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MASTER'S THESIS

Validation of Aerodynamic Non-conformance Definitions

Andreas Öberg

Master of Science in Engineering TechnologySpace Engineering

Luleå University of TechnologyDepartment of Engineering Sciences and Mathematics

Abstract

Non-conformances are effects related to the difference between the nominal design of an aircraft

engine component and the finished manufactured product. At the aerothermodynamics

department at Volvo Aero, a number of definitions are used to classify the non-conformances

and their impact on the engine performance.

The main objective of this thesis has been to validate the defined definitions limits for local non-

conformances (bumps) positioned on the outlet guide vanes of a turbine rear frame, using CFD,

and derive a correlation for calculation the drag coefficient of the bumps. The project was

divided into two parts; a flat plate analysis and a real geometry analysis.

The definition of a local non-conformance is based on the height of the bumps in relation to the

boundary layer thickness at that location. The flow over a flat plate has been studied with and

without bumps at a wide range of Reynolds numbers to see how different bump sizes affects the

shape and size of the boundary layer. The added drag to the plate due to the presence of the

bumps has been calculated and compared to the bump-free cases to see if a correlation was

possible to derive.

From the flat plate simulations it was found that the lower limit of 10 % and the upper limit of 99

% of the defined borders are valid. The lower limit can however be rectified due to an increase of

just 11 % of the boundary layer thickness for bumps with a height of 40 %. A correlation was

derived that calculates the drag coefficient of the bumps with an error of ± 5 % between the

correlation calculation and the CFD results.

The real geometries that were analyzed were representative of the regular vanes and mount vanes

of a turbine rear frame. The boundary layer thickness has been calculated for both nominal vanes

and for vanes with non-conformances (bumps) to determine the effect of the bumps on the

boundary layer and if it’s possible to compare the results with the flat plate.

The boundary layer thickness on the suction peak was found to be 3.1 mm on the regular vane

and 3.3 mm on the mount vane. However, the method used for calculating the boundary layer

thickness was found to be unstable when the flow over the vane separates. The only cases that

are separation free are the 1 mm bumps, which are located at a height of 32 % of the nominal

boundary layer thickness on the regular vane and 30 % on the mount vane. The increase in

boundary layer thickness differs from the flat plate results and a detailed analysis on how the

thickness is calculated needs to be performed. The correlation was tested on the 1 mm bumps

and the drag coefficient calculated to be 0.285 on the regular vane and 0.275 on the mount vane.

This can be compared to a drag coefficient of 0.25 calculated at the department using a similar

geometry and another method. However, the correlation needs to be compared with other real

geometry bump sizes to be considered fully validated.

Contents Preface.............................................................................................................................................. i

List of figures .................................................................................................................................. ii

List of tables ................................................................................................................................... iv

Nomenclature .................................................................................................................................. v

1 Introduction ............................................................................................................................. 1

1.1 Background ...................................................................................................................... 1

1.1.1 Non-conformance definitions ................................................................................... 2

1.2 Purpose of the project ....................................................................................................... 3

1.3 Limitations ....................................................................................................................... 3

1.4 Problem description.......................................................................................................... 4

2 Theory ...................................................................................................................................... 5

2.1 Computational fluid dynamics ......................................................................................... 5

2.1.1 Turbulence modeling ................................................................................................ 6

2.2 Boundary layers................................................................................................................ 9

3 Method ................................................................................................................................... 12

3.1 Simulation approach ....................................................................................................... 12

3.1.1 Software and simulations settings ........................................................................... 13

3.2 Flat plate ......................................................................................................................... 14

3.2.1 Reference case ........................................................................................................ 14

3.2.2 Bumps ..................................................................................................................... 16

3.2.3 Finding a correlation for Cd .................................................................................... 19

3.3 Real geometry ................................................................................................................ 20

3.3.1 Testing the correlation ............................................................................................ 21

3.4 Boundary layers.............................................................................................................. 22

3.4.1 Flat plate.................................................................................................................. 22

3.4.2 Real geometry ......................................................................................................... 23

4 Results and discussion ........................................................................................................... 24

4.1 Flat plate ......................................................................................................................... 24

4.1.1 Reference case ........................................................................................................ 24

4.1.2 Bumps ..................................................................................................................... 29

4.1.3 Drag coefficient ...................................................................................................... 35

4.2 Real geometry ................................................................................................................ 40

4.2.1 Nominal................................................................................................................... 40

4.2.2 Bumps ..................................................................................................................... 42

5 Conclusions ........................................................................................................................... 50

5.1 Validation of the defined borders ................................................................................... 50

5.1.1 Flat plate.................................................................................................................. 50

5.1.2 Real geometry ......................................................................................................... 50

5.2 The correlation ............................................................................................................... 52

6 References ............................................................................................................................. 53

7 List of appendices .................................................................................................................. 54

7.1 Reference case ................................................................................................................ 55

7.2 Flat plate with bump ....................................................................................................... 59

7.3 Correlation data .............................................................................................................. 64

7.4 Real geometry ................................................................................................................ 66

7.5 CFX-Scripts .................................................................................................................... 72

7.6 MATLAB code .............................................................................................................. 82

i

Preface This thesis is the final project of the Master of Science degree in Space Engineering with

specialization in Aerospace Engineering at the Department of Engineering Sciences and

Mathematics, Division of Fluid and Experimental Mechanics at Luleå University of Technology,

Sweden. My examiner at the university was PhD Lars-Göran Westerberg and the work has been

carried out at the Department of Aerothermodynamics at Volvo Aero Corporation in Trollhättan,

Sweden under the supervision of Céline Souillet and Mats Henström.

I would like to express my sincere gratitude to my main supervisor Céline for all the help and

guidance you have provided me with and for the large interest you have shown in the project. I

would also like to thank Hans Mårtensson for your expertise in the subject and Lars Ljungkrona

for you extensive knowledge about codes and numerics. To the other thesis students and interns

that lived in Trollhättan during these months, Markus, François, Mikael, Visakha and Stijn I send

a big thank you for all the good times outside of office hours and for making my stay here very

pleasant. Finally I want to thank Linea for always being there for me and my family for all your

support.

Andreas Öberg

Trollhättan, July 2011

ii

List of figures Figure 1: Some of the components of a jet engine and also Volvo Aero’s commercial component specializations. ..... 1

Figure 2: The two layers in the near-wall region. ......................................................................................................... 9

Figure 3: Boundary layer along a flat plate. ............................................................................................................... 10

Figure 4: Reference case geometry: flat plate. ............................................................................................................ 14

Figure 5: Initial 2D-mesh for the reference case. ....................................................................................................... 15

Figure 6: Standard bump (h/L = 0.318). ..................................................................................................................... 16

Figure 7: Flat plate with bump. ................................................................................................................................... 17

Figure 8: Wide bump (h/L = 0.159). ........................................................................................................................... 17

Figure 9: Aggressive bump (h/L = 0.637). .................................................................................................................. 17

Figure 10: Flat plate with a bump. .............................................................................................................................. 18

Figure 11: Close-up on a bump. .................................................................................................................................. 18

Figure 12: Nominal regular vane. ............................................................................................................................... 20

Figure 13: Nominal mount vane. ................................................................................................................................. 20

Figure 14: Regular vane with 5 mm bump on SS. ....................................................................................................... 21

Figure 15: Mount vane with 5 mm bump on PS. ......................................................................................................... 21

Figure 16: Boundary layer thickness for the reference case. ...................................................................................... 25

Figure 17: Boundary layer thickness at the four Reynolds numbers a) 105 b) 7.5*10

5 c) 2*10

6 d) 10

7. ..................... 26

Figure 18: Skin-friction drag coefficient for the flat plate. ......................................................................................... 26

Figure 19: Skin-friction drag coefficient for a smooth plane surface depending on Reynolds number. ..................... 27

Figure 20: Displacement and momentum thickness for three reference cases. ........................................................... 28

Figure 21: Boundary layer thickness for the 10 % case compared to the reference case. .......................................... 30

Figure 22: Velocity contours for a 10 % bump at Re = 106. ....................................................................................... 30

Figure 23: BLT with bumps at height a) 40 % b) 60 % c) 99 % d) 150 %. ................................................................. 31

Figure 24: Velocity contours for standard bumps at a) 40 %, b) 60 %, c) 99 % and d) 150 % of the BLT at Re = 106.

..................................................................................................................................................................................... 32

Figure 25: Aggressive bump at 40 % showing a) BLT for three Re compared with the reference case and the

standard bump b) Velocity contours for Re = 106. ...................................................................................................... 34

Figure 26: Wide bump at 40 % showing a) BLT for three Re compared with the reference case and the standard

bump b) Velocity contours for Re = 106. ..................................................................................................................... 34

Figure 27: Relation between the drag coefficient and the scale factor based on simulation results. .......................... 35

Figure 28: Drag coefficient depending on Re for aggressive, wide and standard bumps with height 40 %. .............. 36

Figure 29: a as a function of Re. ................................................................................................................................. 37

Figure 30: b as a function of Re. ................................................................................................................................. 38

Figure 31: Relation between the drag coefficient and the scale factor based on the derived correlation. ................. 38

Figure 32: Comparison between correlation and simulation results for three Reynolds numbers. ............................ 39

Figure 33: BLT for nominal regular vane, SS. ............................................................................................................ 40

Figure 34: BLT for nominal regular vane, PS. ............................................................................................................ 40

Figure 35: BLT for nominal mount vane, SS. .............................................................................................................. 41

Figure 36: BLT for nominal mount vane, PS. .............................................................................................................. 41

Figure 37: Local Mach number around the mid-span of the regular vane. ................................................................ 41

Figure 38: Static pressure contours on the a) regular vane b) mount vane. ............................................................... 42

Figure 39: Regular vane, SS, 5 mm. ............................................................................................................................ 43





iii

Figure 44: Regular vane, SS, 5 mm, separation. ......................................................................................................... 44





Figure 49. Mount vane, SS, 3 mm. ............................................................................................................................... 46

Figure 50: Mount vane, SS, 2 mm. .............................................................................................................................. 46

Figure 51: Mount vane, SS, 1 mm. .............................................................................................................................. 46

Figure 52: Mount vane, SS, 3 mm, separation. ........................................................................................................... 46



Figure 55: Mount vane, PS, 5 mm. .............................................................................................................................. 48




Figure 59 Mount vane, PS, 1 mm. ............................................................................................................................... 48


Figure 61: BLT for the reference case at Re = a) 2.5*105, b) 5*10

5, c) 10

6, d) 4*10

6, e) 6*10

6 and f) 8*10

6. ........... 55

Figure 62: DBT for Re a) 105, b) 2.5*10

5, c) 5*10

5, d) 10

6, e) 4*10

6, f) 6*10

6 and g) 8*10

6. .................................... 57

Figure 63: MBTfor Re a) 105, b) 2.5*10

5, c) 5*10

5, d) 10

6, e) 4*10

6, f) 6*10

6 and g) 8*10

6. ..................................... 58

Figure 64: BLT for the four lowest Reynolds numbers with bump height a) 40 % b) 60 % c) 99 % d) 150 %. .......... 59

Figure 65: Displacement boundary thickness for standard bumps with Re > 106 with height 40, 60, 99 and 150 %. 60

Figure 66: Displacement boundary thickness for standard bumps with height 10 %. ................................................ 60

Figure 67: Displacement boundary thickness for aggressive bumps with height 40 %. (Reference included). .......... 61

Figure 68: Displacement boundary thickness for wide bumps with height 40 % (Reference included). ..................... 61

Figure 69: Momentum boundary thickness for standard bumps with Re > 106 with height 40, 60, 99 and 150 %. ... 62

Figure 70: Momentum boundary thickness for bump height 10 %. ............................................................................. 62

Figure 71: Momentum boundary thickness for aggressive bumps with height 40 %. (Reference included). .............. 63

Figure 72: Displacement boundary thickness for aggressive bumps with height 40 %. (Reference included). .......... 63

Figure 73: Displacement thickness, Regular vane, SS, bump height 1-5 mm (Nominal included). ............................. 66

Figure 74: Displacement thickness, Mount vane, SS, bump height 1-3 mm (Nominal included). ............................... 67

Figure 75: Displacement thickness, Mount vane, PS, bump height 1-5 mm (Nominal included). ............................... 68

Figure 76: Momentum thickness, Regular vane, SS, bump height 1-5 mm (Nominal included). ................................. 69

Figure 77: Momentum thickness, Mount vane, SS, bump height 1-3 mm (Nominal included). ................................... 70

Figure 78: Momentum thickness, Mount vane, PS, bump height 1-5 mm (Nominal included). ................................... 71

iv

List of tables Table 1: Non-conformance definitions for local defects. ............................................................................................... 3

Table 2: Inlet velocities for different Reynolds numbers. ............................................................................................ 15

Table 3: Pressure loss and drag coefficients for the grids. ......................................................................................... 16

Table 4: Pressure loss and drag coefficient for the four grids. ................................................................................... 18

Table 5: Flat plate simulation summary. ..................................................................................................................... 24

Table 6: BLT for all Reynolds numbers. ...................................................................................................................... 29

Table 7: The five bump heights (in mm) that were to be created for each Reynolds number. ..................................... 29

Table 8: Percentile increase in boundary layer thickness due to the bumps. .............................................................. 33

Table 9: Percentile increase in boundary layer thickness for the aggressive and wide bumps. .................................. 34

Table 10: Percentile difference in Cd for aggressive and wide bumps compared to the standard bump, for the sim.

data. ............................................................................................................................................................................. 36

Table 11: The a and b coefficients for each Reynolds number. ................................................................................... 37

Table 12: Percentile difference between the correlation and the simulation results. .................................................. 39

Table 13: Boundary layer thickness at the suction peak for the two nominal vanes. .................................................. 42

Table 14: Summary of all bump analyzed and their location. ..................................................................................... 42

Table 15: Reynolds number for the chord and at the suction peak at 90 % span for both regular and mount vane. .. 49

Table 16: Drag coefficient for the real geometry bumps based on the correlation. Scale factor is included for

comparison. ................................................................................................................................................................. 49

Table 17: Bump height (in mm) at x = 19 m for each ReL depending on the intended position in the reference BL. . 64

Table 18: Drag coefficients calculated from the simulation data. ............................................................................... 64

Table 19: Drag coefficients calculated from the correlation....................................................................................... 65

Table 20: Drag coefficient for 40 % bumps with different shape. ............................................................................... 65

v

Nomenclature Abbreviations

BL Boundary layer

BLT Boundary layer thickness

CFD Computational fluid dynamics

DBT Displacement boundary thickness

DNS Direct numerical simulation

LE Leading edge

LES Large eddy simulation

MBT Momentum boundary thickness

OGV Outlet guide vane

PS Pressure side

RANS Reynolds-averaged Navier-Stokes equations

SP Suction peak

SS Suction side

SST Shear stress transport

TE Trailing edge

TEC, TRF Turbine exhaust case, Turbine rear frame

Latin letters

Ab [m2] Frontal area, bump

Ap [m2] Area, plate

Cd [-] Drag coefficient

Cf [-] Friction coefficient

Cp [-] Pressure coefficient

cp [-] Specific heat constant, pressure

cv [-] Specific heat constant, volume

h [m] Bump height

i [J ] Energy

k [m2/s

2] Turbulent kinetic energy

kt [W/mK] Thermal conductivity

L [m2] Length

M [-] Mach number

n [-] Normal direction

Pdyn, Pd [Pa] Dynamic pressure

Pstat, Ps [Pa] Static pressure

Ptot, P0 [Pa] Total pressure

Re [-] Reynolds number

V, u , u, v, w [m/s] Velocity

y+ [-] Dimensionless distance from the

wall to the first node in the

computational grid

vi

Greek letters

[m] Displacement boundary thickness

[m] Boundary layer thickness

ij [-] Kronecker delta

[m2/s

3] Turbulent dissipation

[-] cp/cv

[kg/ms] Dynamics viscosity

t [kg/ms] Turbulent dynamic viscosity

[m] Momentum boundary thickness

[kg/m3] Density

[Pa] Wall shear force

[1/s] Turbulent specific dissipation

Subscripts

in Inlet

out Outlet

wall, w Point on the wall

∞ Freestream

tan Tangential

per Perpendicular

x, y, z Coordinate direction

1

1 Introduction This chapter presents the reader with an introduction to the thesis, including background,

information about non-conformances and the problem description.

1.1 Background

Volvo Aero (VAC) develops and manufactures components for commercial and military aircraft

engines in co-operation with some of the world’s leading engine manufacturers. These include

General Electric, Pratt & Whitney and Rolls-Royce among others. Because of this, VACs

components can be found in 90 % of the world’s large commercial aircrafts. The motto, “Make It

Light” is the core in VACs goal to reduce aircraft emissions by 50 % until 2020, and the

company focuses heavily on developing lightweight solutions for aircraft engine structures and

rotors. Within the areas of specialization for commercial components (Figure 1) Volvo has

established a number of Centers of Excellence (CoE) and Advanced Technology Areas, which

have enabled them to focus on developing optimal advanced technology solutions and being able

to provide strong competence in all engineering disciplines.

Figure 1: Some of the components of a jet engine and also Volvo Aero’s commercial component specializations.

This project has been conducted in one of these CoEs, namely the aero-thermodynamics

department, which is the competence centre for method and technology development within

aerodynamics at VAC. This function is part of all the stages of the product development process

and provides R&D and specialist competence in a large number of disciplines such as aero

acoustics, aeromechanics, fluid dynamics, CFD/Numerics, combustion, heat-transfer, radiation,

performance and experimental verification.

OGV

Fan/compressor structures

Shafts

Vanes

Turbine rear frames

Fan Case

Compressor rotors Combustor structure

LPT-Case

2

Many of the currently ongoing commercial engine projects at VAC are to develop turbine rear

frames (TRFs). The TRF is located behind the low-pressure turbine (LPT) and is one of the parts

used for mounting the engine onto the aircraft wing. The TRF contains a large number of outlet

guide vanes (OGVs) which are used to de-swirl the flow from the LPT so that the air leaving the

engine does so in a straight axial direction.

1.1.1 Non-conformance definitions

During manufacturing, the engine components go through a number of processes, like assembly

and adjustment which will affect the products in different ways. This quite often has the impact

that the finished product does not look like the intended design. Non-conformances (NC) or

geometry defects are effects related to the difference between the nominal (ideal) design of the

components and the actual finished products. These deviations can have an unfortunate impact

on the engine performance resulting in e.g. increased pressure losses, flow separation and

increased swirl angles. The challenge is to determine how much these NC affects the

aerodynamics of the components and if it is possible to relieve manufacturing and design

constraints if they are found not to be detrimental. It will then be possible to reduce

manufacturing and design costs.

The aero-thermodynamics department has spent a lot of time and resources on studying non-

conformances and the process is on-going. The goal in this thesis is to validate the defined limits

of some of the most common NC to identify which defect sizes are acceptable or not and thus

reduce the amount of work spent on analyzing NC at the department.

The department keeps a detailed list over the different types of NC that are commonly

encountered, together with definitions of when the NC can be considered to have a large or small

impact on the aero-parameters as well as the most critical locations to have them on. The most

common of these non-conformances can be seen in Table 1 where the definitions are shown for

local defects. That is, defects that has a local impact on the flow and aren’t large enough to be

considered global defects.

3

Table 1: Non-conformance definitions for local defects.

Type Definition Critical areas Impact on

aero-parameters

Size < 25 % of chord

Bump on vane

Large if h > 99 % of the boundary layer

thickness

Suction peak

Pressure loss Separation

Small if h << than the boundary layer thickness (5-10 % )

Dimple on vane

Large if h > 99 % of boundary layer

thickness

Suction peak Pressure loss

Small if h << than the boundary layer thickness (10-20 %)

1.2 Purpose of the project

The definitions in Table 1 have not been validated but instead been developed by qualified

guessing and experience. It is thus of great interest for the department to know if the definitions

can be considered accurate enough or if they need to be adjusted. Therefore, one the purposes of

this project is to validate the defined borders of the local defects to see if the definitions needs to

be adjusted or not. The second purpose is to develop a correlation for calculating the drag

coefficient of a bump, consisting of variables such as bump height in relation to boundary layer

thickness and the Reynolds number at the bump position. The correlation can then be used in a

non-conformance analysis program currently in development at the department.

1.3 Limitations Due to the limited amount of time available for the project a number of limitations had to be set

to make sure that it would be possible to finish everything in time. Since there are a large number

of different non-conformances that can be studied (welds, surface roughness, bumps, dimples

etc.) it was decided to focus on making a detailed study on the most common one, bumps, seeing

as it would be close to impossible to validate them all during the project.

The definition for the local bumps is defined as a bump height in relation to the boundary layer

thickness at that location. Due to the complex nature of the flow around a vane a more simple

geometry was initially studied to better understand the physics of the flow over a bump. The two

dimensional flow over a flat plate was chosen for the main analysis of the boundary layer and the

h

Flow direction

Flow direction

4

bumps effect on it. The flat plate is commonly used in boundary layer analysis and it was

assumed that the results would correlate well with those of a vane so that just a few simulations

would have to be done for the real geometries.

1.4 Problem description The project can be divided into two parts, one for the flat plate study and one for the real

geometries. These are presented below.

The first part consists of a flat plate study where the flat plate is simulated at a wide range of

Reynolds numbers and the boundary layer thickness is calculated at a certain position along the

plate. Bumps are then created with heights based on a percentile part of the undisturbed

boundary layer thickness. The bumps effects on the boundary layer in the region behind the

bumps are then analyzed. From these simulation results it is possible to determine the added

force to the plate in the flow direction due to the presence of the bumps so that a correlation for

the drag coefficient can be derived.

The second part consists of boundary layer calculations on a real representative geometry to the

vanes in a turbine rear frame. Initially, nominal regular vanes and nominal mount vanes are

analyzed before moving on to the analysis of bumps placed on the suction peak on the suction

side and the region below the suction peak on the pressure side. The idea is to get results that will

correlate reasonable well with the flat plate results. The correlation derived in the first part is

then to be tested on some of the bumps to see if it is able to predict the drag coefficient for

bumps on a vane.

5

2 Theory

In this chapter the reader is given a brief introduction to computational fluid dynamics,

turbulence modeling and boundary layer theory. It is however assumed that the reader has some

knowledge in the subject.

2.1 Computational fluid dynamics

Computational fluid dynamics (CFD) is a computer-based simulation tool for the analysis of

systems involving fluid flow, heat transfer and other related processes. These simulation tools

make use of numerical algorithms to solve the physical process of interest. There are three

distinctive types of numerical solution techniques that can be used, namely, finite difference,

finite element and spectral methods. The CFD code chosen in this project, CFX, uses a special

finite difference formulation called the finite volume method. How this works is that the user

creates a computational grid on the domain consisting of cells (control volumes). The outline of

the method is then that in each control volume the governing equations of fluid flow are

integrated, discretized into a system of algebraic equations and solved with an iterative method.

The iterative method is required because of the complex and non-linear nature of the governing

equations (equations (2.1) – (2.5)) which can be written in the following form, from Versteeg &

Malalasekera (2007).

Continuity:

(2.1)

X-momentum:

(2.2)

Y-momentum:

(2.3)

Z-momentum:

(2.4)

Energy:

(2.5)

Where SM and Si are momentum and energy source terms, and the dissipation function

(equation (2.6)). Equations (2.2) – (2.4) are usually referred to as the Navier-Stokes equations.

(2.6)

The governing equations come from applying the three fundamental physical laws of

conservation of mass, momentum and energy to a control volume. For further information about

these laws, the derivation of the equations and the numerical approach used by CFX, the reader

is referred to standard text books in fluid dynamics and CFD such as Cengel & Cimbala (2006),

Versteeg & Malalasekera (2007) and the CFX User-guide (2009).

6

2.1.1 Turbulence modeling

The turbulent nature of flows makes them much more difficult to calculate than if they are

laminar. This is because of the random and chaotic behavior of turbulence that gives rise to

rotational flow structures, so-called eddies, with a wide range of length and time scales. There

currently exists three ways to calculate turbulence in CFD, direct numerical simulation (DNS),

large eddy simulation (LES) and Reynolds-averaged Navier-Stokes equations (RANS).

The DNS method can use the incompressible form of the turbulent continuity and Navier-Stokes

equations to form a set of four equations with four unknowns. These can then be used to find a

starting point for the simulations, which then develops a transient solution to resolve all the

scales of the motion. This method requires extremely fine computational grids (around 103 grid

points in each coordinate direction) and very small time steps, which makes it too computational

heavy to be used in industrial applications and hence it is more commonly used in fundamental

research in turbulence.

The LES method uses a filtering method on the Navier-Stokes equations to separate the larger

and smaller eddies. The larger eddies are then resolved using unsteady flow simulations while

the smaller scale eddies are modeled with a so called sub-grid model. This method is much less

demanding on computational resources than DNS but it still requires a lot more computer power

than the third method.

The third method, RANS, is the most common way of dealing with turbulence. This method

doesn’t resolve any eddies in the flow but models turbulence by utilizing turbulence models.

This makes the method the most practical to use in engineering applications where it’s often

unnecessary to resolve the details of the turbulent fluctuations, and mean and time-averaged

properties of the flow are considered satisfactory enough. The averaging is done by applying

Reynolds decomposition on the governing equations so that the flow variables are split up into a

steady mean component and a time-varying fluctuating component .

(2.7)

Utilizing this decomposition on the continuity and Navier-Stokes equations (equation (2.1) –

(2.4)) and using tensor notation yields the set of equations depicted below (the energy equation

has been left out since the case simulated in this project is incompressible).

Continuity:

(2.8)

Navier-Stokes:

(2.9)

Where ij is the molecular stress tensor and SM the sum of the body forces. The time-averaging

process introduces extra terms on the right hand side of the Navier-Stokes equations.

7

These terms are usually referred to as the Reynolds stresses and they introduce a new set of

unknowns to the equations. To model these stresses it is possible to use the eddy viscosity

hypothesis (ANSYS Inc, Turbulence and Wall Function Theory, 2009) that propose that the

Reynolds stresses can be related to the mean velocity gradients and the eddy viscosity, analogous

to the relationship between the stress and strain tensors in laminar flow.

(2.10)

Where is the Kronecker delta, the eddy viscosity and the turbulence kinetic energy. A

number of turbulence models have been developed over the years to solve these equations but

the two most commonly used in CFX are the k- and SST k- models, which were the ones

under consideration when choosing turbulence model in this project.

The two models are so called two-equation models, which means that they introduce two new

transport equations that represents the turbulent properties of the flow. Both models use an

equation for the turbulent kinetic energy k but depending on the model they use either a transport

equation for turbulent dissipation or turbulent specific dissipation . The transport equations

used by the models aren’t shown here but they can be viewed in a turbulence modeling textbook

or in ANSYS Inc. - Turbulence and Wall Function Theory (2009). In the subsequent chapters

some general information about the models and their advantages/disadvantages are presented.

2.1.1.1 The k-model

The model is one of the most widely used in the industry and it’s proven to show excellent

performance for a large number of industrial flows. It’s the simplest model for which only initial

and boundary conditions need to be supplied and it’s also one of the least demanding on

computational resources. It does however show poor performance in a variety of important cases

such as some unconfined flows, curved boundary layers, swirling flows, rotating flows and fully

developed flows in non-circular ducts according to Versteeg & Malalasekera (2007). In curved

models the k- model predicts excessive levels of turbulent shear stress, leading to suppression of

separation, which poses a problem in areas such as aerodynamic flows. Some of the deficiencies

of the model can be related to how it calculates in the near-wall region, and hence different wall

treatment methods have been developed over the years to try and solve this issue.

8

2.1.1.2 The SST k-model

The standard k- model was developed for solving the -equations problems when modeling the

flow in the boundary layer region, close to the wall. This makes the -model much better in

predicting adverse pressure gradient boundary layer flows and separation. One of the downsides

of the standard model is its strong sensitivity to freestream conditions, outside the shear layer,

which tends to make it difficult to use in aerodynamic flows.

The SST k- model was designed to deal with this problem by combining the -equation and the

-equation by the use of a blending function. This makes it possible for the model to use the

advantages of the -formulation in the freestream region and the -equation in the boundary

layer region. The model is therefore valid for a great number of flow cases and it gives accurate

predictions of the onset and the amount of flow separation under adverse pressure gradients.

Because of this it is one of the most widely used turbulence models in aerodynamic flows. Some

of the deficiencies of the model are that it in some cases can under-predict pressure losses and be

too conservative in predicting separation. The SST model is recommended for high accuracy

boundary layers simulation by ANSYS Inc. in the CFX User-guide - Turbulence and Wall

Function Theory (2009).

2.1.1.3 The y+ value

The y+ value is the dimensionless distance from the wall to the first node in the computational

grid. It is used to check how fine the grid is in the boundary layer region and gives information

on what turbulence model and wall treatment can be used. The y+ value can be calculated using

equation (2.11) and (2.12).

(2.11)

(2.12)

Where ∆y is the distance from the wall, u the friction velocity at the wall, the local kinematic

viscosity, w the wall shear stress and w the density at the wall.

Experiments and mathematical analysis have shown that the near-wall region can be divided into

two layers, the viscous sub-layer, and the logarithmic layer. In the viscous sub-layer, which is

closest to the wall, the flow is almost laminar and the molecular viscosity is dominant in

momentum and heat transfer while in the logarithmic layer, turbulence is the dominating mixing

process. These layers are illustrated in Figure 2. Between these two layers there exists a region

called the buffer layer where molecular viscosity and turbulence effects are equally important.

9

Figure 2: The two layers in the near-wall region.

There are two approaches that can be used to model the flow in this region, the wall function

method and the low-Re method. The wall function method doesn’t resolve the inner region

(viscous sub-layer and buffer layer) but instead uses empirical formulas to bridge the inner

region between the wall and the logarithmic layer, thus saving a lot of computational resources.

This method requires that the first node is in the logarithmic layer with a lower limit on y+ of 30,

and an upper limit that can extend up to several thousand depending on the Reynolds number.

The low-Re method fully resolves the details of the boundary layer profile. This does however

require that the grid is refined in the direction normal to the surface so that a y+ < 1 can be

achieved. To take full advantage of the capabilities of the method one should try to have between

10 and 20 nodes within the boundary layer to fully resolve it. This approach is much more

computational heavy than using wall functions since is requires a larger number of nodes. If the

details of the boundary layer are of little interest, a wall function approach might be more

suitable to use.

2.2 Boundary layers

Consider fluid flow over a flat plate, like in Figure 3. In the vast region of the flow field away

from the surface, the velocity gradients are very small and friction has little effect on the flow. At

the wall however, the velocity gradients are large and friction has a large impact on the flow due

to the frictional forces retarding the motion of the fluid, and hence a thin layer is formed above

the surface. This thin viscous region is called the boundary layer. At the surface the flow velocity

is zero (the no-slip condition) and as we move away from the surface in the y-direction the

velocity increases until it reaches a point where it equals the freestream velocity u∞. The height

above the wall where this occurs is called the boundary layer thickness and it’s normally

defined as the point above the wall where the velocity equals 99 % of the freestream velocity

(equation (2.13)).

0.99uu (2.13)

10

Figure 3: Boundary layer along a flat plate.

Boundary layers (BL) can be either laminar or turbulent, depending on the Reynolds number

(Re). In the flow over a flat plate there is a transition between the two at approximately Re =

5*105. For lower Reynolds numbers the BL is laminar and the velocity changes uniformly as one

move away from the wall, while for higher Re the boundary layer is turbulent and characterized

by unsteady swirling flows.

The fluid particles in the BL do not always remain in the thin layer which adheres to the body

along the length of the wall (Schlichting, 1979). In some cases when adverse pressure gradients

are present, the flow in the boundary layer can become reversed and the boundary layer increases

its thickness considerably. The consequence of a reversed flow is that the flow separates from the

surface and creates a large wake of recirculating flow downstream of the surface. This will cause

a pressure drop in the region and will increase the pressure drag on the body.

Two commonly used boundary layer properties are the displacement thickness and the

momentum thickness , which can be calculated from equation (2.14) and (2.15).

(2.14)

(2.15)

Where y1 is a point above the boundary layer. The displacement thickness can be thought of as

an index proportional to the “missing mass flow” due to the presence of the BL, but could also

be explained as the imaginary increase in wall thickness, as seen by the outer flow, due to the

effect of the growing boundary layer. The momentum thickness in an index that is proportional

to the decrement in momentum flow due to the presence of the BL. In other words, it is the

height of a hypothetical streamtube that contains the missing momentum flow at freestream

conditions (Andersson, 2007; Cengel & Cimbala, 2006).

11

Over the years a number of correlations have been derived for and for both laminar and

turbulent flows over a flat plate. The most widely used correlations are shown in equations (2.16)

– (2.21).

Laminar flow:

(2.16)

(2.17)

(2.18)

Turbulent flow:

(2.19)

(2.20)

(2.21)

Where x is a point along the plate and Rex the Reynolds number at that point.

Due to the large uncertainties associated with turbulent flow fields the turbulent flow correlations

are less exact than the correlations used for laminar flow and should therefore be treated as more

approximate solutions. They do however provide a good measure of comparison when

performing boundary layer calculations.

12

3 Method

To validate the local non-conformance definition for the bumps the work was split up into

several parts. First an investigation of the flow over a flat plate was conducted (reference case) to

visualize the boundary layers and to find the thickness at a position where the flow was fully

developed. This was done for Reynolds numbers ranging from 105 to 10

7 to see how the

thickness changed with Re.

Since the NC definition that VAC use is defined as a percentage of the boundary layer thickness

(BLT), bumps were created with a height of 10, 40, 60, 99 and 150 % of the BLTs found in the

flat plate simulations. This was done to see how the size and shape of the BL was affected by the

bumps. A correlation for the drag coefficient for the bumps was then derived based on the data

from the bump analysis.

Finally the boundary layers for representative vanes of a TRF were analyzed for both nominal

cases and with bumps so that VAC could be provided with recommended values for maximum

allowed bump sizes on the vanes. The correlation derived from the flat plate simulations was

then tested on some of the bump cases to see how well it predicted the drag coefficient.

3.1 Simulation approach

A similar approach was used during all the simulations to standardize the work. All the flat plate

simulations were very much alike, apart from slight geometry changes and boundary conditions,

which made it possible to keep a lot of things constant during the process.

For each case the steps below were followed.

1. Create the geometry.

2. Create the computational grid. Depending on the inlet boundary condition used in step 3,

modify the distance from the wall to the first node.

3. Define the simulation case with appropriate boundary conditions and simulation settings.

4. Run the calculations. Monitor convergence of the residuals and domain imbalances until

the monitored parameters can be considered to be low enough and steady.

5. Check if the y+ value fulfills the criteria demanded by the turbulence model. If it does,

continue to step 6, otherwise repeat step 2-5.

6. Post-process the results.

More details about each step are presented in the subsequent section.

13

3.1.1 Software and simulations settings

Due to the simple geometry of a flat plate it was constructed in the geometry builder of the

meshing software ANSYS ICEM 12.1. The bumps that were to be placed on the flat plate were

created in MATLAB and then imported as formatted point data files into ICEM. All the mesh

generation was then done with ICEM.

The solver chosen was the commercial software package ANSYS CFX 12.1 where CFX-Pre was

used for defining the simulations and CFX-Solver for running them. In CFX-Pre the simulations

were set up as steady and incompressible and run with the k- SST turbulence model. For

calculating the advection terms in the discrete finite volume equations as well as the turbulence

numerics, 2nd

order high resolution schemes were utilized. Since the boundary layers were

studied in detail it was important to use a turbulence model that could utilize a fine mesh and

calculate well close to the walls. Hence the k- SST model was chosen because of it advantages

compared to the k- model when it comes to wall treatment for complex geometries and its

conservative separation prediction.

The post-processing was carried out in several programs. CFX-Post was mainly used for

exporting data from the CFD-simulations for the BL and drag coefficient calculations. MATLAB

was utilized to deal with the large amount of data needed to calculate the boundary layer

thickness and doing the BL calculations. Microsoft Excel was used for evaluating all the data

relevant to the drag coefficient as well as deriving the correlation.

14

3.2 Flat plate

The flat plate case was divided into two parts. A reference study and a bump study. The

reference case is presented first since it lays the foundation for the bump study.

All the simulations were made in 2D for simplicity. Since CFX is a 3D-solver it can’t work with

2D geometries unless they are converted to 3D. To solve this problem the computational grids

were saved as FLUENT 2D files in ICEM and then extruded with a thickness of 1 element when

imported into CFX (ANSYS Inc, Modeling 2D Problems, 2009).

3.2.1 Reference case

3.2.1.1 Geometry

The geometry for the reference case (Figure 4) was shaped as a rectangle with a height of ten

meters and a length of twenty-five meters, where the bottom side represents the plate and the left

and right hand side where the air would flow through. The large size of the domain was chosen

to allow for the flow to fully develop, the boundary layer to build up and to avoid blockage in the

domain so that the pressure in the free-stream would remain constant. The distances in Figure 4

are not according to scale and are exaggerated to better illustrate the geometry.

3.2.1.2 Mesh

All the computational grids (meshes) were created as structured grids, which made it easy to

adjust the amount of cells in the domain. When defining the edge parameters of the meshes a

hyperbolic mesh law was used, in accordance with recommendations from VAC, with a growth

factor of between 1.1 and 1.2. This has been proven to be an effective method since it gives a

good transition between the cells and avoids large sudden changes in cell size, thus providing a

high quality mesh. It also made it possible to coarsen the grid far away from the plate where the

flow where of less importance.

Length = 25 m

Width = 1 element

OUTLET

TOP WALL

INLET

Height = 10 m

BOTTOM WALL FRONT WALL

BACK WALL

Flow direction

Figure 4: Reference case geometry: flat plate.

15

Figure 5: Initial 2D-mesh for the reference case.

3.2.1.3 Boundary conditions

The leftmost side in Figure 5 was set as an inlet with a velocity Vin depending on the

Reynolds number ReL wanted at a location of x = 19 meters (Table 2). VACs applications

aren’t restricted to just one Re so it was important to study how the BL changed with an

increasing value, but also to be able to find a correlation that would work for a wide

range of Re. The inlet velocity was calculated from equation (3.1).

LV

LRe (3.1)

Table 2: Inlet velocities for different Reynolds numbers.

ReL Vin [m/s]

100’000 0.081

250’000 0.203

500’000 0.407

750’000 0.610

1’000’000 0.813

2’000’000 1.627

4’000’000 3.253

6’000’000 4.880

8’000’000 6.506

10’000’000 8.132

The bottom side of the geometry was given the appearance of a plate by setting it as a

wall with a no-slip boundary condition.

The top side of the domain was set as a wall with a free-slip condition.

The two walls in the cross-flow direction (Figure 4) were both given a symmetry

boundary condition (ANSYS Inc, Modeling 2D Problems, 2009).

For the rightmost side of the domain an outlet boundary condition was set with an

average static pressure of zero Pascal over the whole outlet.

x

y

INLET

WALL

OUTLET

WALL

16

3.2.1.4 Mesh dependency

To be certain that the simulation results were independent of the grid size a mesh dependency

study was performed on the reference case. An initial resolution of 75’000 cells was created and

then multiplied by a factor of 2, 3 and 4 to get four different meshes. Simulations were then run

for all four cases at Re = 106 and variables such as pressure loss Ploss (equation (3.2)) and drag

coefficient Cd (equation (3.3), where Ap is the area of the plate as seen from the y-direction) were

monitored. The results can be seen in Table 3.

100

.

,0,0

indyn

outin

loss

P

PPP (3.2)

pin

x

d

AV

FC

2

2

1

(3.3)

Table 3: Pressure loss and drag coefficients for the grids.

Mesh Cells Measured y+ Ploss [%] Cd [10-3]

Initial (x1) 75’000 <1 1.24741 4.23522

Mid-size (x2) 150’000 <1 1.24735 4.23499

Refined (x3) 225’000 <1 1.24734 4.23495

Extra Refined (x4) 300’000 <1 1.24734 4.23493

Because of the small differences between the grids, it was possible to use the initial grid for the

reference simulations.

3.2.2 Bumps

3.2.2.1 Geometry

The shape of the bumps was chosen as a standard cosine curve (Figure 6), which is a good

approximation of the normal shape of a bump or weld in the flow direction. The cosine curve

data was exported from MATLAB, changed to a formatted point data file, and then imported into

ICEM where its coordinate system was matched to that of the flat plate and the bump moved to

its intended position at x = 19 meters (Figure 7).

Figure 6: Standard bump (h/L = 0.318).

L

h

17

Figure 7: Flat plate with bump.

The shape of the bumps in the flow direction was also of interest since a larger height to length

ration is more prone to cause separation and increase the amount of pressure drag. Therefore two

more types of bumps were investigated, one with double the length and one with half the length

of the standard model (Figure 8 and Figure 9).

Figure 8: Wide bump (h/L = 0.159).

Figure 9: Aggressive bump (h/L = 0.637).

3.2.2.2 Mesh

All the meshes were created in the same way as the reference case with the exception that they

were refined around the bumps. This was done to avoid sharp edges around the bumps and

improve the transition between the cells in this area (Figure 10 and Figure 11). Since the y+

value

is dependent on the wall shear and the friction velocity (equation (2.11)), the distance to the first

node from the plate was adjusted in all cases to ensure a value less than one.

x

y

L

h

L

h

18

Figure 10: Flat plate with a bump.

Figure 11: Close-up on a bump.

3.2.2.3 Mesh dependency

Four different grids were created to be certain that the grid size chosen during all simulations

was independent of the resolution. As previously, two variables were monitored and compared

(Ploss and Cd), and the mesh study was done with a Reynolds number of 106

and a bump height of

40 % of the boundary layer thickness. The results can be seen in Table 4.

Table 4: Pressure loss and drag coefficient for the four grids.

Mesh Cells (approx.) Measured y+ Ploss [%] Cd

Initial (x1) 75’000 <1 1.78852 0.20505

Mid-size (x2) 150’000 <1 1.82145 0.21790

Refined (x3) 225’000 <1 1.82702 0.22026

Extra Refined (x4) 300’000 <1 1.82855 0.22067

Comparing the variation in pressure loss and drag coefficient between the four cases shows that

at a factor of 3 the grid is fine enough to be considered mesh independent. Because of this, the

number of cells aimed at during the mesh creation was approximately 225’000.

19

3.2.3 Finding a correlation for Cd

For VAC it is of great interest to find a correlation between the drag coefficient of the bumps, the

Re and the bump height in relation to the boundary layer thickness.

For the reference case the drag is almost completely consisting of frictional drag while for the

bumps it’s mainly due to pressure. By subtracting the force in the x-direction of the reference

case from the bump case it is possible to find the additional force on the plate due to the presence

of the bumps (equation (3.4)).

(3.4)

It is then possible to calculate the drag coefficient for all bumps in the whole span of Re using

equation (3.5), taken from Andersson (2007),

(3.5)

where the normalization area Ab is the frontal area of the bumps, calculated from hwAb

where w is the width of the bump in the cross-flow direction (1 mm for the flat plate) and h the

bump height.

Plotting these values against the bump height in relation to the BLT h/99 for each Re yields ten

logarithmically shaped curves. It is then assumed that the correlation for the drag coefficient can

be written in the following form,

(3.6)

where a and b are functions of Re and x of h/99. Excel is then used to find values for a and b that

minimize the error between equation (3.6) and the simulations. The results can be seen in chapter

4.1.3.

20

3.3 Real geometry

The real geometries used for studying the boundary layer thickness were the vanes from a

representative TRF (both a regular and a mount vane). The simulation results were all supplied

by VAC so no simulations were performed.

Two geometries were investigated, and each with a set of non-conformances. Initially the BLT,

displacement boundary thickness (DBT) and momentum boundary thickness (MBT) were

calculated at 50 % span (with the method presented in the next chapter) for nominal cases

without geometry defects, for both the suction and pressure side. The most critical location to

have a non-conformance is on is the suction peak, so by finding the boundary layer thickness in

that position makes it possible to decide where in the boundary layer the investigated bumps

were positioned. Five bumps were studied on the regular vane SS, three on the mount vane SS

and finally five bumps on the mount vane PS.

An aerodynamic study of non-conformances on a TRF done by VAC in 2009, where the pressure

losses and the swirl angles had been investigated for the bumps mentioned above, were used for

analyzing the results from the real geometry bump study. Comparisons were done between the

results from the real geometry study and the flat plate study to see if the boundary layer was

affected in a similar way on the vane as it was on the flat plate.

Figure 12 and Figure 13 below shows the nominal vanes and Figure 14 and Figure 15 two of the

geometry defects studied.

Figure 12: Nominal regular vane.

Figure 13: Nominal mount vane.

21

Figure 14: Regular vane with 5 mm bump on SS.

Figure 15: Mount vane with 5 mm bump on PS.

3.3.1 Testing the correlation

The correlation derived from the flat plate simulations was tested on the bumps of the

representative geometries to see if the correlation could be used on real geometries.

Hundreds of bump simulations have been conducted at the aerothermodynamics department on

geometries similar to the one analyzed in this project. From these simulations, Cd values have

been calculated using a method based on force equilibrium for a control volume, resulting in an

equation on the following form.

(3.7)

Where )/cos(tan, xinperin

vvAA is the domain inlet area perpendicular to the flow, hwAb

is

the front area of the bump, outin

PPP,0,0

is the pressure difference between the inlet and

outlet of the domain and Pdyn the maximum dynamic pressure in the zone on the vane where the

bump is located.

It was therefore expected that the correlation results would be of similar magnitude as the Cd

value calculated for bumps at the department, which is 0.25 on the suction side of separation free

vanes.

22

3.4 Boundary layers

Calculating the boundary layer thickness from the simulation data is a quite challenging task

since there is no built-in function or general method available for doing this in CFX. Because of

this a method had to be developed for doing the following,

Create lines normal to the surface of interest in CFX.

Export the velocity profile along the lines together with position.

Use MATLAB to find where

0.99uu and do the integration that yields the

displacement and momentum thicknesses.

3.4.1 Flat plate

Creating lines normal to the flat plate is an easy process due to the fact that they are only

depending on the position along the plate (x-direction) and the normal to the surface (y-

direction). However, since a lot of lines had to be created for a wide span of Reynolds numbers

the process was simplified by creating a script for CFX-Post (Appendix 7.5). In addition to

creating the lines the script exports data for position, static pressure and total pressure along each

line to a file, to be used in the post-processing in MATLAB.

To get comparative results between the flat plate and the real geometries the same method for

calculating the velocity was used. Since the stream situation for the real geometries (vanes) lack

free stream conditions the isentropic velocities along the lines had to be calculated for both cases

(flat plate and vanes) using equation (3.8).

(3.8)

Where equation (3.8) originates from Bernoulli’s equation.

(3.9)

A script was constructed in MATLAB (Appendix 7.6) for post processing all the data from the

flat plate simulations. The script calculates the DBT and MBT and gives an approximate value

for the BLT at each line along the plate and then plots the data to visualize the shape of the

layers. By doing this it was possible to extract the BLT at x = 19 meters for the whole range of

ReL.

The same procedure used for the reference case was also used for the case with bumps. In

MATLAB it is then easy to compare the differences and see how the shape and size of the

boundary layer were affected by the bumps, depending on their height in relation to the reference

thickness.

23

3.4.2 Real geometry

To be able to validate the results from the flat plate simulations it was important to calculate the

boundary layer thicknesses for a real geometry. This was done in a similar way as for the flat

plate but with some large modifications to CFX script.

Creating lines normal to the surface for a curved 3D model involves a lot of additional steps

compared to a straight model that only depends on x- and y-coordinates. The following method

was therefore developed,

At the span of interest, create a surface around the vane and export data for x-, y-, z-

coordinates as well as the normal directions.

Use the coordinates and normal directions for creating planes normal to the surface along

the vane. Transform the normals in the following way, depending on whether the SS or

PS is of interest,

SS:

PS:

Create a contour on each plane with enough levels that one level lands on top of the span

chosen. Then create a polyline along that level and export the coordinates.

Use the coordinates from the polyline to create a regular line spanning from the vane

surface to half the length of the polyline. This to be able to choose the length of the line

and the amount of data points, options that are not available for polylines.

Export the data required to calculate the BLT, DBT and MBT in MATLAB, i.e.

coordinates, static pressure and total pressure.

The simulation results for the real geometry cases were supplied by VAC and had been run in

FLUENT and a variable for total pressure was missing when imported into CFX-Post. To solve

this problem a user defined variable was created from equation (3.10).

(3.10)

The equation comes from Andersson (2007) and is used since the flow over the vane is

compressible. The script created for the representative vanes can be seen in Appendix 7.5.

The MATLAB code only required some slight modifications to how the length to each point

along the lines were calculated as well as the method of finding the maximum velocity

(Appendix 7.6).

24

4 Results and discussion

In this chapter the results from the different phases of the project is presented and discussed. It

also contains a discussion about the methods used, the choice of turbulence model and some

recommendations.

4.1 Flat plate

This section deals with the results from the flat plate and flat plate with bump simulations. For

both cases the results were compared to the equations for ideal laminar and ideal turbulent flow

presented in the theory chapter. Table 5 summarizes all the simulations done on the flat plate (S

= Standard, A = Aggressive, W = Wide, Ref = Reference). The five heights depicted in Table 5

were chosen since it was of interest to study the upper and lower limit of the non-conformance

definition (Table 1) as well as values above the upper limit and values between the two limits.

Table 5: Flat plate simulation summary.

Summary Scale factor

1.5 0.99 0.6 0.4 0.1 -

ReL S A W S A W S A W S A W S A W Ref

100’000 x - - x - - x - - x x x x - - x

250’000 x - - x - - x - - x x x - - - x

500’000 x - - x - - x - - x x x - - - x

750’000 x - - x - - x - - x x x - - - x

1’000’000 x - - x - - x - - x x x x - - x

2’000’000 x - - x - - x - - x x x - - - x

4’000’000 x - - x - - x - - x x x - - - x

6’000’000 x - - x - - x - - x x x - - - x

8’000’000 x - - x - - x - - x x x - - - x

10’000’000 x - - x - - x - - x x x x - - x

As was mentioned in the method chapter it was of great importance to calculate the BLT, DBT

and MBT for a flat plate at a wide range of Re. It was believed that by doing so the effect of a

bump on the BL could be seen and that the results would correlate well with those for a real

geometry. The reason for choosing a flat plate as reference is because it is good for developing a

basic understanding of how the boundary layer develops along a surface and it is known to be a

fairly good approximation for many applications, such as airfoils. Even though the flow over a

vane will differ due to increasing/decreasing pressure gradients and depending on if observing

the SS or PS, the flat plate will show a similar behavior and will aid in the understanding of the

BL development over the vane.

4.1.1 Reference case

Following the method for calculating the BLT for the flat plate yielded the results shown in

Figure 16. The line at the top of the graph is the lowest Re (105) and the bottom one the highest

(107) and as was mentioned earlier they all correspond to a Re at 19 meters in the flow direction.

What is seen is how the boundary layer builds up along the plate and it can be observed that as

25

the Re increases the boundary layer thickness decreases. This was of course expected since the

BLT is inversely proportional to the Reynolds number. However, what wasn’t expected was that

no transition between laminar and turbulent could be seen for either case, since it was anticipated

that the BLT for the lower Re would correlate quite well with equation (2.16) for laminar flows

and that the higher Re would do the same with the 1/7th

power law from Schlichting (1979),

equation (2.19). By looking at some of the BLTs individually it can be observed that they deviate

from these correlations (Figure 17). The exempt is the first portion of the boundary layer for the

lowest Re (Figure 17a) which matches quite well with equation (2.16), and the highest Re

(Figure 17d) that almost match with the power law. The left-out cases can be seen in Appendix

7.1.

Figure 16: Boundary layer thickness for the reference case.

a) Re = 105.

b) Re = 7.5*105.

26

c) Re = 2*106.

d) Re = 107.

Figure 17: Boundary layer thickness at the four Reynolds numbers a) 105 b) 7.5*105 c) 2*106 d) 107.

A small investigation was done to answer the question as to why the transition from laminar to

turbulent couldn’t be seen in the flat plate results. It was expected that by calculating the drag

coefficient Cd (equation (3.3)) for each case and plotting it against the respective Reynolds

number would yield results (Figure 18) that could be compared to the theory of a flat plate, and

that this would give some insight into the source of the error.

Figure 18: Skin-friction drag coefficient for the flat plate.

Comparing the results with Figure 19 from chapter 2-6 in Hoerner (1965) strongly indicated that

the flow over the flat plate was a forced turbulence flow and not a developing flow. After

studying the user manual for CFX in more depth these results could be confirmed, as the

turbulence model in CFX models fully turbulent flow, which wasn’t realized from the start.

Because of this the BLTs for low Re (below 750’000) will be somewhat thicker than what they

should be. However, since VAC most often deal with Re larger than this the deviation for low

values were of less importance.

0.001

0.01

1E+04 1E+05 1E+06 1E+07 1E+08 1E+09 1E+10

Cd

ReL

27

Figure 19: Skin-friction drag coefficient for a smooth plane surface depending on Reynolds number.

Looking at the DBT and MBT for each case shows that they match very well with the correlation

for an ideal turbulent flat plate (equation (2.20) and (2.21). The reason for this is considered to be

that the method used for calculating both the DBT and MBT is much more robust and accurate

than the approximate method used for finding the BLT. Looking at these thicknesses is therefore

a good addition when studying the boundary layer for checking that the simulation results are

correct. Figure 20 below shows the DBT and MBT for the three Reynolds numbers 7.5*105 (a

and b), 2*106 (c and d) and 10

7 (e and f) and the rest are shown in Appendix 7.1.

a) Re = 7.5*105

b) Re = 7.5*105

28

c) Re = 2*106

d) Re = 2*106

e) Re = 107

f) Re = 107

Figure 20: Displacement and momentum thickness for three reference cases.

The boundary layer thickness associated with each Reynolds number at x = 19 meters can be

seen in Table 6. This data was the base for determining the height of the bumps for the different

scale factors (h/99).

29

Table 6: BLT for all Reynolds numbers.

ReL BL thickness [mm] 100’000 471.571 250’000 394.648 500’000 354.515 750’000 331.103

1’000’000 317.725 2’000’000 287.625 4’000’000 260.869 6’000’000 250.836 8’000’000 240.803

10’000’000 237.458

4.1.2 Bumps

As mentioned earlier three types of bumps were created. Initially a standard cosine shaped bump

was studied for the whole Re range and for all five heights. After this an aggressive and a wide

bump was simulated for all Reynolds numbers at a height of 40 % to see how the shape of the

non-conformance in the flow direction would affect the BL. First the results for the standard

cosine bumps are presented.

Utilizing the results from Table 6 the five bump heights could be determined for all Re (Table 7).

The five heights of interest were 10 %, 40 %, 60 %, 99 % and 150 % of the BLT.

Table 7: The five bump heights (in mm) that were to be created for each Reynolds number.

Scale factor

ReL 1.5 0.99 0.6 0.4 0.1

100’000 707.357 466.856 282.943 188.629 47.157

250’000 591.973 390.702 236.789 157.860 39.465

500’000 531.772 350.970 212.709 141.806 35.452

750’000 496.655 327.792 198.662 132.441 33.110

1’000’000 476.588 314.548 190.635 127.090 31.773

2’000’000 431.438 284.749 172.575 115.050 28.763

4’000’000 391.304 258.261 156.522 104.348 26.087

6’000’000 376.254 248.328 150.502 100.334 25.084

8’000’000 361.204 238.394 144.482 96.321 24.080

10’000’000 356.187 235.083 142.475 94.983 23.746

The first height to be tested was 10 %, which was the lower limit of the non-conformance

definition as was mentioned in the introduction. It was expected that these bumps would have

little or no effect on the flow and therefore only three Re (105, 10

6 and 10

7) were initially

simulated. If this was proven to be true then it would be unnecessary to do any further

simulations for that height, which would then save a lot of computational time. Utilizing the

scripts (Appendix 7.5 and 7.6) on the simulation data the following results were found.

30

Figure 21: Boundary layer thickness for the 10 % case compared to the reference case.

Figure 22: Velocity contours for a 10 % bump at Re = 106.

Figure 21 and Figure 22 clearly shows that a bump that is placed at 10 % of the boundary layer

thickness have an insignificant effect on the size and shape of the BL in the region behind the

bump, regardless of the Reynolds number. Figure 21 shows that the BL stabilizes at the same

thickness as the reference case almost immediately after the bump. The definition that VAC uses

to define when the size has a negligible effect on aero parameters (Table 1) is therefore valid.

Looking at the MBT and DBT for the 10 % case gave the same results as for the BLT, a

negligible effect on the thickness. These results are shown in Appendix 7.2.

31

The second height simulated was 40 %. An initial simulation for a Reynolds number of 106

indicated that a difference in BLT compared to the reference case existed. That gave a clear

indication that all heights larger than this would have an impact on the BLT and hence the whole

range of Re were simulated for each remaining case. In the figures below the four lowest

Reynolds numbers have been left out to make it easier for the reader to view the results. They

have instead been included in Appendix 7.2. The reference cases have been included in the

graphs to visualize the increase in thickness better.

a) 40 %.

b) 60 %.

c) 99 %.

d) 150 %.

Figure 23: BLT with bumps at height a) 40 % b) 60 % c) 99 % d) 150 %.

As can be seen in Figure 23 a – c, the boundary layer stabilizes before reaching the outlet for all

cases and it is quite straightforward to see how much the boundary layers have been affected by

the bumps. For the 150 % case (Figure 23d) it nearly stabilizes for the highest Reynolds number

but for the remainder the domain is too short to make this possible. This indicates that the BL

would need a substantial distance behind the bumps to stabilize themselves. If the domain was to

be extended around 10 meters or so it would probably be enough to solve this problem. It should

therefore be noted that the BLT values at the outlet for this height is somewhat larger than what

they should be.

32

In Figure 24 the velocity contours can be seen for the four bump heights at a fixed Re. The dark

blue color in the figures represents negative velocity, i.e. separated flow and it’s easy to see that

the two largest bump heights (99 % and 150 %) induce a large separated region behind them.

a) 40 %.

b) 60 %.

c) 99 %.

d) 150 %.

Figure 24: Velocity contours for standard bumps at a) 40 %, b) 60 %, c) 99 % and d) 150 % of the BLT at Re = 106.

By comparing the BLT for each bump with the corresponding reference case it was possible to

calculate the increased thickness due to the presence of the bumps. The results presented in Table

8 are shown as an increase from the reference case (i.e. a 20 % increase is the same as a 20 %

thicker boundary layer than the reference case) and values at the domain outlet are used.

33

Table 8: Percentile increase in boundary layer thickness due to the bumps.

BLT increase Scale factor

ReL 0.4 0.6 0.99 1.5

100’000 5.8 18.0 48.0 129.5

250’000 8.3 19.3 54.8 125.1

500’000 9.0 21.2 57.2 124.4

750’000 8.1 20.0 55.6 118.1

1’000’000 9.8 20.2 50.6 109.1

2’000’000 12.8 25.4 61.1 114.1

4’000’000 15.2 29.2 62.1 112.8

6’000’000 13.6 28.0 60.6 108.9

8’000’000 13.4 28.2 60.6 107.8

10’000’000 12.4 26.2 54.0 92.8

Average increase [%] 10.8 23.6 56.5 114.3

From the results presented in the table it can be seen that for the lowest scale factor the average

increase in boundary layer thickness is only about 11 %, which indicates that the bump has a

small impact on the BL but not small enough to be considered unimportant. It’s interesting to see

that the increase rises about 50 % between each scale factor, which could indicate that it follows

some sort of power law and that a correlation might be possible to derive from the results. The

large increase in thickness due to the bumps with a size close to or above the BLT tells us that

they will have a large effect on the aero parameters. A large increase of the thicknesses (BLT,

DBT and MBT) will have the unfortunate effect of changing the geometry’s effective shape and

hence reduce the circulation and lift. It will also be highly prone to cause separation on the vane

and give rise to a high pressure drag, consequently increasing the total drag on the vane.

The aggressive and the wide bumps were studied at a height of 40 % and for the full range of Re.

The reason for not analyzing the rest of the bump sizes was because it was assumed that they

would follow the same trend as the standard cosine bumps. In Figure 25 and Figure 26 the BLT

for three Reynolds numbers (105, 10

6 and 10

7) are shown for the two bump types and the

displacement and momentum boundary thicknesses in Appendix 7.2.

34

a) Boundary layer thickness for aggressive bumps at 40 %.

b) Velocity contours, Re = 106.

Figure 25: Aggressive bump at 40 % showing a) BLT for three Re compared with the reference case and the standard

bump b) Velocity contours for Re = 106.

a) Boundary layer thickness for wide bumps at 40 %.

b) Velocity contours, Re = 106.

Figure 26: Wide bump at 40 % showing a) BLT for three Re compared with the reference case and the standard bump

b) Velocity contours for Re = 106.

Comparing the figures shows that the wide bump affects the shape and size on the boundary

layer less than the aggressive one does, and in Table 9 we can see that the wide bump has a very

small increase in BLT compared to the reference case. The aggressive bump however shows

tendencies towards the 60 % case for the standard bumps. The velocity contours tells us that the

separated region behind the bumps is noticeably larger for the aggressive bump. We can

therefore assume that the wide bumps will have a smaller impact on the aero parameters

compared to the standard case and that the aggressive one will have a larger impact.

Table 9: Percentile increase in boundary layer thickness for the aggressive and wide bumps.

BLT increase Scale factor = 0.4

ReL Aggressive Standard Wide

100’000 9.4 5.8 5.1

1’000’000 11.9 9.8 4.6

10’000’000 17.9 12.4 5.4

35

4.1.3 Drag coefficient

Below follows the results from the derivation of the correlation for the drag coefficient. First the

results from the simulation data is presented and then the corresponding correlation. The

correlation has been derived from the simulation data for the standard cosine bump.

4.1.3.1 Simulations

4.1.3.1.1 Standard bump

Using equation (3.4) from the method chapter it was possible to calculate the force induced by

the bumps in the flow direction. This data was then used in equation (3.5) and normalized with

the height of the bumps to find a corresponding drag coefficient for each bump height and Re.

The tables containing the calculated drag coefficients are shown in Appendix 7.3. By plotting the

Cd for each Re against the scale factor h/99 it was possible to see the correlation between the

variables (Figure 27).

Figure 27: Relation between the drag coefficient and the scale factor based on simulation results.

It is clear that the curves in the figure follow a logarithmic shape and that a logarithmic function

can be derived from the data, as was mentioned in the method chapter. More about this in

chapter 4.1.3.2.

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0 0.5 1 1.5 2

Cd

h/99

10^52,5*10^55*10^57,5*10^510^62*10^64*10^66*10^68*10^610^7

36

4.1.3.1.2 Aggressive and wide bumps

Since the aggressive and wide bumps were only evaluated at 40 % of the BLT they were

compared the 40 % case for the standard bump and plotted against the Re.

Figure 28: Drag coefficient depending on Re for aggressive, wide and standard bumps with height 40 %.

In Figure 28 we can see that the aggressive bump follows the same trend as the standard bump

with the exception that the drag coefficient is higher due to the increase in pressure drag. The Cd

for the wide bump on the other hand starts to decline somewhat for Re higher than 106. The

shape and magnitude of the curves can be related to the height to length ratio of the bumps h/L,

where a higher h/L value will increase the amount of pressure drag and hence increase the drag

coefficient, while a lower value will decrease the drag coefficient.

The results from comparing the increase/decrease in Cd for the aggressive and wide bumps with

the standard bump can be seen in Table 10.

Table 10: Percentile difference in Cd for aggressive and wide bumps compared to the standard bump, for the sim. data.

Difference Scale factor = 0.4

ReL Aggressive bump Wide bump

100’000 18.82 -27.84

250’000 22.85 -31.86

500’000 25.54 -33.81

750’000 25.99 -35.15

1’000’000 23.42 -37.07

2’000’000 26.62 -41.50

4’000’000 28.81 -47.20

6’000’000 30.78 -50.85

8’000’000 30.95 -52.95

10’000’000 31.68 -56.07

0.00

0.10

0.20

0.30

0.40

0.50

0 5000000 10000000

Cd

Re

Standard

Aggressive

Wide

37

Clearly, there are large differences between the bumps, with an average increase in Cd of 27 %

for the aggressive and a 41 % decrease for the wide compared to the standard bump.

4.1.3.2 Correlation

Following the results in Figure 27 it was assumed that the correlation would have the form

presented in the method chapter (equation (3.6)). By using the simulation results from the

standard cosine bump study it was possible to find values for the coefficients a and b in equation

(3.6). The solver in Excel was used for guessing values of a and b that minimized the error

between the correlation and the simulation results. The results are presented in Table 11.

Table 11: The a and b coefficients for each Reynolds number.

Coefficients

ReL a b

100’000 0.455424553 0.252791285

250’000 0.454860716 0.221292917

500’000 0.461813834 0.195293037

750’000 0.462009399 0.179833327

1’000’000 0.471259180 0.163746198

2’000’000 0.460685200 0.137872449

4’000’000 0.452094966 0.121280189

6’000’000 0.444300587 0.112079976

8’000’000 0.436480399 0.106156106

10’000’000 0.435992571 0.089439931

It was sought after to find coefficients depending on the Reynolds number, hence a and b were

plotted against the Reynolds number to find suitable values (Figure 29 and Figure 30).

Figure 29: a as a function of Re.

y = -3E-09x + 0.4628

0.43

0.44

0.45

0.46

0.47

0.48

0 5000000 10000000

f(Re)

Re

38

Figure 30: b as a function of Re.

Trendlines were then created in Figure 29 and Figure 30 to find correlations for a and b, and

when these were inserted into equation (3.6) it resulted in the following correlation for Cd.

(4.1)

If we use this correlation with the bump heights and Re calculated for the flat plate it results in

the following figure.

Figure 31: Relation between the drag coefficient and the scale factor based on the derived correlation.

Selecting three of the Reynolds numbers (105, 10

6 and 10

7) in the figure and comparing them

with the corresponding simulation data shows us that the difference between the two is very

small (Figure 32) and that the correlation match the simulation data quite well.

y = -0.035ln(x) + 0.6475

0

0.05

0.1

0.15

0.2

0.25

0.3

0 5000000 10000000

f(Re)

Re

0.200000

0.250000

0.300000

0.350000

0.400000

0.450000

0.500000

0.550000

0.600000

0 0.5 1 1.5 2

Cd

h/99

10^52,5*10^55*10^57,5*10^510^62*10^64*10^66*10^68*10^610^7

39

Figure 32: Comparison between correlation and simulation results for three Reynolds numbers.

Looking at these results even further we can see that the percentile difference between Figure 27

and Figure 31 is within ± 5 % (Table 12), which can be considered to be a quite good match.

Table 12: Percentile difference between the correlation and the simulation results.

Difference Scale factor

ReL 0.4 0.6 0.99 1.5

100’000 -5.58 3.46 2.52 -0.06

250’000 1.49 3.42 3.51 0.65

500’000 1.49 0.86 0.90 -0.64

750’000 1.30 -0.10 -0.22 -0.75

1’000’000 -3.63 -2.86 -2.93 -2.12

2’000’000 0.20 -1.24 -1.58 -0.61

4’000’000 3.42 0.37 -0.60 -0.73

6’000’000 4.37 1.51 -0.12 -0.80

8’000’000 5.13 2.30 0.08 -0.78

10’000’000 0.67 -0.37 1.92 -1.20

The correlation is based on the simulation results from the standard cosine bump study. It will

therefore miss-predict the drag coefficient if used with aggressive or wide bumps since the

correlation is independent of the length in the flow direction. A more detailed analysis needs to

be carried out by e.g. finding a coefficient that the correlation can be multiplied with so that the

correlation can be used with bumps of different h/L.

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0 0.5 1 1.5 2

Cd

h/99

Sim. Re = 10^5

Sim. Re = 10^6

Sim. Re = 10^7

Corr. Re = 10^5

Corr. Re = 10^6

Corr. Re= 10^7

40

4.2 Real geometry

To see how the size and shape of the boundary layer was for a real geometry, the vanes of a

representative Turbine Rear Frame was investigated. As was mentioned in the method chapter,

two models were initially looked at, a nominal regular vane and nominal mount vane. The

modified scripts for CFX-Post and MATLAB were used on the simulation results to calculate the

BLT, DBT and MBT for both the SS and PS of the vanes, and also at different spans.

After this the results from a bump investigation on the suction side of the regular vane and

suction and pressure side on the mount vane are presented and discussed.

4.2.1 Nominal

The BL results for the regular and the mount vane on both the SS and PS can be seen in the

figures below (DBT and MBT in Appendix 7.4). It was assumed that the SS would have a

similar appearance as the flat plate, with the exception that the boundary layers would be thinner

before the suction peak due the increase in velocity and wider after the suction peak due to the

decrease in velocity. For the PS the BL were assumed to show a similar tendency, with the

exception of being thinner near the TE due to the shape of the geometry. It was also suspected

that the stagnation point at the LE would cause problem in the BL calculations, which later

turned out to be correct for some of the cases.

Figure 33: BLT for nominal regular vane, SS.

Figure 34: BLT for nominal regular vane, PS.

41

Figure 35: BLT for nominal mount vane, SS.

Figure 36: BLT for nominal mount vane, PS.

The figures presented above clearly go against what was initially assumed concerning the

boundary layer thickness for the region before the suction peak. The reason for this was unclear

at first but it was realized to be caused by compressible effects. The inlet velocity for the domain

is close to 260 m/s and at the vane it is even higher. In Figure 37 the Mach number is shown at

the mid-span of the regular vane and it is easy to see that the velocity is highly accelerated

around the vane.

Figure 37: Local Mach number around the mid-span of the regular vane.

The increasing temperature near the vane surface due to friction causes the boundary layer to

become thicker when the velocity increases (Chapter 13 and 23 in (Schlichting, 1979)). That is

why the BLT in Figure 33 - Figure 36 is thicker than suspected. The calculated boundary layer

thickness at 50 % span on the suction peak located at 30 % of the chord length can be seen in

Table 13. Also, since the bumps studied for the real geometries were positioned at approximately

90 % span the boundary layer was analyzed in this location and the thicknesses calculated at the

suction peak to see if it there was any difference between the two spans.

42

Table 13: Boundary layer thickness at the suction peak for the two nominal vanes.

BLT [mm] Case

Span Regular vane Mount vane

50 % 3.4 4.0

90 % 3.1 3.3

The reason that the boundary layer thickness is lower at 90 % span was assumed to be due to

different velocities at the suction peak along the vane. This could be confirmed by looking at the

static pressure contours (Figure 38) on both the regular and the mount vane, which also explains

why the boundary layer is somewhat thicker on the mount vane.

a) Regular vane, SS.

b) Mount vane, SS.

Figure 38: Static pressure contours on the a) regular vane b) mount vane.

4.2.2 Bumps

Table 14 shows a summary of the range of bump sizes that were part of the validation work for

the real geometries and also their location.

Table 14: Summary of all bump analyzed and their location.

Regular vane Mount vane

Height [mm] SS PS SS PS

1 x - x x

2 x - x x

3 x - x x

4 x - - x

5 x - - x

The graphs for the DBT and MBT from all cases are presented in Appendix 7.4 and shown next

are the results for the BLT, starting with the regular vane.

43

4.2.2.1 Regular vane

If we look at Figure 39 - Figure 42 we can see that the boundary layer calculations collapse in

the region behind the bump. Comparing these results with the pressure losses in VAC (2009) and

with the negative axial velocities in Figure 44 – Figure 47, we can see large regions of separation

for all four heights. The code used in the boundary layer calculations therefore has large

problems with determining the thicknesses for highly separated regions. However, for the 3 mm

case there is a separation free region about halfway between the bump and the trailing edge of

the vane and for the 2 mm case the flow barely manages to reattach near the trailing edge, which

explains the spikes in Figure 41 and Figure 42 and gives some information of how much the

boundary layer thickness has increased due to the bumps.

Figure 39: Regular vane, SS, 5 mm. Figure 40: Regular vane, SS, 4 mm.

Figure 41: Regular vane, SS, 3 mm.


44


Figure 44: Regular vane, SS, 5 mm, separation.





45

The 1 mm case experiences a small separation behind the bump, but the effect in negligible on

the calculation method. The BLT is however affected at the end of the vane, with an increase of

about 80 % compared to the nominal case. For the 2 mm bump the increase is about 250 %

compared to the nominal case but the second result is unreliable due to the separated flow along

the vane. The 1 mm bump height is approximately 32 % of the BLT at the suction peak for the

nominal case, while for the 2 mm bump it is roughly 65 %. Since the 2 mm bump causes

separation that barely manages to reattach there seems to be a limit of what height can be

allowed on the regular vane of roughly 30 %, to be conservative. It is likely that bumps in the

range 30-60 % of the BLT can be allowed but this needs to be studied in more detail.

As can be noted from the results the percentile increase in boundary layer thickness in the region

behind the 1 and 2 mm bumps compared to nominal is much larger than for bumps located at the

same h/99 on the flat plate. The cause of this can be due to the shape of the vane being curved

and the adverse pressure gradients causing a more rapid increase in boundary layer thickness in

the region behind the bumps. It’s possible that the boundary layer thicknesses from the flat plate

simulations are somewhat underestimated, seeing as the boundary layer thicknesses were

calculated to be thinner than the ideal turbulent boundary layer thicknesses calculated from the

theory. The impact would be that the bumps created and simulated on the flat plate would be in a

smaller percentile part of the BLT, which would imply that the real geometry bumps should be

compared to flat plate bumps located at a higher percentile part of the BLT rather than at the

same percentile part.

It is unlikely that the boundary layer would be affected by the 1 mm bump on the regular vane to

such a large extent compared to the mount vane (presented in the next chapter). Looking at the

increase in displacement boundary thickness (Figure 73e and Figure 74c in Appendix 7.4) and

momentum boundary thickness (Figure 76e and Figure 77c) compared to nominal for both 1 mm

bumps gives some insight into this problem. It can be seen in the figures that the thickness

increase is about the same for both bumps. This tells us that there must be some error in the

calculation method when used on the regular vane since there is such a large difference between

the increases in boundary layer thickness for the two cases. The code used when analyzing the

BLT on the vanes as well as the method in chapter 3.4.2 therefore needs to be investigated

further.

4.2.2.2 Mount vane

It was found that the mount sector is more sensitive to NC since the 3 mm bump (Figure 52)

shows a separated region that is larger than the separation observed for the 5 mm bump on the

regular vane. It was therefore decided not to analyze 4 and 5 mm bump heights. The analysis on

the simulations results from the mount vanes suction side was therefore done for three bump

heights (1, 2 and 3 mm).

46

Figure 49. Mount vane, SS, 3 mm.

Figure 50: Mount vane, SS, 2 mm.

Figure 51: Mount vane, SS, 1 mm.

Clearly the boundary layer calculation falters for the 3 and 2 mm defects while it’s just slightly

affected by the 1 mm bump. Again, comparing the results with the negative axial velocities

(Figure 52 and Figure 53) gives a clearer image as to what happens in the flow.

Figure 52: Mount vane, SS, 3 mm, separation.


47


There is a large difference in separated flow between the 2 and 3 mm cases and the

corresponding results for the same bump heights on the regular vane (Figure 46 and Figure 47).

This further strengthens the conclusions from VAC (2009) that the mount vanes are much more

sensitive to non-conformances. For the 1 mm case the separation is restricted to a few small

areas close to the trailing edge. These are present on the nominal mount vane as well and they

have a negligible effect on the calculations. The spikes in Figure 49 and Figure 50 are due to the

separated flow detaching from the surface behind the bump and then reattaching halfway to the

TE. The flow barely manages to reattach again at the TE for the 2 mm bump while for the 3 mm

bump it fails completely. Looking at the pressure losses from VAC (2009) for 1 and 2 mm shows

that there is a rapid increase in pressure loss between these two heights. It therefore seems as 2

mm bumps are too large to be allowed. The 1 mm bump is in this case approximately 30 % of

the BLT and the 2 mm bump 60 %. It is interesting to note that the percentile increase in

boundary layer thickness for the 1 mm case compared to nominal is just about 18 %, which is

more in line with the flat plate results, and especially the 60 % bump height results. The results

from the boundary layer calculations on the suction side of the mount sector gives an

approximate limit for what bump sizes can be allowed on the suction peak. Because of the mount

vanes sensitivity to non-conformances and due to the rapid increase in pressure loss between 1

mm and 2 mm bumps a height of approximately 30 % of the nominal BLT would be

recommended.

The pressure side on the mount vane was very insensitive to geometry defects and the boundary

layers stabilized before reaching the trailing edge for all five bump heights (Figure 55 - Figure

59). As can be seen in the figures the large bumps greatly increase the boundary layer just behind

the defects but it still manages to stabilize before the trailing edge. By looking at the negative

axial velocity on the vane with a 5 mm bump on it (Figure 60) shows that the flow over the vane

is separation free, with the exempt of a small region just behind the bump. It can therefore be

concluded that bump sizes of up to 5 mm can be allowed on the pressure side of the mount vane

since no visible effects on the aero parameters can be seen.

48

Figure 55: Mount vane, PS, 5 mm.






49

4.2.2.3 Testing the correlation

As was mentioned in the method chapter the correlation derived from the flat plate simulations

was tested on a representative geometry with bumps. The correlation needs the boundary layer

thickness, the bump height and the Reynolds number to be able to calculate the drag coefficient.

The boundary layer thickness was calculated on the suction peak at 90 % span for both the

regular and the mount vane in chapter 4.2.1. The Reynolds number based on the chord was then

calculated from equation (3.1) and from these results the Reynolds number at the suction peak

could be determined (Table 15).

Table 15: Reynolds number for the chord and at the suction peak at 90 % span for both regular and mount vanes.

Case Position Reynolds number

Reg vane Chord, 90 % Span = 0.203 m 3’582’117

Mount vane Chord, 90 % Span = 0.265 m 4’693’553

Reg vane, SP Chord, 90 % Span = 0.075 m 1’325’383

Mount vane, SP Chord, 90 % Span = 0.077 m 1’361’130

Knowing all the data necessary to utilize the correlation the drag coefficient was calculated for

the bumps on the suction side of the vanes.

Table 16: Drag coefficient for the real geometry bumps based on the correlation. Scale factor is included for comparison.

Case

Variable Regular Mount

Bump height [mm] 1 2 3 4 5 1 2 3

h/99 0.324 0.647 0.971 1.295 1.618 0.301 0.602 0.904

Cd 0.285 0.392 0.454 0.499 0.533 0.275 0.381 0.443

Considering that the drag coefficient calculated by the department based on all their simulation

results is 0.25, the results from the correlation shows quite good accuracy for the smallest bumps.

It should however be noted that their results comes from vanes with bumps that are separation

free. The only bumps analyzed here that are separation free are the 1 mm bumps on the regular

and the mount vane. Owing to the fact that the drag coefficients calculated for the bump sizes

above 1 mm are much larger than the ones calculated for the 1 mm bumps, and that no validation

data is available for these, a more thorough investigation needs to be performed on these bump

sizes using the force equilibrium method presented in the method chapter to be able to call the

correlation validated and accurate enough.

50

5 Conclusions

In this chapter the conclusions that can be drawn from the results of the different project parts are

presented. Also included are some recommendations for future work.

5.1 Validation of the defined borders

5.1.1 Flat plate

From the reference simulations done in CFX it was found that the flow over the flat plate was

that of a forced turbulence flow and not of a developing flow. This resulted in that the boundary

layer thickness for lower Reynolds numbers matched the ideal laminar results for a flat plate

from the theory quite poorly. For higher Reynolds numbers the thickness matched the ideal

turbulent flat plate results much better but the boundary layer thicknesses are somewhat thinner

than what they should be according to theory. It is therefore possible that the boundary layer

thicknesses are somewhat under-predicted. However, the displacement and momentum boundary

thicknesses matched the theory results very well and the method used for calculating those can

thus be considered more reliable. The transition model available in the SST turbulence model

would be interesting to use since it would enable one to simulate a developing flow and thus be

able to get a more accurate solution for low Reynolds numbers.

From the flat plate with bump simulations it was realized that the average increase in boundary

layer thickness in the region behind the standard cosine bumps is 11% for bumps with a height of

40 % of the reference boundary layer thickness, 24 % for bumps with a height of 60 %, 57 %

with a height of 99 % and 114 % with a bump height of 150 %. For aggressive bumps at 40 %

height the increase was approximately 13 % and for wide bumps 5 %. The 10 % bumps showed

a negligible effect on the boundary layer and the lower limit in Table 1 can thus be considered

accurate. However, based on the flat plate simulations the boundary layer increase is just above

10 % for bump heights of 40 % so the lower limit can be rectified.

5.1.2 Real geometry

The boundary layers on the suction side and pressure side of real geometries representative of

vanes in a turbine rear frame was analyzed both with and without bumps. The boundary layer

thickness at the suction peak was calculated to be 3.4 mm at 50 % span and 3.1 mm at 90 % span

on the regular vane along with 4 mm at 50 % span and 3.3 mm at 90 % span on the mount vane.

The code used for calculating the boundary layer thickness along the vanes had great difficulties

when highly separated regions were present, as was the case with bumps larger than 1 mm on

both vane types. For the 1 mm bumps the increase in boundary layer thickness in the region

behind the bumps was found to be approximately 80 % on the regular vane and 20 % on the

mount vane. The two 1 mm bumps’ height in relation to the boundary layer thickness was

determined to be 32 % for the regular vane 33 % for the mount vane based on the nominal

thicknesses. Comparing the increases in boundary layer thickness with the results from the flat

plate simulation shows tendencies that the 1 mm bump on the regular vane has a boundary layer

51

thickness increase similar to the 99 % height and above while the bump on the mount vane is

close to 60 %.

There is number of possible reasons for this. One of these are that the vane geometry is curved,

compared to the plate being flat and the flow is therefore subject to adverse pressure gradients in

the region behind the bump which will make the boundary layer grow faster and the flow more

likely to separate from the vane. Another reason is that the boundary layer thicknesses found in

the flat plate reference simulations are, as was mentioned earlier, somewhat under-predicted.

This would make the five h/99 cases in the flat plate study higher than what they truly are even

though their effect on the boundary layer thickness would be correct. It would therefore be

interesting and advantageous to analyze and modify the code used for calculating the boundary

layer thickness so that it would match the correlations from the flat plate theory better.

The boundary layer increase is very large for the 1 mm case on the regular vane, compared to the

mount vane. By looking at the increase in displacement and momentum boundary thickness

between these two cases it was found that they were of similar magnitude. For that reason, the

bumps should have a similar impact on the boundary layer and the increase found for the bump

on the regular vane therefore seems to be over-predicted. The regular vane needs to be analyzed

in more detail to find the cause of the large boundary layer increase and the code used for

calculating the boundary layer thickness needs to be developed further.

Due to the fact that the mount vane is more sensitive to bumps on the suction peak it is

recommended that the bumps allowed on the suction side are considerably lower than 2 mm,

preferably below 1.5 mm since a 2 mm bump shows a large separation near the trailing edge and

almost a doubling in pressure loss compared to the 1 mm bump. A 1.5 mm bump would

correspond to a h/99 of around 45 %. The pressure side is less sensitive to non-conformances

and bump heights of 5 mm has a negligible effect on the boundary layer and on the pressure loss.

On the regular vane the flow separates for 2 mm bumps but manages to reconnect at the trailing

edge. For bumps higher than this the flow doesn’t reconnect and it is therefore recommended that

the bumps allowed on the suction side are below 2 mm, preferably in the range 30-50 % of the

boundary layer thickness. This limit does however need to be investigated further due to the

large uncertainties around the boundary layer calculations.

52

5.2 The correlation The drag coefficient correlation derived from the flat plate simulations is more reliable than the

boundary layer calculations due to the fact that it is calculated using more robust methods. The

correlation is based on the simulation results from the standard cosine bumps and calculates the

drag coefficient with an error of ± 5 % between the correlation calculation and the CFD results,

when used with standard cosine bumps. It was realized that changing the height to length ratio

affected the drag coefficient results from the simulations with an approximate increase of 27 %

for aggressive bumps and a 41 % decrease for wide bumps at 40 % height. Therefore, in its

current state the correlation should be analyzed further and possibly multiplied by a coefficient,

depending on the height to length ratio of the bump analyzed, so that a new correlation can be

derived.

When using the correlation on the real geometry with bumps it was able to determine the drag

coefficient for 1 mm bumps on the suction side of the regular vane and on the mount vane with

quite good accuracy. The calculated drag coefficient for the 1 mm bumps was 0.285 on the

regular vane and 0.275 on the mount vane. These can be compared to the drag coefficient

calculated by the department of 0.25, which are based on a few hundred simulations on a similar

geometry. However, for larger bumps the correlation needs to be further validated seeing as the

drag coefficient increases rapidly with increasing bump height. It would therefore be interesting

to use the force equilibrium method mentioned in the method chapter to realize how well the

correlation predicts the drag coefficient for bumps larger than 1 mm. The correlation derived is

shown below.

53

6 References

Andersson, J. D. (2007). Fundamentals of Aerodynamics, 1st international edition. McGraw-

Hill.

ANSYS Inc. (2009). Modeling 2D Problems. In CFX-Solver Modeling Guide. ANSYS Inc.

ANSYS Inc. (2009). Turbulence and Wall Function Theory. In CFX-Solver Theory Guide.

ANSYS Inc.

Cengel, Y. A., & Cimbala, J. M. (2006). Fluid Mechanics - Fundamentals and Applications, 1st

edition. McGraw-Hill.

Hoerner, S. F. (1965). Fluid-Dynamic Drag. Hoerner Fluid Dynamics.

Schlichting, H. (1979). Boundary-Layer Theory, 7th edition. McGraw-Hill Inc.

VAC. (2009). Aerodynamic study of non-conformances on a TRF. Volvo Aero Corporation.

Versteeg, H. K., & Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics

- The Finite Volume Method, 2nd edition. Pearson Education Limited.

54

7 List of appendices

7.1 Reference case

7.2 Flat plate with bump

7.3 Correlation data

7.4 Real geometry

7.5 CFX Scripts

7.6 MATLAB code

55

7.1 Reference case

Boundary layer thickness

a) Re = 2.5*105.

b) Re = 5*105.

c) Re = 106.

d) Re = 4*106.

e) Re = 6*106.

f) Re = 8*106.

Figure 61: BLT for the reference case at Re = a) 2.5*105, b) 5*105, c) 106, d) 4*106, e) 6*106 and f) 8*106.

56

Displacement boundary thickness

a) Re = 105.

b) Re = 2.5*105.

c) Re = 5*105.

d) Re = 106.

e) Re = 4*106.

f) Re = 6*106.

57

g) Re = 8*106.

Figure 62: DBT for Re a) 105, b) 2.5*105, c) 5*105, d) 106, e) 4*106, f) 6*106 and g) 8*106.

Momentum boundary thickness

a) Re = 105.

b) Re = 2.5*105.

c) Re = 5*105.

d) Re = 106.

58

e) Re = 4*106.

f) Re = 6*106.

g) Re = 8*106.

Figure 63: MBTfor Re a) 105, b) 2.5*105, c) 5*105, d) 106, e) 4*106, f) 6*106 and g) 8*106.

59

7.2 Flat plate with bump

Boundary layer thickness

Standard bump

a) 40 %.

b) 60 %.

c) 99 %.

d) 150 %

Figure 64: BLT for the four lowest Reynolds numbers with bump height a) 40 % b) 60 % c) 99 % d) 150 %.

60


Standard bump

a) 40 %.

b) 60 %.

c) 99 %.

d) 150 %.

Figure 65: Displacement boundary thickness for standard bumps with Re > 106 with height 40, 60, 99 and 150 %.

Figure 66: Displacement boundary thickness for standard bumps with height 10 %.

61

Aggressive bump

Figure 67: Displacement boundary thickness for aggressive bumps with height 40 %. (Reference included).

Wide bump

Figure 68: Displacement boundary thickness for wide bumps with height 40 % (Reference included).

62


Standard bump

a) 40 %.

b) 60 %.

c) 99 %.

d) 150 %.

Figure 69: Momentum boundary thickness for standard bumps with Re > 106 with height 40, 60, 99 and 150 %.

Figure 70: Momentum boundary thickness for bump height 10 %.

63

Aggressive bump

Figure 71: Momentum boundary thickness for aggressive bumps with height 40 %. (Reference included).

Wide bump

Figure 72: Displacement boundary thickness for aggressive bumps with height 40 %. (Reference included).

64

7.3 Correlation data

Table 17: Bump height (in mm) at x = 19 m for each ReL depending on the intended position in the reference BL.

ReL BLT Ref.

[mm]

Bump size

150% 120% 99% 80% 60% 40% 30% 20% 10% 5%

100’000 471.5 707.3 565.8 466.8 377.2 282.9 188.6 141.4 94.3 47.1 23.5

250’000 394.6 591.9 473.5 390.7 315.7 236.7 157.8 118.3 78.9 39.4 19.7

500’000 354.5 531.7 425.4 350.9 283.6 212.7 141.8 106.3 70.9 35.4 17.7

750’000 331.1 496.6 397.3 327.7 264.8 198.6 132.4 99.3 66.2 33.1 16.5

1’000’000 317.7 476.5 381.2 314.5 254.1 190.6 127.0 95.3 63.5 31.7 15.8

2’000’000 287.6 431.4 345.1 284.7 230.1 172.5 115.0 86.2 57.5 28.7 14.3

4’000’000 260.8 391.3 313.0 258.2 208.6 156.5 104.3 78.2 52.1 26.0 13.0

6’000’000 250.8 376.2 301.0 248.3 200.6 150.5 100.3 75.2 50.1 25.0 12.5

8’000’000 240.8 361.2 288.9 238.3 192.6 144.4 96.32 72.2 48.1 24.0 12.0

10’000’000 237.4 356.1 284.9 235.0 189.9 142.4 94.98 71.2 47.4 23.7 11.8

Table 18: Drag coefficients calculated from the simulation data.

Cd - Simulations Bump size

ReL 40 % 60 % 99 % 150 %

100’000 0.252523 0.326292 0.448712 0.562004

250’000 0.263432 0.341819 0.444298 0.544685

500’000 0.284602 0.362053 0.455332 0.541089

750’000 0.297230 0.372024 0.459825 0.535102

1’000’000 0.321220 0.387126 0.471965 0.537653

2’000’000 0.328126 0.390292 0.462694 0.516588

4’000’000 0.333603 0.390398 0.452341 0.501270

6’000’000 0.337287 0.387222 0.444305 0.489745

8’000’000 0.337927 0.383388 0.437516 0.479523

10’000’000 0.354040 0.391675 0.423839 0.472258

65

Table 19: Drag coefficients calculated from the correlation.

Cd - Correlations Scalefactor

ReL 40 % 60 % 99 % 150 %

100’000 0.238423 0.337579 0.460042 0.561656

250’000 0.267359 0.353511 0.459915 0.548202

500’000 0.288838 0.365154 0.459408 0.537616

750’000 0.301092 0.371653 0.458801 0.531111

1’000’000 0.309568 0.376046 0.458152 0.526279

2’000’000 0.328797 0.385439 0.455396 0.513442

4’000’000 0.345026 0.391832 0.449640 0.497606

6’000’000 0.352030 0.393081 0.443782 0.485852

8’000’000 0.355256 0.392225 0.437884 0.475769

10’000’000 0.356412 0.390214 0.431962 0.466602

Table 20: Drag coefficient for 40 % bumps with different shape.

Cd - Simulations Bump height 40 %

Bump shape

ReL Standard Aggressive Wide

100’000 0.25252 0.30004 0.18222

250’000 0.26343 0.32361 0.17951

500’000 0.28460 0.35729 0.18837

750’000 0.29723 0.37451 0.19277

1’000’000 0.32122 0.39646 0.20214

2’000’000 0.32813 0.41547 0.19194

4’00’0000 0.33360 0.42973 0.17615

6’000’000 0.33729 0.44111 0.16578

8’00’0000 0.33793 0.44253 0.15901

10’000’000 0.35404 0.46620 0.15552

66

7.4 Real geometry


Regular vane SS

a) 5 mm.

b) 4 mm.

c) 3 mm.

d) 2 mm.

e) 1 mm.

Figure 73: Displacement thickness, Regular vane, SS, bump height 1-5 mm (Nominal included).

67

Mount vane SS

a) 3 mm.

b) 2 mm.

c) 1 mm.

Figure 74: Displacement thickness, Mount vane, SS, bump height 1-3 mm (Nominal included).

Mount vane PS

a) 5 mm.

b) 4 mm.

68

c) 3 mm

d) 2 mm.

e) 1 mm.

Figure 75: Displacement thickness, Mount vane, PS, bump height 1-5 mm (Nominal included).

69


Regular vane SS

a) 5 mm.

b) 4 mm.

c) 3 mm.

d) 2 mm.

e) 1 mm.

Figure 76: Momentum thickness, Regular vane, SS, bump height 1-5 mm (Nominal included).

70

Mount vane SS

a) 3 mm.

b) 2 mm.

c) 1 mm.

Figure 77: Momentum thickness, Mount vane, SS, bump height 1-3 mm (Nominal included).

Mount vane PS

a) 5 mm.

b) 4 mm.

71

c) 3 mm.

d) 2 mm.

e) 1 mm.

Figure 78: Momentum thickness, Mount vane, PS, bump height 1-5 mm (Nominal included).

72

7.5 CFX-Scripts

Flat plate # Session file started: 2011/02/28 10:56:26

# CFX-12.1 build 2009.10.15-23.00

# To avoid unnecessary file pre-processing and modifications, include

# COMMAND FILE at the top of your session file.

# If it is not included, the file is assumed to be older and will be

# modified for backward compatibility.

COMMAND FILE:

CFX Post Version = 12.0

END

! mkdir "Reference/Reynoldsnmb1" ;

! $count = 0;

! while ($count <= 25) {;

! $LineName = "Reference/Reynoldsnmb1/line_data".$count.".csv";

! print "$count \n";

! $count2 = $count;

#! Used if creating lines on the bump peak

#! if ($count2==19) {

#! $bumpcount=0.1886286; #Modify depending on bump height

#! }

#! else {

#! $bumpcount=0;

! }

!

#Create vertical line with 600 data points

LINE:Line $count

Apply Instancing Transform = On

Colour = 1, 1, 0

Colour Map = Default Colour Map

Colour Mode = Constant

Colour Scale = Linear

Colour Variable = Pressure

Colour Variable Boundary Values = Hybrid

Domain List = /DOMAIN GROUP:All Domains

Instancing Transform = /DEFAULT INSTANCE TRANSFORM:Default Transform

Line Samples = 600

Line Type = Sample

Line Width = 1

Max = 0.0 [Pa]

Min = 0.0 [Pa]

Option = Two Points

Point 1 = $count2 [m], $bumpcount [m], 0 [m]

Point 2 = $count2 [m], 2.5 [m], 0 [m]

Range = Global

OBJECT VIEW TRANSFORM:

Apply Reflection = Off

Apply Rotation = Off

Apply Scale = Off

Apply Translation = Off

73

Principal Axis = Z

Reflection Plane Option = XY Plane

Rotation Angle = 0.0 [degree]

Rotation Axis From = 0 [m], 0 [m], 0 [m]

Rotation Axis To = 0 [m], 0 [m], 0 [m]

Rotation Axis Type = Principal Axis

Scale Vector = 1 , 1 , 1

Translation Vector = 0 [m], 0 [m], 0 [m]

X = 0.0 [m]

Y = 0.0 [m]

Z = 0.0 [m]

END

END

#Export the data from Line2 to a new file

EXPORT:

ANSYS Export Data = Element Heat Flux

ANSYS File Format = ANSYS

ANSYS Reference Temperature = 0.0 [K]

ANSYS Specify Reference Temperature = Off

ANSYS Supplemental HTC = 0.0 [W m^-2 K^-1]

BC Profile Type = Inlet Velocity

Export Connectivity = Off

Export Coord Frame = Global

Export File = $LineName

Export Geometry = On

Export Node Numbers = Off

Export Null Data = On

Export Type = Generic

Export Units System = Current

Export Variable Type = Current

Include File Information = Off

Include Header = On

Location List = /LINE:Line $count

Null Token = null

Overwrite = On

Precision = 8

Separator = " "

Spatial Variables = X,Y,Z

Variable List = Pressure, Total Pressure, Velocity

Vector Brackets = ()

Vector Display = Scalar

END

>export

! $count ++;

! }

# End while loop

74

Real geometry

COMMAND FILE:

CFX Post Version = 12.0

END

! mkdir "Polylines_mount_ss_90" ;

# Start with reading in points from the surface around the blade containing X,Y,Z

# coordinates and normal directions.

! $cnt = 0;

! open(IN,"<blade_data_mount_span90_ss.csv");

! while (defined ($line = <IN>)) {;

! @coord = split(',',$line);

#! $coord[3] = $coord[3] * -1; #Pressure side

! $coord[5] = $coord[5] * -1; #Suction side

! $LineName = "Polylines_mount_ss_90/Linevert".$cnt.".csv";

! $LineName3 = "Polylines_mount_ss_90/Line".$cnt.".csv";

! $LineName4 = "Polylines_mount_ss_90/Line_data".$cnt.".csv";

PLANE:Plane 2


Apply Texture = Off

Blend Texture = On

Bound Radius = 0.5 [m]

Colour = 0.75, 0.75, 0.75






Culling Mode = No Culling

Direction 1 Bound = 1.0 [m]

Direction 1 Orientation = 0 [degree]

Direction 1 Points = 10

Direction 2 Bound = 1.0 [m]

Direction 2 Points = 10


Draw Faces = On

Draw Lines = Off


Invert Plane Bound = Off

Lighting = On

Line Colour = 0, 0, 0

Line Colour Mode = Default

Line Width = 1

Max = 0.0 [Pa]

Min = 0.0 [Pa]

Normal = $coord[5] , $coord[4] , $coord[3]

Option = Point and Normal

Plane Bound = None

Plane Type = Slice

Point = $coord[0] [m], $coord[1] [m], $coord[2] [m]

Point 1 = 0 [m], 0 [m], 0 [m]

Point 2 = 1 [m], 0 [m], 0 [m]

75

Point 3 = 0 [m], 1 [m], 0 [m]

Range = Global

Render Edge Angle = 0 [degree]

Specular Lighting = On

Surface Drawing = Smooth Shading

Texture Angle = 0

Texture Direction = 0 , 1 , 06.32090271e-01

Texture File =

Texture Material = Metal

Texture Position = 0 , 0

Texture Scale = 1

Texture Type = Predefined

Tile Texture = Off

Transform Texture = Off

Transparency = 0.0

X = 0.0 [m]

Y = 0.0 [m]

Z = 0.0 [m]




Apply Scale = Off


Principal Axis = Z








X = 0.0 [m]

Y = 0.0 [m]

Z = 0.0 [m]

END

END

76

CONTOUR:Contour N


Clip Contour = Off



Colour Variable = Span Normalized


Constant Contour Colour = Off

Contour Range = Global

Culling Mode = No Culling


Draw Contours = On

Font = Sans Serif

Fringe Fill = On


Lighting = On

Line Colour = 0, 0, 0

Line Colour Mode = Default

Line Width = 1

Location List = /PLANE:Plane 2

Max = 0.0 [m]

Min = 0.0 [m]

Number of Contours = 19 #(Change so that one contour is above the span of interest)

Show Numbers = Off

Specular Lighting = On

Surface Drawing = Smooth Shading

Text Colour = 0, 0, 0

Text Colour Mode = Default

Text Height = 0.024

Transparency = 0.0

Value List = 0 [m],1 [m]




Apply Scale = Off


Principal Axis = Z








X = 0.0 [m]

Y = 0.0 [m]

Z = 0.0 [m]

END

END

77

POLYLINE:Polylinee


Colour = 0, 1, 0





Colour Variable Boundary Values = Conservative

Contour Level = 13 #(Change to that the contour is on the span of interest)

Contour Name = /CONTOUR:Contour N


Input File =


Line Width = 2

Max = 0.0 [Pa]

Min = 0.0 [Pa]

Option = From Contour

Range = Global




Apply Scale = Off


Principal Axis = Z








X = 0.0 [m]

Y = 0.0 [m]

Z = 0.0 [m]

END

END

USER SCALAR VARIABLE:Vel

Boundary Values = Conservative

Calculate Global Range = On

Expression = sqrt(Velocity u^2+Velocity v^2+Velocity w^2)

Recipe = Expression

Variable to Copy = Pressure

Variable to Gradient = Pressure

END

USER SCALAR VARIABLE:MachNumber



Expression = Vel/sqrt(1.4*287[J kg^-1 K^-1]*Temperature)

Recipe = Expression



END

78

USER SCALAR VARIABLE:TotalPressure



Expression =Pressure*(1+((1.4-1)/2)*MachNumber^2)^(1.4/(1.4-1))

Recipe = Expression



END

EXPORT:


ANSYS Export Locator = /USER SURFACE:User Surface 1








Export File = $LineName3








Include Header = Off

Location List = /POLYLINE:Polylinee

Null Token = null

Overwrite = Off

Precision = 8

Separator = ", "


Variable List =



END

>export

! open(INV,">$LineName"); #Opens the file with the vertical polyline

! open(OUT,"<$LineName3"); #Opens the file LineXX.csv

! $count_rows=0;

! while (defined ($line2 = <OUT>)) {;

! @coord_poly = split(',',$line2);

! if ($coord_poly[2] >= $coord[2]){ #(change to <= for pressure side)

! print INV @coord_poly;

! $count_rows = $count_rows+1;

! }

! }

! close(OUT);

! close(INV);

! open(INV2,"<$LineName");

79

! $count_sort = 0;

! while (defined($line3=<INV2>)){;

! @coord_sort=split(' ',$line3);

! if ($count_sort == 0){

! @coord_one = @coord_sort;

! }

! if ($count_sort == 5){

! @coord_two = @coord_sort;

! }

! $count_sort=$count_sort+1;

! }

! print "Point1: $coord_one[2]\n";

! print "Point5: $coord_two[2]\n";

! close(INV2);


! if ($coord_one[2] < $coord_two[2]){ #(change to < for pressure side)

! $count3 = 0;

! $middle = $count_rows*0.6;

! $middle_roundoff = int($middle);

! while (defined($line4 = <INV3>)) {;

! @coord_vert = split(' ',$line4);

! if ($count3 == 0){

! @coord_blade = @coord_vert;

! }

! if ($count3 == $middle_roundoff) {

! @coord_end = @coord_vert;

! }

! $count3 = $count3+1;

! }

! close(INV3);

! } else {

! $count4 = 0;

! $middle = $count_rows*0.4;

! $middle_roundoff = int($middle);


! while (defined($line5 = <INV4>)) {;

! @coord_vert2 = split(' ',$line5);

! if ($count4 == ($count_rows-1)){

! @coord_blade = @coord_vert2;

! }

! if ($count4 == $middle_roundoff) {

! @coord_end = @coord_vert2;

! }

! $count4 = $count4+1;

! }

! close(INV4);

! }

80

! print "X0= $coord_blade[0]\n";

! print "Y0= $coord_blade[1]\n";

! print "Z0= $coord_blade[2]\n";

! print "XE= $coord_end[0]\n";

! print "YE= $coord_end[1]\n";

! print "ZE= $coord_end[2]\n";

#Creating the line normal to the surface.

LINE:Line 2


Colour = 1, 1, 0








Line Samples = 3500

Line Type = Sample

Line Width = 2

Max = 0.0 [Pa]

Min = 0.0 [Pa]

Option = Two Points

Point 1 = $coord_blade[0] [m], $coord_blade[1] [m], $coord_blade[2] [m]

Point 2 = $coord_end[0] [m], $coord_end[1] [m], $coord_end[2] [m]

Range = Global




Apply Scale = Off


Principal Axis = Z








X = 0.0 [m]

Y = 0.0 [m]

Z = 0.0 [m]

END

END

#Exporting the line normal to the surface.

EXPORT:


ANSYS Export Locator = /USER SURFACE:User Surface 1






81



Export File = $LineName4








Include Header = Off

Location List = /LINE:Line 2

Null Token = null

Overwrite = On

Precision = 8

Separator = " "


Variable List = Pressure, TotalPressure



END

>export

! $cnt = $cnt +1;

! }

! close(IN);

# End while sats

82

7.6 MATLAB code

Flat plate

clear all

close all

%% Beginning of loop for going through all Reynolds numbers in the folder

for it=1:10;

%% Import data from files

format long

% Path to simulation data files

dirName =

cat(2,'/home/yy53418/Simuleringar/Flat_plate/Simulations/Reference/Reynoldsnmb',num2st

r(it))

fileName = cell(31,1);

for n=1:31;

fileName{n,1} = cat(2,dirName,'/line_data',num2str(n-1),'.csv');

end

Delimiter = ' ';

Headerlines = 6;

BL0i = importdata(char(fileName(1)), Delimiter, Headerlines);































83

%% Compile data. Extracts the data from the imported files.

BL0 = BL0i.data;

BL1 = BL1i.data;

BL2 = BL2i.data;

BL3 = BL3i.data;

BL4 = BL4i.data;

BL5 = BL5i.data;

BL6 = BL6i.data;

BL7 = BL7i.data;

BL8 = BL8i.data;

BL9 = BL9i.data;

BL10 = BL10i.data;

BL11 = BL11i.data;

BL12 = BL12i.data;

BL13 = BL13i.data;

BL14 = BL14i.data;

BL15 = BL15i.data;

BL16 = BL16i.data;

BL17 = BL17i.data;

BL18 = BL18i.data;

BL19 = BL19i.data;

BL20 = BL20i.data;

BL21 = BL21i.data;

BL22 = BL22i.data;

BL23 = BL23i.data;

BL24 = BL24i.data;

BL25 = BL25i.data;

BL26 = BL26i.data;

BL27 = BL27i.data;

BL28 = BL28i.data;

BL29 = BL29i.data;

BL30 = BL30i.data;

%% Calculate thicknesses

% Inlet velocity for the domain. (Used for ideal flat plate calculations)

if it==1;

Vf = 0.081323562;

elseif it==2;

Vf = 0.203308905;

elseif it==3;

Vf = 0.40661781;

elseif it==4;

Vf = 0.609926716;

elseif it==5;

Vf = 0.813235621;

elseif it==6;

Vf = 1.626471241;

elseif it==7;

Vf = 3.252942483;

elseif it==8;

Vf = 4.879413724;

elseif it==9;

Vf = 6.505884966;

elseif it==10;

Vf = 8.132356207;

end

84

% X-position depending on which plate length is simulated.

Xpos = (0:1:30)';

% Density

rho = 1.185;

% Viscosity

mu = 1.831*10^-5;

% Kinematic viscosity

nu = mu/rho;

% Boundary layer calculations

clear dbts0 mbts0 lenVel0;

% The location of the maximum pressure along the line.

[C, I] = max(BL0(1:end,5));

% The length between the plate and each point.

len = BL0(1:I,2);

% V/Vmax.

V = sqrt(BL0(1:I,5) - BL0(1,4))/sqrt(BL0(I,5) - BL0(1,4));

% Velocity at the wall = 0.

V(1) = 0;

% Displacement boundary-layer series.

dbts0(1) = (1-V(1))*len(1);

% Momentum boundary-layer series.

mbts0(1) = dbts0(1)*V(1);

% Velocity at each point up to the point of maximum velocity.

lenVel0 = [len, sqrt(2*(BL0(1:I,5) - BL0(1,4))/rho)];

% The approximate position of 0.99*V.

[min_difference, array_position] = min(abs(lenVel0(:,2)-0.99*max(lenVel0(:,2))));

% Calculation of DBT and MBT

for iter = 2:length(V);

dbts0(iter) = (1-V(iter))*(len(iter) - len(iter-1));

mbts0(iter) = dbts0(iter)*V(iter);

end

% The displacement boundary-layer thickness at Xpos = 0.

dbt0 = sum(dbts0)

% The momentum boundary-layer thickness at Xpos = 0.

mbt0 = sum(mbts0)

% The boundary layer thickness at Xpos = 0.

blt0 = lenVel0(array_position,1)

% The code for calculating the blt, dbt and mbt for line 1-29 has been left out

% to save space in the report.

clear dbts30 mbts30;

% The location of the maximum pressure along the line.

[C, I] = max(BL30(1:end,5));

% The length between the plate and each point.

len = BL30(1:I,2);

% V/Vmax.

V = sqrt(BL30(1:I,5) - BL30(1,4))/sqrt(BL30(I,5) - BL30(1,4));


V(1) = 0;


dbts30(1) = (1-V(1))*len(1);


mbts30(1) = dbts30(1)*V(1);

% Velocity at each point up to the point of maximum velocity.

lenVel30 = [len, sqrt(2*(BL30(1:I,5) - BL30(1,4))/rho)];


min_difference, array_position] = min(abs(lenVel30(:,2)-0.99*max(lenVel30(:,2))));

% Calculation of DBT and MBT.

85




end

% The displacement boundary-layer thickness at Xpos = 30.

dbt30 = sum(dbts30)

% The momentum boundary-layer thickness at Xpos = 30.

mbt30 = sum(mbts30)

% The boundary layer thickness at Xpos = 30.


%% Array of the BL, DBL and MBL thicknesses

dbt =

[dbt0;dbt1;dbt2;dbt3;dbt4;dbt5;dbt6;dbt7;dbt8;dbt9;dbt10;dbt11;dbt12;dbt13;dbt14;dbt15

;dbt16;dbt17;dbt18;dbt19;dbt20;dbt21;dbt22;dbt23;dbt24;dbt25;dbt26;dbt27;dbt28;dbt29;d

bt30];

mbt =

[mbt0;mbt1;mbt2;mbt3;mbt4;mbt5;mbt6;mbt7;mbt8;mbt9;mbt10;mbt11;mbt12;mbt13;mbt14;mbt15

;mbt16;mbt17;mbt18;mbt19;mbt20;mbt21;mbt22;mbt23;mbt24;mbt25%mbt26;mbt27;mbt28;mbt29;m

bt30];

blt =

[blt0;blt1;blt2;blt3;blt4;blt5;blt6;blt7;blt8;blt9;blt10;blt11;blt12;blt13;blt14;blt15

;blt16;blt17;blt18;blt19;blt20;blt21;blt22;blt23;blt24;blt25;blt26;blt27;blt28;blt29;b

lt30];

%% Calculation of the BL, DBL and MBL thickness for an ideal flat plate

for i = 1:31

% laminar

delta(i) = 1.7208*sqrt(nu*Xpos(i,1)/Vf);

theta(i) = 0.664*sqrt(nu*Xpos(i,1)/Vf);

blti(i) = 5*sqrt(nu*Xpos(i,1)/Vf);

% turbulent

delta2(i) = 0.04625*Xpos(i,1)*((Vf*Xpos(i,1))/nu)^-0.2;

theta2(i) = 0.036*Xpos(i,1)*((Vf*Xpos(i,1)/nu))^-0.2;

blti2(i) = 0.37*Xpos(i,1)*((Vf*Xpos(i,1)/nu))^-0.2;

end

%% Graphs of the layers along the flat plate

figure(1); clf;

plot(Xpos(:),blt(:),'k',Xpos(:),blti(:),'rx',Xpos(:),blti2(:),'g*','linewidth',2,'mark

ersize',5);

legend('BL thickness','ideal laminar','ideal turbulent',2)

title('Boundary-layer thickness','fontsize',12);

ylabel('Thickness [m]','fontsize',12);

xlabel('Position along plate [m]','fontsize',12);

axis([0 30 0 1])

figure(2); clf;

plot(Xpos(:),dbt(:),'k',Xpos(:),delta(:),'rx',Xpos(:),delta2(:),'g*','linewidth',2,'ma

rkersize',5);

legend('Displacement thickness','ideal laminar','ideal turbulent',2)

title('Displacement boundary-layer thickness','fontsize',12);



axis([0 30 0 0.2])

86

figure(3); clf;

plot(Xpos(:),mbt(:),'k',Xpos(:),theta(:),'rx',Xpos(:),theta2(:),'g*','linewidth',2,'ma

rkersize',5);

legend('Momentum thickness','ideal laminar','ideal turbulent',2)

title('Momentum boundary layer thickness','fontsize',12);



axis([0 30 0 0.16])

%% Save the graphs to the image directory as jpeg files

savePath =

cat(2,'/home/yy53418/Simuleringar/Flat_plate/Images/Reference/Reynoldsnmb',num2str(it)

)

dbtPath = cat(2,savePath,'/DBT');

mbtPath = cat(2,savePath,'/MBT');

bltPath = cat(2,savePath,'/BLT');

saveas(1,bltPath,'jpg')

saveas(2,dbtPath,'jpg')

saveas(3,mbtPath,'jpg')

% Save the boundary layer thickness at 19 meters to a file.

compData = [Vf*Xpos(20,1)/nu, blt19];

save /home/yy53418/Simuleringar/Flat_plate/Simulations/Reference/compRE.txt compData -

append -ascii

blData = blt(:);

dbtData = dbt(:);

mbtData = mbt(:);

% Save the BLT, DBT, MBT along the plate for a file.

if it==1;

save

/home/yy53418/Simuleringar/Flat_plate/Simulations/Reference/Reynoldsnmb1/BLthick.txt

blData dbtData mbtData -append -ascii

end

if it==2;

save



end

if it==3;

save



end

if it==4;

save



end

if it==5;

save



end

if it==6;

save



end

if it==7;

save



87

end

if it==8;

save



end

if it==9;

save



end

if it==10;

save



end

it

%% end of loop

end

88

Real geometry

Same as the code used for the flat plate but with the modification that the maximum pressure

along each line is found using the method shown below.

clear dbts0 mbts0 lenVel0;

% The location of the first maximum pressure along the line.

[C, I] = findpeaks(BL0(1:end,5));

% The length from the blade to each point.

len = sqrt((BL0(1:I(1),1)-BL0(1,1)).^2+(BL0(1:I(1),3)-BL0(1,3)).^2);

% V/Vmax.

V = sqrt(BL0(1:I(1),5) - BL0(1,4))/sqrt(BL0(I(1),5) - BL0(1,4));


V(1) = 0;


dbts0(1) = (1-V(1))*len(1);


mbts0(1) = dbts0(1)*V(1);

% Velocity at each point up to the point of first maximum velocity.

lenVel0 = [len, sqrt(2*(BL0(1:I(1),5) - BL0(1,4))/rho)];


[min_difference, array_position] = min(abs(lenVel0(:,2)-0.99*max(lenVel0(:,2))));

% Calculation of DBT and MBT.




end

% The displacement boundary-layer thickness.

dbt0 = sum(dbts0)

% The momentum boundary-layer thickness.

mbt0 = sum(mbts0)

% The boundary layer thickness.


89

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