Numerical simulation of atmospheric flows using general ... · Numerical simulation of atmospheric...

Budapest University of Technology andEconomics

PhD Thesis

Numerical simulation of atmospheric

flows using general purpose CFD solvers

Author:

Miklos Balogh

Supervisor:

Dr. Gergely Kristof

A thesis submitted in fulfillment of the requirements

for the degree of Doctor of Philosophy

in the

Faculty of Mechanical Engineering

Department of Fluid Mechanics

24 September 2014

Declaration of Authorship

I, Miklos Balogh, hereby declare that this thesis titled, “Numerical simulation of at-

mospheric flows using general purpose CFD solvers” and the work presented in it are

my own. I confirm that:

This work was done wholly or mainly while in candidature for a research degree

at this University.

Where any part of this thesis has previously been submitted for a degree or any

other qualification at this University or at any other institution, this has been

clearly stated.

Where I have consulted the published work of others, this is always clearly at-

tributed.

Where I have quoted from the work of others, the source is always given. With

the exception of such quotations, this thesis is entirely my own work.

Where the thesis is based on work done by myself jointly with others, I have made

clear exactly what was done by others and what I have contributed myself.

I have acknowledged all main sources of help.

Signed:

Date:

i

“I am an old man now, and when I die and go to heaven there are two matters on which I

hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent

motion of fluids. And about the former I am rather optimistic.”

Horace Lamb, 1932

“Turbulence was probably invented by the Devil on the seventh day of Creation when the

Good Lord wasn’t looking.”

Peter Bradshaw, 1994

BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS

Faculty of Mechanical Engineering, Department of Fluid Mechanics

AbstractDoctor of Philosophy

Numerical simulation of atmospheric flows using general purpose CFD

solvers

by Miklos Balogh

The thesis aims to give a comprehensive view about the problems connected to the application of

general purpose CFD solvers for the numerical simulation of atmospheric flows, furthermore to

introduce practicable solutions for these. The problems are compartmentalized into the subjects

of boundary conditions, turbulence modeling, flows in urban canopy layers and the specialized

methodology of atmospheric simulations.

The rough wall functions in general purpose CFD solvers often produce inappropriate results

in the simulation of homogeneous neutral atmospheric boundary layers (ABL), while their com-

patibility with the meteorological formalism is limited. In order to simulate atmospheric flows

properly, modifications are required on both the inlet conditions and wall functions, as well as on

the turbulence model. In the course of model developments, a novel set of boundary conditions

is applied with corresponding source terms forming a novel approach. This is validated against

measurements, which demonstrated its advantages.

The modifications applied on the model for resolving the homogeneous neutral ABL flows are

not appropriate for inhomogeneous boundary layers are developing over complex terrain. Ac-

cordingly, the model generalized for arbitrary ABL with a continuous smooth blending between

the formalisms of the different regimes. The generalized model shows significant improvements

compared to the former approaches.

For the simulation of flows in the urban canopy layer, a scale adaptive hybrid method is developed

for combining the advantages of the implicit methods applied in the meteorological practice, and

the explicit one applied in CFD approaches. The novel hybrid method gives similar results than

the clearly explicit method, but it consuming much less computational resources.

The CFD simulation of atmospheric flows implies several problem specific tasks, such as the

geometrical realization of the complex terrain, the definition of the lateral boundary conditions

and surface parameters, furthermore the mesh generation with respect to the formers. These are

fairly different from the general tasks of the engineering practice. The developed methodology

serves as a guideline for handling these special tasks and solving atmospheric problems using

general purpose solvers.

Acknowledgements

I would like to express my gratitude to my supervisor Dr. Gergely Kristof who gave

me the opportunity to work on this challenging research project, and for supporting me

during the recent years.

It is a pleasure to thank those who made my PhD studies possible at the Department of

Fluid Mechanics of BME, namely Prof. Tamas Lajos and Prof. Janos Vad. for providing

me with a good environment and facilities to complete this project. I am glad to be a

part of this amazing group.

I am heartily thankful to Prof. Carlo Benocci and Dr. Alessandro Parente, whom

consequent guidance enabled me to develop my research skills and knowledge during

our collaboration. I consider it an honor to work with them. The complete development

of this thesis would not have been possible without their support, encouragement and

technical expertise.

I want to express my gratitude to my beloved family always supporting me even in the

most difficult periods of my studies.

Last but not least, I am grateful to all my friends for the support and tolerance.

iv

Contents

Declaration of Authorship i

Abstract iii

Acknowledgements iv

Contents v

List of Figures viii

List of Tables x

Abbreviations xi

Symbols xii

Introduction 1

1 Boundary conditions for atmospheric simulations 3

1.1 Turbulence in neutral, homogeneous ABL . . . . . . . . . . . . . . . . . . 3

1.1.1 The standard two-equation model . . . . . . . . . . . . . . . . . . 5

1.1.2 Inlet profiles for homogeneous ABL . . . . . . . . . . . . . . . . . . 6

1.1.3 Extended formulation and consistent source terms . . . . . . . . . 8

1.1.4 Modified wall treatments . . . . . . . . . . . . . . . . . . . . . . . 10

1.1.5 Notations for model comparisons . . . . . . . . . . . . . . . . . . . 12

1.2 Verification at full scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Geometry and mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Validation at laboratory scale . . . . . . . . . . . . . . . . . . . . . . . . . 16



1.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Validation of the extented formulation . . . . . . . . . . . . . . . . . . . . 18


v

Contents vi


1.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.5 Conclusions on boundary conditions . . . . . . . . . . . . . . . . . . . . . 20

2 Turbulence over complex terrain 22

2.1 Turbulence in boundary layers over complex terrain . . . . . . . . . . . . 22

2.1.1 Transition to non-homogeneous ABL . . . . . . . . . . . . . . . . . 22

2.1.2 Realizability issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Implementation notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Implementation in OpenFOAM . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Implementation in ANSYS-Fluent . . . . . . . . . . . . . . . . . . 26

2.2.3 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.4 Quantitative evaluation . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Validation on a simplified 2D hill at laboratory scale . . . . . . . . . . . . 28



2.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Validation on a simplified 3D hill . . . . . . . . . . . . . . . . . . . . . . . 34



2.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Validation of the extended formulation on the 3D hill . . . . . . . . . . . 38

2.5.1 Computational mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 38


2.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6 Validation on the Askervein Hill . . . . . . . . . . . . . . . . . . . . . . . 40



2.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.7 Computational performances . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.8 Conclusions on ABL over complex terrain . . . . . . . . . . . . . . . . . . 46

3 Modeling flows in urban canopy layers 47

3.1 Flows in urban canopy layer . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.1 Distributed drag force approach . . . . . . . . . . . . . . . . . . . 48

3.1.2 Hybrid method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Validation of the hybrid method . . . . . . . . . . . . . . . . . . . . . . . 50



3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 An example on modeling turbulent flows in an urban canopy layer . . . . 55



3.3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Conclusions on urban canopy layer flows . . . . . . . . . . . . . . . . . . . 62

4 Methodology of atmospheric simulations 63

Contents vii

4.1 Guideline for atmospheric simulations . . . . . . . . . . . . . . . . . . . . 63

4.2 Problem specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.1 Size of the domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.1.1 Numerical analysis of the required domain size . . . . . . 65

4.2.2 Data requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.2.1 Topographic data . . . . . . . . . . . . . . . . . . . . . . 70

4.2.2.2 Wind data . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Geographic conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.1 Overview of map projections . . . . . . . . . . . . . . . . . . . . . 71

4.3.2 Selection of map projection . . . . . . . . . . . . . . . . . . . . . . 72

4.4 Problem specific meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4.1 Terrain manipulation and representation . . . . . . . . . . . . . . . 74

4.4.2 Object meshing concepts . . . . . . . . . . . . . . . . . . . . . . . 75

4.4.2.1 Subdivision concept . . . . . . . . . . . . . . . . . . . . . 76

4.4.2.2 Base mesh concept . . . . . . . . . . . . . . . . . . . . . . 76

4.4.3 Stand-alone mesher . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4.4 Mixed mesher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5 Simplified specification of boundary conditions . . . . . . . . . . . . . . . 78

4.6 Conclusions on the methodological survey . . . . . . . . . . . . . . . . . . 79

5 Conclusions and outlook 81

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Thesis points 86

1. Thesis: Boundary conditions for atmospheric simulations . . . . . . . . . . . 86

2. Thesis: Modeling atmospheric flows over complex terrain . . . . . . . . . . . 87

3. Thesis: Scale adaptive modeling approach for urban flows . . . . . . . . . . 88

4. Thesis: Methodology of atmospheric simulations . . . . . . . . . . . . . . . . 89

Bibliography 90

Appendices 99

A. Derivation of source terms for simple cases . . . . . . . . . . . . . . . . . . 99

B. Derivation of source terms for limited Cµ . . . . . . . . . . . . . . . . . . 100

C. Smooth function for limited Cµ . . . . . . . . . . . . . . . . . . . . . . . . 101

List of Figures

1.1 Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Sketch of the 2D full scale domain . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Debugged wall function and the comprehensive approach . . . . . . . . . . 14

1.4 Richards and Hoxey and the comprehensive approach . . . . . . . . . . . 15

1.5 Sketch of the 2D laboratory scale domain . . . . . . . . . . . . . . . . . . 16

1.6 Inlet and outlet profiles against measurements . . . . . . . . . . . . . . . . 17

1.7 Inlet and outlet profiles against CEDVAL measurements . . . . . . . . . . 19

1.8 Inlet and outlet profiles against ERCOFTAC 69 measurements . . . . . . 20

1.9 Inlet and outlet profiles against TOKYO UNI WT measurements . . . . . 20

2.1 Blending function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Sketch of the domain with the different 2D hills . . . . . . . . . . . . . . . 29

2.3 Comparison of the measured and simulated profiles for a = 3H . . . . . . 31



2.6 Side view of the 3D hill computational domain . . . . . . . . . . . . . . . 34

2.7 Profile comparison for the 3D hill . . . . . . . . . . . . . . . . . . . . . . . 36

2.8 Close-up of the finest mesh at the hill . . . . . . . . . . . . . . . . . . . . 39

2.9 Profile comparison for the 3D hill with extended formulation . . . . . . . 41

2.10 Domain and mesh for the Askervein hill . . . . . . . . . . . . . . . . . . . 42

2.11 Horizontal profiles comparison for Askervein hill . . . . . . . . . . . . . . 44

2.12 Vertical profiles comparison for Askervein hill . . . . . . . . . . . . . . . . 45

3.1 Geometrical setup for the MUST experiment . . . . . . . . . . . . . . . . 51

3.2 Computational meshes for the MUST experiment . . . . . . . . . . . . . . 52

3.3 Inlet profiles for the MUST experiment . . . . . . . . . . . . . . . . . . . 53

3.4 Simulation results for the MUST experiment . . . . . . . . . . . . . . . . 54

3.5 Pressure distribution on the VIP CAR . . . . . . . . . . . . . . . . . . . . 55

3.6 Computational domain and mesh for district XI. . . . . . . . . . . . . . . 56

3.7 Computational mesh for the downtown . . . . . . . . . . . . . . . . . . . . 58

3.8 Inlet profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.9 Inlet profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.10 Normalized velocity at lower levels . . . . . . . . . . . . . . . . . . . . . . 60

3.11 Normalized velocity at higher levels . . . . . . . . . . . . . . . . . . . . . . 61

4.1 Guideline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Terrain elevation of extended domains . . . . . . . . . . . . . . . . . . . . 67

4.3 Terrain elevation of extended domains . . . . . . . . . . . . . . . . . . . . 68

viii

List of Figures ix

4.4 Profiles for first hill height . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Profiles for second hill height . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Profiles for third hill height . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.7 Profiles for fourth hill height . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.8 Comparison of projections I . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.9 Comparison of projections I . . . . . . . . . . . . . . . . . . . . . . . . . . 74

C.1 Simple and Cubic Bezier limiters . . . . . . . . . . . . . . . . . . . . . . . 103

List of Tables

1.1 Coefficients of the standard k-ε model . . . . . . . . . . . . . . . . . . . . 6

1.2 Notations for inlet boundary conditions . . . . . . . . . . . . . . . . . . . 12

1.3 Notations for wall functions . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Notations for models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Domain properties for simple 2D cases . . . . . . . . . . . . . . . . . . . . 18

1.6 Velocity profile parameters for 2D cases . . . . . . . . . . . . . . . . . . . 19

1.7 Turbulent kinetic energy profile parameters for 2D cases . . . . . . . . . . 19

1.8 Correlation coefficients for the velocity and turbulent kinetic energy profiles 19

2.1 Switching options of the ABL k-ε model . . . . . . . . . . . . . . . . . . . 26

2.2 Properties of meshes generated for 2D hills . . . . . . . . . . . . . . . . . 29

2.3 Results of the profile fitting for different hill geometries . . . . . . . . . . 30

2.4 Hit rates for 2D hills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Hit rates values for the different meshes on the 3D hill . . . . . . . . . . . 35

2.6 Hit rates values on 3D hill profiles . . . . . . . . . . . . . . . . . . . . . . 37

2.7 Errors on the separation region for the 3D hill . . . . . . . . . . . . . . . . 38

2.8 Inlet profile parameters for the extended formulation . . . . . . . . . . . . 39

2.9 Hit rates values on 3D hill profiles . . . . . . . . . . . . . . . . . . . . . . 40

2.10 Errors on the separation region for the 3D hill . . . . . . . . . . . . . . . . 40

2.11 Hit rates and relative errors on Askervein hill . . . . . . . . . . . . . . . . 45

2.12 Computational performance . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1 Forces acting on the VIP CAR . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Typical values of canopy parameters . . . . . . . . . . . . . . . . . . . . . 57

4.1 Recommendations for domain extension . . . . . . . . . . . . . . . . . . . 65

4.2 Geometry of extended domains . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 Recommendations for map projection . . . . . . . . . . . . . . . . . . . . 73

x

Abbreviations

ABL Atmospheric Boundary Layer

BC Boundary Conditions

BIA Building Influence Area

CAD Computer Aided Design

CEDVAL Compilation of Experimental Data for VALidation

CFD Computational Fluid Dynamics

DTRA Defence Threat Reduction Agency

DNS Direct Numerical Simulation

ENIAC Electronic Numerical Integrator And Computer

ERCOFTAC European Research Community On Flow,

Turbulence And Combustion

HBL Homogeneous atmospheric Boundary Layer

STD STandarD

MMK Murakami-Mochida-Kondo model

MUST Mock Urban Setting Trial

KL Kato-Launder model

KLY Kato-Launder model with Yap correction

VKI Von Karman Institute

LES Large Eddie Simulation

NACA National Advisory Committee for Aeronautics

NURBS Non-Uniform Rational Bezier Spline

RANS Reynolds Averaged Navier-Stokes

URANS Unsteady Reynolds Averaged Navier-Stokes

xi

Symbols

Roman symbols

xi the ith cartesian coordinate (Einstein notation) m

x streamwise coordinate m

y crosswise coordinate m

z surface normal coordinate m

zi depth of the atmospheric boundary layer m

ui the ith velocity component (Einstein notation) ms−1

U streamwise velocity ms−1

V crosswise velocity ms−1

W surface normal velocity ms−1

u′ streamwise fluctuation velocity ms−1

v′ crosswise fluctuation velocity ms−1

w′ surface normal fluctuation velocity ms−1

Uh horizontal velocity magnitude ms−1

G magnitude of the geostrophic wind ms−1

uτ friction velocity ms−1

U∞ free stream velocity ms−1

Hc canopy layer height m

u+ non-dimensional velocity −

z0 aerodinamical roughness K

E wall function coefficient −

Cs roughness parameter −

ks equivalent sand grain roughness m

k turbulent kinetic energy m2s−2

S modulus of the rate of strain tensor s−1

xii

Symbols xiii

Cµ coefficient of the turbulent viscosity −

C1ε constant of the k − ε model −

C2ε constant of the k − ε model −

C3ε coefficient of the k − ε model (boundary layer orientation) −

C4ε constant of the porous k − ε model −

C5ε constant of the porous k − ε model (mixing length anisotropy) −

Cm scaling coefficient between aerodynamics and sand grain roughness −

Pk turbulent kinetic energy production m2s−3

Pb turbulent kinetic energy production due to buoyancy m2s−3

N blending exponent of the ABL k − ε model −

n number of iteration step −

Greek symbols

δ boundary layer thickness m

ϕ geographical coordinate (latitude)

λ geographical coordinate (longitude)

Ω vorticity s−1

λt solid fraction of the porous k − ε model −

τw wall shear stress Pa

ε turbulent kinetic energy dissipation m2s−3

σk turbulent Prandtl number for k −

σε turbulent Prandtl number for ε −

Sub- and Superscripts

p wall adjacent value of a quantity

w wall referenced quantity

+ non-dimensional quantity

ref reference value of a quantity

I dedicated my work to my family, including the past, present andfuture members, whom I love. . .

xiv

Introduction

The numerical investigation of atmospheric flows were intensively developed, since com-

puters have been applied for solving such complex numerical problems. ENIAC was the

first electronic general purpose computer, and an atmospheric simulation was among

the first successful computations executed on this revolutionary machine. The last few

decades have seen considerable technological advances in computer sciences, which en-

ables the possibility of detailed numerical simulation of micro- and meso-scale atmo-

spheric phenomena. Since the spatial resolution is approaching to the lower scales with

the increasing computational capacity, more and more details should be considered in

order to their influence on the flow field. These details, such as the geometry of build-

ings and structures, generate turbulence and modify the flow patterns on smaller scales.

General purpose CFD solvers handle the geometrical constraints pliantly, although rep-

resenting the physical flavor of atmospheric flows is still a challenge.

The present research focused on the numerical simulation of micro- and meso-scale at-

mospheric flows using general purpose CFD solvers. There is a developing demand in

civil engineering practice for the accurate description of small scale atmospheric phe-

nomenon, inasmuch as the sustainable development is getting more and more important,

hence the constructions should be suitable in environmental and energetics point of view

as well. Several engineering problems are connected to the atmospheric processes, such

as the analysis of optimal wind farm design and city ventilation, calculation of wind

load on buildings and structures, furthermore the prediction of pollutant dispersion.

Consequently, the study deals with the adaptation of commonly used solvers in engi-

neering practice, the development, verification and validation of models and approaches,

furthermore the elaboration of the modeling procedure. The work is divided into four

parts, whose cover the four main problem investigated in this research.

The first problem is focused on the boundary conditions used for the simulation of flows

within the neutral atmospheric boundary layer (ABL). The objective requires physically

correct boundary conditions, whose are representative for the entire ABL. While the

1

Introduction 2

atmosphere is bounded by the ground from below, the lower boundary is always consid-

ered as a rough surface. In the engineering practice, the rough wall functions are used

for modeling the effects of the surface roughness, using the sand grain roughness and the

roughness parameter for characterizing the quality of the surface. On the other hand,

in the description of the neutral atmospheric boundary layer, the quality of the surface

is characterized by the roughness length, according to the meteorological terminology.

For this reason, the application of the rough wall function available in general purpose

solvers together with the atmospheric profiles is problematic, while the fully developed

inlet profiles degenerates along the domain, even in case of flat terrain with homogeneous

roughness, which results inhomogeneous boundary layer. To overcome this problem de-

velopments should be carried out on the rough wall functions and on the turbulence

model, in order to reproduce the homogeneous ABL.

The second problem is focused on the simulation of flows over complex terrain. A

turbulence model developed and applied for resolving the homogeneous, neutral ABL

are not appropriate for inhomogeneous boundary layers develop over complex terrain.

The solution is a generalized turbulence model, which is valid for mixed boundary layers.

The third problem is formed on the simulations of urban canopy layers. In the simula-

tions of urban flows, the geometry of the buildings can be considered in the course of

spatial discretization, but it implies high computational costs, since the required spa-

tial resolution (∼0.1 - 10 m) results unmanageably high number of cells, in particular

for larger domains. The spatial resolution of limited area atmospheric models used for

meteorological purposes (∼100 - 10000 m) does not allow to take into consideration

the geometrical features of the buildings, but the effects of the built-up environment

could be realized in the calculations via parametrizations. These two scale should be

converged to each other, since the computational costs should be reduced, but the impor-

tant flow features should be resolved in order to achieve reasonable results with feasible

computational requirements.

The last main part has several portion, since the simulation of atmospheric flows im-

plies several problem specific task, such as the geometrical representation of the complex

terrain, the setup of boundary conditions, the definition of the surface roughness and

other surface specific parameters, furthermore the mesh generation considering the for-

mers. These tasks are different from the general tasks of the engineering practice, thus

a specialized methodology is required.

Chapter 1

Boundary conditions for

atmospheric simulations

This section presents the theory of turbulence modelling in neutral Atmospheric Bound-

ary Layer (ABL) using a basic two-equation model. The main difficulties of turbulence

modeling in the ABL, using general purpose solvers are introduced, such as ensuring the

streamwise homogeneity of the fully developed ABL. Recommendations can be found in

the literature are discussed in details, together with the solutions and developments are

given to overcome the problems appear in practical modeling.

1.1 Turbulence in neutral, homogeneous ABL

As it is introduced in [1] and [2], the simulation of atmospheric flows over complex ter-

rains is necessary for the estimation of wind load on buildings as well as for the choice

of sites for wind turbines. This topic has been intensively investigated by several re-

search groups (e.g. [3, 4, 5, 6, 7, 8]). The major part of the available investigations was

performed by means of Reynolds-Averaged Navier-Stokes (RANS). Large Eddy Simu-

lation (LES) studies of ABL flows [5, 6, 9, 10, 11, 12, 13] can provide a more accurate

solution for the turbulent flow field with respect to RANS simulations, provided that

the range of resolved turbulent scales is sufficiently large and that the inflow condi-

tions are well characterized. However, LES simulations are significantly more expensive

than RANS [14] and, therefore, practical simulations of ABL flows are still often car-

ried out using RANS in combination with two-equation turbulence models, to provide

fast answers to design questions. Indeed, for this approach to be effective, turbulence

modeling should adequately describe the problem under investigation. However, the

application of two-equation RANS turbulence models with rough wall functions often

3

Chapter 1. Boundary Conditions 4

results in unsatisfactory predictions, due to an inconsistent formulation of the law of

the wall for rough surfaces and the fully developed inlet conditions for ABL simulations

[7, 8, 15, 16, 17, 18]. It means that the inlet profiles deteriorate throughout the compu-

tational domain, namely significant differences appear between the turbulent profiles at

the inlet and outlet sections, as it is shown in Figure 1.1.

0

100

200

300

400

500

0 5 10 15 20

z [

m]

U [ms-2

]

Full-scale U

Inlet RH

Outlet RH

Inlet ESDU

Outlet ESDU

0

100

200

300

400

500

0.5 1 1.5 2 2.5 3

z [

m]

k [m2s

-2]

Full-scale TKE

0

100

200

300

400

500

0.001 0.01 0.1 1 10

z [

m]

ε [m2s

-3]

Full-scale TKED

0.5

0.75

1

1.25

1.5

1.75

2

0 1 2 3 4 5

τ w/τ

w,t

he

ory

[-]

x [km]

Full-scale N-D WSS

Ground RH

Ground ESDU

Figure 1.1: Inhomogeneity shown by the inlet and outlet profiles. Both the inletconditions given in ESDU 85 (Engineering Science Data Unit) [19] and suggested by

Richards and Hoxey [15] are compared.

Due to this deterioration, the target region of the analysis in a practical application is

exposed to the turbulent wind profiles, whose are not appropriate. In a recent work [20],

it has been proposed a new approach ensuring consistency among turbulence model, inlet

and wall boundary conditions for the numerical simulation of neutral ABL flows. This

is accomplished through a reformulation of the wall function based on the aerodynamic


roughness and on the derivation of the kinetic energy inlet profile from the solution of

the turbulent kinetic energy transport equation. Such an approach has been validated

for the simulation of a homogeneous neutral ABL and a ground mounted bluff body

using the commercial CFD code ANSYS-Fluent [20].

1.1.1 The standard two-equation model

In atmospheric applications, the standard k-ε two transport equations turbulence model

is widely used due to its relatively low computational costs and its robustness. The two

transported quantity are the turbulent kinetic energy (denoted by k), and its dissipation

rate (denoted by ε). The former represents the energy in the turbulence, while the latter

determines the scale of this. As described in [21], this model has been shown to be useful

for free-shear layers, as well as wall-bounded and internal flows with relatively small

pressure gradients. The transport equations for incompressible flows can be written in a

general form as Eq. 1.1 for the turbulent kinetic energy and as Eq. 1.2 for its dissipation

rate.∂k

∂t+∂kui∂xi

=∂

∂xj

[(ν +

νtσk

)∂k

∂xj

]+ Pk + Pb − ε+ Sk (1.1)

∂ε

∂t+∂εui∂xi

=∂

∂xj

[(ν +

νtσε

)∂ε

∂xj

]+ C1ε

ε

k(Pk + C3εPb)− C2ε

ε2

k+ Sε (1.2)

In Eq. 1.1 and 1.2 ν and νt are the kinematic and the turbulent viscosity respectively,

Pk is the turbulent kinetic energy production, Pb is the buoyancy term, furthermore Sk

and Sε are arbitrary source terms. In case of the standard k-ε model, the turbulent

viscosity (also called eddy viscosity) is expressed with the transported variables, written

as Eq. 1.3.

νt = Cµk2

ε(1.3)

The turbulent kinetic energy production is expressed with the modulus of the mean rate

of strain tensor S =√

2SijSij , as Eq. 1.4.

Pk = νtS2 (1.4)

The buoyancy term (Pb) expresses the generation of turbulence kinetic energy due to

buoyancy effects. In case of neutral boundary layer this term is negligible, hence the

temperature gradient is zero. The model coefficients of the standard k-ε model are

summarized in Table 1.1. It should be noted that C3ε coefficient can be expressed in

different forms depend on the application. In the present formula W is the vertical

velocity in an atmospheric boundary layer aligned with the gravitational acceleration,

while the denominator is the magnitude of the horizontal velocity. Its value approaches


Cµ σk σε C1ε C2ε C3ε

0.09 1 1.3 1.44 1.92 tanh

(W√

U2 + V 2

)Table 1.1: Coefficients of the standard k-ε model

1 for buoyant shear layers for which the main flow direction is aligned with the direction

of gravity, while in neutral boundary layers it becomes zero.

In the practice of RANS techniques, the model itself cannot stand without wall functions.

The mesh resolution is far coarser than in case of LES or DNS techniques, thus the

effect of the wall on the flow should considered via the application of the law of the

wall, published by Theodor von Karman in 1931 [22]. In k-ε models, the equilibrium

assumption is applied, which assumes that the production (Pkp) and dissipation (εp) of

turbulent kinetic energy are equal at the wall adjacent cell, where index p denotes the

variables defined at its centroid. A special, zero-gradient like boundary condition used

to be imposed at the wall for the turbulent kinetic energy, since the transport equation

is solved for kp by the model in the wall adjacent cell centroid. In such solvers, where

the implementation is accessible (such as in OpenFOAM), one can observe that the law

of the wall is applied implicitly for the velocity. It means that zero velocity imposed

at the wall according to the no-slip condition, and the velocity is adjusted in the wall

adjacent cell centroid throughout the definition of the turbulent viscosity νtp, namely

applying the so called νt wall function.

1.1.2 Inlet profiles for homogeneous ABL

Considering homogeneous, steady state neutral ABL, several assumptions of the stan-

dard k-ε model can be applied. Assuming constant pressure and shear stress, the cross-

wise and wall normal components of the velocity getting to zero. The horizontal deriva-

tives can be neglected due to the horizontal homogeneity, such as the buoyancy term

thanks to the neutral stratification. Considering that the dynamic viscosity of air is

much lower than the eddy viscosity (ν νt), the former could be also neglected. The

Prandtl number of the turbulent kinetic energy σk = 1, thus the transport equations are

simplified to the following ones with z vertical coordinate, and U streamwise velocity.

νt∂U

∂z=τwρ

= u2τ (1.5)

∂

∂z

(νt∂k

∂z

)+ Sk = 0 (1.6)


∂

∂z

(νtσε

∂ε

∂z

)+ (C1ε − C2ε)

ε2

k+ Sε = 0 (1.7)

The inlet boundary conditions proposed by Richards and Hoxey [15] are widely used for

simulating the neutral ABL, written as

U(z) =uτκ

lnz + z0z0

(1.8)

ε(z) =u3τ

κ (z + z0)(1.9)

k =u2τ√Cµ

(1.10)

These boundary conditions in Eq. 1.8 – 1.10 automatically satisfy the 1D k-ε equations

(Eqs. 1.5 – 1.7), if the turbulent Prandtl number of the dissipation rate (σε) is expressed

as

σε =κ2

(C2ε − C1ε)√Cµ

. (1.11)

This formulation is equivalent to using an arbitrary constant σε value and applying

an additional Sε source term in the transport equation of turbulent dissipation rate,

as proposed by Pontiggia et al. [23]. Although the constant turbulent kinetic energy

satisfies the 1D model, it is not proper in real cases neither physical, nor experimental

point of view.

More realistic boundary condition was proposed by Yang et al. [24] for the turbulent

kinetic energy varying with the height, which takes the form

kYA(z) =√AYAln(z + z0) +BYA. (1.12)

Gorle et al. [17] proposed a modification of the constant Cµ and of the turbulent

dissipation Prandtl number, σε, which provided an approximate solution to the system

of equations when using the turbulent kinetic energy inlet profile in the form of Eq. 1.12.

In addition to that, Parente et al. [18] introduced two source terms in the transport

equations for k and ε, respectively, to guarantee an exact solution to the model equations

when using these inlet profiles. More recently, Parente and Benocci [25] developed a

comprehensive approach for the numerical simulation of neutral ABL flows. A novel

profile for turbulent kinetic energy was derived (Eq. 1.13) from the solution of the

turbulent kinetic energy transport equation, resulting in a new set of fully-developed

inlet conditions for the neutral ABL.

kPB(z) = APBln(z + z0) +BPB. (1.13)


The consistency between the inlet profiles and the standard k-ε model was ensured with

the introduction of a universal source term in the transport equation for the turbulent

dissipation rate:

Sε =u4τ

(z + z0)2

((C2 − C1)

√Cµ

κ2− 1

σε

), (1.14)

and the re-definition of the coefficient Cµ as a wall distance dependent value:

Cµ =u4τk2PB

. (1.15)

According to the 1D k-ε equations (Eqs. 1.5 – 1.7) one can derive the Sk and Sε source

terms should be applied for horizontally homogeneous ABL and for arbitrary vertical

distribution of turbulent kinetic energy. The analytical derivative of the logaritmic veloc-

ity profile (Eq. 1.16) results an ε profile (Eq. 1.17), regarding the turbulent equilibrium

(ε = Pk).

U =uτκ

ln

(z + z0z0

)→ ∂U

∂z=

uτκ (z + z0)

(1.16)

ε = Pk = νt

(∂U

∂z

)2

= Cµk2

ε

(uτ

κ (z + z0)

)2

→ ε =

√Cµkuτ

κ (z + z0)(1.17)

The remaining terms of the transport equations yield the following expressions for Sk

and Sε source terms (Eq. 1.18 and 1.19).

Sk = − ∂

∂z

(νt∂k

∂z

)(1.18)

Sε = − 1

σε

∂

∂z

(νt∂ε

∂z

)− (C1ε − C2ε)

ε2

k(1.19)

As it is shown by Parente and Benocci ([18]), the Sk source term is zero using the inlet

turbulent kinetic energy profile suggested by them. Considering the turbulent kinetic

energy profile proposed by Yang et al. [24], the Sk source term can be given analytically

as

Sk = − ∂

∂z

(νt∂k

∂z

)=

uτκA2YA

4 (z + z0) k(1.20)

1.1.3 Extended formulation and consistent source terms

In some cases, the profile fitting of turbulent kinetic energy using only two parameters

results unacceptable deviations from the measured one, thus a four parameter profile is

developed, written in Eq. 1.21.

k(z) = A ln

(z + z0z0

)+B

(z + z0z0

)2

+ C

(z + z0z0

)+D (1.21)


We should note that this formulation can reproduce the PB and RH profiles using the

appropriate parameters:

k(z) = APB ln(z + z0) +BPB → A = APB, B = C = 0, D = BPB +APB ln(z0) (1.22)

k(z) =u2τ√Cµ→ A = B = C = 0, D =

u2τ√Cµ

(1.23)

The new formulation for the turbulent kinetic energy profile implies the following source

terms, in accordance with the derivation can be found in the appendix A..

Sk = − ∂

∂z

(νt∂k

∂z

)= −uτκ

z0

(4B

z + z0z0

+ C

)(1.24)

Sε = − u4τ

(z + z0)2

(1

σε+

√Cµ (C1ε − C2ε)

κ2

)(1.25)

In such cases, where the domain is higher than the boundary layer, the above written

profiles are not appropriate over the boundary layer height. Therefore the vertical

coordinate z should be limited to the boundary layer height (δ) in the profiles. While

the derivatives of the profiles are zero above δ, only ε in Eq. 1.6 and C2εε2

k term in Eq.

1.7 are the non-zero terms, in this manner the equilibrium is not valid anymore. The

source terms over δ turn into 1.26 and 1.27, in order to keep the quantities constant

above δ.

Sk(z > δ) =u3τ

κ (δ + z0)(1.26)

and

Sε(z > δ) =√CµC2ε

(uτ

κ (δ + z0)

)2

(1.27)

Although the definition of Cµ (Eq. 1.15) is mathematically consistent with the model,

its extremely high value causes convergence problems. To avoid numerical problems, Cµ

can be limited, with a smooth transition to the maximized value. This smoothing is

discussed in detail in the Appendix B. together with the derivation of the source terms

required for limited Cµ (Appendix C.).

The extended model is implemented in a form of a comprehensive framework [26], in

which the modeling options as well as the inlet profiles and wall function formulations

are selectable according to the type of analysis. For this, the so called HBL k-ε model

is implemented, which is able to apply all the above discussed formalism according to

the given requirements.

It should be noted, that the profile parameters are calculated by an algorithm applies

non-linear regression for the velocity, and a non-linear least squares fit using Newton-

Raphson method for the turbulent kinetic energy [26]. Both the velocity and the kinetic


energy profile fits are built in one automatic framework implemented in FORTRAN 95.

Using this framework, the four parameter profile fits provide a significant improvement

on the correlation between the measured values and the fitted ones, compared to the

former interpolation formulas.

1.1.4 Modified wall treatments

The limitations related to the RANS simulation of neutral atmospheric boundary layer

(ABL) with commercial CFD codes are due to an inconsistency between the wall func-

tion formulation and the fully developed inflow conditions for ABL [7, 8, 16, 27, 28].

Remedial measures have been proposed in the literature [7]; however, these are generally

code dependent and they do not provide a general solution to the problem. In partic-

ular, such modifications only affect the velocity profile and the effect of roughness on

turbulent quantities is not taken into account, causing an undesired non-homogeneity

of the turbulent quantities throughout the computational domain. For example the for-

mulation applied by Richards and Hoxey [15] reduce the non-homogeneity, but in full

scale a well-documented peak of turbulent kinetic energy develops near to the wall using

their inlet boundary conditions (Eqs. 1.8 – 1.10). Their wall function formulation is

written as Eqs. 1.28 – 1.30, parametrized with the non-dimensional equivalent sand

grain roughness ks+ = ksuτ/ν.

Up =uτκ

ln

(Ez+p

1 + Csks+

), ks ≈ 20z0 (1.28)

εp =C0.75µ k1.5p

κ (zp + z0)(1.29)

Pkp =τ2w

κC0.25µ k0.5p 2zp

ln

(2zp + z0

z0

)(1.30)

A new approach has been recently proposed [18], which addresses and solves the problem.

This approach specifies velocity, turbulent kinetic energy and its dissipation rate in

the wall adjacent cell using a formulation consistent with the velocity and turbulence

inlet profiles. Assuming logarithmic profiles for the velocity, the dissipation ε and the

production term Pk for kp at first inner node (zp) are reformulated following Eqs. 1.31

– 1.33, where the subscript p indicates the cell value of the quantity at the wall adjacent

cell, while z0 is the aerodynamic roughness.

Up =uτκ

ln

(zp + z0z0

)(1.31)


εp =C0.75µ k1.5p

κ (zp + z0)(1.32)

Pkp =τ2w

κC0.25µ k0.5p (zp + z0)

(1.33)

It can be observed that, differently from Richards and Hoxey [15], the production of tur-

bulent kinetic energy at the wall is computed at a location displaced by the aerodynamic

roughness, zp + z0, to ensure consistency with turbulent dissipation rate calculation. In

Parente et al. [18] this formulation was proved to be effective in removing the peak of

turbulent kinetic energy observed at the wall [28]. As far as the numerical implementa-

tion is concerned, the form of the universal law of the wall is preserved:

Up =uτκ

ln(E′z+p

′), (1.34)

through the introduction of a new wall function constant and the non-dimensional wall

distance as

E′ =ν

z0uτ, z+p

′=uτ (z+p + z0)

ν. (1.35)

The non-dimensional distance z+p′

is simply a z+p shifted by the aerodynamic roughness,

whereas the new wall function constant depends on the roughness characteristics of the

surface. The friction velocity is calculated locally as

uτ = C0.25µ k0.5p . (1.36)

The wall function based on the aerodynamic roughness is compared to an alternative

wall function formulation based on the approach proposed by Blocken et al. (2007a) [7].

In their formulation the sand grain roughness ks is written as a function of z0, which

implies extremely large vertical cell size in case of very rough terrain. For example the

aerodynamic roughness in a city centre can exceed the value of 2 meters, which results

an approximate value of 60 meters for ks and larger than 120 meters for the height of the

wall adjacent cell, according to the criterion for ks type wall functions defined as zp > ks.

To overcome this problem, the alternative formulation is derived by imposing first-order

matching between the inlet velocity profile and the velocity imposed by the wall function

at the wall adjacent-cell. In particular, under the assumption that the regime is fully

rough, the modified roughness constant Cs and the non-dimensional equivalent sand

grain roughness ks+ are computed using the physical roughness z0 and the wall distance

of the first cell centroid zp. According to the equivalency of Eq. 1.28 and Eq. 1.31 ks


and Cs are defined as

ks = 0.95zp, ks+ = 0.95zp

+, (1.37)

and

Cs =Ez0

0.95 (zp + z0)− 1

0.95z+p. (1.38)

This approach has the advantage that its application does not require modifications

on the wall functions [1, 2], whereas in some general purpose solver, the range of the

roughness constant is limited, for example in ANSYS-Fluent it should be in the range

of 0.5–1.0.

1.1.5 Notations for model comparisons

Several comparisons was made using different formulations, thus the notations of the

different inlet conditions wall functions, and models are summarized in Table 1.2 – 1.4

in support of better understanding. In table 1.3 the Cm constant is solver dependent,

its value is Cm ≈ 30 for ANSYS-CFX, and Cm ≈ 20 for ANSYS-Fluent as well as for

OpenFOAM, using the default value of Cs.

Notation Inlet BC Formulation

RH Richards and Hoxey Eq. 1.8 – 1.10YA Yang et. al. Eq. 1.8, 1.17, 1.12PB Parente and Benocci Eq. 1.8, 1.17, 1.13BM Balogh Eq. 1.8, 1.17, 1.21

Table 1.2: Notations for inlet boundary conditions

Notation Wall function Formulation Cs ks

RWO OpenFOAM 1.6 original Eqs. 1.28 – 1.30 0.5 Cmz0RWD OpenFOAM 1.6 debugged Eqs. 1.28 – 1.30 0.5 Cmz0RWR Richards and Hoxey Eqs. 1.28 – 1.30 0.5 Cmz0RWC Parente and Benocci Eqs. 1.31 – 1.33 - -RWM Balogh Eqs. 1.28 – 1.30 Eq. 1.38 Eq. 1.37

Table 1.3: Notations for wall functions

1.2 Verification at full scale

The improvements, corrections and modifications on the wall treatments are investigated

with a simple, 2D full scale simulation of the neutral ABL.


Notation Model Sk Sε σε

STD Standard - - 1.3RH Richards and Hoxey - - Eq. 1.11YA Yang et al. Eq. 1.20 Eq. 1.25 1.3PB Parente and Benocci - Eq. 1.25 1.3BM Balogh Eq. 1.24 Eq. 1.25 1.3

Table 1.4: Notations for models

1.2.1 Geometry and mesh

The model geometry and the mesh is already used for the investigation of the homoge-

neous ABL over flat surface [29]. As in a typical and practical application, the modeled

area is 5000 × 500 m, with the resolution of 500 × 50 cells, according to the typical

resolution used for atmospheric simulations in the engineering practice (Fig. 1.2). The

mesh is uniform in streamwise direction, while the cell height is varying between 1 and

36 m, with a cell expansion ratio of 1.076. This vertical resolution results a y+ ≈ 20 ·103

value at the wall adjacent cell centroid.

Top (patch)

Inlet

(patch)

Ground (wall)

Outlet

(zero gradient)

5000 m

500 m

Figure 1.2: Sketch of the 2D full scale domain

1.2.2 Boundary conditions

As in the reference cases of Hargreaves and Wright [29], the Richard and Hoxey boundary

conditions are used, with the values of z0 = 0.01 m, uτ = 0.625 ms−1 and Uref = 10 ms−1

at zref = 6 m. At the top of the domain, the turbulent quantities and the velocity is

imposed, using the values given by the profiles at that height. Different wall formulation

are compared, namely the original wall function (RWO), its debugged version (RWD),

the Richard and Hoxey formulation (RWR), as well as the new (RWM) wall function,

furthermore the comprehensive approach (RWC). The effects of the σε modification and

the additional source term applied in the ε transport equation is also compared.


1.2.3 Results

In the first verification case, the different wall formulation are compared on full scale.

The original wall function (RWO), its debugged version (RWD), and the Richard and

Hoxey formulation (RWR) are compared to the new (RWM) wall functions.

0

100

200

300

400

500

0 5 10 15 20

z [

m]

U [ms-2

]

Full-scale U

Inlet

Outlet RWO

Outlet RWDOutlet RWM σε

0

100

200

300

400

500

0 0.4 0.8 1.2 1.6

z [

m]

k [m2s

-2]

Full-scale TKE

0

100

200

300

400

500

0.001 0.01 0.1 1 10

z [

m]

ε [m2s

-3]

Full-scale TKED

-2

-1

0

1

2

0 1 2 3 4 5

τ w/τ

w,th

eo

ry [

-]

x [km]

Full-scale N-D WSS

Ground RWO

Ground RWDGround RWM σε

Figure 1.3: Evaluation of the debugged wall function (RWD) and the wall functionusing modified roughness parameters (RWM) with modified σε compared to the original

rough wall treatment in OpenFOAM (RWO).

The correction on the available rough wall function is successful, as it is shown in Fig.

1.3. The agreement between the inlet and outlet profiles obtained by RWD is pronounced

in case of the U and ε, and the difference remains below 8% in case of k in the near

wall region. This difference results only a few percent error in the wall shear stress.

The RWM approach gain the best agreement between the inlet and the outlet for all

quantity, since the matching shown by Fig. 1.3 is almost perfect.


0

100

200

300

400

500

0 5 10 15 20

z [

m]

U [ms-2

]

Full-scale U

0

100

200

300

400

500

1 1.1 1.2 1.3 1.4 1.5

z [

m]

k [m2s

-2]

Full-scale TKE

0

100

200

300

400

500

0.001 0.01 0.1 1 10

z [

m]

ε [m2s

-3]

Full-scale TKED

Inlet

Outlet RWROutlet RWM σεOutlet RWC Sε

0.5

0.75

1

1.25

1.5

0 1 2 3 4 5

τ w/τ

w,th

eo

ry [

-]

x [km]

Full-scale N-D WSS

Ground RWRGround RWM σεGround RWC Sε

Figure 1.4: Evaluation of the Richards and Hoxey approach (RWR), the wall functionwith modified roughness parameters (RWM) using modified σε and the comprehensive

approach (RWC) with Sε source term

On Fig. 1.4, the results of the RWR wall function, RWM wall function with modified

σε, furthermore the RWC approach using the Sε source term are compared. Both RWM

and RWC is proved to be better, then RWR formulation. The formers show a very

good agreement between the inlet and outlet section for all quantity, while the RWR

gives acceptable agreement for U and ε and less realizable results for k and τw. The

comparisons between RWM σε and RWC Sε shows that the two approach gives the same

results, thus their equivalency is proved. The Sε approach is more general, hence the

turbulent Prandtl number correction is calculated with the local value of the coefficients.

In this reason, the application of the Sε source is more sufficient than the modified σε,

if Cµ is not constant.


1.3 Validation at laboratory scale

The ABL approach implemented in OpenFOAM and ANSYS-Fluent was verified on

simple 2D laboratory scale case namely the neutral homogenous ABL reproduced by the

atmospheric wind tunnel for University of Hamburg, (CEDVAL A1-1) [30], for which

velocity and turbulence intensity measurements are available.


The 2D domain has the size of 4 × 1 m and the mesh has 400 × 71 cells. The mesh is

uniform in the longitudinal direction with a cell length of 0.01 m, and it is stretched in

the vertical direction with a 6.05 ratio between the last and the first cell, the height of

the wall adjacent cell having a size of 0.005 m. Formerly, a mesh independency study

shown that the mesh coarsening or refining has a negligible effects on the results at this

resolution, namely the results are mesh independent.

Top (patch)

Inlet

(patch)

Ground (wall)

Outlet

(zero gradient)

4 m

1 m

Figure 1.5: Sketch of the 2D laboratory scale domain


The interpolated experimental data of the velocity and turbulent quantities were intro-

duced for boundary condition at inlet and the top boundary, while zero longitudinal

gradient is imposed at the outlet. The fitted data at the inlet correspond to a friction

velocity as uτ = 0.374 ms−1 and a reference velocity as uref = 6.43 ms−1 at zref = 1 m.

The fitting parameters for the reformulated turbulent kinetic energy profile are APB =

-3.82e-2 and BPB = 5.15e-1, while for the Yang profile, AYA = -4.54e-2 and BYA =

2.62e-1, respectively.


1.3.3 Results

Figure 1.6 shows inlet and outlet profiles of velocity, turbulent kinetic energy and dis-

sipation rate. It can be observed that the approach proposed by Parente et al. [20]

ensured homogeneity of velocity and turbulence between inlet and outlet sections of the

domain. Furthermore, it allowed retrieving almost the exact value of the wall shear

stress, at difference of the standard wall function reformulated in terms of the aerody-

namic roughness.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7

z [m

]

U [ms-2

]

CEDVAL A1-1 U

Inlet

OF Outlet

AF Outlet

Exp.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.3 0.4 0.5 0.6 0.7 0.8

z [m

]

k [m2s

-2]

CEDVAL A1-1 TKE

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1 1 10 100

z [m

]

ε [m2s

-3]

CEDVAL A1-1 TKED

0.5

0.75

1

1.25

1.5

0 1 2 3 4 5

τ w/τ

w,t

he

ory

[-]

x [m]

CEDVAL A1-1 N-D WSS

Ground OF

Ground AF

Figure 1.6: Inlet and outlet profiles against measurements (CEDVAL data), O -original rough wall function, C - Comprehensive approach, YA - Yang et al. [24] inlet

profiles, PB - Parente et al. [20] inlet profiles.


1.4 Validation of the extented formulation

The extension of the ABL formulation was validated on 2D laboratory scale cases against

experiments. Three different data set were used for the validation in order to prove

the sufficient fitting of the extended formulation, furthermore the validity of the source

terms. The measured data sets were obtained from wind tunnel experiments applied over

flat rough surfaces. One of them is the formerly introduced CEDVAL A1-1 ([30]) case,

the latters are flat reference cases of wind tunnel simulations over 2D and 3D hills. The

2D hill flat reference data is obtained by Khurshudyan et al. [31] and it is accessible in the

ERCOFTAC database (case 69). The 3D hill flat reference measurements are obtained in

the thermally stratified wind tunnel of The University of Tokyo under neutral conditions,

using three-dimensional laser doppler anemometry (Takahashi et al., [32]).


The simple 2D domains have meshed with uniform longitudinal resolution and increasing

cell size towards the wall, where the size of the wall adjacent cell determined from

y+ ≈ 20. The properties of the domains and meshes are summarized in Table 1.5, where

∆zl/∆zf denotes the grading ratio between the first and last cell height.

Case Length [m] Height [m] Nx Nz ∆zl/∆zf

CEDVAL A1-1 5 1 400 74 33.95ERCOFTAC 69 5 1.6029 400 81 46.67TOKYO UNI WT 5 0.5 400 32 4.2

Table 1.5: Domain properties for simple 2D cases


The interpolated experimental data of the velocity and turbulent quantities were intro-

duced for boundary condition at inlet and the upper boundary of the domain, while zero

longitudinal gradient is imposed at the outlet. The fitting parameters for the extended

profiles are summarized in Table 1.6 and 1.7. The four parameter profile fits provide a

significant improvement on the correlation between the measured values and the fitted

ones, summarized in Table 1.8.


Case uτ [ ms−1] z0 [m] δ [m]

CEDVAL A1-1 3.58e-1 7.11e-4 1.00ERCOFTAC 69 1.82e-1 1.57e-4 1.25TOKYO UNI WT 8.61e-2 5.94e-4 0.51

Table 1.6: Velocity profile parameters for 2D cases

Case A B C D APB BPB

CEDVAL A1-1 9.69e-2 1.99e-7 -6.30e-4 2.03e-1 -3.82e-2 5.15e-1ERCOFTAC 69 1.20e-2 0.33e-8 -0.54e-4 1.14e-1 -5.90e-3 1.15e-1TOKYO UNI WT 1.37e-2 2.23e-8 -5.69e-5 -7.09e-3 -5.25e-3 4.95e-2

Table 1.7: Turbulent kinetic energy profile parameters for 2D cases

Case RU Rk (BM) Rk (PB)

CEDVAL A1-1 0.997 0.822 0.119ERCOFTAC 69 0.999 0.966 0.416TOKYO UNI WT 0.977 0.941 -0.785

Table 1.8: Correlation coefficients for the velocity and turbulent kinetic energy profiles

1.4.3 Results

Figures 1.7–1.9 shows inlet and outlet profiles of velocity, turbulent kinetic energy and

dissipation rate for the three different cases. It can be observed that the extended ap-

proach as well as the one proposed by Parente et al. [20] ensured homogeneity of velocity

and turbulence between inlet and outlet sections of the domain. The new turbulent ki-

netic energy profile formulation with four parameter suitably fit for all measured profiles.

For the ERCOFTAC 69 flat reference case, the homogeneity is guaranteed even with

limited general Cµ value, as well above the boundary layer height, thanks to the source

term transition (Fig. 1.8).

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

z [m

]

U [ms-2]

CEDVAL A1-1 velocity

Inlet BMOutlet BM

Inlet PBOutlet PB

Inlet

0

0.2

0.4

0.6

0.8

1

0.3 0.4 0.5 0.6 0.7 0.8

z [m

]

k [m2s-2]

CEDVAL A1-1 turbulent kinetic energy

0

0.2

0.4

0.6

0.8

1

0.1 1 10 100

z [m

]

ε [m2s-3]

CEDVAL A1-1 turbulent dissipation rate

Figure 1.7: Inlet and outlet profiles against the CEDVAL measurements (PB - Parenteet al. [20] inlet profiles, BM - Balogh inlet profiles).


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5

z [m

]

U [ms-2]

ERCOFTAC 69 velocity

Inlet BMOutlet BM

Inlet PBOutlet PB

Inlet

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.05 0.1 0.15 0.2

z [m

]

k [m2s-2]

ERCOFTAC 69 turbulent kinetic energy

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.0001 0.001 0.01 0.1 1 10

z [m

]

ε [m2s-3]

ERCOFTAC 69 turbulent dissipation rate

Figure 1.8: Inlet and outlet profiles against the ERCOFTAC 69 measurements (PB- Parente et al. [20] inlet profiles, BM - Balogh inlet profiles).

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

z [m

]

U [ms-2]

TOKYO UNI WT velocity

Inlet BMOutlet BM

Inlet PBOutlet PB

Inlet

0

0.1

0.2

0.3

0.4

0.5

0 0.02 0.04 0.06 0.08 0.1

z [m

]

k [m2s-2]

TOKYO UNI WT turbulent kinetic energy

0

0.1

0.2

0.3

0.4

0.5

0.001 0.01 0.1 1

z [m

]

ε [m2s-3]

TOKYO UNI WT turbulent dissipation rate

Figure 1.9: Inlet and outlet profiles against the TOKYO UNI WT measurements (PB- Parente et al. [20] inlet profiles, BM - Balogh inlet profiles).

1.5 Conclusions on boundary conditions

The approach implemented in OpenFOAM and ANSYS-Fluent proved to be sufficient

for reproducing the structure of the fully developed, neutral, homogeneous atmospheric

boundary layer. The consistent set of inlet boundary conditions and wall functions

implemented, together with a k-ε model satisfies the theory. The verification and the

validation cases shows the capabilities of this approach, which proved to be much more

efficient than the approaches formerly published in the literature. The approach is

extended above the boundary layer, with a novel for turbulent kinetic energy profile

formulation, to achieve better agreement between the boundary conditions and the ex-

perimental profiles, within and above the boundary layer. This implies modifications

on the approach, in order to guarantee the stream-wise homogeneity of the fully devel-

oped boundary layer. Accordingly, modified source terms are derived for the transport

equations of turbulent kinetic energy and dissipation rate. The extended model, the

so called HBL k-ε is validated against experimental data obtained from wind tunnel


measurements. Results indicate that the four parameter profile suitably reproduces all

measured profiles, as indicated by correlation coefficients above 0.8 for all cases. Fur-

thermore, the extended model ensures the stream-wise homogeneity of the velocity and

turbulence profiles at the inlet and outlet sections of the domain, even above the bound-

ary layer thickness.

Chapter 2

Turbulence over complex terrain

This chapter discusses the developments carried out for the generalization of the ho-

mogeneous approach introduced in the previous chapter. While the modifications on the

turbulence model applied for retrieving homogeneous neutral atmospheric boundary layer,

those are not appropriate for inhomogeneous cases over complex terrain. In this reason,

the approach is further developed for improve its results over arbitrary terrain topogra-

phy, and enhance the results compared to the common approaches can be found in the

literature.

2.1 Turbulence in boundary layers over complex terrain

The overall approach [1, 2, 26], discussed in the previous chapter, was proved to be

successful for the simulation of unperturbed ABL using ANSYS-FLUENT and Open-

FOAM, but inadequate for separated regions. To overcome this drawback, the concept

of Building Influence Area was adopted (BIA), following Gorle et al. [33], to allow a

gradual transition of the turbulence model parameters from the values compatible with

the unperturbed ABL to the ones adequate for the regions perturbed by the presence

of obstacles. In this approach, the departure from the undisturbed flow conditions was

measured locally by evaluating the deviation of the actual local velocity from the homo-

geneous one.

2.1.1 Transition to non-homogeneous ABL

The further developed model applies smooth, continuous blending between the homo-

geneous (HBL) and inhomogeneous (STD) formalisms, where the rate of the transition

22

Chapter 2. Turbulence over complex terrain 23

is described by a function on power N . The blending is applied on Cµ and the source

terms as

CABLµ = CSTD

µ +(CHBLµ − CSTD

µ

)fNb (Uerr) , (2.1)

SABLε = SHBL

ε fNb (Uerr) , (2.2)

SABLk = SHBL

k fNb (Uerr) , (2.3)

where the superscripts STD, HBL and ABL denotes respectively the standard or in-

homogeneous, the homogeneous, and the universal value of the quantities, while fb is

the blending function depends on the velocity deviation (Uerr) between its local and

homogeneous reference values. The latter is calculated as

Uerr =|U −U ref ||U ref |

. (2.4)

The velocity deviation calculated automatically by the turbulence model in each iteration

step, using the reference velocity Uref corresponds to the unperturbed, homogeneous

velocity. This reference velocity is calculated at the initialization, with the wall distance

as vertical coordinate, using the uτ and z0 values representative for the given cell. In our

simulations the surface roughness is homogeneous and the reference velocity determines

the value of uτ , thus the their values are the same in each cell, but they can be arbitrary

in case of realistic full scale calculations.

Originally, in the adopted BIA concept suggested by Parente et al. [18, 20], the blending

function formulated as Eq. 2.5 applied implicitly the blending exponent N .

fb (Uerr) = 1− UNerr. (2.5)

This formulation gives a decreasing rate of transition with the increment of blending

exponent, furthermore if N = 0 then fb (Uerr) = 1, which yields to the STD value.

This disagrees with the concept, that the rate of transition is in propotion with the

magnitude of the exponent, furthermore an additional input parameter is required in

order to switch off the blending and applying HBL value. A novel sinusoidal blending

function is introduced [34] in order reduce the required input parameters of the model

and make the rate of transition propotional to the blending exponent. Moreover, even

in homogeneous simulations, tiny velocity deviations can be observed near to the wall,

due to the physical imperfectness of the k inlet profiles, with the maximum value of

Uerr = 0.05. This should be taken into account in the blending, therefore the novel

blending function formulated using a threshold limit of Utr = 0.05, and it is written as

fb (Uerr) =1

2− 1

2sin

(min

[max [Uerr − Utr, 0]

1− Utr, 1

]π − 1

2π

). (2.6)


An example is given in figure 2.1 for the smooth transition of Cµ in terms of the velocity

deviation using different blending exponents. In figure 2.1 the sub-figures in the second

line show the behavior of the blending near to the threshold limit of 5%, while the

bottom subfigures show the transition of the original blending function for comparison.

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.2 0.4 0.6 0.8 1

Cµ

Uerr

Sinusoidal blending (CµHBL = 0.03 and Cµ

STD = 0.09)

N = 0.0N = 0.5N = 1.0N = 2.0N = 3.0

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

f b(U

err)

Uerr

Sinusoidal blending function

0.029

0.03

0.031

0.032

0 0.02 0.04 0.06 0.08 0.1

Cµ

Uerr

Sinusoidal blending (CµHBL = 0.03 and Cµ

STD = 0.09), Uerr < 0.1

0.97

0.98

0.99

1

1.01

0 0.02 0.04 0.06 0.08 0.1

f b(U

err)

Uerr

Sinusoidal blending function, Uerr < 0.1

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.2 0.4 0.6 0.8 1

f b(U

err)

Uerr

Original blending (CµHBL = 0.03 and Cµ

STD = 0.09)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

f b(U

err)

Uerr

Original blending function

Figure 2.1: Blending of Cµ (left) using the blending function (right) with differentblending exponents (N).


2.1.2 Realizability issues

This latter approach was implemented in OpenFOAM, with the specific purpose of im-

proving the BIA concept for flows over complex terrains. The entire present approach

was thought and implemented on the standard formulation of k-ε model, but it is also

compatible with the different modifications to the standard k-ε model proposed in the

literature (Launder and Kato [35], Yap [36], Tsuchiya et al. [37]) to improve the simula-

tion of separation and reattachment regions. As the research was aimed to the simulation

of atmospheric flows over hilly terrain or building blocks, these modifications were im-

plemented and tested to attempt to improve the simulation and to assess their influence

on the results.

The Kato-Launder approach is based on the reformulation of the production term writ-

ten as

P STDk = νtS

2 → PKLk = νtSΩ (2.7)

where S is the modulus of the rate of strain tensor (symmetric part of the velocity

gradient tensor) and Ω is the vorticity (antisymmetric part of the velocity gradient

tensor).

The approach proposed by Yap is a correction for separated flows; it corrects the turbu-

lent kinetic energy prediction in the separated regions through an additive source term

in the ε transport equation, defined as

SYapε = 0.83

ε2

k

(k1.5

εC−0.75µ κzw− 1

)(k1.5

εC−0.75µ κzw

)2

(2.8)

where zw is the nearest wall distance. The Yap correction is applied together with the

Kato-Launder approach, as it is suggested by Launder [38], thus it is hereafter denoted

as KLY.

The modification proposed by Tsuchiya et al. (usually reported as MMK model in the

literature) differs from the standard k-ε model by the evaluation of the eddy viscosity

(νt). In this approach, the standard formulation of νt is modified by a multiplier ex-

pressed as a constrained ratio between the vorticity and the modulus of the rate of strain

tensor, in the following form

νMMKt = fwCµ

k2

ε, fw = min

(Ω

S, 1

)(2.9)


2.2 Implementation notes

2.2.1 Implementation in OpenFOAM

The approach and the model improvements discussed in section 1.1 and 2.1 were im-

plemented in OpenFOAM. All the wall function treatments and the source terms for

turbulent dissipation rate are included in the Reynolds-Averaged Navier-Stokes (RANS)

module of the code, following the description given before. To respect the structure of

the code, a new turbulence model is defined and denoted as ABL k-ε model. All of

the modifications are available as options, which can be activated within the RANS

model properties, which uses the setup file called RASProperties in OpenFOAM. The

case selectors are vectors with three digital components (i, j, k) spanning the options

summarized in Table 2.1. For example, the model applied as the STD k-ε model, if all

selectors are set to zero except Pk,sel = (1, 0, 0) and Cµ,sel = (1, 0, 0). The implemented

boundary conditions are also selectable, namely only one library is required for each

variable.

Selector Option for i = 1 switch j = 1 switch k = 1 switch

εsel ε source term SHBLε SYap

ε SABLε

ksel k source term SHBLk - SABL

k

Msel k-ε model version P STDk PKL

k νMMKt

Cµsel Cµ value CSTDµ CHBLµ CABLµ

Table 2.1: Switching options of the ABL k-ε model

2.2.2 Implementation in ANSYS-Fluent

The implementation of the new approach in ANSYS-Fluent is more complicated than in

OpenFOAM, since the manipulation of the models is not straightforward. The modifica-

tions are performed by means of User Defined Functions (UDF). As the implementation

of the standard k-ε model makes impossible to set Cµ as a local parameter, the realizable

k-ε model was used to implement the approach discussed in section 1.1 and 2.1, as it

already takes into account a variable Cµ. The model is first converted to a standard k-ε

model and then it is generalized to match the ABL k-ε model formulation. Importantly,

the definition of the eddy viscosity according to the realizable k-εmodel is overwritten by

introducing a user-specified turbulent viscosity. The implementation in ANSYS-Fluent


resembles the OpenFOAM implementation as much as possible. However, a perfect con-

sistency cannot be guaranteed, as ANSYS-Fluent does not provide access to the source

code, thus the details of the implementation might be different.

2.2.3 Numerical setup

To allow a meaningful comparison between ANSYS-Fluent and OpenFOAM, the nu-

merical settings have been selected to be as similar as possible. In this work, all the

OpenFOAM simulations were carried out with the simpleFoam solver, which is a steady

state, incompressible solver. For the spatial discretization of differential operators, the

Gaussian integration was used with different interpolation schemes. The 2nd order lin-

ear interpolation was applied for gradient terms, the 2nd order upwind interpolation for

divergence terms, while for the Laplacian terms the 2nd order linear interpolation was

used with explicit non-orthogonal correction. The preconditioned conjugate gradient

solver was applied with simplified diagonal-based incomplete Cholesky preconditioner

for pressure, and its bi-conjugate version with incomplete LU preconditioner for velocity

and turbulence. The SIMPLE method was used for pressure-velocity coupling. The

relaxation parameters were set to the default value of 0.3 for pressure and of 0.7 for

the other prognostic variables. In ANSYS-Fluent simulations, the second order upwind

schemes were used for velocity and turbulence, and the second order scheme is ap-

plied for pressure. Some numerical simulations presented divergence when the SIMPLE

method for pressure-velocity coupling was employed. Therefore, the coupled scheme

was used to avoid this problem, with the explicit under-relaxation factors of 0.75 for

both the momentum and pressure. For the turbulent quantities, a value of 0.8 for the

under-relaxation factors appeared satisfactory for reaching the full convergence.

2.2.4 Quantitative evaluation

Several quality metrics can be used for the quantitative evaluation of CFD models.

Generally, the numerical comparison based on the statistical analysis of the measured

and simulated results. The letter is obtained with the interpolation of the simulated

flow field variables to the measurement locations. The most commonly used quality

metrics are the hit rate, the normalized mean square error, fractional bias and correlation

coefficient. For the model validation, the hit rate was used as a quality assessment

metric. This quantity indicates the fraction of N measurement locations where the CFD

results are within a given relative of absolute deviation from the measurement data

[16]. Generally, the relative deviation defined by the uncertainties of the measurements,

otherwise (if the uncertainties are not available) can be set to RD = 25%. According to


former model validation studies (see e.g. Eichhorn [39], Santiago et al. [40] and Goricsan

et al. [41]) the absolute deviation is given as AD = 0.05 for the normalized velocities

(U/Uref , V/Uref , W/Uref ) and AD = 0.017 for the normalized turbulent kinetic energy

(k/U2ref ). Using these deviation values, the hit rate (q) is given as

q =1

n

n∑i=1

Qi, (2.10)

where n is the number of observations, while Qi calculated as

Qi =

1 if|Si −Oi||Oi|

< RD or |S′i −O′i| < AD,

0 otherwise.

(2.11)

In Eq. 2.11 Si denotes the simulated value, while Oi is the observed value of the given

variable in the ith measurement location. The prime symbol denotes the normalized

values of the quantities. The hit rate limit for validation is fixed at q > 66%, as it is

suggested by Schlunzen et al. [42].

2.3 Validation on a simplified 2D hill at laboratory scale

This validation study is the ERCOFTAC 69 case, namely a simplified 2D hill. Hot

Wire measurements are obtained by Khurshudyan et al. [31], smoothed by Trombetti et

al. [43] on three different hill geometries. This case is formerly investigated by Castro

and Apsley [44] moreover Sladek et al. [45] using RANS techniques. The reference

measurements are available for two-dimensional symmetric hills of varying steepness,

with the same hill height (h = 0.117 m).


Three different hills were investigated with the length-height ratio (a/h) of 3, 5, and 8.

All of the hills have rough surface, as well as the flat surfaces in the absence of the hill

with z0 = 1.57 ·10−4 m. The computational domain (Fig. 2.2) is symmetrical to the hill

axis and it measures 80h× 13.7h for each case, with 400× 80 resolution.

The shape of the hills are given parametrically by 2.12 and 2.13.

x (ξ) =1

2ξ

(1 +

a2

ξ2 +m2 (a2 − ξ2)

)(2.12)

y (ξ) =1

2m√a2 − ξ2

(1− a2

ξ2 +m2 (a2 − ξ2)

)(2.13)


Top (patch)

Inlet

(patch)

Ground (wall)

Outlet

(zero gradient)

L= 80h

H=

13

.7h

h=0.117 m

a=3h

a=5h

a=8h

Figure 2.2: Sketch of the domain with the different 2D hills

where m = n+√n2 + 1 is the shape parameter and n = h/a is the average slope. Block

structured meshes are prepared applying three different mesh resolution using 320× 42,

160 × 34 and 80 × 28 cell. The mesh refinement based on the distance from the wall

and from the downwind slope of the hill. The minimum and maximum cell sizes are

summarized in Table 2.2.

Hill Mesh resolution ∆xmin/h ∆xmax/h ∆ymin/h y+

a = 3h fine 0.043 0.692 0.043 ≈ 30a = 5h fine 0.072 1.145 0.043 ≈ 30a = 8h fine 0.115 1.832 0.043 ≈ 30a = 3h medium 0.086 1.339 0.086 ≈ 60a = 5h medium 0.143 2.200 0.086 ≈ 60a = 8h medium 0.230 3.470 0.086 ≈ 60a = 3h coarse 0.172 2.582 0.172 ≈ 120a = 5h coarse 0.287 4.186 0.172 ≈ 120a = 8h coarse 0.459 6.698 0.172 ≈ 120

Table 2.2: Properties of meshes generated for 2D hills


In this validation study, the BM inlet boundary conditions are used with a free-stream

velocity of U∞ = 4 ms−1. For the profile parametrization, the measured profiles at the

far upwind of the different hills were used for the profile fitting, which resulted the values

listed in Table 2.3.


Case uτ z0 A B C D

Flat reference 1.824e-1 1.569e-4 1.196e-2 0.332e-8 -0.539e-4 1.135e-1a = 3h 2.053e-1 4.972e-4 2.502e-2 3.641e-8 -2.010e-4 0.821e-1a = 5h 1.988e-1 3.576e-4 2.614e-2 2.138e-8 -1.602e-4 0.880e-1a = 8h 1.829e-1 1.586e-4 3.025e-2 0.469e-8 -0.765e-4 0.429e-1

Table 2.3: Results of the profile fitting for different hill geometries

2.3.3 Results

The results obtained on different meshes are compared to measurements over the three

different hills. Only the BM and PB boundary conditions are investigated. The compre-

hensive approach is used, without applying realizability constraints, such as the Kato-

Launder model with Yap correction, or the MMK model. Both the comprehensive

approach and BM boundary conditions are used with general Cµ limited by the value of

0.09, applying the sinusoidal blending with N = 3. As a quantitative comparison, the

hit rate results are summarized in Table 2.4. The hit rate values are fairly low in case

Hill Mesh Inlet U [%] W [%] k [%]

a = 3h coarse BM 57.58 79.84 94.38a = 5h coarse BM 65.31 96.56 100.00a = 8h coarse BM 65.78 100.00 100.00a = 3h medium BM 65.70 83.59 94.22a = 5h medium BM 71.09 100.00 100.00a = 8h medium BM 75.31 100.00 100.00a = 3h fine BM 74.22 89.22 94.77a = 5h fine BM 79.22 100.00 100.00a = 8h fine BM 85.78 100.00 100.00a = 3h fine PB 73.91 88.59 95.08a = 5h fine PB 74.69 100.00 100.00a = 8h fine PB 77.73 100.00 100.00

Table 2.4: Hit rates for 2D hills

of the steepest hill, because of the poor performance of the k-ε model in case of strong

separation. The quality of the results is increasing with the mesh resolution, the hit

rates values are above the validation acceptance level for all the variables for the fine

mesh. The simulations using the BM profile shows higher hit rate values for the velocity

and similar values for turbulent kinetic energy.

Figures 2.3 – 2.5 show the results of the simulations obtained on different meshes, using

the comprehensive approach and the modified wall treatment. The simulated velocity

agrees well with the measurements at the far absence of the hill furthermore at the

upwind side of the hill, but largish differences can be observed in the region of strong


0

1

2

3

4

5

6

-3 -2 -1 0 1 2 3 4 5 6

z/h

[-]

x/a [-]

Streamwise Velocity (a = 3h)

0

1

2

3

4

5

6

-3 -2 -1 0 1 2 3 4 5 6

z/h

[-]

x/a [-]

Turbulent Kinetic Energy (a = 3h)

-1

0

1

2

3

4

5

6

-3 -2 -1 0 1 2 3 4 5 6

τ w/τ

w,th

eory

[-]

x/a [-]

Wall Shear Stress (a = 3h)

Exp.Sim. coarse (BM)

Sim. medium (BM)Sim. fine (BM)Sim. fine (PB)

Figure 2.3: Comparison of the measured and simulated profiles for a = 3H

separation, which is confined only for the steepest hill (a = 3h). The results are getting

better and better with the resolution, as it is expected. The turbulent kinetic energy

prediction shows a behavior similar to the velocity, although its differences are more


pronounced in the stagnation region. While the measured values of the wall shear stress

are extracted from the profile, their reliability is not guaranteed. For this reason, these

values are used only for a qualitative comparison.

0

1

2

3

4

5

6

-2 0 2 4 6

z/h

[-]

x/a [-]


0

1

2

3

4

5

6

-2 0 2 4 6

z/h

[-]

x/a [-]


-1

0

1

2

3

4

5

6

-2 0 2 4 6

τ w/τ

w,th

eory

[-]

x/a [-]






0

1

2

3

4

5

6

-2 0 2 4 6

z/h

[-]

x/a [-]


0

1

2

3

4

5

6

-2 0 2 4 6

z/h

[-]

x/a [-]


-1

0

1

2

3

4

5

6

-2 0 2 4 6

τ w/τ

w,th

eory

[-]

x/a [-]






2.4 Validation on a simplified 3D hill

For the investigation of the approach on more complex cases, simulations were performed

on more complex geometries. The implemented modifications and the new models were

tested against the measurements are obtained in the thermally stratified wind tunnel of

The University of Tokyo, using three-dimensional laser doppler anemometry (Takahashi

et al., [32]).


The model is an axisymmetric hill, whose shape is defined as Eq. 2.14, with the radius

at the hill base of rmax = 0.42 m and with the height at the hill-top hmax = 0.2 m.

h(r) =

hmax1

2

[1 + cos

(πr

rmax

)]if r < rmax,

0 otherwise.

(2.14)

The hill model is positioned 2 m downstream of the inlet of the test section of the wind

tunnel, which has a size of 6 × 2.2 × 1.8 m. The computational domain contains the

entire test section. The origin of the model coordinate system is set to the x = 0, y = 0

position. The sketch of the domain at the symmetry plane (parallel to the streamwise

direction) is shown in Figure 2.6. The computational mesh is composed of 200×87×60

Top (smooth wall)

Inlet

(patch)

Ground (rough wall)

Outlet

(zero gradient)

L= 6 m

H=

1.8

m

h=0.2 m

d=0.84 m

Figure 2.6: Side view of the 3D hill computational domain

cells, resulting in 1.044 million hexahedral elements refined horizontally at the near-field

of the hill. A grid sensitivity analysis was carried out to estimate the solution error

associated to the discretization. The investigation was performed using two additional

grids with a uniform coarsening ratio (r = h2/h1 = h3/h2 = 1.5) in each coordinate

direction, as prescribed by Roache [46], resulting in 303240 and 93717 cells, respectively.



The hill and the floor of the wind tunnel are modeled as rough walls with the same

physical roughness as z0 = 1.22e − 3 m, while smooth wall boundary conditions were

applied on its ceiling and side walls. At the downwind side, a pressure outlet boundary

condition was used. The interpolated experimental data of the velocity and turbulent

quantities are introduced at inlet, while zero longitudinal gradient is imposed at the

outlet. The fitted data at the inlet correspond to a friction velocity as uτ = 0.0923 ms−1,

a boundary layer height as δ = 0.67 m, and a free stream velocity as u∞ = 1.42 ms−1.

The fitting parameters for the reformulated turbulent kinetic energy profile are APB =

-3.82e-2 and BPB = 5.15e-1, while for the Yang profile, AYA = -4.54e-2 and BYA =

2.62e-1, respectively.

2.4.3 Results

The hit rate for velocity and turbulent kinetic energy measurements was employed as

monitoring parameter for the selection of the grid using OpenFOAM. The results of

the quantitative evaluation summarized in Table 2.5 indicated a very slight variation

of hit rates between the set of meshes; therefore, the medium-size grid was chosen as

it provided the best compromise between the computational cost and model accuracy.

Figure 2.7 shows the measured and computed velocity and turbulent kinetic energy pro-

Case Streamwise velocity [%] Turbulent kinetic energy [%]

Coarse 77.66 55.84Medium 83.76 57.87Fine 81.73 64.97

Table 2.5: Hit rates values for the different meshes on the 3D hill

files along the longitudinal direction, provided by the comprehensive approach with a

blending exponent N=3 using the medium grid with the standard production term for

turbulent kinetic energy. The value N = 3 for the blending exponent was chosen as

it provided the best compromise between velocity and turbulent kinetic energy predic-

tions. It can be observed that, for the simulated case, the effect of the turbulence model

formulation is not significant for the velocity predictions provided by ANSYS-Fluent,

whereas a substantial improvement is observed for OpenFOAM applying the proposed

approach, especially for velocity prediction. As far as turbulent kinetic energy is con-

cerned, OpenFOAM simulations always present a large underestimation of k downstream

of the hill, whereas ANSYS-Fluent provides calculated values more in agreement with

the measurements. However, this effect is due to the large overestimation of the size of


the separation bubble by ANSYS-Fluent, which results in erroneous velocity predictions

but turbulent kinetic energy levels closer to the measured ones.

0

1

2

3

4

-4 -2 0 2 4 6 8

z/H

[-]

x/H [-]

Streamwise Velocity

0

1

2

3

4

-4 -2 0 2 4 6 8

z/H

[-]

x/H [-]

Turbulent Kinetic Energy

-2

0

2

4

6

8

10

-4 -2 0 2 4 6 8

τ w/τ

w,th

eory

[-]

x/h [-]

Wall Shear Stress

exp.OpenFOAM COpenFOAM O

Fluent CFluent O

Figure 2.7: Streamwise velocity (top), turbulent kinetic energy profiles (middle), andwall shear stresses (bottom) at the symmetry plane against measurements obtained on

the 3D hill at laboratory scale.


A comparison of the wall shear stress as a function of the non-dimensional longitudinal

coordinate is given in Figure 2.7 (bottom), where the circles denote the values extracted

from measurements. The comparison shows that the simulations are in fair accordance

with the observations in the separated region, whereas an important deviation can be

remarked at the first measurement location at the top of the hill. The latter can be

caused by the difference between the roughness elements mounted on the flat part of the

wind tunnel and the hill surface. The wall shear stress obtained by ANSYS-Fluent better

reproduces the experimental data, especially as the distance from the hill increases. The

hit rate results presented in Table 2.6, quantitatively support the conclusion that both

the velocity and turbulent kinetic energy prediction is significantly improved by the

comprehensive approach.

Case Streamwise velocity [%] Turbulent kinetic energy [%]

OpenFOAM C 83.76 57.87OpenFOAM C KLY 81.73 55.33OpenFOAM C MMK 81.73 56.35OpenFOAM O 70.05 36.55ANSYS-Fluent C 74.11 65.99ANSYS-Fluent C KLY 73.10 65.99ANSYS-Fluent C MMK 73.10 67.01ANSYS-Fluent O 62.94 55.33

Table 2.6: Hit rates values for simulations on the 3D hill

The computational results consistently show significant differences between OpenFOAM

and ANSYS-Fluent, concerning the reproduction of the separation bubble. Table 2.7

shows that the location of separation point for all ANSYS-Fluent simulations is closer

to the measured one, although the wake length is consequently overestimated. When

the comprehensive approach is applied in OpenFOAM, the prediction of wake length

is far better than for ANSYS-Fluent, as indicated by the error values listed in Table

2.7. It should be noted, that the “measured” wall shear stress values extracted from the

measured profiles using a forward difference approach, thus their reliability is question-

able. Although the simulated wall shear results are comparable in this way, the values

extracted from the profiles are not reproduce exactly the measurable values.

The length of the wake is highly affected by the location of the separation point, and

the former is erroneously predicted in ANSYS-Fluent, while the latter shows better

agreement with the measurements. This can be explained by the differences between

turbulence models. While the velocity at the hilltop is quiet similar for OpenFOAM

and ANSYS-Fluent (only a few percent difference can be observed), the turbulent ki-

netic energy near to the surface is overestimated by OpenFOAM and underestimated by

ANSYS-Fluent. It is interesting to observe that when the original k-ε model is applied


Case Separation point location [%] Wake length [%]

OpenFOAM C 26.30 -11.88OpenFOAM C KLY 26.30 -7.88OpenFOAM C MMK 22.30 -3.87OpenFOAM O 14.29 36.18ANSYS-Fluent C 18.29 32.18ANSYS-Fluent C KLY 14.29 48.20ANSYS-Fluent C MMK 14.29 40.19ANSYS-Fluent O 10.28 72.23

Table 2.7: Errors on the separation region for the 3D hill, normalized by length of thewake extracted from the measurements. Negative and positive values sign the under-

and over-estimation respectively

for both solvers, the separation point results are more comparable. When the turbu-

lence model modifications are applied, however, the wake length results improve for the

OpenFOAM implementation whereas a less pronounced improvement is observed for

ANSYS-Fluent results (Table 2.7). This can be explained by taking into account that

full access to the source code is possible in OpenFOAM whereas in ANSYS-Fluent the

modifications are carried out by means of user-defined functions which are ”interpreted”

by the code. This interpretation step, which is not controllable by the user, is likely to be

the cause of the discrepancies between the two codes, being their performances compara-

ble for the base k-ε case. While it is difficult to provide a definitive explanation for such

discrepancies, it should be recalled that several literature studies (e.g. Bechmann [5])

has indicated that ANSYS-Fluent solution schemes are generally very dissipative, which

can explain the overestimation of the recirculation region and the differences between

ANSYS-Fluent and OpenFOAM.

2.5 Validation of the extended formulation on the 3D hill

In order to validate the extended formulation and the four parameter (BM) profile, a

comparison has been made between the simulation results obtained with the current and

the former (PB) boundary conditions.

2.5.1 Computational mesh

Meanwhile a more advance meshing tool is developed, whose details will be discussed in

subsection 4.4.4, therefore a new set of meshes was used for validation.

This new meshing method calculates the wall distance fields as well, based on the true

vertical distance from the ground surface, therefore the turbulence model is adopted


0

0.1

0.2

0.3

0.4

-0.4 -0.2 0 0.2 0.4z

[m]

x [m]

Figure 2.8: Cross sectional close-up at the symmetry plane from the finest computa-tional mesh.

and further developed accordingly. The finest mesh is composed of 186× 132× 41 cells,

resulting in 1006632 hexahedral elements refined horizontally at the near-field of the

hill, shown in Figure 2.8. Grid sensitivity analysis was carried out again to estimate the

solution error associated to the discretization. The investigation was performed using

two additional grids with a uniform coarsening ratio (r = h2/h1 = h3/h2 = 1.5) in each

coordinate direction, resulting in 387450 and 136242 cells respectively. The resolution

of these meshes are comparable with the former ones, but their structure is slightly

different, especially in the near wall region.


The profiles of the extended formulation for the velocity and turbulent quantities were

introduced for boundary condition at the inlet. The BM profile parameters are obtained

with the framework introduced in section 1.1.3, while the PB profile parameters are

derived from the ones used in the former simulations on the 3D hill. As it was mentioned

before, the extended formulation can reproduce the PB profiles, considering Eq. 1.22.

The same boundary conditions were applied at the bottom, top and side walls, and at

the outlet than in the former simulations for the 3D hill. The fitting parameters for the

extended profiles are summarized in Table 2.8.

Inlet uτ z0 δ A B C D

BM 8.61e-2 5.94e-4 0.51 1.37e-2 2.23e-8 -5.69e-5 -7.09e-3PB 8.61e-2 5.94e-4 0.51 -5.25e-3 0.0 0.0 8.85e-2

Table 2.8: Inlet profile parameters for the extended formulation


2.5.3 Results

As we can see in Figure 2.9 and in Table 2.9, the extended formulation with the BM

profile performs better on every meshes for the velocity, although its turbulent kinetic

energy prediction is weaker on the coarse and medium mesh, compared to the PB profiles.

The hit rate values for the velocity on the medium mesh is higher than in the former

simulations.

Mesh Inlet Streamwise velocity [%] Turbulent kinetic energy [%]

coarse PB 82.74 55.84medium PB 83.76 55.84fine PB 75.13 58.88coarse BM 83.25 53.30medium BM 85.28 53.30fine BM 77.66 58.38

Table 2.9: Hit rates values for simulations on the 3D hill using the extended formu-lation

Mesh Inlet Separation point location [%] Wake length [%]

coarse PB 48.99 -50.84medium PB 25.59 -7.57fine PB 9.78 24.97coarse BM 45.21 -47.06medium BM 25.59 -3.72fine BM 9.78 24.97

Table 2.10: Errors on the separation region for the 3D hill, normalized by lengthof the wake extracted from the measurements. Negative and positive values sign the

under- and over-estimation respectively

The prediction of the separation point and the wake length, is better for the coarse and

medium mesh, and similar on the fine mesh in the cases, where the BM profiles were

used (see Table 2.10).

2.6 Validation on the Askervein Hill

In practical atmospheric applications the surface is generally not as regular as in the

previous examples; therefore, simulations were performed on more complex geometry to

validate the approach and, possibly, propose modifications. In particular, the full scale

measurements obtained over the Askervein hill (Taylor and Teunissen, [47, 48]) were

chosen to this purpose. This is a popular case study for validating CFD models for

ABL simulations and it was, therefore, investigated formerly by several authors, such as


0

1

2

3

4

-4 -2 0 2 4 6 8

z/h

[-]

x/h [-]

Streamwise Velocity

0

1

2

3

4

-4 -2 0 2 4 6 8

z/h

[-]

x/h [-]

Turbulent Kinetic Energy

-1

0

1

2

3

4

5

6

7

-4 -2 0 2 4 6 8

τ w/τ

w,th

eory

[-]

x/h [-]

Wall Shear Stress

exp.sim. coarse (BM)

sim. medium (BM)sim. fine (BM)sim. fine (PB)

Figure 2.9: Streamwise velocity (top), turbulent kinetic energy profiles (middle), andwall shear stresses (bottom) at the symmetry plane against measurements obtained

using the extended formulation.

Raithby et al. [49], Kim and Patel [50], Castro et al. [3, 4] and Rodrigues [51], applying

different turbulence models.



The Askervein hill has a nearly elliptical form with major and minor axis of 2000 and

1000 m. The height of the hill is 116 m, and its slopes range from 12 to 25%. The

surrounding area is flat at the upwind side of the hill, and it is hilly at the downwind

side. For describing the surface coverage, the aerodynamic roughness was taken as

z0 = 3.53e−2 m, based on the data measured at the reference mast. The computational

domain (Fig. 2.10) has dimensions 6000× 6000× 1000 m with an origin located at the

hill top (HT). Similarly to a previous study (Rodrigues, [51]), the mesh was generated

with 97× 111× 30 hexahedral elements using 15 m maximum resolution on the hill and

1 m for the first cell height. The refinement the grid at the hill is shown in Fig. 2.10.

Figure 2.10: Domain around the Askervein hill (left) shaded by the terrain elevationand the computational mesh refined at the hill (right).


Assuming that the terrain does not affect the flow on the top of the domain, the same

velocity inlet boundary condition was used as in the inlet, defining U , k, and ε. At

the downwind side of the domain pressure outlet and at the lateral sides of the domain

symmetry boundary condition was used. Measurements had been obtained at almost

neutral condition, stable wind direction and relatively high wind speed (Mickle et al.,

[52]). The measurement campaign was carried out using sonic anemometers at 10 m

above the surface, in every must along a given line (denoted as line-A in Fig. 2.10)

and at the hill top, thus the velocity components, such as the turbulent fluctuations

for the three directions are available for these locations. The wind direction for the

examined case was 210 degrees, which determines the mesh orientation. Namely, the

inlet is fixed on the western side of the computational domain and the outlet is the

eastern side. On its southern and northern side the symmetry boundary condition are


defined. On the top of the domain, the uniform values, corresponding to the fitted

profiles are imposed. The profile fitting is based on the data of Mickle et al. [52]. They

correspond to uτ = 0.661ms−1 and uref = 9.11ms−1 at zref = 10 m at the inlet. The

fitting parameters for the definition of inlet turbulent kinetic energy are: APB = −0.351

and BPB = 2.61. The wall treatments and modifications are equivalent to the simulation

of the simplified 3D hill.

2.6.3 Results

The comparison between computed and measured horizontal profiles is shown in Fig.

2.11: the horizontal component (Uh), vertical velocity (W ) and turbulent kinetic energy

are presented, where the RMS of the velocity is used to characterize the uncertainty.

The agreement between measured and computed velocity can be considered sufficiently

satisfactory. Although the comprehensive approach overestimates the horizontal velocity

at the far upwind of the hill summit (19.4% for OpenFOAM and 12.1% for ANSYS-

Fluent), its prediction at the top of the hill (Figs. 2.11 and 2.12) and along the downwind

of the hill is fairly good (underestimation of 2.1% for OpenFOAM and 5.5% for ANSYS-

Fluent). The simulation using the original approach provides also good agreement,

especially in the far upwind area, although the agreement is worse at the top of the hill

and in the downwind region. As for turbulent kinetic energy, the results obtained with

the comprehensive approach are significantly better than the ones given by the original

approach, even at the top of the hill. The differences between the CFD solvers are

significant: OpenFOAM results are better for velocity, whereas ANSYS-Fluent provides

a more accurate prediction of turbulent kinetic energy.

Table 2.11 shows the hit rate of the horizontal velocity and the turbulent kinetic en-

ergy for the OpenFOAM and ANSYS-Fluent codes, furthermore the relative errors of

the fractional speed-up ratio (FSR) at 30 m above the ground. This latter quantity

characterizes the acceleration effects of the terrain as the increment of the velocity:

FSR =u (zl)− uref (zl)

uref (zl), (2.15)

where u(zl) is the velocity at a given level (zl), while uref (zl) is the undisturbed velocity

at the same level. A hit rate value of 100% for velocity is obtained for the OpenFOAM

simulation using the comprehensive approach with the MMK model, with a correspond-

ing hit rate of about 44% for turbulent kinetic energy. Higher hit rate values for k is

obtained only using ANSYS-Fluent in combination with the Kato-Launder correction.

However, for this case, the hit rate value for velocity is far below 100%, i.e. 81%. The


0

4

8

12

16

20

-1000 -750 -500 -250 0 250 500

Uh [m

s-1

]

distance from HT [m]

meas.

OpenFOAM C

OpenFOAM O

Fluent C

Fluent O

-4

-2

0

2

4

6

-1000 -750 -500 -250 0 250 500

W [m

s-1

]


0

1.5

3

4.5

6

-1000 -750 -500 -250 0 250 500

k [m

2s

-2]


Figure 2.11: Comparison of simulated and measured horizontal and vertical streamvelocity (Uh and W ), and turbulent kinetic energy (k) along line-A. Notations: C-comprehensive approach (N = 3) using PB inlet profiles, O-modified rough wall formu-

lation using RH inlet profiles.

relative errors of FSR shows, that OpenFOAM could be an effective tool for wind turbine

sitting.

In general, the comprehensive approach affects the performances of both codes for

the full-scale simulations; the single exception of the velocity predictions provided by


0

10

20

30

40

0 4 8 12 16 20

z [

m]

U [ms-1

]

0

10

20

30

40

0 2 4 6 8

dis

tan

ce

fro

m t

he

gro

un

d [

m]

k [m2s

-2]

meas.

OpenFOAM C

OpenFOAM O

Fluent C

Fluent O

Figure 2.12: Comparison of simulated and measured vertical profiles (U and k), atthe hill summit (HT).

Case Hit rates onUh [%]

Hit rates on k[%]

Relative error onFSR at 30 m [%]

OpenFOAM C 93.75 87.50 5.38OpenFOAM C KLY 93.75 87.50 3.42OpenFOAM C MMK 100.00 87.50 4.31OpenFOAM O 87.50 62.50 -4.41ANSYS-Fluent C 81.25 93.75 8.83ANSYS-Fluent C KLY 81.25 93.75 8.71ANSYS-Fluent C MMK 81.25 93.75 7.31ANSYS-Fluent O 81.25 93.75 9.54

Table 2.11: Hit rates and relative errors on fractional speed-up ratio (FSR) for sim-ulations on the Askervein hill.

ANSYS-Fluent.

2.7 Computational performances

To have a complete comparison between the ANSYS-Fluent and OpenFOAM solvers, the

computational costs of the calculations are monitored. The calculations are executed on

the same computer and platform, namely on one core of an Intel Core 2 Quad computer

(Q6600 3 GHz, 8 GB RAM). Due to convergence problems observed in ANSYS-Fluent,

the numerical setup was different. Although in OpenFOAM the simple velocity-pressure

coupling was used, at difference of to the coupled scheme used in ANSYS-Fluent, the

time requirements of the simulations were quite similar (Table 2.12). However, the

memory usage is two times higher for ANSYS-Fluent (∼ 1.37 Gb on Askervein and

∼ 1.5 Gb on the 3D hill) than for OpenFOAM (∼ 660 Mb on Askervein and ∼ 780

Mb on the 3D hill), due to the higher memory allocation related to the coupled solver.


Surprisingly, at full scale, the comprehensive approach converged faster than the original

approach, for both solvers.

Case 3D hill (lab. scale) Askervein (full scale)

OpenFOAM C 12367/2000 5093/500OpenFOAM C KLY 18999/2000 5892/500OpenFOAM C MMK 12738/2000 5548/500OpenFOAM O 18008/2000 5204/500ANSYS-Fluent C 14451/2000 4947/500ANSYS-Fluent C KLY 21254/2000 5075/500ANSYS-Fluent C MMK 18418/2000 4506/500ANSYS-Fluent O 20881/2000 5209/500

Table 2.12: Computational performances (time in seconds/number of steps is neededfor full convergence).

2.8 Conclusions on ABL over complex terrain

A recent approach for the simulation of the neutral ABL was applied for the simulation

of flows over complex terrains. The approach was implemented in OpenFOAM and

validated against experimental data obtained on different cases with varying complexity,

namely on a 2D empty fetch, a 2D simplified hill, a 3D sinusoidal hill at wind tunnel

scale, furthermore on the full-scale Askervein Hill. The results indicate the potential of

the proposed approach for the numerical simulation of ABL flows over complex terrains.

In particular, the validation study on the 3D cases pointed out the major influence of

appropriate inlet and wall conditions on the results. Moreover, the impact of turbulence

model corrections, i.e. MMK model appeared to be relevant for full-scale simulations.

In general the results indicate satisfactory predictions for velocity, with hit rate values

around 80%, whereas the performances of the model for turbulent kinetic energy appears

inadequate in such cases, where strong separation appeard. This trend is observed for

all the simulated cases, codes and modification of k-ε leads, which indicate an intrinsic

limitation of linear k-ε. Therefore, the implementation of non-linear k-ε turbulence

model is currently under investigation, as it could have the potential of better performing

in the recirculation zones of the flow, as it is reported by different authors such as

Papageorgakis and Assanis [53] and Lun et al. [54]. The successful implementation of

the present model proves its flexibility and generality. The OpenFOAM toolbox proved

to be a very useful tool for ABL flows; compared with an available commercial code, it

yielded better results with comparable numerical effort and even better performance in

memory requirements.

Chapter 3

Modeling flows in urban canopy

layers

This chapter introduces a scale adaptive approach for modeling urban flows as a com-

bination of the advantages of meso- and micro scale approaches. The substance of this

method is that the porous drag force approach is applied in the inconsequential regions

for describe the effects of the buildings and the vegetation, while the geometry of building

blocks, or either a single building can be considered with mesh refinement in the regions,

whose are more important in the course of the analysis.

3.1 Flows in urban canopy layer

The detailed description of turbulence in the atmospheric boundary layer (ABL) is

essential with regard to the dispersion and heat transfer processes, both within and

above urban canopies. The realization of the accurate description is very difficult due

to the complex structure of an urban canopy layer, which is a varied system of bluff

obstacles including trees and buildings. The examination of the urban climate, including

the ventilation of the city, is possible by using statistical methods, such as roughness

parameter mapping ([55]) or using dynamical methods. Though the detailed numerical

simulation of flows in an urban canopy layer is possible by using CFD techniques, the

higher numerical cost of the spatial discretization of the complex geometries makes

this unrealizable in practice for very large domains. Finding the balance between the

numerical cost and reasonable results is very important with regard to the civic design

and environmental protection.

47

Chapter 3. Modeling flows in urban canopy layers 48

A potential technique, which keeps the balance, and furthermore, conforms to the varia-

tion of surface coverage, could be a hybrid method, such as a combination of the explicit

and implicit description of the flow properties in an urban canopy. For simulating tur-

bulent flows in the mostly exposed areas, (target area of the analysis, e.g. a parts of

the downtown area) where we need the most detailed results, the explicit modeling of

the buildings could be used, resolving their details with the computational mesh. In the

marginal parts of the examination area, the distributed drag force approach ([56], [57]),

as an implicit technique, could be applied with much lower numerical costs, although

resulting in a lower resolution.

The efficient modeling of the flows in the atmospheric boundary layer, including tur-

bulence, is feasible by solving the unsteady or steady state Reynolds-Averaged Navier-

Stokes (URANS and RANS) equation due to the relatively low computational cost and

reasonable accuracy. One of the most common turbulence model used in microscale

investigations is the k-ε two equation model both for unsteady (URANS) and steady

(RANS) simulations ([29]). The main goal of this study was to develop an efficient hy-

brid method for simulating turbulent urban flows with CFD techniques, which combine

the advantages of the micro- and meso-scale approaches.

3.1.1 Distributed drag force approach

Recently, the application of CFD techniques is increasing in the field of micrometeo-

rology, thus the knowledge of the flow structure and turbulence in an urban canopy

layer is rapidly developing as well. Many scientists are working on the development of

a distributed drag force approach, both for vegetated canopies (e.g., Green [56]; Liu et

al. [57]) and for building arrays (Lien and Yee, [58, 59]; Lien et al., [60, 61]; Carissimo

and MacDonald, [62]), therefore, it already has a solid theoretical background. This

approach has already been used in practice for the urbanization of weather prediction

models (Hamdi and Masson, [63]) to implicitly take into account the effect of buildings.

The essence of the drag force approach is an additional drag term in the momentum

equation and two other terms in the transport equation of the turbulent kinetic energy

(k) and the turbulent dissipation rate (ε). The drag term of the momentum equation is

composed of the viscous and the form drags, while the value of the viscous component is

much lower than the form component. Therefore, the former could be neglected. With

this simplification, the source term of the momentum equation has a general form of

Si = −ρCdAiUui, (3.1)


where ρ is the density of air, Cd is the drag coefficient, Ai is the frontal area per unit vol-

ume normal to the ith direction, U is the velocity magnitude, while ui its ithcomponent.

In case of vegetated canopies, the obstacles (e.g., branches and leafs) convert the kinetic

energy of the flow into wake turbulences with a smaller length scale than the shear-

generated turbulence. Therefore, the canopy yields a net turbulent kinetic energy loss

(Green et al., [64]) instead of enhancing the wake production. This could be modeled

with a source term in the following form,

Sk = ρCdAf(βpU

3 − βdUk), (3.2)

where Af is the total frontal area per unit volume, βp constant is a fraction of the mean

flow producuce kinetic energy and βd is an empirical constant for short-circuiting the

turbulent cascade (Green, [56]). The simplest model of the turbulent dissipation rate

source term is based on the Kolgomorov’s relation, which yields

Sε = C4εε

kSk. (3.3)

In Eqn. 3.3, C4ε is a constant. This relation was improved by Liu et al. [57] providing

a better fit to wind tunnel data. Accordingly, an alternative model could be defined as

a more general form, which reads as

Sε = ρCdAf

(C4ε

ε

kβpU

3 − C5εεβdU), (3.4)

where the new constant C5ε express the mixing length anisotropy, if it is not equal to

C4ε. Otherwise, the alternative model of Eqn. 3.4 turns into the simpler one (Eqn

3.3) as it is noticed by Sanz [65]. It should be mentioned, that the source term of

the turbulent dissipation rate is required in those microscale simulations that are based

on two-equation turbulence models. In mesoscale models these terms can be neglected

(Otte et al., 2004). The coefficients in Eqs. 3.1 - 3.4 depend on the type of the turbulence

model applied and the characteristics of the canopy layer. The relations between these

constants and the constants of the k-ε model, together with the characteristics of the

vegetated canopy were suggested by Sanz [65] and were analyzed by Katul et al. [66] and

by Sanz and Katul [67]. In present studies, the source terms of momentum, turbulent

kinetic energy, and turbulent dissipation rate were modeled in the above presented forms,

applying the alternative model for describing Sε. In vegetated areas, the properties of

the drag terms were based on the works of Balczo et al. [68], while in building arrays, the

drag coefficient Cd was calculated as a function based on the volumetric porosity of the

obstacle arrays applied by Coirier and Kim [69] for similar applications. After adapting


this function to our model, the drag coefficient could be defined in the following way

Cd(z,∆z,Hc, λt) =

min (∆z,Hc + 0.5δz − zc)

∆z (1− λt), if zc + ∆z < H,

0 otherwise,

(3.5)

where λt is the total solid volume per unit volume composed by the volume of the

buildings and the vegetation, Hc is the canopy height, zc is the height of the cell centroid

above the ground, and ∆z is the height of the cell.

3.1.2 A novel hybrid method

In the simulations of urban flows, the geometry of the buildings can be considered in

the course of spatial discretization, but it implies high computational costs, since the

required spatial resolution (∼0.1 - 10 m) results unmanageably high number of cells,

in particular for larger domains. The computational costs should be reduced, but the

important flow features should be resolved in order to achieve reasonable results with

feasible computational requirements. The goal of our method is to provide a mesh, which

allows the separation of the regions with different surface coverage, furthermore which

provide continuous transition between the coarser mesh applied in marginal regions and

the finest applied in the target area of the analysis can be considered in details [70, 71].

For this purpose, the target regions of the detailed analysis should be define for the

mesh refinement, but the geometry of marginal regions should be simplified in order to

prepare a coarser mesh.

3.2 Validation of the hybrid method

This section discusses the validation of the novel hybrid method. For this purpose a

benchmark study, namely the MUST (Mock Urban Setting Trial) data set were used

[72, 73]. This data set has been provided by the Defence Threat Reduction Agency

(DTRA). The objective of the test was to acquire meteorological and dispersion data sets

at near full-scale for the development and validation of obstacle-scale resolving models.

The experiment was designed to overcome the scaling and measurement limitations of

laboratory experiments and characterization difficulties presented by real urban settings.

For the validation, the test case with the wind direction of 0 degree is chosen, using the

up-scaled wind tunnel data set and the simulation results as reference [74].



In order to validate the hybrid method, different geometries and meshes are prepared for

the implicit, hybrid, and explicit concept. In the original MUST wind tunnel experiment

([74]) 120 obstacles of the full scale experiment were modeled at 1:75 scale, while the

geometrical details of the simulations, namely the number of modeled obstacles were

method dependent. The sketch of the modeled objects and measurement locations are

shown in Figure 3.1. In case of the implicit method, when clearly the porosity drag force

approach was used, a simple domain is prepared without obstacles, with a fairly coarse

mesh, although the mesh is refined in the inner region (thick dashed line in Fig. 3.1),

where the drag terms are applied, as it is shown in Figure 3.2 top. In case of the explicit

method, all of the obstacles are resolved by the mesh (Fig. 3.2 bottom right). In case of

the hybrid method, the area of interest is located around the main-tower, thus only the

black colored containers in Figure 3.1 are resolved and modeled (see Fig. 3.2 bottom

left). The size of the domain was identical for each method, namely 620, 644 and 60 m

in x, y and z directions respectively.

-100

-50

0

50

100

-100 -50 0 50 100

y [m

]

x [m]

MUST domain (0 degree case)

Inner regionVIP CAR

Main towerWT profiles

Figure 3.1: Top view of the examination area (inner region, bounded by thick dashedline), where the gray and black shapes shows the obstacle arrangement. The red circlesdenote the sampling locations for the quantitative comparisons, the blue cross denotes

the location of the main-tower for the canopy profile comparisons.

The computational mesh is generated from hexahedral elements and it is refined in the

inner region. For the containers, a block-structured mesh is prepared, applying mesh

refinement at the edges and corners of the obstacles with a minimum cell size of 0.2

m. Non-structured mesh is applied around the inner region, with a smooth cell size


expansion towards the lateral boundaries, where the horizontal cell size is maximized to

20 m. The meshes are graded vertically as well, with 0.2 m cell height at the ground

surface, and at the roof level of the obstacles. In this way, the three mesh prepared

for the implicit, hybrid and explicit method contains 89096, 1039056 and 5080320 cells

respectively. The meshes are shown in Figure 3.2.

Figure 3.2: Isometric view of surface meshes on the solid surfaces. The simple mesh(top) is used for the implicit method, the denser mesh (bottom left) for the hybrid

method and the densest (bottom right) for the explicit method


The ground surface of the computational domain is modeled as rough walls with an

identical physical roughness of z0 = 1.439e − 2 m, while smooth wall boundary con-

ditions were applied on the surface of the obstacles. This roughness obtained by the

comprehensive fitting algorithm, and differs from the one derived in [74] (that value is

z0 = 1.65e − 2 m). At the downwind side, a pressure outlet boundary condition was

used. The interpolated experimental data of the velocity and turbulent quantities are


introduced at inlet, while zero longitudinal gradient is imposed at the outlet. The fitted

data at the inlet correspond to a friction velocity as uτ = 0.467 ms−1 a reference velocity

of uref = 7.0 ms−1 at zref = 7.62 m. The four fitting parameters for turbulent kinetic

energy profile are ABM = −3.177e − 1, BBM = −8.177e − 8, CBM = 4.652e − 4 and

DBM = 2.912 respectively. The agreement between the fitted and measured profiles can

be seen in Figure 3.3.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.00 2.00 4.00 6.00 8.00 10.00

z/H

c [-

]

U [ms-1]

ExperimentInlet

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.00 1.00 2.00 3.00 4.00 5.00

z/H

c [-

]

k [m-2s-2]

ExperimentInlet

Figure 3.3: Fitted and measured velocity and turbulent kinetic energy profiles for theapproach flow

3.2.3 Results

Simulations are carried out on the three different meshes, applying the explicit, implicit

and hybrid method as well. In case of the hybrid and implicit method, the source terms

of the porous drag force approach is applied in the canopy layer, in those regions, where

the effects of the buildings are modeled. In order to improve the simulation results,

the Kato-Launder modification ([35]) is also applied, namely two different simulations

were done for each cases. Figure 3.4 shows the results of the different approaches and

models at the main tower, and the layer averaged results for the entire canopy. The

simulated values are compared to measurements and former simulation results of Ketzel

[73], obtained using the MISKAM microscale wind field and dispersion model. The

results of the explicit method is fairly similar to the MISKAM results at the main

tower location for the velocity, although the turbulent kinetic energy differs over the

canopy layer. In the quantitative comparison the hit-rate values using all measurement

locations are 75.44% and 77.21% for the horizontal and vertical velocity components,

while 64.84% for the turbulent kinetic energy. These values are above the validation

limit for the velocity components. The results obtained with the hybrid and explicit


method are agrees well, which prove that the concept is operable, however, the hybrid

approach requires much less computational resources, namely one in five. The layer

averaged velocity results show good agreement among the three different approaches

within the canopy layer, although slight differences can be observed above, which proves

that implicit approach are able to represent the effects of the canopy at larger scales.

These differences can be explained with the friction effects layer appears in the different

methods at the top of the canopy. The wall shear stresses on the roof of the containers

are fairly important in the development of the flow above the canopy height, but their

effects are not present in the implicit and hybrid method similarly.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.00 2.00 4.00 6.00 8.00 10.00

z/H

c [-

]

U [ms-1]

Main tower

ExperimentMiskamExplicit

Explicit KLHybridHybrid

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.00 1.00 2.00 3.00 4.00 5.00

z/H

c [-

]

k [m-2s-2]

Main tower

ExperimentMiskamExplicit

Explicit KLHybrid

Hybrid KL

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.00 2.00 4.00 6.00 8.00 10.00

z/H

c [-

]

U [ms-1]

Cell layer average

ExplicitExplicit KL

ImplicitImplicit KL

HybridHybrid KL

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.00 1.00 2.00 3.00 4.00 5.00

z/H

c [-

]

k [m-2s-2]

Cell layer average

ExplicitExplicit KL

ImplicitImplicit KL

HybridHybrid KL

Figure 3.4: Simulated and measured velocity and turbulent kinetic energy profiles atthe main tower, and layer averaged results in the canopy layer.

Additionally, the forces acting on the VIP CAR are extracted, in order to compare the

predictions of the different methods, summarized in Table 3.1. Although the hybrid


method overestimates the force magnitude, the difference is below 10%.

Method Fx [N] Fy [N] Fz [N] |F | [N] ∆F [%]

Explicit 179.76 -0.65 60.96 189.82 0Explicit KL 180.04 -1.07 59.86 189.73 -0.04Hybrid 190.67 1.38 82.74 207.85 9.50Hybrid KL 191.67 0.95 85.46 209.86 10.56

Table 3.1: Force components, magnitude and differences on the VIP CAR.

The surface pressure distributions on the VIP CAR are quite similar for each method,

as it is shown in Figure 3.5. One can conclude that using the hybrid method similar

results can be achieved in the area of interest, but with much lower computational costs.

Figure 3.5: Relative static pressure distribution on the VIP CAR.

3.3 An example on modeling turbulent flows in an urban

canopy layer

In this section, we introduce a typical and practical application of CFD techniques

in modeling turbulent atmospheric flows in urban areas. In this study ([70, 71]), the

examination area is the 11th district of Budapest (the capital of Hungary), where a

diversified landscape could be found, ranging from rural to downtown areas. Since this

region is also surrounded by diversified regions, the examination area was extended


with a relaxation zone [75] for simplifying the setup of the lateral boundary conditions

(discussed in the next chapter in section 4.5). While the aim of the investigation was

the analysis of the city ventillation of the district, the target areas were identified as the

possiple wind pathes of this region. In order to consider this aim, the mesh refinement

is applied at the wider roads and avenues, and at open areas such as parks and squares.

The coherent building blocks and denser canopes are considered as separated regions

with homogeneous surface coverage, in this way the narrow streets are not considered

explicitly.


The dimensions of the examination area in the x, y, and z directions were 9155, 7150,

and 1800 m, respectively, while the computational domain extended by the relaxation

zone (Fig. 3.6) is twice the original.

Figure 3.6: Top view of the computational domain colored by the elevation. Theregions with similar surface coverage are highlighted (fine solid lines) in the examinationarea (thick dashed red line), which is extended with the relaxation zone (thick solidblack line). The circles denote the sampling points of different canopy profiles used

later.

The geometrical setup of the computational domain was based on the SRTM (Shuttle

Radar Topography Mission) elevation database and a raster graphical map. It contains

surface coverage and building cluster data in a simplified, type dependent form. The


elevation data could be used after a coordinate transformation from WGS84 to a Carte-

sian frame of reference, namely the Uniform National Projection system (Hungarian

abbreviation: EOV). Nevertheless, the surface coverage data needed a pre-processing

procedure for efficient use in modeling. As a result of these, the properties of the canopy

layer such as the canopy layer height, the frontal area and solid fraction of the obstacles

are available at each region of the computational domain. Since the domain contains

several regions, we only introduce some typical parameters of urban type canopies (Ta-

ble 3.2). Due to the finite volume method used in our simulation system, the spatial

Parameter Suburban (C) Block of flats (B) Downtown (A)

H [m] 7 20 15λt [-] 0.3 0.4 0.6Af [m2m−3] 0.55 0.63 0.77

Table 3.2: Typical values of the total canopy height H, the solid fraction λt, and thefrontal area per unit volume Af in the urban type canopy, from different parts of the

examination area.

discretization of the domain, based on the computational mesh, is a system of wedge

cells. This mesh was generated from the elevation data and the polygons are bounding

the areas with the same type of surface coverage. The cells are triangular wedges with

vertical orientation and their lateral edge length is varying between 8 and 160 meters

in the examination area, growing up to 1000 meters in the relaxation zone, with a cell

growth rate of 1.3. For open fields, such as wider streets, parks, and the Danube River,

the mesh size is the minimum possible. Therefore, the impact of these could be described

at a higher resolution, as it can be seen in Fig. 3.7. Vertically, the cells are ordered in

layers and their height is increasing from 3 to 530 meters with the distance from the

ground surface. The layers follow the terrain near the ground and become flat at the

upper boundary with a continous transition.


At the inlet boundaries the inlet boundary conditions proposed by Richards and Hoxey

[15] are applied for the velocity components, turbulent kinetic energy, and dissipation

shown in Fig. 3.8. These were defined using the reference surface coverage condition

(z0 = 0.01 m) and reference flow conditions (Uref = 3 ms−1 at zref = 10 m) extracted

from the measurements of the Hungarian Meteorological Service. The wind climate of

the district could be examined by averaging the results of the simulations executed with

different wind directions, weighted by wind direction probabilities from the measured

data set. In the present example, eight runs were performed with different boundary


Figure 3.7: The structure of the computational mesh in the downtown area of the11th district, where the lighter areas show the building arrays, and the darker onesdenote uncovered or pourly covered areas such as streets, squares, and open fields,

together with vegetated areas.

conditions for the primary wind directions having the same velocity magnitude. Assum-

ing that the terrain does not affect the flow on the top of the domain, the same velocity

inlet boundary condition was used as in the inlet, defining U , k, and ε at the height of

the upper boundary.

0

20

40

60

80

100

0 0.5 1 1.5 2

z/z r

ef [-

]

U/Uref [-]

Velocity magnitude

0

50

100

150

200

0 0.5 1 1.5 2

z/z r

ef [-

]

k/kref [-]

Turbulent kinetic energy

0

50

100

150

200

0.001 0.01 0.1 1

z/z r

ef [-

]

ε/εref [-]

Turbulent dissipation rate

Figure 3.8: Vertical profiles at the inlet boundaries, where the velocity magnitude,turbulent kinetic energy, as well as dissipation rate are normalized by the reference

values observed at 10 meters above the ground.

3.3.3 Results and discussion

The simulations were steady state runs, while both the non-hydrostatic and Coriolis

force effects were neglected. Therefore, only the source terms of the porous drag model

and the non-reflective diffusion were enabled. The solver was used with second order

upwind schemes for the spatial discretization of the momentum equation, the turbulent

kinetic energy and turbulent dissipation rate transport equations, and a second order


scheme for the pressure equation. For better numerical stability, the SIMPLE pressure-

velocity coupling was used, along with the node based Green-Gauss gradient scheme.

With these settings, the simulations for the eight wind directions took only six hours

using a Quad-Core computer.

Vertical profiles of the weighted average of the velocity magnitude and the turbulent

shear stress, plotted along lines selected from different canopy regions from sampling

points A, B, and C in Fig. 1, are verified using analytical canopy profiles. The analytical

canopy profiles were published by Finnigan and Belcher [76], calculating the velocity as

U(z) =

uτκ

ln

(z −Hc + d

z0

), if z > H

U(Hc)eβ(z−Hc)

lm , if z ≤ H, (3.6)

and the shear stress as

τ(z) = l2m

(∂U(z)

∂z

)2

. (3.7)

where U(z) and τ(z) are the velocity and the shear stress magnitudes, uτ is the friction

velocity, κ is the von Karman constant, z is the height above the ground, Hc is the

height of the canopy, U(Hc) is the velocity magnitude at the top of the canopy, d is

the displacement height, z0 is the roughness height, lm is the mixing length, and β is a

constant of the profile. Note that uτ , z0 , d, and lm are functions of the canopy density.

These fall within the range of the analytical results, for low and high canopy densities

0

5

10

15

20

0 1 2 3 4 5 6

z/H

c [-

]

U/U(Hc) [-]

Velocity magnitude

open woodlanddense canopydowntown (A)

block of flats (B)suburban (C)

0

5

10

15

20

0 0.2 0.4 0.6 0.8

z/H

c [-

]

τ/τref [-]

Shear-stress

Figure 3.9: Normalized velocity and shear stress profiles as a function of the dimen-sionless height z/Hc, where the normalization is based on the velocity at the top of the

canopy U(Hc), and τref = ρu2τ .


(Fig. 3.9), although some differences could be found in the shape of the profiles farther

on the ground surface. The reason of these differences could be that the analytical

profiles were calculated for flat surface, while our simulations were applied on complex

terrain.

Figure 3.10: Distribution of the velocity magnitude and turbulence intensity scaledby reference values (unperturbed velocity and turbulence intensities) in the examination

area, at 3, 10 and 20 meters above the ground.

The averaged flow fields, which characterize the different regions of the district concern-

ing the ventilation, are calculated from the results of the different cases weighted by the

probability of the case dependent wind direction, both for the velocity magnitude and

turbulent quantities. The average velocity magnitude as a quantitative parameter pre-

dicts the ventilation of the different regions, while the turbulent intensity contains useful


information about the turbulent fluctuations. These were plotted at different heights

above the ground, namely at 10 and 30 meters. The velocity and turbulent intensity

fields are also scaled by the reference values of those calculated from the inlet profiles

taking into consideration the local elevation. Since the inlet profiles are defined for an

undisturbed free flow over a smooth surface, the scaled fields express the impact of the

topography of the examination area and the topology of the canopy layer.

Near the surface, as shown in Figure 3.10, over open areas, such as streets, squares,

parks, as well as over the Danube River, the velocity magnitudes are significantly higher

than the reference values due to the horizontal displacement of dense regions where the

flow is moderated by obstacles. Moving away from the surface, the velocity is increasing

as an effect of the canopy blocking, although above the canopy layer height, the impact

of the drag is still realizable (Fig. 3.11).

Figure 3.11: Distribution of the velocity magnitude and turbulence intensity scaledby reference values (unperturbed velocity and turbulence intensities) in the examination

area, at 30 and 50 meters above the ground.

The scaled turbulence intensity has a local minimum near the surface (Fig. 3.10), since

the reference turbulence intensity has the maximum next to the wall. The reference

profiles were calculated with free flow conditions, thus, only the wall has an impact

on its turbulent properties. At 30 meters above the ground, shown in Figure 3.11, the

turbulence develops at the boundaries of the blocks, mainly where the canopy properties


change suddenly. This effect is stronger on those side of the canopy blocks where the

gradient of the porous drag has a high positive value in the streamwise direction of the

locally dominant wind.

3.4 Conclusions on urban canopy layer flows

A scale adaptive hybrid modeling method is developed for combining the advantages of

the parametrization schemes applied in the meteorological practice, and the explicit ge-

ometrical modeling, resolving the flow features applied in CFD approaches. The porous

drag force approach specialized for atmospheric flows is implemented in the ANSYS-

Fluent general purpose solver, in the form of additional source terms in the transport

equations of momentum, turbulent kinetic energy and its dissipation rate. The source

terms are computed based on the local cell values of the field variables, furthermore the

canopy parameters specified automatically in each cells at the initialization [70, 71]. The

scale adaptive hybrid method is validated against up-scaled wind tunnel measurements

and it was applied together with the clearly explicit and implicit approaches. The layer

averaged velocity results show good agreement among the three different approaches

within the canopy layer, although slight differences can be observed above. The local

velocity results of the hybrid and explicit method is similar in the target area, however,

the hybrid approach requires much less computational resources.

A practical application was executed as well, which demonstrates the capabilities of

the CFD based approach in fields of the urban climatology and pollution control. The

source term of the parameterization was applied within the areas where obstacles were

found, thus, the impacts of the street canyons could also be considered. The properties

of the canopy layer were also changed according to their type. The averaged results of

the simulation were verified with analytical canopy profiles in representative points and

good qualitative agreement has been found.

The results of the current study are useful for the further development in modeling

stratified canopy layers, including the effects of the heat island phenomena and thermal

convection. The realization of these requires the adaptation of the surface parametriza-

tion schemes modeling heat and mass transfer and storage in the urban canopy layer

(see e.g. the work of Balogh [77] and Vu et al. [78]) with higher resolution.

Chapter 4

Methodology of atmospheric

simulations

In this chapter, the methodological survey of the atmospheric simulations is given. All

the problem specific tasks are discussed, including the numerical model representation

of the geometry and the surface coverage, boundary condition specification, furthermore

problem specific meshing.

4.1 Guideline for atmospheric simulations

The new boundary conditions and the further developed turbulence model allow to anal-

ize atmospheric flows using general purpose CFD solvers, but the simulation of this flows

implies several problem specific task. The geometrical realization of the complex terrain,

the definition of the surface roughness and other surface specific parameter, furthermore

the mesh generation are different from the general tasks in engineering practice. The

methodology of the atmospheric simulations is developed and applied succesfully in sev-

eral studies. Mesh generation methods are developed for the efficient and automatic

meshing of domains with complex terrain contains separated regions with different sur-

face coverage, producing task specific meshes. In the so called relaxation zone, which

frames the investigational region, the relief as well as the surface coverage parameters

are connected with a smooth transition to the terrain height and parameters defined

at the boundaries of the domain. This simplifies the setup of the boundary conditions,

futhermore improves the numerical stability and the convergence of the calculations. A

guideline is given for the complex, multy-step progress of atmospheric simulations in

Figure 4.1. The methodology follows the engineering prectice, but it is adapted for the

63

Chapter 4. Methodology for atmospheric simulations 64

no

M3.2. Executing simulation(s)

Data for domain geometry

Is there any objectin the domain?

yes

M1.1. Specification of the problem (type of investigation) - Target domain (area of interest) - Location: φ, λ - Size: Lx, Ly, H - Extension: ΔLx, ΔLy, ΔH - Data sets: surface coverage, terrain elevation, objects, wind climate

φ, λ, Lx+ΔLx, Ly+ΔLy, H+ΔH

M1.2. Geographic conversion (if needed) - WGS84 to Cartesian based on - Location: φ, λ - Extended size: Lx+ΔLx, Ly+ΔLy, H+ΔH

M2.1. Terrain representation - Terrain manipulation - Crop, merge, resize, resample - Landscaping elevation at the inlet - Landscaping elevation around planned objects - Surface fitting - Format conversions

M2.2.a. Object mesher - Base mesh concept - Create a mesh paying respect to the objects in the domain with flat lower boundary (using meshing software such as Gambit or Salome) - Inprint the terrain elevation on the mesh taking care of the objects - Subdivision concept - via snappyHexMesh

M3.1. Setup simulation(s) - Setting up inlet conditions - Setting up wall functions

M2.2.b. Stand alone mesher - Creating surface mesh - Define poligons are bounding regions with the same surface coverage - Create parametric surface from the elevation data - Generate surface mesh - Creating volume mesh - extrude surface mesh

M3.3. Post-processing

M1. Preparations

M2. Meshing

M3. Simulation

wind datamesh

terrain filesparametric surface

Figure 4.1: Flow chart of atmospheric CFD analysis

specificity of atmospheric problems. In the following sections, the details of the specific

tasks are provided.


4.2 Problem specification

In order to determine the important properties and attributes of the simulations, the

type of investigation should be defined first, according to point M1.1 of the guideline

in Figure 4.1. Various atmospheric problems used to be investigated via numerical

simulations in wind engineering practice, such as wind climate calculations for wind

energy assessment and urban confort and urban ventillation, wind load calculations on

buildings and structures, furthermore calculations of air pollution dispersion.

4.2.1 Size of the domain

The type of investigation determines the key features of target domain (can be considered

as the area of interest), namely its geographical location described by latitude (φ) and

longitude (λ) of its centre, its horizontal size given in x and y direction (Lx and Ly),

furthermore its height (H). For the reliability of the wind field in the target domain, the

comptational domain should be enlarged in order to model the effects of the sorrounding

area, furthermore to allows the simplification on the definition of boundary conditions

(discussed in Section 4.5). The magnitude of the extension is based on the variations of

the terrain elevation. Table 4.1 summarize the recommendations for domain extensions,

according to the slope criterion ∆ze/∆L ≤ 10%, where the variation of terrain elevation

defined as ∆ze = |zt − zmin|max, with zt heights sampled at the target domain.

Landscape ∆ze [m] ∆Lx and ∆Ly [m] ∆H [m]

Flat 0 0 0Undulated 0 - 300 5000 750Hilly 300 - 600 10000 1350Mountainous 600 < 10∆ze 2.5∆ze

Table 4.1: Recommendations for domain extension as a function of the variation ofterrain elevation.

4.2.1.1 Numerical analysis of the required domain size

The effects of the domain size extension are investigated by a series of numerical analysis.

Four different elevation heights are investigated, with a reference and four extended

domains. The base geometry was given as a simplified array of sinusoidal hills with the

maximal heights of Hmax = 5, 10, 15, 20 m. The hill array was aligned in an even tile


raster with a size of Lt = Lt,x = Lt,y = 40 m, whilst the terrain elevation defined as

ze =

Hmax

2[1 + cos (πR)] , if R ≤ 1.0,

0, otherwise,(4.1)

where R is the scaled distance from the center of the tile, denoted with the coordinates

xc and yc.

R =2.2√

(x− xc)2 + (y − yc)2Lt

. (4.2)

The area of interest, namely the target domain is selected as the four hill tiles in the

middle of the domains. Different extension sizes applied for the comparisons of each hill

height based on the variations of the terrain elevation. The size of the extensions was

calculated as Eqn. 4.3, with the ratios of Se = 0, 4, 6, 16.

Le =

Lt (1 + Se∆ze/∆L) , if Se > 0,

0, otherwise,(4.3)

For the reference case, the base geometry is applied with the maximum extension, while

for the test cases, the base elevation is blended with a smooth transition to flat surface in

the so called relaxation zone, whose length (∆Lr) is measured from the lateral boundaries

(inlet and outlet). The extensions are applied only in the x-direction, inasmuch this was

chosen as the main stream direction. In y-direction, the domains were periodic. The

height of the domain is limited to 200 m in order to reduce the computational costs,

since several simulations had to be done for the comprehensive investigation. Naturally,

the required height of the domain is much higher than this value, but the aim of the

current analysis was the investigation of the effects of lateral extension. The geometrical

properties of the extended domains for the highest hill array are summarized in Table

4.2 and their terrain elevation is shown in Figure 4.2.

Case Se [-] ∆Lr [m] ∆Le [m] Lx + ∆Le [m] Ly [m] H + ∆H [m]

Reference 16 0 360 800 80 200Extended 1 0 0 0 80 80 200Extended 2 4 40 80 240 80 200Extended 3 8 120 160 400 80 200Extended 4 16 320 360 800 80 200

Table 4.2: Geometrical properties of the different extended domains in case of thehighest hills (Hmax = 20 m).

The computational mesh is generated using a new meshing method (discussed in section

4.4.2.2) calculates the wall distance fields as well, based on the true vertical distance

from the ground surface. The horizontal resolution of the mesh was equidistant in x and


y/L y

[-]

x/Lx [-]

Terrain elevation [m], Hmax = 20 [m] (reference case)

-1

0

1

-10 -8 -6 -4 -2 0 2 4 6 8 10 0 5 10 15 20

y/L y

[-]

x/Lx [-]

Terrain elevation, Hmax = 20 [m] (extended case 1)

-1

0

1

-10 -8 -6 -4 -2 0 2 4 6 8 10 0 5 10 15 20

y/L y

[-]

x/Lx [-]


-1

0

1

-10 -8 -6 -4 -2 0 2 4 6 8 10 0 5 10 15 20

y/L y

[-]

x/Lx [-]


-1

0

1

-10 -8 -6 -4 -2 0 2 4 6 8 10 0 5 10 15 20

y/L y

[-]

x/Lx [-]


-1

0

1

-10 -8 -6 -4 -2 0 2 4 6 8 10 0 5 10 15 20

Figure 4.2: Terrain elevation of extended domains for the highest hill. The reddashed line shows the boundaries of the target area, whilst the black dashed line sign

the boundaries of the non-relaxed geometry.

y directions with the value of 1.6 m, whilst the mesh was graded vertically with a cell

expansion ratio of 1.2, using the cell height of 0.4 m for the wall adjacent cell. Figure

4.3 shows an example for the structure of the mesh on a cross sectional plane.

For the simulations, the latest version of the ABL k-ε model were used, imposing the

Richard and Hoxey boundary conditions at the inlet, with the values of z0 = 0.05 m and

Uref = 3 ms−1 at zref = 10 m. At the top of the domain, the turbulent quantities and the

velocity is imposed, using the values given by the profiles at that height. The simulations

are carried out on the geometries for the four different hill height and five different


0

20

40

60

80

100

120

140

160

-160 -120 -80 -40 0 40 80 120 160

z [m

]

x [m]

Cross section of the mesh at y = 20 m, Hh = 20 [m] (extended case 2)

Figure 4.3: Terrain elevation of extended domains for the highest hill. The reddashed line shows the boundaries of the target area, whilst the black dashed line sign

the boundaries of the non-relaxed geometry.

extension cases including the reference case. The results of the simulations are sampled

at eight different layer, in 121 points at each layer within the target area. The layers were

defined based on the wall distance, from 1 m to 150 m, with an increasing space between

them. Hit rates are calculated using the sampled velocity components and turbulent

kinetic energy values of the extended cases, and comparing them to the values of the

reference case. As it was expected, the hit rates values were converged with the extension

size towards 100% for each variable, and they reach the 100% at least at the largest

extension, even for the highest hill. In order to present a qualitative comparison between

the different extensions, velocity and turbulent kinetic energy profiles are extracted in

the middle of the target area shown in Figures 4.4–4.7.

0

50

100

150

200

0 1 2 3 4 5

z [m

]

U [ms-1]

Hmax = 5 m

Ref.Ext. 1Ext. 2Ext. 3Ext. 4

0

50

100

150

200

0 0.3 0.6 0.9 1.2 1.5

z [m

]

V [ms-1]

Hmax = 5 m

0

50

100

150

200

-0.1 0 0.1 0.2 0.3

z [m

]

W [ms-1]

Hmax = 5 m

0

50

100

150

200

0 0.1 0.2 0.3 0.4

z [m

]

k [m2 s-2]

Hmax = 5 m

Figure 4.4: Velocity components and turbulent kinetic energy profiles extracted atthe middle of the target area (Hmax = 5 m).

In case of lower hills, one can observe only slight differences between the results obtained

on the reference and the extended domains. The differences getting more and more


0

50

100

150

200

0 1 2 3 4 5

z [m

]

U [ms-1]

Hmax = 10 m


0

50

100

150

200

0 0.3 0.6 0.9 1.2 1.5

z [m

]

V [ms-1]

Hmax = 10 m

0

50

100

150

200

-0.1 0 0.1 0.2 0.3

z [m

]

W [ms-1]

Hmax = 10 m

0

50

100

150

200

0 0.1 0.2 0.3 0.4

z [m

]

k [m2 s-2]

Hmax = 10 m


0

50

100

150

200

0 1 2 3 4 5

z [m

]

U [ms-1]

Hmax = 15 m


0

50

100

150

200

0 0.3 0.6 0.9 1.2 1.5

z [m

]

V [ms-1]

Hmax = 15 m

0

50

100

150

200

-0.1 0 0.1 0.2 0.3

z [m

]

W [ms-1]

Hmax = 15 m

0

50

100

150

200

0 0.1 0.2 0.3 0.4

z [m

]

k [m2 s-2]

Hmax = 15 m


0

50

100

150

200

0 1 2 3 4 5

z [m

]

U [ms-1]

Hmax = 20 m


0

50

100

150

200

0 0.3 0.6 0.9 1.2 1.5

z [m

]

V [ms-1]

Hmax = 20 m

0

50

100

150

200

-0.1 0 0.1 0.2 0.3

z [m

]

W [ms-1]

Hmax = 20 m

0

50

100

150

200

0 0.1 0.2 0.3 0.4

z [m

]

k [m2 s-2]

Hmax = 20 m


pronounced with the increasing hill heights, especially the turbulent kinetic energy shows

larger discrepancies. While the hit rates are reached the 100% and the tested extension

sizes are far below the recommended ones, we can conclude that the recommended values

allow the correct numerical analysis of flows over various terrain elevation.

4.2.2 Data requirements

The preparation of the computational domain for atmospheric simulations is a special

problem, since one should consider both the terrain elevation and the surface coverage


data, respect to their effects on the flow features in the boundary layer. It is a complex

process to generate a base surface of the domain in a suitable coordinate system and

format from the available geographical databases.

4.2.2.1 Topographic data

The geometry of the terrain surface can be reconstructed from raw data available in

databases such as USGS (U.S. Geological Survey) and SRTM (Shuttle Radar Topogra-

phy Mission). The USGS is a science organization that provides near global topographic

maps, such as historical, geographical, geological maps, furthermore maps about land

cover and water resurces. The SRTM database provides near global topographic maps of

Earth called Digital Elevation Models (DEMs), produced by NASA originally and were

processed at the Jet Propulsion Laboratory and are being distributed through the USGS

EROS Data Center. The SRTM v4.1 terrain elevation data [79] can be freely downloaded

from the website of CGIAR-CSI (Consultative Group on International Agricultural Re-

search - Consortium for Spatial Information). The void regions of the database are filled

via interpolation [80], thus the complete dataset covers the Earth surface from between

60 north and 56 south latitude with data points with 3 arc-second (approximately 90

meters) spatial resolution. The data points are stored as WGS84 (World Geodetic Sys-

tem 1984) geographical coordinates in GeoTiff and ArchInfo ASCII files. Both filetype

are accessible on the website, in 5 × 5 tiles (e.g. the Carpatian-Basin covered by the

SRTM-40-03 and SRTM-41-03 tiles).

4.2.2.2 Wind data

The wind field related informations are used to be provided by the contractor, typically

not in a standardised form. The source of data can be varied, namely the source could

be a measured data set, a predicted data set, or it could be originated from reanalyses.

The format of the data is varied as well, it could be given as raw data, or in the form

of wind statistics (wind rose, parameters of a Weibull fit). The turbulent properties are

typically missing, but these can be estimated from the wind engineering standards (e.g.

from ESDU 85 [19]).

4.3 Geographic conversion

Considering to the guideline (point M1.2 in Figure 4.1), the suitable reference frame

should be formed in order to obtain the geometry of the ground surface, if it is not


provided by the contractor. Since the terrain elevation and the surface coverage data

stored in a format of WGS84, and this spherical coordinate system can only be used to

measure angles, not distances or areas, the data should be converted to Cartesian frame

of reference. The choice of the appropriate projection is strongly depends on the size

and location of the target area used in the simulation. In the following, a short overview

is given about the most common projections, which can be used in the progress of

atmospheric simulations. Suggestions are given as well for the selection in some typical

engineering problem.

4.3.1 Overview of map projections

Every flat mapping projection misrepresents the surface of the Earth. There is no

map can rival a globe in truly representing the surface of the entire Earth in a planar

coordinate system (USGS [81]).

• UTM (Universal Transverse Mercator) projection is used to define horizontal, po-

sitions world-wide by dividing the surface of the Earth into 6 zones, each mapped

by the Transverse Mercator projection with a central meridian in the center of

the zone. UTM zone numbers designate 6 degree longitudinal strips extending

from 80 South latitude to 84 North latitude. UTM zone characters designate

8 zones extending north and south from the equator. This projection is in the

family of Transverse Mercator projections are used for many quadrangle maps,

such maps can be joined at their edges only if they are in the same zone with one

central meridian. Also used for mapping large areas that are mainly north–south

in extent. Distances are true only along the central meridian selected by the map-

maker or else along two lines parallel to it, but all distances, directions, shapes,

and areas are reasonably accurate within 15 of the central meridian. Distortion

of distances, directions, and size of areas increases rapidly outside the 15 band.

Because the map is conformal, however, shapes and angles within any small area

(such as that shown by a USGS topographic map) are essentially true. Graticule

spacing increases away from central meridian. Equator is straight. Other paral-

lels are complex curves concave toward nearest pole. Central meridian and each

meridian 90 from it are straight. Other meridians are complex curves concave

toward central meridian.

• EOV (Uniform National Projection system) is a conformal cylindrical projection

in transversal position used uniformly for the Hungarian civilian base maps and,

in general, for spatial informatics. It is optimal for large and medium scale topo-

graphic mapping and for engineering survey.


• STG (STereoGraphic) projection could be used to map large continent-sized areas

of similar extent in all directions, typically in geophysics to solve spherical geom-

etry problems. The directions are true only from center point of projection, the

scale increases away from center point. Any straight line through center point is

a great circle. Distortion of areas and large shapes increases away from center

point. The stereographic maps are conformal and perspective but not equal area

or equidistant.

• LCE (Lambert Cylindrical Equal area) projection is undistorted along its standard

parallel, but distortion increases rapidly with the distance. Like any cylindrical

projection, it stretches parallels increasingly away from the equator. The poles

accrue infinite distortion, becoming lines instead of points.

• LCC (Lambert Conformal Conic) projection is used to show a country or region

that is mainly east-west in extent. One of the most widely used map projections

in the United States today. Retains conformal. Distances are true only along

standard parallels; reasonably accurate elsewhere in limited regions. Directions

reasonably accurate. Distortion of shapes and areas is minimal at, but increases

away from standard parallels. Shapes on large-scale maps of small areas essentially

true. Map is conformal but not perspective, equal area, or equidistant.

• LAE (Lambert Azimuthal Equal area) projection is suited for regions extending

equally in all directions from center points, such as Asia and Pacific Ocean. The

areas on the map are shown in true proportion to the same areas on the Earth.

Quadrangles (bounded by two meridians and two parallels) at the same latitude

are uniform in area. The directions are true only from center point, the scale

decreases gradually away from center point. Distortion of shapes increases away

from center point. Any straight line drawn through center point is on a great

circle. These maps are equal area but not conformal, perspective, or equidistant.

For showing the differences between the projections, the border of Budapest is also

plotted in the relative coordinate system in Figure 4.8, where coordinates are offset with

the projected reference coordinate respectively. As a graphical comparison of different

planar coordinate systems, the region of Hungarian capital is shown in Figure 4.9, using

the original (WGS84) topographic data and its projected form obtained with an in house

code.

4.3.2 Selection of map projection

The accuracy and the practicability of the data resulted by the simulations is an im-

portant aspect. The accuracy depends on the geometrical representation of the domain,


-20000

-15000

-10000

-5000

0

5000

10000

15000

20000

-20000 -15000 -10000 -5000 0 5000 10000 15000 20000

y -

y 0 [m

]

x - x0 [m]

Lambert Conformal ConicLambert Azimuthal Equal AreaLambert Cylindrical Equal Area

StereographicUTMEOV

Figure 4.8: Comparison of geographical projections in relative coordinate system. Areference coordinate defined for allow distortion comparison.

therefore a suitable projection should be selected with regards to the geographic location

and the domain size. In the other hand, the post-processing is much more straightfor-

ward, if the simulation results are directly available in a standard coordinate system

(engineering standard, e.g. EOV in Hungary), namely no conversion is needed for ac-

cessing wind data at an arbitrary location of the target domain. The recommendations

based on these essentials summarized in Table 4.3. It should be noted that a software is

developed for the automatic conversion of the above introduced map projections. This

software is implemented in FORTAN95 with a command line interface implemented as

a Bash shell script.

Region

Suitable extent Equatorial Middle latitude Polar

hemisphere LAE, STG LAE, STG LAE, STGcontinent/ocean LAE, STG, UTM LAE, STG, LCC, UTM LAE, STG

region/sea LAE, STG, UTM LAE, STG, LCC, UTM LAE, STGcountry LCE, UTM LCC, UTM, EOV -locality LCE, UTM LCC, UTM, EOV -

Table 4.3: Recommendations for map projection with respect to the location and do-main size. The red text highlights the global optimum, the green denotes the optimum

for Hungary.


φ [

°]

λ [°]

World Geodetic System 1984

47.3

47.4

47.5

47.6

47.7

19.0 19.1 19.2 19.3

100

200

300

400

500

y [

km

]

x [km]

Lambert Conic Conformal

-16

-8

0

8

16

-16 -8 0 8 16

100

200

300

400

500

y [

km

]

x [km]

Lambert Azimuthal Equal Area

-16

-8

0

8

16

-16 -8 0 8 16

100

200

300

400

500

y [

km

]

x [km]

Lambert Cylindrical Equal Area

6928

6936

6944

6952

6960

-16 -8 0 8 16

100

200

300

400

500

y [

km

]

x [km]

Stereographic

-16

-8

0

8

16

-16 -8 0 8 16

100

200

300

400

500

y [

km

]

x [km]

Universal Transverse Mercator (UTM)

5248

5256

5264

5272

344 352 360 368 376

100

200

300

400

500

y [

km

]

x [km]

Uniform National Projection System (EOV)

224

232

240

248

640 648 656 664 672

100

200

300

400

500

Figure 4.9: Comparison of geographical projections.

4.4 Problem specific meshing

The geometry of the mesh is based on the topographic data (terrain elevation, surface

coverage) and on the objects, whose are important in simulation point of view. Neces-

sarily, the object of the investigation could not be neglected, but other objects could be

relevant as well, if the flow field is significantly affected by them. The typical objects of

investigation are buildings or structures such as billboards and wind turbines.

4.4.1 Terrain manipulation and representation

Nevertheless, the projected terrain elevation data should be prepared for meshing as

it is shown at the point M2.1 in Figure 4.1. If a meshing software is used for mesh

generation, the topographic data should be converted to a suitable CAD format. Before

the format conversion, some required modification should be provided on the geometry,

listed below.


• Merge terrain elevation tiles to have a full extent includes the whole extended

domain.

• Crop the terrain elevation tile if it exceeds imoderately the extended domain.

• Resize or resample the terrain elevation data with respect to the required mesh

size.

• Landscaping:

– Relaxing the terrain at the inlet section of the domain, which enables the

simplified specification of the boundary conditions (discussed in the next sub-

section).

– Relaxing the terrain in the sorrounding area of objects will be placed in the

domain to avoid object distortion.

• With respect to the further usage, fit a parametric surface to the data points:

– Inverse weighted interpolation (Sheppard’s method)

– NURBS (Non-Uniform Rational Bezier Spline)

– Cubic spline interpolation

• With respect to the further usage, convert the terrain elevation data to suitable

CAD format, namely from ArcInfo ASCII Grid format to ASCII or binary STL

(STereoLithography) format. This conversion requires the triangulation of the

surfaces.

The above written progress implemented in FORTRAN 95, with a command line inter-

face implemented in Bash shell.

At this point the meshing progress branches out, according to the problem. If complex

objects are placed in the domain, a body fitted mesh should be created taking into

consideration these objects, otherwise the so-called stand-alone mesher is used. Both of

them are discussed in the following.

4.4.2 Object meshing concepts

As it is outlined in point M2.2.a of the flowchart, there are two basic concept for meshing

domains include complex objects, namely the mesh subdivision concept and a novel

method, the so called base mesh concept.


4.4.2.1 Subdivision concept

The former method based on the application supplied with the OpenFOAM software

package. This automatic utility generates 3-dimensional meshes from hexahedra (hex)

and split-hexahedra (split-hex) elements from triangulated surface geometries in Stere-

olithography (STL) format. The mesh approximate the surface by an iterative refining

of the starting mesh and morphing the resulting split-hex mesh to the surface. Its details

are discussed in the OpenFOAM user documentation [82]. The mesh generated with this

method for wind-turbine siting in a former publication [83].

4.4.2.2 Base mesh concept

As it is turned out from the name of this concept, the first step of the progress is to

create a suitable background (base) mesh using a meshing software. Several software

are available for mesh generation, the base mesh can be obtained by an open-source

application such as the BlockMesh utility of OpenFOAM, or the Salome platform, but

even a commercial software can be applied, such as the ANSYS-Gambit or the ANSYS-

ICEM software. The output format of the base mesh should be provide as a polyMesh

(native format of OpenFOAM), or as an ANSYS-Fluent mesh file (with .msh extension).

The base mesh is formed to have flat lower boundary while contains all objects should be

taken into consideration in the simulations. The base mesh could be arbitrary, namely

the type of elements does not affect the progress, but their quality is important.

The key of the base mesh concept is to in-print the terrain elevation into the mesh [75]

generated in the first step, with the modification of the node coordinates with respect

to the objects and the relief. Therefore the bounding polygon and the landscaping

size of the objects should be defined to avoid the distortion of them (as it is shown in

section 4.4.1). Using the landscaped terrain elevation data, the surface elevation can

be determined directly for each node location via interpolation based on fast Bi-Cubic

spline algorithm. From the interpolated value, the new location of the given node can

be calculated using the simple algebraic equation written as

zn =

(zo − zbm)(H − zt − zbm)

H − zbm+ zt + zbm, if zo > zbm,

zo + zt, otherwise,(4.4)

where zn and zo is the vertical coordinate of the new, and original node position, zt is

the terrain elevation interpolated to the horizontal coordinates of the node (xo, yo), H

is the height of the domain and zbm is the thickness of the boundary layer mesh. Below

zbm, the nodes are offset with the terrain elevation, therefore in this layer the mesh is


ground conforming, above zbm the nodes turn to conform the upper boundary. The mesh

used in the investigations discussed in section 3.2 generated by this application as well

as meshes used in a former studies [84, 85, 86, 87, 88, 88, 89]. This meshing application

is implemented in FORTRAN 95 as well, with a command line interface (Bash shell).

4.4.3 Stand-alone mesher

The previous node offset method is well suited for regions with lower slope, but for

steeper terrain it results cells with high distortion (high non-orthogonality). To overcome

this problem a stand-alone meshing application is developed (point M2.2.b), which is

able to create high-quality meshes with triangular wedges with low non-orthogonality.

The steps of the mesh generation are listed below.

• Surface mesh generation

– Create a parametric surface from the elevation data.

– Create parametric representation for polygons are bounding the regions of

different surface coverages.

– Apply sizing: create size map in the parametric space based on the distance

from the sizing sources (bounding polygons) and from highly curved regions.

– Create a non structured surface mesh on the parametric surface by means of

delaunay triangulation and Bowyer-Wattson algorithm [90].

– Smooth the surface mesh by means of a modified Laplacian smoothing apply

an alternative attractive-repulsive function [91].

– Project the surface mesh from the parametric to the Cartesian space.

• Volume mesh generation

– Calculate surface normals at the node locations based on the normals of the

triangles which connected to the node.

– Extrude the surface mesh cell layer by layer with triangle normals turns to

be vertical in certain steps and with increasing cell layer thickness.

This meshing application is implemented in FORTRAN 95 as well, with a command line

interface (Bash shell).


4.4.4 Mixed mesher

The basic idea behind this concept is to combine the base mesh concept and the stand-

alone mesher. The mesh generation process is devided into two steps. In the first step

of the process, similarly to the base mesh concept, a suitable background (base) mesh

is created, while in the second one, the vertices of the base mesh are repositioned.

In this particular case the blockMesh utility of OpenFOAM is used for the former objec-

tive. This utility generates block-structured meshes using hexahedral elements, based on

a dictionary (blockMeshDict), which contains all the required information for the mesh

generation process. The properties of the hexahedral mesh are pre-calculated based on

the geometrical and sizing constraits defined by the simulation problem. These proper-

ties, namely the number of cells and the size of the domain in each coordinate direction

are stored in the blockMeshDict file, from which the blockMesh utility generates the

mesh as one block with uniform cell size.

In the second step, the application generates a graded surface mesh, refined at the

critical regions, then the surface mesh extruded similarly to the stand-alone mesher.

The application reads the uniform hexahedral mesh prepared in the first steps and refers

its vertices based on their coordinates. The coordinates of the vertices then overwritten

by the corresponding ones calculated in this step. Additionally, the progress creates

the field contains the true vertical wall distance for each cell, which is effectively in

accordance with the theory of atmospheric boundary layers. The turbulence model is

modified in order to read this field, instead of calculates a wall distance by itself. This

modification is required also in those cases, where not only the ground surface defined

as wall boundary condition, but other walls are also present in the domain (e.g. in

case of the 3D sinusoidal hill), hence the wall distance calculation of the model cannot

distinguish between the walls, thus the calculated value given by the solver would be

inappropriate.

This method creates quality meshes from hexahedral elements with lower skewness com-

pared to the base mesh concept, and it is implemented in FORTRAN 95 as well, with

a command line interface (Bash shell). The meshes used in the investigations discussed

in section 2.5 and subsection 4.2.1.1 generated by this application.

4.5 Simplified specification of boundary conditions

In order to simplify the specification of the boundary conditions, modifications should

be applied on the domain. Both the elevation and surface coverage data could be relaxed


in the space to their reference values along the relaxation zone [75] from the edge of the

examination area to the lateral boundaries with the use of a dumping function Eqn.

4.5. The reference value for the elevation was its spatial average at the lateral sides

of the domain, while for the specification of the relaxed surface coverage parameters,

the properties of the open grassland were used as a reference. For this reason, identical

vertical profiles could be defined at every inlet boundary, calculated with the reference

surface coverage properties.

σ(r) =1 + cos(r)

2, r ∈ [0, π]. (4.5)

The dumping coefficient σ(r) is a function of the normalized distance (r) from the

closest lateral boundary. In the relaxation zone, the elevation and the characteristics of

the canopy continuously approach a reference value defined on the boundary by Eqn.

4.6.

φ(r) = σ(r)φref + (1− σ(r))φexm, (4.6)

where φexm and φref are the values of the relaxed parameter at the nearest part of the

examination area and at the lateral boundary respectively. The realization of the non-

reflective boundary conditions could also be obtained by using additional source terms

(Bodony, [92]) in the relaxation zone as an analogy of Eqn. 4.6, written in a general

form as

S(r, φ) = −σ(r)

∆t(ρφ− ρrefφref ) , (4.7)

where S(r, φ) is the non-reflective diffusion source term, ∆t is the time-step size, ρ and

φ are the current, ρref and φref are the reference values of the fluid density and the

field variable of the transport equation, respectively. The field variable is the velocity in

the source term of the momentum equation, the enthalpy in the energy equation, and

φ equals 1 in the continuity equation. Note that in steady simulations, the value of the

time-step size could be replaced by a time scale, while in incompressible cases ρ is equal

to ρref .

4.6 Conclusions on the methodological survey

In this chapter, the methodology of atmospheric simulations were discussed. It has

several portion, since the simulation of atmospheric flows implies several problem spe-

cific task, such as the geometrical representation of the complex terrain, the setup of

boundary conditions, the definition of the surface roughness and other surface specific

parameters, furthermore the mesh generation considering the formers. These tasks are


different from the general tasks of the engineering practice, thus a specialized method-

ology is developed.

In order to simplify the setup of the lateral boundary conditions, the accommodation

of the domain extension with the so-called relaxation zone is proposed. In this zone,

which frames the target area, the relief as well as the surface coverage parameters are

connected with a smooth transition to the constant terrain height and surface parameters

defined at the lateral boundaries of the domain. It does not only facilitate the setup, but

improves the numerical stability and convergence of the calculations [70, 71]. According

to facilitate the preparation of domain geometry, recommendations are given for the

extended size of the computational domain and for the selection of map projection

with respect to the location and size of the target area, furthermore the features and

properties of the relief. The recommended size of the extension is corroborated by a series

of numerical simulation, while the recommendations for the applicable map projections

are based on practical considerations. Mesh generation methods are developed for the

efficient and automatic meshing for domains with complex terrain geometry, and with

separated regions with different surface coverage, producing task specific meshes. The

terrain relaxation is built in this mesh generation framework [75]. The methodology

for the preparation and execution of atmospheric simulations is developed and applied

successfully in different case studies (see e.g. [1, 2, 34, 71]). This methodology serve as

a guideline for solving atmospheric problems in the engineering practice.

Chapter 5

Conclusions and outlook

This research was focused on the numerical simulation of micro- and meso-scale at-

mospheric flows using general purpose CFD solvers. The study incorporated with the

adaptation of commonly used solvers in engineering practice, the verification and vali-

dation of models and approaches, furthermore the elaboration of the modeling procedure,

which includes geometrical pre-processing, meshing as well as the specification of bound-

ary conditions. Accordingly, the research was devided into four unit, whose cover the four

main investigated problems are concluded in the following. The methods, approaches and

models developed to overcome these problems are discussed together with their results.

5.1 Conclusions

To overcome the problems originated from the parametrization of rough wall functions,

an alternative formulation is derived for the calculation of roughness parameters by

imposing first-order matching between the velocity given by the inlet profile and the wall

function at the wall adjacent cell. Under the assumption that the regime is fully rough,

a modified roughness constant and equivalent sand grain roughness are calculated using

the physical roughness and the wall distance of the first cell centroid. This approach has

the advantage that its application does not require modifications on the wall function,

thus it can be applied directly in most of the general purpose CFD solver. It should be

noted that its application could be limited by the given roughness constant range of the

solver [1, 2]. The deteoriation of the inlet profiles in streamwise direction yielded the

motivation to the development and implementation of a consistent set of inlet and wall

boundary conditions. The STD k-ε turbulence model is taken as a basis and modified

according to make it valid for the entire homogeneous neutral atmospheric boundary

layer (HBL), namely to ensure the conformity between the model, the wall functions

81

Chapter 5. Conclusions and outlook 82

and the inlet boundary conditions [1, 2]. This approach is extended above the boundary

layer thickness, with a novel profile formulation including a four parameter turbulent

kinetic energy profile to achieve better correspondence between the boundary conditions

and experimental profiles. The consequential source terms of the extended approach are

derived for ensuring the validity of the turbulence model within and above the boundary

layer [26]. The boundary conditions, wall functions and the implemented HBL k-ε

model are validated against theoretical profiles at full scale, and against measurements

at laboratory scale from three different data sets obtained via wind tunnel experiments.

Results indicate that the novel set of boundary conditions reproduces the measured

profiles for all examined cases, as indicated by correlation coefficients above 0.8. This

improvement is pronounced compared to the formerly suggested boundary conditions,

where the correlation coefficients of the turbulent kinetic energy profiles had the values

of 0.119, 0.416 and -0.785 by turns, and whose are improved by the new method to 0.822,

0.966 and 0.941, respectively. Moreover, the extended model ensures the stream-wise

homogeneity of the velocity and turbulence profiles at the inlet and outlet sections of

the domain, even above the boundary layer thickness [26].

In order to simulate inhomogeneous ABL flows develops over complex terrain, the HBL

k-ε model is further developed, to achieve that its formalism is applied in those regions,

where the boundary layer is considered as homogeneous, but in case of streamwise in-

homogeneities caused by the complex terrain, the STD k-ε model is applied. The new

model, the so-called ABL k-ε model provides continuous smooth blending between the

formalisms, which based on the normalized velocity difference between its simulated

and theoretical value. The rate of transition is described by a sinusoidal power function

with a supplemental threshold offset, which required in order to avoid pseudo blending

caused by small numerical discrepancies next to the wall in former blending functions.

The optimal value for the exhibitor N of this function is determined in the course of

different flow simulations over complex terrain. Out of the examined integer values, the

best results were obtained with N = 3 [1, 2, 34]. The ABL k-ε model is validated against

measurements at laboratory scale and full scale as well, using the optimal blending ex-

hibitor. The calculated velocity agreed well with the measurements at both laboratory

and full scale, while reasonable agreement is found for the turbulent kinetic energy. In

order to measure the improvement produced by the ABL k-ε, its results are compared to

results obtained by the formalisms can be found in the literature. These are proved one-

self to be significantly better than the results obtained by using the formerly suggested

formalisms. The improvement on the hit rate was around 10–15% for the velocity and

10–20% for the turbulent kinetic energy, thanks to the application of the new approach

[1, 2, 34]. According to the recommendation can be found in the literature, the Kato-

Launder modification, the Yap correction, furthermore the Murakami-Mochida-Kondo


(MMK) model are implemented to enhance the performance of the ABL k-ε model in

the simulations of flow where stagnation and separation appears. The effects of these

are investigated on the cases used for the former validation. At laboratory scale, the

application of these modifications decreased the hit rates with 1–2% for both the ve-

locity and turbulent kinetic energy. At full scale, the application of the Kato-Launder

modification with the Yap correction are not produce remarkable improvement, but the

MMK model resulted a significant improvement reaching the 100% hit rate value for

the velocity [1, 2]. In order to further extend the examination of the ABL k-ε model, a

quantitative comparison is carried out between the results obtained by ANSYS-Fluent

and OpenFOAM general purpose solvers. The OpenFOAM solver provided significantly

better results with similar computational costs. The simulations, using the original in-

built models of the solvers, gives similar quantitative results, although these are worse

than the results obtained with the new approach. As it is identified, the differences

are originated from the way of the implementation. While the OpenFOAM is an open

source C++ library, the implementation of a model is straightforward. Contrarily the

ANSYS-Fluent commercial solver can be only modified via user defined functions, more-

over, the source of the model formalisms are not accessible, therefore the potential of

the strict implementation is limited [1, 2].

Dealing with the third problem, a scale adaptive hybrid modeling method is developed

for combining the advantages of the parametrization schemes applied in the meteorologi-

cal practice, and the explicit geometrical modeling, resolving its features applied in CFD

approaches. The substance of this method is that the porous drag force approach is ap-

plied in the marginal regions for simulate the effects of the buildings and the vegetation,

while the geometry of building blocks, or even some selected buildings can be considered

explicitly via mesh refinement in the regions, where the flow features are important in

the analysis point of view. A special meshing procedure is developed, which allows the

separation of the regions with different surface coverage, furthermore provide continuous

transition between the coarser mesh applied in marginal regions and the finest applied

in the target area of the analysis [70, 71]. The porous drag force approach specialized

for atmospheric flows is implemented in the ANSYS-Fluent general purpose solver, in

the form of additional source terms in the transport equations of momentum, turbulent

kinetic energy and its dissipation rate. The source terms are computed based on the

local cell values of the field variables, furthermore the canopy parameters specified au-

tomatically in each cells at the initialization [70, 71]. The scale adaptive hybrid method

is validated against up-scaled wind tunnel measurements, where it was applied together

with the clearly explicit and implicit approaches. The results of the clearly explicit

method are above the validation limit (66%) for both streamwise and vertical velocity

components with the hit rate of 75.44% and 77.21%, respectively. The resulted hit rate


for turbulent kinetic energy just fall below the limit with 64.84%. The explicit results

are fairly similar to the former simulation results obtained with the MISKAM solver at

the measurement tower placed in the middle of the target area, although their agree-

ment with the measurements is not satisfactory. The local velocity results of the hybrid

and explicit method is similar in the target area, however, the hybrid approach requires

much less computational resources, namely one in five. The layer averaged velocity

results show good agreement among the three different approaches within the canopy

layer, although slight differences can be observed above, which proves that implicit ap-

proach are able to represent the effects of the canopy at larger scales. The approach

verified against theoretical profiles in a practical application as well, which demonstrate

the its potential in field of real engineering problems [70, 71].

For the problem specific tasks of atmospheric simulations are brought up from the fourth

problem, the methodology of their procedure is developed. In order to simplify the setup

of the lateral boundary conditions, the accommodation of the domain extension with

the so-called relaxation zone is proposed. In this zone, which frames the target area, the

relief as well as the surface coverage parameters are connected with a smooth transition

to the constant terrain height and surface parameters defined at the lateral boundaries

of the domain. It does not only facilitate the setup, but improves the numerical stability

and convergence of the calculations [70, 71, 75]. According to facilitate the preparation

of domain geometry, recommendations are given for the extended size of the compu-

tational domain and for the selection of map projection with respect to the location

and size of the target area, furthermore the features and properties of the relief. The

recommended size of the extension is corroborated by a series of numerical simulation,

while the recommendations for the applicable map projections are based on practical

considerations. Mesh generation methods are developed for the efficient and automatic

meshing for domains with complex terrain geometry, and with separated regions with

different surface coverage, producing task specific meshes. The terrain relaxation is built

in this mesh generation framework [70, 71, 75]. The methodology for the preparation

and execution of atmospheric simulations is developed and applied successfully in dif-

ferent case studies (see e.g. [1, 2, 34, 71]). This methodology serve as a guideline for

solving atmospheric problems in the engineering practice.

5.2 Outlook

In general the presented results of the ABL k-ε turbulence model indicate satisfactory

predictions for velocity, with hit rate values around 80%, whereas the performances of

the model for turbulent kinetic energy appears inadequate, being the typical obtained


hit rate values for k are around 60%. This trend is observed for all the simulated

cases, codes and modification of k-ε leads, which indicate an intrinsic limitation of linear

k-ε. Therefore, the implementation of non-linear k-ε turbulence model is should be

investigated, as it could have the potential of better performing in the recirculation zones

of the flow, as it is reported by different authors such as Papageorgakis and Assanis [53]

and Lun et al. [54].

Although the developments on the turbulence model and boundary conditions for neu-

tral ABL allows us to obtain more and more reliable results using general purpose CFD

solvers, the numerical simulations of phenomena connected to the atmospheric stratifi-

cation is not fully resolved. The non-hydrostatic approach published by Kristof et al.

[93] implies further developments on the turbulence model and boundary conditions in

order to enables the comprehensive approach to simulate stably and unstably stratified

atmospheric problems as well.

The results of the study focusing on flows in the urban canopy layer are useful for the

further development in modeling stratified canopy layers, including the effects of the

heat island phenomena and thermal convection. The realization of these requires the

adaptation of the surface parametrization schemes modeling heat and mass transfer and

storage in the urban canopy layer (see e.g. the work of Balogh [77] and Vu et al. [78])

with higher resolution.

Chapter 6

Thesis points

The new scientific results of the present research composed in thesis points are listed in

the following.

1. Thesis: Boundary conditions for atmospheric simulations

A novel wall function parametrization is derived for rough walls can be used in at-

mospheric simulations. A former approach is extended with a realizable set of inlet

conditions, including a novel turbulent kinetic energy profile. The source terms required

for the simulation of homogeneous neutral ABL is derived. The turbulence model, which

applies the extended approach and called HBL k-ε is implemented in the OpenFOAM

general purpose CFD solver and validated against measurements, together with the

formerly suggested approaches for comparison [1, 2].

a) In order to overcome the problems originated from the parametrization of rough

wall functions, alternative formulation is derived for the calculation of the rough-

ness constant and the sand grain roughness. This approach has the advantage that

its application does not require modifications on the wall function.

b) The consistent set of inlet boundary conditions and wall functions are implemented

in the OpenFOAM simulation system. The STD k-ε turbulence model in Open-

FOAM is modified according to make it valid within the homogeneous neutral

atmospheric boundary layer (HBL), namely to ensure the conformity between the

model and the boundary conditions [1, 2]. This former approach is extended above

the boundary layer thickness, with a novel inlet profile formulation including a four

parameter turbulent kinetic energy profile. The consequential source terms of the

extended approach are derived for ensuring the validity of the turbulence model

86

Chapter 6. Thesis points 87

within and above the boundary layer [26]. The extended turbulence model is called

as HBL k-ε.

c) The boundary conditions, wall functions and the implemented HBL k-ε model

are validated against theoretical profiles at full scale, and against measurements

at laboratory scale. Results indicate that the novel set of boundary conditions

reproduces the measured profiles for all examined cases, as indicated by correlation

coefficients above 0.8. This improvement is pronounced compared to the formerly

suggested boundary conditions. Moreover, the extended model ensures the stream-

wise homogeneity of the velocity and turbulence profiles, as it is shown by the

matching at the inlet and outlet sections of the domains, even above the boundary

layer thickness [26].

2. Thesis: Modeling atmospheric flows over complex terrain

The HBL k-ε model is generalized in order to simulate both homogeneous and inhomoge-

neous ABL flows over complex terrain. A novel sinusoidal blending function formulated

for the universal model called ABL k-ε. The ABL k-ε model are validated against

measurements. The recommended modification on the turbulence model for flows with

stagnation and separation, can be found in the literature, are implemented into the

generalized model, and the effectiveness of these are investigated on different cases.

a) The HBL k-ε turbulence model is generalized to achieve that in the homogeneous

regions of the boundary layer the formalism of HBL model is applied, and it is con-

tinuously blended to the STD model formalism in those inhomogeneous regions,

where the velocity field is strongly affected by the complex geometry. The gener-

alized, so-called ABL k-ε model applies smooth blending between the formalisms,

which based on the normalized velocity difference between its simulated and refer-

ence value. The rate of the transition is described by a sinusoidal power function.

The optimal value for the exhibitor of the power function N is determined in the

course of different flow simulations over complex terrain. Out of the examined

integer values, the best results were obtained with N = 3 [1, 2, 34].

b) The ABL k-ε model is validated against measurements at laboratory scale and full

scale, using the optimal blending exhibitor. The results are compared to the ones

obtained by the formerly suggested approaches. The calculated velocity agreed

well with the measurements at both laboratory and full scale, while reasonable

agreement is found for the turbulent kinetic energy. The present results is proved


oneself to be significantly better than the results obtained by using the former

approaches [1, 2, 34].

c) According to the recommendation can be found in the literature, the Kato-Launder

modification, the Yap correction, furthermore the MMK model are implemented to

enhance the performance of the ABL k-ε model in the simulations of flows, where

stagnation or separation appears. The effectiveness of these are investigated in

detail on 3D cases at both laboratory and full scale. The investigation shown

that the application of the Kato-Launder modification and the Yap correction are

not produce remarkable improvement, but the MMK model resulted a significant

improvement at full scale [1, 2].

3. Thesis: Scale adaptive modeling approach for urban flows

A novel scale adaptive, hybrid method is developed for the simulations of atmospheric

flows in urban canopy layers, implemented in a general purpose solver, validated against

measurements and verified on a practical engineering application.

a) The novel hybrid method is developed, which combines the advantages of the im-

plicit porous drag force approach, applied at larger scales, and the explicit modeling

of the geometry, resolving its features applied in CFD approaches. The substance

of this method is that the porous drag force approach is applied in the marginal

regions of the domain for describe the effects of the buildings and the vegetation,

while the geometrical features of the urban environment are considered via mesh

refinement in the target area, which is the most important in the aspect of the

analysis. A special meshing procedure is developed, which allows the separation of

the regions with different surface coverage, furthermore provide continuous tran-

sition between the coarser mesh applied in marginal regions and the finest applied

in the target area of the analysis [70, 71].

b) The porous drag force approach specialized for atmospheric flows, and imple-

mented in the ANSYS-FLUENT general purpose solver. The scale adaptive hybrid

method is validated against up-scaled wind tunnel measurements, and applied to-

gether with the clearly explicit and implicit approaches for comparison. The layer

averaged velocity results showed good agreement among the three different ap-

proaches within the canopy layer, while slight differences are observed above. The

velocity results of the hybrid and explicit method were matching in the target

area, however, the hybrid approach requires much less computational resources.

The novel method verified against theoretical profiles in a practical application,


which demonstrates the potential of the approach in field of real wind engineering

problems [70, 71].

4. Thesis: Methodology of atmospheric simulations

The methodology of atmospheric CFD simulations is developed, which focuses on those

problem specific tasks, whose are different from the general tasks in the engineering

practice. These tasks are the geometrical realization of the complex terrain, the defini-

tion of the lateral boundary conditions and surface parameters, furthermore the mesh

generation process and turbulence modeling.

a) For the simplified setup of the lateral boundary conditions, the usage of a do-

main extension with the so-called relaxation zone is proposed. In this zone, which

frames the target area, the relief as well as the surface coverage parameters are

connected with smooth transition to the constant terrain height and surface pa-

rameters defined at the lateral boundaries of the domain. It does not only facilitate

the setup, but improves the numerical stability and convergence of the calculations.

[70, 71, 75].

b) Recommendations are given for the extended size of the computational domain and

for the selection of map projection with respect to the location and size of the target

area, furthermore the features and properties of the relief. The recommended

size of the extension is corroborated by a series of numerical simulation, while

the recommendations for the applicable map projections are based on practical

considerations.

c) Mesh generation methods are developed for the efficient and automatic meshing of

domains with complex terrain geometry, and with separated regions with different

surface coverage. The terrain smoothing and the formation of the relaxation zone

is built in this mesh generation framework [70, 71, 75].

d) The methodology for the preparation and execution of atmospheric simulations is

developed and applied successfully in different case studies (see e.g. [1, 2, 34, 71]),

using novel approaches for turbulence modeling. This methodology serves as a

guideline for solving atmospheric problems in the engineering practice.

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Appendices

A. Derivation of source terms for simple cases

In this section the source terms are derived for the cases, where Cµ is in the proper

range, therefore it is consistent with the model. Step 1 (from 1.18 and 1.19):

− Sk =∂

∂z

(νt∂k

∂z

)(1)

− Sε =1

σε

∂

∂z

(νt∂ε

∂z

)+ (C1ε − C2ε)

ε2

k(2)

Step 2 (variables):

Cµ =u4τk2

(3)

νt = Cµk2

ε= uτκ (z + z0) (4)

ε =√Cµ

kuτκ (z + z0)

=u3τ

κ (z + z0)(5)

Step 3 (derivatives): The first and second order derivative of this function with respect

to the height can be given analytically in Eq. 6 and 7.

∂k

∂z=

A

z + z0+

2B (z + z0)

z20+C

z0(6)

∂2k

∂z2= − A

(z + z0)2 +

2B

z20(7)

∂νt∂z

= uτκ (8)

∂ε

∂z= − u3τ

κ (z + z0)2 (9)

99

Appendix B. Source terms for limited Cµ 100

Step 4 (terms one by one):

νt∂k

∂z= uτκ

(A+ 2B

(z + z0z0

)2

+ Cz + z0z0

)(10)

∂

∂z

(νt∂k

∂z

)=uτκ

z0

(4B

z + z0z0

+ C

)(11)

νt∂ε

∂z= − u4τ

(z + z0)(12)

∂

∂z

(νt∂ε

∂z

)=

u4τ

(z + z0)2 (13)

ε2

k=

u6τ

kκ2 (z + z0)2 =

√Cµu

4τ

κ2 (z + z0)2 (14)

Step 5 (source terms):

Sk = − ∂

∂z

(νt∂k

∂z

)= −uτκ

z0

(4B

z + z0z0

+ C

)(15)

Sε = − u4τ

(z + z0)2

(1

σε+

√Cµ (C1ε − C2ε)

κ2

)(16)

B. Derivation of source terms for limited Cµ

In this chapter the source terms are derived for that cases, where Cµ is out of the proper

range, therefore it is limited. Step 1 (expansion of source terms using the chain rule)

from 1.18 and 1.19:

− Sk =∂νt∂z

∂k

∂z+ νt

∂2k

∂z2(17)

− Sε =1

σε

∂νt∂z

∂ε

∂z+νtσε

∂2ε

∂z2+ (C1ε − C2ε)

ε2

k(18)

Step 2 (νt):

νt = Cµk2

ε=√Cµk

κ (z + z0)

uτ(19)

∂νt∂z

=1

2

k2

ε

∂Cµ∂z

+κ√Cµ

uτ

((z + z0)

∂k

∂z+ k

)(20)

Step 3 (ε):

ε =√Cµ

kuτκ (z + z0)

(21)

∂ε

∂z=

ε

2Cµ

∂Cµ∂z

+ε

k

∂k

∂z− ε

(z + z0)(22)

Appendix C. Smooth function for limited Cµ 101

∂2ε

∂z2=

1

ε

(∂ε

∂z

)2

+ ε∂fε (z)

∂z(23)

∂fε (z)

∂z=

1

2

[∂2Cµ∂z2

− 1

Cµ

(∂Cµ∂z

)2]

+1

k

[∂2k

∂z2− ε

k

(∂k

∂z

)2]

+1

(z + z0)(24)

Step 4 (substitution):

Sk = −∂νt∂z

∂k

∂z− νt

∂2k

∂z2(25)

Sε =1

σε

∂νt∂z

∂ε

∂z+νtσε

∂2ε

∂z2− (C1ε − C2ε)

ε2

k(26)

C. Smooth function for limited Cµ

The smooth transition of Cµ function to its upper limit is settled by using a cubic Bezier

curve. This is a commonly used parametric function in cumputer aided design (CAD)

to model smooth curves. In our case a cubic Bezier curve is used. Important property

of the cubic Bezier curve is that the function continous as well as its first and second

derivatives, furthermore the curve defined by only four control points. Generally the

Bezier curve (n = 3) is defined with its basis functions (Bernstein polinomials) Ji, and

with its control points Bi for 0 ≤ t ≤ 1 parameter:

P (t) =

n∑i=0

BiJn,i(t), Jn,i =n!

i!(n− i)!ti(1− t)n−i, (27)

where the cubic basis functions are defined as

J3,0 = (1− t)3, J3,1 = 3t(1− t)2, J3,2 = 3t2(1− t), J3,3 = t3. (28)

Generally, the derivatives of the curve are calculated with the derivatives of the basis

functions:

P ′(t) =

n∑i=0

BiJ′n,i(t), P

′′(t) =

n∑i=0

BiJ′′n,i(t) (29)

where J ′n,i(t) and J ′′n,i(t) are given analitically as

J ′3,0 = −3(1− t)2, J ′3,1 = 3(1− t)2 − 6t(1− t), J ′3,2 = 6t(1− t)− 3t2, J ′3,3 = 3t2, (30)

J ′′3,0 = 6(1− t), J ′′3,1 = 6t− 12(1− t), J ′′3,2 = 6(1− t)− 12t, J ′′3,3 = 6t. (31)


In order to determine the control points, the endpoint values and derivatives can be

used asP (0) = B0, P (1) = B3,

P ′(0) = 3(B1 −B0), P′(1) = 3(B3 −B2),

P ′′(0) = 6(B0 − 2B1 + B2), P′′(1) = 6(B3 − 2B2 + B1).

(32)

whereP (0) = [z(0), Cµ(0)]T , P (1) = [z(1), Cµ(1)]T ,

P ′(0) = [z(0), Cµ(0)′]T , P ′(1) = [z(1), Cµ(1)′]T ,

P ′′(0) = [z(0), Cµ(0)′′]T , P ′′(1) = [z(1), Cµ(1)′′]T .

(33)

In order to simplify the determination of t parameter based on the known z coordinate,

uniform ∆z step size can be choosen for the control points, thus

∆z =z(1)− z(0)

3, zi = z(0) + i(∆z), t =

z − z(0)

∆z. (34)

For avoiding overshooting of the limited value, the optimal z(0) coordinate should be

found under the conditions:

C1 : P ′(0)y = 3(y1 − y0) = (x3 − x0)[∂y

∂x

]x0

, (35)

C2 : P ′′(1)y = 6(y3 − 2y2 + y1) = 0, (36)

C3 : P ′(1)y = 3(y3 − y2) = 0. (37)

where x denotes z and y denotes Cµ. Using these conditions, one can express y1 from

C1, and y2 from C3:

y1 = y0 +x3 − x0

3

[∂y

∂x

]x0

, (38)

y2 = y3. (39)

Substituting to C2:

y0 +x3 − x0

3

[∂y

∂x

]x0

− y3 = 0. (40)

Rearranging to x0:

x0 = x3 − 3(y3 − y0)[∂y

∂x

]−1x0

. (41)

While both

[∂y

∂x

]x0

and y0 are functions of x0, this problem turns to be a non-linear

one, therefore the Newton-Raphson iterative root finding method is applied as

f(x) = x3 − x− 3(y3 − y)

(∂y

∂x

)−1(42)


f ′(x) = 2− 3(y3 − y)∂2y

∂x2

(∂y

∂x

)−2(43)

xi+1 = xi− f(x)

f ′(x)(44)

Considering that x = z and y = Cµ, the formulation turns:

f(z) = zt − z − 3(Cµt − Cµ)

(∂Cµ∂z

)−1, (45)

f ′(z) = 2− 3(Cµt − Cµ)∂2Cµ∂z2

(∂Cµ∂z

)−2, (46)

zi+1 = zi − f(z)

f ′(z). (47)

where Cµt denotes the limit and zt = min (δ, zmax), futhermore

Cµ =u4τk2, (48)

∂Cµ∂z

= −2u4τ1

k3∂k

∂z= −2Cµ

1

k

∂k

∂z, (49)

∂2Cµ∂z2

= −2u4τ

(− 3

k4

(∂k

∂z

)2

+1

k3∂2k

∂z2

)= 2Cµ

(3

k2

(∂k

∂z

)2

− 1

k

∂2k

∂z2

). (50)

Figure C.1 shows simply limited functions of Cµ for the ERCOFTAC 69 case.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.04 0.05 0.06 0.07 0.08 0.09 0.1

z [m

]

Cµ [-]

Cµ

Simple limitedSmooth limited

Figure C.1: Simple and Cubic Bezier limiters for ERCOFTAC 69 case, with an upperlimit of 0.09

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