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WIND EFFECT ON SUPER-TALL BUILDINGS
USING COMPUTATIONAL FLUID DYNAMICS AND STRUCTURAL DYNAMICS
by
Bilal Assaad
A Thesis Submitted to the Faculty of
The College of Engineering and Computer Science
In Partial Fulfillment of the Requirements for the Degree of
Master of Science
Florida Atlantic University
Boca Raton, Florida
May 2015
Copyright 2015 by Bilal Assaad
ii
ACKNOWLEDGEMENTS
The author wishes to express sincere gratitude to his committee members for all of their guidance
and support in the development of this manuscript. Special thanks to my thesis advisor, Dr. M.
Arockiasamy, Professor and Director, Center for Infrastructure and Constructed Facilities, Department of
Civil, Environmental and Geomatics Engineering. Dr. M. Arockiasamy provided the author with incessant
support, persistence, patience and encouragement during the evolution of this thesis. The author is grateful
to Dr. Yan Yong, and Dr. Panagiotis Scarlatos for serving as fair evaluators of the work presented herein.
Thankfulness must also be extended to Dr. Chaouki Ghenai for the enlightenment provided to the author
with his precise and comprehensive expertise in the subject matter of Computational Fluid Dynamics. Last
but not least, Mr. Sijal Ahmed must be acknowledged as a brilliant tutor in the manipulation of the
advanced software package utilized for the analysis of the work presented in this publication.
iv
ABSTRACT
Author:
Bilal Assaad
Title:
Wind Effect on Super-Tall Buildings Using Computational Fluid Dynamics and
Structural Dynamics
Institution:
Florida Atlantic University
Thesis Advisor:
Dr. Madasamy Arockiasamy
Degree:
Master of Science
Year:
2015
Super-tall buildings located in high velocity wind regions are highly vulnerable to large lateral
loads. Designing for these structures must be done with great engineering judgment by structural
professionals. Present methods of evaluating these loads are typically by the use of American Society of
Civil Engineers 7-10 standard, field measurements or scaled wind tunnel models. With the rise of high
performance computing nodes, an emerging method based on the numerical approach of Computational
Fluid Dynamics has created an additional layer of analysis and loading prediction alternative to
conventional methods. The present document uses turbulence modeling and numerical algorithms by means
of Reynolds-averaged Navier-Stokes and Large Eddy Simulation equations applied to a square prismatic
prototype structure in which its dynamic properties have also been investigated. With proper modeling of
the atmospheric boundary layer flow, these numerical techniques reveal important aerodynamic properties
and enhance flow visualization to structural engineers in a virtual environment.
v
DEDICATION
This manuscript is in part and as whole dedicated to my humble and unconditionally loving
family. Their relentless support during this journey has provided me with the courage to develop,
overcome, and realize this level of intellectuality. I would like to particularly dedicate this thesis to my
Father and Mother for nurturing a family with genuine love, regardless of the circumstances and hardships
they have encountered and surpassed in their destined lives. I will forever be indebted to their kindness,
exemplary courage, and love. This document is also dedicated to the people that have embraced me as a
friend, brother and/or son and provided me with the opportunity to reach this accomplishment. Lastly, the
work presented herein is dedicated to my loving partner that kindly brightens my heart, mind, and soul.
vi
WIND EFFECT ON SUPER-TALL BUILDINGS
USING COMPUTATIONAL FLUID DYNAMICS AND STRUCTURAL DYNAMICS
LIST OF TABLES .......................................................................................................................................... x
LIST OF FIGURES ........................................................................................................................................xi
CHAPTER 1: INTRODUCTION.................................................................................................................... 1
1.1.
Wind Engineering Tools ................................................................................................................... 1
1.2.
Study Objectives ............................................................................................................................... 4
CHAPTER 2: LITERATURE REVIEW......................................................................................................... 5
2.1.
Overview ........................................................................................................................................... 5
2.2.
Wind Effect on Structure .................................................................................................................. 5
2.3.
Wind Damaged Structures ................................................................................................................ 6
2.4.
Computational Wind Engineering..................................................................................................... 9
2.4.1. Microscale and CFD ................................................................................................................ 11
2.4.2. Reduced-scale Wind Tunnel Testing and CWE ...................................................................... 12
2.5.
Wind Tunnel Measurements ........................................................................................................... 12
2.6.
Overview of Tall Buildings............................................................................................................. 14
2.6.1. Factors Affecting Growth, Height, and Structural Form ......................................................... 14
2.6.2. Criteria for the Definition of Tall Buildings ............................................................................ 16
2.7.
Wind Loading on Tall Buildings..................................................................................................... 18
CHAPTER 3: ELEMENTARY PRINCIPLES OF WIND ENGINEERING................................................ 19
3.1.
Forces Acting in the Free Atmosphere............................................................................................ 19
3.1.1. Pressure Gradient and the Coriolis Force ................................................................................ 19
3.2.
Geostrophic Wind, Gradient Wind, and Frictional Effects ............................................................. 20
3.3.
Atmospheric Boundary Layer ......................................................................................................... 21
3.3.1. Mean Wind Profiles................................................................................................................. 21
3.3.1.1. The Logarithmic Law ..................................................................................................... 22
3.3.1.2. The Power Law ............................................................................................................... 23
3.3.2. Turbulence ............................................................................................................................... 24
3.3.2.1. Turbulence Intensities ..................................................................................................... 25
3.3.2.2. Integral Turbulent Length Scale ..................................................................................... 27
vii
CHAPTER 4: THEORY OF COMPUTATIONAL FLUID DYNAMICS ................................................... 29
4.1.
Overview ......................................................................................................................................... 29
4.2.
Computational Fluid Dynamics ...................................................................................................... 30
4.3.
Governing Equations of Fluid Dynamics ........................................................................................ 31
4.3.1. Infinitesimal Fluid Element ..................................................................................................... 31
4.3.2. The Continuity Equation ......................................................................................................... 31
4.3.3. The Momentum Equation ........................................................................................................ 34
4.3.4. The Energy Equation ............................................................................................................... 35
4.4.
Turbulence and Modeling for CFD ................................................................................................. 36
4.4.1. Turbulent Scales ...................................................................................................................... 38
4.4.2. Energy Spectrum ..................................................................................................................... 38
4.4.3. Transition from Laminar to Turbulent Flow............................................................................ 40
4.4.4. Turbulent Flow Calculations ................................................................................................... 40
4.4.5. RANS Equations and Classical Models .................................................................................. 41
4.4.5.1. κ-ε Model ........................................................................................................................ 43
4.4.5.2. κ-ω Model ....................................................................................................................... 43
4.4.5.3. SST κ-ω Model ............................................................................................................... 44
4.4.6. Large Eddy Simulation ............................................................................................................ 45
4.4.6.1. Smagorinsky-Lilly Model ............................................................................................... 47
4.4.6.2. Remarks on LES ............................................................................................................. 47
4.4.7. Typical Meshing Information .................................................................................................. 48
4.4.8. Near Wall Treatment ............................................................................................................... 50
4.4.9. Discretization Techniques and Algorithms.............................................................................. 51
CHAPTER 5: THEORY OF STRUCTURAL DYNAMICS ........................................................................ 54
5.1.
Basic Concepts of Vibration ........................................................................................................... 54
5.2.
Degrees of Freedom and Classification of Vibration ...................................................................... 55
5.3.
Equation of Motion and Natural Frequency .................................................................................... 56
5.4.
Property of Matrices for Vibrating Systems ................................................................................... 58
5.4.1. Flexibility and Stiffness Matrix ............................................................................................... 58
5.4.2. Mass Matrix ............................................................................................................................. 59
5.5.
Eigenproblem Formulation, Natural Vibration Frequencies and Modes......................................... 60
CHAPTER 6: CFD ANALYSIS OF WIND FLOW USING RANS & LES FORMULATIONS ................ 62
6.1.
2D & 3D Setup of Model in CFD ................................................................................................... 62
6.2.
Grid Independent Study .................................................................................................................. 64
6.3.
Model Validation ............................................................................................................................ 68
6.4.
Discussion of Results Based on 2D Plan Model ............................................................................. 71
viii
6.5.
3D LES Model ................................................................................................................................ 73
6.6.
Pressures Acting on Slender Structure ............................................................................................ 75
CHAPTER 7: BASIC STRUCTURAL AND DYNAMIC ANALYSIS OF TALL BUILDING ................. 77
7.1.
Overview of Idealized Example Structure ...................................................................................... 77
7.2.
Assigned Loading and Section Properties to Structural Members .................................................. 78
7.3.
Idealized Structure Discretization ................................................................................................... 81
7.4.
Vibration Modes of Modeled Structure .......................................................................................... 82
7.5.
Deflections of Super-Tall Structure ................................................................................................ 85
CHAPTER 8: CONCLUSIONS & FUTURE RESEARCH.......................................................................... 88
8.1.
Final Remarks ................................................................................................................................. 88
8.2.
Recommendations for Future Research .......................................................................................... 89
APPENDICES............................................................................................................................................... 91
Appendix A................................................................................................................................................ 92
Appendix B .............................................................................................................................................. 102
Appendix C .............................................................................................................................................. 109
Appendix D.............................................................................................................................................. 111
REFERENCES ............................................................................................................................................ 120
ix
LIST OF TABLES
Table 1: Roughness lengths and Surface drag coefficients per ASCE 7-10 .................................................. 23
Table 2: Power law exponents and gradient height per ASCE 7-10 Standard .............................................. 23
Table 3: Experimental data and derived quantities for various cross sections .............................................. 69
Table 4: Model input criteria for 2D plan CFD ............................................................................................. 72
Table 5: Assumed loads acting on structure .................................................................................................. 79
Table 6: Cross-section and properties of members along the height of the structure .................................... 80
x
LIST OF FIGURES
Figure 1: Typical wind tunnel setup of a model high rise building ................................................................. 3
Figure 2: (a) Moment of collapse of Cooling Tower 2A, U.K. (b) Collapse of midsection of Tacoma
Narrows Bridge, WA ........................................................................................................................... 7
Figure 3: (a) Shiten'noji Pagoda collapse in 1937 and (b) Present day rebuilt structure ................................. 8
Figure 4: (a) One Indiana Square tornado induced damage (b) Remodeled façade of building ...................... 9
Figure 5: (a) Woolworth Building, NY (b) Empire State Building, NY ....................................................... 15
Figure 6: Comparison of tall, supertall and megatall building height criteria ............................................... 17
Figure 7: World's ten tallest buildings according to 'height to architectural top' as of November 2014 ....... 18
Figure 8: Coriolis force results in wind being deflected owing to the rotation of the Earth .......................... 20
Figure 9: Wind profile in different boundary layers ...................................................................................... 22
Figure 10: Comparison of the logarithmic and power law for mean velocity profile for z0 = 0.02 m
and α = 0.128 ..................................................................................................................................... 24
Figure 11: Typical point velocity measurements in turbulent flow ............................................................... 25
Figure 12: Longitudinal turbulence intensity for rural terrain (z0 = 0.04 m) and suburban terrain (z0 =
0.15 m) ............................................................................................................................................... 26
Figure 13: Variation of turbulent length scale as height increases per AIJ code ........................................... 28
Figure 14: Three dimensions of fluid dynamics ............................................................................................ 29
Figure 15: Infinitesimal fluid element model of flow ................................................................................... 31
Figure 16: Model of infinitesimally small element fixed in space including mass flux diagram .................. 32
Figure 17: Model used for the derivation of the x-component of momentum equation ................................ 34
Figure 18: von Kármán vortices representing turbulent flow characteristics forming in clouds
flowing past a volcano ....................................................................................................................... 37
Figure 19: Representation of cascade process with a spectrum eddies ......................................................... 38
Figure 20: Spectrum for turbulent kinetic energy, κ...................................................................................... 39
Figure 21: Transition of flow in a pipe from laminar to turbulent................................................................. 40
Figure 22: Increase of computational cost per type of turbulence modeling ................................................. 41
Figure 23: Spectrum of velocity .................................................................................................................... 46
Figure 24: Typical 2D and 3D Mesh Shapes................................................................................................. 49
Figure 25: Typical geometry meshing process .............................................................................................. 49
Figure 26: Log-Law of the Wall .................................................................................................................... 50
Figure 27: Discretization of grid points......................................................................................................... 51
Figure 28: Idealized multi-story building with five degrees of freedom ....................................................... 56
xi
Figure 29: Model of a simple SDOF and free-body diagram of system ........................................................ 57
Figure 30: Flexibility coefficients for a beam ............................................................................................... 59
Figure 31: Lumped mass coefficients for structural elements with distributed mass .................................... 59
Figure 32: 3D Boundary conditions .............................................................................................................. 63
Figure 33: 2D Plan boundary conditions ....................................................................................................... 63
Figure 34: 2D Elevation boundary conditions ............................................................................................... 64
Figure 35: 157 K Nodes quad-mesh for 2D plan view of structure ............................................................... 65
Figure 36: Quad-mesh for 2D elevation of structure ..................................................................................... 66
Figure 37: Boundary layer development along wind tunnel section ............................................................. 67
Figure 38: Variation of mean pressure coefficient along the windward face for grid analysis ..................... 68
Figure 39: Drag coefficient curve for 44K SST-κω transient model ............................................................. 69
Figure 40: Lift coefficient curve for 44K SST-κω transient model ............................................................... 70
Figure 41: Mean velocity magnitude along centerline .................................................................................. 71
Figure 42: Variation of mean pressure coefficient along faces ..................................................................... 72
Figure 43: 3D mesh with 7.7 M nodes .......................................................................................................... 73
Figure 44: Wind profile generated in computer domain................................................................................ 74
Figure 45: Present LES windward contours compared to experimental data ................................................ 75
Figure 46: Windward estimated pressure variation ....................................................................................... 76
Figure 47: Framing model of 67-story tower used in present study .............................................................. 78
Figure 48: First three modes of vibration of tall structure ............................................................................. 83
Figure 49: Fourth to sixth modes of vibration of tall structure...................................................................... 84
Figure 50: Deflections experienced by the loading acting on super-tall structure......................................... 86
xii
NOMENCLATURE
Atmospheric pressure
x
p
Stiffness matrix
k
Boundary layer value of first grid pt.
y
Surface shear stress
Csd
T
Acceleration
Standard Deviation
Coef. dependent on terrain roughness
Temperature
Constant Velocity
U
Terrain exposure constant
Time
t
Coriolis parameter
f
Density
Density of air
a
Displacement
Distance
Dynamic viscosity
Elevation or height
Equivalent height
Flexibility matrix
Force
Frequency
Friction velocity
Integral length scale in i -th direction
Kinematic velocity
Kinematic viscosity
Kinetic energy dissipation
Latitude
Local grid scale
Mass matrix
Mean wind speed
Mixing length for subgrid scales
Total internal energy
Turbulence frequency
Turbulence Intensity in i -th
direction
Turbulent kinetic energy
x
d
z
Velocity
Viscous damping
von Kármán’s constant
z
a
F
S ij
Roughness coefficient of terrain z0
Stiffness matrix
k
k
Surface shear stress
Csd
Temperature
T
Shear stress
v
Spring constant or stiffness
m
Standard Deviation
V
Ls
Terrain exposure constant
Time
t
Total internal energy
Natural circular frequency
Radius
r
Rate-of-strain tensor
S ij
Velocity
Roughness coefficient of terrain
z0
Spring constant or stiffness
k
Rate-of-strain tensor
Turbulence frequency
Turbulence Intensity in i -th
direction
Turbulent kinetic energy
Shear stress
x
c
Radius
0
Modal matrix
TIi
0
r
Natural circular frequency
f
u*
Li
e
k
xiii
e
TIi
Viscous damping
x
c
von Kármán’s constant
k
CHAPTER 1
INTRODUCTION
The development of new construction techniques in the 20th century has created structures that are
flexible, low in damping, and relatively light in weight which therefore exposes the structure to the effect of
wind acting upon it. Wind engineering has been the field with the aim of primarily developing tools to
better understand the action of the fluid on the structure with origins that could be traced back to the 1960s.
The development of knowledge found in the present literature regarding this subject has lead structural
engineers to design and ensure the performance of the structure subjected to the action of wind to be within
adequate limits during the lifetime of the structure in structural safety and serviceability criteria (Simiu and
Scanlan 1978).
1.1.
Wind Engineering Tools
Computational Wind Engineering (CWE) is primarily defined as the use of Computational Fluid
Dynamics (CFD) for wind engineering applications, although it also includes other approaches of computer
modelling and in the broadest sense also field and wind tunnel measurements supporting CWE model and
development. Wind engineering itself is best described as “the rational treatment of interactions between
wind in the atmospheric boundary layer and man and his works on the surface of Earth.” (Cermak 1975).
In spite of these difficulties, in the past decades and driven by the pioneering studies, CWE has
undergone a successful transition from an emerging field into an increasingly established field in wind
engineering research, practice, and education. This transition and the success of CWE are illustrated by: (1)
the establishment of CWE as an individual research and application area in wind engineering with its own
successful conference series, (2) the increasingly wide range of topics covered in CWE, ranging from
pedestrian-level wind conditions over natural ventilation of buildings and wind loads on buildings and
bridges to sports aerodynamics, and (3) the history of review and overview papers in CWE.
CWE/CFD has some particular advantages over experimental (full scale or reduced scale) testing.
It can provide detailed information on the relevant flow variables in the whole calculation domain, under
1
well-controlled conditions and without similarity constraints. The accuracy and reliability of CFD
simulations are of concern and solution verification and validation studies are essential. Therefore,
experiments remain indispensable for CWE.
As already previously mentioned, CWE is complementary to other, more traditional areas of wind
engineering, such as full scale on-site experimentation and reduced scale wind tunnel testing. Each
approach has its specific advantages and disadvantages. CFD has some particular advantages over
experimental testing (full scale or reduced scale), especially the fact that it provides detailed information on
the relevant flow variables in the whole calculation domain (whole flow field data) under well controlled
conditions and without similarity constraints.
The advancement of high-speed digital computer combined with the accurate numerical
algorithms for solving physical problems on these computers revolutionized the way we study and practice
fluid dynamics today. It has introduced a fundamentally important new third approach in fluid dynamics—
the approach of computational fluid dynamics.
Computational fluid dynamics results are analogous to wind tunnel results obtained in a laboratory
in which they represent sets of data for given flow configurations at different Reynolds numbers. In other
words, a computer program with CFD code imbedded into it acts like a “virtual wind tunnel.” Having
access to the program allows the user to carry out some interesting experiments with it. Experiments you
perform with the computer program are numerical experiments instead of physical ones.
Tallness is a relative matter, and tall buildings cannot be defined in specific terms related just to
height or to the number of floors. The tallness of a building is a matter of a person’s or community’s
circumstance and their consequent perception; therefore a measurable definition of a tall building cannot be
universally applied (Stafford Smith and Coull 1991). From the structural engineer’s point of view, a tall
building may be defined as one that because of its height, it is affected by lateral forces due to wind and
earthquake actions to an extent that they play a critical role in the structural design (Stafford Smith and
Coull 1991).
Tall buildings are subject to resonant wind loads. The primary source of the along-wind motion is
the pressure fluctuations in the windward and leeward faces, which are affected by the nature of the
turbulent wind and the interaction with the structure itself. The across-wind motion is caused by the
2
fluctuating separation in the shear layers of the fluid. Torsional motion is caused by the imbalance of
pressure distribution on the faces of the building which could be due to the varying angles of the wind
direction, interference effect of neighboring buildings, or due to varying building geometry and eccentricity
of the center of mass. Kareem (1985) has performed an experimental study in which it was concluded that
tall buildings are subject to greater magnitudes of the across-wind and torsional response when compared to
the along-wind response in terms of serviceability and limit state requirements.
The present engineer uses design codes and procedures to properly account for wind loading on
structures. The American Society of Civil Engineers Standard ASCE 7-10 gives provisions for the design
of Main Wind Force Resisting Systems (MWFRS) for building with common geometric shapes in different
types of exposures. Many present structures with complex geometries do not fall in the categories specified
by the code, and therefore it refers the engineer to implement the use of physical model and testing of the
project in boundary layer wind tunnel facilities. Model scale testing is the most common and widely used
engineering tool. Wind tunnel testing can predict wind-induced effects on structures that address some of
the difficulties in specifying the effect in codes.
Figure 1: Typical wind tunnel setup of a model high rise building
(Credit: RWDI Inc., Canada)
The concerns pertaining the limited provisions by standards and codes, the cost of physical testing
and scarce wind tunnel facilities, have encouraged researchers to adopt hardware and software technology
to investigate the potential of numerical modelling using the theory of computational fluid dynamics. This
provides and additional tool to the engineer and is also an alternative to common practice. The use of these
numerical methods is easily accessible and is expected to gain popularity in the practice of structural design
3
for wind loading on structures. This will result in more resilient and more sustainable systems by allowing
the engineer to adapt aerodynamic and a smart structure design.
The most significant parameter concerning the extent of the dynamic effect a load causes on a
structure is the natural period of vibration (Tedesco et al. 1999). The natural period of vibration simply is
the time required for the structure to go through one complete cycle of vibration (i.e. units of seconds). The
natural frequency is defined as the number of cycles the structure undergoes per second (i.e. units of cycles
per second or Hertz). If the load acting on a structure acts on a sufficiently large time period, the load can
be considered to be static. However, if the load acting on a structure is close to the natural period of the
structure, it will induce a dynamic response. It must be noted that the stresses, strains, and deflections are
generally more severe when loads of given amplitude acts dynamically. Wind loads are dynamic in nature
and they must be accounted for as essential loading criteria to ensure proper design of any given structure.
In short, wind loads are a function of velocity, height of structure, and shape and stiffness characteristics of
the structure.
1.2.
Study Objectives
The main objective of this research was to investigate the strategies that can be adapted in the
computational evaluation and assessment of wind loads on tall buildings under turbulent wind flows. The
present study explored the numeric of the aerodynamics of a square slender building. The investigation was
carried using two-dimensional and three-dimensional study of the structure using CFD. Moreover, a basic
structural dynamic analysis was carried on a prototype building designed by the author. The building
analysis was deemed satisfactory based on industry deflection recommendations.
4
CHAPTER 2
LITERATURE REVIEW
2.1.
Overview
Wind is composed of a multitude of eddies of varying sizes and rotational characteristics carried
along in a general stream of air moving relative to the earth’s surface. These eddies give wind its gusty or
turbulent character. Wind interaction with surface features gives rise to the gustiness of strong winds in the
lower levels of the atmosphere. The average wind speed over a time period of the order of ten minutes or
more, tends to increase with height, while the gustiness tends to decrease with height. There are several
different phenomena giving rise to dynamic response of structures in wind. These include buffeting, vortex
shedding, galloping and flutter. Slender structures can be sensitive to dynamic response in line with the
wind direction as a consequence of turbulence buffeting. Transverse or cross-wind response is more likely
to arise from vortex shedding or galloping but may also result from excitation by turbulence buffeting.
Flutter is a coupled motion, often being a combination of bending and torsion, and can result in instability.
For building structures flutter and galloping are generally not an issue. An important problem associated
with wind induced motion of buildings is concerned with human response to vibration and perception of
motion. Humans are very sensitive to vibration to the extent that motions may feel uncomfortable even if
they correspond to relatively low levels of stress and strain. Therefore, for most tall buildings serviceability
considerations govern the design and not strength issues (Mendis et al. 2007).
2.2.
Wind Effect on Structure
The greatest probability of damage to structures has been presented by Davenport (1963) to be the
case of strong winds with neutral atmospheric conditions. Davenport suggests that structural response to
repeated loads of successive gusts is an important factor in the design of tall buildings. Repeated loading
may lead to fatigue, failure, foundation settling, excessive deflections causing cracking to building
elements, or induced motion that may affect the comfort of the occupants of the structure. A building can
be considered to have failed if it becomes unserviceable due to the action of repeated loads or the action of
5
a single large load of great magnitude. It is very important that the fluctuating loads caused by wind on a
structure play an important role in the design and analysis of tall buildings, especially structures with large
aspect ratios (Reinhold 1977).
The primary concern for a structural engineer when studying wind phenomena, around a building,
is the mean velocity profile of the wind. Moreover, two aspects of turbulent flows are of interest to the
engineer: (a) the state of turbulence of the natural wind approaching a structure, and (b) the local turbulence
provoked in the wind by the structure itself. Most structures in civil engineering present bluff forms, in
wind engineering studies we focus on the bluff-body aerodynamics aspects of the wind and structure
interaction. This has led the industry to further research on the details of flow effects around bluff bodies
such as tall buildings. This finally leads to the interest of the engineer in the study of the development of
body pressures by the flow acting around a structure (Simiu and Scanlan 1978).
2.3.
Wind Damaged Structures
Damage to buildings and other structures by windstorms has been a fact of life for human beings
from the time they moved out of cave dwellings to the present day. Trial and error has played an important
part in the development of construction techniques and roof shapes for small residential buildings, which
have usually suffered the most damage during severe winds. In past centuries, heavy masonry construction,
as used for important community buildings such as churches and temples, was seen, by intuition, as the
solution to resist wind forces. For other types of construction, windstorm damage was generally seen as an
‘act of god’, as it is still viewed today by many insurance companies.
The nineteenth century was important as it saw the introduction of steel and reinforced concrete as
construction materials, and the beginnings of stress analysis methods for the design of structures. The latter
was developed further in the twentieth century, especially in the second half, with the development of
computer methods. During the last two centuries, major structural failures due to wind action have occurred
periodically, and provoked much interest in wind forces by engineers. Long-span bridges often produced
the most spectacular of the failures, with the Brighton Chain Pier, England in 1836, the Tay Bridge,
Scotland in 1879, and Tacoma Narrows Bridge, Washington State, U.S.A. in 1940 being the most notable,
with the dynamic action of wind playing a major role (Holmes 2007).
6
Other large structures have experienced failures as well – for example, the collapse of the
Ferrybridge cooling towers in the United Kingdom in 1965, and the permanent deformation of the columns
of the Great Plains Life Building in Lubbock, Texas during a tornado in 1970. These events were notable,
not only as events in themselves, but also for the part they played as a stimulus to the development of
research into wind loading in their respective countries.
(b)
(a)
Figure 2: (a) Moment of collapse of Cooling Tower 2A, U.K. (b) Collapse of midsection of Tacoma Narrows
Bridge, WA
(Credit: Shellard 1965, and Bashford and Thompsons 1940)
Some major windstorms, which have caused large scale damage to residential buildings, as well as
some engineered structures, are also important for the part they played in promoting research and
understanding of wind loads on structures. The effects of Hurricane Andrew in Florida proved to be the
costliest natural disaster in the state’s history. Andrew made landfall near Homestead, Florida on August
24, 1992 as a Category 5 hurricane. Strong winds from the hurricane affected four southeastern counties of
the state in which it damaged or destroyed 730,000 houses and buildings. The hurricane caused about $25
billion in damage and 44 deaths.
The first ‘tall buildings’ to appear in Japan might be the traditional wooden pagodas which are
seen in historic Japanese cities such as Nara and Kyoto. Strong typhoons could cause damage to pagodas.
The 5-story, 47.8 m (157.8 ft.) high Shiten’noji Pagoda collapsed due to typhoon Muroto on September 21,
1934. The maximum peak gust speed was estimated to be more than 60 m/s (134.2 mph) and was
accompanied by a high tidal wave of more than 4 meters (13.1 ft.). Thus, the history of the development of
design and construction methods for tall buildings was a record of fights with strong winds. There are many
7
wind-related problems in construction of tall buildings, but the main problem for engineers is their
capability of resistance to wind forces, because higher altitudes mean higher wind speeds, and consequently
higher wind forces (Tamura 2009).
(a)
(b)
Figure 3: (a) Shiten'noji Pagoda collapse in 1937 and (b) Present day rebuilt structure
As well as damage to buildings produced by direct wind forces – either overloads caused by
overstressing under peak loads, or fatigue damage under fluctuating loads of a lower level, a major cause in
severe wind storms is flying debris. Penetration of the building envelope by flying ‘missiles’ has a number
of undesirable results: high internal pressures threatening the building structure, wind and rain penetration
of the inside of the building, the generation of additional flying debris, and the possibility of flying missiles
inside the building endangering the occupants. The area of a building most vulnerable to impact by missiles
is the windward wall region, although impacts can also occur on the roof and side walls. As the air
approaches the windward wall, its horizontal velocity reduces rapidly. Heavier objects in the flow with
higher inertia will probably continue with their velocity little changed until they impact on the wall. Lighter
and smaller objects may lose velocity in this region or even be swept around the building with the flow if
they are not directed at the stagnation point (Holmes 2007).
One Indiana Square is a 36-story (504 ft) tall building located in downtown Indianapolis, Indiana.
The building went exterior remodeling after damage by tornado-strength winds reaching speeds exceeding
130 km/h (80.7 mph) that occurred on April 2, 2006. This particular storm brought winds sufficient to
cause severe damage to the façade and structural elements of 16 out of 36 stories of the tower causing
millions of dollars in monetary loss and the closing of streets and businesses for several days. The nature of
8
the damage prompted debate about whether the damage was caused by tornado, downburst, or extreme
straight wind conditions. The recorded wind speed was very close to typical design wind speed for
buildings as recommended per ASCE 7-10 national standard. The recommended speeds are dependent
upon geographic locations in which the region of southeast Florida has the highest wind speed values. After
the 2006 damage, design of a new façade with curtain wall to be installed over the existing façade was
released in 2007 by the integrated design firm Gensler. The new façade after the re-cladding process
essentially put another layer of skin around the building’s exterior face by expanding it by 18 in. around its
perimeter (Yilmaz and Duffin 2014).
(a)
(b)
Figure 4: (a) One Indiana Square tornado induced damage (b) Remodeled façade of building
(Credit: Chriss Barrett via CTBUH)
2.4.
Computational Wind Engineering
The historical starting point of CWE could be situated around 1963 when Smagorinsky developed
one of the first successful approaches to Large Eddy Simulation (LES), the Smagorinsky-Lilly model,
which is still intensively used in many areas of fluid mechanics today. The main research area of
Smagorinsky was Numerical Weather Prediction applied at the meteorological macroscale. Of particular
importance for CWE were the pioneering studies by Meroney and his co-workers in which a hybrid
approach was pursued for the systematic comparison of numerical simulations with dedicated wind tunnel
measurements in atmospheric boundary layer wind tunnel (Meroney and Yamada 1971, Yamada and
Meroney 1972, Derickson and Meroney 1977). In Aerospace Engineering, the T3 group at the Los Alamos
9
National Laboratories in 1963 first used computers to model the 2D swirling flow around an object using
the vorticity stream function method, followed by the first 3D application by Hess and Smith (1967) using
the so called panel method. Driven by these early achievements, early efforts in CWE focused on the
determination and analysis of wind velocity and pressure field around buildings (Blocken 2014).
The difference in time between the earliest CFD developments in the 1950s and the later
application of CFD in CWE for wind velocity and pressure fields around buildings is attributed to the
specific difficulties associated with the flow around bluff bodies with sharp edges. Murakami (1998)
diligently outlined some of the difficulties encountered in CWE: (1) high Reynolds numbers in wind
engineering applications, necessitating high grid resolutions, especially near wall regions as well as
accurate wall functions, (2) the complex nature of the 3D flow field with impingement, separation and
vortex shedding, (3) the numerical difficulties associated with flow at sharp corners and consequences of
discretization schemes, and (4) the inflow and outflow boundary conditions which are particularly
challenging for LES. These difficulties were directly linked to limitations in physical modelling and in
computational requirements at those times, but many of those limitations are still to some extent present
today.
CWE is complementary to other, more traditional areas of wind engineering, such as full scale onsite experimentation and reduced scale wind tunnel testing. Each approach has its specific advantages and
disadvantages. The main advantage of on-site measurements is that they are able to capture the real
complexity of the problem under study. To name a few, important disadvantage are that they are not fully
controllable due to the inherently variable meteorological conditions, that they are not possible state in the
design stages of the building, and that usually only point measurements are performed. The latter
disadvantage also hold true for wind tunnel measurements. Techniques such as Particle Image Velocimetry
(PIV) and Laser-Induced Fluorescence (LIF) in principle allow planar or even full 3D data to be obtained in
wind tunnel tests, but the cost is considerably higher and application for complicated geometries can be
hampered by laser-light shielding by the obstructions constituting the model. Another disadvantage is the
required adherence to similarity criteria in reduced scale testing, which can limit the extent and the range of
problems that can be studied in wind tunnels.
10
In addition, it is widely recognized that the results of CFD simulations can be very sensitive to the
wide range of computational parameters that have to be set by the modeler. For typical simulations, the user
has to select target variables, the approximate form of the governing equations, the turbulence model, the
computational domain, the computational grid, the boundary conditions, the discretization schemes, the
convergence criteria, etc. Therefore this expresses the need for best practice guidelines for CWE. CWE has
grown to a strongly established field in wind engineering research, practice and education. It is employed
daily by probably thousands of researchers, practitioners and teachers all over the world.
2.4.1. Microscale and CFD
At the microscale, the flow around surface mounted obstacles such as buildings is explicitly
resolved, i.e. these obstacles are represented with their actual shape. Yamada and Meroney (1972) studied
2D airflow over a square surface mounted obstacle in a stratified atmosphere, both in the wind tunnel and
with CFD. Hirt and Cook (1972) calculated 3D flow around structures and over rough terrain. CFD
simulations around 3D buildings started with fundamental studies of isolated buildings, often with a cubical
shape, to analyze the velocity pressure fields (Murakami and Mochida 1988, 1989; Baskaran and
Stathopoulos 1989, 1992). Together with later studies they laid the foundations for the current best practice
guidelines by focusing on the importance of grid resolution, the influence of boundary conditions on the
numerical results, and by comparing the performance of various types of turbulence models in steady
RANS simulations. Also steady RANS versus LES studies were performed (Blocken 2014).
In the past, especially the deficiencies of the steady RANS approach with the standard κ-ε model
(Jones and Launder 1972) for wind flow around buildings were addressed. These include the stagnation
point anomaly overestimation of turbulent kinetic energy near the frontal corner and the resulting
underestimation of the size and separation and recirculation regions on the roof and the side faces, and the
underestimation of turbulent kinetic energy in the wake resulting in an overestimation of the size of the
cavity zone and wake. Various revised linear and non-linear κ-ε models and also second-moment closure
models were developed and tested and showed improved performance for several parts of the flow flied.
However the main limitation of steady RANS modelling remained: its incapability to model the inherently
transient features of the flow field such as the separation and recirculation downstream of windward edges
and vortex shedding in the wake. These large-scale features can be explicitly resolved by LES. The studies
11
by Murakami et al. (1987), later by Murakami et al. (1990, 1992) illustrated the intrinsically superior
performance of LES compared to RANS. Nevertheless, LES entails specific disadvantages that are not easy
to overcome, including the strongly increased computational requirements and the difficulty in specifying
appropriate time-dependent inlet and wall boundary conditions (Blocken 2014).
2.4.2. Reduced-scale Wind Tunnel Testing and CWE
In the past decades, often statements have been made that CFD would replace reduced scale wind
tunnel testing and that it would be the numerical wind tunnel. Many scholars such as Castro and Graham
(1999) and Stathopoulos (2002) convincingly denounced the label without recognizing the important
complementary value and potential of CWE.
The complementary aspects of wind tunnel testing and CWE are multifold. Wind tunnel testing
can provide the indispensable high-quality validation data needed for CWE, and CWE can supplement
wind tunnel testing by providing whole flow field data on all relevant parameters. Leitl and Meroney
(1997) indicated the value of CFD to design wind tunnel experiments by using numerical codes that can
help design and setup wind tunnel experiments which can reduce the time required to optimize a physical
model and expensive pre-runs in a wind tunnel. Moonen et al. (2007) developed a series of new indicators
for wind tunnel test section flow quality and applied CFD to illustrate the effectiveness of these indicators.
This approach was adopted by Calautit et al. (2014) for further development of design methodologies of
closed-loop subsonic wind tunnels (Blocken 2014).
2.5.
Wind Tunnel Measurements
Wind tunnel tests are powerful tools that give engineers the ability to estimate the nature and
intensity of wind forces acting on complex structures such as tall buildings. Wind tunnel testing is
especially useful when the surrounding terrain and the shape of the structure causes complex wind flows
that are not fully addressed by simplified codes (Samali et al. 2004). Many studies have been performed in
the measurements of wind loads on structures by either using full-scale measurements or by wind tunnel
model studies. As technology has advanced, the estimation of these forces have increased in reliability.
Wind loads are particularly important for flexible structures such as tall-buildings with low damping.
Typically, wind tunnel measurements are performed in boundary-layer wind tunnels that are
capable of developing flow conditions that meet these conditions (Taranath 2005):
12
1) The natural atmospheric boundary layer is modeled as such to account for the variation of
wind speed with height
2) The length scale of atmospheric turbulence is approximately the same scale as of that of the
building
3) The model building and surrounding topography are geometrically similar to the full-scale
4) The pressure gradient in the longitudinal direction is accounted for
5) Reynolds number effects on pressures and forces are kept to a minimum
6) Response characteristics of the instrumentation are consistent with the measurements to be
taken
The wind tunnels have generally these test-section dimensions: width of 9 to 12 feet, height of 8 to
10 feet, and length of 75 to 100 feet. Wind speeds than can be generated in these tunnels can range from 25
to 100 miles per hour (Taranath 2005). Typically there are two types of test models being used to conduct
studies: the first one is the rigid High Frequency Base Balance Model (HFBBM), and the second being the
Aeroelastic Model (AM). The models can be used independently or combined to obtain design loads for a
structure. The HFBBM measures overall fluctuating loads for the determination of dynamic responses. The
aeroelastic model is employed for direct measurements of loads, deflections and accelerations when the
lateral motions of a building are considered to have a large influence on the loading produced by the wind.
Numerous techniques are used in these wind tunnels to generate the turbulence and atmospheric boundary
layer by using tools such as spires and grids. In long wind tunnel sections, turbulent boundary layer is
generated by providing roughness elements in the approaching flow. Although these techniques are
considered to be appropriate, there are concerns in whether the wind turbulence is appropriately modeled.
Typically the scaling used to account for all these variables varies in the order of 1:400 to 1:600 for urban
environments (Taranath 2005).
Reinhold (1977) investigated several problems associated with the measurements of fluctuating
wind loads on tall structures using a number of building orientation and configurations. The author
generated atmospheric winds over urban areas in a short-test section tunnel and presented the results in the
dissertation document. In the study, a simple square prism was used because of its simplicity. Reinhold’s
study extended measurements of these random loads at multiple levels that improved with the respect of the
13
placement of pressure transducers along the model structure. It must be noted that most complete wind
tunnel tests and reports which have been conducted in the past that are of aid to design engineers are often
considered proprietary and are almost never published (Reinhold 1977).
2.6.
Overview of Tall Buildings
Tall towers and building have fascinated mankind from the beginning of civilization, their
construction being initially for defense and subsequently for ecclesiastical purposes. The growth in modern
tall building construction which began in the late part of the 19th century has been largely for commercial
and residential purposes. Tall commercial buildings are primarily a response to the demand by business
activities to be as close to each other, and to the city center, as possible, thereby putting intense pressure on
the available land space. Tall commercial buildings are frequently developed in city centers as prestige
symbols for corporate organizations. Furthermore, the business and tourist community has fuelled a need
for more frequently city center hotel accommodations such as high rises.
The rapid growth of the urban population and the consequent pressure of limited space have
influenced city residential developments. The high cost of land , the desire to avoid continuous urban
sprawl, and the need to preserve important agricultural production have all contributed to drive residential
buildings vertically. Also, some topographical conditions make tall buildings the only feasible solution for
housing needs such as the ones encountered in Hong Kong and Rio de Janeiro.
2.6.1. Factors Affecting Growth, Height, and Structural Form
The feasibility and desirability of high-rise structures have always depended on the available
materials, the level of construction technology, and the state of development of the services necessary for
the use of buildings. Significant advances have occurred with the advent of a new material, construction
facility, or form of service.
The socioeconomic problems that followed industrialization in the nineteenth century coupled
with an increasing demand for space in U.S. cities created a strong stimulus to tall building construction.
The growth could not have been sustained without two major technical innovations that occurred in that
century:
1) The development of higher strength and structurally more efficient materials, wrought iron
and thereafter steel.
14
2) The introduction of the elevator
For the first time, this made upper stories as attractive to rent as the lower ones and made the taller
buildings financially viable. The new materials allowed the development of lightweight skeletal structures
permitting buildings of greater height and with larger interior open spaces and windows. Improved design
methods and construction techniques allowed the maximum height of steel frame structures to reach a
height of 60 stories with the construction of the Woolworth Building in 1913. This golden age of
skyscraper construction culminated in 1931 with its crowning glory, the Empire State Building, whose 102story brace steel frame reached a height of 1250 ft. (381 m).
(a)
(b)
Figure 5: (a) Woolworth Building, NY (b) Empire State Building, NY
(Credit: The Pictorial News Co. and Virginia University)
Reinforced concrete construction began around the turn of the 20th Century but it has only been
used for the construction of multistory buildings approximately after the end of World War I. The inherent
advantages of the composite material which could be readily formed to simultaneously satisfy both
aesthetic and load-carrying requirements were not fully appreciated by then due to limited design
knowledge of the material. The economic depression of the 1930s put a hold to the great skyscraper era and
it was only after some years passed after World War II that the construction of high-rise buildings
recommenced with new structural and architectural solutions.
Different structural systems have gradually evolved for residential and office buildings, reflecting
their differing functional requirements. In modern office buildings, the need to satisfy the differing
15
requirements of individual clients for floor space arrangements led to the provision of large column-free
open areas to accommodate flexibility in planning. Other architectural features of commercial buildings
that have influenced structural form are the large entrances and open lobby areas at ground level, the
multistory atriums, and the high-level restaurants and viewing galleries that may require more extensive
elevator systems and associated sky lobbies. A residential building’s basic functional requirement is the
provision of self-contained individual dwelling units, separated by substantial partitions that provide
adequate acoustic and fire insulation. Because partitions are repeated from floor to floor, modern designs
have utilized them in a structural capacity leading to the shear wall, cross wall, or infilled-frame forms of
construction.
The trends to exposed structure and architectural cutouts, and the provision of setbacks at upper
levels to meet daylight requirements have also been features of modern architecture. The requirement to
provide adequately stiff and strong structures led to the development of a new generation of structural
framing such as braced frames, framed-tube and hull-core structures, wall-frame systems, and outriggerbraced structures (Stafford Smith and Coull 1991). The latest generation of buildings with their more varied
and irregular external architectural treatment has led to a hybrid double and sometimes triple combinations
of the structural forms for modern buildings.
2.6.2. Criteria for the Definition of Tall Buildings
The Council on Tall Buildings and Urban Habitat (CTBUH) has developed a guideline to define
what constitutes a “tall building” that exhibits some element of height in one of these three categories:
1) Height relative to context: it is not just about height but about the context in which it exists. A
20-story building may not be considered a tall building in a high-rise city such as New York
or Hong Kong, but in a provincial city or suburb this may be distinctly taller than the urban
norm.
2) Proportion: a tall building is not just about height but also proportion. There are a number of
buildings which are slender enough to give appearance of a tall building against the
background of a low urban environment. On the other hand, there are numerous large
footprint which are quite tall but their floor area rules them out as being classified as a tall
building.
16
3) Tall building technologies: If a building contains technologies which may be attributed as
being a product of tallness such as high speed elevators and wind bracing, then this building
can be classified as a tall building.
The number of floors if a poor indicator of defining a tall building due to the changing nature of
floor to floor height between different buildings uses. A building of perhaps 15 or more stories, or over 50
m (165 ft.) in height could be used as a threshold for considering it a “tall building.” However, the CTBUH
defines a “supertall” building over 300 meters (984 ft.) in height, and a “megatall” as a building over 600
meters (1,968 ft.) in which it recognizes building height in three categories. As of August 2014 there exists
82 supertall and 2 megatall buildings that have been completed and are presently occupied (CTBUH 2014).
Figure 6: Comparison of tall, supertall and megatall building height criteria
(Credit: CTBUH)
17
Figure 7: World's ten tallest buildings according to 'height to architectural top' as of November 2014
(Credit: CTBUH)
2.7.
Wind Loading on Tall Buildings
Wind is a phenomenon of great complexity arising from the interaction of wind with structures.
Simple quasi-static treatment of wind loading, which is universally applied to design of typical low to
medium-rise structures, can be very conservative for design of very tall buildings. Important factors in wind
design of tall buildings are dynamic response (effects of resonance, acceleration, damping, structural
stiffness), interference from other structures, wind directionality, and cross wind response. Mendis et al.
(2007) considered a number of key factors associated with the design of tall buildings to the effects of wind
loading. The general design requirements for structural strength and serviceability assume particular
importance in the case of tall building design. Significant dynamic response can result from both buffeting
and cross-wind loading excitation mechanisms. Serviceability with respect to occupier perception of lateral
vibration response can govern the design. The authors have suggested a specific purpose-designed damping
system in order to reduce these vibrations to acceptable levels. Dynamic response levels also play an
important role in the detailed design of façade systems. State of the art boundary layer wind tunnel testing,
for determining global and local force coefficients and the effects of wind directionality, topographical
features and nearby structures on structural response are identified to be quite useful to tall building design.
The emerging use of CFD codes, particularly at the concept design stage, is also noted as assuming
increasing importance in the design of tall buildings. The authors have suggested that the design criteria for
lateral wind loads shall consider stability against overturning, uplift and or sliding of the structure as a
whole, strength of the structural components of the building, and serviceability so as to restrict the
interstorey and overall deflections within acceptable limits.
18
CHAPTER 3
ELEMENTARY PRINCIPLES OF WIND ENGINEERING
3.1.
Forces Acting in the Free Atmosphere
Wind is air movement relative to the earth, driven by several different forces, especially pressure
differences in the atmosphere, which themselves are produced by differential solar heating of different parts
of the earth’s surface, and forces generated by the rotation of the earth. The differences in solar radiation
between the poles and the equator produce temperature and pressure differences. These, together with the
effects of the earth’s rotation, set up large-scale circulation systems in the atmosphere, with both horizontal
and vertical orientations (Holmes 2007).
Severe tropical cyclones such as hurricanes and typhoons generate extremely strong winds over
some parts of the tropical oceans and coastal regions both north and south of the equator. For these types of
severe storms, the wind is highly turbulent or gusty. The turbulence is produced by eddies or vortices
within the air flow, which are generated by frictional interaction at ground level or shearing action between
air moving in opposite directions with respect to altitude (Holmes 2007).
The two most important forces acting in the free atmosphere, i.e. above the frictional effects of the earth’s
boundary layer, are the pressure gradient and the Coriolis force.
3.1.1. Pressure Gradient and the Coriolis Force
Based on the principles of fluid mechanics, at a point in a fluid in which there is a pressure
gradient, ∂p/∂x, in a given direction, x, in a Cartesian coordinate system, there is a resulting force per unit
mass given by:
1
Pressure gradient per unit mass =
a
p
x
(3.1)
where ρa is the density of air and p is the atmospheric pressure. This force acts form a high-pressure region
to a low-pressure region.
19
The Coriolis force is an apparent force due to the rotation of the earth. It acts to the right of the
direction of motion in the northern hemisphere and to the left of the velocity vector in the case of the
southern hemisphere; at the equator, the Coriolis force is zero. Within about 5° of the equatorial region, the
Coriolis is negligible in magnitude thus explaining why tropical cyclones do not form within this region
(Holmes 2007).
Figure 8: Coriolis force results in wind being deflected owing to the rotation of the Earth
Credit: The Atmosphere, 8th Edition, Lutgens and Tarbuck (2001)
3.2.
Geostrophic Wind, Gradient Wind, and Frictional Effects
Steady flow under equal and opposite values of the pressure gradient and the Coriolis force is
known as balanced geostrophic flow. Equating the pressure gradient force per unit mass and the Coriolis
force per unit mass given by f∙U, we obtain:
1 p
U
a f x
(3.2)
This is the equation for the geostrophic wind speed, which is directly proportional to the
magnitude of the pressure gradient. In the northern hemisphere the high pressure is to the right of an
observer facing the flow direction; in the southern hemisphere, the high pressure is on the left. This results
in a counter-clockwise rotation of winds around a low pressure in the northern hemisphere, and a clockwise
rotation in the southern half. Rotation about a low-pressure center is known as a cyclone to meteorologists,
which usually produces strong winds (Holmes 2007).
Near the center of tropical cyclones, the centrifugal force acting on the air particles cannot be
neglected due to the significant curvature of the isobars. For flows around a low-pressure center, i.e.
cyclone, the centrifugal force acts in the same direction as the Coriolis force and opposite to the pressure
20
gradient force. The equation of motion for a unit mass of air moving at a constant velocity, U, for a cyclone
is:
U2
1 p
f U
0
r
a r
(3.3)
The quadratic equation represents the gradient wind speed formulation. This equation has two
theoretical solutions, but if the pressure gradient is zero then U must also be zero so that the solution, for a
cyclone, becomes:
U
f r
2
f 2r 2
r p
4
a r
(3.4)
where f is the Coriolis parameter ( = 2Ω sin λ), λ is the latitude, and r the radius from the storm center. The
term under the square root is always positive, therefore the wind speed in a cyclone is only limited by the
pressure gradient, i.e. cyclones are associated with strong winds.
As we approach the earth’s ground surface, frictional forces gradually play a larger role through the shear
between layers of air in the atmospheric boundary layer. The frictional force acts in opposite direction to
that of the flow (Holmes 2007).
3.3.
Atmospheric Boundary Layer
As the earth’s surface is approached, the frictional forces play an important role in the balance of
forces on the moving air. For larger storms such as extra-tropical depressions, this zone extends up to 500
to 1,000 m height. The region of frictional influence is called the atmospheric boundary layer and it is
similar in many respects to the turbulent boundary layer on a flat plate at high wind speeds.
3.3.1. Mean Wind Profiles
The ABL characteristics vary with the conditions of the terrain considered. Design standards such
as the ASCE 7-10 take into consideration and factor the effect of surrounding structures and terrain in the
load conditions of the structure of interest. There are four different types of terrain that will affect shape
and thickness of the boundary layer as seen in Figure 9.
21
Marine
Suburban
Flat/Open
Urban
Figure 9: Wind profile in different boundary layers
3.3.1.1. The Logarithmic Law
In strong wind conditions, the most accurate mathematical expression is the logarithmic law for
wind profiles. The logarithmic law was originally derived for the turbulent boundary layer on a flat plate by
Prandtl; however it has been found to be valid in an unmodified form in strong wind conditions in the
atmospheric boundary layer near the surface (Holmes 2002). The logarithmic law describes the variation
with height and surface roughness of strong mean speeds with averaging times of 10-min to 1-hr in straight
winds. After mathematical derivation, its expression may be written as:
ln
V ( z ) Vzref
ln
z
z0
zref
(3.5)
z0
where V ( z ) and Vzref is the mean wind speeds at elevation z and zref respectively, and zref is a reference
elevation, and z0 is an empirical measure of the surface roughness called roughness length. Another
measure of terrain roughness is the surface drag coefficient, Csd, which is a non-dimensional surface shear
stress defined as:
k
Csd
ln 33
z0
2
(3.6)
where k is known as von Karman’s constant, and has been found experimentally to have a value of ≈ 0.4.
This value is typically based by the mean wind speed measured at a height zref of 10 m (32.8 ft.). Table 1
(Simiu 2011) gives the appropriate value of roughness length and surface drag coefficient, for various types
of terrain types adapted from ASCE 7-10.
22
Table 1: Roughness lengths and Surface drag coefficients per ASCE 7-10
(Credit: Simiu 2011)
On a final note, the logarithmic law has some mathematical constraints which may cause
problems: first, the logarithms of negative numbers to not exist, and secondly, it is less easy to integrate. To
avoid some of these problems, wind engineers have often preferred the use of the power law.
3.3.1.2.
The Power Law
The power law has no theoretical basis but is easily integrated over height – a convenient property
when wishing to determine bending moments at the base of a tall structure. To relate the mean wind speed
at any height z with that at 10 m (32.8 ft.) the power law can be represented as:
V ( z ) Vzref
z
z
ref
1/
(3.7)
The exponent α will change with terrain roughness, with height range, and upon averaging time.
The power law applied to 3 sec. gust wind profiles has the same form as Equation 3.7. Table 2 shows
power law exponents and gradient heights specified by the ASCE 7-93 Standard for sustained wind speeds
(including fastest-mile wind speeds) and for ASCE 7-10 Standard for 3-sec gusts (Simiu 2011).
Table 2: Power law exponents and gradient height per ASCE 7-10 Standard
(Credit: Simiu 2011)
23
Figure 10 shows a matching of the two laws for a height range of 100 m using the previous
equations where the average height in the range over which matching is required (i.e. 50 m). It is clear that
the two are relatively close, and the power law can be adequately used for engineering purposes (Holmes
2007).
Height, z (m)
100
80
60
40
20
0
0.6
0.8
1.0
1.2
1.4
V/Vz ref
Logarithmic law
Power Law
Figure 10: Comparison of the logarithmic and power law for mean velocity profile for z0 = 0.02 m and α = 0.128
3.3.2. Turbulence
The general level of turbulence or ‘gustiness’ in the wind speed, such as that it can be measured by
its standard deviation, or root-mean-square. First we subtract out the steady or mean component, then
quantify the resulting deviations. Since both positive and negative deviations can occur, we first square the
deviations before averaging them, and finally take the square root to give a quantity with the units of wind
speed. Mathematically, the formula for standard deviation can be expressed as:
2
T
1
u U (t ) U dt
T 0
1
2
(3.8)
where U(t) is the total velocity component in the direction of the mean wind equal to U u(t ) , where u(t)
is the ‘longitudinal turbulence component (i.e. in the mean wind direction). Other components of
turbulence in the lateral horizontal direction is denoted by v(t) and the vertical direction by w(t) are
quantified by their standard deviations v and w respectively.
24
3.3.2.1.
Turbulence Intensities
The ratio of the standard deviation of each fluctuating component to the mean value is known as
the turbulence intensity (TI) of that component. Figure 11 illustrates a typical time dependent measurement
of wind velocities measured in the atmospheric boundary layer. The velocity in the figure is decomposed
into a steady mean value with a fluctuating component (Versteeg and Malalasekera 2007).
u
u(t)
t
Figure 11: Typical point velocity measurements in turbulent flow
Therefore the equation for turbulence intensity can be represented as:
TIu
u
U
(longitudinal ) ; TI v
v
U
(lateral ) ; TI w
w
U
(vertical )
(3.9)
It has been measured that near the ground by winds produced by large depression systems the
standard deviation of the longitudinal wind is approximately equal to 2.5 u* , where u* is the friction
velocity. Alternatively, the turbulence intensity TIu can be expressed as the following equation:
TI u
u*
2u*
z
0.4 ln z0
1
z
ln
z0
(3.10)
For a rural terrain with a roughness length ( z0 ) of 0.04 m the various longitudinal turbulence
intensities for increasing height above ground is demonstrated in Figure 12, thus it can be concluded that
the turbulence intensity above ground decreases as the height increases.
25
300
Height, z (m)
250
200
150
100
50
0
10%
15%
20%
25%
30%
35%
40%
Turbulence Intensity (Longitudonal)
Rural Terrain
Suburban Terrain
Figure 12: Longitudinal turbulence intensity for rural terrain (z0 = 0.04 m) and suburban terrain (z0 = 0.15 m)
The lateral and vertical turbulence components are generally lower than the corresponding
longitudinal value. For well-developed boundary layer winds, simple relationships between standard
deviation and the friction velocity u* have been developed. For well-developed boundary layer winds,
simple relationships between standard deviation and the friction velocity u* have been developed.
Therefore, the standard deviation for the lateral velocity, v, is approximately equal to 2.20 u* and the
vertical, w, component is given by approximately 1.35 u* . The equivalent turbulent intensity equations for
TIv and TIw can be shown to be:
TI v
0.88
0.55
; TI w
z
z
ln
ln
z
0
z0
(3.11)
The ASCE 7-10 code uses a single formulation for the calculation of turbulence intensity based on
the variable present in gust-effect factor equation. The intensity of turbulence at height z is defined as:
1/6
33
Iz c
z
1/6
10
or I z c
z
(3.12)
where z is the equivalent height of the structure at 0.6h and not less than zmin. The coefficient c is given by
the Table 26.9-1: Terrain Exposure Constants found in the ASCE 7-10 publication. The equation on the
left represents imperial units, and on the right the international system.
26
3.3.2.2.
Integral Turbulent Length Scale
The velocity fluctuations in a flow passing a point may be considered to be caused by an overall
eddy consisting of a superposition of component eddies transported by the mean wind. Each component
eddy is viewed as causing, at that point, a periodic fluctuation with frequency f. Integral turbulence length
scales are measures of the spatial extents of the overall turbulent eddy.
The integral turbulence scale Lu is an indicator of the extent to which an overall eddy is associated
with the longitudinal wind speed fluctuation u will engulf a structure in the along-wind direction, and will
thus affect at the same time both its windward and leeward sides. If Lu is large in relation to the along-wind
dimension of the structure, the gust will engulf both sides. The scales Lv and Lw are measures of the lateral
and vertical spatial extent of the fluctuating longitudinal component u of the wind speed. Mathematically
the integral turbulent length Lu is defined as follows:
L
x
u
1
u2
R
u1u2
( x)dx
(3.13)
0
in which u1=u(x1,y1,z1,t), u2=u(x1+x,y1,z1,t), and the denominator is the variance of the longitudinal velocity
fluctuations, a statistic that for a given elevation z is the same throughout the flow. The integrand is the
cross covariance of the signals u1 and u2. The integral length is a measure of the average eddy size (Simiu
2011). Measurements show that Lu increases with height above ground and as the terrain roughness
decreases.
The ASCE 7-10 code uses a single formulation for the calculation of the integral length scale of
turbulence at the equivalent height. The length scale at height z is defined as:
33
10
Lz or Lz
z
z
(3.14)
The coefficients of the above equations are given by the Table 26.9-1: Terrain Exposure Constants found
in the ASCE 7-10 publication. The equation on the left represents U.S. customary units, and on the right the
SI system.
The Architectural Institute of Japan defines the turbulence length scale to be height dependent but
defined independently of the terrain categories (AIJ 2006).
27
z 0.5
100
Lz 30
100
30m z z g
(3.15)
z 30m
where z is the height above ground measured in meters, and zg determines the exposure factor as defined in
AIJ’s code Table A6.3. A graphical representation of this formulation can be seen in Figure 13. Based on
AIJ, the turbulent length
300
Height, z (m)
250
200
150
100
50
0
0
50
100
150
200
250
300
350
Turbulent Length Scale, Lu (m)
Figure 13: Variation of turbulent length scale as height increases per AIJ code
28
CHAPTER 4
THEORY OF COMPUTATIONAL FLUID DYNAMICS
4.1.
Overview
In the seventeenth century, the foundations of experimental fluid dynamics were laid in France
and England. The eighteenth and nineteenth centuries saw the gradual development of theoretical fluid
dynamics primarily in Europe. As a result, throughout most of the twentieth century the study and practice
of fluid dynamics involved the use of pure theory on the one hand and pure experiment in the other hand. If
a person were learning fluid dynamics in the 1960s, the individual would have been operating in the “twoworld approach” of theory and experiment. As seen in Figure 14, computational fluid dynamics (CFD) is
today an equal partner with pure theory and pure experiment in the analysis and solution of fluid dynamics
problems. CFD provides this new third approach which nicely and synergistically complements the other
two approaches of pure theory and pure experiment; however, it will never replace either of these
approaches (Anderson 1995). There will always be need for theory and experiment. The advancement of
CFD rests upon a proper balance of all three approaches, with computational fluid dynamics helping
understand and interpret the results of pure theory and experiment, and vice versa.
Comp. Fluid
Dynamics
Pure Exp.
Pure Theory
Figure 14: Three dimensions of fluid dynamics
The development of CFD started in the 1970s in which the type of computers and algorithms that
existed at that time limited most practical solutions to basically two-dimensional flows. However the real
world in which fluids exists is three-dimensional. It was only until the 1990s that computer machines
29
increased in storage and speed capacity which has allowed CFD to operate in a three-dimensional
environment. However a great deal of human and computer resources are still frequently needed to
successfully carry out numerically intensive solutions for applications like flow over a complete speed
racing car. These solutions have become more and more prevalent within industry and government
facilities to an extent that some three-dimensional flow solutions have attained standard guidelines as a tool
during the design process of machinery (Anderson 1995). CFD is playing a strong role as a design tool in
which it has become a powerful influence in the way fluid dynamicists and aerodynamicists are engineering
products. There are various types of applications that CFD can be employed as a design tool in several
industrial applications such as:
4.2.
1.
Automobile and Engine
2.
Manufacturing
3.
Civil Engineering
4.
Environmental Engineering
5.
Naval Architecture
Computational Fluid Dynamics
The physical aspects of any fluid flow are based upon three fundamental principles in which they
can be expressed in terms of mathematical equations with a general form of either integral or partial
differential equations.
1.
Mass is conserved (Continuity Equation)
2.
Newton’s second law (Momentum Equation, F = ma)
3.
Energy is conserved (First law of thermodynamics)
Anderson (1995) defines CFD as “the art of replacing the integrals or the partial derivatives in the
above equations with discretized algebraic forms, which in turn are solved to obtain numbers for the flow
field values at discrete points in space and/or time.” The end product of CFD is a collection of numbers.
The tool that has allowed the practical growth of this topic is the high-speed digital computer because it
processes the manipulation of thousands and millions of numbers.
30
4.3.
Governing Equations of Fluid Dynamics
All of CFD, in one form or another is based on the fundamental governing equations of fluid
dynamics—the continuity, momentum, and energy equations. These equations speak physics (Anderson
1995). They are mathematical statements of three fundamental physical principles upon which all of fluid
dynamics is based. It is important for the reader to feel comfortable with these equations before continuing
further with applications of CFD to a specific problem. The governing equations can be obtained in various
different forms such as conservation and nonconservation form.
4.3.1.
Infinitesimal Fluid Element
Consider the general flow field as represented in the streamlines in Figure 15. Imagine an
infinitesimally small fluid element in the flow with a differential volume dV. The fluid element is
infinitesimal in the same sense of differential calculus however, it is large enough to contain a huge number
of molecules so that it can be viewed as a continuous medium. The element may be moving along a
streamline with a velocity vector V equal to the flow velocity at each point. The fundamental physical
principles are applied to just the infinitesimally small fluid element itself. This application leads directly to
the fundamental equations in partial differential equation (PDE) form. The particular partial differential
equations obtained directly from the moving fluid element are called nonconservation forms of the
equations (Anderson 1995).
(a) Infinitesimal fluid element fixed in space with the
fluid moving through it
(b) Infinitesimal fluid element moving along a
streamline with the velocity V equal to the local flow at
each point
Figure 15: Infinitesimal fluid element model of flow
4.3.2.
The Continuity Equation
In this section we will treat the following case for the physical principle: mass is conserved.
Consider the flow model shown in Figure 15 (a). This fluid element is fixed in space and has the fluid
moving through it. We will adopt a Cartesian coordinate system where the velocity and density are
31
functions of space (x, y, z) and time (t). There is mass flow passing through this fixed element. Consider the
left and right faces of the element which are perpendicular to the x axis as seen in Figure 16.
Figure 16: Model of infinitesimally small element fixed in space including mass flux diagram
The mass flow through the left face with area dy dz is, (ρu) dy dz. Since the velocity and density
are functions of spatial location the values of the mass flux across the right face will be different than that
of the left; therefore, the mass flow can be expressed as u ( u) / x x dy dz across the right face. In
similar fashion the expression can be represented for the faces perpendicular to the y and z axes. Note that
u, v, and w are positive by convention in the positive x, y, and z directions, respectively. Hence the net mass
flow out of the element is given by
( u ) ( v) ( w)
Net mass flow
dx dy dz
y
z
x
(4.1)
The total mass of fluid in the infinitesimally small element is ρ (dx dy dz); hence the time rate of
increase of mass inside the element is given by
Time rate of mass increase
(dx dy dz )
t
(4.2)
32
The physical principle that mass is conserved can be expressed in words as follows: the net mass
flow out of the element must equal the time rate of decrease of mass inside the element. In equation terms
this means
( u ) ( v) ( w)
0
t x
y
z
( V) 0
t
or
(4.3)
where in Cartesian coordinates, the vector operator nabla, , is defined as
j k
i
x
y
z
(4.4)
Equation 4.3 is a partial differential equation form of the continuity condition. This represents an
unsteady, three-dimensional mass conservation at a point in a compressible fluid. The infinitesimally small
aspect of the element is why the equation is directly obtained in this form. The fact that the element was
fixed in space leads to the specific form given by the equation 4.3 which is called the conservation form.
Consider the flow model shown in Figure 15 (b); an infinitesimally small fluid element moving
with the flow. This fluid element has fixed mass, but in general its shape and volume will change as it
moves downstream. Denote the fixed mass and variable volume of this moving fluid element by m and δV,
respectively. Since the mass in conserved we can state that the time rate of change of the mass of the fluid
element is zero as the element moves along with the flow. Therefore the equation for this condition
becomes
D
V 0
Dt
(4.5)
where D/Dt is the substantial derivative, which is physically the time rate of change following a moving
fluid element and it can be generally written as
D
( V)
Dt t
(4.6)
/ t is called the local derivative, which is physically the time rate of change at a fixed point; V is
called the convective derivative, which is physically the time rate of change due to the movement of the
fluid element from one location to another in the flow field where the flow properties are spatially different.
33
Equation 4.6 states physically that the density of the fluid element is changing as the element
sweeps past a point in the flow because at that point the flow-field density itself may be fluctuating with
time, e.g. local derivative, and because the fluid element is simply on its way to another point in the flowfield where density may be different, e.g. convective derivative.
Equation 4.5 is a partial differential equation form of the continuity equation which was derived
on the basis of an infinitesimally small fluid element moving with the flow. The fact that the element is
moving with the flow leads to the specific differential equation form given by the aforementioned equation
which is called the nonconservation form.
4.3.3.
The Momentum Equation
The momentum equation is based on Newton’s second law of motion pertaining to the behavior of
objects for which all the existing forces acting on the body are unbalanced. The physical principle of the
law states that the net force acting on a body is dependent upon two variables—the net force acting on the
body and the mass of the object. Newton’s second law can also be expressed as the rate of change of
momentum of a fluid particle equals the sum of the forces acting on the particle (Anderson 1995).
Figure 17: Model used for the derivation of the x-component of momentum equation
(Credit: Anderson 1995)
The element experiences two types of forces based on Newton’s principle. The first being body
forces, which act directly on the volumetric mass of the fluid element. Typically these forces act at distance
and examples of these are: electric, magnetic, and gravitational forces. The second type is surface forces,
which act directly on the surface of the fluid element. They are due to two sources: (a) the pressure
34
distribution acting on the surface which is imposed by the outside fluid surrounding the fluid element, and
(b) the shear and normal stress distributions acting on the surface which is imposed by the outside fluid
pushing on the surface by means of friction (Anderson 1995). The shear and normal stresses in a fluid are
related to the time rate of change of the deformation of the fluid element.
The scalar equations presented below are called the Navier-Stokes equations which were derived
by Frenchman M. Navier and Englishman G. Stokes who independently obtained the equations in the first
half of the nineteenth century. These equations are partial differential equations obtained directly from an
application of the physical principle of the infinitesimal fluid element that is moving with the flow which
take the nonconservation form.
xx yx zx
Du
fx
Dt
x
x
y
z
(4.7a)
xy yy zy
Dv
fy
Dt
y
x
y
z
(4.7b)
xz yz zz
Dw
fz
z
x
y
z
Dt
(4.7c)
4.3.4.
The Energy Equation
This is the third physical principle as outlined before which is based on the first law of
thermodynamics. We present the flow model of an infinitesimally small fluid element moving with the
flow. When applied to the flow model of a fluid element moving with the flow, the first law states
Rate of change of energy = Net flux of heat + Rate of work done on element
inside fluid element
into element
due to body and surface forces
(A)
=
(B)
+
(C)
(4.8)
By evaluating term C, it can be shown that the rate of doing work by a force exerted on a moving
body is equal to the product of the force and the component of velocity in the direction of the force. In total,
the net rate of work done on the moving fluid is the sum of the surface force contribution in the x, y, and z
directions, as well as body force contribution. By evaluating term B, the net flux of heat into the element is
due to (1) volumetric heating such as absorption or emission of radiation and (2) heat transfer across the
surface due to temperature gradients. Term A denotes the time rate of change of energy in the fluid
element. The total energy of a given molecule is the sum of its electronic, vibrational, rotational, and
translational energies; the total energy of the each atom is the sum of its translational and electronic energy
35
(Anderson 1995). The internal energy of the gas systems simply is the energy of each molecule or atom
summed over all the molecules or atoms in the system which is the physical significance of the internal
energy that appears in the first law of thermodynamics. Term A concerns the energy of a moving fluid
element which has two contributions to its energy:
1.
The internal energy due to random molecular motion, e (per unit mass).
2.
The kinetic energy due to translational motion of the fluid element which simply put is V2/2.
Hence, the moving fluid has both internal and kinetic energy and the sum of these two is the total
energy. The final form of the energy equation can be presented in nonconservation terms which is based on
the total energy e + V2/2.
D V2
T T T (up ) (vp ) ( wp )
e
q k
k
k
2
x x y y z z
x
y
z
Dt
(u xx ) (u yx ) (u zx ) (v xy ) (v yy ) (v zy ) ( w xz ) ( w yz )
x
y
z
x
y
z
x
y
( w zz )
f V
z
(4.9)
All of theoretical and computational fluid dynamics is based upon the equations presented in the
aforementioned section. It is essential that the reader is familiar with them and that their physical
significance is understood. Moreover, there are numerous references in which the reader can rely on for
further clarification of the different equations in which computational fluid dynamics is based upon.
4.4.
Turbulence and Modeling for CFD
Almost all fluid flow which we encounter in daily life is turbulent. Typical examples are flow
around cars, airplanes, and buildings. The boundary layers and the wakes around and after these bluff
bodies are turbulent in nature. In turbulent flows we divide the velocities in one time-averaged part, v ,
which is independent of time (i.e. when the mean flow is steady), and one fluctuating part v' so that
v v v' .
36
Figure 18: von Kármán vortices representing turbulent flow characteristics forming in clouds flowing past a
volcano
(Credit: Gary Davies)
Turbulent flows have no specific definition, however it has a number of characteristics that we use
to describe its properties, such as:
1.
Irregularity: turbulent flow is irregular and chaotic; they may seem random, but they are
indeed governed by the Navier-Stokes equations. The flow consists of a spectrum of different
scales, also referred to as eddy sizes. Turbulent eddies exists in a certain region of space for a
certain time and that it undergoes into dissipation. These eddies have velocity and length
scales associated with it.
2.
Diffusivity: in these types of flows, diffusivity increases. The turbulence increases the
exchange of momentum in boundary layers, and reduces separation at bluff bodies.
3.
Large Reynolds Numbers: turbulent flows occur at high Reynolds number. For example, the
transition to turbulent flow in pipes occurs at Re ≈ 2,300 and in boundary layers at Re ≈
500,000.
4.
Three-dimensional: turbulent flow is always unsteady and exists in three dimensional space.
However, we can treat the flow as two-dimensional when the equations are time averaged.
5.
Dissipation: turbulent flow is dissipative, which means that its kinetic energy in the small
eddies are transformed into thermal energy. The largest eddies extract their energy from the
mean flow which transfers the energy to smaller eddies in a cascade process.
6.
Continuum: even though we have small turbulent scales in the flow they are much larger
than the molecular scale, therefore we can treat the flow as a continuum (Davidson 2015).
37
Figure 19: Representation of cascade process with a spectrum eddies
(Credit: Davidson 2015)
4.4.1.
Turbulent Scales
The largest scales of the order of the flow geometry with length scale 0 and velocity scale v0 .
These scales extract kinetic energy from the mean flow which has a time scale comparable to the large
scales. Part of the kinetic energy of the large scales is lost to slightly smaller scales with which the large
scales interact. The kinetic energy is in this way transferred from the largest scale to the smallest scale
through a method referred to as cascade process. The kinetic energy dissipation is denoted by ε which is
energy per unit time and unit mass (ε = m2/s3). The dissipation is proportional to the kinematic velocity, ν,
times the fluctuating velocity gradient to the power of two. The friction forces exist at all scales but they
are largest at the small eddies. The smallest scales where dissipation occurs are called the Kolmogorov
scales whose velocity scale is denoted by v , length scale by and time scale . It is assumed that these
scales are determined by viscosity, ν, and dissipation, ε (Davidson 2015).
4.4.2.
Energy Spectrum
As mentioned previously, the turbulence fluctuations are composed of a wide range of eddie
scales. The turbulent scales are distributed over a range of scales which extends from the largest scales
which interact with the mean flow to the smallest scales where dissipation occurs. In wavenumber space the
energy of eddies can be expressed as
E ( ) d
(4.10)
38
where the above equation expresses the contribution from the scales with wavenumber between κ + dκ to
the turbulent kinetic energy κ. The dimension of wavenumber is one divided by length (e.g. m-1); thus we
can think of the wavenumber as inversely proportional to the eddy’s diameter. The kinetic energy is
obtained by integrating over the whole wavenumber space. In other words, we compute the kinetic energy
by first sorting all eddies by size, then computing the energy of each eddy size, and finally sum the kinetic
energy for all eddy sizes. The kinetic energy is the sum of the kinetic energy of the three fluctuating
components of velocity
k
2
2
1 '2
1
u1 u 2' u3' ui' ui'
2
2
(4.11)
The spectrum of E is shown in Figure 20. Region I, II and III correspond to the following:
I. Large eddies which carry most of the energy in which the eddies’ velocity and length scale
are v0 and 0 , respectively.
II. Dissipation range where eddies are small and isotropic. It is in this region is where true
dissipation occurs.
III. Inertial subrange requires that the Reynolds number is high and the flow is fully turbulent.
The turbulence in this region is also isotropic.
As a final note, the ratio of velocity, length, and time scales of the energy-containing eddies to the
Energy of eddies
Kolmogorov eddies increases with increasing Reynolds number (Davidson 2015)
Turbulent kinetic energy
Figure 20: Spectrum for turbulent kinetic energy, κ
(Credit: Davidson 2015)
39
4.4.3.
Transition from Laminar to Turbulent Flow
Osborne Reynolds, a British scientist, was the first to distinguish the difference between the
classification of laminar and turbulent flow in a pipe. Reynolds observed that for small flow rates a drop of
dye remained a well-defined line as it flowed along the fluid (i.e. laminar flow). For intermediate flow rates
the drop fluctuated in time and space and bursts of irregular behavior appear along the fluid flow (i.e.
transition flow). For large flow rates, the drop of dye almost immediately became blurred and spread across
with randomness (i.e. turbulent flow) (Munson et al. 2006).
In general terms, flows are classified as laminar or turbulent. The important parameters that help
us distinguish these two types of flows is largely based on the Reynolds number and their critical values
depend on the specific flow situation involved. For flow in a pipe the Reynolds number must be less than
2,300 for laminar flow, and greater than 4,000 for turbulent flow. The region in between is known as
transition flow. For flow along a flat plate, the transition between laminar to turbulent occurs at a Reynolds
number approximately to 500,000.
Figure 21: Transition of flow in a pipe from laminar to turbulent
(Credit: Munson et al. 2006)
4.4.4.
Turbulent Flow Calculations
Turbulence causes the appearance in the flow of eddies with a wide range of length and time
scales that interact dynamically complex way. A substantial amount of research effort has been dedicated to
the development of numerical methods to capture the important effects due to turbulence. These methods
can be grouped in three main categories (Versteeg and Malalasekera 2007).
1.
Reynolds-averaged Navier-Stokes (RANS) equations: the numerical attention is focused on
the “mean flow” and the effects of turbulence on the mean flow properties. The Navier-Stokes
40
equations are time averaged prior to the application of numerical techniques. Extra terms
appear in the time-averaged flow equations to represent the interaction between turbulent
fluctuations. These terms are modelled with two “classical” models – κ-ε and κ-ω
formulation. The computer requirements required to arrive at reasonable accurate flow
computations are modest.
2.
Large Eddy Simulation (LES): this is considered to be an intermediate form of turbulence
calculations which tracks the behavior of the larger eddies. This method involves space
filtering of the unsteady Navier-Stokes equations prior to the start of computations, which
allows larger eddies to be accounted for and excludes smaller eddies. The effects on the
resolved flow due to the smallest unresolved eddies are included by means of sub-grid scale
modelling. The computer demands are greatly increased due to the solution of unsteady flow
equations.
3.
Direct Numerical Simulation (DNS): this type of simulation compute the mean flow and
“all” turbulent velocity fluctuations. The unsteady Navier-Stokes equations are solved on
spatial grids that are very fine to resolve the Kolmogorov length scale at which energy
dissipation takes place and with very small time steps to capture the period of the fastest
fluctuations. These calculations are extremely demanding on computer resources.
DNS
LES
RANS
Figure 22: Increase of computational cost per type of turbulence modeling
(Credit: psc.edu)
4.4.5.
RANS Equations and Classical Models
For most engineering studies, it is unnecessary to resolve the details of turbulent fluctuations. CFD
engineers are almost always satisfied with the information about the time-averaged (mean) properties of the
flow of a fluid, such as pressures, velocities, stresses, etc. The majority of turbulent computations are
carried out based on the procedure presented on the Reynolds-averaged Navier-Stokes equations. In order
41
to be able to compute turbulent flows with RANS equations, the general purpose CFD code must have wide
applicability, be simple, and economical to simulate. The most common RANS turbulence models are
classified on the basis of the number of additional transport equations that need to be solved along with the
RANS flow equations. Listed below, there are four types of extra transport equations models (Versteeg and
Malalasekera 2007):
1.
Zero equations (e.g. Mixing length model)
2.
One equation (e.g. Spalart-Allmaras model)
3.
Two equations (e.g. κ-ε model, κ-ω model)
4.
Seven equations (e.g. Reynolds stress model)
In Reynolds averaging, the solution variables in the instantaneous Navier-stokes equations are decomposed
into the mean (i.e. ensemble-averaged or time-averaged) and fluctuating components:
'
(4.12)
where denotes the mean and ϕ’ the fluctuating velocity components in the x, y, and z directions (i = 1, 2,
3, respectively). Likewise the same formulation as Equation 4.12 applies for scalar quantities such as
pressure, energy, or species concentration (ANSYS 2014).
Substituting expressions of this form for the flow variables into the instantaneous continuity and
momentum equations and taking a time average yields the ensemble-averaged momentum equations which
can be written in Cartesian tensor form as
ui 0
t xi
p
ui ui u j
t
x j
xi x j
(4.13)
ui u j 2 ul
ij
ui' u 'j
x j xi 3 xl x j
(4.14)
Equations 4.13 and 4.14 are called the Reynolds-averaged Navier-Stokes (RANS) equations. They
have the same general form as the instantaneous Navier-stokes equations, with the velocities and other
variables now representing the ensemble or time-averaged values. Additional terms now appear that
represent the effects of turbulence. These Reynolds stresses, u i' u 'j , must be modeled in order to close
Equation 4.14.
42
4.4.5.1.
κ-ε Model
The κ-ε Model focuses on the mechanisms that affect the turbulent kinetic energy. It must be noted
that the symbol κ represents the turbulent kinetic energy, and ε represent the turbulent dissipation rate. The
two-equation turbulence model allows the determination of both, a turbulent length and time scale by
solving two separate transport equations. The κ-ε model falls within a class of models that has become the
workhorse for practical engineering problems since it was proposed by Launder and Spalding (1974).
(Versteeg and Malalasekera 2007).
The standard κ-ε model is based on the transport equations for the turbulence kinetic energy (κ)
and its dissipation rate (ε). The equation for κ was derived from exactness, however the model transport
equation for ε was obtained by physical reasoning and is of little resemblance to its exact counterpart. The
transport equations for the turbulence kinetic energy, κ, and its rate of dissipation, ε, are represented as
follows:
ui t
t
xi
x j
ui t
t
xi
x j
G Gb YM S
x j
2
S
C
G
C
G
C
1
3 b
2
x j
(4.14)
(4.15)
In these equations, Gκ, represents the generation of turbulence kinetic energy due to the mean
velocity gradients calculated by referring to the model of turbulent production in the κ-ε models. Gb is the
generation of turbulence kinetic energy due to buoyancy. YM represents the contribution of fluctuating
dilatation in compressible turbulence to the overall dissipation rate. C1ε, C2ε, Cμ, are constants equal to 1.44,
1.92, 0.09, respectively; σκ, σε are the Prandtl numbers for κ and ε with values of 1.0 and 1.3, respectively.
These values have been determined from fundamental experiments for turbulent flow which were
frequently encountered in shear flows like boundary layers. They have been found to be work fairly well
for a wide range of wall-bounded and free shear flows (ANSYS 2014).
4.4.5.2.
κ-ω Model
The most prominent alternative to the previous model is the κ-ω model which was proposed by
Wilcox (1988). This model uses the turbulence frequency ω = ε/κ as the second variable; ω can also be
referred to as the specific dissipation rate.
43
ui G Y S
t
xi
x j x j
(4.16)
ui G Y S
t
xi
x j x j
(4.17)
In the above equations, Gκ, represents the generation of turbulence kinetic energy due to mean
velocity gradients. Gω represents the generation of ω. Γκ and Γω represent the effective diffusivity of κ and
ω, respectively. Yk represents the dissipation of κ and Yω the dissipation of ω due to turbulence. These
terms have specific models associated with them that are left to the reader to reference to. Lastly, Sκ and Sω
are user defined source terms.
The κ-ω attracted attention because integration to the wall does not require wall damping
functions in low Reynolds number applications. The value of κ at the wall is set to zero and the frequency
ω tends to infinity at the wall, however we normally specify a very large value. Practical experience with
this model has proved that the results do not depend entirely on the precise details of this condition
(Versteeg and Malalasekera 2007).
4.4.5.3.
SST κ-ω Model
The shear-stress transport (SST) κ-ω model was developed by Menter (1994) to effectively
combine the robust and accurate formulation in the near-wall region with the freestream independent of the
κ-ε model in the far field. The SST κ-ω model is equivalent to the standard model, but it includes the
following refinements:
1.
The standard κ-ω and the transformed κ-ε model are multiplied by a blending function and
then they are summed together. The function is designed to have a value of one in the nearwall region, and zero away from the surface.
2.
The SST model includes a damped cross-diffusion derivative term in the ω equation.
3.
The definition of the turbulent viscosity is modified to account for the transport of shear
stress.
4.
The constants in these models have different values.
The major way in which the shear-stress transport (SST) model differs from the standard model
are as follows: (1) gradual change from the standard κ-ω model in the inner region of the boundary layer to
44
a high Reynolds number of the κ-ε model in the outer part of the boundary layer, and (2) modified turbulent
viscosity formulation to account for the transport effects of the principal turbulent shear stress. These
features make the SST κ-ω model more accurate and reliable for a wider class of flows.
ui G Y S
t
xi
x j x j
(4.18)
u j G Y D S
t
x j
x j x j
(4.19)
In the above equations, Gκ, represents the generation of turbulence kinetic energy due to mean
velocity gradients. Gω represents the generation of ω. Γκ and Γω represent the effective diffusivity of κ and
ω, respectively. Yk represents the dissipation of κ and Yω the dissipation of ω due to turbulence. Dω
represents the cross-diffusion term. These terms have specific models associated with them that are left to
the reader to reference to. Lastly, Sκ and Sω are user defined source terms (ANSYS 2014).
The field of turbulence modelling provides abundant research activities for the CFD and
engineering communities. The RANS formulations presented in this section are widely available in
commercially available computer codes. Industry experts have widely used these concepts and have
produced useful results in spite of observations relating to its limited capabilities (Versteeg and
Malalasekera 2007). In the present study the SST κ-ω is utilized for the modeling of the wind flow around
the bluff body.
4.4.6.
Large Eddy Simulation
A different approach to the computation of turbulent flows accepts that the larger eddies need to
be computed for each problem with a time-dependent simulation. The universal behavior of the smaller
eddies should be easier to capture with a compact model. Instead of time-averaging, LES uses spatial
filtering to separate the larger from smaller eddies. This method uses the selection of a filtering function
and a certain cutoff length scale with the aim of resolving in an unsteady flow computation all those eddies
that have a larger length scale than the cutoff dimension. During the spatial filtering, information related to
the smaller eddies below the cutoff length is destroyed. This interaction effects between larger and smaller
eddies give rise to sub-grid-scale (SGS) stresses. This is the key concept of the large eddy simulation (LES)
approach to the numerical treatment and solution of turbulence in fluids (Versteeg and Malalasekera 2007).
45
Energy of eddies
Turbulent kinetic energy
Figure 23: Spectrum of velocity
(Credit: Davidson 2015)
The rationale behind LES can be interpreted as: (1) momentum, mass, energy and other scalars are
transported by larger eddies, (2) larger eddies are problem-dependent; they are governed by the geometry
and boundary conditions of the flow, (3) smaller eddies do not depend on geometry, they are isotropic. By
resolving the large eddies the must use finer meshes than those used in RANS models. LES has to be
typically run for long duration of time to obtain stable statistics for the flow being modeled (ANSYS 2014).
Therefore the computational time is longer and high-performance computing (HPC) is mandatory for LES
simulations. The disadvantage of LES lies in the high resolution requirement for wall boundary layers since
near the wall even larger eddies become small and require a Reynolds number dependent resolution. LES is
typically limited to Reynolds numbers in the range of 104 – 105 and smaller computational domains.
A substantial portion of ij is attributed to convective momentum transport due to interactions
between the unresolved eddies which are referred to as subgrid-scale stresses. The subgrid-scale turbulence
models usually employ the Boussinesq hypothesis (Hinze 1975). We compute the subgrid-scale turbulent
stresses from
1
3
ij kk ij 2 t S ij
(4.20)
where ij is the subgrid-scale turbulent viscosity. S ij is the rate-of-strain tensor for the resolved scale define
by
S ij
1 u i u j
2 x j xi
(4.21)
46
For compressible flows, we must introduce Favre (density-weighted) filtering operator. The compressible
form of the subgrid stress tensor is defined by splitting it into its isotropic and deviatoric parts
1
3
1
3
ij ij kk ij kk ij
(4.22)
where the deviatoric part of the subgrid-scale stress tensor is modeled using the compressible form of the
Smagorinsky model (ANSYS 2014)
1
3
1
3
ij kk ij 2t Sij Skk ij
4.4.6.1.
(4.23)
Smagorinsky-Lilly Model
This model was first developed by Smagorinsky (1963). In this formulation, the eddy-viscosity is
modeled as
t L2s S
(4.24)
where Ls is the mixing length for subgrid scales and S 2 S ij S ij and Ls can be computed using
Ls min( d , Cs )
(4.25)
where κ is the von Kármán constant ≈ 0.41, d is the distance to the closest wall, Cs is the Smagorinsky
constant, and Δ is the local grid scale based on the element volume. The researcher Lilly derived a value of
Cs = 0.23 for homogenous isotropic turbulence in the inertial subrange.
It must be noted that Cs is not a “universal constant.” The fact that CS has different values is due to
the mean flow strain or shear that gave an indication that the behavior of small eddies is not as universal as
first hypothesized. A different Cs value of around 0.1 has been found to yield very good results for a wide
variety of flows and this value is used in present computer codes. Further formulation of this theory can be
found in computational fluid dynamic textbooks.
4.4.6.2.
Remarks on LES
LES has been established in the 1960s, however the computational power to process this model
has not been made available until recently for industrially relevant problems. The inherent unsteady nature
of LES leads to the much higher computational demand than those needed by traditional RANS techniques.
LES is good at resolving certain time dependent features of turbulence with no additional equation since it
is inherent in its own formulation. LES gives a deeper insight into the mean flow and statistics of the
47
resolved fluctuations. The incorporation of LES codes in commercially available softwares has only been
accessible recently to the engineering community. The pace of development and research based on this
model will increase as computing resources become more powerful. Engineers will gain awareness of the
advantages of the LES method to turbulence modelling as more meaningful data is published (Versteeg and
Malalasekera 2007).
4.4.7. Typical Meshing Information
The partial differential equations that govern fluid flow are not usually responsive to analytical
solutions, except for very simple cases. Therefore, in order to analyze fluid flows, flow domains are split
into smaller subdomains. The governing equations are then discretized and solved inside each of these
subdomains. Typically, one of three methods is used to solve the approximate version of the system of
equations: finite volumes, finite elements, or finite differences. Care must be taken to ensure proper
continuity of solution across the common interfaces between two subdomains, so that the approximate
solutions inside various portions can be put together to give a complete picture of fluid flow in the entire
domain. The subdomains are often called nodes, elements, or cells. The collection of all elements or cells is
called a mesh or grid. There mainly are three classifications of meshes that depend on the connectivity of
the nodes:
1.
Structured Meshes: these are characterized by regular connectivity that can be expressed as
arrays. These elements are usually quadrilaterals in 2D and hexahedra in 3D.
2.
Unstructured Meshes: there are characterized by irregular connectivity and cannot be
expressed as an array in a computer. This allows for the use of any variation of geometric
mesh shapes that can fit the model geometry.
3.
Hybrid meshes: there are meshes that contain partially structured and unstructured meshes
around the geometry being investigated.
Meshes can be classified as either 2D or 3D. Common elements for 2-dimensional meshes are
rectangles and triangles. In 3-dimensional geometry, common elements are hexahedral, tetrahedral, square
prisms, and triangular prisms as seen Figure 24.
48
Figure 24: Typical 2D and 3D Mesh Shapes
(Credit: Bakker 2006)
Figure 25 illustrates a typical flow chart in the geometry and meshing generation process that must
be followed for CFD modeling. The user must first create the geometry using a computer aided design
package and the must transfer it to a meshing software. The user then must create the appropriate mesh
based upon the geometry requirements. Once all items are satisfied, the user must export the mesh to a
CFD solver. Explain mesh flow chart.
Figure 25: Typical geometry meshing process
(Credit: ANSYS 2014)
49
4.4.8. Near Wall Treatment
It is critical to capture boundary layer near wall properly. In order to do that, the mesh should be
generated in such a manner that it captures the boundary layer properly. For turbulent flows, calculation of
the y+ value of the first interior grid point helps achieve the capture of the boundary layer. This
dimensionless distance is defined as
y
u* y
(4.26)
where u* is the friction velocity given by
u*
(4.27)
The wall shear stress is usually determined after the simulation has been completed and usually
the engineers must assume a value and then check it with the simulation results. The role of the wall
function becomes unrealistic when the flow velocity increases and a more refined method must be used
such as the three-layer form of the log-law of the wall
y
u 3.05 5ln y
5.5 1/ lny
for 5
for 5 < y + 30
(4.28)
for y 30
+
Figure 26: Log-Law of the Wall
(Credit: McDonough 2007)
50
4.4.9. Discretization Techniques and Algorithms
In essence, discretization is the process by which a closed-form mathematical expression, which
are viewed as having an infinite continuum of values through a domain, can be approximated by analogous
expressions which prescribe values at only a finite number of discrete points or volumes in computational
domain. Numerical solutions can give answers at only discrete points in the domain also referred to as grid
points or nodes as see. Figure 27 represents a typical discretization that may be encountered in a fluid
domain. Uniform spacing such as Δx = Δy is typically used in CFD to allow for the simplification of
numerical computations, saves computer storage, and usually gives results with greater accuracy. This type
of grid system is also referred to as structure grid which reflects a consistent geometrical regularity in the
computational domain.
Figure 27: Discretization of grid points
(Credit: Anderson 1995)
The finite volume method (FVM) is a discretization method which is well suited for the numerical
simulation of various types of conservation laws. It is widely used in engineering fields such as fluid
mechanics. The finite volume method may be used in arbitrary geometries that can lead to robust schemes.
An additional feature of the FVM is that the numerical flux is conserved between neighboring cells which
in fact is a very important feature when modeling fluid mechanics problems (Eymard et al. 2003).
It must be noted that in most CFD applications, a first order accuracy is not sufficient to represent
the numerical techniques. The first order accuracy consists of a finite representation of a partial derivative
and the remaining terms in that equation give the truncation error. The truncation error tells the formulation
what is being neglected in the approximation. A more common technique is the use of the second-order
51
accuracy equation that enhances the numerical approximation as shown in Equation 4.29. The information
used comes from both sides of the grid located at a specific coordinate falling between two adjacent points.
The truncation error in the formulation involves second-order accuracy (Anderson 1995).
ui 1, j ui 1, j
u
O ( x ) 2
2 x
x i , j
(4.29)
There are two common types of errors in all CFD techniques that are difficult to be avoided. These
are: (a) discretization error, (b) round-off error. As mentioned in the previous paragraph, discretization
errors are caused by truncation error for the difference equation plus any additional error introduced by the
numerical treatment of boundary conditions. Round-off error is introduced due to the repetitive number of
calculations which the computer is constantly rounding the numbers to some particular significant figure
such as the limits of double precision (Anderson 1995).
There are four different numerical schemes that can be adopted as algorithms for the solution of
CFD problems. These are:
1) Semi-Implicit Method for Pressure-Linked Equations (SIMPLE)
2) SIMPLE-Revised (Patankar 1980) or SIMPLE-Consistent (Van Doormal and Raithby 1984)
3) Pressure Implicit with Splitting of Operators (PISO) by Issa (1986)
4) Coupled
The SIMPLE algorithm gives a method of calculating pressure and velocities for a given flow.
This method is iterative, and when other scalars are coupled to the momentum equations the calculation
need to be performed sequentially. The SIMPLER algorithm developed by Patankar (1980) is an improved
version of the previous. In this algorithm the discretized continuity equation is used to derive a discretized
equation for pressure instead of the pressure correction equations as in SIMPLE. Therefore the intermediate
pressure field is obtained directly without the use of correction factors. However, velocities are still
obtained using corrections. The PISO algorithm is a pressure-velocity computational procedure developed
for non-iterative analysis of unsteady compressible flows. However, it has also been successfully adapted
for the iterative solution of steady state problems. PISO involves one predictor step and two corrector steps
and may be seen as an extension of SIMPLE (Versteeg and Malalasekera 2007). The corrector steps
involve neighbor and skewness correction.
52
Coupled algorithms offer advantages over the segregated approach. The coupled method obtains a
robust and efficient single phase implementation for steady-state flows, with superior performance when
compared to the previous schemes. The pressure-based algorithm offers an alternative over the other
systems. For transient flows, using the coupled algorithm is a proper choice when the quality of the mesh is
poor, or if large time steps are used (ANSYS 2014). It is left to the reader to further explore the formulation
for each of these numerical algorithms as they are readily available in many literature references.
53
CHAPTER 5
THEORY OF STRUCTURAL DYNAMICS
5.1.
Basic Concepts of Vibration
Humans became interested in vibration when the first musical instruments were discovered. Since
then, we have applied a critical investigation in the study of the “vibration phenomena.” Galileo discovered
the relationship between the length of a pendulum and its frequency. Galileo also found the resonance of
two bodies which were connected by an energy transfer medium and tuned to the same natural frequency.
Many mathematicians such as Bernoulli, D’Alembert, etc. contributed to the development of vibration
theory. Sauveur coined the term fundamental for the lowest frequency and harmonics for the others. (Rao
1990). Whenever the natural frequency of vibration of a structure coincides with the frequency of an
external excitation load, it produces the phenomenon known as resonance. This phenomenon leads to
excessive deflections and can cause catastrophic failures (Rao 1990). In its broadest term, vibrations or
oscillations can be defined as any motion that repeats itself after an interval of time.
The theory of vibration focuses on the study of the motions of bodies and the forces associated
with them. A vibrating system includes means of having three types of energy. The first being potential
energy, the second being kinetic energy, and third by means of gradual loss of energy. In general, the
vibration of a system involves the transfer of its potential to kinetic energy, and vice-versa. The damping of
the system causes energy dissipation in each cycle of vibration. In order for a system to vibrate infinitely,
the system would have no means of damping or energy dissipation due to friction.
A structural dynamics problem differs greatly from its static equivalent. One difference between
dynamic and static analysis is that dynamic loading is time-varying in nature and the structural response to
that excitement also is time-dependent. In addition, the occurrence of inertia forces is another important
factor that distinguishes dynamic from static analysis. Therefore, there is no single solution for this
problem. The engineer must investigate a solution over a given period of time to fully evaluate the
54
structural response and to design accordingly. Dynamic analysis is more computationally intensive than
static analysis (Tedesco et al. 1999)
5.2.
Degrees of Freedom and Classification of Vibration
The minimum number of independent coordinates required to determine completely the position
of all parts of a system an any given instant defines the degree of freedom (DOF) of the system. For
example, a simple pendulum swinging in a plane represents a single degree of freedom system. Two
pendulums connected to each other is an example of two degrees of freedom; three pendulums connected to
each other is an example of three degrees of freedom and so on. A large number of practical systems can be
described with a specific and deterministic number of degrees of freedom. However some systems such as
a continuous elastic member can be said to have an infinite number of degrees of freedom. Systems with a
finite number of degrees of freedom are called lumped parameter/mass systems, and those with infinite
number of degrees of freedom are termed as continuous/distributed systems (Rao 1990). A continuous
model will exhibit the mathematical formulation of a system of partial differential equations. For lumped
systems, the mathematical formulation gives a set of ordinary differential equations (one for each DOF).
Most structures exhibit at least several degrees of freedom. The number of DOF that a structure
contains is equivalent to the number of independent spatial coordinates that can describe its geometric
configuration and motion (Tedesco et al. 1999). For each degree of freedom, there will be an equivalent
number of equations of motion and natural frequencies associated with it. A single node of a structural
element represented in a three dimensional space has a up to six degrees of freedom; three in translation
and three in rotation. The number of degrees of freedom can be reduced based on the boundary condition of
the node, such as fixed, pinned, or roller support.
Buildings can be idealized as lumped parameter systems with multiples degrees of freedom (i.e.
based on the number of stories it contains) as seen in Figure 28. For each DOF exhibited by a structure,
there corresponds a natural frequency of vibration. For each natural frequency the structure will vibrate to a
particular mode of vibration.
55
Figure 28: Idealized multi-story building with five degrees of freedom
(Credit: Rao 1990)
The mass or inertia element is assumed to be a rigid body that can gain or lose kinetic energy
whenever the velocity of the body fluctuates. Based on Newton’s second law of motion, multiplying the
mass by the acceleration of the body gives an equivalent force acting on the body. Work is equal to the
force multiplied by the displacement in the direction of the force acting on the system. The work done is
stored by the system in the form of kinetic energy of the mass. By assuming that the mass of the frame is
negligible compared to the mass of the floor system, the building can be treated as multi-degree of freedom
system as previously stated. The masses at the various floors represent the mass elements, and the vertical
columns are comparable to spring elements which contribute stiffness.
MDOF systems require matrix formulations to clarify the problem and to create a systematic
manner to conduct the response calculations. Matrices provide a convenient format for organizing the
calculations required to analyze MDOF systems. Matrix notations create the option for the engineer to use a
procedure to be able to program digital computers to arrive at solutions. There are three main physical
properties important to every MDOF systems: the mass, stiffness/flexibility, and damping matrices. If the
mode of superposition method is employed for the dynamic analysis, the system must be solved as an
eigenproblem (Tedesco et al. 1999).
5.3.
Equation of Motion and Natural Frequency
The mathematical expression that defines the dynamic equilibrium of a system is referred to as the
equation of motion of a structure. An important conclusion that can be derived from the solution of this
56
equation is the displacement-time history of a structure that is subject to a specific time-varying dynamic
load. The basic components of a vibrating system include: mass, stiffness/flexibility, damping, and forcing.
Damping is the energy loss of the system and forcing is the source of loading.
Figure 29: Model of a simple SDOF and free-body diagram of system
The mechanical model for a simple single degree of freedom (SDOF) systems and its
corresponding free-body diagram is depicted in Figure 29. It consists of a rigid body of of mass m, which is
limited to translate in only one direction, in this case the x-axis direction. A spring of stiffness k fixed at the
west end and attached to the body provides an elastic resistance to displacement. Energy dissipating system
is represented by a damper having its own particular coefficient c. The time-varying load being applied to
the systems is represented by F(t).
The motion of the mass is resisted by the force FS that develops in the spring and it is defined by
FS kx
(5.1)
A conservative system (i.e. no dissipation of energy) will continue to oscillate indefinitely even
after the applied load is removed. However, structures in the real world experience energy dissipation due
to the effects of friction or damping that prevents the structure from behaving as a conservative system.
Damping is a very complex phenomenon for which a number of analytical models have been developed to
describe its behavior. The most common model is the linear viscous dashpot model (Craig 1981). The
damping force is proportional to the velocity of the mass and is given by
FD cx
(5.2)
where c is the viscous damping coefficient having units of pounds-seconds per inch (lbf∙sec/in) or newtonseconds per meter (N∙sec/m).
The inertia force is equivalent to the product of the mass and the acceleration. The negative sign in
the below equation represents the inertial force acting against the acceleration of the mass.
57
FI mx
(5.3)
By using D’Alembert’s principle of dynamic equilibrium, the equations of motion of a SDOF and MDOF
systems can be established. For a given body with a constant mass, the rate of change of momentum is
equal to the product of the mass and its acceleration. By referring to the free-body diagram given in Figure
29, the expression for dynamic equilibrium can be expressed as
mx cx kx F (t )
(5.4)
If we divide equation by m we can obtain the natural circular frequency, ω0, of a system with the
units of radians per second. To obtain the natural frequency, f, of the system we further divide ω0 by 2π as
seen in the below equation. The units for natural frequency is Hertz (Hz) or cycles per second. The natural
frequency of a structure plays an important role in vibration analysis (Tedesco et al. 1999).
f
5.4.
0
k/m
2
2
(5.5)
Property of Matrices for Vibrating Systems
In this section the author briefly discusses the flexibility/stiffness and mass matrices since these
matrices are of relevant significance in structural dynamic theory. The document also briefly discusses the
eigenproblem in vibration analysis.
5.4.1. Flexibility and Stiffness Matrix
The displacement and forces that act upon a structure can be related to one another by using
flexibility or stiffness properties. First, the discussion focuses on the concept of flexibility but then move on
the stiffness since the unique relationship between these methods will be evident.
Consider the structural beam illustrated in Figure 30. If we apply a load F2 at the second point of
the beam, deflections at point 1, 2 and 3 will be x1, x2, x3 respectively. The same concept can be done for
points 1 and 3 and the following expression can be arrived at in matrix form
x1 a11
x2 a21
x a
3 31
a12
a22
a32
a13 F1
a23 F2
a33 F3
(5.6)
The matrix (a) is known as the flexibility matrix that contains the influence coefficients that relate
the displacements, x, relative to the applied loads, P. For elastic systems, the flexibility matrix is
symmetric.
58
Figure 30: Flexibility coefficients for a beam
It can be easily demonstrated that the inverse of the flexibility matrix results in the stiffness matrix
[k] since the stiffness matrix relate the forces {F} to the deformations {x}. The force vector {F} contains
forces and moments, and the deformation vector {x} contains translations and rotations.
k a
F k x
1
(5.7)
The stiffness can be expressed in verbal terms as the force or moment required to produce a unit
displacement or rotation at a given point in a structure if displacement or rotation is stopped from occurring
at all other nodes. The nodes of interest correspond to the defined DOF of the system.
5.4.2. Mass Matrix
The mass matrix will generally be diagonal in nature and it includes the inertia effect associated
with the translational DOF. The inertial influence for rational DOF may also be included. The lumped mass
matrix for an element of a three-dimensional frame possess coefficients which are equal to one half of the
total inertia of the beam segment. A mass matrix for a uniform beam can be obtained in greater complexity
by including the matrix of axial effects, the matrix for torsional effects, and the matrix for flexural effects
(Paz 2004).
x1
F1 m1 0 0
F 0 0 x
F 0 0 m
n xn
n
(5.8)
m1
mL
2
and
m2
Figure 31: Lumped mass coefficients for structural elements with distributed mass
59
mL
2
5.5.
Eigenproblem Formulation, Natural Vibration Frequencies and Modes
Analysis of a vibrating system with MDOF is required to go the solution of the standard
eigenproblem which can be expressed as
DA A
(5.9)
where [D] is a square symmetric matrix, {A} is the solution vector, and λ a scalar quantity equivalent to ω2.
{A} must not be a null vector to allow for Equation 5.9 to be satisfied. The solution is given by the
eigenvalue λr and corresponding eigenvector {A}r where r represents the rth solution. So each solution
consists of an eigenpair with solutions given arrived to be ((λ1,{A}1), …, ((λn,{A}n)) where λ is arranged
from smallest to largest numbers.
In the analysis of structures for vibration, [D] is knowns as to the dynamic matrix and is usually
formulated by the stiffness matrix. The n eigenvalues of the dynamic matrix represent the natural
frequencies of the structure, and the corresponding eigenvectors represent the modes of vibration as normal
or principal. The eigenvector can be multiplied by any chosen constant cr. Equation 5.8 is satisfied if we
multiply both sides and a more general form can be arrived at by introducing the modal vector {Φ}r
corresponding to λr equivalent to cr{A}r. The relative values of the elements in the matrix remain
unmodified and the eigenvector {A}r is normalized by the constant. Therefore by considering all λn’s, the n
number of modal vectors {Φ}r from the modal matrix that defines all the normal modes of vibration of the
system. (Tedesco et al. 1999). The modal matrix [Φ] can be written in terms of the vectors as
nn
11 22
1 , 2 , , n 21 22
n1 n 2
1n
2 n
nn
(5.10)
For a system with a large quantity of degrees of freedom, obtaining the solution for the
eigenproblem must be accomplished via computerized numerical eigensolvers to extract the mode shapes
and natural frequencies of the structure. We must also note that a MDOF vibrating systems possess an
important property referred to as orthogonality. The orthogonality of the eigenvectors system is with
respect to the stiffness and mass matrix. For modes with different frequencies, where ωs ≠ ωr, it follows
60
s m r 0
T
s k r 0
T
(5.11)
These orthogonality conditions of normal modes are the basic theory that leads to the modal
analysis of a vibrating system; the formulation of the aforementioned system is left for the reader to
explore.
61
CHAPTER 6
CFD ANALYSIS OF WIND FLOW USING RANS & LES FORMULATIONS
6.1.
2D & 3D Setup of Model in CFD
The computational domain defines the region where the flow field is computed numerically used
the assigned models. It should be large enough to accommodate all relevant flow features that will have
potential effects in altering the characteristics of the flow field (Franke et al. 2007). The building geometry
for the computational domain was based on wind tunnel scaling for the study of prototype structures. In the
case of our model, the prototype was a square structure dimensioned as 40 meters (B) by 40 meters (W)
(130 feet × 130 feet) in plan form, and had height of 300 meters (H) (≈ 1,000 feet) in order for it to qualify
as a super-tall building. The structure had a height to base ratio of 7.5 which made it fall in the category of
a slender structure. Typically, slender structures are greatly affected by lateral loading and extra diligence
must be taken by the engineer when designing such structures that are known to be sensitive to lateral
loading.
The wind tunnel scale was chosen to be 1:400 as that is typical in industry practice to use this
factor between prototype and model studies. In the case of the chosen building, this reduced its dimensions
to 0.1 × 0.1 × 0.75 meters. The constant ‘B’ throughout the study was computed to be equivalent to 0.1
meters based on the scaling ratio. The aforementioned constant was used as a common variable to describe
all geometric properties of the model.
CFD problems must be defined by initial and boundary conditions for meaningful numerical
output. It is important for the engineer to understand their vital role in the numerical algorithm. The most
common boundary conditions used are: inlet, outlet, wall, and symmetry. The distribution of the flow
variables must be specified at the inlet along the upstream direction. Outlet boundary conditions are
selected at the downstream direction and distant enough from geometrical disturbances so that the flow can
reach fully developed state. The wall boundary conditions is the most commonly used in confined fluid
flow problems in which the no-slip condition is associated with it. Symmetry is regarded as a condition
62
where normal velocities are set to zero, and the values of all other properties just outside the domain are
equated to their values at the nearest node inside the domain. Figure 32 represents a sketch of the proposed
computational domain dimensions and boundary conditions used for the CFD analysis performed in this
study. The three dimensional boundary conditions were also used as a basis for the configurations used for
the two dimensional studies.
Figure 32: 3D Boundary conditions
The plan form of the geometric properties and boundary conditions can be seen in Figure 33. The
overall computational domain was 21B × by 47B where the structure was centered at 10.5B from the inlet
and 10.5B from the south wall condition. A distance of 36B in the downstream direction was chosen to
allow for the reduction of the disturbances experienced by the flow. The north and south edge were
assigned the wall boundary conditions to simulate a wind tunnel cross section.
Figure 33: 2D Plan boundary conditions
63
The elevation form of the geometric properties and boundary conditions can be seen in Figure 34.
The overall computational domain was 18.75B × by 47B where the structure was centered at 10.5B from
the inlet. Again, a distance of 36B in the downstream direction was chosen to allow for the reduction of the
disturbances experienced by the flow. A symmetry condition was applied to the top edge of the
computational domain.
Figure 34: 2D Elevation boundary conditions
6.2.
Grid Independent Study
A grid sensitivity analysis for the model was performed using different number of nodes in
increasing multiples of two. This type of analysis must be performed to reduce the influence of the number
of nodes on the computational results since the solution must be independent of the mesh resolution in the
computational domain. It is good practice to run this study before a more global analysis of the system is
completed. The building geometry for the computational domain was based on wind tunnel scaling for the
study of prototype structures. In the case of our model, the prototype was a square structure dimensioned as
40 meters (B) by 40 meters (W) (130 feet × 130 feet) in plan form, and had roof height of 300 meters (H)
(≈ 1,000 feet) in order for it to qualify as a super-tall building. The structure had a height to base ratio of
7.5 which falls in the category of a slender type structure. Typically, slender structures are greatly affected
by lateral loading and extra diligence must be taken by the engineer when designing such structures that are
known to be sensitive to lateral loading.
The wind tunnel scale was chosen to be 1:400 as that is typical in industry practice to use this
factor between prototype and model. In the case of the chosen building, this reduced its dimensions to 0.1 ×
0.1 × 0.75 meters. The constant ‘B’ throughout the chapter was assigned to be equivalent to 0.1 meters
which was used as a common variable for all geometric properties of the model. Figure 35 displays typical
64
meshing arrangement used for the simulation. As evident, the mesh was concentrated in the immediate
surroundings of the structure to capture the flow separation and consequent fluid behavior. The mesh was
created using the software ICEM CFD in which it allows for the user to have greater control of the meshing
criteria to be utilized for any type of geometries, whether they are regular or irregular shape. ICEM CFD
allows the user to have control of the mesh by utilizing the concept of blocking.
For the simulation involving 157,000 nodes, the model used a very fine grid cell in the near-wall
region and ensured a non-dimensional wall distance y+ to be less than 5 which was well within the inner
sublayer. A bias ratio 1.05 was applied to the mesh between successive grid points to ensure solution
convergence for the numerical methods applied. The blocking of the geometric model in ICEM CFD
allowed the author to place 100 grid points in the upstream direction, and 360 grid points in the
downstream directions. The transverse directions were also assigned 100 grid points. The bias were applied
to the nodes approaching the walls of the square structure. They were also applied in the direction of the
north and south wall of the wind tunnel. The building structure was assigned 100 grid points along the four
faces of the building where were also biased to have a greater concentration along the sharp edges of the
building. The combination of the aforementioned arrangements generated a structure quadrilateral mesh
with near perfect orthogonality. This allowed the CFD model to perform with one less possible source of
error in the computation of results.
Figure 35: 157 K Nodes quad-mesh for 2D plan view of structure
For the grid analysis of 78,000 and 44,000 the same concept as outlined above was applied. The
only difference was in the reduction of the grid points along the faces of the building, the upstream and
65
downstream flow, and transverse directions. This introduced an increase in aspect ratios between the
rectangular mesh elements.
A similar principle was applied for the meshing of the two dimensional elevation study of the
wind flow around the structure. The upstream was divided into 100 nodes, and the downstream was divided
into 360 nodes. Along the height of the building, 150 nodes were specified to better capture the wind
pressures acting in that direction. Directly above the building, the number of nodes was reduced to 41
points. The final meshing resulted in 90,000 nodes with rectangular shapes and it is displayed in Figure 36.
ICEM CFD evaluation of the orthogonality of the meshes proved to be near perfect (i.e. close to a value of
unity).
Figure 36: Quad-mesh for 2D elevation of structure
The grid independent study was validated using two different data outputs from the model. The
first was based on the concept of the development of fluid flow along a “pipe.” A wind tunnel can
essentially be seen as a pipe when a cross section of the tunnel is taken along the longitudinal direction.
The region where the fluid enters the pipe is known as the entrance region. The fluid in this experiment
entered the tunnel with a uniform velocity. As the fluid moved through the pipe, viscous effects caused the
air to stick to the wind tunnel wall therefore a boundary layer was formed along this edge. Figure 37
represents this behavior that occurred and was captured in the CFD model. At the inlet, the normalized
velocity magnitude was constant. However at a distance of 45B from the inlet, the normalized velocity
magnitude taken along a height of 0.2 meters from the face of the south tunnel wall, displayed the
characteristical behavior of “pipe” flows. The boundary layer formed at 45B was captured for the fully
developed flow. That was made possible due to the fine mesh resolution near the wall that was specifically
66
designed by the author. Based on the figure, the boundary layer for three different meshing followed the
same profile. Moreover, the 78 K nodes model was almost identical to the 157 K nodes model thus
indicating that the solution has reached an acceptable grid independence and the engineer can opt to go
with the lowest amount of nodes (i.e. 44 K) to continue the study.
V/Vstream (unitless)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
0.05
0.1
0.15
0.2
Position perpendicular to South wall (m)
Inlet
44 K Nodes
78 K Nodes
157 K Nodes
Figure 37: Boundary layer development along wind tunnel section
The second data set chosen to ensure the grid convergence of the study was chosen to be the mean
pressure coefficient (Cp) acting along the windward face of the structure as presented in Figure 38.
Cp
p p
0.5V2
(6.1)
The pressure coefficient describes the relative pressure throughout a flow field in fluid dynamics
and it is a dimensionless number. In engineering modeling of fluid dynamics problems, this number is of
paramount importance since a model can be tested at a wind tunnel scale where the Cp values can be
determined at critical locations, and these same coefficients can be used with confidence to predict the fluid
pressure at these same critical locations of the full-scale structure.
When comparing the mean Cp values at y/B = 0.5 for the cases with lesser nodes, the percent error
between the values of mean Cp decreases as the number of nodes increases. In other words, the error
between mean Cp at 157 K and 78 K is approximately 2%, and the error between mean Cp 157 K and 44 K
is almost 7%. Thus, giving a clear indication of grid convergence as the number of nodes increased. For the
purpose of this study, a 44 K node analysis was adopted as an adequate meshing option.
67
Mean Pressure
Coefficient (Cp)
1.5
1
0.5
0
-0.5
-1
-1.5
0
0.2
0.4
0.6
0.8
1
y/B
44 K Nodes
78 K Nodes
157 K Nodes
Figure 38: Variation of mean pressure coefficient along the windward face for grid analysis
6.3.
Model Validation
Validation of a model being investigated by a CFD user is imperative to perform. Validation is
seen as the process of determining the degree to which a model is an accurate representation of the real
world from the perspective of the intended model uses. The best way to achieve comfort with a CFD model
is to make a direct comparison to published or experimental data. In the case of this study, numerous
physical studies have been performed for the fluid flow around square geometries.
Any object where fluid is moving around it will experience drag, which can be simply defined as a
net force in the direction of flow due to the pressure and shear forces acting on the surface of the obstacle.
Typically an object with any given shape can be defined by its drag coefficient, CD, where mathematically
it can be written as
1
FD CD U 2 A
2
(6.2)
The drag coefficient is also a function of dimensional parameters such as Reynolds number, Mach number,
Froude number, and relative roughness. A square cylinder has been found out to be a CD value of
approximately 2.10 derived from experimental studies as reported in Table 3.
68
Table 3: Experimental data and derived quantities for various cross sections
(Credit: Ahlborn et al. 2002)
The Reynolds number (Re) is considered to be the most famous non-dimensional number in fluid
mechanics. This number demonstrated the relationship of the combined variables to determine whether the
flow was in laminar or turbulent state. The ratio between the inertial forces and the friction force acting on
the volume of a fluid is expressed mathematically as:
Re
VL VL
(6.3)
The validation of the model was performed with the fluid behaving in a turbulent manner since its
Reynolds number was in the order of 2.0 × 105. The details of the model set up can be found in Appendix D
as it contains all the means and methods used as input parameters for the model in ANSYS Fluent. Figure
39 presents a time-dependent plot of the coefficient of drag that the square structure possesses. The
comparison of the coefficient of drag computed by the numerical CFD model was in very close agreement
to experimental data.
2.20
Coefficient of Drag
2.10
2.00
1.90
1.80
1.70
1.60
1.50
1.40
0.00
0.20
0.40
0.60
0.80
1.00
Flow time (sec)
Figure 39: Drag coefficient curve for 44K SST-κω transient model
69
The CFD model was also programmed to measure the coefficient of lift, which by definition acts
Coefficient of Lift
in perpendicular direction to the drag forces.
2.50
2.00
1.50
1.00
0.50
0.00
-0.50
-1.00
-1.50
-2.00
-2.50
0.00
0.20
0.40
0.60
0.80
1.00
Flow time (sec)
Figure 40: Lift coefficient curve for 44K SST-κω transient model
An additional validation analysis was performed based on the numerical output of the CFD
analysis based upon the stream line velocity of the fluid along the centerline of the structure as plotted in
Figure 41. It can be seen that the κ-ε predictions by Franke and Rodi (1991) show an unrealistically large
recirculation zone and therefore the discrepancy is noted. The present study utilized the SST-κω model
showed a large improvement in the performance of the formulation. The upstream velocity is predicted
accurately up to the stagnation point that the square structure experiences near its midpoint in the windward
face. The model also shows significant performance along the downstream flow. The SST-κω formulation
captures the immediate wake of the flow velocity occurring within a distance of 1B from the leeward face.
The normalized velocity follows relatively closely the slope as presented by the experimental study
performed by Durão et al. (1998) and Lyn et al. (1995). The fine resolution of meshes in the two
dimensional plane offered adequacy for the CFD model.
70
1.2
1.0
0.8
U/Uo
0.6
0.4
0.2
0.0
-0.2
-0.4
-2
-1
0
1
2
3
x/B
4
5
6
7
Durão et al. (1988)
Lyn et al. (1995)
Franke and Rodi - Stnd. κ-ε (1991)
Present - SST κ-ω - Transient 3 sec
8
Figure 41: Mean velocity magnitude along centerline
It can be stated with confidence that the use of the SST κ-ω was of adequate choice for the
numerical computation of the wind flow patterns acting near the structure.
6.4.
Discussion of Results Based on 2D Plan Model
A multiple level analysis was performed for sections located along the height of the structure in
the wind tunnel CFD model of the slender building. The parameters used for the analysis along the height
are presented in Table 4. The chosen section corresponded to points of interest along the height of the
building. The Reynolds number was based by the characteristic length, L, equivalent to 0.1 meters (B) and
the corresponding wind speeds at the respective section cut heights levels. The wind profile was computed
using the power law with the exponent 1/α = 1/7. The velocity scale was proportioned to the following 1:4
ratio from the prototype wind profile for urban/suburban terrain located in Miami, Florida. The data
presented in Table 4 was deemed acceptable inputs based on computations and scaling assumptions for the
present study.
71
Table 4: Model input criteria for 2D plan CFD
SECTION
CFD INPUT FOR ANALYSIS
Turbulence
Reynolds
Wind Speed (m/s)
Turb. Length Scale (m)
Intensity (%)
No.
23.9
1.31E+05
19.6
0.250
No.
%H
Height (m)
Cut 1
3.33%
0.0250
Cut 2
10.0%
0.0750
22.9
18.9
0.250
1.53E+05
Cut 3
30.0%
0.2250
26.8
15.6
0.433
1.79E+05
Cut 4
60.0%
0.4500
29.6
14.1
0.613
1.98E+05
Cut 5
91.7%
0.6875
31.4
13.2
0.758
2.10E+05
A second property was compared to previous experimental studies and CFD analysis performed
by Kawamoto (1997) in which the coefficient of pressures were measured along the four faces of a square
prism. In order for the results in this study to be comparable, the present model was matched to the
Reynolds number used by the work performed by Kawamoto (1997).
1.5
1
Mean Pressure
Coefficient (Cp)
0.5
1
2
0/4
3
0
-0.5
-1
-1.5
-2
-2.5
0
1
2
3
Coordinate of Square Prism (d/B)
Kawamoto - Stnd. κ-ε (1997)
4
Kawamoto - Experimental (1997)
Present - SST κ-ω - Transient 3 sec
Figure 42: Variation of mean pressure coefficient along faces
Figure 42 is a chart that compared the present SST κ-ω model mean pressure coefficients along the
windward, leeward and suctions faces of the pyramid. The windward face was represented by the 0 to 1
normalized coordinate system, the leeward face was identified by 2 to 3. Comparing the results to the
experimental and numerical model, the present study provided good agreement with the published
literature. The negative coefficient of pressures followed the same profile although their magnitude was
relatively larger.
72
6.5.
3D LES Model
LES is a multi-scale computational approach that offers a comprehensive way of capturing
unsteady flows. The use of LES as a wind load evaluation tool has been improved significantly in recent
years through the use of numerical techniques. LES holds promise to becoming the future of computational
wind engineering for which turbulent flow is of fundamental importance. In this study the SmagorinskyLilly subgrid-scale was employed.
The discretization of convective terms the central difference scheme gives adequate accuracy
when compared to upwind schemes. The bounded central difference was adapted to minimize the effect of
oscillations arising from high Reynolds number flows in the wake regions. For temporal discretization,
second order schemes are recommended by the CWE community. The PISO algorithm was used for
pressure-velocity coupling and is typically used in LES simulations. This algorithm is based on the higher
degree of the relationship approximations between corrections for velocity and pressure. Figure 43 was a
representation on the meshing criteria used for this model. The same concept as previously presented was
utilized in the creating of this computational domain. The mesh was concentrated around the slender
building structure.
Figure 43: 3D mesh with 7.7 M nodes
The simulation has been carried out by utilizing a newly released High Performance Computer
(HPC) based on Linux Operating System available at Florida Atlantic University. The computations have
73
been performed using 24 physical central processing units (CPUs). The time step chosen for the model was
2 × 10-4 seconds with 25 sub-iterations per time step. A strict convergence criteria was applied to the model
equivalent to 10-5 to ensure convergence of the solution.
In addition to the above specifications, a user defined function (UDF) based on the computer C
language was written by the author to simulate the desired wind profile. The profile was in accordance to
the assumptions mentioned previously. The code was written to accommodate the three dimensional model
and is made available in Appendix C.
Figure 44: Wind profile generated in computer domain
Figure 44 is the computer output of the velocity profile as written by the code developed by the
author. It was based on the power law and carried an exponent of 1/7 as per the prototype model conditions
and ASCE 7-10 recommendations.
The LES model was ran with the HPC facility for 17 days and at a very fine time step and subiterations. No grid independent study was performed since the set-up of the analysis was based to similar
work performed by Dagnew (2012) based on a rectangular building with a ratio of width to length of 2:1.
74
Figure 45: Present LES windward contours compared to experimental data
(Credit: Simiu and Scanlan 1996)
The windward contours have been graphically post-processed from the LES model of this present
study. In Figure 45 the left image represents the contours of the study which was compared to previously
published wind tunnel measurements and the respective pressure coefficient contours acting on a slender
square slender body (Simiu and Scanlan 1996). The results have good agreement as it can be seen from the
approximate 0.9 coefficient of pressure predicted by CFD to the experimental study. The higher
magnitudes are highlighted with red colors, and the lower ones with blue colors. The pressure variation
along the face of the CFD model truly represents the wind behavior acting on the slender body.
6.6.
Pressures Acting on Slender Structure
Similarity requires that the reduced frequencies and the Reynolds numbers be the same in the
laboratory and in the prototype. This should hold true regardless of the nature of the frequencies involved,
or of the densities being considered. There are three fundamental requirements concerning mass, length,
and time in which three fixed choices of scale can be made (e.g. length scale, velocity scale, and density
scale).
In principle, for similarity between prototype and wind tunnel flows to be achieved, the respective
Reynolds numbers must be the same, also known as the Reynolds number similarity. In wind tunnels, the
fluid used is air at atmospheric pressure, and Reynolds number similarity is violated. Bodies with sharp
75
edges are assumed to be “independent” of Reynolds number. It is assumed that flows around these bodies
are similar at full scale and in the model scale.
1,000
900
800
Height (ft)
700
600
500
400
300
200
100
0
0
50
100
150
200
250
Pressure (psf)
3D LES
ASCE 7-10
SAP2000
2D SST KW
Figure 46: Windward estimated pressure variation
Hand calculations were carried based on ASCE 7-10 Main Wind Force Resisting System
(MWFRS) and presented on Appendix A. The profile of the pressure generated on the windward face can
be seen in Figure 46. The structural analysis software SAP2000 contained a built in wind loading
evaluation tool and the criteria outlined in Appendix A were given as an input. A parabolic pressure profile
was generated by the structural software as is typical in for these types of tall structures. Two different CFD
techniques were utilized to derive the pressures acting along the windward face of the building, the first
was using the two dimensional SST κ-ω formulation, and the second was based on the three dimensional
LES. The maximum pressure acting on the face of the building using ASCE 7-10 hand and software
method were determined to be approximately 150 lb/ft2. However, the two dimensional SST κ-ω CFD
model estimated the pressure to be approximately 220 lb/ft2, which is approximately 46% larger than the
ASCE 7-10 provisions. The three dimensional LES model presented a result of approximately 160 lb/ft2 or
7% greater than the code. The LES profile seen in the above figure represents a reasonable representation
of the true flow and pressure variation acting along the building face. The two-dimensional CFD model is
an over simplified analysis and generates results that overestimate the wind loads on the tall building
model.
76
CHAPTER 7
BASIC STRUCTURAL AND DYNAMIC ANALYSIS OF TALL BUILDING
7.1.
Overview of Idealized Example Structure
The structure being considered in this study was a reinforced concrete structure consisting of 67-
stories. The overall plan dimensions of the building were 129.5 by 129.5 feet (40 by 40 meters) containing
7 bays in the horizontal x and y directions. The tower was 1,010 feet (300+ meters) high above the street
level. The façade was assumed to be a glass and aluminum curtain wall system. The tower floors consist of
reinforced concrete beams spanning from the cores to the façade with a concrete two-way slab deck. The
column spacing was at 18.5 feet center-to-center. The typical floor-to-floor height assigned was 15 feet (4.6
meters). However the first story had a floor height of 20 feet (6.1 meters) to produce a typical architectural
features of a tall building. Figure 47 illustrates the typical framing plan of the structure in different views.
The wind resistance structure consisted of the core walls cantilevering from the base of the building to its
full height. In these systems typically 85-95% of the lateral load is carried by the shear wall.
(a) Discretized FEM model 67-Story
tower
(b) Floor plan framing
Figure 47: Framing model of 67-story tower used in present study
77
(c) East elevation framing (typ.)
(d) North elevation framing (typ.)
Figure 47 (cont’d): Framing model of 67-story tower used in present study
The frame was designed as moment resisting which consists of horizontal and vertical components
that are rigidly connected together in a planar grid form. This system primarily resists through the flexural
stiffness of the members. A point of contra flexure is usually located near the midpoint of the beams and
columns. The deformation of the frame is typically induced by shear-sway and partially by column
shortening. Some advantages of using this system is the flexibility it provides to architectural planning.
The size of the members used in a moment resisting frame is often controlled by stiffness
characteristics rather than strength. The stiffness of elements greatly control acceptable lateral drifts for a
tall structure. The structures response is mainly a function of column and beam stiffness. Moreover, in
moment resisting frames the size of the column members tend to decrease in size as higher levels are
reached in the building which is in proportion to the lateral shear. Advances in computer technology have
allowed for better analysis of indeterminate moment frames. Optimization techniques when used also
determine the most efficient distribution of material for a given deformation limit analysis.
Reinforced concrete frames have positive aspect in which the connections are casted
monolithically which are well suited for moment resisting systems. Advances in testing has led to
conclusions in improved concrete characteristics, additional reinforcement requirements for the
improvement of ductility, and better frame forming techniques to be used in the field.
7.2.
Assigned Loading and Section Properties to Structural Members
The loading applied on the analysis of this building was based on ASCE 7-10 Minimum Design
Loads on Buildings. The building was assumed to exist in Miami, FL in an urban/suburban region. The
function of the building was assumed to be an office high-rise located near the downtown area of this
78
hurricane prone city. Further details for the assumptions used for the design of the building can be found in
Appendix A.
Table 5: Assumed loads acting on structure
Description
Type
Magnitude (psf)
DL
Member Size x Unit Weight
SDL
10
Live Load
LL
50 (Floor) ; 100 (Lobbies) ; 80 (Corridors)
Roof Load
RL
20
Wind Loads
WL
Generated per ASCE 7
Dead Load
Superimposed Dead
Based on the ASCE7/ACI318 there are typically seven load combinations to be used for the
strength design method. Structures, components, and foundations are required to be designed so that their
strength is greater or equal than the effects of the factored load in the following combinations:
1.
1.4D
2.
1.2D + 1.6L + 0.5(Lr or S or R)
3.
1.2D + 1.6(Lr or S or R) + (L or 0.5W)
4.
1.2D + 1.0W + L + 0.5(Lr or S or R)
5.
1.2D + 1.0E + L + 0.2S
6.
0.9D + 1.0W
7.
0.9D + 1.0E
There are a few exceptions and conditions that must be met for each case. The information can be
readily found in published literature and codes. For the case of this structure and the application of the
loading presented in the previous table, there were 74 problem specific combinations that were analyzed for
the structure. The large quantity of combinations is generated due to the study of the wind effect in multiple
directions acting along the building. This provides the analysis a comprehensive overview of the loads
acting on the whole structure. Based on this information, the members can be designed to account for the
worst case bending, shearing, and torsional conditions of the loading being experienced by the structure.
Based on the discussion of the previous section, the concrete building was designed taking the
state of the art guidelines into consideration. The building was subdivided into five categories where
different section properties to the members of the structure where assigned. The first category was
concentrated along the base region of the building where most of the bending and shearing loading is
79
concentrated. The fifth category was located on the last seven floors of the structure where the members
tend to be smaller in dimension.
Table 6: Cross-section and properties of members along the height of the structure
Column Design Shear Wall Beam Design Slab Design Header Beam
Cat.
Floor No. Elevation B x W
(ft)
1
2
3
4
5
(in)
f'c
Thick.
f'c
B xH
f'c
Thick.
f' c
B xW
f' c
(psi)
(in)
(psi)
(in)
(psi)
(in)
(psi)
(in)
(psi)
1
20
50 x 50 10,000
24
10,000 24 x 34 5,000
8
5,000 24 x 48 10,000
15
230
50 x 50 10,000
24
10,000 24 x 34 5,000
8
5,000 24 x 48 10,000
16
245
48 x 48 10,000
24
10,000 24 x 34 5,000
8
5,000 24 x 48 10,000
30
455
48 x 48 10,000
24
10,000 24 x 34 5,000
8
5,000 24 x 48 10,000
31
470
44 x 44 10,000
24
10,000 24 x 34 5,000
8
5,000 24 x 48 10,000
45
680
44 x 44 10,000
24
10,000 24 x 34 5,000
8
5,000 24 x 48 10,000
46
695
40 x 40
8,000
24
10,000 20 x 34 5,000
8
5,000 24 x 48 10,000
60
905
40 x 40
8,000
24
10,000 20 x 34 5,000
8
5,000 24 x 48 10,000
61
920
36 x 36
6,000
24
10,000 20 x 34 5,000
8
5,000 24 x 48 10,000
67
1010
36 x 36
6,000
24
10,000 20 x 34 5,000
8
5,000 24 x 48 10,000
Table 6 illustrates that columns had the greatest cross section along the height located within the
section named Category 1. The final column design was arrived using an iterative procedure and verified
against the ACI318-08 code requirements using SAP2000 Version 16 as the structural analysis software.
Moreover, the columns had to be design using concrete with relatively high compressive strength (e.g. f’c =
10,000 psi). The cross section of the columns along the Category 5 levels was significantly reduced by
approximately 48% when comparing it to the base floor levels. The compressive strength of the concrete
was also reduced by 40%. The interior and exterior columns were assumed to be equivalent in dimensions
throughout all category levels to simplify computational design time. The reinforcement specified for this
model was ASTM A615 Grade 60 and Grade 75 steel. Grade 60 steel was used in the design of beams and
slabs. Grade 75 steel was used for the design of columns and shear walls.
Beam design along the height of the building was concluded to be almost uniform since the span
of the bays were equal in both longitudinal and transverse direction. The only difference being that
throughout Category 1 and 3, the width of the beam had to be 24 inches (610 mm) wide to accommodate
the larger shearing forces acting at those elevations imposed by the lateral loading acting on the structure.
The slab design remained constant along all levels of the building and was assumed to provide rigid
diaphragm action on the structure. The shear wall was designed to be 24 inches thick and have a
80
compressive strength of 10,000 psi along the full height of the structure. A final element, header beams,
was used to ensure proper framing, stiffness, and interaction among the shear walls in the building. They
were placed in such a manner that they framed physically to all corresponding core walls.
7.3.
Idealized Structure Discretization
Beams are the most common type of structural components used in civil engineering. A beam can
be thought of as a bar-like structural element whose function is to support transverse loadings and
effectively transfer the forces to the supports. A beam is significantly larger in magnitude in the
longitudinal direction, than the two cross-section dimensions. A longitudinal plane passes through the beam
axis. A beam resists loads mainly through bending action. Moments acting on the beam produce
compressive stresses on one side, and tensile stresses on the opposite. These two surfaces are separated by a
neutral axis/surface of zero stress. The interaction of these stresses causes an internal bending moment
which is the mechanism that transports the loads to the support. Beams must also support shearing forces
acting upon them to be properly used as a structural component in a system. Columns are vertical members
that carry significant importance as component of a structure. Stacked one on top of the other, they receive
loads that must be finally transmitted to the foundation system. Columns can be considered to be a type of
beam with a different function. Columns must resist axial and flexural loads to be effective.
Beams and columns are analyzed as frame elements. A frame element is modeled as a straight bar
with an assigned cross-section which can deform in the axial and perpendicular directions. The bar is
capable of carrying axial and transverse loads, and moments. A frame element possesses the properties of
truss and bema elements which is encountered in most of real world structural elements. The formulation of
frame elements is based on the equations for beam elements developed by the finite element method. Many
commercial softwares, such as SAP2000, use the theory of frame elements for actual engineering
applications such as the analysis of tall buildings. Frame elements are applicable for the analysis of skeletal
space frames in three dimensions and were therefore used in this study. The formulation of such theory is
left to the reader to explore, and they can be found in finite element textbooks such as Logan (2012).
A plate is considered to be a two-dimensional extension of a beam in simple bending. Beams and
plates support transverse loads to their plan and through bending action. A plate is considered to be flat,
and a shell is considered to be curved, therefore a plate is a shell with an infinite radius of curvature.
81
Kirchhoff plate theory is considered to be the classical derivation of equations in this analytical concept.
For the structural model in this study, slabs were assigned the property of shell-type behavior which
accounts for in-plane membrane stiffness and out-of-plane plate bending stiffness that are provided in the
section. This was deemed adequate since it can receive horizontal loading due to wind action, and can
account for bending caused by gravity loading. The slab was modeled as a thin shell element. The shear
walls were also considered to fall within this acceptable theory, however a thick shell element was assigned
to this structural section.
In structural engineering, a diaphragm is a structural system used to transfer lateral lads to shear
walls primarily through in-plane shear stresses. The diaphragm of a structure usually has an additional
function in where it can also support gravity loading. Diaphragms can also provide lateral support to walls
and parapets. Typically, roofs and floors participate in the distribution of lateral forces to vertical elements
up to the limit of its strength. For a slab to function as a diaphragm, horizontal elements must be
interconnected to transfer shear with relatively stiff connections. Rigid diaphragm distributes the horizontal
forces to the vertical resisting elements in direct proportion the relative rigidities. This element is assumed
to not deform and therefore will cause the vertical elements deflect the same amount. A rigid diaphragm
assumes infinite in-plane stiffness of floors, and therefore reduces the stiffness matrix. The super-tall
building was analyzed using the rigid diaphragm approach for the lateral loads induced by the wind acting
along the faces of the structure.
7.4.
Vibration Modes of Modeled Structure
A normal mode of a system is a pattern of motion in which all parts of the system moves in
sinusoidal pattern with the same frequency and a fixed phase relation. The frequencies of the normal modes
of a system are referred to as its natural frequencies. A building has a set of normal modes that mainly
depend on its structure stiffness, material’s mass, and boundary conditions. Every building has a number of
ways, or modes, in which it can vibrate naturally. For each mode, the structure can oscillate naturally to and
fro with a particular distorted shape known as its mode shape. The building can be made to sway at other
frequencies of vibration with different mode shapes than its fundamental (lowest) frequency when the
loading occurs at different levels. One of the main functions that affect the stiffness of a building is its
height. Therefore, taller buildings tend to be more flexible and susceptible to lower natural frequencies. The
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results presented on Figure 48 and Figure 49 were arrived using the eigenvector modal analysis code
imbedded in SAP2000 which follows the theory presented in Chapter 5. The eigenvector analysis
determines the undamped free-vibration mode shapes and frequencies of the system. The analysis was
limited by the user to six modes of vibration.
(a) Mode 1 – f = 0.1563 Hz
(b) Mode 2 – f = 0.1595 Hz
(c) Mode 3 – f = 0.2949 Hz
Figure 48: First three modes of vibration of tall structure
Generally speaking, the modes of vibration can be understood in the following simplified terms.
The first mode is a translational in direction. The second mode is also translational. The third mode is a
torsional mode. The rest of the higher modes are typically a combination of the first three. For the study of
83
a building, the first five to six modes give plenty of information to the structural engineer in terms of design
criteria.
(a) Mode 4 – f = 0.5573 Hz
(b) Mode 5 – f = 0.5938 Hz
(c) Mode 6 – f = 0.8391 Hz
Figure 49: Fourth to sixth modes of vibration of tall structure
In order to mitigate vibrations, the engineer can make several modifications in the design of the
building to avoid possible resonance between the structure and the lateral loading. If the structure vibrates
at frequency which humans are sensitive to, the engineer can alter the structures period of vibration by
altering its mass or stiffness. This can be achieved by: (a) stiffening structural members, (b) increasing the
number of columns, and (c) reduce the decking material to a lighter weight component. Moreover, all
84
structures have some inherent damping. Further method can be employed in which the damping can be
increased thus dissipating energy when the structure moves. However, the employment of mechanical
damping devices comes at a great financial cost.
7.5.
Deflections of Super-Tall Structure
There are mainly four types of analysis than can be performed in a framed systems such as of a tall
structure:
1.
First Order Elastic Analysis
2.
Second Order Elastic Analysis
3.
First Order Plastic Analysis
4.
Second Order Plastic Analysis (Advanced analysis)
The work presented in this thesis was performed using the first analysis method listed above
which is considered the most basic and fundamental analysis. In a first order analysis the internal forces,
and the displacements of the structure are evaluated in relation to the initial, undeformed shape of the
building. This method is mostly used in the analysis of current structures. The stresses, strains, and
displacements may be obtained using established methods such as force method or displacement method. In
the case of first order analysis the deformations are proportional to the applied loads.
The loading used for the analysis was described previously. The wind load generated by the
structural analysis software was based upon the ASCE 7-10 code and design criteria presented in Appendix
A. The wind loads were applied on the structure as exposure form extents of rigid diaphragms. The loads
varied in a parabolic shape along the height of the structure on the windward face. The wind loading
generated by the software implemented the four different cases for the Main Wind Force Resisting System
presented in the ASCE 7-10 code. Case 1 considered the full design wind pressure acting on the projected
area perpendicular to each principal axis of the structure. Case 2 considered 75% of the design wind
pressure acting on the structure in conjunction of a torsional moment. Case 3 considers a similar criteria to
Case 1 but the loading acts simultaneously. Case 4 considers similar criteria to Case 2 but the loading acts
simultaneously at 75% of the specified value.
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(c) Deflection in x-direction
(a) Deflection in x-direction
(b) Deflection in y-direction
(d) Deflection in y-direction
Figure 50: Deflections experienced by the loading acting on super-tall structure
The analysis of the structure was determined to have a total of 5,236 number of joints in which 76
of them were properly restrained with the fixed condition. The model contained 9,913 frame elements
representing the columns and beams. The quantity of shell elements was counted to be 5,165. The software
assembled all the matrices and determined that there was 15,681 number of equilibrium equations with
386,244 number of non-zero stiffness terms. Based on this large number of members, the only feasible
method at arriving at a solution to the matrix analysis of the structure was by using computer hardware
coupled with appropriate software to perform these demanding computations.
Wind drift limits for building design are on typically in the order of H/400. These limits generally
are sufficient to minimize damage to the structure (Taranath 2005). In the case of the present study, it was
verified that the magnitudes of the displacement were within the prescribed limit as shown in Figure 50.
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The maximum limit imposed was computed to be 30 inches. After analyzing the software outputs, it was
determined that the maximum deflection occurred in the lateral y-direction having a magnitude of
approximately 24 inches, which is within the allowable limit. As a note, there was a number of different
deflection modes not presented in this thesis due to the large number of output generated by the software
analysis. Some results included torsional deflections of the floor systems; they were separately verified to
still abide by the H/400 criteria to ensure that the model had acceptable deflections throughout all the 16
load cases and 74 combinations that were analyzed during the evaluation of the tall building structure.
87
CHAPTER 8
CONCLUSIONS & FUTURE RESEARCH
8.1.
Final Remarks
The advancements of computational wind engineering is making the evaluation of wind loads
acting on structures more attractive to researchers due to the numerical evaluation that can be performed
using modern day computational technologies. In general, the use of computational fluid dynamics allows
the engineer to utilize the techniques developed by subject pioneers to study the accuracy of RANS and
LES when combined to building aerodynamics, or bluff bodies, which is distinguished by flow separation
at building corners and high Reynolds numbers. CFD provides continuous visualization of aerodynamic
data on the entire surface areas of the building instead of the discrete point measurements extracted from
experimental wind tunnel analysis. In addition, CFD can provide detailed flow visualizations for the wind
field around the building which information on flow separation and reattachment, recirculation zone, and
vortex shedding can be extracted.
It is imperative to ensure that the accuracy of the simulation is validated rigorously to identify its
positive and negative implications. It is expected that the numerical approach with further developments in
the practical use and analysis of CFD will allow engineers to overcome the existing obstacles. Therefore
this tool might be incorporated in the design process of building engineering whether at the predesign
phases or final phases. This will provide the industry with a new set of tools to use for the aerodynamic
considerations in the building design stages. Presently, most studies are still conducted using wind tunnel
experimental techniques due to the acceptable results that industry engineers have become comfortable
with for the use of loading design purposes of tall buildings and other structures.
The current state of the art of computational wind engineering has been reviewed by several
researchers. Key findings indicate that turbulence modeling, boundary conditions, high Reynolds numbers
turbulent flow, and computational cost are essential parameters when arriving at appropriate computational
predictions. The advance in computational machines and development of parallel computing has enabled
88
the use of CFD for industrial applications such as in tall building design. Generating the atmospheric
boundary layer in the computational domain is a critical step for the wind load evaluation process. Making
data of model and full scale available are very important for the validation of the CFD results. Creating a
time-dependent analysis model can create a time-history of pressure fluctuations similar to what wind
tunnel data provide engineers with. Generally speaking, mean static pressure values showed comparative
agreement with published data and industry accepted design codes.
It must be noted that CFD results are greatly dependent on the input of the mean wind parameters
such as the velocity profile and turbulence intensities in the relevant three dimensional coordinate system.
The engineer must be attentive in defining these properties accurately. This has great implications in the
convergence of the simulation and the time required for the computer machine to generate qualitative and
quantitative aerodynamic data which can be used by the design engineer.
Generally speaking, it can be fairly said that CFD simulations can be used as an alternative tool for
wind pressure loading for at least pre-design stages of a tall structure project. The results of employing this
technique should be taken with caution to ensure that the designer is still abiding by design codes. With
proper boundary conditions applied to the model, and the advancement of computational resources could
become useful for wind loading studies in the future. The present study would like to point out that the
main limitation of using LES is that the machines required to numerically compute this model are
expensive, and the time it takes to arrive at a final computational iteration is more time consuming than
present boundary wind tunnel techniques. Performing RANS simulations can significantly decrease the
computational cost and time to arrive at reasonable results.
8.2.
Recommendations for Future Research
Based on the work presented in this thesis, the author would like to suggest future venues for
research to increase the current state-of-the-art knowledge in this subject:
By performing CFD parametric studies and assimilating the data with experimental wind
tunnel models, the present theory could be enhanced to lead to new mathematical formulation
to better predict CFD data output to real-world values.
89
A complete fluid-structure interaction between the structural body (i.e. tall building) could be
investigated by assigning the structural properties to the object and utilizing dynamic meshes
to evaluate the response of the fluid and the structure through a desirable time domain.
Further investigation can be carried by utilizing different numerical techniques in a simplified
model to allow researchers to identify and couple the best present methods that could provide
improved accuracy in results.
As the computational power provided by new technologies and hardware become available in
the foreseeable market, full-scale modeling at high Reynolds numbers could be researched
with more computationally expensive techniques such as DNS.
90
APPENDICES
91
Appendix A
Wind Loading Computations of Prototype Building per ASCE 7-10
92
93
94
95
96
97
98
99
100
101
Appendix B
ICEM CFD Meshing Script for 2D Plan Geometry
102
103
104
105
106
107
108
Appendix C
User Defined Function for 3D Wind Profile in C-language
109
110
Appendix D
ANSYS Fluent 2D Simulation Criteria of CFD Model
111
112
113
114
115
116
117
118
119
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