Aerodynamic Design of the NASA Rotor 67 for Non Uniform Inﬂow Due to Boundary Layer Ingestion Master Thesis Report

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Rotor 67 for Non Uniform Inflow Due

to Boundary Layer Ingestion

Master Thesis Report

Faculty of Aerospace Engineering

Department of Energy Technology

Done By:

Tan Yi Yun Raynold

Thesis Supervisors:

Dr Matteo Pini

Dr. Arvind Gangoli Rao

Dr. Bjorn Laumert

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Executive Summary

As the demand to improve the fuel efficiency of current commercial aircraft increases, new commer-cial airliner concepts such as the Blended Wing Body has been researched on and studied in various aspects over the years as an efficient alternative to the conventional transport configuration. One particular aspect of the Blended Wing Body is the use of the propulsive fuselage concept. In this concept, the fuselage boundary layer is ingested by the engine and this is aimed at producing benefits such as improved fuel efficiency, reduced ram drag as well as lower structural weight of the engine. During the ingestion process, the low momentum boundary layer is re-energized by the propulsion system before exiting into the atmosphere. In this way, the ingested flow does not contribute to the wake deficit and hence, the overall drag of the aircraft is reduced. Since thrust equal drag in steady and level flight, and power is equal to thrust multiplied by velocity, the reduction in drag implies a reduction in the power required to drive the vehicle.

In essence, the ingestion of the boundary layer which leads to a lower inlet stagnation pressure rep-resents a direct thermodynamic penalty. However, the momentum deficit captured by the engine represents a drag reduction to the aircraft. In this way, the propulsion system performance suffers a decrease in engine efficiency while the aircraft drag is reduced in proportion to the amount of boundary layer flow that is ingested. Therefore, a trade-off exists between the increase in aircraft drag reduction and the decrease in engine performance as more boundary layer is consumed. Another important concern is the significant flow distortion which can lead to increased vibration and fatigue of the fan and compressor blades in particular. This flow distortion is characterised by the distortion coefficient, a standard widely used in the aircraft engine industry. While it was found that the ingestion of the boundary layer can provide a decrease in fuel burn of several percentages, the benefits of boundary layer ingestion have shown to be very sensitive to the magnitude of the fan and duct losses. Hence, it is crucial that fan designers are able to design new rotor blades that are able to withstand theses flow distortion while ensuring that engine performance degradation is kept to a minimum in order to maximise the overall gain in fuel efficiency.

The main aim of this research is therefore to understand and analyse the rotor performances un-der both uniform and non uniform inflow condition. This will then provide insights into the main fluid mechanism affecting rotor performances under such conditions. As such, the early phase of this research was focused on the development of an in-house blade modeller which was then later used in the parametrisation and reconstruction of the NASA Rotor 67. Other than the development of the Blade Modeller, this research was also focused on the coupling of an open-source meshing software, SALOME to the Blade modeller which will then allow the user to achieve automated meshing needed for the design optimization process.

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perfor-mance as a whole but very few literatures were found which pry into the detailed fluid mechanism of boundary layer ingestion influencing the rotor performance. Detailed analysis of the blade to blade domain under non uniform inflow condition were not found. Therefore, the main highlight of this thesis will be on the detailed analysis of the blade to blade domain as well as the overall rotor perfor-mance under non uniform inflow condition.

The framework of this report is as follows; The first part of the report will focus on blade parametri-sation as well as the process of blade reconstruction. This is followed by the procedures and approach taken to carry out the simulations using the open source Computational Fluid Dynamics (CFD) Tools, SU2. The CFD simulations and analyses conducted in this thesis are divided into two main stages. The first stage will deal with the computational analysis of the 2D fitted profiles in order to validate the profile fitting process. The second stage will deal with the comparison of the 3D simulation results under both uniform and non-uniform inflow.

From the 2D results, it was found that the 2D fitted and Actual point coordinate sources of profile 1265 and 1865 have the closest fit in terms of blade loading, Mach/Pressure Contour as well as their stagnation pressure loss results. However, in the case of profile 2265 and 1565, it was found that the 2d fitted and surface fitted profile have the closest fit in terms of their blade loading, Mach/Pressure Contour and stagnation pressure loss results. Therefore, this result suggests that it is difficult to pinpoint a particular source of error for the blade fitting procedures. However, it can be seen that minor deviation in the leading/trailing edge position and curvature of the blade profile can bring about a significant shift in the shockwave position. It can also be concluded that the BSpline surface generation algorithm gives a better fit to the 2d fitted profiles closer to the tip radius than the hub radius. It should be pointed out that the current blade geometry was generated based on a previous version of the Blade Modeller using a different technique for the generation of BSpline surfaces as well as the flaring angle. The conclusion made from this results are based on the older version of the Blade Modeller.

The 3D simulation results composed of two parts. The first part of this result will looks into the comparison of results between the case of a uniform and non uniform inflow boundary condition. It was found that the effect of the non uniform inflow is an increase in entropy production and a decrease in isentropic efficiency. The decrease in efficiency is mainly a result of the change in inlet incidence angle as the inlet total pressure varies. Consequently, this has an effect on the blade loading, shock position as well as the trailing edge wake pattern. For the comparison of the results between the case of a rotor with casing and without casing under uniform inflow, it was found that the case of the rotor without casing exhibits a reduction in isentropic efficiency. This is mainly because of the increased tip leakage losses as compared to the case of a rotor with casing. However, it is noted that the pressure ratio over the unducted rotor is slightly higher as compared to the ducted rotor. In this simulation, the total duct length is relatively short and therefore the pressure induced by the shroud does not significantly add to the total pressure ratio.

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Acknowledgements

The thesis work presented in this report would not have been possible without the help of a number of people. Firstly, I would like to extend my greatest gratitude to my daily supervisor, Ir. Salvatore Vitale for his endless support in helping me to get started with this thesis work and also providing me with the technical support and advice throughout my thesis work. In addition, he was instrumental in motivating me and providing valuable feedback on my work.

Secondly, I would like to thank Professor Dr. Arvind G. Rao and Dr. Matteo Pini for their help in the formulation of this thesis topic and providing me the opportunity to carry out this work at TU Delft. The opportunity to perform my thesis work at the TU Delft has enabled me to learn more about the techniques used in Computational Fluid Dynamics (CFD) to obtain a series of results. In particular, I was exposed to the opportunity to learn more about SU2, SALOME and python during the course of this work.

Thirdly, I would like to thank my fellow friend and MSc student, Roel de Koning for introducing me to the Blade Modeller which he has started developing and also providing me the opportunity to work on both the developmental aspect and usage of the Blade Modeller. He was very helpful in sharing his knowledge and I am truly grateful to him for his relentless support and time invested on my thesis work.

Last but not least, I would like to thank all my fellow master students, friends and family for their patience, motivation and encouragement which have kept me going despite all the difficulties faced in recent times.

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List of Figures

2.1 BWB with podded engines . . . 4

2.2 BWB with embedded engines . . . 4

2.3 Closer look on the Embedded Engine Concept [1] . . . 5

2.4 Podded versus Embedded Engine Concept [1] . . . 6

2.5 LHS terms of Power Balance Equation [2] . . . 7

2.6 RHS terms of Power Balance Equation [2] . . . 7

2.7 Control Volume of aircraft with and without BLI configuration [3] . . . 8

2.8 Control Volume Analysis of a Engine with Boundary Layer Ingestion [4] . . . 9

2.9 Enthalpy - Entropy Diagram of a Non Ideal Brayton Cycle . . . 11

2.10 Plot of total pressure rise efficiency versus flow quantity coefficient [5] . . . 13

2.11 Comparison of the Influence of distortion pattern amplitude on the stability margin between experimental and computational results [6] . . . 14

2.12 Definitions used in characterising 2D Blade Profile . . . 15

2.13 Camberline Parameterization [7] . . . 16

2.14 Effect of w on camberline shape [7] . . . . 16

2.15 Use of two control points to adjust camberline [7] . . . 17

2.16 Blade Thickness Definition [8] . . . 17

2.17 Influence of arctangent stretching parameter on points distribution along ’u’ parameter [7] . . . 18

2.18 Construction of Blade Profile Suction/Pressure Side [7] . . . 18

2.19 Stacking of a 3D Axial Turbine Blade [8] . . . 18

2.20 End wall losses [9] . . . 20

2.21 Secondary flow losses [9] . . . 20

2.22 Tip leakage losses [9] . . . 21

2.23 Type of Inlet Distortion . . . 22

2.24 Definitions for P_av and P_av,low . . . 23

4.1 Original NASA Rotor 67 . . . 28

4.2 Sliced NASA Rotor 67 . . . 28

4.3 Sliced 2D Profiles of the NASA Rotor 67 . . . 29

4.4 Plot of Original Point Coordinates . . . 30

4.5 Plot of Re-ordered Points Coordinates . . . 30

4.6 Plot of Re-ordered Points Coordinates with sep. suction and pressure side indexes . . 31

4.7 Plot of Re-ordered Points with camberline . . . 31

4.8 Illustration of projection algorithm method . . . 31

4.9 Illustration of geometry fitting algorithm method . . . 32

4.10 Illustration of profile before optimization . . . 33

4.11 Illustration of profile after optimization . . . 33

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4.13 Illustration of full profiles . . . 34

4.14 Original NASA Rotor 67 . . . 34

4.15 Reconstructed NASA Rotor 67 . . . 34

4.16 Angle Distribution of NASA Rotor 67 . . . 35

4.17 Thickness Distribution of Suction Side NASA Rotor 67 . . . 36

4.18 Thickness Distribution of Pressure Side NASA Rotor 67 . . . 36

4.19 Thickness Distribution of Pressure Side NASA Rotor 67 . . . 37

5.1 Concept of Radial Equilibrium [10] . . . 41

5.2 Calculation procedure for profile 1865 outlet boundary condition . . . 42

5.3 Calculation procedure for profile 1265, 1565 and 2265 outlet boundary condition . . . 43

5.4 Illustration of Mesh Zone . . . 44

5.5 Overall mesh topology . . . 45

5.6 Mesh region near leading edge . . . 45

5.7 Mesh region near trailing edge . . . 45

5.8 Plot of Density Residuals versus Number of Iterations . . . 47

5.9 Comparison of Pressure contour at different domain length . . . 48

5.10 Comparison of Mach contour at different domain length . . . 48

5.11 Comparison of 1865 Mach Contour between 2D fitted and Actual Point Profile . . . . 49

5.12 Comparison of 1865 Pressure Contour between 2D fitted and Actual Point Profile . . . 49

5.13 Comparison of 1865 Mach Contour between 2D fitted and Surface fitted Profile . . . 50

5.14 Comparison of 1865 Pressure Contour between 2D fitted and Surface fitted Profile . . 50

5.15 Comparison of 1865 Mach Contour between Surface fitted Profile and Actual point profile 50 5.16 Comparison of 1865 Press. Contour between Surf. fitted Profile and Actual point profile 50 5.17 Comparison of 1865 Blade loading between 2D fitted Profile and Actual point profile . 51 5.18 Comparison of 1865 Blade loading between 2D fitted Profile and Surface fitted profile 51 5.19 Comparison of 1865 Blade loading between Surface fitted Profile and Actual point profile 51 5.20 Comparison of 1865 Blade Profiles from 3 Coordinate sources . . . 51

5.21 1865 Blade Profile leading edge zoom . . . 52

5.22 1865 Blade Profile Trailing edge zoom . . . 52

5.24 Comparison of 1265 Pressure Contour between 2D fitted and Actual Point Profile . . 52

5.25 Comparison of 1265 Mach Contour between 2D fitted and Surface fitted Profile . . . . 53

5.27 Comparison of 1265 Mach Contour between Actual point and Surface fitted Profile . . 53

5.28 Comparison of 1265 Pressure Contour between Actual point and Surface fitted Profile 53 5.29 Comparison of 1265 Blade loading between 2D fitted and Actual point profile . . . 54

5.30 Comparison of 1265 Blade loading between 2D fitted and Surface fitted profile . . . . 54

5.31 Comparison of 1265 Blade loading between Surface fitted and Actual point profile . . 54

5.32 Comparison of 1265 Blade Profiles from 3 Coordinate sources . . . 54

5.40 Comparison of 1565 Blade loading between 2D fitted and Surface Fitted profile . . . . 56

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5.44 1565 Blade Profile trailing edge zoom . . . 57

5.52 Comparison of 2265 Blade loading between 2D fitted and Surface Fitted profile . . . . 59

5.54 Comparison of 2265 Blade Profiles from 3 Coordinate sources . . . 60

5.56 2265 Blade Profile trailing edge zoom . . . 60

5.57 Enthalpy-Entropy diagram of a Stator Blade . . . 61

5.58 Area Distribution for various blade profiles . . . 62

6.1 Features of a good quality 2D mesh . . . 63

6.2 2D Mesh Generated by Salome . . . 64

6.3 Leading edge zoom on Salome 2D Mesh . . . 64

6.4 Trailing edge zoom on Salome 2D Mesh . . . 64

6.5 Boundary Layer zoom on Salome 2D Mesh . . . 64

6.6 Comparison of Profile 1265 Salome and UMG 2 Mesh Mach Contour . . . 65

6.7 Comparison of Profile 1265 Salome and UMG 2 Mesh Pressure Contour . . . 65

6.8 Comparison of Profile 1265 Salome and UMG 2 Mesh Blade Loading . . . 65

6.9 Features of a good quality 3D mesh . . . 66

6.10 Fine Discretized Mesh across the Leading/Trailing edge and the Top/Bottom Blade Surfaces . . . 66

6.11 Clipped Vertical View of NASA Rotor 67 Internal Mesh Domain . . . 67

6.12 Clipped Horizontal View of NASA Rotor 67 Internal Mesh Domain . . . 67

6.13 3D mesh of Centrifugal Rotor with tip clearance . . . 67

6.14 Clipped Vertical View of Centrifugal Rotor Internal Mesh Domain . . . 68

6.15 Clipped Cross-Section View of Centrifugal Rotor Internal Mesh Domain . . . 68

6.16 Mach Streamlines of Wing Profile . . . 68

6.17 Top view of Mach contour of Wing Domain . . . 68

6.18 Pressure Coefficient Plot of Salome Meshed Wing Slice . . . 69

6.19 SU2 Mesh Format . . . 70

7.1 Spanwise Total Pressure at Inlet/Outlet Boundary . . . 72

7.2 Spanwise Total Temperature at Inlet/Outlet Boundary . . . 72

7.3 Spanwise Static Pressure at Inlet/Outlet Boundary . . . 72

7.4 Measurement Points at Inlet and Outlet Boundary . . . 72

7.5 Total Pressure Contour at the Inlet and Outlet Plane of the Inlet Configuration . . . . 73

7.6 Velocity Contour at the Inlet and Outlet Plane of the Inlet Configuration . . . 74

7.7 Sampled Points along the Outlet Plane of inlet configuration . . . 76

7.8 Plot of total pressure versus angular position for varying radial position . . . 77

7.9 Plot of velocity versus angular position for varying radial position . . . 77

7.10 Calculation of distortion intensity at mid radius . . . 78

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7.13 Meridional View of Single Blade . . . 81

7.14 20% Span blade loading Uniform Inflow . . . 82

7.15 20% Span blade loading Non Uniform Inflow . . . 82

7.20 20% Span Uniform Inflow Blade to Blade Domain Velocity Vector . . . 83

7.21 20% Span Non Uniform Inflow Blade to Blade Domain Velocity Vector . . . 83

7.26 20% Span Uniform Inflow Blade to Blade Domain Entropy Change . . . 85

7.27 20% Span Non Uniform Inflow Blade to Blade Domain Entropy Change . . . 85

7.32 20% Span Uniform Inflow Blade to Blade Domain Static Pressure . . . 86

7.33 20% Span Non Uniform Inflow Blade to Blade Domain Static Pressure . . . 86

7.38 Visualisation of Uniform Inflow Streamlines . . . 88

7.39 Visualisation of Non Uniform Inflow Streamlines . . . 89

7.40 20% Span Blade Loading Comparison . . . 91

7.45 Circumferential Average Total Pressure Versus Normalized Span Position at Inlet . . . 93

7.46 Circumferential Average Total Pressure Versus Normalized Span Position at Outlet . . 93

7.47 Visualisation of Rotor Streamlines (with casing and Uniform Inflow) . . . 93

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List of Tables

5.1 Design and Geometry Data . . . 38

5.2 Atmospheric Data . . . 38

5.3 Calculated BC Data at tip radius . . . 40

5.4 Calculated Inlet BC Data at other radial position . . . 40

5.5 Inlet and Outlet BC Data at R = 0.1265, 0.1565, 0.1865 & 0.2265 . . . . 43

5.6 Common Parameters for Numerical Methods . . . 46

5.7 Flow Numerical Methods . . . 46

5.8 Turbulent Numerical Methods . . . 47

5.9 Comparison of Stagnation Pressure Loss Coefficient between different profile types . . 61

7.1 BWB Flight Operating Condition . . . 75

7.2 Mass flow Averaged Total Condition Across Inlet Configuration . . . 75

7.3 Comparison of Outlet Plane Total Condition vs. Aft Section Total Condition without BLI . . . 76

7.4 Comparison of Inlet Plane Total Condition vs. Aft Section Total Condition without BLI 76 7.5 Performance of the Rotor under Non Uniform Inflow Boundary Condition . . . 90

7.6 Performance of the Rotor under Uniform Inflow Boundary Condition . . . 90

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Nomenclature

(∆pc

p )i Circumferential distortion intensity

(Q

)_comp Compressor Pressure Ratio (θi)¯ Circumferential distortion extend

(Pav)i Circumferential Averaged Static Pressure

for a given ring

βin Inlet Blade Metal Angle βout Outlet Blade Metal Angle

δloss stagnation pressure loss coefficient

˙

Ea Wake Streamwise Kinetic Energy

Deposi-tion Rate ˙

Ep Wake Pressure Defect Work Rate

˙

Ev Wake Transverse Kinetic Energy

Deposi-tion Rate ˙

Ew Wave Pressure Work and Kinetic Energy

Outflow Rate

ηcomp Adiabatic Compressor Efficiency ηp Propulsive Efficiency

ηth Thermal Efficiency

ηtt Total to Total Isentropic Efficiency ηturb Adiabatic Turbine Efficiency

Q1

nD3 Flow Quantity coefficient

γ Stagger Angle

λf reevortex Free vortex constant µ Dynamic Viscosity of Air Φ_duct Inlet Duct Dissipation

Φf uselage Jet Dissipation

Φ_jet Jet Dissipation Φ_shock Shock Dissipation

Φ_total Total viscous dissipation rate Φ_vortex Vortex Dissipation

Φwake Wake Dissipation ρ Density

τs Temperature Ratio τw Wall shear stress alocal Local Speed of Sound

Cθ Absolute Tangential Velocity Cax Axial Chord Length

DA Airframe Drag

Fengine Propulsive force provided by the engine P01,rel Relative Stagnation Pressure

Pk Net Propulsor Mechanical Energy Flow

Rate

Ps2 Outlet Static Pressure

Ps Net Propulsor Shaft Power PT2 Outlet Total Pressure

Pv Net Pressure Volume Power Q1 Volumetric Flow Rate

T01,rel Relative Stagnation Temperature

U∞ Free Stream Velocity

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V0 Inlet Velocity

V2 Exit Velocity

W ˙h Potential Energy Rate

Wθ Relative Tangential Velocity xle Leading edge axial position xte Trailing edge axial position yle Leading edge tangential position yte Trailing edge tangential position

ALE Arbitrary Lagrangian Eulerian

BLI Boundary Layer Ingestion BWB Blended Wing Body

CFD Computational Fluid Dynamics FPR Fan Pressure Ratio

FW-H Ffowcs Williams - Hawking H Blade Height

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1 |

Introduction

In Europe, the Advisory Council for Aeronautical Research has established ambitious targets to sub-stantially reduce fuel consumption, C02, NOx and noise emission by 2020 under the Clean Sky project. These requirements demand a novel configuration for air vehicles, which suggest a highly integrated components for optimum overall performance. Improved propulsion system architecture are of pri-marily importance in achieving these goals [11].

According to recent estimates, a 1% reduction on fuel burn can result in savings of 1 million per engine year. Other than fuel consumption, noise also plays an important role in the commercial engine business with growing tighter regulations. Increased noise emission can result in curfew at air-ports, which increases the overall operational costs due to reduced airport capacity in a given period of time. With regards to engine design, fuel consumption, emissions, noise, drag and weight plays a vital role in the overall aircraft performance. Propulsion airframe integration (PAI) is also an important element in meeting future aviation goals.

Hybrid wing body has become increasingly popular as a result of increased aerodynamic efficiency as compared to conventional airframe design. Such a configuration blurs the division between fuselage and wings. The Blended Wing Body (BWB) is one such configuration, where the propulsion system is mounted on the upper aft surface of the fuselage. PAI on a BWB can be achieved by either pod-ding the engines or embedpod-ding them onto the aft fuselage section. While podded engines offer the advantage of ingesting uniform flow into the engine, this concept is however associated with increased structural mass, drag and consequently increased fuel consumption. Embedding the engines into the fuselage implies the ingestion of the boundary layer. This concept has shown to provide reduction in the overall fuel burn percentages as a result of reduced momentum deficit.

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2 |

Literature Review

In this chapter, a review of the most relevant literature is summarised and presented. The aim of this chapter will be to provide the reader a brief introduction of concepts that will be used throughout this entire report. The first sub-section of this chapter will touch on propulsion airframe integration. The second sub-section will present a cycle analysis on the effect of boundary layer ingestion on the thermodynamic and propulsive efficiency. The third sub-section of this report will focus on blade parametrisation , construction, loss mechanism as well as the performance characterisation of the rotor blade. Finally, the last sub-section of this chapter will touch on the capabilities and function of SU2 (Stanford University Unstructured), a CFD tool that was used throughout this research work.

2.1 Propulsion Airframe Integration (PAI)

Propulsion airframe integration (PAI) is an important element to consider when meeting modern aviation requirement of reduced emissions, noise and fuel consumption. In the case of the Blended Wing Body concept, PAI can be achieved by either podding the engines onto the aft fuselage or embedding the engine into the fuselage.

Figure 2.1: BWB with podded engines

Figure 2.2: BWB with embedded engines

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A closer look at the embedded engine concept is shown in figure 2.3. The main advantage of this concept is the lower propulsion system thrust line which implies lower balance requirement and there-fore smaller trim drag. The absence of the pylons also leads to a reduction in both the structural weight as well as the wetted surface area. Finally, embedded engine configuration features shielded intakes which leads to lower noise production. Embedded configurations also give rise to size limitation which impose constraint on the selection of an optimum thermodynamic cycle for a given flight profile. Most importantly, The main drawback of the embedded engine concept is the significant flow distortion into the engine, resulting in both engine stability issues and performance loss. The elevated distortion level also leads to increased maintenance to address possible high cycle fatigue and operability with the compression system. Taken these limitations together, they have a significant impact on the engine subsystem performance. Lastly, the use of an embedded engine design leads to a more complex and integrated design for the aft fuselage section.

Figure 2.3: Closer look on the Embedded Engine Concept [1]

A mathematical analysis of an embedded engine configuration with BLI versus podded engine by A.Plas [1] has shown that an embedded engine configuration requires lesser power as compared to the podded engine case with no BLI. This analysis is made on the assumption that the total weight of either configuration is the same. This is mainly because the inlet velocity into the propulsion system for the embedded engine concept is lesser than the podded engine concept.

In the podded engine concept, the flow entering the engine has a freestream velocity of U∞. The engine then accelerates the flow to a velocity of Uj. For an ideally expanded nozzle, the momentum

excess created by the jet balances the momentum deficit created by the drag of the airframe (D_A).

Fengine = ˙m(Uj− U∞) = ˙m(U∞− Uw) = DA (2.1)

The rate of (kinetic) energy added to the flow by the engine is given as follows:

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Figure 2.4: Podded versus Embedded Engine Concept [1]

then used as the inlet velocity into the engine. The propulsive force provided by the engine balances the momentum deficit caused by the airframe drag:

Fengine= ˙m(Uj− Uw) = ˙m(U∞− Uw) = DA (2.3)

The mechanical power required to produce this force in the presence of boundary layer ingestion is then formulated as such:

PBLI = ˙ m 2(U 2 j − Uw2) = ˙ m 2(Uj− Uw)(Uj + Uw) = Fengine 2 (Uw+ U∞) (2.4) Comparing equation 2.2 and 2.4, it can be seen that the embedded engine concept with BLI requires lesser mechanical power since Uw < Uj.

Power Balance Method

Another way of investigating the benefit of boundary layer ingestion is to use the power balance method developed by Drela [2]. This method is based on the conservation of total power. The rate of energy entering into the system equates the rate of energy leaving the system. Specifically, the sum of the total mechanical power supply is equal to the sum of the total viscous dissipation rate, Φtotal

and the mechanical energy flow rate out of the control volume (CV), ˙ε. The total mechanical power

supply is made up of three terms; the net propulsor shaft power P_s, the net pressure volume power,

Pv power and the net propulsor mechanical energy flow rate into the CV, Pk. Mathematically, the

power balance equation can be expressed as follows:

Ps+ Pv+ Pk = ˙ε + Φtotal (2.5)

The first term on the R.H.S represents the energy flow out of the control volume while the second term represents the energy loss within the control volume. The mechanical energy outflow rate, ˙ε can

be decomposed into 5 main terms; the potential energy rate, W ˙h, the wake streamwise kinetic energy deposition rate, ˙Ea, the wake transverse kinetic energy deposition rate, ˙Ev, the wake pressure defect

work rate, ˙Ep and the wave pressure work and kinetic outflow rate, ˙Ew. Taken all these together, the

power balance equation can be expressed as follows:

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An illustration of the control volume representing all the different terms in equation 2.6 is shown in figure 2.5 and 2.6. The definition of each terms given in equation 2.6 is presented below:

Figure 2.5: LHS terms of Power Balance Equation [2]

Figure 2.6: RHS terms of Power Balance Equation [2]

• P_s The net propulsor shaft power is defined as the integrated force multiplied by velocity on all moving surface

• P_v The net pressure volume power is a volumetric mechanical power provided by the fluid expanding against atmospheric pressure.

• P_k The net propulsor mechanical flow rate is the net pressure work and kinetic energy flow rate across the SB. This is to account for power sources whose moving blade is not covered by SB.

• W ˙h The potential energy rate is the power consumption needed to increase the aircraft’s potential energy and become a power source during descent.

• ˙Ea The wake streamwise kinetic energy deposition rate term is the rate of streamwise kinetic

energy being deposited in the flow out of the control volume through the treffez plane (see figure 2.6), also known as the outlet plane of the CV.

• ˙Ev The wake transverse kinetic energy deposition rate term is the rate of transverse kinetic

energy being deposited in the flow out of the control volume through the treffez plane (see figure 2.6).

• ˙Ep The wake pressure defect work rate is the rate of pressure work done of the fluid crossing the

treffez plane at some pressure P different from the ambient P∞.

• ˙Ew The wave pressure work and kinetic energy outflow rate is the rate of pressure work and

kinetic energy deposition crossing the side cylinder.

• Φ The viscous dissipation rate is the rate at which the kinetic energy of the flow is converted into heat inside the control volume

For a simplified aircraft configuration, the total dissipation comes from the following sources; the jet dissipation Φ_jet, the wake dissipation Φ_wake, the fuselage dissipation Φ_{f uselage} and the vortex dissipation Φvortex. This is illustrated in figure 2.7. In other more complex cases, the shock dissipation

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Figure 2.7: Control Volume of aircraft with and without BLI configuration [3]

In actual situation, all form of kinetic energy outflow are converted into dissipations downstream of the aircraft. Using this fact, the dissipation term in the power balance equation can be separated into various sources as follows:

Ps+ Pk=

W ˙h + ( ˙Ep+ Φvortex)+

+ ( ˙Ew+ Φwave) + (Φf use− Pv,f use)

+ ( ˙Ea,wake+ ˙Ep,wake+ Φwake− Pv,wake)

+ ( ˙Ea,jet+ ˙Ep,jet+ Φjet− Pv,jet)

(2.7)

Defining the mechanical energy loss as follows:

Φ∗ =XE +˙ XΦ −XPV (2.8)

The power balance equation can be simplified as follows:

Ps+ Pk = W ˙h + (Φ∗)vortex+ (Φ∗)wave+ (Φ∗)f use+ (Φ∗)wake+ (Φ∗)jet (2.9)

Considering that the flight configuration of both system in figure 2.7 is level (therefore, the flight path angle is equal to zero); W ˙h = W V∞sinγ = 0 and that no components are moving relative to the control volume, P_s= 0. The power balance equation of a non BLI configuration can then be expressed as shown in equation 2.10:

Pk = (Φ∗)vortex+ (Φ∗)wave+ (Φ∗)f use+ (Φ∗)wake+ (Φ∗)jet (2.10)

As for the BLI configuration, the net propulsive power is the sum of the mechanical flow power without the jet dissipation, (P_K−(Φ∗)jet). The (Φ∗)wake is reduced by a factor, fBLI as a result of the ingested

wake by the engine. The remaining dissipation term from the fuselage, wave pressure as well as the vortex source are approximately the same for both the BLI and non BLI concept. The power balance equation of an BLI configuration can be expressed as shown in equation 2.11:

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2.2 Cycle Analysis of the Influence of BLI

In this section, the influence of boundary layer ingestion on the size of the propulsion system, ther-modynamic as well as propulsive efficiency will be discussed. It should be pointed out that the exact inflow condition is dependent on the aircraft geometry, placement of the propulsor as well as the flight condition. The concept of inlet recovery ratio will be introduced as it is an indicator of the amount of boundary layer ingested.

Figure 2.8: Control Volume Analysis of a Engine with Boundary Layer Ingestion [4]

Y

loss

= PT1

PT0

(2.12) A greater ingestion of the boundary layer implies a lower inlet recovery ratio. In order to understand the influence of boundary layer ingestion on engine design and cycle performance, the following math-ematical relations between inlet recovery ratio and related design parameters were derived below. The thrust produced by an engine is given by the following expression:

T = ˙m(Vexit− Vinlet) = ˙m(V2− V0) (2.13) Assuming incompressible flow, the bernouli equation can be used to calculate the exit velocity as follows: PT2 = Ps,2+ ρV2 2 2 (2.14) V2 = s 2 ρPT2 − Ps,2 (2.15)

Defining the fan pressure ratio as:

F P R = PT2

PT1

(2.16) The exit velocity can be expressed as:

V2=

s

2

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Substituting the expression for the inlet recovery ratio into the exit velocity: V2 = s 2 ρF P R ∗ Y loss ∗P_T₀ − Ps,2 (2.18)

The total thrust produced can then be expressed as follows:

T = ˙m ∗ ( s 2 ρF P R ∗ Y loss ∗PT0 − Ps− V0) (2.19)

Defining the mass flow as:

˙

m = ρApV1 =

ρπD2_pV1

4 (2.20)

The total thrust produced can finally be expressed as follows:

T = ρπD 2 pV1 4 ∗ ( s 2 ρF P R ∗ Y loss ∗P_T₀ − P_s,2− V₀) (2.21)

Considering that the total thrust produced has to be constant in steady and level flight, the following relation can be observed:

• As the inlet recovery ratio decreases, the fan diameter has to increase in order to produce the same thrust. This implies an increase in the structural mass of the propulsion system.

• As the inlet recovery ratio decreases, the exit velocity also reduces which implies a reduction in the specific thrust.

From this relation, it can be said that an embedded propulsion system with lower pressure recovery tends to suffer from a lower specific thrust and therefore require a larger fan diameter to achieve the same level of thrust. As such, more horsepower is required in order to drive the same fan pressure ratio with greater airflow. This additional power has to come from a larger core airflow and hence increased fuel flow for the same combustor exit temperature. Therefore, the result of a reduced inlet recovery ratio is a larger core to power the propulsion system.

Boundary layer ingestion also has an influence on both the thermal and propulsive efficiency. The propulsive efficiency is related to the level of wasted kinetic energy trailed by the wake of the propulsion system. It is a measure of how effective the net work output is converted to thrust power.

ηp =

P ropulsiveP ower

Rateof P rod.of P ropulsiveKE =

T V∞ 1

2m(V˙ j2− V∞2)

(2.22) Substituting equation 2.13 into equation 2.22,

ηp = ˙ m(Vj− V∞)V∞ 1 2m(V˙ j− V∞)(Vj+ V∞) = 2 1 + Vj V∞ (2.23)

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From equation 2.24, it can be seen that a reduction in the inlet recovery ratio leads to a reduction in the exit velocity and consequently an increase in the overall propulsive efficiency. Strictly speaking, equation 2.23 is not valid for engine configurations that incorporate boundary layer ingestion. This is because the incoming inlet velocity is not equal to the free-stream velocity but to a different lower incoming velocity. Denoting the inlet velocity into the engine as V_in, the propulsive efficiency of an engine configuration using BLI can be re-expressed as follows:

(ηp)BLI = ˙ m(Vj − Vin)V∞ 1 2m(V˙ j− Vin)(Vj + Vin) (2.25) (η_p)_BLI = 2(Vj− Vin)V∞ (Vj − Vin)(Vj+ Vin) = 2V∞ (Vj+ Vin) = 2 Vj V∞ + Vin V∞ (2.26)

Substituting the expression for the exit velocity, the propulsive efficiency can then be expressed as follows: (ηp)BLI = 2 q 2 ρF P R∗ Q loss∗PT0−Ps,2 V∞ + Vin V∞ (2.27)

In equation 2.27, a similar conclusion can be reached as in equation 2.24. However, it is noted that the increase in propulsive efficiency is greater as compared to equation 2.24 as more boundary layer is ingested since both terms in the denominator of equation 2.27 reduces.

The thermal efficiency is a measure of how effective the thermodynamic cycle converts the ther-mal energy to net work output. It is a ratio of the work output to the heat input. Considering the illustration of a non ideal brayton cycle given below, the thermal efficiency of a jet engine can be expressed as follows:

Figure 2.9: Enthalpy - Entropy Diagram of a Non Ideal Brayton Cycle

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The adiabatic turbine and compressor efficiency can be expressed as follows: ηturb = h4− h5 h4− h5s = T4− T5 T4− T5s (2.29) ηcomp = h3s− h0 h3− h0 = T3s− T0 T3− T0 (2.30) Using equation 2.29 and 2.30, T5 and T3 can be derived respectively as follows:

T4− T5 = ηturbT4(1 − T5s T4 ) = ηturbT4(1 − ( Y ) k−1 k turb) (2.31) T5 = T4− ηturbT4(1 − T5s T4 ) = T4− ηturbT4(1 − ( Y ) k−1 k turb) (2.32) T3− T0 = T3s− T0 ηcomp = T0( T3s T0 − 1) ηcomp = T0(( Q ) k−1 k comp− 1) ηcomp (2.33) T3= T0+ T3s− T0 ηcomp = T₀+T0( T3s T0 − 1) ηcomp = T₀+T0(( Q₎ k−1 k comp− 1) ηcomp (2.34) Substituting expressions for T5 and T3 into equation 2.28 and using the following assumption for the isentropic pressure ratio and temperature ratio.

(Y)comp= 1 (Q )_turb = Y (2.35) τs = T3s T0 = (Y))k−1k (2.36)

As such, the thermal efficiency of a non ideal brayton cycle can be expressed as follows:

ηth= 1 − T4− ηturbT4(1 − (Q) k−1 k turb− T0 T4− T0(1 +_η_comp1 ((Q) k−1 k comp− 1)) (2.37)

By having a common denominator for all the terms in equation 2.37 and then perform the following operation:

• Multiply the equation by ηcomp

T0 on both the numerator and denominator terms • Factorise the numerator term and denominator term by (1 − 1

τs) and ηcomp respectively

Then the thermal efficiency of the non ideal brayton cycle can be re-expressed as follows:

ηth = (1 −_τ1 s)( T4 T0ηturbηcomp− τs) ηcomp(T_T4₀ − 1) − τs+ 1 (2.38) The influence of BLI on cycle performance is expressed directly via a reduction in the adiabatic compressor efficiency, η_cwith the following relation:

(ηcomp)BLI = ηBLI ∗ ηcomp (2.39)

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ηBLI =

1 − τ_s(k−1_k )(1 − (Q

)_loss)

τs− 1

(2.40) Substituting equation 2.39 into equation 2.38 results in an expression which relates the thermal effi-ciency of a non ideal Brayton cycle to the inlet recovery ratio. It can be seen that the inlet recovery ratio has a strong negative effect on the overall thermal efficiency. It should be pointed out that the loss in thermal efficiency is more significant than the gain in propulsive efficiency [4]. As a result, the overall impact is a reduction in the overall efficiency of the propulsion system.

A related experiment which was conducted by NASA on the influence of wall boundary layer on the performance of an axial flow fan rotor [5] had shown that fan peak efficiency η_H reduces as much as 2.5 % when a 1₈inch spoiler was used. At the peak efficiency flow quantity coefficient Q1

nD3, the measured displacement boundary layer thickness was found to be 0.069 inch and 0.060 inch for the inner and outer surface respectively. The flow quantity coefficient is defined as ratio of the volumet-ric flow rate Q₁ to the product of the rotation speed n multiplied by the cube of the diameter D. At the low blade loadings,(that is at high value of Q1

nD3), the thickest boundary layer reduced the efficiency approximately 8 %. It was concluded that the overall loss in efficiency may possibly be re-duced by decreasing the blade pitch angle in the boundary layer to conform to the upstream velocity profile. Essentially, the increase in the boundary layer thickness leads to a reduction of the axial velocity, resulting in an increase in the flow incidence angle. Therefore, the blade pitch angle should be reduce to better match the flow direction. The results of this experiment is illustrated in figure 2.10.

Figure 2.10: Plot of total pressure rise efficiency versus flow quantity coefficient [5]

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the total pressure, W is the circumferential velocity component and δ is the ratio of the compressor entry total pressure to the standard sea level static pressure. It can be seen that the stability margin of all the compressors tested decreases as the distortion amplitude increases.

Figure 2.11: Comparison of the Influence of distortion pattern amplitude on the stability margin between experimental and computational results [6]

Finally, the overall influence of boundary layer ingestion on the engine design and cycle performance can be summarised as presented below. An increase in ingestion of the fuselage boundary layer into the engine leads to the following outcome:

• Reduced specific thrust

• Reduced thrust to weight ratio (since more mass flow is required to achieve the same thrust level)

• Increased fan diameter • Increased core size

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2.3 Blade Parameterization, Construction and Performance

Char-acterization

After introducing the concept of boundary layer ingestion and analysing the influence of BLI on engine design and cycle performance, this section of the report will present a brief introduction on the parametrisation and construction of axial blades (used in the Blade Modeller) as well as the parameters used to characterise the performance of turbomachinery cascade.

2.3.1 2D and 3D Axial Blade Parameterization and Construction

The parameterisation of a complex 3D axial blade such as the NASA Rotor 67 starts from the parametrisation of a 2D Profile. After the construction of 2D profiles at various span positions, the profiles are then stacked together to form a 3D blade shape. Depending on the type of stacking distribution, the blade can be shaped to have lean, sweep or both characteristics. An illustration of the commonly used definition for blade profile geometry is shown in figure 2.12.

Figure 2.12: Definitions used in characterising 2D Blade Profile

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the performance of the geometry. Most design of shapes rely on curves. In particular, Beizer and BSpline curves are most suited for shape parameterization [8]. These curves have simple mathematical formulation and their characteristics are strongly coupled to the underlying polygon of control points [8].

Figure 2.13: Camberline Parameterization [7] Figure 2.14: Effect of w on camberline shape [7] The parameterization of a 2D Blade profile begins with the definition of the camberline. This is illustrated in figure 2.13. The camberline can be described by the following parameters; the inlet blade metal angle βin, the exit blade metal angle βout, the axial chord length Cax and finally the

stagger angle γ. The leading edge position is first specified by the user. Using the axial chord length and the stagger angle, the trailing edge position can be derived. The intersection of the inlet and outlet line tangential to the inlet and outlet blade metal angle defines the position of the mid control point. For this camberline, the points located at the LE, TE as well as the middle position defines the control points of the beizer curve. Mathematically, the position of the trailing edge point can be derived as follows:

xte = xle+ Cax (2.41)

yte = yle− Caxtan γ (2.42)

By adjusting the weight w of the mid control point, the desired curvature of the beizer curve can be obtained. This is illustrated in figure 2.14. The other possibility to construct the camberline curve in the Blade Modeller is to use four control points instead. In this method, the position of the two mid control points are controlled by the t1 and t2 parameter which moves the control points linearly along the inner and outer line tangential to the inlet and outlet blade metal angle.

Mathematically, the location of these mid control points can be calculated as shown in equation 2.43 [7].

xp1= xle+ t1(xmid− xle) (2.43a) yp1= yle+ t1(ymid− yle) (2.43b) xp2= xmid+ (1 − t2)(xte− xmid) (2.43c) yp2= xmid+ (1 − t2)(yte− ymid) (2.43d)

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Figure 2.15: Use of two control points to adjust camberline [7]

Figure 2.16: Blade Thickness Definition [8]

curve. The thickness distribution is specified by providing a list of points for which a 2nd degree BSpline curve can be constructed. It should be pointed out that these points specifically control the thickness and not the curvature of the profile. The x values of these points represent a normalized coordinate system ranging from 0 to 1, where 0 and 1 represent the LE and TE position respectively. The y values would then represent the thickness. Optional parameters such as the number of points that control the thickness as well as a thickness factor is included in the blade modeller. The thickness factor is a parameter which scales the thickness value of all the distributed points along the camberline. The distribution of these points along the camberline can be specified by using an arctangent or simply an equispaced function. The arctangent method enables more points to be specified near the leading and trailing edge position. In addition, a stretching factor parameter is available which en-ables the user to control the actual spacing distribution of the points. Mathematically, the arctangent relation between a point distributed along the normalized axial coordinate and the ’u’ parameter is expressed as follows:

u = arctan (bx) (2.44)

Where ’b’ is the stretching parameter. A larger value of the stretching factor results in more points being distributed near the leading and trailing edge position. The variation of the points distributed along the ’u’ parameter with different stretching factor is illustrated in figure 2.17. Once the distri-bution of the points have been determined, the distributed points would then offset perpendicularly from the camberline with reference to the thickness value specified earlier. This process is illustrated in figure 2.18.

For both the suction and pressure side curves, a B-Spline curve is used to represent the thickness distribution. The distributed points along the curve represent the thickness value used to shape the pressure and suction side curve. Therefore, by controlling the control points of the B-Spine curve, the user will be able to directly control the thickness distribution of either the suction or pressure side. Finally, the last aspect to consider is the blade profile’s leading and trailing edge shape. Within the blade modeller, the edge can be constructed by using either the radius values of the B-Spline curve (see figure 2.18) or a circular arc. The circular arc is constructed using an additional B-Spline curve based on three control points. In order to obtain the 2D blade profile, the circular arc is then connected to both the suction and pressure side curves taking G1 continuity into account.

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Figure 2.17: Influence of arctangent stretching parameter on points distribution along

’u’ parameter [7]

Figure 2.18: Construction of Blade Profile Suc-tion/Pressure Side [7]

construction of a B-Spline surface along the span-wise direction. Within the blade modeller, the blade can be either stacked through the center of gravity of the mid control profile or through the leading edge of each control profile. During the generation of the B spline surfaces for the suction/pressure side as well as the LE/TE circular arc surfaces, the total number of control points along each control profile as well as the degree of all the spanwise B-Spline curves has to be kept constant. Some other examples of blade stacking can be found in the paper of Tom Verstraete [8]. In his paper, the stacking process of an axial turbine is being demonstrated. This is illustrated in figure 2.19

Figure 2.19: Stacking of a 3D Axial Turbine Blade [8]

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After the blade profiles are stacked accordingly, the next step is to generate B-Spline surfaces for the pressure/suction side as well as the LE/TE edges. The generation of the B-Spline surfaces de-pends on a number of parameters. They are the degree, multiplicities as well as the knots. They control the continuity of the span-wise surface as well as the accuracy with which the surface approxi-mates the individual control profile. A selection of higher degree leads to a higher continuity but lower accuracy. Intermediate profiles are constructed based on interpolation using the generated span-wise distribution of the various design parameters (blade inlet/outlet metal angles, stagger angle, leading edge position, axial chord, thickness distribution, etc).

For any typical turbomachinery blade, the hub and tip control profiles may be flared. Flaring is defined as the inclination of the blade profile with respect to a certain reference axis. In the case of the NASA Rotor 67, the control profiles at the hub and tip location are flared with respect to the axial direction (x-axis). Flaring is controlled in the Blade Modeller using a 2D meridional channel.

2.3.2 Performance characterisation of Turbomachinery Cascade

Firstly, a short description of the various loss mechanisms and flow phenomenon that initiate these losses in turbomachinery cascade will be explained. Then, the parameters used to characterize the performance of turbomachinery cascade will be described in detail.

Loss Mechanism

A review of the most common type of losses are presented below:

• Profile Losses: The profile loss consists of a combination of the wall shear stress losses and the trailing edge mixing losses. Since both losses can hardly be separated in experimental results, they are often combined under the term known as the profile losses. Wall shear stress losses originate in the boundary layer. Trailing edge mixing losses occur when the boundary layer mixes with the free stream. These losses can be minimised by increasing the surface quality and decreasing the trailing edge radius. However, there are manufacturing limitations to both solutions.

• Shock Losses: Shock losses can occur especially in fans that operate in transonic flow field. It contributes greatly to the overall entropy rise and has the same order of magnitude of losses as that of the profile loss.

• EndWall/Secondary Losses Secondary losses are generated by the build up of the endwall boundary layer, secondary flow as well as the tip clearance. Secondary flow can be described as flow that flows at right angle to the intended primary flow. They are mostly caused by a pressure gradient that is generated between the pressure and suction side of two adjacent blades within a flow channel. This pressure gradient is a function of the blade loading present on each blade. A positive pressure gradient gives rise to an increase in the boundary layer thickness. Secondary flow generates vibration, noise and flutter because of the unsteady pressure field between the blades and the rotor-stator interaction.

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Other than these major classification of losses in turbomachinery, it is prudent to look into the various mechanisms that initiate these losses. The two main flow phenomenon concerned are separation and flow deviation.

• Separation: Separation is characterized by the flow completely detaching from the blade profile. Prior to separation, the boundary layer grows very thick. At the onset of the separation bubble, the fluid particles located at the wall has no velocity relative to the blade. In the bubble itself, the fluid is turned to flow in the opposite direction due to the adverse pressure gradient, leading to an area of recirculation. Separation can occur in various area of the blade, although it most commonly occurs on the aft suction side of the blade. Other than the usual separation at the trailing edge of the airfoil, leading edge separation is another uncommon but localised phenomenon. It can occur after an excessive high suction peak on the suction side of the airfoil or an incorrect blade metal inlet angle onto the pressure side of the airfoil. In the case of the suction side, the flow tends to reattach mainly because of the strong acceleration on the suction side. On the pressure side, such a separation will lead to an extensive re-circulation zone. Separation in transonic fan is promoted by the shock boundary layer interaction due to high pressure gradient across the shockwave. Detailed analysis reveals a significant growth of the boundary layer in the shock region. If the shock is strong enough, separation can occur.

• Flow Deviation: Flow deviation represents the discrepancies in angle between the inlet/outlet flow and its respective blade metal angle. However, deviation is not easy to predict by analytical expression. As the incidence angle increases, the flow over the blade deteriorates until the point of stall. It is possible to achieve un-disturbed flow over the blade with zero degree incidences when operating at design rotation speed.

An illustration of some of the loss mechanisms described earlier is shown in figure 2.20, 2.21 and 2.22.

Figure 2.20: End wall losses [9] Figure 2.21: Secondary flow losses [9] After a discussion on the various loss mechanism, the next part of this subsection will describe the various parameters that characterise the performance of turbomachinery cascade. Other than that, the distortion coefficient parameter that is used to characterised flow distortion will be explained as well.

Characterization of Turbomachinery Performance

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Figure 2.22: Tip leakage losses [9]

noted that the last terms in the denominator can be either the static pressure at the inlet or outlet respectively. δloss= P01,rel− P02,rel P01,rel− P2 (2.45a) δloss= P01,rel− P02,rel P01,rel− P1 (2.45b)

The bar sign that appears above each quantities implies that mass averaged values have been used for both the inlet and outlet condition. The total to total isentropic efficiency of the fan (NASA Rotor 67) can be defined as follows:

(ηtt)f an=

h02ss− h01

h02− h01

(2.46) Where ss defines the total condition for which the process is isentropic. h01 represents the total condition at the inlet of the rotor, h₀₂ represents the total condition at the outlet of the rotor. h_02ss represents the isentropic total condition at the outlet of the rotor. Since the kinetic energy of the flow after the rotor is being used, the total to total efficiency is used instead of the total to static efficiency. Another useful measure of the rotor efficiency is to compute the entropy change across the rotor. The entropy change across the rotor can be derived starting from the second law of thermodynamics:

T ds = dh − dp

ρ (2.47)

Considering the process of a perfect gas passing through the rotor at constant relative stagnation enthalpy (h₀₁_rel = h₀₂_rel), the second law of thermodynamics can be expressed as follows:

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Where the negative sign in T ds disappears since the relative stagnation pressure is decreasing through the rotor. It should be pointed out that since the rotation speed is constant, a constant relative stagnation enthalpy implies a constant rothalpy across the rotor. Substituting equation 2.45 and using the ideal gas law relation, the entropy change across the rotor can be expressed as follows:

∆s_rotor= p01,rel− p02,rel

ρ01,relT01,rel = R(p01,rel− p02,rel) P01,rel = Rδloss(P01,rel− P1) P01,rel (2.50) ∆s_rotor= Rδ_loss(1 − P1 P01,rel ) (2.51) Inlet Distortion

Inlet total pressure distortion can be classified as either circumferential or radial with respect to the machine axis. Circumferential distortions are considered to be more critical than radial distortions as it produces a disturbance normal to the airfoil motion, resulting in a change in the angle of attack and stall margin. An illustration of both type of distortion is shown in figure 2.23.

Figure 2.23: Type of Inlet Distortion

In order to assess the circumferential type inlet distortion, the distortion analysis methodology de-scribed by the Society of Automative Engineers document ARP 142019 was used. The concept of the circumferential distortion intensity as well as the circumferential distortion extent are being being introduced. The circumferential distortion extend for a given ring, (θi)¯, is the angular distance in

degrees for which the local total pressure falls below the ring average total pressure, where i refers to a single ring along the contour of the circle (See figure 2.24).

(θi)¯= θ2i− θ1i (2.52)

The circumferential distortion intensity, (∆pc

p )i, which quantifies the amount of total pressure variation

around the ring is computed as follows:

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(P_av)_i= 1 360 Z 360 0 P (θ)idθ = 1 N ∗ dθ n=N X n=1 P (θ)idθ (2.54)

N is the total number of sampling points taken over a 360◦ sector and dθ is the angular size of each sampling point. (P_av,low)i = 1 (θ_i)¯ Z θ2,i θ1,i P (θ)idθ = 1 (θ_i)¯ N2dθ X N1dθ P (θ)idθ (2.55) N1 is the starting index position where the total pressure falls below the (Pav)i and N2 is the index position where the total pressure rises above the (Pav)ivalue. A low total pressure region occurs when

the local total pressure P (θ)_i lies below the ring average total pressure, P_av,i. If two low total pressure regions occur within 25◦ of each other, it is treated as a single once-per-revolution low total pressure region. If two low total pressure regions are separated by more than 25◦, the circumferential distortion intensity is taken to be the maximum of the two. An illustration of the various definitions presented above is shown in figure 2.24.

Figure 2.24: Definitions for P_av and P_av,low

2.4 Capabilities and Function of SU

2

In this section, a brief introduction on the capabilities and function of the Stanford University Un-structured (SU2) CFD Code will be described. This is a tool that was used throughout this research work. SU2 is a computational analysis and design software that is being developed to solve complex multi-physics analysis and optimization task using arbitrary unstructured meshes. Fundamentally, it is able to discretize and solve problems described by Partial Differential Equation (PDE) as well as PDE constrained optimization problems such as optimal shape design.

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with an adjoint solver that computes the gradient of the response. Solutions from the adjoint solver are then used to compute the functional driven mesh adaptation techniques [13]. In SU2_{, both the} continuous and discrete adjoint equations have been implemented. In the continuous approach, the adjoint equations are derived from the governing partial differential equation and then subsequently discretized. Wheras in the discrete approach, the adjoint equations are directly derived from the discretized governing equation[13].

The SU2 suite is composed of several analysis modules as described below [13]:

• SU2 _{CFD: The main Partial Differential Equation solution module, which is primarily an Euler}

or Reynolds Navier Stoke Solver, also includes the adjoint equations for many of the supported governing equation system.

• SU2 DDC: The Domain Decomposition Code is used for computation which involves multiple

processors. The specified volumetric grid is partitioned for use with several other core tools in parallel.

• SU2 _{MAC: The Mesh Adaptation Code is used to refine unstructured computational meshes}

so as to improve the accuracy of the predictions. Based on the analysis of converged flow, adjoint or linearized solution, grid adaptation is applied to strategically refine the mesh about the key flow features.

• SU2 _{GPC: The Gradient Projection Code is used in the calculation of sensitivities for the}

purpose of optimization and uncertainty quantification. It uses the surface sensitivities computed using SU2 CFD, the flow solution and the definition of the geometric design variable to evaluate the derivative of a particular function.

• SU2 _{MDC: The Mesh Deformation Code is used to perturb an existing unstructured volume}

mesh to a new surface geometry. It computes the geometric deformation of a surface within the computation mesh and the surrounding volumetric grid.

• SU2 _{PBC: The Periodic Boundary Code is used for the solutions of PDE with periodic}

bound-ary condition. A new mesh containing the proper communication structure between the periodic faces is generated.

• SU2 _{SMC: The Sliding Mesh Code enables solution on meshes that slide past each other. Ghost}

cells in the computation domain are created for performing simulations with sliding surfaces. A new multi zone mesh containing the proper communication structure between the sliding interfaces is generated.

SU2 is composed of several groups of python scripts which enable the coupling of the various modules for performing complex activities including design optimization and adaptive grid refinement.

• High Fidelity Analysis Script: These scripts are designed to enhance the flexibility of the SU2 framework. They simplify the execution of parallel tasks, grid adaptation or the interface with other softwares.

• Optimal Shape Design Script: These scripts are designed to automate the shape design process

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Brief Description of Modelling Equations in SU2

Several form of the Reynold Average Navier Stokes (RANS) equation have been implemented in SU2_. They include the compressible, incompressible, Arbitrary Lagrangian-Eulerian,etc. As a subset of the RANS equation, both the laminar Navier Stokes as well as the inviscid Euler equations are also available by disabling turbulence modelling and removing viscosity respectively. In the case of turbu-lence modelling, both the Spalart-Allmaras (S-A) model as well as the Menter Shear Stress Transport (SST) model have been implemented. The Menter SST turbulence model is a two equation model that blends the traditional k − ω and k − model. For the modelling of laminar to turbulence transition, the γ −Reθt transition model which is an adaptation of the model of Langtry and Menter [14] have

been implemented. This model is an correlation based model that augments the Spalart-Allmaras turbulence model with two equations. For simulations that involve steady rotation of aerodynamic bodies, it is advantageous to transform the system of flow equation into a reference frame (a.k.a ro-tating frame) that rotates with the body of interest [13]. As such, a flow field that is unsteady when viewed from the inertial frame can be solved in a steady manner without the need for grid motion. This is achieved by applying a transformation of the RANS equation from the inertial reference frame to the rotating frame. In the case that the rotation of the aerodynamic bodies includes unsteady flow phenomenon, solutions which includes grid motion is necessary. Therefore, the Arbitrary Lagrangian-Eulerian (ALE) formulation is also implemented in SU2.

For wave related equations, both the Ffowcs Williams - Hawking (FW-H) equation and the linear elasticity equation was implemented for aeroacoustic and linear elastic small deformation problems respectively. Other physical models such as the heat equation, Gauss’s law as well as the plasma equation have also been implemented in SU2. The Gauss law is part of the Maxwell equations and is used to relate the distribution of electric charge to the resulting electric field [13]. The plasma flow equation is tightly coupled with the gauss law for the calculation of electric field generated by any local separation of charge[13].