since local refinements propagate outwards along thecoordinate lines.

Unstructured mesh : If the internal structure of the grid is abandoned, the result is anunstructured mesh. In this arrangement, the control volumes can be of any shape and thereare no restrictions on the number of cells meeting in a point. The computational domaincan therefore be meshed very easily and the meshing process can be largely automatedregardless of the geometric complexity of the problem. Moreover, the refinements can bedefined locally leading to a very efficient mesh in terms of cell count. These advantagescome at a cost. Due to the lack of internal coordinate structure, the neighbours of everycell need to be explicitly listed and the resulting system matrix has to be sorted. Thesolvers for unstructured grids are usually slower than their structured counterparts.

Due to the complexity of the turbine geometry considered here, the unstructured meshingapproach was chosen. The meshing process is described in more detail in section 3.2.

22

3.1 Computational Fluid Dynamics Theory

3.1.4 Finite Approximation

Once the grid has been created, a suitable finite approximation of the terms used in thediscretization of the governing equations must be chosen. The Finite Volume Method requires anapproximation of surface and volume integrals across the control volumes and their boundaries.Since the variables are computed at the cell centers their values on the control volume boundariesmust be interpolated.

3.1.5 Solution Method

Finally, the solution method itself must be chosen to compute the solution of the set of non-linear algebraic equations resulting from the discretisation process. The appropriate solutiontechnique is problem dependent but given the usually very large matrices involved in modernCFD, the equations are practically never solved directly but instead approximated by an iterativetechnique. Unsteady viscous flow problems are solved with so-called time-marching methods inwhich the solution to the elliptic problem is computed at every time step. For steady flows, apseudo-timestepping scheme is often used and run until the steady state has been reached.In most successful steady-state solvers, multigrid algorithms are an essential part of the solutionprocedure. The HYDRA solver used in this work also incorporates a multigrid method [24].The multigrid method is used to accelerate the convergence of an iterative solution of a linearequation system and is based on the observation that the high frequency errors are quicklydamped while low frequency errors require many iterations to disappear. The fundamentalconcept of the multigrid method is to use a sequence of successively coarser grids in the iterationprocedure. The solution is periodically restricted (transferred from the fine to the coarse gridlevel) transforming formerly low frequency errors to high frequency errors that are efficientlydamped, and prolonged (transferred back to the fine grid) to obtain the accurate solution. Thisway, all error modes are quickly eliminated considerably accelerating the convergence [11, pp.107]. Details to solver configuration used in this work are given in section 3.3.

3.1.6 Errors

Numerical solution are always approximate solutions of the underlying mathematical models.If the results of a numerical simulation are to be meaningful, it is paramount that the possiblesources of error within the simulation are known and assessed and that the different types oferrors are distinguished and mitigated.Modelling errors are errors stemming from the modelling applied in the derivation of themathematical model. These errors may be very large in complex flows that require extensivemodelling to be tractable. Turbulence and combustion modelling are typical sources of modellingerrors. Even if the underlying equations are exactly solved, the solution is not a perfectrepresentation of reality. Therefore, validation against experimental data or data from moreaccurate models such as fully featured CFD or DNS solutions of turbulence is an essential stepin a numerical study.Furthermore, the discretisation of the equations introduce a discretisation error or truncationerror that reduces as the grid spacing is reduced. The order of the approximations used in thenumerical scheme are a measure for accuracy.When the discretised equations are solved iteratively, the difference between the iterative solutionand the exact solution is termed convergence error or iteration error. This error is usuallyestimated using the norm of the solution residual. Usually a reduction by 3-4 orders of magnitude

23


of the residual normalised by an initial value is an indicator that the solution is sufficientlyconverged [11, p. 118]. The convergence of the computations performed in this work is consideredin more detail in section 3.3.2.

In order for a numerical method to be able to yield a reasonable result, it must possess certainproperties related to the errors above. If the truncation error vanishes with vanishing grid spacingthe discretised equation will become exact in the limit of zero grid spacing and the method isconsistent. Furthermore, it must be stable which implies that small errors are not amplifiedin the solution process thus avoiding divergence. A numerical scheme that is both consistentand stable is convergent. The stability of numerical schemes is routinely analysed, often usingthe von Neumann method, but for very complicated methods, stability is impossible to provemathematically [11, p. 32]. Therefore, convergence of a method is usually empirically checkedwith a refinement or mesh independence study in which the equations are solved on successivelyfiner grids until the solution is independent of the mesh size.

3.2 Mesh Generation

In the mesh generation process, the aim is to create a computational mesh that is both fine enoughto resolve the important flow features and coarse enough to keep simulation times manageable.The final mesh used in the computation will therefore always represent a trade-off between thesetwo objectives.

3.2.1 CAD Cleaning

Before the grid generation itself, the CAD geometry needs to be cleaned and prepared for efficientmeshing. These steps are performed using the proprietary CAD software package NX 9.0. Thecleaning process includes the removal of details that are modelled such as the individual coolingholes on the blades that are replaced by cooling strips. Figure 3.2 shows the original CAD modelof the NGV1 with the cooling holes (left) and the surfaces extracted from the patched modelshowing the location of the cooling strips (right).Geometrical features that are either too small to be resolved in the computation or have negligibleimpact on the overall flow field are smoothed out in the cleaning process. A detail that wasremoved is a small backward facing step in the casing between stator and rotor in the first stageshown on the left in Fig. 3.3. Other relatively small features are kept, in spite of the increasedcell count due to the high level of refinement needed to resolve them, because they are expectedto have a noticeable influence on the flow field. An example is the casing wall step betweenstator and rotor in stage 2 shown on the right in Fig. 3.3.Furthermore, the surfaces need to be tagged (named) and grouped according to the requiredlevel of surface refinement to accelerate the setup of the meshing process. After cleaning, thesurfaces are exported as parasolids (NX file format) that can then be loaded into the meshingtool, tessellated and saved as tessellation files. During the parasolid import, care must be takento ensure that the surfaces and edges are tessellated with sufficient resolution to capture theimportant features and avoid bumps in the final mesh (around 100 triangles per coordinatedirection was found to be sufficient).

24

3.2 Mesh Generation

(a) Fully-featured model. The individual cool-ing holes as well as the entire inner core ofthe blade are resolved.

(b) Cooling strip model. The white lines indi-cate the cooling strip locations, the blue ar-rows show the cooling flow exit angle match-ing the cooling hole orientation in the fully-featured model.

Figure 3.2: NGV1 blade suction side surface comparing the fully featured model (left) to thecooling strip model used in this work (right). The flow is from left to right.

3.2.2 Meshing

The mesh is created using the commercial Boxer suite (Version 3.8.0.20), an automated, hybridmeshing tool. The aim to reproduce the test rig geometry as accurately as possible includingseal cavities, trailing edge slots and rotor blade tip geometry disqualifies a structured meshingtool for this work since the setup and mesh optimization process for such complex geometriesis exceedingly time-consuming. The hybrid approach with Boxer allows for a simpler, moreselective local refinement of critical regions while maintaining a good accuracy in the near wallregion using boundary layer extrusions.

The mesh is generated in three steps. First, the entire fluid volume is partitioned into cubes alongthe coordinate directions (in this case a cylindrical coordinate system around the turbine shaft ischosen) already including user specified local mesh refinement where the volume difference oftwo adjacent cubes is never greater than a factor of eight. This mesh is called the octree mesh.In the next step, a body-fitted mesh is created by projecting the nodes of the octree mesh ontothe surface of the actual geometry and introducing tetrahedral cells to remove hanging nodeslying on faces of adjacent cells. The last step consists of the extrusion of the structured near-wallboundary layer cells into the hybrid mesh and subsequent mesh optimization. The basic octreemesh as well as the resulting body fitted mesh (with boundary layers) mesh is shown in Fig. 3.4.To minimize the mesh size, only a single blade pitch is meshed for each row and a periodicboundary condition is automatically applied to the circumferential boundaries. The backgroundoctree cell size was defined by setting 40 circumferential cells per stator pitch and analogously 20cells per rotor pitch. The duct pitch was set identical to the rotor.

25


Figure 3.3: Sketch of the turbine geometry from the inlet (left) to the NGV2 exit (right) showinggeometrical details of the casing wall that were removed (· · ·) or kept (−). On the hub wall thegeometry of the seal leakage cavities is shown.

(a) Basic octree mesh showing overall mesh re-finement.

(b) Final body-fitted mesh with boundary lay-ers.

Figure 3.4: Midspan through the ROT1 mesh at the leading edge.

3.2.3 Local Mesh Refinement

The meshing tool Boxer does not include an automatic mesh refinement algorithm so that allrefinements need to be specified by the user. In order to set the required refinement levels, thecritical regions of the flow domain need to be identified. In the following section, the refinementmethodology is presented and the impact of local refinement illustrated based on concreteexamples.

The first step is surface and edge refinement that is defined independently for specific geometricalfeatures that are imported individually. Surface refinement can be applied globally for a given

26

3.2 Mesh Generation

surface, which can be very costly in terms of cell count, or locally using so-called polylinesto confine the refinement. The chosen refinement is then spread locally into the fluid domainsurrounding the surface to ensure a smooth transition to coarser regions of the mesh. Therefinements are applied by defining refinement levels that correspond to the number of divisionsof the background cells in each coordinate direction. An increase of the refinement level by onedivides every cell into eight smaller cells.

In order to adequately resolve the blade surface itself as well as the complex vortical structuresappearing around the blades and vanes, the blade surface refinement level is set to 2. The filletsconnecting the blades to the hub wall (and casing for the NGVs) require extra refinement due tothe high level of curvature and the necessity to obtain a good resolution of the horseshoe vortexdeveloping along the geometry. Due to the high level of streamline curvature in the vicinity ofthe blade leading edge, this region is also refined to level 3. A critical region of the NGV mesh isthe trailing edge slot due to the small width of the slot itself and the high surface curvature atthe lip and trailing edge. A good resolution of the flow in this area is especially important sinceit has a large impact on the exit flow angle. From a more practical standpoint, it was found thatthe solver failed to converge at all if this region is inadequately resolved. The slot inlet boundary,inner slot walls as well as trailing edge of the vanes are therefore refined to levels 4 and 5. Themixing planes between the domains are also refined in order to resolve the complex wake flowsand ensure an adequate coupling with subsequent rows. The rimseal cavities, similarly to thetrailing edge slots, must be refined to resolve the sharp bends and small surfaces. A refinementlevel of 2-3 is chosen for the cavities. Finally, the rotor tip gap and winglet are heavily refined(level 4-5) to ensure a good resolution of the tip leakage and squealer flows as they are a majorsource of aerodynamic losses in a turbine stage [8].

(a) No volume refinement. (b) Level 2 volume refinement.

Figure 3.5: Comparison of absolute Mach number distributions on a midspan cut in the NGV1wake region close to the trailing edge overlaid with the mesh with and without mesh volumerefinement. The flow is from left to right. The Mach number is subsonic in the entire domainand increases from blue to red.

27


Mesh refinement can also be defined using volumes created in the CAD program that force arefinement of the entire mesh contained to the specified level. Such a volume refinement wasdefined for the blade wakes for both stator and rotor rows. Figure 3.5 contrasts the absoluteMach number distributions around the NGV1 trailing edge computed with and without volumerefinement of the wake region. The comparison clearly shows the impact of the refinement: Onthe coarse mesh (Fig. 3.5(a)) the increased numerical diffusion, aggravated by the fact that theflow in this region is poorly aligned with the cells, smears the sharp velocity gradients in thewake and leads to an unphysical widening of the wake region. The wake is much smoother whenthe mesh is sufficiently refined (Fig. 3.5(b)).

3.2.4 Boundary Layer ResolutionThe last important step in the meshing process is the boundary layer mesh creation. The distancefrom the first mesh node to the wall as well as the wall-normal expansion ratio into the flow iscritical for an accurate resolution of the boundary layer and therefore has to be well controlled.The first layer height was therefore individually adjusted for each surface depending on the freestream velocities to maintain at most y+ ≈ 2 on the aerofoil and y+ < 4 in the entire domain(apart from singular peaks due to sharp corners). The expansion ratio was set to 1.2 throughoutthe mesh. The choice of boundary layer resolution is motivated by detailed studies of the coolingstrip model and its interaction with the boundary layer in section 5.2. The y+-distribution onthe two turbine stages is shown in Fig. 3.6.

Figure 3.6: y+-distribution for the final mesh of the full HPT.

28

3.3 HYDRA Configuration

(a) Midspan cut through the first two zones (stage 1) of the final mesh showing the refinements aroundthe blades and in the wakes.

(b) Close-up of the mesh in the rotor tip clear-ance (y-normal crinkle cut) for the small tipclearance case showing the boundary layer reso-lution. The distance from squealer tip to casingis 0.33 mm (small tip clearance).

Figure 3.7: Selected cuts through the final mesh of the full HPT.

3.3 HYDRA ConfigurationThe computations in this work are carried out using the Rolls-Royce in-house CFD solver suiteHYDRA (version 7.12). HYDRA is a general purpose viscous CFD code for both structured andunstructured grids using a parallelised multigrid method for faster convergence. In this section,the pre-processing procedure and solver configurations used in this work are presented followedby a description of the reference boundary conditions.

3.3.1 Pre-Processing and MultigridDuring pre-processing, the individual meshes created for each blade row are merged to a singlemulti-zone mesh. An initial guess for the flow is created from a 2D-through-flow computation ofthe rig and applied uniformly over the full blade pitch.

29


The coarser grid levels used in the multigrid method are automatically created in the pre-processorby successively collapsing adjacent cells. The best practise recommendation to use 4 multigridlevels was followed.Within the multigrid method implemented in HYDRA, the "w"-scheme was chosen, in which thesolution is restricted to the coarsest level, prolonged to the second-finest and restricted againto the coarsest level before returning to the finest grid level. This scheme is more costly periteration but requires fewer iterations for convergence and leads to lower residuals.A full multigrid (FMG) startup procedure is used when starting the iteration from the initialguess. The equations are first solved on the coarsest grid level only and the finer levels are thensuccessively added to the multigrid iteration scheme. This procedure is followed only once fora given mesh; All subsequent simulations on the same mesh with varied boundary conditionsare run starting from the converged baseline computation with the full "w"-scheme. Since thevariations are relatively small, using a restart from the converged reference case reduces thenumber of iterations needed to convergence by half compared to starting from the initial guesswith the FMG startup.

3.3.2 Convergence CriteriaThe convergence of the solution is monitored via the flow residuals averaged over the entiredomain. The residuals are computed separately for the main flow equations (RANS equations)and the turbulence equations (k and ω equations), combined to a single RMS (root-mean-squared)value respectively. For details see [7].

0 200 400 60010−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

Iterations

Residua

ls

Main FlowTurbulence RMS

(a) Full HPT computation at reference conditions frominitial guess with FMG start-up.

0 100 200 30010−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

Iterations

Residua

ls

NGV1 (SR)Q-2D

(b) Mean flow residuals for a single row NGV1(SR) computation and quasi-2D computation(Q2D) at midspan. Both started from theinitial guess with FMG startup.

Figure 3.8: Typical convergence history diagrams for a full HPT computation (left) and preminiarystudies on the NGV1 (right).

A typical convergence history for a full HPT computation with FMG startup is shown inFig. 3.8(a). The dents and abrupt changes in the residual slope indicate the addition of the next

30

3.3 HYDRA Configuration

0 50 100 150 200 250 300 350 400 450 500 550 600−1−0.5

00.5

1 ·10−6

Iterations

C−C

0 50 100 150 200 250 300 350 400 450 500 550 600−0.4−0.2

00.20.4

Iterations

η H−ηH

0 50 100 150 200 250 300 350 400 450 500 550 600−0.4−0.2

00.20.4

Iterations

η P−ηP

0 50 100 150 200 250 300 350 400 450 500 550 600−0.4−0.2

00.20.4

Iteration

η s−ηs

Figure 3.9: Convergence history of the NGV1 outlet capacity and efficiency monitors. Theoverbar signifies an average over the last 200 iterations.

finer grid level to the multigrid iteration loop during the first 120 iterations. Once a pseudo-periodic regime is reached, the residuals for the main flow have fallen by 2-3 orders of magnituderelative to the initial guess. Usually, 3-4 orders of magnitude in residual reduction is regardedas converged [11]. The residual reduction reached here is nevertheless acceptable consideringthe quality of the initial guess and the fact that this study aims to quantify delta-values, i.e.,changes from one simulation to the next, instead of absolute values.

Figure 3.8(b) shows typical convergence histories of the main flow residuals for computationsperformed for the cooling strip model calibration and boundary layer study (see sections 5.1and 5.2). In contrast to the full HPT computations, the latter converge very well and fast, theresiduals plateauing within 200 iterations after dropping by over 4 to 6 orders of magnitude. This

31


Figure 3.10: Meridional view of the computational domain (without exit duct) showing inlets(green arrows), cooling strips (light blue arrows) and wall speed (red is stationary, dark blue isrotating). The flow is from left to right.

result is not surprising since these studies are restricted to the NGV1 domain that, although itcontains film cooling and a trailing edge slot, does not include the rotor-stator coupling and theinherent unsteadiness this entails. The convergence history shows that the results are trustworthy.The full HPT computations converge much more slowly. The residuals reaching a pseudo-periodicregime only after roughly 300 iterations. The efficiencies require even more time to converge.This becomes apparent looking at the convergence histories from the capacity and efficiencymonitors in Fig. 3.9. Once all objectives have reached a relatively periodic regime (after about 400iterations) the simulation is run for another 200 iteration over which the performance parametersare averaged. To cross-check the convergence, the efficiency drift was computed by averaging overthe last 100 iterations and comparing the values to the first average. The efficiency drift is wellbelow 0.02% for all simulations. The capacity drift is negligible given the convergence history.

3.4 Reference caseIn order to model the flow through the test rig, the appropriate boundary conditions have tobe set. In the following section, the boundary conditions for the reference operating point arepresented onto which the subsequent parametric variations are applied. Figure 3.10 shows themerdional view of the test rig presenting the location of the different inlets (main inlet, 4 rimsealleakage inlets, 2 trailing edge slots) as well as the cooling strips on the first stage. The ductextending to the right and the main outlet are not shown. The different boundary conditions arepresented in more detail below and are summarized in Tabs. 3.1 and 3.2.

Main Inlet and outlet

The target inlet total pressure and temperature are known and are uniformly imposed directlyonto the entire inlet plane via a pressure inlet boundary condition. The main inlet mass flowis thus floating and part of the solution. Due to compressibility, the flow velocity itself cannotbe specified at the inlet but the inflow angles are set to match the geometry of the rig. Theinlet radial flow angles at hub and casing are adjusted to be parallel to the walls and linearlyinterpolated in between (see Fig. 3.10 on the left). The inlet tangential flow angle was set tozero corresponding to a swirl-free inlet. The inlet turbulence parameters are set according tobest practice with an inlet turbulence intensity of I = 5% due to the absence of both swirler and

32

3.4 Reference case

combustor and an turbulence integral length scale of Lturb = 4 mm corresponding to 10% of theinlet duct height.

At the outlet, the static pressure is imposed on a non-reflecting boundary. The value for theoutlet static pressure is taken from preliminary rig measurements. The outlet is located 4-5 bladeheights downstream of the second rotor. This arrangement has two advantages. On the one hand,it decouples the outlet boundary and the stage 2 rotor domain since, especially in swirling flowsas they occur in turbomachinery, the flow may be influenced by a pressure outlet located tooclose. On the other hand, it extends the duct domain up to the location of a downstream staticpressure rake in the rig allowing a direct comparison between CFD and experimental results.The additional computational cost is manageable since the total number of cells in the duct meshrepresents roughly 1.5% of the total mesh count.

Film Cooling

PS01

PS02

PS03

PS04

SS01

SS02

SS03

SS05

SS04

TE slot

(a) NGV1

SS02

SS01

PS02PS01

PS04

(b) ROT1

Figure 3.11: Schematic of the cooling geometryof the first turbine stage.

The film cooling is modelled using a coolingstrip model. The mass flow split between thecooling rows is taken from the preliminary de-sign predictions validated by the fully-featuredcomputations. More details on the coolingstrip model are given in section 2.2.4. Theflow splits for the individual cooling rows or-dered by row and blade side are separatelysummarized in Tab. 3.2. Some cooling rowsare split into two individual strips at midspan(subscript u for upper and l for lower) to ac-count for different exit angles on the upperand lower blade halves.

The cooling geometry is schematically shownon a midspan cut in Fig. 3.11. Not repre-sented are the cooling strips PS03 and PStipfor ROT1, located near the blade tip and visi-ble on Fig 3.10, as well as the dustholes leadingto the squealer on the ROT1 tip. The coolingsystem geometries as well as the blades arewarped.

Rimseal leakages and trailing edge slots

To model the secondary air system inlets, it was chosen to use a mass flow inlet boundary condition(rimseal leakages and trailing edge slots) to ensure that the flow split between the different inletsis correctly represented. The imposed mass flow rates correspond to the preliminary designpredictions. The total temperature for the secondary inlet flows is set to 300 K while the totalpressure is floating and is automatically adjusted to match the imposed flow rate. The inflow atthe trailing edge slots is defined normal to the inlet boundary. To account for the shaft rotation,the tangential velocity at the rimseals is set to the wall velocities on the rotor and stator siderespectively and linearly interpolated in between (swirling flow mass inlet). The inlet turbulence

33


intensity is set to 20% for the trailing edge slots and 30% for the rimseal leakages according tobest practice. The turbulence integral lengthscale was set to 10% of the smaller inlet dimension.

Solid walls

Following usual practice in turbine CFD, the heat losses across the walls are neglected incomparison to the changes in temperature due to work extraction. All walls are therefore definedas adiabatic, viscous (no-slip) walls.

Rotor-stator coupling

The turbine model is divided into 5 individual zones (2 stator zones, 2 rotor zones and theexit duct) of which the two rotor zones are set to a fixed rotation speed around the machineaxis corresponding to 8000− 9000 rpm. The rotor-stator coupling is achieved using the mixingplane approach. In this approximation, all flow quantities are spatially averaged ("mixed") incircumferential direction at the interface between two zones rotating at different speeds. Thevelocity fluctuations are also transferred across the mixing plane in the form of increased turbulentkinetic energy.

34

3.4 Reference case

Boundary SpecificationMain inlet (NGV1) Swirl-free pressure inlet:

Ptot = 159 000 Pa, uniformTtot = 450, 00 K, uniformuθ = 0, uniformur set to match hub an casing orientation.Turbulence: I = 5%, Lturb = 4.0 mm

Main outlet (DUCT) Non-reflecting pressure outlet:Pstat = 28 000 Pa, uniform

NGV1 TE slot Mass inlet:m = 3.240 %W40Ttot = 300 K, uniformInflow normal to boundary.Turbulence: I = 20%, Lturb = 0.06 mm

NGV2 TE slot Mass inlet:m = 1.474 %W40Ttot = 300 K, uniformInflow set normal boundary.Turbulence: I = 20%, Lturb = 0.06 mm

ROT1 rimseal Swirling flow mass inlet:m = 1.976 %W40Ttot = 300 K, uniformuθ set to match hub rotation.Turbulence: I = 30%, Lturb = 0.15 mm

NGV2 rimseal Swirling flow mass inlet:m = 2.359 %W40Ttot = 300 K, uniformuθ set to match hub rotation.Turbulence: I = 30%, Lturb = 0.20 mm

ROT2 rimseal Swirling flow mass inlet:(upstream) m = 1.179 %W40

Ttot = 300 K, uniformuθ set to match hub rotation.Turbulence: I = 30%, Lturb = 0.20 mm

ROT2 rimseal Swirling flow mass inlet:(downstream) m = 0.590 %W40

Ttot = 300 K, uniformuθ set to match hub rotation.Turbulence: I = 30%, Lturb = 0.80 mm

Table 3.1: Reference boundary conditions for reduced power case. All mass flows are given as apercentage of W40, the expected main inlet mass flow from a 2D throughflow solution.

35


Zone Row ρc [kg/m3] Area [mm2] Flow split [%] m [%W40]H01S SS01u 2.207 1.990 3.97 0.255

SS01l 2.189 1.991 3.18 0.204SS02u 2.144 1.990 6.18 0.397SS02l 2.118 2.274 4.67 0.300SS03 1.986 4.555 15.81 1.014SS04 1.828 4.365 17.52 1.124SS05 1.704 4.846 20.98 1.346PS01u 2.210 1.991 2.84 0.182PS01l 2.210 2.274 3.08 0.198PS02u 2.211 2.004 2.65 0.170PS02l 2.208 1.990 3.25 0.208PS03 2.200 3.999 7.06 0.453PS04 2.288 3.980 8.80 0.346

Σ H01S 100.00 6.196H01R SS01 1.102 2.962 14.49 0.556

SS02 0.922 3.337 26.18 1.004PS01 1.218 2.962 10.24 0.393PS02 0.985 4.038 14.73 0.565PStip 0.983 0.987 6.04 0.232PS03 0.983 0.790 4.05 0.155PS04 0.854 3.949 20.41 0.783Dustholes 1.031 0.617 3.86 0.148

Σ H01R 100.00 3.834

Table 3.2: Reference boundary conditions for cooling flows (excluding TE slots). All mass flowsare given as a percentage of W40, the expected main inlet mass flow from a 2D throughflowsolution.

36

4 Sensitivity Quantification

4.1 Theory of Sensitivity AnalysisThe aim of CFD is to model and predict the flow and derived features of interest for a particularoperating point with a fixed geometry and boundary conditions. The choice of boundary conditionsdepends on the purpose of the simulation, whether a specific operating point is analysed (suchas the design conditions to benchmark the model) or a comparison to a laboratory experimentis sought thus applying the exact boundary conditions from the experiment. These boundaryconditions are always a simplification of the underlying physics and do not fully represent theanalysed system. This can be due to a lack of experimental data for a specific input that has tobe set using engineering judgement. When experimental data is available, these present variationsdue to varying degrees of intrinsic unsteadiness in the underlying physical processes that translateinto fluctuations of the input variables that cannot be taken into account in the computationalmodel. Moreover, the the computational model itself may require a simplification of the boundaryconditions (spatial/temporal averaging, reduced number of inputs, etc). Therefore, a single runof the CFD code will yield only one set of outputs for the chosen input configuration, givingno information as to how much this output is dependent on the particular value of a giveninput (sensitivity) and how variable it is with respect to the fluctuations and uncertainties in thephysical inputs (uncertainty of inputs).

Physical process

Input parameters Turbine Performance

Sensitivity Analysis

CFD Model

Figure 4.1: Schematic of the basic principle of sensitivity analysis to obtain a more completepicture of the variations of the turbine performance by running the CFD model multiple timesin the input parameter space. The statistical analysis of the computed points then allows forinferences in between with varying degrees of uncertainty (shaded areas).

In broad terms, sensitivity analysis is "the study of how the uncertainty in the output of amodel (numerical or otherwise) can be apportioned to different sources of uncertainty in themodel input" [31]. Closely related and usually run in tandem with a sensitivity analysis isuncertainty quantification that aims to quantify the uncertainties present in the inputs and howthey propagate through the model and influence the outputs. In this work, the emphasis is on thesensitivity analysis to extract and quantify the main effects of the considered input parameters

37


in the range observed in the test rig and to identify their important interactions. The aim isto determine the most influential parameters that need to be controlled in the experiment andcreate an assessment basis for the comparison of test data with future computer simulations. Inthe following section, the methods of sensitivity analysis are presented and the approach takenin this work is detailed.

4.1.1 DefinitionsFor the purpose of sensitivity analysis, the details of the CFD code used are not important andthe high-fidelity computation is treated as a generic "black box" model that for a given vector ofinput parameters X = (X1, . . . , XN ) produces one or more outputs Y . We have

Y = f(X) . (4.1)

In general, the vector of input parameters Xi for the sensitivity analysis is not necessarily thesame as for the computational model itself but is a (usually small) subset of the total numberof inputs. In fact, especially when the number of inputs is large, a sensitivity analysis wouldquickly become infeasible if all inputs were considered. Therefore, a judicious choice of the inputparameters relevant to the study must be made by the practitioner. In this work, the number ofinput parameters was set to 12 (see section 4.2).

4.1.2 Local versus Global MethodsLocal sensitivity analysis involves the computation of local gradients of the output Y with respectto the inputs Xi at a fixed reference point X0. These analyses are conceptually simple and,especially for linear models and when only small perturbations are considered, usually lead touseful sensitivity estimations. Important draw-backs include the possibly flawed analysis for amodel of unknown linearity and the fact that they do not explore the entire input space. Globalmethods on the other hand try to quantify the model sensitivity with respect to the full range ofeach input parameter with methods valid for linear and non-linear models alike [31]. In this work,the more general global approach was chosen since non-linear effects cannot be excluded a priori.

4.1.3 Surrogate ModelsSensitivity analyses often rely on variance based approaches using Sobol indices to quantify therelative importance of each input factor [31]. For a global variance based sensitivity analysisto be accurate, a large number of evaluations of the numerical model are necessary (tens ofthousands of evaluations are not uncommon). As 3D CFD computations are extremely expensive,the computation of the output for a single input combination taking several hours on multiplecores, such an analysis is impossible due to strong limitations in the amount of simulationsthat can be carried out. Instead, a surrogate model (or metamodel) is constructed, based on arelatively small amount of evaluations of the expensive model to emulate its behaviour. Thismodel is much cheaper to evaluate and is the basis of the sensitivity analysis.

There are several types of surrogate models that differ greatly in terms of complexity andrequirements regarding the sampling/training points. Simpson et al. [33] present a survey ofsurrogate modelling techniques along with recommendations regarding the choice of the methodmost appropriate to the task at hand. They find that problems with a very large number of

38

4.1 Theory of Sensitivity Analysis

inputs (10 000+) neural networks are most advantageous in spite of the high computational costof training the network. If the underlying model is deterministic (i.e., two simulations with thesame inputs yield the same outputs) and the number of inputs is moderate (< 50), the authorsrecommend the use of a Kriging model due to its flexibility. For problems with fewer than 10inputs, the well-established Response Surface Methodology (RSM) usually based on lower-orderpolynomial fitting functions is suitable.Following these recommendations, a Kriging model is used to create a surrogate model for theefficiency and capacity based on the deterministic CFD computations. The resulting Krigingresponse surface is then used directly to quantify the input parameter sensitivities.

The Kriging model is an interpolation scheme based on a method first developed by DanielKrige in 1951 to estimate the most likely gold distribution in an area based on a small numberof bores and subsequently formalised by Georges Matheron in 1963 [21]. The method yields aprediction for the output at a given location using a linear combination of known outputs in theneighbourhood including their spatial correlations and using maximum likelihood estimation todetermine the optimal weights. Mathematically, the Kriging model is the best linear unbiasedestimator of a random variable based on known values of the variable in the neighbourhood ofthe point of interest [29]. The advantage of Kriging model over approximating response surfaces,warranting the additional complexity, is that the method is in fact an interpolation and the valuesof the estimated variable are exactly reproduced at locations where they are known, matchingthe deterministic nature of the underlying high-fidelity simulation.The details of the procedure to fit the Kriging model to the data is outside of the scope of thiswork. The model-fitting itself is carried out using the existing Gaussian Process Emulator (GPE)toolbox (the Gaussian random process is used as a correlation function) in the software packageiSight (Version 5.6).

4.1.4 Sampling Strategy: Design of Experiments (DoE)In order to build the surrogate model, the CFD model has to be evaluated a certain number oftimes to represent the design space. For a given number of sampling points, i.e., combinations ofinput parameters in their respective ranges, the quality of the interpolation is highly dependenton the sampling strategy adopted. A few common sampling techniques are reviewed in thefollowing section.

Random Sampling (RS): This sampling technique creates a matrix of random (or pseudo-random) samples for each input variable according to a predefined probability densityfunction. Monte Carlo Sampling is the standard sampling technique for variance basedsensitivity analysis. Unfortunately, to obtain reliable statistics especially when many inputparameters are involved, a large amount of samples is required. When only few samplesare taken, larger holes are liable to appear at random locations the input space where nosamples are taken thus noticeably reducing the accuracy of the response surface predictionsin this area.

One Factor at a Time (OAT) Sampling: OAT sampling is the most intuitive systematic sam-pling technique that consists of changing only one element of the input parameter vectorin every evaluation of the model. This sampling technique is often used in local sensitivityanalyses and seems especially attractive in contexts with an obvious baseline or referencecase to which the results of the varied inputs can easily be related.Nevertheless, OAT designs are inefficient when more than one factor is considered due to

39


the "curse of dimensionality": OAT designs are non-explorative by construction, in factwhen 12 input parameters are considered, this type of design covers less than 0.1% of theparameter space [30]. Furthermore, a crucial element of sensitivity analysis, namely theidentification of parameter interaction, cannot be performed based on a OAT experimentaldesign [31].Alternatives to the basic OAT design have been put forward to overcome some of theshortcomings such as the Morris Method or Elementary Effects Method [25].

Full/Fractional Factorial Design (FD): In Factorial Designs the number of sampling points isdictated by the product of the number of levels for each parameter (usually 2-3) commonlyleading for two levels to 2n−p samples where 1/2p is the fraction of samples left out toreduce the sample size (p = 0 and p > 0 correspond to full factorial and fractional factorialdesigns respectively). If quadratic effects are to be estimated, at least 3 levels per parameterare required leading to even higher sample counts so that alternatives are used such as theCentral Composite Design (CCD) where the 2-level FD is augmented by center and starpoints between the FD sample points [33].

Optimal Latin Hypercube: Latin Hypercube Sampling (LHS) and its extension Optimal LHS(OLHS) belong to the class of so-called space-filling sampling designs introduced by McKayet al. in 1979 [22]. Stemming from the reasoning of stratified sampling in which the samplespace is partitioned into a set of disjoint strata and samples are randomly chosen for eachstratum and each parameter, LHS ensures that the entire input space is represented inthe sample even for small sample sizes. The prior probability density function of theinput parameter is also considered by choosing the strata in such a way that each has thesame marginal probability. Park [26] proposed the OLHS method optimizing the standardLHS in terms of the Integrated Mean Squared Error (IMSE) or in terms of entropy. Animportant feature of LHS is that all parameters are varied at each sample point allowingfor the estimation of both main and interaction effects.

The different sampling techniques are exemplified for ni = 2 input parameters in Fig. 4.2. Whilethe number of samples increases exponentially with the number of considered input parametersni for OAT and FF sampling (Figs. 4.2(a) and 4.2(b)), it is independent of ni for randomsampling and OLHS. Fig. 4.2(c) clearly shows the lack of samples in the bottom right cornerand for 0.16 < X2 < 0.26 that is avoided in the OLH sampling (Fig. 4.2(d)). Moreover, therandom sampling contains point clusters that are problematic for the computation of the Krigingweights since the model prefers well spaced design points and may fail if this condition is notmet according to the documentation of the iSight-tool.

In this work, 12 input parameters are investigated using Kriging interpolation as a surrogatemodel. OLHS is chosen to create the sample matrix given the shortcomings of OAT designs andthe prohibitive number of evaluations of the high-fidelity model to adequately cover the designspace using random sampling as well as full factorial and related designs. Furthermore, Simpsonet al. recommend the use of IMSE-optimal space-filling designs such as the OLHS designs forKriging models [33].Before the sample matrix is created, the number N of samples must be set. This choice isnot obvious and several recommendations exist in literature depending on the number of inputparameters ni, sometimes output parameters no and the types of interactions that are examined.The reference for the iSight-tool used to create the sample matrix recommends a minimum ofN = 2ni + 1 = 25 samples. Since non-linear interactions are expected for the efficiencies and

40

4.1 Theory of Sensitivity Analysis

0 0.2 0.4 0.6 0.8 100.20.40.60.8

1

X1

X2

(a) OAT sampling (N = 2ni).

0 0.2 0.4 0.6 0.8 100.20.40.60.8

1

X1

X2

(b) 2-level FF sampling (N = 2ni);Filled circles: fractional factorialsampling (N = 2ni−1).

0 0.2 0.4 0.6 0.8 100.20.40.60.8

1

X1

X2

(c) RS (N = 40, uniform probabilitydistribution).

0 0.2 0.4 0.6 0.8 100.20.40.60.8

1

X1

X2

(d) OLHS (N = 40, uniform prob-ability distribution) created iniSight.

Figure 4.2: Comparison of different sampling techniques for ni = 2 input variables. For RS andOLHS the number of sampling points was set to N = 40.

the computational expense of a single run can be considerably reduced using restarts from thebaseline, it was chosen to set the number of samples to N = 40.

4.1.5 Quantification of Main Effects

The main effect Ei of a given parameter Xi across the parameter range is given by the averagegradient

Ei = ∆Y∆Xi

, (4.2)

where all input parameters but Xi are held constant at the average of the respective parameterrange.

Since the continuous Kriging response surface is available, the main effects can be visualized byplotting the variation of the output over the normalized parameter range of input Xi, all otherparameters held constant.

41


4.1.6 Interaction Effects

The main effects consider only how the objective varies with respect to a single input parameters.If the sensitivities are additive, i.e., there are no interactions, the main effects are an exhaustivedescription of the dependence and the effect of several input variations can be computed bysumming the main effects of each input parameter.If on the other hand, as it is often the case, the input parameters exhibit interactions, the maineffects are not sufficient to understand the behaviour of the system and the interactions need tobe considered.

The Kriging response surface can also be used to quantify the interaction effects by computingSobol indices for the parameters involved. Unfortunately, the iSight-tool used for the sensitivityanalysis does not contain this functionality and the interactions can only be assessed usingcontour plots showing the co-influence of two parameters at a time. Therefore, the analysis ofthe interaction effects is limited.

4.2 Variation MatrixBefore the sensitivity study is carried out, the parameters that are liable to influence the outputvariable are chosen in close consultation with the test engineers operating the rig. The parametersconsidered in this work are the inlet distributions of total pressure and temperature as well asthe cooling and leakage flow rates for each individual row. The parameter ranges are defined bycomparing several preliminary measurements from the rig and computing the observed variability.In the following, the parameters and their value ranges are described in more detail before theDoE-Matrix is generated with N = 40 samples using the OLHS technique in iSight.

4.2.1 Inlet Total Temperature and Pressure Distributions

In any experimental setup, it is technically impossible to produce a uniform block profile fortotal temperature and pressure at the turbine inlet. Preliminary measurements were performedto quantify the variations.

Figure 4.3(a) shows two temperature measurements in the inlet plenum taken 4 hours apartduring operation (corresponding to half a typical test day). The measurements are made using 4radial rakes equally spaced on the annulus circumference. Although the average total temperatureis consistently less than 1% off target, the measurement reveals two important effects. On the onehand, the first measurement exhibits a noticeable circumferential variation with the temperatureat rake C being consistently lower than at the other locations. This temporarily increasedheat loss can be partly explained by piping running close to the casing and making a betterinsulation impossible at the location of rake C. During operation, the circumferential temperaturedistribution becomes more uniform as the entire rig heats up and the annulus average increasesto within less than 0.5% of the target value. On the other hand, there is an apparent, systematicdrop in total temperature towards the walls. Due to heat losses across the walls that are especiallyimportant towards the casing (outer wall), the fluid is 3% colder there than the bulk during thefirst measurement. These variations also decrease during operation but remain noticeable evenduring the second measurement.For the sensitivity study, the average maximum temperature variation is considered (firstmeasurement point at the casing wall). This leads to a parameter range of ±2.2% of the target

42

4.2 Variation Matrix

total temperature (450 K) corresponding to 10 K in absolute terms.

For the inlet total pressure, a similar analysis was undertaken. Figure 4.3(b) shows two pressuremeasurements (not taken simultaneously with the temperature measurements above) with typicalvariations around the average of the whole day that itself is 0.1% off target. While the annularaverages fluctuate within 0.1% of the daily average, the radial variations are about 3 timessmaller (±0.04%) and random with the exception of a systematic total pressure drop towardsthe casing. The circumferential variation is small since it is of the same amplitude as the radialvariation over a quarter of the annulus.Overall, the pressure fluctuations of 0.04% of the target total pressure (159 kPa), correspondingto about 64 Pa in absolute terms, are very small. Nevertheless, the variations are significantcompared to the inlet dynamic pressure q∞ that is the appropriate parameter for the comparisondue to the very low inlet velocities (Ma∞ = 0.05). The inlet dynamic pressure computed usingvalues averaged over the inlet plane (subscript ∞), yields

q∞ = 12ρ∞a

2∞Ma2

∞ = 276 Pa . (4.3)

The pressure fluctuations a the inlet therefore correspond to roughly 25% of the inlet dynamicpressure and are not negligible. In order to ensure that the sensitivity analysis yields reliableresults, the variation was doubled to 128 Pa.

In order to qualitatively represent the inlet radial variations with a minimum of independentparameters, it was chosen to model the temperature and pressure profiles with three independentparameters each as the superposition of variations at the hub, the casing and at midspan leadingto a step profile. To ensure comparability between the simulations, the mass flow weighted areaaveraged values for total temperature and pressure must be kept constant equal to the baseline.The circumferential variations as well as the transient changes observed in the rig are notconsidered in this study but could be of interest for future work.

Mathematically, we require ∫ 1

0Φ(r)m(r) · r dr = 0 , (4.4)

where Φ(r) is the temperature or pressure variation profile around the average, m(r) is the knownmass flow per unit radius taken from preliminary through-flow computations shown in Fig. 4.4(b)and r the annulus span varying from r = 0 at the hub to r = 1 at the casing.

Instead of using simple step functions as base functions φ(r) for the variations that may lead toconvergence issues, the following continuous functions are used.

HUB : Φ1(r) = a1 + b1π

tan−1 [100π · (r − 0.3)]

(4.5)

CEN : Φ2(r) ={a2 + b2

π tan−1 [100π · (r − 0.3)]

r ≤ 0.5a2 − b2

π tan−1 [100π · (r − 0.7)]

r > 0.5(4.6)

CAS : Φ3(r) = a3 −b3π

tan−1 [100π · (r − 0.7)]

(4.7)

For each simulation, the maximum variations at hub, case and midspan are set according to the

43


−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.50

20

40

60

80

100

∆Ttot to target [%]

%span

Rake ARake BRake CRake DMP2MP1

(a) Two measurement points (MP) taken for the inlet total temperature at a 4 hour interval duringoperation.

−0.1 −5 · 10−2 0 5 · 10−2 0.1 0.15 0.2 0.250

20

40

60

80

100

∆Ptot to target [%]

%span Rake A

Rake BRake CRake DMP1MP2Avg. over day

(b) Two measurement points taken for the inlet total pressure showing typical deviations from thetarget and the average over the day.

Figure 4.3: Total temperature (top) and total pressure (bottom) rake measurementstaken at 4 equally spaced circumferential locations in the inlet plenum. Both quantitiesare measured in the same axial plane, the temperature and pressure rakes are shiftedby 45◦ relative to each other. Averages (shown as thick lines) are area weighted overthe annulus.

44


−15 −10 −5 0 5 10 150

20

40

60

80

100

Temperature variation [K]

%span

T1T2T3TΣ

(a) Inlet total temperature: Base profiles Ti and sum TΣ for(∆THUB,∆TCAS,∆TCEN) = (1 K, 3 K,−6 K)

0 0.5 10

20

40

60

80

100

m/mavg

%span

(b) Inlet mass flow rate per streamtuberelative to average mass flow rate

Figure 4.4: Inlet boundary conditions for total temperature.

DoE-matrix and the parameters (ai, bi) are computed such that Eq. (4.4) is fulfilled as well aseach imposed variation:

Φ1(0) = ∆ΦHUB ,

Φ2(0.5) = ∆ΦCEN , (4.8)Φ3(1) = ∆ΦCAS .

The three variation profiles are then summed and added to the baseline. Figure 4.4(a) showsan example of the base as well as the final inlet total temperature profiles for the variations(∆THUB,∆TCAS,∆TCEN) = (1 K, 3 K,−6 K). The requirement of a constant mass flow weightedarea average makes it impossible to exactly set the variations on the final profile. The originalvariations must be substituted with the de facto variations from this profile since they carryphysical meaning, even though this leads to a less optimal sample distribution. Although theparameter range is slightly broadened it still remains physically plausible since the original rangewas the average variations in temperature and pressure. Table 4.1 summarizes the final parameterranges for the total temperature and pressure distributions relative to the reference case.

4.2.2 Cooling Flows and Secondary Air System

The cooling flows and the secondary air system are fed by external compressors that provide airto chambers around the test section that distribute it to the cooling holes, slots and seals. Theactual flow through each inlet is then governed by the geometry of the particular part that may

45


HUB CEN CAS∆Ttot [K] [−13.7, 13.2 ] [−17.5, 14.8 ] [−14.1, 12.3 ]∆Ptot [Pa] [−156, 186 ] [−174, 202 ] [−213, 197 ]

Table 4.1: Final total temperature and pressure ranges for the inlet variations at hub (HUB),midspan (CEN) and casing (CAS).

exhibit a certain deviation from the design geometry.

To quantify this variability, different pressure differences were applied over the blades of eachcooled row and the mass flow through the cooling holes measured. The blade-to-blade capacityvariations due to manufacturing were are largest for the NGV1 which is why the data for this rowis used to set the range for the sensitivity study. Figure 4.5 shows the mass flow characteristics(blade capacity, Cb) of the 34 NGV1 blades for pressure ratios ranging from 1.1 to 2.1 relative toatmospheric pressure (99.40 kPa) as well as the average over all blades. The blade capacity iscomputed for each blade individually based on the cooling flow mass flow rate. The characteristicsshow a typical behaviour over the pressure range. For low pressure ratios the capacity increasessharply with the inlet pressure. Towards the highest tested pressure ratios, the curves flattenout as the flow comes close to choking conditions where the mass flow reaches a maximumand becomes independent of the inlet pressure. The plot shows considerable variability in themanufactured parts that is due to the fact that the blades are milled and sintered instead ofcast like production engine parts since manufacturing using casting would be too expensive forthe small number of units. The measured capacities deviate from the mean by 20% over theentire pressure range (which would lead to their rejection for a production engine). Furthermore,the blades seem to follow a normal distribution about the mean. Assuming that the blades ofdifferent rows exhibit a similar scatter, a variation range of 20% of the reference value is chosenfor the three cooling flows on NGV1, ROT1 and NGV2.Inside the NGV1 blade, two separate cavities provide the cooling flow feed. The front cavityfeeds the leading edge impingement and film cooling and the back cavity the trailing edge slot aswell as the last cooling row on the pressure side. Although the mass flows through each cavitycan vary independently, it was chosen to vary both flows together in order to reduce the numberof parameters for the sensitivity study.

The seals are similarly connected to the secondary air system. There is currently no experimentaldata on the seal flow rate variations but it can be assumed that they are of the same order asthe fluctuations observed in the cooling flows. Since the seal flows are small in comparison to thecooling flows in the first stage, the variations were set to 50% of the reference value.

Due to an error in the iSight set up, the input range for the ROT1 rimseal leakage relative tothe reference condition was set to XROT1RSL ∈ [0.5, 1.18] instead of extending to 1.5. Since theparameter value XROT1RSL = 1 is covered, this has only a marginal impact on the results as theinfluence of both increase and decrease of the parameter are considered and the absolute rangesfor the rimseal leakage flows are the result of educated guesswork.

46


1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.20.4

0.6

0.8

1

1.2

1.4

P/Pa

Normalized

blad

ecapa

cityCb

Individual bladesAverage

Figure 4.5: Measurements of flow capacity through all 34 NGV1 blades for different pressuredifferences relative to atmospheric pressure (Pa = 99.4 kPa). Blade capacity normalized withvalue at P/Pa = 1.60 corresponding to the reduced case inlet conditions.

4.2.3 Rotor Tip ClearanceIn the actual engine, the rotor tip clearance changes continuously due, e.g., to thermal expansionduring the warm up cycle and blade deterioration over its lifetime. The test rig is designedwith an exchangeable rotor casing allowing two different tip clearances to be tested. Bothtip clearances are considered in the sensitivity study: A small or design clearance of 0.33 mmcorresponding to the new blades and a large tip clearance of 0.66 mm. Figure 4.6 shows a cutthrough the ROT2 tip gap and casing superimposing the two clearances tested. The meshes werebuilt such that the tip clearance can be altered independently for each stage making it possibleto quantify the impact of the clearance on turbine performance for each stage independently.

ROT2

Casing

Tip Gap

Figure 4.6: Cut through the ROT2 domain in y-normal direction at the location of the tip gap.The two tested tip clearances are superimposed: Small (design) tip clearance (−) and large tipclearance (−−).

47

5 Model Calibration and Validation

Before the parametric studies can be undertaken with the turbine model, it must be calibratedand tested for accuracy. This is typically done by comparing numerical results with experimentaldata for the same setup. Unfortunately, no experimental data is available for the test rig due todelays in the testing procedure; Instead, a fully-featured CFD computation including details ofthe film cooling holes is used. This makes the validation of the model more challenging whichis why the model scrutiny was undertaken more carefully. The following sections present thecalibration of the cooling strip model using the fully featured results and a boundary layer studyto assess the required resolution of the near wall region in view of the interactions betweenboundary layer and film cooling model.

5.1 Cooling Strip Model Calibration

As described in section 2.2.4, most parameters in the cooling strip model directly reflect thegeometry of the actual cooling holes. In spite of this fidelity, the introduced simplifications leadto a fundamentally different flow field downstream of the cooling rows. Instead of individual jetsprotruding into the main flow, mixing and spreading in spanwise direction coalescing into a filmonly within a few cooling hole diameters of the injection point, the coolant is introduced evenlyalong the blade span. The mixing processes downstream of the injection point are thereforelikely to be different both in terms of timescales and spreading of coolant into the main flow.It is important to reproduce the mixing rates at least approximately since crucial performanceparameters like the capacity are sensitive to the blockage induced by the coolant flow that inturn is linked to the spatial temperature distribution via the density. The penetration depth lcis therefore used to calibrate the cooling strip model to the fully-featured computations of theNGV1 domain under the same conditions.

5.1.1 Single Row Computations on the NGV1

The model calibration on the NGV1 domain was performed using a mesh with a boundarylayer resolution comparable to the fully-featured computation. Using this mesh, single rowcomputations were performed for penetration depths in the range 0.5 mm ≤ lc ≤ 2 mm, thelower bound corresponding to the physical penetration depth measured in the fully-featuredsolution and the upper bound corresponding a value slightly beyond the model’s best practicevalue of 1.5 mm. For all single row computations, the scaled residuals of the converged solutionshad fallen by 5-6 orders of magnitude and reached a plateau.

Figure 5.1 shows the adiabatic wall temperature on the NGV1 from the fully-featured computation(left) and a computation using the cooling strip model with lc = 1.5 mm (right). The comparisonshows the qualitative differences between the flows produced by the two film cooling models.While the fully-featured computation clearly shows the gradual mixing of the individual coolant

49


(a) Fully featured CFD (b) Cooling strip model, lc = 1.5 mm

Figure 5.1: Adiabatic wall temperature distributions (in Kelvin) on the NGV1.

(a) FF (b) lc = 0.5mm (c) lc = 2.0mm

Figure 5.2: Contours of total temperature (in Kelvin) just before the trailing edge slot lip. Theflow is into the plane, suction side to the left, pressure side to the right of the surface.

flows, the cooling strip model expectedly leads to an overall much more uniform temperaturedistribution on the blade. The hot streaks appearing on the first cooling strip from the leadingedge and on the outer edges of some of the downstream strips are due to the imperfect distributionof the source terms while the target mass flow rate for each row is ensured.In order to assess the influence of the film cooling on the main flow, Fig. 5.2 plots contours ofthe total temperature in a plane normal to the machine axis far downstream of the cooling stripsfor different values of lc. The evolution on the pressure side is very similar albeit a slightly toosmall midspan bulge for high lc. On the suction side, the differences are much more pronounced.While a small penetration depth leads to heavily reduced mixing close to the wall and subsequentsharp gradients to the main flow (5.2(b)), a large value of lc leads to premature coolant dilution(5.2(c)) and a larger protrusion of the mixing region into the main flow. Since the influence ofthe penetration depth is small on the pressure side, the calibration will focus on the suction side,which is the more sensitive surface.

50


A typical parameter used in the assessment of a film cooling setup is the adiabatic film coolingefficiency (sometimes called effectiveness) η defined as

η = T∞ − TawT∞ − Tc

, (5.1)

where T∞, Tc, and Taw are the respective temperatures of the free stream, the coolant and the(adiabatic) wall.

For high-speed flows, this definition is of limited utility since the temperature changes considerablywith the Mach number. The alternative is to recast Eq. (5.1) in terms of stagnation temperatures2

(subscript 0) [14]:ηT = T∞,tot − Taw

T∞,tot − Tc,tot, (5.2)

where the adiabatic wall temperature is equal to the stagnation temperature of the fluid at restat the wall.

The adiabatic wall temperature and hence the cooling effectiveness is directly related to themixing rate of the coolant with the main flow and allows for a quantitative comparison of thecooling model performance for different penetration depths. Figure 5.3(a) shows the evolution ofthe cooling effectiveness ηT along the NGV1 suction side. The values are averaged over 10 evenlyspaced spanwise locations in the range of 40-50% span to mitigate the influence of the apparentspanwise variability. This averaging process is the reason why the average cooling effectivenessof the fully featured computation is relatively low although the local effectiveness is unity at thecooling holes themselves. Figure 5.3(b) shows the average as well as the scatter of ηT for thecooling strip model and the fully-featured computation.The plot clearly shows the location of the cooling rows coinciding with the local peaks in coolingeffectiveness as well as the gradual decrease of effectiveness downstream of the injection pointsdue to turbulent mixing. A closer look at the regions close to the cooling rows emphasizes theintense mixing in the fully-featured computations leading to a very quick drop in effectivenessdownstream of the injection which is not captured by the cooling strip model that exhibits muchslower mixing rates. Furthermore, non-physical wiggles appear close to the injection points.This is a known problem in film cooling models relying on volume source terms [17]. Heidmannet al. attribute it to the Gibbs phenomenon [13] that is an unavoidable consequence of thediscontinuous application of source terms.

Downstream of the last cooling row, ηT varies much more smoothly in all cases. While therealistic penetration depth lc = 0.5 mm leads to a dramatic overprediction of ηT by 25% overvirtually the entire blade surface, the largest tested penetration depth of lc = 2.0 mm consistentlyunderpredicts the cooling effectiveness by the same amount. The variation of effectiveness overthe blade on the other hand seems qualitatively similar for all values of lc but the smallest. Infact, ηT linearly decreases after the last cooling row for lc ≥ 1.0 mm whereas for lc = 0.5 mm itfirst stays constant over a third of the blade length before decreasing much more rapidly thanthe other profiles towards the trailing edge.A closer examination of the latter case shows that the differences are heavily mesh dependent. In

2In flight applications it is more common to relate the effectiveness to the recovery temperature [14]

51


0 5 10 15 20 25 30 35 40 45 50 55 60

0

0.2

0.4

0.6

0.8

1

Distance from leading edge on suction side [mm]

Coo

lingeff

ectiv

enessη T

CSM, lc = 0.5 mmCSM, lc = 1.0 mmCSM, lc = 1.5 mmCSM, lc = 2.0 mmFF

(a) Penetration depth lc varied over the entire range considered (0.5 mm ≤ lc ≤ 2.0 mm).

0 5 10 15 20 25 30 35 40 45 50 55 60

0

0.2

0.4

0.6

0.8

1

Distance from leading edge on suction side [mm]

Coo

lingeff

ectiv

enessη T

FFFF scatterCSM, lc = 1.15 mmCSM scatter

(b) Best fit CSM (lc = 1.15 mm). Shaded area corresponds to the respective scatter over the range40-50% span.

Figure 5.3: Comparison on the cooling effectiveness on the NGV1 suction side between the coolingstrip model (CSM) and the fully-featured computation (FF). All profiles are an average over therange 40-50% span.

fact, when lc = 0.5 mm, the coolant injection is located entirely inside the near wall structuredprism layers of the mesh (protruding 0.67 mm into the flow). At first, the coolant stays inthe prism layer and mixes very little but then, as it diffuses outwards, reaches the tetrahedralmesh with increased numerical diffusion leading to the rapid drop in cooling efficiency towardsthe trailing edge. The diffusion is accelerated by the large cross-stream temperature gradient(ηT = 0.94 when the tetrahedral mesh is reached). All other penetration depths protrude wellinto the tetrahedral mesh and are subjected to the increased diffusion immediately. Notingthat the cooling strip model itself creates a flow situation in which the injected fluid is simply

52


zx

y

(a) Radial cut through NGV1 domain.

0 20 40 60 80 100

380

400

420

440

460

% NGV pitchSp

an-averagedT

tot[K

]

FFCSM, lc = 1.15 mm

(b) Span-averaged total temperature over the NGV1 pitch.

Figure 5.4: Comparison of the span-averaged total temperature distributions over the NGV1pitch between trailing edge and outlet in the NGV1 domain for the cooling strip model (CSM)and fully-featured (FF) computations.

convected downstream with little cross-stream diffusion, the mesh-induced numerical diffusionseems to be beneficial and leads to a more realistic mixing rate.

The best match between cooling strip model and fully-featured model was found for a penetrationdepth of lc,opt = 1.15 mm, shown in Fig. 5.3(b).Comparing the span-averaged total temperature over the NGV pitch for the fully-featured modeland the cooling strip model (Fig. 5.4) confirms that the mixing between film cooling and mainflow is well captured. The slight shift in the profiles near the minimum (corresponding to thewake) is due to the trailing edge slot. In fact, the jet emanating from the trailing edge slot in thefully-featured simulation, due to the large boundary layers that have grown inside the blade core,has reduced momentum and quickly mixes with the main flow. In the model used in this study,the core is not included and, as can be seen in Fig. 5.4(a), the mass inlet boundary for the slotis located relatively close to the slot lip. Since no boundary layer profile can be input for suchan inlet, the resulting layer at the exit is thin leading to higher jet momentum, less mixing andtherefore a lower temperature on the pressure side of the wake.

Ultimately, the aim is to compute the performance parameters for the turbine. Figure 5.5 showsthe capacity change relative to the value for lc,opt over the penetration depth indicating that ifthe penetration depth is chosen too low, the capacity is noticeably underpredicted with the CSM.For higher penetration depths, the capacity quickly becomes independent of the penetrationdepth. The dotted line represents the penetration depth lc,opt that is sufficiently large to avoidsignificant mesh effects.

5.1.2 Conclusion

The calibration study for the cooling strip model has shown that the mass injection with volumesource terms spread over the blade span leads to a qualitatively different flow field than the

53


0.5 1 1.15 1.5 2

−0.6

−0.4

−0.2

0

lc [mm]

∆C/Cl c,o

pt[%

]

Figure 5.5: Capacity change relative to the value computed with lc = 1.15 mm versus penetrationdepth.

injection through cooling holes. Moreover, the turbulent mixing of the coolant stream with themain flow is highly dependent on the penetration depth lc of the source terms. Nevertheless itwas found that setting lc = 1.15 mm leads to a very good match of the mixing characteristicsbetween the fully-featured computation and the cooling strip model, exemplified by the coolingeffectiveness on the blade and the total temperature distribution in the wake of the NGV.The study also yielded important guidelines for the boundary layer mesh generation in conjunctionwith the cooling strip model. In order to compute realistic mixing rates, the penetration depthmust be chosen such as to cover the full structured near-wall mesh and protrude into theunstructured main mesh in order to aid the cross-stream mixing.

5.2 Boundary Layer StudyThe thin boundary layer grids close to solid walls are a crucial part of a high-quality CFD meshsince a large proportion of the losses occur due to very high velocity gradients induced by theno-slip condition. A common approach in turbine RANS CFD is the use of high Reynolds numberturbulence models in conjunction with wall functions to save cells in the boundary layer.As we have seen in the calibration study for the cooling strip model, the use of volume sourceterms has a major impact on the boundary layer flow. This situation is not considered in thevalidation process for the turbulence model used in this work (Menter SST k-ω model). Althoughlow Reynolds number models can still be expected to yield good results if the near-wall mesh isappropriate, especially the accuracy of wall functions cannot be taken for granted and must beexamined. These considerations make a more thorough analysis of the boundary layer mesh andits interaction with the cooling strip model an important aspect of increasing the level of trust inthe final model and determining how well the boundary layer must be resolved to adequatelycapture the film cooling effects.

Quasi 2D subdomain

There is no method of a priori determination (i.e., before actually simulating the flow) of theboundary layer mesh parameters required to accurately compute the near-wall flow. In orderto reduce the computational cost of this study, the boundary layer mesh is analysed only forthe stator row of the first stage. The NGV1 is a good candidate for a representative boundarylayer study since it includes film cooling as well as high velocities, thus leading to restrictive

54

5.2 Boundary Layer Study

(a) Subdomain with full NGV1 geometry delim-ited by the red and blue surfaces (inviscidwalls).

zx

y

(b) Top view of the subdomain. Flow is fromleft to right.

Figure 5.6: Quasi 2D subdomain for boundary layer study.

requirements on the non-dimensional wall distance y+.

The full NGV1 domain with a cell count in the order of 5 million cells for a moderate boundarylayer resolution is impractical for this study, not only because of the large computation timesbut also because of the difficulty to identify general characteristics due to the complexity of 3Dflow. Furthermore, the most important region in terms of analysing the interaction between theboundary layers and the cooling strip model is the blade surface making a 2D study of a bladeslice ideal. The meshing tool used in this work, due to the underlying octree mesh strategy, isnot capable of creating purely 2D meshes. Instead, a quasi-2D mesh is created for a thin radialslice of the NGV1 domain (spanning 5% of the blade height, centered on 45.1% span) close tothe blade midspan where the geometry exhibits very little spanwise variation. The advantage ofthis approach in comparison to using another meshing tool to create a 2D mesh is that the samemeshing strategy is used for the mesh study as for the final mesh and the mesh refinements thatare not related to the wall normal boundary layer resolution are maintained. The computationaldomain for the study is shown in Fig. 5.6.

The main parameters considered in the following study are the initial layer height (thus y+) aswell as the wall-normal expansion ratio in the boundary layer. The aim is to determine the bestcompromise between high accuracy and low cell count and to get a better understanding of theeffect of the cooling strip model on the boundary layer flow. The influence of the mesh resolutionwas studied for 5 different meshes that only differ in their boundary layer resolution. The meshparameters are summarized in Tab. 5.1 and Fig. 5.7 shows the near-wall resolution for eachmesh.Meshes 1-3 differ only with respect to the expansion ratio ranging from 1.1 to 1.3 (1.2 is acommon recommendation in literature [11]) with y+ ≈ 2 on the suction side surface (the mostcritical surface where the velocities are highest). Although the y+-value is not ideal, it is stillacceptable for low Reynolds number models placing 3-4 cells in the viscous sublayer [3]. Mesh4 is a very coarse and computationally cheap mesh for a high Reynolds number model withy+ > 20 on most of the blade. The first mesh point is thus fully in the logarithmic region of theboundary layer and wall functions are applied everywhere. Mesh 5 is an extremely fine mesh fullyresolving the boundary layer (y+ < 1 everywhere) for a low Reynolds number model without wall

55


Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 5Number of layers 15 21 38 10 50Expansion ratio 1.3 1.2 1.1 1.1 1.15Initial layer height 3µm 3µm 3µm 60µm 0.1% SCS 3

Total layer height 0.50 mm 0.67 mm 1.08 mm 0.96 mm 0.81 mmy+ on blade surface ≈ 2 ≈ 2 ≈ 2 mostly

over 20< 1

# of cells (· 106) 0.361 0.405 0.564 0.312 1.28

Table 5.1: Mesh parameters for the boundary layer study.

(a) Mesh 1 (b) Mesh 2 (c) Mesh 3 (d) Mesh 4 (e) Mesh 5

Figure 5.7: Cut through boundary layer meshes. All meshes are shown at the same scale.

functions. This mesh is expected to yield the most trustworthy results and will be used to assessthe quality of the other meshes. Due to its prohibitive cell count is impossible to apply to the fullHPT. In order to achieve the extremely low y+-values of mesh 5, the surface refinement of theblade had to be increased by one level leading to a further increase of the number of cells to wellover one million for the small domain considered. The total number of layers for each mesh waschosen such that: i) the transition from structured prism cells at the wall to the unstructuredtetrahedral transition cells to the main flow is as smooth as possible in terms of cell volumesand ii) owing to the results of the film cooling calibration, the total height of the structuredboundary layer mesh is chosen smaller than the film cooling penetration depth lc = 1.15 mm.

In the quasi-2D model, all boundaries are defined the same way as in the full NGV model withthe exception of the exit plane which is set to a pressure outlet using mixing plane data froma full HPT calculation. The computational domain is a slice of the three-dimensional blade sothat the upper and lower surfaces are not completely identical precluding the use of periodicboundary conditions. The blade slice is taken close to the blade midspan where the flow hasa negligible radial component so that the upper and lower boundaries can be set to inviscidwalls without noticeably influencing the flow. The cooling flows were scaled to the new geometrywhile maintaining their streamwise injection angle and fluid properties. It should be noted thatresults from this study cannot be directly compared to the fully-featured computation since thequasi-2D model does not capture the radial contraction present in the full annulus that alters the

3Surface cell size (SCS): In order to achieve this level of resolution the initial layer height needs to be givenrelative to the surface cell size instead of being directly specified.

56


−40 −30 −20 −10 0 10 20 30 40 50 6010−2

10−1

100

101

102

Distance from LE [mm]

y+

Meshes 1-3Mesh 4Mesh 5

Figure 5.8: y+-distributions on the NGV1 surface without film cooling. The distance to the LEis measured positive on the SS and negative on the PS. The curve in the center correspondsto meshes 1-3 that have virtually identical y+-distributions apart from a discrepancy at 15 mmfrom the leading edge on the SS.

overall flow field. Nevertheless, the results concerning the required boundary layer resolution andthe interactions with the cooling strip model can be used as guidelines for the full HPT mesh.

5.2.1 Boundary Layer Flow without Film Cooling

As a first step, the flow is computed with the cooling strip model turned off to assess the apriori mesh adequacy and to produce a useful baseline for the subsequent computations withfilm cooling since the turbulence model is expected to perform well for this case.

Figure 5.8 shows the y+-distributions along the blade surface at the radial middle of the domain(45.1% span) for the different meshes. Since meshes 1, 2 and 3 differ in the wall-normal expansionratio but have the same initial cell height, the profiles are nearly identical so that the total scatterof the three curves is plotted instead of the individual curves for clarity. The vertical dottedlines indicating the location of two specific points along the suction side are included for futurereference (A, −−, 20.0 mm from LE and B, −.−, 40.1 mm from LE).

The plot shows that the respective requirements for y+ are indeed fulfilled for most of the bladesurface. The relatively low y+-values in a small region close to the leading edge for the coarsemesh (4) are no major issue since the wall functions in this case apply blending functions thattake into account the lower y+-values. More crucial is the quality of mesh 5 as it will serve asa reference for subsequent simulations with film cooling. For this mesh, the requirement of y+

below unity is consistently fulfilled so that it can be expected to yield accurate results.Since aerofoil computations aim to assess blade performance, two parameters are typically usedto analyse the results. The pressure coefficient cp is a measure of the pressure distribution on

57


the blade and relates to the lift force as well as the blade loading. It is defined as

cp = pstat,wqref

, (5.3)

where pstat,w is the local static pressure on the blade surface and qref the reference dynamicpressure computed using the average density at the domain inlet ρref and a reference velocityUref :

qref = 12ρrefU

2ref . (5.4)

Here, the maximum absolute velocity reached in the domain without film cooling is used as areference.

The computation of the exact wall shear stress distributions on the blade are a challenge for anyviscous computation. Yet the wall shear stress is crucial in terms of the viscous drag experiencedby the blade but also regarding the boundary layer structure and identifying separation. Inanalogy to the pressure coefficient, the friction coefficient cf is defined as

cf = τwqref

. (5.5)

Figure 5.9 shows the evolution of the pressure and friction coefficients over the blade surface.The presentation is analogous to Fig. 5.8. The predictions for meshes 1-3 and 5 are in very closeagreement. This is an important result since the collapse of the profiles increases the trust in thesolution overall and also shows that, in terms of calculating the correct wall shear stress andpressure distributions, the requirement of y+ < 1 might be too restrictive, at least for the simplecase without cooling. Although the results for mesh 4 also follow the trends predicted on theother grids, the data is extremely noisy especially on the suction side. Although the flow aroundthe blade nose is difficult to compute due to strong surface curvature, no full explanation couldbe found for noise of such amplitude in the wall shear stresses that is inherited to y+ via thefriction velocity. The converged solution for mesh 4 showed similar residual levels as the othercases thus excluding insufficient convergence as a possible cause. The results for the coarse meshare therefore to be considered with caution.

5.2.2 Boundary Layer Flow with Film Cooling

When the cooling strip model is turned on, the boundary layer flow is heavily disrupted close tothe injection points, especially since the volume source terms are introduced discontinuously. Asexpected, the profiles are less smooth in the case with film cooling, as seen in Fig. 5.10, but thematch is still nearly perfect for cp in spite of slight undershoots for mesh 5 in the direct vicinityof the cooling strips on the suction side. The region close to the last two cooling strips on thesuction side also exhibits large discrepancies between the cf profiles computed on meshes 1-3 andmesh 5 that are very similar otherwise. The noise in the results for mesh 4 are also amplifiedwhen the cooling strips are added.The comparison between the pressure coefficients in 5.9 and 5.10 shows that, especially on thepressure side, the cooling strips have a negligible effect on the overall pressure distribution on theblade dictated by the main flow, apart from the region immediately downstream of the injectionpoints.

58


−40 −30 −20 −10 0 10 20 30 40 50 601

1.21.41.61.8


c pMeshes 1-3Mesh 4Mesh 5

−40 −30 −20 −10 0 10 20 30 40 50 600

2

4

6 ·10−3


c f


Figure 5.9: Results with cooling strip model turned off. Locations A and B (vertical dotted lines)for reference.

−40 −30 −20 −10 0 10 20 30 40 50 601

1.21.41.61.8


c p


−40 −30 −20 −10 0 10 20 30 40 50 600

2

4

6 ·10−3


c f


Figure 5.10: Results with cooling strip model turned on. Locations A and B (vertical dottedlines) for reference.

59


The profiles of pressure and friction coefficients, though suggesting that a mesh with superiorboundary layer resolution should be employed instead of mesh 4, do not allow for a clean cutdecision which of the three remaining candidates is best suited to capture the impact of thecooling flow on the overall boundary layer structure. Since Mesh 3 has over 50% more cells thanmesh 1, implying considerable differences in computation time when applied to the full HPT, it isimportant to assess whether the coarser mesh 1 can be used in spite of its high expansion ratio.

In order to complete the picture, the boundary layer structure up to the free stream is consideredin further detail at the two specific locations A and B. To avoid interpolation errors, the flowquantities are extracted directly at the grid nodes. Since the mesh is unstructured outside theboundary layer, the nodes may not lie exactly in wall-normal direction of the considered surfacepoint. The streamwise deviation is at most half a cell diameter which is in the order of 0.1 mmfor the cells close to the boundary layer in the interblade passage and thus negligible in termsof streamwise boundary layer evolution. The profiles are normalized using the throat widthlref = 10.3 mm, the maximum absolute velocity over the entire NGV domain in the case withoutcooling Uref as well as a reference dynamic pressure qref.

zx

y

LE

A

B

Figure 5.11: Mid-span cut of the domain showingthe approximate locations A and B as well asthe locations of the cooling strips (black).

The first location (location A) is on the suctionside, just before the injection point of the CSSS05 at a distance of 20.0 mm from the LE.This point is located sufficiently far from theprevious cooling strip to give the flow timeto smooth out local non-physical effects dueto the volume sources while still presenting astrong interaction between the boundary layerand the coolant flow.The second location (location B) is chosen atthe locus of the section exhibiting the highestMach number on the blade (40.1 mm from theleading edge on the suction side) where theskin friction is close to its peak and the flowgradients are especially high. At the sametime this location is further away from the lastcooling strip.

In a first step, we consider the velocity profiles across the boundary layers. The profiles withfilm cooling extracted at locations A and B are plotted in Fig. 5.12 with the profile obtained onmesh 5 without film cooling as a reference. While the profiles at location A are displayed upto the middle of the interblade passage, the profiles at location B span the full passage to thetrailing edge wake.

The boundary layers at both locations exhibit similar characteristics. The injection of coolantleads to a reduction of the maximum free stream velocity through momentum transfer from themain flow to the coolant that needs to be accelerated. This reduction is more pronounced atlocation A (10%) than at location B (3%). The depth up to which the film cooling influences themain flow on the suction side is about 25% of the throat width at both locations and is consistentwith the total temperature contours in Fig. 5.2 close to the trailing edge. Looking at the velocity

60


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−4

10−3

10−2

10−1

100

U/Uref

y/l r

ef

Mesh 1Mesh 2Mesh 3Mesh 4Mesh 5Mesh 5 (NC)

(a) Location A: 20.0 mm from the LE on the suction side of NGV1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−4

10−3

10−2

10−1

100

U/Uref

y/l

ref


(b) Location B: 40.1 mm from the LE on the suction side of NGV1.

Figure 5.12: Boundary layer velocity profiles at locations A and B on the suction side of NGV1for the 5 considered meshes (symbols) including the velocity profile for mesh 5 at the samelocation without film cooling (NC). The dotted line indicates the film cooling penetration depthlc.

61


profiles close to the wake at location B, the small dent due to the trailing edge slot flow is visibleas well as the overall reduction of the flow speed in the wake when film cooling is present.Comparing the profiles in the viscous sublayer (the first few nodes from the wall) shows that thevelocity gradient at the surface is considerably reduced at location A explaining the reduced skinfriction in this area (see Fig. 5.10(b)), suggesting that the boundary layer is still adjusting tothe coolant injection. Further downstream at location B, the profiles with and without coolanthave become very similar in the near-wall region. All low Reynolds number meshes (1-3 and 5)collapse in this region showing that the viscous sublayer is well resolved even with y+ ≈ 2.Moving into the logarithmic region, the differences between the meshes become apparent. Theprofiles split into two distinct groups, mesh 3 and 4 showing a very similar but consistently lowervelocity than meshes 1,2 and 5. This could be due to the total thickness of the boundary layerprism mesh compared to the penetration depth lc = 1.15 mm. In fact, the near wall structuredlayers in meshes 3 and 4 cover over 80% of the penetration depth while this value is between40 and 70% for the other meshes. This impression is reinforced by the fact that the profiles ofmesh 1, having the thinnest structured region, is the least influenced by the coolant injection.

In the absence of experimental data to validate the simulations, another strategy must be soughtto assess whether the boundary layer solutions are mesh dependent or not. To this end, weconsider the boundary layer distributions of the compressible turbulent kinetic energy kc definedin analogy to the standard kinetic energy k (Eq. (2.22)) as

kc = 12ρu

′iu′i ≈

12ρ u

′iu′i = ρk , (5.6)

to take into account the density variation across the boundary layer due to large spatial differencesin both temperature and velocity. The (compressible) turbulent kinetic energy is a typicalparameter used to judge whether the boundary layer is adequately resolved since its distributionin the boundary layer is well understood. The compressible turbulent kinetic energy profiles atlocations A and B are shown in Fig. 5.13(b).

The turbulent kinetic energy is energy transferred from the main flow to turbulence by the meanflow gradients. Therefore, the profiles exhibit a peak very close to the wall (y+ ≈ 10) wherethe velocity gradients are highest. In the viscous sublayer, the molecular viscosity dominatesthe flow and damps the turbulent fluctuations and the kc-profile consequently sharply dropsto zero. Towards the free stream, the flow gradients quickly become less pronounced and, ina typical turbulent boundary layer, the turbulent kinetic energy smoothly approaches its freestream value [27].

At location A, all profiles except mesh 4 collapse and reproduce the expected distributionsexactly differing only slightly in the absolute value of kc reached at the maximum. The highReynolds number mesh overpredicts the compressible kinetic energy for a large part of the nearwall region. Interestingly, the results of mesh 2 are more similar to those of mesh 5 than mesh 3that theoretically has a better boundary layer resolution.At location B, the near-wall profiles are nearly identical for all meshes, which is to be expectedsince the velocity profiles have become very similar. Again, mesh 2 has the best agreementwith mesh 5. The profiles of meshes 3-5 exhibit a non-physical "bump" or local maximum incompressible turbulent kinetic energy precisely at the boundary layer edge where the structuredprism layers interface with the tetrahedral transition mesh to the main flow. The fact that the

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6·10−3

10−4

10−3

10−2

10−1

100

kc/qref

y/l r

efMesh 1Mesh 2Mesh 3Mesh 4Mesh 5Mesh 5 (NC)

(a) Location A: 20.0 mm from the LE on the suction side of NGV1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6·10−3

10−4

10−3

10−2

10−1

100

kc/qref

y/l

ref


(b) Location B: 40.1 mm from the LE on the suction side of NGV1.

Figure 5.13: Boundary layer profiles for the compressible turbulent kinetic energy kc at locationsA and B on the suction side of NGV1 for the 5 considered meshes (symbols) including the profilefor mesh 5 at the same location without film cooling (NC). The dotted line indicates the filmcooling penetration depth lc.

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peaks are very sharp and lie on the connecting nodes between the two mesh types shows thatthe phenomenon is a mesh-induced numerical artefact. The fact that meshes 1 and 2 do notexhibit a similar peak although the main flow meshes are similar for all meshes confirm thatthe observed effect is due to the film cooling penetration depth being too close to the thicknessof the structured prism region for meshes 3-5. It is interesting to observe that the mesh effectsat the edge of the boundary layer mesh occur only downstream of the last cooling strip on thesuction side. It is plausible that the combination of the relatively large penetration depth relativeto the structured mesh thickness with the reduced favourable pressure gradient, keeping theboundary layers thin close to the blade nose, leads to the non-physical effects downstream of thelast cooling row.

5.2.3 Conclusion

The thorough analysis of the interaction between the boundary layer mesh and the coolingstrip model has revealed the inadequacy of the high Reynolds number mesh 4 to capture thecomplicated flow physics, most probably mainly due to the wall functions that are not validatedfor flow situations with film cooling. The study suggests that the use of a low Reynolds numbermesh is advisable for an adequate resolution of the boundary layer in conjunction with the coolingstrip model. The typical requirement of y+ < 1 on all surfaces (mesh 5) on the other hand is toorestrictive and leads to an unnecessarily high cell count. In fact, mesh 2 with y+ ≈ 2 and a cellcount corresponding to only a third of that of mesh 5 yields very similar results especially inthe near-wall region. The expansion ratio had more surprising effects since a reduction to 1.1led to worse performance (mesh 3). An increase of the expansion ratio from 1.2 to 1.3 had onlylittle effect on the overall cell count (-10%) and the resulting profiles. Since mesh 2 was found toperform slightly better and an expansion ratio of 1.2 is a common recommendation in literature,the conservative choice to use mesh 2 as a guideline for the full HPT mesh was made.

5.3 Mesh Independence Study

Using the results from the calibration of the cooling strip model and the boundary layer study, thefinal mesh is created individually for each row and subsequently merged into a single multi-zonemesh during pre-processing. The mesh details for each row are summarized in Tab. 5.2. Therequirement to adequately resolve the boundary layers and the wake regions leads to relativelyhigh cell counts, especially for the blades and vanes of the second stage.

STAGE 1 STAGE 2ROW NGV ROT NGV ROT DUCT Σ# of cells (· 106) 7.76 6.48 13.2 16.8 0.63 46.3% of total 16.7 % 14.0 % 28.6 % 35.1 % 1.36 % 100.0 %

Table 5.2: Full HPT mesh summary

The quality as well as the resolution of the mesh used for a simulation is crucial for its accuracy.If the mesh is too coarse, it might fail to capture important flow features or incur high levels ofnumerical diffusion. On the other hand if the mesh is unnecessarily fine, the computational costsquickly become prohibitive. Therefore, it is important to conduct a mesh refinement study to

64

5.3 Mesh Independence Study

assess the quality of the mesh resolution with respect to the quantity of interest. In this study,the efficiency was chosen as reference performance parameter to assess the grid independencebecause it is difficult to compute and is dependent on the full computational domain.

The grids for the refinement study are generated on the basis of the final mesh by systematicallyrefining and coarsening the mesh. This is done by changing the background cell size (BCS) thatthen inherits the refinement to all other mesh regions. Given the already large mesh, the studyis carried out on the first stage of the turbine only to save computation time. The results can betransferred to the second stage since the flow is essentially similar and the meshing parametersettings are comparable.

Mesh Coarse Baseline Fine FinerBCS (NGV/ROT) 20/10 40/20 50/25 60/30# of cells (· 106) 4.70 14.2 21.1 29.3% of baseline 33 % 100 % 148 % 206 %Extrapolated full HPT mesh (· 106) 15.3 46.3 68.5 95.4

Table 5.3: Global refinement study

Coarse Baseline Fine Finer

−0.2

0

0.2

0.4

∆η[%

]

ηHηPηs

Figure 5.14: Change in computed efficiencies for the coarser and finer meshes. Changes given inpercent relative to respective baseline value.

The three grids studied in addition to the baseline grid are summarized in Tab. 5.3 and theresults of the refinement study for the three efficiencies are shown in Fig. 5.3. We observe thatthe values for all efficiencies increase as the mesh is refined due to the reduction in numericaldiffusion and reach mesh independence only for the two finest meshes tested. The efficiencies atmesh independence lie 0.2 to 0.3 % higher than the values computed for the baseline mesh andthe subsequent variation between the two finest meshes is below 0.05% of the baseline value forall three efficiencies.

Although the baseline mesh is not sufficiently refined to exclude mesh effects on the outputvariables, it was nevertheless chosen for the subsequent sensitivity study since the additionalcomputational cost of running a simulation of the full HPT on a mesh yielding a fully meshindependent solution was too large in comparison with the expected accuracy gain. This choiceis motivated by the fact that the aim of the sensitivity study is not the computation of absolutevalues for which a mesh independent solution is necessary, but rather the analysis of delta-values,i.e., changes in the parameters of interest around a baseline caused by changes in the boundary

65


conditions. Even though the absolute values for the efficiencies are not mesh independentand must be considered with care, the sensitivity results gained with the baseline mesh aretrustworthy.

66

6 Results and Discussion

6.1 Results of the Kriging Response Surface Based Analysis

Once the response vectors Yi (i = 1, . . . , N) have been obtained by evaluating the high-fidelitymodel for the N samples of the OLHS matrix, they are loaded into the GPE toolbox in iSightfor the subsequent sensitivity analysis.

The optimal weights assigned to each sample point during the interpolation are evaluated usingan optimisation procedure. Several locally optimal combinations of weights exist; Random searchsteps are therefore included in the optimisation. Since the optimisation procedure converges to apotentially different local optimum at every run, the Kriging response surface is not unique. Whenthe optimisation converges adequately, the corresponding response surfaces exhibit the sametrends but exhibit slight differences in the absolute values, especially towards the domain edges.For the capacity, the absolute values of the main effects differ by up to 5% of the amplitude ofthe deltas for different Kriging fits. For the efficiencies, especially in cases with strong non-lineareffects, the differences reach up to 33% of the amplitude of the deltas. The variations fromone response surface fit to the next occurred more frequently for the enthalpy and power basedefficiencies that exhibit stronger non-linearities and mainly at the edges of the domain wherethe uncertainty of the Kriging model increases since fewer exact data points are available in thevicinity for the estimation of the response. The resulting uncertainty in the computed maineffects is shown with errorbars in the corresponding bar charts.

6.1.1 Kriging Cross-validation

In order to assess the quality of the model fit, the results are checked using the leave-one-outcross-validation, i.e., the exact response value is compared to the value estimated by the Krigingmodel created from the full dataset excluding the point in question. This procedure is repeatedfor each point and the result is analysed in a scatterplot of the exact values versus the predictionsshown in Fig. 6.1. The better the approximation, the closer the points to the line y = x plottedonto the scatter. The plots show that all variables are well predicted by the interpolationsalthough the prediction error for the NGV1 outlet capacity is significantly smaller than for theefficiencies for which the quality of the model fit is similar. This is already an indication ofnon-linear sensitivities of the efficiencies with respect to the input parameters. Correspondingly,the coefficient of determination4 (R2) lies above 0.9 for all interpolations and as high as 0.993 forthe NGV1 outlet capacity showing the overall good predictive capabilities of the Kriging model.The plots also indicate the overall scatter of the computed values for the efficiencies that is 1-2

4The coefficient of determination is a statistical measure of the proportion of the variance in the dependentvariable that is predictable from the independent variable. In other words, R2 measures how well the model(here, the predicted efficiencies) approximates the exact data points. The coefficient ranges from 0 (the modelcannot predict any part of the observed variance, the prediction and the exact values are uncorrelated) to 1(the model fully predicts the observed variance and the predicted efficiencies equal the exact efficiencies).

67


orders of magnitude larger than the efficiency drift in a single computation. This shows that theefficiency drift is negligible, the simulations are sufficiently converged.

−0.4 −0.2 0 0.2 0.4

−0.4

−0.2

0

0.2

0.4

predicted ∆C [%]

exact

∆C

[%]

(a) Outlet Capacity, R2 = 0.993.

−0.4 −0.2 0 0.2 0.4−0.4

−0.2

0

0.2

0.4

predicted ∆ηH [%]exact

∆η H

[%]

(b) Enthalpy based efficiency, R2 = 0.928.

−0.4 −0.2 0 0.2 0.4−0.4

−0.2

0

0.2

0.4

predicted ∆ηP [%]

exact

∆η P

[%]

(c) Power based efficiency, R2 = 0.904.

−0.2 0 0.2

−0.2

0

0.2

predicted ∆ηs [%]

exact

∆η s

[%]

(d) Entropy based efficiency, R2 = 0.902.

Figure 6.1: Cross-validation plots for each output variable including the corresponding R2-value.The percentage delta-values are computed relative to the reference configuration.

6.1.2 Capacity Sensitivity Analysis

Using the Kriging response surface, the sensitivities of the turbine performance parameters onthe considered input parameters can be directly analysed. In the following section, the maineffects and interactions are analysed for the NGV1 outlet capacity.

68


Main effects

The main effect diagrams for the NGV1 outlet capacity are shown in Fig. 6.2. For clarity, theplots for the individual outputs are divided in the effects related to the cooling and rimsealleakage mass flow rates (left) and the effects due to the inlet distributions of total temperatureand pressure (right), both shown with the same scale for comparison. The corresponding barcharts below show the magnitude of the main effects ordered by importance as well as thevariation in the different Kriging fits (errorbars).

The main effect diagrams for the NGV1 outlet capacity lead to two conclusions. First, all maineffects are straight lines indicating a linear response of the capacity to variations of the consideredinput parameters. The errorbars emphasize that the uncertainty is small regarding these trends.Secondly, the bar charts show that the capacity is largely independent of variations in the radialdistribution of total temperature and pressure at the inlet in the considered range. This is notsurprising given the fact that the inlet temperature and pressure profiles were constructed tomaintain the mass flow weighted area average over the inlet surface. The changes in mass flowas well as losses across the NGV induced by the variations in the inlet distributions are smalland the total temperature and pressure averages taken at the NGV1 outlet surface are virtuallyunaltered. The corresponding main effects are well under 0.05% capacity change over the entireparameter range.

All cooling and secondary inlet mass flows have a negative main effect, i.e., a decrease of the massflow rate leads to an increase in the NGV1 outlet capacity. The sensitivity is also a function ofthe geometrical proximity of the respective secondary inlet to the NGV1 outlet surface where thecapacity is computed: The further away the inlet is from the NGV1 outlet surface, the smallerthe effect. The strongest main effects have a similar magnitude, the ROT1 rimseal leakage andfilm cooling flow rates, impacting the capacity by roughly 0.4% over the parameter range.The sensitivity of the capacity to the NGV1 cooling flow rates and the NGV2 rimseal leakageare similar and weaker with about 0.2% capacity change. Both the NGV2 cooling flow rate (TEslot) and especially the ROT2 rimseal leakage flow rate that is injected far downstream haveonly a small impact on the NGV1 outlet capacity.

The sensitivity of the capacity to the secondary inlet flow rates is directly linked to the maininlet mass flow that is floating, i.e., not a boundary condition but part of the solution. The massflow entering the flow domain through the secondary inlets essentially acts as a blockage reducingthe overall area available for the main flow and thus the amount of air sucked into the turbinevia the main inlet. Since the air temperature and density experiences dramatic changes throughthe turbine, the effects of downstream mass flow inlets are washed-out and damped before theyreach the main inlet explaining the reduced sensitivity of the capacity to the flow rates at inletslocated further downstream. The sensitivities must of course also be seen relative to the absolutemagnitude of the change in mass flow rate that is very different depending on the inlet.The NGV1 cooling flow rate, while contributing by far the largest secondary mass flow ratein absolute terms, is seen to have only a moderate negative main effect on the capacity. Thissurprising effect is due to the fact that the NGV1 cooling flow is injected upstream of the planeat which the capacity is computed. Although the main inlet mass flow rate changes considerablywith the NGV1 cooling flow, the total flow rate at the NGV1 outlet plane changes only little sincethe changes in inlet and NGV1 cooling flow rates balance each other. The weaker negative maineffect is therefore in large parts due to the losses across the NGV that increase with the cooling

69


0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

0.2

Normalized input parameter range

∆C

[%]

COOL NGV1COOL ROT1COOL NGV2RSL ROT1RSL NGV2RSL ROT2

0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

0.2


∆C

[%]

Ttot HUBTtot CENTtot CASPtot HUBPtot CENPtot CAS

0 0.1 0.2 0.3 0.4 0.5

RSL ROT2

COOL NGV2

COOL NGV1

RSL NGV2

COOL ROT1

RSL ROT1

Main effects for C [%]0 0.2 0.4

Ttot CAS

Ttot HUB

Ptot CEN

Ptot CAS

Ttot CEN

Ptot HUB

Main effects for C [%]

Figure 6.2: Main effects diagrams for the NGV1 outlet capacity C. The plot on the left showsthe main effects of the cooling (COOL) as well as the rimseal leakage mass flow rates (RSL)respectively for each row. The plot on the right shows the main effects of the total temperatureand pressure inlet profiles. Relative to the baseline, the cooling flows are symmetrically varied by20%, the rimseal flows by 50%. The total temperature and pressure variations are summarizedin Tab. 4.1.The bar charts present the magnitude of the main effects for each input parameter group orderedby importance corresponding to the diagrams above and include the errorbar indicating theuncertainty in the Kriging fit. The bar colours (light grey and dark grey) correspond to positiveand negative absolute main effects respectively.

70


COOL ROT1

CO

OL

NG

V1

(a) The contours for the capacity cover therange of ∆C = ±0.3%.

CO

OL

NG

V1

RSL NGV2

(b) The contours for the capacity cover therange of ∆C = ±0.4%.

Figure 6.3: Input parameter interactions on NGV1 outlet capacity over the full parameter ranges.The capacity increases from blue to red.

flow rate. If the same analysis would be conducted using the NGV1 inlet capacity (computed atthe main inlet plane) instead of the outlet capacity, the NGV1 cooling flow rate would be thedominant influence on the inlet capacity by including the considerable reduction of main inletmass flow rate with increasing NGV1 cooling flow rate.

Interaction

Figure 6.3 shows two typical contour plots of the co-influence of two parameters with largeindividual main effects on the NGV1 outlet capacity. The contours show a linear and additivedependency of the capacity on the considered input parameters that exhibit no interaction. Thecombination of the different input parameters can therefore lead to large additive changes of thecapacity. For parameters with weak main effects, the interaction plots are not meaningful sincethe numerical noise due to the uncertainties in the Kriging model cannot be distinguished fromthe physical parameter interaction.

Conclusion

The analysis of the Kriging response surface for the NGV1 outlet capacity showed little dependencyon the total temperature and pressure distributions at the inlet and a linear negative dependencyon the rimseal leakage and cooling mass flow rates. As expected, the dependence is weaker if thesecondary inlet is further away from the NGV1 outlet surface, where the capacity is computed,due to damping through the turbine. The influence of the secondary inlets is mainly due to thechanges in inlet mass flow rate induced by the variation of downstream blockage.

The ROT1 rimseal leakage flow, located directly after the NGV1 outlet plane, has the largestimpact on the capacity. The impact of the other secondary inlets is similarly proportional to themagnitude of the imposed change of mass flow rate at the inlet as well as inversely proportionalto the distance to the NGV1 outlet plane. The only exception is the NGV1 cooling flow that

71


only has a moderate influence due to the fact that it is injected upstream of the NGV1 outletplace. The observed negative main effect of this flow is due to increased losses across the NGVmore than the change in mass flow rate.The co-influence plots show no noticeable interaction between the input parameters whoseinfluences are linearly additive.

6.1.3 Efficiency Sensitivity Analysis

In the following section, the main effects and interactions are analysed for the turbine efficiency.Since the enthalpy and power based efficiencies exhibit nearly identical sensitivities, only theresult for the enthalpy based efficiency is shown.

Main Effects

The efficiencies, compared to the capacity, present more complex main effect diagrams, asshown in Figs. 6.4 and 6.5 for the enthalpy and entropy based efficiencies respectively. Thefirst observation is that both the enthalpy and the entropy based efficiencies show very similarsensitivities overall, although the differences are more pronounced than between the enthalpy andpower based efficiencies. The differences stem, in part, from the considerably higher uncertaintyin the Kriging fits for the enthalpy and power based efficiencies than for the entropy based values,especially for the cooling flows.

Considering first the secondary inlet flow rates, similarly to the capacity, most of the parametersstudied appear to have negative main effects on the efficiency, apart from the ROT1 cooling massflow rate. The sensitivities are largely linear with the notable exception of the STG1 cooling flowrates (ROT1 and NGV1) and the ROT1 rimseal leakage flow rate.While the main effect diagrams for the efficiencies agree on the overall sensitivity trends, thereis a major discrepancy in the profile for the ROT1 cooling flow rate. While the entropy basedefficiency shows a linear dependence on this parameter, the diagram for the enthalpy basedefficiency suggests a non-linear and even non-monotonic dependence with an extremum aroundthe middle of the input range (the plots for the enthalpy and power based efficiencies both showthis trend). The particular circumstances of the ROT1 cooling flow are considered in more detailbelow.

The sensitivities of all efficiencies to the secondary flow rates in the second turbine stage (NGV2cooling flow, NGV2 and ROT2 rimseal leakages) are in very close agreement regarding bothtrends and absolute values for the main effects predicting a 0.25% and 0.18% change in efficiencyover the parameter range for the NGV2 and ROT2 rimseals respectively. The relative influenceof the rimseals is again proportional to the respective absolute mass flow rates. The NGV2cooling flow (TE slot) has negligible influence on the efficiency.The good agreement observed here is due to the linearity of the dependencies. Only the main effectof the ROT2 rimseal flow rate on the enthalpy based efficiency exhibits a certain non-linearitythat is accompanied by a larger uncertainty. The high sensitivity to the rimseal leakage flowrate in contrast to the TE slot flow rate is due to the fact that the former are more disruptiveof the flow: The rimseal flows enter the main gas path at an angle, create noticeable blockageand interact with the endwall flows and vortical structures around the blade hubs. Furthermore,they lead an increased negative incidence of the flow on the hub of the blades of the subsequentrotor row. The TE slot flow on the other hand is injected in flow direction and therefore even

72


0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

0.2


∆η H

[%]


0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

0.2


∆η H

[%]


0 0.1 0.2 0.3 0.4

COOL NGV2

RSL ROT1

COOL ROT1

RSL ROT2

RSL NGV2

COOL NGV1

Main effects for ηH [%]0 0.2 0.4

Ptot CAS

Ttot CAS

Ptot HUB

Ttot HUB

Ptot CEN

Ttot CEN

Main effects for ηH [%]

Figure 6.4: Main effect diagrams for enthalpy based efficiency. The plots on the left show themain effects of the cooling (COOL) as well as the rimseal leakage mass flow rates (RSL) for eachrow. The plot on the right shows the main effects of the total temperature and pressure inletprofiles. Relative to the baseline, the cooling flows are symmetrically varied by 20%, the rimsealflows by 50%. The total temperature and pressure variations are summarized in Tab. 4.1.The bar charts present the magnitude of the main effects for each input parameter group orderedby importance corresponding to the diagrams above and include the errorbar indicating theuncertainty in the Kriging fit. The bar colours (light grey and dark grey) correspond to positiveand negative absolute main effects respectively.

73


0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

0.2


∆η s

[%]


0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

0.2


∆η s

[%]


0 0.1 0.2 0.3 0.4

COOL NGV2

COOL ROT1

RSL ROT1

RSL ROT2

COOL NGV1

RSL NGV2

Main effects for ηs [%]0 0.2 0.4

Ttot CAS

Ptot CAS

Ptot HUB

Ttot HUB

Ptot CEN

Ttot CEN

Main effects for ηs [%]

Figure 6.5: Main effect diagrams for the entropy based efficiency. The plot on the left shows themain effects of the cooling (COOL) as well as the rimseal leakages mass flow rates (RSL) for eachrow. The plot on the right shows the main effects of the total temperature and pressure inletprofiles. Relative to the baseline, the cooling flows are symmetrically varied by 20%, the rimsealflows by 50%. The total temperature and pressure variations are summarized in Tab. 4.1.The bar charts present the magnitude of the main effects for each input parameter group orderedby importance corresponding to the diagrams above and include the errorbar indicating theuncertainty in the Kriging fit. The bar colours (light grey and dark grey) correspond to positiveand negative absolute main effects respectively.

74


considerable variations of the mass flow rate do not alter the overall flow field. The sensitivitiesof the NGV1 and the NGV2 cooling flow rates are very different since the cooling mass flow alsoincludes film cooling that is absent on the NGV2. It is to be expected that the NGV1 TE slotby itself has a similar weak main effect.

The secondary inlet flow rates in the first stage have more complicated sensitivities. The NGV1cooling flow rate has a strong influence on both efficiencies, with a main effect over the fullparameter range of 0.35% ηH but only 0.25% ηs and a more pronounced non-linearity in the firstcase. The ROT1 secondary flow rates also have a non-linear but weaker influence leading to, atmost, an efficiency change of around 0.05%. Due to the strong non-linearity and non-monotonicity,the integral values for the main effects over the full parameter range are not representative forthis parameter. Instead the curve in the main effects diagram should be considered.

The main effect diagram for the inlet total temperature and pressure distributions shows verygood agreement for all efficiencies that appear to be insensitive to profile variations at the casingwall and largely insensitive to pressure fluctuations at the hub wall. A slight linear and negativemain effect appears to exist for the temperature profile at the hub.Variations of the temperature and pressure profiles at midspan (CEN) on the other hand lead toa much stronger and non-linear response. Their main effect is positive and of the same orderof magnitude as the stronger main effects of the flow rates at the secondary inlets with about0.18% η (for all three efficiencies) over the considered range.

The positive main effect of increasing the total temperature and pressure at the centerline islinked to the loss distribution on the rotor blades. In fact, the flow passing the rotor close to thehub and tip experiences considerably more losses than the flow at the centerline. At the hub,the losses are high due to the strong vortices and boundary layer interaction as well as the flowdisruption brought about by the rimseal leakage that also leads to non-optimal incidence on theblade. At the tip, large vortices with corresponding losses are fuelled by the tip leakage flow.Concentrating a larger proportion of the inlet total pressure on the centerline leads to a highermass flow rate in this region that does not noticeably smooth out before reaching the first rotorrow. The proportion of the total flow rate passing the rotor (and doing work) at the centerlineis therefore higher. Increasing the centerline inlet total temperature at the expense of the huband tip regions on the other hand has a much smaller effect on the mass flow distribution. Thevariation in fact leads to a decrease of the centerline mass flow due to the reduced density of theair at higher temperature. Nevertheless the temperature increase outweighs the decrease in massflow. In this case, the flow at the rotor centerline is slightly reduced but contains considerablymore energy that is then converted into shaft power.Both mechanisms therefore lead to more work being done by the fluid on the rotor centerlinewhere the losses are smaller leading to an increase in shaft power and thus efficiency.

The weak main effect of the total temperature at the hub is more difficult to analyse. Themagnitude of the effect either points to a minor mechanism or to the balance of positive andnegative effects. In both cases the analysis is difficult and only tentative explanations can begiven within the present study. The observed main effect might be due to the concentration offlow energy at the hub, a region with especially high losses, but this theory needs to be backedby further studies.

75


A thermodynamic analysis of the turbine system suggests that an increase in the cooling flowrate leads to a decrease in total temperature and thus usable energy in the main flow. Theconsequence is a drop of efficiency. At the same time, an increase in cooling flow rate leads toincreased mixing with the losses this entails. This analysis is in line with the sensitivities ofthe secondary flow rates that show a negative main effect and are therefore detrimental to theturbine efficiency.With this background, the results for the ROT1 cooling flow rate seem intriguing since theyexhibit a trend opposite to all other secondary flow rates and show that an increased cooling flowrate is beneficial to the efficiency. This result is indeed correct within the considered system ofthe turbine isolated from the rest of the engine: From the perspective of the first stage rotor, anincrease in cooling air means a higher flow rate leading to a higher work output that outweighsthe increased losses. The aerodynamic efficiency therefore rises. The isolated turbine analysisis biased: The increase in cooling flow comes at no thermodynamic cost since it ignores theconsiderable windage and pumping losses induced by the supply of the cooling air to the turbinerotor blades. Therefore, while an increased ROT1 cooling flow rate is beneficial to the purelyaerodynamic turbine efficiency, an analysis of the full engine cycle efficiency would yield theexpected result that the overall effect of the ROT1 cooling flow is detrimental to overall efficiency.In principle, all secondary flows introduced before a rotor stage have a similar effect providinga higher flow rate and increase work output. This notwithstanding, the positive effect on theefficiency is only seen for the ROT1 cooling flow because for the other flows, the negative effectsoutweigh the benefits of the increased flow rates. Because the ROT1 cooling flow is bled directlyonto the blade it does not disrupt the flow only leading to additional mixing losses. The othersecondary flows on the other hand, e.g. the ROT1 rimseal flow, not only lead to increased mixinglosses and blockage in the main gas path but are also detrimental to the ROT1 work output byinducing a negative incidence flow on the rotor blade. It is probable that the insensitivity of theefficiencies to the NGV2 TE slot flow is due to the cancellation of the positive effects (increasedwork output at ROT2) by the increased mixing losses.

Interactions

The sensitivity interactions for the efficiencies are more pronounced than for the capacity. Thisis expected given the stronger non-linearities in the main effect diagrams.

If one parameter dominates the other in terms of its main effect, the interaction is usually weak,as it is the case for the midspan total temperature and pressure variation and the NGV1 coolingmass flow (the latter shown in Fig. 6.6 for the enthalpy and the entropy based efficiencies). TheNGV1 cooling flow rate is in fact one of the most influential parameters while the total pressureand temperature variations have only moderate main effects. The comparison of the two plotsclearly shows how the NGV1 cooling flow main effect, that is predicted to be considerably largerfor ηH , is more dominant in the left figure than in the right.

In cases where two parameters have similar main effects, the interactions can be more pronouncedsuch as for the total pressure and total temperature variations at midspan shown in Fig. 6.7 forthe enthalpy and entropy based efficiencies. Two localized extrema appear for both efficiencies inagreement, a minimum for low midspan total temperatures and a maximum for high midspanvalues for both parameters. The non-linearity of the co-influence is apparent since the isolinesare heavily curved. The stronger the non-linearity, the more difficult it is to capture in thesurrogate model with a given number of data points and the more the approximations for the

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CO

OL

NG

V1

Ptot CEN

(a) ηH , the contours span ∆η = ±0.25%C

OO

L N

GV

1

Ptot CEN

(b) ηs, the contours span ∆η = ±0.15%

Figure 6.6: Co-influence of the of the midspan total pressure variation and the NGV1 coolingmass flow rates.

Ptot CEN

Ttot C

EN

(a) ηH , the contours span ∆η = ±0.16%

Ttot C

EN

Ptot CEN

(b) ηs, the contours span ∆η = ±0.22%

Figure 6.7: Co-influence of the of the total pressure variation and total temperature variation atmidspan.

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different efficiencies show discrepancies. The contours in the top left corner in Fig.6.7 are atypical example of variations appearing at the domain extremities due to the uncertainty in theKriging interpolation.

For two parameters with opposing effects the interaction plots are not meaningful due to therelative amplification of numerical noise, analogously to the situation for the capacity.

Conclusion

The sensitivity analysis of the three efficiencies considered in this work yielded identical resultsfor the enthalpy and power based efficiencies due to the similarity of their definitions and verysimilar trends for the entropy based efficiency. Contrary to the results for the NGV1 outletcapacity, the sensitivities for the efficiencies exhibit considerable non-linearities, especially withrespect to the secondary flow rates of the first turbine stage.

While the efficiency showed very little dependence on the inlet total temperature and pressuredistributions overall, a variation at midspan of both total temperature and pressure has acomparatively strong positive main effect on the efficiencies both lead to more work transfer atthe rotor midspan region that incurs lower losses.The efficiencies showed an overall negative sensitivity to the secondary flow rates which is expectedsince these flows have a lower total temperature that decreases the overall total temperature ofthe flow and also increase the mixing losses. An exception is the ROT1 cooling flow rate thatis seen to have a weak positive main effect because it leads to an increased work output of therotor with only weak adverse effects. The isolated analysis of the turbine ignores whole enginecycle efficiency that would show a detrimental effect of the ROT1 cooling flow on the efficiency.The rimseal leakage flow rates have a considerable influence on the efficiencies due to the factthat they disrupt the flow and interact with the boundary layers and vortices at the hub wallwhile the TE slots have little impact on the overall flow structure.For the second turbine stage, the sensitivities for the different efficiencies show a very goodagreement whereas they show considerable discrepancies for the first stage which may be linkedto the non-linearity of the dependencies.

The interactions between the input parameters are more pronounced for the efficiencies than forthe NGV1 outlet capacity. While the influences of the input parameters are largely linear andadditive for the capacity, they exhibit partly non-linear interactions for the efficiencies. This is anindication of the complexity of the underlying physical processes involved in the turbine efficiency.The capacity variations in this study on the other hand are simpler and well explained.

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6.2 Rotor Tip Clearance Sensitivity

6.2 Rotor Tip Clearance SensitivityThe rotor tip clearance variation has a dramatic effect on both NGV1 outlet capacity and turbineefficiency. The results of the three computations with varied tip clearance compared to thebaseline configuration (the baseline is the design tip clearance, i.e. S/S) are summarized inFig. 6.8 (S and L refer to the small (0.33 mm) and the large (0.66 mm) clearances respectively).

L/S S/L L/L−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Tip Clearance (ROT1/ROT2)

∆[%

]

CηHηPηs

Figure 6.8: Variations of the performance parameterswith the rotor tip clearance.

Doubling the rotor tip clearance on bothturbine stages leads to roughly 1% effi-ciency loss over the entire HPT, a muchlarger single effect than the other param-eter variations investigated in this work.An increase of the tip clearance leads toa rapid increase of tip leakage flow thatdoes not participate in the work transferprocess reducing the amount of extractedwork and thus the efficiency.The variation of the tip clearance for eachstage independently gives two indications.On the one hand, the effects of tip clear-ance variation on the different rows seemto be linear and additive and on the otherhand, the efficiency is roughly twice as sen-sitive to variations of the rotor tip clear-ance on the first stage as on the secondstage. The ROT2 blades are 65% longerthan the ROT1 blades. The change fromsmall to large tip clearance corresponds toaround 1% and 0.6% of the blade heightsfor ROT1 and ROT2 respectively. Theefficiency of both stages therefore dropsby 0.6% for every 1% of the blade height that the tip clearance is increased. The results forthe efficiencies are in line with previous computations made in the preliminary design phase.The fact that the impact of increasing the first stage rotor tip clearance has roughly the samemagnitude for the capacity and the efficiencies is also in line with expectations.

The NGV1 outlet capacity depends mainly on the ROT1 tip clearance exhibiting a negligibledependence on the ROT2 clearance. This is expected given the linearity of the results suggestinga relative decoupling of the two stages. When the tip clearance of ROT1 is increased, the axialflow area and thus the amount of flow sucked though the machine increases since the NGV1 isnot choked. Once choking conditions are reached in the NGV1, its capacity becomes largelyindependent of the downstream configuration.

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6.3 Implications for the Rig MeasurementsThe input parameter ranges for the sensitivity analysis were defined using preliminary test datafrom the rig and the computed performance fluctuations are therefore representative of thevariations expected during the experimental campaign.

Tab. 6.1 and 6.2 summarize the sensitivity results in terms of standardized variations to aid thecomparison with existing and future data.

∆C per 1% W40 ∆ηH ,∆ηP per 1% W40 ∆ηs per 1% W40COOL NGV1 −0.05 % −0.09 % −0.06 %COOL ROT1 −0.25 % −0.02 % −0.04 %COOL NGV2 −0.11 % −− −−RSL ROT1 −0.22 % 0.02 % 0.05 %RSL NGV2 −0.11 % −0.11 % −0.10 %RSL ROT2 −− −0.10 % −0.10 %

Table 6.1: Summary of the main effects on the on the capacity and efficiencies in percent ofthe reference value for each input parameter relative to an absolute flow rate change of 1%W40. Negligible main effects (absolute values below 0.01% per 1% W40) are not shown andindicated with −−. The reference mass flow W40 is the expected main inlet mass flow from a2D throughflow solution.

∆C/mm TC ∆ηH ,∆ηP /mm TC ∆ηs/mm TCROT1 only 1.71 % −1.95 % −1.68 %ROT2 only 0.04 % −1.04 % −1.09 %ROT1 and ROT2 1.81 % −2.79 % −2.63 %

Table 6.2: Summary of the influence of tip clearance (TC) increase on the capacity and efficienciesextrapolated to a standard variation of 1 mm. The value for the enthalpy and power basedefficiency is the average of the two values.

The results of the analysis have several implications for the measurements on the rig. On theone hand, the influence of the total temperature and pressure fluctuations was seen to be verysmall apart from the centerline region where both total temperature and pressure increases hada positive main effect on the efficiency. On the other hand, the secondary flow rates have aconsiderable impact on both capacity and efficiency that must be taken into account.

Considering again the measured inlet profiles for the total temperature and pressure (Figs. 4.3(a)and 4.3(b)), the target temperature is perfectly matched at the centerline and the pressurefluctuations at that location are more erratic and a clear pressure bulge seems improbable duringan actual experiment. The sensitivity of the efficiency to fluctuations in this region is thereforenot expected to play a large role during efficiency measurements. Especially the considerabledrop in total temperature toward the casing, while being undesirable, is not critical in terms ofmeasurement accuracy since variations in this region are found to have no measurable impact onany of the considered performance parameters.

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6.3 Implications for the Rig Measurements

The situation is different for the flow rates at the secondary inlets that are seen to have considerableimpact on both capacity and efficiency. While the tested variations in the cooling flow rates forROT1 and NGV2 are larger than the actual variation in the rig and are therefore less critical,the cooling flow rates of the NGV1 occurring in the rig cover the entire range considered in thesensitivity analysis. Given that the cooling flow rate variations are dictated by the hardware(precluding major improvements without considerable costs) and have a large impact on theturbine performance, care must be taken in the subsequent analysis of the test data.The influence of the rimseal leakage flow rates is also considerable but a definitive assessmentof the absolute capacity and efficiency deltas they induce is pending until the actual flow ratevariations are known. In the meantime, the sensitivity results indicate that precise knowledgeof the flow rates in the secondary air system is required for a accurate prediction of turbineperformance.

The rotor tip clearance, in contrast to the other parameters considered in the sensitivity analysis,is not a fluctuating quantity. The results are therefore not a measure of the uncertainty to beconsidered when analysing the test data but are important for a comparison with preliminarydesign predictions and subsequently with the rig measurements.

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7 Concluding Remarks

7.1 Summary

The sensitivity of the performance of a 3D steady state CFD model of a high-pressure turbinetest rig to variations in the inlet boundary conditions as well as rotor tip clearance was analysedand quantified.

The steady state 3D CFD computation was set up and run for the full turbine consisting of twohigh-pressure stages and an exit duct using the hybrid meshing tool BOXER and the in-houseviscous solver HYDRA. The inlets from the representative secondary air system (rimseal leakages)as well as the trailing edge slots were included in high fidelity whereas the film cooling present onthe blades of first turbine stage was replaced by a cooling strip model to reduce the computationalcost.Due to the lack of experimental data to validate the final model, detailed studies were carried outon the first stage NGV to calibrate the cooling strip model to existing fully-featured computationsas well as to investigate the interaction between the cooling strip model and the boundary layerresolution on the blades. The resulting parameter settings and meshing recommendations werethe basis for the full turbine mesh with a total of 46 million cells.

The performance parameters considered in this work are the first stage NGV capacity measuredat the outlet plane of the NGV domain and the turbine efficiency for which three alternativedefinitions are compared. The inlet boundary conditions for which variations were considered arethe inlet total temperature and pressure distributions as well as the cooling and rimseal leakageinlet mass flow rates for each individual row. The parameter ranges were chosen to match thefluctuations measured in preliminary rig tests. The rotor tip clearance was studied for the caseof a small (design) clearance and a large clearance.

Due to the size of the final model, the sensitivity analysis was not directly carried out with thehigh-fidelity CFD model with the exception of the rotor tip clearance. For all other parameters,40 sample points covering the full input space were chosen using the optimal latin hypercubesampling method and computed with the CFD model. Subsequently, a surrogate model wascreated by fitting the Kriging model to the output data that was then directly analysed to extractthe sensitivity information. The creation of the Kriging response surface and the subsequentanalysis was carried out using the proprietary software package iSight.

The capacity as well as the turbine efficiency were found to be very dependent on the rotor tipclearance, the clearance of the first stage having the strongest impact. Compared to the designtip clearance, the large tip clearance led to a 0.6% increase in capacity and an efficiency drop of0.9%. The capacity change can be attributed to 98% to the first stage while it is responsible forroughly 60% of the efficiency drop.

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7 Concluding Remarks

The variations in the total temperature and pressure distributions were found to have a negligibleeffect on both capacity and efficiency overall. The only major exception is the midspan region,where both the total temperature and the total pressure have a positive main effect on theefficiency (0.09% change in efficiency at the limits of the parameter range). These effects are dueto the corresponding shift in the radial distribution of flow energy that is concentrated in themidspan region. This is beneficial to turbine efficiency since the rotor blades extract the flowenergy with less losses at the centerline than at hub and tip.The mass flow rates at the secondary inlets were found to have an overall strong negative maineffect on both capacity and efficiency. While the sensitivities of the capacity are all linear, theefficiency sensitivities also exhibit non-linearities especially for the first turbine stage.Rimseal leakage flow rate variations of 50% of the nominal value lead to capacity changes of0.01− 0.22% and efficiency changes of 0.02− 0.13% that are proportional to the absolute valueof the respective mass flows. The capacity changes are furthermore inversely proportional tothe distance of the inlet to the NGV1 outlet plane. Variations of the cooling flows by 20% havesimilar effects on the capacity (changes of 0.02− 0.19%). The impact of the NGV1 cooling flowrate, the largest secondary flow in absolute terms, is mitigated because the injected cooling flowis by definition included in the capacity computation at the domain outlet. The main effect inthis case is mainly due to increased mixing losses with the cooling flow rate.The effects of the cooling flow variations on the efficiency are the only case where the differentefficiency definitions showed considerable discrepancies in the computed sensitivities. Whilethere is agreement that the NGV2 trailing edge slot has negligible influence, the enthalpy basedefficiency was found to be 50% more sensitive to NGV1 cooling flow variations (0.15%) than theentropy based efficiency (0.1%).The ROT1 cooling mass flow rate was found to have a moderately weak positive main effect(0.05%) on the efficiencies. This surprising result is due to the fact that the considerable windageand pumping losses occurring during the supply of the cooling air to the rotor are not consideredin the isolated turbine analysis in which the positive impact of the mass flow increase in terms ofrotor power output outweighs the detrimental effects of increased mixing losses.

In view of the measurements on the test rig, the sensitivity quantification carried out in thiswork yields several insights. The observed variations in the inlet total temperature and pressuredistributions are not critical since the regions where large variations actually occur (hub andespecially casing wall) have little to no influence on turbine performance while the centerlineregion where variations have a large impact exhibit very little variation in the rig. The influenceof the cooling and rimseal flow rate variations is considerable. If the flow rates cannot be adjusted(in the case of the film cooling flows they are due to hardware variations), the strong sensitivitiesindicate that an exact measurement of these flow rates is crucial.

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7.2 Future work

7.2 Future workThe sensitivity study carried out in this work is a first step in understanding the flow in the testrig. Nevertheless, a complete understanding of the physical processes involved is still out of reachso that there is room for further investigations. This section points to some unsolved issues thatmay be taken up in future research.

Validation with experimental data: To this point, the CFD model used for the sensitivity studyhas only been validated using numerical data from a fully-featured computation. Althoughthe model was created following the best engineering judgement and many sources of errorand inaccuracy such as local and global grid refinement, the cooling strip model and theboundary layer resolution were studied in detail, the final comparison to experimental datais still outstanding.

Numerical grid: The grid used in this work is suspected to be the source of some of the numericaldifficulties notably the transition from structured boundary layer mesh to unstructuredmain flow mesh and the structured layers near the wall themselves. A better grid might becreated with a fully structured approach or using a different meshing algorithm for theunstructured grid.

Unsteady simulation: All simulations performed in this work were steady state due to theprohibitive computational cost of unsteady simulations. The flow in a high-pressure turbineis highly unsteady and this is one of the reasons for the difficulty to further reduce theresiduals and reach better convergence. Especially the mixing layers between the rimsealsand the main gas path are intrinsically unsteady and consistently had the highest flowresiduals of the entire domain. An unsteady simulation is very costly in comparison tosteady state but may yield more accurate results especially considering the level of accuracyexpected from the CFD solutions. The capture of unsteadiness is not an easy task andseveral methods exist ranging from URANS via LES to DNS. Whether such a simulation isreliable strongly depends on the method used.

Uncertainty quantification and sensitivity analysis: In this work, only a sensitivity analysiswas carried out. An uncertainty quantification based on experimental data at the operatingpoint would complete the picture begun by this work.The sensitivity analysis itself can be taken further by considering different types of inputparameters. Interesting candidates are total temperature and pressure variations along theannulus (this work was concerned with the radial distributions only) requiring part-annuluscomputations.

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