A NEW WIND-FORMULATION FOR HYDRODYNAMIC MODELS

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A NEW WIND-FORMULATION FOR HYDRODYNAMIC MODELS

J.S. Bokhorst

MSc Thesis

Supervisors: August 2003

Prof. Dr. Ir. G.S. Stelling

Dr. J.D. Pietrzak Delft University of Technology

Dr. Ir. A. van Mazijk Faculty of Civil Engineering and Geosciences

Ir. A.W. Dollee, RIZA Section of Fluid Mechanics

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A NEW WIND-FORMULATION FOR HYDRODYNAMIC MODELS

J.S. Bokhorst

ABSTRACT

A new wind formulation for hydrodynamic models is proposed, in which the wind stress term is not only dependent on the squared wind velocity, but on the wind velocity relative to the water. This should result in physically more realistic simulations.

Two different implementations for the new formulation have been tested in depth- averaged 1-dimensional and 2-dimensional models. The first implementation is fully explicit, while the second is semi-implicit. The semi-implicit one turns out to be more stable in shallow waters and is therefore preferable to the explicit implementation.

In very shallow water the new formulation clearly leads to different results. The 1D experiments show that the eigenfrequencies damp out much faster when the new formulation is used. The cause of this appears to be that the flow velocity is included in the wind stress, which leads to extra friction. In 2D experiments the new formulation leads to a smaller wind setup on very shallow areas surrounded by channels (e.g. tidal flats). This is because in a stationary situation a constant flow remains, so the relative wind velocity is smaller and the wind stress term decreases. The new formulation leads to more (numerical) stability, especially in cases where flooding and drying are important.

In 3D experiments - not tested during this research - the effects of the new formulation will probably be even stronger, since the flow velocity at the surface is usually larger than the depth-averaged velocity.

The new formulation introduces a factor α, which is the relation between the wind velocity at the surface and the wind at 10 m height (u_{surf ace} = αu₁₀, 0 < α < 1).

However, the exact value of α is yet unknown; further research on this parameter is needed. The effect of this parameter is that the influence of the flow velocity on the wind is reinforced.

Finally, the influence of the grid size is tested here in the case of a small domain. It is found that water levels are underestimated when a grid is chosen that is too coarse.

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List of Figures

2.1 Definition of q_down and q_up . . . 14

2.2 Staggered grid (Arakawa C) . . . 17

3.1 Overview of modelled area . . . 20

3.2 Detail of the eastern part of the Ketelmeer . . . 21

3.3 Bottom profile of the eastern part of the Ketelmeer in the IJsselmeer-model 21 3.4 Results at eastern bank of Ketelmeer . . . 22

4.1 Wind stress also dependent on flow velocity . . . 27

4.2 Logarithmic velocity profile . . . 30

5.1 Setup of the experiment - 1D closed basin . . . 32

5.2 Closed basin 1D . . . 32

5.3 Closed basin in 2D . . . 34

5.4 Setup of the experiment for shore . . . 35

5.5 Shore in 1D . . . 35

5.6 Results for wind 2 (explicit) with different time steps . . . 36

5.7 Wind 2 and 3 converge for a very small time step . . . 36

5.8 1 Dimensional closed basin - side view . . . 37

5.9 Drying in 1D, cross sections; results with wind 3 . . . 38

5.10 Results for the different wind formulations using method 2 . . . 38

5.11 2D Flooding and drying - top view . . . 39

5.12 Results for 2D flooding and drying . . . 40

5.13 Tidal flat model . . . 41

5.14 Results for tidal flat with channels . . . 42

5.15 Results for verification with Delft3D . . . 43

5.16 Velocity profile in a 3D-computation . . . 44

5.17 Setup of the experiment: 1D closed basin - side view . . . 45

5.18 Surface elevation at end of basin . . . 45

5.19 Tidal flat model . . . 46

5.20 2D results for different values of α . . . . 46

6.1 Setup of experiment 1 . . . 48

6.2 Water level at station 1 using different grid sizes, test 1 . . . 48

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6.11 Depth for different grid sizes . . . 53

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List of Symbols

C Chezy coefficient

C_B bottom friction coefficient (= g/C²) C_d drag coefficient

d depth below reference

f Coriolis parameter

F_wind wind term in the momentum equation g acceleration due to gravity

h water depth (h = ζ + d)

k time step

L length scale associated with the mixing process m numerical grid point in x-direction

n numerical grid point in y-direction q discharge (= u ∗ h)

t time

u component of depth mean current in x-direction u_∗ friction velocity

u_rel wind velocity relative to flow velocity (= u_w− u) u_w wind velocity at 10m height (=u₁₀) in x-direction v component of depth mean current in y-direction v_w wind velocity at 10m height (=v₁₀) in y-direction x coordinate in x-direction

y coordinate in y-direction

z height above surface

z₀ surface roughness length

α multiplication factor to get u_{surf ace} from u₁₀ ζ water elevation above reference

κ Von Karman constant

ν eddy-viscosity coefficient

Ψ effect of stability of the air column of the wind velocity

ρ density

σ Courant number (u∆t/∆x)

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Chapter 1

Introduction

1.1 Background

Storms that come ashore across wide, shallow, continental shelves cause severe changes in the sea level at the coast; these are called storm surges (Stewart, 2000). Basically, these surges can be divided into two types: surges due to (midlatitude) storms and surges caused by tropical cyclones.

Midlatitude storms are, amongst others, the cause of the storm surges around the North Sea. They can last for a few days during which they bring about a water level elevation of several meters. In combination with the astronomical (spring-)tide, water levels may exceed dike heights, dikes may fail and flooding can take place which can have disastrous consequences. Very infamous is the 1953 North Sea storm surge which flooded great parts of Zeeland in the Netherlands, as a result of which 1835 people died.

Tropical cyclones, also referred to as hurricanes or typhoons, can be even more devastating. Though the wind of the storm itself can bring about a lot of damage and casualties, the major loss is due to the associated storm surge (Frank and Husain, 1971). When the cyclone reaches its point of landfall, the storm surge causes water to be piled up against the coastline. This can cause coastal inundation, which may result in high numbers of casualties (Lighthill, 1998). The Bay of Bengal is probably the most vulnerable area in the world; over the last century hundreds of thousands people have died here as a result of cyclone-induced storm surges. The November 1970 storm alone killed over half a million Bangladesh’ people. Areas around the Gulf of Mexico, Australia and Japan are also frequently struck by cyclones.

The midlatitude storm surges are mainly a result of the wind stress on the water surface, which - if the storm lasts for a while - causes a gradient in the surface and thereby causes a higher water level near the shore. In combination with the tidal waves, this can lead to extreme water levels. Wave setup, caused by radiation stress in breaking waves, also contributes to the storm surge. Apart from these phenomena, the water level is dependent on wind waves and hydrostatic pressure.

In the case of tropical cyclones the wind field is much more variable. Again, strong winds acting on shallow water form the primary generating mechanism, but there is less time for the water to reach a state of equilibrium with the wind stress than in midlatitude storms. On the other hand the cyclone is accompanied by a wave which originates in deeper waters by the inverse barometer effect. In deep water these waves have a relatively small amplitude, but since they take the form of long waves, their amplitude increases as they reach the coastal shallow waters (Roy, Kabir, Mandal, and Haque, 1999). This is

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called shoaling, the same effect as occurs in tidal waves. Channelling by local bathymetry and reflection from the coast also contribute to substantial amplification of the surge height (Jelesnianski and Holland, 2003).

Casualties caused by storm surges are often a result of lack of warning and insufficient preparation. Therefore, a lot of effort is being put into trying to forecast storm surges.

Over the last decades scientists have been developing computer-models with which they try to simulate these surges and predict the sea water levels. The (geographic) division between the storm surge phenomena (i.e. midlatitude storms versus tropical cyclones) is also reflected in modelling; the various modelling groups are usually concentrated in only one of the areas (Bode and Hardy, 1997).

In Europe a lot of storm surge modelling has been done and noteworthy developments have been made, which have eventually lead to the development of storm surge warning systems in the U.K., the Netherlands and Denmark (Bode et al., 1997).

U.S. researchers primarily focus on hurricane surge prediction. The National Hurri- cane Center (NHC), a department of the NOAA, uses amongst others the SLOSH (Sea, Lake, and Overland Surges for Hurricanes) model to predict cyclone-induced storm surges (Jelesnianski, Chen, and Schaffer, 1992). This model covers all of the U.S.’s East and Gulf coastline, as well as several islands and great parts of the coasts of the Peo- ples Republic of China and India. SLOSH does not include stratification, river inflow, rain and tides (Stewart, 2000). Tides seem to be an important parameter in storm surge modelling, but it would be inappropriate to include in the SLOSH-model, because the time of landfall (the moment when the cyclone come ashore) can currently only be predicted with a normal six-hour error, which is roughly the difference between a high and low astronomical tide (Jelesnianski and Holland, 2003). Tides can be superimposed afterwards; unfortunately this omits the nonlinear interaction. The point of landfall is crucial to determining which areas are in danger. However, the forecast of the cyclone track is still very hard, and where this is inaccurate, SLOSH model results will be inaccurate. The SLOSH model, therefore, is best used for defining the potential maximum surge for a location (NOAA, 2003). Tests have shown that the SLOSH model has an overall accuracy of ± 20% (Stewart, 2000). More advanced models have been developed, but the unpredictability of the cyclone track is still the main problem.

WL | Delft Hydraulics has modelled cyclones in Vietnam and India with the use of Delft3D, a software package mainly used to calculate midlatitude storm-surges. So, Delft3D has also been used for tropical-cyclone induced storm surge computations, but according to Capel (2001) good results have only been obtained after calibrations with the observed data. Some of these calibrations did not make any sense physically (Capel, 2001). The errors might occur because wave effects are not included in the computations, due to the coarse grid, or as a result of a physically incomplete wind-formulation.

The surface stress is the main driving force that generates the storm surge. A realistic representation of this stress in hydrodynamic models is therefore very important.

However, ”in the shallow waters on the continental shelf, the effects of surface stress are so profound, yet so complex physically, that the results depend critically on the manner in which τ_s is specified” (Bode et al., 1997, p.319). Much research is aimed at creating a better understanding of the air-sea interaction physics, together with the coupling of surge and wave models. Falconer and Chen (1991) signify a need to review the representation of wind stress effects in existing two-dimensional models.

Though currently a lot of attention is being paid to find a correct and physically justifiable formulation for the wind drag coefficient, all formulations are dependent on

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the squared wind velocity, while they omit the influence of the flowing water itself.

However, it seems more logical that the wind stress should be dependent on the wind velocity relative to the flow velocity. For most situations this does not make much of a difference, because the flow velocity is relatively very small. But it might be one of the reasons why the models often produce wrong results on very shallow areas, such as inundation areas, where the velocity of the water is not negligible with respect to the wind velocity.

Therefore, a new wind stress-formulation will be defined in this report, which makes the wind stress also dependable on the flow velocity of the water.

1.2 Objective and approach

A new wind-formulation is introduced, which makes wind stress not only dependent on the wind velocity but on the wind velocity relative to the flow velocity. The new formulation is implemented in two manners: fully explicit and semi-implicit. In depth- averaged 1D and 2D-models the influence of the new formulation on the results is tested by setting up several theoretical test cases.

An existing model of the IJsselmeer will be used as a case study, to see the influence of the wind on very shallow waters in small domain models. A simplified schematization of this model will be inserted in the 2D-model, so that an investigation of the influence of the new formulation on the IJsselmeer-model (and other storm surge models) can be made.

Because the influence of grid size is still not completely clear, a grid size sensitivity analysis is performed as well.

1.3 Report outline

In chapter 2, the hydrodynamic models that have been used during this research are described. The shallow-water equations which they solve are given as well as the numerical method that is used to solve them. Chapter 3 describes the IJsselmeer-model, which is used as a case study. Chapter 4 focuses on the different wind-formulations.

First, the usual formulation will be given followed by the newly proposed formulations.

In chapter 5 the experiments that are used to test the different wind formulations are described and the results of these tests are given. The grid size sensitivity analysis is reported in chapter 6, followed by the general discussion in chapter 7.

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Chapter 2

Hydrodynamic models

2.1 Introduction

This chapter provides a brief description of the hydrodynamic models used in this research. The shallow-water equations that these models solve are given, as well as the numerical method that is used to solve them. First, a 1-dimensional model is described, followed by a model that solves the 2-dimensional equations. In both these programs the wind stress has not yet been implemented. They are written in Fortran by Prof. Dr. Ir.

G.S. Stelling as prototype programs to test the effects of certain alterations. The source code for the 1-dimensional model (including the adjustments made during this research) is included in appendix A. The 2-dimensional model is considerably larger, therefore only the computational part where wind is implemented is given in appendix B. The 1-dimensional model will later be used to test the influence of the wind-formulations on 1D shores and basins, while the 2-dimensional model can be used to compute the wind- influence on more complex (2D) topographies like tidal flats. Finally, the hydrodynamic software package WAQUA is described.

2.2 1D Model

2.2.1 Shallow-water equations

The 1D shallow-water equations that will be solved by this program are the continuity equation (2.1) and the momentum equation (2.2) (Stelling, 2002):

∂ζ

∂t +∂(uh)

∂x = 0 (2.1)

∂u

∂t + u∂u

∂x+ g∂ζ

∂x+ gu|u|

C²h = F_wind (2.2)

where: u depth mean current

ζ water elevation above reference g acceleration due to gravity C Chezy coefficient

h water depth (h = ζ + d) d depth below reference

F_wind wind term (defined in chapter 4)

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2.2.2 Numerical solution

The discretization of (2.1) and (2.2) is given by:

ζ_m^k+1− ζ_m^k

∆t +h^k_mu^k+1_m − h^k_m−1u^k+1_m−1

∆x = 0 (2.3)

u^k+1_m − u^k_m

∆t + advec^k+ gζ_m+1^k+1 − ζ_m^k+1

∆x + gu^k+1_m |u^k_m|

C²h = 0 (2.4)

where in this case for the advection term (advec^k) the following approximation is used:

advec^k= q^k_upu^k_m+1− q_down^k u^k_m− (q^k_up− q_down^k )u^k_m 0.5 ∗ (h_m+ h_m+1) ∗ ∆x

q_up= q_m+ q_m+1 2 q_down= q_m−1+ q_m

2

(2.5)

where: q discharge (= u ∗ h) k numerical time step

m numerical grid point (in x-direction)

+ - + - + -

m m+1

m-1

qm -1 qm qm +1

q_down q_up

m-1 m-½ m m+ ½ m+1 m+1½

Figure 2.1: Definition of q_down and q_up

Figure (2.1) shows the numerical grid for this 1D-model. The ”+” points represent the water level ζ, while velocities u are defined in the ”-” points. In the computer program, this information must be stored in arrays - which do not have entries at half points - so both m as m + 1/2 are referred to as m in the Fortran code. Obviously, this does not influence the results.

Stelling (2002) shows that the use of this advection approximation guarantees not only mass conservation, but also positive water levels and a correct momentum balance near discontinuities. Also the process of flooding and drying can be modelled accurately;

the approach of positive water levels avoids the necessity of separate flooding and drying procedures (Stelling, 2002).

Equations (2.3) and (2.4) are solved by first rewriting the momentum equation (2.4) to:

u^k+1_m = α_m+ β_mζ_m+1^k+1 + γ_mζ_m^k+1, (2.6)

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where

α_m = (u^k_m− ∆t ∗ advec^k)/λ_m β_m = (−g∆t

∆x)/λ_m γ_m = (+g∆t

∆x)/λ_m λ_m = 1 + ∆tC_B

d u^k_m C_B = g

C²

Substitution of (2.6) in (2.3) then gives:

a_mζ_m−1^k+1 + b_mζ_m^k+1+ c_mζ_m+1^k+1 = d_m (2.7) where

a_m = −γ_m−1h^k_m−1β_m−1, b_m = 1

∆t− h^k_m−1β_m−1+ h^k_mγ_m, c_m = h^k_mβ_m,

d_m = ζ_m^k

∆t+ h^k_mα_m− h^k_m−1− h^k_mα_m.

This creates a tri-diagonal system of m equations, with m unknowns (all ζ_m^k). The

”Thomas algorithm” as described by Godunov and Ryabenki (1964) and Isaacson and Keller (1966), a simple recursive method also known as the ”double sweep method”, is used to solve this tri-diagonal coefficient matrix (Stelling, 1984). This function first reduces the tri-diagonal system of equations to a bi-diagonal system:

ζ_m+ X_mζ_m+1^k+1 = Y_m, for m = 1, 2, . . . , m_max− 1

ζ_m= Y_m, for m = m_max (2.8)

where the coefficients X_m and Y_m are defined by the following (recursion) formulae:

X_m= c_m b_m Y_m= d_m b_m





 for m = 1

X_m = c_m b_m− a_mX_m−1 Y_m = −a_mY_m−1+ d_m

b_m− a_mX_m−1









for m = 2, . . . , m_max

(2.9)

The final solution of (2.7) is then obtained by backward substitution of (2.9) as follows:

ζ_m^k+1= Y_m, for m = m_max

ζ_m^k+1= Y_m− X_mζ_m+1^k+1, for m = 1, . . . , m_max− 1 (2.10) Now all ζ_m^k+1 (the water levels on the new time step) are known, thus all u^k+1_m can be computed using equation (2.6).

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2.3 2D Model

2.3.1 Shallow water equations Continuity equation:

∂ζ

∂t +∂(uh)

∂x +∂(vh)

∂y = 0 (2.11)

Momentum equations:

∂u

∂t + u∂u

∂x+ v∂u

∂y − f v + g∂ζ

∂x+ gu√

u²+ v²

C²h = F_wind,x+ ν(∂²u

∂x² +∂²u

∂y²) (2.12)

∂v

∂t + u∂v

∂x+ v∂v

∂y+ f u + g∂ζ

∂y + gv√ u²+ v²

C²h = F_wind,y+ ν(∂²v

∂x² + ∂²v

∂y²) (2.13) where: u, v components of depth mean current ~u

ζ water elevation above reference f Coriolis parameter

g acceleration due to gravity C Chezy coefficient

h water depth (h = ζ + d) d depth below reference

F_wind wind stress term (defined in section 4) ν eddy-viscosity coefficient

In the 2D-model as used during this research, the forces due to the viscosity and Coriolis have been neglected. The force due to the wind stress (F_wind) hasn’t been implemented in the computer program at this stage; it will be specified and implemented in chapter 4.

2.3.2 Numerical solution

Discretization in space: The staggered grid

For the discretization of the shallow water equations, the modelled area is covered by a rectangular grid. The primitive unknown variables (u,v and ζ) are arranged on a staggered Arakawa C-grid, see figure (2.2) (Vreugdenhil, 1994).

The semi-discretization (i.e. discretized in space, but not in time) in this model is as follows (omitting the advection and bottom friction for now and assuming the bottom depth to be constant in time):

dζ

dt +h_m,nu_m+1/2,n− h_m−1,nu_m−1/2,n

∆x +h_m,nv_m,n+1/2− h_m,n−1v_m,n−1/2

∆y = 0 (2.14)

du_m+1/2,n

dt + gζ_m+1,n− ζ_m,n

∆x = 0 (2.15)

dv_m,n+1/2

dt + gζ_m,n+1− ζ_m,n

∆y = 0 (2.16)

Again, since the computer does not use half steps in the x-direction, both m and m+1/2 are called m in the numerical model, see figure (2.2). Of course, this does not influence the results.

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ζ

ζ ζ

ζ

ζ ζ

ζ

ζ ζ

ζ

ζ ζ

u u

u

u u

u

u u

u v

v

m m+½ m+1

m-½ m-1

n+½

n

n-½

n-1 n+1

m+1½

m m+1

m-1 n

n-1

Figure 2.2: Staggered grid (Arakawa C)

Discretization in time: The ADI-method

If the 2-dimensional equations would be solved fully implicitly, this would give a very large system of equations, which should be solved at each time step. This would take a lot of computational time. This problem can be avoided by the use of the Alternating- Direction Implicit (ADI) method. This method reduces the matrix to a tri-diagonal system (which can be solved very quickly with the double-sweep method), while it still retains the major advantages of a fully implicit method (Vreugdenhil, 1994).

The ADI-method splits each time step into two half time steps. In the first half time step the rows (u) are solved implicitly, while the columns (v) are computed explicitly:

ζ_m,n^k+1/2− ζ_m,n^k

0.5 ∗ ∆t + hu^k+1/2_m+1/2,n− u^k+1/2_m−1/2,n

∆x + hv_m,n+1/2^k − v_m,n−1/2^k

∆y = 0 (2.17)

u^k+1/2_m+1/2,n− u^k_m+1/2,n

0.5 ∗ ∆t + gζ_m+1,n^k+1/2 − ζ_m,n^k+1/2

∆x = 0 (2.18)

v^k+1/2_m,n+1/2− v^k_m,n+1/2

0.5 ∗ ∆t + gζ_m,n+1^k − ζ_m,n^k

∆y = 0 (2.19)

In the second half time step, the columns are solved implicitly, while the rows remain explicit.

ζ_m,n^k+1− ζ_m,n^k+1/2

0.5 ∗ ∆t + hu^k+1/2_m+1/2,n− u^k+1/2_m−1/2,n

∆x + hv_m,n+1/2^k+1 − v_m,n−1/2^k+1

∆y = 0 (2.20)

u^k+1_m+1/2,n− u^k+1/2_m+1/2,n

0.5 ∗ ∆t + gζ_m+1,n^k+1/2 − ζ_m,n^k+1/2

∆x = 0 (2.21)

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v^k+1_m,n+1/2− v^k+1/2_m,n+1/2

0.5 ∗ ∆t + gζ_m,n+1^k+1 − ζ_m,n^k+1

∆y = 0 (2.22)

2.4 WAQUA

WAQUA is a two-dimensional water movement and water quality simulation system (Rijkswaterstaat, 2002). The WAQUA software is based on SIMONA, a flexible concept for the development of modelling software. In the SIMONA-environment also TRIWAQ has been developed, which is the three-dimensional version of WAQUA. WAQUA is commonly used by Rijkswaterstaat (RIKZ, RIZA) and KNMI for models regarding the North Sea, the coastal areas, and the Dutch estuaries.

WAQUA can handle rectilinear and curvilinear models. For the rectilinear models it solves the 2D shallow water equations (2.11), (2.12) and (2.13). The numerical method that is used is slightly different as the one described in section 2.3.2.

A full description of the terms and equations can be found in the User’s Guide WAQUA, which also denotes some aspects of the numerical solution (Rijkswaterstaat, 2002). For the complete numerical method this guide also refers to Stelling (1984), On the Construction of Computational Methods for Shallow Water Flow.

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Chapter 3

The IJsselmeer-model

3.1 Introduction

As a result of the high-water problems in the winters of 1993 and 1995, the Dutch government decided to speed up the process of dike raising in the area of the IJsselmeer (Lake IJssel). In order to determine to what extent the dikes should be raised, it is important to accurately predict the expected water levels during a similar storm. The water level is the main parameter to determine the necessary dike height (Bak, 1999).

The RIZA, Institute for Inland Water Management and Waste Water Treatment, uses the 2D hydrodynamic software package WAQUA, as described in section (2.4), to model this area.

3.2 Model-description

The area that is included in the WAQUA IJsselmeer-model covers the entire IJsselmeer and the Ketelmeer, Zwarte Meer and Vossemeer, as shown in figure 3.1. For the purpose of this research, we are specifically interested in the eastern part of the Ketelmeer, since this is a very shallow area that becomes flooded during a western storm; severe (westerly) wind also induces a setup in this area. The following parts, located within the Ketelmeer, have been incorporated in the model: the inundation area Kampereiland, the Ramspol barrier, and silt deposit area ’het IJsseloog’ (Bak, 1999), see figure 3.2.

The bottom profile of the eastern part of the Ketelmeer as it is modelled is shown in figure 3.3.

3.3 Setup of the experiment

In order to see the flow pattern and water levels in the Ketelmeer-area during severe wind events, a 48 hour westerly storm with a constant velocity of 20 m/s has been simulated with the software package WAQUA. This value and direction has been chosen because it is the same as the maximum average wind velocity over one hour as in the most severe 1993 storm.

The initial waterlevel over the entire model is NAP -0.35m. There are two rivers flowing into the system: the IJssel which has a constant runoff of 395 m³/s during this simulation, flowing directly into the Ketelmeer and the Vecht with a runoff of 59

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Figure 3.1: Overview of modelled area

m³/s which flows into the Zwarte Meer. In the Afsluitdijk are two outlets, with a total discharge similar to the river runoffs.

The Ramspol barrier is closed during the simulation.

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Figure 3.2: Detail of the eastern part of the Ketelmeer

Figure 3.3: Bottom profile of the eastern part of the Ketelmeer in the IJsselmeer-model

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3.4 Results

Within a few hours, the complete eastern part of the Ketelmeer is being flooded. After a few initial oscillations, the system reaches a stationary situation, see figure 3.4(c).

Figure 3.4(b) shows the flow pattern in stationary condition; though the water level remains stable, a continuous flow remains, in which water is being dragged onshore at the shallow parts and flowing back through the deeper channels. This illustrates that the influence of the wind stress per unit mass is stronger in shallow areas.

(a) Water level after 48 hours

1.82 1.825 1.83 1.835 1.84 1.845 1.85 1.855 1.86

x 10⁵ 5.09

5.1 5.11 5.12 5.13 5.14

5.15x 10⁵ Velocity profile

(b) Flow pattern after 48 hours

0 2 4 6 8 10 12 14 16

0 0.1 0.2 0.3 0.4 0.5 0.6

Water level at station 1

Time in hours

Water level in m

(c) Water level at station 1

Figure 3.4: Results at eastern bank of Ketelmeer

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3.5 Discussion

Unfortunately, there are few field data available in the Ketelmeer area. Ramspol is the only place at which water levels have been measured in the past. The maximum water level here during the January 1993 storm was NAP +0.60 m. This is slightly more than computed by this simulation. However, in this computation only one value for the wind has been chosen over the entire model, so the field data cannot really be compared to the results of this simulation.

RIZA, on the other hand, has made computations in which they simulated the wind fields as well as possible. The version of WAQUA that they used could only handle a constant drag coefficient. This coefficient has been determined by a best fit to the complete data set. However, according to the RIZA the drag coefficient should actually be decreased in the eastern area of the Ketelmeer. So the water level in the Ketelmeer area is slightly over-estimated with the use of the drag coefficient that fits best over the entire model.

In the course of this report a shortcoming of the wind formulation will be pointed out, which might be the cause of the over-prediction of the water levels in very shallow areas like the eastern part of the Ketelmeer. An alternative wind formulation will be introduced, which might lead to a smaller setup in this area, without making it necessary to adjust the drag coefficient in the Ketelmeer area.

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Chapter 4

Wind formulations

4.1 Introduction

In the 1D and 2D-model, as described in Chapter 2, the wind stress term has not yet been implemented. Therefore, the usual wind-formulation, as used in most of today’s hydrodynamic models like Waqua, Sobek and Delft3D, is first implemented in the models. In this report this will be referred to as ’wind 1’. After the implementation of wind 1, two methods for the implementation of a new wind formulation are discussed.

Their physical background is explained and the manner in which they are implemented is shown. The implementations are given in formula-notation; the exact Fortran code can be found in the source codes in appendices A and B.

Please note that the term F_wind, which is used throughout this report, represents the complete wind term in the momentum equation (see formulae (2.2), (2.12) and (2.13)).

This is the force on the water due to the wind (τ_wind) divided by the mass on which it acts. Its unit is m/s².

4.2 The wind formulations

4.2.1 Wind 1: The usual formulation Stress term

The commonly used wind stress term is the semi-empirical formulation

F_wind = C_d ρ_air ρ_water

u_w|u_w|

h , (4.1)

where u_w is the wind velocity at a height of 10 m, h is the water depth, and C_dthe drag coefficient (Wu, 1982).

Drag coefficient

The drag coefficient does not have a constant value. On dimensional grounds, Charnock (1955) derived an implicit relationship between the wind and the roughness. Generally,

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the following empirical relation is used (Smith and Banke, 1975):

C_d=









C_dÂ, for u_w ≤ uÂ_w, C_dÂ+ (C_d^B− C_dÂ)û_u^wB^−uÂ^w

w−uÂ_w, for uÂ_w ≤ u_w ≤ u^B_w, C_dÂ, for u^B_w ≤ u_w,

(4.2)

where:

C_d^A,C_d^B are user specified drag coefficients, u^A_w,u^B_w are user specified wind speeds.

A lot of research on the right values for C_d has been done. According to the quasi- linear theory of Janssen (1991), the stress of airflow over surface gravity waves depends on both wind speed and the wave-induced stress. Janssen (1992) also found experimental evidence for the effect of surface waves on the wind stress.

Mastenbroek, Burgers and Janssen (1993) modelled three storm surges in the North Sea. They found that the use of the Smith and Banke relation underestimated the surges by 20%, while calculations with a wave-dependent drag reproduced the overall level of the surges within a few percent.

Implementation

The wind-stress term is added to the right part of the momentum-equation (2.2). Hence, the equation becomes:

∂u

∂t + u∂u

∂x+ g∂ζ

∂x+ gu|u|

C²h = C_d ρ_air ρ_water

u_w|u_w|

h (4.3)

Just as in section (2.2.2), the numerical approximation of (4.3) can be written in the form:

u^k+1_m = α_m+ β_mζ_m+1^k+1 + γ_mζ_m^k+1 (4.4)

but in this case the parameter α_m has changed from (u^k_m− ∆t ∗ advec^k)/λ_m into:

α_m = (u^k_m− ∆t ∗ advec^k+ ∆t ∗ C_d ρ_air ρ_water

u^k_w|u^k_w| h )/λ_m λ_m = 1 + ∆tC_B

h |u^k_m|

(4.5)

The complete derivation of these parameters is given in appendix C.

4.2.2 Wind 2: Using relative wind speed Introduction

As Falconer and Chen (1991) indicate, the common formulation that most models use only includes the effects of surface wind stress in the form of a quadratic friction law related to the wind velocity, with no consideration of the effects of this wind stress on the velocity profile and the advective terms.

As an alternative to the common formulation, in this research a wind stress term will be introduced that is not only dependent on the wind velocity but also on the flow velocity at the surface, thus the wind velocity in the stress term will be replaced by the velocity relative to the water.

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Why this new formulation

It seems logical that the influence of wind stress will reduce if the water is flowing in the same direction and it will improve when the water flow and wind are in opposite directions. This is illustrated in figure 4.1.

(a) Less wind stress (b) More wind stress

Figure 4.1: Wind stress also dependent on flow velocity

Stress term

The variable u_w in the stress term will be replaced by the relative velocity u_w− u. In this report only depth-averaged models are used, so the water velocity u is the depth- averaged velocity in a specific point. In a 3D-model this would be the velocity at the surface (i.e. the velocity in the upper layer).

F_wind= C_d ρ_air ρ_water

u^k_rel|u^k_rel| h u^k_rel= u_w− u^k_m

(4.6)

Implementation

The relative wind stress in this case will be calculated using the u^k, the flow velocity on the last time step. This means that all parameters - and the value of the complete stress term - are known. Thus, the stress term can be added to the momentum equation just as in wind1. Now the parameter α_m becomes (see appendix C for the derivation):

α_m= (u^k_m− ∆t ∗ advec^k+ ∆t ∗ C_d ρ_air ρ_water

u^k_rel|u^k_rel| h )/λ_m λ_m= 1 + ∆tC_B

h |u^k_m| u^k_rel= u_w− u^k_m

(4.7)

4.2.3 Wind 3: Relative, semi-implicitly solved Stress term

Because the wind stress in Wind 2 was calculated with the flow velocity at the previous time step, the wind stress was a known, external factor, that could be added to the right hand of the momentum equation. It could be better to use the flow velocity on the new time step (u^k+1). However, if both relative terms would be taken on k + 1, the solution would become too complex. Therefore, similar to the bottom stress term,

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only the first part of the quadratic term will be taken implicitly, while the other one is chosen explicitly:

u^k+1_rel |u^k_rel| h u^k_rel= u_w− u^k_m u^k+1_rel = u_w− u^k+1_m

(4.8)

Implementation

In this case, the implementation is a bit more complex. The value of u^k+1 (and u^k+1_rel ) is unknown, so the value of the stress term is not yet familiar. The numerical parameters now become (see appendix C):

α_m= (u^k_m− ∆t ∗ advec^k+ ∆t ∗ C_d ρ_air ρ_water

|u^k_rel|u_w h )/λ_m λ_m= 1 + ∆tC_B

h |u^k_m| + ∆t ∗ C_d ρ_air ρ_water

|u^k_rel| h u^k_rel= u_w− u^k_m

(4.9)

4.2.4 Limitation of the water depth

If the water depth becomes smaller, the wind stress term becomes larger. When h ↓ 0, the stress term grows infinitely large, which can cause the model to become unstable. In order to prevent this, a minimum depth should be introduced. Therefore, a minimum water depth of 1.0 · 10⁻³ is used in the wind stress implementations. This makes the model more stable in the process of flooding and drying. This will be demonstrated in section 5.5.

4.3 Correcting the wind speed to the surface

4.3.1 Introduction

In the new formulation the flow velocity is subtracted from the wind velocity. However, the wind that is used for the computation is the wind at 10 m above surface, while it would actually be necessary to subtract the flow velocity from the wind velocity at the surface. Unfortunately, a valid formulation for the relation between the wind at 10 m height and the wind at the surface is still unknown.

The wind speed is generally assumed to have a logarithmic profile over the height. In field experiments almost all wind data are measured at 10 m. Laboratory experiments have been carried out by Wu (1969) in which he measured the wind speed at 0.1 m and derived wind shear stress coefficients from these data. He reported a major discrepancy between field and laboratory coefficients.

Safaie (1984) later proposed a relationship between the wind and drag coefficients at 0.1 m and 10 m to adjust the data from the laboratory experiments, under the assumption of a logarithmic profile. The results of the adjusted data were remarkably close to the field data.

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According to the Shore Protection Manual (1984), the logarithmic profile (equation (4.10)) is valid up to an elevation of 100 m:

u_z u_∗ = 1

κ[ln z

z₀ − Ψ(z

L)] (4.10)

where: u_z wind speed at elevation of z m.

u_∗ friction velocity κ von Karman constant z height above surface

z₀ surface roughness length, z₀ = αu²_∗/g, α=Charnock parameter.

Ψ effect of stability of the air column on the wind velocity L length scale associated with the mixing process

Johnson (1999) demonstrated that, under the assumption of neutral atmospheric stratification (i.e. no air-sea temperature difference), this equation can be simplified to:

u₁₀

u_z = ln[¹⁰_z

0] ln[_z^z

0] (4.11)

which shows that the wind speed correction factor is only dependent on the sea roughness. Johnson, Hojstrup, Vested, and Larsen (1998) found through dimensional analysis that the sea roughness itself is dependent on the wave age through the Charnock parameter. The correlation between the Charnock parameter and the wave age has been derived from field data out of the 1996 ASGAMAGE experiment by Oost, Komen, Jacobs, and van Oort (2002).

The Shore Protection Manual (1984) suggests the following simple expression to correct the wind speed.

u₁₀ u_z = (10

z )^1/7 (4.12)

Johnson (1999) also proved that for most typical values of wave age, this relationship leads to errors below 10%.

Using this expression, the relation between u_0.1 and u₁₀ would become

u_0.1 = 0.52u₁₀ (4.13)

but it must be noted that Johnson only validated the expression for heights between 2 and 60 m. It is unclear how heights below 2 m should be treated. Theoretically the wind speed will be zero at the surface itself. But when there is wind over the water, wind waves will be created too, which makes it questionable at which height the wind speed exactly should be taken, see also figure 4.2. Further research is needed to find a generally suitable relation between u₁₀ and u_{surf ace}.

4.3.2 Formulation

A factor α (0 < α < 1) is introduced to take into account the difference between the velocity at 10m height and the velocity at the surface, where u_{surf ace} = αu₁₀. This

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u₁₀ z

10 m

0 m

u_surface

Figure 4.2: Logarithmic velocity profile

adjustment will be tested only using wind formulation 3. The wind stress term then becomes:

1 α²

u^k+1_rel∗|u^k_rel∗| h u^k_rel∗= αu₁₀− u^k_m

u^k+1_rel∗ = αu₁₀− u^k+1_m

(4.14)

With the use of α in this formulation the influence of the flow velocity becomes stronger.

Therefore, it is expected the effects of the new formulation will become stronger when this factor is applied. If the flow velocity is zero, the factor equals itself out and the old wind stress formulation remains.

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Chapter 5

Experiments

5.1 Introduction

A number of simulations have been performed to test the influence of the different wind formulations. In this chapter both the setup of the experiments as well as the test results are reported.

The first experiment is a 1D-model of a closed basin. This is used to find out whether the wind has been implemented correctly (the model verification is described in section 5.8). This model also shows the effects of the new wind-formulation on startup-behavior.

Next, the same simulation is performed in the 2D-model to test whether this gives the same results. After that, a hypothetical shore with a constant slope is modelled in the 1D-model, in which the wind drags the water onto the slope. It may be interesting to see how the different wind-formulations handle this.

Subsequently, two tests (the first in 1D, the second 2D) are performed with very shallow basins, in which drying will occur. This is a good check on the stability of the different formulations. The last simulation is a rough schematization of the Ketelmeer- area in 2D: a large shallow area with two deep channels on the side.

Finally, the influence of the factor α, as introduced in section 4.3, is tested.

In all experiments a constant drag coefficient of 0.0043 has been chosen; the air den- sity (ρ_air) is 1.225 kg/m², the water density (ρ_water) is 1000 kg/m², the Chezy factor (C) is 60 m^1/2/s and the acceleration due to gravity (g) is 9.81 m/s².

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5.2 A closed basin in 1D

5.2.1 Setup of the experiment

In this experiment a basin is modelled with the 1D-hydrodynamic model. The basin has a length of 10 km. A grid size (∆x) of 100 m is chosen, so the basin consists of 100 grid points.

u_wind

Figure 5.1: Setup of the experiment - 1D closed basin

First, this experiment will be used to find out whether the wind has been implemented correctly. The results are later verified by an analytical solution and by a model run in Delft3D, see section 5.8. Subsequently, the model will test the influence different wind-formulations on startup behavior.

The following parameters have been used:

∆t = 10 s

∆x = 100 m

mmax= 100 (number of grid points in x-direction) depth = 5 m

u_w = 30 m/s

5.2.2 Results

0 1 2 3 4 5 6 7 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

time in hrs

surface elevation in m

Surface elevation in station at m=100

wind 1 − traditional formulation wind 2 & 3 − new formulation

Figure 5.2: Closed basin 1D

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Figure 5.2 shows the water level during the simulation for wind 1 and 2, i.e. the usual wind formulation and the new one. The results for wind 3 are identical to the results of wind 2.

When the model is started, a gradient in the water surface arises, which balances out the wind stress. Since the model is in rest at the start, the water first starts oscillating around the steady-state solution. This oscillation slowly damps out.

During the oscillation the velocity becomes alternately positive and negative. When it is positive the wind and water have the same direction. The wind stress term is then smaller with the new formulation. Therefore the wind stress on the water is smaller. On the other hand, when the water flows in the opposite direction of the wind velocity, the wind stress term is larger with the new formulation. In this case the wind stress slows down the water, and because this frictional force is larger with the new formulation, the oscillations in the basin damp out quicker when this formulation is used, as can be seen in figure 5.2.

So one characteristic of the new formulation is that the startup behavior damps out more quickly. This is a good characteristic, because it means that less computational time is needed to see the desired results.

Because it is a 1D depth-averaged model, all flow velocities are zero in the stationary situation and the wind stress is then the same for all formulations. Therefore, the final wind setup is the same for all formulations.

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5.3 A closed basin in 2D

The first experiment to be performed in 2D, is almost similar to the closed basin in 1D (section 5.2). The results of both experiments can be used to see whether both models produce similar results. A squared basin (10 x 10 km) is chosen, so the results for different wind directions can also be compared.

∆t = 10 s

∆x = 100 m

∆y = 100 m

mmax= 100 (number of grid points in x-direction) nmax = 100 (number of grid points in y-direction) depth = 5 m

u_w = 30 m/s (wind velocity in x-direction) v_w = 0 m/s (wind velocity in y-direction)

5.3.2 Results

Results are shown in figure 5.3. The water level displacement is practically the same as in

0 1 2 3 4 5 6 7 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Surface elevation in station (100,2)

time in hrs

Surface elevation in m

wind 1 − traditional formulation wind 2 & 3 − new formulation

Figure 5.3: Closed basin in 2D

the 1D-model, so the two models conform with one another very well. Small differences are probably caused by a different definition of the advection-terms. Letting the wind blow from different directions (north, east, south, west) leads to the same results (but on other sides of the basin of course). Therefore it is likely that the implementation in different directions has been done correctly. Again, wind 3 gives the same results as wind 2.

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5.4 Shore

The next experiment is a 1D coastal model, see figure 5.4. The experiment is almost identical to the first one (section 5.2), but this time the depth is not a constant 5 m, but it changes linearly from 5 meters below to 5 meters above reference. The model is closed at the right boundary and has an open boundary on the left hand side, with ζ₁ = 0 m as the boundary condition.

u_wind

5.0 m

Figure 5.4: Setup of the experiment for shore

The following parameters are used:

∆t = 1 s

∆x = 100 m mmax= 100 u_w = 30 m/s

5.4.2 Results

Figure 5.5 shows the results of this experiment. It shows the water elevation at m = 50, which is exactly in the middle of the model, where the water depth is zero when the model is started.

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1 1.2

time (hrs)

surface elevation (m)

Water level in m=50

wind 1 wind 2 wind 3

(a) surface elevation in m=50

0.4 0.6 0.8 1 1.2 1.4 1.6

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

time (hrs)

surface elevation (m)

Water level in m=50

wind 1 wind 2 wind 3

(b) detail of (a)

Figure 5.5: Shore in 1D

A NEW WIND-FORMULATION FOR HYDRODYNAMIC MODELS