Ariane5 Transoinc Base Flow Analysis CFD

Cranfield University

Shamoon Jamshed

An Investigation of the Ariane5 Base Flow Using RANS

Methods

School of Engineering

Computational Fluid Dynamics

MSc Thesis

Academic Year 2008-09

Supervisor: Dr. B. Thornber

September 2009

Cranfield University

School of Engineering

Computational Fluid Dynamics

MSc Thesis

Academic Year 2008-09

Shamoon Jamshed

An Investigation of the Ariane5 Base Flow Using RANS

Methods

Supervisor: Dr. B. Thornber

September 2009

This thesis is submitted in partial fulfilment of the requirements for the degree of

Master of Science

c©Cranfield University, 2009. All rights reserved. No part of this publication may be

reproduced without the written permission of the copyright holder

Abstract

Base flow buffeting has been the subject of much research. It gained importance for Ariane

V after the first launch of the mission A 501. At launch the Vulcain engine is responsible

to produce an enormous amount of thrust. The transonic free-stream flow from the outside

of the main engine interacts with the high velocity supersonic flow forming a shear layer.

These shear layers produce high side force on the nozzle impeding the control of the rocket.

This could also generate three dimensional flow separation inside the nozzle as well.

Although buffeting is an unsteady phenomenon, but due to limited time of three months

and a complex geometry to deal with, two types of simulations have been run, i.e inviscid

and RANS. Initially unsteady simulations with inviscid, explicit-density-based solver, were

performed with first order discretisation in space and 4th order Runge Kutta method in time.

All theses simulations were based on based on hardware configuration of 8 processors with

64-bit 3 GHz each on the supercomputer Astral, but it was found uneconomical to run in

this limited span of time. It was concluded that for the coarse grid the k − ω model had

given better performance both when compared with the experiment and when analysed on

the basis of contour plots. The S-A model had given better performance for the adapted

grid. The inviscid model was not suitable to simulate such a highly turbulent and massively

separated flow. Therefore unsteady simulations are recommended for this case.

Keywords: Ariane5, base flow, buffeting, turbulence models, supercomputer, Astral,

physical time, RANS, inviscid, viscosity, shear layer.

ii

Acknowledgements

I would like to thank first of all to my parents whose prayers led me to complete my thesis

successfully. I would like to thank the Cranfield University whose assistance in every sort

(library, IT etc) made this thesis possible. I am thankful to my supervisor Dr Ben Thorn-

ber whose logical advices at every step made this (apparently looking impossible) thesis

possible. He never let me down at any moment and made me to keep up the spirit. I am

thankful to Dr Evgeniy Shapiro, the course director, who was also helpful at every moment

although he was not supervising this thesis. I am thankful to Dr Drikakis and his whole

team of FMCS for valuable suggestions and knowledge not only in the thesis but for the

whole course of CFD. Thanks to Dr Ludeke from NLR whose expertise in this area acted

as a guide for us. I am thankful to Mr Zeeshan Rana (a PhD student in FMCS) who helped

me in using LATEX which I used as a typesetter for my thesis report. Lastly I would like

to thank my organization, Institute of Space Technology, (IST) Pakistan, who took care of

the training of her employees and gave me an opportunity to gain expertise in this area of

scientific computation and research.

Shamoon Jamshed

iii

Contents

Abstract i

Acknowledgements ii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Nomenclature xiii

1 Introduction 1

1.1 Ariane history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Ariane5 Flight 501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Thesis layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Literature Review 6

2.1 Experimental investigation . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Theoretical and analytical analysis of base flow buffeting . . . . . . 7

2.1.2 Measurement of side loads . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Computational studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Zonal-Detached-Eddy-Simulation of a Two-Dimensional and Ax-

isymmetric Separating/Reattaching Flow . . . . . . . . . . . . . . 17

iv

CONTENTS v

2.2.2 Turbulence study . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3 Unsteadiness of an axisymmetric separating flow . . . . . . . . . . 21

2.2.4 Extra Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . 22

3 Grid Generation 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Structured solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Algebraic grid technique . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.2 Elliptic solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Coarse grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Pressure far-field . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.2 Pressure outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.3 Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.4 Pressure inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Governing Equations and Turbulence Modelling 36

4.1 The Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.2 Conservation of mass: continuity equation . . . . . . . . . . . . . . 37

4.1.3 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . 37

4.1.4 Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.5 Closure problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Turbulence at a glance . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Reynolds Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Mean and fluctuating parts . . . . . . . . . . . . . . . . . . . . . . 43

4.3.2 Boussinesq approximation . . . . . . . . . . . . . . . . . . . . . . 44

vi CONTENTS

4.3.3 The Spalart-Allmaras model . . . . . . . . . . . . . . . . . . . . . 44

4.3.4 The k − ε model . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.5 The k − ω model . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Numerical Methods 48

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Density based solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.1 The Roe-FDS solver . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.2 AUSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2.3 Low-diffusion Flux Difference Splitting . . . . . . . . . . . . . . . 51

5.3 Discretisation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3.2 Temporal discretisation . . . . . . . . . . . . . . . . . . . . . . . . 53

6 Results and Discussion 58

6.1 Inviscid and RANS Simulations . . . . . . . . . . . . . . . . . . . . . . . 58

6.2 Flow characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.3 Quantitative comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.3.1 Pressure Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.3.2 The recirculation bubble . . . . . . . . . . . . . . . . . . . . . . . 65

6.3.3 Turbulent kinetic energy . . . . . . . . . . . . . . . . . . . . . . . 69

6.3.4 Axial and lateral velocity profiles . . . . . . . . . . . . . . . . . . 77

6.3.5 Drag coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3.6 Stagnation pressure at the tip . . . . . . . . . . . . . . . . . . . . . 82

6.4 Qualitative comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.4.1 Vorticity magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.4.2 Stream traces (Path-lines) . . . . . . . . . . . . . . . . . . . . . . 87

CONTENTS vii

7 Conclusions and Future Work 90

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

A 93

A.1 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 93

B 97

B.1 Spalart Allmaras model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

B.2 k − ε model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

B.3 k − ω model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

C 100

C.1 Convergence history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Bibliography 102

List of Figures

1.1 Ariane family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Effect of Jet Mach Number on the base pressure[29] . . . . . . . . . . . . . 9

2.2 Effect of base pressure on Jet-shear layer envelope[29] . . . . . . . . . . . 9

2.3 Position of kulites on the nozzle[29] . . . . . . . . . . . . . . . . . . . . . 10

2.4 Frequency spectrum, Mach 0.8 with jet, with exhaust pipes (left) and with-

out exhaust pipes (right) [29] . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Integrated forces with no protuberance at Mach 0.8 [29] . . . . . . . . . . . 11

2.6 Effect of 5 deg yaw (left) and 5 deg pitch (right) to integrated forces at Mach

0.8 with jet but without exhaust pipes.[29] . . . . . . . . . . . . . . . . . . 12

2.7 Table of the different test cases conducted by NLR [9] . . . . . . . . . . . . 13

2.8 Model in SST with zoomed view of the nozzle [9] . . . . . . . . . . . . . . 13

2.9 High loads observed in test #2, [9] . . . . . . . . . . . . . . . . . . . . . . 14

2.10 Acoustic array spectra, [9] . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.11 Spectrum for the attenuation devices [9] . . . . . . . . . . . . . . . . . . . 16

2.12 Spectra for truncated nozzle cases, [9] . . . . . . . . . . . . . . . . . . . . 17

2.13 Backward facing step and Axisymmetric geometries . . . . . . . . . . . . . 18

2.14 Grid information [22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.15 Comparison of the results between Spalart Allmaras and DES models [21] . 20

2.16 Bakward facing step, LES [21] . . . . . . . . . . . . . . . . . . . . . . . . 21

viii

LIST OF FIGURES ix

2.17 PSD of the buffet pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 View of the grid-symmetry plane is visible . . . . . . . . . . . . . . . . . . 26

3.2 Zoomed view of the EPC nozzle region . . . . . . . . . . . . . . . . . . . 26

3.3 Dimension of the grid in meters (pressure-far-field is not shown) . . . . . . 27

3.4 Single and multiple domains . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Bulging domain is the upper wall of the inside nozzle . . . . . . . . . . . . 29

3.6 Domains at the boosters’ end after running elliptic solver . . . . . . . . . . 31

3.7 Original grid from ESTEC with 2390208 hexahedral cells based on y+=1

and near wall cell size 1×10−06 . . . . . . . . . . . . . . . . . . . . . . . . 34

3.8 Coarsened grid with 2241792 hexahedral cells based on y+=100 and near

wall cell size 1×10−04 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 A schematic of a typical turbulent kinetic energy spectrum for homoge-

neous turbulence plotted with logarithmic scales [26] . . . . . . . . . . . . 41

6.1 Separation in the axial direction . . . . . . . . . . . . . . . . . . . . . . . 60

6.2 Separation indicated by stream lines in the lateral (cross-stream) planes . . 60

6.3 Contours of vorticity magnitude in the lateral (cross-stream) planes . . . . . 60

6.4 Adapted grid region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.5 Position of pressure taps used in the experiment [30] . . . . . . . . . . . . 62

6.6 The recirculation on the left side, viewed from the top, effect of various

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.7 The recirculation on the right side, viewed from the top, effect of various

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.8 The recirculation on the left side (viewed from the top) of the base region,

effect of grid adaptation with (a) k − ω, (b) k − ε and (c) S-A model . . . . 67

6.9 The recirculation on the right side (viewed from the top) of the base region,

effect of grid adaptation with (a) k − ω, (b) k − ε and (c) S-A model . . . . 68

x LIST OF FIGURES

6.10 Sections along the base region . . . . . . . . . . . . . . . . . . . . . . . . 69

6.11 Set A: Turbulent kinetic energy behaviour for RANS models at (a) z= 0.036

m , (b)z= 0.027 m , (c)z= 0.009 m (d)z= 0 . . . . . . . . . . . . . . . . . . 71

6.12 Set B:Turbulent kinetic energy behaviour for RANS models at (a) z= -

0.018m , (b)z= -0.026m , (c)z=-0.045m (d)z=-0.06m . . . . . . . . . . . . 72

6.13 TKE plots for k − ε model with adapted grid: comparison with original

(coarse grid) is also shown . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.14 TKE plots for k − ω model with adapted grid: comparison with original

(coarse grid) is also shown . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.15 Set A : Turbulent viscosity distribution at various sections in the base region

using with all of the three turbulence models at (a) z = 0.036 m , (b)z = 0.027

m , (c)z = 0.009 m (d) z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.16 Set B : Turbulent viscosity distribution at various sections in the base region

with all of the three turbulence models (a) z = -0.018m , (b) z = -0.026m ,

(c)z = -0.045m (d) z= -0.06m . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.17 Set A : Axial velocity profiles at various sections in the base region using

with all of the three turbulence models at (a) z = 0.036 m , (b)z = 0.027 m ,

(c)z = 0.009 m (d) z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.18 Set B : Axial velocity profiles at various sections in the base region with all

of the three turbulence models (a) z = -0.018m , (b) z = -0.026m , (c)z =

-0.045m (d) z= -0.06m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.19 Set A : Lateral velocity profiles at various sections in the base region using

with all of the three turbulence models at (a) z = 0.036 m , (b)z = 0.027 m ,

(c)z = 0.009 m (d) z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.20 Set B : Lateral velocity profiles at various sections in the base region with

all of the three turbulence models (a) z = -0.018m , (b) z = -0.026m , (c)z =

-0.045m (d) z= -0.06m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

LIST OF FIGURES xi

6.21 Contours of Mach number for Inviscid and k − ε cases . . . . . . . . . . . 84

6.22 Contours of Mach number for k − ω and S-A cases . . . . . . . . . . . . . 85

6.23 Contours of vorticity magnitude a) k − ω, (b) k − ε and (c) S-A model . . . 86

6.24 3D-path-lines coloured by z-velocity showing unstability in the flow field

(a) Inviscid (b) k − ω, (c) k − ε and (d) S-A model . . . . . . . . . . . . . 88

6.25 Oil flow on the symmetry surface coloured by z-velocity (a) Inviscid (b)

k − ω, (c) k − ε and (d) S-A model . . . . . . . . . . . . . . . . . . . . . . 89

C.1 Set A: Residual history for coarse grid, (a) Inviscid (b) k − ω . . . . . . . . 100

C.2 Set B: Residual history for coarse grid, (a) k − ε (b) S-A model . . . . . . . 101

C.3 Residual history for adapted grid, (a) Inviscid (b) k − ω, (c) k − ε and (d)

S-A model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

List of Tables

3.1 Grid Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1 Density based solver attributes Source: Fluent 6.3 User Guide section 25.5.5 55

5.2 Time statistics for invisicd case with 8 processors . . . . . . . . . . . . . . 57

6.1 Adapted grid information . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 Pressure coefficients at four positions with coarse-grid . . . . . . . . . . . 63

6.3 Pressure coefficients at four positions with adapted grid . . . . . . . . . . . 64

6.4 Coefficient of drag for original and adapted grids . . . . . . . . . . . . . . 82

6.5 Total pressure at the tip of the main tank . . . . . . . . . . . . . . . . . . . 82

xii

Nomenclature

Units

The units which are used in the nomenclature have the following meanings:

M mass

L length

T time

θ temperature

Acronyms

ESA European Space Agency

GTO Geostationary Transfer Orbit

ATV Automated Transfer Vehicles

SRI Abbreviation in French for Inertial Reference Frame

OBC On-Board Computer

IBR Inquiry Board Report

NLR National Research Laboratories

CFD Computational Fluid Dynamics

XLES Extra Large Eddy Simulation

EPC Etage Pricipal Cryotechnique (French name for Central Cryogenic Engine)

HSWT High Speed Wind Tunnel

xiii

xiv Nomenclature

DES Detached Eddy Simulation

ZDES Zonal Detached Eddy Simulation

RANS Reynolds Averaged Navier Stokes

LES Large Eddy Simulation

S-A Spalart Allmaras

AUSM Advanced Upstream Splitting Method

URANS Unsteady Reynolds Averaged Navier Stokes

MUSCL Monotone Upwind Scheme for Conservation Laws

NS Navier Stokes

SGS Sub-grid Scale

CFL Courant Fiedrichs Lewy

PSD Power Spectral Density

TFI Trans Finite Interpolation

PDE Partial Differential Equation

DNS Direct Numerical Simulation

RNG Re-normalisation group

SST Shear Stress Transport

FDS Flux Difference Splitting

QUICK Quadratic Upwind Interpolation for Convective Kinetics

FAS Full Approximation Storage

TKE Turbulent Kinetic Energy

URF Under Relaxation Factor

Latin letters

Symbols Units Description

x,y,z L Cartesian coordinates

l L length scale in DES

Nomenclature xv

d L First cell height near the wall

CDES - Constant in DES = 0.65

y+ - Dimensionless cell height

V L3 Volume of fluid

S L2 Surface area

p ML−1T−2 Pressure

T θ Temperature

u, v, w LT−1 Components of velocity

Pr - Prandtl number

Sij T−1 Strain rate

c′ LT−1 Modified acoustic velocity

n - Time level

Cp - Pressure Coefficient

Re - Reynolds number

R L2T−2θ−1 Gas constant

i,j,k - Normal vectors to x,y and z directions

V LT−1 Velocity Magnitude

g LT−2 Gravity vector

E ML2T−2 Energy

q MT−3 Heat diffusion flux

W,F, G - Vector of convective fluxes in the ξ, η, and ζ directions

A - Flux Jacobian

K - Eigenvalue matrix

xvi Nomenclature

Greek letters

Symbols Units Description

∆ ML−1T−1 Sub-grid Stress

ξ, η, ζ L Curvilinear Coordinates

ρ ML−3 Density of the fluid

ε L2T−3 Turbulent dissipation rate

ω T−1 Specific dissipation rate

ρ ML−3 Density

τ MLT−2 Shear stress, characteristic time scale and pseudo time derivative

κ M−2T−2θ Thermal conductivity and von-Karman constant = 0.4187

γ - Specific heat ratio

νl ML−1T−1 Laminar viscosity

µ0 ML−1T−1 Reference viscosity for air (1.7894× 10−5)

ν L2T−1 Kinematic viscosity

η L Length scale of small eddies

φ - Flow parameter

φ - Mean component of the turbulent flow parameter

φ′ - Fluctuating component of the turbulent flow parameter

ρ, ρ′ ML−3 Mean and fluctuating components of density

δ - Kronecker delta

ν L2T−1 Eddy viscosity variable

λtay L Taylor micro-scale

σ, σ∗ - constants used in k − ω model

β, β∗ - Constant used in k − ε model

Γ - Preconditioning matrix

ρT ML−3T−1 Time derivative of density

β L−2T−2 Compressibility factor

Nomenclature xvii

α - (1− βU2r )

Λ - Diagonalizing matrix

ρp L2T 2 Density derivative with respect to pressure

λmax - Maximum of the eigenvalue

Chapter 1

Introduction

1.1 Ariane history

Four decades ago, the European scientists and engineers felt a need to develop their own

access to space. For this purpose, in 1973, Ariane Launch system came under operation. It

was a joint venture of the European Space Agency (ESA) and the French National Space

Agency CNERS, who was the prime contributor to the ESA. The first flight with Ariane1

took place on December 24th 1979 which launched several European and non-European

satellites. It had a payload of 1800 kg and access to the Geostationary Transfer Orbit

(GTO)1. But this capacity was soon found to be insufficient for the future satellites. In

early 1980s Ariane2 was launched which had a payload of 2200 kg and it also had access

to the GTO. In the same year Ariane3 was launched which could carry a payload of 2,700

kg and could launch two space-crafts at a time [8]. Ariane4 was an immense improvement

in the history of Ariane family. Although the project started in April 1982 it made its first

flight in 1988. It had many variants from then its capability of carrying 4,800 kg to GTO

and adaptation to different missions and payloads made it versatile from previous versions

of Ariane. Ariane4 proved its reliability with 74 consecutive successful flights from Jan-

1A geostationary transfer orbit is used to move a satellite from low Earth orbit (LEO) into a geostationaryorbit.

1

2 Chapter 1. Introduction

Figure 1.1: Ariane family

uary 1995 to February 2003 and strengthened Europes position in the market in spite of the

stiff international competition specially with the USA and Russia. Ariane5 proved to be

a milestone in the history of ESA. The project began in 1987 with the aim of developing

a stronger and different architecture rocket. This version of Ariane5 had highly equipped

electrical and computer systems. It had two solid-fuel boosters which provided 90% of the

thrust at lift-off. Moreover, it had a cryogenic core stage whose function was to provide the

remaining thrust for the first part of the flight up to the upper stage separation. To further

enhance its lift capability, Ariane5 is now equipped with a cryogenic upper stage powered

by the Ariane4 cryogenic engine [8].

Now Ariane5 is able to place heavy loads in GTO and is ideally suited for launching

Automated Transfer Vehicles (ATV) towards the International Space Station. Besides GTO

Ariane5 has the capability for circular and polar circular, inclined orbits and escape missions

as well. Figure 1.1 shows the five versions of the Ariane project.

1.2. Ariane5 Flight 501 3

1.2 Ariane5 Flight 501

The Ariane5 Flight 501 took place on June 4th 1996. Unfortunately, it was not a successful

flight and saw the failure just after 44 seconds of its take off. The main reason that had

been reported in the literature was the malfunction in the control software which veered off

the rocket of its flight path. This also led to the generation of intense level of aerodynamic

forces which caused the rocket to disintegrate. Mark Dowson [7] has highlighted the main

reasons for this catastrophe in his note. His comments were based on the report by the

Investigation Board who worked on the causes of the failure of Flight 501 [13]. Dowson

highlighted the events during the disaster as follows:

1. Problem in the Inertial Reference System (SRI) indicated by an unexpected value,

which is obtained from the launcher’s horizontal velocity. The backup of this system

failed just after 37 seconds of the launch.

2. Due to the same reason an identical SRI failed also after 72 milliseconds of the first

event.

3. The On Board Computer (OBC), which takes input from SRI, thus misinterpreted

the data and commanded full nozzle deflections, both the solid boosters and the main

Vulcain engine.

4. These deflections created side loads (buffet loads) which caused ultimately the boost-

ers to separate and thus disintegrated the whole system just after 40 seconds of the

launch.

The clause highlighting the fact that the aerodynamic side-loads caused the nozzles

to vibrate would be the main core of our thesis, as will be discussed shortly. Another

anomaly had been pointed out in the Inquiry Board Report was the unusual variations in the

hydraulic pressure of the actuators of the main engine nozzle. This was observed during

H0 + 22 seconds. However, this has been mutually accepted by the board that the main

4 Chapter 1. Introduction

problem occurred was due to the software because the SRI used was the same as used

in Ariane4. It has been explained that the part of the software in the SRI is used for the

realignment purpose in the Ariane4. Nevertheless, it had no need for that in Ariane5 but

it continued to work even 40 seconds after lift-off. This caused the trajectory of Ariane5 a

more pronounced pitch than Ariane4 resulting an increase in the magnitude of horizontal

velocity and thus error in the software of SRI.

The following recommendations were given by the board:

1. “Although the failure was due to a systematic software design error, mechanisms can

be introduced to mitigate this type of problem .... ” (IBR page 5)”.

2. “Do not allow any sensor ... to stop sending best effort data.” (IBR Recommendation

3, page 13 ).

3. “...perform complete, closed-loop, system testing. Complete simulations must take

place before any mission .... ” (IBR Recommendation 4, page 13).

As this report was issued in 1996 later on many wind tunnel tests were conducted and it

was found that the aerodynamic loads also had a contribution in the failure.

1.3 Motivation

Although experiments are still in use today but sophisticated high performance computing

and robust numerical techniques has simplified the things. As engineers are much worried

about the budget and cost, CFD has solved this problem to a higher extent. After the failure

of Flight 501 many wind tunnel tests were conducted by NLR [9]. These tests although

helped in investigating many factors behind the cause of side-loads but they took a lot of

time and money. Therefore, many people started working on the CFD of the base flow of

Ariane5.

1.4. Thesis layout 5

Although the work that we are going to present has already been done,the main dif-

ference is the CFD code used. Dr Ludeke [15] has employed Flower code by DLR while

Maseland [16] has done this research using X-LES (a new LES approach).

1.4 Thesis layout

Chapter 2 discusses the previous experimental and CFD research conducted in this area.

Chapter 3 discusses the grid generation, the problems faced, the solvers, the reason to switch

to coarse grid and the boundary conditions employed. Chapter 4 discusses the governing

equations that are involved in the solution process of CFD. Chapter 5 discusses numeri-

cal methods used, explicit and implicit methods, reason for selecting explicit time step-

ping method and preconditioning. Chapter 6 is the results and discussion, the steady and

unsteady cases, adapted grid, contours and graphs, Power spectral density and frequency

curves. The last chapters include the conclusions and future recommendations including

grid improvization, simulations at different angles of attack and use of higher order turbu-

lence methods. Finally, Chapter 7 discussed conclusions and recommended future work.

After that the appendices are attached.

Chapter 2

Literature Review

The first mission of Ariane5 Flight 501 led to a failure on June 4, 1996, due to a malfunc-

tion in the control software. The rocket veered off its flight path 37 seconds after launch

and was destroyed by its automated self-destruct system. It was the combined effect of the

aerodynamic forces which caused the core of the vehicle to disintegrate [? ]. These risk

factors should be considered before any flight. These factors include mechanical, electri-

cal, structural, aerodynamics and controls. This thesis focuses on the aerodynamic factors

involved in the launching of the space-crafts. Aerodynamic factors are not only concerned

with the local atmosphere, significant analysis is required for re-entry of the vehicles; a

special field called aerothermodynamics has a vital role there. Aerothermodynamics covers

a wide range like [18]:

• External aerodynamics of aerospace vehicles, which cover the transition from high al-

titude free molecular flow to continuum flow as vehicles enter planetary atmospheres.

• Unsteady flow effects due to aerodynamic buffeting and fluttering of structural ele-

ments.

• Internal aerodynamics of aerospace vehicles covering the design of propulsion engine

inlets, propulsive exhaust nozzles, engine flow control valves, manifolds and vents.

6

2.1. Experimental investigation 7

• Micro-aerodynamics which involve the assessment of local flow effects in gaps be-

tween thermal-protection tiles, and at steps between structural elements and at cor-

ners, for example, at aerodynamic control surface hinges.

• Chemical reactions in combustion chambers and in the shock layer of aerospace ve-

hicles during entry into planetary atmospheres.

• Multiphase flows in non-equilibrium, chemically frozen or the conditions where con-

tamination and debris are of concern.

• Rocket-engine exhaust-plume flow impingement effects on spacecraft surfaces in-

volving forces moments, heating and contamination.

The major focus here will be unsteady effects due to aerodynamic buffeting. During

the launch of space vehicles there is a contribution of forces from external flow which

usually peaks in transonic flow and interacts with the internal flow from nozzles (both the

main engine and boosters). This can cause significant side loads on the nozzle and create

instability. The phenomenon is commonly known as buffeting.

2.1 Experimental investigation

2.1.1 Theoretical and analytical analysis of base flow buffeting

The thesis will have a focus around the paper about the experimental and theoretical analysis

of Ariane5 base flow buffeting [29]. This paper details that the nozzle design should be

in such a way the buffeting must be reduced if not completely removed. The author has

discussed that the issue of the side loads on the nozzle that cause buffeting was somewhat

acceptable but in later versions it was found that the forces have significant effect especially

when the nozzle design changes (e.g. bell-shaped). This was obviously due to the dual

fact that large nozzles could have large lengths and three-dimensional flow separation in the

nozzle.

8 Chapter 2. Literature Review

The author mentioned some rigorous conclusions from the theoretical analysis per-

formed by Zumwalt [2] on a model of the flow in the annular base (rear-facing step) of a

circular engine nacelle with subsonic conditions but the exhaust from the jet is supersonic.

The conclusion was that a decrease in the base pressure affects two things (i) jet-Mach num-

ber is increased Fig 2.1 (ii) the jet-shear-layer envelope shrinks and the shear-layer comes

closer to the nozzle Fig 2.2. Conversely, the increase in pressure is due to the increase in

nozzle deflection angle and increase in the upstream boundary layer thickness. This anal-

ysis is related to the experimental work by Rossiter [20] who discussed the acoustic and

shear-layer effects for cavity flow.

Wong suggests that the main source of the base flow buffeting in Ariane5 is the free

shear-layer emanating from the central body below the central cryogenic engine (EPC).

The instability consists of Kelvin-Helmholtz vortices which interact with the forthcoming

acoustic waves from the edge the EPC Vulcain engine nozzle. Those incoming acoustic

waves itself generated by the impingement of the vortices. This interaction amplifies the

disturbance and produces more oscillatory waves and as result a feed-back loop forms.

The initial experimental data was extracted from three different resources which were

PHST Wind tunnel in National Aerospace Laboratory (NLR), Swedish Defense Research

Agency (FOI) and French National Aerospace Research Establishment (ONERA). These

experiments were conducted on a 1:76.5 scale model. The Reynolds number based on

the stream-wise length-scale was 1 × 107 and the Mach number ranged from 0.4 to 0.9.

More than 40 kulites (pressure transducers) were distributed uniformly over the Ariane5

base for measurements as shown in Fig 2.3. The results for the pressure fluctuations were

compared with the experiment and found in good agreement. The axial (X direction), side

(Y direction) and lateral forces (Z direction) were measured and discussed with varying

configurations e.g. inclusion or absence of the exhaust pipes on the nozzle and of the Helium

tank. It was found that the inclusion of the exhaust pipe adds to the force in the Y direction

while the absence of pipe on the nozzle increases the cavity depth thereby causing of the


Figure 2.1: Effect of Jet Mach Number on the base pressure[29]

Figure 2.2: Effect of base pressure on Jet-shear layer envelope[29]


Figure 2.3: Position of kulites on the nozzle[29]

impingement of shear-layer on or near to the nozzle, Fig 2.4. Free stream Mach 0.8 and

absence of the Helium tank showed that the peaks became sharp in Y-direction due to the

wake region that came due to the absence of the tank Fig 2.5. Results were also discussed

for the roll, yaw and pitch cases without the exhaust pipe on the nozzle and at Mach 0.8, it

was found that the forces in Y and Z directions amplified. Effect of the Y-force was more

significant in yaw as shown in Fig 2.6.

2.1.2 Measurement of side loads

Side loads on nozzle

Heiu et al [10] has studied the interaction of transonic flow and flow separation in over-

expanded nozzle experimentally. The measurements were taken at Mach numbers between

0.5 and 0.82 with especially designed strain-gauge balances. The case was studied for the

interaction between transonic buffeting and nozzle flow separation by first studying the

base flow alone, nozzle flow alone and then combination of both flows. Two dual-bell

nozzles with different emergent length sizes were considered and the influence of Mach

number and pressure ratios was investigated. Inertia correction method was adopted for the


Figure 2.4: Frequency spectrum, Mach 0.8 with jet, with exhaust pipes (left) and withoutexhaust pipes (right) [29]

Figure 2.5: Integrated forces with no protuberance at Mach 0.8 [29]


Figure 2.6: Effect of 5 deg yaw (left) and 5 deg pitch (right) to integrated forces at Mach0.8 with jet but without exhaust pipes.[29]

calibration of the instruments. It was found that it is possible to measure direct loads on the

nozzle with transonic buffeting. The results suggest that at least for dual-bell nozzle some

interactions between the internal separated flow and transonic external flow are possible and

the root-mean-square value of the side loads seems higher than the sum of the two sources

considered separately. However, the author did not highlight the reasons of generation of

peaks in large nozzle case explicitly.

Measurements of loads on Ariane5 base flow

NLR E.G.M. Geurts [9] wrote a detailed report on the steady and unsteady pressure mea-

surements on the nozzle surface. This report is based upon the different tests conducted

during 1999 to 2001 for Ariane5. Here only a brief account is given and the details can be

peeked in the actual reference. The tests were based on calculating the pressures thereby

transforming them into dimensionless force coefficients. These loads were then compared

with the experimental data for Ariane5.

Basic requirement was the calculation of steady and unsteady pressure data because it


Figure 2.7: Table of the different test cases conducted by NLR [9]

Figure 2.8: Model in SST with zoomed view of the nozzle [9]

was thought (by the author) that the numerical methods are not worthy to trust. From these

pressure the objective was to calculate the steady-unsteady side loads and then the non-

dimensional coefficients. This was to provide EADS sufficient ground to develop aft-body

elements of the launch vehicle that minimize buffeting. A number of kulites (144 on the

base only) were used and 18 additional for the Vulcain 2 configuration over the He Tank,

booster modules and high pressure vessels. Fig 2.7 shows the tests’ detail and Fig 2.8 shows

the wind tunnel model and the kulites can be seen on the nozzle surface.

Wind Tunnel facility was provided by the DNW foundation in the Netherlands which

had an HSWT (High Speed Wind Tunnel) of Mach range 0.2 to 1.3, while for supersonic

tests Supersonic Wind Tunnel (SST) was used. The later was a blow-down type tunnel

with Mach ranging from 1.2 to 4. The test #1 showed high transverse loads and high


(a) Ariane5 Evolution + Vulcain 2: base character-ization

(b) Ariane5 + Vulcain 2: attenuation devices

Figure 2.9: High loads observed in test #2, [9]

recirculation. Test #2 was conducted in 1999 with Vulcain2 plus launcher. High levels of

transverse loads were observed there as well. It was suggested that these could be lowered

by using skirts mounted on central body on the engine frame. It was also pointed out that

the recirculation in the base area and the vortices near the boosters can cause the buffeting.

The rear connection struts were also suspected to involve in this phenomenon. This was the

reason for test #4, but it was not complete because the reference measurements did not give

satisfactory results. This was evident from the high peaks formed at Mach 0.7 than in the

test #2 as shown in Fig. 2.9a and 2.9b. It was said that these peaks could be due to some

acoustic resource whose reason was not clear at that time, thus test #5 was conducted. Test

#5 was conducted with the following proposals:

1. Streamline fairing

2. New rear-connection struts

3. A new longer twin-sting support if the streamline fairing was found to show an effect.

Additionally, acoustic arrays were employed to test the acoustic signals.These had the time

signals which were recorded for 20 seconds per 6400 Hz. For the four types of twin-sting


(a) Spectrum for straight twin-sting support (b) Spectrum for Z-shaped twin-sting support

Figure 2.10: Acoustic array spectra, [9]

supports it was said that, “With the wedge support a dipole was observed that radiates

towards the wall.” Although, the term “dipole” was not clearly defined, it was concluded

that the acoustic effect was considerable. The blunt body support was found to reduce the

pressure on the walls. However, the Z-shaped beam support was found suitable for reducing

acoustic. This is shown in Fig. 2.10a and Fig. 2.10b. The numbers in the legend indicate

the four types of the wedge-supports used. The important issue was that the peaks observed

in test #4 were not present in tests #2 and #3. It was suggested that this was probably

because of the flow shedding from the base area of the nozzle and its impingement on the

wedged shaped support. Secondly, the roughness of the material could also have had a role

there; the model used in tests #2 and #3 was partially painted with dull black, while in

this case it was fully painted black. Additional roughness could be due to the plasticine

which was used for attenuation devices. It is interesting to note that this phenomenon was

observed in test #2 and #3 but not well-pronounced because the tests were conducted at

Mach 0.9.

Other tests included the use of attenuation devices which was in the optimisation phase.

The following devices were suggested:

• Fairings on the rear-connection struts

• Fences on boosters


• Vortex generators on the boosters

A “nominal” configuration was defined which was selected in each of these three types

of devices on the basis of some parameters (flight angles, Mach number). As shown in Fig.

2.11, it was found that the nominal configuration gives better performance as compared

to the other configurations. The verification phase consisted of the nozzle tests. One test

Figure 2.11: Spectrum for the attenuation devices [9]

included the used of truncated nozzle with a disk and compared to the results of Vulcain2

nozzle. The nozzle configurations can be found in [9]. It was found that inclusion of disk

produces high loads, Fig. 2.12a and Fig. 2.12b. The verification phase also included the

tests with the skirts. The skirts were used on the central engine body and they were found

to have a weak but overall positive effect. However, it was reported that the choice of skirt

needs more analysis.

2.2. Computational studies 17

(a) Spectrum for truncated nozzle (b) Spectrum for truncated nozzle with extensiondisk

Figure 2.12: Spectra for truncated nozzle cases, [9]

2.2 Computational studies

2.2.1 Zonal-Detached-Eddy-Simulation of a Two-Dimensional and Ax-

isymmetric Separating/Reattaching Flow

Sebastien Deck [22] performed the Zonal Detached Eddy Simulation (ZDES) which is no

different from DES (Detached Eddy Simulation)except that in ZDES the user specifies the

regions of RANS (Reynolds Averaged Navier Stokes) and LES (Large Eddy Simulation).

The way it switches from RANS to LES is similar to the DES in which a distance is adjusted

through the relation:

` = min(d, CDES∆) (2.2.1)

where, d is the grid scale size, and l is the length scale, CDES is a constant whose value

according to Spalart Allmaras [25] is 0.65 and ∆ is the sub-grid scale size. It is now user

dependent to refine the regions of interest without being worrying about the boundary layer

properties.


Numerical method

The solver used was FLU3M developed by ONERA for multi-blocked structured grid. Spa-

tial scheme was AUSM , which is basically a flux vector splitting scheme. Time derivatives

were evaluated using backward step and second order accurate scheme of Gear.

Geometry and grid sizes

Two types of geometries were used as shown in Fig 2.13a and 2.13b. The axisymmet-

ric configuration was composed of an after-body with a diameter equal to 100 mm and a

120 mm long extension which provides an length-to-dia ratio of 1.2. For the backward

facing step (the author has called this a 2D geometry because of constant depth) a diam-

eter,equivalent diameter, of 40 mm and a height of 30 mm were taken. The length in the

cross-stream direction was 4H. Free stream Mach number was 0.702 and Re = 1.2 × 106,

based on the fore-body diameter. The initial external boundary layer thickness was obtained

by modeling the necessary length for the fore-body to have a thickness-to-dia ratio of 0.2.

The computational size for the two geometries is shown in Fig 2.14.

(a) Backward facing step (b) Axisymmetric body

Figure 2.13: Backward facing step and Axisymmetric geometries


Figure 2.14: Grid information [22]

2.2.2 Turbulence study

Sebastien Deck [21] also studied turbulence with its application in Space Launch Vehi-

cle. He used three different approaches i.e. Unsteady Reynolds Average Navier-Stokes

(URANS), DES and LES. The code FLU3M from ONERA was used. The code has the

ability to solve NS equations on a multi-block structured grid. Second order upwind finite

volume scheme was adopted for discretisation in space with MUSCL (Monotone Upwind

Scheme for Conservative Laws). Time discretisation was done with second order scheme

of Gear with LU factorisation of sparse matrix systems. The DES and steady RANS (with

SA model) simulation was performed for the buffeting analysis over and aft-body as shown

in Fig 2.15. In the SA computation, it was observed that the incoming subsonic turbulent

boundary layer separates at the base corner and the free shear-layer has formed in the wake.

Furthermore, this expansion meets with a recompression downstream of the base which re-

aligns the flow. This can be seen by the formation of a low-pressure region downstream of

the base which is characterized by low-speed recirculating flow. Thus, a free shear mixing

region forms due to the interaction of recirculating region and the inviscid external flow. In

addition, a recompression shock was observed through the supersonic jet. The most impor-

tant point is the difference between the SA and the DES simulations. In the SA case, the

flow leads to the reattachment before the base of the emerging nozzle (emerging nozzle is

like a small pipe), while in the DES the flow reattaches to the over expanded jet. For the

experimental point of view, the reattachment length is approximately equal to the emerging


Figure 2.15: Comparison of the results between Spalart Allmaras and DES models [21]

nozzle length.

The LES simulations were performed for Mach 0.7 and Reynolds number based on the

radius of the cylinder (0.05 m) was 6 × 105. To test the influence of the inflow, two types

of LES simulations were analysed. In the first one (LES1), no fluctuations were imposed

in the inflow plane whereas in the second one (LES2), random fluctuations were imposed.

For both cases, the number of grid points were about 2.5 million with 180 points in the

azimuthal direction. For both computations, the CPU cost was about 180 hours of NEC

SX5 supercomputer time to simulate a physical time of 0.1s which with a Strouhal number

of 0.2, represents 50 times the expected shedding period of the wake. The recirculation

length was found to be 2.66 times radius of the cylinder as shown in Fig 2.16. Pressure

was found to be constant on the base for both computations. Spectral analysis with Fast

Fourier Transform with LES 2 showed good agreement with experiment. The author made

the conclusion that although these methods are not universally accurate. The LES is good

in comparison to experiment but is still expensive for industrial applications. Therefore a


Figure 2.16: Bakward facing step, LES [21]

compromise must be made between URANS and DES.

2.2.3 Unsteadiness of an axisymmetric separating flow

Another study by Deck [6] has been investigated numerically in detail for axisymmetric

separating-reattaching flow Fig 2.13b. The author has mentioned, in detail, the references

of researchers who have thoroughly analysed this area of research. However, the author’s

study comprised of the Zonal Detached Eddy Simulation and its comparison with the ex-

perimental data of ONERA, conducted on an axisymmetric base flow configuration. Details

of the experimental setup can be found in Dussauge et al [19]. The flow Reynolds number

(based on the base flow diameter) was 1.1×106. The author reported that past literature was

less informative for the analysis of asymmetric separating/reattaching flow in comparison

with the 2D backward-facing-step case. However, with regard to shear layer stability and its

strong growth rate it was found that the Reynolds averaged data on asymmetric geometry

gave similar results to the analyses done with two-dimensional backward facing step.

A deep investigation of spectral analysis of pressure fluctuations showed that the fre-


quency contributions behave differently depending on their location of the recirculation

bubble. It was also observed from the statistical data that the peak near a normalized fre-

quency of orderO(0.08) showed shear-layer flapping (periodic growth/decay of the bubble)

phenomenon near the step.

It was concluded that the separated bubble dynamics depend on very complex inter-

actions of large eddies formed in the upstream free shear layer with the wall in the reat-

tachment length. Secondary corner vortices experience a cycle of growth and decay in the

vicinity of the step and a link was observed between frequencies of the secondary vortex

shedding and its flapping motion which led to the conclusion that different types of same

motion are involved. This was due to the presence of large scale coherent motion. The

evidence was supported with two-point correlation analysis of the pressure signals. They

showed that the flow is dominated by large scale asymmetric eddies at the flapping and

vortex shedding frequencies whose indications were found in the buffet-load-spectrum.

The beginning of the large-scale motions was also discussed. It was concluded that

the space-time characteristics of wall-pressure fluctuations on the aft-body indicates strong

feedback disturbances originating from the centre of the recirculating bubble rather than the

reattachment point. The most energetic upstream disturbances are restricted in the region

half of the non-dimensional recirculation length. This was reported by the author, along

with many references, that this phenomenon could be the possible cause of instability in

the flow. Although it was an in-depth study of the base-flow, the nozzle flow which was

modelled in the examination, was not analysed in detail. The reason the author stated was

the length of the aft-body cylinder was long enough that the separated flow from the base

would not affect the exhaust from the jet.

2.2.4 Extra Large Eddy Simulation

A new technique has been developed during the past 10 years using a similar model like

DES and it has been named by the researchers as Extra-Large Eddy Simulation or X-LES.


The so called simulation was developed to predict the flow structures with massive sep-

arations and broad-band spectrum of the aerodynamic loading. X-LES is also a hybrid

technique like the DES which combines LES solution in the separated flow region and the

RANS in the near-wall region. The main difference from the DES is that in X-LES a sepa-

rate SGS (Sub-grid Stress) model is defined along with the RANS while in DES the RANS

is made to behave like SGS model. X-LES is supposed to consists of one set of equations

obtained by time-filtering of Navier-Stokes Equation.

Maseland [16] has studied the effect of dynamic load on the wind tunnel model of Ar-

iane5 and studied in detail the load predictions calculated with X-LES simulations. The

model considered was a clean geometry of Ariane5 (clean means without external piping

around Vulcain engine and without Helium tank). This model was of the same dimensions

as of the experiments, however, it was not mentioned explicitly where the experimental data

was taken from. Unsteady conditions were applied with free-stream Mach number of 0.73

and Reynolds number of 6 million based on the height of the wind tunnel model. It was

mentioned that the physical time span was determined through the lowest dominant fre-

quency for the separated flow in the vicinity of the nozzle. It was reported that according to

the literature 40 periodic cycles must be calculated in order to capture flow physics for hy-

brid LES-RANS, however, Maseland performed 6 periodic cycles and found to be sufficient

to capture the flow physics.

Initially steady simulations were performed with RANS, after that, 4 periodic cycles

with 64 times step per period were run in X-LES case to damp out the unsteadiness. Sub-

sequently, 6 periodic cycles were computed in X-LES with 256 time steps per period. CFL

number based on the final time step was found to be 2.

The experimental data was compared in terms of PSD (Power Spectral Density) of the

buffet pressure at a single point on the nozzle. For comparison in X-LES the PSDs were

compared for the same time span. The experimental data was window based and the author

made a conclusion that the results of CFD can be compared with experimental in terms of


Figure 2.17: PSD of the buffet pressure

PSDs for a selected window. Typical peaks were observed both in experiment and CFD at

the same time as shown in Fig 2.17.

It was concluded that X-LES is feasible for predicting unsteady loads on nozzles and it

is also suitable in terms of, turn-around times, to support the design process. The results

were compared with experiment both qualitatively and quantitatively and it was found that

although qualitative gives better prediction (if flow physics is considered) but numerical

results in terms of PSD could give better prediction if plotted in the same reference window.

Chapter 3

Grid Generation

3.1 Introduction

Grid generation is the basic building block of CFD. It is a skill that needs mastering in order

to produce good quality grid. That is why grid generation is more likely to be an art rather

than science. Sometimes, while handling complex geometries unstructured grids are usually

preferred due to there flexibility of usage and less time consumption. In this case, creating a

structured grid is really difficult. It is generally accepted that solutions on a structured grid

are better than the unstructured grid.

The geometry of Ariane5 apparently looks simple to handle with. However, when it

comes to the narrow region between its boosters and the main nozzle, the real challenges

arise, especially when y+ = 1 is required. This task, however, was completed by the ES-

TEC who provided structured grid for Ariane5. This grid was multi-block with 2.4 million

hexahedral cells. For this study Gridgen V15 was used to read the mesh file and the neces-

sary boundary conditions were applied. As the original mesh was made by ESTEC in their

in-house code, several problems arose in creating an appropriate mesh which is acceptable

by Gridgen and readable by Fluent 6.3 later. A plan view of the grid with the geometry and

symmetry plane and zoomed view of nozzle are shown in Fig 3.1 and Fig 3.2 respectively.

25

26 Chapter 3. Grid Generation

Figure 3.1: View of the grid-symmetry plane is visible

Figure 3.2: Zoomed view of the EPC nozzle region

3.1. Introduction 27

Figure 3.3: Dimension of the grid in meters (pressure-far-field is not shown)

According to the Gantt chart planned for the thesis “Grid Study” had been allocated

one week. The term “Grid Study” had been used because it was expected that the grid from

ESTEC would be free of any ambiguities and Fluent 6.3, with no harm, would automatically

accept it. Later on it was realized that there are certain problems in the geometry that needed

modification for a proper workable file for Fluent 6.3. This work took more than one month.

Fig 3.3 shows the extent of the grid.

The first problem was in those places where multiple faces existed at the interfaces.

Figs 3.4a and 3.4b show an example of the front block where there is one domain overlap-

ping the three sides of the block and in the second figure there are six separate domains i.e.

two domains on one side of the block. The mesh with overlapping domains was initially

exported, after applying the boundary conditions, to Fluent 6.3 but it was found that Fluent

6.3 is not accepting as a part of fluid and recognizing them as wall. The original grid had

65 blocks, 355 domains, and 574 connectors. The work was not so simple so the case was

initiated in a way of bottom-down approach. This consisted of splitting the domains and


(a) A single overlapping domain (b) Multiple domains in the adjacent block

Figure 3.4: Single and multiple domains

then, if found necessary, splitting the blocks. It was also realized that the geometry of Ari-

ane5 had itself certain defects. One of them was found in the booster region that the inner

wall of the booster-nozzle was appearing out of the booster wall like a bulging spot on the

outer. This is shown in Fig 3.5 where the black arrow indicates the lower wall of the nozzle

coming out of the upper surface.

After this investigation and by hiding the outer booster walls it was realized that this

fault is basically due to the reason that the nozzle is not maintaining symmetry across its

profile. This is simple to understand that if you take a profile which contains a down-hill

and you rotate it 180, you should get the same profile at each degree of rotation. However,

in this case it joined like a straight line from the inlet of the nozzle to the outlet at 90 of

rotation. The problem was sorted out that Gridgen is unable to follow the profile till 180

and is unable to create a true nozzle shape. The solution found was that a circle should be

generated at those locations which had a sudden profile change (beginning, middle and the

end of throat). This technique was successful as expected and in this way the whole nozzle

domain was divided into three domains. This applied to both of the booster nozzles. But

unfortunately, this was not the end. Initially inside the nozzle there was another small block.

So when the new geometry of the nozzle had been defined; and the block was recreated, the

3.2. Structured solver 29

Figure 3.5: Bulging domain is the upper wall of the inside nozzle

divergent surface of the nozzle hit the inside block. This created negative Jacobians. This

problem was solved by using Stretch (Block) command. The Stretch command in Gridgen

is very helpful to deal with these kinds of situations where you don’t need to delete the

blocks or domains and recreate them from scratch.

3.2 Structured solver

Gridgen has a special feature to improve the mesh quality. This feature lies under the

Solver (Structured) option. It has built-in Elliptic solver and smoothens cells in a way

to be perfectly aligned with respect to the flow and with respect to the geometry. The

procedure is iterative however, so one should be careful in running a certain number of

iterations because running the solver many times can spoil the grid. There is another option

that initially if you find a grid which has abnormalities such as grid points passing through

the boundary or cells are mixing up with each other, you can first initialize the grid using

Transfinite Interpolation or (TFI). This basically creates an algebraic grid. Now further


improvement can be obtained via running Elliptic solver. These two techniques, algebraic

and the elliptic grid generation are discussed in the below.

3.2.1 Algebraic grid technique

Before the advent of elliptic solvers algebraic techniques were very popular. The reason was

simple that these techniques are simple to implement and easy to solve. An algebraic grid

is used to relate the grid points in the computational domain to those in the physical domain

Hoffmann [11]. This is achieved by using an interpolation scheme between the specified

boundary grid points to generate the interior grid points. Usually Trans-Finite Interpolation

(TFI) is used for this purpose [14]. Besides there computational speed, the matrices are

easy to handle analytically thus avoiding numerical errors. Clustering the grid points in the

regions near the wall, for example, can be easily applied. Apart, from that they have certain

disadvantages as well. They create problems if there are discontinuities at the boundary and

there is less control over the grid smoothness and skewness.

3.2.2 Elliptic solver

In Partial Differential Equations (PDEs) techniques, a system of PDEs is solved for the

location of grid points in physical space whereas the computational domain is rectangular

shape with uniform grid spacing. These methods are commonly categorized as parabolic,

elliptic, and hyperbolic based on the forms of the equations used. In elliptic solver a system

of Laplace or Poissons Equation (which are obviously elliptic) is introduced which is solved

for the coordinates of the grid points in the physical domain. These equations can then be

solved using an iterative scheme such as Gauss-Seidel. Elliptic solvers are useful because

they provide a smooth grid point distribution, i.e. if a discontinuity is present at the bound-

ary (sharp corner for example) it will be smoothed out inside the domain. Besides, a demerit

of using an elliptic solver is that the computational time is large and as the solver is based

on the solution of PDEs the equations must be computed numerically. Fig 3.6 shows the

3.2. Structured solver 31

Figure 3.6: Domains at the boosters’ end after running elliptic solver

two domains at the end of the booster nozzles. These two are shown specially because the

solver was run many times upon them. It should be noted that it is not necessary sometimes

that if the solver smoothens the grid the block quality will also get improved. Therefore,

it is useful to run Block Structured Solver as well. The data can be examined at the same

time in Gridgen while you run the solver in a block. The parameters that can be checked

are usually Positive, Positive skew, Negative and Negative Skew. The derivation of these

Jacobians can be found in Appendix A.1. Simply speaking, Jacobians are the ratio between

areas of the cells in computational and physical domains in 2D and between volumes in 3D

respectively. It is necessary that all the Jacobians should be positive. The Grid that was

initially used with 2.4 Million cells had many positive skewed elements (over 100), after a

lot of modification explained above, the grid was tried to export to Fluent 6.3 to see how

it works. In Fluent 6.3, the grid check was failed reporting many left-handed faces. For a

workable grid all faces must be right handed. As there is no way in Fluent 6.3 to see these

problematic cells in Fluent 6.3, T-Grid was used. T-Grid is generally used for hybrid mesh-


Grid type No of cells No of nodes

Coarse 2241792 2327293Fine 2390308 2479497

Table 3.1: Grid Information

ing, but it is useful to see the mesh quality (aspect ratio, face-handedness, etc.) as well. The

T-Grid showed easily all the left-handed faces and so after some improvements in Gridgen

again, the final mesh was ready to work with for Fluent 6.3.

3.3 Coarse grid

Fluent 6.3 was happy to work with the grid but with Inviscid solver case only. For the

viscous simulation it showed certain warnings like temperature limited to 1K in 3 cells and

pressure limited to 1 Pa in 3 cells. These warnings normally disappear after sometime but

as they persist for a long time the simulations were stopped to examine the corrupted cells.

The cells were on the booster surfaces, so it was decided to use the coarse grid, which had

improved cell shapes in the critical regions between the boostersand the main tank. The new

grid was modified with the assistance of my course-fellow A.Atzori. This new grid was split

into more blocks (108) for easiness in handling domains. It contained 2.23 Million cells.

After few more improvements and applying the necessary boundary conditions the new grid

was first checked in TGrid and passed there, although it had some high aspect ratio elements

though. Finally the grid check was also passed by Fluent 6.3.

The difference between the two types of grid is shown in Figs 3.7 and 3.8. The coarse

grid was the modification of the fine grid. The original cell size near the wall was of order

of magnitude 10−06 based on the y+ of 1 and Re 11 × 1006. It was then coarsened to

10−04 which, with the same Re, gave y+ of 100. The calculations were based on flat-plate

correlation. Although this is not appreciative but it is a plausible value for this kind of

3.4. Boundary conditions 33

challenging grid, especially when unsteady simulations are required. Later on, in Fluent

6.3, grid adaptation was employed but it would not be discussed here due to its close link

with the coarse-grid results.

3.4 Boundary conditions

Five types of boundary conditions were applied into the grid. These were pressure far-field,

pressure outlet, pressure inlet, wall and symmetry.

3.4.1 Pressure far-field

The far-field consisted of the entire outside domain which has less influence of the flow field

of the rocket. This was specified to put free-stream pressure, free-stream Mach number and

the flow direction. The free stream pressure was 6.04 × 104 Pa, free-stream Mach number

0.8 and free-stream temperature was 278 K [30].

3.4.2 Pressure outlet

The pressure outlet was applied at the far end of the grid. Here the same pressure as far-field

was applied. Rest of the options were kept as default.

3.4.3 Wall

No-slip boundary condition was applied on the wall. The walls were named according to

the different parts of Ariane5, i.e. boosters, main tank, outer nozzle wall, inner nozzle wall

and booster nozzle wall.


Figure 3.7: Original grid from ESTEC with 2390208 hexahedral cells based on y+=1 andnear wall cell size 1×10−06

Figure 3.8: Coarsened grid with 2241792 hexahedral cells based on y+=100 and near wallcell size 1×10−04

3.4. Boundary conditions 35

3.4.4 Pressure inlet

The pressure inlet was applied on the nozzle inlets. This condition is used for the internal

flow problems as specified in many Fluent 6.3 manual and tutorials. The pressure Inlet

boundary condition opens with two options: 1) Initial pressure 2) Supersonic/Initial gauge

pressure.

1. Initial pressure : This was the pressure as it was specified in the wind tunnel [30] . It

was 35 bar for the Vulcain nozzle and 30 bar for the booster nozzles. The pressure

was same for all the nozzles and it was equal to 320 K.

2. Supersonic/Initial gauge pressure : The supersonic/initial gauge pressure is required

by the Fluent 6.3 to initialize the solution, so it was put 22% less than the inlet pressure

as it is normally taken in the manual and tutorials.

Within each of these boundary conditions, Fluent 6.3 has got some options to choose for

turbulence specifications. These include intensity and length scale, Intensity and viscosity

ratio,Intensity and hydraulic diameter, Intensity and kinetic energy. For this case the inten-

sity and viscosity was selected, as suggested by Fluent 6.3 for k − ε and k − ω models.

However,it was kept same for all of the turbulence models for consistency. Usually the

intensity in experiments is kept very low, so a suitable value was taken as 2% and a value

of 2 for turbulent viscosity, this was in the the recommended range of 1-10 for the flows

which involve separation and high turbulence. It should be noted that these value were not

calculated and they were just an approximation.

Chapter 4

Governing Equations and Turbulence

Modelling

4.1 The Navier-Stokes Equations

4.1.1 Background

The Navier-Stokes (N-S) equations has a vital role in Fluid Dynamics. These equations

are partial differential equations and exact solution of these equations has not been found

yet. Thus these equations are solved numerically and this is the essence of CFD. A brief

overview is described in the this section.

A lot of literature is available on the basic theory of N-S. The main idea begins from the

three basic conservation laws, viz. conservations of mass, conservation of momentum and

the conservation of energy. Altogether, these equations form a set which is then referred

to as Navier-Stokes equations. There are six unknowns in these equations which are: three

components of velocity, the density, the pressure and the temperature. Conservation equa-

tions are five (one for mass, three for momentum and energy), so one additional equation

is provided by means of equation of state which will be discussed shortly. The problem

36

4.1. The Navier-Stokes Equations 37

formerly arise due to the unavailability of variables is known as the closure problem which

is solved by introducing the equation of state.

4.1.2 Conservation of mass: continuity equation

The continuity equation is based on the assumption that the mass is conserved. This is done

by considering a control volume fixed in space through which mass is conserved. So the

principal states that: The net mass flow out of the control surface must be equal to time rate

of decrease of mass inside the control volume.

Mathematically it can be written as,

∂

∂t

∫V

ρdV +

∫S

V.dS = 0 (4.1.1)

where,ρ is the density, V is the volume of the fluid, V is the velocity of the fluid and dS is

the differential area of the control surface.

4.1.3 Conservation of momentum

The conservation of momentum has the basic principle of second law of motion behind it

that is F = ma applied to a fixed volume in space. This force is the net force acting on the

fluid element and it is the sum of body and surface forces.

This can be mathematically written as,

∂

∂t

∫V

ρVdV = −∫S

p.dS−∫S

ρVV.dS +

∫S

τ .dS +

∫V

ρgdV (4.1.2)

where τ represents here the viscous stresses. This shows that the change of momentum

inside the control volume is equal to the net change of flux through the surface of the

control volume, change of momentum due to stresses and the change of momentum due to

the body forces (gravity, magnetic etc).

38 Chapter 4. Governing Equations and Turbulence Modelling

4.1.4 Energy equation

The energy equation is based on the first law of thermodynamics i.e. energy is conserved.

The energy equation is given as:

∂

∂t

∫V

ρEdV = −∫S

pV.dS−∫S

ρEV.dS +

∫S

τV.dS−∫S

qV.dS +

∫V

V.ρgdV

(4.1.3)

where E is the total energy per unit volume.

This equation implies that the net energy variation inside the control volume is the sum of

the flux of total energy, the heat flux q and the work done by the stresses and volumetric

forces. The heat flux q can be calculated using Fourier’s law

q = κ∇T (4.1.4)

here κ is the thermal conductivity of the fluid and∇T is the temperature gradient.

4.1.5 Closure problem

The above equations form a set of five which are called the N-S equations. The system of

equations can be closed by defining the equation of state

p = ρRT (4.1.5)

where R is the Gas constant (J/kgK). Its value is 287.1 J/kgK for air. There also exists one

more relation between pressure and the total energy via equation:

p = (γ − 1) ρ

(E − v2

2

)(4.1.6)

The only thing left now is the physical viscosity which is the sum of the laminar and

turbulent viscosities. The turbulent viscosity is defined using turbulence models while the

4.2. Turbulence 39

laminar viscosity is defined using Sutherland’s law:

µl = µ0

(T

To

) 32(To + 110.4κ

T + 110.4κ

)(4.1.7)

where the µ0 = 1.7894× 10−5 kg/(m.s) for a reference temperature T0 = 288.16 K and

the thermal conductivity coefficient κ can be expressed for a calorically perfect gas as:

κ =

(γ

γ − 1

)( µ

Pr

)(4.1.8)

where Pr is the Prandtl number. The governing equations can be written in compact form in

one-dimension as:∂

∂t

∫V

WdV +

∫S

[F−G].dS =

∫V

HdV (4.1.9)

where

W =

ρ

ρu

ρv

ρw

ρE

,F =

ρv

ρvu+ pi

ρvv + pj

ρvw + pk

ρvE + pv

G =

0

τxi

τyi

τzi

τijV + q

,H =

ρgxi

ρgyi

ρgzi

V.ρg + ρq

(4.1.10)

where, V = [u, v, w] and g = [gx, gy, gz].

4.2 Turbulence

4.2.1 Turbulence at a glance

No flow in this world is perfectly smooth and turbulence is always present in nature in every

flow to some extent. Generally, high speed flows are turbulent in nature and thus, an under-

standing of turbulence is very important in order to fully understand flow physics. In the


present thesis the main motivation is to study the base flow buffeting phenomenon which is

obviously turbulent. This is because that flow has a lot of turbulent energy when separation

occurs, and this can cause severe effects upon an object like a rocket if proper measures are

not taken to avoid or reduce it. This has been already experienced in the Ariane5 Flight 501.

This section is mainly on the theory behind turbulence and its modelling.

Even today scientists have not developed a full understanding of this complex flow of na-

ture. Computationally turbulence is more difficult to predict accurately as well. Therefore,

modelling approaches are adopted generally. Turbulence has different scales and energy

dissipation. The length scales are called “Turbulent Length Scales”.

Turbulent flows are usually comprised of eddies. these eddies can be as large as the the

size of the characteristic length of the body (e.g diameter in case of sphere) or they can be

very very small (just a few mm). These all eddies contain turbulent kinetic energy which

depends on their length scales. This turbulent energy can be best understood through energy

cascade diagram. Fig 4.1 shows the kinetic energy versus the wave number in space. For a

subsequently high Reynolds number the energy level can be split into three main domains.

The largest of these energy scales correspond to low wave numbers and are called as Integral

length scales or Energy-containing scales. Since the Reynolds number is very large based on

the largest eddy size l there is no role of viscosity as such in the whole cascade. However,

the cascade stops when the eddy size becomes so small that Re becomes of the order of

unity. This is the point where viscous effects become important. At high levels of wave

numbers (or low Re) eddies begin to dissipate energies. This range corresponds to the

Dissipative scales where the eddies dissipate energy by the action of viscosity. The region

is also commonly known as the Kolmogrov scale and the length of the eddies is associated

as ηk. There is one more important region that lies in between the energy-containing scales

and Kolmogrov scales which is called the sub-Inertial region. It has the intermediate energy

as compared to the two extreme scales, however, the length scale corresponds to the Taylor

micro scales λtay. In this region the energy is proportional to k−5/3. A few important things

4.2. Turbulence 41

Figure 4.1: A schematic of a typical turbulent kinetic energy spectrum for homogeneousturbulence plotted with logarithmic scales [26]

need concentration here. Firstly, each eddy has a turn-around time which is the time of their

survival (after that they break-up into smaller eddies) [5]. So in large eddies three types of

scales can be defined:

• Characteristic length , l

• Characteristic time scale, τ ≈ lu

• Energy Scale ε ≈ u3

l

For a statistically steady flow these conditions must match the dissipation rate of energy

at the smallest scales. Otherwise, there must be some accumulation at some intermediate

scale. The rate of dissipation of energy at the smallest scales is given as:

ε ∼ νSijSij (4.2.1)

where Sij is the rate of strain associated with the smallest eddies, as Sij ∼ v/η, where

v is the velocity of small eddies and η is the length scale of small eddies, so that,

ε ∼ ν(v2/η2

)(4.2.2)


Ideally this rate must match with the energy of the large scales for a statistically steady

flow which implies,

u3/l ∼ ν(v2/η2

)(4.2.3)

but as for small Re of the order of unity vη ∼ 1, the following results are obtained

η ∼(ν3/ε

)1/4 (4.2.4)

and

v ∼ (νε)1/4 (4.2.5)

Internal and external heat transfer mechanisms involving wall friction, energy losses,

increased mixing and separated rate, reduction in tendency to separate and reduction in

pressure drag due to small recirculation zones are some of the practical consequences of

turbulence.

4.3 Reynolds Averaging

Although modern computer generations have solved many problems in CFD, but still sci-

entists and engineers are not fully confident in trusting the results of CFD. The reason is

that the Direct Numerical Simulation (DNS) is the highest possible technique which can

give us everything (from Large to Kolmogrov scales). But we still have to wait 10-20 years

to get more powerful generation of computers to simulate DNS with less computational

effort. LES is a good alternate but still expensive. The Reynolds Average Navier Stokes

(RANS) provides a reasonable solution which gives quite good results if used intelligently.

The advantage of using RANS is that engineers are mostly interested in mean quantities

and using RANS the governing equations are simplified to solve only for the mean flow.

Another computational advantage is that even in unsteady flows the time step is determined

4.3. Reynolds Averaging 43

by the global unsteadiness (time stepping) rather than by the turbulence itself.

4.3.1 Mean and fluctuating parts

The turbulent flow quantities are usually described as the sum of the mean and fluctuating

quantities. This can be written as

φ = φ+ φ′ (4.3.1)

where φ is any flow parameter, velocity, pressure, density etc.

This substitution is then applied to the instantaneous Navier-Stokes equations to obtain

RANS. The formulations are given below:

For incompressible flow the continuity equation will be written as

∂ρ

∂t+∂ρui∂xi

= 0 (4.3.2)

and the momentum equation,

ρDuiDt

= − ∂p

∂xi+

∂

∂xj

[µ

(∂ui∂xj

+∂uj∂xi− 2

3δij∂ui∂xi

)]+

∂

∂xj

(−ρu′iu′j

)(4.3.3)

For variable density flows ρ = ρ+ ρ′ the continuity equation can be written as

∂ρui + ρu′i∂t

= 0 (4.3.4)

Now Favre-averaging is applied with density-weighted average, u = u + u′, where u′ 6= 0

and u′ = u− u. Thus, ui = ρuiρ

and ρu′i = 0, so

ρui = ρ(u+ u′) = ρu (4.3.5)


The RANS will remain the same except that the ui is replaced by ui. This implies,

The continuity equation∂ρ

∂t+∂ρui∂xi

= 0 (4.3.6)

and the momentum equation

ρDuiDt

= − ∂p

∂xi+

∂

∂xj

[µ

(∂ui∂xj

+∂uj∂xi− 2

3δij∂ui∂xi

)]+

∂

∂xj

(−ρu′iu′j

)(4.3.7)

The stress terms form the closure problem which are modelled. How they are modelled?

The answer to this is discussed in next section.

4.3.2 Boussinesq approximation

The Boussinesq assumption is the simplest method to model the stresses in RANS. It uses

the assumption that the Reynolds stresses are related to the mean velocity gradients through

the equation:

−ρu′iu′j = µt

(∂ui∂xj

+∂uj∂xi

)(4.3.8)

where µt is the turbulent viscosity or eddy viscosity. It has been used in Spalart-

Allmaras (S-A) and k − ε models. For S-A one extra equation for turbulent viscosity is

used, while for k − ε two equations each for k and ε are solved. Thus, Boussinesq is a zero

equation model. The model has, however, poor prediction of stresses in turbulent channel

flows and predicts dissipative behaviour in transitional flows.

4.3.3 The Spalart-Allmaras model

The one-equation models are based on the Boussinesq approximation. They determine the

Reynolds stresses via the Boussinesq hypothesis and then solve the RANS equations for

the mean quantities. The Spalart-Allmaras model was developed by P.Spalart [24]. It is a


one equation model for the eddy viscosity. Originally, SA model is for the low Reynolds

number flows where the boundary layer is fully resolved. In Fluent 6.3, wall-functions are

used in regions where the mesh resolution is not fine enough. The main equations for the

model consists of the the turbulent production and destruction terms. The equation for S-A

model can be written as [3]:

Dν

Dt= Cb1 (1− ft2) Sν

+1

σ

(∇.(ν + ν)∇ν + Cb2(∇ν)2

)+

(Cw1fw −

Cb1κ2

ft2

)(ν

d

)2

+ ft1∆V2 (4.3.9)

The terms on the right hand side represents the eddy-viscosity production, conservative

dissipation, non-conservative dissipation, the near-wall turbulence destruction, transition

damping of production, and transition source of turbulence. The ν is the laminar viscosity

and d is the distance closest to the wall. The turbulent viscosity is given as:

νt = fwν (4.3.10)

Rest of the constants and coefficients are given in Appendix B.1.

Wall boundary conditions

The initial value of ν is usually taken as 10 % of the laminar viscosity. The same value

is used at inflow boundaries. At outflow boundaries, it is extrapolated from the interior

of the computational domain. At solid walls, it is appropriate to set to ν = 0 and hence

νt = 0. When the mesh is fine enough to resolve the laminar sublayer, the wall shear stress

is obtained from the laminar stress-strain relationship: uuτ

= uτyν

. If the mesh is too coarse

to resolve the laminar sublayer, it is assumed that the centroid of the wall-adjacent cell falls

within the logarithmic region of the boundary layer, and the law-of-the-wall is employed:


uuτ

= 1κ

lnE(uτyν

). where u is the velocity parallel to the wall, uτ is the shear velocity, y is

the distance from the wall, κ is the von Karman constant (0.4187), and E = 9.793.

4.3.4 The k − ε model

The k − ε model is a two equation model solving one equation for turbulent kinetic energy

k and one for the energy dissipation ε. It was originally developed by Launder and Spalding

[12] and then modified by Shih et al as realizable [23]. Another version of this model

was presented by Yakhot [4] as RNG (Renormailization Group). Fluent 6.3 suggests that

the realizable usually gives better prediction for the flows involving high pressure gradient

flows, recirculating and high gradient flows and flows involving round and planar jets. The

general equation for the k-ε model is given as

∂

∂t(ρk) +

∂

∂xi(ρkui) =

∂

∂xj

[(µ+

µtσk

)∂k

∂xj

]+ Pk + Pb − ρε

∂

∂t(ρε) +

∂

∂xi(ρεui) =

∂

∂xj

[(µ+

µtσε

)∂ε

∂xj

]+ C1ε

ε

k(Pk + C3εPb)− C2ερ

ε2

k+ Sε (4.3.11)

In these equations the right hand side represents the conservative diffusion, eddy-viscosity

production and dissipation, respectively and in the ε equation the term Sε denotes the user-

defined source term. The rest of the terms are given in Appendix B.2.

The realizable model

The realizable has given the name because it satisfies certain mathematical constraints on

the Reynolds stress compatible with the turbulent flow physics. The model provides the

eddy viscosity modification by providing a new term called Cµ and it also provides a new

formulation of ε based on the dynamic equation of the mean-square vorticity fluctuation.

One demerit of this model is that it is not suitable for multi-zones (involving one moving


and one stationary) due to the production of non-physical turbulent viscosities.

Enhanced Wall Treatment

The enhanced wall treatment was tuned in to provide wall-function. As the mesh was not

very fine near the wall to capture the y+ of 1, this function was enabled to solve the laminar

sub-layer. However, Fluent 6.3 suggests that there is nothing to worry about even if it is in

the viscous sub-layer where y+ < 4 to 5.

4.3.5 The k − ω model

The k − ω model solves one equation for k and another for specific dissipation rate ω. The

model used in this thesis was standard k − ω by Wilcox [27]. It had been used because it

was found good for mixing layers, jets and free shear flows. The transport equation for this

model is,

∂k

∂t+ uj

∂k

∂xj= τij

∂ui∂xj− β∗kω +

∂

∂xj

[(ν + σ∗νT )

∂k

∂xj

]∂ω

∂t+ uj

∂ω

∂xj= α

ω

kτij∂Ui∂xj− βω2 +

∂

∂xj

[(ν + σνT )

∂ω

∂xj

](4.3.12)

The constants have been given in Appendix B.3. The terms on the right hand side represent

conservative diffusion, eddy viscosity production and dissipation, respectively.

Chapter 5

Numerical Methods

5.1 Introduction

This section is not explicitly written to describe the common numerical methods used in

CFD, rather it includes the numerical methods used within the simulations performed for

this thesis.

5.2 Density based solver

5.2.1 The Roe-FDS solver

Fluent 6.3 has three kind of density-based (compressible) solvers, Roe-FDS (Flux Dif-

ference Splitting), Advanced Upstream Splitting Method (AUSM) and low-diffusion Roe-

scheme (for LES only). These methods are used to calculate convective type fluxes. Here,

for the sake of brevity only Roe-FDS will be discussed in detail. Beginning from the com-

pact form of the governing equations which is:

Γ∂

∂t

∫V

QdV +

∮[F−G] .dA =

∫V

HdV (5.2.1)

48

5.2. Density based solver 49

where Q = [p, u, v, w, T ]T and Γ is the preconditioning matrix given as

Θ 0 0 0 ρT

Θu ρ 0 0 ρTu

Θv 0 ρ 0 ρTv

Θw 0 0 ρ ρTw

ΘH− 1 ρu ρv ρw ρTH + ρCp

(5.2.2)

where Θ =(

1U2r− ρT

ρCp

), ρT is the time derivative of density and Cp is the specific heat at

constant pressure. Ur is the reference velocity. According to the Roe scheme the inviscid

flux vector F in Eq. 5.2.1 is calculated through upwind flux-difference splitting scheme.

The reference velocity Ur appearing in the above equation is chosen locally such that the

eigenvalues of the system remain well conditioned with respect to the convective and diffu-

sive time scales.

The resultant eigenvalues of the preconditioned system Eq. 5.2.1 are given as,

u, u, u, u′ + c′ u′ − c′ (5.2.3)

where u′ = u (1− α), c′ =√α2u2 + U2

r , α = (1− βU2r ) /2 and β =

(ρP + ρT

ρCp

). For an

ideal gas β = (γRT )−1, so as Ur = 0, α = 0, at sonic speeds and the eigen values adapt

the form u± c. However, if Ur → 0 then α → 0.5 and then all the eigen values become of

the order of u. For constant-density flows, however, the value of α = 0.5 and it becomes

independent of Ur. As said before, the flux appearing in Eq. 5.2.1 is calculated through

upwind differencing scheme. The upwinding means that the information is propagated in

the direction speed and direction of the eigen values. Now the flux vector F is evaluated at

each cell face and split according to the direction of the eigen values. The flux at each face

can be written as,

50 Chapter 5. Numerical Methods

F =1

2(FR + FL)− 1

2ΓA (UR −UL) (5.2.4)

where A, is the matrix of the eigen values, U is the matrix of primitive variables that is

U = [p, u, v, w, T ]T . The subscripts L and R indicate the left and right hand side of the

cell-face. The matrix A, as it is diagonalizable, can be given as:

A = KΛK−1 (5.2.5)

where K is the matrix of eigen values in the diagonal. K matrices are the eigen vectors.

Also, A = ∂F∂U

.

If 1U2r

= ρp, where ρp is the density derivative with respect to pressure, this condition

becomes non-preconditioning and this reduces Γ to ∂F/∂U.

Eq. 5.2.4 can be viewed as a second-order central difference plus an added matrix dis-

sipation. This term is responsible for producing an upwinding of the convected variables,

and of pressure and flux velocity in supersonic flow and also provides the pressure-velocity

coupling. This is desired for stability and efficient convergence of low-speed and incom-

pressible flows.

5.2.2 AUSM

Th AUSM scheme employs flux-vector splitting rather flux-difference splitting. AUSM

stands for Advection Upstream Splitting Method. AUSM provides fine resolution of shocks,

and no oscillations for discontinuities. But it is a new scheme and Roe is recommended for

most of the cases that’s why it was not used in the calculations.

5.3. Discretisation schemes 51

5.2.3 Low-diffusion Flux Difference Splitting

This option is available only for LES simulations. It has been used employing to lower the

dissipation in LES. A mixed scheme with central difference and second-order upwind (Roe-

FDS) is adopted with low-Mach preconditioning. Fluent 6.3 has the following suggestions

about this scheme.

“The low diffusion discretisation must be used only for subsonic flows. For high Mach

number flows, you should switch to the second-order upwind scheme. The low diffusion dis-

cretisation is only available with the implicit-time formulation (dual-time-stepping). When

running LES with the explicit time formulation, the second-order upwind scheme will be

required.”

5.3 Discretisation schemes

5.3.1 Spatial discretization

In spatial discretisation Fluent 6.3 calculates the cell faces values through the cell centre

values. This is accomplished through the upwinding scheme. Fluent 6.3 has different types

of upwinding schemes. Upwinding means that the cell faces values are computed from the

direction of flow of the upstream or “upwind” velocity. In the compressible flow cases this

direction depends upon the direction of the characteristic eigen values as discussed before.

In Fluent 6.3 there are various upwind schemes like first-order, second order, power law and

QUICK (Quadratic Upwind Interpolation for Convective Kinetics). In this thesis, only first

and second order schemes were employed and the reason of not using higher order schemes

was simply the time constraint.


First order upwind

In First order upwind scheme the values at the cell faces are identical to the cell centre value

at the upstream. This is because the face values are determined through the assumption that

the cell centre values is average and remains same throughout the cell.

Second order upwind

In this case the flux is computed using cell centred values which itself obtained through

Taylor series expansion. This expansion makes the flux high order (2nd order truncation

error). The flux is computed using the formula:

φf = φ+∇φ.~r (5.3.1)

where, φ is the value of the flux at the cell centre, ∇φ is the flux gradient in the upstream

cell and ~r is the displacement vector from the upstream cell centroid to the face centroid.

Under relaxation factors

The transport equations (energy and turbulence models) are solved in Fluent 6.3 using an

iterative approach. This evokes a new term called the under relaxation factor. Simple to

understand, in an iterative procedure it is usually desirable that the proceeding value of a

term decreases from the previous value in subsequent iterations and converged to a desired

level of accuracy. However, if this is not the case, the value would increase and solution

could diverged. These URFs avoids the the solution to diverge. Too small values can cause

the solution to converge very slowly, while too much large values can cause oscillations or

divergent iterative solutions so its a game of experience to use appropriate URFs.


5.3.2 Temporal discretisation

When a pressure based or density based solver is selected in Fluent 6.3 it asks for Im-

plicit(only for pressure based) or explicit independent of the steady or unsteady solver. This

means that the solver is still based on time-marching calculations (based on Global time

stepping). This is explained further.

Explicit

The explicit time stepping evaluates the flux at the current time level. This can be shown as:

φn+1 = φn + ∆tF(φn) (5.3.2)

∆t is the time step. It depends on the CFL (Courant-Friedrichs-Lewy) condition which

can be given as In Fluent 6.3 3-stage Runge Kutta scheme is used for explicit time stepping

in steady case while a 4th order Runge Kutta scheme is used for explicit case for an unsteady

case.

∆t =2CFL.V

Σ(λmaxf Af )(5.3.3)

where the subscript f indicates the face values, λmax is the maximum of the eigen values

in Eq. 5.2.3. It is desirable for the stability of the solution that all the cells in the domain

must be of the same time-step, which is the minimum of all the local time steps within a cell.

This method is called Global Time Stepping. Fluent 6.3 suggest that the explicit scheme

is more stable and accurate than implicit because it is computationally less expensive and

can be used to capture the transient behaviour of shocks and moving waves. The main

difference between the steady explicit and unsteady explicit is of the local and global time

stepping respectively.


Implicit time stepping

The Implicit time stepping technique is not as straightforward as the explicit scheme but

one advantage of this scheme is that it is unconditionally stable. It is an iterative procedure

and require a lot of computational power. This is often called a dual-time stepping because

it is simultaneously dealing with two types of time derivatives in the governing equation Eq.

5.2.1. This is given as:

∂

∂t

∫V

WdV + Γ∂

∂τ

∫V

QdV +

∮[F−G] .dA =

∫V

HdV (5.3.4)

where, τ is the pseudo time derivative. In steady simulation, the convergence would mean

that the solution is converged to steady state or τ → ∞. t denotes the physical time. In

this equation the time derivative can be discretised using first or second order backward

Euler schemes. It is important to note that the τ is determined by the CFL condition. The

pseudo-time-derivative is driven to zero at each physical time level by a series of inner

iterations using either the implicit or explicit time-marching algorithm. Throughout the

(inner) iterations in pseudo-time, W n and W n−1, where super script n denotes time level,

are held constant andW k is computed fromQk. As τ →∞, the solution at the next physical

time level W n+1 is given by W (Qk), where k is any time step. Table 5.1 summarizes the

density based solver attributes available in Fluent 6.3.

Comments

In this thesis final decision was for the explicit formulation. The details are evident from

Table 5.2, where the cases have been checked for inviscid simulations for a physical time of

1 ms. From the table it can be seen that the implicit solver with the implicit time stepping is

expected to take a lot of time as compared to the explicit-explicit combination with the same

number of processors. This is mainly due to the reason that in explicit formulation, each


Solution Method Density Based Solver-Explicit formulation

Density Based Solver-Implicit formulation

Steady

-3-stage Runge Kutta --local time step -local time step-time-derivative precondi-tioning

-time-derivative precondi-tioning

FAS

Unsteady-ExplicitTime Stepping

-4-stage Runge- Kutta Not applicable- global time step- no time-derivative pre-conditioning- No FAS

Unsteady-ImplicitTime Stepping(dual timestepping)Firstorder

-dual-time formulation -dual-time formulation-Physical time:1st orderEuler backward

-Physical time:1st orderEuler backward

-preconditioned pseudo-time derivative


-inner iteration: explicitpseudo-time marching, 3-stage Runge-Kutta

- inner iteration: implicitpseudo-time marching

Unsteady-ImplicitTime Stepping(dual timestepping)Secondorder

-dual-time formulation -dual-time formulation-Physical time:1st orderEuler backward

-Physical time:2nd orderEuler backward



-inner iteration: explicitpseudo-time marching, 3-stage Runge-Kutta

- inner iteration: implicitpseudo-time marching

Table 5.1: Density based solver attributes Source: Fluent 6.3 User Guide section 25.5.5


iteration represents each time step while in implicit case the solution needs a certain number

of iterations to converge within each time step and moreover, you don’t know that (if time

step size is constant) the solution may not converge within a certain time step specially for

this complex geometry case where there are hundreds of uncertainties associated with the

flow and the grid as well. Thus it was totally uneconomical to go for the implicit case. Its

worthy to note that all the simulations (both steady and unsteady) were performed on Astral

super computer which has 872 Woodcrest CPUs each with 3 GHz. The Fluent 6.3 licences

were available for 70 CPUs out of which per user limit was 16. Therefore, the unsteady

results were performed with 8 processors on Astral, considering two cases running at a time

(e.g one for inviscid and one for RANS).

Fast convergence

Fast convergence is an appreciative thing in CFD. There are certain methods in Fluent 6.3

which can be used to achieve this. A few are discussed below.

Local time stepping It is a method by which the solution at each control volume is ad-

vanced in time with respect to the cell time step, defined by the local stability limit of

the time-stepping scheme (through the Courant number). This method is meaningless

for unsteady simulations.

Residuals smoothing Implicit residual smoothing is employed in Fluent 6.3. It uses Lapla-

cian operator to smooth the residuals. The level of smoothing is 0.5 which was used

as default for explicit cases.

Full-Approximation Storage (FAS) multigrid This option is available with density-based

solvers. The grids are made coarser by merging the cells surrounding a common node.

This method accelerates the convergence, and fortunately the unevenness in the cell

shape does not have an effect upon the discretisation.


Cas

eD

ensi

ty-

base

dso

lver

Tim

efo

r-m

ulat

ion

CFL

Tim

e-st

epsi

zeTi

me

per

time-

step

Est

imat

edtim

eto

sim

ulat

e1

ms

Rem

arks

Invi

scid

Impl

icit

Impl

icit

0.8

1.30

6e-0

827

2sec

5742

hrs

=23

9.25

days

long

est

Invi

scid

Exp

licit

Exp

licit

0.8

8.7e

-09

3.22

sec

the

itera

tions

will

take

102

hrs=

4da

ysto

sim

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e1

mse

c

Per

itera

tion

spee

dw

asfa

st

Invi

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Exp

licit

Impl

icit

0.6

1.30

6e-

08(b

yde

faul

t)

101s

ec21

48hr

s=89

.5da

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sim

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mse

c

Con

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was

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Kif

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fully

(with

the

pre-

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atea

chtim

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ep

Invi

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Exp

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Impl

icit

0.8

1.30

6e-0

898

.520

95hr

s=87

days

tosi

m-

ulat

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ms

Con

verg

ence

was

fast

but

itson

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Kif

itco

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ges

atea

chtim

est

ep

Tabl

e5.

2:Ti

me

stat

istic

sfo

rinv

isic

dca

sew

ith8

proc

esso

rs

Chapter 6

Results and Discussion

6.1 Inviscid and RANS Simulations

At the beginning of the thesis it was planned to conduct detailed unsteady simulation in-

cluding DES and ILES of the vehicle base-flow. But as the Table 5.2 in Chapter 5 indicates

that even the inviscid simulations were time consuming, the simulations were restricted to

RANS and steady inviscid cases. The results obtained from these simulations are the topic

of discussion of the present chapter.

6.2 Flow characteristics

At launch the Vulcain engine is responsible to produce an enormous amount of thrust. The

transonic free-stream flow from the outside of the main engine interacts with the high ve-

locity supersonic flow forming a shear layer. These shear layers produce high side force on

the nozzle impeding the control of the rocket. This could also generate three dimensional

flow separation inside the nozzle as well. This phenomenon, as seen in wind tunnel tests,

is unsteady in nature. It has been observeds in these results that even though RANS and

inviscid captured the phenomenon quite well (as will be discussed soon). The flow separa-

58

6.3. Quantitative comparisons 59

Turbulence model No of cells

k − ε 3851386k − ω 3882606S-A 3882606

Table 6.1: Adapted grid information

tion can be seen in Fig 6.1 and Fig 6.2 where the first one shows the separated flow along

the plane in the axial direction through the centre of the rocket. The second one shows the

stream traces on the lateral planes. Vorticity magnitude is also shown along these planes

indicating the complex nature of the flow Fig 6.3.

6.3 Quantitative comparisons

Before discussing the quantitative comparisons it should be noted that the results of adapted

grid are also shown here. Fluent 6.3 and older versions have the capability to refine or

coarsen the grid on the regions of interest where high accuracy or flow details are required.

Taking this advantage the grid was adapted in the base flow region. This was done using the

contours of vorticity magnitude which was a plausible choice. It is interesting to note that

because different models did not have the same accuracy level therefore adaptation gave

different number of grid points in each case. This is as shown Table 6.1.

Fig 6.4 shows the region of adaptation. Here only one grid is shown because it was

similar in the other two cases as well. Note the regions of high vorticity value (of the order

of 1004) near the walls have been adapted. Fluent use hanging-nodes to add the cells (also

called as non-conformal meshing).

60 Chapter 6. Results and Discussion

Figure 6.1: Separation in the axial direction

Figure 6.2: Separation indicated by stream lines in the lateral (cross-stream) planes

Figure 6.3: Contours of vorticity magnitude in the lateral (cross-stream) planes


Figure 6.4: Adapted grid region

6.3.1 Pressure Coefficient

The only experimental data available (for steady measurements) was for pressure taps at

four positions which are shown in Fig 6.5. The data obtained for these taps for inviscid and

RANS cases with the original coarse grid and the adapted one are shown in Tables 6.2 and

6.3. Its important to note that Cp values were not consistent for a particular model, i.e, for a

single model the P1 and P2 were in good agreement with the experiment while for the same

model the results for P3 and P4 are far away. However, for P3 and P4 the results are not

more than 10% for both coarse and adopted grids except for the inviscid case where they

are more than 50%. Therefore, the judgement is based on averaged values taken for these

four pressure taps. This shows that for the coarse grid k − ω model is better and near to the

experimental values while for the adapted grid the S-A model results are better. This could

be due to the reason that S-A gives optimal performance when the mesh is fine enough to

resolve the laminar sublayer as mentioned in Chapter 4 Sec 4.3.3.


Figure 6.5: Position of pressure taps used in the experiment [30]


Coa

rse

grid

Cp-

Exp

Cp-

Invi

scid

%er

ror

Cp-

S-A

%er

ror

Cp-k−ε

%er

ror

Cp-k−ω

%er

ror

P1-0

.168

-0.1

661.

78-0

.052

468

.8-0

.077

653

.8-0

.109

35.1

1P2

-0.1

15-0

.025

78.2

-0.0

539

53.1

5-0

.085

225

.9-0

.123

6.96

P3-0

.260

-0.1

2950

.38

-0.2

641.

5-0

.261

0.38

-0.2

502

3.77

P4-0

.25

-0.2

8514

-0.2

676.

8-0

.257

2.8

0.25

120.

48

Ave

rage

erro

rval

ues

36.0

932

.56

20.7

211

.58

Tabl

e6.

2:Pr

essu

reco

effic

ient

sat

four

posi

tions

with

coar

se-g

rid


Adapted

grid

Cp-E

xpC

p-S-A%

errorC

p-k−ε

%error

Cp-k−ω

%error

P1-0.168

-0.08847.62

-0.02982.73

-0.06660.71

P2-0.115

-0.08525.7

-0.03966.09

-0.06444.35

P3-0.260

-0.2703.80

-0.2513.62

-0.2494.23

P4-0.25

-0.2729.8

-0.25040.15

-0.22510.00

Averaged

errorvalues21.73

38.14829.83

Table6.3:

Pressurecoefficients

atfourpositionsw

ithadapted

grid


6.3.2 The recirculation bubble

The recirculation bubble forms where there are negative velocity gradients in the flow field.

So that the flow retards rather than accelerate and this is not desirable. Fig 6.6 shows the

results of the recirculation regions obtained via four models in Fluent 6.3 on the left side

at x = -0.03 m. It can be seen that with k − ω, the velocity direction changes twice thus

indicating a double recirculation region. Fig 6.7 shows the recirculation zone on the right

side x= 0.03 m (looking from the top) of the base region which indicates a similar pattern.

The region after recirculation where the line was terminated must have shown similar values

for velocities, but as can be seen in Fig 6.22 in Sec 6.4, the plumes are different for all the

cases. This is reason for this high magnitude of difference in the values of the axial velocity.

It is evident that the S-A gives higher values which indicates that the plume from the booster

is expanding more rapidly than the other two cases.

Effect of grid adaptation upon recirculation bubble

Fig 6.8 shows the recirculation region calculations for the adapted grid. The results are

compared with the coarse grid. It can be seen that the difference is 15 m/s in k − ω while

25 m/s for k − ε in the recirculation region. In both cases the adapted grid results are more

than the original grid results. For S-A case this difference is even smaller indicating that the

grid adaptation did not effect the recirculation in S-A model case. A similar result can be

see in Fig 6.9, where the negative velocity is more than the original grid. Ideally the results

should be symmetric but there is some difference in the profiles. However, the results from

the original grid differ almost by the same order of magnitude. The kinks have become

sharper. The difference is not very well pronounced for the S-A model as seen for the left

side region as well.


Figure 6.6: The recirculation on the left side, viewed from the top, effect of various models

Figure 6.7: The recirculation on the right side, viewed from the top, effect of various models


(a)

(b)

(c)

Figure 6.8: The recirculation on the left side (viewed from the top) of the base region, effectof grid adaptation with (a) k − ω, (b) k − ε and (c) S-A model


(a)

(b)

(c)

Figure 6.9: The recirculation on the right side (viewed from the top) of the base region,effect of grid adaptation with (a) k − ω, (b) k − ε and (c) S-A model


Figure 6.10: Sections along the base region

6.3.3 Turbulent kinetic energy

The Turbulent Kinetic Energy (TKE) was compared using k − ε and k − ω only as S-A

model does not employ TKE. A general form of this parameter is given as:

k =1

2(u′u′ + v′v′ + w′w′) (6.3.1)

For a comprehensive sort of comparison the plots were shown by cutting different sections

along x axis at constant z locations. The sections are shown in Fig 6.10. Fig 6.11 and 6.12

show the plots of TKE at different sections along these sections.

Figs 6.11 and 6.12 shows some interesting features of the flow. It should be noted that

unfortunately there was no experimental data available to compare with, so these results did

not have have a solid ground to make conclusion. But whatever judgement was made, truly

depended upon the plausible physics , logic and the overall performance of the turbulence

models. It can be seen that moving across the regions where the nozzle begins the peaks are


symmetric on both sides. This behaviour is continued in both the models until the middle

z = 0 is reached. After that, due to complex nature of the flow, asymmetry is observed.

Results of TKE with adapted grid

The results for the adapted grid are shown in Figs 6.13 and 6.14. These are along the main

regions of the were the flow was adapted, i.e along the four regions.

These graphs show a comparison between the adapted grid and the original coarse grid.

Fig 6.13 shows that in half of the region the original grid has higher value while for half of

the region the original grid values are higher. The behaviour is somehow different in case of

k−ω where the adapted grid remained dominated over the coarse gird results at all sections.

The asymmetry in the profile can still be seen in both the cases at the sections after z = 0

m.

Turbulent viscosity

Figs 6.15 and 6.16 show the effect of different models on the turbulent viscosity. As it is

known that the S-A model solves an equation for turbulent viscosity it has better perfor-

mance as compared to the k − ε and k − ω models. However, the same problem persists

from sections z = 0 to z = −0.036m. It can be concluded that S-A model is better than the

other two models as far as the symmetric behaviour is concerned.

Wilcox [28] in his book mentioned that the equations involving the turbulent dissipation

rate ε did not give good results for separated flows. This is true for this case as well where

k − ε showed the smeared and asymmetric behaviour of TKE. The TKE equation is used

for the evaluation of the eddy viscosity therefore the behaviour is similar for the turbulent

viscosity profiles, which gives the reason for its asymmetry as well.


(a) (b)

(c) (d)

Figure 6.11: Set A: Turbulent kinetic energy behaviour for RANS models at (a) z= 0.036m , (b)z= 0.027 m , (c)z= 0.009 m (d)z= 0


(a) (b)

(c) (d)

Figure 6.12: Set B:Turbulent kinetic energy behaviour for RANS models at (a) z= -0.018m, (b)z= -0.026m , (c)z=-0.045m (d)z=-0.06m


(a) (b)

(c) (d)

Figure 6.13: TKE plots for k−εmodel with adapted grid: comparison with original (coarsegrid) is also shown


(a) (b)

(c) (d)

Figure 6.14: TKE plots for k−ω model with adapted grid: comparison with original (coarsegrid) is also shown


(a) (b)

(c) (d)

Figure 6.15: Set A : Turbulent viscosity distribution at various sections in the base regionusing with all of the three turbulence models at (a) z = 0.036 m , (b)z = 0.027 m , (c)z =0.009 m (d) z = 0


(a) (b)

(c) (d)

Figure 6.16: Set B : Turbulent viscosity distribution at various sections in the base regionwith all of the three turbulence models (a) z = -0.018m , (b) z = -0.026m , (c)z = -0.045m(d) z= -0.06m


6.3.4 Axial and lateral velocity profiles

As the behaviour of TKE and turbulent viscosity was unexpectedly asymmetric, it was

found plausible to see the velocity profiles because they are involved in the NS equations

themselves and not merely an output of the turbulence models. The plots were taken along

the same sections as before. They were were taken for z-velocity (axial velocity) and x-

velocity (lateral velocity).

Axial velocity profile

The axial velocity profile Figs 6.17 and 6.18 show a considerably symmetric behaviour for

all of the turbulence models. Note that the graphs represent some values at the centre (from

z = 0 to z = -0.036m) because they are at locations where the sections intersected the main

Vulcain nozzle. Moreover, Tecplot also joined some of the points blindly resulting in curves

that could be misleading. These results, however, are not so important from the thesis point

of view. The main region of interest is the base region. For all the models the results are

symmetric. However, it is quite difficult to suggest that which model is behaving better

because all models have a similar behaviour. But it should be noted that the k − ε model is

giving higher axial velocity in all cases in comparison to the other two models.

Lateral velocity profiles

The profiles for the lateral velocity were also taken for the lateral velocity (x-direction

velocity) to understand the behaviour of the models Fig 6.19 and 6.20. The plots show

a mirror symmetry about the centre. This is obvious because the positive x-coordinate is

towards the right hand side. The results are in good agreement, however, the performance

of S-A is different in from the rest.


(a) (b)

(c) (d)

Figure 6.17: Set A : Axial velocity profiles at various sections in the base region using withall of the three turbulence models at (a) z = 0.036 m , (b)z = 0.027 m , (c)z = 0.009 m (d) z= 0


(a) (b)

(c) (d)

Figure 6.18: Set B : Axial velocity profiles at various sections in the base region with allof the three turbulence models (a) z = -0.018m , (b) z = -0.026m , (c)z = -0.045m (d) z=-0.06m


(a) (b)

(c) (d)

Figure 6.19: Set A : Lateral velocity profiles at various sections in the base region usingwith all of the three turbulence models at (a) z = 0.036 m , (b)z = 0.027 m , (c)z = 0.009 m(d) z = 0


(a) (b)

(c) (d)

Figure 6.20: Set B : Lateral velocity profiles at various sections in the base region with allof the three turbulence models (a) z = -0.018m , (b) z = -0.026m , (c)z = -0.045m (d) z=-0.06m


Turbulence model Original grid Adapted grid

Inviscid 0.3783 N/Ak − ω 0.4694 0.4726k − ε 0.4657 0.4733S-A 0.5379 0.5547

Table 6.4: Coefficient of drag for original and adapted grids

Turbulence model Total Pressure

k − ω 1.26 ×105 Pak − ε 1.31 ×105 PaS-A 1.26 ×105 Pa

Table 6.5: Total pressure at the tip of the main tank

6.3.5 Drag coefficients

The drag coefficients were calculated based on the half of the reference area mentioned in

[17]. The half area was taken because half of the vehicle was modelled. Table 6.4 shows

the values for both the grids.

6.3.6 Stagnation pressure at the tip

Although this seems quite unusual to look at the stagnation pressure at the tip of the main

tank. This is discussed here to just see any significant difference among the performance

of these models in calculating this simple parameter. Interestingly, it has been found that

k − ω model differs slightly than the other two models where the other two models predict

the same value. This is given in Table 6.5.

6.4. Qualitative comparisons 83

6.4 Qualitative comparisons

The physics becomes much more interesting when the subsonic flow merges with the su-

personic flow of the nozzle. This phenomenon was observed in all the cases, inviscid, and

the three turbulence models. Figs 6.21 and 6.22 show the flow behaviour showing both sub-

sonic region and the supersonic flow out of the nozzle. Anderson [1] has discussed in detail

these type of flows. This is observed in case of under-expanded nozzles in which the exit

pressure is greater than the outside atmospheric pressure and thus the flow wants to expand

further. When gas from nozzle exhausts into the outside stagnant air, the pressure across

the boundary must be preserved. Hence the jet boundary pressure must be equal to the free-

stream pressure throughout the length of the jet and this is evident as well from the Figures

of Mach contours shown below. As the exit pressure is less than atmospheric pressure, this

generates an expansion fan at the exit which when interacts with the free-boundary form

an oblique shock and this process is repeated leading to the formation of diamond shape

patterns. These are equally captured by the Euler and RANS models. The reason of Invis-

cid case to capture this phenomenon is because of the numerical dissipation and not only

because of that the shock formation is a viscous dissipation phenomenon.

6.4.1 Vorticity magnitude

Flow visualization can be better visualized by looking at the vorticity magnitude contours.

Fig 6.23 shows the vorticity magnitude for the RANS case. For the k − ω there is a big

region of separation adjacent to the convergent portion of the Vulcain nozzle. While for

k − ε and S-A the behaviour is not much different. However one can see that with change

in models the plume from the Vulcain nozzle becomes wider. The non-flooded contours

have been shown here for proper flow visualization. This type of flow behaviour was also

observed by Dr Ludeke in his paper [15]. However the main feature discussed there was the

plume and not the base flow itself.


(a) Inviscid

(b) k − ε

Figure 6.21: Contours of Mach number for Inviscid and k − ε cases


(a) k − ω

(b) S-A

Figure 6.22: Contours of Mach number for k − ω and S-A cases


(a)

(b)

(c)

Figure 6.23: Contours of vorticity magnitude a) k − ω, (b) k − ε and (c) S-A model


6.4.2 Stream traces (Path-lines)

The inviscid shows interesting patterns of the recirculating flow and gives an asymmetric

flow behaviour Fig 6.24. The randomness of the flow field indicates the instability of this

solver and continuous fluctuations in the residuals App C.1 Fig C.1 indicate the unsteady

nature of the flow. The spaghetti like patterns representing the big vortex core around the

Vulcain and small vortices near the protuberance on the boosters’ surface and at the end of

the boosters’ wall where diverging portion of booster nozzle starts can be seen on both sides

for all the turbulence cases. Note that the path lines are coloured by z-velocity which could

be misleading because it is taken positive along the negative z-direction. For the turbulence

models, the behaviour is sufficiently symmetric though.

Fig 6.25 shows the oil flow patterns on the symmetry plane coloured by z-velocity. The

behaviour is similar to as was observed in three-dimensional path-lines case. The flow is

again random in inviscid case. It can be seen that the shear-layer, in case of k − ω hits the

nozzle at the divergent section protrusion leaving a gap in the region adjacent to the conver-

gent region. For k − ε the path-lines are fairly attached to the nozzle. However, the swirls

are quite large in magnitude. S-A model does not show a well-pronounced swirl as the two

other models but small vorticity can be seen in all cases near the boosters’ protuberances

and their lower region.


(a) (b)

(c) (d)

Figure 6.24: 3D-path-lines coloured by z-velocity showing unstability in the flow field (a)Inviscid (b) k − ω, (c) k − ε and (d) S-A model


(a) (b)

(c) (d)

Figure 6.25: Oil flow on the symmetry surface coloured by z-velocity (a) Inviscid (b) k−ω,(c) k − ε and (d) S-A model

Chapter 7

Conclusions and Future Work

7.1 Conclusion

In this study our objective was to study the base flow phenomenon of the Ariane5 rocket.

Through CFD, no doubt, it was a challenging task to achieve. In this thesis the performance

of different turbulence models available in Fluent 6.3 was tested. The best-suited versions

of the models, based on the flow problem, were selected for analysis. Although, it is not

a hard and fast rule to stick to these models, other versions and variations, in constants for

example, can be tried in future.

A comprehensive grid study including modification of the grid from ESTEC was also

done. It was repaired at many places and took a considerable time to achieve a good quality

grid.

Initially unsteady simulations were run for inviscid case with first order discretisation in

space and 4th order Runge Kutta method in time. A set of experiments was performed to

examine the computational time per time step using several different time stepping methods

and it was realized that based on the current hardware, explicit scheme was taking the

least time to simulate 1ms (7 days of CPU clock), hence it was found impossible to run

an unsteady simulation using Fluent 6.3. It was concluded that explicit solver with explicit

90

7.2. Future work 91

time formulation was taking the least time (7 days for 1ms) on 8 processors of each 3 Ghz

of the 872 CPU-based supercomputer Astral to simulate 1ms. However, this objective was

not achieved due to thesis time constraints.

The steady results were based on the inviscid and RANS simulations. The performance

was checked based on the overall performance of the models as the experimental data was

limited. Quantitative results suggested that the inviscid gave the worst performance and was

not successful in capturing the flow field very well. The path lines also showed a random

behaviour and one can see the residuals which indicate the unsteady nature of the flow, thus

an unsteady simulation could give better results. Quantitative performance of the turbulence

models was difficult to judge as well. However, Cp values showed better results for k−ω in

comparison to the experimental value while for the adapted grid the S-A model performed

well, although the errors were still of the order of 20%.

It was concluded that k − ω and S-A models’ results showed less discrepancy for the

adapted grid. The adapted grid should have given much better performance with respect

to the k − ω (as it did in the coarse grid)but as it has been mentioned that the grid was

adapted with respect to the gradients of vorticity magnitude, which could cause difference

in the results. This is because every model did not capture the same level of the vorticity

magnitude.

On the basis of grid refinement study, it was concluded that for the coarse grid the k−ω

model had given better performance both when compared with the experiment as well as

when analysed on qualitative basis. The S-A model had given better performance for the

adapted grid. The inviscid model was not suitable to simulate such a highly turbulent and

massively separated flow. Therefore unsteady simulations are suitable for this case.

7.2 Future work

Nothing in this world is perfect so there is a huge margin for improvement. Beginning from

the grid, it needs a lot of modification especially with regard to geometry and booster and

92 Chapter 7. Conclusions and Future Work

nozzle regions. An unstructured grid can also be tried. For a detailed buffeting analysis,

unsteady simulations are required. This can be continued from the steady simulations done

so far. More options in turbulence models can be tried such as k − ω SST and k − ε RNG.

Discretisation schemes of higher order (third order MUSCL) can be tried as well. Con-

vergence criteria can be increased for accuracy. Performing of DES and LES simulations

for this complex flow would also be a challenging task. Further investigation with angle of

attack, study of drag coefficients and different free stream Mach numbers would also be an

interesting field of research.

Appendix A

A.1 Coordinate transformation

Body-fitted are coordinates that are stream-lined along the geometry. Body-fitted grid is

suitable for the cases involving complex geometry. This is in contrast to Cartesian (rectan-

gular) grid which is not suitable for complex geometries and usually acquired for the simple

geometries usually rectangular in shape.

The computational [ξ, η, ζ, τ ] and Cartesian coordinates [x, y, z, t] are related in 3D as:

τ = t (A.1)

ξ = ξ(t, x, y, z) (A.2)

η = η(t, x, y, z) (A.3)

ζ = ζ(t, x, y, z) (A.4)

Applying the chain rule give the following expression and substituting partial derivatives

as, for example,∂ξ

∂x= ξx

we have,

93

94 Chapter A.

∂

∂t=

∂

∂τ+ ξt

∂

∂ξ+ ηt

∂

∂η+ ζt

∂

∂ζ(A.5)

∂

∂x= ξx

∂

∂ξ+ ηx

∂

∂η+ ζx

∂

∂ζ(A.6)

∂

∂y= ξy

∂

∂ξ+ ηy

∂

∂η+ ζy

∂

∂ζ(A.7)

∂

∂z= ξz

∂

∂ξ+ ηz

∂

∂η+ ζz

∂

∂ζ(A.8)

The expressions that concatenate between the physical space and computational state should

be a partial differential equation which can be written as:

dt =dt

dτdτ +

dt

dξdξ +

dt

dηdη +

dt

dζdζ (A.9)

as,dt

dτ= 1

anddt

dξ=dt

dη=dt

dζ= 0

so,

dt = dτ (A.10)

Following the same logic as for the differential equation for time we can write for x, y and

z coordinates the differential equation as,

dx =dx

dτdτ +

dx

dξdξ +

dx

dηdη +

dx

dζdζ (A.11)

A.1. Coordinate transformation 95

dy =dy

dτdτ +

dy

dξdξ +

dy

dηdη +

dy

dζdζ (A.12)

dz =dz

dτdτ +

dz

dξdξ +

dz

dηdη +

dz

dζdζ (A.13)

In matrix form, the above equations can be written as

dt

dx

dy

dz

=

1 0 0 0

xτ xξ xη xζ

yτ yξ yη yζ

zτ zξ zη zζ

dτ

dξ

dη

dζ

(A.14)

Reversing the role of variables

dτ = dt (A.15)

dξ =dξ

dtdt+

dξ

dxdx+

dξ

dydy +

dξ

dzdz (A.16)

dη =dη

dtdt+

dη

dxdx+

dη

dydy +

dη

dzdz (A.17)

dζ =dζ

dtdt+

dζ

dxdx+

dζ

dydy +

dζ

dzdz (A.18)

In matrix form dτ

dξ

dη

dζ

=

1 0 0 0

ξt ξx ξy ξz

ηt ηx ηy ηz

ζt ζx ζy ζz

dt

dx

dy

dz

(A.19)

comparing Eq (A.14) and Eq (A.19)

96 Chapter A.

1 0 0 0

ξt ξx ξy ξz

ηt ηx ηy ηz

ζt ζx ζy ζz

=

1 0 0 0

xτ xξ xη xζ

yτ yξ yη yζ

zτ zξ zη zζ

−1

(A.20)

Appendix B

B.1 Spalart Allmaras model

The constants and the coefficients of the Spalart Allmaras model in Eq 4.3.9 are given

below:

νt = νfv1, fv1 =χ3

χ3 + C3v1

, χ =ν

ν

S ≡ S +ν

κ2d2fv2, fv2 = 1− χ

1 + χfv1

fw = g

[1 + C6

w3

g6 + C6w3

]1/6

, g = r + Cw2(r6 − r), r ≡ ν

Sκ2d2

ft1 = Ct1gt exp

(−Ct2

ω2t

∆U2[d2 + g2

t d2t ]

)ft2 = Ct3 exp

(−Ct4χ2

); S =

√2ΩijΩij (B.1)

where Ω is given as:

Ωij =1

2

(∂ui∂xj− ∂uj∂xi

)(B.2)

97

98 Chapter B.

and the constants are:σ = 2/3

Cb1 = 0.1355

Cb2 = 0.622

κ = 0.41

Cw1 = Cb1/κ2 + (1 + Cb2)/σ

Cw2 = 0.3

Cw3 = 2

Cv1 = 7.1

Ct1 = 1

Ct2 = 2

Ct3 = 1.1

Ct4 = 2

(B.3)

B.2 k − ε model

The constants and coefficients for the k − ε model Eq 4.3.11 are given as:

Turbulent viscosity

µt = ρCµk2

ε(B.1)

k Production

Pk = −ρu′iu′j∂uj∂xi

Pk = µtS2 (B.2)

where S is the modulus of the mean rate-of-strain tensor which can be given as

S ≡√

2SijSij (B.3)

Buoyancy

Pb = βgiµtPrt

∂T

∂xi(B.4)

B.3. k − ω model 99

where Prt is the turbulent Prandtl number for energy and gi is the component of the gravi-

tational vector in the ith direction. For the standard and realizable models, the default value

of Prt is 0.85. The coefficient of thermal expansion, β is

β = −1

ρ

(∂ρ

∂T

)p

(B.5)

B.3 k − ω model

The constants the k − ω model Eq 4.3.12 are given as:

α = 59

β = 340

β∗ = 9100

σ = 12

σ∗ = 12

ε = β∗ωk

(B.1)

Appendix C

C.1 Convergence history

The residuals for different cases for coarse grid are shown in Fig C.1 and Fig C.2.

(a) (b)

Figure C.1: Set A: Residual history for coarse grid, (a) Inviscid (b) k − ω

For the adapted grid the residual history is shown in Fig C.3

100

C.1. Convergence history 101

(a) (b)

Figure C.2: Set B: Residual history for coarse grid, (a) k − ε (b) S-A model

(a) (b)

(c)

Figure C.3: Residual history for adapted grid, (a) Inviscid (b) k − ω, (c) k − ε and (d) S-Amodel

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Documents

Ariane5 Transoinc Base Flow Analysis CFD