The UMIST-N Near-Wall Treatment Applied to Periodic Channel Flow

The UMIST-N Near-Wall
Treatment Applied to Periodic
Channel Flow
Submitted for the Degree of
Master of Philosophy
by
Bryn Richards
Department of Mechanical, Aerospace
and Manufacturing Engineering
University of Manchester
Institute of Science and Technology
2005

Declaration
No portion of the work referred to in this thesis has been submitted in sup-
port of an application for another degree or qualification of this or any other
university or other institution of learning.
i

Acknowledgements
I would like to thank my supervisor, Dr. A. P. Watkins for his guidance
throughout this project and for his willingness to be interrupted and to invest
time with me whenever I have sought his attention. His style as a supervisor
allows flexibility, independence, and self-actualisation. Our discussions have
generated possibilities more than they have limited them.
I would like to thank Professor Dominique Laurence for his involvement. He
has continually made connections that have identified further sources of input
into the project and further applicability to the work of others.
Dr. T. J. Craft has offered access to his unparalleled power of explanation
and keen investigative ability whenever I have approached his open door.
Dr. Simon Gant has offered patient and careful explanations on innumerable
occasions. I greatly admire the enthusiasm, stewardship, and excellence that
he brings to every facet of his professional life.
Many others throughout the department have acted as sources of inspiration
and support. I will continue to benefit from the diverse perspectives and
experiences of the many interesting people whom I have had the good fortune
to meet during my time at UMIST.
I gratefully acknowledge the support of the DESIDER project in partially
funding this research.
ii

Abstract
This thesis assesses the performance of the UMIST-N subgrid near-wall treat-
ment when applied to periodic flow. The thesis also assesses the use of the
k-ω turbulence model with the subgrid approach. Based on this work, the
approach appears to be applicable to time-variant flow. Further research is
required to improve the implementation of the k-ω model.
UMIST-N solves simplified transport equations near a solid boundary at a
lower computational cost than that of a low-Reynolds-number treatment.
In applying the method to periodic channel flow, none of the approaches
considered performed in an exemplary manner, but the subgrid exhibited no
apparent failing when compared to the low-Reynolds-number results. It did
offer an improvement upon the logarithmic law of the wall.
This work highlights a numerical difficulty in applying the subgrid solution
as a boundary layer to the main grid when the k-ω model is used. This has
manifest itself in the results as an enhanced propensity toward a discontinuity
in calculated profiles at the subgrid / main grid interface under certain flow
conditions.
A range of steady channel flow data has been compiled and presented in this
thesis. Empirical correlations are offered which identify general tendencies
in the data and may provide a useful tool for researchers engaged in the
computation of flows.
iii

Contents
1 Introduction & Literature Survey 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Wall Functions & the Subgrid Approach . . . . . . . . . . . . 7
1.4 Relevance to Large Eddy Simulation . . . . . . . . . . . . . . 11
1.5 Periodic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Study Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Turbulence Models 17
2.1 Reynolds Averaging . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The k-ε Model . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 The Low-Reynolds-Number k-ε Model . . . . . . . . . 22
2.2.2 Yap Correction . . . . . . . . . . . . . . . . . . . . . . 24
2.3 The k-ω Model . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 The Logarithmic Law of the Wall . . . . . . . . . . . . . . . . 26
3 Channel Flow 31
3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 The k-ε Model . . . . . . . . . . . . . . . . . . . . . . 33
iv

3.1.2 The k-ω Model . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Flow Characterisation . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Steady Channel Flow Data . . . . . . . . . . . . . . . . . . . . 37
3.4.1 Empirical Profile for U + . . . . . . . . . . . . . . . . . 39
3.4.2 Empirical Profile for − 〈uv〉 + . . . . . . . . . . . . . . 39
3.4.3 Empirical Profile for k + . . . . . . . . . . . . . . . . . 40
3.4.4 Empirical Profile for 〈u 2 〉 + . . . . . . . . . . . . . . . . 41
3.4.5 Empirical Profile for 〈v 2 〉 + . . . . . . . . . . . . . . . . 42
3.4.6 Near-Wall Behaviour . . . . . . . . . . . . . . . . . . . 43
3.5 Local Nondimensionalisation . . . . . . . . . . . . . . . . . . . 45
4 Numerical Implementation 48
4.1 The Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Volume Integral Form . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.1 Wall Boundaries on k-ε . . . . . . . . . . . . . . . . . . 55
4.4.2 Wall Boundaries on k-ω . . . . . . . . . . . . . . . . . 55
4.4.3 The Logarithmic Law of the Wall . . . . . . . . . . . . 56
4.4.4 The Subgrid Approach . . . . . . . . . . . . . . . . . . 60
4.5 Under-Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Results 66
5.1 Steady Flow Results . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Prescribed Periodic Pressure Gradient . . . . . . . . . . . . . 71
5.3 Prescribed Periodic Bulk Flow Rate . . . . . . . . . . . . . . . 75
6 Conclusions & Suggestions for Future Work 83
v

List of Figures
2.1 The log-law compared to experiments in channel flow . . . . . 30
4.1 The low-Reynolds-number grid . . . . . . . . . . . . . . . . . 49
4.2 The high-Reynolds-number grid . . . . . . . . . . . . . . . . . 57
4.3 The subgrid mesh, adapted from Gant [21] . . . . . . . . . . . 60
3.1 Reichardt’s law to estimate U + . . . . . . . . . . . . . . . . . 100
3.2 Reichardt’s law applied to − 〈uv〉 + . . . . . . . . . . . . . . . 101
3.3 A revised profile for − 〈uv〉 + . . . . . . . . . . . . . . . . . . . 102
3.4 A profile for k + . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.5 A profile for 〈uu〉 + . . . . . . . . . . . . . . . . . . . . . . . . 104
3.6 A profile for 〈vv〉 + . . . . . . . . . . . . . . . . . . . . . . . . 105
3.7 Near-wall behaviour of flow parameters . . . . . . . . . . . . . 106
3.8 y ∗ vs. y + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.9 U ∗ superimposed on U + . . . . . . . . . . . . . . . . . . . . . 108
3.10 y ∗ v2 vs. y + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.11 U ∗ v2 superimposed on U + . . . . . . . . . . . . . . . . . . . . . 110
5.1 〈U〉 + vs y/δ in the steady flow case . . . . . . . . . . . . . . . 111
5.2 k + vs y/δ in the steady flow case . . . . . . . . . . . . . . . . 112
5.3 〈U〉 + vs y + in the steady flow case . . . . . . . . . . . . . . . 113
5.4 k + vs y + in the steady flow case . . . . . . . . . . . . . . . . . 114
vi

5.5 Bulk flow variation in the periodic pressure case . . . . . . . . 115
5.6 Wall shear stress variation in the periodic pressure case . . . . 116
5.7 Variables with time at y/δ = 0.1 (prescribed
5.8 Variables with time at y/δ = 0.2 (prescribed
5.9 Variables with time at y/δ = 0.5 (prescribed
5.10 Variables with time at y/δ = 0.9 (prescribed
5.11 〈U〉 vs y/δ snapshots through time (prescribed
5.12 〈U〉 vs y/δ snapshots through time (prescribed
5.13 k vs y/δ snapshots through time (prescribed
5.14 k vs y/δ snapshots through time (prescribed
5.15 〈U〉 vs y + snapshots through time (prescribed
5.16 〈U〉 vs y + snapshots through time (prescribed
5.17 k vs y + snapshots through time (prescribed
5.18 k vs y + snapshots through time (prescribed
∂〈P 〉
) . . . . . . . 117
∂x
∂〈P 〉
) . . . . . . . 118
∂x
∂〈P 〉
) . . . . . . . 119
∂x
∂〈P 〉
) . . . . . . . 120
∂x
∂〈P 〉
) - part 1 . 121
∂x
∂〈P 〉
) - part 2 . 122
∂x
∂〈P 〉
) - part 1 . . 123
∂x
∂〈P 〉
) - part 2 . . 124
∂x
∂〈P 〉
) - part 1 . 125
∂x
∂〈P 〉
) - part 2 . 126
∂x
∂〈P 〉
) - part 1 . . . 127
∂x
∂〈P 〉
) - part 2 . . . 128
∂x
5.19 Pressure variation in the periodic bulk flow . . . . . . . . . . . 129
5.20 Wall shear stress variation in the periodic bulk flow case . . . 130
5.21 Variables with time at y/δ = 0.1 (prescribed U) . . . . . . . . 131
5.22 Variables with time at y/δ = 0.2 (prescribed U) . . . . . . . . 132
5.23 Variables with time at y/δ = 0.5 (prescribed U) . . . . . . . . 133
5.24 Variables with time at y/δ = 0.9 (prescribed U) . . . . . . . . 134
5.25 〈U〉 vs y/δ snapshots through time (prescribed U) - part 1 . . 135
5.26 〈U〉 vs y/δ snapshots through time (prescribed U) - part 2 . . 136
5.27 k vs y/δ snapshots through time (prescribed U) - part 1 . . . 137
5.28 k vs y/δ snapshots through time (prescribed U) - part 2 . . . 138
5.29 〈U〉 vs y + snapshots through time (prescribed U) - part 1 . . . 139
5.30 〈U〉 vs y + snapshots through time (prescribed U) - part 2 . . . 140
5.31 k vs y + snapshots through time (prescribed U) - part 1 . . . . 141
vii

5.32 k vs y + snapshots through time (prescribed U) - part 2 . . . . 142
viii

List of Tables
2.1 Constants in the standard k-ε model [33] . . . . . . . . . . . . 22
2.2 Constants in the Wilcox 1988 k-ω model [89] . . . . . . . . . . 26
2.3 Log-law constants [83] . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Near-wall flow regimes adapted from [53] . . . . . . . . . . . . 28
3.1 Qualitative behaviour of terms in Equation 3.41 . . . . . . . . 41
3.2 Qualitative behaviour of terms in Equation 3.42 . . . . . . . . 42
4.1 The notation for discretised values . . . . . . . . . . . . . . . 50
4.2 Under-relaxation factors used in the main grid . . . . . . . . . 65
4.3 Under-relaxation factors used in the subgrid . . . . . . . . . . 65
5.1 Configurations of turbulence models . . . . . . . . . . . . . . . 67
ix

Chapter 1
Introduction & Literature
Survey
1.1 Background
Computational Fluid Dynamics (CFD) is a field of study that seeks to simu-
late and predict fluid flow using computers. CFD may occasionally be used
to investigate the physical behaviour of flow, but the usual goal is to provide
an input to engineering analysis and design. Following from this goal, the
thrust of most CFD development is to continually reduce the extent of the
tradeoff between the predictive accuracy and the computational affordability
of CFD. The field of CFD has developed quickly in recent decades because
of the rapid advance and increasing accessibility of computer technology.
Having first made CFD possible, then practical as an engineering tool, this
ongoing advancement increasingly drives a trend toward more complex cal-
culations. Recognising this, researchers in CFD tend to direct their efforts
toward enhancing the predictive accuracy per unit cost of CFD treatments
1

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 2
that are ever more complex.
The type of flow considered in this thesis is incompressible and Newtonian.
On scales that are several orders of magnitude larger than the molecular
scale, the flow is governed by the Navier-Stokes equations. These equations
model a flow field as a continuum. CFD codes discretise the flow domain
to allow variables to be represented numerically. The most common type of
discretisation is the Finite Volume method, which is applied in this work. The
governing equations of fluid flow identify a large degree of interconnectedness
between the various properties of a flow field. Because of this, most CFD
codes use an iterative approach. This involves traversing the flow field and
everywhere calculating flow parameters based on local information. The
solution converges as this process is repeated with updated local information.
In general, a flow field is a complex system with mechanisms for internal
feedback. Because of this, a flow field can exhibit chaotic behaviour. This
means that, although fluid flow is deterministic, it may appear to include
randomness because of its extreme physical complexity. A flow is classed as
turbulent when it behaves chaotically. This is a subjective definition, and
there does exist ‘transitional flow’ where the appropriateness of classing the
flow as turbulent is unclear. However, most flows of relevance to engineering
applications are clearly turbulent, and most CFD work is aimed at predicting
the behaviour of turbulent flow. Most commonly, this is accomplished by
the adoption of a turbulence model that makes use of certain assumptions to
approximate flow behaviour using more tractable equations. This precludes
the possibility of predicting every feature of the flow, but the parameters
providing the greatest engineering relevance may be estimated at a greatly
reduced computational cost.

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 3
One approach to turbulence modelling is the Reynolds Averaged Navier-
Stokes (RANS) type. RANS models use time- or ensemble-averaged Navier-
Stokes equations to calculate the mean values of flow parameters. 1 The
fluctuating components of these parameters are modelled, rather than being
fully resolved. Conceptually, this amounts to solving a flow as though it
were laminar, but with the addition of modelled turbulence superimposed
over the bulk flow behaviour. This modelled turbulence affects the bulk flow
according to the details of the model.
The most popular RANS models are the Eddy-Viscosity Models (EVMs).
EVMs model the affect of turbulence on bulk flow via the concept of turbulent
viscosity. Local turbulence is presumed to manifest itself as an increase in
the effective viscosity of the fluid. The physical justification for this concept
is that turbulence entails greater interaction between fluid particles. This
leads to a greater exchange of energy between adjacent parcels of fluid. In
terms of the momentum equations, this interaction produces an effect that
is analogous to increased viscosity.
The EVM may be extended by the incorporation of non-linear terms [77, 14]
This is the non-linear type of EVM. An alternative to EVMs is presented
by the Reynolds Stress Transport models [69, 15]. Here, the products and
squares of root-mean-square velocity fluctuations (Reynolds stresses) are cal-
culated as being convected and diffused through the flow field according to
their own transport equations. This thesis deals with the application of linear
EVM RANS.
1 Ensemble averaging is associated with periodic flow problems, and time averaging is
associated with steady problems.

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 4
Another type of CFD is Direct Numerical Simulation (DNS) [30]. DNS in-
volves numerical approximations to discretize the flow field, but no turbu-
lence model is used. Instead, DNS resolves all the quantities associated
with a flow, including small-scale turbulent fluctuations. DNS is very com-
putationally expensive, and is only employed for problems involving simple
geometries and low Reynolds numbers. Although there is a potential for
error in DNS studies, DNS results are usually taken to represent true fluid
behaviour, when compared against modelled results. Because DNS provides
more complete information than do experiments, DNS data are often used
to validate turbulence models.
A popular test case for CFD is channel flow. The geometrical simplicity
of channel flow allows solutions to be achieved at minimal computational
expense and also reduces the complexity of the CFD code, minimizing the
potential for errors in coding. This thesis deals with channel flow. In addi-
tion to the steady-flow case, a variable pressure gradient is applied to test
the impact of periodic fluctuations on the solution method. The solution of
channel flow leads to a system of equations that is parabolic, meaning that
they may be solved using only local and upstream information. In the case
of steady flow, the flow field is also statistically stationary, meaning that flow
statistics such as mean flow and turbulence levels are invariant with time.
In the case where a variable pressure gradient is applied, the flow field is
not strictly stationary, but a converged solution produces results which are
statistically stationary with respect to a given phase angle of pressure gradi-
ent fluctuation. Thus, ensemble averaging of the Navier-Stokes equations is
used.

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 5
1.2 Turbulence Models
RANS models are based upon the idea of filtering turbulence from the gov-
erning equations of a flow so that it may be treated separately. This is due
to Reynolds [59]. In EVMs, the problem of knowing the effect of turbulence
on the mean flow is made tractable by employing the idea of a turbulent
viscosity, based on the work of Boussinesq [6]. Turbulent viscosity may be
calculated in a number of different ways.
The simplest approach is to specify the turbulent viscosity at a given location
based on known local quantities. Models based on this approach are called
‘algebraic models’. Taylor [80] and Prandtl [54] have proposed an algebraic
model in which turbulent viscosity is calculated as a function of a length scale
and a local mean velocity gradient. The length scale is specified as a function
of wall-distance. Unfortunately, an appropriate length scale can be difficult
to obtain in complex geometries. Another limitation of algebraic models is
that the dependence of turbulent viscosity on a local mean velocity gradient
is unrealistic in many types of flow.
To remove the latter limitation of algebraic models, ‘one-equation’ models
track an additional turbulent quantity through the solution of an additional
transport equation. The most popular choice of the additional quantity is
the turbulent kinetic energy per unit mass, k (usually referred to simply as
turbulent kinetic energy). Prandtl [55] proposed this approach. Spalart et
al. [73] have proposed directly solving a transport equation for turbulent
viscosity. A limitation that is common to all one-equation models is that a
length scale is still required to fully specify the modelled flow.

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 6
To remove the dependence on a length scale, ‘two-equation’ models intro-
duce an additional transported quantity. Various choices exist for the second
quantity. Kolmogorov [31] suggested k 1
2 /l, where l is the mixing length. This
quantity has later been dubbed ω. Chou [9] proposed modelling the rate of
dissipation of turbulence, ε. In terms of dimensional analysis, this amounts
to modelling k 3
2 /l. However, it may be argued that turbulent dissipation,
which involves the conversion of turbulent energy to heat, takes place on a
physically much smaller scale than turbulence itself, so that ε and k 3
2 /l may
not be the same thing [91].
Other choices for the transported quantities bear mentioning. Rotta [63]
and Spalding [74] proposed l as the second quantity. Rotta [64], Rodi &
Spalding [62], and Ng & Spalding [46] proposed models using the product kl.
Spalding [75], Wilcox & Rubesin [88], and Robinson et al. [61] have proposed
ω 2 . Coakley [10] proposed k 1
2 and ω as the two transported quantities. These
two-equation models are compared and discussed by Wilcox [91].
The most popular choice of two-equation model is the k-ε model of Launder
& Sharma [33]. This is a revised tuning of the k-ε model of Jones & Launder
[27]. These papers follow the work of Chou [9], Davidov [19], and Harlow &
Nakayama [23]. The work of Jones & Launder introduced a modification to k-
ε modelling that allowed the model to be applied in the near-wall region. The
model of Launder & Sharma is commonly called the ‘standard’ k-ε model.
Yap [92] improved the performance of the standard k-ε model in impinging
and recirculating flow. Yap’s modification, the ‘Yap correction’ is a popular
addition to the model.
The k-ω model is a popular alternative to the k-ε model. Following the

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 7
work of Kolmogorov [31] and Saffman [65], Wilcox [89] proposed the most
well-known k-ω model. The model does not require the same damping terms
employed by the standard k-ε model in order to be effective in the near-wall
region, but boundary conditions are more difficult to apply. The principal
difficulty associated with the k-ω model is in specifying a value for ω in free-
stream turbulence [40]. Speziale et al. [76], Menter [41], Peng et al. [52],
and Wilcox [90] have proposed modifications to the k-ω that further improve
its performance at the cost of adding much the same degree of complexity
found in the standard k-ε model.
The SST 2 model of Menter [42] may be thought of as a hybrid k-ε / k-ω
approach. It employes a k-ω model near solid boundaries and a k-ε model
elsewhere. The SST model is appealing because the principal strength of
the k-ω model is its simplicity and relative accuracy in the near-wall region,
while the k-ε model is generally more effective in free-stream flow. The SST is
implemented by the use of a blending function to provide a smooth transition
between the two models.
1.3 Wall Functions & the Subgrid Approach
Near solid boundaries, the gradients of turbulence quantities become large.
Numerically, this means that greater storage and computational demands are
placed on a CFD code that performs calculations in the near-wall region. To
avoid this, wall functions are often employed. A wall function is a solution
method that provides a means of characterising turbulence at some point,
2 Shear Stress Transport

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 8
P away from the wall. The calculations performed by the wall function
are simpler than applying a full CFD approach between the wall and the
point P . Thus, a computational savings is achieved. The CFD code then
calculates the remaining flow field, taking the wall-function output at point P
as a boundary condition, replacing the actual wall boundary condition. The
disadvantage of wall functions is that they cannot provide the same level of
accuracy as a full CFD treatment near the wall, except in certain flows for
which they were explicitly designed. 3
The most common type of wall function is that which relies on the ‘logarith-
mic law of the wall’. This empirical equation relates velocity to wall-normal
distance within a certain near-wall region. Launder & Spalding [34] have
produced a log-law-based wall-function that is a popular default in many
industrial applications. In addition to offering a means of calculating near-
wall velocity, turbulent kinetic energy, k is calculated by tracking the rates
of production and dissipation of this quantity, averaged analytically over the
near-wall region. This wall function acts as a one-equation model in the
sense that it only resolves production and dissipation of one quantity. When
used as a boundary condition on two-equation models, the second turbulence
quantity is calculated from k and the length scale. Launder & Spalding offer
a means of calculating ε. Wilcox [91] offers a similar wall function for the
k-ω model.
Various other wall treatments exist. These may be thought of as falling on
a spectrum between a costly but relatively accurate full CFD solution and,
as the other polar extreme, a set of simple algebraic expressions resulting
3 The great majority of wall functions have been designed for steady channel flow,
including the wall function of Launder & Spalding [34] employed in this work.

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 9
from an analytical treatment of some presumed near-wall behaviour. Be-
tween these two extremes, various alternatives exist. Broadly, these may be
classified as either attempting to extend the applicability of standard wall
functions through the incorporation of more complex mathematics and more
accurate empirical and theoretical information or attempting to simplify CFD
near the wall to achieve accurate near-wall solutions at less computational
expense than the full CFD treatment.
In the realm of extending the applicability of analytical wall functions, Amano
[3] presented a wall function that finds average production and destruction
of ε, in addition to k. Thus, the use of the length scale to calculate ε is
avoided. Smith [72] and Craft et al. [17] have proposed wall functions using
numerical, rather than analytical methods to obtain near-wall averaged val-
ues within the wall function. This allows the use of more complex equations
for turbulence quantities in the wall function, more closely approximating
the RANS equations that would be solved by a full CFD treatment. Viegas
& Rubesin [84] offered a wall function for compressible flow.
Much progress has been made at UMIST in reducing the computational
cost of CFD near solid boundaries. This approach may still be classified
as a wall function treatment, in the sense that the main CFD code sees
a ‘black box’ calculation scheme offering values for flow quantities in the
near-wall region obtained at a reduced computational cost. However, the
approach is not analytical, and instead may be described as using a simplified
CFD calculation applied to the near-wall region. Thus, a potential exists
for this simplified CFD approach to offer enhanced accuracy over other wall
functions, although the implementation is generally more costly and complex.

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 10
A precursor to simplified CFD wall functions is the PSL 4 approach of Ia-
covides [25]. This is a modification to the full CFD near-wall solution that
constrains the CFD code to ignore the coupling between velocity and pressure
in the near-wall region. Instead, the static pressure distribution is taken as
fixed in this region. This results in significant computational savings. Craft
et al. [18] extended this approach by encapsulating the near-wall CFD cal-
culation as an independent code using its own subgrid to perform simplified
CFD calculations in the near-wall region. Thus, the simplified CFD approach
was encapsulated as a wall function. This approach is called UMIST-N 5 . As
with PSL, the assumption of constant pressure distribution results in equa-
tions which are parabolic and may be solved using simplified methods. A
momentum equation is not employed for wall-normal velocity. Instead, lo-
cal conservation of mass is employed in the near-wall region to determine
wall-normal velocities from wall-parallel velocities.
UMIST-N offers a simplified near-wall treatment, relative to a full CFD solu-
tion, but it resolves a two-equation turbulence model near the wall. UMIST-
N returns near-wall averaged production and destruction of two turbulence
parameters to the main grid, eliminating the use of a length scale. The
development of UMIST-N is detailed in the PhD thesis of Gant [21]. The
k-ε model has been employed in UMIST-N, including the non-linear EVM
k-ε model of Craft et al. [14]. UMIST-N has been applied to channel flow,
impinging jet, spinning disk, and Ahmed body flow, and has been adapted
to non-orthogonal coordinate systems [16, 21]. All flows considered thus far
have been steady in time.
4 Parabolic Sub-Layer
5 Unified Modelling through Integrated Sublayer Treatment - a N umerical approach

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 11
1.4 Relevance to Large Eddy Simulation
Large Eddy Simulation (LES) is a modelling approach that allows large-
scale turbulent fluctuations to remain represented within the Navier-Stokes
equations while small-scale turbulent fluctuations are filtered out statistically
and treated separately. Thus it can be thought of as offering a compromise
between DNS and RANS. The LES approach involves the complexities of
resolving large-scale turbulence, of modelling small-scale turbulence, and of
handling the interaction between these two scales of turbulence. However,
LES provides a potential for greater predictive accuracy than any RANS
method.
LES and RANS approaches are far from alien, and hybrid calculations have
been undertaken. Labourasse & Sagaut [32] have run LES within an overall
RANS calculation. This provided a solution that exhibited the robustness
of a RANS method with some additional accuracy derived from the use of
LES. Quéméré et al. [56] have run RANS and LES calculations alongside one
another in different zones within a flow domain. These hybrid investigations
highlight the complimentary strengths of RANS and LES in some flows.
LES and RANS face analogous tradeoffs in the treatment of flow near solid
boundaries. Performing a detailed LES calculation to resolve the turbulence
near a wall is very computationally expensive. In most LES calculations,
a wall function based on the logarithmic law of the wall is used to specify
boundary conditions at a finite distance away from the wall. Balaras et al. [5]
improved upon this by obtaining wall shear stress from a near-wall subgrid
result within an LES calculation. The subgrid employed an algebraic model
to obtain wall-parallel velocity and thus obtain a wall shear stress to act as

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 12
a boundary condition on the LES solution.
The UMIST-N approach entails a more complex calculation than that used
by Balaras et al. and may offer improved results if used in LES. Certainly,
a potential exists for UMIST-N to provide value in LES calculations. In
the future, advanced wall functions may help to alleviate the well-known
challenge in LES of adequately resolving the grid in regions where turbulence
is generated by ‘driving mechanisms’.
Because LES calculations resolve large-scale turbulent structures, any local
region of an LES calculation space can be exposed to rapid fluctuations as
eddies shift and move through the flow field. Therefore, the application of
UMIST-N to periodic flow represents an important first step in assessing its
suitability to LES.
1.5 Periodic Flow
Periodic flow refers to an arrangement in which the flow field varies smoothly
and cyclicly as a function of time. This is sometimes referred to as oscillating
flow. However, some researchers assert that the term ‘oscillating’ implies
reciprocation, meaning that the direction of the bulk flow (and not merely
its magnitude) is changing with time. This distinction is of questionable
value. In this work, ‘periodic’ is used as the generic term. Furthermore,
the term ‘periodic boundary conditions’ refers to boundary conditions that
vary with time, driving periodic flow. This is distinct from the arrangement
where values at an output boundary are copied to an input boundary in a
loop. This type of boundary condition is also termed ‘periodic’, but these do

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 13
not appear in the present work.
Periodic flow provides an interesting test case, because it offers insight into
some of the more illusive physical behaviours of a flow even in relatively
simple geometries. Various experimentalists have investigated periodic flow
in pipe and channel geometries. Sarpkaya [66] investigated periodically- and
randomly-pulsed flow in a pipe from the standpoint of understanding the
conditions under-which the flow transitioned to turbulence. Ohmi & Iguchi
[48] performed an investigation in a similar vein, and Ohmi et al. [49, 50] also
investigated higher Reynolds numbers. Tu & Ramaprian [82, 57] performed
widely-recognised experimental work into periodic pipe flow at a rather high
Reynolds number. More recently, Tardu et al. [79] have investigated periodic
channel flow. Hino et al. [24] performed an investigation of periodic flow at
approximately the same time as Tu & Ramaprian, but their use of a rectan-
gular duct geometry may have impacted the receptiveness of the modelling
community to their results.
Other interesting studies bear mentioning. Shemer et al. [68] offered a com-
parison between laminar and turbulent flows at the same Reynolds number
and under the same periodic conditions. Sleath [71] and Jensen et al. [26]
investigated the impact of various surface roughnesses. Siginer [70] investi-
gated periodic pipe flow using a non-Newtonian fluid. Scotti & Piomelli [67]
have applied LES to periodic channel flow at a range of oscillation frequen-
cies. Lee et al. [36] and Walther et al. [85] have investigated heat transfer
in periodic flow. Lodahl et al. [38] have investigated periodic pipe flow with
a periodic applied electric current.
Furthermore, internal combustion engine experiments offer insight into pe-

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 14
riodic flow. Ahmadi-Befrui et al. [2] published mean velocity readings at
various locations within a cylinder through a piston stroke. Tabaczynski [78]
has investigated reciprocating flow in engines. Also, the Society of Automo-
tive Engineers has published an extensive collection of results.
One very notable DNS study of periodic flow is that of Kawamura & Homma
[29], which investigates channel flow at a low Reynolds number driven by a
periodic pressure gradient. The periodic flow results presented in this thesis
are compared against the DNS results of Kawamura & Homma.
Various researcher have investigated the use of turbulence models in predict-
ing periodic flow. Cotton & Ismael [12] applied the standard k-ε model to
periodic pipe flow. Cotton et al. [13] and Addad [1] have investigated the use
of Reynolds Stress Transport models in predicting periodic flow. Watkins [86]
produced an early investigation into the use of CFD in internal combustion
engines, and various other studies have followed [60, 35]. Naitoh & Kuwahara
[45] applied LES to engine flow.
The relevance of wall-functions to periodic flow has received less attention.
The UMIST-N subgrid wall function has not been applied to periodic flow
prior to this work. Standard wall functions are generally applied to internal
combustion engine studies.
1.6 Study Objectives
The primary objective of this study is to assess the applicability of the
UMIST-N subgrid near-wall treatment to a periodic flow problem, relative to

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 15
other popular approaches. The secondary objective is to experiment with the
use of the k-ω model in the subgrid solution scheme. This work represents
the first application of the UMIST-N approach to time-variant flow and the
first use of a k-ω model within the subgrid calculation.
These objectives are met through the analysis of the logarithmic law of the
wall, the standard low-Reynolds number k-ε treatment, a k-ε subgrid treat-
ment, and a k-ω subgrid treatment in steady channel flow and in periodically
variable channel flow. The results of this study are compared against the de-
tailed DNS data of Kim et al. [30] in the case of steady flow and Kawamura
& Homma [29] in the case of periodic channel flow.
A tertiary objective that arose over the course of the project was to inves-
tigate the study of channel flow in general, so as to provide some input to
other CFD efforts that make use of channel flow data. This objective is met
through a detailed background discussion of channel flow, a compilation of
some useful experimental and DNS results concerning channel flow, and the
revision and presentation of a set of analytical profiles to characterise the ex-
pected behaviours of flow parameters in a channel as a function of Reynolds
number.
1.7 Thesis Outline
The various turbulence models employed in this work are discussed in Chap-
ter 2. This includes a more detailed discussion of the RANS approach and
a presentation of the k-ε and k-ω models. The differences between the high-
and low-Reynolds-number k-ε models are highlighted. The logarithmic law

CHAPTER 1. INTRODUCTION & LITERATURE SURVEY 16
of the wall is introduced.
Chapter 3 contains a detailed discussion of channel flow. The governing equa-
tions associated with the various turbulence models are adapted to channel
flow. The chapter also presents various experimental and DNS channel flow
results arising from other studies. Correlations are presented to identify
trends in the data that may be useful for the design and testing of CFD
codes. Alternative nondimensionalisation schemes are also investigated.
Chapter 4 contains a discussion of the numerical implementation of the tur-
bulence models. This includes a discussion of the Finite Volume method and
the particular grid arrangements used in this work. Boundary conditions are
also discussed, including the implementation of the logarithmic law of the
wall. Since the novelty of UMIST-N is essentially in its numerical treatment
of the boundary layer problem, the approach is detailed in this chapter.
Chapter 5 presents an analysis of the results computed for the various config-
urations considered. This includes steady channel flow, periodic flow results
compared to data with the same driving pressure gradient, and periodic flow
results compared to data with the same bulk flow rate.
Chapter 6 offers conclusions and suggestions for future work.

Chapter 2
Turbulence Models
The Reynolds Averaged Navier Stokes equations are obtained for continuity,
conservation of momentum, and generic transport. The assumptions of the
EVM are employed where appropriate. Then, specific transport equations
are introduced for the turbulence parameters tracked by the k-ε and k-ω
models. The logarithmic law of the wall is introduced.
2.1 Reynolds Averaging
Consider a flow field containing an incompressible fluid with constant proper-
ties: density (ρ), dynamic viscosity (µ), and kinematic viscosity (ν = µ
). In ρ
Cartesian coordinates, the flow field extends in three orthogonal directions,
x, y, and z. The flow field may vary in time, t. A pressure field, P (x, y, z, t)
and a velocity field, U (x, y, z, t) = (U, V, W ) T are associated with the flow.
This flow field is governed by continuity and the conservation of momentum.
17

CHAPTER 2. TURBULENCE MODELS 18
Furthermore, it is assumed that a passive scalar, having no effect on the fluid
properties, may be convected and diffused through the flow field according
to a transport equation.
Neglecting body forces such as gravity, the Navier-Stokes equations governing
incompressible, constant-property Newtonian fluid flow in Cartesian coordi-
nates are
Continuity : ▽ · U = 0 (2.1)
Momentum : DU
Dt
T ransport :
1 = − ▽ P + ▽ · (ν ▽ U) (2.2)
ρ
DΦ
Dt = ▽ · (γ ▽ Φ) + Sφ (2.3)
where Φ (x, y, z, t) represents a scalar quantity transported within the flow
field. γ is the diffusivity of Φ in the fluid. In turbulence modelling, γ is
usually replaced with ν , where σ is the effective Prandtl number of Φ in the
σ
fluid. Sφ (x, y, z, t) in Equation 2.3 is a net source of Φ.
RANS modelling involves time- or ensemble-averaged Navier-Stokes equa-
tions. The aim of Reynolds averaging is to allow separate tracking of mean
flow and turbulent fluctuations. Because time-dependent flows are studied
in this thesis, ensemble averaging is appropriate. Conceptually, an ensemble
average is the mean of the instantaneous values of a parameter through a
large number of repeated experiments. Ensemble averaging is similar to time
averaging in that it separates turbulent fluctuations from the bulk flow by de-
composing the velocity field into a mean component (〈U〉) and a fluctuating
velocity component (u), such that U = 〈U〉 + u. 1 Similarly, Φ = 〈Φ〉 + φ.
1 Other popular notations include an over-bar for mean velocity U and a prime on
fluctuating velocity (u ′ ).

CHAPTER 2. TURBULENCE MODELS 19
The RANS equations are
D
Dt
Continuity : ▽ · 〈U〉 = 0 (2.4)
Momentum :
T ransport :
D〈U〉
Dt
D〈Φ〉
Dt
= −1 ▽ 〈P 〉 + ▽ · (ν ▽ 〈U〉)
ρ
∂ 〈uu〉 ∂ 〈vu〉 ∂ 〈wu〉
− − −
∂x ∂y ∂z
ν
= ▽ · ▽ 〈Φ〉 + Sφ
σ
∂ 〈uφ〉 ∂ 〈vφ〉 ∂ 〈wφ〉
− − −
∂x ∂y ∂z
may be referred to as the mean material derivative and is defined by
D
Dt
(2.5)
(2.6)
∂
≡ + 〈U〉 · ▽ (2.7)
∂t
It can be shown [53] that the mean material derivative is related to the
material derivative according to
DΘ
=
Dt
D 〈Θ〉
Dt
for a quantity Θ = 〈Θ〉 + θ.
∂ ∂ ∂
+ 〈uθ〉 + 〈vθ〉 + 〈wθ〉 (2.8)
∂x ∂y ∂z
For convenience, Equation 2.5 may be re-written as
D 〈U i〉 ∂ 〈P 〉
= −1 +
Dt ρ ∂xi ∂
ν
∂xj ∂ 〈U
i〉
−
∂xj ∂
ujui ∂xj
(2.9)
where each side of the equation is a vector in i comprised of terms that are
summed in j.
By decomposing velocities into mean and fluctuating components, RANS
models seek to treat separately the physical details of turbulence. The fluctu-
ating velocity components which are associated with turbulence are therefore

CHAPTER 2. TURBULENCE MODELS 20
considered unknown, and must be modelled. The EVM models the unknown
−
ujui term in Equation 2.9 by
−
∂ 〈U
ujui =
i〉
νt +
∂xj ∂
U j
−
∂xi 2
3 kδij (2.10)
where k is turbulent kinetic energy. νt is turbulent viscosity, an additional
viscosity arising as a result of turbulence. δij is the Kronecker delta, defined
such that
⎧
⎨ 1 i = j
δij =
⎩ 0 i = j
(2.11)
Essentially, the EVM replaces ν with (ν + νt) in Equations 2.5 & 2.6. In
summary, the governing equations of the RANS EVM are
Continuity : ▽ · 〈U〉 = 0 (2.12)
Momentum : D〈U〉
Dt
T ransport :
= − 1
ρ ▽ 〈P 〉 + ▽ · ((ν + νt) ▽ 〈U〉) (2.13)
D〈Φ〉
Dt = ▽ · ν+νt
σ
▽ 〈Φ〉 + Sφ
(2.14)
Thus the RANS and EVM method reduces the problem of calculating chaotic
fluctuating velocity components to a problem of specifying an unknown local
parameter, the turublent viscosity, νt. A variety of approaches exist for
modelling νt. The approaches considered in this thesis are the k-ε model and
the k-ω model.

CHAPTER 2. TURBULENCE MODELS 21
2.2 The k-ε Model
The high-Reynolds number version of the standard k-ε model of Launder &
Sharma [33] defines turbulent viscosity, νt as
2 k
νt = Cµ
ε
(2.15)
where k is turbulent kinetic energy, and ε is the rate of dissipation of k. Cµ
is a constant given in Table 2.1.
The transport equation for k is based on Equation 2.14 (with 〈Φ〉 = k). The
source term consists of production, Pk and dissipation, ε:
Dk ν + νt
= ▽ ·
▽ k + Pk − ε (2.16)
Dt σk
Pk is
Pk = −aij 〈Sij〉 (2.17)
where the right hand side of the equation is summed over all permutations
of i and j to obtain Pk.
aij is defined as
〈Sij〉 is defined as
aij = u iu j
〈Sij〉 ≡ 1
∂ 〈U i〉
2 ∂xj − 2
3 kδij
+ ∂ U j
∂xi
Applying the EVM (Equation 2.10) to Equation 2.17,
(2.18)
(2.19)
Pk = 2νt 〈Sij〉 〈Sij〉 (2.20)

CHAPTER 2. TURBULENCE MODELS 22
Equation 2.17 is an exact expression for Pk, within the assumptions of the
RANS approach. Equation 2.20 is exact within the assumptions of the RANS
EVM. However, ε is modelled. In k-ε models, ε is obtained through an addi-
tional transport equation. In high-Reynolds-number version of the standard
k-ε model of Launder & Sharma [33] the additional transport equation is
Dε ν +
νt
ε
2 ε
= ▽ ·
▽ ε + Cε1 Pk − Cε2
(2.21)
Dt σε
k k
production destruction
The constants appearing in the standard k-ε model are given in Table 2.1.
Table 2.1: Constants in the standard k-ε model [33]
Cµ σk σε Cε1 Cε2
0.09 1.0 1.3 1.44 1.92
2.2.1 The Low-Reynolds-Number k-ε Model
If the k-ε model is applied in the near-wall region, viscous corrections are
required in order to produce reasonable results. Viscous corrections to the k-ε
model involve two modifications. One is the incorporation of so-called viscous
damping terms. Also, a transport equation is solved for a new parameter, ˜ε
rather than ε.
Because ε tends to a finite value at a wall, the wall boundary condition on
ε is difficult to specify. Therefore, the ε transport equation is replaced by a
new transport equation of an alternative parameter, ˜ε. ˜ε is defined in such a

CHAPTER 2. TURBULENCE MODELS 23
way that
The difference between ˜ε and ε is defined as ˆε:
From Equations 2.22 and 2.23, it follows that
˜ε| y=0 = 0 (2.22)
ε = ˜ε + ˆε (2.23)
ˆε| y=0 = ε| y=0
(2.24)
The equation for νt is modified to use ˜ε for convenience. Also, damping
terms exist in the equation for νt and the transport equation for ˜ε. These
damping terms are a function of the turbulent Reynolds number, ˜
Ret. ˜
Ret
is a Reynolds number based on local turbulence quantities (in this case, k
and ˜ε). The tilde is a reminder that, for convenience, the turbulent Reynolds
number used in the low-Reynolds-number standard k-ε model uses ˜ε rather
than ε. ˜ Ret is defined as
˜Ret ≡ k2
˜εν
(2.25)
In the low-Reynolds-number version of the standard k-ε model [33], the trans-
port equation of ˜ε is
D˜ε ν + νt
= ▽ ·
Dt σε
where
▽ ˜ε + Cε1f1
E = 2ννt
˜ε
Pk − Cε2f2
k
∂ 2 〈U〉
∂y 2
2
2 ˜ε
+ E (2.26)
k
(2.27)
f1 = 1 (2.28)
−Ret ˜
f2 = 1 − 0.3e 2
(2.29)

CHAPTER 2. TURBULENCE MODELS 24
ˆε is required in order to calculate ε for use in the k transport equation
(Equation 2.16). ˆε is defined as
ˆε = 2ν
∂ √ 2 k
∂y
(2.30)
Thus, ˆε does not require a transport equation, but may be solved from local
quantities.
νt is modified according to
where
2.2.2 Yap Correction
νt = Cµfµ
fµ = exp
⎡
2 k
˜ε
⎣ −2.5
1 + ˜ Ret
50
⎤
(2.31)
⎦
(2.32)
The ‘Yap correction’ refers to an additional source term in the ˜ε transport
equation of the low-Reynolds-number k-ε model. The Yap correction was
originally introduced by Yap [92] to improve the performance of the k-ε
model in impinging and recirculating flows. Yap correction is often implicitly
included in standard k-ε modelling, so it has been included in the present
UMIST-N wall function for completeness.
Using Yap correction, the ˜ε transport equation becomes
2
D˜ε ν + νt
˜ε
˜ε
= ▽ ·
▽ ˜ε + Cε1f1 Pk − Cε2f2 + E + Y (2.33)
Dt σε
k
k
with the additional Y source term defined as
⎛⎡
Y = max ⎝⎣0.83
k 3
2
− 1
2.5˜εy
k 3
2
2.5˜εy
2 2 ˜ε
k
⎤ ⎞
⎦ , 0⎠
(2.34)

CHAPTER 2. TURBULENCE MODELS 25
y is the distance from the wall.
2.3 The k-ω Model
The k-ω model is like the k-ε model in being dubbed a two-equation model,
because it demands the solution of two additional transport equations in
order to characterise turbulence. Like most two-equation models, the k-ω
model tracks turbulent kinetic energy, k. However, the dissipation rate, ε is
not tracked using its own transport equation, as in the k-ε model. Instead, a
transport equation is solved for ω. ω is sometimes referred to as the specific
dissipation rate.
The Wilcox 1988 k-ω model [89] uses the following transport equation for k:
Dk ν + νt
= ▽ ·
▽ k + Pk − ωkβ
Dt σkω
∗
(2.35)
so that ω is defined as
ω ≡ ε
kβ ∗
(2.36)
Pk is defined in the same way as for the k-ε model, as shown in Equation
2.20.
The equation for νt becomes
νt = γ ∗
The transport equation for ω is
Dω ν + νt
= ▽ ·
Dt σω
k
ω
▽ ω + γ
ω
Pk − βω
k
2
(2.37)
(2.38)

CHAPTER 2. TURBULENCE MODELS 26
The constants appearing in the Wilcox 1988 k-ω model are given in Table
2.2.
Table 2.2: Constants in the Wilcox 1988 k-ω model [89]
β β ∗ γ γ ∗ σkω σω
0.075 0.09 5
9 1 2 2
2.4 The Logarithmic Law of the Wall
The wall function approach spares the CFD code from resolving the be-
haviour of the flow in the near-wall region. The standard wall function
approach is to assume that the the flow behaviour at a certain distance away
from the wall will match the logarithmic law of the wall. The governing
equations are solved in the bulk of the flow regime, with the logarithmic law
of the wall providing a boundary condition near the wall.
Wall functions are often used with the high-Reynolds-number standard k-ε
model. The wall function approach is essentially an alternative to the low-
Reynolds-number approach, providing computational savings at the cost of
reduced accuracy. Wall functions may be applied to the k-ω model, but,
since the advantages of the k-ω model are primarily in its behaviour near the
wall, this is seldom done in practice. Wall functions on k, ε, and wall-parallel
velocity, 〈U〉 will be discussed here from a conceptual standpoint. A detailed
discussion of the implementation of log law boundary conditions is left until
Chapter 4.

CHAPTER 2. TURBULENCE MODELS 27
The logarithmic law of the wall assumes a logarithmic relationship between
velocity and displacement away from a solid boundary. These quantities are
expressed nondimensionally as U + and y + respectively. (See the discussion
in Chapter 3 on nondimensionalisation.) This relationship is due to von
Kármán [28]. The relationship is
U + = 1
κ ln Ey +
(2.39)
where κ is von Kármán’s constant. E is a function of wall roughness, and the
smooth-wall value is used in this thesis. Log-law constants appear in Table
2.3. The wall roughness constant, E is not to be confused with the E term
in the ˜ε transport equation.
Table 2.3: Log-law constants [83]
κ 0.4187
E 9.793
Table 2.4 offers a qualitative appraisal of the behaviour of flow with respect
to the nondimensional wall distance, y + . This highlights the qualitatively
different behaviours exhibited by a fluid as a solid boundary is approached.
The log law applies approximately where y + > 30. Where y + < 5, flow is
characterised by Prandtl’s [54] hypothesis, that U + = y + very near the wall.
The buffer layer is a region (5 < y + < 30) where neither of these assumptions
holds. In specifying a low-Reynolds-number solution, it is important that the
gradients nearest the wall fall within the viscous sublayer (y + < 5). In a high-
Reynolds-number CFD treatment, it is important that the log-law boundary
conditions are applied at a location where y + > 30.
When used as a boundary condition on a high-Reynolds-number k-ε treat-

CHAPTER 2. TURBULENCE MODELS 28
Table 2.4: Near-wall flow regimes adapted from [53]
y + < 5 The viscous sublayer. The flow is essentially laminar.
5 < y + < 30 The buffer layer. The log law overestimates U + .
y + > 30 The log law region. The log law is valid.
ment, the log law’s effect on the x momentum equation can be thought of as
a source resulting from shear stress at the wall (τw). The relationship can be
shown by considering the definition of U + . (See Chapter 3.)
U + = 〈U〉
Uτ =
Uτ
τw
ρ
(2.40)
(2.41)
Following Launder & Spalding [34], C 1/4
µ k 1/2 may be taken as a velocity scale.
The log law estimates the parameter Uτ by
Therefore,
Uτ = U 2 τ
Uτ
Uτ = C 1/4
µ k 1/2
= τw/ρ 〈U〉
=
1/2 U +
C 1/4
µ k
τw = ρC1/4 µ k1/2 〈U〉
U +
(2.42)
(2.43)
(2.44)
The shear stress source in the equation of 〈U〉 is thus known from Equation
2.44, Equation 2.39, and the previous iteration value of 〈U〉 at the point
where the log law is applied. See Chapter 4 for a discussion of the numerical
implementation of the log law.

CHAPTER 2. TURBULENCE MODELS 29
In applying a wall-function boundary condition to k, the goal is not simply
to prescribe a value of k at the near-wall cell node. Rather, better accuracy
is achieved by applying sources to the k equation at the near-wall cell that
represent cell-averaged production and dissipation terms (Pk and ε). Various
methods exist for estimating these quantities. The Launder & Spalding [34]
wall function is employed in this work. In this wall function,
∂ 〈U〉
Pk = τw
∂y
(k)
ε = ρCµ
2
∂
〈U〉
∂y
τw
(2.45)
(2.46)
The transport equation for ε does not make use of ε. Instead, ε is prescribed
in the near-wall cell from k according to the mixing length hypothesis,
C 3/4
3/2 k
µ = κy (2.47)
ε
The log law appears in Figure 2.1, plotted together with DNS and experi-
mental data at various Reynolds numbers. It can be seen that the log law
provides reasonable agreement with DNS and experimental results beyond
the buffer layer (y + > 30), and particularly when y + is not excessively large.
The log law produces good results when applied as a boundary condition in
a code whose near-wall cell extends to a height that fully encompasses the
buffer layer. Since y + is a function of the flow rate as well as y, the optimum
choice of near-wall cell size may be influenced by the flow rate considered.
However, it is usually sufficient to choose a large enough near-wall cell to
fully encompass the buffer layer for the lowest flow rate anticipated.

CHAPTER 2. TURBULENCE MODELS 30
U +
35
30
25
20
15
10
5
0
10 0
10 1
+
+
+
+
the log-law
DNS: Reτ = 180 [30]
DNS: Reτ = 395 [44]
DNS: Reτ = 584 [44]
◦ Exp.: Reτ = 708 [87]
▽ Exp.: Reτ = 921 [47]
+
10 2
+
+
+
+
+
+
+
x +
x xxx
x
x
x
x +
x +
+
10 3
y +
10 4
Exp.: Reτ = 1017 [47]
+ Exp.: Reτ = 1655 [47]
⋄ Exp.: Reτ = 2340 [11]
× Exp.: Reτ = 4800 [11]
△ Exp.: Reτ = 8150 [11]
Figure 2.1: The log-law compared to experiments in channel flow

Chapter 3
Channel Flow
Channel flow refers to an arrangement where fluid flows between two parallel
walls. Consider walls that are a distance 2δ apart. Flow is driven by a
pressure gradient in the x direction and solved from y = 0 at a wall to y = δ
at the axis of symmetry between the two walls. The flow field is long in
the third dimension, so that the flow is independent of z. The flow is also
independent of x. Even in unsteady channel flow, the flow is taken to develop
through time but not through the length of the flow direction, x.
3.1 Governing Equations
For ease of explanation, the RANS equations for continuity and conservation
of momentum (Equations 2.4 and 2.5) may be expanded into scalar form.
31

CHAPTER 3. CHANNEL FLOW 32
Removing all velocity gradients in x and z, this yields
Continuity :
X − momentum :
Y − momentum :
∂〈U〉
∂t
∂〈V 〉
∂t
+ 〈V 〉 ∂〈U〉
∂y
+ 〈V 〉 ∂〈V 〉
∂y
∂〈V 〉
∂y
1 ∂〈P 〉 ∂
= − + ρ ∂x ∂y
1 ∂〈P 〉 ∂
= − + ρ ∂y ∂y
= 0 (3.1)
ν ∂〈U〉
− ∂y
∂〈uv〉
(3.2)
∂y
∂〈V 〉
ν − ∂y
∂〈v2 〉
(3.3)
∂y
A boundary condition affecting continuity, Equation 3.1 is that of zero flow
through a solid boundary
〈V 〉| y=0 = 0 (3.4)
Therefore, Equation 3.1 produces the result that V = 0 throughout the flow
field. The momentum equations become
X − momentum :
∂〈U〉
∂t
1 ∂〈P 〉 ∂
= − + ρ ∂x ∂y
ν ∂〈U〉
∂y
Y − momentum : 0 = − 1 ∂〈P 〉
ρ ∂y − ∂〈v2 〉
∂y
− ∂〈uv〉
∂y
(3.5)
(3.6)
It is notable, at this stage, that convection has been eliminated from the
momentum equations.
Equation 3.6 can be integrated with respect to y and the following boundary
conditions applied
v 2 y=0 = 0 (3.7)
〈P 〉| y=0 = Pw
(3.8)
where Pw is pressure at the wall. Velocities, including turbulent fluctuations,
are zero at the wall, so there are no pressure fluctuations. Therefore, Pw
is not shown as a mean value. Furthermore, since the flow is driven by a
pressure gradient in x, the pressure at the wall must only vary as a function

CHAPTER 3. CHANNEL FLOW 33
of x and time, hence Pw(x, t). Integrating Equation 3.6 with respect to y
and applying the appropriate boundary conditions yields
ρ v 2 + [〈P 〉 − Pw(x, t)] = 0 (3.9)
Equation 3.9 can be differentiated with respect to x to give
Thus
∂〈P 〉
∂x
∂ 〈P 〉
∂x
is not a function of y.
= ∂
∂x Pw(x, t) (3.10)
In the above analysis, continuity has served to simplify the momentum equa-
tions, and y-momentum has yielded the insight that 〈P 〉 varies only in the x
direction and in time. The remaining x-momentum equation is solved in the
CFD code. Applying the EVM to Equation 3.5, this becomes:
∂ 〈U〉 ∂ 〈P 〉 ∂ ∂ 〈U〉
= −1 + (ν + νt)
∂t ρ ∂x ∂y
∂y
(3.11)
In summary, channel flow is governed by a single momentum equation con-
taining terms for fluid acceleration, shear stress, and a driving pressure gra-
dient. Furthermore, convection does not take place in any channel flow, since
wall-normal velocity is zero throughout and all gradients in the wall-parallel
direction are zero.
3.1.1 The k-ε Model
In channel flow, the transport equations in the k-ε model (Equations 2.16 &
2.33) become
k :
˜ε : ∂ ˜ε
∂t
∂k ∂ ν+νt
= ∂t ∂y σk
∂ ν+νt ∂ ˜ε
˜ε
= + Cε1f1
∂y σε ∂y
k
∂k
∂y
+ Pk − ε (3.12)
˜ε 2
Pk − Cε2f2 + E + Y (3.13)
k

CHAPTER 3. CHANNEL FLOW 34
E and Y in Equation 3.13 are unaffected by application to channel flow ge-
ometry. (See Equations 2.27 & 2.34.) However, Pk (Equation 2.20) becomes
3.1.2 The k-ω Model
Pk = νt
∂ 〈U〉
The k-ω transport equations (Equations 2.35 & 2.38) for channel flow are
k :
ω : ∂ω
∂t
∂k
∂t
∂y
∂ ν+νt ∂k
= ∂y σkω ∂y
∂ ν+νt ∂ω
= ∂y σω ∂y
with Pk as defined in Equation 3.14.
3.2 Flow Characterisation
2
(3.14)
+ Pk − ωkβ ∗ (3.15)
+ γ
ω
Pk − βω k
2 (3.16)
Steady channel flow is usually characterised by a quantity called the friction
velocity Reynolds number, Reτ. Unsteady channel flow is usually charac-
terised by a time-mean Reτ and an amplitude of oscillation expressed as a
proportion of this time-mean value. To explain Reτ, it is necessary to further
analyze steady channel flow.
For steady flow, ∂〈U〉
∂t
= 0 in Equation 3.11 and d〈U〉
dy is not a partial derivative.
In this case, the x-momentum equation (3.5) indicates a balance between the
force acting on the fluid because of the driving pressure gradient and a shear
force resulting from friction. The shear stress is a function of velocity, and it

CHAPTER 3. CHANNEL FLOW 35
varies only in y. The shear stress may be expressed as
so Equation 3.5 can be rewritten as
d 〈U〉
τ(y) = ρν − ρ 〈uv〉 (3.17)
dy
dτ
dy
It follows from Equation 3.18 that dτ
= d 〈P 〉
dx
(3.18)
dy is constant. Let τw be the shear stress
at the wall, such that τ| y=0 = τw. Also, τ| y=δ = 0 since d〈U〉
dy
y=δ
= 0 and
〈uv〉| y=δ = 0. With these boundary conditions on τ, Equation 3.18 may be
integrated to yield
d 〈P 〉
dx
= −τw
δ
(3.19)
Reτ may be defined from τw, the geometry of the channel, and the properties
of the fluid according to
Reτ =
δ τw
ν ρ
It is of interest, based on Equations 3.19 and 3.20, that only one of
(3.20)
d〈P 〉
, τw, dx
or Reτ must be specified in order to completely define the flow. Usually, Reτ
is the quoted parameter.
Another Reynolds number is the bulk Reynolds number, based on average
velocity
Re♭ =
where U indicates a spatial average.
U (2δ)
ν
A further Reynolds number is that based on free stream velocity
Re0 = U0δ
ν
(3.21)
(3.22)

CHAPTER 3. CHANNEL FLOW 36
Re♭ and Re0 are often quoted by experimentalists, who favour these quanti-
ties because they can be readily and directly measured.
3.3 Nondimensionalisation
Viscous velocity and length scales can be defined such that the Reynolds
number based on these scales is equal to 1. The velocity scale is
and the length scale is
Uτ =
τw
ρ
ρ
δv = ν
τw
The friction velocity Reynolds number may be expressed as
Reτ = Uτδ
ν
= δ
δv
(3.23)
(3.24)
(3.25)
Flow variables may be nondimensionalised by the viscous scales in Equations
3.24 and 3.23. By convention, U + , the nondimensional average U velocity is
not shown with braces – 〈〉.
y + = y
δν
Uτ
= y
ν
U + = 〈U〉
Uτ
〈uv〉 + = 〈uv〉
U 2 τ
k + = k
U 2 τ
(3.26)
(3.27)
(3.28)
(3.29)

CHAPTER 3. CHANNEL FLOW 37
u 2 + = 〈u 2 〉
U 2 τ
v 2 + = 〈v 2 〉
U 2 τ
Sometimes the driving pressure gradient is nondimensionalised by
P + x =
∂〈P 〉
∂x
2
(Uτ ) ρ
δ
(3.30)
(3.31)
(3.32)
Note that 〈u 2 〉 and 〈v 2 〉 may be nondimensionalised as 〈u2 〉
k
also notable that the definitions of Reτ and y + give
y + |y=δ = Reτ
3.4 Steady Channel Flow Data
and 〈v2 〉
. It is
k
(3.33)
Based on the wealth of experimental and DNS results available concerning
the behaviour of steady channel flows, a set of empirical profiles have been
developed to obtain estimates of U + , − 〈uv〉 + , k + , 〈u 2 〉 + , and 〈v 2 〉 + . These
results are compared to DNS and experimental data in the following sections.
Much of the data were found in the AGARD Aerospace Database 1 . The
contributors of specific data sets are cited in the graph legends 2 .
1 Advisory Group for Aerospace Research and Development, France
2 Another excellent source of experimental information may be found at
http://www.efluids.com. Furthermore, at http://torroja.dmt.vpm.es Dr. Javier Jimenez
(School of Aeronautics, University of Madrid) has published full flow fields for channel
flow at Reτ = 180 and Reτ = 550

CHAPTER 3. CHANNEL FLOW 38
It is intended that these profiles will be used as an aid in developing new
turbulence models or numerical approaches. One particular application is as
a means of setting initial conditions or inlet boundary conditions for a code,
to facilitate easier convergence during debugging. A further application is
to apply these relations within the code as a means of verifying the internal
calculations of the turbulence model.
The profiles shown below are designed to match fully turbulent flow and also
laminar/turbulent transitional flow. The profiles are reasonably accurate, as
compared to the variation generally seen between different sets of experimen-
tal results. It is notable that, in general, flow parameters are strong functions
of Reynolds number. While experimental and DNS results are only available
at a selection of discrete Reynolds numbers, these empirical profiles offer
a means of interpolation and allow codes to be validated at intermediate
Reynolds numbers.
One limitation of these emperical profiles is that they do not adequately cap-
ture the limiting shape of turbulence quantities as y + → 0. The principal
concern in developing the profiles was to accurately account for the magni-
tude and location of peak values. Therefore, separate near-wall profiles are
also presented.
〈u 2 〉 + and 〈v 2 〉 + are found by multiplying empirical profiles for 〈u2 〉 +
〈v 2 〉 +
k +
k +
and
by the empirical profile for k + . The reason why 〈u 2 〉 + and 〈v 2 〉 + are
normalised by k + rather than being fit directly is that this process reduces the
Reynolds number dependence of the resulting empirical profiles, with most
of the Reynolds number dependence in 〈u 2 〉 + and 〈v 2 〉 + being accounted for
in k + . The complexity of the empirical profiles is therefore reduced.

CHAPTER 3. CHANNEL FLOW 39
3.4.1 Empirical Profile for U +
Reichardt’s log law [58] states that
U + = 1
κ ln 1.0 + 0.4y +
+7.8 1 − exp − y+
+ y
− exp −
11 11
y+
3
(3.34)
This is plotted against experimental and DNS data in Figure 3.1. Reichardt’s
law follows DNS results very closely near the wall. As y + increases, Re-
ichardt’s law approaches the log law, while experimental values of U + tend
to fall above the log law for high y + .
3.4.2 Empirical Profile for − 〈uv〉 +
Based on Reichardt’s law, a profile can be derived for − 〈uv〉 + . This was
originally done by Alexander Davroux and Dominique Laurence at Electricité
de France.
dτ
dy is a constant, τ| y=0 = τw and τ| y=δ = 0. Therefore, the equation for τ (y)
in a channel is
τ (y) = τw
1 − y
δ
This can be inserted into Equation 3.17 to give
− 〈uv〉 = τw
ρ
In nondimensional form, this becomes
1 − y
δ
− 〈uv〉 +
= 1 − y
δ
−
− ν dU
dy
dU +
dy +
(3.35)
(3.36)
(3.37)

CHAPTER 3. CHANNEL FLOW 40
Substituting Equation 3.34 into Equation 3.37 gives
− 〈uv〉 + =
1 − y
0.976
−
δ 1 + 0.4y +
+ 0.709 exp − y+
11
−0.709 exp − y+
+ 0.234y
3
+
∗ exp − y+
3
into Equation 3.38 gives
− 〈uv〉 + = 1 − y+
0.976
−
Reτ 1 + 0.4y +
− 0.709 exp − y+
11
+ 0.709 − 0.234y +
exp − y+
3
Substituting y+
Reτ
for y
δ
(3.38)
(3.39)
Figure 3.2 shows the performance of Reichardt’s law when used to calcu-
late − 〈uv〉 + by the above technique. One limitation of this profile is that
− 〈uv〉 + < 0 for low values of y + , particularly when Reτ is small. To im-
prove upon this, the (0.709 − 0.234y + ) component of Equation 3.39 can be
modified to alter the behaviour at low y +
. Furthermore, can be
0.976
1+0.4y +
adjusted so that the profile produces a value of zero when y + = 0. The result
of these manipulations is
− 〈uv〉 + = 1 − y+
1
−
Reτ 1 + 0.4y +
+ 0.709 − 0.18y +
exp − y+
3
− 0.709 exp
Figure 3.3 plots Equation 3.40 at various values of Reτ.
3.4.3 Empirical Profile for k +
− y+
11
(3.40)
The following profile for k + is adapted from a profile originally fitted by
Alexander Davroux and Dominique Laurence at Electricité de France. The

CHAPTER 3. CHANNEL FLOW 41
profile is
k + =
1
Reτ 2
0.07 + 0.05
∗
1600
y +
2
∗ exp − y+
7
+4.5 1 − exp − y+
+ −1
4y
+ 1
20 Reτ
+ 2
y
∗ 1 − exp −
3
Figure 3.4 shows this profile for k + plotted against DNS results.
(3.41)
Table 3.1 offers qualitative descriptions of the contributions of the various
terms in Equation 3.41. The final term ensures that k + = 0 at the wall, but
the shape of the profile in the very near-wall region does not closely match
DNS data. An analysis of the near-wall behaviour of turbulence quantities
will follow.
Table 3.1: Qualitative behaviour of terms in Equation 3.41
Reτ
1600
0.07 + 0.05
4.5 1 − exp
∗ (y + ) 2
∗ exp − y+
7
4y + −1
+ 1
− y+
20
1 − exp −
y +
3
Reτ
2
3.4.4 Empirical Profile for u 2 +
The following formula for 〈u2 〉 +
k +
fits the peak value of k +
helps to roughly capture the curve
at high Reynolds numbers
forces a value of zero at the wall
was adapted from a profile fitted by Alexan-
der Davroux and Dominique Laurence at Electricité de France.

CHAPTER 3. CHANNEL FLOW 42
〈u 2 〉 +
k +
+ 1/4
y
= 1.46 − 0.542 ∗ +
Reτ
⎛
y +
⎜
⎝
1.5 +
7
2 y +
7
⎞
⎟
⎠ (3.42)
Figure 3.5 shows the profile for 〈u 2 〉 + obtained by multiplying Equation 3.42
by Equation 3.41. This is plotted against DNS results.
Table 3.2 offers qualitative descriptions of the contributions of the various
terms in Equation 3.42.
Table 3.2: Qualitative behaviour of terms in Equation 3.42
„ y +
«
7
“ ”
y + 2
1.5+ 7
y + 1/4
−0.542 ∗
Reτ
fits the peak value of 〈u2 〉 +
k +
shifts the curve at higher y + to account for some
Reynolds number dependence of 〈u2 〉 +
3.4.5 Empirical Profile for v 2 +
This profile for 〈v2 〉 +
k +
was originally fitted by Alexander Davroux and Do-
minique Laurence at Electricité de France. The profile is
〈v 2 〉 +
k +
where
=
y + 2
+
20
y
0.08 ∗ 2 + 0.8 ∗ 1λR + λR ∗ 0.06 ∗ + 1
y +
Reτ
2 + 20
+ 2
+ 2
−1
y y
∗ 1 − exp −
∗ 1 − exp −
(3.43)
20
3
λR = exp − 100
Reτ
k +

CHAPTER 3. CHANNEL FLOW 43
Figure 3.6 shows the resulting profile for 〈v 2 〉 + plotted against DNS results.
3.4.6 Near-Wall Behaviour
Various researchers have analysed the limiting behaviour of turbulence pa-
rameters near a wall [4, 8, 22, 39, 43, 81]. Following Hanjalic & Launder [22],
fluctuating velocity components can be represented in the following way as
y → 0.
u + rms = a ∗ y + y +
+ O
2
v + rms = b ∗ y +
2 y +
+ O
3
(3.44)
where a and b are constants, and u + rms and v + rms are root-mean-square aver-
aged values.
These relationships are supported by the DNS results of Kim et. al. [30].
As Suga [77] points out:
2
u + 2
= a ∗ y +
2 y +
+ O
4
2
v + 2
= b ∗ y +
4 y +
+ O
6
〈uv〉 + = 〈ab〉 ∗ y +
3 y +
+ O
4
(3.45)
Thus the near-wall behaviour of 〈u 2 〉 is expected to be dominated by a curve

CHAPTER 3. CHANNEL FLOW 44
of the form (y + ) 2 ; 〈v 2 〉 by (y + ) 4 ; -〈uv〉 by (y + ) 3 . Specifically, as y + → 0,
where,
〈u2 〉 +
(y + 2
)
→ A
〈v2 〉 +
(y + 4
)
→ B
− 〈uv〉 +
(y + 3
)
→ C (3.46)
A = a 2
B = b 2
C = − 〈ab〉 (3.47)
Based on the definition of k + , it can be shown that [22]
where γ is a constant in,
k + = 1 2
a
2
+ γ 2 ∗ y +
2 y +
+ O
3
2
w = γ 2 ∗ y +
2 y +
+ O
4
Thus, k + is dominated by (y + ) 2 near the wall and, as y + → 0,
where,
(3.48)
(3.49)
k +
(y + 2 → D (3.50)
)
D = 1 2
a
2
+ γ 2
Figure 3.7 shows the near-wall behaviours of 〈u2 〉 +
(3.51)
(y + ) 2 , 〈v2 〉 +
(y + ) 4 , −〈uv〉+
(y + ) 3 , and k+
(y + ) 2 ,
respectively. These graphs are expected to tend to horizontal lines as y + → 0.

CHAPTER 3. CHANNEL FLOW 45
The profiles quoted above for 〈u 2 〉 + , 〈v 2 〉 + , − 〈uv〉 + , and k + near the wall do
not adequately satisfy the relationships in 3.46 and 3.50. Therefore, in the
interest of completeness, values of A, B, C, and D are found as functions of
Reynolds number based on DNS results 3 . [30, 44]
The functions are:
A = 1.060 × 10 −4 Reτ + (0.1107)
B = 1.757 × 10 −7 Reτ + 4.546 × 10 −5
C = 9.005 × 10 −7 Reτ + 5.855 × 10 −4
D = 9.362 × 10 −5 Reτ + (0.0680) (3.52)
The Reynolds-number dependent near-wall proportionalities indicated above
are consistent with the DNS results of Antonia & Kim [4], who analyzed
the near-wall behaviour of u + , v + , − 〈uv〉 + , and other parameters at two
Reynolds numbers.
3.5 Local Nondimensionalisation
When examining computed results, it is often more straightforward to nor-
malise y by k, rather than by τw. In a complex, multidimensional mesh, it is
not possible, in general, to relate modelled flow parameters to friction at a
corresponding point along the wall.
When y is normalised by k, the customary notation is y ∗ . The method of
3 It is notable that the relationships A = a 2 , B = b 2 , and C = 〈ab〉 do not imply
C = √ AB.

CHAPTER 3. CHANNEL FLOW 46
obtaining y ∗ is
y ∗ can be related to y + by
y ∗ = y√ k
ν
Likewise, 〈U〉 can be normalised by k.
(3.53)
y ∗ = y +√ k + (3.54)
U ∗ = 〈U〉
√ k
U ∗ =
U +
√ k +
(3.55)
(3.56)
y + is plotted against y ∗ for various values of Reτ in Figure 3.8. U + and U ∗ at
various Reynold’s numbers are plotted in Figure 3.9. It can be seen that U +
and U ∗ can differ significantly. This may be caused by the fact that k + is not
increasing monotonically. An interesting feature, visible in Figure 3.9, is that
U ∗ approaches a constant as y → 0. This happens because, as y → 0, U +
becomes proportional to y + and k + becomes proportional to (y + ) 2 . Thus,
U +
√ k + = U ∗ approaches a constant.
Another choice of variable against which to normalise y and U is 〈v 2 〉. 〈v 2 〉
has the same units as k. Since most of the local variability of k results from
turbulent fluctuations in the mean flow direction (〈u 2 〉), local nondimension-
alisation based on 〈v 2 〉 may be expected to offer smoother profiles. We can
define
y ∗
v 2 = y 〈v 2 〉
ν
U ∗ 〈U〉
v2 = =
〈v2 〉
= y +
〈v2 〉 +
U +
〈v 2 〉 +
(3.57)
(3.58)

CHAPTER 3. CHANNEL FLOW 47
y + is plotted against y ∗
v 2 for various values of Reτ in Figure 3.10. There is
reasonable agreement between y + and y ∗
v 2. U + and U ∗
v 2 at various Reynold’s
numbers are plotted in Figure 3.11. The 〈v 2 〉 normalising approach suffers
from the fact that 〈v 2 〉 + tends to zero as y → 0 faster than U + does, so that
U ∗
v 2 becomes large. As y → 0, U + is expected to become proportional to y +
and 〈v2 〉 + becomes proportional to (y + ) 4 U , so +
√
〈v2 +
〉
1
y + .
becomes proportional to

Chapter 4
Numerical Implementation
This work uses a code derived from PASSABLE. 1 Prior to this work, the
code was modified for periodic flow by Addad [1]. A finite volume method
is used to discretise the equations governing the flow. More information on
the numerical implementation of CFD can be found in books. [83],[20]
4.1 The Mesh
As discussed in Chapter 3, the flow field is a one-dimensional expanse with
a wall at y = 0 and symmetry at y = δ. This is modelled as a channel ’slice’
of finite thickness. The slice is divided into finite volumes, called ‘cells’.
The slice is one cell thick in x and is divided in y according to the degree
of numerical accuracy required. This is shown schematically in Figure 4.1.
The actual grid used for low-Reynolds-number solutions within this work
1 PArabolic Solution Scheme Applied to Boundary Layer Equations
48

CHAPTER 4. NUMERICAL IMPLEMENTATION 49
contains 98 cells. Each cell is 5% larger than its wall-facing neighbour. This
geometrical feature is referred to as an expansion factor of 1.05.
†
d
y
U P
symmetry
nodes
vertexes
wall
Figure 4.1: The low-Reynolds-number grid
Within each cell is a ‘node’, where parameters associated with the cell are
tracked. Cell boundaries, called ‘vertexes’, are centred between nodes. Ex-
ceptions exist at y = 0 and y = δ. At y = δ, a half-cell exists, with the
final node placed at the symmetry plane. At y = 0, a node is placed at
the wall with no associated cell. This represents a new modification to the
PASSABLE code, allowing easier implementation of wall function boundary
conditions. Previously, the code employed a similar mesh style at y = 0 to
the mesh at y = δ.
Table 4.1 shows the notation used in this thesis to refer to discretised values.
Values at vertexes are not stored in the code, but may be calculated via
linear interpolation from adjacent nodal values.

CHAPTER 4. NUMERICAL IMPLEMENTATION 50
Table 4.1: The notation for discretised values
current node subscript P
node above subscript N
node below subscript S
vertex above subscript n
vertex below subscript s
previous time step superscript t − 1
cell height ∆yp
node-to-node dist. ∆yn,s
time step ∆t
4.2 Volume Integral Form
The finite volume discretisation method is convenient because quantities are
conserved within each finite volume and are therefore necessarily conserved
throughout the flow field. In order to employ the finite volume method, it is
first necessary to integrate the differential equations governing the flow with
respect to volume. The volume-integrated forms of the equations are pre-
sented below. Note that, in channel flow geometry, integrating with respect
to volume is equivalent to integrating with respect to y.
〈U〉 :
k :
˜ε :
∂〈U〉
∂k
∂t
∂〈P 〉
∂〈U〉
dy + (ν + νt) ∂x ∂y
ν+νt ∂k
∂t dy = − 1
ρ
dy =
∂ ˜ε dy = ∂t
ω : ∂ω dy = ∂t
− Cε2f2
ν+νt
σk
ν+νt ∂ ˜ε
σω
σε
˜ε 2
(4.1)
∂y + Pkdy − εdy (4.2)
∂y +
˜ε
Cε1f1 Pkdy k
dy + k
Edy + Y dy (4.3)
∂ω
∂y + γ
ω Pkdy − k
βω2dy (4.4)

CHAPTER 4. NUMERICAL IMPLEMENTATION 51
Before performing any dimensional analysis on the above equations, one must
actually integrate with respect to volume, applying limits of 0 to 1 on the in-
tegrations in x and z. It is also notable that the PASSABLE code uses a grid
that is unit length in z but not unit length in x (∆x = 1). This detail must
be remembered when working with the code, particularly when specifying
volume-integrated source terms. Furthermore, in the PASSABLE implemen-
tation, Equations 4.1-4.4 are multiplied through by ρ. This convention is
maintained in the subgrid.
4.3 Discretisation
The process of discretisation involves approximating the governing differen-
tial equations by algebraic equations so that they can be solved by a com-
puter.
A fully implicit scheme is used in this work. This means that the previous
time-step values of adjacent nodes are not used. In contrast, a fully explicit
scheme would use the previous time-step values, but not the current time-
step values of adjacent nodes. The fully implicit method ensures that the
solution can converge at large time steps.
Central differencing is used in this work to approximate derivatives. Other
differencing schemes include PLDS 2 [51] and QUICK 3 [37]. These alternative
schemes involve higher-order approximations. The QUICK scheme is also
upstream-biased, in the sense that it makes use of more information on the
2 Power-Law Differencing Scheme
3 Quadratic Upstream Interpolation for Convective Kinetics

CHAPTER 4. NUMERICAL IMPLEMENTATION 52
upstream side of the derivative than on the downstream side. (The scheme is
influenced by the local direction of 〈U〉.) This is physically reasonable, since
quantities are convected downstream. Because of the simple geometry and
the lack of convective terms in the governing equations, central differencing
was felt to be an effective choice in this work.
Within the CFD code, a system of algebraic equations is compiled to repre-
sent each of the momentum equations or transport equations to be solved.
A set of algebraic equations representing one differential equation is solved
using an iterative procedure 4 before the next differential equation is treated.
The various differential equations are coupled to one-another, so the entire
process of solving each differential equation in sequence must be repeated
iteratively until the overall solution converges. 5
The discretised momentum equation for 〈U〉 is
where
〈U〉 P − 〈U〉 t−1
P
∆t
∆yp = − 1 ∂ 〈P 〉
ρ
∂ 〈U〉
∂y
∂ 〈U〉
∂x ∆yp + (ν + νt) n
∂ 〈U〉
− (ν + νt) s
n
∂y
= 〈U〉 N − 〈U〉 P
∆yn
s
∂ 〈U〉
∂y
n
(4.5)
(4.6)
∂y
=
s
〈U〉 P − 〈U〉 S
∆ys
(4.7)
and (ν + νt) n and (ν + νt) s are found using linear interpolation between
nodes.
4 The iterative solution procedure used for the solution of algebraic equations in this
study is the Tri-Diagonal Matrix Algorithm (TDMA). This procedure is appropriate to
parabolic systems of equations.
5 An iterative solution algorithm is said to be converged when the solution that it
produces is observed to undergo no significant change with successive iterations.

CHAPTER 4. NUMERICAL IMPLEMENTATION 53
The discretised transport equation for k is
kP − k t−1
P
∆t ∆yp =
ν + νt
∂k
∂y
where
σk
n
+Pk∆yp − ε∆yp
Pk = νt
n
〈U〉n − 〈U〉 s
∆yp
and 〈U〉 n and 〈U〉 s are interpolated values.
ν + νt
−
2
σk
s
∂k
∂y s
(4.8)
(4.9)
Discretised transport equations for ˜ε and ω can be found in a similar manner.
Some further equations whose discretised forms are noteworthy are
⎛
⎞2
∂〈U〉 ∂〈U〉
− ∂y
∂y
E = 2ννt ⎝ n
s ⎠
∆yp
⎛
ˆε = 2ν ⎝
√k n
−
∆yp
√k s
⎞
⎠
2
(4.10)
(4.11)
Thus, an algebraic equation may be generated for each node P . The discre-
tised differential equations are expressed in the code in the form
(AP − SP ) φp = ANφN + ASφS + SU
(4.12)
where φ is the unknown parameter from the original differential equation.
AP , AN and AS are coefficients on nodal values. Previous time step infor-
mation is included as a source. The source is split into two terms, SU and
SP φp for reasons of numerical stability. It is advantageous to have a large
coefficient on φp, so negative quantities are sometimes moved from SU into
SP (dividing by φp) to artificially increase this coefficient. At other times,
sources involve a product of φp, and the use of SP is a natural choice.

CHAPTER 4. NUMERICAL IMPLEMENTATION 54
The fully implicit scheme also requires information on φ t−1
p . This does not
appear explicitly in Equation 4.12 because it is included in the source term,
SU.
As an example of the expressions of the coefficients presented in Equation
4.12, the coefficients on 〈U〉 are
AN = (ν + νt) n
∆yn
AS = (ν + νt) s
∆ys
AP = (ν + νt) n
∆yn
= AN + AS
SP = − 1
∆t ∆yp
SU =
〈U〉 t−1
P
∆t
+ (ν + νt) s
∆ys
∆yp −
4.4 Boundary Conditions
1
ρ
∂ 〈P 〉
∆yp
∂x
(4.13)
At y = δ, a symmetry boundary condition is employed. The governing
equations are not solved at y = δ. Values of 〈U〉, k, ˜ε, and ω are copied to
the symmetry plane node from the nearest adjacent node. At this adjacent
node, the coefficients are adjusted as follows for each of 〈U〉, k, ˜ε, and ω:
AN = 0
AP = AS (4.14)
This ensures that all gradients in y are zero at the symmetry plane.
At the wall, velocities are zero. This creates large gradients in the near-

CHAPTER 4. NUMERICAL IMPLEMENTATION 55
wall region. Because CFD uses discretisation, the storage and computational
requirements associated with accurately representing large gradients are high.
This results in a great potential for computation-saving approaches at the
wall. Many methods exist for applying wall boundary conditions to CFD
codes, and these may be model-specific. Boundary conditions on the wall
are discussed below.
4.4.1 Wall Boundaries on k-ε
In the low-reynolds-number standard k-ε model, ˜ε is defined as being equal
to zero at the wall. Furthermore, k = 0 at the wall, arising from the fact that
k represents turbulent kinetic energy and all velocities, including turbulent
fluctuations, are zero at the wall. In the low-Reynolds-number model, these
quantities are simply prescribed to be zero at the wall. Transport equations
are not solved at the wall, but are solved in the wall-adjacent cell. Because
of the large gradients involved, finer grid resolution is required near the wall,
as shown in Figure 4.1.
4.4.2 Wall Boundaries on k-ω
As with the k-ε model, 〈U〉 = 0 and k = 0 at the wall, and transport
equations may be solved up to the wall-adjacent cell. However, ω → ∞ as
y → 0, so it is not numerically possible to specify ω at the wall. Wilcox [89]
notes that the limiting behaviour of ω at the wall is
ω → 6ν
βy 2 as y → 0 (4.15)

CHAPTER 4. NUMERICAL IMPLEMENTATION 56
Thus, while ω is infinite at the wall, finite values may be specified near the
wall according to Equation 4.15. However, the gradient of ω near the wall
is excessively large (approximately two orders of magnitude greater than
gradients on ˜ε), so it is not practical to solve the ω transport equation in the
near wall region. The approach recommended by Wilcox [91] is to specify
ω according to the limiting behaviour suggested in Equation 4.15 within the
first 7 to 10 cells, where y + < 2.5. Transport equations on 〈U〉 and k use
specified values of ω in this region, and the transport equation of ω is only
solved outside of this very near-wall region.
4.4.3 The Logarithmic Law of the Wall
A wall function mesh incorporates a near-wall cell that is large enough to
fully encompass the buffer layer and extend into the region where the log
law is valid. This is shown schematically in Figure 4.2. Because flows of low
Reynolds number are considered in this thesis, the buffer layer is very large.
The near-wall node location was set to y/δ = 20% or y + = 36.0. The main
grid of PASSABLE uses vertexes centred between nodes, so the near-wall
node is positioned far from the centre of its cell. The remaining grid was
made up of 18 nodes. An expansion factor of 1.00 was used for these 18 cells.
The total number of grid cells used with wall functions was thus much less
than with the low-Reynolds-number solution.
Numerically, the log law is implemented by severing the link to the wall
in the near-wall cell, and then manipulating the 〈U〉, k, ε source terms in
the near-wall cell to reflect the log law. A full discussion of the numerical
implementation of log laws in CFD can be found in the book of Versteeg &

CHAPTER 4. NUMERICAL IMPLEMENTATION 57
Malalasekera [83].
U +
P
nience:
wall
function
applied
Figure 4.2: The high-Reynolds-number grid
is calculated from the log law (Equation 2.39), repeated here for conve-
U +
P
1
=
κ ln Ey +
p
(4.16)
where y + p is obtained from the C 1/4
µ k 1/2 velocity scale suggested by Launder
& Spalding [34]:
y + p = C1/4 µ k 1/2
p yp
ν
yp is the distance of the near-wall cell node from the wall.
Wall shear stress is obtained from Equation 2.44. It is
− τw
ρ
τw = ρC1/4 µ k 1/2
p 〈U〉 P
U +
P
(4.17)
(4.18)
from Equation 4.18 may be added to the x-momentum equation as a

CHAPTER 4. NUMERICAL IMPLEMENTATION 58
source. 6 Since τw contains a 〈U〉 P term, the SP source is used:
SP = S ′ P − C1/4 µ k 1/2
p
U + p
(4.19)
where S ′ P is the value of SP before the log-law term is added (containing only
the time-dependence term).
Also, AS is modified to cut the link to the wall and prevent a second (in-
accurate) accounting of τw via the normal functioning of the x-momentum
equation. The modified terms are
AS = 0
AP = AN (4.20)
This modification is common to all wall function treatments on 〈U〉, k, ε,
and ω.
Average production and dissipation of k in the near-wall cell (
Pk p and
(ε) p ) are calculated from Equations 2.45 and 2.46. In descretised form, these
become
Substituting Equation 4.18,
〈U〉P
Pk = τw
p
yp
(kp)
(ε) p = ρCµ
2
〈U〉P
Pk p
τw
yp
ρC1/4 µ k
= 1/2
p (〈U〉 P ) 2
U +
P
(ε) p = C 3/4
µ k 3/2
· yp
U +
P
yp
(4.21)
(4.22)
(4.23)
(4.24)
6 In general, one would integrate τw with respect to the surface area over which it acts
before applying the result as a source to the x-momentum equation. In channel flow
geometry, τw acts over a surface of unit area.

CHAPTER 4. NUMERICAL IMPLEMENTATION 59
1
1
Pk dy − (ε)p dy
ρ p ρ
must be added to the transport equation of k
in the near-wall cell as a source. k may be factored out of the equation for
(ε) p , and the remaining equation (including the k 1/2 term) may be placed in
the SP source. This amounts to modelling k 3/2
p in the equation for (ε) p as
1/2
kp · k old
p
, where k old
p is the previous iteration value of kp.
The terms in the k equation are modified as follows:
SP = S ′
C
P −
3/4
µ k 1/2
U +
p
P
∆yp
ρ yp
C 1/4
µ k 1/2
p (〈U〉 P ) 2
SU = S ′ U +
U +
P
· yp
∆yp
(4.25)
In treating the transport equation of ε, the value of ε is simply prescribed in
the near-wall cell according to the mixing length hypothesis (Equation 2.47):
εp = C3/4 µ k3/2 κyp
(4.26)
By making SP and SU large, other terms in the transport equation of ε may
be made numerically insignificant in the near-wall cell, allowing a prescribed
value to be specified in SU. The source terms are set as follows:
SP = −G
C
SU = G ·
3/4
µ k3/2
κyp
(4.27)
where G is a very large number, chosen to be several orders of magnitude
greater than any other terms that will appear in the transport equation of ε.

CHAPTER 4. NUMERICAL IMPLEMENTATION 60
4.4.4 The Subgrid Approach
The subgrid implementation employed in this work begins with the same grid
used for a log-law wall function, discussed above. This is referred to as the
‘main grid’. Within the wall-adjacent cell of the main grid, the subgrid code
generates a subgrid mesh, as shown schematically in Figure 4.3. Within
the subgrid mesh, 50 nodes were used, with an expansion factor of 1.10.
The subgrid mesh is similar to the low-Reynolds-number mesh shown in
Figure 4.1, but it fills only the near-wall region corresponding to the wall-
adjacent cell of the main grid. The subgrid mesh differs from the main
grid mesh in the choice of node locations. While the main grid uses cell
vertexes centred between nodes, the subgrid coded in this work uses nodes
centred between vertexes. 7 This affects way in which linear interpolations
are calculated within the code.
main grid
node
subgrid
node
subgrid
region
Figure 4.3: The subgrid mesh, adapted from Gant [21]
7 This is due only to programmer preference.

CHAPTER 4. NUMERICAL IMPLEMENTATION 61
The subgrid values are updated within the main iteration loop of the CFD
code. After 〈U〉, k, and ε or ω in the main grid have been updated in one
iteration, the subgrid update function is called. This function is essentially
the main iteration loop of another, embedded CFD code that performs one
iteration to solve for 〈U〉, k, and ˜ε or ω in the subgrid. Before the subgrid
calculations are performed, data are taken from the main grid to act as
boundary conditions on the subgrid. After the subgrid is updated, subgrid-
averaged values are extracted in order to act as wall function inputs to the
next main grid iteration. Thus, there is a cyclic exchange of information.
The data required as input to the subgrid calculation are the values of 〈U〉,
k, and ε or ω at the outer extent of the subgrid (where the subgrid ends and
the main grid transport equations begin). Transport equations are solved in
every subgrid cell (except in the case of the k-ω model, where the very-near-
wall values of ω are prescribed, as discussed above). At the wall node, zero
values of 〈U〉, k, and ˜ε are set. At the node placed farthest from the wall,
at the outer-most point on the subgrid, the values of 〈U〉, k, and ˜ε or ω are
set as being equal to the corresponding values linearly interpolated from the
two nearest main grid nodes.
The boundary conditions on the main grid are more complex. 〈U〉 receives its
boundary condition as in the above section on log-law wall functions, except
that the wall shear stress, τw is obtained from the application of Newton’s
law of viscosity at the subgrid’s near-wall cell node:
∂ 〈U〉
τw = ρν
∂y
y=0
Taking s1 as the near-wall subgrid cell node and discretising ∂〈U〉
∂y
y=0
(4.28)
within

CHAPTER 4. NUMERICAL IMPLEMENTATION 62
the subgrid, SU of the main grid near-wall cell node is modified as follows:
SU = S ′
〈U〉s1
U − ν
(4.29)
To obtain the main grid boundary condition on k, volume-weighted subgrid
averages of turbulent production (Pk) and dissipation (ε = ˜ε+ˆε) are required.
These are readily obtained after the subgrid governing equations have been
solved. These values may be integrated with respect to the main grid near-
wall cell volume and then incorporated as a source in the main grid near-wall
cell equation for k:
For the k-ω model
ys1
SU = S ′ U + Pk − ε (4.30)
ε = (ωk)β ∗
(4.31)
Note that, in the above equation for ε, the quotient is averaged. If the
individual terms are averaged separately and then multiplied, the boundary
condition will not be correct. Terms may only be averaged separately where
the net source applied to an equation is a linear combination of those terms.
The boundary condition on ε is obtained in a very similar manner to the
boundary condition on k. Rather than prescribing a value of ε in the main
grid near-wall cell, the terms leading to the production and distruction of ˜ε
are averaged throughout the subgrid, leading to a net production of ˜ε to be
applied to the main grid. (In the main grid node, ˜ε may be taken as being
synonymous with ε, since ˆε → 0 away from solid boundaries.) The modified
source term applied to the main grid near-wall cell is
SU = S ′ U + Cε1
f1
˜ε
Pk − Cε2 f2
k
˜ε 2
− E − Y (4.32)
k

CHAPTER 4. NUMERICAL IMPLEMENTATION 63
Unfortunately, a similar approach was found to be impossible in specifying
the boundary condition on ω. This is thought to be due to the near-wall be-
haviour of ω. Because ω approaches infinity near the wall, subgrid-averaged
production and destruction values are difficult to obtain. Therefore, ω is
specified at the main grid near-wall cell node according to linear interpola-
tion between the two closest subgrid nodes. Given an interpolated subgrid
ω value, ωsg, the source terms on ω in the main grid are
where G is a large number.
SP = −G
SU = G · (ωsg) (4.33)
The UMIST-N subgrid wall function simplifies near-wall calculation, as com-
pared to a low-Reynolds-number treatment, by removing pressure correction
and the wall-normal momentum equation from the solution procedure. Thus,
the flow calculation is reduced to a parabolic problem. Since PASSABLE
is a parabolic code with wall-normal velocity constrained to be zero, the
differences between the subgrid wall function approach and a low-Reynolds-
number treatment of steady channel flow may be expected to be less sig-
nificant, both in terms of accuracy and computational cost, than would be
found in other flow geometries.
One key difference remaining between the subgrid and low-Reynolds-number
calculations is the necessity of applying subgrid results to the main grid as
wall-function-type boundary conditions, rather than solving the governing
equations in an uninterrupted manner through the near-wall region. Fur-
thermore, a potential exists for instability to be introduced by the lag that
exists between a main grid update and a subgrid update. This lag comes

CHAPTER 4. NUMERICAL IMPLEMENTATION 64
about by the way in which main grid and subgrid solutions are updated in
alternation, with data from each being passed as boundary conditions to the
other, in turn. Finally, the subgrid allows a very different level of grid re-
finement to be employed within the subgrid than in the main grid. Thus,
extensive near-wall grid refinement may be achieved without the numerical
problems that are usually associated with a single grid employing a very large
expansion factor.
4.5 Under-Relaxation
Under-relaxation is a technique that involves hindering the rate of conver-
gence of the CFD code in order to dampen spurious effects and reduce the
risk of overshooting the converged solution.
Generally, after an updated solution to one of the governing differential equa-
tions has been obtained, the resulting calculated local values of a quantity φ
will replace the previous values, according to
φ new = φ calc
(4.34)
When under-relaxation is used, an under-relaxation factor, α may be applied,
and the updated values of φ will be the weighted average of φ calc and φ old ,
according to
φ new = αφ calc + (1 − α) φ old
(4.35)

CHAPTER 4. NUMERICAL IMPLEMENTATION 65
In the discretised transport equations, under-relaxation is implemented by
(AP ) ur = 1
α AP
(SU) ur = SU + (1 − α) AP φp
(4.36)
The under-relaxation factors used in this thesis are shown in Tables 4.2 and
4.3.
Table 4.2: Under-relaxation factors used in the main grid
〈U〉 0.8
k 0.5
ε 1.0
˜ε 1.0
ω 1.0
Table 4.3: Under-relaxation factors used in the subgrid
〈U〉 0.7
k 0.7
˜ε 0.7
ω 0.7

Chapter 5
Results
The performance of the UMIST-N near-wall subgrid treatment was tested
in channel flow. The PASSABLE code was used to implement a parabolic
solution method to solve this flow domain. For this work, the PASSABLE
code was extended to allow high-Reynolds-number (wall function and sub-
grid) solutions. Also, a separate implementation of the turbulence models
and a separate solver were written to be employed in solving the subgrid
region near the wall (extending from y/δ = 0 to y/δ = 0.2).
The models implemented in the subgrid are the standard k-ε model [33] with
Yap correction [92] and the 1988 k-ω model [89]. PASSABLE supports a
range of turbulence models for the solution of the main grid. The models
employed in this work were the low-Reynolds-number k-ε model with Yap
correction, the 1988 k-ω model, and the high-Reynolds-number k-ε model
(newly added). The configurations of turbulence models used in this work
are summarised descriptively in Table 5.1.
66

CHAPTER 5. RESULTS 67
Table 5.1: Configurations of turbulence models
Low-Re k-ε The subgrid was not employed and the low-Reynolds-
number standard k-ε model was used.
k-ε with log law The subgrid was not employed and the high-Reynolds-
number standard k-ε model was used with a
log-law boundary condition near the wall.
Subgrid k-ε The high-Reynolds-number standard k-ε model
was used in the main grid with a near-wall boundary
condition derived from a subgrid solution. The subgrid
employed the low-Reynolds-number standard k-ε model.
Subgrid k-ω The 1988 k-ω model was used in the main grid with a
near-wall boundary condition derived from a subgrid
solution. The subgrid also employed the 1988 k-ω model.
Three test cases were implemented: steady channel flow, channel flow driven
by a periodically variable pressure gradient, and channel flow constrained to
exhibit a periodically variable bulk flow rate. The two periodic cases differ
in that the former matches the pressure gradient fluctuation of the DNS
study of Kawamura & Homma [29] while the latter matches the bulk flow
variation presented in that DNS result. The steady flow case was performed
at a nominal friction velocity Reynolds number of Reτ = 180. 1 The periodic
flow cases were configured such that the time average flow would conform to
1 The boundary conditions on the code were configured such that 180 would be a correct
value of Reτ based on a theoretical analysis. The actual Reynolds number obtained was
different, as discussed below. This is what is meant by ‘nominal’ Reτ .

CHAPTER 5. RESULTS 68
a nominal Reτ of 180.
5.1 Steady Flow Results
Although the principal goal of this work is to test the performance of the
UMIST-N method in periodic flow, the steady flow case offers a simpler test
case in which to verify the code.
Steady channel flow results for the k-ε and k-ω models are widely available.
The UMIST-N approach with the standard k-ε model was applied to channel
flow using the TEAM code by Gant during the early development of UMIST-
N [21]. The application of the UMIST-N approach with a k-ω model to
steady channel flow is new.
All the model configurations shown in Table 5.1 were applied to steady chan-
nel flow and the results are presented here. Figures 5.1 & 5.2 show U + and
k + (respectively) plotted against y/δ. Figures 5.3 & 5.4 show a better view
of near-wall values plotted against y + on logarithmic scales. The results are
compared the analytical profiles from Chapter 3 and to the data of Kim et
al. [30], who employed the same Reτ.
The pressure gradient driving the flow was chosen to produce a nominal Reτ
of 180. By applying the definition of Reτ (Equation 3.25), a value of Uτ was
obtained that was used in nondimensionalisation. From Equations 3.23 &
3.19, a corresponding
∂〈P 〉
∂x
was specified. This ∂〈P 〉
∂x
was allowed to drive the
flow, with the actual calculated flow rate being a function of the turbulence
model used.

CHAPTER 5. RESULTS 69
The solution procedure employed involved initialising flow variables to zero,
applying the driving pressure gradient, and iteratively updating the solution
until convergence was achieved. Gradients in time ( ∂
∂t
terms) were included
in the solution, so that unconverged results may have been regarded con-
ceptually as the transient response of the flow to a step change in pressure.
However, transient results are not shown. ∂
∂t
terms do not affect the con-
verged result in steady flow and were included primarily for debugging.
In Figure 5.1, the k-ε solutions overpredict peak velocity and bulk flow, while
the k-ω model underpredicts. Figure 5.3 shows that the 〈U〉 + results nearer
to the wall are similar for each model and match the DNS result closely.
Equation 3.17 can be evaluated at y = 0 to obtain
d 〈U〉 y=0
τw = ρν
dy
d〈P 〉
dx by
Recalling Equation 3.19, τw is related to
d 〈P 〉
τw = −δ
dx
(5.1)
(5.2)
Therefore, close agreement on 〈U〉 + near the wall is to be expected from the
prescription of
d〈P 〉
dx . The deviation in 〈U〉+ occurs farther from the wall and
is likely due to the prediction of νt.
The subgrid solution produces results that are very similar to those of the
low-Reynolds-number approach. This is to be expected, since the internal
subgrid calculation is identical to a standard low-Reynolds-number calcu-
lation in a parabolic solution scheme. The unique aspect of the subgrid
approach as compared to a low-Reynolds-number parabolic solution is in the
transferral of information between the subgrid and the main grid via mutu-
ally dependent boundary conditions. However, this complexity is unlikely to

CHAPTER 5. RESULTS 70
affect the solution of steady channel flow. This assumption is supported by
the smoothness of the subgrid solution on both U + and k + in the vicinity
of y/δ = 0.2, where the subgrid and the main grid meet. The small differ-
ence observed between the subgrid and low-Reynolds-number k-ε solutions
is likely due to differences in grid refinement. Although the subgrid solution
uses fewer cells in total (summing the subgrid and the main grid cells) than
the low-Reynolds-number approach, it offers enhanced near-wall refinement
within the subgrid, with a coarser grid employed away from the wall. The
fact that the subgrid solution produced a velocity profile that is very slightly
nearer to the DNS result highlights a secondary benefit of the UMIST-N
approach, which is that the application of boundary conditions at a subgrid
/ main grid boundary allows an opportunity for extensive and sudden vari-
ation in grid refinement without some of the numerical challenges usually
associated with inconsistent cell sizes.
The logarithmic law of the wall produces excellent results in steady channel
flow. This is to be expected, since it was designed and tuned with this par-
ticular flow configuration in mind. Figures 5.1 & 5.3 indicate nearly perfect
agreement on velocity. The discrepancy in k + seen in Figures 5.2 & 5.4 is
likely the result of that variable’s definition within the k-ε model. Although
k is intended to model turbulent kinetic energy, the model constants have
been chosen with the primary aim of producing good agreement with exper-
iments in the prediction of velocities. k is related to velocity via the EVM.
Functionally, then, it may be convenient to conceptualise k not as turbulent
kinetic energy but simply as that parameter which is required by the EVM
to produce desirable values of velocity. With this in mind, the log law’s pre-
diction of k + near the wall in channel flow must match a hypothetical ideal
k-ε solution that reproduces the DNS result for U + . It must not necessarily

CHAPTER 5. RESULTS 71
match the DNS result for k + .
The subgrid approach with the k-ω turbulence model performs rather well. It
can be seen to underpredict U + in Figures 5.1 & 5.3, although by a relatively
small margin. Away from the wall, the value of k in the k-ω model offers a
good approximation to the DNS result, as seen in Figures 5.2 & 5.4.
All of the solutions underpredict turbulent kinetic energy near the wall, as
is particularly apparent in Figure 5.4. Tracing from the symmetry plane of
the channel towards the wall, the point at which the modelled solutions tend
to depart from the DNS result is y + ≈ 40. This corresponds roughly to the
outer extent of the buffer layer (see Table 2.4).
Overall, the steady channel flow results appear consistent with prior work
and offer credibility to the following calculations involving periodic flow.
5.2 Prescribed Periodic Pressure Gradient
The flow was subjected to the same periodically variable pressure gradient
employed by Kawamura & Homma [29] in their DNS study. This pressure
gradient was defined by
∂ +
〈P 〉
2π
= 1 + 6 sin
∂x
6 t+
where
∂〈P 〉
∂x
(5.3)
and t (time) are nondimensionalised by
+
∂ 〈P 〉
∂x
=
∂〈P 〉
∂x
2
ρ(Uτ ) ss
(5.4)
t + = (Uτ) ss t
δ
δ
(5.5)

CHAPTER 5. RESULTS 72
(Uτ) ss is calculated from (Reτ) ss = 180. This implies that
+
∂〈P 〉
∂x
ss
= 1.
Because of Equation 5.2, this also suggests a time-mean value of τw corre-
sponding to the steady flow case considered above.
To generate the periodic flow results, a converged steady flow solution was
first reached. Then, pressure gradient variation was introduced according to
Equation 5.3. To ensure convergence, 1000 time steps were used per period in
the periodic flow case. At each time step, the code was run until convergence
was achieved. However, this convergence was intermediate in the sense that
it represented an estimate of the solution using the given ∂
∂t
information. To
obtain results representing the effect of long-term periodicity, this process was
repeated through several periods until only negligible changes were observed
in subsequent periods. Thus, start-up effects were eliminated.
Figures 5.5 & 5.6 show the prescribed pressure gradient, the calculated bulk
flow rate, and the calculated wall shear stress plotted with DNS results. The
graphs show values plotted from periods 0 to 2. This is not meant to imply
that the first two periods of generated output are plotted. These graphs are
produced from a final, converged period plotted twice for visual clarity.
Figure 5.5 shows a marked underprediction of the amplitude of variation
in bulk flow for all of the models employed. Figure 5.6 also indicates that
wall shear stress is underpredicted. This is consistent with the findings of
Addad [1], and, as he pointed out, the relationship between τw, U, and
is manifest in the x-momentum equation governing the flow (Equation 3.5).
Employing the definition of the shear stress, τ(y) from Equation 3.17, the
x-momentum equation is
∂ 〈U〉
∂t
∂ 〈P 〉
= −1
ρ ∂x
1 dτ
+
ρ dy
∂〈P 〉
∂x
(5.6)

CHAPTER 5. RESULTS 73
Integrating with respect to y,
δ
∂
1 ∂ 〈P 〉
〈U〉 dy = −
y|
∂t
ρ ∂x
y=δ
τ
y=0 +
ρ
0
τ=0
τ=τw
Dividing by δ and noting that 1
δ
〈U〉 dy = U, Equation 5.7 becomes
δ 0
dU
dt
+ τw
ρδ
∂ 〈P 〉
= −1
ρ ∂x
(5.7)
(5.8)
Thus, in unsteady channel flow, the driving pressure gradient applies energy
to both overcome a wall shear stress and to accelerate the flow. The fact that
peaks and troughs in U are underpredicted in Figure 5.5 indicates that dU
dt is
underestimated by the turbulence models. τw also displays a lower amplitude
of variation than expected, as seen in Figure 5.6. This is consistent with
(Equation 5.1). Together, these
Figure 5.5, since τw is a function of ∂〈U〉
∂y
results suggest that the energy is added to and removed from the flow at a
less than realistic rate.
The equality in Equation 5.8 appears to be compromised when using these
turbulence models in periodic flow. In terms of the x-momentum equation
(Equation 3.5), this suggests that the models are not as well tuned to predict
〈uv〉 in periodic channel flow as in steady channel flow. Errors in predicting
〈uv〉 produce deviations in 〈U〉, particularly farther from the wall, as seen in
Figures 5.11 & 5.12. This effects the dU
dt
y=0
term in Equation 5.8.
Figures 5.7, 5.8, 5.9 & 5.10 show flow variables plotted as a function of phase
angle at various locations throughout the channel (y/δ = 0.1, 0.2, 0.5 & 0.9,
respectively).
Results for the k-ε log law do not appear in Figure 5.7, because y/δ = 0.1
is outside the calculated flow field when the log law is used. In Figure 5.8,

CHAPTER 5. RESULTS 74
where y/δ = 0.2, the log law produces results for k + that are in phase with the
results for 〈U〉 + . This is to be expected from the log law equations. Further
from the wall, where the k-ε model equations are applied, the generated
result is improved, but still offers little encouragement as to the applicability
of the log law to this flow.
Examining turbulent kinetic energy, k + in Figures 5.7, 5.8, 5.9 & 5.10, it
appears that the subgrid k-ε treatment closely matches the low-Reynolds-
number k-ε solution for y/δ ≤ 0.2, within the subgrid region. Outside of the
subgrid region (Figures 5.9 & 5.10), the subgrid solution appears to capture
the effects of variations in
∂〈P 〉
∂x
slightly less effectively than the low-Reynolds-
number counterpart. This suggests that the process of averaging the subgrid
solution to be applied as a boundary condition to the main grid introduces
some discernible measure of inaccuracy. The subgrid solution can therefore
produce different results than a calculation that is continuous throughout the
flow field (standard low-Reynolds-number treatment), even when the same
turbulence model is used in each case.
The k-ω solution produces and amplitude of oscillation of τw that more closely
matches the DNS than that produced by the k-ε subgrid solution (Figure
5.6). Correspondingly, the k-ω model produces fluctuations in bulk velocity
that are also slightly improved (Figure 5.5). This is an indication of the
applicability of the k-ω model to boundary layer flow. The k-ω model exhibits
a less prominent peak in turbulent kinetic energy than is produced by the
k-ε model (Figures 5.7, 5.8, 5.9 & 5.10). This may be compounded by the
general tendency of the k-ω model to offer lower values of k, as observed in
the steady flow results.

CHAPTER 5. RESULTS 75
Figures 5.11, 5.12, 5.13 & 5.14 offer snapshots showing 〈U〉 + and k + , respec-
tively, throughout the channel at various discrete phase positions. Figures
5.15, 5.16, 5.17 & 5.18 show the same again, but plotted against y + rather
than y/δ.
Because of the general inaccuracy of all the models, it is difficult to make
comparisons based on Figures 5.11 to 5.18. Figures 5.13 & 5.14 again high-
light the tendency of the log law solution of k + to generate spurious results
at y/δ = 0.2 where the log law is applied. Furthermore, Figures 5.17 & 5.18
show that all the models produced an underprediction of k + near the wall.
This is similar to the underprediction observed in the steady flow case.
To better facilitate comparisons between the models, and following the pe-
riodic channel flow investigation of Addad [1], a further test case was con-
sidered in which the modelled flows are constrained to exhibit the same bulk
flow rates shown in the DNS study. These results are presented below.
5.3 Prescribed Periodic Bulk Flow Rate
The code was configured to match the bulk flow from the DNS study of
Kawamura & Homma [29], rather than the
∂〈P 〉
∂x
data. Figure 5.19 shows
the resulting pressure variations compared to the pressure variation in the
DNS study. As expected, the resultant pressure variation is higher. The
amplitude on the fluctuation in
∂〈P 〉
∂x is greater than the DNS amplitude by
approximately 100%. Correspondingly, wall shear stress is increased.
As seen in Figure 5.20, the k-ε low-Reynolds-number and subgrid results now

CHAPTER 5. RESULTS 76
more closely resemble the DNS data for τw, except that they exhibit a double-
peak behaviour. τw produced by the k-ω subgrid matches the troughs in the
DNS data effectively, but the peak values are disappointingly overestimated.
The asymmetry in τw suggests that the k-ω model predicts excessively steep
profiles of 〈U〉 + near the wall for most phase angles, as can be seen in Figures
5.31 & 5.32.
The further figures mirror the presentation format employed in the previous
section. Figures 5.21, 5.22, 5.23 & 5.24 display flow variables through time
at y/δ locations of 0.1, 0.2, 0.5 & 0.9, respectively. Figures 5.25, 5.26, 5.27
& 5.28 show snapshots of the channel at various phase angles, and Figures
5.29, 5.30, 5.31 & 5.32 show these against a logarithmic y + scale.
Figures 5.21, 5.22, 5.23 & 5.24 show a tendency for the low-Reynolds-number
k-ε model to exhibit a delay in predicting an increase in k + when the flow
is accelerating. When it appears (for example, at a phase angle of 7π/6
in Figure 5.23), the anticipated increase in k + is sudden and sharp. At
y/δ ≤ 0.5, the peak value of k + through time predicted by the low-Reynolds-
number k-ε model (and indeed by all of the models considered in this work) is
lower than the DNS result, but k + is actually overpredicted throughout most
of the portion of the cycle during which the flow is decelerating. In Figure
5.24, with y/δ = 0.9, the peak value of k + is overpredicted by all except the
k-ω subgrid model and the overprediction of k + during deceleration of the
flow is very pronounced in all of the modelled results.
The use of the UMIST-N subgrid approach with the k-ε model does not ap-
pear to alter the phase position of the suden increase in k + noted above. How-
ever, the subgrid approach does appear to soften this effect. The maximum

CHAPTER 5. RESULTS 77
gradient of k + seen in Figures 5.23 & 5.24 is lower than for the low-Reynolds-
number k-ε model. Furthermore, the extent of over- and underprediction of
k + is diminished throughout the cycle. One reason for this may be that
diffusion is not accounted for across the subgrid boundary. Although the
subgrid is constrained to match the main grid value of k + at y/δ = 0.2, there
is no diffusion of k into the main grid from the subgrid. The only information
transmitted from the subgrid to the main grid in the calculation of the model
equation of k is subgrid-averaged Pk and ε.
Two other potential sources of discrepancy between subgrid and low-Reynolds-
number treatments are detalied here. Firstly, greater numerical errors may
occur in the subgrid solution as a result of the averaging process that is used
to apply subgrid results to the main grid as boundary conditions. Secondly,
the subgrid solution may be configured to offer greater near-wall grid refine-
ment because of the fact that the subgrid cell sizes may be set independently
of the main grid cell sizes. In this work, the subgrid solution did employ
smaller cells near the wall than were used in the low-Reynolds-number treat-
ment, because the use of a high-Reynolds-number model in the main grid
allowed the use of fewer nodes in total. However, suitable grid refinement
was employed in all cases so that the computed results may be assumed to
be independent of cell size. It must be highlighted that the potential of
the subgrid treatment to achieve greater near-wall grid refinement without
impacting the refinement of the main grid is a powerful feature of UMIST-N.
Notably, any difference between the subgrid and low-Reynolds-number k-ε
profiles is less discernable in Figures 5.21 & 5.22, where y/δ ≤ 0.2. This
region falls within the subgrid itself, and the calculation there is substan-
tially similar to a standard low-Reynolds-number treatment. The difference

CHAPTER 5. RESULTS 78
between the subgrid and standard low-Reynolds-number treatments is in the
averaging used within the subgrid to apply subgrid results as a boundary con-
dition to the main grid. Therefore, the impact of using a subgrid treatment
manifests itself beyond y/δ = 0.2.
The subgrid k-ω model predicts k + admirably well in Figures 5.21, 5.22, 5.23
& 5.24. It does not exhibit the sudden increase in k + noted above, and actual
predicted values of k + appear to be at least as accurate as those of other
models at each of the traverse points shown. However, careful examination
of the DNS data suggests that a slightly greater slope of k + with time is
to be expected when k + is increasing than when it is decreasing. At lower
values of y/δ, the k-ω model appears to fail to capture this effect, suggesting
a qualitative error. Such a criticism would be harsh, however, in light of
Figure 5.24 (y/δ = 0.9), in which the k-ω model does exhibit a greater slope
on the increase of k + than on the decrease. On the whole, the model appears
to offer good qualitative and quantitative accuracy.
The favourable performance of the k-ω subgrid in predicting k + is most ap-
parent at large y + , as seen in Figures 5.27, 5.28, 5.31 & 5.32. The k-ω
prediction matches the DNS almost perfectly for y = δ and phase angles of
5π/4 and 3π/2, and the model produces good results at y = δ for other phase
angles. Near the wall, the behaviour of k + is more accurately reproduced by
the k-ω subgrid for phase angles of π/2 and 3π/4 (favourable pressure gradi-
ent) than by the k-ε subgrid. However, the k-ω model’s near-wall predictions
are less favourable at phase angles of 3π/2 and 7π/4, under the influence of
an adverse pressure gradient. At phase angles of 0 and π, there is little
discernible difference among the near-wall predictions of the various models.

CHAPTER 5. RESULTS 79
The log law offers predictions of k + in Figures 5.23 & 5.24 that are reasonable.
The log law solution has also been improved greatly by the prescription of U
rather than
∂〈P 〉
. It must once again be noted that the log law only produces
∂x
interesting results away from y/δ = 0.2, where it is applied for this particular
flow.
A ‘kinked’ region of large change in gradient tends to exist in many of the
profiles of 〈U〉 + . This phenomenon can be seen most clearly in Figures 5.25
& 5.26. It is produced by all of the models except for the log law treatment,
and is particularly prevalent in the profiles produced by the low-Reynolds-
number k-ε model. A kink indicates an underprediction of the diffusion of
〈U〉 + within the vicinity of the kink. Deferring to the assumptions of the
EVM, this would suggest a local underprediction of k + . This corollary can
in fact be seen in the figures. Comparing phase angles of 3π/4, π, and 5π/4,
suggests that the location of a kink in 〈U〉 + corresponds to a location of
maximum underprediction of k + . This may be seen by examining Figures
5.25 & 5.26 for velocity and 5.27 & 5.28 for k + .
The same phenomenon of a kinked velocity profile can be observed indirectly
in Figures 5.21, 5.22, 5.23 & 5.24, where 〈U〉 + predicted by the low-Reynolds-
number k-ε model appears to lead the DNS result in phase closer to the wall,
whereas it lags the DNS in phase at higher y/δ. This apparent peculiarity is
in fact a symptom of the changing slope of the 〈U〉 + profile produced by the
k-ε model at different phase angles.
All of the models produce flatter velocity profiles than the DNS, as seen in
Figures 5.25, 5.26, 5.29 & 5.30. This is particularly true just after a phase
angle of π, when the flow is subjected to an adverse pressure gradient and a

CHAPTER 5. RESULTS 80
sudden increase in k + occurs for y/δ ≥ 0.5 (Figures 5.23 & 5.24). When the
pressure gradient diminishes and becomes adverse, the effect on velocity is
expected to manifest itself first near the wall and then propagate outwards.
This happens because the laminar layer near the wall contains less kinetic
energy and therefore less momentum. All of the modelled results exhibit
this preferential slowing of the fluid near the wall when the pressure gradient
becomes adverse, but they do so later in phase and to a lesser extent than
what is seen in the DNS data. This can be seen in Figures 5.25 & 5.26.
It is interesting to compare the snapshots for the phase angles of π/2, 3π/4,
and π found in Figures 5.29 & 5.30. The excessive flatness of the 〈U〉 +
profiles is apparent in all of these snapshots, but particularly at the phase
angle of π, where the flattening occurs at a lower value of y + for all the models
than at the phase angle of π/2. Between these two, the snapshot at phase
angle 3π/4 shows the subgrid k-ω model already flattening at lower y + , while
the low-Reynolds-number k-ε model persists in flattening only at higher y + .
The subgrid k-ε modelled results follow the low-Reynolds-number solution
for lower y + , but the 〈U〉 + profile flattens at y/δ ≈ 0.2 earlier to join the
subgrid k-ω results for higher y + . The point at which the subgrid solution
flattens corresponds roughly to the outer extent of the subgrid. Thus, at this
transitional snapshot in time, when the 〈U〉 + profile is undergoing a change
of shape, the solutions appears to be particularly sensitive to the averaging
procedure used to calculate main grid boundary condition from the subgrid
solution.
An interesting feature of the UMIST-N subgrid approach is highlighted by
the k-ω prediction of k + at phase angles of π/2 and 3π/4. Here, the k +
profiles predicted by the subgrid k-ω model exhibit a double peak (two local

CHAPTER 5. RESULTS 81
maxima). This is particularly visible in Figure 5.31, but Figure 5.27 shows
that the local minimum between these two peaks lies at y/δ = 0.2, the limit
of the subgrid region and also the point where boundary conditions from the
subgrid are applied to the main grid and visa versa. Close inspection of all
the k + profiles produced by the k-ω subgrid reveals minor discontinuities at
this y/δ location for other phase angles also. This is a result of the way
in which the subgrid solution of ω was applied as a boundary condition to
the main grid. The UMIST-N approach to k and ε boundary conditions
is to calculate subgrid averaged production and dissipation terms for these
parameters, which are then applied as source terms within the near-wall cell
in the main grid solution. However, ω tends to infinity at the wall, and
source terms on ω become numerically unwieldy in near-wall cells. Thus,
subgrid averaged production of ω could not be obtained and instead the
subgrid profile of ω was interpolated to the main grid node and applied there
as a fixed value. 2 This boundary condition appears to be less effective than
the average source approach, and, until a more satisfactory solution to the ω
boundary condition problem can be found, it advises against the applicability
of the UMIST-N approach to a k-ω model solution. This aside, the subgrid
k-ω results appear satisfactory when considered as a whole.
In addition to the above difficulty in the prescription of an ω boundary
condition, further difficulties were encountered in attempting to run the k-ω
model in the subgrid with a high-Reynolds-number k-ε model in the main
grid. Poor results were seen to arise from the difference in the values of k
produced by the two models under similar conditions. In general, the k-
ω model produces lower values of k than the k-ε model, as noted in the
2 See Chapter 4 for more information.

CHAPTER 5. RESULTS 82
steady flow results. In order to reconcile the two models, it was proposed
that a blending function similar to that used by the SST model [42] may be
required. The development effort was stopped because the added complexity
would have compromised the relevance of the UMIST-N approach.
The UMIST-N approach has been successfully applied to periodic flow. The
subgrid calculation offers discernibly different results than the standard low-
Reynolds-number k-ε model, but neither may readily be identified as supe-
rior. Therefore, the UMIST-N approach is applicable to periodic flow. The
potential breadth of applicability of the approach is further indicated by its
application to a k-ω model solution, although the challenge of obtaining main
grid boundary conditions with the k-ω model is highlighted by these results.

Chapter 6
Conclusions & Suggestions for
Future Work
In the present study, the UMIST-N near-wall subgrid treatment has been
applied to time-variant flow. The cases considered have been periodic and
steady channel flow. Results have been matched to DNS data on driving
pressure gradient and on bulk flow rate. The logarithmic law of the wall, the
k-ε subgrid solution and the low-Reynolds-number k-ε solution were sub-
stantially similar in the steady case, as expected. In the unsteady case, the
subgrid produced results that were much improved over the log law, bear-
ing more similarity to the low-Reynolds-number results. The subgrid and
low-Reynolds-number results were discernibly different, however, and some
of the features to be considered when implementing a subgrid treatment were
highlighted.
The use of the k-ω turbulence model within the UMIST-N subgrid was also
83

CHAPTER 6. CONCLUSIONS & SUGGESTIONS FOR FUTURE WORK84
tried in each of the above test cases. The model produced good results overall,
but its performance was hindered by spurious behaviour at the interface
between the subgrid and the main grid. Numerical challenges associated
with the implementation of the k-ω model within a subgrid approach have
been revealed during this work and were highlighted in the results.
Further steady channel flow data from various sources were also compiled
within this thesis and emperical profiles were presented to identify trends.
These profiles could serve as a tool to CFD researchers who are engaged in
early development or debugging.
The UMIST-N approach has been shown to be applicable to periodic flow.
The potential breadth of applicability of the approach is further indicated by
its application to a k-ω model solution, despite challenges that were encoun-
tered. Further investigation into the usefulness of the UMIST-N approach
in the computation of time-variant industrial flows appears justified.
With particular reference to industrial applicability, one potential avenue
for the further development of UMIST-N is towards its use in Large Eddy
Simulation. Most LES calculations rely on wall functions, and hybrid com-
binations of RANS and LES are not unknown. Therefore, a potential exists
for UMIST-N to offer a benefit to LES. The satisfactory performance of the
subgrid method in periodic flow within this study suggests that UMIST-N
may be suitable when exposed to the rapid local oscillations that can be
associated with the movement of large-scale turbulent structures in LES.
As a tool for industrial modelling, UMIST-N offers an intermediate choice,
in terms of accuracy and computational cost, between a standard wall func-
tion boundary and a low-Reynolds-number treatment. This choice could be

CHAPTER 6. CONCLUSIONS & SUGGESTIONS FOR FUTURE WORK85
further optimised or simply expanded by the incorporation of other turbu-
lence models within the subgrid. This work has applied the k-ω model to the
subgrid. A further attempt was made to mix a k-ω subgrid with a k-ε main
grid, but the development effort was stopped when it became clear that dif-
ferences in the predicted values of k would necessitate the use of a blending
function similar to that used by the SST [42].
Building upon the use of k-ω, future work could be directed towards the
use of simpler turbulence models, particularly a one-equation model and an
algebraic model within the subgrid. If this can be done while retaining a
k-ε treatment in the main grid, then the adapted subgrid would offer an
additional level of flexibility in the cost/accuracy tradeoff associated with it.
Furthermore, the UMIST-N wall function could be applied to a further va-
riety of flows. Most notably, higher Reynolds numbers could be considered.
There is a difficulty in obtaining DNS results at higher Reynolds numbers,
but perhaps the completion of this work will allow the subgrid approach to
be confidently compared to experiments for periodic flow at higher Reynolds
numbers. Also, this work applies the subgrid treatment to periodic flow at
a relatively low frequency of oscillation and without sufficient amplitude to
generate flow reversal. A higher frequency and a larger amplitude would
provide an interesting test case, provided that suitable data can be found
for comparison. Such a test case would advise as to the applicability of
UMIST-N to the flow inside engines.
In a two-dimensional grid, the UMIST-N method involves simultaneously
storing subgrid results throughout the flow field. In addition to a storage
cost, this presents an added complexity in the implementation of the method.

CHAPTER 6. CONCLUSIONS & SUGGESTIONS FOR FUTURE WORK86
An attempt could be made to reduce this storage requirement and allow the
subgrid treatment on a given main grid cell to run as a self-contained function.
If this can be done without a prohibitive sacrifice of accuracy, it could further
broaden the industrial usefulness of UMIST-N.

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Figures
99

FIGURES 100
U +
35
30
25
20
15
10
5
0
10 0
10 1
+
+
+
+
Reichardt’s law
DNS: Reτ = 180 [30]
DNS: Reτ = 395 [44]
DNS: Reτ = 584 [44]
◦ Exp.: Reτ = 708 [87]
▽ Exp.: Reτ = 921 [47]
+
10 2
+
+
+
+
+
+
+
x +
x
x xxx
x
x
x +
x +
+
10 3
y +
10 4
Exp.: Reτ = 1017 [47]
+ Exp.: Reτ = 1655 [47]
⋄ Exp.: Reτ = 2340 [11]
× Exp.: Reτ = 4800 [11]
△ Exp.: Reτ = 8150 [11]
Figure 3.1: Reichardt’s law to estimate U +

FIGURES 101
− 〈uv〉 +
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
+ + +
+ +
+
+
+
+
+
+
+
10 1
Reichardt: Reτ = 180
Reichardt: Reτ = 584
Reichardt: Reτ = 1017
Reichardt: Reτ = 1655
+
+
x
x
x
+
+
+
++++++++++++++++++++++++++++++++
x x x x
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
x
x
x
x
x
10 2
x
x
y +
10 3
DNS: Reτ = 180 [30]
+ DNS: Reτ = 584 [44]
⋄ Exp.: Reτ = 1017 [47]
× Exp.: Reτ = 1655 [47]
Figure 3.2: Reichardt’s law applied to − 〈uv〉 +

FIGURES 102
− 〈uv〉 +
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
+ + +
+ +
+
+
+
+
+
+
+
10 1
Profile: Reτ = 180
Profile: Reτ = 584
Profile: Reτ = 1017
Profile: Reτ = 1655
+
+
x
x
x
+
+
+
++++++++++++++++++++++++++++++++
x x x x
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
x
x
x
x
x
10 2
x
x
y +
10 3
DNS: Reτ = 180 [30]
+ DNS: Reτ = 584 [44]
⋄ Exp.: Reτ = 1017 [47]
× Exp.: Reτ = 1655 [47]
Figure 3.3: A revised profile for − 〈uv〉 +

FIGURES 103
k +
5
4
3
2
1
0
10 0
+
+
+
+
+
+
+
+
+
+
10 1
Profile: Reτ = 180
Profile: Reτ = 395
Profile: Reτ = 584
+
+
+ + +
+++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++
+
10 2
Figure 3.4: A profile for k +
10 3
y +
DNS: Reτ = 180 [30]
+ DNS: Reτ = 395 [44]
⋄ DNS: Reτ = 584 [44]

FIGURES 104
〈uu〉 9
+
8
7
6
5
4
3
2
1
0
10 0
+
10 1
Profile: Reτ = 180
Profile: Reτ = 395
Profile: Reτ = 584
+
+
+
+
+
+
+
+
+
10 2
+
+
+
+
+
10 3
y +
DNS: Reτ = 180 [30]
+ DNS: Reτ = 395 [44]
⋄ DNS: Reτ = 584 [44]
Figure 3.5: A profile for 〈uu〉 +

FIGURES 105
1.4
〈vv〉 +
1.2
1.0
0.8
0.6
0.4
0.2
0.0
10 0
+
10 1
Profile: Reτ = 180
Profile: Reτ = 395
Profile: Reτ = 584
+
+
+
+
+
+
+ +
+
10 2
+
+
+
+
+
10 3
y +
DNS: Reτ = 180 [30]
+ DNS: Reτ = 395 [44]
⋄ DNS: Reτ = 584 [44]
Figure 3.6: A profile for 〈vv〉 +

FIGURES 106
0.20
k +
“
y +” 2
0.15
0.10
0.05
0.20
〈uu〉 +
“
y +” 2
0.00
0 2 4 6 8 10
0.15
0.10
0.05
y +
0.00
0 2 4 6 8 10
y +
−〈uv〉 +
“
y +” 3
〈vv〉 +
“
y +” 4
0.0020
0.0015
0.0010
0.0005
0.0000
0 2 4 6 8 10
0.00020
0.00015
0.00010
0.00005
y +
0.00000
0 2 4 6 8 10
▽ DNS: Reτ = 180 [30]
⋄ DNS: Reτ = 395 [44]
△ DNS: Reτ = 584 [44]
Figure 3.7: Near-wall behaviour of flow parameters
y +

FIGURES 107
y ∗
y ∗
600
500
400
300
200
100
0
0 100 200 300 400 500 600
100
80
60
40
20
y +
0
0 20 40 60 80 100
y +
▽ DNS: Reτ = 180 [30]
⋄ DNS: Reτ = 395 [44]
△ DNS: Reτ = 584 [44]
y ∗ = y +
Figure 3.8: y ∗ vs. y +

FIGURES 108
U ∗
or
U +
25
20
15
10
5
++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++
+++
xxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxx x xxxxxxxx
0 + x
0 100 200 300 400 500 600
y∗ or y +
U + vs. y + :
▽ DNS: Reτ = 180 [30]
⋄ DNS: Reτ = 395 [44]
△ DNS: Reτ = 584 [44]
U ∗ vs. y ∗ :
✄✂ ✁ DNS: Reτ = 180 [30]
+ DNS: Reτ = 395 [44]
x DNS: Reτ = 584 [44]
Figure 3.9: U ∗ superimposed on U +

FIGURES 109
y ∗ v2
y ∗ v2
600
500
400
300
200
100
0
0 100 200 300 400 500 600
100
80
60
40
20
y +
0
0 20 40 60 80 100
y +
▽ DNS: Reτ = 180 [30]
⋄ DNS: Reτ = 395 [44]
△ DNS: Reτ = 584 [44]
y ∗ v2 = y +
Figure 3.10: y ∗ v2 vs. y +

FIGURES 110
U ∗ v2
or
U +
35
30
25
20
15
10
x
+
x
+
+
+
+
+
+
+
++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++
x
x
x
x
x
x
x
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
5
0 + x
0 100 200 300 400 500 600
y∗ v2 or y +
U + vs. y + :
▽ DNS: Reτ = 180 [30]
⋄ DNS: Reτ = 395 [44]
△ DNS: Reτ = 584 [44]
U ∗ v2 vs. y ∗ v2:
✄✂ ✁ DNS: Reτ = 180 [30]
+ DNS: Reτ = 395 [44]
x DNS: Reτ = 584 [44]
Figure 3.11: U ∗ v2 superimposed on U +

FIGURES 111
〈U〉 +
〈U〉 +
20
18
16
14
12
10
8
6
4
2
0
0 0.25 0.5 0.75 1
20
18
16
14
12
10
8
6
4
2
a
y/δ
0
0 0.25 0.5 0.75 1
c
y/δ
〈U〉 +
20
18
16
14
12
10
8
6
4
2
0
0 0.25 0.5 0.75 1
a {
b{
c {
Figure 5.1: 〈U〉 + vs y/δ in the steady flow case
Low-Re k-ε
Subgrid k-ε
b
y/δ
k-ε with log law
Subgrid k-ε
Subgrid k-ε
Subgrid k-ω
✄✂ ✁ DNS [30]
Reichardt’s law [58]

FIGURES 112
k +
k +
5
4
3
2
1
0
0 0.25 0.5 0.75 1
5
4
3
2
1
a
y/δ
0
0 0.25 0.5 0.75 1
c
y/δ
k +
5
4
3
2
1
0
0 0.25 0.5 0.75 1
a {
b{
c {
Figure 5.2: k + vs y/δ in the steady flow case
Low-Re k-ε
Subgrid k-ε
b
y/δ
k-ε with log law
Subgrid k-ε
Subgrid k-ε
Subgrid k-ω
✄✂ ✁ DNS [30]
analytical profile

FIGURES 113
〈U〉 +
〈U〉 +
20
18
16
14
12
10
8
6
4
2
10 0 0
20
18
16
14
12
10
8
6
4
2
10 0 0
10 1
10 1
c
a
10 2
10 2
y +
y +
.
〈U〉 +
20
18
16
14
12
10
8
6
4
2
10 0 0
a {
b{
c {
Figure 5.3: 〈U〉 + vs y + in the steady flow case
10 1
Low-Re k-ε
Subgrid k-ε
b
10 2
k-ε with log law
Subgrid k-ε
Subgrid k-ε
Subgrid k-ω
y +
✄✂ ✁ DNS [30]
Reichardt’s law [58]

FIGURES 114
k +
k +
5
4
3
2
1
10 0 0
5
4
3
2
1
10 0 0
10 1
10 1
a
c
10 2
10 2
y +
y +
k +
5
4
3
2
1
10 0 0
a {
b{
c {
Figure 5.4: k + vs y + in the steady flow case
10 1
Low-Re k-ε
Subgrid k-ε
b
10 2
y +
k-ε with log law
Subgrid k-ε
Subgrid k-ε
Subgrid k-ω
✄✂ ✁ DNS [30]
analytical profile

FIGURES 115
U
U ss
U
U ss
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
-0.5 -0.5
-1.0 -1.0
-1.5 -1.5
-2.0 -2.0
0.0 0.5 1.0 1.5 2.0
a
period
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
-0.5 -0.5
-1.0 -1.0
-1.5 -1.5
-2.0 -2.0
0.0 0.5 1.0 1.5 2.0
a {
Low-Re k-ε
Subgrid k-ε
b{
✄✂ ✁ DNS [29] prescribed
b
period
k-ε log law
Subgrid k-ε
Subgrid k-ω
∂〈P 〉
∂x
( ∂〈P 〉
∂x )
( ∂〈P 〉
∂x ) ss
( ∂〈P 〉
( ∂〈P 〉
∂x ) ss
Figure 5.5: Bulk flow variation in the periodic pressure case
∂x )
× 10 −1
× 10 −1

FIGURES 116
τw
(τw) ss
τw
(τw) ss
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
-0.5 -0.5
-1.0 -1.0
-1.5 -1.5
-2.0 -2.0
0.0 0.5 1.0 1.5 2.0
a
period
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
-0.5 -0.5
-1.0 -1.0
-1.5 -1.5
-2.0 -2.0
0.0 0.5 1.0 1.5 2.0
a {
Low-Re k-ε
Subgrid k-ε
b{
✄✂ ✁ DNS [29] prescribed
b
period
k-ε log law
Subgrid k-ε
Subgrid k-ω
∂〈P 〉
∂x
( ∂〈P 〉
( ∂〈P 〉
∂x ) ss
∂x )
( ∂〈P 〉
( ∂〈P 〉
∂x ) ss
∂x )
Figure 5.6: Wall shear stress variation in the periodic pressure case
× 10 −1
× 10 −1

FIGURES 117
〈U〉
(Uτ ) ss
k
(U 2 τ ) ss
40
35
30
25
20
15
10
5
0
0 π/2 π 3π/2 2π
10
9
8
7
6
5
4
3
2
1
phase angle
0
0 π/2 π 3π/2 2π
Low-Re k-ε
Subgrid k-ε
phase angle
Subgrid k-ω
✄✂ ✁ DNS [29]
Figure 5.7: Variables with time at y/δ = 0.1 (prescribed
∂〈P 〉
∂x )

FIGURES 118
〈U〉
(Uτ ) ss
k
(U 2 τ ) ss
40
35
30
25
20
15
10
5
0
0 π/2 π 3π/2 2π
10
9
8
7
6
5
4
3
2
1
phase angle
0
0 π/2 π 3π/2 2π
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
phase angle
Subgrid k-ω
k-ε log law
Figure 5.8: Variables with time at y/δ = 0.2 (prescribed
∂〈P 〉
∂x )

FIGURES 119
〈U〉
(Uτ ) ss
k
(U 2 τ ) ss
40
35
30
25
20
15
10
5
0
0 π/2 π 3π/2 2π
10
9
8
7
6
5
4
3
2
1
phase angle
0
0 π/2 π 3π/2 2π
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
phase angle
Subgrid k-ω
k-ε log law
Figure 5.9: Variables with time at y/δ = 0.5 (prescribed
∂〈P 〉
∂x )

FIGURES 120
〈U〉
(Uτ ) ss
k
(U 2 τ ) ss
40
35
30
25
20
15
10
5
0
0 π/2 π 3π/2 2π
10
9
8
7
6
5
4
3
2
1
phase angle
0
0 π/2 π 3π/2 2π
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
phase angle
Subgrid k-ω
k-ε log law
Figure 5.10: Variables with time at y/δ = 0.9 (prescribed
∂〈P 〉
∂x )

FIGURES 121
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
phase angle = 0
0
0.00 0.20 0.40 0.60 0.80
30
20
10
y/δ
phase angle = π
2
0
0.00 0.20 0.40 0.60 0.80
y/δ
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
phase angle = π
4
0
0.00 0.20 0.40 0.60 0.80
30
20
10
Subgrid k-ω
k-ε log law
Figure 5.11: 〈U〉 vs y/δ snapshots through time (prescribed
0
0.00 0.20 0.40 0.60 0.80
y/δ
phase angle = 3π
4
∂〈P 〉
) - part 1
∂x
y/δ

FIGURES 122
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
phase angle = π
0
0.00 0.20 0.40 0.60 0.80
30
20
10
y/δ
phase angle = 3π
2
0
0.00 0.20 0.40 0.60 0.80
y/δ
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
phase angle = 5π
4
0
0.00 0.20 0.40 0.60 0.80
30
20
10
Subgrid k-ω
k-ε log law
Figure 5.12: 〈U〉 vs y/δ snapshots through time (prescribed
0
0.00 0.20 0.40 0.60 0.80
y/δ
phase angle = 7π
4
∂〈P 〉
) - part 2
∂x
y/δ

FIGURES 123
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
phase angle = 0
0
0.00 0.20 0.40 0.60 0.80
5
y/δ
phase angle = π
2
0
0.00 0.20 0.40 0.60 0.80
y/δ
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
k
(U 2 τ ) ss
k
(U 2 τ ) ss
Subgrid k-ω
k-ε log law
Figure 5.13: k vs y/δ snapshots through time (prescribed
5
phase angle = π
4
0
0.00 0.20 0.40 0.60 0.80
5
0
0.00 0.20 0.40 0.60 0.80
y/δ
phase angle = 3π
4
∂〈P 〉
) - part 1
∂x
y/δ

FIGURES 124
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
phase angle = π
0
0.00 0.20 0.40 0.60 0.80
5
y/δ
phase angle = 3π
2
0
0.00 0.20 0.40 0.60 0.80
y/δ
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
k
(U 2 τ ) ss
k
(U 2 τ ) ss
Subgrid k-ω
k-ε log law
Figure 5.14: k vs y/δ snapshots through time (prescribed
5
phase angle = 5π
4
0
0.00 0.20 0.40 0.60 0.80
5
0
0.00 0.20 0.40 0.60 0.80
y/δ
phase angle = 7π
4
∂〈P 〉
) - part 2
∂x
y/δ

FIGURES 125
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
10 0
0
30
20
10
10 0
0
phase angle = 0
10 1
10 2
y +
phase angle = π
2
10 1
10 2
y +
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
10 0
0
30
20
10
10 0
0
Subgrid k-ω
k-ε log law
Figure 5.15: 〈U〉 vs y + snapshots through time (prescribed
phase angle = π
4
10 1
10 1
10 2
10 2
y +
phase angle = 3π
4
∂〈P 〉
) - part 1
∂x
y +

FIGURES 126
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
10 0
0
30
20
10
10 0
0
phase angle = π
10 1
10 2
y +
phase angle = 3π
2
10 1
10 2
y +
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
10 0
0
30
20
10
10 0
0
Subgrid k-ω
k-ε log law
Figure 5.16: 〈U〉 vs y + snapshots through time (prescribed
phase angle = 5π
4
10 1
10 1
10 2
10 2
y +
phase angle = 7π
4
∂〈P 〉
) - part 2
∂x
y +

FIGURES 127
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
10 0
0
5
10 0
0
phase angle = 0
10 1
10 2
y +
phase angle = π
2
10 1
10 2
y +
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
10 0
0
10 0
0
Subgrid k-ω
k-ε log law
Figure 5.17: k vs y + snapshots through time (prescribed
5
phase angle = π
4
10 1
10 1
10 2
10 2
y +
phase angle = 3π
4
∂〈P 〉
) - part 1
∂x
y +

FIGURES 128
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
10 0
0
5
10 0
0
phase angle = π
10 1
10 2
y +
phase angle = 3π
2
10 1
10 2
y +
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
10 0
0
10 0
0
Subgrid k-ω
k-ε log law
Figure 5.18: k vs y + snapshots through time (prescribed
5
phase angle = 5π
4
10 1
10 1
10 2
10 2
y +
phase angle = 7π
4
∂〈P 〉
) - part 2
∂x
y +

FIGURES 129
( ∂〈P 〉
∂x )
( ∂〈P 〉
∂x ) ss
( ∂〈P 〉
∂x )
( ∂〈P 〉
∂x ) ss
25.0 25.0
20.0 20.0
15.0 15.0
10.0 10.0
5.0 5.0
0.0 0.0
-5.0 -5.0
-10.0 -10.0
-15.0 -15.0
-20.0 -20.0
0.0 0.5 1.0 1.5 2.0
a
period
25.0 25.0
20.0 20.0
15.0 15.0
10.0 10.0
5.0 5.0
0.0 0.0
-5.0 -5.0
-10.0 -10.0
-15.0 -15.0
-20.0 -20.0
0.0 0.5 1.0 1.5 2.0
a {
Low-Re k-ε
Subgrid k-ε
DNS
∂〈P 〉
∂x [29]
b{
b
period
k-ε log law
Subgrid k-ε
Subgrid k-ω
Figure 5.19: Pressure variation in the periodic bulk flow

FIGURES 130
τw
(τw) ss
τw
(τw) ss
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
-0.5 -0.5
-1.0 -1.0
-1.5 -1.5
-2.0 -2.0
0.0 0.5 1.0 1.5 2.0
a
period
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
-0.5 -0.5
-1.0 -1.0
-1.5 -1.5
-2.0 -2.0
0.0 0.5 1.0 1.5 2.0
a {
Low-Re k-ε
Subgrid k-ε
b{
✄✂ ✁ DNS τw [29] DNS
b
period
k-ε log law
Subgrid k-ε
Subgrid k-ω
∂〈P 〉
∂x [29]
( ∂〈P 〉
( ∂〈P 〉
∂x ) ss
∂x )
( ∂〈P 〉
( ∂〈P 〉
∂x ) ss
∂x )
Figure 5.20: Wall shear stress variation in the periodic bulk flow case
× 10 −1
× 10 −1

FIGURES 131
〈U〉
(Uτ ) ss
k
(U 2 τ ) ss
40
35
30
25
20
15
10
5
0
0 π/2 π 3π/2 2π
10
9
8
7
6
5
4
3
2
1
phase angle
0
0 π/2 π 3π/2 2π
Low-Re k-ε
Subgrid k-ε
phase angle
Subgrid k-ω
✄✂ ✁ DNS [29]
Figure 5.21: Variables with time at y/δ = 0.1 (prescribed U)

FIGURES 132
〈U〉
(Uτ ) ss
k
(U 2 τ ) ss
40
35
30
25
20
15
10
5
0
0 π/2 π 3π/2 2π
10
9
8
7
6
5
4
3
2
1
phase angle
0
0 π/2 π 3π/2 2π
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
phase angle
Subgrid k-ω
k-ε log law
Figure 5.22: Variables with time at y/δ = 0.2 (prescribed U)

FIGURES 133
〈U〉
(Uτ ) ss
k
(U 2 τ ) ss
40
35
30
25
20
15
10
5
0
0 π/2 π 3π/2 2π
10
9
8
7
6
5
4
3
2
1
phase angle
0
0 π/2 π 3π/2 2π
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
phase angle
Subgrid k-ω
k-ε log law
Figure 5.23: Variables with time at y/δ = 0.5 (prescribed U)

FIGURES 134
〈U〉
(Uτ ) ss
k
(U 2 τ ) ss
40
35
30
25
20
15
10
5
0
0 π/2 π 3π/2 2π
10
9
8
7
6
5
4
3
2
1
phase angle
0
0 π/2 π 3π/2 2π
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
phase angle
Subgrid k-ω
k-ε log law
Figure 5.24: Variables with time at y/δ = 0.9 (prescribed U)

FIGURES 135
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
phase angle = 0
0
0.00 0.20 0.40 0.60 0.80
30
20
10
y/δ
phase angle = π
2
0
0.00 0.20 0.40 0.60 0.80
y/δ
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
phase angle = π
4
0
0.00 0.20 0.40 0.60 0.80
30
20
10
Subgrid k-ω
k-ε log law
0
0.00 0.20 0.40 0.60 0.80
y/δ
phase angle = 3π
4
Figure 5.25: 〈U〉 vs y/δ snapshots through time (prescribed U) - part 1
y/δ

FIGURES 136
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
phase angle = π
0
0.00 0.20 0.40 0.60 0.80
30
20
10
y/δ
phase angle = 3π
2
0
0.00 0.20 0.40 0.60 0.80
y/δ
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
phase angle = 5π
4
0
0.00 0.20 0.40 0.60 0.80
30
20
10
Subgrid k-ω
k-ε log law
0
0.00 0.20 0.40 0.60 0.80
y/δ
phase angle = 7π
4
Figure 5.26: 〈U〉 vs y/δ snapshots through time (prescribed U) - part 2
y/δ

FIGURES 137
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
phase angle = 0
0
0.00 0.20 0.40 0.60 0.80
5
y/δ
phase angle = π
2
0
0.00 0.20 0.40 0.60 0.80
y/δ
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
k
(U 2 τ ) ss
k
(U 2 τ ) ss
Subgrid k-ω
k-ε log law
5
phase angle = π
4
0
0.00 0.20 0.40 0.60 0.80
5
0
0.00 0.20 0.40 0.60 0.80
y/δ
phase angle = 3π
4
Figure 5.27: k vs y/δ snapshots through time (prescribed U) - part 1
y/δ

FIGURES 138
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
phase angle = π
0
0.00 0.20 0.40 0.60 0.80
5
y/δ
phase angle = 3π
2
0
0.00 0.20 0.40 0.60 0.80
y/δ
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
k
(U 2 τ ) ss
k
(U 2 τ ) ss
Subgrid k-ω
k-ε log law
5
phase angle = 5π
4
0
0.00 0.20 0.40 0.60 0.80
5
0
0.00 0.20 0.40 0.60 0.80
y/δ
phase angle = 7π
4
Figure 5.28: k vs y/δ snapshots through time (prescribed U) - part 2
y/δ

FIGURES 139
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
10 0
0
30
20
10
10 0
0
phase angle = 0
10 1
10 2
y +
phase angle = π
2
10 1
10 2
y +
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
10 0
0
30
20
10
10 0
0
Subgrid k-ω
k-ε log law
phase angle = π
4
10 1
10 1
10 2
10 2
y +
phase angle = 3π
4
Figure 5.29: 〈U〉 vs y + snapshots through time (prescribed U) - part 1
y +

FIGURES 140
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
10 0
0
30
20
10
10 0
0
phase angle = π
10 1
10 2
y +
phase angle = 3π
2
10 1
10 2
y +
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
〈U〉
(Uτ ) ss
〈U〉
(Uτ ) ss
30
20
10
10 0
0
30
20
10
10 0
0
Subgrid k-ω
k-ε log law
phase angle = 5π
4
10 1
10 1
10 2
10 2
y +
phase angle = 7π
4
Figure 5.30: 〈U〉 vs y + snapshots through time (prescribed U) - part 2
y +

FIGURES 141
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
10 0
0
5
10 0
0
phase angle = 0
10 1
10 2
y +
phase angle = π
2
10 1
10 2
y +
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
10 0
0
10 0
0
Subgrid k-ω
k-ε log law
5
phase angle = π
4
10 1
10 1
10 2
10 2
y +
phase angle = 3π
4
Figure 5.31: k vs y + snapshots through time (prescribed U) - part 1
y +

FIGURES 142
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
10 0
0
5
10 0
0
phase angle = π
10 1
10 2
y +
phase angle = 3π
2
10 1
10 2
y +
Low-Re k-ε
Subgrid k-ε
✄✂ ✁ DNS [29]
k
(U 2 τ ) ss
k
(U 2 τ ) ss
5
10 0
0
10 0
0
Subgrid k-ω
k-ε log law
5
phase angle = 5π
4
10 1
10 1
10 2
10 2
y +
phase angle = 7π
4
Figure 5.32: k vs y + snapshots through time (prescribed U) - part 2
y +