Flexible Modular Wind Tunnel for Renewables Research
1. 1
Department of Mechanical Engineering
FACULTY OF ENGINEERING AND DESIGN
Final Year Project: Flexible, Modular Wind Tunnel
Contraction for Renewables Research
Author: Anthony Man
13th
May 2020
Supervisor: Dr. Anna Young
Assessor: Professor Chris Bowen
Word Count: 11950 (Excluding tables, appendices, references, equations and captions)
2. 2
Abstract
Airflow through seventy five wind tunnel contraction geometries was analysed using
computational fluid dynamics (CFD) as part of a parametric study. These were elliptical three-
dimensional contractions. Three flow features were optimised: streamwise velocity
nonuniformity, turbulent viscosity ratio and area occupied by vortices. The flow features were
assessed at three planes: contraction outlet, test section outlet and the test section plane
equidistant between these two. It was found that a less concave contraction reduced all three
flow features, as this mitigates the growth of Görtler vortices. In addition, velocity
nonuniformity was further reduced by setting the contraction roof and floor to contract as
similarly as possible to the side walls. This reduces asymmetry, crossflows and hence improves
streamwise velocity. An optimum contraction geometry was solved using CFD and successfully
validated the parametric study results. Several phases of work were completed prior to the
parametric study, to provide an appropriate mesh and set-up settings for it. A 𝑦+
of 1 at the
contraction outlet and Menter’s Shear Stress Transport turbulence model was used for all CFD
simulations in the parametric study.
3. 3
Acknowledgements
I would like to express my appreciation to Dr. Mauro Carnevale for his constructive suggestions
in computational fluid dynamics for this project. I would also like to express my gratitude to
Professor Chris Bowen for his logistical recommendations and for assessing my work.
Special thanks should be given to Dr. Anna Young, my final year project supervisor for her
professional guidance during the entire course of the project.
Finally, I would like to thank my parents and sister for their continued support and
encouragement during my education.
4. 4
Table of Contents
List of Abbreviations.......................................................................................................................8
Nomenclature..................................................................................................................................9
1 Introduction.............................................................................................................................11
1.1 Background..........................................................................................................................11
1.2 Aim and Objectives.........................................................................................................12
2 Literature Review....................................................................................................................13
2.1 Overview .........................................................................................................................13
2.2 Nature of the Design Problem........................................................................................13
2.3 Axisymmetric Contraction Design.................................................................................14
2.4 Three-Dimensional Contraction Design........................................................................14
2.5 Swirling Flows.................................................................................................................15
3 Present Approach ...................................................................................................................17
3.1 Overview .........................................................................................................................17
3.2 Design Criteria ................................................................................................................17
3.3 Geometry Constraints.....................................................................................................17
3.4 Velocity Constraints........................................................................................................18
4 General Methods ....................................................................................................................19
4.1 Overview .........................................................................................................................19
4.2 Workflow.........................................................................................................................19
4.3 Wall Profile Geometry ....................................................................................................19
4.4 DefineEllipse and EllipseParams...................................................................................20
4.5 CAD ................................................................................................................................22
4.6 Meshing..........................................................................................................................22
4.6.1 O-Grid.........................................................................................................................22
4.6.2 Turbulent Flow....................................................................................................... 23
4.6.3 𝑦 + and Inflation Layer .......................................................................................... 23
4.6.4 Element Growth Rate.............................................................................................24
4.6.5 Boundary Layer Thickness.....................................................................................25
4.6.6 Mesh Refinement ...................................................................................................26
4.7 Setup...............................................................................................................................26
4.8 Solving............................................................................................................................ 27
5 First Pass Model.....................................................................................................................28
5.1 First Pass Geometry .......................................................................................................28
5. 5
5.2 First Pass Mesh...............................................................................................................29
5.3 First Pass Setup and Solve ..............................................................................................31
5.4 First Pass Post-Processing and Discussion ....................................................................31
5.4.1 Velocity........................................................................................................................31
5.4.2 Vortices....................................................................................................................31
5.5 Validation Model ........................................................................................................... 32
6 Second Pass Model ................................................................................................................34
6.1 Second Pass Geometry...................................................................................................34
6.1.1 Bend Addition ............................................................................................................34
6.1.2 Second Pass Contraction ...........................................................................................34
6.1.3 Second Pass Test Section ........................................................................................... 35
6.1.4 Second Pass CAD ....................................................................................................... 35
6.2 Second Pass Meshing..................................................................................................... 36
6.3 Second Pass Setup..........................................................................................................38
6.4 Second Pass Solving....................................................................................................... 39
6.5 Second Pass Post-Processing and Discussion............................................................... 39
6.5.1 Recirculation Regions ................................................................................................ 39
6.5.2 Asymmetry of Results .............................................................................................41
6.5.3 Mesh Assessment ...................................................................................................42
6.5.4 Other Results..........................................................................................................44
6.6 Mesh Convergence Study ..............................................................................................44
7 Parametric Study Contraction Sampling..............................................................................48
7.1 Input Parameter Multiplier Ranges...............................................................................48
7.2 Latin Hypercube Sampling............................................................................................48
7.3 Macros............................................................................................................................49
8 Parametric Study CFD...........................................................................................................50
8.1 Parametric Study Meshing ............................................................................................50
8.1.1 Adaptation from Second Pass Mesh..........................................................................50
8.1.2 𝑦 + Calculations.........................................................................................................50
8.1.3 Inflation Layer Calculations.......................................................................................50
8.1.4 Results.........................................................................................................................50
8.2 Parametric Study Set-Up ...............................................................................................52
8.2.1 Turbulence Model Change ........................................................................................52
8.2.2 Outlet Boundary Condition Change......................................................................52
8.3 Parametric Study Solving...............................................................................................52
9 Parametric Study Results and Discussion ............................................................................ 53
7. 7
Appendix 7 Second Pass Model Input Parameter Calculations ...............................................85
Appendix 8 Second Pass Model Inflation Layer Thickness Calculations.............................86
Appendix 9 Second Pass Model Convergence Plots .................................................................87
Appendix 10 Input Parameter Multipliers ..............................................................................89
Appendix 11 Multiplier Combinations of Samples.................................................................90
Appendix 12 DefineEllipse Modified for Parametric Study....................................................92
Appendix 13 Examples of Sampled Contraction Profiles .......................................................94
Appendix 14 InventorParams ..................................................................................................95
Appendix 15 𝑦 + Calculation for Parametric Study Models...................................................97
Appendix 15.1 Equations ........................................................................................................97
Appendix 15.2 Calculation......................................................................................................97
Appendix 16 Inflation Layer Calculation for Parametric Models...........................................99
Appendix 17 Mass Flow Rate Calculation for Test Section Outlet in Parametric Study Models
100
Appendix 18 Velocity Nonuniformity Equations................................................................... 101
Appendix 18.1 𝑦-velocity Nonuniformity.............................................................................. 101
Appendix 18.2 𝑧-velocity Nonuniformity ............................................................................. 101
Appendix 19 Contraction Outlet Velocity Nonuniformity Results.......................................103
Appendix 20 TSMP Velocity Nonuniformity Results.............................................................105
Appendix 21 Far Outlet Velocity Nonuniformity Results .....................................................107
Appendix 22 Parametric Study - 𝑀𝐻𝑏 vs 𝑀𝑇𝑏 vs 𝜙𝑥 Surface Plots ..................................... 109
Appendix 23 Parametric Study - 𝑀𝐻𝑏 vs 𝑀𝑇𝑐 vs 𝜙𝑥 Surface Plots....................................... 110
Appendix 24 Parametric Study - 𝑀𝐻𝑐 vs 𝑀𝑇𝑏 vs 𝜙𝑥 Surface Plots........................................ 111
Appendix 25 Parametric Study - 𝑀𝐻𝑐 vs 𝑀𝑇𝑐 vs 𝜙𝑥 Surface Plots........................................112
Appendix 26 Parametric Study - 𝑀𝑇𝑏 vs 𝑀𝑇𝑐 vs 𝜙𝑥 Surface Plots........................................113
Appendix 27 Turbulent Viscosity Ratio Formulation............................................................ 114
Appendix 28 Turbulent Viscosity Results ...............................................................................115
Appendix 29 Dimensionless Vortex Area Formulation ..........................................................117
Appendix 30 Vortex Area Results ........................................................................................... 118
8. 8
List of Abbreviations
CAD Computer aided design
CFD Computational fluid dynamics
DNS Direct Numerical Simulation
LHS Latin hypercube sampling
PSP Project Scoping and Planning Report
RANS Reynolds-averaged Navier-Stokes Equations
RMS Root mean squared
SSC Stratford’s Separation Criterion
SST Shear Stress Transport
TSMP Test section middle plane
9. 9
Nomenclature
Symbol Definition
𝑎 Duct height
𝐴 𝑉 Vortex area
𝐴 𝑉𝑠𝑢𝑚
Total vortex area
𝐴 𝑑 Dimensionless vortex area
𝐴 𝑛 Area of element
𝐴 𝑡 Total planar area
𝐴1 Contraction inlet area
𝐴2 Contraction outlet area
𝑏 Duct width
𝑏1 Contraction inlet width
𝑏2 Contraction outlet width
𝐶 𝑝𝑒 Pressure coefficient at velocity overshoot
𝐶 𝑝𝑖 Pressure coefficient at velocity undershoot
𝑐𝑓 Average skin friction coefficient
𝑐1 Contraction inlet height
𝑐2 Contraction outlet height
𝐷ℎ Hydraulic diameter
𝑔 Inflation layer element growth rate
𝑔 𝑓 Freestream element growth rate
𝐻 𝑏
Point along the contraction width where the curves used to model the
roof and floor meet
𝐻𝑐
Point along the contraction height where the curves used to model the
side walls meet
𝐼 Turbulence intensity
𝑘 Turbulent kinetic energy
𝐿 Overall length of contraction
𝐿 𝑐ℎ Characteristic length
𝑛 Inflation layer counter, element counter
𝑁 Total number of inflation layers, total number of elements
𝑃
Point along the contraction length where the curves used to model the
side walls meet and where the curves used to model the roof and floor
meet
𝑅𝑒 Reynolds number
𝑅𝑒 Averaged Reynolds number
𝑅𝑒 𝑥 Local Reynolds number
𝑟𝐴 Area contraction ratio
𝑟𝑏 Roof contraction ratio
𝑟𝑐 Side wall contraction ratio
𝑇 Total time duration
𝑇𝑏 Maximum slope of roof and floor curves
𝑇𝑐 Maximum slope of side wall curves
𝑈1,∞ Velocity far upstream of contraction inlet
𝑈2,∞ Velocity far downstream of contraction outlet
𝑢 𝑥 Area-averaged mean 𝑥-velocity
𝑢 𝑥 Instantaneous 𝑥-velocity
𝑢 𝜏 Shear velocity
𝑢∞ Mean freestream velocity
10. 10
Symbol Definition
𝑢1,∞ Mean freestream velocity at contraction inlet
𝑢2,∞ Mean freestream velocity at contraction outlet
𝑉 Target velocity
𝑉𝑒 Velocity overshoot
𝑉𝑖 Velocity undershoot
𝑥 Streamwise distance
𝑦+ Y-plus
𝑦 𝐻 Total inflation layer thickness
𝑦𝑛 Height of current inflation layer
𝑦𝑝 Distance from wall to centre of an element
𝑦1 Height of first element adjacent to the wall
𝛿 Boundary layer thickness
𝛿 𝑚𝑎𝑥 Maximum boundary layer thickness
𝜀 Turbulent kinetic energy dissipation rate
𝜇 Dynamic viscosity
𝜇 𝑡 Turbulent viscosity
𝜇 𝑡 𝜇⁄ Turbulent viscosity ratio
𝜌 Fluid density
𝜏 𝑤 Wall shear stress
𝜐 Kinematic viscosity
𝜙 Nonuniformity
𝜔 Specific turbulent kinetic energy dissipation rate
Subscripts
Symbol Definition
𝑏 Roof-floor contraction profile
𝑐 Side-walls contraction profile
𝑅𝑀𝑆 Root mean squared
𝑥 𝑥-direction component
𝑦 𝑦-direction component
𝑧 𝑧-direction component
1 At contraction inlet
2 At contraction outlet
Superscripts
Symbol Definition
∗ Modified input parameter
′ Fluctuating value about the mean
11. 11
1 Introduction
1.1 Background
In most wind tunnels, a contraction is situated immediately upstream of the test section and an
expansion is situated immediately downstream of the test section.
Morel (1975) states there are two ways in which a contraction improves airflow quality in the
test section. Firstly, the contraction accelerates airflow, reducing mean velocity nonuniformities
to produce a near even velocity profile at test section entrance. Secondly, the contraction
‘streamlines’ the airflow, reducing turbulent velocity fluctuations.
A gust generator of rectangular cross-section has been built to facilitate aerodynamic and
hydrodynamic research of wind and tidal blades, respectively. A set of motor-powered vanes in
the gust generator creates sinusoidal waves in the fluid passing through, simulating natural
wind and water motion more accurately. The gust generator should be positioned immediately
downstream of the contraction, such that pure sinusoidal waves are incident on the blade in the
test section.
Figure 1. Diagram of a closed-loop open-jet wind tunnel, adapted from Chegg Inc (2019)
The closed-loop open-jet wind tunnel at the University of Bath currently has a contraction and
collector fitted, both of circular cross-section. However, they can be unbolted from the rest of
the wind tunnel. Therefore, a contraction and collector of rectangular cross-section, also of
“modular” nature were proposed. This would allow the gust generator to perfectly interface with
the new contraction outlet. The test section shall have an open top and bottom, with side walls.
This project covers the design optimisation of the contraction.
To produce an effective contraction, the design shall optimise the quality of airflow passing
through it and the test section. A literature review was conducted to explore how good airflow
can be achieved in a contraction and subsequently in the test section.
12. 12
1.2 Aim and Objectives
At the start of the project, a Project Scoping and Planning Report (PSP) was produced. The PSP
documents an initial literature review, aims and objectives. Over time, the aim and objectives
have been revised to suit the project timeframe. The current aim and objectives are shown below.
Aim: To design a contraction, both modular and with rectangular cross-section for the
University of Bath’s closed-loop open-jet wind tunnel
Objective 1: Develop a selection of contraction slope profiles for a parametric study that
achieve the prescribed inlet and outlet section geometry
Objective 2: Using computational fluid dynamics (CFD), investigate slope profile
influence on velocity nonuniformity, turbulent flow and corner vortices
Objective 3: Analyse CFD results by calculating performance criterion, such as
maximum velocity nonuniformity and distortion parameters, e.g. flow area loss
The objectives are based on the literature review and justified in Section 3.
13. 13
2 Literature Review
2.1 Overview
This section combines the initial literature review conducted as part of the PSP and the study
of further literature while undertaking the project.
Much of the research conducted on wind tunnels was completed before 1980, when many were
being built. CFD software did not exist so wind tunnel geometry optimisation previously
depended on experimental results. There is a lack of knowledge on wind tunnel contractions of
rectangular cross-section, as earlier results showed that corners can enhance boundary layer
separation effects. However, modern CFD software can now be used to model fluid flow through
contraction geometries without the requirement of experimental set ups.
Although some of this literature review focuses on axisymmetric contractions, the results from
these have been found to be analogous to three-dimensional contractions.
2.2 Nature of the Design Problem
As the reducing cross-sectional area of a wind tunnel contraction accelerates flow, the pressure
gradients on the walls are generally favourable. However, two regions of adverse pressure
gradient exist near the contraction inlet and outlet. These are caused by changes in wall
curvature.
In an axisymmetric or two-dimensional contraction, Morel (1975, 1977) explains that these
regions of adverse pressure gradient create a “band” of local minimum velocity near the inlet
and local maximum velocity near the exit. Downie, Jordinson and Barnes (1984) identified that
for a three-dimensional contraction, the band reduces to an undershoot of local minimum
velocity and overshoot of local maximum velocity in each corner near the inlet and outlet,
respectively. These velocity extrema cause streamlines to “curve”, resulting in an exit velocity
profile that is always nonuniform.
These regions of adverse pressure gradient can result in boundary layer separation if high
enough. Localized separation can cause boundary layer thickening, while large scale separation
causes flow unsteadiness. Both lead to further velocity nonuniformity at the contraction outlet.
This can be avoided by effective contraction design (Watmuff, 1986).
Papers on the topic of contraction design often define four main design criteria: exit flow
uniformity, avoidance of flow separation, minimum exit boundary layer thickness and
minimum contraction length. The first three constitute good flow quality while minimum
contraction length is often a wish. For three-dimensional contractions, a fifth design criterion
is added: minimum swirling flow.
Exit flow uniformity describes how uniform the velocity is at the contraction outlet plane and
should be as uniform as possible in the streamwise direction. The other design criterion aim to
increase flow uniformity. Flow separation refers to the detachment of a boundary layer from a
surface. This is unwanted as the flow can take the form of eddies and vortices after separating,
reducing flow uniformity. Exit boundary layer thickness is the thickness of the boundary layer
at the outlet plane. Boundary layers are inherently nonuniform in streamwise velocity as the
velocity increases from zero to the mainstream value in this region. Therefore, keeping the
boundary layer thickness to a minimum reduces velocity nonuniformity in the streamwise
direction. Swirling flows are unwanted, as they exhibit velocity components in non-streamwise
directions and contribute to another form of velocity nonuniformity. Finally, reduction in
14. 14
contraction length reduces cost and space. Both of these can be vital to the assembly and
construction of a wind tunnel.
2.3 Axisymmetric Contraction Design
Morel (1975) developed design charts for a range of axisymmetric contraction geometries, which
are intended to serve as practical guidelines for axisymmetric contraction design. A number of
axisymmetric profiles were modelled using a pair of matching cubic curves and flows were
analysed numerically.
Stratford’s Separation Criterion (SSC) has been used extensively to predict flow separation in
turbulent boundary layers (Stratford, 1958). 𝐶 𝑝𝑖 and 𝐶 𝑝𝑒 are defined as the pressure coefficient
at the velocity undershoot and overshoot, respectively. By applying SSC, Morel (1975) noticed
that using a maximum tolerable value for 𝐶 𝑝𝑖 to avoid flow separation near the inlet and
maximum tolerable value for 𝐶 𝑝𝑒 to achieve a prescribed velocity nonuniformity was found to
achieve the shortest contraction and satisfy the four main design criteria to a “reasonable
degree”:
𝐶 𝑝𝑖 ≡ 1 − (
𝑉𝑖
𝑈1,∞
)
2
(1)
𝐶 𝑝𝑒 ≡ 1 − (
𝑈2,∞
𝑉𝑒
)
2
(2)
𝑉𝑖 and 𝑉𝑒 are the minima velocity near the inlet and maxima velocity near the outlet, respectively.
𝑈1,∞ is the velocity far upstream of the entrance plane and 𝑈2,∞ is the velocity far downstream
of the exit plane.
2.4 Three-Dimensional Contraction Design
Downie, Jordinson and Barnes (1984) demonstrated a finite difference method for analysing
incompressible, inviscid flow through three-dimensional contractions with rectangular cross-
sections. Contraction profiles were created using pairs of matching elliptic arcs, which allowed
independent control over the position and magnitude of the maximum slope. This is an
improvement on using cubic curves.
Downie, Jordinson and Barnes concluded that the velocity undershoots and overshoots as
described in Section 2.2 indicated that pressure distributions are nonuniform in the plane
normal to the central axis and therefore, crossflows must be present. These are components of
flow normal to the mainstream. Crossflows are undesirable as they directly reduce velocity
uniformity. Further results showed that crossflows can be reduced by minimising the velocity
extrema or reducing asymmetry. Unwanted additional regions of adverse pressure gradient can
also be created if the axial position of the max slope of the side walls is different to those on the
roof and floor. Su (1991) conducted further numerical analysis on three-dimensional contraction
flow phenomena. He found that Morel’s philosophy (1975) of calculating maximum tolerable
values of 𝐶 𝑝𝑖 and 𝐶 𝑝𝑒 can be applied to corner velocity undershoots and overshoots for three-
dimensional contractions.
15. 15
Morel (1975) and Downie, Jordinson and Barnes (1984) noted the regions of adverse pressure
gradient mainly depend on wall curvature. By making the magnitude of the maximum slope
large, the curvature in the regions where adverse pressure gradients exist can be reduced. This
was studied further in this project and considered a higher priority than optimising the overall
contraction length.
Literature describing the use of CFD for three-dimensional contraction design has been rare to
find. However, one such paper written by Abdelhamed et al. (2014) documents the use of CFD
and design optimization to redesign an open-jet wind tunnel contraction. Contraction profiles
were produced using a different approach. Bezier curves were used, which are parametric curves,
defined by control points (Weisstein, 2020). These give high flexibility in creating curves of
varied complexity. However, ellipses are simpler to model and proved to be effective in
modelling the contraction profiles in the study by Downie, Jordinson and Barnes (1984).
CFD was used by Abdelhamed et al. as it allows insight into minute flow details, which cannot
be captured using flow bench tests. In addition, numerical methods for contraction design often
solve equations that assume inviscid flow, such as Laplace’s equation. This assumption could
not be applied, as significant boundary layer thickness was expected in the contraction.
Therefore, Navier-Stokes Equations in the form of Reynolds-averaged Navier-Stokes Equations
(RANS) were solved efficiently in CFD and were also solved in this project.
2.5 Swirling Flows
The work of Abdelhamed et al. (2014) includes the additional design criterion of minimising
vortex effects. Görtler (1955) discovered when boundary layer thickness is comparable to the
radius of wall curvature on a concave wall, such as the first half of a contraction, centrifugal
effects create a pressure variation across the boundary layer, resulting in Görtler instability and
the formation of Görtler vortices. The size of the vortices can be mitigated by reducing the
boundary layer thickness or increasing the wall radius of curvature (Saric, 1994). As vortices
disrupt primary flow uniformity, limiting them was considered an additional design criterion in
this project.
Vortices can be identified using a variety of methods. The 𝜆2-criterion by Jeong and Hussain
(1995) and Q-criterion by Hunt, Wray and Moin (1988) are among industry standard as they
have been validated extensively. A vortex is defined as a connected region where 𝜆2 velocity is
negative when using the 𝜆2-criterion while the same applies for a connected region of positive
Q when using the Q-criterion. The definition of 𝜆2 velocity and Q are outside the scope of this
report but curious individuals can understand more about them in the provided references.
17. 17
3 Present Approach
3.1 Overview
A parametric study is attempted here, to scope the geometric parameters and their required
magnitudes to optimise flow quality through the contraction. CFD was used to provide accurate
flow details with appropriate settings, while facilitating automation in obtaining results.
Using the method by Downie, Jordinson and Barnes (1984), all contractions were modelled with
elliptic equations, allowing independent control over the magnitude and position of maximum
slope. Morel (1975) and Downie, Jordinson and Barnes (1984) discovered that the regions of
adverse pressure gradient govern the values of 𝐶 𝑝𝑖, 𝐶 𝑝𝑒, velocity undershoot and overshoot,
which are all related to crossflow and velocity nonuniformity. The regions of adverse pressure
gradient are hugely dependent on the regions of high wall curvature, which depend on the
magnitude and position of maximum slope. Therefore, priority was set on varying the
contraction slope on all walls in the parametric study. Length could be optimised according to
the optimum maximum wall slopes in further work.
To evenly contract the flow, both side walls were modelled with the same profile. Similarly, the
roof and floor of the contraction shared the same profile. Axial position of maximum slope of
side walls was always equal to those on the roof and floor to avoid creating additional regions
of adverse pressure gradient, as recommended by Downie, Jordinson and Barnes (1984).
There are no openly accessible publications on the optimisation of three-dimensional
contractions modelled by explicit equations using a parametric study and CFD. Therefore, this
project is a novel study. The pursuit of more efficient renewable energy generation is especially
relevant in our time. The new contraction will be used to facilitate future studies in renewable
energy generation and to provide more data on flow quality in rectangular contractions.
3.2 Design Criteria
Although the design criteria in Section 2.2 is commonly found in contraction design, the flow
at the central portion of the contraction outlet plane was deemed significantly more important
than the flow at the boundary. Therefore, a modified design criteria was established for this
project: Exit flow velocity uniformity, minimum turbulence (which addresses the impact of
crossflow) and minimum vortex effects. Each design criterion was quantified to compare
different contraction geometry designs in the parametric study.
3.3 Geometry Constraints
Figure 2 is a plan view of the existing wind tunnel. The new contraction, test section and
collector must all fit within bubbles M and C in Figure 2. This is a length of 4145 𝑚𝑚.
The new contraction must have inlet area dimensions of 1418 𝑚𝑚 × 1418 𝑚𝑚, governed by
the dimensions of the duct upstream. Similarly, the contraction outlet must be 1003 𝑚𝑚 in
height and 340 𝑚𝑚 in width. These are the same dimensions as the gust generator.
From a plan view of a contraction, the roof contraction ratio can be defined by:
𝑟𝑏 =
𝐼𝑛𝑙𝑒𝑡 𝑤𝑖𝑑𝑡ℎ, 𝑏1
𝑂𝑢𝑡𝑙𝑒𝑡 𝑤𝑖𝑑𝑡ℎ, 𝑏2
(3)
18. 18
This equates to
1418
340
≈ 4.17. Likewise, from a side view, the side wall contraction ratio can be
defined by:
𝑟𝑐 =
𝐼𝑛𝑙𝑒𝑡 ℎ𝑒𝑖𝑔ℎ𝑡, 𝑐1
𝑂𝑢𝑡𝑙𝑒𝑡 ℎ𝑒𝑖𝑔ℎ𝑡, 𝑐2
(4)
This equates to
1418
1003
≈ 1.41.
Finally, an area contraction ratio can be defined:
𝑟𝐴 =
𝐼𝑛𝑙𝑒𝑡 𝐴𝑟𝑒𝑎, 𝐴1
𝑂𝑢𝑡𝑙𝑒𝑡 𝐴𝑟𝑒𝑎, 𝐴2
(5)
𝐴1 = 1.418 × 1.418 = 2.01𝑚2
and 𝐴2 = 1.003 × 0.34 = 0.34𝑚2
. Therefore:
𝑟𝐴 =
2.01
0.34
≈ 5.91
There is freedom in the way that the contraction contracts the fluid flow area from the
prescribed inlet to outlet dimensions. This project investigates the best contraction profile
shape to achieve this area contraction ratio, while optimising the design criteria.
3.4 Velocity Constraints
A prescribed velocity of 30 𝑚/𝑠 has been assigned to the contraction outlet for all simulations
in this project, as the majority of wind tunnel tests will be conducted using this velocity. For
incompressible flow, the continuity equation is as follows:
𝑢1,∞ 𝐴1 = 𝑢2,∞ 𝐴2 (6)
𝑢1,∞ and 𝑢2,∞ are the mean freestream velocities at the inlet and outlet, respectively. 𝐴1 is the
inlet area and 𝐴2 is the outlet area as defined and calculated in Section 3.3.
By rearranging Equation (6), 𝑢1,∞ can be found:
𝑢1,∞ =
𝑢2,∞ 𝐴2
𝐴1
=
(30)(0.34)
(2.01)
= 5.09 𝑚/𝑠
Hence an average inlet freestream velocity of 5.09 𝑚/𝑠 was used for all simulations.
19. 19
4 General Methods
4.1 Overview
This section covers the workflow of the project and explains the methods used that apply to all
simulations conducted as part of this project. Underlying fluid dynamics and information on
mesh practices is included in this section, to justify some of these methods.
4.2 Workflow
The project workflow can be split into five phases. The first phase involved conducting CFD on
a first pass model. Some assumptions were made to simplify the geometry in order to test
fundamental CFD settings for more detailed models later on. Validation of the mesh used in the
first pass model took place in the second phase. The same mesh was used to discretize a
contraction detailed by Downie, Jordinson and Barnes (1984) and the results were compared
with those published in the paper. The third phase involved conducting CFD on a second pass
model. The bend upstream of the contraction was incorporated and the test section downstream
was changed to an open-jet to model the domain more accurately. These first three phases were
used to prepare appropriate CFD settings and an effective mesh for the parametric study. Hence
mostly qualitative results were evaluated up to this stage. 75 different contraction geometries
were sampled in the fourth phase as part of the parametric study and the process of CFD solving
undertaken in the third phase was repeated for each sample. Using results from the parametric
study, CFD was conducted on an optimised contraction in the final phase.
4.3 Wall Profile Geometry
All contraction profiles consist of four wall profiles: the roof, floor and two side walls. The roof
profile refers to the wall profile that is observed at a plan view of the contraction, while the side
wall profile refers to the wall profile that is observed from a side view. The view from either two
side walls share the same wall profile as the roof and floor evenly contract together. Similarly,
the roof and floor share the same wall profile as the two side walls evenly contract together.
Viewing the contraction from underneath shows the same geometry as from above. For
simplicity, this report will refer to the plan and bottom views as “roof-floor profile” and the two
side wall views as “side-walls profile”.
Figure 3 shows the general composition of the contraction geometry in this project, using an
arbitrary example of a contraction profile. The roof-floor profile is represented by the inner
profile and the side-walls profile is represented by the outer profile. Although the wall profiles
would be viewed on different geometry planes, the two profiles have been collapsed onto one
figure for comparison and clarity.
Bubble 1 points to one half of a side wall modelled by a pair of ellipse curves, joined at the
position of maximum slope, marked with a small circle. The other half labelled bubble 2 is
created by mirroring the first half at the central axis. Combining the two creates a full side wall
profile, which can be used to model both side walls. The roof and floor are modelled using the
same method but with a different pair of mirrored ellipse curves. This are labelled by bubbles 3
and 4.
20. 20
Figure 3. General contraction structure
4.4 DefineEllipse and EllipseParams
A MATLAB script shown in Appendix 1 called DefineEllipse, programmed to compute ellipse
coefficients, which are used to formulate ellipse equations to model contraction profiles was
provided by Dr. Anna Young. Figure 4 shows a graphical representation of the input parameters
required in DefineEllipse, with their definitions detailed in Table 1. The contraction inlet and
outlet dimensions are also required in the script.
Similar to Figure 3, the roof-floor profile is shown by the inner profile, while the side-walls
profile is shown by the outer profile. Both wall profiles are collapsed onto one dimensionless
plot of size. As the roof and floor contract to 0.34 𝑚 while the side walls contract to 1 𝑚 at the
contraction exit plane, the inner profile in dimensionless contraction profile plots in this project
are always the roof-floor profile, while the outer profiles are always the side wall profile. See
Appendix 2 for the default expressions for these parameters, defined by Downie, Jordinson and
Barnes (1984).
3
2
1
4
21. 21
Figure 4. DefineEllipse input parameters
Table 1. Ellipse Input Parameter Definitions
Symbol Definition
𝐿 Overall length of contraction
𝑃
Point along the contraction length where the curves used to model the side
walls meet and where the curves used to model the roof and floor meet
𝐻 𝑏
Point along the contraction width where the curves used to model the roof
and floor meet
𝐻𝑐
Point along the contraction height where the curves used to model the side
walls meet
𝑇𝑏 Maximum slope of the roof and floor curves
𝑇𝑐 Maximum slope of the side wall curves
Having read the input parameter values, the programme executes a function called
EllipseParams, containing equations presented in the paper by Downie, Jordinson and Barnes
(1984) to calculate ellipse equation coefficients for modelling the contraction profile, using
ellipse curves that satisfy the input parameters. The initial unmodified code is shown in
Appendix 1. See Appendix 2 for more details on how the input parameters are used to calculate
the ellipse coefficients and how the ellipse coefficients are assembled into ellipse curve
equations.
With some modifications, the code was used to produce ellipse coefficients for the first pass
(Section 5), validation (Section 5.5) and second pass models (Section 6). Further changes were
made to the code to sample ellipse coefficients for the parametric study (Section 7).
22. 22
4.5 CAD
The geometry of all CFD models was produced using CAD. Using input parameter values
defined by the required geometry for each model, ellipse coefficients were produced in the
DefineEllipse script through the EllipseParams function (Section 4.4). These coefficients were
then formulated into ellipse equations and used to model the contractions in Autodesk Inventor.
The CAD models were then imported into SpaceClaim, where named selections for meshing
and solving were specified. Afterwards, the SpaceClaim models were imported into ANSYS
Meshing.
It is worth noting that the DefineEllipse and CAD processes model specifically the fluid domain
“sculpted” by the walls. That is the fluid occupied by the contraction, not the contraction walls.
4.6 Meshing
4.6.1 O-Grid
Geometry is often discretized using two regions of meshes before undertaking a CFD simulation:
the freestream mesh and inflation layer. The freestream mesh is the mesh that discretizes the
freestream, while the inflation layer is the mesh that discretizes the boundary layer. The domain
is the flow field subject to CFD analysis. A structured O-grid was used to mesh all domains.
There are several advantages in using a structured mesh over an unstructured mesh for this
project. Firstly, aside from the contraction, the domains can be considered noncomplex with
limited curved geometry, which suits the use of a structured mesh (Bakker, 2006). Secondly, a
structured mesh should align better in the flow direction as the flows in all contractions
predominantly travel in one direction – streamwise through the contraction, leading to more
accurate results and better convergence (Ali, Tucker and Shahpar, 2016). They also require less
memory and computation time than unstructured meshes. Lastly, structured meshes allow for
a much higher degree of control over mesh design and refinement (Envenio, 2017).
As the gradients of fluid characteristics such as velocity and viscosity are often high near the
walls, the inflation layer normally uses a much finer grid compared to the freestream mesh to
capture those changes more accurately. The element size must therefore grow from the wall.
An O-grid mesh allows elements to grow steadily from the inflation layer to the freestream in
all directions while maintaining a sensible growth rate (Tu, Yeoh and Liu, 2018, p.151).
Figure 5. The O-grid at the first pass model inlet
23. 23
Hexahedral elements were used for the structured mesh as they often produce more accurate
results than tetrahedral elements for wall-bounded flows. Orthogonal grids in the wall-normal
direction can be maintained in wall-bounded flows using hexahedral elements as their faces can
be kept close to 90˚ (Johnen, Weill and Remacle, 2017).
4.6.2 Turbulent Flow
Ubiquitous in fluid mechanics, the Reynolds Number, 𝑅𝑒 describes the ratio of inertial forces to
viscous forces:
𝑅𝑒 =
𝜌𝑢∞ 𝐿 𝑐
𝜇
=
𝑢∞ 𝐿 𝑐ℎ
𝜐 (7)
𝜌 = fluid density, 𝑢∞ = freestream velocity, 𝐿 𝑐ℎ = characteristic length, 𝜇 = dynamic viscosity
and 𝜐 = kinematic viscosity.
Air, a Newtonian fluid, is expected to be the only fluid to flow through the contraction. The
average Reynolds Number for Newtonian flow in a rectangular duct is given by the following
equation (Tosun, Under and Ozgen, 1988):
𝑅𝑒 =
𝑢∞ 𝐷ℎ
𝜐
(8)
The hydraulic equivalent diameter, 𝐷ℎ is given by the following:
𝐷ℎ =
4𝑎𝑏
2(𝑎 + 𝑏)
(9)
𝑎 = duct height and 𝑏 = duct width.
For most duct applications, internal flow is considered laminar when 𝑅𝑒 < 2300 and fully
turbulent when 𝑅𝑒 > 10000 (Schlichting and Gersten, 2017, p.416). Calculations in Appendix 3
show that average Reynolds numbers at the inlet and outlet are greater than 10000. Therefore,
it was assumed that most or all of the internal flow in the contraction is fully turbulent. If a
turbulent freestream flow is considered, the boundary layer must also be turbulent (Bahrami,
2009).
4.6.3 𝑦+
and Inflation Layer
The wall 𝑦+
is a dimensionless distance, similar to the local Reynolds number. It determines
whether flow in the wall-adjacent cells is laminar or turbulent, therefore indicating the part of
the turbulent boundary layer that they resolve (Ariff, Salim and Cheah, 2009).
𝑦+
=
𝑢 𝜏 𝑦𝑝
𝜈 (10)
𝑢 𝜏 = shear velocity, 𝑦𝑝 = distance from the wall to the centre of the element and 𝜈 = kinematic
viscosity of the fluid. The shear velocity is a velocity scale, that represents the shear stress
between the fluid layers in the boundary layer (Pokrajac et al., 2006) and is defined by the
following equation:
𝑢 𝜏 = √
𝜏 𝑤
𝜌 (11)
24. 24
𝜏 𝑤 = wall shear stress. This is the shear stress exerted by the wall onto the fluid layer. 𝜌 = fluid
density. The term 𝑦+
normally refers to the 𝑦+
value for elements immediately adjacent to the
wall.
There are three regions in a turbulent boundary layer (rightmost profile in Figure 6). The viscous
sublayer normally exists at 𝑦+
< 5. This is where the flow is assumed to be laminar and viscous
stresses are predominant. 5 < 𝑦+
< 30 is the buffer region where both viscous and turbulent
stresses occur. Lastly, 30 < 𝑦+
< 200 is known as the log-law region (turbulent region in the
figure), where turbulent stresses dominate.
Figure 6. Boundary layer profiles (Frei, 2017)
Figure 7 shows an element next to a wall, with 𝑦𝑝 marked as defined before. A new distance
called 𝑦1 can be defined as the height of the first element from the wall, which is 𝑦𝑝 doubled:
𝑦1 = 2𝑦𝑝 (12)
Figure 7. An element next to a wall
As the adverse pressure gradient regions (Section 3.1) exist in the boundary layer, 𝑦+
of 1 was
aimed for in all simulations to ensure the laminar sublayer was solved (Murad, 2018). Using this
value of 𝑦+
, calculations were undertaken to determine the required first element height, 𝑦1
from the wall.
4.6.4 Element Growth Rate
The relative thickness of adjacent inflation layers is defined by the growth rate, 𝑔:
𝑔 =
𝑦 𝑛+1
𝑦𝑛
(13)
𝑦𝑛 = height of the current layer considered and 𝑦 𝑛+1 = height of the next layer.
25. 25
It is advised to have an inflation layer growth rate between 1 and 1.2 (Elmekawy, 2018). This
ensures that the elements grow from the wall in a controlled manner, capturing high gradients
near the wall and reducing accuracy where appropriate as the gradients reduce with distance
from the wall, to relieve computational effort (Lanfrit, 2005).
Figure 8. Inflation layer growth rate
If 𝑁 = total number of inflation layers, the total inflation layer thickness, 𝑦 𝐻 can be given by the
following summation of the first layer thickness:
𝑦 𝐻 = ∑ 𝑦𝑛
𝑁
𝑛=1
∑ 𝑦𝑛
𝑁
𝑛=1
= 𝑦1 + 𝑦2 + 𝑦3 + ⋯ + 𝑦 𝑁
𝑦1 + 𝑦2 + 𝑦3 + ⋯ + 𝑦 𝑁 = 𝑦1 + 𝑔𝑦1 + 𝑔2
𝑦1 + ⋯ + 𝑔(𝑁−1)
𝑦1 (14)
See Appendix 4 for the derivation of this summation.
4.6.5 Boundary Layer Thickness
The growth of a boundary layer on duct walls was approximated to be the same as found on a
flat plate. Schlichting (1979, p.638) states that the thickness of a turbulent boundary layer over
a flat plate, 𝛿 can be empirically approximated as:
𝛿 ≈
0.37𝑥
𝑅𝑒 𝑥
1
5
(15)
𝑥 = distance travelled by the fluid along the duct wall and 𝑅𝑒 𝑥 = local Reynolds number.
𝑅𝑒 𝑥 =
𝑢∞ 𝑥
𝜈 (16)
𝑢∞ = freestream velocity and 𝜈 = kinematic fluid viscosity.
26. 26
To ensure that the total inflation layer thickness covered the entire boundary layer, the total
inflation layer thickness was set greater than the highest value of boundary layer thickness. That
is:
𝑦 𝐻 ≥ 𝛿 𝑚𝑎𝑥 (17)
𝛿 𝑚𝑎𝑥 was assumed to be the boundary layer thickness 𝛿 at the domain outlet, as the fluid there
would have travelled the farthest, hence building up the thickest boundary layer.
Equation (14) is a geometric series and can be solved using the following equation:
𝑦 𝐻 = 𝑦1 (
1 − 𝑔 𝑁
1 − 𝑔
) (18)
Given the required 𝑦 𝐻 from the estimated highest boundary layer thickness in Equation (17) and
with knowledge of 𝑦1 and 𝑔, Equation (18) can be rearranged into the following form to set the
required number of inflation layers, 𝑁 as the subject:
𝑁 =
log (1 − (
𝑦 𝐻
𝑦1
) (1 − 𝑔))
log(𝑔)
(19)
This equation was solved for all simulations to find the required number of inflation layers. See
Appendix 4 for the derivation of this equation.
4.6.6 Mesh Refinement
Areas in the freestream mesh that were expected to possess high gradients were also refined.
Similar to the inflation layer growth rate, it is recommended to limit the growth rate of elements
in the freestream to between 1 and 1.2 for the same reasons of maintaining accuracy in areas of
high gradient.
4.7 Setup
After meshing, the models were imported into CFX or Fluent setup. Algorithms for solving were
chosen, the turbulence model was decided upon and boundary conditions were assigned.
As subsonic velocity was expected, a pressure-based solver was used in all simulations. These
are well suited for incompressible and subsonic flows (Menter et al., 2004). In addition, all
simulations were chosen to run as steady state. This is because flow parameters, such as velocity,
temperature and viscosity were not expected to change with time during full operation of the
wind tunnel. Incompressible continuous fluid of air at 25˚𝐶 was the chosen fluid in all
simulations. 25˚𝐶 is the Standard Ambient Temperature and is a close approximation to the
actual conditions that would occur in the wind tunnel laboratory, giving a density of
1.185 𝑘𝑔/𝑚3
(Rogers and Mayhew, 1995, p.16).
27. 27
For any turbulent CFD simulation, a turbulence model is often chosen to close the Navier-
Stokes and Reynolds stress equations that describe turbulent flow. Choice of turbulence model
is often based on their strengths and weaknesses in modelling certain flow regions and
phenomena (Tu, Yeoh and Liu, 2018, p.264). RANS models were used for all simulations, as the
other options – Direct Numerical Simulation (DNS) and Eddy Simulation were decidedly
unfeasible. DNS is not a turbulence model. Instead it is the process of solving the Navier-Stokes
equations directly, which requires very high computational resources. Eddy Simulation often
produces more accurate results for turbulence in comparison to RANS. However, higher mesh
resolutions and longer run times are required as they can only run as transient cases (Solmaz,
2012). RANS models are designed to be widely applicable, simple and computationally
economical (Tu, Yeoh and Liu, 2018, p.264).
The no-slip condition assumes that fluid at a solid boundary has zero velocity relative to the
boundary (Day, 2004). This assumption is applicable to viscous flows and was therefore
assigned to the walls of all simulations, except the validation model, which modelled inviscid
flow.
All simulations used second order or high resolution spatial discretization schemes. Despite
first-order discretization generally yielding better convergence, ANSYS recommends the use of
second-order discretization or higher to obtain more accurate results when using a hexahedral
mesh.
4.8 Solving
Table 2 shows the root mean squared (RMS) residual targets set for all simulations. The target
values were chosen for good convergence; a compromise between quality of convergence and
computational effort (Kuron, 2015). Monitor points were also chosen for some of the
simulations and are detailed later in this report.
Table 2. RMS Residual Targets
Residual Target Value
Continuity 1e-4
𝑥-velocity 1e-6
𝑦-velocity 1e-6
𝑧-velocity 1e-6
Turbulent kinetic energy, 𝑘 1e-5
Turbulent kinetic energy dissipation rate, 𝜀 or specific
turbulent kinetic energy dissipation rate, 𝜔
1e-5
Energy 1e-7
28. 28
5 First Pass Model
5.1 First Pass Geometry
A first attempt of a contraction model was created and solved in CFD. Several assumptions and
geometry modifications were made to simplify the model.
The inlet and outlet area were set as 1418 𝑚𝑚 × 1418 𝑚𝑚 and 1003 𝑚𝑚 × 340 𝑚𝑚
respectively in DefineEllipse as specified in Section 3.3. However, the input parameters (see
Section 4.4) were set to arbitrary values that created a contraction which would fit into the
existing wind tunnel, shown in Table 3. The resulting dimensionless contraction profile is shown
in Figure 9.
Table 3. First Pass Model Input Parameter Values
Input Parameter Value
𝐿 1.21
𝑃 0.91
𝐻 𝑏 0.27
𝐻𝑐 0.45
𝑇𝑏 1.10
𝑇𝑐 0.55
Figure 9. Dimensionless contraction profile for first pass model
A straight duct of 3 𝑚 was attached to the contraction inlet to test for boundary layer thickness.
Another straight duct of 3 𝑚 was attached to the contraction outlet to test for vortex formation.
The model had completely-walled internal flow, with one defined inlet and outlet. This
simplified boundary conditions for a first attempt.
Roof-floor profileSide-walls profile
29. 29
Figure 10. From top to bottom: isometric view, plan view, and side view of the first pass model
5.2 First Pass Mesh
Consistent with Section 4.6, an O-grid consisting of hexahedral elements was used to mesh the
entire domain. The freestream mesh was refined at the contraction domain due to high velocity
gradients expected. The mesh refinement was kept to a growth rate below 1.2, as recommended
in Section 4.6.4.
See Appendix 5 and Appendix 6 for 𝑦+
and inflation layer calculations undertaken. The
following equation assumes that the boundary layer by the wall behaves like one on a flat plate
and was used to calculate the average skin friction coefficient, which was used to calculate 𝑦1
(Monaghan, 1953):
𝑐𝑓 =
0.074
𝑅𝑒
1
5
(20)
30. 30
Table 4 summarises the mesh parameter values. See Figure 11 for the full mesh.
Table 4. First Pass Model Mesh Parameter Values
Symbol Definition Value
𝑅𝑒 Average Reynolds number 5.9 × 105
𝑦+ y-plus 1
𝑦𝑝
Distance between first cell element
centre and nearest wall
1.7 × 10−5
𝑚
𝑦1
Height of the first cell element from the
wall
3.4 × 10−5
𝑚
𝑔 Inflation layer growth rate 1.2
𝑔 𝑓 Highest freestream growth rate 1.2
𝑦 𝐻 Total inflation layer thickness 100 𝑚𝑚
𝑁 Number of inflation layers 35
Number of nodes 442694
Number of elements 434600
Figure 11. Top left: full mesh with view of inlet, Bottom left: full mesh with view of outlet,
Right: O-grid of outlet
31. 31
5.3 First Pass Setup and Solve
Standard 𝑘 − 𝜀 model, a type of RANS model was used as it is considered robust and stable for
many applications. This turbulence model has been widely validated and seemed a sensible
choice for a first pass test (Tu, Yeoh and Liu, 2018, p.267). When using a 𝑘 − 𝜀 model, turbulent
kinetic energy, 𝑘 and kinetic energy dissipation rate, 𝜀 are solved at discretized points in the
domain. These are then used to calculate turbulent viscosity 𝜇 𝑡 and Reynolds stress, which is
used to solve RANS at each discretized point (Versteeg and Malalasekeera, pp.72 - 80). As 𝑘 − 𝜀
models do not resolve flow in the boundary layer accurately, a wall function was used in
conjunction. Wall functions are empirical equations used to satisfy the nature of flow in the
near wall region. This is the region between the wall and mainstream turbulent flow (Liu, 2016).
See Section 4.7 for all other set up settings.
Minimum number of iterations was set to 1000 and the RMS residual targets were as shown in
Table 2. The simulation achieved the required RMS residual targets within the minimum
number of iterations.
5.4 First Pass Post-Processing and Discussion
5.4.1 Velocity
Figure 12 shows a velocity contour plot of the side-walls profile through the contraction centre.
A half plane is shown for clarity and the plane is truncated at both ends to focus on the
contraction.
Figure 12. Velocity contour plot of side view plane through centre of contraction (units m/s)
The contour plot shows that the prescribed mean inlet and outlet velocities of 5.09 𝑚/𝑠 and
30 𝑚/𝑠 respectively have been achieved. Additionally, the velocity gradient with respect to
streamwise direction, 𝑥 is much greater at contraction centre, as expected.
The deviation of the 5 𝑚/𝑠 contour from the wall with 𝑥 suggests that the boundary layer
thickens with distance from the inlet, as expected. The same can be observed with the 30 𝑚/𝑠
contour. Therefore, it was deduced that the mesh resolves the near wall region.
5.4.2 Vortices
Vortices were observed near each corner as shown in Figure 13, justified by Abdelhamed et al.
(2014) in Section 2.5. Four vortex cores were identified, resulting from two planes of symmetry
in the contraction. The top and bottom walls contract at the same rate with 𝑥, as do the two
side walls. Each corner of the contraction experiences disturbance from one side wall and one
roof-floor wall. Therefore, similar crossflow phenomena would occur and create a vortex of the
same strength at each corner.
32. 32
Vortex strength was plotted in the form of Q-criterion in Figure 13. It is worth noting that vortex
cross-sectional area grows from the contraction outlet. This supports the findings of
Abdelhamed et al. (2014) in Section 2.5 regarding Görtler instability originating from
contraction geometry creating vortices.
Figure 13. First pass model corner vortices
5.5 Validation Model
For conservatism, the mesh was further validated using available experimental data. Figure 14
shows velocity isosurfaces for a typical contraction published by Downie, Jordinson and Barnes
(1984). The two-dimensional contraction had wall and roof contraction ratios of 1.38 and 2.89,
respectively. The fluid was assumed inviscid. Equivalent to the first pass model, maximum slope
on each contraction surface occurred at the same 𝑥 position.
Geometry of the first pass model was modified to satisfy the contraction ratios and the dynamic
viscosity of the fluid was reduced to 1 × 10−15
𝑘𝑔/𝑚𝑠 to simulate inviscid fluid. No information
was given regarding contraction input parameter values (Figure 4) so input parameter values
used in the first pass model were reimplemented (Section 5.1). The first pass model mesh
(Section 5.2), set-up and solve options (Section 5.3) were also reused.
33. 33
Figure 14. Velocity isosurfaces of a typical two-dimensional contraction by Downie, Jordinson
and Barnes (1984), Top: roof-floor profile, Bottom: side-walls profile
Figure 15. Velocity isosurfaces of the validation model, Top: roof-floor profile, Bottom: side-
walls profile
34. 34
Comparison of Figure 14 and Figure 15 shows similar results. The velocity minimum has been
captured by the validation model, while the velocity maximum is not exactly the same as the
experimental plots. The roof-floor and side-walls profile of the validation model have a velocity
overshoot of 40.4 and 40 𝑚/𝑠, respectively. In comparison, both experimental profiles have an
overshoot of 43 𝑚/𝑠. This may be due to the implementation of different input parameters
(Figure 4). Nonetheless, the results of the validation model match well with the experimental
data. Therefore, the first pass mesh was kept for the next phase of the project.
6 Second Pass Model
6.1 Second Pass Geometry
6.1.1 Bend Addition
The domain was evaluated to model the contraction area more accurately for the second pass
model. A bend exists immediately upstream of the contraction, as seen in Figure 2. The bend
has incorporated vanes to guide the airflow. It was thought that the bend may have an effect on
the airflow distribution reaching the contraction inlet and should therefore be included in the
domain.
Several options were considered. The bend could be modelled with its vanes. Another was to
take the approach of the first pass model and assume that the flow through the bend was
adequately uniform, such that it could be modelled as a straight duct. Instead, the bend was
included with no vanes. This presented a worst case scenario whereby unguided air was likely
to separate at the inside turning due to change in wall direction. The effect this would have on
flow distribution at the contraction was studied. It was assumed that a contraction geometry
which performed best under this worst case should also perform best under guided flow at the
bend.
6.1.2 Second Pass Contraction
The overall contraction length was changed to the value of the existing contraction: 1130.3 𝑚𝑚
(Figure 2), as this was considered more realistic. Other input parameters to DefineEllipse were
set to the default values defined by equations stipulated by Downie, Jordinson and Barnes (1984)
(Appendix 2.1 and Appendix 2.2). All calculated input parameter values are shown in Table 5
and their calculations in Appendix 7. See Section 4.4 for their definitions. The resulting
dimensionless contraction profile is shown in Figure 16.
Table 5. Second Pass Model Input Parameters
Input Parameter Value
𝐿 0.80
𝑃 0.40
𝐻 𝑏 0.19
𝐻𝑐 0.42
𝑇𝑏 1.91
𝑇𝑐 0.73
35. 35
Figure 16. Second pass model contraction profile
6.1.3 Second Pass Test Section
As stated in Section 1.1, the test section is planned to have an open top and bottom with side
walls. Therefore, it was modelled as such here. The domain outlet effectively became three
surfaces of the test section: the top, bottom and far outlet as shown in Figure 17. As the collector
had not been designed yet, the length of the test section was set to the current test section
length of 1 𝑚 as shown in Figure 2.
Figure 17. Top outlet, bottom outlet and far outlet
6.1.4 Second Pass CAD
Figure 18 shows the resulting geometry of the second pass model. The outside edge of the bend
was extended by 2282 𝑚𝑚. This is approximately the distance to the centre of the straight run
of duct leading to the bend. It was assumed that the flow would be adequately uniform at this
point, given the fluid has travelled some distance from the bend before.
Roof-floor profileSide-walls profile
36. 36
Figure 18. Top left: plan view, Bottom left: view from the contraction side (outside track),
right: isometric view of second pass model geometry
6.2 Second Pass Meshing
Given its success, features of the first pass model mesh were replicated, such as the structured
O-grid and hexahedral elements. However, the mesh was also adapted to suit the different
geometry.
In accordance with Section 4.6.6, the mesh at expected areas of high gradient were refined. This
included the bend which would cause a sudden change in flow direction, the inside track after
the bend, where boundary layer separation was expected and the test section, where the flow
would be affected by surrounding pressure. The contraction was kept refined due to expected
high values of velocity gradient there.
Due to the student license node limit, growth rate of elements in the freestream was set to a
maximum of 1.4 at the outside track between the bend and contraction. This is ill-advised as
explained in Section 4.6.6. However, this region was expected to have low gradients. Therefore,
it seemed appropriate to compromise the mesh here. A solution to the node limitation is to
adopt a half model if the results are symmetrical with respect to the 𝑋 − 𝑍 plane slicing through
the centre of the domain. This was explored in post-processing, documented in Section 6.5.2.
As the fluid properties are the same and there are no changes in mean velocity and hydraulic
diameters at the contraction inlet and outlet, the same 𝑦+
calculations used for the first pass
model were applicable to the second pass model when aiming for a 𝑦+
of 1. However, the total
inflation layer thickness, 𝑦 𝐻 and number of inflation layers required, 𝑁 were different as the
length of the domain changed. Calculations for 𝑦 𝐻 and 𝑁 are shown in Appendix 8. Table 6
provides a summary of the mesh parameter values. The second pass model mesh is shown in
Figure 19 and Figure 20.
37. 37
Table 6. Second Pass Model Mesh Parameter Values
Symbol Definition Value
𝑅𝑒 Average Reynolds number 5.9 × 105
𝑦+ y-plus 1
𝑦𝑝
Distance between first cell element
centre and nearest wall
1.7 × 10−5
𝑚
𝑦1
Height of the first cell element from the
wall
3.4 × 10−5
𝑚
𝑔 Inflation layer growth rate 1.2
𝑔 𝑓 Highest freestream growth rate 1.4
𝑦 𝐻 Total inflation layer thickness 58.4 𝑚𝑚
𝑁 Number of inflation layers 32
Number of nodes 5.31 × 105
Number of elements 5.09 × 105
38. 38
Figure 20. Isometric views of second pass model domain
6.3 Second Pass Setup
As the top and bottom test section surfaces were changed from solid walls to open outlets, the
fluid can now leave the domain from three outlets: the top, bottom and far outlet. Therefore,
the velocity at the far outlet is now indeterminate. To solve this, the three outlets were changed
to the outflow boundary condition. This boundary condition is used to model flow where details
of flow velocity and pressure are unknown. Fluent extrapolates the required information from
the interior and assigns zero diffusion across the boundary face. This is most accurate if the flow
is fully developed (Tu, Yeoh and Liu, 2018, p.118). All other set up settings were the same as
those used in the first pass model (Section 5.3).
39. 39
6.4 Second Pass Solving
Monitor points are variables that are tracked every solver iteration and serve as another
indication of solution convergence. Five monitor points were added: Mass flow rate and
turbulent intensity at far outlet, standard deviation of 𝑥-velocity at contraction outlet and
average 𝑦-velocity at Point 1 on the top outlet (-0.6, 2.2, 0.00115) and Point 2 on the bottom
outlet (-0.6, 1.2, 0.00115) (Figure 21). These two coordinates were chosen as high 𝑦-velocities
were expected due to the immediate pressure difference as air escapes the contraction. A variety
of monitor points were used to yield various sources of convergence results.
Figure 21. Monitor point locations
The monitor points were considered converged if the value at current iteration step had a
residual of less than 0.01% of the value at the last iteration step. As the convergence residuals
detailed in Table 2 did not meet their targets after 5000 iterations, the monitor points were
used as the convergence criteria instead. Fluctuations in convergence residuals were likely due
to poor mesh quality. All convergence plots are shown in Appendix 9.
6.5 Second Pass Post-Processing and Discussion
6.5.1 Recirculation Regions
Figure 22 shows iso-volumes of 𝑥-velocity greater than 0.1 𝑚/𝑠, whereby reversed flow is
denoted positive. Two recirculation regions can be seen in the domain. Both are immediately
downstream of the bend, at the inside and outside track.
Reversed flow at the outside track is less threatening to 𝑥-velocity uniformity at the contraction,
as it is further away, has a lower maximum magnitude of 0.9 𝑚/𝑠 and has more travel distance
to restore 𝑥 -velocity uniformity. Conversely, the recirculation region at the inside track
protrudes into the contraction, threatening uniformity at the contraction outlet. The greatest
𝑥-velocity here is 5 𝑚/𝑠. Fortunately, the velocity restores to 95% 𝑥-velocity uniformity at the
contraction outlet as seen in Figure 23. Figure 24 shows a clipped isovolume of Q-criterion >
100. There is a clear vortex column at the inside track due the recirculation.
40. 40
Figure 22. Isovolume of regions with 𝑥-velocity greater than 0.1 𝑚/𝑠
Figure 23. 𝑥-velocity contours at contraction inlet plane (top) and contraction outlet plane
(bottom)
41. 41
Figure 24. Vortex column of inner track recirculation region, Top: isometric top view, Bottom:
bottom view
6.5.2 Asymmetry of Results
As previously stated in Section 6.2, a way to reduce computation effort is to simplify the domain
as a half model and to use Neumann boundary conditions at the symmetry plane. However, it
was found that the results are not symmetrical at the symmetry plane (𝑋 − 𝑍 plane at 𝑦 = 0).
An example of the asymmetric results can be seen in Figure 25. Velocity in 𝑦 and 𝑧-direction at
the contraction outlet are displayed as contour plots. The results are mostly symmetrical about
the 𝑋 − 𝑍 plane, but differences can be observed, especially at the minima and maxima regions.
Asymmetry may be due to inaccuracies resulting from mesh quality. It may be argued that using
a half model allows a finer mesh to be used, thereby restoring accuracy. However, a half model
was ultimately not pursued due to the time limitations of the project. A full model was preferred
to allow the “full picture” to be investigated.
42. 42
Figure 25. Second pass model 𝑦-velocity contour plot (left) and 𝑧-velocity contour plot (right)
at contraction outlet plane
6.5.3 Mesh Assessment
It is evident from Figure 26 that high gradients consistently exist near the contraction outlet
plane. An 𝑋 − 𝑌 plane slice through the contraction centre at 𝑧 = 0 has been taken and colour
maps of
𝑑𝑢 𝑥
𝑑𝑧
,
𝑑𝑢 𝑦
𝑑𝑧
𝑑𝑢 𝑧
𝑑𝑧
have been plotted. In all three plots, the highest gradients are located near
the contraction outlet, marked with a black line. Therefore, the mesh should be refined in this
region to capture gradients more accurately. Only velocity components in the 𝑧-direction are
shown as they have the greatest range. This may be due to the bend causing a large difference
in velocity downstream across the 𝑧-axis. It is clear from Figure 26 that the gradients have
significantly less magnitude near the bend.
The geometry was split into cell zones in set-up to allow independent mesh refinement of
regions. An observation can be made regarding the mesh matching between cell zones. On the
right hand side of each plot in Figure 26, it is clear that the meshes are not conformal across cell
zones, as there are gradient discontinuities. This is a source of discretization error, as results are
interpolated across the boundary in a nonconformal mesh. In a conformal mesh, the algorithms
are executed across the boundary in the same way as if the two bodies are a single component
(ANSYS, 2014). This shall be rectified in the parametric study.
43. 43
Figure 26. Top:
𝑑𝑢 𝑥
𝑑𝑧
colour map, middle:
𝑑𝑢 𝑦
𝑑𝑧
colour map, bottom:
𝑑𝑢 𝑧
𝑑𝑧
colour map
Figure 27 shows a colour map of 𝑦-velocity at the contraction outlet plane. The magnitude of 𝑦-
velocity matches the shape of the O-grid inflation layer. This may be due to the truncation of
the last two inflation layers, negating the blending of flow physics between the inflation layer
and free stream. The mesh has a clear influence on the results, producing an additional source
of unwanted discretization error. This shall be corrected in the parametric study.
44. 44
Figure 27. 𝑦-velocity colour map at contraction outlet
6.5.4 Other Results
The mean velocity increased from 5.09 𝑚/𝑠 at the inlet to 30 𝑚/𝑠 at the outlet as expected. Due
to the large recirculation region at the outside track, the velocity distribution has been shifted,
with a high magnitude velocity region of −7 𝑚/𝑠 between the two recirculation regions. Flow
in the corner at the outside track experiences low convection in comparison to the inside track,
where pressure is continually driving the flow forward. This velocity unbalance perpetuates
towards the contraction outlet but recovers by the time flow reaches the contraction outlet
plane.
The absence of corner vortices was concerning. They were firmly present in the first pass model
results (Section 5.4) and voiced by Abdelhamed et al. (2014) as a flow feature that must be
monitored in three-dimensional contractions. The addition of the bend may have upset the
velocity conditions at the contraction required to produce them as it is evident that the bend
causes velocity asymmetry as discussed in Section 6.5.2.
6.6 Mesh Convergence Study
A mesh convergence study involves repeating a simulation using different levels of mesh
refinement and is normally highly recommended to undertake in CFD simulations (Tu, Yeoh
and Liu, 2018, p.230). Firstly, they allow convergence of flow results to be observed when mesh
refinement increases, as the discretization becomes a more accurate representation of the
domain, provided that an accurate choice of boundary conditions and turbulence model are
made. Secondly, they expose the level of compromise between computation effort and accuracy.
Sometimes it is worth using a coarser mesh to sacrifice some accuracy if there is sufficient
reduction in computation time.
The second pass mesh was coarsened at five levels, whereby each refinement level contains
approximately 1.3 × the node count of the previous level. The results of the initial mesh (Section
45. 45
6.2) was used as the most refined case. Table 7 details the results of the study. Final converged
values of all five monitor points were plotted against mesh refinement level, as shown in Figure
28 to Figure 32 and outliers marked with red crosses were omitted from the curves of best fit.
The convergence criteria for all refinement levels was the same as the second pass model
(Section 6.4).
Table 7. Mesh Convergence Study Results
Refinement
level
Node Count Element Count
Solution Completion
Time
1 1.68e+5 1.58e+5 358 mins
2 2.37e+5 2.24e+5 152 mins
3 3.38e+5 3.22e+5 232 mins
4 4.48e+5 4.29e+5 297 mins
5 5.31e+5 5.09e+5 360 mins
Figure 29 and Figure 32 are concerning, as the monitor points diverge as refinement is increased.
This occurs across all refinement levels and is an indication of poor choice of boundary
conditions or turbulence model. On the other hand, Figure 28, Figure 30 and Figure 31 show
convergence.
It is evident from Table 7 that poor set-up choices exist, as the solution time does not converge
as refinement is increased. The set-up choices shall be revised for the parametric study. However,
it can be concluded that the mesh used in refinement level 5, identical to the second pass model
mesh provides the highest accuracy. This mesh will be adapted upon in the parametric study.
Figure 28. Final converged value of mass flow rate at far outlet vs mesh refinement level
46. 46
Figure 29. Final converged value of maximum turbulence intensity at far outlet vs mesh
refinement level
Figure 30. Final converged value of standard deviation of 𝑥-velocity at contraction outlet vs
mesh refinement level
47. 47
Figure 31. Final converged value of average 𝑦-velocity at Point 1 vs mesh refinement level
Figure 32. Final converged value of average 𝑦-velocity at Point 2 vs mesh refinement level
48. 48
7 Parametric Study Contraction Sampling
7.1 Input Parameter Multiplier Ranges
As detailed in Section 1.2, Objective 1 of this project is to develop a selection of contraction
profiles for a parametric study that meet the geometry and velocity conditions outlined in
Section 3.3 and 3.4. Up to this point, geometry, mesh and CFD settings have been investigated
in preparation for creating models in this parametric study.
To develop a selection of contraction profiles, the range of values that the input parameters
detailed in Section 4.4 can take were explored. As there are six input parameters, constraints
were placed to “shrink” the parameter space. Although no constraints would be ideal as a global
optimum may be sought, allocated time for the project meant that only a limited number of
simulations could be run.
Derived from the literature review, priority was set on varying the magnitude and position of
the contraction wall slopes, as these variables strongly affect the adverse pressure gradient
regions. The optimal wall slopes can then be used to calculate the minimum length which does
not compromise flow quality. Additionally, the 𝑥-position of the curve meeting point, 𝑃 was
kept constant at the middle of the overall contraction length, whereby 𝑃 = 0.5𝐿.
To vary all other input parameters, a range of multiplier values were scoped. The multiplier
multiplies the default input parameter value, to give a modified value used to create the ellipse
curves. For example, let 𝑀 𝐻𝑏 be the multiplier for 𝐻 𝑏:
𝐻 𝑏
∗
= 𝐻 𝑏 𝑀 𝐻𝑏 (21)
𝐻 𝑏
∗
is modified 𝐻 𝑏 and would be read by DefineEllipse and EllipseParams instead of 𝐻 𝑏. The
superscript * is given to input parameters modified by its multiplier. See Appendix 10 for further
details. Using DefineEllipse, the following range of multipliers with their input parameter
shown in Table 8 were found compatible to give ellipse profiles:
Table 8. Input Parameter Multiplier Ranges
Input Parameter Scoped Multiplier Range
𝐻 𝑏 1 < 𝑀 𝐻𝑏 < 2.25
𝐻𝑐 0.9 < 𝑀 𝐻𝑐 < 1.15
𝑇𝑏 0.5 < 𝑀 𝑇𝑏 < 4.5
𝑇𝑐 0.5 < 𝑀 𝑇𝑐 < 4.5
7.2 Latin Hypercube Sampling
To sample various combinations of multiplier values in order to create different contraction
profiles, Latin hypercube sampling (LHS) was used. LHS is a statistical method that recreates
an input multidimensional distribution with a finite number of samples. Each input distribution,
which are the multiplier ranges in this problem, is divided into equal intervals.
A Latin square is a square grid that only contains one sample randomly placed in each row and
column. In LHS, this concept is expanded to an arbitrary number of dimensions, where each
sample is the only one in each interval containing it (Iman, Helton and Campbell, 1981). LHS is
especially useful when the number of samples is limited, as it can recreate input distributions
49. 49
efficiently and covers a wide range of the parameter space. Furthermore, it negates the aliasing
of results which uniform sampling may cause (Santner, Williams and Notz, 2018, pp. 145-200).
Figure 33. Latin square (Santner, Williams and Notz, 2018)
LHS of the parameter space was conducted using the Design of Experiments tool in ANSYS
Workbench. 75 samples were created; this number was based on the amount of project time
allocated for the parametric study. Appendix 11 shows the multiplier combinations for each
sample. The multiplier combinations were input into a modified version of DefineEllipse
(Appendix 12) to produce a MATLAB figure plot for each sample. Some examples are shown in
Appendix 13.
7.3 Macros
A new script called InventorParams was created and exports a .csv file of the ellipse curve
coefficients for each sample. More information on what the coefficients are can be found in
Appendix 2.1 and Appendix 2.2. The code for InventorParams is shown in Appendix 14.
Using the existing second pass model geometry in Autodesk Inventor, a macro coded in Visual
Basic Application was created to automatically import a csv file containing ellipse coefficient
values, into Inventor to update the model geometry in accordance with the new coefficients.
The macro also repeated the operation for all 75 samples. Geometry of the domain was kept the
same as the second pass model, with the exception of different contraction geometry in each
sample.
Lastly, a Python script was written to automate the creation of named selections in each sample,
after the geometry is imported into SpaceClaim. The named selections were used for assigning
a structured mesh to edges and boundary conditions to faces. The script also repeated the
operation for all 75 samples.
50. 50
8 Parametric Study CFD
8.1 Parametric Study Meshing
8.1.1 Adaptation from Second Pass Mesh
The second pass model mesh performed well to give overall results of velocity acceleration and
recirculation. However, the results mapped the mesh in the near wall region as a result of poor
blending between the inflation layer and the freestream. Additionally, the mesh was found to
be nonconformal between cell zones. The former was corrected using a lower 𝑦+
for wall
adjacent elements and thicker inflation layers, while the latter was mitigated by using shared
topology in SpaceClaim. Regions of high gradient were observed at the contraction outlet plane
as documented in Section 6.5.3. As the gradients were not as high in the test section as expected,
the refinement was reduced here and increased near the contraction outlet plane by a factor of
3.
8.1.2 𝑦+
Calculations
The Drew, Koo and McAdams relation (1930) is widely used to calculate the average skin friction
coefficient for turbulent flow in a smooth duct within the range of 3 × 103
< 𝑅𝑒 < 3 × 106
:
𝑐𝑓 = 0.0014 + 0.125𝑅𝑒
−0.32
(22)
Although the relation is based on round pipe calculations, it can also be used for non-circular
conduits by using the hydraulic diameter, 𝐷ℎ as the characteristic length in the Reynold’s
number calculation.
As high gradients were present at the contraction outlet plane in the second pass model (Figure
26), the 𝑦+
calculations were adapted to ensure a value of 1 at this plane. The average Reynold’s
number and mean velocity at the contraction outlet plane were used.
As the contraction is a duct, Equation (22) was thought to be more accurate for the case than
Equation (20), which is an approximation on a flat plate. The average Reynold’s number at
contraction outlet plane is 1.01 × 106
and therefore falls within the required range. With the
assumption that the contraction walls will be sufficiently smooth, the relation was used for the
𝑦+
calculations, shown in Appendix 15.
8.1.3 Inflation Layer Calculations
For a more conservative calculation, a distance of 10 𝑚 was added to the maximum travel length
of the fluid in calculating the boundary layer thickness at far outlet. This increased the thickness
of the required inflation layer with the aim of improving blending. To account for a thicker
boundary layer under the node count limitation, the inflation layer growth rate, 𝑔 was increased
to 1.25. Contrary to Elmekawy’s recommendations (2018), this compromised a small amount of
accuracy in the viscous sublayer but allowed for better blending with the freestream. The
calculations are shown in Appendix 16.
8.1.4 Results
All other mesh settings were the same as those used in the second pass model. All calculation
results for the parametric study mesh are shown in Table 9.
51. 51
Table 9. Parametric Study Mesh Parameter Values
Symbol Definition Value
𝑅𝑒 Average Reynolds number 1.01 × 106
𝑦+ y-plus 1
𝑦𝑝
Distance between first cell element
centre and nearest wall
1.4 × 10−5
𝑚
𝑦1
Height of the first cell element from the
wall
2.8 × 10−5
𝑚
𝑔 Inflation layer growth rate 1.25
𝑔 𝑓 Highest freestream growth rate 1.4
𝑦 𝐻 Total inflation layer thickness 188 𝑚𝑚
𝑁 Number of inflation layers 30
Number of nodes 5.01 × 105
Number of elements 4.91 × 105
The same mesh was applied to all samples, ensuring consistency. However, as the contraction
geometry was different for all samples, a source of discretization error occurred whereby the
mesh quality was better for some samples than others. This is unavoidable if the same mesh is
to be used. An example of a meshed sample is shown in Figure 34.
Figure 34. Example of a parametric study mesh (Sample_1), top: plan view, bottom: side view
from outside track
52. 52
8.2 Parametric Study Set-Up
8.2.1 Turbulence Model Change
It was discovered that wall functions used in 𝑘 − 𝜀 models in CFX and Fluent do not resolve
elements below a 𝑦+
of 11. This is because they are empirical equations that do not model the
viscous sublayer well. It was important to resolve up to the wall, as some of the gradients in the
near wall region were very large in the second pass model, especially at the contraction outlet
plane (Figure 26).
The turbulence model was changed to Menter’s Shear Stress Transport (SST) model (1994), a
type of 𝑘 − 𝜔 model that combines the 𝑘 − 𝜀 model and 𝑘 − 𝜔 model. Instead of solving 𝜀 at
discretized points, specific turbulent kinetic energy dissipation rate, 𝜔 is solved instead in 𝑘 −
𝜔 models. The 𝑘 − 𝜀 formulation is used in the freestream and 𝑘 − 𝜔 formulation is used in the
boundary layer, replacing the wall function to accurately resolve up to the wall. Versteeg and
Malalasekeera (2007, p.91) explain that due to its high fidelity at the wall, SST is especially useful
when predicting separation and has good free stream and boundary layer results.
8.2.2 Outlet Boundary Condition Change
The outflow boundary condition used in the second pass model is only fully accurate if the flow
is fully developed, as explained in Section 6.3. To better represent the situation, the top, bottom
and far outlets were combined into one outlet, called the “test section outlet” and a mass flow
rate boundary condition was assigned to it. The target mass flow rate was 12.5 𝑘𝑔/𝑠 , as
calculated in Appendix 17. All other set up settings were the same as those used in the second
pass model.
8.3 Parametric Study Solving
Monitor points in the second pass model were reused for all samples as they proved effective as
indications of convergence. The samples were run for a minimum of 1000 iterations and all
samples met the residual targets detailed in Table 2 after the 1000 iterations. The monitor
points were used as an additional source of convergence assurance.
53. 53
9 Parametric Study Results and Discussion
9.1 Correction Checks
To be consistent with first and second pass model results, the following spot checks on a number
of sample results were made: appropriate boundary layer to freestream blending, mesh
conformality, and appropriate boundary layer growth. The new set-up was able to resolve
Görtler vortices. The adaptations described in Section 8.1.1 were successful in producing
expected flow features.
9.2 Results Planes
The following were optimised in accordance with the design criteria (Section 3.2) and
Objectives 2 and 3 (Section 1.2): exit flow uniformity, minimum turbulence and minimum
vortex effects. The design criteria was assessed at three locations: contraction outlet plane, far
outlet plane and test section middle plane (TSMP). The TSMP is the 𝑌 − 𝑍 plane located
equidistant between the other two planes. Illustrated in Figure 35, flow at these planes were
considered critical for the performance of the contraction and should give a good overall picture
of the flow passing through the test section.
The variation of four multipliers across the samples culminated six pairs of multiplier
combinations. These are listed in Table 10. Each design criterion was quantified for each sample
and plotted on the 𝑧-axis against the six pairs of multiplier combinations to produce six surface
plots at each results plane. There were 75 plotted points on each plot, each sample representing
a point.
Figure 35. Parametric study planes (test section is shaded grey)
54. 54
Table 10. Pairs of Multiplier Combinations
𝑥-axis 𝑦-axis
𝑀 𝐻𝑏 𝑀 𝐻𝑐
𝑀 𝐻𝑏 𝑀 𝑇𝑏
𝑀 𝐻𝑏 𝑀 𝑇𝑐
𝑀 𝐻𝑐 𝑀 𝑇𝑏
𝑀 𝐻𝑐 𝑀 𝑇𝑐
𝑀 𝑇𝑏 𝑀 𝑇𝑐
9.3 Surface Fitting
For all three-dimensional surfaces plotted, the settings shown in Figure 36 were used:
Figure 36. Surface plot settings
Lowess is a scatterplot smoothing algorithm that divides a plot into regions and computes an
average value for the dependent variable in each region (Cleveland, 1981). Linear regression is
then performed to produce a three-dimensional surface through these average values. The
“Span” governs the percentage of data points influencing a regional average value. For example,
a span of 30% means that an average value is influenced by 30% of all data points that are closest
to the region. The span was set to this value to produce plots that were neither overfitted, where
the curve meets all the data points in a jagged manner, or underfitted, where the curve fails to
capture enough detail and overgeneralises the relationship.
Figure 37. Two-dimensional example of regression fit quality, adapted from Johnson (2013)
The “polynomial” governs the method that the surface uses to connect between the regional
average points. For example, choosing “linear” builds the surface using discrete flat surfaces.
Linear was used as higher degree polynomials can overfit the data.
Outliers can distort the values of regional average points. To overcome this problem, the
smoothing can be augmented by using a robust version of Lowess. Bisquare is a version that
assigns a weighting to each data point, depending on how far they are from the fitted surface.
55. 55
Afterwards, the surface is refitted, using local regression and the weights (Cleveland, 1979). As
a number of outliers were observed on many of the surface plots, bisquare was enabled.
R-square is a measure of the proportion of variance for a dependent variable that is explained
by the independent variables in a regression model (Bar-Gera, 2015). For example, an R-square
of 0.4 means that the fit explains 40% of the total variation in the dependent variable. The
adjusted R-square is used to account for using more than one independent variable, such as in
three-dimensional plots. For all surface plots, the adjusted R-square was evaluated to assess the
level of influence that the multipliers had on the dependent variable.
9.4 Velocity Nonuniformity
9.4.1 Velocity Nonuniformity Equations
Reducing velocity nonuniformity optimises velocity uniformity. To quantify velocity
nonuniformity, an analogy from turbulence intensity was made. At any point in space and time,
instantaneous velocity, 𝑢 can be decomposed into a time-averaged mean velocity 𝑢 and
fluctuating velocity 𝑢′
in turbulent flow (Cebeci, 2013):
𝑢 = 𝑢 + 𝑢′ (23)
This is illustrated in Figure 38.
Figure 38. Turbulent velocity, adapted from Burt (2010)
𝑢 can be calculated as:
𝑢 = lim
𝑇→∞
1
𝑇
∫ 𝑢 𝑑𝑡
𝑇
0
(24)
𝑇 = total time duration. The RMS of the velocity fluctuations with time, 𝑢 𝑅𝑀𝑆 can be calculated
using the following:
56. 56
𝑢 𝑅𝑀𝑆 = √ lim
𝑇→∞
1
𝑇
∫ 𝑢′2
𝑑𝑡
𝑇
0
(25)
Finally, turbulence intensity, 𝐼 is defined as:
𝐼 ≡
𝑢 𝑅𝑀𝑆
𝑢
(26)
By using area as the independent variable instead of time, the following analogous equations to
assess 𝑥-velocity nonuniformity were constructed:
𝑢 𝑥 = 𝑢 𝑥 + 𝑢 𝑥
′ (27)
𝑢 𝑥 =
1
𝐴 𝑡
∑ 𝑢 𝑥 𝐴 𝑛
𝑁
𝑛=1
(28)
𝑢 𝑥 𝑅𝑀𝑆
= √
1
𝐴 𝑡
∑ 𝑢 𝑥
′ 2
𝐴 𝑛
𝑁
𝑛=1
(29)
𝜙 𝑥 =
𝑢 𝑥 𝑅𝑀𝑆
𝑉
(30)
Table 11. Velocity Nonuniformity Nomenclature
Symbol Definition
𝑢 𝑥 Instantaneous 𝑥-velocity
𝑢 𝑥 Area-averaged mean 𝑥-velocity
𝑢 𝑥
′ Fluctuating 𝑥-velocity
𝐴 𝑡 Total planar area
𝑛 Element counter
𝑁 Total number of elements
𝐴 𝑛 Area of element
𝑢 𝑥 𝑅𝑀𝑆 RMS of 𝑥-velocity fluctuations with respect to area
𝑉 Target velocity of 30 𝑚/𝑠
𝜙 𝑥 𝑥-velocity nonuniformity
The main differences between Equations (23) to (26) Equations (27) to (30) is the use of
summations instead of integrals in the latter. The area of a plane is finite and discretized into 𝑁
number of elements, while Equation (24) is generalised for infinite time. 𝑥 -velocity
nonuniformity, 𝜙 𝑥 is the ratio between the RMS 𝑥-velocity fluctuation, 𝑢 𝑥 𝑅𝑀𝑆
and the target
mean velocity of 30 𝑚/𝑠, 𝑉. The same analogy can be applied to 𝑦-velocity and 𝑧-velocity, as
shown in Appendix 18, producing 𝑦-velocity nonuniformity, 𝜙 𝑦 and 𝑧-velocity nonuniformity
𝜙 𝑧.
57. 57
9.4.2 Velocity Nonuniformity Results
Only velocity results of elements within the central 90% area of the results planes were used in
these calculations as the focus was on uniformity in the freestream. For each sample, 𝜙 𝑥, 𝜙 𝑦 and
𝜙 𝑧 were calculated at the results planes shown in Figure 35. See Appendix 19, Appendix 20 and
Appendix 21 for the nonuniformity results at the contraction outlet, TSMP and far outlet planes
respectively.
It was discovered that surface plots containing 𝜙 𝑦 and 𝜙 𝑧 as the dependent variable had poor
adjusted R-square values of 0.1 or below. Hence the multipliers had very little influence on 𝜙 𝑦
and 𝜙 𝑧. Therefore, these plots were not analysed further and are not included in this report.
Some surface plots containing 𝜙 𝑥 as the dependent variable had some respectable adjusted R-
square values above 0.5. Therefore, sensitivity analysis was conducted on them to scope
multiplier ranges that optimised 𝜙 𝑥. MHb vs MHc vs 𝜙 𝑥 at the results planes are shown in Figure
39 to Figure 41 as examples of the surface plots produced. See Table 12 for the location of all 𝜙 𝑥
surface plots in this report.
Table 12. Velocity Nonuniformity Surface Plot Location
Plots Caption or Location
𝑀 𝐻𝑏 vs 𝑀 𝐻𝑐 vs 𝜙 𝑥 Figure 39 to Figure 41
𝑀 𝐻𝑏 vs 𝑀 𝑇𝑏 vs 𝜙 𝑥 Appendix 22
𝑀 𝐻𝑏 vs 𝑀 𝑇𝑐 vs 𝜙 𝑥 Appendix 23
𝑀 𝐻𝑐 vs 𝑀 𝑇𝑏 vs 𝜙 𝑥 Appendix 24
𝑀 𝐻𝑐 vs 𝑀 𝑇𝑐 vs 𝜙 𝑥 Appendix 25
𝑀 𝑇𝑏 vs 𝑀 𝑇𝑐 vs 𝜙 𝑥 Appendix 26
To ensure robustness, an acceptance range was implemented to all variables, equal to a quarter
of the total range observed in the results, shown in Table 13. This limits changes in 𝜙 𝑥 across a
sufficiently large area of the two multipliers plotted. Optimisation results for 𝜙 𝑥 at contraction
outlet, TSMP and far outlet are tabulated in Table 14, Table 15 and Table 16 respectively.
Table 13. Acceptance Range of Variables
Variable Total Range Acceptance Range
𝜙 𝑥 0.017 0.004 or less
𝑀 𝐻𝑏 1.250 0.300 or more
𝑀 𝐻𝑐 0.250 0.060 or more
𝑀 𝑇𝑏 4.000 1.000 or more
𝑀 𝑇𝑐 4.000 1.000 or more
