numerical investigation of a transonic nozzle guide …1070786/fulltext01.pdf · the combustion...

NUMERICAL INVESTIGATION OF A

TRANSONIC NOZZLE GUIDE VANE

UNDER ELEVATED LOADING

DANILO BOCCADAMO

Master of Science Thesis

Stockholm, Sweden, 2016

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NUMERICAL INVESTIGATION OF A TRANSONIC NOZZLE

GUIDE VANE UNDER ELEVATED LOADING

Danilo Boccadamo

Master of Science Thesis

EGI_2016-093 MSC EKV1168

Department of Energy Technology

Division of Heat and Power Technology

Royal Institute of Technology

100 44 Stockholm, Sweden

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Master of Science Thesis EGI_2016-093 MSC EKV1168

Numerical investigation of a transonic nozzle

guide vane under elevated loading

Danilo Boccadamo

Approved Examiner

Assoc. Prof. Paul Petrie-Repar

Supervisor

Dr. Jens Fridh,

Tek. Lic. Nenad Glodic Commissioner

Contact person

ABSTRACT

Despite many new investigations over the last years, there is no indication that alternative

energy conversion technologies will overtake the place of turbomachinery. Hence thermal

turbines are still the most dominant movers for electricity generation.

Although this leadership in the energy production does not seem to be in danger, the current

drivers in turbomachinery industry are to work towards both less fuel consumption and less

pollution. In order to meet the future economic and environmental goals, researchers press

towards highly loaded vanes and blades. This has to be performed at maintained or improved

aerodynamic performances. Increased performances and blade loading lead in turn to increased

velocities and larger regions of supersonic fluid velocities and consequently general increasing

of shock intensities. The biggest problem dealing with supersonic flow and high shock

intensities is that the boundary layer, when walking through these regions, experiences strong

pressure gradients and intense shock-boundary layer interaction. This may lead the blade to

stall meaning detachment of both boundary layer and cooling-film from the wall. These effects

can evidently lead to catastrophic consequences since nowadays the materials used in

turbomachinery applications have temperature strengths much lower than those coming from

the combustion chamber. This thanks to very complex blade and vane cooling systems.

There are even other features that may take benefit from increased velocities such as an

attenuation in the boundary layer growth and the static pressure distribution on the blade

surface. For helping researchers studying these new geometries, a cold air annular test rig

designed by “Siemens Industrial Turbomachinery AB”, it has been built and placed at “Division

of Heat and Power Technology” at KTH.

The present thesis has the goal to provide a numerical model for CFD calculations, optimized

for boundary layer studies, able to give a good prediction of detachment of the boundary layer

and losses for different working cases. A previous model was provided with a commercial

software for both ideal vane and real test rig. Recover of results and adaptions of the model

were performed with a new version of the same software starting from the previous model. A

comparison between numerical and experimental results have shown a good match for the

subsonic and transonic case. Instead, problems were met for the supersonic case. Many attempts of different boundary condition at the inlet have been run. No reliable solution has been reached

with realistic pressure profile at inlet while realistic results have been found using the mass flow

rate as Inlet boundary condition. At the end, an analysis of shock and detachment is provided

in terms of density gradient and static entropy distribution through the blade passage. Future

works may aim to solve the “supersonic problem”.

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AKNOWLEDGEMENTS

This thesis comes at the end of an experience which I will always bring in my heart. The last

year has allowed me grow from both professional and human point of view and I would feel

guilty to not express my gratitude to some people which have made all of this possible.

First of all I would like to express my gratitude to my supervisors Jens Fridh and Nenad Glodic

who have spent their time to teach me and support me during this work.

I must express the same gratitude to my two examiners Paul-Petrie Repar and Alessandro

Talamelli. The latter in particular for the kind advices he gave me after the presentation to help

me improving my work.

Thanks to my family, Mum, Dad, Debi and Carletto, who gave me support and believed in me

without questioning my choices but driving me instead to choose my dreams.

Sincere gratitude also to my new “Swedish” friends, most of all Andrea. We met each other

almost by incident and you didn’t even like me. However you have been the nicest flatmate and

I really hope our friendship can last for the rest of our life.

Thanks to “Testina” (Filippo), you shared this adventure with me since the beginning and

without you this year would have been much worse. I’m happy to continue to ruin your plans

whenever you want to go clubbing in “Opera” or in “Pineta”……Sboccing like no tomorrow.

Thanks to my dear friends Lucia and Igor who once again have demonstrated what a true

friendship is, no matter of the distance or your Wi-Fi agreement (Igor maybe it’s time to upgrade

it).

Finally thanks to my sweet girlfriend Chiara, who for one year fought against all the people

who were saying the love cannot survive for one year abroad. To you is my deepest and

sincerest gratitude.

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TABLE OF CONTENTS

ABSTRACT…………………………………………………………………………………………..... 4

TABLE OF CONTENTS……………………….…………………………………………………….. 6

LIST OF FIGURES…………………….……………………………………………………………... 8

LIST OF TABLES…………………….………………………………………………………………. 11

NOMENCLATURE…………………….……………………………………………………………... 12

1

INTRODUCTION…………………….……………………………………………..…………………. 14

1.1 Background………………………………………………………............................. 14

1.2 Outcomes of Former Analysis………………………………………………………. 16

1.3 Theoretical Background…………………………………………………………….. 18

1.3.1 Turbulence and Turbulent Boundary Layer…..…………………... 18

1.3.2 Secondary flow……………………………………………………. 22

1.3.3 Aerodynamic Losses……………………………………………… 23

1.3.4 Shockwave Losses and Boundary Layer Interaction……………… 26

1.4 State-of-the-art in External Cooling………………………………………………… 28

1.5 ANSYS ICEM CFD & CFX………………………………………………………... 32

2 MOTIVATION AND OBJECTIVES………………….…………………………………………. 33

2.1 Motivation………………………………………………………............................... 33

2.2 Objectives………………………………………………………................................ 33

3 METHODOLOGY………………….……………………………………………………………... 35

3.1 Research Methodology………………………………………………………............ 35

3.1.1 Literature Study……………………………………………............ 35

3.1.2 Recover of Previous Numerical Model…………………………… 35

3.1.3 Mesh Refinement…………………………………………….......... 35

3.1.4 Matching of Experimental Data…………………………………... 35

3.1.5 Data Analysis……………………………………………................ 36

3.2 Research Limitations……………………………………………............................... 36

4 ANNULAR SECTOR CASCADE FACILITY AT KTH….…………………………………….. 37

4.1 Brief Description of the Rig………………………………........................................ 37

4.2 NGV Geometry……………………………………................................................... 38

5 CFD MODELLING….…………………………………………………………………………….. 41

5.1 Geometry……………………………………............................................................. 41

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5.1.1 Refinement of the Mesh……………………………………........... 42

5.2 Physics of the Model……………………………………........................................... 45

5.3 Mathematical model for turbulence...……………………………………................. 45

5.4 Near-wall treatment ………………………………………………………………… 47

5.5 Boundary Conditions ……………………………………………………………….. 47

6 RESULTS…………………………………………………………………………………………... 49

7 CONCLUSIONS and FUTURE WORK…………………………………………………………. 74

8 REFERENCES……………………………………………………………………………………... 76

APPENDIX A: Mach number comparison ………………………………………………………….. 80

APPENDIX B: density and total pressure …………………………………………………………… 82

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LIST OF FIGURES

Figure 1.1 Modern industrial gas turbine SGT5-8000H (Courtesy of Siemens

Industrial Turbomachinery AB) 14

Figure 1.2 Increase of thermal cycle efficiency through pressure ratio and

turbine inlet temperature over last decades (Birch, 2000).

15

Figure 1.3 Logarithmic law for the boundary layer (Nezu et al., 2000) 20

Figure 1.4 Streamlines showing separation lines in a near endwall plane of linear

blade passage (Acharya and Mahmood, 2006) 22

Figure 1.5 Passage and horseshoe vortices (Sharma and Butler, 1987)

23

Figure 1.6 Spanwise loss distribution at different Mach numbers (Perdichizzi

1989). 27

Figure 1.7 Spanwise exit flow angle distribution at different Mach numbers

(Perdichizzi 1989). 27

Figure 1.8 Schematic of cooling-film configurations on a vane (Bogard, 2006). 28

Figure 1.9 Mixing process of mainstream and coolant jets (Wilfred and Fottner,

1994).

29

Figure 1.10 Flow field in the area of a single cooling jet in cross flow. 29

Figure 1.11 Boundary layer pattern for different momentum-flux ratios (Roux,

2004). 30

Figure 4.1 Scheme of the ASC Arrangement (Glodic, 2008). 37

Figure 4.2 ASC radial view (Saha, 2014). 38

Figure 4.3 Axial cross section of ASC (Saha, 2014). 38

Figure 4.4 Grid of measurement points (Saha, 2014). 39

Figure 4.5 Scheme of profile geometric parameters (Saha R., 2014 40

Figure 5.1 General view of former mesh (model from Schäfer, 2009). 41

Figure 5.2 Detailed view of the blade mesh around vane 0 in former model

(model from Schäfer, 2009). 42

Figure 5.3 View of the entire refined mesh. 43

Figure 5.4 Detailed view of the blade mesh around vane 0 in new model. 43

Figure 5.5 Inlet geometry (up) and cascade geometry (down). 44

Figure 5.6 Detailed view of the mesh quality. 44

Figure 5.7 Detailed view of the angle quality. 45

Figure 5.8 Plot of directed inlet flow (Schäfer, 2009). 48

Figure 5.9 Plot of velocity direction at the inlet (numerical model). 48

Figure 6.1 Experimental stream wise speed at -55.7 % for different grids (Saha

R., 2014).

49

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Figure 6.2 Normalised numerical total pressure profile at -55.7 %. 50

Figure 6.3 Total pressure distribution at Mach 0.95 from (Lilienberg L., 2016). 51

Figure 6.4a Total pressure distribution at Mach 0.95 numerical result. Level gap

10 kPa. 51

Figure 6.4b Total pressure distribution at Mach 0.95 numerical result. Level gap

1 kPa. 52

Figure 6.5 Mass averaged kinetic energy loss at Mach 0.95. 52

Figure 6.6 Area averaged yaw angle at Mach 0.95. 53

Figure 6.7 Static pressure at 15% of the span at Mach 0.95. 54



Figure 6.10 Convergence history of physical parameters Mach 0.95. 56

Figure 6.11 Total pressure distribution at Mach 1.05 from (Lilienberg L., 2016). 57

Figure 6.12 Total pressure distribution at Mach 1.05 numerical result. 57







Figure 6.19 Total pressure distribution at Mach 1.15 experimental result. 62

Figure 6.20 Total pressure distribution at Mach 1.15 numerical result. 62



Figure 6.23 Convergence history of physical parameters Mach 1.15 with MFR at

inlet 65

Figure 6.24 Pressure profile at Station 1 for MRF at inlet Mach 1.1596. 65

Figure 6.25 Total pressure distribution at Mach 1.15 numerical result 66






Figure 6.31 Static pressure at 15% of the span: comparison between Miso = 0.95

and Miso = 1.15.

69


and Miso = 1.15. 70

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and Miso = 1.15. 70

Figure 6.34 Mass averaged kinetic energy losses: comparison between Miso =

0.95 and Miso = 1.15. 71

Figure 6.35 Blade-to-Blade density gradient (left) and static entropy distribution

(right) at 50% of span, Miso=1.15. 72


(right) at 15% of span, Miso=1.15. 72


(right) at 85% of span, Miso=1.15 73

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LIST OF TABLES Table 4.1 Position of measurement points

39

Table 4.2 Vane design parameters (Lilienberg, 2016 and Schäfer, 2009). 40

Table 6.1 Experimental measures from previous campaign (Saha R., Fridh J.,

2015). 47

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NOMENCLATURE Latin Symbols

m Mass flow [kg/s]

u Velocity [m/s]

h Specific Enthalpy [kJ/kg]

Y Mass-flux ratio [%]

M Mach Number [-]

T Temperature [K]

p Pressure [Pa]

R Gas constant [J/(kg*K)]

BR Blowing ration [-]

MR Momentum flux ratio [-]

DR Density ratio [-]

C Chord [m]

Greek Symbols

η Efficiency [-]

φ Velocity coefficient [-]

χ Specific heat capacity ratio [-]

ρ Density [kg/m3]

ϕ Radial traverse angles [°]

Subscription

c Coolant

iso Isentropic

st Static

1 Upstream

2 Downstream

m Mainstream

t Total

ax Axial

Abbreviations

ASC Annular Sector Cascade

TIT Turbine Inlet Temperature

CFD Computational Fluid Dynamics

LE Leading Edge

TE Trailing Edge

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SH Shower Head

SS Suction Side

PS Pressure Side

HPT High Pressure Turbine

NGV Nozzle Guide Vane

MFR Mass Flow Rate

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1 INTRODUCTION

1.1 Background

The enormous growth in industrialization and transportation has caused distinctive public sense

of responsibility about the use of fossil-based fuel and the resulting impact on carbon dioxide

emission and climate change. Despite many new technologies and attractive decentralization

scenarios lead researchers to study alternative energy-conversion systems, actually there are no

indicators that such technologies will overtake the place of turbo machinery if also taken into

account that turbines are even an important part of “more green” electricity plants like small-

scale Combined Heat and Power plants (CHP) and combined cycle plants for future de-

centralized power generation (Fridh, 2012). Many other industrial fields may require the use of

turbomachinery systems, whenever the recovery of energy from a highly charged flow is

needed. Indeed gas turbines are used in power generation units (Figure 1.1) for jet propulsion

(aircraft engines), and in marine propulsion. Today the transportation industries also need to

cut down the amount of carbon dioxide emission and this determines the importance of an

efficiency improvement for matching new environmental goals.

Figure 1.1: Modern industrial gas turbine SGT5-8000H (Courtesy of Siemens Industrial

Turbomachinery AB)

For gas turbines, the fact that the process of energy production involves the combustion of fuel

is the main reason to make these machines as much effective as possible, meaning to maximize

efficiency and energy production and minimize the fuel consumption. By knowing that such

machines are physically based on the Brayton (or Joule) Cycle, a good improvement of the

thermal efficiency in modern gas turbines is achieved by increasing the compressor pressure

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ratio and the “Turbine Inlet Temperature” (TIT). Today the overall pressure ratio, for example

in Trent XWB engine, is already 50:1 (Roll-Royce, 2014) and the trend shown in Figure 1.2

indicates that it will increase even further. For recent aero engines and high performance

industrial gas turbine engines, the TIT is typically equal to or even higher than 1850 K

(Siddique, 2011) and it is higher for military engines with shorter intervals between turbines

overhauls. The temperature level, however, is much less than the stoichiometric combustion

temperature of ≈2500 K, thus there is a continuous push by the industry to increase the

temperature and the thermal efficiency.

The main issue for this feature is that no material nowadays can resist to such temperatures

(temperature strength of the materials used for turbines without occurring of thermal stress

problems is around 1500 K) but nevertheless, due to expensive external and internal vane

cooling systems, it is possible to achieve a TIT that is beyond the melting point. For the power

generation industry the TIT should be as high as possible and the cooling flow (taken from the

compressor) as low as possible in order to achieve high thermal cycle efficiency. On the other

hand, for aircraft engines, a high bypass level is advantageous in order to obtain a higher thrust

with lower specific fuel consumption. In both cases, the efficiency improvement is the key

driver, either by decreasing the cooling or by decreasing overall losses.

Figure 1.2: Increase of thermal cycle efficiency through pressure ratio and turbine inlet temperature

over last decades (Birch, 2000).

In both applications, gas turbine designers have to face many interdisciplinary aspects during

the design process, where the high pressure turbine (HPT) is perhaps the most challenging

component to design due to the fact that it pushes the limits of aerodynamics, heat transfer,

cooling mechanisms and structural criteria in an environment which is extremely hot, corrosive

and unsteady. The HPT is the component where the increased efficiency has the most

significant influence on the overall efficiency as the downstream losses are substantially

affected by the pre-history of the flow. Typically, for one gas turbine unit of roughly 50 MW

with an effective efficiency of 38%, only a 0.1% of improvement in turbine efficiency provides

fuel saving of about 300 tons per year (Mamaev, 2013).

For the aviation industry it is also necessary to reduce as much as possible the weight of the

engine and this give rise even to structural goals. By looking at the trend of development of

modern gas turbines one would say that it is possible to reach such targets. The trend indicates

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that the first stage vane, in the future, will become shorter since the pressure ratio of the

compressor is increasing. At the same time there is a tendency to reduce the number of blades,

which leads in turn to less cost due to the less material required and less cooling demand.

Unluckily, it should be considered also that the gap among blades would increase and the flow

would be less guided through the nozzle guide vane (NGV), affecting negatively the

aerodynamic performances.

The consequential features of the NGV, in the future, will be a very short height and a long

chord blade with complex intensive cooling.

The most used way for reducing the blade temperature is the cooling-film method. This method

implies that a well-designed mass flow rate of cold air is taken from a determined stage of the

compressor, carried by pipes to the turbine and blown by mean of many small holes opportunely

located on the turbine blades and walls. In this way blades and walls surfaces are protected from

the hot gases, coming from the combustion chamber, by a cool air film. This solution is usually

applied although some pressure losses are present along the pipes carrying the cool air. Beside

this method there exist other ways for cooling the blades such as the heat shield which protects

mechanical parts from heat radiation. One of the main characteristics of the cooling-film is that

its functionality is strictly related to the boundary layer. The current drivers in turbo machinery

are to work towards highly loaded vanes and blades in order to meet new future economic and

environmental goals, which evidently become stricter and stricter with the technological

development. This requires in turn to constantly improving the aerodynamic performances

since highly loaded blades means increased velocities and larger regions of supersonic flow,

and consequently a general increase of shock intensities.

Since the boundary layer is strongly affected when passing through these regions of intense

shocks, this may lead to separation of it together with the cooling-film, leading to catastrophic

consequences.

However, to make the idea of highly loaded vane reasonable and fascinating there are certainly

even other features of the flow that may take benefit from the higher speed such as the

attenuation in boundary layer growth or the static pressure distribution along the blade. These

features can lead to better wake characteristics and a less base area although the thin boundary

layer may implies that the vortices inside are small and strong and slightly higher contribution

of energy from the boundary layer is delivered in the wake when detachment happens

1.2 Outcomes of Former Analysis

Former studies on the sector rig have been focused on the characterisation of aerodynamic

losses and thermal characteristic of the blades.

The influence of different tailboard configurations on the flow field was investigated by

Gafurov in 2008. Therefore full three-dimensional steady state simulations have been

performed for the new design of the Annular Sector Cascade (ASC). Two different

configurations, namely the initial case (cascade case) and the ideal case (periodic case) have been used. In the initial model different sidewall configurations were tested to gain better

knowledge about the flow field.

The outcome of this thesis was that the main factors influencing the flow periodicity are the

convergent-divergent (CD) nozzle effect, the tangential pressure gradient and the critical areas.

The impact of the single factors depends on the tailboard’s inclination angle and on the test-rig

load. With inclination angles larger than approximately 75° a premature chocking, downstream

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of the test section, makes impossible to reach the operating point. Instead, inclination angles

less than 75° lead to CD nozzle effect. With an inclination angle of approximately 75° one gets

the best periodicity at the given flow conditions. For the initial case Gafurov came to the result

that the flow periodicity is satisfying without the tailboards, especially in the main passages

FP2 and FP3.

Another investigation was made by Glodic in 2008 where measurements with a five-hole probe

have been performed for cooled and uncooled vane configurations in order to gain information

about how the downstream flow field and secondary flow phenomena are affected by the

cooling-film. Glodic concluded that secondary flows are strongly emphasized in the shroud

region, where a shroud passage vortex is located. Further investigations concluded that the

losses increment due to the cooling-film is predominant in the shroud region, where the

secondary flow vortices are strongly affected by the cooling-film. In this work also the influence

of a heat shield have been investigated. It came out that an influence of this thermal shield can

be observed close to the endwalls, especially for the case with cooling-film.

Rubensdörffer 2006, studied the influence of a heat shield with and without additional cooling.

He concluded that the heat shield in front of the airfoil changes the secondary flow field and

therewith the hub endwall heat transfer which increases.

By studying the configuration where additional cooling air comes from under the heat shield, it

has been found to have only a minor influence on the flow field. This cooling air displaces the

generated vortices only a little. For this case it is prevented that air from the main flow intrudes

under the heat shield.

Schäfer, in 2009, implemented a heat shield in an existing numerical model and performed

extensive parametric study, for different boundary layer conditions, by calculating typical flow

parameters and loss coefficients and compared the results with existing results of a model

without heat shield. His main conclusions were about the flow periodicity of the test cascade

which decreases in general from hub to tip and going downstream. He found no big deviation

between the isentropic Mach number curves for the cascade with and without the heat shield

but the losses were higher with the heat shield in both cascade and periodic case.

Puetz 2010 performed deep measurements on the transonic ASC about the upstream flow field,

blade loading, losses and secondary flow, aiming to discover and provide reasons and suggest

modification of the inlet section. Then the influence of the cooling on the secondary flow has

been studied experimentally by measuring pitch, yaw angle and losses. He concluded that a

change of the location of the turbulence grid can improve the inlet flow field to a satisfying

level: the results show smooth velocity distribution and do not indicate boundary layer

separation at any point. The study has shown also that the injection of cooling-film has a strong

effect on the development of secondary flow. This is especially true in the endwall region,

where leading edge cooling influences the development of the horseshoe vortex.

Saha in 2014 has performed an aerodynamic investigation of the leading edge contouring and

external cooling on the cascade geometry. Saha compared a LE fillet with the baseline case and

concluded that LE fillet has no significant influence on the flow and secondary losses of the

investigated NGV. His measurements have shown no evidence of SS flow separation and the

load distribution had no noticeable effects. Saha even has observed that in general the fileted

case shifts the losses cores towards the midspan direction. Yet the investigation has revealed

that the inlet prehistory of the flow may shift the losses cores and therefore it affects the next

blade row.

Regarding the external cooling system, Saha has observed a continuous increasing of the profile

losses with increasing mass-flux ratio for all kinds of cooling configuration studied. The TE

cooling and SS cooling have the strongest impact on losses in the central portion of the blade

and in general the SS cooling have a stronger impact than the PS cooling.

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Saha and Fridh in 2015 have explored the flow field and aerodynamic losses for high exit Mach

condition. They have carried experimental investigations with increasing mass flow rate and

studied the flow field for exit Mach condition from 0.90 to 1.33. They have concluded that

maximum inlet total pressure was found as 336.103 kPa with PF2 (power of the second fan) =

100% and no downstream blockage (SV9 = 100%) when mass flow was 5kg/s and downstream

static pressure was 115.191 kPa. The maximum inlet total pressure was found instead as

342.412 kPa with PF2 = 100% and with downstream blockage (SV9 = 18%) when mass flow

was 5 kg/s and downstream static pressure was 166.495 kPa.

They also have concluded that the Reynolds number can be maximized by increasing the

downstream static pressure to ~166 kPa with mass flow 5 kg/s but however the Mach number

starts to go down due to the blockage. They explored chocking conditions for mass flow around

3.53 – 3.63 kg/s, pressure ratio of 2.39 – 2.49 and isentropic exit Mach number 1.19 – 1.22.

In these conditions the downstream static pressure remains unaffected by the increase of

upstream mass flow and beyond the choke condition (mass flow 3.63 – 4.03 kg/s) the total

pressure continues to increase but there is no change in the downstream static pressure, which

instead starts to increase for mass flows over 4.03 kg/s, meaning that the flow behaves as a CD

nozzle.

1.3 Theoretical Background

Presentation and discussion of results involve the knowledge of some aerodynamic issues which

play an important role in the studied phenomena. In particular, approaching the results section,

one must know the fundamentals about boundary layer and turbulence phenomenon as well as

compressible flow aerodynamics. Those are not treated deeply here because they are subjects

far too wide for a short description. Here following are described only the main features.

1.3.1 Brief description of Turbulence and Turbulent Boundary Layer

At a certain critical Reynolds Number (Re) the flow, which was before laminar, starts to change

its behaviour and undergo a process called transition. The transition consists in the growing of

a small disturbance which cannot be damped anymore by the flow field when the critical Re is

reached, until the disturbance is spread through the whole domain, which is when the flow can

be called turbulent. There is no definition of turbulent flow, but it is recognised to have a number

of characteristic features (Davidson, 2015):

Irregularity, turbulent flows are irregular, random and chaotic and consist of a spectrum of different scales (eddies size) which goes from the largest scale of the order

of the flow geometry to the smallest eddies which are dissipated by viscous forces into

internal energy. Even if turbulence is chaotic, it is deterministic and it is described by

the Navier-Stokes equations.

Diffusivity, in turbulent flows diffusivity increases, meaning that the spreading rate of boundary layers increases as the flow becomes turbulent. The phenomenon of the

entrainment is raised which increases the exchange of momentum and thereby allows

to delay boundary layer separation. However higher diffusivity means also a higher wall

friction.

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Large Reynolds Number, flows require a Reynolds number high enough for the

perturbation to not be damped.

Three-Dimensional, turbulence is always a 3D phenomenon. However, when the equations are time averaged, the flow is usually treated as two-dimensional.

Dissipation, turbulent flows are dissipative. At the smallest (dissipative) eddy scale the kinetic energy is transformed into internal energy. The largest eddies extract the kinetic

energy from the mean flow. Then the large eddies break into smaller eddies and the

same process undergo until the smallest scales where the kinetic energy is dissipated.

This process is called energy cascade process.

Continuum, even if a wide range of scales is present, the scales are much larger than the molecular scale therefore we can treat the flow as continuum.

The largest scales are of the order of the flow geometry, with length scale l and velocity scale

U, and are those who extract kinetic energy from the mean flow (for which the timescale is of

the same order of the one of the largest scale) as shown below.

𝜕𝑈

𝜕𝑦= 𝒪(𝑇−1) = 𝒪(𝑈/𝑙) (1.1)

Through the cascade process the kinetic energy is in this way transferred from the largest scale

to the smaller ones. At the smallest scale, the viscous forces become too large and the kinetic

energy is dissipated. Certainly the viscous forces are present at all the scales as well as not all

the kinetic energy is transferred from the largest scale to the smallest. However, the smaller the

eddy the higher are the viscous forces and it is assumed that most of the energy (say 90%) that

is extracted by the largest scale is at the end dissipated at the smallest one. The smallest scale

at which dissipation occurs is called Kolmogorov scale: the velocity scale v, the length scale η

and the time scale τ. By indicating the dissipation with ε ([m2/s3]) and the viscosity with υ

([m2/s2]) they can be derived with an easy dimensional analysis which gives:

Velocity scale: 𝑣 = (𝜈𝜀)1/4

Length scale: 𝜂 = (𝜈3

𝜀)1/4

Time scale: 𝜏 = (𝜈

𝜀)

1

2

When flowing close to a wall the flow behaves in a particular way. In fact at the walls the no-

slip condition is effective, meaning that the fluid particle at the surface cannot slip on it and

therefore has the same velocity of the surface. This does not mean that the flow molecules have

0 speed on the surface but that the average of the speeds of the molecules constituting the fluid

particle is 0 (if using the wall-fixed frame of reference).

This condition generates a region of the flow close to the wall in which the velocity goes from

0 at the wall up to the free stream velocity. From the diffusion equation we know that the free

stream velocity is physically reached at infinite far from the wall so that many definitions of

boundary layer have been given during history depending on the feature studied. Examples of

these “wrong” definitions are “...the region in which the flow velocity goes from 0 to 99%

of the free stream velocity” or “…the region where the viscous effects are relevant”. However,

- 20 -

none of these many definitions are universally accepted and an exact definition of boundary

layer does not exist yet.

Whatever definition of boundary layer is used, its thickness is usually called δ. Depending on

the Reynolds Number the boundary layer may behave either in a laminar or turbulent way. The

general structure of the boundary layer it is different for the two cases. However, due to the

nature of this thesis (where a cool-air test-rig is used to study the NGV), the Reynold it remains

very high so that the boundary layer along the blade may be assumed to become turbulent very

soon.

In order to approximate the shape of the “Turbulent boundary layer” (von Kármán, 1930)

proposed the so-called “law of the wall” in which the boundary layer itself can be divided again

into two parts, namely the inner layer and the outer layer, which follow a general logarithmic

formulation.

𝑢+ =1

𝑘𝑙𝑛 𝑦+ + 𝐶+ (1.2)

With:

𝑦+ =𝑦𝑢𝜏

𝜐 is the dimensionless wall coordinate, that is the distance normal to the wall,

much useful in CFD since the position of the first node from the wall allow us to

understand the degree of description of the boundary layer

𝑢𝜏 = √𝜏𝜔

𝜌 is the friction velocity,

𝑢+ =𝑢

𝑢𝜏 is the dimensionless velocity, which is the velocity parallel to the wall as a

function of y,

𝜅 is the Von Kármán constant, which is a dimensionless constant often used in turbulent

modelling and considered to be universal (κ ≈ 0.40),

𝜏𝜔is the wall shear stress,

𝜌 is the density of the flow,

𝐶+ is a constant

Figure 1.3: Logarithmic law for the boundary layer (Nezu et al., 2000)

- 21 -

The inner viscous layer can be divided in turn into 3 different sublayers known as the viscous

sublayer, the buffer layer and the log-law region (Figure 1.3). The boundary layer and

especially the viscous sublayer are responsible for all the shear forces acting on a wall.

Since the beginning of the studies about the boundary layer (Prandtl, 1904) is well known that

the most important assumption is that the pressure distribution throughout the boundary layer

in the direction normal to the surface remains constant. (Schlichting et al., 2006) show that

inside the boundary layer the gradient of the static pressure perpendicular to the wall vanishes;

ps is constant, the static pressure is imposed onto the boundary layer. This differs from the total

pressure since can be expressed as (1.3) which surely can change within the boundary layer.

𝑝𝑡 = 𝑝𝑠 +1

2𝜌𝑢2 (1.3)

Even if usually is not the case, boundary layer separation can be detected also in turbine

cascades. Boundary layer detachment from a wall may occur mainly for two reasons:

Highly curved surface, the negative pressure coefficient increases with the curvature and if the latter is too high then the less pressure normal to the surface and the inertia of

the flow particles may lead to separation of the boundary layer (an example of this is

the separation at a sharp edge of a surface, );

The boundary layer has not enough kinetic energy for penetrating the adverse (positive)

pressure gradient, in which the losses usually increase and can drain out energy from

the flow (this is also the case for the flow passing through shock lines, which can be

seen physically as very strong adverse pressure gradients)

What happen is that in in a region close to the wall a back flow is present. Depending on the

general geometry and flow characteristics, after the boundary layer (which has just met the back

flow) detaches from the wall, can either re-attach to the wall (forming a recirculation bubble)

or goes further away from it. Even if the turbines accelerate the relative speed of the flow since

is present a strong favourable pressure gradient (the natural behaviour of the flow is to stream

towards lower pressure), detachment may occur in small region of increasing static pressure

where usually a recirculation bubble is formed whose increase dissipation of energy and thus

losses are raised.

As it is clear the separation of boundary layer is strongly affected by the surrounding flow field.

The risks of separation are smaller if the flow is accelerated due to the increasing in kinetic

energy. Also the turbulence can affect in a positive way the separation, as a turbulent boundary

layer characterized by a much higher energy level and the strong fluctuations generate the so call “entrainment” which can guide the boundary layer to the wall again. An important feature

of the flow in NGVs is that shock waves can be presents which strongly affect negatively the

separation leading in the most of the cases to the formation of recirculation bubble. Furthermore

the injection of cooling-film may be cause of separation of boundary layer in turbines but this

will be elaborated better in section 1.5. Anyway the separation of boundary layer always causes

rising losses due to the mixing and changes in the effective geometry in which the flow is

flowing.

- 22 -

1.3.2 Secondary Flow

“The term ‘secondary flow’ is usually associated with the small velocity components along the

blade passage and along the span of the blades which are produced by the turning of a shear

flow in a cascade.” (Glynn D.R. and Marsh H., 1980)

In particular this is possible to detect, as stated by Lakshminarayana and Horlok (1963), by the

fact that the deflection of the low momentum fluid equal to the deflection of the mainstream

would not be sufficient to obtain the balance between pressure gradients and centrifugal forces.

The consequence is an overturning of the low momentum fluid with respect to the mainstream.

Figure 1.4: Streamlines showing separation lines in a near endwall plane of linear blade passage

(Acharya and Mahmood, 2006)

Since the very strong and complex relationship between mainstream and secondary flow, even

with the most advanced computers is almost impossible to achieve an accurate simulation of

the flow with a CFD technique. The main knowledge about secondary flow is of experimental

derivation. When the flow approach the NGV the flow field can be divided in 2 regions: the firs

is the LE flow region while the second is the blade passage flow region. The latter can in turn

be separated in 2 more regions: the one around the midspan, far from the endwalls, and the one

of the secondary flow close to the endwalls. (Figure 1.4)

When the boundary layer in the LE flow hits the NGV, it forms a “horseshoe vortex”. This

vortex continues to grow in the NGV passage influencing the mainstream and the secondary

flow.

- 23 -

The flow closer to the endwalls has lower velocity but it is subjected to the same pressure

gradients as the main flow across the NGV. This makes the deviation of this slower part of flow

higher. The higher deviation of the flow indicates the zone where secondary flow is present.

There are even other flow phenomena which affects the passage. The total flow results in a

much more complex flow which consists of crossflows from PS to the SS at the TE, complex

passage vortex and counter rotating secondary vortices behind the TE (Saha R., 2014) (Figure

1.5)

Figure 1.5: Passage and horseshoe vortices (Sharma and Butler, 1987)

There have been numerous studies conducted on linear cascades with high exit velocities, but

these experiments neglect the 3-dimensional pressure gradients which are present in real life

machines. In fact, turning non uniform flows into linear cascades creates 3-dimensional flows

with the flow angle varying along the blade span. This phenomenon becomes even more

important in annular cascade machines where non-uniformity of the flow is a starting parameter

(Glynn and Marsh, 1980).

1.3.3 Aerodynamic Losses

Losses can be defined as the difference between the ideal and real case when analysing the

energy at the end of a physical process (Saha R., 2014). It means physically the existence of a

process which cannot be reversed and where useful energy is wasted.

There exist different sources of aerodynamic losses but, even if not all the loss-creating

mechanisms are fully understood yet (Bartl J., 2010), the largest part is considered coming from

the mixing of the flow and the boundary layer viscosity. These are usually the result of a

frictional interaction either between the flow and the wall or within the fluid itself. Former

studies about NGVs confirm that the largest losses arise from the boundary layer interaction

- 24 -

with the endwalls and endwall vortices which in turn have a big influence on the downstream

flow.

A non-dimensional loss coefficient is common to be used in aerodynamic, which needs to be

adapted for different cases of study. For NGVs with the uncooled condition the loss coefficient

is defined through:

𝜁 = 1 −𝐴𝑐𝑡𝑢𝑎𝑙 𝑒𝑥𝑖𝑡 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑜𝑤

𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑘𝑖𝑛. 𝑒𝑛. 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑖𝑛 𝑠𝑡𝑟𝑒𝑎𝑚

While for the cooled case the energy injected in the system must be considered:

𝜁 = 1 −𝐴𝑐𝑡𝑢𝑎𝑙 𝑒𝑥𝑖𝑡 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑜𝑤

𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑘𝑖𝑛. 𝑒𝑛. 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑖𝑛 𝑠𝑡𝑟𝑒𝑎𝑚 + 𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑘𝑖𝑛. 𝑒𝑛. 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑜𝑙𝑎𝑛𝑡

The efficiency of the cooled system can be expressed as a ratio between the kinetic energies of

coolant and main flow after the cascade and the kinetic energies of coolant and main flow before

the cascade, namely:

𝜂 =0.5(𝑚1 +𝑚𝑐)𝑢2

2

𝑚1ℎ1,𝑖𝑠𝑜 +𝑚𝑐ℎ𝑐,𝑖𝑠𝑜

Where m1 is the main gas mass flow, mc is the coolant mass flow and h represents the enthalpies.

Through the definition of efficiency, we can express the loss coefficient as:

𝜁 = 1 − 𝜂 = 1 − 𝜑21 + 𝑌

1 +𝑌ℎ𝑐,𝑖𝑠𝑜ℎ1,𝑖𝑠𝑜

Where Y is the mass-flux ratio between the coolant and the main flow and φ is the velocity

coefficient which can be represented as:

𝜑 =𝑢2𝑢2,𝑖𝑠𝑜

=𝑀2𝑀2,𝑖𝑠𝑜

√𝑇𝑚𝑖𝑥𝑇1

Where u2 stands for the mixed flow average velocity at the outlet while the “iso” subscription

represents the isentropic values. If we represent the Mach number with M and the temperature

with T, with the help of isentropic relations we can express some useful relations as:

𝑀22

𝑀2,𝑖𝑠𝑜2 =

(1 −𝑝2𝑠𝑡𝑝2)

𝑘−1𝑘

(1 −𝑝2𝑠𝑡𝑝1)

𝑘−1𝑘

- 25 -

ℎ𝑐,𝑖𝑠𝑜ℎ1,𝑖𝑠𝑜

=1 − (

𝑝2𝑠𝑡𝑝𝑐)

𝑘−1𝑘

1 − (𝑝2𝑠𝑡𝑝1)

𝑘−1𝑘

∗𝑇𝑐𝑇1

It must be considered that in real life the losses and efficiencies are dependent on many

parameters which are very difficult to model and simulate. Therefore, they are usually

expressed as quantified terms. In order to do that we need to pose few assumptions:

The flow is isothermal, no temperature difference between the coolant and the main

flow,

The gas constant R and the specific heat capacity χ are the same for the coolant and the main flow,

The mixing between the coolant and the main flow is perfect.

These allow us to express the loss equation for the cooled case as:

𝜁 = 1 −

(1 + 𝑌) (1 − (𝑝2𝑠𝑡𝑝2)

𝑘−1𝑘)

(1 − (𝑝2𝑠𝑡𝑝1)

𝑘−1𝑘 ) + 𝑌(1 − (

𝑝2𝑠𝑡𝑝𝑐)

𝑘−1𝑘 )

From the experimental point of view is possible to note that the uncooled case can be recovered

by setting the flux ratio Y to 0 but this not necessarily applies because of the mentioned

assumptions (Saha R., 2014). In this case we usually speak of primary or enthalpy loss

coefficient.

𝜁 = 1 −(𝑝2𝑠𝑡𝑝2)

𝑘−1𝑘− (𝑝2𝑠𝑡𝑝1)

𝑘−1𝑘

1 − (𝑝2𝑠𝑡𝑝1)

𝑘−1𝑘

Nowadays this loss is divided into endwall loss, profile loss and leakage loss (Denton, 1993)

which are separate and independent. The endwall losses are accountable in the secondary flows

in the boundary layers close to the endwalls. The profile losses are instead created in the

boundary layer but far from the endwalls. Finally the tip leakage is present over the tip of the

blades of rotor and stator. Physically the more or the less relative significance of these losses

are dependent on the characteristic of the machine upon which are studied but usually they are

accountable for 1/3 of the total aerodynamic losses (Denton, 1993). Furthermore, more losses

are in turn generated by the interaction between these losses (Saha R., 2014).

In turbomachinery the determination of the loss coefficient is dependent on several parameters

like geometry parameters, for instance the inlet angle, pitch-to-chord ratio, TE thickness etc.

and flow parameters like Reynolds number and compressibility.

- 26 -

In order to get a precise value of all the contributions, measurements in a real-life machine

would be required. Difficulties arise for this aspect since the operating temperature and the

accessibility of a turbine are problems very difficult to deal with. In this sense experimental

cold-flow models are developed with proper scaling conditions in order to create a flow which

resemble as much as possible the real one.

1.3.4 Shock Waves Losses and Boundary Layer Interaction

Compressible flows started to receive major attention with the advent of jet propulsion and

high-speed flight during World War II and the fundamentals as well as the applications became

classic (calorically perfect gas, exact solutions of one dimensional flows and approximate

solutions for two or three dimensional) (Anderson J.D., 1990).

From roughly 1960 the new era of modern compressible flows characterized by:

1. The necessity of dealing with high-temperature, chemically reacting gases associated to

hypersonic flows and rocket engines.

2. The rise of CFD, which allowed a new way of understanding physical problems,

complementing to the previous existing way of pure experiment and theory.

The modern compressible flow, nowadays is a mutually supportive mixture of classical

analyses along with computational techniques.

With the always increasing speed of aircrafts and turbines, the development of these machines

seems to have reached a limit given by the sound barrier which means the formation of shocks

in the flow and increasing losses. Even if a very wide field of studies have been carried out on

this topic, the physics of the phenomenon is still not fully understood.

For an ideal flow (without the boundary layer) when it approaches an object at velocities above

the speed of sound, a shock is generated, which may be reflected or met by a solid surface. The

shock is the consequence of a strong deceleration of the fluid particles because of an obstacle

on their way and across it the pressure gradient is so high that the increase of pressure can be

considered discontinuous. In a real flow the boundary layer is a subsonic region therefore it

cannot go through a discontinuous pressure change. This results in a channelization of the

irregular pressure change upstream in the subsonic boundary layer. This generates a divergence

in the streamlines which will cause compression waves in the supersonic part of the fluid

(Bodony and Smith, 1986).

In case of supersonic NGVs a characteristic distribution of losses take place. In 1989

Perdichizzi raised the Mach number of a linear cascade from 0.3 up to 1.55. He found that in

the subsonic region the secondary losses are nearly uniformly distributed in the top/bottom

quarter of the span while decrease around the midspan because of the passage vortex. The

higher losses at the tip and hub region are due to the fading corner vortex and to the shear

stresses at the endwalls. In the subsonic part the increasing Mach number causes the losses to decrease since at higher velocities decrease also the expansion of the flow, causing mixing (and

thus secondary losses) to reduce. This is also followed by a reduction of the flow deviation.

- 27 -

Figure 1.6: Spanwise loss distribution at different Mach numbers (Perdichizzi 1989).

Reaching the supersonic region the losses cores increase in magnitude and move more and more

towards the endwalls while they decrease in magnitude in the midspan passage (Figure 1.6).

This is due mainly to that fact that the shock losses occurs mostly in the boundary layer while

the passage vortex is decreasing. The loss cores stop moving when the chocking condition is

reached. Leading the flow to higher Mach numbers, the flow deviation angle decrease in both

over and under turning (Figure 1.7). This is caused by the fact that the primary velocities

increase more than the secondary ones. It can be observed that at high velocities is generated a

small over turning close to the endwalls associated to the increased role of the corner vortex.

Figure 1.7: Spanwise exit flow angle distribution at different Mach numbers (Perdichizzi 1989).

- 28 -

1.4 State-of-the-art in External Cooling

In order to deal with the hot gas temperature, a huge amount of aggressive cooling both internal

and external to the blade is required. The amount of cooling used in the HPT can reach

sometimes the 20 to 30% and it comes with a severe penalty on the thermodynamic efficiency

unless the firing temperature is sufficiently high to outweigh the losses (Bunker, 2006). The

most common and very efficient external cooling system is the so-called cooling-film where

cold air is sucked from the last stages of the compressor, taken directly to the turbine bypassing

the combustion chamber (it needs to be compressed but is not heated up in the combustion

chamber, which causes a thermodynamic loss) and then blown through tiny holes (located in

strategic points of the surface of the turbine blade and endwalls) which generates a protecting

film layer of cooling between the hot gases and the component external surfaces aiming to

protect the surface not only in the immediate region of injection but also in the downstream

surface (Figure 1.8).

However, it must be considered that although the huge thermal benefits, the cooling-film brings

the unwanted phenomenon that mixing process between coolant and mainstream reduces the

aerodynamic efficiency of the turbine whereas the increasing in TIT should lead to an increase

in efficiency. The aerodynamics of the cooling-film as both influence on losses and heat

transfer. This is the reason why is a critical system which needs to be designed properly.

Turbulence level, the approaching boundary layer, holes geometry and pattern have strong

influence on both aerodynamic and cooling performances. In turn the cooling-film may have

effects on transition and shocks structure, adding even more complexity to this research field.

In order to optimize the cooling system, the cooling flow must change its direction starting

being tangent to the wall as soon as possible to form a protective film coverage around the blade

surface (Figure 1.9). The mixing process of the coolant and mainstream in the boundary layer

is three-dimensional, especially in those regions close to the cooling holes (Figure 1.10) which

of course is very critical. As the jet exits the hole and emerges into the free-stream, it undergoes

a bending towards the surface due to the local variation of pressure in the vicinity of the jet.

Figure 1.8: Schematic of cooling-film configurations on a vane (Bogard, 2006).

- 29 -

Consequently, the flow accelerates above and around the jet and decelerate upstream and

downstream of the jet. The deceleration of the upstream flow results in a three-dimensional

separation of the external boundary layer forming a HS vortex that wraps around the jet. The

low pressure zone beneath the jet, in certain cases, can be strong enough to give a zone of

reversed flow.

The bending of the jet, together with the strong shear on the sides of the jets, strengthens the

pair of counter rotating vortices. Thus a kidney shaped vortex is formed, which can entrain the

mainstream fluid towards the surface which is cooled. Immediately after the bending of the jet

completes, pressure forces can be small and the jet follows the local flow. After some distance

downstream, the cooling jets can merge to build a blanket of cooling flow. More than one row

of coolant injection is used because coolant coming from upstream can help in forming the

blanket and in reducing the influence of the kidney-shaped vortex in size and momentum. This

allows to reduce the associated mixing with the main flow. Again looking at Figure 1.9, 1.10

and 1.11, is evident that the success of the injection depends on the injection angle which has a

strong influence on the boundary layer just after the injection hole.

Figure 1.9: Mixing process of mainstream and coolant jets (Wilfred and Fottner, 1994).

Cooling-film aerodynamic (and thus even the heat transfer) may be influenced by a lot of

parameters. The most important ones, used to explore the characteristics of the cooling-film

associated with the injection cooling air in relation to the mainstream flow are the mass-flux

ratio (Y), blowing ratio (BR), momentum-flux ration (MR) and density ration (DR).

Figure 1.10: Flow field in the area of a single cooling jet in cross flow.

- 30 -

Mass-flux ratio, 𝑌 =�̇�𝑐

�̇�𝑝

Blowing ration, 𝐵𝑅 =𝜌𝑐𝑢𝑐

𝜌𝑓𝑠𝑢𝑓𝑠

Momentum-flux ratio, 𝑀𝑅 =𝜌𝑐𝑢𝑐

2

𝜌𝑓𝑠𝑢𝑓𝑠2

Density ratio, 𝐷𝑅 =𝜌𝑐

𝜌𝑓𝑠

Where c refers to cooling air, fs refers to the local free stream and p to the primary flow. The

BR can be described with the quantities of the upstream flow field or local quantities at the

different cooling rows. It is a measure of the mass-flux injected into the boundary layer and

provides detailed information about the local velocity ratios, which are useful for the mixing

process whereas the momentum-flux ratio determines the dynamics of the flow field and thus

penetration of the jet into the mainstream. Therefore, momentum-flux ratio is relevant for the

behaviour of the coolant just after it has been blown from the injection hole (Figure 1.11).

Hence, the momentum-flux ratio value indicates whether the coolant jet is attached to the

surface or detached. The coolant can remain attached to the surface, it can detach and reattach,

or it can lift-off completely for high momentum-flux ratio values.

It is important to scale both velocity and density fields to match realistic engine conditions since

in real gas turbine engine, there is a temperature difference between the mainstream and the

cooling that leads to a DR of about two or more. It is difficult to match the temperature ratio

required to achieve the engine representation conditions in a lab experiment. One methodical

solution is to use a heavier foreign gas, for example CO2, CO or SF6 in order to simulate colder

injection flow to match the DR in heat transfer and aerodynamics effects. Consequently, the

use of foreign gases in cooling-film experiments has been commonly applied as a tracer gas.

Figure 1.11: boundary layer pattern for different momentum-flux ratios (Roux, 2004).

A great deal of understanding can be achieved by using foreign gases to act as a tracer gas

(Jones, 1999 and Goldstein, 1971). The most common foreign gas used in literature is CO2 that

has a DR=1.53 at standard conditions. It should be pointed out that air (k = 1.4) and CO2 (k =

1.3) behave differently from a thermodynamic point of view so the question that the DR can

have an impact on the loss measurement of a cooled blade can be raised.

- 31 -

In general, external cooling is used on the leading edge, suction side, pressure side, trailing edge

and on the endwalls. The endwall cooling can be on platforms inside or outside of the passages.

The LE of the turbine’s first stages experiences the highest thermal loads within the engine and

thus a considerable amount of compressor bleed air is used for managing the high temperature

in this region. Usually, multiple rows of coolant holes are used in a staggered manner in the LE

zone, this is sometimes known as Shower Head (SH) cooling. The SH cooling is very critical

due to the fact that it not only protects the LE region but also affect the aerodynamics and heat

transfer over the entire airfoil surface. Unfortunately, the cooling-film effectiveness is very poor

around the LE because of the nature of the complex flow in the region, such as the lack of

boundary layer in the stagnation line, strong pressure gradients, highly curved streamlines, high

strain rate of the fluid, multiple cooling-film interactions, the high turbulence level and large

length scale and coolant injection nearly opposite to the main flow direction. Because of all of

these phenomena the actual stagnation line is not fixed in turbines and it is very unsteady in

location and hence an extra row cooling is usually put on each side of the expected stagnation

line. The design of such a cooling-film system is one of the most challenging features for turbine

engineers, starting from the good calculation of the LE edge stagnation line that, regarding the

SH cooling, will regulate whether the coolant will go to the PS or SS of the vane. It is possible

to find in literature investigations about changes of loss level due to the coolant injection

(Osnaghi et al., 1997, Kolen et al., 1995, Yamamoto et al., 1991). An increasing of the loss

level can be detected due to the SH cooling with injection from the LE (against the main stream)

and PS. However a small amount of injection at the LE, if well designed, can decrease the loss

by energizing the local low momentum fluid. Still there are tests which detected reduction of

losses due to SS injection. It must be considered that all such tests just mentioned strongly

depend on the coolant injection configuration used during the experimentation. In general it has

been reported that the SH cooling can either cause decrease of losses (Reiss et al., 2000) or

boundary layer separation in LE region, resulting thus in both higher aerodynamic losses and

augmentation of the local heat transfer (Bunker, 2005). It has also been reported that SS

cooling-film has much greater impact on capacity than PS cooling-film (Hambidge et al., 2012)

and that SS and rear part of endwalls contribute most to mixing losses (Bunker, 2005). Usually

the post-throat injection shows higher thermodynamic losses compared to a combination of pre-

and post-throat injection (Stephan et al, 2010) and aerodynamic losses due to mixing are

significantly greater than those due to oblique TE shock waves (Jackson et al., 2000).

Losses can be reduced due to an optimum TE coolant injection. In fact this increases the base

pressure level and thereby a reduction of losses is achieved (Sieverding et al., 1996, Kapteijn

et al., 1996). Evidently the mass flow injection must be properly designed since, from 0% up

to near 3% of TE injection, the loss level increases and start to decrease again around 5% (Uzol

et al., 2001). Of course this depends also on the studied geometry which can be affected more

or less by the same TE injection configuration.

But it must be considered that the cooling location plays an important role in choosing the

amount and configuration of the coolant injection and this runs often against the cycle efficiency

the system is spread along the whole machine and affects the thermodynamic in all of it.

Therefore, although previous investigations show better aerodynamic performances with a high

rate of TE cooling, in the end it is not practical from a cycle efficiency point of view.

- 32 -

1.5 ANSYS ICEM CFD & ANSYS CFX

Although experimental investigations give quite good results of fluid flow investigations,

numerical techniques are widely used. This is mainly due to the possibility of running a lot of

simulations cases and reveal flow details that are difficult to experimentally measure, provided

that the numerical model is adequately validated. Thanks to this nowadays is possible to save

money by choosing which experiments are to be performed after finding interesting cases

between wide ranges of cases computed with numerical techniques

ANSYS ICEM CFX is a universal mesh generator and pre-processor for analysis including

FEA, CFD and other CAE applications. One of its essential advantages is the direct CAD

interface. This is available for all common CAD-programs. This is a big advantage for the

present thesis where the CAD model is given as an input. All commercial meshing programs

feature such interfaces.

Through the direct CAD Interfaces boundary conditions and grid size specifications can be

applied in the CAD system, and carried all the way through to the solver. Even throughout

design modifications the whole information is preserved with the original CAD model which

makes the parametric modelling much simpler. The ANSYS ICEM CFD Hexa meshing tool,

which is used in this thesis, is based on a global block topology. It provides a top down approach

to generate the grid. ICEM CFD Hexa is a semi-automated meshing module and presents

generation of multi-block structured or unstructured hexahedral volume meshes. In this tool the

user can adjust the blockings to the underlying CAD geometry by himself. Body fitted internal

or external O-Grids can be generated by the system automatically. The grid is projected onto

the underlying CAD geometry automatically.

In this thesis the Shear Stress Transport (SST) for turbulence model is used. The advantage of

this turbulence model is that it combines the k-ω-model and the k-ε-model. The k-ω-model has

advantages in the area close to the wall while the k-ε-model hast advantages in the area far from

the wall. The SST model was developed by Menter (Menter, 2003).

Both k-ω-model and the k-ε-model belong to a particular RANS modelling named 2-equations

models. Since CFX solve the modelled math instead of the real equations of fluid dynamics,

when high accuracy is requested the problem of what kinds of issues are involved with these

simplifications must be posed. Generally, the 2-equations models do not take into account the

turbulence damping (both positive and negative) due to the curvatures, which here are strongly

present. The reason of this is implicit into the equations.

- 33 -

2 MOTIVATION and OBJECTIVES

2.1 Motivations

Designing a turbo machine today is a very complex challenge. On one hand the aim is it to

increase the TIT and the vane loading on the other hand there is a desire to reduce the fuel

consumption and the number of cooled parts. Cooling hot parts in turbo machinery is attended

by aerodynamic losses and increasing costs. It is possible to reduce the required cooling by

using a “heat shield”, which can be applied on the cascade in different configurations. Still

when the load increases a very complex flow take place through the turbine. New cooling

configurations cannot result efficient if the boundary layer detaches from the blade, making the

blade walls experiencing the high temperature and causing structural damages. Regarding this

subject, one can find in literature that studies with test data and physical description of the

phenomenon lack of sufficient explanations. Not many in-deep studies on boundary layer in

highly loaded turbines blades have been performed till now since the attention of many

researchers has been focused on increasing the TIT with new materials and cooling

configurations.

It is well known that the aerodynamic features of the cooling aspects are extremely complex

and a good understanding and prediction of the flow in which the external cooling is blown is

essential in order to be able to further increase turbine efficiency. Very detailed studies were

reported in literature regarding cooling systems on SS, PS, TE and SH cooling. The

aerodynamic effects of the external cooling have been studied in depth and many injection

configurations were developed.

Based on the above a new ASC for an existing research rig, used for aerodynamic cold flow

measurements has been designed and tested at HPT in KTH. The project is conducted in

collaboration with Industry. The new sector contains 4 passages from the 1st nozzle guide vane

row in an industrial gas turbine.

CFD calculations have given deep knowledge of the flow field in the past and are now used

again to give deep knowledge of the interaction between the boundary layer region and

shockwaves. In comparison to test rig measurements those calculations are less costly, much

less labour intensive and easier to adapt to new boundary conditions. In addition, also geometry

adjustments can be realised much faster.

2.2 Objectives

The main goals of the present study aim to provide deep knowledge of the boundary layer region

in terms of pressure distribution and velocity field, focusing on features which may show

detachment of both boundary layer and cooling-film. This can be achieved by means of some steps:

Recover of the numerical model from former studies on the test rig;

Dealing with the mesh properties of the cascade geometry trying to optimize (make more resolved) the numerical model;

- 34 -

Numerically studying the behaviour of the flow in the experimental geometry, in highly

loaded blade condition, with a commercial software, by performing a parametric study

of the key parameters of the flow such as the pressure profile (both static and total),

velocity profile and vorticity distribution;

Try to predict, both qualitatively and quantitatively, how the boundary layer, and thus the cooling-film, behave around the blade in terms of detachment from the blade wall;

Comparison of the numerical results with experimental data taken with the ASC, at the Heat and Power Lab at the Department of Energy Technology.

- 35 -

3 METHODOLOGY

3.1 Research Methodology

Since the work comes from different studies on different subjects about the test rig, in order to

reach the objectives set at the beginning a plan was needed to allow to focus on different

problems without make any confusion.

3.1.1 Literature Study

The work on this thesis started with an extensive literature study to achieve the knowledge and

know-how about the cascade aerodynamics and existing system for improving the aerodynamic

and thermic losses. Papers about boundary layer separation have been reed. Most of them are

previous thesis or can be found in ASME Journals, many of them are written regarding the same

test rig studied in this thesis.

3.1.2 Recover of Previous Numerical Model

Much time has been spent working with the CFD software ANSYS ICEM and ANSYS CFX,

which have been chosen for this thesis due to the fact that the starting points are files computed

with these two software. A coarse-mesh (≈ 700’000 nodes) was created by former master thesis

student at HPT (Schäfer L., 2009) using ANSYS ICEM (Ver.11.0) and the flow was studied

with ANSYS CFX (Ver.11.0). The numerical model of those simulations have been recovered

with the new version of the software (Ver. 16.0) and made working again. This involves also

the old boundary conditions files.

Many solutions run with the starting mesh have been analysed for both validate the previous

model and achieving knowledge about how to extract useful information.

3.1.3 Mesh refinement

Then an in-deep study for the mesh adaption it has been carried on focusing mainly on possible

ways of modelling the most difficult region of the flow (the contouring and the trailing edge of

the blades). The main objective was to bring at least 10 cells into the boundary layer. After

many attempts a mesh has been achieved with roughly 5 million of nodes.

3.1.4 Matching of Experimental Data

With the refined mesh simulations have been run in order to match experimental results of high

relevance. As a key parameters the Miso at 136.5 % Cax and the pressure profile along the span

at -51.7% Cax have been used. For this mesh it is important to mention that the computer used

for the calculations have not enough RAM to run the simulations in double precision. To be

- 36 -

sure that the simulation result has physical meaning, some physical parameters (such as Blade

Loading and MFR) were kept monitored during the simulation.

3.1.5 Data Analysis

After all the validation of the numerical model, an analysis has been performed to study the

losses in the blade at different Mach numbers. The results will give an idea to researchers about

if more improvements shall be made on the geometry mounted into the test rig.

3.2 Research Limitations

As said above, in the present research the problem of periodicity is posed. This implies that

only results on vane 0 can be considered reliable during the experimentation in the test rig.

Here, the conditions are, in general, different from those in a real turbine. The test rig is scaled

in terms of geometry and Mach number but the Reynolds number similarity cannot be reached.

For the losses, the Mach number similarity is of highest importance at the high Reynolds

number developed in the rig. On the other hand, for heat transfer studies the Reynolds number

is more important. The high energy carried by the flow in the real engine means also that the

level of turbulence is most probably different if compared between the experimental and real

case.

The profile loss values increase with the increase of turbulent intensity whereas the secondary

loss is less sensitive (Gregory-Smith D.G. et. al, 1992 and Mamaev B.I., 2010). For the cooled

vane case, the temperature ratio is approximately unity as well as the DR is unity. In real engines

DR is usually greater than unity. Hence the experimental condition of the flow field created

inside the test rig are to be considered only a simulation of the real engine. Nevertheless, the

investigations carried on at the KTH facility have been performed for comparison purposes so

that the relative changes should still hold true for these kind of investigations.

- 37 -

4 ANNULAR SECTOR CASCADE

FACILITY AT KTH

4.1 Brief Description of the Rig

The first version of the annular sector cascade test facility, which is installed at the Heat and

Power Technology Department at KTH, was built in 1998 in cooperation with Siemens

Industrial Turbomachinery AB in Finspång. It has been used in several experimental

investigations of steady and unsteady effects of cooling-film. The main goals of these

investigations are to increase the gas turbine efficiency and provide extensive validation data

for calculations. The test rig facility comprises of settling chamber, intermediary section, a

turbulent grid, a test-section and an outlet. In Figure 4.1 the “Annular Sector Cascade” is

marked in red. The intermediary section is used to get smooth transition from round shape of

settling chamber to the sector shape

Figure 4.1: Scheme of the ASC Arrangement (Glodic, 2008).

A screw compressor provides the air to the facility. This compressor is driven by a 1 MW

electric motor. The maximum mass flow of the compressor is 4.7 kg/s at 4 bars. Since the

temperature from the compressor is relatively high (approximately 180ºC) an air cooling system

is installed. This cools the air down to 40ºC (313 K). A more detailed depiction of the test

facility at KTH, including the wind tunnel arrangement, test section configuration and

instrumentation can be found in Roux, 2004 and Glodic, 2008.

Evidently, the 1MW of energy consumption needed for running the compressor of the sector

rig is the main reason why the facility is wanted to be used only for validating the numerical

results. In fact, it would be too costly to avoid numerical simulations and to acquire directly

experimental data and the possibility of easily change the geometry plays only a second role.

- 38 -

4.2 NGV Geometry

It is important to focus on the annular section (Figure 4.2) which is the part re-created into the

numerical model. There is a turbulence grid, which is responsible for producing different inlet

pressure profiles. In this study it has been simulated a 2% of turbulence given by a parallel bar

turbulence grid. After the turbulence grid the flow hits the ASC which has an opening of 36°

and consists of three NGVs and two sidewalls.

Figure 4.2: ASC radial view (Saha, 2014).

During the post-processing of the numerical simulations the data are extracted from the same

stations at which are located the sensors in the real test rig. All the measurement points up or

downstream are shown in Figure 4.3 and the locations of these points are presented in Table

4.1

Figure 4.3: Axial cross section of ASC (Saha, 2014).

- 39 -

Table 4.1: Position of measurement points

Measurement points Location (reference at LE hub) Notes

Turbulence grid -264% Cax,hub ≈ 2% turbulence intensity

1 -55.7 Cax,hub Upstream traverse location

2 107.7 Cax,hub Downstream traverse location

3 136.5 Cax,hub Hub pressure taps

The downstream measurements in the real test rig are taken at station 2, with an L-probe, and

at station 3 with 31 static pressure taps positioned on the hub. The L-probe is mounted on a

traverse mechanism which span the whole area behind NGV 0. This lead to 1443 measurement

points, 39 tangential and 37 radial covering 6% - 96% of the span (Figure 4.4).

Figure 4.4: Grid of measurement points (Saha, 2014).

Note that unlike the convention Figure 4.4 shows the measurement points looking from the

downstream. This means the left side of NGV 0 is the SS and the right side is the PS.

The geometry of the blades is the result of former studies whose aim was to reduce the number

of blades and the secondary losses coming from the fact that the flow is less guided (Figure

4.5). Respect to the initial geometry the main improvement has been made with the optimization

of the LE contouring which was a main breakthrough in the reduction of losses coming from

the horseshoe vortex and corner vortices.

- 40 -

Figure 4.5: Scheme of profile geometric parameters (Saha R., 2014).

The existing version of the testing facility is designed to produce similar flow conditions as

they may be found in a first stator row of a highly loaded (transonic) stationary high-pressure

gas turbine. The geometry has three blades and four passages with high angle of attack. In table

4.1 it is possible to find the main design parameters for the existing NGVs.

Table 4.2: Vane design parameters (Lilienberg, 2016 and Schäfer, 2009).

Design Parameter Denomination Value Unit

True chord at midspan C 0.1292 m

Axial chord at midspan Cax 0.0665 m

Axial chord at hub radius Cax,hub 0.0625 m

Pitch-to-chord ratio at midspan (TE) S/C 0,826 -

Hub radius at exit RTE,hub 0.6153 m

Outer radius at exit RTE,tip 0.6751 m

Tip-to-hub ratio at exit RTE,tip/ RTE,hub 1,097 -

Aspect ratio based on TE vane

height

HTE/C 0,463 -

Inlet metal angle - 90 deg

Reference effective exit angle α2 16,05 deg

Stagger angle x 33.3 deg

LE radius - 0.0138 m

TE radius - 0.0014 m

Uncovered turning angle - 19 deg

The blades in the test rig have also cooling holes covering four different regions: shower head

(SH), suction side (SS), pressure side (PS) and trailing edge (TE). However, these holes have

not been simulated into the numerical model since the purpose of this thesis is to check the

interaction between boundary layer and shockwaves for which the presence of the cooling-film

can be neglected thanks to the initial assumption of isothermal flow (see chap. 1.3.3).

- 41 -

5 CFD MODELLING

5.1 Geometry

The Computational Fluid Dynamics (CFD) tool is used in this thesis to simulate the flow in the

studied geometry. Since the equations of motion are modelled and solved, all the physical

aspects of the flow are considered so not only the velocity components are defined but also

other features like the heat transfer and the turbulence percentage. With the boundary conditions

we can simulate different kinds of the incoming flow at the inlet.

For this thesis the CFD process entails the following steps:

Recover the geometry from former studies (Schäfer L., 2009);

Blocking;

Generating the mesh;

Define the physics of the model;

Solve the CFD problem;

Analyse the results.

The importance of the first 4 steps is easy to understand: the quality of the results (level of

similarity between the model and the real case) is mostly determined here. One has to take care

that the density of the nodes is good enough in each region of the flow. The number of cells

must be both higher in those regions where high resolution is required and smaller in those of

no interest. This in order to reduce as much as possible the CPU time. Solving the CFD problem

and demonstrating the results is mostly done automatically. The phases of the recovery of the

geometry and the computation of a new mesh have been carried out by using ANSYS ICEM,

while for solving the CFD problem and visualising the results ANSYS CFX and Matlab have

been used.

Figure 5.1: General view of former mesh (model from Schäfer, 2009).

- 42 -

For the purpose of investigating the losses on highly loaded blades, the model from former

study shown in Figure 5.1 and 5.2 has been used as a starting point. The model was a coarse

mesh with 792865 node and the lowest y+ ≈ 293 for the case Miso=0.9. The lowest y+ was on

the SS of the NGV 0 at about 50% of the true chord.

Figure 5.2: Detailed view of the blade mesh around vane 0 in former model (model from Schäfer,

2009).

5.1.1 Refinement of the Mesh

The quality and the refinement level of the mesh play a major role in the CDF process. The

perfect target that must be followed is to have non-skewed volumes which means that the corner

angles of the cells must be as close as possible to 90°. However for many geometries it is almost

impossible to achieve non-skewed cells so that the quality of the mesh must always be judged

either if good enough or not. The refinement and the quality of the mesh affect the resolution

of the flow field. Especially in the near-wall area the mesh has to be highly refined due to the

fact that the no-slip condition is applied on the wall and this generates strong gradients in the

distribution of the physical parameters.

The first target was to make the y+ value low enough in the region of interest of the blade surface

which is the SS of the second half-chord. The final value of y+ has been judged a good

compromise between the accuracy of boundary layer description and size of the mesh (in terms

of number of nodes). In fact the final mesh resulted in roughly 5 million nodes and the y+ value,

for the high subsonic case (Miso = 0.95), goes from 3 at the LE to the range of 40-60 at 80% -

90% of Cax and down to 10 at the TE.

For numerical reasons (which will be explained in the paragraph 5.3.1), in order to have a very

good description of the boundary layer, it would have been better to bring the y+ value below

10 (close to 1). Unfortunately, due to the nature of the commercial software used to build the

mesh it has been not possible to satisfy this requirement. However, this issue is taken into

account inside the solver which therefore is capable to guarantee a very good accuracy of the

boundary layer. Moreover it has been achieved a satisfying y+ distribution along the whole PS.

Note however that the region of interest is the second half-chord of the SS. Details of the refined

mesh are shown in Figure 5.3 and 5.4.

- 43 -

Figure 5.3: View of the entire refined mesh.

Figure 5.4: Detailed view of the blade mesh around vane 0 in new model.

The quality of the mesh has been checked and improved by smoothing modules using

determinant, angle and volumes as reference parameters. Indeed judging the mesh only by

volumes (no negative volumes) does not ensure the requirement on non-skewed cells. High

skewed cells are still deemed as negative volumes in the ANSYS CFX-Solver while in ANSYS

ICEM these volumes are not detected as negative. Therefore, the angles and determinant as well

have been used as quality criteria. The entire geometry is divided into two main part: the Inlet,

from the turbulence grid to the end of the heat shield, and the Cascade, from after the heat shield

to the Outlet (Figure 5.5).

- 44 -

Figure 5.5: Inlet geometry (up) and cascade geometry (down).

The Inlet part was the easiest part to model, therefore the mesh is of higher quality. For this part

the volumes range from 0.012 to 140 and the determinant 3x3x3 goes from 0.659 to 1. Particular

attention have been spent on the interface region for this part for which the model was a bit

trickier to be achieved.

The mesh quality regarding volumes and determinant 3x3x3 is shown below (Figure 5.6). As it

can be seen, the range of the 3x3x3 determinant lasts from 0.32 to 1.

Figure 5.6: Detailed view of the mesh quality.

Low values (<0.4) in this criterion mean that highly skewed cells can be found in the area on

the hub and shroud close to the right endwall. In this area the cells are skewed by the reason

- 45 -

that here there is a non-well matching of the right wall at the TE in the geometry. Instead of

fixing it by CAD software it has been chosen to leave it like this since it is probably due to a

real non-well matching into the ASC. A lot of effort was put into this problem in order to reach

values bigger than zero. Also the angle is a criterion of importance for which a range between

4.95° and 90° has been achieved (Figure 5.7).

Figure 5.7: Detailed view of the angle quality.

5.2 Physics of the model

In the ANSYS CFX pre-processor the fluid flowing through the test section is modelled as an

adiabatic compressible flow. In fact we set the simulation to solve an “Air Idea Gas” with the

“Total Energy” model for the heat transfer. The boundary walls, the heat shield and the vanes

are defined as smooth adiabatic no-slip surfaces.

In this project the k-ω based Shear Stress Transport (SST) model is used for the turbulence

modelling. The choice this turbulence model has been made on the base that it is commonly

known as the most accurate 2-equations model. This model was designed to give a highly

accurate predictions of the onset and the amount of flow separation under adverse pressure

gradients by the inclusion of transport effects into the formulation of the eddy-viscosity. This

results in a major improvement in terms of flow separation predictions (Bardina et al., 1997).

5.3 Mathematical model for turbulence

As said in previous chapter (see 1.3.1) a turbulent flow is characterized by swirling structures

which span a wide range of scales (Johansson & Wallin, 2012). If we consider the ratio between

the largest and the smallest scale, it is possible to demonstrate that this ration depends on the

Reynolds number and can be approximated as Re3/4. This holds for only one dimension of the

flow therefore, for a DNS simulation, a grid which aims to describe all the turbulence scales in

a 3D domain must have a number of Re9/4, which is evidently too high (Re at least in the order

of 104). In order to reduce the required number of nodes and consequently saving CPU time,

different models of the equations of motion have been developed for turbulent flow.

The mathematical model implemented in ANSYS CFX describes the turbulent flow by means

of RANS, for which the turbulence is seen as an average velocity field where all the fluctuations

have been damp out due to the action of an additional viscosity known as turbulent viscosity or

eddy viscosity. From the mathematical point of view this assumption generates an extra term in

the Navier-Stokes equations, the Reynolds Stress, which is unknown and must be modelled by

- 46 -

means of a turbulence model which consists in the derivation of two new independent equations

which allow to close the mathematical problem. For this reason these models are also known

as 2-equations models

The most common turbulence models of this kind are the k-ε and the k-ω. When added to the

set of RANS these models account for the transport of the turbulence parameters. As the name

may suggests the former model adds one equation for the turbulent kinetic energy “k”, and one

equation for the turbulence dissipation “ε”. The second model instead replaces the ε-equation

with one for the specific turbulent dissipation “ω” (𝜔 ≝𝜀

𝑘).

They are quite similar since both rely on the assumption that the Reynolds Stress term is related

to the gradient of the mean velocity (strain) through the turbulent viscosity, according to the

Boussinesq hypothesis (empirical)

−𝑢𝑖′𝑢𝑗′̅̅ ̅̅ ̅̅⏟ 𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠𝑆𝑡𝑟𝑒𝑠𝑠

= 𝜈𝑡⏟𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦

∗ 𝑓 (𝜕𝑈𝑖𝜕𝑥𝑗)

⏟ 𝑆𝑡𝑟𝑎𝑖𝑛

, 𝑖, 𝑗 ∈ {1,2,3}

The main difference in the behaviour of the two models is that the k-ε model is reported to have

a general good behaviour but it starts to give huge numerical problems in the ε-equation when

both terms k and ε go to zero and it is not applicable to flows under adverse pressure gradients.

On the other hand, the k-ω model may provide highly accurate prediction of flow separation

caused by adverse pressure gradient if compared to the previous one (Davidson, 2015).

Therefore, the advantage of the ω-equation is in the near wall treatment for low-Reynolds

number computation where it is more accurate than the ε-equation in predicting the boundary

layer separation. In this thesis a hybrid k-ω based SST model is used, since it combines the

advantages of both models as it shifts form k-ω to k-ε, and vice versa, depending of the distance

from the wall (Valentini, 2016).

The SST k-ω formulation is shown below:

Turbulent Kinetic Energy:

𝜕𝑘

𝜕𝑡+𝜕(𝑈𝑗𝑘)

𝜕𝑥𝑗= 𝑃𝑘 − 𝛽

∗𝑘𝜔 +𝜕

𝜕𝑥𝑗[(𝑣 +

𝑣𝑡𝜎𝑘3)𝜕𝑘

𝜕𝑥𝑗]

Specific Dissipation Rate:

𝜕𝜔

𝜕𝑡+𝜕(𝑈𝑗𝜔)

𝜕𝑥𝑗= 𝛼3

𝜔

𝑘𝑃𝑘 − 𝛽3𝜔

2 +𝜕

𝜕𝑥𝑗[(𝑣 +

𝑣𝑡𝜎𝜔3)𝜕𝜔

𝜕𝑥𝑗] + 2(1 − 𝐹1)

1

𝜎𝜔2𝜔

𝜕𝑘

𝜕𝑥𝑗

𝜕𝜔

𝜕𝑥𝑖

where 𝑃𝑘 is the term responsible for the production of turbulence and depends on the Reynolds

stress (which is modelled as a function of 𝑣𝑡 ). 𝐹1 ∈ [0,1] is called blending function and represent the term of the equation which accounts for the position with respect to the wall. Thus, this term is the one responsible for shifting the model

from k-ε (F1 = 0) to the k-ω (F1 = 1). The blending function is formulated empirically.

The constants which appear in the model (β*, β3, σk3, … ) are linear combinations of the

coefficients appearing in the formulations of k-ε and k-ω through the blending function, i.e.

taken ϕ as a general constant it holds 𝜙𝑆𝑆𝑇 = 𝐹1𝜙𝑘−𝜔 + (1 − 𝐹1)𝜙𝑘−𝜀.

- 47 -

5.4 Near-wall treatment

As said in chapter 5.1.1, the SST model for turbulence require a very good resolution of the

boundary layer as the model switch from k-ε to k-ω based on the distance from the wall.

Although the target y+ has been set taking into account this feature, it is difficult to ensure that

the final value will respect the initial requirement. This due to its dependence on the Re.

When solving the near-wall flow by means of a SST model, two different approach may be

used.

The firs one is the low Reynolds number method which solves the boundary layer in all its

physical characteristics in the usual way, meaning that it lets the flow to develop node-by-node

starting from the no-slip boundary condition at the wall. However, for an accurate solution of

the simulation, it is required a very resolved grid in the boundary layer region.

The second method is the wall function method which allow instead to reduce the numerical

error given by the previous method when applied to boundary layers which are not satisfyingly

resolved. In this case the boundary layer does not depend on the Reynolds number but instead

an empirical “wall function” is applied to describe the boundary layer.

Evidently the boundary layer is not described in the same way in the whole geometry, which

may be a source of inaccuracy when choosing one method or the other for the whole domain.

For the SST model, the wall function in automatically switched to “automatic”, meaning that,

when solving the near-wall flow, the solver processor computes the y+ and chooses which

method is better to use for describing the boundary layer. If the y+ is small enough, the boundary

layer is well resolved, otherwise wall functions are used. In this way the software provide the

highest possible grade of accuracy.

5.5 Boundary Conditions

In this investigation many boundary conditions have been tried. Finally, given the complexity

of the model, a configuration with MFR or pressure profile plus flow direction at the Inlet and

Pressure at the Outlet was choose to be the most reliable at Miso3 =0.95. Some changes has been

made for higher loads since the presence of shockwaves make the model much less robust. A

classical trick used to stabilize the simulation in presence of shockwave is to solve the mass

equation twice for each iteration.

Note that previous investigations were performed with pressure profile and direction at the Inlet

and MFR at the Outlet. However this configuration was found to give an overflow error if

applied to the new mesh.

The velocity components at the inlet given as an input of the problem. This in order to guarantee

the similarity between numerical model and real case. In accordance with the real near-wall

inlet flow, the direction of the flow is given to make the flow tangent to the surfaces. Thus, an

angle of 44° is given at the shroud which corresponds to the wall angle (Figure 5.8), and the

angle value is linearly distributed over the span (the angle is 0° at hub). This is done to avoid

local separation at the shroud and have a smoother flow field. Moreover in this study it has been

simulated an inlet boundary condition which simulates the presence of a “parallel bars”

turbulence grid. This turbulence grid was judged to provide the most realistic in-flow condition

with a turbulence level of 2%.

- 48 -

Figure 5.8: Plot of directed inlet flow (Schäfer, 2009).

Figure 5.9: plot of velocity direction at the inlet.

- 49 -

6 RESULTS

In order to provide a reliable measurement of the losses on the NGV, different load conditions

for cold-air uncooled test rig have been simulated numerically. The purpose of this is to provide

deeper knowledge about the differences of the flow in subsonic, transonic and supersonic

conditions. The losses are calculated for 1 pitch at 107.7 Cax,hub. The flow around the NGVs in

a real engine is supposed to be identical while, for the investigated geometry, several studies

were performed questioning the periodicity of the flow. Therefore, only the results around NGV

0 are shown, since for the investigated geometry it has been shown that the hypothesis of

periodic flow holds only for NGV 0.

The simulations have been run in order to have the possibility to compare the extracted data

with experimental results (Lilienberg L., 2016). On this purpose, measurements of the yaw

angle and total pressure are shown to infer the validity of the numerical results. Unfortunately

it is known that the results presented by Lilienberg were reliable only for the subsonic case

while the cases at higher load were not very accurate as they needed further treatment.

Each solution has been run with an inlet boundary condition which simulates the presence just

before the inlet of a parallel bars turbulence grid. Former studies (Saha R., 2014) have studied

the behaviour of the total pressure profile for different turbulence grids. Figure 6.1 shows the

velocity profiles for different turbulence grid: note that the total pressure follows exactly the

same behaviour.

Figure 6.1: Experimental stream wise speed at -55.7 % for different grids (Saha R., 2014).

The pressure profile along the span must match the one given by the parallel bar grid (pbg, blue

lines in Figure 6.1). As a reference for the pressure at inlet and outlet in the numerical model,

the results from previous campaign of measures are taken (Saha R., Fridh J., 2015) and the one

used are shown in Table 6.1.

- 50 -

GP401 [kPa] MFR [kg/s] PR Miso GP404 [kPa] Patm

173.9941 2.4861 1.699 0.9042 101.6482 102.1206

212.5139 3.1466 2.0751 1.0768 97.9154 102.1124

228.7477 3.4221 2.2717 1.1493 97.9153 102.094

Table 6.1: Experimental measures from previous campaign (Saha R., Fridh J., 2015).

The first simulation was performed at Mach 0.95 which is the case for which reliable

experimental results are available (Lilienberg L., 2016).

As a matter of fact it is important to confirm the similarity in the pressure profile at -55.7%

(Figure 6.2). From this result we have proven the similarity of the inlet flow between in

numerical model and the experimental one.

Figure 6.2: Normalised numerical total pressure profile at -55.7 %.

Once the similarity of the upstream flow has been confirmed the comparison of the results

between numerical model and experimental case may begin. The Total pressure distribution

(Figure 6.3, 6.4a and 6.4b) is mostly uniform in the vane passage while two loss cores can be

observed at both tip and hub region around the TE. It can also be seen how the boundary layer

remains thin around the vane surface. The two vortex cores are the result of the development

of the horse-shoe vortices generated at the leading edge by the boundary layer flowing along

the hub and the shroud.

Note that from figure 6.3 and 6.4a one may say that little shift of the total pressure is present.

However, when analysing these kind of results, one must take into account the plot limitations.

In our case, we can confirm the similarity of the total pressure distribution by reducing the gap

between different levels in the contour plot. Unfortunately this reduce drastically our

possibilities to show what really happens when strong gradients are present inside the domain

as shown in Figure 6.4.

- 51 -

Figure 6.3: Total pressure distribution at Mach 0.95 from (Lilienberg L., 2016).

Figure 6.4a: Total pressure distribution at Mach 0.95 numerical result. Level gap 10 kPa.

- 52 -

Figure 6.4b: Total pressure distribution at Mach 0.95 numerical result. Level gap 1 kPa.

When looking at the mass averaged losses (Figure 6.5) and the area averaged exit flow angle

(Figure 6.6) it is possible to recognize a connection between the loss cores and the low pressure

regions at the tip and hub.

Figure 6.5: Mass averaged kinetic energy loss at Mach 0.95.

- 53 -

Figure 6.6: Area averaged yaw angle at Mach 0.95.

As shown in Figure 6.5 the kinetic energy loss matches quite well the experimental data. The

big difference can be seen close to the tip and hub region, where the numerical case shows an

opposite behaviour with respect to the experiment. This is mainly due to the very complex

structure of the flow inside vortices which makes the experimental probe to “lose” data. In fact

the velocity direction inside a vortex is characterized by a high randomness, and the

experimental probe cannot catch all the features of the flow. This result in a falsified measure

of the losses which of course must be higher in the vortex region. However, it is good for future

studies to have a better knowledge about this difference.

The area averaged yaw angle is instead a bit less than the experimental measurement even if

the behaviour is the same. The numerical curve is shifted backward of approximately 2°. In the

first place this behaviour was associated to a possible error during the extraction of the speed

components and computation of this parameter. After a deep check of the post-processing it has

been concluded that no error has been made. However, the experimental plot finds credits also

in previous experimental studies (Saha R., 2014) where an accuracy of 0.01° has been claimed.

Yet it is impossible to know if an error is present in the standardized post-processing of the

experimental data. Thus, further studies may be necessary to understand the accuracy of the

model and that of the instruments used during the experiment.

In order to understand where the shocks act on the blade surface to increase the losses, it is good

to have a measurement of the static pressure on the blade surface for different spans (Figure 6.7

6.8 and 6.9). In this case it is interesting to focus on the comparison of the PS of NGV 0 with

PS on NGV +1 and the SS of NGV 0 with SS of NGV -1.

- 54 -

Figure 6.7: Static pressure at 15% of the span at Mach 0.95.


- 55 -


At each span almost no differences can be found between the PS of NGV 0 and NGV +1. The

only difference is that after the low peak around 97% Cax, NGV +1 seems to compensate less

than NGV 0 while going higher with the span. It is instead possible to recognize on the SS of

NGV 0 and NGV -1 a strange behaviour of the pressure around 70% Cax at 15% of the span

which moves backwards with the increasing span. The geometry of the problem suggests that

this is consistent with the presence of a weak TE shock which start from the TE of the frontal

blade and hit the surface. The behaviour of the static pressure along the SS of NGV +1 confirms

that the periodicity of the solution holds only from the PS of NGV +1 to the SS of NGV -1.

The reliability of the solution is also suggested by the convergence history of the domain

(Figure 6.10) which does not show any instability. The very complex structure of the grid makes

the plot of the residues of no meaning as they will never reach convergence in some points of

the domain, which is why the success of the numerical iterative process it has been judged either

good or not based on parameters of physical interest which are for us the force acting on the

blades and the mass flow rate. Note that the blade loading is referred to the z-component of the

total resultant of the forces, therefore without taking into account the curvature of the geometry.

In fact the blades are gradually tilted with respect to the z-axis which is the reason why in the

plot they appear visibly different.

The solution has been achieved solving a coarse mesh first and using the result as initial

condition for the resolved mesh. The step between the two grids is visible around the 380th

iteration (Figure 6.10).

- 56 -

Figure 6.10: Convergence history of physical parameters Mach 0.95.

The second run was performed at Mach 1.05. Here the shocks start to influence the flow in a

non-negligible way.

As expected the boundary layer thickness increase as the shocks start to appear on the blades

which is visible in the sketch of the total pressure (Figure 6.11 and 6.12). The horse-shoe

vortices are less influenced as they move faster through the vane passage so that also the losses

cores move closer to the two extremes of the span in particular the one closer to the hub. In

general there is a good match of the total pressure distribution between experimental case and

numerical case.

Indeed it is possible to notice that between the experimental and numerical result, the behaviour

is more or less the same but everything is shifted from 5 to 10 kPa down in the numerical one.

This may have different explanations, first of all the reliability of the experimental results and

how they are analysed. Furthermore there is not a very good matching of the isentropic Mach

number since a value of 1.04 has been calculated for the numerical case while the experimental

one was set to 1.08.

However the difference is very small and the very good similarity between the two cases is

confirmed by further analysis of the losses and yaw angle.

- 57 -

Figure 6.11: Total pressure distribution at Mach 1.05 from (Lilienberg L., 2016).

Figure 6.12: Total pressure distribution at Mach 1.05 numerical result.

By looking at the losses distribution (Figure 6.13), the same difference as the subsonic case is

noticeable: to a positive peak in the numerical distribution, due to the presence of the vortices,

a ditch is present in the experimental curve. Here the same discussion as the subsonic case

holds, thus it has more sense to detect an increment of the losses in the presence of a vortex in

the passage.

By analysing the midspan instead, one finds that the two curves follow practically the same

path, which is again an index of the good matching of the two flow fields.

- 58 -

Figure 6.13: Mass averaged kinetic energy loss at Mach 1.05.

As the velocity increases there is a general raise in the exit flow angle (Figure 6.14). This is due

to the passage vortex becoming less relevant and the suction side of the horseshoe vortex more

dominant. Again, although the curves are almost exactly the same, one can find that the

numerical yaw angle is shifted backward of roughly 2° through the whole span. This confirm

the presence of the same error as the subsonic case.


- 59 -

From the static pressure distribution along the blade (Figure 6.15, 6.16 and 6.17) it is possible

to see clearly the effect of the shocks which still are not stable on the blade surfaces.



- 60 -


The behaviour of the pressure on the PS is more or less the same as the case Miso=0.95 while is

clearly visible an oscillation on the SS for NGV 0 and NGV -1 around 70% Cax, which moves

backward with increasing span. Like before, the geometry suggests that a shock starting from

the TE of NGV +1 may hit the SS of NGV 0. The fact that the disturbance moves backward

with increasing span is justified by the difference in pitch of 9.47 between one blade and the

other. Moreover the intensity of the shock is stronger at lower span.

Again the study of the convergence reliability is given by the parameters of physical interest

(Figure 6.18).

The transonic region is well known as the most problematic flow to reach convergence. A

common way to help the numerical model to reach convergence is to solve the continuity

equation twice for each iteration. However this implies a not negligible increment of the CPU

time. In fact, a solution for this load case has been reached in more than 15 hours (only the fined

mesh) in 4 parallel partitions while for the previous case the needed time was under the 8 hours

in 4 partitions.

- 61 -


When the “line” between transonic and supersonic region is overcome, at Mach higher than

1.10, the effects of the shocks are clearly visible on the model. In fact a very strange behaviour

is detected for the convergence. By using the reference pressure of 228 kPa as boundary

conditions what was achieve is a mass flow rate of 3.442 (roughly the same) but with a Miso =

1.2847.

When trying to reduce the Mach, by reducing the inlet pressure at about 217 kPa, it has been

achieve a much lower mass flow rate, 3.2803 kg/s, and a Miso = 1.2389.

This makes clear the presence of numerical problems due to the presence of a stable shock from

the TE. Indeed the value of pressure used at the inlet is very similar to the one used for transonic

region. However a comparison of results for the same mass flow rate is given.

- 62 -

Figure 6.19: Total pressure distribution at Mach 1.15 experimental result.


- 63 -


As it is possible to see in Figure 6.20 the total pressure assumes a quite different distribution.

Unlike the experimental case, the numerical model tends to increase the shroud side vortex with

a general total pressure around 227 kPa which is more than 10 kPa lower than the experimental

case.

This happens even if Figure 6.21 suggests a good physical convergence. When trying to reduce

the pressure at the inlet it is possible to achieve an even more confirmed convergence (Figure

6.22), but still unreliable for the information which the study starts from.

- 64 -


At this point it is a good thing try to understand what is, in the model, which provides such a

strong problem in solving the flow domain. After many attempts it has been seen that the inlet

boundary condition plays a major role in the convergence of the solution.

A good starting result has been achieved using the mass flow rate as inlet boundary condition

instead of the pressure profile. Indeed, the convergence of the physical parameters (Figure 6.23)

makes interest to study this solution.

Since using a mass flow rate boundary condition means to consider no boundary layer at the

inlet, and to let it develop inside, now the total pressure profile at Station 1 (Figure 6.24) is

much more similar to the one detected using a parallel-plates turbulence grid.

- 65 -

Figure 6.23: Convergence history of physical parameters Mach 1.1596 with MFR at inlet.

Figure 6.24: Pressure profile at Station 1 for MRF at inlet Mach 1.1596.

- 66 -

With these boundary conditions the mass flow rate of 3.4221 kg/s was respected between inlet

and outlet with a Miso = 1.1596.

Although it was known that the experimental results needed a bit more post-processing, as we

can see from the numerical results shown below (Figure 6.25) the distribution of the total

pressure at Station 2 is now much more similar to the real (Figure 6.19) one respect to the case

in which the right pressure profile was used.



- 67 -

Figure 6.27 Mass averaged kinetic energy loss at Mach 1.15.

Again looking at the yaw angle one can see from span 0.3 on that the behaviour of the curves

is the same with the general 2° backward shift which has characterized all the load cases

analysed so far. Both numerical and experimental curves are characterized by a general

increment of the yaw angle at lower span. This is probably due to a detachment of the boundary

layer in both cases. However, the different behaviour of the curves may be explained by two

main phenomena:

It is very difficult to rely on the numerical flow after detachment has occurred, due to the strong turbulence level of this kind of flow (after detachment the flow is fully

turbulent and the CFD modelling damps out all the turbulences)

Even if averaged, the area spanning from pitch -0.5 and +0.5 it is not very big, and the

high randomness of the turbulent flow may result in different values of the averaged

parameters taken in this section.

From Figure 6.28, 6.29 and 6.30 it is possible to detect the presence of the shocks hitting the

SS of the blades. Close to the hub, the flow seems to be slightly more accelerated on the SS of

NGV-1 than on NGV 0. This, together with the adverse pressure gradient given by the reflected

shock, seems to lead the flow to a detachment. The region of the blade which experiences this detached flow is very small as the flow is well guided and the value of the static pressure

suggests that further along the blade the boundary layer has re-attached. Furthermore, one may

notice the strong adverse pressure gradient close to the trailing edge followed by a flat region,

which is an index of another detachment occurring on the SS. This distribution, together with

the geometry of the flow, suggest the presence of a strong normal shock on the SS. However,

in order to confirm the presence of the shock, an analysis of the density gradients and entropy

may be needed.

- 68 -

At increasing span the distribution of static pressure on the SS around 70% Cax starts to match

again for the two vanes.

Regarding the PS, the distribution of static pressure along the span is completely similar to the

slower cases but for the curves shifted upward.



- 69 -


In order to make clearer how the situation changes by increasing the load on the vanes, a

comparison between the subsonic case and supersonic one is provided in terms of static pressure

along the blade (Figure 6.31, 6.32 and 6.33). The Mach number distribution is directly

dependent on the static pressure so that it is not strictly useful to the discussion and it is not

shown here. However, the plot are available in appendix A.

Figure 6.31: Static pressure at 15% of the span: comparison between Miso = 0.95 and Miso = 1.15.

- 70 -



Going from 15% to 85% of the span it is possible to see that the shock coming from the vane

in front and reflected on the surface (discontinuity at 65-75% of Cax) it seems to have almost

the same effect at each span and no huge differences can be detected between the two load

conditions. This can be attributed to the fact that at this position the flow is still very much

guided by the vane which makes easier for the boundary layer to remain attached even passing

through stronger shocks. Instead, a great difference is detected in the final part of the blade. In

- 71 -

fact, the strong adverse gradient for the supersonic case it may indicate the presence of a shock

occurring on the SS. Moreover it is possible to see that at 15% of the span the static pressure

has a clear stable region, from roughly 93% Cax on, which may be an indication of possible

boundary layer detachment from the surface. This hypothesis finds credits in the contour of the

total pressure (Figure 6.25) and in the losses distribution (Figure 6.34). The hub-side core of

losses move upwards and became much bigger. This is consistent with the hypothesis of

detachment.

One may notice that the losses due to the upper vortex are lower in the supersonic case. This

may be due to the fact that, when increasing the speed, horse-shoe vortex is less influenced by

the rest of the flow. This has the double effect that the vortex remains small and close to the

shroud wall-

Figure 6.34: Mass averaged kinetic energy losses: comparison between Miso = 0.95 and Miso = 1.15.

Of particular interest is to look at the density gradient distribution and static entropy along the

blade-to-blade section. The static entropy is usually the parameters that one should study when

searching for a detachment as the separation of boundary layer leads to a strong increment of

turbulence level, end therefore of the entropy, close to the surface of the blade.

Figure 6.34, 6.35 and 6.36 (left hand side) show a clearly visible shock starting from the TE of

NGV +1 but the interaction with the boundary layer (right hand side) it does not seem to be

intense through the blade passage. It is instead much stronger the shock occurring on the surface

due to the speed of the flow. Close to the end of the chord, the boundary layer meets a very

strong adverse pressure (and density) gradient which may lead to separation. As it is suggested

by previous considerations about total and static pressure and kinetic energy, the region where

the boundary layer is more affected is close to the hub. In particular at 15% of span (Figure

6.36) the strong gradient it is not straight normal to surface as in the rest of the span but it is

curved. This is usually associate to a huge losses growth and it is consistent with a detachment

of the boundary layer.

This finds credits by looking at the static entropy which allows to recognize the boundary layer

region as the entropy is much higher inside the boundary than in the free stream. One may see

- 72 -

how, going from shroud to hub, the shock affects the boundary layer more and more until

detachment occurs close to the hub.

Figure 6.35: Blade-to-Blade density gradient (left) and static entropy distribution (right) at 50% of

span, Miso=1.15.


span, Miso=1.15.

- 73 -


span, Miso=1.15.

Plots of the density distribution and total pressure distribution throughout the vane passage are

available in appendix B.

- 74 -

7 CONCLUSION and FUTURE WORK

In this thesis a help for improving and validating existing CFD methods was presented

supported by the comparison with experimental results. This may help future study on the sector

rig from many point of view, first of all the possibility to have many data in the cheapest way.

The results were achieved by observing the response of the CFD model to different load

condition and different boundary condition and adapting the mathematical model for dealing

with the shocks which, by being nothing but very strong gradients, are the main problem source

for the convergence.

At the end of the data analysis the following conclusions could be drawn.

A good and reliable solution has been achieved for the subsonic and transonic cases. The behaviour of the flow in the numerical results match quite well the experiments

even if a general shift backward of the yaw angle of about 1.5/2 degrees has been

detected in the numerical result.

Despite the convention, for which the most difficult solution to be achieved is the one at transonic regime, the very complex structure of the shocks at supersonic regime, with

high gradients spread out through the domain, made the physical convergence a very

difficult target to be achieved. For this reason the solution at Miso = 1.15 it is not a very

accurate one.

After the shocks are stable on the domain, both the coarse mesh and the refined mesh

have shown high sensibility to pressure changes at inlet when specifying the pressure

profile.

Reliable solution for supersonic case has been achieved by specifying the mass flow and turbulence intensity at inlet, thus by letting develop the pressure profile at inlet by

itself.

At increasing load, the effect of shock reflected on the surface from frontal blade does not lead to detachment of boundary layer since the flow is still well guided through the

vane. Once may notice that the effect of the reflected shock, on the blade surface, has a

backward trend for increasing span. This is explained by the curvature of the geometry

as different blades has different pitch.

Close to the blade-end a normal shock on the suction side is develop. The shock seems to be more effective al lower span leading to stable detachment of the boundary layer.

This phenomena may be explained in two ways:

o The cylindrical shape of the geometry as the pitch is smaller at the hub and

bigger at the shroud in absolute value.

o By judging on the total pressure profile, the flow goes faster close to the shroud,

index of a more energized flow which may help the boundary layer staying

attached.

A nice set of data has been extracted by the numerical model which will help researchers

to better understand the behaviour of the flow throughout the NGV.

Even if the data of main interest are shown in this thesis, it is possible to extract many more data for many type of studies about NGVs.

The presented results can inspire a wide range of studies, either experimental or numerical.

- 75 -

The quality of the mesh can definitely be improved, which could provide more

reliability to the solution. In particular the commercial software used to create the mesh

does not allow to apparently to apply the nodes in the best way in some regions.

Fortunately these region are well far from the measurement zone but still they can be

source of non-convergence of the model.

A good way to localize what makes the convergence difficult for the supersonic cases is to try to run the model with different mathematical models. To this purpose it would

be good, in principle, to run the model with other software which provide standard and

confirmed models of the equations of motion instead of build a new mathematical

model.

By having access to more data about losses, static and total pressure at supersonic regime, the developing of new e more efficient geometries is reasonable, which

furthermore could be justified by the cheap cost of changing the geometry of the blades.

This involve the meshing process to be re-done but the method used in this thesis should

be robust to changes, even if improvements may be of help.

Some attention must be paid to identify the reason why all yaw angle plots are not matching for some translation factor.

- 76 -

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- 80 -

APPENDIX A: Mach number comparison

Figure A.1: Mach number at 15% of the span: comparison between Miso = 0.95 and Miso = 1.15.


- 81 -


The plots of the Mach number around NGV 0, for both subsonic and supersonic case, show the

presence of a strong velocity gradient close to the TE. The effects of a normal shock occurring

on the SS are visible in particular at lower span.

Another particular of interest is the appearing of what it seems to be a TE shock in both the load

cases.

- 82 -

APPENDIX B: density and total pressure

At supersonic conditions it may be of interest to have a look also at the total pressure distribution

combined with the density. This just to have an idea of the magnitude of these parameters

through the vane, which, together with the results shown in this thesis, may be of help to the

designers and researchers for possible improvements of the blade geometry.

Figure B.1: Blade-to-Blade density (left) and total pressure (right) at 15% of span, Miso=1.15.


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