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Master thesis and internship[BR]- Master's Thesis : Loss Sources and

Magnitudes Breakdown in an Axial Low Pressure Compressor Blade Row[BR]-

Internship (linked to master's thesis)

Auteur : Crutzen, Gilles

Promoteur(s) : Hillewaert, Koen

Faculté : Faculté des Sciences appliquées

Diplôme : Master en ingénieur civil en aérospatiale, à finalité spécialisée en "aerospace engineering"

Année académique : 2019-2020

URI/URL : http://hdl.handle.net/2268.2/10177

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Master Thesis

Loss Sources and MagnitudesBreakdown in an Axial Low

Pressure Compressor Blade Row

Thesis submitted to the Applied Science Faculty of Liège University in partialfulfilment of the requirements for the degree of Master in Aerospace Engineering.

Internship supervisorMARICHAL Yves, PhD

Faculty advisorHILLEWAERT Koen, PhD

CRUTZEN Gilles

Liège, Belgium

August 2020

Abstract

The transition to more environmentally friendly and economically efficient aircraft en-gines involves a reduction of aerodynamic loss inside compressors. On the one hand,a qualitative breakdown enables to identify, understand and separate the different losssources. On the other hand, a quantitative breakdown can be used for design iterationloop and trade-off study.

A decomposition of loss in terms of sources or mechanisms is chosen based on existingpapers in the literature. The decomposition of loss sources is used as basis for the qual-itative breakdown. Each source is related to existing models for the computation of thecorresponding loss magnitude. Focusing on two-dimensional loss through a single rowwith a very basic blade geometry and with a subsonic inlet mach number, the existingmodels are compared to each other. Profile boundary layer loss, wake mixing loss andprofile shock loss models are computed and related to the overall reference. The referenceloss is computed from Computational Fluid Dynamics. The results are used to performthe quantitative breakdown. The total pressure loss corresponding to each loss sourceis plotted for different inlet mach numbers. The corresponding loss percentage for eachsource is also computed.

This work is expected to be extended to a general tool for preliminary design stages ofcompressors. The main limitations come from the assumptions, the simulations condi-tions and the error due to the interaction between the wake integration and the passageshocks. The perspectives for future works are the use of more representative blade geome-tries and flow conditions while also expanding the scope of the work to three-dimensionalloss computation.

Keywords - loss breakdown, loss models, profile loss, shock loss, axial compressor.

Acknowledgement

This research topic was proposed by Safran Aero Booster in the scope of an internshiplinked to a master thesis at Liège University. I am grateful for their trust and I thankthem for their very warm welcome. I would like to thank my supervisor Yves Marichaland my colleagues from Safran for their advice and their support during my internship.Despite the unusual sanitary and economical conditions, I thank the managers for thevalidation of my work.

My gratitude goes also to Liège University and its academic community for these fiveyears of formation as an aerospace engineer. I acquired a technical background and de-veloped a set of soft skills that will be a basis for my future career. Special thanks tomy master thesis faculty advisor Koen Hillewaert for the support during the final part ofmy work. In particular, he helped me to redefine the scope of my work after the sanitarycomplications and he supervised my work.

I would like to thank my fellow aerodynamicist interns at Safran for the good time wehad there and for the after-work activities. Many thanks to my friends for these greatfive years at Liège University and thanks to my co-teleworking partners which made thework during the lock-down much more pleasant. Special thanks to François for the goodtime working together.

Finally, I thank my parents and family for the support during these five years and for thepropitious environment. I would never have gone this far without them. Special thanksto my brother Lucas for his support and forbearance during the lock-down period.

Table of contents

1 Introduction 1

2 Axial compressors 32.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Gas turbine cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Performance characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Velocity triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Multistage design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Identification of loss 103.1 Total pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Exergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Boundary layer characteristics 144.1 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Integral thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Displacement thickness . . . . . . . . . . . . . . . . . . . . . . . . 164.2.2 Momentum thickness . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.3 Energy thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2.4 Pseudo-energy thickness . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Shape factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Decomposition of loss 195.1 Existing breakdown review . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.1.1 Howell (1945) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.1.2 Jennions (1993) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.1.3 Cumpsty (1989) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.1.4 Denton (1993) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.1.5 Arntz (2019) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Practical loss breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Loss mechanisms 246.1 Viscous dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.1.1 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.1.2 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.1.3 Vortex friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.2 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 External loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.3.1 Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.3.2 Windage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.4 Other loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.4.1 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.4.2 Unsteadiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7 Loss sources 287.1 Blade profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7.1.1 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.1.2 Wake mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.1.3 Shock loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.2 Endwall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3 Tip clearance leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.4 Cavity leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.5 Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Loss computation models 398.1 Profile boundary layer and wake mixing . . . . . . . . . . . . . . . . . . . 39

8.1.1 Lieblein’s incompressible model . . . . . . . . . . . . . . . . . . . . 408.1.2 Lieblein’s incompressible model with mixing . . . . . . . . . . . . . . 428.1.3 König’s compressible model . . . . . . . . . . . . . . . . . . . . . . 428.1.4 Stewart’s compressible model with mixing . . . . . . . . . . . . . . . 438.1.5 Wake mixing by conservation . . . . . . . . . . . . . . . . . . . . . 44

8.2 Profile shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468.2.1 Koch and Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478.2.2 Freeman and Cumpsty . . . . . . . . . . . . . . . . . . . . . . . . . 478.2.3 Manfredi and Fontaneto . . . . . . . . . . . . . . . . . . . . . . . . 48

8.3 Endwall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.3.1 Koch and Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.3.2 Hanley, Manfredi and Fontaneto . . . . . . . . . . . . . . . . . . . . 50

8.4 Tip clearance leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518.4.1 Rains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518.4.2 Storer and Cumpsty . . . . . . . . . . . . . . . . . . . . . . . . . . 528.4.3 Lakshminarayana . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.4.4 Manfredi and Fontaneto . . . . . . . . . . . . . . . . . . . . . . . . 54

8.5 Cavity leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.5.1 Wellborn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.5.2 Denton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

9 Two-dimensional loss computation 579.1 Computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.2 Mesh and simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.3 Wake integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

10 Validation and discussion 6210.1 Mesh resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.2 CFD reference loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6310.3 Blade profile boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . 6510.4 Blade wake mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6710.5 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

11 Two-dimensional loss breakdown 7411.1 Chosen models review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

11.2 Breakdown graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7611.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7711.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

12 Conclusion 79

13 Perspectives 80

A Three-dimensional version of mixing loss computation and rotor extension 82

List of figures

1 Illustration of a turbojet engine [5]. . . . . . . . . . . . . . . . . . . . . . 42 Illustration of a turbofan engine [6]. . . . . . . . . . . . . . . . . . . . . . 43 Illustration of a rotor and stator blade row in the axial-radial plane. . . . 54 Temperature-entropy diagram of the Brayton’s cycle. Adapted from [7]. . 65 Compressor performance characteristics map [8]. . . . . . . . . . . . . . . 76 Compressor rotor velocity triangle [9]. . . . . . . . . . . . . . . . . . . . . 87 Multistage compressor. Adapted from [12]. . . . . . . . . . . . . . . . . . 98 Compression process on enthalpy-entropy diagram. Adapted from [14]. . 119 Laminar boundary layer. Adapted from [21]. . . . . . . . . . . . . . . . . 1510 Transition from laminar to turbulent boundary layer. Adapted from [22]. 1511 Illustration of blade profile, secondary flows and endwall loss from Howell

[25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2012 Practical loss breakdown with corresponding loss mechanisms and loss

sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2313 Conservation of the tangential component across an oblique shock. Adapted

from [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614 Total pressure ratio across normal shock in terms of the upstream mach

number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2715 Development of surface boundary layers and wake about blade section.

Adapted from [29]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2916 Velocity profile before and after complete mixing of the wake. Adapted

from [29]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017 Shock structure in the blade-to-blade plane of a typical stator blade row

with subsonic inlet. Adapted from [10]. . . . . . . . . . . . . . . . . . . . 3118 Supersonic pocket shock model [32]. . . . . . . . . . . . . . . . . . . . . . 3219 Supersonic two-shock model for sharp leading edge blades [32]. . . . . . . 3220 Shock structure in the blade-to-blade plane of a rotor blade row with su-

personic inlet. Adapted from [30]. . . . . . . . . . . . . . . . . . . . . . . 3321 Boundary layer and passage generation at endwall. Adapted from [37]. . 3522 Rotor blade hub stall region. Adapted from [10] and [38]. . . . . . . . . . 3623 Rotor tip gap leakage flow and vortex generation. Adapted from [14]. . . 3724 Illustration of the rotating stall regions. Part-span stall case. Adapted

from [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3825 Cavity flow and injection mixing. Adapted from [41]. . . . . . . . . . . . 3926 Illustration of the mixing computation methodology. . . . . . . . . . . . . 4527 Control volume for Freeman and Cumpsty ’s model. Adapted from [48]. . 4728 Illustration of Storer and Cumpsty mixing simple model [56]. . . . . . . . 5329 Hub shrouded stator blade design used by Wellborn [41]. . . . . . . . . . 5530 Mixing of a low mass flow rate stream with the main stream [14]. . . . . 5631 Control volume of the two-dimensional simulation. . . . . . . . . . . . . . 5832 Mesh of the control volume near the blade section using GMSH. . . . . . . 5933 Integration plane and integration line at the blade trailing edge. . . . . . 6034 Velocity profile in the wake just after the trailing edge. . . . . . . . . . . 6135 Mach number profile in the wake just after the trailing edge. . . . . . . . 6136 Density profile in the wake just after the trailing edge. . . . . . . . . . . 62

37 Overall reference loss from CFD. . . . . . . . . . . . . . . . . . . . . . . 6338 Shocks at different inlet mach numbers. Solidity = 0.5. . . . . . . . . . . 6439 Shocks at different inlet mach numbers. Solidity = 1. . . . . . . . . . . . 6440 Illustration of Lieblein’s approximation for the definition of the total pres-

sure loss coefficient with respect to the exact energy-based definition. . . 6541 Blade profile boundary layer loss models comparison with respect to the

overall CFD loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6642 Blade profile boundary layer loss models relative errors with respect to the

overall CFD loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6743 Lieblein’s models version with and without mixing loss. Solidity = 0.5. . 6744 Mixing loss results from Lieblein and from conservation methodology. So-

lidity = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6845 Influence of solidity on mixing loss computed with conservation methodology. 6846 Comparison between König ’s model with and without additional mixing

loss from conservation methodology. Solidity = 0.5. . . . . . . . . . . . . 6947 Blade profile boundary layer and wake mixing loss models comparison with

respect to the overall CFD loss. . . . . . . . . . . . . . . . . . . . . . . . 7048 Blade profile boundary layer and wake mixing loss models relative errors

with respect to the overall CFD loss. . . . . . . . . . . . . . . . . . . . . 7149 Influence of solidity on maximum suction side mach number. . . . . . . . 7250 Koch and Smith’s model for shock loss and the influence of solidity. . . . 7251 Blade profile boundary layer, wake mixing and shock loss models compar-

ison with respect to the overall CFD loss. . . . . . . . . . . . . . . . . . . 7352 Blade profile boundary layer, wake mixing and shock loss models relative

errors with respect to the overall CFD loss. . . . . . . . . . . . . . . . . . 7453 Summarise of all König ’s extensions. . . . . . . . . . . . . . . . . . . . . 7554 Summarise of all König ’s extensions relative error. . . . . . . . . . . . . . 7655 Absolute total pressure loss breakdown usingKönig ’s andKoch and Smith’s

loss models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7756 Relative loss breakdown using König ’s and Koch and Smith’s loss models. 77

List of tables

1 Loss breakdown according to Howell (1945) [26] [25]. Subsonic conditions. 202 Loss breakdown according to Jennions (1993) [26]. Transonic conditions. 213 Loss breakdown from Cumpsty (1989) [10]. . . . . . . . . . . . . . . . . . 214 Loss breakdown from Denton (1993) [14]. . . . . . . . . . . . . . . . . . . 225 Loss breakdown according to Arntz (2019) [18]. . . . . . . . . . . . . . . 226 Summary of profile boundary layer and wake mixing loss models. . . . . 407 Summary of profile shock loss models. . . . . . . . . . . . . . . . . . . . . 468 Summary of endwall loss models. . . . . . . . . . . . . . . . . . . . . . . 499 Summary of tip clearance leakage loss models. . . . . . . . . . . . . . . . 5110 Summary of cavity leakage loss models. . . . . . . . . . . . . . . . . . . . 5511 Summary of the simulation parameters specified in SU2. . . . . . . . . . . 5912 Summary of the model used for the quantitative loss breakdown. . . . . . 74

Nomenclature

β Velocity angle with respect to axialdirection

χ Blade incidence angle

δ1 Displacement thickness

δ2 Momentum thickness

δ3 Energy thickness

δ∗3 Pseudo-energy thickness

ε Exergy

η Efficiency

θ Wake momentum-thickness parame-ter

ν Kinematic viscosity

ν∗ Tangential force thickness

ω Total pressure loss coefficient

Ωs Streamtube contraction ratio

ω Mass averaged total pressure loss co-efficient

ρ Density

σ Blade solidity

ξ Entropy loss coefficient

a Speed of sound

c Chord length

Cp Specific heat capacity at constantpressure

D Diffusion factor

h Enthalpy

ha Tangential force thickness

H12 Shape factor - displacement thick-ness

H32 Shape factor - energy thickness

H∗32 Shape factor - pseudo-energy thick-ness

p Pressure

s Entropy

T Temperature

t Blade thickness

U Linear velocity of the blade

u Normal velocity component

V Velocity

v Tangential velocity component

Y Blade spacing

γ Heat capacity ratio

µ Dynamic viscosity

Ω Hub rotation speed

τ Viscous shear stress

CFD Computational Fluid Dynamics

Re Reynolds number

Subscripts:

∞ Free-stream quantity

1 Station upstream of blade row or upstream of the shock

2 Station just downstream of blade row or downstream of the shock

3 Station after complete mixing occurs

is Isentropic state/process

LE Leading edge

p Polytropic state/process

s Static state

t Total state

TE Trailing edge

x Axial direction

y Azimuthal direction

z Radial direction

Superscripts:

- Mass-averaged quantity

1 Introduction

Economical and ecological challenges align themselves in the aviation sector. The ulti-mate economical goal has always been the fuel consumption reduction while the ecologicalobjective is the reduction of emissions. The increase of an aircraft efficiency has an effectmore than proportional on the fuel consumption. In fact, an efficiency increase reducesthe drag and therefore the required engine thrust. Consequently, the fuel consumptionand the total weight of the aircraft are reduced so that the required lift is also reduced.It follows that all aircraft design parameters are scaled down which again reduces theweight and fuel consumption of the aircraft. From an ecological point of view, a fuelconsumption reduction means an emission reduction. This explains why maximum ef-ficiencies are expected from aircraft manufacturers for each component. The demandfor more environmentally friendly aircraft engine is especially important in a context ofecological transition. This leads to significant research on performance optimisation inorder to increase the efficiency as much as possible.

A theoretical breakdown of the loss in turbomachines is very useful since it enables toseparate the loss components from each other. Understanding the physics behind a lossis very helpful when studying loss models. More accurate results enable relevant designmodifications and an increase of efficiency. A quantitative breakdown of loss for eachsource provides additional information. In particular, the link with the source gives anidea of which percentage of the loss is modified as one design parameter is modified. It istherefore really interesting to know all the loss percentages when performing a trade-offon all these design parameters.

Throughout the twentieth century, many research was made on the loss estimation inturbomachines. These estimations were often based on geometrical design parametersand experimental data. There is a main problem corresponding to this approach. Foreach significant design modification, an experimental campaign was required to tunethe correlation to the new design. Significant changes in the design are mainly bladegeometry or blade family modifications. With the appearance of Computational FluidDynamics (CFD), experimental campaigns were progressively replaced by numerical sim-ulations. CFD offers solutions in many different configurations for reduced cost and time.However, many other problems have to be considered such as consistency, stability andconvergence criteria. Even if correlations gave satisfactory results for the design of air-craft engine, there has been major modifications in the flight conditions that reduce theaccuracy of the results. The main aspect is the increase of the compressor relative inletmach number throughout the years. In addition to the general increase in rotation speedfor the whole engine, low pressure compressor have also been coupled with the fan insome engine design. The low pressure compressor angular rotation speed has thus beendrastically increased. This led to an important increase in relative inlet mach numberfor the rotor stages. The compressibility effect and the appearance of shocks required amodification of the correlation. There has always been a lack of generality in the esti-mation of loss in turbomachine. In particular, correlations are not fully consistent withthe physics behind the loss because they require a fitting with experimental results. Theempirical factors that are used are not based on loss mechanisms or on physics at all.

1

With the appearance of CFD came also the access to the whole computational domain.It gives the opportunity to base the loss estimation on these simulations. The goal wasthen not to predict the loss from empirical data but rather to model the real physicsbehind the loss. The formulation of the models is more complex than the correlationsbut since it is not based on experimental data, the models are more general and havea link with the physics. Since experimental data acquisition represents important costsand time, CFD and loss models have been used increasingly over the years.

In the scope of this thesis, a simple loss breakdown computation tool based on lossmodels is introduced. It only represents a first step as only the two-dimensional losscomponent is taken into account, a very simple blade geometry is used and only subsonicinlet mach number cases is computed and validated. Models are however presented forall loss sources. Moreover, this thesis develops the analysis behind the tool, but the im-plementation of a practical interface is not performed. The tool is expected to be usedfor all compressors geometries, during the preliminary design stages. Furthermore, thetool has the potential to be extended to a full aircraft engine and to be linked to a designiteration loop.

In order to be consistent, a precise definition of the theoretical loss breakdown is requiredbefore computing the corresponding magnitudes. It enables to have a good knowledgeof the source definition and physics behind each loss magnitude. Therefore, before goinginto the details of the loss computation, a particular attention is payed to the loss sourceand mechanism definitions.

In this thesis, the axial compressor design and characteristics are first introduced. It givesa first insight of the physics behind the following of the analysis. Then, some backgroundabout the method of identification of the loss is developed and one approach is chosen.The boundary layer characteristics that are used in models are introduced and explained.The existing theoretical loss breakdowns are enumerated and described. Based on thisliterature review, a theoretical loss decomposition is chosen for the practical loss com-putation. Using first a phenomenal approach, the loss mechanisms are described beforedeveloping the real loss source from a physical point of view. For each loss source, theexisting models for the computation of the corresponding loss magnitude are developed.The scope of the two-dimensional simulation is detailed and the different two-dimensionalmodels are compared to each other. Once a model is chosen for each two-dimensional losssource, the loss breakdown is performed. The conclusions are drawn and perspectives aresuggested for further work on the subject.

It is important to note that even if the three-dimensional loss components were not com-puted and analysed, the corresponding loss sources and models were developed anyway.The three-dimensional extension of the present loss breakdown tool could thus be basedon the work that was already made in this thesis. In addition to this, the present work islimited to the study of the two-dimensional part of blade row with a subsonic inlet machnumber. The supersonic version mainly differs from the subsonic one because of shocksappearing at the blade leading edge.

2

2 Axial compressors

Before going into the details of loss inside compressors, the axial compressor is firstintroduced. Going from the turbojet to the turbofan, the existence and the characteristicsof the low pressure compressor is explained using the general gas turbine and propulsiontheory [1] [2] [3].

2.1 Design

Axial compressors have applications in different sectors. They are mainly used in gasturbines for the propulsion of aircraft and ships as well as small-scale electricity powergeneration units. There are also many other applications in the industrial sector. Theaerospace propulsion sector is the most important one. By extending the basic turbojetengine shown in Figure 1, many different engines have been designed depending on theapplication. The main parameter that has to be taken into account is the design flightmach number. For example, military aircraft have been modified in order to be efficientat high mach number. Additional after-burn technology was mounted on the engine forspecific manoeuvre. On the contrary, civil aircraft fly subsonic and need to increase theefficiency at lower speeds. In particular, the take-off is the most critical part of the flight.In this case, the turbojet was adapted into a turbofan engine. The main modification isan extended first rotor blade row at the flow inlet called a "fan". Moreover, one part of theinlet flow does not go through the following part of the compressor and the combustionchamber but is directly ejected. The fan acts just like a propeller and increases thepropulsion efficiency at low speed and especially at take-off. Because of the large radiusdifference between the fan and the compressor, different shaft rotation speeds have tobe used. In fact, if the same rotation speed was used, the large radius would induce aincreased relative speed and it would cause blade stall near the blade tip and reduce theefficiency. The angular rotation speed of the fan is reduced to avoid this stall and thecompressor is decoupled in different parts. The decoupling of compressors in turbofan hasbeen designed in different ways depending on the engine manufacturer. One design choicewas to separate the compressor and turbine into low-pressure and high-pressure parts.Each compressor part has a given rotation speed and is linked to its corresponding turbinepart using different shafts. The fan is then linked to the low-pressure compressor that hasa lower rotation speed. An illustration of the turbofan design that was described is shownin Figure 2. Note that decoupling technologies based on gearboxes can also be found inthe industry in order to tune the rotation speed. This technology is very promising andis expected to become the future of turbofan engines [4].

3

Figure 1: Illustration of a turbojet engine [5].

Figure 2: Illustration of a turbofan engine [6].

The low-pressure stage is driven by the low-pressure turbine using the shaft. In the com-pressor, the shaft surface that is in contact with the flow is called the "hub". The angularrotation speed of the hub is noted Ω. The rotor blades are mounted on the hub and arethen rotating at the same angular rotation speed. The air flows between the rotating huband the casing across the rotor and stator blade rows. The casing is fixed and so are thestator blade rows. There is therefore a large relative speed between the rotor/hub andthe stator/casing. In order to avoid any contact between these two component groups, aconstant gap has to be maintained. Depending on the design choice, different methodsexist to separate the rotating part from the static one. A typical design is an unshroudedrotor at the casing but a shrouded stator at the hub. In this way, the rotor is held by thehub and the stator is held by the casing. For the unshrouded rotor, there is a gap betweenthe blade tip and the casing called the "rotor tip clearance". For the shrouded stator,the separation is ensured by a "cavity". The rotor and stator blade row are illustrated

4

in the axial-radial plane in Figure 3.

Again, it is reminded that the loss breakdown performed in the present work is limited tothe two-dimensional loss part of a blade row with a subsonic inlet mach number. Three-dimensional effects are the most important near the hub and casing while they are lessimportant near the region at the mean radius located just between the hub and casing.The mean radius line located at the same distance from the hub and the casing is calledthe "pitchline". Even if there will always be residual three-dimensional effect, the two-dimensional analysis can be associated to this line. In the design process of an engine,it is customary to begin with the calculation at the pitchline and then to perform theadaptation to account for three-dimensional effect. It is thus logical to begin with a two-dimensional study before considering the overall real loss breakdown with a distributionalong the span.

Figure 3: Illustration of a rotor and stator blade row in the axial-radial plane.

2.2 Gas turbine cycle

The gas turbine cycle is based on Brayton’s cycle. As shown if Figure 4 on the Brayton’sdiagram, the ideal/theoretical cycle consists in different phases:

· 1-2: Adiabatic compression

· 2-3: Constant pressure combustion

· 3-4: Adiabatic expansion

· 4-1: Return to initial/atmospheric condition

In practice, the cycle is actually different because of loss contributions and irreversibilities.These ones consist respectively for each phase in:

· 1-2’: Entropy rise during the compression

5

· 2’-3’: Loss in static pressure during the combustion

· 3’-4’: Entropy rise during the expansion

The difference between the theoretical and actual cycles is represented by the efficiencyof the gas turbine. Of course, the main goal is to increase the efficiency by reducing theentropy rise in the compressor and turbine and reduce the pressure loss in the combustionchamber. The theoretical cycle corresponds to an efficiency of 1 and is the maximumavailable performance for a given cycle. The different efficiency definitions are developedin Section 3.3.

Figure 4: Temperature-entropy diagram of the Brayton’s cycle. Adapted from [7].

2.3 Performance characteristics

The main goal of a compressor is to maximise the pressure ratio while having the max-imum efficiency to reduce the required input work. The compressor performance char-acteristics map gives a graphical relation between the pressure ratio, the mass flow rateand the efficiency. A compressor map is illustrated in Figure 5.

The surge line corresponds to the line for which the compressor blade rows stall and theflow reaches a limit in compression for a given mass flow rate. Just like an aircraft wingstalls, the blade adverse pressure gradient has become too large. In compressors, theadverse pressure gradient is especially important because the basic goal is to increasethe pressure of the flow. The natural flow direction goes however from high pressure tolow pressure. At one point, the force generated by the pressure gradient overcomes theactual flow inside the compressor and the compressor enters in a surge. This phenomenon

6

happens at high pressure ratio but at low mass flow rate.

The choke line corresponds to all the points at which the mass flow inside the compressorhas reached a maximum for a given pressure ratio. Choke of the flow occurs when one partof the compressor flow reaches sonic condition and that the flow passage is obstructed.Therefore, the mass flow rate cannot be increased until the pressure ratio is increased.A pressure ratio increase enables a more important flow passage and an increase in themass flow rate at which the flow is choked.

The available points of the compressor are located between the surge and choke lines. Foreach mass flow rate value, one corresponding pressure ratio exists for which the efficiencyis maximum. The maximum efficiency line corresponds to all the points of the compressormap providing the best efficiency. The main goal is to stay on the maximum efficiency linewhile increasing both the pressure ratio and mass flow rate. This is why the maximumefficiency line is also the design target. The design line corresponds to the minimum losscharacteristic of the compressor and is therefore the optimal performance location. Thisdesign or "operating line" is the location on the map on which the performance analysisare typically performed. In this way, it is also the line on which the loss breakdown shouldbe performed.

Figure 5: Compressor performance characteristics map [8].

2.4 Velocity triangle

The drivers for the flow compression are the rotor blade rows. Their rotation speedmodifies the flow direction and velocity. The velocity triangle modification across a rotorblade row in the blade-to-blade (axial-azimutal) plane is shown in Figure 6. The fluidenters the control volume at radius, r1, with tangential velocity Vw1 and leaves at radius,r2 with tangential velocity Vw2. The other variables are:

· V1 and V2: the absolute velocities at the inlet and outlet respectively.

7

· Vf1 and Vf2 are the axial flow velocities at the inlet and outlet respectively.

· Vw1 and Vw2 are the swirl velocities at the inlet and outlet respectively.

· Vr1 and Vr2 are the blade-relative velocities at the inlet and outlet respectively.

· U is the linear velocity of the blade.

· α is the guide vane angle and β is the blade angle.

Figure 6: Compressor rotor velocity triangle [9].

Momentum is exchanged between the rotor blade and the fluid. The work input comingfrom the turbine is transmitted by the compressor rotor to the flow. This increases theflow total pressure. Since the passage area and the velocity is limited by the surroundinghub and casing and the obstructing flow, the increase in total pressure naturally increasesthe static pressure. During the process, one part of the total pressure is lost through dif-ferent mechanisms and due to different sources that will be covered in this thesis.

In the case of a stator, the linear blade velocity U is equal to zero and there is theoreticallyno work or momentum exchange. In practice, there is however a total pressure loss. Therotor case can be studied by using a coordinate system linked to the rotating blade. Thisremoves the contribution of the rotation and enables the total pressure loss analysis.

2.5 Multistage design

The main goal for compressors is to increase the pressure ratio between the inlet and theoutlet. As shown in Figure 7, typical axial compressors are composed of several stages.Each stage is composed of a rotor blade row followed by a stator blade row. The rotorslinked to the same shaft are rotating at the same angular speed but they have different

8

radius and therefore different mach numbers at the blade tip. Additional stationary bladerows are added, mainly at inlet and outlet of the compressor, to ensure uniformity in theflow and to specify desired flow angles.The design pressure ratio has been increased throughout the year while the number ofstages has been reduced. The reduction of the number of stages enables to have morecompact compressors and reduce the weight of the engine. In the aerospace industry,pressure ratios have increased from 12.5 in the late 1950s to more than 40 in 2000. Thecorresponding number of stage was of 17 in the late 1950s and has been reduced to 10stages in 1980. These data are coming from General Electrics [10] and [11].

Figure 7: Multistage compressor. Adapted from [12].

Since the density continuously increases through the compressor, the volumetric flow ratedecreases. In order to match this decrease, the flow path cross section is reduced. Themotivation of this matching is to keep a constant distribution along the different bladerows and therefore keep approximately the same flow angle [13]. Because of the largenumber of stages and in order to optimise the interaction between the different bladerows, this matching of the stages is very important. The perfect multistage matching ofthe compressor can only be achieved at one design point. The goal will therefore be tostay as much as possible near the design for the whole compressor. However, in off-designconditions corresponding for example to a change in the compressor rotation speed, thismatching is not possible.

Different technologies are available in order to reduce the effect of a mismatch. "Dischargevalves" are used to extract one part of the flow to adapt the flow rate and even avoidchoke in the stage. "Variable stator blades" are used in order to match continuously theangle of the flow to the design point.

Moreover, the variation of the flow path cross section changes the mach number at thetip of the compressor blade. The separation of the compressor into different parts linkedto different shafts with different angular rotation speeds also contributes to an increase ofthe multistage compressor efficiency in turbofan engines. This design characteristic wasalready discussed in Section 2.1.

9

3 Identification of loss

A great deal of research focusing on loss in turbomachines has been conducted throughoutthe twentieth century. Even if most of the authors agree about the mechanisms throughwhich loss arises, there is still a great confusion concerning its very source. The same dis-agreement lies in a potential breakdown of the loss. In this section, the different methodsof characterising loss are described and analysed for practical application. Whether theyare described in a microscopic or macroscopic way, loss can be characterised using severalthermodynamic definitions. The main concepts are based on total pressure loss and totalpressure loss coefficient, efficiency drop, entropy generation or exergy loss. The followingexpressions of loss coefficient are given for compressor case. The turbine expressions canbe found in the reference paper of Denton (1993) [14].

3.1 Total pressure

Pressure is the most common thermodynamic variable used to describe loss [15]. Com-puting relative total pressure at the inlet and at the outlet gives the drop in total pressureacross the blade row and a measure of loss. A more theoretical computation is to calcu-late the difference in total pressure at the outlet but with respect to the ideal isentropictotal pressure at the same location. In general, total pressure drop as a measure of lossis used for practical reasons, when it is possible.

The total pressure corresponds to the pressure that would be obtained by an isentropic"stagnation" of the flow. Total quantities are equal to stagnation quantities if a zerogravity head is considered. This assumption is made in the scope of this thesis becausethe potential energy linked to gravity is negligible. Total pressure can be related to totaltemperature using the isentropic relations:

pt = ps

(TtTs

)γ/(γ−1)

(3.1)

TtTs

=

(1 +

γ − 1

2M2

)(3.2)

In the absence of external work, total pressure is only conserved between two stations ifthere is no loss. In this way, total pressure variation on a streamline is only due to loss.This concept is the basic idea of the definition of loss as a total pressure drop.

In the incompressible case, from simplified Bernoulli equation [16], total pressure can bewritten:

pt = ps +1

2ρV 2 (3.3)

10

The latter equation is used for the definition of non-dimensional variables (Equation 3.6).Another parameter can be defined when dealing with pressure loss, the mass weightedaverage total pressure, performed on the azimuthal coordinate:

pt =

∫pt ρV dy∫ρV dy

(3.4)

From this definition, considering a uniform inlet pressure, total pressure drop betweenstation 1 and 2 is defined as:

∆pt = pt1 − pt2 (3.5)

In order to work with non-dimensional variable, it is also convenient to define a losscoefficient based on Bernoulli (Equation 3.3). From Figure 8, recalling taking the case ofa compressor, this total pressure loss coefficient is given as:

ω2 =1− pt2

pt1

1− ps2pt1

=1− pt2

pt2∞

1− ps2pt2∞

(3.6)

It is important to note that the total pressures have to be taken in a relative frame at-tached to the component for which the loss is computed. This remark becomes of highestimportance when dealing with high speed mechanisms such as rotor compressor bladerows.

Since the pressure-based methodology requires a mass-averaged value, it is used whensuch integration is relevant and conceivable. Other methodologies are preferred whensuch an average is not possible.

Figure 8: Compression process on enthalpy-entropy diagram. Adapted from [14].

11

3.2 Entropy

In thermodynamics, entropy is a probability measurement of occurrence of a process.Low entropy configurations have low probabilities while high entropy represents statesthat have a high probability of occurrence. Moreover, the second law of thermodynamicsstates that "the total entropy of an isolated system can never decrease over time, and isconstant if and only if all processes are reversible". From these definitions, entropy can beunderstood as a variable measuring irreversibility of a process which is a measure of loss[17]. In fact, an irreversible work cannot be recovered and is therefore not useful anymore.

In practice, entropy increase is a measure of heat transfer and irreversible work generatedduring a process. Mathematically it is expressed as:

ds =dq + dWirr

T(3.7)

For an ideal gas, it can be expressed after thermodynamic development [13] [14] as apractical expression:

∆s = Cp ln (Tt2/Tt1)−R ln (pt2/pt1) (3.8)

which is related to the total pressures developed in section 3.1 as expected.

In order to work with non-dimensional variables, entropy rise can be expressed using aloss coefficient or an isentropic efficiency. Based on the enthalpy-entropy diagram shownin Figure 8, the entropy loss coefficient is expressed as:

ξ =T2∆s

ht1 − hs1(3.9)

This approach of characterising loss is based on the irreversibility of the process and isthus more relevant in a theoretical environment rather than a practical one. Moreover ithas been shown that the link with total pressure can also be made and the approachesare interconnected. Furthermore, it is a general approach that can be used in an integralform whatever the number of dimensions involved in the control volume. Finally, becausethe total entropy rise is the sum of all the component entropy variations, it enables anintegrated approach when dealing with complex mechanisms such as turbomachines.

3.3 Efficiency

Isentropic efficiency : The use of efficiency in turbomachinery is very common. Itsdefinition goes from one single element efficiency to a full engine overall efficiency. Inthe case of a compressor and considering similar pressure rise for both compressors, thegeneral expression of efficiency [10] is expressed as:

η =Work into ideal compressorWork into actual compressor

(3.10)

12

Based on this expression, there are different ways of computing efficiencies. The com-parison with an adiabatic and reversible compression is very useful. These isentropiccompressors are linked to the isentropic efficiency as:

ηis =Wis

Wactual(3.11)

A work input in an adiabatic environment means an increase in stagnation enthalpy.Neglecting the distinction between stagnation and static enthalpy leads to:

ηis =h02is − h01

h02 − h01

≈ h2is − h1

h2 − h1

(3.12)

If an ideal gas is considered, which is often the case in compressor aerodynamics, theisentropic efficiency can be written from the definition of the isentropic temperatureratio:

T2,is

T1

=

(p2

p1

)(γ−1)/γ)

(3.13)

Using the previous equation of isentropic efficiency (Equation 3.12) leads to a simpleuseful expression:

ηis ≈

(pt2pt1

)(γ−1)/γ

− 1(Tt2Tt1

)− 1

(3.14)

From Equation 3.12, using the same assumption for stagnation and static enthalpy, theisentropic efficiency can also be estimated [14] as:

ηis ≈ 1− h2 − h2is

∆h= 1− T2∆s

∆h(3.15)

Polytropic efficiency : Compressor stages from same quality but with different pressureratios have different isentropic efficiencies [10]. When studying multi-stage compressors,the polytropic efficiency is more relevant as it removes this confusion. In fact, stages withdifferent pressure ratios but with same quality have the same polytropic efficiencies. Thisefficiency is often referred as "small-stage efficiency". Mathematically, it is based on theratio between the enthalpy rise of an isentropic process and a real one. It is expressed as:

ηp =dhisdh

(3.16)

If the isentropic efficiency is assumed to be constant over a finite change in pressure, thepolytropic relation between pressure and temperature ratios is expressed as:

Tt2Tt1

=

(pt2pt1

)γ − 1

ηpγ (3.17)

13

Isolating the polytropic efficiency leads to:

ηp =γ − 1

γ

ln

(pt2pt1

)ln

(Tt2Tt1

) (3.18)

Finally, the relation between polytropic and isentropic efficiencies is given by the relation:

ηis =

(pt2pt1

)γ − 1

γ − 1

(pt2pt1

)γ − 1

ηpγ − 1

(3.19)

3.4 ExergyExergy of a system is theoretically defined as "the maximum shaft work that could bedone by the composite of the system and a specified reference environment that is assumedto be infinite, in equilibrium, and ultimately to enclose all other systems" [17]. In hisrecent paper published in 2019 [18], Arntz derived balance equations for exergy. Applyingthem to components of propulsion systems gives a measure of loss and performance. Thedefinition of exergy used in his paper is based on Adrian (1999) [19], which definesexergy as the part of energy that can be recovered for a useful purpose (e.g. shaft work).Another way of expressing exergy is by subtracting the anergy from the total mass specificenthalpy. Anergy is defined as the lost work, i.e. the work that cannot be recovered forthe required purpose. Mathematically, exergy is defined as:

ε = δhi − T∞δs (3.20)

where the term T∞δs is the expression of anergy. T∞ is the temperature at the referencestate which is usually the state of the flow far from the aerodynamic system.

Even if the approach has not been widely used in the literature yet, the methodologydraws all the attention for future work. In fact, this equation-based model is more suit-able for the integration into a computer code than the approaches presented in previoussections. Another issue in using exergy as a measurement for loss is that it does not takeleakage flow into account. Even if leakage does not represent the highest loss effect, itshould be taken into account for a general loss computation model.

4 Boundary layer characteristicsLoss in turbomachines is strongly linked to the change in momentum in the flow. In par-ticular, the flow is strongly modified in the boundary layer region and the correspondingwake. It is therefore important to define the different boundary layer characteristics thatare used in the loss computation. Note that all developments in this section are valid forwakes but the boundary layer case will be presented first. This section was based on thebook of Schlichting and Gersten, "Boundary Layer Theory" [20].

14

4.1 Boundary layer

In aerodynamics, the definition of a boundary layer is the layer of fluid in the immediatevicinity of a surface where the effects of viscosity are significant. In turbomachinery,boundary layers arise from the blades and wall surfaces. An illustration of the develop-ment of a laminar boundary layer is shown in Figure 9. V0 represents the uniform velocitybefore the solid surface. V∞ is the free-stream velocity while V (y) is the velocity profilegoing from 0 at the surface to V∞ far from the surface. The boundary layer limit is oftendefined at 99 % of the free-stream velocity.

Figure 9: Laminar boundary layer. Adapted from [21].

At some point, the smooth laminar boundary layer becomes turbulent as shown in Figure10. The turbulent boundary layer is thicker than the laminar boundary layer. The slopeof the velocity profile near the surface in the turbulent boundary layer is higher thanin the laminar boundary layer. The turbulent boundary layer is also more stable todetachment because there is more inertia of the flow near the wall. The friction at thewall has therefore less effect and does not detach the flow.

Figure 10: Transition from laminar to turbulent boundary layer. Adapted from [22].

The criterion for the beginning of the transition from laminar to turbulent is given bythe Reynolds number. It is defined as the ratio between the inertia and viscous forces ina flow. Mathematically, the Reynolds number is expressed as:

Re =V L

ν(4.1)

15

Where L is the reference length and ν is the kinematic viscosity. The range of Reynoldsnumber value for the transition from laminar to turbulent is from 2000 to 13000 dependingon the conditions.

4.2 Integral thicknesses

The basic integral characteristics are integral values computed based on the velocity anddensity profile of the boundary layer. The density profile is taken into account if thecompressibility effects are considered. The integration line is typically perpendicular tothe flow main direction. In the case of a boundary layer, the integration is performedfrom 0 to ∞ and in the case of a wake, the integration is performed from −∞ to ∞.The expressions are defined such that out of the boundary layer, the integrated termsare equal to zero. In this way, the integrand is non-zero in the boundary layer and theintegration is practically performed inside the boundary layer (i.e. from the surface to 99% of the free-stream velocity, for simplicity).

4.2.1 Displacement thickness

The first characteristic is the displacement thickness δ1. In the compressible case, it iscomputed as:

δ1 =

∫ (1− ρV

ρ∞V∞

)dy (4.2)

In the incompressible case, the displacement thickness is defined as:

δ1 =

∫ (1− V

V∞

)dy (4.3)

As already mentioned earlier, the integral is computed over the whole line but sinceV = V∞ and ρ = ρ∞ out of the velocity profile, the integral is actually limited to theboundary layer region. Physically, the displacement thickness is the normal distance toa reference plane representing the lower edge of a hypothetical inviscid fluid of uniformvelocity V∞ that has the same flow rate as occurs in the real fluid with the boundary layer.In practice, the displacement thickness represents the offset of the streamlines outside theboundary layer due to the flow blockage inside the boundary layer. In other words, itrepresents the displacement of the inviscid flow along the line at which the integrationis performed. It is a very important characteristic when studying the flow outside theboundary layer. In fact, the conservation formula will have to take into account thereduction of section in the free-stream region due to this blockage [23].

4.2.2 Momentum thickness

The second characteristic is the momentum thickness or momentum displacement thick-ness in the boundary layer. It is defined in the compressible case as:

δ2 =

∫ρV

ρ∞V∞

(1− V

V∞

)dy (4.4)

16

In the incompressible case, the displacement thickness can be written as:

δ2 =

∫V

V∞

(1− V

V∞

)dy (4.5)

Physically, the momentum thickness is the normal distance to a reference plane repre-senting the lower edge of a hypothetical inviscid fluid of uniform velocity V∞ that hasthe same momentum flow rate as occurs in the real fluid with the boundary layer. Inpractice, the momentum thickness definition corresponds to the integrated influence ofthe wall shear stress from the beginning of the development of the boundary layer untilthe location of the integration. Such a definition enables to introduce the momentumloss in the actual flow because of the presence of the boundary layer as ρv2δ2 [24]. Theconcept of momentum loss is used in many models to compute the loss related to theboundary layer.

4.2.3 Energy thickness

The third characteristic corresponds to the energy thickness or energy displacement thick-ness. For the compressible case, it is defined as:

δ3 =

∫ρV

ρ∞V∞

(1−

(V

V∞

)2)dy (4.6)

Again, the incompressible case is also given. In this case, the energy thickness is definedas:

δ3 =

∫V

V∞

(1−

(V

V∞

)2)dy (4.7)

From a physical point of view, just like the two previous characteristics, the energythickness has a signification. It is the normal distance to a reference plane representingthe lower edge of a hypothetical inviscid fluid of uniform velocity V∞ that has the sameenergy flow rate as occurs in the real fluid with the boundary layer. This characteristic isespecially interesting since the mass flux of the loss is proportional to the energy thickness[13]. In this way, it seems the most appropriate characteristic for loss computation.However, existing models were shown to prefer the momentum thickness rather than theenergy thickness for the loss computation.

4.2.4 Pseudo-energy thickness

A variant for the definition of the energy-based displacement thickness is the pseudo-energy thickness or pseudo-energy displacement thickness. It is mathematically definedin the incompressible case as:

δ∗3 =

∫ (V

V∞

)2 (1− V

V∞

)dy (4.8)

It has not the same physical meaning as the energy thickness but it is used in modelsfor the computation of profile loss. The pseudo-energy thickness is related to the energythickness and momentum thickness as:

δ∗3 = δ3 − δ2 (4.9)

17

4.3 Shape factors

Boundary layer and wake shape factors are parameters that are defined as ratios of bound-ary layer and wake characteristics. Their physical meanings depend on the characteristicsused for the definition.

The first shape factor links the displacement thickness with the momentum thickness andis defined as:

H12 =δ1

δ2

(4.10)

In the boundary layer and wake region, it physically represents the shape of the velocityprofile. In particular, it gives an information on the slope of the profile near y = 0 (i.e.near the wall for the boundary layer and in the centre of the profile for the wake). Thisinformation is used to evaluate the turbulence of the boundary layer and wake. In fact,if the shape factor is greater but close to 1, it means that the boundary layer or wakeis very turbulent. Theoretically, a shape factor of 1 represents a velocity profile with aninfinite slope and therefore a velocity profile with a right angle. It was already explainedpreviously that the turbulent velocity profile has a higher slope at the wall than thelaminar velocity profile. As the shape factor increases, the velocity profile correspondsto a laminar flows. A conventional value of laminar shape factor comes from Blasiusboundary layer with H12 = 2.59. However, because of the overlapping range of value forlaminar and turbulent shape factors, the flow regime can never be exactly determinedusing the shape factor. Moreover, there is a large range of value in which the regime canbe considered as transitional.

In a similar way, the shape factor with respect to the energy displacement can be definedas:

H32 =δ3

δ2

(4.11)

Physically, it represents the stagnation-point flow value. It gives an information on theattached or detached condition of the flow. A conventional value for separation occurs atH32 = 1.515. While there is a minimum limit for H12, there is a maximum limit for thissecond shape factor H32 = 2 corresponding again to the theoretical limit of a velocityprofile with an infinite slope at the wall.

Finally, the last shape factor corresponding to the pseudo-energy thickness is defined as:

H∗32 =δ∗3δ2

(4.12)

Shape factors are used in many models because of their non-dimensional property andtheir physical meaning. In particular, they are used in profile loss models where the wakecharacteristics are first computed and the loss is then expressed in terms of integratedthicknesses and shape factors.

Even if the loss is directly related to the energy displacement thickness, research andmodels have always been focused on the momentum displacement thickness [13]. Inthis way, it is interesting to express the energy shape factor in terms of the momentum

18

shape factor. These two shape factors defined in this section can be related by theapproximation:

H32 ≈4H12

3H12 − 1(4.13)

Using this, it is also possible to approximate the pseudo-energy thickness shape factor.First, using Equation 4.9 and dividing all the terms by δ2, the two shape factors can berelated by:

H∗32 = H32 − 1 (4.14)

Then, using Equation 4.13 leads to:

H∗32 ≈H12 + 1

3H12 − 1(4.15)

5 Decomposition of loss

Depending on the approach and the point of view of researchers, many different lossbreakdowns have been suggested in the literature. The two main approaches are eitherbased on the loss mechanisms or the loss sources. The approach based on mechanismsidentifies the different loss components as they occur in the passage while the source-basedapproach identifies the upstream elements generating these mechanisms downstream. Inthis section, the existing theoretical loss breakdowns are enumerated and shortly analysedin terms of the chosen approach. Some authors also gave a first estimation of the relativeimportance of each loss component with respect to the overall loss. Finally, a practicalbreakdown is chosen by justifying the better approach for the present application.

The loss sources and loss mechanisms are described more in details in following sectionsbased on the literature review that is performed in this section. Based on the chosentheoretical breakdown, the loss components are then computed in the final sections ofthis thesis.

5.1 Existing breakdown review

The literature is reviewed and existing theoretical breakdowns of loss inside compressorsare enumerated. The work of authors from 1945 to 2019 is chronologically presented.

5.1.1 Howell (1945)

In his paper about fluid dynamics of axial compressors, Howell [25] studied the loss in acompressor and gave a first qualitative breakdown. His research was made on a low speedcompressor. Therefore, there is no shock loss effect. He also estimates the quantitativedistribution of each loss source with respect to the overall loss. The loss breakdown fromthe point of view of Howell is given for the design point in Table 1 with the correspondingpercentages.

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Loss source % Total loss

Blade profile 39

Endwall 20

Secondary flows 41

Table 1: Loss breakdown according toHowell (1945) [26] [25]. Sub-sonic conditions.

Blade profile

39%

Endwall20%

Secondary flows

41%

For Howell, profile loss is due to the blade boundary layer, secondary loss is mainlythe vortices generated at the blade trailing edge and endwall loss is generated by theinteraction of the flow with the hub and casing. Endwall loss is also called "annulus loss"depending on the author. An illustration for these three loss components was given bythe author and is given in Figure 11.

Figure 11: Illustration of blade profile, secondary flows and endwall loss from Howell [25].

The approach of Howell is based on the loss sources and not on the loss mechanisms.His approach was later used as a reference by other authors. These authors also basedtheir loss decomposition on the sources [26]. In particular, his loss breakdown graph interms of the flow coefficient has been used as a reference in many articles (See Figure 81in [25]).

5.1.2 Jennions (1993)

An extension of Howell ’s breakdown for transonic compressor was performed by Jennions.Using numerical simulation in a transonic compressor rotor, he showed a breakdown ofloss based on entropy generation. Table 2 shows that the importance of shock at designpoint represents about one third of the total loss. This breakdown comes as an alterna-tive version of Table 1 for transonic compressors. The approach is also based on the losssources.

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Loss source % Total loss

Blade profile 35

Shocks 30

Endwall 15

Secondary flows 20

Table 2: Loss breakdown according toJennions (1993) [26]. Tran-sonic conditions.

Blade profile

35%

Shocks30%

Endwall

15%Secondary flows

20%

5.1.3 Cumpsty (1989)

"Compressor Aerodynamics" by Cumpsty [10] is a summary book covering almost everycharacteristics of compressors. In particular, Cumpsty also developed loss concepts incompressors. His results are based on a breakdown given in Table 3. Drag at solidsurface may be seen as the drag on an unstaggered and uncambered blade. Mixingloss arises both from the previous drag interaction with the free stream flow and fromvarious three-dimensional effects. Shear work dissipation is present wherever there isa velocity gradient in the flow. The definition of these loss components is too generaland the identification of loss using this breakdown is not appropriate for a practical use.Cumpsty’s breakdown is more based on loss mechanisms than loss sources. It will beshown in the next sections that an approach focusing on loss source will be preferred inthe scope of this master thesis.

Loss mechanism/source

Drag at solid surface

Mixing

Shock loss

Shear work

Table 3: Loss breakdown from Cumpsty (1989) [10].

5.1.4 Denton (1993)

Using a similar approach as those of Howell and Jennions (i.e. based on the loss sources),Denton defined a new loss breakdown in more details [14]. The most important modifi-cation is that Denton separated the "Secondary loss" into several loss sources. As it isshown in Table 4, there is only a theoretical breakdown available, but his paper gives avery good summary of loss computation methods. In fact, his breakdown is defined ina way that is compatible with the separate and independent computation. This comesfrom the decomposition based on loss sources rather than on loss mechanisms.

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Loss source

Blade profile boundary layer

Mixing

Shocks

Heat transfer

Tip leakage flow

Endwall

Table 4: Loss breakdown from Denton (1993) [14].

5.1.5 Arntz (2019)

The breakdown suggested by Arntz is based on loss mechanisms [18]. Moreover, it isrelated to his definition of loss as a waste of exergy (See section 3.4). Each loss mechanismgiven in Table 5 is related to a term in the equations of the first and second laws ofthermodynamics. As it was already told previously, the integration of this approach intoa computer code can be very interesting. However, in the case of a breakdown analysis,all the mechanisms are too interconnected and far to difficult to separate from each other.The approach of Denton based on the "sources-approach" of Howell and Jennions willbe preferred in the present work.

Loss type Loss mechanisms

Mechanical

Wakes and Jets

Vortices

Boundary pressure work

ThermalOutflow

Air displacement

Irreversibilities

Shock waves

Viscous dissipation

Mixing

Table 5: Loss breakdown according to Arntz (2019) [18].

5.2 Practical loss breakdown

In order to analyse the different loss components separately, a practical breakdown hasto be chosen. As a reminder, Denton’s approach is an extension of the one of Howelland Jennions. These three approaches are based on the loss sources whereas Arntz ’s andCumpsty ’s point of view are based on mechanisms. The approach of Denton is chosenbecause it is the most physical and source-oriented point of view with the higher level of

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details. In fact, the loss computation is much more physical when based on the source.Only considering the mechanism would be more general but much less adapted for appli-cations. Based on this approach, the chosen loss sources breakdown is shown in Figure 12.

This chosen breakdown enables to consider three main contributions: two-dimensional,three-dimensional and shocks independently. The 2D/3D separation is not fully consis-tent since there is always interaction between the three dimensions, but it is assumedthat this interaction is very low and that such a breakdown is relevant. In particular,the two mixing contributions are very difficult to separate from each other. This choiceis made for simplicity since it enables for example the use of existing two dimensionalmodels. Again, this theoretical breakdown is arbitrary and is only considered relevantin the scope of this work. More advanced breakdowns are required for further analysiswith extended loss source considerations. The different loss sources models that will bedeveloped in details are: blade profile (profile boundary layer, wake mixing and shocks),endwalls, tip clearance leakage flow and cavity flow.

This thesis focuses on two-dimensional loss breakdown. Thanks to the theoretical 2D/3Dseparation, it is thus easy to compute independently the loss from two-dimensionalsources. The shock loss that will be considered in this work is the shocks arising in theblade-to-blade plane only. This shock loss will be referred as profile shock in the following.

The three two-dimensional loss components that are computed and analysed in detailsare all arising from the presence of the blade profile. This profile loss is separated into:Blade profile boundary layer, blade wake mixing with the main flow and blade profileshock waves.

Mechanism

Source

Blade Profile Endwall Tip Clearance Leakage Cavity leakage

Boundary layer

Mixing

Vortex friction

Shocks

Leakage Flow

Profile Boundary layer

Blade wake Mixing

Blade profile Shock

Endwall Boundary layer

Secondary flow Mixing

Tip clearance vortex Mixing

Cavity flow Mixing

Endwall vortex Friction

Tip clearance vortex Friction

Cavity flow Friction

Tip clearance flow Leakage

Cavity flow Leakage

Figure 12: Practical loss breakdown with corresponding loss mechanisms and loss sources.

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6 Loss mechanisms

It was shown in the previous section that the loss breakdown could be performed followingan approach based on the source or on the mechanism generating the loss. Even if theapproach based on loss mechanisms was not chosen for the future loss computation, it isimportant to understand the main phenomenon generating loss as it occurs in the passage.The following phenomenon comes from many different loss sources that are developed inSection 7. The loss mechanisms described in this section are partially based on the lossapproach suggested by Arntz in Table 5.

6.1 Viscous dissipation

Viscous dissipation in flows arises from interaction between fluid particles. Entropy cre-ation due to viscous shear is generated by shear strain. In turbomachinery, the mainsources of viscous dissipation are boundary layers and mixing coming from wakes, vor-tices and flow injections (leakage jets). The loss generated by viscous dissipation comesfrom the irreversible conversion from kinetic energy into thermal energy.

6.1.1 Boundary layer

Velocity profiles are generated by no-slip conditions at the boundaries of solid surfaces.It creates a growing boundary layer generating viscous dissipation. Considering a two-dimensional case for simplicity, shearing between particles is generated from the differencein velocity. Shearing and dissipation are only present in viscous flows. For a laminar flow,the viscous shear stress is expressed as:

τlam = µ∂u

∂y(6.1)

which is directly proportional to the velocity gradient [16]. The higher the velocitygradient, the higher the viscous shear stress and viscous dissipation. In the turbulentcase, an additional turbulent shear stress has to be taken into account as:

τ = µ∂u

∂y− ρ u′v′ = τlam + τturb (6.2)

While laminar boundary layers are smoother and generate less shear stress, turbulentboundary layers have increased loss due to the appearance of eddies and unsteadiness.For turbulent boundary layers, the Reynolds decomposition is the separation of the flowproperties into a main part and a fluctuating part. The additional term u′v′ in theturbulent boundary layer equation is the Reynolds shear stress term. It expresses thefluctuation of the stress around the mean value. It is unknown a priori and requires aturbulence model in order to be expressed in terms of flow variables and derivatives [23].

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6.1.2 Mixing

Flow mixing in turbomachines comes from blade wakes and from different three-dimensionalsecondary flows. Depending on the technology that is used in the engine, secondary flowshave various components in non-axial directions. The fluid particles interact with theaxial free stream flow and generate viscous dissipation through shear strain. As mixingoccurs, the velocity gradients and the transverse velocity components are reduced until aperfectly uniform flow is obtained. In a uniform flow, there is no mixing between the fluidparticles and the velocity profile does not change anymore. The loss in a boundary layerarises in the exact same way as in mixing processes which is again the viscous interactionbetween particles with different velocity vectors.

Flow mixing processes are extremely difficult to predict without CFD because of theunsteadiness and complexity of the particle interactions. The entropy rise through mixingcan however be analysed through a control volume approach. It is assumed that theupstream condition is known and that the downstream condition will be uniform, recallingthat mixing will occur until the flow is perfectly uniform. The mass, momentum andenergy balance equations are then applied between the two boundaries in order to havethe uniform flow at the outlet. A full mathematical development of the methodologyis presented in Appendix A. A more systematic way of computing mixing loss is usingCFD simulations. Furthermore, depending on the source of the mixing loss, additionalcorrelations, models and formulations will be developed in this master thesis.

6.1.3 Vortex friction

Depending on the location of the source, vortex formation can generate different losscomponents. A vortex generated inside a free-steam flow will generate mixing throughshear strain and viscous dissipation. However, a vortex generated near a solid surface willhave additional features. The interaction with both the surface and the surface boundarylayer must be taken into account. The so-called endwall loss accounts for the interactionbetween the vortex generated at the blade ends and the hub/casing depending on theconfiguration (Stator/rotor). Tip clearance leakage flow is also a vortex source and thefriction of this vortex on the casing as well as the interaction with the boundary layer isan important feature of tip clearance leakage flow loss.

6.2 Shocks

From a mechanism-based point of view, a shock wave is an irreversible process duringwhich the mach number drops instantaneously. Because oblique shocks can be defined asnormal shock for the velocity component normal to the shock plane, both concepts canbe summarised as normal shocks. In fact, the tangential velocity component across anoblique shock is not modified as illustrated in Figure 13.

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Figure 13: Conservation of the tangential component across an oblique shock. Adaptedfrom [27].

Considering a perfect gas, the Rankine-Hugoniot relations [27] can be used for the calcu-lation of the variables before and after a normal shock:

M22 =

(γ − 1)M21 + 2

2γM21 + 1− γ

(6.3)

p2

p1

= 1 +2γ

γ + 1(M2

1 − 1) (6.4)

ρ2

ρ1

=(γ + 1)M2

1

(γ − 1)M21 + 2

(6.5)

T2

T1

= 1 + 2γ − 1

(γ + 1)2

(γM21 + 1)

M21

(M21 − 1) (6.6)

Following the above explanation, these relations can be used to compute directly a normalshock or they can be adapted to compute the normal component of an oblique shock using:

Mn1 =u1

a1

= M1 sin(σ) (6.7)

Mn2 =u2

a2

= M2 sin(σ − δ) (6.8)

For the application of compressors, there is much interest in the total pressure ratio linkedto the shocks. In this way, the ratio can be computed from:

pt2pt1

=

((γ + 1)M2

(γ − 1)M2 + 2

)γ/(γ−1) ((γ + 1)

2γM2 − (γ − 1)

)1/(γ−1)

(6.9)

The total pressure ratio across a normal shock for upstream mach number going from 1to 4 is given in Figure 14.

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Figure 14: Total pressure ratio across normal shock in terms of the upstream mach num-ber.

The source of shock waves is described in the part of the section dedicated to the back-tracking of the origin on the loss mechanism. Moreover, the shock structure is alsodetailed in this Section 7.1.3 from a source-based point of view.

6.3 External loss

6.3.1 Leakage

Leakage loss, without considering the other loss induced by the leakage flow, is a macro-scopic loss that is not related to viscous effects. The main goal of a compressor is to in-crease total pressure and this is done by having the highest mass flow rate going throughthe blade rows. More information can be found in Section 2. Impermeability of thestages is never obtained, mainly because of the stator-rotor relative motion. One part ofthe flow leaks through the gaps and is not compressed. Because at constant efficiency,the pressure ratio of the compressor is proportional to the mass flow rate, an increasedleakage induces a reduced compressor pressure ratio [14]. Moreover, an analogy can bemade between a leakage flow and the trailing edge flow from the tip of an aircraft wing[28]. The leakage reduces the lift force produced by the blade and represents a reductionin performance of the turbomachine.

The leakage generates additional friction, secondary flows and mixing depending on thesource of the leakage flow. The different sources of leakage in compressor are rotor tipclearance and stator cavities. These loss sources are developed in details in the sectiondedicated to the backtracking of the loss origin (Section 7). The leakage loss calculationis difficult and the use of empirical formula is often necessary.

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6.3.2 Windage

Windage loss accounts for all the friction loss contributions that have not been taken intoaccount in the other loss sources. It mainly consists in rotating disc friction loss. Forexample, the bearing system maintaining the rotating hub axis in a steady position willinduce friction. This kind of mechanical loss is however never considered when studyingaerodynamic performance in turbomachines. Nevertheless, this loss has to be consideredwhen evaluating the input and output work of a compressor and a turbine. In fact, anincrease in mechanical friction requires an increase in turbine input work. This reducesthe global efficiency of the engine.

6.4 Other loss

In addition to the main loss sources enumerated in the previous section, there are alsoother contributions which have less influence on the overall loss. This loss will not becomputed in details but is mentioned for information and for the sake of generality.

6.4.1 Heat transfer

Entropy creation by heat transfer from the turbomachine to its surrounding is usuallyvery small and compressors are generally assumed to be adiabatic [14]. The analysis ofheat transfer loss is only relevant in the presence of blade cooling systems. Since bladecooling is typically used for turbines, such a mechanism is not taken into account in thescope of this master thesis focusing on compressors.

6.4.2 Unsteadiness

As they convect through downstream rows, the vortices and wakes generated by previousloss sources mix together in an unsteady environment. It was shown that the unsteadinessof the processes increases the entropy generation [14]. In a similar way, any periodicmotion of the position of a shock also increases the amount of entropy generated. Ina more general way, secondary flows increase the unsteadiness along the span direction.This induces change in lift and drag coefficient which also lead to increased loss.

7 Loss sources

At the microscopic level, the loss is mainly arising from viscous dissipation coming fromboundary layers and mixing. However, working at a macroscopic level reveals that thereare many different loss sources interconnecting and accumulating. In this section, thedifferent loss sources generating loss mechanisms are backtracked and identified. A firstdescription and illustration is given to understand the link with the loss mechanism andthe interaction between the flows. The sources are based on the breakdown from Denton[14]. Note that this section is only dedicated to a first introduction to the differentsources, a more detailed calculation is made in following sections for chosen loss sources.

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7.1 Blade profile

Blade profile loss is related to the growing boundary layer and wake development abouta blade section and mixing with the free stream flow. Furthermore, blade profile is thesource of the shock mechanism in the blade-to-blade plane. These three loss sources aredeveloped successively.

7.1.1 Boundary layer

Figure 15 illustrates the change in velocity profile just before and after the passage of ablade section row. The boundary layer grows along the blade due to the momentum lossalong the blade. Profile loss can often be assumed to be calculated from two-dimensionalvelocity profiles. The third component of the velocity (i.e. in the blade span direction)is often very small and can be neglected. This assumption is acceptable as soon as thesection is taken far from the hub and casing and only if the compressor is working closeto the design point. It follows that when studying profile loss, an infinitely long bladeis considered. However, for highly three-dimensional compressor stages, this assumptionis not acceptable. In particular, wide chord length and high solidity blades have higherthree-dimensional effects.

Figure 15: Development of surface boundary layers and wake about blade section.Adapted from [29].

7.1.2 Wake mixing

The blade boundary layer grows into a wake after the trailing edge. The velocity profileinteracts with the main flow. As illustrated in Figure 16, the wake just after the trailingedge dissipates with the free stream flow. This mixing arises from viscous dissipation dueto the viscous shear stress generated by the velocity profile and flow viscosity. After com-plete mixing of the flow, considering a two-dimensional flow, the velocity profile becomes

29

uniform. In theory, an infinite mixing length is required in order to have a fully uniformflow after a blade row.

The blade boundary layer growth and wake mixing effects are strongly linked togetherand the loss studies are also strongly bounded. If there is no wake there will not be anymixing and if there is no wake it means there was no boundary layer and no blade profile.

Figure 16: Velocity profile before and after complete mixing of the wake. Adapted from[29].

7.1.3 Shock loss

Higher mach numbers in turbomachines have two main key advantages [30]. Firstly, ahigh relative mach number implies a higher mass flow rate per unit area and thus anincreased engine density (i.e. reduced diameter). This first advantage is especially im-portant in aircraft application where the diameter of the engine is directly proportionalto the installed drag of the engine as well as the total mass of the compressor. Reducingthe fuel consumption by reducing the drag and the mass is one of the main challenges forengine manufacturers. Secondly, it also implies higher blade rotation speed and thus anincreased blade work input. This generates higher pressure ratios, which is the main goalof compressors. These two advantages justify why transonic compressors are currentlyconsidered as a key option for future aircraft engines [31]. However, shocks also generatemuch greater loss. Consequently, studying shocks inside compressors in order to have theright trade-off between this loss and the above advantages is meaningful.

In aerodynamic flows around wings and in turbomachinery, it is customary to assumeincompressible flow when the inlet mach number is smaller than 0.3 everywhere in theflow. On the contrary, if the flow does exceed this limit somewhere in the flow, the flow isconsidered compressible. Depending on the mach distribution in the flow, the followingscenarios are possible [27]:

30

· Subsonic flow, M < 1 everywhere in the flow: No shock wave.

· Transonic flow, M ∈ [0.8; 1.2]: Shock waves appear.

· Supersonic flow, M ∈ [1; 3]: Shock waves are generally present.

· Hypersonic flow, M > 3: Huge viscous dissipation in boundary layers. Dissociationof molecules and other chemical effects.

The above scenarios are not directly related to the inlet mach number. A compressorwith an subsonic inlet mach number can go supersonic through the blade row and thusencounter shock systems. This is typically the case for compressor blade rows at highincidence. The main acceleration occurs around the blade leading edge and the flow isfaster at the suction side of the blade than at the pressure side.

Shock structures generally begin at the blade vicinity and propagate depending on theflow angles and thermodynamic variables. Depending on the blade rotation speed andinlet mach number, different shocks can appear. Across a blade row, as the inlet machnumber reaches a high enough value, there will typically be a choked flow in the bladepassage resulting in a shock. This passage shock appears for high subsonic inlet machnumber and supersonic flow conditions. The mach number increases and reaches the su-personic condition at the sonic line because of the reduced passage area across the bladerow. This shock can appear even if the inlet mach number is lower than 1. Figure 17 illus-trates the choked flow in the blade passage with the inlet and outlet subsonic conditions.As the flow goes from a supersonic condition to subsonic one, the shock is strong. It isalso in accordance with the large shock angle since strong shocks are normal to the surface.

Figure 17: Shock structure in the blade-to-blade plane of a typical stator blade row withsubsonic inlet. Adapted from [10].

In high subsonic mach number inlet flow, an additional shock structure before the passageshock can form. The so-called supersonic pocket model is illustrated in Figure 18.

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Figure 18: Supersonic pocket shock model [32].

If the inlet mach number is higher than 1, there will be at least one normal shock. Theshock structure details will then depend on the blade type. For conventional blades, thetwo-shock model is applicable. There is a detached oblique shock at the blade leadingedge and a normal shock at the passage [33]. For modern blade with sharp leading edges,the two-shock model is modified since the oblique shock at the leading edge can be ne-glected. There will be a normal shock at the passage and an oblique or a normal shockat the passage entrance [34]. This two-shock model is illustrated in Figure 19.

Figure 19: Supersonic two-shock model for sharp leading edge blades [32].

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Moreover, in the case of a typical transonic rotor blade row, the inlet mach number willbe supersonic and an expansion wave will appear at the leading edge and then a passageshock will form. An illustration of the passage shock wave and expansion waves in thecase of this transonic rotor blade row is shown in Figure 20.

Figure 20: Shock structure in the blade-to-blade plane of a rotor blade row with super-sonic inlet. Adapted from [30].

Shocks are often represented as two-dimensional in the blade-to-blade plane because ofthe symmetry of the blade along the span. In turbomachines, the blade geometry is how-ever never fully symmetrical along the span. Moreover, the inlet mach number dependson the radial location. The angular rotation speed is constant but the relative inlet machnumber depends on the position along the span. It follows that, especially for rotors,the shock structure will never be similar along the span height. Moreover, the secondaryflows due to the presence of the hub and casing as well as the rotor tip clearance andcavity also have an impact on the shock structure. Because of the geometry, the differentinlet mach number and the secondary flows, shocks are actually three-dimensional. Itis therefore not relevant anymore to study shock loss in the blade-to-blade plane, but itis still a conventional way of illustrating and studying the flow [10]. The blade-to-bladeplane is still used when computing two-dimensional loss.

Three-dimensional effects first have an impact on the inlet mach number as told above.These effects change the shock intensity, the shock structure and therefore the corre-sponding loss. Moreover, they are also influenced by other three-dimensional features

33

such as the inclination of the shock and blockage. In fact, most anomalous shock be-haviours were found to be linked to the surface obliquity of the shock structure [35].

The entropy produced by shocks in the blade-to-blade plane is due to the viscous dissipa-tion across the shock and the interaction with the blade boundary layer [32]. Moreover,the blade boundary layer also has a direct influence on the shock process. In fact, shockdoes not appear in the boundary layer region due to the subsonic flow close to the bound-ary. Moreover, the displacement thickness of the streamlines reduces the available areafor the free stream flow and the mach number increases. This strong link between profileloss and shock sometimes leads to combined analysis. For example, some shock modelsuse profile loss correlation for the identification of unknown variables.

7.2 Endwall

Endwall regions are the most important but least well understood parts of compressors[10]. In turbomachines, endwalls are the hub and the casing. They are referred as annuluswalls since they are annular geometries. The main source of endwall loss comes fromsecondary flows. Secondary flow refers at all flow features that are not intended or usefulin the compressor. In particular, endwall migration results in complex separations, vortexstructures and interaction with and between vortices that are unpredictable without usingCFD simulations. Different types of secondary flows coexist in turbomachines [36]:

· Endwall migration, due to the turning of the annulus or the hub wall boundarylayer through the cascade and leading to the passage vortex.

· Corner stall, at the corner of the annulus or the hub wall and the blade suctionsurface. This is due to interaction of the secondary flows with the boundary layeron the blade.

· Horse shoe vortex, a separation system upstream of the leading edge leading to acollection of streamwise vortex tubes covering the intersection of blade and endwall,and interacting with the endwall migration.

· Mainstream secondary flows, arising due to the trailing vortices caused by radialvariation in circulation.

Endwall secondary flows, corner stall and horse shoe vortex are attributed primarily tothe existence of the annulus and hub wall boundary layers. Consequently, the study andunderstanding of secondary flows is very useful when analysing endwall loss.

Endwall loss is the most difficult loss component to predict [14]. These predictions arebased on correlation due to lack of existing satisfactory models. In addition to the linkwith secondary flows, another difficulty is to separate the endwall loss from the rotor tipleakage loss. The later is described more in details in section 7.3 while this section focuseson other endwall loss.

There are two main sources of loss when adding the endwall at the blade ends. Firstly, aboundary layer grows and generates shear stress and loss over the whole hub and casing

34

surface. Secondly, a vortex is generated at the blade ends and additional friction anddissipation is induced (Mixing and vortex friction). The presence of the latter vortex isdue to the endwall boundary layer. The flow in the boundary layer is less resistant topressure gradients, in particular the blade-to-blade one. In fact, the low speed at the sur-face reduces the inertia of the flow and makes it less resistant to separation. At the bladeleading edge, pressure gradients along with the curved trajectory imposed by the bladechannel turning generate a force perpendicular to the flow. It results in a global motionfrom the pressure side to the suction side, called the "Endwall migration". The resultingcross-passage motion from pressure to suction side results in the formation of a passagevortex, to counter-rotating vortices within the blade channel. Figure 21 illustrates thepassage vortex generation and motion.

The strength and size of passage vortex is directly proportional to blade turning, whichmakes it more important in turbines. However, in the case of compressors, because of thepassage vortex combined with large blade-to-blade pressure gradients, low momentumflow accumulates at the suction side of the blade near the endwall. Corner vortices arecreated by the superposition of the endwall boundary layer and the boundary layer gener-ated about the blade at the suction side. All these secondary flows at hub generate a lowtotal pressure region with increased adverse pressure regions. At final extends, throughcombined boundary layer growth, this results in a "corner flow separation" and the so-called "hub stall" phenomenon. The hub stall plays a crucial role in compressor efficiencydue to its large contribution to flow blockage and flow loss. An illustration of the cor-ner flow generation and the resulting separation region and hub stall is given in Figure 22.

Figure 21: Boundary layer and passage generation at endwall. Adapted from [37].

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Figure 22: Rotor blade hub stall region. Adapted from [10] and [38].

7.3 Tip clearance leakage

The relative speed between the rotating rotor blade and the casing is very important inturbomachinery. For mechanical reasons, there needs to be a clearance between the rotortip and the casing to avoid mechanical wear of the engine. From an aerodynamic pointof view, the clearance needs to be as small as possible to reduce the leakage flow throughclearance space. However, taking a mechanical point of view, the clearance should belarge enough to enable the elongation of the rotor blade at larger rotation speed dueto centrifugal forces. The rotor tip clearance represents a trade-off between mechanicalcompatibility and aerodynamic efficiency. Research and experimentation have been madein order to obtain an optimal clearance that increases the performance of the compressor[39]. However, the optimal clearance was generally found to be smaller than the minimumlimit due to mechanical constraints.

Entropy creation due to leakage flow at rotor blade tip has different sources. First, theleakage flow in itself reduces the pressure ratio as some particles do not go throughoutthe blade row. Moreover, taking the case of unshrouded rotor blade, it is shown in Figure23 that the leakage flow generates secondary flows behind the gap. The most importantone is a vortex generated by the difference between the flow that is coming from theblade row and the leakage flow. Flow mixing occurs everywhere in the flow but its mainloss sources come from this tip leakage flow vortex. In addition to mixing, the locationof the tip leakage vortex also generates large vortex friction due to the casing and theinteraction with the casing boundary layer.

Clearance flow loss has large interaction with other loss sources. The main interactions arewith the endwall loss and the mixing loss. In fact, clearance changes the flow condition

36

at the endwall vicinity and thus has a large impact on endwall loss. As the presentwork is dedicated to perspective of blade optimisation, the largest interest is at the bladelocation. When studying the blade efficiency at the endwall, there will be four options:Stator base with cavity flow, Stator tip, Rotor base, Rotor tip with clearance flow. Eachoption has to be studied independently and in particular the interaction of the secondaryflows. Moreover, the vortex also increases the mixing with the main flow. Tip leakageflow will be shown to be a large contributor to mixing in turbomachines. All observationsmade for unshrouded blades in the scope of this thesis can be extended to shrouded blades[14].

Figure 23: Rotor tip gap leakage flow and vortex generation. Adapted from [14].

The tip clearance leakage flow loss is also linked to an instability phenomenon called"rotating stall" [10]. As mass flow rate in a compressor is reduced, the pressure ratio in-creases. At some point called the "surge point", the pressure rise reaches a maximum andthen decreases. The reduction can either be progressive or abrupt depending on the flowfeatures. In both cases, the axial velocity distribution is no more uniform/axisymmetric.The blade row is divided in two flow regions as shown in Figure 24. The cells are stalledregions forming near the casing and rotating from one blade passage to another, in anunsteady environment. The flow tries to adapt to the reduces mass flow rate but thereduction is not axisymmetric. Instead, the cells are the location where there is a largemass flow rate reduction while the unstalled region is just slightly influenced. The flowparameters influence the number, shape and size of the cells as well as their rotation.

One of the driving parameters of rotating stall features is the tip clearance [40]. As theleakage flow generates vortex components moving upstream of the blade (Figure 23),tip clearance leakage is the driver for the non-axisymmetric mass flow rate distribution.Consequently, tip leakage flow is strongly linked to rotating stall instabilities.

37

Figure 24: Illustration of the rotating stall regions. Part-span stall case. Adapted from[10].

7.4 Cavity leakage

The cavity is designed in order to separate the stator from the rotating hub. The cav-ity is designed to reduce the mass flow rate but cavity flow can not be avoided. Thereis an extraction/injection from/into the main flow as shown in Figure 25. Cavity flowgenerates different loss mechanisms. The leakage flow goes from the pressure side to thesuction side. This does not create leakage loss but the flow in itself generates two mainmechanisms.

First, the interaction between the injected flow and the main flow induces mixing. As itis shown in Figure 25, the angle between the two flows is about 90°. This large differ-ence in velocity vectors components creates large interactions between the fluid particles.Thermal mixing is also present even if it could be neglected. Both flows are not at thesame temperature because the injected part already went trough the blade row while theother one did not.

From a loss perspective, the cavity flow is also a source of friction. Because the flow insidethe cavity needs to be as small as possible, cavity designers included labyrinth seal-teethas shown in Figure 25. Just as the clearance for rotor tip leakage flow, the seal-toothclearance must be as small as possible while avoiding any collision between the hub andthe stator. Increasing the number of teeth and reducing the clearance reduces the cavityflow but also increases the friction. The cavity is designed following a trade-off betweenthe cavity flow mass rate, the minimum clearance and the friction loss.

38

Figure 25: Cavity flow and injection mixing. Adapted from [41].

7.5 Distortion

Any partial admission or non-uniform flow distribution across a blade row induces ad-ditional loss. Firstly, this asymmetry has the first effect of generating unsteadiness inthe flow. Secondly, the non-uniformity automatically generates mixing in the flow as itcreates larger velocity profiles. Even larger than velocity profiles, some distortion casescan lead to jet flows drastically increasing the mixing loss. Moreover, the interactionbetween the different compressor or turbine stages will suffer of additional loss as thedistortion increases.

8 Loss computation models

The models found in the literature for the computation of the different loss sources aredeveloped in this section. Depending on the assumption and the applicability, differentmodels are presented. In this section, all loss source models are presented. However,since this work focuses on blade profile loss, only the two-dimensional loss source modelswill be computed and validated.

8.1 Profile boundary layer and wake mixing

Blade profile boundary layer and wake mixing is the first loss source that is studied. Theseloss sources are studied together because the existing loss models are interconnected. Theloss is mainly due to the presence of the blade in the flow and arises from the boundarylayer growth at the blade surface. Blade profile boundary layer loss percentage variesfrom 35% to 39% of total loss depending on the point of view of Howell or Jennions.These important percentages justify a careful analysis of this source.

39

Incompressible and compressible models are presented. If available, different versions ofthe models with and without the consideration of the wake mixing are developed. Then,an additional model for the computation of the wake mixing independently is presentedusing conservation principles. All these models and their corresponding assumptions anddetails are given in Table 6.

Model Details

Lieblein 1 Profile boundary layer, incompressible

Lieblein 2 Profile boundary layer and wake mixing, incompressible

König Profile boundary layer, compressible

Stewart Profile boundary layer and wake mixing, compressible

Conservation Wake mixing, compressible

Table 6: Summary of profile boundary layer and wake mixing loss models.

8.1.1 Lieblein’s incompressible model

Compressibility has a strong influence on compressor efficiency. There are however tworeasons to compute profile loss using an incompressible model. Firstly, all extended mod-els are based on an incompressible model. Secondly, it enables to have an idea of theinfluence of compressibility on loss.

In 1953, Lieblein, Schwenk, and Broderick [42] published a paper in which they developeda method to estimate two-dimensional loss and limiting blade loading in axial compres-sors by using the definition of a diffusion factor. The diffusion factor is used as separationcriterion since it was shown to be proportional to the gradient of the momentum thick-ness (Equation 4.4). The definition is based on the general momentum boundary layerequation:

dδ2

dx=

τ

ρV 2− (H12 + 2)

δ2

V

dV

dx(8.1)

The basic definition of the diffusion factor is local and depends on the blade geometry.In fact using the local momentum thickness and the local velocity and velocity gradient,the local diffusion factor is defined as:

D = −δ2

V

dV

dx(8.2)

Lieblein approximated the mean diffusion factor for conventional compressor blades. Af-ter several assumptions detailed in [42] that are valid for typical geometries and aftersimplifications made in the case of commonly used blades, the diffusion factor is ex-pressed as:

D =

(1− V2

V1

)+

∆Vθ2σV1

(8.3)

Where ∆Vθ is the tangential velocity increment. It follows that the diffusion factor canbe estimated a priori for a given blade family based on the velocity triangle. In practice

40

the loss for a single blade is computed from the diffusion factor and the result is thentransformed to be converted into the loss in the full cascade.

This first paper was dedicated to the prediction of loss using correlation and experimentaldata. However, considering the approach of this master thesis, a model is preferred to acorrelation. In their paper of 1956, Lieblein and Roudebush [29] presented a theoreticalmodel for low-speed two-dimensional flow. The model is based on the above definitionof the diffusion factor and is extended from the 1953 paper. Lieblein’s profile loss modelrequires geometrical information as chord length and solidity. Moreover, the inlet ve-locity, inlet angle, inlet total pressure and outlet velocity angle are also required. Themodel also includes a version taking the mixing of the wake with the free stream flowinto account.

The global approach of Lieblein’s model is based on the integration of the wake velocityprofile for an incompressible flow across a two-dimensional cascade of compressor blade.Based on the wake characteristics defined in Section 4, the model is defined using a totalpressure loss coefficient. The different steps of the methodology are described hereafterincluding the corresponding mathematical equations. The full mathematical developmentand extended results and analysis are found in Lieblein’s paper [29].

In this second approach computing directly the loss coefficient without using the diffusionfactor explicitly, the first step consists in integrating the displacement and momentumwake characteristics at station 2, just after the trailing edge. The displacement thick-ness, momentum thickness and the form factors are defined in Section 4. Based on themomentum thickness definition, the blade chord length, the solidity and the outlet flowangle, the wake momentum-thickness parameter is defined by Lieblein as:

θ =

(δ2

c

)σ

cosβ2

(8.4)

At this point, Lieblein gave two definitions of the loss coefficient. The most accuratedefinition based on the energy thickness leads to a mass averaged total pressure losscoefficient defined as:

ω = θH32(

1− θH12

)3 (8.5)

Lieblein then used the approximation based on the displacement thickness rather thanthe energy thickness (Equation 4.13) to define his loss coefficient:

ωlieb = 2 θ

2H12

3H12 − 1(1− θH12

)3 (8.6)

The total pressure loss can be obtained from the definition of the total pressure losscoefficient, based on the inlet velocity and inlet/outlet flow angles:

∆plieb = ωlieb1

2ρV 2

1

(cosβ1

cosβ2

)2

(8.7)

41

Following these steps gives the total pressure loss for incompressible condition across ablade row, based on inlet values, geometry and wake integration. Again, this model isbased on the second approach of Lieblein. This approach does not explicitly computethe diffusion factor but directly computes the loss coefficient. In more recent papers, thismodel has been extended to take into account different additional flow features.

8.1.2 Lieblein’s incompressible model with mixing

Lieblein and Roudebush extended their two-dimensional profile loss model to accountfor the mixing loss arising from the interaction of the wake with the free stream flow.Theoretically this complete loss is measured at a station sufficiently far downstream ofthe blade row so that the flow becomes uniform again as shown in Figure 16. After amathematical development given in the appendices of their paper [29], they obtained anew formulation of the total pressure loss coefficient taking this mixing into account. Theapproach and the mathematical development are based on the flow momentum conser-vation in the axial and tangential directions, the incompressible Bernoulli equation andthe mass flow conservation.

ωlieb,∞ =2 θ

(1− θH12)2

(1 +

θ

2

[H2

12 − sin2β2

(H12 −

1

1− θH12

)2])

(8.8)

The total pressure loss after complete mixing can be obtained using the same formulationas Equation 8.7. Again the definition of the total pressure loss coefficient was made usingthe inlet flow conditions.

∆plieb,∞ = ωlieb,∞1

2ρV 2

1

(cosβ1

cosβ2

)2

(8.9)

Two different versions of the same authors are available. The first one does not takemixing into account while the second one does. It will be one of the two methods usedfor evaluating and extracting the mixing part of the loss.

8.1.3 König’s compressible model

In 1996, an extension of Lieblein’s model was developed by König, Hennecke, and Fottner[43] to account for compressibility effect. The profile loss definition is based on Lieblein’sconcept of diffusion factor. The equivalent diffusion factor taking the compressibilityeffect into account is given by:

Deq =1

Ωs

ρ2

ρ1

sinβ2

sinβ1

VmaxV1

(8.10)

The definition is based on the streamtube contraction ratio Ωs which is the velocity-density ratio. Based on König ’s definition of the equivalent diffusion factor, an extension

42

of Lieblein’s total pressure loss coefficient (Equation 8.6) can be defined using the bound-ary layer theory developed in Section 8.1.1 as [32]:

ωkon = 2 θρ1

ρ2

Ω2s

2H12

3H12 − 1(1− θH12

)3

(1 +

γ − 1

γM2

2

) 1γ − 1

(8.11)

So, the model that is used in this thesis does not compute the equivalent diffusion factorbut directly computes the loss coefficient using boundary layer thicknesses. Despite thesimplicity of König ’s model, its definition is based on physical modelling using boundarylayer theory and is proven in the literature to have a good predictability. In fact, themodel was validated over conventional blades and control diffusion airfoils [43].

8.1.4 Stewart’s compressible model with mixing

The effect of compressibility on the loss downstream of two-dimensional blade row is stud-ied by Stewart in his paper of 1955 [44]. In particular, Stewart presented a computationmethod for the profile loss after complete mixing of the flow. The method is based onthe definition of his compressible total pressure loss coefficient:

ωstew =

∫ 1

0

[1−

(ptpt∞

)] (ρV

ρ∞V∞

)d( yY

)[1−

(pspt∞

)]∫ 1

0

(ρV

ρ∞V∞

)d( yY

) (8.12)

From this definition, Stewart expressed the quantities in terms of basic boundary layercharacteristics using the compressible versions of Section 4.

The methodology for the computation of the total pressure loss after complete mixingof the flow first consists in the definition of several parameters at station 2. Then theflow parameters at station 2 are used to evaluate the flow at station 3. Finally the totalpressure ratio is evaluated at station 3. First, the A function is defined as:

A∞,2 =γ − 1

γ + 1M2∞,2 (8.13)

Using the thickness of the trailing edge tTE and the compressible boundary layer charac-teristics, the C function is given by:

C =

(1− A∞,2)γ + 1

2γ+ cos2β2 (1− δ1 − tTE − δ2) M2

∞,2

cosβ2 (1− δ1 − tTE) M∞,2(8.14)

Finally, the D function is defined as:

D = M∞,2 sinβ2

(1− δ1 − tTE − δ2

1− δ1 − tTE

)(8.15)

43

The C and D functions are known since they are evaluated at station 2, just after theblade row. In fact, this location is where the wake integration is performed. The flowvariables and distributions are known at this station. Based on these functions at station2, the flow axial mach number at station 3 (i.e. after complete mixing) is evaluated as:

Mx,3 =γ C

γ + 1−

√(γ C

γ + 1

)2

− 1 +γ − 1

γ + 1D2 (8.16)

Once the axial mach number is obtained at station 3, the density ratio is obtained from:

(ρ

ρt

)3

=

(1− γ − 1

γ + 1

[D2 +M2

x,3

]) 1γ − 1

(8.17)

Finally, the pressure ratio between the station after complete mixing 3 and the inletstation 1 is calculated as:

(pt3pt1

)stew

=

(ρ

ρt

)∞,2

M∞,2 cosβ2 (1− δ1 − tTE)(ρ

ρt

)3

Mx,3

(8.18)

And the total pressure loss is computed using:

∆pstew,∞ = pt1

[1−

(pt3pt1

)stew

](8.19)

8.1.5 Wake mixing by conservation

In order to get a better idea of the importance of mixing loss with respect to the over-all profile loss process, a methodology is presented in order to compute this mixing lossindependently [45]. The methodology is based on the mass, momentum and energy con-servation. As it was already mentioned several times, the wake mixing with the freestream flow occurs from station 2 to station 3, until the flow becomes uniform (See Fig-ure 16). The velocity, pressure and temperature profiles are known at station 2 and thewake integration is performed at this location. The conserved quantities are thus com-puted at this station and the conservation principle is then applied from station 2 tostation 3. The variables at station 3 are computed assuming the flow is uniform (i.e. theflow is completely mixed). The methodology that is applied for the profile loss mixing istwo-dimensional. The general three-dimensional extension is given in Appendix A withthe relative rotating speed taken into account in the case of a rotor. Further developmentscan be found depending on the section variation and the flow conditions [45].

The conserved quantities are first computed at station 2. As shown in Figure 26, this inletboundary is chosen perpendicular to the axial direction. The mass flow rate will thenbe computed using the axial component of the velocity since it is perpendicular to theintegration plane. The upper and lower surfaces are periodic since there is only one bladethat is considered. In fact, the compressor is assumed to be periodically axisymmetric.Moreover, the annulus walls are assumed to be frictionless so that there is no influence on

44

the two-dimensional computation methodology. Furthermore, both the inlet and outletsurfaces are of length S, taking a unit width for simplicity. The length S corresponds tothe pitch i.e. the blade spacing Y .

Figure 26: Illustration of the mixing computation methodology.

The mass flow rate as well as the momentum in the two dimensions and the energyconstants are obtained following:

m =

∫2

ρ Vx dS

A =

∫2

[ρ V 2

x + p]dS

B =

∫2

ρ Vx Vy dS

D =

∫2

ρ Vx H dS

Using the mass flow rate conservation, the tangential momentum component as well asthe energy constant are directly obtained from:

V y3 =B

m

H3 =D

m

Guessing the value for the first iteration as the inlet velocity vx3 = vx1, the iteration on

45

this variable is performed following:

ρ3 =m

S V x3

p3 =A− m V x3

S

T 3 =p3

R ρ3

h3 = cp T 3

V(i+1)

x3 =

√2

(H3 − h3 −

1

2V

2

y3

)After convergence of the above loop on vx3, a value for each variable at station 3 isobtained and the total pressure after complete mixing is given by:

pt3 = p3

(1 +

γ − 1

2M2

) γγ − 1

(8.20)

Finally, the total pressure loss can be computed knowing all the uniform flow character-istics at station 3.

This methodology is based on assumptions and on a simple design. Moreover, it doesnot take the complex flow accelerations and interactions into account. A paper writtenby Rose and Harvey [46] presented a methodology for extracting wake mixing loss. Theirpaper took the total pressure and temperature deficit of the wake into account but alsothe acceleration prior to mixing. In top of that, the approach also accounts for theinteraction of blade profile wakes with downstream blade rows. The latter aspect is notstudied in this thesis. Rose and Harvey ’s paper content goes beyond the scope of thiswork.

8.2 Profile shockDepending on the shock location in the blade-to-blade plane, different shock models arepresented. It is important to recall that many shock models are developed for rotorblade row. However, since this work focuses on blade rows with a subsonic inlet machnumber, some models presented hereafter will not be implemented. The models and thecorresponding shock that is studied are given in Table 7.

Model Details

Koch and Smith Leading edge and passage shock

Freeman and Cumpsty Leading edge shock

Manfredi and Fontaneto Koch and Smith’s extension

Table 7: Summary of profile shock loss models.

46

8.2.1 Koch and SmithIn their summary paper of 1976 about the loss sources and magnitudes in compressors,Koch and Smith [47] developed a section about shock loss. Their approach is based on aseparation of shock loss into different parts related to their source. Firstly, the entropygeneration is due to the blade nose bluntness. They correlated the entropy generationdepending on the inlet mach number and the leading edge thickness tLE as:

∆s

R= −ln

[1− tLE

Y cosβ1

[1.28 (M1 − 1) + 0.96 (M1 − 1)2

]](8.21)

Secondly, the loss due to the passage shock is computed using the oblique shock relations(Rankine-Hugoniot relations at Section 6.2). This passage shock is generated by the ac-celeration of a highly subsonic inlet flow or a flow with a supersonic inlet mach number.If the exit mach number is subsonic, Koch and Smith’s model is based on the assumptionthat the loss is equivalent to the entropy rise of an oblique shock generating a variationof mach number from a representative value to unity. If the exit mach number is higherthan 1, then the assumption of the equivalent entropy rise is related to a change of machnumber through an oblique shock from a representative value to the real exit value. Therepresentative mach number is typically the maximum suction side mach number.

The passage oblique shock loss can be expressed in terms of entropy rise as:

∆s = cv ln([

1 +2γ

γ + 1(M2

1 − 1)

] [(γ − 1)M2

1 + 2

(γ + 1)M21

]γ)(8.22)

Again, this expression is valid for normal shocks but can be adapted to oblique shock bytaking the component normal to the shock plane.

8.2.2 Freeman and CumpstyIn 1989, Freeman and Cumpsty [48] developed a simple shock model for supersonic bladeinlet. Based on conservation of stagnation enthalpy, mass and momentum, this one-dimensional model is able to predict the loss generated in this inlet region. The controlvolume corresponding to this model is illustrated in Figure 27.

Figure 27: Control volume for Freeman and Cumpsty ’s model. Adapted from [48].

47

Mathematically, the three simplified conservation equations were expressed by Freemanand Cumpsty assuming a thin leading edge and very small blade camber. First, the massconservation between the inlet and the blade spacing is expressed as:

ρ1V1s

cosχ1

cosβ1 = ρ1V1(Y − t) (8.23)

The stagnation of enthalpy is conserved following:

T1

[1 +

γ − 1

2M2

1

]= T2

[1 +

γ − 1

2M2

2

](8.24)

Finally, neglecting higher order terms, the conservation of momentum in the blade chorddirection is given by:

p1 Y + p1γM21

Y

cosχ1

cosβ1cos(χ1 − β1) = p2 Y + p2γM21 (Y − t) (8.25)

The assumptions that were made were shown to have a very low influence taking againthe case of a low cambered blade. After algebraic manipulations, the three conservationequations result into:[1 +

γ − 1

2M2

2

]−1/21 + γM2

2 (1− t/Y )

M2(1− t/Y )=

[1 +

γ − 1

2M2

1

]−1/2cosχ1

cosβ1+ γM2

1 cos(β1 − χ1)

M1

(8.26)In order to apply this principle to practical shock loss computation, Freeman and Cumpstyfinally obtained a correlation giving a reasonable approximation. The correlation is basedon the shock loss in total pressure across a normal shock. The Rankine-Hugoniot relationsdefined in Section 6.2 are used for the computation of the normal shock. The correlationis then computed as:(

∆pt(pt1 − p1)

)rel

=

(∆pt

(pt1 − p1)

)normal

+ [0.026 + 0.0018 (χ1 − 65)] (β1 − χ1) (8.27)

This relation is only valid for shock in the inlet region. This model can therefore onlybe applied in the case of a compressor with a supersonic inlet and a blunt leading edge.As explained in Section 7.1.3, modern blade design have sharp leading edges so that theoblique shock at the leading edge can typically be neglected.

8.2.3 Manfredi and Fontaneto

In a recent paper from the Von Karman institute for Fluid Dynamics, Marco Manfredi[32] used an approach similar to Koch and Smith’s [47]. Focusing on shock loss withsupersonic inlet, the method is performed in two steps:

1. Computation of the suction side maximum mach number.

2. Computing the loss using oblique shock relations.The suction side maximum mach number can be computed using profile loss models orusing CFD. In Marco Manfredi ’s paper, since König ’s model was chosen for profile losscomputation, it was also chosen for the suction side maximum mach number computation.However, in the scope of this paper, CFD will be preferred as the goal is not to predictbut rather to extract loss.

48

8.3 Endwall

Loss in the endwall region is very complex to compute. Two main models are describedand one extension is also presented. The main difference between the models comesfrom the availability of a span-wise distribution of loss. The endwall models and thecorresponding characteristics are given in Table 8.

Model Details

Koch and Smith Span-wise averaged value

Hanley Basic cascade flow, incompressible

Manfredi and Fontaneto Hanley ’s extension, compressible

Table 8: Summary of endwall loss models.

8.3.1 Koch and Smith

Koch and Smith’s paper of 1976 was already mentioned earlier [47]. It also include apart related to endwall loss. In particular, the authors presented a model for the endwallloss computation without the influence of tip leakage loss. The paper is based on theexperimental results of Smith [49].

The model relates the real flow efficiency as a function of the flow efficiency without theendwalls η, the hub and casing displacement thickness δ1 and tangential force thicknessν∗ and the annulus height ha:

η = η1− (δ1h + δ1c)

ha

1− (ν∗h + ν∗c )

ha

(8.28)

The tangential force thickness corresponds to the tangential component of the blade forcereduction from its free-stream value due to the presence of the boundary layer. Usingmean values, the efficiency can be expressed as:

η = η1− 2δ1

Y(Y/ha)

1− 2ν∗

2δ1

2δ1

Y(Y/ha)

(8.29)

In this equation, (Y/ha) is the weighted average rotor and stator mean-diameter staggeredspacing/annulus height ratio, the weighting function being the blade row inlet dynamichead. The main disadvantage of this model lies in the absence of radial loss distribution.This approach only gives a span-wise averaged value of endwall loss.

49

8.3.2 Hanley, Manfredi and Fontaneto

In order to obtain a model for endwall loss with radial distribution and without tip leakageloss effect, Manfredi and Fontaneto [32] extended Hanley ’s model [50]. This latter modelis one of the few models that can be extended to give a radial distribution of endwallloss. Several updates of this model were suggested by Manfredi and Fontaneto:

· Radial distribution formulation extension

· Extension to annular blade rows

· Compressibility taken into account

· Modification of the velocity profile law

Hanley ’s model is based on the inlet axial and tangential velocity profiles. Once theseprofiles are known, the outlet velocity profile is estimated using semi-empirical relationsand boundary layer theory. The basic assumptions of this first model were:

1. Basic cascade flow

2. No radial pressure gradient

3. No tip clearance

4. Incompressible flow

5. Skewed and fully turbulent endwall boundary layer

The extension of the model performed by Manfredi and Fontaneto enables to write thetotal pressure loss coefficient including profile and endwall loss as:

ωp,ew =pt1 − pt2pt1 − p1

= 1− ∆p1

2ρV 2

− V 22

V 21

(8.30)

This modification enables to apply the model to the annular blade row location as ex-pected. Because the cascade is considered simple, the endwall boundary layer, the freestream zones and the velocity profile are clearly identified. Using the inlet dynamic headq1, the loss coefficient can be rewritten as:

ωp,ew = 1− ∆p

q1

V 21,∞cos

2(β1 − β1,∞)

V 2x1

− V 2x2 cos

2(β1 − β1,∞)

V 2x1 cos

2(β1 − β2,∞)(8.31)

Then, as it was mentioned earlier, the boundary layer theory and two semi-empirical rela-tions are used to solve the above equation. The radial distribution lies in the stream-wisecomponent of the outlet boundary layer velocity Vx2. It is described by two-dimensionalturbulent boundary layer models. It is shown in [32] that once the inlet axial boundarylayer displacement thickness, the span-wise averaged static pressure rise across the bladerow ∆P and the outlet velocity free-stream velocity V2∞ are known, the outlet boundarylayer velocity Vx2 can be found. The computation of V2∞ is discussed in [51]. The span-wise averaged static pressure rise across the blade row ∆P is related to the blade loading

50

and can therefore be computed from blade profile models.

The application of Hanley ’s model is limited to the endwall viscous boundary layer whereit is found to be generally valid. Out of this region, the second assumption assuming noradial pressure gradient is not valid. Moreover, the model is strongly linked to the accu-racy of the boundary layer identification.

A further extension of the model consists in taking the compressibility effect into ac-count when using the semi-empirical relations. This modification was discussed [32] andvalidated [51] by Manfredi and Fontaneto. As a reminder, since the assumption of theabsence of tip clearance was made, this approach cannot be applied in the case of a rotorcasing.

8.4 Tip clearance leakageTwo main approaches are used for the computation of total pressure loss due to the tipclearance flow [52] [32]. While the first one is based on the loss of kinetic energy acrossthe leakage flow, the second one relies on potential flow vortex models. The latter modelis based on the induced velocity at the trailing edge vortex. All the models and thecorresponding details are summarised in Table 9.

Model Details

Rains Viscous relation, kinetic energy based

Storer and Cumpsty Kinetic energy based

Lakshminarayana 1 Potential vortex based, semi-empirical formulation

Lakshminarayana 2 Potential vortex based, well based physical formulation

Manfredi and Fontaneto Combination of both Lakshminarayana’s models

Table 9: Summary of tip clearance leakage loss models.

8.4.1 Rains

In his PhD thesis of 1954, Rains [53] was one of the first authors to really focus on thetip clearance leakage flow. In particular, unlike older papers, his approach was not basedon the lifting line theory used on typical wings. By taking into account the relative bladerotation with respect to the casing, his approach was based on the computation of themomentum flux across the clearance leakage. The leakage velocity was then computedtaking viscosity into account [32]. Rains developed a rational model giving a relativelygood approximation of loss related to this leakage flow. After a mathematical developmentfully described in his thesis, Rains ’ model can be written as a drop in efficiency due tothe tip clearance leakage flow:

∆η =

2

(δtR0

)(c0

S0

)ρ3

0

[S0

c0

ψ′0ρ0

cosβ∞,0 +cos4β∗0ρ2

0

I

(δt

2δ1

)]ρ ψ′ (1− κ) cos3β∗0

(8.32)

where the different terms appearing in the formula are defined as:

51

· δt : Tip clearance

· R0 et Ri : respectively, external and internal radius.

· c0 : Blade chord length at the tip.

· S0 : Blade spacing.

· ρ0 : Clearance density.

· ψ′0 : Head loss without tip clearance.

· β∞,0 : Mean flow angle at the clearance.

· β∗0 : Blade angle at the clearance.

· ρ : Integrated mass flow rate coefficient.

· ψ′ : Work coefficient.

And where the following function is defined as:

I

(δt

2δ1

)= 1−

[(δt

2δ1

)− 1

4

(δt

2δ1

)2]2

κ =Ri

R0

Based on the work of Rains, Vavra’s version [54] presented a simple expression of thedrag coefficient in the form of a semi-empirical relation based on experimental results[55].

The main disadvantage of the approach based on the kinetic energy corresponds to thelack of radial distribution. In fact, models based on this theory only give a mean valueused for the overall loss computation. In the scope of performing a three-dimensional lossbreakdown, the access to a radial distribution is mandatory.

8.4.2 Storer and Cumpsty

Following the same kinetic energy based approach as Rains ’, Storer and Cumpsty de-veloped a model based on flow mixing [56]. The basic idea is a simple model for thecomputation of mixing loss between the leakage flow and the main flow. Figure 28 illus-trates the simple model based on kinetic energy. Defining χg as the area of tip clearancegap divided by main stream flow area, the corresponding total pressure loss is given by:

∆Pt =1

2ρV 2

E

[χg sinζ

(2 + χg sinζ − 2 sinζ

(1 + χg sinζ)2

)](8.33)

This simple model is then applied to a compressor cascade case and adapted to accountfor the design variables. The disadvantage of this method is the same as for Rains ’ model.There is no formulation that gives a radial distribution of the loss.

52

Figure 28: Illustration of Storer and Cumpsty mixing simple model [56].

In order to apply this model to a real compressor cascade, additional formulations arespecified. Assuming that the inlet and outlet axial velocities are equal enables to write:

VE = V1cosβ1

cosβ2

(8.34)

Moreover, the term χg can be expressed using the tip gap height δ0, the blade height hb,the row stagger angle γ and a discharge coefficient Dc defined as the ratio of the real flowand the ideal flow rate:

χg = Dc

σ

(δ0

c

)(hbc

)cosγ

(8.35)

Finally, the leakage jet angle is computed in practice as a weighted mean value along theblade chamber line l as:

ζ =

∫ c

0

VLζdl∫ c

0

VLdl

(8.36)

Where the integrated values are local values.

8.4.3 Lakshminarayana

Based on the potential vortex approach that was first adopted by Betz [57], Lakshmi-narayana and Horlock developed [58] and improved [59] [60] a similar model. WhereBetz used the lifting line theory to estimate the induced velocity at the blade tip, Lak-shminarayana and Horlock adapted the model taking the viscous effect into account. Aspointed out by Denton [14], any model computing loss without taking viscous effect intoaccount should be avoided if possible. Loss can be computed and estimated withoutconsidering it, but viscous models will always be preferred.

53

The basic idea lies in the computation of the circulation at the blade tip clearance andthe use of this circulation to identify the vortex generation and their evolution in theaxial direction. The vortex circulation amplitude Γvortex is computed as a proportion ofthe uniform blade circulation at the blade tip Γblade as:

Γvortex = (1− κ) Γblade (8.37)

Where κ is a parameter that represents the amount of the lift that is lost at the blade tipclearance. Since it is very complex to model the latter parameter, it is computed fromempirical data using correlations.

Based on this work, Lakshminarayana [61] developed two models for the tip leakage losscomputation. The first one gives a simple formulation of an overall stage efficiency lostin terms of design parameters. The semi-empirical formulation of this first model is givenby:

∆η =0.7 λ ψ

cosβm

[1 + 10

√φ

ψ

λA

cosβm

](8.38)

In this equation, the different design parameters are:

· λ = τ/h : Non-dimensionalized tip clearance

· ψ = 2∆P0,is

ρV 2 : Load coefficient

· βm : Mean flow angle

· φ : Flow coefficient

· A = h/c : Blades aspect ratio

The second model developed by Lakshminarayana gives a radial distribution of loss due tothe tip clearance leakage flow. Using an incompressible laminar flow solution but takingviscosity into account, the author gave a method to compute the local loss coefficient(Equations 26 and 29 in [61]). By averaging the local loss coefficient at each radiusheights, Lakshminarayana obtained a total pressure loss distribution along the span.

8.4.4 Manfredi and Fontaneto

After having studied both Lakshminarayana’s models [61], Manfredi and Fontaneto [32][51] chose to use a combination of the models for the computation of tip clearance totalpressure loss. While the first model was shown to give a good prediction of the span-averaged total pressure loss, the second one offered an effective qualitative distribution ofloss along the span. The authors decided to use the first model as a quantitative methodto compute the overall loss due to tip leakage flow and to use the second one in order togive the span-wise distribution of this loss.

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8.5 Cavity leakage

Two main models are presented for cavity leakage flow loss in this section. The first modelcomputes all the features of the leakage flow. It enables a computation of total pressureloss based on the different flow features in the cavity. The second model computes themixing of the leakage flow with the main flow based on leakage flow characteristics. Themodels that are considered in this section are given in table 10.

Model Details

Wellborn Leakage flow model

Denton Simple mixing model

Table 10: Summary of cavity leakage loss models.

8.5.1 Wellborn

The complexity of cavity leakage flow computation comes from the various seal teethexisting geometries. In fact, the shrouded stator can be designed in many different waysdepending on the technology. Thus, there is not one good way of modelling cavity flowsin a general way. Wellborn reviewed existing research on the influence of stator shroudseal teeth on compressor performance [41]. In 2000, Wellborn developed a simple one-dimensional model based on the flow characteristics at the interface between the main flowand the cavity and the flow characteristics of the main stream [62]. This model estimatesthe leakage mass flow rate, the temperature rise and angular momentum increase acrossthe cavity. An illustration of the cavity design used by Wellborn for the development ofits model is given in Figure 29. The author also gives the qualitative influence on eachdesign parameters of the cavity on the performance of the labyrinth seal.

Figure 29: Hub shrouded stator blade design used by Wellborn [41].

It is recalled that the one-dimensional model aims at estimating the leakage mass flowrate, the temperature and angular momentum increase. The "knife to knife" approachof Tipton [63] is used to estimate the flow characteristics across the cavity. In particular,the leakage mass flow and the temperature are obtained from this first model component.

55

The latter model depends on aerodynamic and design geometry variables. The secondpart of the one-dimensional model aims at obtaining the change in tangential momen-tum and total temperature. The angular momentum is obtained from a control volumeapproach. A momentum balance is performed as described in Wellborn’s paper in orderto obtain the angular momentum equation. From this, the tangential velocity is knownand the total temperature can be computed.

This simple one-dimensional leakage cavity model can be used as a basis for cavity leak-age loss computation. In fact, once the cavity flow characteristics are known, the totalpressure loss across the cavity can easily be computed such as the leakage loss. Moreover,the computation of the mixing of the leakage flow with the main stream also requires thecharacteristics of the leakage flow.

8.5.2 Denton

As one important part of the loss arising from the cavity leakage flow corresponds to themixing with the main flow, it is interesting to have an independent model computingthis loss component. In the Appendix 2 of Denton’s paper [14], a theory for the entropyrise computation due to a mixing of two streams is developed. The author also gives aformulation for the case when one stream has a relatively small mass flow rate [64]. Thelatter case is shown in Figure 30. In this case the entropy rise is computed as:

∆s = Cpmc

mm

[(1 +

γ − 1

2M2

m

)Ttc − TtmTtm

+ (γ − 1)M2m

(1− Vc cosα

Vm

)](8.39)

Figure 30: Mixing of a low mass flow rate stream with the main stream [14].

Once all the characteristics of both flows are known, the entropy rise can be computedan then converted into a total pressure loss. However, the model relies on leakage flowparameters that are not necessarily known. Moreover, it does not take the leakage loss andthe friction inside the cavity into account. Therefore, its use is mainly complementaryfor the computation of the mixing part of the cavity leakage flow. The leakage flowparameters thus need to be computed from another model or from CFD.

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9 Two-dimensional loss computation

In order to perform a quantitative two-dimensional loss breakdown, the models are ap-plied in order to compute the corresponding loss. The computation is detailed in thissection by developing the computational domain, the mesh used for the computationand the simulation software and parameters. Since the wake integration is one key stepof the profile loss computation, the methodology is given in more details. Again, it isrecalled that starting from this point, only the two-dimensional components of the lossare computed and analysed.

9.1 Computational domain

In order to perform the simulation of the two-dimensional case, a control volume and thecorresponding boundary conditions were chosen. This simulation is assumed to be per-formed at a constant radius height in a compressor. Moreover, three-dimensional effectsare completely neglected. The results of the two-dimensional simulation are close to theones of the three-dimensional simulation at half the radius height between the hub andcasing. In fact, this is the location where the tree-dimensional effects are typically thelowest.

A representative illustration of the computational domain is shown in Figure 31. Thesimulation was performed using a NACA 65-210 blade row [65]. This airfoil was used inthe past by Lieblein in his validation of profile loss model [42]. Moreover, similar NACA-65 blades were generally used in the literature when studying the loss or when evaluatingthe performance of blades [10]. However, it is clear that this airfoil is not representativefor compressor blades since it has a really low camber and no stagger angle. This choice ofairfoil is just for the validation of the models but the results are clearly not representativefor a real compressor. The chord length was chosen to be equal to a unit for simplicity.An important design parameter in such a two-dimensional problem is the solidity. Thesolidity is modified using the blade spacing Y since both the chord and blade spacing arelinked by the solidity as:

σ =c

Y(9.1)

Two typical solidity cases were studied in the scope of this work: σ = 0.5 , σ = 1 orrespectively Y = 2 , Y = 1 since c = 1.

In order to simulate only one blade of the row, a periodic condition was set betweenthe top and bottom boundaries of the control volume. It is therefore assumed that thecompressor is periodically axisymmetric at the compressor radius height. Even if asym-metrical aspects discussed earlier are very significant in compressors (see Section 7.3),the symmetrical assumption is generally made when estimating the performance at on-design conditions. Off-design characteristics are generally studied separately with thecorresponding instabilities (e.g. distortion instabilities phenomena).

The inlet boundary condition is set by a uniform velocity profile and an angle of attack.The distance from the inlet to the leading edge is large enough to be independent of theresults. This condition is not fully consistent with real compressors. In fact, as shown

57

in Figure 7, almost every compressor stage is preceded by another stage. This previousstage modifies the flow and has a wake. The wake carries turbulence and a non-uniformvelocity profile with secondary flows. In an optimal simulation, the inlet condition of theblade row should take the wake of the previous stage into account. However, in the scopeof this work, the condition was limited to a uniform condition. The outlet boundarycondition is specified by a constant pressure. Moreover, the boundary is far enough fromthe trailing edge to enable the complete mixing of the flow. This aspect is importantsince the wake mixing needs to be studied. In order to obtain the CFD results of themixing, the control volume must be chosen adequately.

Figure 31: Control volume of the two-dimensional simulation.

Again, this computational domain choice is not representative for real compressors andis just a first base for the validation of the models that are expected to be used in futuredesign methodologies. Real compressor blades have a higher camber and stagger anglewhile there are also interactions with other blade rows upstream and downstream theblade.

9.2 Mesh and simulation

The meshing of the control volume is performed using the GMSH software [66]. Thissoftware is an open source two- and three-dimensional finite element mesh generator.An unstructured mesh was chosen for simplicity and for convergence. The grid in theboundary layer region is made of rectangular elements whereas out of the boundary layerregion, triangular elements are used. The mesh is illustrated in Figure 32. Two zoomsare made at the interest location of the leading edge and trailing edge. It important tonote that the grid resolution at the leading and trailing edge could clearly be increased.It is an important required improvement for future research.

58

Figure 32: Mesh of the control volume near the blade section using GMSH.

The simulation is then performed using Stanford University Unstructured (SU2) [67].This open source software solves the partial differential equations (PDE) for many differ-ent applications including CFD. Among the different options that were available, Table11 summarises the choices that were specified in the software.

Parameter Choice

Physical governing equations Reynolds-averaged Navier–Stokes (RANS)

Turbulence model Spalart–Allmaras (SA)

Convective numerical method Jameson-Schmidt-Turkel (JST)

Numerical method for spatial gradients Weighted Least Squares (WLS)

Table 11: Summary of the simulation parameters specified in SU2.

After convergence of the simulation, the pressure drop across the blade row is computeddirectly from the CFD. Then, the flow characteristics distribution in the plane are ex-ported to Paraview which is an open source visualisation and post-processing software[68]. A cut is performed just after the trailing edge for the wake integration required inthe profile loss computation models. The cut was performed at a distance c

1000from the

trailing edge. The inlet and other required flow characteristics are also imported. From

59

this, a Python code is developed in order to model the loss and compute the correspond-ing total pressure drop. Finally, graphs comparing the models and the CFD are plotted.

The simulations were made for both solidity values, for inlet mach numbers going fromM = 0.3 to M = 0.8. This choice was made in order to analyse the subsonic and highsubsonic cases. This enables to observe the effect on compressibility and shocks appear-ance at higher subsonic mach numbers. Again, the case of supersonic inlet mach numberwas not studied in the scope of this work.

It is important to note that the flow detachment was not studied by varying the angleof attack. In this work, the angle of attack of the flow with respect to the blade wasset to β1 = 1° for all the simulations. In fact, this work does not focus on off-designconfigurations. The analysis in terms of angle of attack with an observation of flowdetachment also goes beyond the scope of this thesis. However, this off-design studyrepresents an important perspective of this work.

9.3 Wake integration

The methodology of almost all profile loss models is based on an integration of the wakejust after the trailing edge at the mid-span section of the blade. The distance between thetrailing edge and the integration plane was arbitrarily chosen to be equal to 10−4 timethe chord length (1 meter in this simulation). Using blade geometry, inlet values andintegrated values of the wake enables to compute the total pressure loss from the modelsdeveloped earlier. The integration plane in the case of a three-dimensional simulationand the corresponding line in the two-dimensional case are illustrated in Figure 33. It isrecalled that in the scope of this work, only the two-dimensional simulation is computedand analysed.

Figure 33: Integration plane and integration line at the blade trailing edge.

60

In the general case of a boundary layer integration, the limits of the integration lime areoften chosen at 99% of the free-stream velocity for simplicity. In fact, this enables to havea precise definition concerning the location of the boundary layer. In the case of a bladerow, the integration line is relatively small since other blades are located next to the onethat is studied. The integration is thus performed on the whole line using the definitionof the boundary layer and wake characteristics developed in Section 4.

Typical velocity profiles for the integration of the wake just after the trailing edge is shownin Figure 34 for two inlet mach numbers. The corresponding mach number profiles areshown in Figure 35. For the compressibility effects that are taken into account in someloss models, the density profiles are shown in Figure 36 for the two corresponding inletmach numbers. The trailing edge is located at y = 0.5. The centre of the wake is thereforealso located at the centre of the azimuth coordinate. Obviously, the wake correspondsto a reduction of the velocity and mach numbers in the profiles. It can also be seen thatthe velocities, mach numbers and densities are the same at y = 0 and y = 1. This is acondition imposed by the periodic boundaries developed in Section 9.1.

(a) Inlet Mach number = 0.5 (b) Inlet Mach number = 0.7

Figure 34: Velocity profile in the wake just after the trailing edge.

(a) Inlet Mach number = 0.5 (b) Inlet Mach number = 0.7

Figure 35: Mach number profile in the wake just after the trailing edge.

61

(a) Inlet Mach number = 0.5 (b) Inlet Mach number = 0.7

Figure 36: Density profile in the wake just after the trailing edge.

10 Validation and discussion

The loss source have been presented and the corresponding models for the computationhave been detailed. The control volume and the simulation parameters have been given.In this section, the reference results of the simulations are presented and analysed. First,the overall loss from the CFD is plotted in order to have a reference to compare withmodels results. For each loss source, the model presenting the better agreement withCFD is taken for the two-dimensional loss breakdown analysis. Then, each loss sourcesare added (i.e. profile boundary layer loss, wake mixing loss and profile shock loss)in order to recover the overall two-dimensional loss. Again, the three-dimensional losshas not been computed in the scope of this thesis and is therefore not discussed. Themathematical details and the references of the models that are used in this section arefound in Section 8.

10.1 Mesh resolution

Before analysing the results of the simulation, it is mandatory to discuss the quality ofthe mesh. The quality of the mesh is one of the most important aspects of numericalsimulation. Even if all the boundary and flow conditions are correctly specified, a badmesh could generate very inaccurate results. In particular, the first thing that has to bechecked is the size of the cell at the wall.

In particular, the y+ criterion is used to check that the flow in the boundary layer has agood resolution. A good resolution enables to capture the information inside the boundarylayer. In order to have the y+ in the desired range, one has to adapt the grid spacing.y+ is defined as:

y+ =y u∗ν

(10.1)

62

In this equation, y is the distance to the wall, ν is the kinematic viscosity and u∗ is thefriction velocity defined as:

u∗ =

√τwρ

(10.2)

The wall shear stress is denoted by τw. Depending on the value of y+, we can evaluate ifthe mesh size was small enough. In the different simulations that were performed in thescope of this master thesis, the maximum y+ value at the wall was found to be equal to4.7. This means that the cells are still located in the viscous sublayer (y+ < 5) [27]. It isan acceptable result and the grid resolution is relatively good. However, it can be notedthat a y+ at the wall closer to 1 would be preferred in more advanced analysis.

10.2 CFD reference loss

In order to compare the different models with the CFD results, the reference loss fromthe numerical simulation is first computed. This reference loss corresponds to the overalltotal pressure drop between the inlet and the outlet of the blade row. The overall lossthus accounts for the profile boundary layer loss but also the shock loss and the lossarising from the wake mixing.

Recalling that the simulation was performed for two different blade solidity cases, theCFD reference loss is given in Figure 37. Before going into the details of each losssource, it can be seen that there is a large increase of the total loss as the inlet machnumber reaches high subsonic values. This large increase is expected to come from thecompressibility effects and shocks. In fact, shocks appear at different inlet mach numbersdepending on the solidity. The two cases are shown in Figure 38 and Figure 39.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

2000

4000

6000

8000

10000

12000

14000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Overall CFD - Solidity = 0.5Overall CFD - Solidity = 1

Figure 37: Overall reference loss from CFD.

63

(a) Mach number = 0.6 (b) Mach number = 0.75 (c) Mach number = 0.8

Figure 38: Shocks at different inlet mach numbers. Solidity = 0.5.

(a) Mach number = 0.6 (b) Mach number = 0.7 (c) Mach number = 0.75

Figure 39: Shocks at different inlet mach numbers. Solidity = 1.

For all inlet mach numbers, the overall loss is higher when the blade solidity increases.Moreover, the large increase described previously takes place for a smaller inlet machnumber in the case of a larger solidity. These two last observations are linked to theeffect of solidity on loss. A larger solidity will induce higher friction of the flow on theblade. It could be noted that at smaller solidity, when analysing real compressor blades,the effect of separation could overcome the reduction of friction and increase the bladetotal profile loss [69]. Lieblein obtained a similar loss distribution graph in terms of themach number [29]. When studying the effect of the inlet mach number on the bladeperformance, Cumpsty concluded that the rapid rise in loss was caused by a shock waveand a massive separation [10].

The relative error percentage with respect to this CFD overall reference loss is computedas:

ε =∆pt −∆pt,CFD

∆pt,CFD· 100 (10.3)

It is used on the following discussion of each loss source magnitude with respect to theoverall reference loss.

64

10.3 Blade profile boundary layer

The first loss source that is studied is the profile boundary layer loss. An incompressibleand a compressible model have been presented earlier for the computation of this corre-sponding loss source. Respectively, Lieblein’s and König ’s models are both based on theintegration of the wake just after the trailing edge. Typical velocity, mach number anddensity profiles have been presented in Section 9.3. Following the methodology presentedin Section 9.2, the two loss distributions in terms of the inlet mach number are computed.As detailed in Section 8.1.1, Lieblein used an approximation for the definition of his losscoefficient. The difference introduced by his approximation is illustrated in Figure 40while the approximation is expressed in Equation 4.13. Since the relative error of theapproximation never exceeds 0.2 %, the two formulations are relatively similar and bothcan be used in the present application. In this section, Lieblein’s version is used.

Figure 40: Illustration of Lieblein’s approximation for the definition of the total pressureloss coefficient with respect to the exact energy-based definition.

Figure 41 shows the results for the two solidity cases that are studied. A comparison withthe reference loss is shown in the same graph. However, it should be noted that the twomodels only consider boundary profile loss while the CFD reference loss is consideringthe overall loss. Therefore, it is expected not to have a perfect agreement between theresults. In order to get more information about the difference between the results in arelative point of view, the relative error between the model results and the overall CFDresults are shown in Figure 42.

From an absolute quantitative point of view, it can be seen in Figure 41 that both modelsshow a good agreement with the reference loss at low mach number. However, when themach number increases, there is a larger difference between the results. From a relativepoint of view, the error goes from 0% to approximately 20% at low mach number while itclimbs from 80% to 100% for highly subsonic inlet mach number. This first observationcan be explained by the appearance of other loss sources at higher mach number. At low

65

mach number, the low error of both models shows that the main part of loss comes fromthis boundary layer profile source.

In particular, Lieblein’s model shows a very good agreement at low mach number. On thecontrary, König ’s model is more accurate at higher mach numbers. This is explained bythe fundamental assumptions behind both models. On the first hand, Lieblein’s modelis an incompressible model. It is expected to be less accurate when the mach numberis increased because of the compressibility effects. On the other hand, König ’s model iscompressible and should give better results in general. It is however not exactly the casefor the relative error. The relative error is in fact slightly higher for König ’s model atlower mach number. However, the absolute error remains very low for both models atlow mach numbers.

In terms of the influence of the solidity on the models, it is shown that König ’s modelgives proportional results. In an absolute point of view, at high mach numbers, the over-all CFD loss increases drastically. It can be seen that despite the small drop at a machnumber of 0.8, the general results of the model have increased in a proportional way withrespect to the overall loss.

It is important to note that since the methodology of both models is based on a wakeintegration, the results are strongly linked to its basic profile. As the mach number in-creases and the solidity also increases, it was shown that the velocity profiles becomemore complex and thus more difficult to integrate in a general and consistent way. Inparticular, the shock appearance increases the complexity of integrating the wake velocityprofile. This is one reason to explain the small drop at a mach number of 0.8.

To conclude, even if it does not give the best relative error at low mach number, König ’smodel is chosen for the profile boundary layer loss results. In fact, the absolute error atlower mach number is very low and it has better absolute and relative results at highermach number. In particular, the results are better in the typical range of modern statorinlet mach number values (i.e. Mach 0.6 to 0.8 [70]).

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

500

1000

1500

2000

2500

3000

3500

4000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Lieblein ModelKönig ModelOverall CFD

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

2000

4000

6000

8000

10000

12000

14000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Lieblein ModelKönig ModelOverall CFD

(b) Solidity = 1

Figure 41: Blade profile boundary layer loss models comparison with respect to the overallCFD loss.

66

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

20

40

60

80

100Re

lativ

e er

ror [

%]

Lieblein ModelKönig Model

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

20

40

60

80

100

Rela

tive

erro

r [%

]

Lieblein ModelKönig Model

(b) Solidity = 1

Figure 42: Blade profile boundary layer loss models relative errors with respect to theoverall CFD loss.

10.4 Blade wake mixing

There are two ways of computing the blade wake mixing independently. The first methoddetailed in Section 8.1.2 consists in using the extension of Lieblein’s model that takesthe mixing of the wake into account. The mixing part of the loss is the difference intotal pressure loss between the two models of the same author. The two models arecompared in Figure 43 for one solidity case. The second method developed in Section8.1.5 was presented in details using a conservation methodology of mass, momentum andenergy. The two mixing loss results in function of the inlet mach number are shown inFigure 44. It is observed that at lower mach number, both models give relatively similarresults. However, once the mach number increases the mixing loss according to theconservation model increases drastically while the mixing from Lieblein’s model remainsmuch lower. This difference mainly comes from the incompressible assumption at thebases of Lieblein’s model.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

200

400

600

800

Tota

l pre

ssur

e lo

ss

P t [P

a]

Lieblein (mixing) ModelLieblein Model

Figure 43: Lieblein’s models version with and without mixing loss. Solidity = 0.5.

67

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

100

200

300

400

500

600

700

800

Tota

l pre

ssur

e lo

ss

P t [P

a]

Mixing from Lieblein ModelMixing from conservation

Figure 44: Mixing loss results from Lieblein and from conservation methodology. Solidity= 0.5.

Figure 45 shows the influence of solidity on the mixing loss computed from the conser-vation methodology. It is observed that the loss increase at high mach number is evenhigher for a solidity of 1. It follows the observation that the overall CFD loss is also muchlarger for the high solidity case.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

500

1000

1500

2000

2500

3000

3500

Tota

l pre

ssur

e lo

ss

P t [P

a]

Mixing - Solidity = 0.5Mixing - Solidity = 1

Figure 45: Influence of solidity on mixing loss computed with conservation methodology.

The goal of this section is to compare the reference CFD loss with models taking theprofile boundary layer loss and the wake mixing into account. In previous sections, onlyLieblein’s incompressible model with mixing and Stewart ’s compressible model are takingmixing loss into account. Since König ’s model was chosen to represent the profile bound-ary loss, it is interesting to combine the latter model with the conservation methodologydeveloped earlier to account for the mixing of the wake. The results is shown in Figure

68

46 where the total pressure loss computed from König ’s model with and without theconservation mixing methodology is plotted. The resulting combined model is comparedto Lieblein’s with mixing and Stewart ’s models in the following.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

500

1000

1500

2000

2500

3000

3500

4000To

tal p

ress

ure

loss

P t

[Pa]

König modelKönig model with mixingOverall CFD

Figure 46: Comparison between König ’s model with and without additional mixing lossfrom conservation methodology. Solidity = 0.5.

The three models taking the profile boundary layer and the wake mixing are comparedwith respect to the overall CFD reference loss:

· Lieblein’s incompressible, version with mixing

· Stewart ’s compressible model

· König ’s model + conservation mixing model

Just like for the profile boundary layer loss, the absolute comparison with the wake mix-ing taken into account is shown in Figure 47 while the relative error with respect to thereference loss is given in Figure 48.

The three models show absolute results in good agreement with the overall loss at lowmach number. However, just like the profile boundary layer loss, the absolute error in-creases as the inlet mach number increases in both solidity cases.

Lieblein’s mixing model behaves like the basic model and still gives very good results atlow speed for both cases. The latter model however gives the lowest accuracy at highmach number by underestimating the loss. Again, this is clearly linked to the incom-pressibility assumption on which all Lieblein’s models are based.

Stewart ’s model gives the least accurate results in the first solidity case by overestimat-ing the loss at low mach number. Moreover, Stewart ’s model shows very different resultsdepending on the solidity. On the first case with a solidity of 0.5, the relative error is

69

very large at low mach number. In fact, the error even exceeds 100% for some values.In this same case, the model gives better results at high mach number. However, for thesecond solidity case of 1, the observations are inverted. The results are very good at lowmach number and worst at high mach number. This large sensitivity in a qualitative waywith respect to the solidity cannot be explained in a direct way. The lack of consistencyrequires huge caution when dealing with Stewart ’s model. The latter results will clearlynot be used to perform the loss breakdown.

Despite giving the worst relative error in the second solidity case, König ’s with mixingmodel gives general good results. The absolute results are consistent in both soliditycases and are in good agreement with the overall loss. In terms of relative error, theresults are getting better as the mach number increases. In particular, just like for theprofile boundary layer loss, König ’s with mixing model gives the best results in the in-terest range of a typical stator inlet mach number. The relative error is bad at low machnumber but since the absolute error is low, it has less impact on the results.

At this point, as profile boundary loss and wake mixing are considered, the modifiedKönig ’s model is chosen to account for mixing. As König ’s model was chosen for theprofile boundary layer, this increases the consistency of the approach.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

500

1000

1500

2000

2500

3000

3500

4000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Lieblein (mixing) ModelStewart ModelKönig Model + mixingOverall CFD

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

2000

4000

6000

8000

10000

12000

14000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Lieblein (mixing) ModelStewart ModelKönig Model + mixingOverall CFD

(b) Solidity = 1

Figure 47: Blade profile boundary layer and wake mixing loss models comparison withrespect to the overall CFD loss.

70

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

20

40

60

80

100

120Re

lativ

e er

ror [

%]

Lieblein (mixing) ModelStewart ModelKönig Model + mixing

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

20

40

60

80

100

120

Rela

tive

erro

r [%

]

Lieblein (mixing) ModelStewart ModelKönig Model + mixing

(b) Solidity = 1

Figure 48: Blade profile boundary layer and wake mixing loss models relative errors withrespect to the overall CFD loss.

10.5 Shocks

Shock waves are the last sources of loss in two-dimensional problem cases. It is remindedthat the scope of the work was focused on blade row with subsonic inlet mach numbers.Therefore, only passage shock needs to be studied while leading edge shocks are not ap-pearing. Illustration of passage shocks have been presented in Figure 38 and Figure 39 forthe two solidity cases. Based on Koch and Smith’s model for passage shock computation,the corresponding loss is computed from oblique shock relations. As it was described inthe model, the methodology requires the maximum suction side mach number. In thescope of this work, it was chosen to identify this maximum value directly from CFD. Theresulting maximum suction side number in terms of the inlet mach number for the twosolidity cases are shown in Figure 49. The maximum suction side mach number increaseswith the inlet mach number as expected. The flow reaches local supersonic conditionas the maximum mach number goes over Mmax = 1. Over this value, a passage shockappears. The supersonic flow is reached for a lower inlet mach number in the case ofa higher solidity. Moreover, for a given inlet mach number, the maximum suction sidemach number is always higher for the latter case.

71

0.3 0.4 0.5 0.6 0.7 0.8Inlet mach Number M [-]

0.4

0.6

0.8

1.0

1.2

1.4

Max

imum

suct

ion

side

mac

h nu

mbe

r Mm

ax [-

]

Solidity = 0.5Solidity = 1

Figure 49: Influence of solidity on maximum suction side mach number.

Using the results of the maximum suction side mach number, the shock loss can becomputed using Koch and Smith’s methodology. The resulting total pressure loss infunction of the inlet mach number is given in Figure 50. Again, there is obviously noshock loss as the maximum suction side mach number remains subsonic. Just as forthe maximum mach number, the total pressure shock loss increases with the inlet machnumber. It can also be observed that the loss is much more important in the high soliditycase.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

500

1000

1500

2000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Shock - Koch and Smith - Solidity = 0.5Shock - Koch and Smith - Solidity = 1

Figure 50: Koch and Smith’s model for shock loss and the influence of solidity.

Even if one model was already chosen for the previous two loss sources, three modifiedmodels are compared for the sake of generality:

· Lieblein’s incompressible, version with mixing + Koch and Smith’s shock loss

72

· Stewart ’s compressible model + Koch and Smith’s shock loss

· König ’s model + conservation mixing model + Koch and Smith’s shock loss

For the three above modified models, in a similar way to the two previous sections, theabsolute results are shown in Figure 51 while the relative error with respect to the overallloss is given in Figure 52. It is the first time both the models and the CFD overall losscan be really compared. In fact, with the addition of the shock component, the threemodels account for the three loss components that make the overall loss.

It is important to note that since shock loss only appears for inlet mach number around0.65, the observations made in the previous section are the same for mach number lowerthan this inlet mach number. In fact, it can be seen in Figure 50 that for both soliditycases, the total pressure shock loss is equal to zero under an inlet mach number of 0.65and the results will thus not be modified. The observation will be focused on high machnumbers.

The modification to account for shock loss is the same for the three models using Kochand Smith’s model. Therefore, the relative difference between the three models is stay-ing proportional. Lieblein’s model still gives the best results at low mach number whilegetting worse with an increasing inlet mach number. Stewart ’s model is still very sensi-tive to the solidity and therefore not consistent to use for quantitative breakdown purpose.

Since König ’s model with mixing was underestimating the loss at high inlet mach num-bers, the addition of the shock component does increase the precision of the results.Despite the fact that it does not give the better results at low mach number and that atsome isolated points at low mach number, Stewart ’s model gives better results, König ’sextended model is chosen again because it gives the better results for both solidity casein the range of inlet mach number of a typical stator blade row.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

500

1000

1500

2000

2500

3000

3500

4000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Lieblein (mixing) Model + shockStewart Model + shockKönig Model + mixing + shockOverall CFD

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

2000

4000

6000

8000

10000

12000

14000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Lieblein (mixing) Model + shockStewart Model + shockKönig Model + mixing + shockOverall CFD

(b) Solidity = 1

Figure 51: Blade profile boundary layer, wake mixing and shock loss models comparisonwith respect to the overall CFD loss.

73

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

20

40

60

80

100

120Re

lativ

e er

ror [

%]

Lieblein (mixing) Model + shockStewart Model + shockKönig Model + mixing + shock

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

20

40

60

80

100

120

Rela

tive

erro

r [%

]

Lieblein (mixing) Model + shockStewart Model + shockKönig Model + mixing + shock

(b) Solidity = 1

Figure 52: Blade profile boundary layer, wake mixing and shock loss models relativeerrors with respect to the overall CFD loss.

11 Two-dimensional loss breakdown

After the computation of the two-dimensional loss using different loss models dependingon different assumptions, one set of model was chosen to perform the two-dimensionalloss breakdown. These models are summarised and reviewed before being used for thebreakdown.

11.1 Chosen models review

Profile boundary layer loss was chosen to be modelled using König ’s approach. It is acompressible model that gives the best results in the high subsonic mach number region.The wake mixing was chosen to be modelled using an approach based on mass, momen-tum and energy conservation. It is computed independently of the other loss source andcan therefore directly be added to König ’s profile boundary layer to build an extension ofhis model to account for blade wake mixing loss. Finally, Koch and Smith’s shock modelis used to add the last layer of loss to the previous model in order to account for all thetwo-dimensional loss sources. This combined loss model enables a separation into all theloss sources independently. This is very useful for a loss breakdown computation. Thedifferent models used for the loss source computation are shown in Table 12.

Loss source Model

Profile Boundary Layer König ’s model

Wake mixing Conservation methodology

Profile Shock Koch and Smith’s model

Table 12: Summary of the model used for the quantitative loss breakdown.

74

In order to get a preview of the loss breakdown, the different extensions of König ’s modelare first compared to each other. Each extension uses the previous version and adds oneloss source layer to finally get to the overall loss. The extensions are recalled as:

· König ’s model

· König ’s model + mixing methodology

· König ’s model + mixing methodology + Koch and Smith’s shock loss

The models are compared from an absolute point of view in Figure 53 while the relativeerror with respect to the overall CFD loss is shown in Figure 54.

Obviously, whenever a loss source is added, the total loss increases. At low inlet machnumber, the addition of mixing tends to increase the relative error because the absoluteloss is overestimated. At high mach number, the error is reduced because the loss isunderestimated. As shock loss is introduced, the model precision only increases sinceshock does not influence the loss at low mach number.

The set of models is used to perform the quantitative loss breakdown. Even if it doesnot perfectly fit the overall CFD reference, it was shown to give the best results. Inparticular, all these models gave the best results in inlet mach regions where the absoluteloss is high, i.e. where the loss breakdown has to give the best results. Moreover, it alsogave the best agreement in the region of a typical stator blade row.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

500

1000

1500

2000

2500

3000

3500

4000

Tota

l pre

ssur

e lo

ss

P t [P

a]

König Model + mixing + shockKönig Model + mixingKönig ModelOverall CFD

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

2000

4000

6000

8000

10000

12000

14000

Tota

l pre

ssur

e lo

ss

P t [P

a]

König Model + mixing + shockKönig Model + mixingKönig ModelOverall CFD

(b) Solidity = 1

Figure 53: Summarise of all König ’s extensions.

75

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

20

40

60

80

100Re

lativ

e er

ror [

%]

König Model + mixing + shockKönig Model + mixingKönig Model

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

20

40

60

80

100

Rela

tive

erro

r [%

]

König Model + mixing + shockKönig Model + mixingKönig Model

(b) Solidity = 1

Figure 54: Summarise of all König ’s extensions relative error.

11.2 Breakdown graphs

Using the models detailed in the previous section, the quantitative breakdown of thetwo-dimensional loss is performed. The absolute loss corresponding to each loss source isdisplayed in Figure 55 for both solidity cases. The corresponding percentage contributionof each loss source to the overall loss is shown in Figure 56. It can be seen that for thesolidity of 1, the result for the inlet mach number of 0.8 was not taken into account forthe breakdown. This point was shown to be incorrect because of the wake integrationfor the profile boundary layer loss. This represents one limitation of the approach. Thisaspect is briefly described in Section 11.4.

At low inlet mach number, there is no shock loss and the profile boundary layer domi-nates the loss. Mixing loss represents only about 20% of the total loss. This percentage isrelatively constant while shock loss is not present. While the percentage stays constant,the overall loss at low inlet mach number almost doubles from mach number between 0.3and 0.6. This means that mixing loss also double in this same range.

At high subsonic mach numbers, shocks loss appears. The corresponding loss proportionremains around 25% of total loss. Obviously, this proportion would be much larger if therotor case with increased relative mach number was considered. As soon as shock lossappears, the absolute overall loss increase is much larger. The loss goes from 500 Pa to3000 Pa for the first solidity case, which corresponds to a factor of 6 between mach valuesof 0.7 and 0.8. For the higher solidity case, the loss increases from about 1000 Pa to 9000Pa in the range of mach numbers between 0.65 and 0.77, which corresponds to an increaseof factor 9. Solidity increase has a large influence on loss absolute value. The overall lossis much larger when the solidity increases and even more with an increasing inlet machnumber. As it was already mentioned in Section 10.2, this is due to the increase of theratio between friction surface and passage area.

Solidity has especially a large influence on mixing loss. It is shown that at high machnumber, mixing loss is much larger for the high solidity case. While for the low soliditycase, the percentage of mixing stays bellow 30%, the mixing loss proportion in the high

76

solidity case reaches values of almost 40%.

To summarise, over the whole inlet mach number range that is considered, profile bound-ary layer loss is the most important source. Mixing loss is always present and becomesespecially important at high mach number in high solidity cases. Shock loss is only ap-pearing at high mach number and remains relatively low because the scope was limitedto subsonic inlet mach numbers.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

500

1000

1500

2000

2500

3000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Profile boundary layerWake mixingShocks

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7Mach Number M [-]

0

2000

4000

6000

8000

Tota

l pre

ssur

e lo

ss

P t [P

a]

Profile boundary layerWake mixingShocks

(b) Solidity = 1

Figure 55: Absolute total pressure loss breakdown using König ’s and Koch and Smith’sloss models.

0.3 0.4 0.5 0.6 0.7 0.8Mach Number M [-]

0

20

40

60

80

100

Loss

per

cent

age

[%]

Profile boundary layerWake mixingShocks

(a) Solidity = 0.5

0.3 0.4 0.5 0.6 0.7Mach Number M [-]

0

20

40

60

80

100

Loss

per

cent

age

[%]

Profile boundary layerWake mixingShocks

(b) Solidity = 1

Figure 56: Relative loss breakdown using König ’s and Koch and Smith’s loss models.

11.3 Applications

Once the two-dimensional breakdown is performed, the practical applications that coulduse the breakdown results, the tool and general methodology are discussed. It is recalledthat the goal was to establish a set of models for the quantification of each loss sourcebased on the previous loss decomposition that was made.

77

Profile boundary layer and mixing loss magnitudes are difficult to reduce. Blade passageis the main driver for the fluid compression and the direct consequence is the appear-ance of profile loss. While secondary flow loss can be reduced by optimising most of thedesign parameter, any change in the blade profile will also have an impact on the bladecompression. Thus, this trade-off reduces the flexibility of the optimisation. Shock lossdue to the leading edge bluntness can be optimised by modifying the inlet blade sectionshape and leading edge thickness. However, passage shock is a direct consequence ofblockage between the blades and is mainly driven by the solidity. To conclude, for agiven inlet mach number, profile loss (Profile boundary layer, wake mixing and passageshock) cannot be avoided and is difficult to optimise.

In this way, the extraction of two-dimensional profile loss out of the overall three-dimensional loss enables to show the remaining loss that can be optimised. Again, theuse of such a tool is especially interesting in the preliminary design stage. It is interestingfor an engineer working on the design of a compressor to have an idea of the loss mag-nitude on which he can work and also the magnitude that is directly linked to the bladedesign. Moreover, the engineer working on the trade-off between compression and loss inthe blade passage also needs the magnitude and percentage of the loss.

11.4 Limitations

Limitations come both from the assumptions that were made for the scope of this thesisand from other external factors. It is important to recall all these limitations in orderto be consistent and to have an idea of the interesting improvements and perspectives ofthe methodology.

The limitations coming from the assumptions have already been enumerated earlier. Thescope of this work was limited to a single two-dimensional blade row with a subsonicrelative inlet mach number. Moreover, the rotor-stator interaction was not taken intoaccount since a uniform inlet was considered for simplicity.

Other limitations come from the approach that was used for profile loss computation.The wake integration can only be performed if there is an access to the integration planeand can only be accurate if this plane is clear of any disturbances. The main potentialdisturbance in two-dimensional cases is passage shock. In fact, one point of the graphwas estimated to be incorrect because of the wake integration. The most probable sourceof error in this case was the interaction with the passage shock and the following shockstructure. In practice, this limitation will therefore appear at higher mach numbers. Inthree-dimensional simulations, other disturbances appear and reduce the accuracy of theintegration. In particular, secondary flows are more important near the endwall and willmodify the velocity profile of the wake just after the trailing edge whatever the inletmach number. Even if this limitation was not encountered during this work because itwas focused on two-dimensional simulations, the limits of the approach have to be keptin mind for future work.

78

Finally, even if a set of models was chosen for the loss breakdown, it is recalled thatKönig ’s model did not gave the best results all over the inlet mach number range. Thus,even if it was preferred to other models, another extension of existing model or combina-tion of models could be used to give even better results over the whole range.

12 Conclusion

In this thesis, the axial compressor was first introduced by justifying the design and char-acteristics. The different methods for identifying loss were enumerated and described.The boundary layer characteristics were introduced. After choosing the total pressuredrop as loss identification, the existing theoretical loss decomposition was presented. Apractical loss breakdown based on loss sources was chosen. The loss mechanisms andloss sources were successively defined and described in order to be consistent with thefollowing loss computation. The different existing models for each loss source were de-scribed. The two-dimensional simulation was detailed with the corresponding parametersand methodology. Then, the models were implemented and compared to each other inorder to finally chose one loss model for each loss source. After having chosen the models,the two-dimensional loss breakdown graphs were shown and the application and limita-tions were specified.

Total pressure drop as a measure of loss was chosen because of the physics behind theconcept and the abundant use of this variable in the literature. Moreover, the strong linkbetween total pressure loss and entropy generation was mathematically developed andphysically explained. It enabled the use of both variables equally when dealing with lossmodels.

The theoretical breakdown was performed by specifying both the source and mechanismapproach. The sources were used for the loss computation since it is easier to find thelocation and to separate sources from each other that interact together downstream. Theexisting loss models were in fact also based on loss source. The loss mechanisms approachis however expected to have applications in computer codes and simulations.

It was chosen to compare two solidity cases with a range of inlet mach number in thetypical range of the stator case. Only one point was chosen to be removed from the graphbecause of the wrong wake integration due to shock appearance at the trailing edge. Theother simulations and loss computations were considered to be consistent and relevant inthe scope of this master thesis.

The simulation was limited to two dimensions, only a very simple geometry was usedand the mesh was relatively coarse. These limitations were due to a lack of compu-tational power and access to existing three-dimensional simulations on real compressorblades. This aspect clearly represents the most important perspective for future research.

After comparison of the different loss source models, a set of models was chosen for thequantitative breakdown. Profile boundary layer loss was chosen to be computed using the

79

compressible model of König. Wake mixing loss was modelled by conservation method-ology based on mass, momentum and energy. Finally passage shock loss was computedusing the oblique shock relations following the model of Koch and Smith.

To conclude, the two-dimensional loss magnitude breakdown graphs and the correspond-ing loss sources definitions enable to extract the first part of loss out of a low pressurecompressor blade row. The resulting tool can already be used for preliminary designpurpose but is expected to be extended in order to be used in every design phase of aentire compressor.

13 Perspectives

This master thesis was strongly limited by the lack of access to simulations and com-putational resource. The scope of the work was limited to a very simple blade row in asubsonic uniform flow, only the two-dimensional part was simulated and the mesh wasrelatively coarse. The main goal was to validate the approach for the generation of ageneral loss computational tool. In order to achieve this goal for general cases, manyextensions of the present work are required. In particular, three-dimensional simulationsmust be computed and validated, real compressor blade rows and stages must be sim-ulated, advanced mesh must be used and additional flow features must be taken intoaccount.

Only the two-dimensional loss computation was performed in the scope of this thesisbut the loss models were developed for each loss source. In this way, the extension ofthe work to the third dimension only requires a three-dimensional simulation. Then, thethree-dimensional case could be validated and the breakdown could be performed. Aninteresting perspective of this extension would be to have a radial distribution of the lossalong the span direction. In this way, the loss breakdown could also be represented interms of the position on the blade height. Then, the mean values could be computed inorder to recover the total loss breakdown in three dimensions.

The extension to real compressor blade row is another very important perspective. Firstly,a real compressor blade should be used with a camber and stagger angle that is morerepresentative. The blade that was used was just a first step for the validation of themodels. Moreover, the rotor case still needs to be extended with the corresponding lead-ing edge shocks and additional specific compressibility features. The rotor applicationrequires to work in a relative frame by taking the relative inlet mach number into ac-count. Secondly, the interaction between the stator-rotor and rotor-stator blade rowsdownstream of the flow still need to be considered. This would first consist in changingthe inlet condition or directly simulate the different blade rows simultaneously. However,this extension is expected to add many difficulties into the analysis. In particular, theappearance of unsteady phenomenon and secondary flow interaction would drasticallyincrease the complexity of the flow features.

The extension of new versions for profile boundary layer loss models giving better resultsover the whole inlet mach number range would be an interesting improvement of the

80

tool. The chosen model did not give the best results at low mach number but was themost accurate in the high mach number range. Therefore, an extension of the modelfor a better accuracy at low mach number would be interesting. In addition to this, thecombined mixing and boundary layer profile loss are overestimated at low mach number.An extension with better results would thus be preferable.

It would also be interesting to analyse and discuss the applicability of the method indifferent configurations. For example, studying different blade section types could drawadditional conclusions for the tool. Moreover, an analysis in terms of the angle of attackwould reveal the performance of the method in off-design conditions, with large separa-tion and increased mixing of the wake with the main flow. In fact, the dependence ofloss on both the mach number and the incidence angle is of importance. Finally, whendealing with three-dimensional simulations, it would be interesting to perform sensitivityanalysis in terms of the three-dimensional design parameters. In particular, it would berelevant to analyse the sensitivity of the loss breakdown in terms of the clearance leakageflow and the cavity injection flow by tuning respectively the rotor clearance spacing andthe stator cavity flow.

81

AppendixesA Three-dimensional version of mixing loss computa-

tion and rotor extensionIn this general case, Vn is the velocity component normal to the integration plane, i.e.the inlet plane of the control volume used to perform the conservation.

Constant computation:

m =

∫2

ρ Vn dS

A =

∫2

[ρ Vx Vn + p ] dS

B =

∫2

ρ Vn Vy dS

C =

∫2

ρ Vn Vz dS

D =

∫2

ρ Vn H dS

Direct results:

V y3 =B

m

V z3 =C

m

H3 =D

mGuess: V x3 = Vx1

Loop on V x3:

ρ3 =m

S V x3

p3 =A− m V x3

S

T 3 =p3

R ρ3

h3 = cp T 3

V(i+1)

x3 =

√2

(H3 − h3 −

1

2V

2

y3 −1

2V

2

z3

)

82

The extension of the mixing loss model also accounts for the relative speed in the case ofa rotor. In this case, the total pressure drop is computed as following.

The radius:r = zmin +

zmax − zminz/Hhc

The relative tangential velocity taking the angular rotation speed Ω into account:

Wt = Vt − Ω r

For the notation:Vr = Vy

The relative velocity:V =

√V 2x + V 2

r +W 2t

The speed of sound:a =

√γ R T

The relative mach number:Mr =

V

a

The relative total pressure after complete mixing:

ptr3 = p3

(1 +

γ − 1

2M2

r

) γγ−1

83

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