3. iii
The influence of offshore windmill foundations on the
stability of large bed forms
____________________________________________________________
Andreas Stengel Hansen, June 16., 2015
5. Preface
The report at hand is the Master’s Thesis of Andreas Myrhøj Stengel Hansen, s092855, con-
ducted at the Department of Mechanical Engineering, in the section of Fluid Mechanics,
Coastal and Maritime Engineering, at the Technical University of Denmark. The thesis is the
product of a full semester of work during the spring of 2015 and is accredited by 32.5 ECTS
points. The official supervisor of the project have been Associate Professor David R. Fuhrman.
I would like to extend my gratitude to my supervisor for patiently and skillfully directing me
towards my final result. Without his guidance and valid input this project could not have
been possible.
Also, a speciale thanks to Ph.d.-student Jonatan Margalit who has been a great help in my
work with MIKE 21. Finally, I would like to thank Kasper Kærgaard and Asger Bendix
Hansen (DHI) for help regarding MIKE 21.
Andreas Myrhøj Stengel Hansen
Kongens Lyngby
June 2015
v
7. Abstract
The demand for sustainable energy sources has increased the past decades and the offshore
wind turbine industry has been growing continuously since the installation of the first offshore
wind farm back in 1991. The offshore wind turbines have in later years been installed in more
remote areas away from coastlines to cause less impact visibility and physically. In areas
where there is abundant sand, strong currents are able to move sediment. The formation of
sand banks and elongated sand ridges occur within these areas, generated from the interac-
tion between the water motion and seabed. Considerations were initiated at the late 60’s
regarding large active seabed configurations, and were given a widely used term, large scale
bedforms. The term covers sand bars and sand ridges with wave heights in the same order
of magnitude as the water depth. The first observations lead to further investigations on the
stability of large scale bedforms. Many conducted analytical linear stability analyses, which
could usually not be verified since the amount of field observations was limited at that time.
Scientists improved the linear stability analysis with morphological models, which remains a
highly difficult discipline.
The investigations has been conducted using MIKE 21, developed by the Danish Hydraulic
Institute. The numerical modelling system solves shallow water equations and sediment trans-
port formulae, hence solving the hydrodynamics continuously with the sediment transport
field, which generally governs the stability of large scale bedforms. The sediment transport
field is modelled using a classical approach based on the bed shear stress. The models devel-
oped have considered a steady uniform flow. The uniform flow is obtained by a sloping bed and
a total resistance described by the Chezy number. The large scale bedform is considered in its
most reduced form, and implemented into MIKE 21 as sinusoidal perturbations on the seabed.
Firstly, the perturbation is implemented as a single wave length perturbation. It is found that
the transition from a plane bed to a perturbed bed cannot be described perfectly in MIKE 21.
The disadvantage is clearly seen from a morphological model giving extensively high amount
of erosion on the lee side of the perturbation. Thus, a periodic perturbed bed is implemented
and the most unstable combination of wave numbers giving the fastest growth is determined
and successfully compared with an analytical approach.
A morphological module is activated in MIKE 21 and a verification of the morphological
capacity is carried out based on a flume experiment, which generates alternating bars. Based
on the findings from the morphological verification, a field scale model is performed giving an
equilibrium seabed profile with limitations. The field scale morphological model is thereafter
subjected to the presence of wind turbines. The investigated internal distances are 200 m, 400
m, 800 m. It is found that the impact from a monopile structure causes a delay in formation
of large scale bedforms within an offshore wind farm, depending on the internal distance
between the wind turbines. The formation of large scale bedforms has restricted growth if the
internal spacing is small. An impact study shows that certain large scale bedforms, depending
on the wave height, are able to penetrate an offshore wind farm without changing shape and
dimensions.
vii
9. Resumé
Kravene til vedvarende energi kilder er steget de seneste år og vindmølle industrien har
tilsvarende vokset med installation af den første offshore vindmølle farm tilbage i 1991.
Offshore vind turbiner er blevet installeret i mere afsidesliggende områder placeret væk fra
kystlinjer for at skabe mindre lokal påvirkning både visuelt og fysisk. I områder med over-
skydende mængder sand, kan stærke strømme flytte sedimenter. Etablering af sand banker
og langstrakte sandrygge opstår i disse områder, ved interaktionen mellem vand bevægelse og
havbund. Betragtninger i de sene 60’ere blev påbegyndt vedrørende store aktive havbunds
konfigurationer, og fik tilnavnet stor skala sandformer. Termen dækker sand banker and
sandrygge med bølgehøjder i samme størelses orden som vanddybden. De første observationer
førte til yderligere undersøgelser om stor skala sandformer. Mange udførte analytiske lineære
stabilitets analyser, som generelt ikke kunne verificeres, da mængden af felt observationer var
begrænset på det tidspunkt. Videnskabsmænd videreudviklede den lineære stabilitets analy-
ses med morfologiske modeller, som stadig anses for værende en vanskelig disciplin.
Undersøgelserne er blevet udført i MIKE 21, udviklet af Dansk Hydraulisk Institut. Det nu-
meriske model system løser lavt vands ligninger og sediment transport ligninger, således ved at
løse hydrodynamikken løbende med sediment transport feltet, som generelt afgør stabiliteten
af stor skala sandformer. Sediment transport feltet er modeleret ved en klassisk betragtning af
bund friktionen. De udviklede modeller betragtes for en ensartet strøm. Den ensartet strøm
opnås ved at implementere en bundhældning, beskrevet med Chezy tallet. Stor skala sandfor-
men betragtes i den mest reducerede form, og implementeres i MIKE 21 som en sinusvarieret
perturbation på havbunden.
Først, implementeres perturbationen som en enkelt bølgelængde pertubation. Det viste sig
at transitionen fra en flad bund til en bølget bund ikke kunne beskrives perfekt i MIKE 21.
Ulempen var tydelig fra en morfologisk model, der resulterede i store erosions rater på bagsi-
den af pertubationen. Således bliver en periodisk pertuberet bund implementeret og den mest
ustabile kombination af bølgetal, som giver den hurtigste vækst er bestemt og med success
sammenlignet med en analytisk model.
Et morfologisk modul aktiveres i MIKE 21 og en verifikation af den morfologiske kapacitet er
udført baseret på et rand-eksperiment, der resulterer i alternerende sandformer. Baseret på
resultaterne fra verifikations morfologien, udføres en felt skala model som giver et ligevægts
havbunds profil med begrænsninger. Den felt skala morfologiske model er herefter udsat for
introduktionen af vind turbiner. Undersøgelser af interne afstande mellem vind turbinerne er
200 m, 400 m, 800 m. Resultater viser at kollisionen med en monopæl struktur forsinker etab-
leringen af stor skala sand former, afhængig af den interne afstand mellem vind turbinerne.
Dannelsen af stor skala sand former har begrænset vækst hvis den interne afstand er lille. Et
kollisions studie viser at visse stor skala sand former, afhængig af bølgehøjde, er i stand til at
penetrere en offshore vindmølle farm uden at ændre form og dimensioner.
ix
13. xiii CONTENTS
I Technical data on Horns Rev 1 Offshore Wind Farm 131
J Amplitude plots for OWF morphological models 133
K MIKE 21 file generator in matlab 135
15. Chapter 1
Introduction
Through the past years, offshore wind farms have become more popular with increasing de-
mands for sustainable energy sources. Offshore wind farms consist of a large number of wind
turbines installed together in a mesh in an offshore field. The first ever commercial offshore
wind farm installed was Vindeby in 1991. Vindeby has a capacity of approximately 5 MW
distributed over 11 turbines, [MTHoejgaard, 1990]. It is located approximately 2 km from
the coastline of Lolland on 5 meters water depth and was one of the first wind farms to be
installed globally. Today, Europe has remained as leader within development and installation
of offshore wind farms and as of January 2015 the total capacity of offshore wind farms in
Europe has reached 8,045 MW, [The European Wind Energy Association, 2014]. 63.3% of
the capacity is located in the North Sea.
The offshore wind turbines are typically supported by an array of support structures such as
monopiles, jacket structures and gravity based foundations among others. The average water
depth within the areas of installation is around 23 m and the most commonly used support
structure is monopiles. 91% of installed wind turbines are supported by monopiles, [The Eu-
ropean Wind Energy Association, 2014]. The monopile is easy to install, cost-efficient and
can be used for water depths of up to 30 meters. For now, monopiles with a diameter up to
6 meter are installed into the seabed.
Unprotected monopiles are exposed to the process of scour, where the surrounding sediment
is transported away from the pile by the flow. The scour process may conflict with the struc-
tural capacity, leading to failure of the vertical or lateral support of the wind turbines. The
seabed is consequently responsible for the structural integrity of the structure. In addition to
the local scour phenomena, the location of the monopiles must also be evaluated in terms of a
morphological active seabed. In areas where potential wind farms can or already are installed,
the seabed might interact with the water motion and generate large sand banks, which may
change the water depth and structural capacity of the installation area.
Field investigations of sand banks have shown a high variety of forms. Their generation re-
quires a source of mobile sediment, which originates from either the local seabed or from coast
erosion, [Dyer and Huntley, 1999]. Most of the large bed configurations appear to have been
created during the post-glacial rise in sea level, but they have been modified in shape and size
with changing currents and waves. Sand banks are present in a wide range of water depths.
They can be found in the mouth of estuaries, adjacent to headlands and beaches, on the
exposed shelf and close to the continental shelf edge. It has been observed that active moving
sand banks are located within shallow waters. There is evidence that sand banks are formed
1
16. CHAPTER 1. INTRODUCTION 2
on shallower water and preserved under conditions of rising sea level, [Dyer and Huntley, 1999].
A highly active seabed can be seen in the North Sea close to the Thames Estuary. In Fig. 1.1
a satellite photo of the Thames Estuary with a white marked rectangle showing the location
of London Array Offshore Wind Farm is showed.
Fig. 1.1a shows the location of the wind farm and Fig. 1.1b shows a closeup of the wind
farm. The aerial photo shows large scale bedforms of high variety. The formation of bedforms
consistently arise from the mechanical interaction between sediment and flow field and does
surprisingly show a high degree of order.
(a) Landsat 8 satellite picture: London Array.
World’s largest offshore wind farm.
(b) Zoom on satellite picture. White dots are wind
turbines.
Figure 1.1: Satellite photo of London Array Offshore Wind Farm, [www.space.com, 2013].
To describe the process of formation of large scale bedforms, it is necessary to have a detailed
description of the interaction between the flow field and sediment transport field, which is
governed by the flow and the bed shear stresses. This has motivated a large number of in-
vestigations on the stability of such bedforms. The first attempts in describing the stability
of large scale bedforms were carried out by establishing analytical models. Recent stability
studies take advantage of numerical models to describe the dynamics of large scale bedforms.
Many linear stability analyses have been conducted, but not many have had success regard-
ing the morphodynamics, which is still a complex discipline in fluid dynamics. [Madsen, 2002]
successfully uses a non-dimensional model to describe the stability and morphological devel-
opment of large scale bedforms.
This motivates the present work, which deals with numerical modelling of flow and sediment
transport in an alluvial bed. The aim is to improve the understanding of large scale bedforms
regarding growth and morphological development in proximity of offshore wind turbines. The
length scales within the field of large scale bedforms makes it difficult to perform physical
model tests, which is why the use of a commercial numerical model will be very advantageous.
1.1 Objective of thesis
The aim of the present thesis is to evaluate the stability of large scale bedforms under influ-
ence of offshore wind turbines. A depth averaged numerical model developed by the Danish
Hydraulic Institute (DHI) is used in the study. The model is MIKE 21, which has been tested
17. 3 CHAPTER 1. INTRODUCTION
throughout a large variety of coastal modelling scenarios. It is not within the scope of the
thesis to explain the comprehensive numerical model in detail, but rather use the commercial
model to provide an accurate tool for modelling the stability of large scale bedforms. The
large scale bedform is reduced to a sinusoidal shape considered as the most simple form. The
most unstable bedform is to be determined and subjected to a morphological model. The in-
fluence of a wind turbine is considered by introducing sub-grid structures at the seabed. The
objective is not to study the local scour phenomena, but rather the mechanisms governing
the growth and migration of large scale bedforms in vicinity of offshore wind turbines.
1.2 Methodology
The numerical modelling of a flow field and a sediment transport field can be approached
either by open source software, which is flexible in terms of modifications of code making it
ideal for research purposes, or by commercial software, which is easier in practical use and
can produce faster results. The disadvantage of using commercial software will be the closed
source principle, where it can be difficult to identify exactly which equations are solved to
produce the solution. The benefits of using commercial software are many if the user is aware
of the structure of the code in general. A great part of the project is used for validation of the
MIKE 21 Flexible Mesh model. On the basis of the governing mechanisms, which in general
considers horizontal actions, a depth averaged numerical model can be applied instead of a
more comprehensive Computational Fluid Dynamics (CFD) model.
To achieve the goals mentioned, the thesis can be divided into a series of steps. Firstly, the
commercial numerical model will be verified for a simple flow. The verification investigates
whether the numerical model is able to produce a uniform flow by using simple flow consid-
erations or not. Results provided by [Madsen, 2002] will be taken into consideration when
evaluating the most unstable bedform. The comparison will on the other hand only serve
as a guiding since the parameter study by [Madsen, 2002] is performed with different non
dimensional parameters. Based on a Linear Stability Analysis the most unstable bedform is
found and subjected to a morphological model. The morphological model will be evaluated
with implemented wind turbines.
1.3 Outline
Chapter 2 presents the state of the art, which is a brief overview of the previously conducted
work regarding large scale bedforms. The first attempts to describe the stability of bedforms
is presented. The chapter includes methodology on the work regarding linear stability analy-
ses. An important flume experiment conducted by [Fujita and Muramoto, 1985] is presented
together with recent studies on morphological models able to predict river morphological
changes. The few studies on influence on bed topography and wind field from offshore wind
farms are presented as well. The chapter serves as a literature study and aims to involve the
reader in the previous work within the field of large scale bedforms.
Chapter 3 presents the theoretical background of the growth of large scale bedforms. A brief
overview of the mechanics governing the growth rate for an unstable migrating bedform is
given. The assumed bedform shape is used to determine the unstable growth rate and mi-
gration frequency. The basis for an investigation of the growth through a Linear Stability
Analysis is covered.
18. CHAPTER 1. INTRODUCTION 4
Chapter 4 explains how the MIKE 21 numerical model works and how it is used in assessment
of the unstable growth of a large scale bedform. The chapter is divided into three parts; a
mesh generation part, a hydrodynamic part and a sediment transport part. The three parts
are in principle the cornerstones of commercial MIKE 21 numerical model.
Chapter 5 presents a dimensional analysis which constitutes the basis of how the results are
interpreted in the whole thesis. The steps of a dimensional analysis are covered and the rele-
vant physical quantities are reduced in terms of independent variables.
Chapter 6 presents a stability analysis for a steady current. A mesh analysis is performed,
which makes the basis of the domain discretisation in the entire thesis. Hereafter, a flatbed
model is created and is used to verify whether a uniform flow can be generated in MIKE 21.
It is evaluated whether the uniformity of the flow is kept such that the net sediment transport
through a control box is approximately zero or not. Firstly, a simple single doubly periodic
bedform is evaluated in terms of a linearity check, stability curves and a predicted growth
rate. Secondly, a similar study is conducted for a doubly periodic bedform reaching in the
whole domain. Lastly, the flow and sediment transport field is evaluated.
Chapter 7 presents the morphological calculations. A flume experiment by [Fujita and Mu-
ramoto, 1985] forming alternating bars is implemented into MIKE 21. A morphological model
is used to verify the morphological model in MIKE 21 by comparison with the flume experi-
ment. The most successfull model is brought to field scale and a similar morphological model
is performed.
Chapter 8 presents the morphological development from the influence of wind turbines. The
previous morphological field model is subjected to wind turbine units in the whole reach of
the domain. It is evaluated whether large scale bedforms can emerge from the seabed or not.
An impact study is conducted, where only half of the domain is subjected to wind turbines.
The impact from the migrating bedforms on the offshore wind farm is studied.
Finally, conclusions based on the findings in the thesis will be drawn in Chapter 9.
1.4 Reference system
A reference system is presented to prepare the reader for the terminology and coordinate
system. A Cartesian coordinate system (x, y, z) is used. The origin of the coordinate system
is located in the center of the domain. The x-axis is directed in the flow direction and the
y-axis is normal to the flow direction x. The z-axis is a normal to the flat horizontal seabed
19. 5 CHAPTER 1. INTRODUCTION
MWL
Seabed
η
z
x
h
D
Lx
A
Figure 1.2: Reference system
In Fig. 1.2 an arbitrary perturbed seabed is shown, here expressed as a sinusoidal function.
The crests refer to the apex of the bed level and the troughs refer to the minimum bed level
for a periodic perturbed bed. The variation of the bed level is expressed as h(x, y, t), such
that the bed level may vary in both the flow direction and transverse flow direction. The
periodic bed is characterised by an amplitude, A, a streamwise wave length, Lx, defined from
the wave number, kx = 2π
Lx
, and a transverse wave length, Ly, defined from the transverse
wave number, ky = 2π
Ly
. The total water depth is defined by, D, which includes the surface
elevation, η.
21. Chapter 2
State of the art
This chapter is a short summary of previous findings within the field of large scale bedform
investigations. The chapter works as a literature study and the most prominent studies are
highlighted in the following. Studies regarding work on Linear Stability Analysis is included,
which will be explained further in Section 3, and morphological modelling of large scale
bedforms in rivers. Furthermore, the work on influence on the stability due to presence of
resistance in the flow will be reviewed. The chapter is divided into the fundamental work done
on large scale bedforms and work related to influence of resistance in flow on the stability of
large scale bedforms.
2.1 The undisturbed flow field
Many scientists in the late 60’s took on the task of describing the formation of alternate bars
in rivers using depth-averaged momentum equations. However, the first approaches were lim-
ited in terms of included physical effects, especially the helical motion, which is described in
more detail in Section 4.3.4.
[Hansen, 1967] was among the first scientists to describe the instability of a river in terms of
stability of bedforms. He studied the effect of formation of alternate bars in rivers, but did
not take the helical motion into account. This left the direction of the bed shear stress into
the same direction as the velocity vector. Hansen perturbed the governing equations, in which
he took the transverse wavelength equal to twice the width of the river. He concluded that
the growth velocity of the alternate bars were always positive. After [Hansen, 1967], others
found that the instability of rivers was limited to an interval, meaning rivers were not always
forming alternate bars.
[Olesen, 1983] conducted a linear pertubation analysis on the flow and bed topography in
a straight alluvial river. A great difference from [Hansen, 1967] was that a secondary flow
was included, relating the bed shear stress direction to the curvature of the depth averaged
streamline. Due to the relatively large migration velocity, [Olesen, 1983] stated that the in-
cipient meandering could not be directly related to occurrence of alternate bars, which is a
result of a stability analysis. [Olesen, 1983] found that the stability analysis was very much
dependent on the depth to width ratio and the analysis showed that the wave lengths of al-
ternate bars was 3-4 times the width of the channel, which agreed with laboratory experiments.
Jonathan M. Nelson has done extensive work within the field of stability of bedforms and
7
22. CHAPTER 2. STATE OF THE ART 8
has compared and predicted wave lengths from a linear stability analysis with wave lengths
determined from morphological models succesfully. The salient conclusions from the work
presented in [Nelson, 1989] was that alternate bars arise spontaneously in initially flat-bedded
straight channels due to a basic instability within the governing equations for the flow and
sediment transport field. A linear stability analysis could predict the initially fastest growing
wavelength. In [Nelson, 1990], Nelson again focuses on the prediction of wave lengths by a
linear stability analysis compared with wave lengths predicted by a morphological model. A
morphological model was developed, which was used to simulate results from a flume exper-
iment conducted by [Fujita and Muramoto, 1985]. Initially, the periodic bed was disturbed
with a small hump and the bed evolved in a similar manner as the conducted flume experiment.
The work by Erik Madsen in the thesis [Madsen, 2002] is highly relevant to the present work.
Madsen studied large scale bedforms in both a steady current and a wave-current environ-
ment. Similar to the work by Nelson, Madsen conducted linear stability analyses, which were
used to define the length scale for the fastest growth rate. The length scale was found through
a non-dimensional numerical model and implemented into a morphological model. Madsen
also reproduced the flume experiment by [Fujita and Muramoto, 1985]. The morphological
models were successful and the equilibrium dimensions of bedforms were predicted developing
from a bump on a flat bed.
Similar to the studies conducted by [Enggrob and Tjerry, 1999] and [Hibma, 2004], a numer-
ical simulation of estuarine and river morphology was carried out at DHI. The MIKE 21C
(Curvilinear) model was used in the simulations of braiding rivers. The simulations carried
out considered the previously conducted case by [Enggrob and Tjerry, 1999] with a finer
resolution and longer domain. The results were very realistic in comparison with satellite
photos of the actual rivers, in terms of longitudinal length scales. Overall, the MIKE 21C
model produced realistic natural morphological features and can be used for long term impact
assessments and predict future bed levels. However, the model represents the nature, but the
final meandering pattern is one of many. In addition to the above, it is underlined that mor-
phological studies is a highly complex discipline. The main reason may be assigned to the
interrelation among mechanisms that act at different spatial and temporal scales, [J. Vested
and Dubinski, 2014].
2.2 Influence on stability due to resistance in flow
The stability of large scale bedforms have been studied in great detail. Due to increasing
demands to renewable energy, more wind farms are to be built. Therefore, the influence the
wind farm has on bed level is very important to establish. Unprotected monopiles are ex-
posed to the process of scour, where the sediment surrounding the pile is transported away
from the pile by the flow. The local scour may be critical for the structural capacity of the
monopile. The scour process has been studied extensively in the past, see e.g. [Fredsøe and
Sumer, 2002]. In terms of stability of large scale bedforms it must be distinguished between
local scour effects and influence on the large scale bedforms from e.g. monopiles. In this thesis
the local scour effect is not assigned special attention.
In [Damgaard et al., 2013], the transmission of wave energy passing an offshore wind farm is
studied, which is related to the study presented in this thesis. It is stated that three effects
can change the flow field, which is the A) the energy dissipation due to drag resistance, B)
23. 9 CHAPTER 2. STATE OF THE ART
wave reflection/diffraction from structures, and C) the effect of a modified wind field inside
and on the lee side of the wind farm. The effects have been modelled with the MIKE 21
Spectral Wave model (SW). Firstly, it was found that the dissipation of the wave energy due
to surface friction and vortex shedding was negligible. Secondly, for moderate wind speed of
U10 = 10 m/s, the local reduction of the wave height 2 km downwind, is generated 1/3 from
reflection/diffraction and 2/3 from reduced wind shear.
As a part of the Seabed Wind Farm Interaction project, [DTU et al., 2012], DHI carried out
a study of the potential morphological impacts of offshore wind farms on large scale bed-
forms, [Hansen, 2012]. The study mainly focused on the question regarding the possibility
to predict the observed natural bed development and how large impact a large offshore wind
farm has on the surrounding morphology. The aim was to develop the natural sand banks in
the Outer Thames Estuary. The modelled bed level changes were typically larger than the
observed. The impact on the sediment transport from the Offshore Wind Farm (OWF), was in
general due to changes in the current field. There were changes of up to 10% on the sediment
transport field, from the impact of gravity based foundations. The gravity based foundations
were representing the OWF, seen as a worst case scenario. The model could be used to pre-
dict the natural observed bed level changes in some areas, but generally overestimated the
increased bed level. Due to the rather erroneous predictions of natural bed levels, [Hansen,
2012] concluded that future research should focus on a better understanding of the physical
processes which controls the natural morphodynamics of large scale bedforms. The study
carried out involved many physical phenomena, and it was not evident which specific physical
effect governed the stability of the large scale bedforms.
25. Chapter 3
Theory - Large-scale bedforms
Bedforms are a result of interactions between water motion and an alluvial bed. The large
scale bedforms observed in the field highly varies in shape and size. If the irregular natural
bedforms are reduced in terms of shape and size a rather simple periodic description of the
large scale bedform may be studied through numerical models. In this chapter, the governing
mechanism of the growth of large scale bedforms will be explained. Two types of perturba-
tions are introduced and the growth rate is determined. The growth rate is to be compared
with the numerically obtained growth rate from MIKE 21.
3.1 Growth rate for an unstable migrating bedform
Bedforms can obtain an equilibrium shape at locations where the sandy bed is subjected to
constant flow conditions over a period of time. Usually, a seabed contains many different
length scales of different bed forms. If the flow conditions are kept constant over a period,
certain bedforms will eventually become much larger than the remaining length scales. A re-
peating pattern of bedforms can be observed in the field and the bed level may be described as
being periodic, [Madsen, 2002]. However, if the flow conditions change, the governing length
scales may redistribute the equilibrium bedform and with frequently changing flow conditions
the bedforms will never reach an equilibrium. The bedforms are generally generated if the
bed is unstable. An unstable bed might occur during different combinations of water depth,
current velocity, sediment grain size etc.
The term ”large scale bedforms” is used to differ between small scale bedforms, such as ripples
and dunes, to large sand bars or sand waves. The bedforms are characterised by horizontal
length scales and can attain wave lengths of up to hundreds of meters and amplitudes as
great as half the water depth. In Figure 3.1 a map of large scale bedforms in the North Sea
is shown.
11
26. CHAPTER 3. THEORY - LARGE-SCALE BEDFORMS 12
Figure 3.1: Map of sand bank distribution in the North Sea showing classification,
[www.geo.uu.nl, 2011].
The North Sea map shows that many large scale bedforms have been formed the past decades,
by the changing sea state. The considerations in this thesis are based on simplified theoretical
approaches, in which many physical effects have been discarded. The anticipated governing
mechanisms for the growth of large scale bedforms are considered. The investigation of large
scale bedforms throughout this thesis considers a pure current model where the effects of
waves and coriolis force are dismissed in the considerations.
Anticipated simplified shapes of large scale bedforms are introduced to investigate their
growth. This means that the bed topography is perturbed by an assumed bedform shape
and takes the form of a sinussoidal periodic function, with characteristics describing the mi-
gration and growth of the bedform. Two perturbed bed level cases are to be studied. A single
wavelength perturbation on the flat bed and a periodic bed level are presented in the following
sections.
3.1.1 Single wavelength perturbed bed
A single periodic bedform migrating in the pure downstream direction has an initial amplitude,
A, which grows exponentially in time:
h(x, y, t) =
A
2
eΩt
(cos(ωt − kxx) + 1) cos(kyy) (3.1)
The bedform migrates with time t, kx = 2π
Lx
is the wave number in the streamwise direction
x, in which Lx is the wavelength. ky = 2π
Ly
is the wave number in the direction normal to the
flow direction y, in which Ly is the wavelength. Ω is a parameter describing the amplifica-
tion of the bedform amplitude A, describing the exponential growth. The migration velocity,
c = ω/kx, of the bedform in the pure x-direction is characterised by the angular propagation
frequency, ω.
In Fig. 3.2a a three dimensional surface plot of the bedform at the time t = 0 is shown. In Fig.
3.2a two snapshots along the centerline, y = 0, showing the initial bedform and the bedform
27. 13 CHAPTER 3. THEORY - LARGE-SCALE BEDFORMS
after some time. The bedform is normalised with the amplitude A and the wavenumbers kx
and ky are equal to one. A positive unstable temporal growth has been assumed, by Ω = 0.2ω.
(a) Three dimensional shape of large scale
bedform at time t = 0.
−3 −2 −1 0 1 2 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
kxx
h/A
t = 0
t = 2π
ω /8
(b) Initial bedform and bedform after some
time, where temporal growth has been as-
sumed.
Figure 3.2: Three dimensional bedform and migrating bedform.
The assumed single wave length perturbation is a doubly periodic function and is shifted
upwards within the first bracket, read +1. This is done to prevent a steep gradient when
the flow approaches the perturbation from the flat bed. Thus, a smooth transition from
the flat bed to the perturbed bed is provided. In terms of a linear stability analysis, the
growth parameter, Ω, is governing the exponential growth in time. When the Linear Stability
Analysis (LSA) is performed later on, the initial growth is determined from MIKE 21 and can
analytically be determined from the time derivative of Eq. (3.1), thus:
∂h
∂t
=
1
2
AetΩ
Ω cos(kyy)(1 + cos(kxx − tω)) +
1
2
AetΩ
ω cos(kyy) sin(kxx − tω) (3.2)
At the time t = 0 Eq. (3.2) becomes:
∂h0
∂t
=
1
2
AΩ(1 + cos(kxx)) cos(kyy) +
1
2
Aω cos(kyy) sin(kxx) (3.3)
The rate of the unstable growth can be determined from the time derivative at the origin of
the perturbation, hence x = 0 and y = 0. From Eq. (3.3) the rate of growth, Ω, is:
Ω =
∂h0
∂t
1
A
(3.4)
To establish a description of the unstable growth, the migration velocity, ω, must be deter-
mined as well. The parameter may be found from the derivative with respect to the streamwise
direction, x, of Eq. (3.3):
∂
∂x
∂h0
∂t
=
1
2
Akxω cos(kxx) cos(kyy) −
1
2
AkxΩ cos(kyy) sin(kxx) (3.5)
Rearranging Eq. (3.5) the migration frequency, ω, is:
ω = 2
∂
∂x
∂h0
∂t
1
kxA
(3.6)
28. CHAPTER 3. THEORY - LARGE-SCALE BEDFORMS 14
When the migration frequency and the unstable growth rate are determined, an analytical
approach to the growth can be established from Eq. (3.3), which will be utilised later on. A
combination of kx = 2π
Lx
and ky = 2π
Ly
giving the largest unstable growth rate should match
the wave number observed in field investigations. This is a purely theoretical assumption as
the sea state might change from time to time.
3.1.2 Doubly periodic perturbed bed
For the perturbed bed in the whole reach of the domain, another doubly-periodic bedform mi-
grating in the pure x-direction is considered. The unstable exponential growth and migration
has the form:
h(x, y, t) = AeΩt
cos(ωt − kxx) cos(kyy) (3.7)
ω is the angular propagation frequency describing the migration velocity of the bedform,
c = ω/kx. A is the initial amplitude of the bedform, which growth may be described by the
unstable growth rate, Ω, with respect to the time, t. kx and ky describe the wavelengths in
the streamwise and transverse direction of the flow, hence kx = 2π/Lx and ky = 2π/Ly.
Using the same procedure as for the single doubly-periodic bedform, the unstable growth rate
and angular frequency can be found by determining the time derivative and the derivative
with respect to the x-direction. The time derivation at the origin of the bedform reads:
∂h0
∂t
= AΩ cos(kxx) cos(kyy) + Aω cos(kyy) sin(kxx) (3.8)
The unstable growth is then found from the origin:
Ω =
∂h0
∂t
1
A
(3.9)
The angular frequency is:
ω =
∂
∂x
∂h0
∂t
1
kxA
(3.10)
3.2 Growth rate for large scale bedforms
The fundamental equation governing the sediment transport rates is the growth rate described
in the previous sections. The change in bed level may be evaluated from the sediment transport
field and especially the continuity equation for the sediment. The bed level change in one
dimension is, [Fredsøe and Deigaard, 2012]:
∂h
∂t
= −
1
1 − n
∂qTx
∂x
(3.11)
Expressing the net outflow, which must equal the change in bed level, when the correction
for the bed porosity, n, is taken into account. h is the bed level and qTx is the net sediment
transport rate in the x-direction, see Fig. 3.3.
29. 15 CHAPTER 3. THEORY - LARGE-SCALE BEDFORMS
qT qT + ∂qT
∂x dx
dx
∂h
∂t
Figure 3.3: Sediment flux through a control box over an erodible bed.
The calculation of the continuity equation, and hereby the development of the bed, requires
numerical solving, which will be explained in Section 4.3. The continuity equation of the
sediment may also be expressed in the two-dimensional case, where the sediment transport
rate normal to the flow direction is considered as well, thus:
∂h
∂t
= −
1
1 − n
∂qTx
∂x
+
∂qTy
∂y
(3.12)
Eq. (3.12) is also known as the Exner equation and is used in MIKE 21 to update the bed
level. From Eq. (3.11) and (3.12) the total sediment transport rate, qT , reads:
qT = qB + qS (3.13)
In which qB and qS denote the bed load transport and suspended load transport, respectively.
3.3 Linear Stability Analysis (LSA)
For investigating the most unstable growth of large scale bedforms, a Linear Stability Analysis,
(LSA) has been widely used in former studies. The stability analysis originates from a basic
flow problem, for which the flow, sediment transport and bed level is in an equilibrium state.
In Fig. 3.4, a perturbation is superposed to the flat bed case. The bed disturbance will evolve
into a series of bedforms, [Madsen, 2002]. The development of series of bedforms is caused
by an unstable system consisting of interactions between the flow field, the erodible bed level
and the sediment transport field. The bed level does not remain plane due to instabilities,
which is amplified if the growth rate is unstable. In order to investigate the phenomena of an
unstable bed level, an LSA has proved to be easily implemented and evaluated.
SWL
Seabed
Figure 3.4: Single perturbation on a flat bed.
30. CHAPTER 3. THEORY - LARGE-SCALE BEDFORMS 16
The LSA has its starting point within an equilibrium system composed by the flow field, bed
level and sediment transport field. In principle, the perturbation may be superposed to the
basic fields of variables. In Fig. 3.4, an infinitesimal sinusoidal perturbation is superposed to
the flat bed as a disturbance to the flow field and sediment transport field. The perturbations
are often chosen as periodic functions, as these are the lowest order in a Fourier expansion of
any perturbation. If the system proves unstable, the perturbation will be amplified. If the
system is stable, it will decay and try to resume its plane bed condition.
When the bed is perturbed with small pertubations, the influence on the flow field and
sediment transport field is also small. To establish a relation between the unperturbed flat
bed case and the disturbed bed, a generic perturbed quantity K is computed by a Taylor
expansion series of the unperturbed bed as:
K = K0 + aK1 + bO(ε2
) + cO(ε3
) + ... + nO(εn
) (3.14)
Where a, b, c, n are constants and the generic quantity, K, can be set equal to the flow velocity,
sediment transport rate or rate of bed level change etc. ε is a small ordering parameter. The
Taylor expansion contains higher order terms which can be neglected by assuming only linear
behaviour for small perturbations. Eq. (3.14) is reduced to the following, where a is small:
K = K0 + aK1 (3.15)
By reducing the Taylor expansion, it is now assumed that the flow field, sediment transport
field and bed level vary linearly with the same order of magnitude as the introduced per-
turbation. For this assumption to be valid, the perturbation must be small and must not
have any abrupt changes. Then, the streamlines are led smoothly over the perturbation. I.e.
a large amplitude compared to the wavelength of the perturbation would raise problems at
the foot of the perturbation, causing the streamlines not to be lead smoothly. Hence, the
linear behaviour would no longer be applicable. When the perturbation is kept small, the
total sediment transport rate, qT , can be written as a sum of the zero order state and the
contribution from the perturbed case:
qT (x) = q0 + q1(x) (3.16)
In which q0 is the transport rate for the unperturbed case corresponding to a flat bed condition.
The term q1(x) contains the transport rate due to disturbance of the flat bed and may vary
along the streamwise direction x. The same considerations can be carried out for the bed
level changes. The rate of bed level change can be expressed similar to Eq. (3.15):
∂h
∂t
=
∂h0
∂t
+
∂h1
∂t
(3.17)
Where ∂h0
∂t denotes the rate of bed level change for flat bed conditions. ∂h1
∂t denotes the rate
of bed level change due to the introduced perturbation. In general, if the flow conditions
are kept uniform, the zero order solution ∂h0
∂t will attain values a lot smaller than that of the
perturbed case. The resulting rate of bed level change for the perturbed case can be expressed
as:
∂h1
∂t
=
∂h
∂t
−
∂h0
∂t
(3.18)
The flow scenarios created in this thesis are uniform, however it is not possible to create a
perfectly uniform flow and it will be evaluated whether the the zero order solution can be
neglected.
31. Chapter 4
Numerical Modelling in MIKE 21
In this chapter, the implementation of a numerical model will be explained. The numerical
model used for the present study is a MIKE 21 model, which is a part of the commercial
software ”Mike by DHI”, developed by the Danish Hydraulic Institute (DHI).
Usually, numerical fluid systems are applied when fluid motion is to be solved. Fluid motion
is governed by the Navier-Stokes equation, also known as the momentum equation, which is
coupled with the continuity equation and solved for velocity and pressure fields. The deriva-
tion of the equations is based on basic laws of conservation of mass, momentum and energy,
but have no known analytical solution. Thus, a numerical model is used to obtain an approx-
imated solution for a finite number of elements, defined by the refinement of a computational
mesh. The procedure is therefore to discretise the domain of interest and create a computa-
tional mesh inside. The equations must then be discretised and solved in each mesh element.
By using numerical schemes, a set of linear equations can be established and solved for a
numerical solution.
The applied numerical model is explained and may be divided into three main parts; the Mesh
Generation part, the Hydrodynamic Module and the Sand Transport Module, accounting for
the non-cohesive sediment transport. The three parts of MIKE 21 make it possible for the
numerical model to solve the equations for the conservation of mass and momentum. The
chapter is written in a way for the reader to understand how the numerical model is set up
in MIKE 21. The chapter will not provide an in-depth explanation of MIKE 21, as it is quite
a comprehensive numerical tool.
4.1 Mesh generation
The Bathymetry editor and the Mesh Generator provides an environment for creating, editing
and presenting detailed 2D bathymetries. A Matlab programme is established in order to
speed up the process, such that the mesh file is created automatically, see Appendix K.
The mesh generator is used to construct meshes that consist of quadrangular elements. The
type of elements are chosen due to the rather simple domain evaluated in the model and the
fact that the computational time is reduced significantly compared to a model using triangu-
lar mesh elements. Typically, rectangular mesh elements are used for simple geometries. As
the domain becomes more complex a rectangular meshing may not be suitable.
17
32. CHAPTER 4. NUMERICAL MODELLING IN MIKE 21 18
The quadrangular mesh generation may be done by either a simple algebraic method using a
boxing technique, or an algebraic grid generator using transfinite interpolation. The boxing
technique may cause skewed grids, hence the transfinite method is preferred, [DHI, 2014c].
4.1.1 Transfinite Interpolation
The algebraic grid generator uses a transfinite interpolation to avoid skewed mesh elements.
The scheme uses relative coordinates along four polylines constituting the polygon to be
meshed, see Figure 4.1, [DHI, 2014c]. The main idea is to break every closed polygon into
quadrangles by defining grid points in a reference coordinate system, (ξ, η). Different to the
algebraic box method, the mesh element sides are not always parallel to the polylines as this
may cause skewed grid. The transfinite method is used, however the models built up in this
thesis only considers rather simple geometries and therefore no difference in the two methods
occur when meshing.
Figure 4.1: Notation for transfinite interpolation grid generation, [DHI, 2014c]
A coordinate system (ξ, η) is created parallel to the polylines as shown in the figure, in which
a ◦ and a • corresponds to a vertex and a node in the following, see Fig. 4.2. ξ and η
vary by a parameterisation along the polylines between 0 and 1, defining the distances in the
coordinatesystem (ξ, η). The polylines are defining the boundaries. The intersection of ’North’
and ’East’ corresponds to (ξ, η) = (1, 1), ’South’ and ’West’ corresponds to (ξ, η) = (0, 0) and
so forth. The grid within the boundaries, called the domain, may then be defined through
the expressions:
x(ξ, η) =(1 − ξ)x(0, η) + ξx(1, η) + (1 − η)x(ξ, 0) + ηx(ξ, 1)− (4.1)
(1 − ξ)(1 − η)x(0, 0) − ξ(1 − η)x(1, 0)−
(1 − ξ)ηx(0, 1) − ξηx(1, 1)
y(ξ, η) =(1 − ξ)y(0, η) + ξy(1, η) + (1 − η)y(ξ, 0) + ηy(ξ, 1)− (4.2)
(1 − ξ)(1 − η)y(0, 0) − ξ(1 − η)y(1, 0)−
(1 − ξ)ηy(0, 1) − ξηy(1, 1)
The parameterised (ξ, η) is formulating a set of grid points in a coordinate system (x, y), which
corresponds to a coordinate system aligned with the flow direction of the domain in this case.
33. 19 CHAPTER 4. NUMERICAL MODELLING IN MIKE 21
Figure 4.2: Polygon with original vertices along with the interpolated vertices. Opposing
polylines have identical number of total vertices, [DHI, 2014c]
In Figure 4.2 it is seen that the original vertices are located at the end of a piecewise linear
polyline. The interpolated vertices are shown with a gray centered on the linear polyline.
Now, vertex pairs on opposing polylines define the gridpoints. However, it should be noted
that the vertex pairs are not connected with a straight line by default due to piecewise linear
polylines. The grid points may now be defined from the maximum grid element side length,
e.g. the maximum side length in the direction of ξ is denoted by ∆s and is used to define
the non-dimensional mesh element side length ∆ξ. The maximum length of the ’North’ and
’South’ polylines is denoted lnorth and lsouth and the non-dimensional maximum side length
reads
∆ξ =
max(lnorth, lsouth)
∆s
(4.3)
The same procedure is used in the direction of η.
The chosen number of gridpoints is evaluated in Section 4.1, based on a convergence anal-
ysis. The final step is to perform a linear bathymetry interpolation for matlab generated
bathymetry coordinates. See Appendix K.
4.2 MIKE 21 Flow Model FM
The present section will elaborate on the most important parts of the MIKE 21 Hydrodynamic
Module (HD). As the HD module is quite a comprehensive model, an in-depth explanation is
not given, but the physical, mathematical and numerical background is briefly covered. An
in-depth description of the model is found in the scientific documentation of MIKE zero, [DHI,
2014a].
In general, a numerical modelling system is based on numerical solution of the two dimensional
incompressible Reynolds averaged Navier-Stokes equations. For the hydrodynamics, the two
horizontal Navier-Stokes equations and the continuity equation are to be solved. Firstly, the
governing equations in fluid dynamics are presented. Secondly, the implementation of the
governing equations in MIKE 21 is explained.
34. CHAPTER 4. NUMERICAL MODELLING IN MIKE 21 20
4.2.1 General flow equations
Fluid flow is governed by the continuity equation coupled with the Navier-Stokes equation,
also known as the momentum equation. The density of the fluid is considered constant, as the
flow is incompressible. A Cartesian coordinate system is used with axes oriented so that the
direction of the normal to the still fluid surface and the direction of z coincide. The direction
of x is parallel to the flow direction and y is in the direction of the normal to x.
Continuity equation
The continuity equation for an incompressible fluid, can be written in Cartesian notation, as:
∂ui
∂xi
= 0 (4.4)
Where xi = (x, y, z) is the Cartesian coordinates and ui = (u, v, w) being the the velocity
components.
Momentum equation
The momentum is described by the Navier-Stokes equation, where Si is a source of momentum,
[Wilcox, 2004].
∂ui
∂t
+
∂uiuj
∂xj
=
−1
ρ
∂p
∂xi
+
∂
∂xj
∂ui
∂xj
+
∂uj
∂xi
+ Si (4.5)
The left hand side of the equation represents the inertia of the flow. The first term is the
unsteady acceleration of the flow and the second term is the convection of momentum. The
inertia of the flow is driven by external forces; pressure gradients, body forces and viscous
forces which is represented by the right hand side of Eq. 4.5. The source of momentum will
typically include the gravitational force.
4.2.2 2D flow equations in MIKE 21
The hydrodynamic module in MIKE 21 solves the water levels and flows in estuaries, bays
and coastal areas, by simulation of 2D flows in a one layer fluid model. A one layer fluid
model means it is vertically homogeneous.
The Navier Stokes equation in (4.5) is typically solved in complex numerical models known as
Computational Fluid Dynamics, in which the computational demands are very time consum-
ing. A rather simple version of the equations listed above is used in MIKE 21. The continuity
and momentum equation in Eq. (4.4) and (4.5) may be written for the two-dimensional case
by integration of the horizontal momentum equations and the continuity equation over the
total water depth D = D0 + η. These equations are known as the Shallow Water Equations
with included viscous terms. A derivation of the general Shallow Water Equations (SWE) is
given in [Hadi et al., 2012]. In MIKE 21 the 2D shallow water equations takes the form:
35. 21 CHAPTER 4. NUMERICAL MODELLING IN MIKE 21
continuity:
∂D
∂t
+
∂Du
∂x
+
∂Dv
∂y
= 0 (4.6)
momentum in x:
∂Du
∂t
+
∂Du2
∂x
+
∂Dvu
∂y
= − gD
∂η
∂x
−
τbx
ρ0
+
∂
∂x
(DTxx) +
∂
∂y
(DTxy)
(4.7)
momentum in y:
∂Dv
∂t
+
∂Duv
∂x
+
∂Dv2
∂y
= − gD
∂η
∂y
−
τby
ρ0
+
∂
∂x
(DTxy) +
∂
∂y
(DTyy)
(4.8)
Where ρ0 is the density of the water, D is the water depth, u and v are the depth averaged
velocities in the x and y direction, η is the surface elevation, τb is the bed shear stress and Tij
is the lateral shear stresses.
The overbar represents the depth averaged value, hence:
Du =
η
−D0
udz , Dv =
η
−D0
vdz (4.9)
The lateral shear stresses, Tij, include viscous shear forces [DHI, 2014a]. They are estimated
using an eddy viscosity concept based on the depth averaged velocity gradients of the flow:
Txx = 2A
∂u
∂x
, Txy = A
∂u
∂y
+
∂v
∂x
, Tyy = 2A
∂v
∂y
(4.10)
Where A is the horizontal eddy viscosity.
4.2.3 Bed resistance
The bed shear stress can be evaluated from considerations of a steady, uniform and mildly
sloping channel with a fixed bed. The bed shear stress is given by:
τb = ρU2
f = ρgDS (4.11)
Where ρ is the density of the water, g is the gravitational acceleration, D is the water depth
and S is the slope of water surface. The vertical distribution of τ reads:
τ(z) = τb 1 −
z
D
(4.12)
Where z is the depth corresponding to the reference system in Figure 1.2. For a turbulent
flow the shear stress is related to the velocity gradient of the flow by:
τ
ρ
= νT
du
dz
(4.13)
Which for an open channel flow is generally accepted to vary as
νT = κUf z 1 −
z
D
(4.14)
Where the Von Karman constant usually is κ = 0.4. Combining and integrating Eq. (4.12)
and (4.13) gives the logarithmic profile:
u
Uf
=
1
κ
ln
30z
kN
(4.15)
36. CHAPTER 4. NUMERICAL MODELLING IN MIKE 21 22
The mean velocity of the open channel flow is described by V = 1
D
D
0 udz which leads to:
V
Uf
= 6 + 2.5 · ln
D
kN
(4.16)
Known as the flow resistance formula, [Fredsøe and Deigaard, 2012]. In MIKE 21 the bottom
friction velocity is described by:
Uf = cf |ub|2 (4.17)
Where |ub| = (ub, vb) is the depth averaged velocity above the bottom in the two dimensional
model and cf is a non-dimensional drag coefficient which is used to define the resistance or
drag due to an obstacle in water. The bed shear stress τb = (τbx, τby) is determined by a
quadratic friction law, [DHI, 2014a] as:
τb
ρ
= cf ub|ub| (4.18)
The drag coefficient may be expressed in terms of the Chezy number, C, or from the Manning
number, M by the following expressions:
cf =
g
C2
, cf =
g
(MD1/6)2
(4.19)
The Chezy number is used to formulate the bed resistance combined with the median grain
size d50. The Chezy number corresponds to a total resistance of the bed. The grain roughness
kN = 2.5 · d50 accounts for skin friction. The remaining resistance from used Chezy number
corresponds to form drag. The bottom friction velocity can be described from the Chezy
number by:
Uf =
U
√
g
C
(4.20)
The Chezy formulation is used in the numerical models as the Manning number depends on
the water depth and could cause different resistances due to non-uniformity of the flow.
Numerical formulation
The hydrodynamic module in MIKE 21 uses an Alternating Direction Implicit (ADI) tech-
nique to integrate the continuity and momentum equations given in (4.6), (4.7) and (4.8) in
the space-time domain. The governing equations are partially differential equations and can-
not be solved directly, which is why a cell-centered finite volume method is used to discretize
the equations, [DHI, 2014a]. The integral form of the system of SWE’s can in general form
be written as:
∂U
∂t
+ · F(U) = S(U) (4.21)
Where U is the vector of conserved variables. F is a flux vector function containing the depth
averaged velocities, a part of the gravity term and viscous terms. S is the vector of source
terms containing a part of the gravity term and the bottom shear stress. The general flux
integral is rewritten by the Gauss’s theorem and the discrete finite volume flux integral can
be solved by an approximate Riemann Solver, [DHI, 2014a].
37. 23 CHAPTER 4. NUMERICAL MODELLING IN MIKE 21
Time integration
For the 2D hydrodynamic simulation performed in MIKE 21, both a low order and a higher
order method can be used for the time integration. The low order is a simple first order Euler
method description and the higher order uses the second order Runge Kutta method.
The Courant number
The time step is governed by the Courant number of the simulation. The number represents
the ratio between the physical flow velocity and the numerical solution velocity:
C =
u
∆x
∆t
=
u∆t
∆x
≤ Cmax (4.22)
The criteria in the above equation ensures that the propagation of the information of the
physical flow can be described by the numerical solution. If C 1, a particle will be able to
travel more than a cell distance during a time step, which might result in instabilities. For
the simulations conducted in this thesis the Courant number is kept at Cmax = 0.8 in the
numerical models. The time step interval input and the Courant number then determines the
final time step used in the simulation.
4.2.4 Boundary conditions
MIKE 21 differentiates between closed and open boundaries and both are needed in the model.
An open channel flow is created, which in terms of boundary conditions make need for closed
boundaries along the side wall and open boundaries at the inlet and outlet of the domain.
For closed boundaries at the side walls the normal fluxes are forced to be zero for all variables,
thus the momentum equations take on full slip conditions along the side walls. If a no-slip
condition is wanted, the tangential velocity components are set to zero as well.
For the open boundaries, a flux, velocity, discharge or flather condition may be applied at
the inlet (upstream boundary condition). A ghost cell technique is used, creating a cell on
the outside of the domain, and the value at the boundary of first cell is then interpolated
between the two. In most of the simulations a discharge boundary condition is applied at the
inlet boundary of the domain. At the outlet (downstream boundary condition) a water level
is typically set. With this setup of boundary conditions, the flow is now driven by a discharge
and a water level.
It is also possible to drive the flow with two water level boundaries, but this setup fixes the
water surface and will not give reliable results. Therefore, a velocity/discharge BC and a
water level BC are always used for the inlet and outlet boundaries, respectively.
The flather condition may be used to create a periodic simulation, meaning that the inlet and
outlet boundary is connected. The flather condition requires a flat bathymetry as the grid
point at the inlet and outlet are connected. When a flather condition is applied, the flow is
not driven by a discharge and a water level, and a body force should be established in another
way. A wind forcing may be applied for this purpose. The use of a periodic boundary is
discussed later on. As the Advection-Dispersion formulation for the Sand Transport cannot
by periodic, a fully periodic domain cannot be achieved in terms of sediment calculations.
38. CHAPTER 4. NUMERICAL MODELLING IN MIKE 21 24
4.3 Sediment transport in MIKE 21
The sediment transport rates are calculated continuously with the HD module. For the pure
current case the transport rates are calculated on the basis of sediment transport formulae
derived from either empirical or deterministic principles, [DHI, 2014b].
The modelling of sediment transport is divided into a bed load and a suspended load com-
ponent. In general, the governing transport description is described by an equilibrium or
a non-equilibrium transport description, which depends on what situation one wants to de-
scribe. The bed load is mainly initiated by the bed shear stress, depending on the stream
power per unit area and can be described by an equilibrium transport description. However,
the suspended load needs some time to be initiated and adapt to the concentration profile,
which requires memory effects. This can best be described by non-equilibrium models, which
typically account for the phase lag between the transport and the flow. A non-equilibrium
transport description is used for all simulations.
The sand transport module (ST) in MIKE 21 is based on a classical approach using the
bottom shear stress to determine when sediment grains start to move, also known as incipient
motion. The forces acting on the sediment grain can be divided into driving and stabilising
forces. When the driving forces exceed the stabilising, the sediment grain is mobilised. The
stability of the sediment grain was described by Shields, [Fredsøe and Deigaard, 2012], and
the considerations of the stability of sediment grains lead to a non-dimensional term relating
the driving and stabilising forces, known as Shields parameter:
θ =
τb
ρgd(s − 1)
=
U 2
f
gd(s − 1)
(4.23)
Where τb is the component of the bottom shear stress tangential to the bed, ρ is the fluid
density, d is the diameter of the sediment grain and s is the relative density of the grain,
which is s = 2.65. Uf is the skin friction velocity. The value s = 2.65 corresponds to
quartz-based sediment. If the Shields parameter is greater than the critical Shields value, θc,
sediment grains are mobilised. The critical Shields parameter is θc = 0.045 by default in the
ST module. As earlier stated, the sediment transport is split up into two components, the
bed load and the suspended load:
• Bed load transport rate, qB: Movement of grains close to sea bed
• Suspended load transport rate, qS: Transport of grains in suspension in the water
column
A derivation of the Shields parameter is given in Appendix A.
4.3.1 Bed load transport
The bed load transport in MIKE 21 can be determined from either a model by ”Engelund
and Hansen”, ”Van-Rijn”, ”Engelund and Fredsøe” or ”Meyer-Peter Müller”. Throughout this
thesis, the transport rate is determined from the ”Engelund and Fredsøe” model. The non-
dimensional bed load transport rate is expressed by a non-dimensional parameter, ΦB:
ΦB =
qB
(s − 1)gd3
(4.24)
39. 25 CHAPTER 4. NUMERICAL MODELLING IN MIKE 21
Considerations of the driving forces during bed load transport are only dependent on the
velocity of the flow relative to the particle velocity. This lead to an established equilibrium
of the agitating forces and stabilising forces consisting of drag and lift, and dynamic friction
respectively. The equilibrium establishment leads to a formulation of the mean transport
velocity, [Fredsøe and Deigaard, 2012]:
UB = 10Uf 1 − 0.7
θc
θ
(4.25)
It is assumed that a fraction, p, of the grains in a single layer of grains on the bed are moving
during sediment transport. The bed load transport is then described by:
qB =
π
6
d3 p
d2
UB (4.26)
The total number of surface grains per unit area is 1/d2, thus the number of moving grains
may be, n = p/d2. With a limiting value of p to be p = 1, the expression for p reads:
p = 1 +
π
6 µd
θ − θc
4 −1/4
, θ θc (4.27)
The dynamic coefficient of friction, µd, is assumed equal to µd = 0.51, [DHI, 2014b]. The non-
dimensional skin shear stress θ is defined as in Eq. (4.23). Analogue to the flow resistance
formula in Eq. (4.16) the friction velocity is determined from the assumption of a logarithmic
velocity profile:
uf =
V
6 + 2.5 · ln D
kN
(4.28)
By substituting Eq. (4.27) into the non-dimensional transport rate in Eq. (4.26) the bed load
formula, known as the Engelund-Fredsøe formulation, takes the form of
qB 5p
√
θ − 0.7 θc (s − 1)gd3 (4.29)
The Engelund-Fredsøe formulation of the bed load transport is chosen in MIKE 21.
4.3.2 Suspended load transport
The total suspended load transport is computed by the velocity profile and the concentration
profile, [DHI, 2014b]:
qS =
D
b
ucdz , ΦS =
qS
(s − 1)gd3
(4.30)
Where the velocity profile is described by Eq. (4.15), where b = 2d z D is a reference
level. The concentration profile is given by the Vanoni distribution:
c(z) = cb
D − z
z
b
D − b
Z
(4.31)
In which cb is a reference concentration and Z is the Rouse parameter defined by Z = ws
κUf
,
where ws is the settling velocity of the grain. The reference concentration near the bed is
determined from an empirical relation obtained by Zyserman and Fredsøe, [DHI, 2014b]:
40. CHAPTER 4. NUMERICAL MODELLING IN MIKE 21 26
cb =
0.331(θ − θc)1.75
1 + 0.331
0.46 (θ − θc)1.75
(4.32)
4.3.3 Morphology
A morphological model is a combined hydrodynamic and sediment transport model. The
hydrodynamic flow field is updated continuously with the changing bed bathymetry. Mor-
phological models may be divided into two categories, coupled and uncoupled models. In
coupled models, the governing equations for the flow and sediment transport are merged into
a set of equations. In uncoupled models the hydrodynamics are solved prior to the sediment
transport, i.e. the flow field is solved for a certain time step prior to the sediment transport
equations. From the net sediment transport rates a new bed level is computed, which will
influence hydrodynamics in the next time step and so forth. In MIKE 21, an uncoupled model
is used.
Sediment Continuity and morphological bed update
The most fundamental parts of the morphological model are the sediment continuity and the
morphological bed update. The essential parameter in morphology for determining the future
bed level changes is the rate of bed level change, ∂h
∂t . This parameter is also referred to as
the growth rate. This parameter is computed for every mesh element center. In principle,
the formulation of the parameter is rather simple as it is based on the sediment continuity
equation, also known as the Exner Equation, [DHI, 2014b]:
− (1 − n)
∂h
∂t
=
∂Sx
∂x
+
∂Sy
∂y
− ∆S (4.33)
In which
n Bed porosity
h Bed level
t Time
Sx Bed load or total sediment transport in x-direction
Sy Bed load or total sediment transport in y-direction
x, y Horizontal Cartesian coordinate
∆S Sediment sink or source rate
The sink source term, ∆S, attains a zero value when the equilibrium sediment transport
description is used. However, when the non-equilibrium transport description is applied, an
advection-dispersion (AD) equation is to be solved, which can be written as:
∆S = Φ0(η0)ws(c − ce) (4.34)
Where η0 is normalised no slip level above bed, Φ0 is a unit profile function for the sediment
concentration, ws is the settling velocity for the suspended sediment, c is the depth-averaged
sediment concentration and ce is the depth-averaged equilibrium concentration. The sink
source expresses that sediment deposits if the actual concentration exceeds the equilibrium
concentration, and opposite if less than equilibrium. When the bed load transport is deter-
mined, the resulting rate of bed level change can be expressed.
41. 27 CHAPTER 4. NUMERICAL MODELLING IN MIKE 21
The bed level is updated continuously at every HD-simulation time step based on the rate of
bed level change, dh
dt . The new bed level is obtained by:
hnew = hold +
1
1 − n
∂h
∂t
∆tHD (4.35)
4.3.4 Helical Flow module
The flow velocity and the concentration profile vary over the water depth. In general, the ve-
locity is described by a logarithmic profile in the streamwise direction and a secondary flow in
the transverse direction. This is due to the flow in a river bend. The reason for the secondary
current is an imbalance of the centripetal force, which will start an outward motion of the
water near the water surface and inward motion near the bed. The effect of the transverse
secondary velocity profile can be included in MIKE 21 separately.
In theory, the strength of the helical motion is related to the flow curvature and the bed
resistance. Considering the governing equation for the radial motion as:
∂vr
∂t
+ vθ
∂vr
∂s
+ vr
∂vr
∂r
+ w
∂vr
∂z
−
v2
θ
r
= −g
∂h
∂r
+
1
ρ
∂τr
∂z
(4.36)
Where the centrifugal force on the LHS is included per unit mass as
v2
θ
r and h is the surface
elevation, [Fuhrman, 2013]. A definition sketch showing the orientation of the flow directions
is given in Fig. 4.3a.
B
D
τS + ∂τS
∂z
dz
τr + ∂τr
∂z
dz
P + ∂P
∂r
dr
τS
τr
P
dθ
∆r ∆S
∆z
r
r S
z
u
v
(a) Three dimensional shape of large scale bedform at
time t = 0.
r
z z
vr
r
τ
pressurecentrifugal force
dr
(b) Initial bedform and bedform
after some time, where temporal
growth has been assumed.
Figure 4.3: Helical flow sketches
Taking Eq. (4.36) to leading order yields:
∂vr
∂t
−
v2
θ
r
= −g
∂h
∂r
+
1
ρ
∂τr
∂z
(4.37)
Assuming a steady flow and defining an eddy viscosity, ε, the radial bed shear stress is :
τr = ρε
∂vr
∂z
(4.38)
42. CHAPTER 4. NUMERICAL MODELLING IN MIKE 21 28
The near bed velocities are small and the net pressure caused by superelevation of the surface
must be balanced by a radial shear stress:
g
∂h
∂r
=
1
ρ
∂τr
∂z
(4.39)
The superelevation of the water surface, see Fig. 4.3b, generates a negative bed shear stress.
This is directly linked to the velocity gradient, causing near bed flow towards the bend center
(inner bank). Due to the radial velocity component, the near bed velocity vector is different
from the longitudinal direction. This means that the helical flow pattern can be considered as
the sum of the main streamwise flow and a circulation normal to the main flow. The helical
flow forms an angle, δ, with the longitudinal flow direction, marked with a dashed line in Fig.
4.3a, and Rozovskii (1957) gives:
tan(δ) = −β
D
Rs
(4.40)
In which D is the local water depth, Rs is the radius of curvature of flow stream lines and
β is a constant, which for channels with arbitrary cross sections takes the value interval of
β = 7 − 11, [Fuhrman, 2013].
The parameter β is defined by:
β = α
2
κ2
1 −
√
g
κC
(4.41)
κ is the Von Kárman constant equal to 0.4, g is the gravitational acceleration, C is the Chezy
number and α is a calibration constant. An approximate value of β is 10. It must be noted
that for a lower value of C (increasing flow resistance) the constant β decreases, giving smaller
deviation in the direction of the bed shear stress. The input constant α is set to a default
value of 1 in the model, when the helical module is active.
The MIKE 21 HD module is a 2D model and the helical motion is handled separately in
the ST module. In the fully three dimensional description in MIKE 3 the HD module will
automatically include this effect, which is why the implementation of the secondary flow in
the ST module is described as a pseudo 3D description, [DHI, 2014b].
4.4 Setup of MIKE 21 model
In this section the typical setup of a numerical model in MIKE 21 is shown. The HD module
and ST module are also presented.
Before entering the HD module and ST module, a time step interval and a number of time
steps should be chosen. The time step is governed by the Courant number, as explained in
Section 4.2.3. The Courant number ensures that the time step is small enough to capture the
physical propagation of information through the flow. The numerical model synchronizes the
time step of the different models. The model always start at time step zero and the amount
of time steps defines the simulation time which must be chosen in order to achieve stable
numerical results.
43. 29 CHAPTER 4. NUMERICAL MODELLING IN MIKE 21
The input parameters for the MIKE 21 model are provided in Tab. 4.1. The parameters shown
are used in the Linear Stability Analysis. When morphology is applied, some parameters have
to be adjusted. This will be explained later on.
Table 4.1: Setup in MIKE 21 HD and ST module.
HD setup Parameter Value/Setting
Simulation Period 10 hrs
Time Step Interval 120sec
No. of Time Steps 300
Solution Technique
Time Integration: Higher order
Space discretization: Higher order
Minimum Time Step: 0.01sec
Maximum Time Step: 60sec
Boundary Conditions
Inlet: Discharge Q
Outlet: Water level η
Sidewalls: Zero normal velocity
Horizontal Eddy Viscosity Smagorinsky formulation: constant value of 0.28
Bed Resistance Chezy: C = 52
ST setup
Model Type
Pure current
Transport description: Non Equilibrium
Helical motion - on/off
Solution Technique
Time Integration: Higher order
Space discretization: Higher order
Boundary Conditions
Inlet: Equilibrium Conditions
Outlet: Zero Gradient
Sediment Properties Constant d50 =0.2mm
Result files HD.dfsu, ST.dfsu
Elapsed Simulation Time About 30-60mins - 3.33GHz Intel i5, 4GB ram
45. Chapter 5
Dimensional analysis
In the following chapters, the governing equations of the flow and sediment transport are
analysed in accordance with a dimensional analysis utilising the Buckingham’s Pi-theorem.
From [Larsen and Brorson, 2003] a stepwise method is applied to the governing equations. The
dimensional analysis offers a method for reducing complex physical problems to the simplest
form prior to obtaining a quantitative answer, [Sonin, 2001]. In general, the dimensional
analysis can be seen as a similarity study, in which the similarity refers to the equivalence
between two phenomena that are actually different.
5.1 The steps of dimensional analysis
A dimensional analysis is often applied when results from physical model experiments are to
be analysed. During the dimensional analysis the model parameters of the system are to be
reduced to fewer dimensionless quantities. The dimension of a system refers to the type of a
given physical quantity, e.g. if movement of a body is to be described the dimensions used are
often mass, length and time, known as the fundamental dimensions. Dimension and units are
related but different concepts. Physical quantities are measured in units and a dimension of
a physical quantity is independent of particular units. For example, grams and kilograms are
units of the dimension mass. Five different fundamental dimensions are length, mass, time,
temperature and charge. All other dimensions known are products and powers of the five
dimensions mentioned. Any physical quantity Q0 in a well defined physical process or event
may be known as a dependent variable. The most important step in dimensional analysis is
to identify the independent variables Q1...Qn, that determine the value of Q0 as:
Q0 = f(Q1, Q2, ..., Qn) (5.1)
The set of chosen independent variables is complete once the value of the members are specified
and no other quantity can affect the value of the dependent variable, Q0. From the complete
set of independent variables Q1..Qn a complete dimensionally independent subset Q1...Qk,
k n, can be established and the dimension of each of the remaining independent variables are
expressed in terms of Qk+1...Qn. This well known method is also known as Buckingham’s π-
theorem in which the dependent variable Π0 is formulated in terms of the reduced independent
variables n − k, as Π1...Πn−k, [Sonin, 2001]:
Π0 = f(Π1, Π2, ..., Πn−k) (5.2)
This equation is the end result of the dimensional analysis and the system has been reduced
31
46. CHAPTER 5. DIMENSIONAL ANALYSIS 32
with k variables. For the governing equations studied in this thesis the fundamental dimen-
sions are length, mass and time, denoted as [L], [M] and [T], respectively.
5.2 Steady uniform flow
The shallow equations and the governing equations for the sediment transport field introduced
in Chapter 4 contains several dependent variables, which can be used in the analysis of large
scale bedforms. In the following chapters, the main dependent variable examined is the
unstable growth rate, which was explained in Chapter 3. The unstable growth rate can
be found from the rate of bed level change as shown in Eq. (3.9), and repeated here for
convenience:
Ω =
∂h
∂t
1
A
(5.3)
The unstable growth rate ∂h/∂t has the fundamental dimensions length, mass and time. The
growth rate is a function of the following independent variables.
∂h
∂t
= f(U, D, C, g, ρ, ρs, d50, kx, ky, A, νt) (5.4)
The units of the independent variables are listed as
Table 5.1: Units of independent variables
Variable Unit Variable Unit
[U] m/s [d50] m
[D] m [kx] 1/m
[C]
√
m/s [ky] 1/m
[g] m/s2 [A] m
[ρ] kg/m3 [νt] m2/s
[ρs] kg/m3
In which U is the mean flow velocity, D is the water depth, C is the Chezy number, g is
the gravitational acceleration, ρ is the density of the fluid, ρs is the density of the sediment
grain, d50 is the median grain size of the sediment, kx is the streamwise wave length, ky is
the transverse wave length, A is the amplitude of the pertubation, νt is the eddy viscosity.
The dimensional considerations is then used in next step where the dimensions are made
dimensionless. Some of the independent variables listed in Eq. (5.4) are constants and can
be left out of the analysis. E.g. the density of the fluid and the sediment grain are kept
constant throughout the thesis. Now the problem is left with the fundamental dimensions
length and time. Firstly the independent variables only containing the length dimension is
made non-dimensional with the water depth D. Thus Eq. (5.4) takes the form:
∂h
∂t
= f U, C, g,
d50
D
, kxD, kyD,
A
D
, νt (5.5)
The length dimension is still kept within the remaining independent variables and will be
made non-dimensional by using both the water depth, D, and the gravitational acceleration,
g, containing the fundamental dimension, time. Hence, Eq. (5.5) becomes:
∂h
∂t
= f
U
√
gD
,
C
√
g
,
d50
D
, kxD, kyD,
A
D
,
νt
UD
(5.6)
47. 33 CHAPTER 5. DIMENSIONAL ANALYSIS
The dependent variable ∂h
∂t has the dimension m/s and can be made non-dimensional with the
term 1/
√
gD. By considering the linearity of ∂h
∂t with the amplitude A, the proportionality
can be utilised to rewrite Eq. (5.6) to:
1
√
gD
∂h
∂t
=
A
D
f
U
√
gD
,
C
√
g
,
d50
D
, kxD, kyD,
νt
UD
(5.7)
The LHS is rewritten and the final non-dimensional equation reads:
Ω =
∂h
∂t
D
g
1
A
= f
U
√
gD
,
C
√
g
,
d50
D
, kxD, kyD,
νt
UD
(5.8)
Which is similar to the unstable growth rate in Eq. (5.3). Some of the independent variables
presented in the dimensional analysis are kept constant throughout the thesis for simplicity,
which makes the unstable growth rate only depending on few variables. The flow properties
are kept constant as well as the sediment transport properties, which makes the unstable
growth a function of only the introduced pertubation, in terms of horizontal dimensions. Eq.
(5.8) can then be written as:
Ω =
∂h
∂t
D
g
1
A
= f (kxD, kyD) (5.9)
The numerical models thus need the input parameters, the Froude number, Fr = U√
gD
, the
non-dimensional Chezy number, C∗ = C√
g , the non-dimensional amplitude of the introduced
pertubation, A/D and the non-dimensional median grain size, d50
D .
A Linear Stability Analysis can now be carried out in terms of the horizontal dimensions of
the pertubation, by keeping flow and sediment transport properties constant. As mentioned
in Chapter 3, a certain combination of kx and ky gives the largest unstable growth rate, Ω,
which should correspond to the wavenumber features observed in field.
49. Chapter 6
Stability analysis - steady current
In the present chapter a Linear Stability Analysis (LSA) is presented. The stability analysis
is carried out in order to investigated the combination of the wavenumbers kx and ky, which
gives the largest unstable growth rate. The morphological computations carried out in Chap-
ter 7 are performed and evaluated on the basis of the LSA. Firstly, the model is validated in
terms of a uniform steady flow, a flat bed sediment transport field, and a mesh convergence
analysis. Secondly, two different stability studies are conducted; a single perturbation model
and a periodic perturbation model.
6.1 Mesh analysis
Before performing any of the investigations, a mesh analysis is conducted in order to obtain a
numerical modelling grid which ensures results with adequate accuracy. In numerical models
it is often necessary to settle a compromise between simulation time and desired relative
error. The meshing procedure is described in Section 4.1 and the mesh resolution for the
numerical model will be evaluated in the following. In the present LSA, the desired result
from the numerical models is the rate of bed level change of the introduced bed perturbation.
The mesh convergence analysis is carried out on the basis of increasing the amount of cells
in the numerical domain. Fig. 6.1 shows a sketch of the domain with a single wavelength
perturbation.
x
y
Lx
Figure 6.1: Sketched model subjected to a mesh analysis. The domain length is LD = 5 · Lx
and width is WD = Ly.
An arbitrary combination of non-dimensional wavenumbers kxD and kyD are chosen to 0.016
and 0.063, respectively. This corresponds to wavelengths of Lx = 4000 m and Ly = 1000 m.
35
50. CHAPTER 6. STABILITY ANALYSIS - STEADY CURRENT 36
For the periodic perturbed model a streamwise wavenumber of kxD = 0.021, corresponding
to Lx = 3000 m is used. The wavelengths are divided into an equal number of gridpoints, n.
The simplest shape of mesh elements is rectangular elements, which will have different side
lengths when the number of gridpoints is defined as:
n =
Lx
∆x
=
Ly
∆y
(6.1)
Where ∆x and ∆y are sidelengths of the rectangular mesh cell. Firstly, the mesh analysis is
carried out for the single perturbed bed and secondly for the periodic perturbed bed. Both a
single wavelength perturbation and a periodic perturbed bed are subjected to a mesh analysis
as both models are investigated in the LSA. Fig. 6.2 shows a single wavelength perturbation
with n = 51 gridpoints in each direction.
Figure 6.2: Domain with 51 gridpoints in each direction defined by transfinite quandrangular
mesh elements. Single perturbation with one wavelength in each direction.
In Fig. 6.3 the results of the mesh analysis is shown.
20 40 60 80 100 120 140 160 180 200
0
0.5
1
1.5
2
2.5
x 10
−3
Gridpoints, n [#]
GrowthrateΩ[-]
Single perturbation
Periodic perturbed bed
Figure 6.3: Mesh analysis for a single wavelength perturbation model and a periodic per-
turbed bed model. Non-dimensional wavenumbers, kxD = 0.016; kyD = 0.063 and kxD =
0.021; kyD = 0.063, for single and periodic perturbed model, respectively.
It may be seen that the growth rate converges when the number of gridpoints increases to
about 100, with a corresponding relative error of 6% and 3%, respectively for the single and
periodic perturbed model. The relative error is computed based on an assumption that the
results are fully converged for 201 gridpoints. Based on a compromise between a small relative
51. 37 CHAPTER 6. STABILITY ANALYSIS - STEADY CURRENT
error and the elapsed computational time, a refinement of the numerical mesh resolution of
51 gridpoints is selected. The compromise is also based on the available CPU power and
allocated time for the project. With a refinement of 51 gridpoints, most of the numerical
models for the LSA use approximately 30-60 minutes simulation time.
6.2 Model validation
A significant feature in this thesis is that all numerical models include uniform open channel
flows. With a perfect uniform flow, the zero order of the rate of bed level change will not have
to be considered since the value is significantly smaller than that of the perturbed case. This
saves computational time and a uniform flow will in general be easier to interpret. The flow
field is verified for a steady uniform sloping open channel flow. The uniformity of the flow
has got zero rate of bed level change within the sediment transport field, because in theory
the inlet of the net sediment transport equals the outlet.
Table 6.1: Sloping open channel flow validation - model input parameters
Physical parameters Non-dimensional parameters
Flow velocity, U [m/s] 1 Fr 0.1
Chezy number, C [m1/2/s] 52 C∗ 16.6
Water depth, D [m] 10 -
Energy grad., I [-] 3.7e-5 -
Grain size, d50 [mm] 0.2 d50/D 2e − 5
Length domain, LD [m] 24000 LD/D 2400
Width domain, WD[m] 1000 WD/D 100
θ 1.12
A MIKE 21 HD and ST model setup corresponding to Tab. 4.1 is used. The hydrodynamic
conditions are shown in Tab. 6.1. The model is investigated to see whether MIKE 21 can
reproduce a completely uniform flow. The growth rate, Ω, will be directly related to the
uniformity of the flow. From the mesh analysis conducted and explained in Section 6.1, a
gridresolution of 51 gridpoints per wavelength is used. In this model, an arbitrary wave length
of Lx = 3000 m was implemented even though the amplitude is zero. The domain length was
chosen arbitrarily to LD = Lx · 8. As explained in Section 4.4, the flow is driven with a
discharge at the upstream boundary and a water level at the downstream boundary. The
setup of the sloping uniform flow is sketched in Fig. 6.4:
x
z Q
D
S
Figure 6.4: Sloping open channel flow. Flatbed conditions.
In Fig. 6.5 the results from the flat bed model is shown.
52. CHAPTER 6. STABILITY ANALYSIS - STEADY CURRENT 38
0 0.5 1 1.5 2
x 10
4
−0.5
0
0.5
η[m]
Surface Elevation, η(x)
0 0.5 1 1.5 2
x 10
4
−10.5
−10
−9.5
−9
h[m]
Bed level, h(x)
0 0.5 1 1.5 2
x 10
4
1
1.1
U[m/s]
Mean flow velocity, U (x)
0 0.5 1 1.5 2
x 10
4
0
5
10
D[m]
Total water depth, D(x)
0 0.5 1 1.5 2
x 10
4
0
1
2
x 10
−6
x [m]
∂h
∂t[m/day]
Rate of bed level change, ∂ h
∂ t
(x)
Figure 6.5: Uniform flow for a sloping flat bed.
From Fig. 6.5 it is seen that the MIKE 21 HD model is very much able to reproduce a uniform
flow by using a sloped model, as the streamwise mean flow velocity, U, is U = 0.999m/s and
the water depth is constant through the entire domain.
For uniform open channel flows, where the water depth is the same in every cross section
downstream, the flow velocity will also be the same in every cross section. This means that
the energy loss, ∆H, can be found from the bed slope, S. Chezy formulated an expression of
the uniformity of an open channel flow from the Darcy-Weisbach equation as:
V = C RhS (6.2)
In which C is the Chezy number, which can be found emperically, and Rh is the hydraulic
radius, which for a wide river equals the water depth D. For the uniform flat bed model the
rate of bed level change, defining the zero order growth rate, is within the O(-7). It must
also be noticed that the inlet and outlet boundaries affect the growth rate, ∂h
∂t , which will be
important when evaluating the growth of an introduced perturbation in the domain. However,
the boundaries are only affected 1000 m downstream and upstream at the inlet and outlet,
respectively.
53. 39 CHAPTER 6. STABILITY ANALYSIS - STEADY CURRENT
6.3 Single doubly periodic bedform
In this section a single doubly periodic bedform is introduced to the uniform sloping model.
The system is sketched in Fig. 6.6 and 6.7. The procedure of this study is similar to the
investigations carried out by [Madsen, 2002]. The numerical model used in the investigations
was explained in Section 3 and 4. The calculations are performed by a MIKE 21 HD and ST
module, which was tested out in Section 6.2. The analytical result of the unstable growth will
be compared to the unstable growth obtained from the numerical model.
The most simple case of the stability of a large scale bedform would be a single wavelength
perturbation subjected to a uniform flow. In theory, a given small sinusoidal bed perturbation
superposed to the flat bed case should provide a linear response within the flow and sediment
transport fields, hence the unstable growth rate. A similar study can be carried out in a
numerical model, known as a linearity study.
Firstly, as described in Section 3, the studied perturbation is the single wavelength bedform
described by:
h(x, y, 0) =
A
2
(cos(kxx) + 1) cos(kyy) (6.3)
The uniform flow with perturbed bed is sketched in Fig. 6.6. As for the validation model in
Section 6.2, the hydrodynamic flow conditions are as described in Tab 6.1.
x
z Q
D
S
A
Figure 6.6: Single wavelength pertubation introduced to the sloping uniform flow model.
The bathymetry of an arbitrary single wavelength perturbation is shown in Fig. 6.7.
Figure 6.7: Bathymetry for a single wavelength perturbation, A = 320 mm, Lx = 1000 m,
Ly = 1000 m, LD = 5 · Lx. Origin located in (x, y) = (500000, 0).
To verify the linear response within the unstable growth rate, different streamwise wavelengths
are investigated. The study of the linearity of the growth rate with respect to the amplitude
of the perturbation is in the following referred to as a linearity study.
54. CHAPTER 6. STABILITY ANALYSIS - STEADY CURRENT 40
6.3.1 Linearity study - single wavelength perturbation
For the single wavelength perturbation a linearity study is carried out for a transverse wave-
length of Ly = 1000 m, corresponding to the non-dimensional wavelength kyD = 0.063. The
streamwise wavelength is varied with wavelengths shown in Tab. 6.2. The helical motion is
not included in the linearity study.
Table 6.2: Parameters for single wavelength linearity study
Run C∗ d50/D θ Lx [m] Ly [m] Φb Φs Helical motion
A1 16.6 2e − 5 1.12 2500 1000 on on off
A2 16.6 2e − 5 1.12 3000 1000 on on off
A3 16.6 2e − 5 1.12 4000 1000 on on off
A4 16.6 2e − 5 1.12 5000 1000 on on off
[Madsen, 2002] found that amplitudes up to approximately 1% of the water depth resulted
in a linear response. Accordingly, an introduced perturbation with an initial amplitude of
10mm, which corresponds to 0.1% of the water depth, is superposed to the flat bed. The
amplitude is then increased to A/D = {0.2%, 0.4%, 0.8%, ..., 51.2%}. The meshing of the
domain corresponds to 51 gridpoints per wavelength in accordance with the mesh convergence
analysis, conducted in Section 6.1. In Fig. 6.8 the results from the linearity study are shown.
0 10 20 30 40 50
0
2
4
6
8
x 10
−3
A/D [%]
∂h
∂t[m/day]
Lx = 2500
Lx = 3000
Lx = 4000
Lx = 5000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
x 10
−4
A/D [%]
∂h
∂t[m/day]
Lx = 2500
Lx = 3000
Lx = 4000
Lx = 5000
Figure 6.8: Linearity study for single wavelength pertubation, with investigated wavelengths
corresponding to Tab. 6.2. Top figure: full A/D test-interval. Lower figure: Zoom within the
A/D test-interval of A/D = [0, 0.01]
In Fig. 6.8 the rate of bed level change, ∂h
∂t , is plotted against the amplitude A/D. The
largest amplitude tested out in the linearity study is 5.120 m and the smallest is 10 mm.
55. 41 CHAPTER 6. STABILITY ANALYSIS - STEADY CURRENT
According to [Madsen, 2002], the linear response should be fulfilled for perturbations of up
to approximately 1% of the water depth. From the lower figure however, it is seen that the
linearity is disputable as the results do not produce fully straight lines. The simplest way to
express the linearity is to determine the non-dimensional growth rate Ω = ∂h
∂t
D
g
1
A , which
within the linear regime should be constant. From the flatbed model in Section 6.2 it is found
that the zero order growth rate is an O(-3) smaller than the obtained growth rates for the
perturbed models. This is why the zero order is neglected in Eq. (3.18), and the growth rate
is determined as:
∂h1
∂t
=
∂h
∂t
−
∂h0
∂t
=
∂h
∂t
(6.4)
In Tab. 6.3, the results shows the linearity of the study. The linear interval of A/D is shown
for the different wavelengths, marked with bold graphic.
Table 6.3: Results from linearity study of single wavelength perturbed bed.
Amplitude A1 A2 A3 A4
A/D [-] Lx = 2500 m Lx = 3000 m Lx = 4000 m Lx = 5000 m
Ω ·10−8 [-] Ω·10−8[-] Ω·10−8[-] Ω·10−8[-]
0.0010 0.502 0.428 2.239 0.270
0.0020 0.849 0.818 1.047 0.764
0.0040 1.442 1.709 1.087 1.182
0.0080 1.638 1.521 1.294 1.094
0.0160 1.497 1.478 1.294 1.217
0.0320 1.525 1.509 1.293 1.117
0.0640 1.550 1.498 1.262 1.051
0.1280 1.593 1.473 1.172 0.936
0.2560 1.665 1.401 0.977 0.709
0.5120 1.700 1.119 0.572 0.336
From the table it is seen that the small perturbations with amplitudes of up to 1% of the
water depth are not linear and the results in general does not show linear behaviour. As
the amplitude is increased it can be seen that an increasing non-linear behaviour is observed.
This was expected and will be investigated further in the following.
The perturbed bed system was made simple by only considering a single wavelength pertur-
bation. It seems, from considering the results, that the problem has been made even more
complex. The obvious next step after a linearity check is an investigation of the fastest growing
bedform wavelength even though the linearity is not fulfilled.
6.3.2 Stability curves - single wavelength perturbation
The disadvantage of using numerical models for an LSA is that for every point in the pa-
rameter space a computation has to be carried out. This is time consuming in comparison
with an analytical method. Due to the computational time for one run, a discretisation of the
wavelength interval is chosen such that a stability curve can be obtained from around 10 runs.
The procedure is to maintain an initial amplitude, A/D, and a transverse wavenumber, kyD,
and vary the streamwise wavenumber, kxD. From the linearity study carried out in Section
6.3.1, an amplitude within the most linear part of the linearity check is chosen. For simplicity,
