Numerical study of the effects of wind-tunnel conditions on a passenger vehicle in crosswinds

Numerical study of the effects of wind-tunnel conditions on a

passenger vehicle in crosswinds

Numerisk studie av effekterna av

vindtunnelförhållanden på en personbil i sidovind

SAHIL GUPTA VIVEK J SHAH

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

Numerical study of the effects of wind-tunnel conditions on a passenger vehicle in

crosswinds

SAHIL GUPTA, VIVEK J SHAH

Master of Science in Vehicle Engineering Date: October 15, 2020

Supervisor: Alessandro Talamelli, Stefan Wallin Examiner: Philipp Schlatter

School of Engineering Sciences

Swedish title: Numerisk studie av effekterna av vindtunnelförhållanden på en personbil i sidovind

Abstract

Wind tunnel tests is a necessity when it comes to study of aerodynamic flow field around vehicle in its development process. However, there are certain limitations on physical wind tunnel tests during vehicle development process.

This includes high running costs and interference effects due to blockage and boundary conditions.

Hence, one of the main objective of this thesis is to study the effect of these boundary conditions and interference by a physical wind tunnel numer- ically in a virtual wind tunnel in straightline and steady crosswinds scenario.

Furthermore, accuracy of different wind tunnel correction methods is studied.

Lastly, effects of rounding of A and C pillars on aerodynamic flow field is investigated.

The open scenario is used as a baseline for comparison. Blockage ratio of virtual wind tunnel is 8%. Windsor model with two different backlight angles have been used for this study.

It has been observed that solid and wake blockage contribute to an increase in drag in wind tunnel. Furthermore, lift coefficient increases and drag coeffi- cient increases when floor is stationary when compared to moving floor. It is also noted that different correction methods are suitable based on flow charac- teristics and scenarios considered. It is difficult to understand which method to apply without extensively understanding the flow characteristics.

Lastly, rounding of A-pillar results in reduction of drag and lift coeffi- cient for both models. Under yawed condition, rounding of A-pillar results in reduction of yaw moment coefficient. Rounding of C-pillar results in in- crease in drag coefficient and increase in lift coefficient for Windsor model with 25^◦slant angle. However, when the same model is kept at yaw, rounding of C-pillar result in reduction in drag coefficient, decrease in lift coefficient and yaw moment coefficient.

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Sammanfattning

Vindtunneltester är nödvändiga när man studerar det aerodynamiska flödet runt ett fordon under utvecklingsprocessen. Fysiska vindtunneltester för ut- veckling av fordonhar dock begränsningar som höga driftskostnader och infly- tandet av interferenseffekter på grund av blockage och randvillkor.

Ett av de huvudsakliga målen för detta examensarbete är därför att studera effekterna av randvillkor och interferens-effekter genom att studera dessa fy- sikaliska effekter numeriskt i en virtuell vindtunnel. Detta studeras både för vinden rakt framifrån och i ett scenario med sidovind. Dessutom har vi stude- rat hur noggranna olika vindtunnel- korrektioner är. Vi har slutligen utvärderat hur rundade A och C stolpar påverkar det aerodynamiska flödet.

Den virtuella vindtunneln har ett blockage av 8% och har jämförts med ett öppet scenarium med minimal blockage. För denna studien har Windsor modellen använts med två olika bakvinklar.

Det har observerats att både geometrisk blockage och blockage från vaken bidrar till en ökad motståndskoefficient i vindtunneln. Det kan även konsta- teras att nedåtkraften minskar och motståndskoefficienten ökar när golvet är stationärt jämfört med ett rullande golv. Det noteras att lämpligheten för olika korrektionsmetoder är beroende av flödets karaktäristik och typ av scenario.

Det är svårt att avgöra vilken metod som bör användas utan en djupgående förståelse av flödes karaktäristik.

Det kunde också konstateras att rundade A stolpar resulterar i en redu- cerad motståndskoefficient och nedåtkraft för båda modellerna. Vid sidovind resulterar rundade A stolpar i en reduktion av girmomentet. Rundade C stolpar resulterar i ett ökat motstånd och reducerad nedåtkraft för Windsor modellen med 25 graders bakvinkel. När samma modell används vid sidovind resulterar rundade C stolpar i ett reducerat motstånd samt ökad nedåtkraft och girmo- ment.

Acknowledgements

We would like to thank our supervisors Dr. Stefan Wallin and Dr. Alessandro Talamelli for their continuous guidance and advice throughout the thesis and the time they devoted to us. This work was carried out smoothly due to their expertise and guidance.

The thesis benefited from computing resources at the Centre for Parallel Computers (PDC) at KTH, Stockholm. Computer supports at PDC (with Mr.

Jing Gong and other support staff) is greatly acknowledged.

Last but not the least, we would like to thank our families who gave us the opportunity to study at one of the best engineering schools in the world.

To conclude, this thesis is an important step in our lives and we have gained profusely, be it technical knowledge or working in a team. This thesis was a small experience for us to see how things work in a professional life. We are excited and hopeful of what future holds for us.

Contents

1 Introduction 1

1.1 Outline of this thesis . . . 2

2 Fundamental definitions 3 2.1 Aerodynamic loads . . . 3

2.2 Blockage ratio . . . 5

2.3 Duplex vehicle . . . 6

3 Background and objective 7 3.1 Wind tunnel blockage and blockage corrections method . . . . 7

3.1.1 Maskell correction for bluff bodies . . . 8

3.1.2 DNW Maskell correction method . . . 9

3.1.3 Maskell III correction method . . . 9

3.1.4 Thom and Herriot’s correction method . . . 10

3.1.5 Modified Thom and Herriot’s correction method . . . 11

3.1.6 Mercker’s correction method . . . 11

3.2 Effect of presence of moving ground . . . 12

3.3 Basic car shapes under crosswind in wind tunnel . . . 13

3.4 Objective of this thesis . . . 15

4 Methodology 16 4.1 Geometries considered . . . 16

4.2 Scenarios simulated . . . 17

4.3 Computational domain . . . 19

4.4 Orientation of Windsor model with respect to wind tunnel . . 20

4.5 CFD Setup . . . 23

4.5.1 Boundary conditions . . . 23

4.5.2 Boundary condition analysis . . . 24

4.5.3 Solver . . . 25

4.5.4 Reynolds Averaged Navier Stokes (RANS) . . . 26

v

5.2 Grid independence test . . . 33

5.3 Grid selection for 3D meshing . . . 36

5.3.1 Grid selection for Windsor_squareback model . . . . 37

5.3.2 Grid selection for Windsor_25 model . . . 40

5.4 Meshing in 3D . . . 43

6 Results and discussion 47 6.1 Blockage effects . . . 47

6.2 Effect of moving ground . . . 58

6.2.1 Effect of moving ground under steady crosswinds . . . 65

6.3 Vehicle under steady crosswinds . . . 70

6.3.1 Blockage correction for Windsor models with sharp edges . . . 82

6.4 Rounding of A-pillar . . . 83

6.4.1 Rounding of A-pillar in straight line . . . 84

6.4.2 Rounding of A-pillar in yaw . . . 87

6.4.3 Blockage correction for Windsor models with rounded A-pillars . . . 91

6.5 Rounding of C-pillar . . . 92

6.5.1 Rounding C-pillar in straight line . . . 93

6.5.2 Rounding of C-pillar in yaw . . . 99

6.5.3 Blockage correction for Windsor models with rounded C-pillars . . . 105

7 Conclusion and future works 106 7.1 Conclusion . . . 106

7.2 Suggestions for future works . . . 108

Bibliography 110

List of Figures

2.1 Aerodynamic loads based on coordinate axis . . . 4

4.1 Windsor models used in this study . . . 18

4.2 Computational domain with blockage ratio of 0.06% . . . 19

4.3 Computational domain with blockage ratio of 8% . . . 20

4.4 Frame of reference for model in straight line (frame of refer- ence 1 in black) and for model in yaw (frame of reference 2 in red) . . . 21

4.5 Orientation of Windsor model with respect to wind tunnel in straight line scenario . . . 22

4.6 Orientation of Windsor model with respect to wind tunnel in yawed scenario . . . 23

5.1 Location of edges and refinement boxes which are referred in table 5.1 . . . 30

5.2 Overall mesh in 2-D for Windsor_25 . . . 31

5.3 Mesh in RB1 and RB2 around Windsor_25 model . . . 31

5.4 Mesh in RB3 at the rear of Windsor_25 model . . . 32

5.5 Grid independence test Windsor_squareback model . . . 34

5.6 Grid independence test Windsor_25 model . . . 35

5.7 Y+ for meshes with 65000 cells (Red) and 160,000 (Black) for Windsor_squareback model . . . 36

5.8 Fully resolved boundary layer . . . 36

5.9 Pressure probes around Windsor_squareback in 2-D . . . 37

5.10 Difference in pressure coefficient along WSQ 1 probe in x- direction between mesh with 160000 cells and 65000 cells . . 38

5.11 Difference in pressure coefficient along WSQ 2 probe in x- direction between mesh with 160000 cells and 65000 cells . . 39

5.12 Difference in pressure coefficient along WSQ 3 probe in x- direction between mesh with 160000 cells and 65000 cells . . 39

vii

5.13 Pressure probes around Windsor_25 model in 2-D . . . 40

5.14 Difference in pressure coefficient along W25 1 probe in x- direction between mesh with 160000 cells and 65000 cells . . 41

5.15 Difference in pressure coefficient along W25 2 probe in x- direction between mesh with 160000 cells and 65000 cells . . 41

5.16 Difference in pressure coefficient along W25 3 probe in x- direction between mesh with 160000 cells and 65000 cells . . 42

5.17 Difference in pressure coefficient along W25 4 probe in x- direction between mesh with 160000 cells and 65000 cells . . 42

5.18 Difference in pressure coefficient along W25 5 probe in x- direction between mesh with 160000 cells and 65000 cells . . 43

5.19 Overall mesh in 3-D for open case . . . 44

5.20 Overall mesh in 3-D for wind tunnel with blockage ratio of 8% 44 5.21 Mesh around Windsor_25 model in 3-D . . . 45

5.22 Mesh in spanwise direction of Windsor_25 model in 3-D . . . 45

5.23 Mesh on the rear of Windsor_25 model in 3-D . . . 46

5.24 Surface contour of Y+ around Windsor_squareback model . . 46

6.1 Location of Probes . . . 48

6.2 Velocity variation along X1 probe . . . 50

6.3 Velocity variation along Y2 probe . . . 51

6.4 Velocity variation along Y1 probe . . . 53

6.5 Velocity variation along Y2 probe . . . 54

6.6 Pressure contour along base of the model in straightline sce- nario for Windsor_squareback model . . . 55

6.7 Pressure contour along base of the model at 5^◦yaw angle for Windsor_squareback model . . . 56

6.8 Pressure contour along base of the model in straightline sce- nario for Windsor_25 model . . . 57

6.9 Pressure contour along base of the model at 5^◦ yaw angle for Windsor_25 model . . . 58

6.10 Variation of x-component of velocity along X2 probe . . . 60

6.11 Variation of velocity along X3 probe . . . 62

6.12 Variation of pressure coefficient along X3 probe . . . 63

6.13 Variation of x-component of velocity along Y3 probe . . . 64

6.14 Velocity contour along z=0.195m extending from (-0.375,-0.05) to (2.8,0.29) in xy plane for Windsor_squareback kept at 5^◦ yaw angle in wind tunnel with 8% blockage ratio (case 4) . . . 66

LIST OF FIGURES ix

6.15 Velocity contour along z=0.195m extending from (-0.375,-0.05) to (2.8,0.29) in xy plane for Windsor_25 kept at 5^◦yaw angle in wind tunnel with 8% blockage ratio (case 4) . . . 67 6.16 Velocity contour along x=1.5m extending from (-0.05,-0.555)

to (0.88,0.945) in (y,z) direction for Windsor_squareback kept at 5^◦yaw angle in wind tunnel with 8% blockage ratio (case 4) 68 6.17 Velocity contour along x=1.5m extending from (-0.05,-0.555)

to (0.88,0.945) in (y,z) direction for Windsor_25 kept at 5^◦ yaw angle in wind tunnel with 8% blockage ratio (case 4) . . . 69 6.18 Change in drag and lift coefficients at 5^◦and 10^◦yaw angles

with respect to 0^◦yaw angle in 8% blockage ratio as a function of yaw angles for Windsor_squareback and Windsor_25 models 72 6.19 Change in side force and yaw moment coefficients at 5^◦and

10^◦yaw angles with respect to 0^◦yaw angle in 8% blockage ratio as a function of yaw angles for Windsor_squareback and Windsor_25 models . . . 73 6.20 Pressure contour along plane x=1.1m extending from (-0.05,-

0.555) to (0.88,0.945) in (y,z) direction for Windsor_squareback kept in wind tunnel with 8% blockage ratio . . . 74 6.21 Pressure contour along x=1.1m extending from (-0.05,-0.555)

to (0.88,0.945) in (y,z) direction for Windsor_25 kept in wind tunnel with 8% blockage ratio . . . 75 6.22 Location of lateral probes for Windsor_25 model at 10^◦ yaw

angle in wind tunnel with 8% blockage ratio (case 4) . . . 76 6.23 Variation of pressure coefficient along Z2 probe . . . 77 6.24 Variation of pressure coefficient along Z3 probe . . . 78 6.25 Velocity contour along y=0m extending from (-0.836,-0.429)

to (4.9,0.78) in (x,z) direction for Windsor_squareback kept in wind tunnel with 8% blockage ratio . . . 80 6.26 Velocity contour along y=0m extending from (-0.836,-0.429)

to (4.9,0.78) in (x,z) direction for Windsor_25 kept in wind tunnel with 8% blockage ratio . . . 81 6.27 Rounded A-pillar . . . 84 6.28 Flow structures for Windsor_squareback at 0^◦ yaw angle in

wind tunnel with 8% blockage ratio (case 2) corresponding to Q = 0.0017 . . . 86 6.29 Velocity contour along y=0.25m extending from (-0.22, -0.195)

to (0.94,0.585) in (x,z) direction for Windsor_squareback at 0^◦ yaw angle in wind tunnel with 8% blockage ratio (case 2) . . . 87

6.30 Flow structures for Windsor_squareback at 10^◦ yaw angle in wind tunnel with 8% blockage ratio (case 4) corresponding to Q = 0.0017 . . . 89 6.31 Velocity contour along y=0.25m extending from (-0.22, -0.195)

to (0.94,0.585) in (x,z) direction for Windsor_squareback at 10^◦ yaw angle in wind tunnel with 8% blockage ratio (case 4) 90 6.32 Rounded C-pillar . . . 92 6.34 Flow structures for Windsor_25 at 0^◦yaw angle in wind tunnel

with 8% blockage ratio corresponding to Q = 0.0017 . . . 94 6.33 Velocity contour along z=0.195m extending from (-0.375,-0.05)

to (2.3,0.29) in (x,y) direction for Windsor_25 at 0^◦yaw angle in wind tunnel with 8% blockage ratio . . . 95 6.35 Velocity streamlines superimposed on pressure contour along

x=1.1m extending from (-0.05,-0.555) to (0.88,0.945) in (y,z) direction for Windsor_25 at 0^◦ yaw angle in wind tunnel with 8% blockage (case 2) . . . 96 6.36 Velocity contour along y=0.25m extending from (-0.22, -0.195)

to (0.94,0.585) in (x,z) direction for Windsor_25 at 0^◦yaw an- gle in wind tunnel with 8% blockage ratio (case 2) . . . 97 6.37 Surface pressure contour of Windsor_25 model at 0^◦yaw an-

gle in wind tunnel with 8% blockage ratio (case 2) . . . 98 6.38 Location of lateral Z4 and Z5 probes . . . 100 6.39 Variation of pressure coefficient along Z4 probe when the model

is kept at 10^◦ yaw angle . . . 101 6.40 Surface pressure contour of Windsor_25 model at 10^◦yaw an-

gle in wind tunnel with 8% blockage ratio (case 4) . . . 102 6.41 Variation of pressure coefficient along Z5 probe when the model

is kept at 10^◦ yaw angle . . . 103 6.42 Flow structures for Windsor_25 at 10^◦yaw angle in wind tun-

nel with 8% blockage ratio (case 4) corresponding to Q = 0.0017 . . . 103 6.43 Velocity contour along y=0.25m extending from (-0.22, -0.195)

to (0.94,0.585) in (x,z) direction for Windsor_25 at 10^◦ yaw angle in wind tunnel with 8% blockage ratio (case 4) . . . 104

List of Tables

4.1 Boundary conditions used in simulations . . . 24 4.2 Comparison of different boundary condition in 2D . . . 25 4.3 Selected solution methods in Ansys 18.1 . . . 26 5.1 Proportionality factors based on cell size for different edges

and faces . . . 32 6.1 Change in drag and lift coefficient for Windsor_squareback

and Windsor_25 model in wind tunnel with 8% blockage ratio with respect to open case . . . 48 6.2 Change in drag and lift coefficient for Windsor_squareback

and Windsor_25 model in wind tunnel with 8% blockage ratio with moving ground (case 2) and stationary ground (case 3) with respect to open case (case 1) in straight line scenario . . . 59 6.3 Change in drag and lift coefficient for Windsor_squareback

and Windsor_25 model in wind tunnel with 8% blockage ra- tio with moving ground and stationary ground with respect to open case in at 5^◦ yaw angle . . . 65 6.4 Coordinates of Z2 and Z3 probes for Windsor models at 10^◦

yaw angle in wind tunnel with 8% blockage ratio (case 4) . . . 71 6.5 Comparison of drag coefficient correction for Windsor_squareback

and Windsor_25 model with sharp edges for blockage ratio of 8% . . . 82 6.6 Change in aerodynamic coefficients for rounded A-pillars with

respect to sharp A-pillar for Windsor_squareback and Wind- sor_25 model at 0^◦yaw angle in wind tunnel with 8% blockage ratio (case 2) . . . 85

xi

6.7 Change in aerodynamic coefficients for rounded A-pillars with respect to sharp A-pillar for Windsor_squareback and Wind- sor_25 model at 10^◦yaw angle in wind tunnel with 8% block- age ratio (case 4) . . . 88 6.8 Comparison of drag coefficient corrections for Windsor_squareback

and Windsor_25 model with rounded A pillar for blockage ra- tio of 8% . . . 91 6.9 Change in aerodynamic coefficients for rounded C-pillar with

respect to sharp C-pillar for Windsor_25 model in straight line case with 8% blockage ratio (case 2) . . . 93 6.10 Change in aerodynamic coefficients for rounded C-pillar with

respect to sharp C-pillar for Windsor_25 model at 10^◦ yaw angle with 8% blockage ratio (case 4) . . . 99 6.11 Coordinates of Z2 and Z3 probes for Windsor models at 10^◦

yaw angle in wind tunnel with 8% blockage ratio (case 4) . . . 100 6.12 Comparison of drag coefficient corrections for Windsor_25

model with rounded C pillar for blockage ratio of 8% . . . 105

Nomenclature

β Yaw angle (^◦)

∆Cd_M Increment in drag due to wake distortion ν Kinematic viscosity

ρ Density of oncoming air (_m^kg3) θ Empirical blockage factor A Frontal area of the vehicle A_dm,side Duplex vehicle side area (m³) A_dm Duplex vehicle frontal area (m²) A_dt Duplex tunnel cross section area (m²) A_F Duplex vehicle projected frontal area (m²) BR Blockage ratio

C_pitch Pitch moment coefficient C_roll Roll moment coefficient C_side Side force coefficient C_yaw Yaw moment coefficient Cd Drag coefficient

Cd_CM Maskell corrected drag coefficient Cd_i Induced drag coefficient

Cd_o Profile drag coefficient

xiii

Cd_uw Uncorrected wind axis drag coefficient Cdw Corrected wind axis drag coefficient CF D Computational fluid dynamics Cl Lift coefficient

Cl_front Front lift coefficient Cl_rear Rear lift coefficient Cp_b Base pressure coefficient Cp Pressure coefficient F_d Drag force (N) F_l Lift force (N) F_s Side force (N)

h_m Height of vehicle model (m) h_t Height of the test section (m)

l₀ Length of wind tunnel test section (m) lb Wheelbase of the vehicle (m)

l_m Length of vehicle model (m)

l_p Projected length of the vehicle model (m) LES Large Eddy Simulation

M_p Pitch moment of the vehicle (N-m) M_r Roll moment of the vehicle (N-m) M_y Yaw moment of the vehicle (N-m) p Local static pressure (Pa)

p∞ Static pressure free stream (Pa) Q Dynamic pressure

NOMENCLATURE xv

q Uncorrected dynamic pressure (Pa) qc Corrected dynamic pressure (Pa)

RAN S Reynolds Average Navier Stokes Equation Re Reynolds number

ROC Radius of curvature

T Thompson’s tunnel shape factor V∞ Velocity of oncoming air (^m_s) V_dm Duplex vehicle volume (m³) w_m Width of vehicle model (m) w_t Width of the test section (m)

Introduction

Automotive wind tunnel tests are employed to simulate the outdoor conditions to study the aerodynamic flow field around the vehicle body. This flow field is responsible for generating aerodynamic load and moment. Furthermore, it is also used to examine the influence of different vehicle shapes and designs on the above mentioned parameters. These parameters are of prime importance in any ground vehicle development as it has first order influence on the top speed, handling characteristics and most importantly, fuel economy of the vehicle.

However, these wind tunnels are not able to produce the exact same sce- nario as on the open road. Significant differences can also be observed in the results obtained from different wind tunnels for the same model under similar conditions. The magnitude of interference effect of wind tunnel on the results has been a major area of research for aerodynamicists since the inception of wind tunnel testing techniques. In reality, these disturbances can arise due to reasons such as test section boundary conditions, disturbances from measur- ing equipment, blockage ratio and irregularities in air flow due to turbulence, non-uniformity and unsteadiness.

A considerable amount of comparison and correlation tests have been con- ducted in ground vehicle aerodynamics in order to minimize the effect of such interference on the results. For instance, use of slotted walls in wind tunnel can minimize these effects upto a certain extent. However, the development of different wind tunnel correction methods over the past years has helped to minimize the wind tunnel interference effectively. It is to be noted that there is no standard wind tunnel correction method which is used in everyday testing or has proven to be most accurate.

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CHAPTER 1. INTRODUCTION 2

In today’s world, vehicle manufacturers have increased competition amongst themselves due to increased and more specialized demands from their cus- tomers for a diversely attributed vehicle. New development tools and working methodology has been developed and adopted in the recent years to make man- ufacturing cost, time and labour effective. For instance, virtual testing tools such as Computational Fluid Dynamics (CFD) itself does provide vehicle man- ufacturers the ability to assess design details and examine more scenarios at a very early stage in vehicle development, in a significantly less amount of time as compared to physical testing. This creates more room for fault detection and correction at an early stage avoiding wastage of resources. As far as vehi- cle aerodynamics is concerned, even though CFD results are still required to be validated by physical tests, they can be very useful and convenient in study- ing wind tunnel interference effect on various aerodynamics parameters.This can provide a deeper insight into individual wind tunnel parameters and its correction component.

1.1 Outline of this thesis

The thesis is structured as follows. The first chapter list and defines some of the fundamental definitions which are used in this study. The second chapter titled ’Background and previous work’ discusses some of of the precedent works done related to wind tunnel blockage, blockage correction methods, wind tunnel boundary conditions and vehicle shapes and design.

A brief description of computational domains, CFD setup and a compre- hensive meshing strategy is presented in the next few chapters.

The results obtained are discussed in different section of the chapter titled

’Results and discussions’. Effect of blockage due to wind tunnel is presented first which is followed by effect moving moving and stationary ground in wind tunnel. Furthermore, steady state crosswind effects and rounding of A and C pillar are discussed. The accuracy of different blockage correction methods in different scenarios is presented at the end of respective sections.

The last part of the thesis deals with some concluding remarks and sug- gestions on future works based on the current study.

Fundamental definitions

2.1 Aerodynamic loads

As Katz [1] mentioned, in continuum frame work, the aerodynamic force is an integral of stresses which can be decomposed in the following:

• Pressure: It acts in a direction normal to the vehicle surface which is responsible for vehicle’s lift and partly drag.

• Shear force: Forces such as friction force acts parallel to the vehicle surface are responsible for only drag.

The resultant of the aerodynamic force arising due to above mentioned stresses can be divided into various components based on coordinate system of the vehicle. Figure 2.1 locates the coordinate system based on which three force and three moment coefficients are defined. The drag force acts parallel to vehicle motion in positive x-direction. Lift force acts in a direction nor- mal (perpendicular) to the vehicle surface along positive y-direction. It is to be noted that downforce acts opposite to lift force along negative y-direction.

Lastly, side force acts along positive z direction.

3

CHAPTER 2. FUNDAMENTAL DEFINITIONS 4

Figure 2.1: Aerodynamic loads based on coordinate axis

However, these components of aerodynamic force and moment are normal- ized into non-dimensional quantities based upon dynamic pressure Q which is defined in equation 2.1.

Q = 0.5ρV_∞² (2.1)

Similar normalization based upon dynamic pressure (Q) is applied to ob- tain pressure coefficient (Cp) which is defined in equation 6.12.

C_p = p − p∞

0.5 ∗ ρ ∗ V_∞² = p − p∞

Q (2.2)

Drag, lift and side force coefficients are defined in equations 2.3, 2.4, 2.5 respectively.

C_d= 2F_d

ρV_∞²A (2.3)

Cl = 2Fl

ρV_∞²A (2.4)

C_side= 2F_s

ρV_∞²A (2.5)

Similarly yaw, roll and pitch moment coefficients are defined in equations 2.6, 2.7, 2.8 respectively.

C_yaw = 2M_y

ρV_∞²Alb (2.6)

C_roll = 2M_r

ρV_∞²Alb (2.7)

C_pitch = 2M_p

ρV_∞²Alb (2.8)

These components of aerodynamic force and moment are strictly depen- dent on Reynolds number which is defined as

Re = inertialf orces

viscousf orces (2.9)

The non dimensionless expression for Reynolds number is described in equation 2.10.

Re = ρV_∞L

µ (2.10)

As Wood [2] mentioned in his study, vehicle width (wm) is considered as reference length (L) for commercial vehicles. An equivalent reference length is the square root of the area of the vehicle (A) which is used in this study. Hence Reynolds number for ground vehicles based on square root of the vehicle is given by equation 2.11.

Re = ρV∞A^0.5

µ (2.11)

2.2 Blockage ratio

While the real bodies are exposed to infinite space, for instance vehicle body running on the road in natural environment, experimental wind tunnel tests are conducted in a limited space, walls of which have boundary layer present. The aerodynamic flow is strongly influenced by the limited space. This influence is governed by blockage ratio.

Blockage ratio is defined as the ratio of frontal area of test model and cross section of the wind tunnel testing chamber and is given by equation 2.12.

CHAPTER 2. FUNDAMENTAL DEFINITIONS 6

BR = A

h_tw_t (2.12)

2.3 Duplex vehicle

To enable to the use of aeronautical blockage corrections methods in auto- motive testing a duplex model is created. In this model, the vehicle and the wind tunnel are mirrored about the ground plane by keeping the floor as part of the model setup. A duplex model is hence created at the center of the du- plex tunnel. Through this method the angle of attack measured for an aircraft is translated to the yaw angle experienced by a vehicle [3]. All the blockage corrections methods used in this thesis have been developed for aeronautical application and hence use duplex model is used for drag correction.

Background and objective

3.1 Wind tunnel blockage and blockage cor- rections method

Yang and Schenkel [4] in their paper designed a virtual wind tunnel in CFD with different blockage ratio with the aim of finding blockage effect on drag coefficient and applying blockage correction methods to overcome the error in the results. Comparison was made between blockage free value for drag and the one with blockage. The simulations were conducted with fixed ground with inviscid wall conditions. Soderblom and Elofsson [3] performed numer- ical analysis on two different truck cab shapes. Their main focus was to evalu- ate the changes in the flow field by the blockage for different cab designs, and applying different blockage correction methods to the cab designs. Interest- ingly, when applying different blockage correction methods, it was observed that there was a significant difference in the corrected drag coefficient values.

Cabs of similar design and shape showed good agreement between blockage corrected results and open road conditions. However, for more detailed un- derstanding of different blockage correction methods, Cooper [5] studied how the blockage correction methods were derived and how they are modified for its application ground vehicles. Das [6] had used LES to evaluate the block- age effects on a heavy duty truck model at 0^◦ and 10^◦ yaw angle. He applied Maskell and Mercker correction methods and compared the performance of the two against blockage free drag coefficient value.

The constraints imposed by the wind tunnel walls on the flow around the vehicle body has an impact on drag and lift characteristics of the body. Both open and closed wind tunnels are used for various studies around the world, but

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CHAPTER 3. BACKGROUND AND OBJECTIVE 8

closed test section wind tunnels are predominantly used in automotive testing.

A virtual closed test section wind tunnel is used in this thesis, hence all the methods discussed are mainly for closed test section application. To apply the correction methods developed for aeronautical applications to vehicle models, the wind tunnel and the vehicle model are mirrored about the ground plane.

A duplex vehicle model is thus created which is located at the centre of the duplex tunnel [7]. The blockage correction methods discussed here assume that the flow is invariant which implies that the separation point will remain unchanged in the presence of wind tunnel walls [3] [7]. For correction of lift coefficient, Wickern [8] in his paper pointed out that the lift coefficient corrections are negligible in passenger vehicles. One of the test was conducted using a production car with a rear wing which was 1.4 m high, which is very high for a car. This was done to increase upwash from the wing and get an idea about the percentage of corrections needed for lift coefficient. The upwash generated from the rear wing interfered with wind tunnel walls which led to blockage and had effect on lift coefficient. From the study, it was concluded that for such extreme level of upwash, only a minor correction (<2%) was needed for lift coefficient which showed that for passenger cars without wings this effect was much smaller. Hence, in this thesis corrections are made to drag coefficient only.

3.1.1 Maskell correction for bluff bodies

Maskell developed one of the first blockage correction methods for separated flows. In this method conservation of momentum and physical arguments by wind tunnel measurements were used to derive the theory for wake blockage generated by normal flat plates.

q_c

q = 1 + 1

2Cdo(A_dm

C_dt ) − 1

Cp_b(Cduw− Cdi− Cdo)A_dm

A_dt (3.1) Equation 3.1 becomes blockage correction for streamlined bodies, if the last term in the above equation is omitted. The last term represents correction due to flow separation. The inverse of base pressure coefficient (Cpb) can be estimated by the below mentioned empirical blockage factor as stated by Cooper [7], Soderblom and Eloffson [3] in their respective works .

θ = −5

2 (3.2)

The induced and surface drag are neglected in this method and equation 3.1 is reduced to equation 3.3 which is known as Maskell correction formula.

q_c

q = [1 + θCd_uw(A_dm

A_dt )] (3.3)

The simplifications made to the original formula has an influence on the the correction as it is only applied to the drag resulting from flow separation and not considering the induced and surface drag [7].

Calculating and correcting the dynamic pressure using equation 3.3, the corrected drag coefficient is calculated using the formula:

Cd_w = [Cd_uw

q_c/q ] (3.4)

3.1.2 DNW Maskell correction method

DNW and NRC wind tunnels developed a correction method [9] based on correction method proposed by Maskell. The method takes into account the vehicle length i.e. surface drag along with drag due to flow separation. The dynamic pressure correction term is calculated using equation 3.5

q_c

q = [1 + (l_m l₀ + 5

2)Cd_uw(A_dm

A_dt )] (3.5)

The corrected drag coefficient is calculated in the same way using equation 3.4.

3.1.3 Maskell III correction method

Hackett and Cooper improved the correction method proposed by Maskell. In this method the drag coefficient is corrected in two steps. In the first step, before applying the correction to dynamic pressure, drag increment due to wake distortion (∆CdM) is removed. In the second step, the drag increment due to wake distortion is added to the corrected drag coefficient. Hackett and Cooper adopted a different approach to calculate the blockage factor which is more suitable for ground vehicles.

CHAPTER 3. BACKGROUND AND OBJECTIVE 10

θ = 0.96 + 1.94e^{(−0.12hm/wm)} (3.6) Drag increment due to wake distortion is calculated using equation 3.7

4 CdM = Cduw( 1

1 + x +1 −√ 1 + 4x

2x ), x = θA_dm

A_dt (3.7) In the next step the drag increment due to wake distortion calculated from equation 3.7 is subtracted from the drag coefficient calculated (CDcm) in equa- tion 3.4.

Cd_CM − 4Cd_M = −1 +p1 + 4Cd_uwθA_dm/A_dt

2θAdm/Adt (3.8)

q_c

q = 1 + θCd_uw(A_dm/A_dt) − 4Cd_M (3.9) The drag correction term can be corrected using equation 3.10. The drag increment due to wake distortion is added to the drag correction term as men- tioned before.

Cd_w = Cd_uw

1 + θ(A_dm/A_dt)(Cd_CM − 4Cd_M)+ 4Cd_M (3.10)

3.1.4 Thom and Herriot’s correction method

Thom and Herriot developed a correction method based on potential flow the- ory [10]. As stated by Soderblom and Elofsson [3], this method was originally developed for aeronautical applications but later adapted for ground vehicles.

The dynamic pressure correction is given by:

q_c

q = [1 + T (V_dm A

3 2

dt

) + 1

4Cduw(A_dm

A_dt )]² (3.11)

In equation 3.11, the second and third terms represent solid and wake blockage respectively. Furthermore, tunnel shape factor is defined as

T = 0.36(w_t h_t + h_t

w_t) (3.12)

Lastly, correction to the drag coefficient is made by the following respec- tive equations:

Cd_w =

Cduw− T (^Vdm

A

3 2 dt

)Cduw qc

q

(3.13)

3.1.5 Modified Thom and Herriot’s correction method

Since equations 3.13 contains volume dependent terms which are related to wake buoyancy that Hackett [11] proved incorrect, Cooper [10] suggested a modified formula for drag coefficient correction which improved the accuracy of the present Thom and Herriot’s correction method.

Cd_w = Cd_uw −¹₄Cd²_uw(^A_A^dm

dt)

qc

q

(3.14)

3.1.6 Mercker’s correction method

A wind tunnel correction method was developed my Mercker [12] in which solid blockage correction was based on the potential flow theories of Glauert [13] and Lock [14] whereas the wake blockage correction was based on Mer- cker [12] and Thom [15]. The dynamic pressure correction is given by

q_c

q = [1+2T

√π( A_F

pL_pV_dm)(V_dm A

3 2

dt

)+(A_dm A_dt )(1

4Cd_uw(β = 0)+η( A_F

A_dm)]² (3.15)

where projected frontal area of the vehicle is given by

A_F = A_dmcosβ + A_dm,sidesinβ (3.16) and projected length of the vehicle is defined by

CHAPTER 3. BACKGROUND AND OBJECTIVE 12

lp = lmcosβ + wmsinβ (3.17) The drag coefficient correction is thus given by following respective equa- tion:

Cd_W = Cd_uw +¹₄Cd²_uw(^A_A^dm

dt)

qc

q

(3.18)

3.2 Effect of presence of moving ground

Experimental tests were conducted by many researchers to find the changes in flow field and its effect on the aerodynamic load when ground is moving in a wind tunnel. Bearman et al.[16] conducted experimental tests on a 1/3 scale model to find the effects on lift and drag coefficient when the floor is moving or fixed, while using different rear upsweep angles. They also conducted tests to measure the wake structure behind the wheel when the floor was moving or fixed. It was concluded that lift was more sensitive floor movement than drag, with 30% reducing in lift coefficient and 8% reduction in drag coeffi- cient when the floor was moving. Changes were found in the wake structure during different floor movement boundary conditions. It was noted that the wake expanded faster when the floor was stationary and the near wake was in- sensitive to floor movement. They also pointed out that wake behind the wheel changed considerably when the floor was moving and the wheel is rotating.

Fago et al. [17] experimentally investigated a simple vehicle body with two different lengths (full scale and 1:5 scale) with different ground clearance over a moving and fixed ground. They also pointed out different methods of sim- ulating relative motion between the ground and the vehicle in a wind tunnel.

From their work, it was concluded that at low ground clearance, moving floor had a large effect on both lift and drag coefficient for both lengths ( full scale and 1:5 scale). Just like in the case with Bearman [16], the lift was more sen- sitive to floor movement. The effect on drag coefficient above 25mm ground clearance was negligible for the model used in the experiment. Hence they concluded that moving floor is essential for detailed analysis of vehicle aero- dynamics especially for the vehicle where ground clearance is low. Kessler and Wallis [18] gave a broad and detailed explanation about wind tunnel testing in general and different techniques to test with moving ground or eliminating the

boundary layer from the ground plane.

Krajnovic and Davidson [19] also studied the effect of moving and station- ary ground effects in wind tunnel. They performed two large eddy simulations (LES), both instantaneous and time averaged on an Ahmed body. The results were similar to that of Fago [17] and Bearman [16]. It was reported that floor motion reduced drag coefficient by 8% and lift coefficient by 16%. The reduc- tion is drag coefficient was attributed to increased surface pressure on the rear slanted and vertical surfaces. Slanted surface produced the largest differences in the flow structures, velocity and stress profiles. Flow in the wake region except close to the ground was found to be insensitive to the floor motion.

3.3 Basic car shapes under crosswind in wind tunnel

Ahmed et al. [20] showed the variation of drag coefficient as a function of backlight angle for a Ahmed body. The results were based on the pressure data. Base drag was found to be reduced linearly an increase in backlight angle. This was attributed to the linear variation of base pressure coefficient and base area with backlight angle. Consequently, the drag developed over the backlight surface was reported to be increased with square of the backlight angle.

In the work of Gilhaus and Renn [21], a simplified 3/8-scale model was used to evaluate influence of various shape parameters on vehicle drag and driving stability related coefficients. It was reported that yaw moment coef- ficient decreased as backlight angle decreased. Furthermore, rolling moment coefficient was found to increase with increased yaw angle. Interestingly, a well rounded A pillar was observed to contribute towards a reduced yawing moment. On the other hand, radiused C/D pillars were found to increase the yaw moment. Howell [22] conducted a similar study which focused on vehicle shapes to lower the both drag coefficient and yaw moment coefficient. Notice- ably, the planform curvature was found to decease the yaw moment coefficient.

To add on, it was also shown that yawing moment coefficient increased with backlight slant angle. This observation is similar to Gilhaus and Renn [21].

However, the hatchback version was found to have a worse yaw moment co- efficient than notchback whereas the contrary was found by Gilhaus and Renn [21].

CHAPTER 3. BACKGROUND AND OBJECTIVE 14

Baker [23] presented a comprehensive set of data for mean aerodynamic forces and moments for variety of ground vehicle types. In one of the data sets for small cars and vans taken from Emmelmann [24] and Takanami et al.

[25], yawing moment coefficient were found to reach a peak at yaw angles of 20^◦-30^◦. Yawing moment coefficient then became negative for higher yaw an- gles. In another study by Cogotti [26], the effects of ground simulation during yaw for a simplified car model shape was experimentally investigated. Aero- dynamics coefficients were observed to increased more in yaw as compared to stationary ground at 200 mm of ground clearance. Furthermore, higher drag was produced for active ground simulation at yaw angles greater than 15 de- grees. Interestingly, a lower aerodynamic drag coefficient was reported over the entire yaw range for active ground simulation with a ground clearance of 100 mm. A general reduction in rear axle lift was also observed due to ground simulations in yaw conditions.

Howell and Le Good [27] conducted a study in which influence of back- light aspect ratio was investigated on vortex and base drag. Drag was observed to be varied parabolically with backlight angle of upto 20^◦. Furthermore, drag minimum was reduced. Also, with increasing aspect ratio, backlight angle for minimum drag was increased. Vortex drag was found to be same for all back- light whereas base drag was reduced with increasing backlight angle. Lastly, lift was found to vary linearly with backlight angle. The lift slope was in- creased with a reduction in aspect ratio. Howell and Fuller [28] explored the relationship between lift and lateral aerodynamic characteristics experimen- tally using Windsor vehicle model as a representation of road vehicles. A reduction in lift associated with lower yawing moment was identified. Side force and rolling moments were found to increase with decrease in yawing moment. This was attributed to the reduced pressure on windward real pillar and increased pressure on leeward rear pillar.

Gohlke et al. [29] performed an experimental study on influence of shape changes on aerodynamic forces, moments and flow structures of a 1/5th model of a mini-van type vehicle in yaw conditions in steady crosswinds. Rounding of A-pillar resulted in a decreased yaw moment. Strength and size of the vor- tex due to rounding of A-pillar was significantly reduced. Surprisingly, rear side force was strongly increased with an increase in boot lid angle. This cre- ated a reduction of nealry 6% in yawing moment.It was further concluded that A-pillar is an important feature for cross wind sensitivity as along with its lo- cal effect, it can also influence forces on rear side through its vortex. Mansor and Passmore [30] also experimentally investigated the effect of rear slant an-

gle on a Davis model for crosswind stability simulations. 0^◦ and 40^◦ slant angle produced highest side force derivative whereas slant angle of 20^◦ pro- duced larger yawing moment derivative. Model with larger value of positive yaw moment derivative produced higher yaw rate and was more sensitive to crosswind sensitivity.

3.4 Objective of this thesis

As described in previous sections, numerous studies have been conducted with respect to blockage correction methods, interference effects, boundary condi- tions and vehicle geometric shape changes.

The main goal of this thesis is to numerically study the effect of wind tun- nel interference effects on a simple vehicle geometry with two different back angles in a steady state environment. Furthermore, these effects are also eval- uated in yaw conditions to simulate the effect of crosswinds.

To add on, different correction methods are employed for every scenario to accommodate virtual wind tunnel interference effect. This objective is of interest because it reveals the variation in accuracy of these correction methods in straight line and yawed conditions.

Apart from the main objectives in the work presented, a side study is also conducted to understand the effect of having a rounding A and C pillars on the aerodynamic flow field individually. Traditional Reynolds Averaged Navier Stokes (RANS) equation is employed by means of steady state k-omega SST turbulence model for all the numerical simulations in this work.

Chapter 4

Methodology

4.1 Geometries considered

In this thesis, a generic car model known as Windsor vehicle model is selected to ease the computation since it simplifies the actual vehicle geometry,and is able to give accurate results. Furthermore, it is a reference model widely used in research for ground vehicle aerodynamics. Windsor model represents 1/4th scale model of small hatch back car as mentioned by Howell and Le Good [27]. As compared to Ahmed body, Windsor model represents a realistic front which is ideal for studying the crosswind scenario [27].

Two versions of rear designs are studied to evaluate the effect of backlight angle in crosswind scenario. A standard Windsor model with a square back geometry is selected to represent the square back passenger car. It is termed as Windsor_squareback which can be seen in figure 4.1a. Second Windsor model with a backlight angle of 25^◦ is termed as Windsor_25. It is depicted in figure 4.1b. 25^◦backlight angle is selected because in a study conducted by Howell and Le Good [27], data recorded for 25^◦backlight angle was found to be inconsistent with the other data. Henceforth, it is of interest to explore the phenomenon taking place at this angle.

The main dimensions are as follows:

Length (L or l^m) 1.045 m Width (W or w^m) 0.39 m

Height (H or hm) 0.29 m Frontal area 0.113 m²

For Windsor_25 model, the length of the constant rear slant is considered

16

to be 0.222m. This gives an aspect ratio of 1.75 [27]. In order to minimize leading edge separation, front edges are rounded off to a radii of 0.05m. How- ever, the roof leading edge is rounded off to a radii of 0.2m as can be seen in figure 4.1

4.2 Scenarios simulated

The present work involves study of multiple case scenarios with different phys- ical conditions in the wind tunnel. Following are the cases and their descrip- tion:

• Case 1: Model is kept in an open condition and entire ground is given a moving boundary condition.

• Case 2: Model is kept in a wind tunnel with high blockage ratio of 8%

and only a small section of the ground is given a moving boundary con- dition.

• Case 3: Model is kept in a wind tunnel with high blockage ratio of 8%

and the entire ground is given a stationary boundary condition.

• Case 4: Model is kept at a yaw angle in a wind tunnel with high blockage ratio of 8% and only a small section of the ground is given a moving boundary condition. The ground is aligned with the wind tunnel in order to maintain simplicity while running the simulations.

• Case 5: Model is kept at a yaw angle in an open condition and entire ground is given a moving boundary condition. The ground is aligned with the wind tunnel in order to maintain simplicity while running the simulations.

For simulating steady state crosswinds, vehicle is yawed. Two yaw angles of 5^◦ and 10^◦ are chosen and compared with 0^◦ yaw angle in wind tunnel with 8% blockage ratio. The choice of yaw angles is based on the inputs from industry as they are the commonly used angles for a wind tunnel testing.

CHAPTER 4. METHODOLOGY 18

(a) Windsor_squareback model

(b) Windsor_25 model

Figure 4.1: Windsor models used in this study

4.3 Computational domain

The computational domain is adjusted according to the scenario taken. For cases 1 and 5 which simulate the open scenario, the computational domain is shown in figure 4.2.The domain size is similar to that used by Favre [31].

The domain is 11L in the stream wise direction and the height is 10H. This corresponds to a blockage ratio of 0.06%. The ground clearance used between the vehicle and the ground is 0.05m.

To add on, the top wall is constructed at an angle with the ground as can be seen in figure 4.2. This is attributed to the fact that presence of an inclination drives the flow towards the top wall and a pressure outlet boundary condition can be used for simulating an open condition.

Figure 4.2: Computational domain with blockage ratio of 0.06%

The computational domain for rest of the cases which simulate the actual wind tunnel scenario are depicted in figure 4.3. Since these cases represent an actual wind tunnel scenario, the blockage ratio of the wind tunnel has been increased. Present domain gives a blockage ratio of 8% which is near to the actual Volvo wind tunnel. However, in the actual wind tunnel, devices and methods are employed to reduce the blockage effect. To include these devices and methods is out of the scope of this thesis. Furthermore, it has a moving

CHAPTER 4. METHODOLOGY 20

ground of area of 5.3m x 1m which represents the Volvo Wind Tunnel moving belt [32]. Principle dimensions for the wind tunnel are as follows:

Length (l0) 11.495 m Width (wt) 1.5 m Height (ht) 0.93 m Frontal Area 1.395 m²

Figure 4.3: Computational domain with blockage ratio of 8%

4.4 Orientation of Windsor model with respect to wind tunnel

Figure 4.4 depicts the frame of reference for the model in straight line sce- nario (frame of reference 1) as well as for the model in yaw (frame of ref- erence 2). Understanding this frame of reference is important because even when the model is kept at a yaw angle, the components of aerodynamic force are calculated with respect to main frame of reference, which in this thesis is frame of reference 1 in a numerical computation. Hence a proper transforma- tion between components of aerodynamic force should be made from frame of reference 1 to frame of reference 2 during the yawed condition.

Figure 4.4: Frame of reference for model in straight line (frame of reference 1 in black) and for model in yaw (frame of reference 2 in red)

Figure 4.5 depicts the location and orientation of Windsor mode with re- spect to the wind tunnel with 8% blockage ratio in straight line scenario. It can be seen that Windsor model is symmetric about z=0.195m and not about the origin (z=0m). This information is useful for understanding the effects of steady state crosswinds in section 6.3. Furthermore, vehicle model begins at x=0m upto x=1.045m. Similarly it extends from y=0m to y=0.29m in height.

CHAPTER 4. METHODOLOGY 22

(a) Frontal area of the vehicle

(b) Stream wise location of Windsor model

Figure 4.5: Orientation of Windsor model with respect to wind tunnel in straight line scenario

Figure 4.6 depicts the location of the Windor_25 model at 10^◦yaw angle kept in wind tunnel with 8% blockage ratio.

(a) Frontal area of the vehicle at 10^◦yaw

(b) Top view of Windsor model at 10^◦yaw

Figure 4.6: Orientation of Windsor model with respect to wind tunnel in yawed scenario

4.5 CFD Setup

4.5.1 Boundary conditions

The boundary conditions used in the simulations are summarized in table 4.1.

The computation is considered to be fully turbulent. The inlet velocity is cho- sen to be 27m/s. This gives a Reynolds number (defined in equation 2.11) to be 6.2x10⁵.

The moving ground moves at a velocity of 27m/s in the same direction as

CHAPTER 4. METHODOLOGY 24

Table 4.1: Boundary conditions used in simulations

Boundary Boundary condition

Inlet Velocity inlet, V^∞= 27m/s

Outlet Pressure outlet

Side walls Symmetry (open case), slip wall (wind tunnel) Top Pressure outlet (open case), slip wall (wind tunnel) Moving ground Prescribed velocity = 27m/s, no slip wall Stationary ground No slip wall (wind tunnel)

oncoming air such that relative speed between moving ground and air is 0m/s.

It is worth mentioning that ground (moving and stationary) is treated as no slip wall as boundary layer is employed throughout the ground. The vehicle is kept stationary in all the cases.

The outlet of the virtual wind tunnel is kept as pressure outlet with gauge pressure as 0 Pa. It is also to be noted that top wall of the virtual wind tunnel in the open conditions is also kept as pressure outlet to avoid wall blockage. The same top wall is given a slip boundary condition for the wind tunnel with 8%

blockage ratio. This is more appropriate as boundary layer is not employed on this wall.

Lastly, the side walls are treated as symmetry for open case as no boundary layer is employed. The same walls are treated as slip walls in case of wind tunnel with blockage ratio of 8%.

4.5.2 Boundary condition analysis

Comparison of different wind tunnel ground boundary condition is made. Ta- ble 4.2 depicts the results when top wall given as slip boundary condition in 2D. It is worth mentioning that slip boundary condition is more appropriate

as compared to no slip or symmetry condition since the boundary layer is not employed on the top surface of the domain as stated earlier.

Table 4.2: Comparison of different boundary condition in 2D 2-D boundary conditions analysis

Boundary condition Cl Cd

Top wall = Slip Complete moving

ground

0.775 0.423

Wind tunnel moving ground

0.772 0.422

Stationary ground 0.851 0.416

It is observed that while the results of complete moving ground and wind tunnel moving ground condition do not differ by a significant margin, station- ary ground does create a significant difference in the results. Lift coefficient is observed to increase when having a stationary ground as compared to a mov- ing ground.

4.5.3 Solver

A coupled solver is chosen due to the fact that it scales linearly with cells counts. The detailed solution methods and spatial discretization schemes used are defined in table 4.3.

A standard initialization is made from the inlet of the domain. A reference frame relative to cell zone is selected. Initial values are not listed in the report as it is dependent on different scenarios and vehicle model.

CHAPTER 4. METHODOLOGY 26

Table 4.3: Selected solution methods in Ansys 18.1 Spatial discretization scheme Gradient Least square cell based

Pressure Second order

Momentum Second order upwind

Turbulent kinetic energy Second order upwind Specific dissipation rate Second order upwind

4.5.4 Reynolds Averaged Navier Stokes (RANS)

Reynolds Averaged Navier Stokes equations are time averaged equations for describing the motion of the fluid. They are mainly used for describing turbu- lent flows. The concept behind RANS equations is Reynolds decomposition where an instantaneous quantity is divided into time averaged and fluctuating quantity.

u_i = u + u^f (4.1)

where uj is instantaneous quantity, u is the time averaged quantity and u^f is the fluctuating quantity.

The governing equations for the Navier Stokes equations are the momen- tum equation (4.2) and the continuity equation (4.3) in Cartesian tensor form are written as:

∂u_i

∂t + uj

∂u_i

∂x_j = −1 ρ

∂p

∂x_i + ν ∂²u_i

∂x_j∂x_j + ρfi (4.2) where ρfi are the body forces, and ν ^∂

2ui

∂xj∂xj are the viscous stresses.

∂u_i

∂x_i = 0. (4.3)

By substituting equation 4.1 in equation 4.2, RANS equations for incom- pressible flows as shown in equation 4.4 are obtained.

ρu_j∂u_i

∂x_j = ρf_j+ ∂

∂x_j[−pδ_ij + µ(∂u_i

∂x_j +∂u_j

∂x_i) − ρu^f_iu^f_j] (4.4)

It is seen that additional term which represent correlation between the fluc- tuating terms have been introduced , which are required to be modeled. The additional term introduced is the Reynold’s stress term (ρu^fiu^f_j). Turbulence modelling provides the solution to the closure problem for the open system of equation which are represented above by modeling the unknown terms repre- sented in the RANS equation in terms of mean flow quantity.

4.5.5 Turbulence model

k-omega (k-ω) SST model is a two equation eddy-viscosity model. The shear stress transport (SST) uses the k-ω formulation in the inner parts of the bound- ary layer all the way to the viscous sub-layer hence it could be used as low-Re turbulence model, and it uses k- in the free stream helping overcome the prob- lems encountered in k-ω turbulence model which made the model sensitive to the inlet turbulence properties.

The two equation model use two additional transport equations to show- case the turbulent properties of the flow. The transported variable which is used most often is the turbulent kinetic energy term (k). The second transport variable in this case is specific turbulence dissipation rate (ω).

The transport equation for turbulent kinetic energy (k) is written as:

∂k

∂t + u_j ∂k

∂x_j = P − β^?kω + ∂

∂x_j[(ν + σ_kν_T ∂k

∂x_j] (4.5)

where P is production of turbulent kinetic energy, ν is kinematic viscosity , νT is the kinetic eddy viscosity and β^?is the closure coefficient whose value is 0.09 which is determined experimentally.

The transport equation for specific turbulence dissipation rate (ω) is written as:

∂ω

∂t + u_j ∂ω

∂x_j = αS²− βω²+ ∂

∂x_j[(ν + σ_ων_T ∂ω

∂x_j] + 2(1 − F₁)σ_ω21 ω

∂k

∂x_i

∂ω

∂x_i (4.6)

CHAPTER 4. METHODOLOGY 28

where α, β, σω2are the closure coefficients which are determined experi- mentally. For better understanding of the closure coefficient reader is advised to have a look at [33].

K-ω SST turbulence model is used for simulating all the scenarios under steady state conditions. Even though it is well known that for vehicle aero- dynamic simulations, k-epsilon turbulence model produce results which are in agreement with actual physical wind tunnel testing, k-ω SST turbulence model is known for its ability to accurately capture the separated flow and its good behaviour in the adverse pressure gradient. To add on, unlike k- model, k-ω SST model does not use a wall function. The boundary layer is resolved all the way till the viscous sub-layer which helps in capturing the flow fluctu- ations in the boundary layer accurately which have significant impact on the drag and lift.

Meshing strategy

The meshes considered in this study are triangular mesh in 2D and tetrahedral mesh in 3D with prism layer present on the surface of the Windsor model and on the ground of the domain. These meshes are composed in ANSYS Mesher.

Around the vehicle model, typical Y+ value is considered to be 1. On the ground, a higher Y+ value of 30 is considered. It is attributed to the reason that ground is a viscous wall region where Y+< 50. Furthermore, viscous sub layer is present around the vehicle model where there is a domination of large viscous stresses and Y+< 5 [31].

5.1 Meshing in 2D

Figures 5.2 and 5.3 depicts the overall mesh in the domain and the mesh around the Windsor model including the boundary layer respectively. To keep the mesh refined around the vehicle model, three refinement boxes are created near the vehicle which are depicted in figure 5.1a. Refinement box 1 (RB1) is the coarser refinement region around the vehicle. The finer refinement at the rear is refinement box 2 (RB2) which captures the rear wake. Meshes in these two refinement boxes can be visualized in figure 5.3. In figure 5.4, the rearmost refinement zone is the refinement box 3 (RB3).

29

CHAPTER 5. MESHING STRATEGY 30

(a) Refinement boxes

(b) Location of edges

Figure 5.1: Location of edges and refinement boxes which are referred in table 5.1

Figure 5.2: Overall mesh in 2-D for Windsor_25

Figure 5.3: Mesh in RB1 and RB2 around Windsor_25 model

CHAPTER 5. MESHING STRATEGY 32

Figure 5.4: Mesh in RB3 at the rear of Windsor_25 model

Furthermore, at the curvatures of the Windsor model, mesh is more refined and aspect ratio is targeted to be 1 to capture the flow variation around the corner more accurately. To add on, at the rear of the vehicle, which is the slant and vertical surface, mesh is finest in order to capture the separation accurately since it will influence lift and drag coefficient. Similar meshing strategy is employed for Windsor_squareback model.

Based on the base cell size, concept of proportionality is used to refine various regions. The proportionality factors for different regions are defined in table 5.1. The nomenclature of edges and refinement boxes are given in figure 5.1. Apart from the mentioned edges and refinement boxes, rest all the areas are meshed with proportionality factor of 1 with respect to the base size.

Table 5.1: Proportionality factors based on cell size for different edges and faces

Edges/Faces Base size RB1 RB2 RB3 Edge 1 Edge 2

Proportion x 7.0x 3.0x 8.0x 1.4x 4.6x

Edges/Faces Edge 3 Edge 4 Edge 5 Edge 6 Edge 7 Edge 8

Proportion 3.0x 1.4x 7.0x 3.0x 1.4x 3.5x

5.2 Grid independence test

Grid independence test is conducted in 2D mesh for the open case (case 1) only. Figure 5.5 depicts the grid independence test for Windsor_squareback model. During the test, the first cell thickness for all the cases is kept the same with Y+ below 1 all around the model as shown in figure 5.7. The mesh around it is made finer to understand the effect of mesh on the results. A Y+ of < 1 is required for good convergence and accuracy in k-omega SST model which is used for this test. As a result, the boundary layer is fully resolved all the way up to the model surface as seen in figure 5.8.

It is observed that the value converges at 155,000 cells for lift coefficient (Cl) and drag coefficient (Cd). Similar test is performed for the Windsor_25 as represented in figure 5.6. Similar trend is observed for both Cl and Cd. The values shows convergence at 160,000 cells.

CHAPTER 5. MESHING STRATEGY 34

(a) Variation of drag coefficient with respect to number of cells

(b) Variation of lift coefficient with respect to number of cells Figure 5.5: Grid independence test Windsor_squareback model