6th Annual International Workshop & Expo on Sumatra Tsunami Disaster & Recovery 2011
in Conjunction with
4th South China Sea Tsunami Workshop
3D numerical simulations of the effect of large
roughness elements on the propagation of solitary
waves
Stefan Leschka1)* and Hocine Oumeraci2)
1)
DHI-NTU Water & Environment Research Centre & Education Hub
200 Pandan Loop, #05-03 Pantech 21, Singapore, 128388
2)
Leichtweiss-Institute for Hydraulic Engineering & Water Ressources, TU Braunschweig
Beethovenstr. 51a, 38106 Braunschweig, Germany
*
Email: [email protected]
Recent non-linear shallow water (NLSW) models usually
use the quadratic friction law with Manning’s coefficient to
describe the surface roughness of the bottom. The
consideration of the effect of large roughness elements such
as buildings and tree vegetation, which are too small to be
resolved by the grid of the bottom topography, is still purely
empirical by often using Manning’s roughness estimated
from previous studies. This approach, however, is not
physically sound and may thus result in very large
uncertainties in the assessment of inundation hazards. A
more physically-based approach is to determine prediction
formulae for the hydraulic resistance of large roughness
elements of different shapes, sizes and configurations which
can then be directly implemented in NLSW models. Such
prediction formulae can be determined on the basis of
systematic 3D numerical simulations and laboratory studies
using a set of configurations of well-parameterized large
roughness elements of different shapes and sizes.
Results in this paper represent one of the first steps
towards the development such an approach, as it relates
numerical results to the influence of an artificial reef and an
emerged cylinder on the propagation of solitary waves based
the numerical simulation using a two- and a threedimensional two-phase Reynolds-averaged Navier-Stokes
(RANS) model and the Volume of Fluid (VOF) method.
Good agreement with physical experimental data and
empirical relations found in literature have been achieved.
I. INTRODUCTION
Seabed topographies and morphologies can cause large
changes in wave height and direction of travel. Shoals can
focus waves, in some cases more than doubling wave
height behind a shoal. Other bathymetric features can
reduce wave heights. The magnitude of these changes is
particularly sensitive to wave period and direction and
how the wave energy is spread in frequency and direction.
In addition, wave interaction with the bottom can cause
wave attenuation. Flooding waves on land, like tsunamis,
underlie even more processes, where roughness plays a
major role in wave propagation. The correct
implementation of these effects is essential for the
prediction of tsunami wave height and inundation.
This leads to the question, how detailed the topography
has to be included in a model calculation so that one still
can expect reasonable results. Non-linear shallow water
(NLSW) wave models account for this, while from its
physical sense, only Boussinesq type models are most
suitable to assess near-shore hydrodynamics which
includes wave breaking. The solution of the governing
equations is only available at certain points of the
discretized area, but it has to take into account the wave
transformation effects mentioned above, which also take
place in between them. The number of these points is
limited due to the availability of computational resources.
Vortices with smaller diameter than the distance of the
calculation points have also to be taken into account. They
arise due to flow disturbances, e.g. sea bottom shape
represented by the bathymetry data included in the model
and bottom surface represented by roughness parameter,
usually based on Chézy or Manning values [1].
Manning values assume fully rough turbulent
conditions. Adaptations to the Manning values have been
investigated to account for large obstacles, involving drag
and drag interaction, e.g. [2; 3]. Large scale
approximations have been proposed, e.g. [4; 5], for city
areas [6] and incorporating land use classes [7]. For the
case of non-submerged roughness elements their area
presented to the flow changes with the flow depth and the
value of any roughness coefficient must mirror this. As
Manning models were formulated on the basis of
boundary (i.e. surface) roughness it is not directly
applicable as they stand to a ground surface with
vegetations and buildings [8].
Run-up measurements from post-tsunami field surveys
suggested that small-scale local bathymetry features, like
vegetation and buildings, affect the run-up height to the
first order [9; 10; 11], which makes predictive modeling
of the tsunami inundation very difficult. These
circumstances are quite crucial for tsunami risk
management, taking into account the dramatic increase of
pressure due to urbanization.
Large roughness elements are defined as obstacles of
large size, but smaller than the grid size, e.g. rocks,
submerged breakwaters, or natural bed features where the
water depth suddenly changes. Because long-wave theory
is based on the assumption that velocity distribution is
almost uniform vertically, it is difficult to reproduce the
complicated phenomena involved. If an appropriate water
depth and a proper roughness coefficient are used, one
would expect that simulation can reproduce the tsunami
6th Annual International Workshop & Expo on Sumatra Tsunami Disaster & Recovery 2011
in Conjunction with
4th South China Sea Tsunami Workshop
run-up and flooding process correctly. However, the
relationship between the roughness coefficient and the
bottom topography is still not well understood [12].
Physical and numerical experiments to assess the
influence of roughness elements on river flow have been
performed in the last decades, e.g. [13; 14]. Field surveys
after tsunamis lead to important observations regarding
damages and their relation to local bathymetric features,
e.g. [11; 15; 16; 17]. Most recently, approaches to deal
with the travelling flood waves and their attenuation due
to emerged and submerged vegetation, represented by
idealized shapes, lead to first correlations of vegetation
density, submergence depth and Reynolds number and
lead to drag and bulk drag coefficients [18; 19], while [20]
used parameterized mangroves. Drag in densly populated
city areas can be related to the gap between the roughness
elements [21].
To fully describe the interaction of large roughness
elements and waves, one has to account further for vortex
loses and inertia losses. Inertia coefficients have been
proposed e.g. in [22; 23; 24]. Drag and inertial
coefficients vary with time [25] as vortices develop in the
wake of large roughness elements and thus are dependent
upon the duration of the flow [26]. Ref. [27] extended the
shallow water equations by including parameters for
volume of vegetation and buildings, assuming hydrostatic
pressure. Ref. [28] developed drag and inertia coefficients
for submerged breakwaters, using a modified KeuleganCarpenter number [29]. Information on vortex losses in
the field of tsunami is very limited. While investigating
tsunami propagation on a submerged breakwater, [30]
observed a pair of vortices appearing behind the edges of
the breakwater. Losses have not been quantified.
In order to investigate energy development for large
roughness configurations for unsteady flow conditions,
one has to consider friction, drag, inertia and vortices in
the close surrounding of the roughness elements using
numerical or physical experiments, analyzing 3D data.
Surface elevation and depth averaged velocities have to be
recorded in farer distances, because they have to be
represented by a modified Boussinesq equation. Friction is
widely described using the quadratic friction law, based
on the Manning roughness coefficient. Drag and inertia
forces are additive terms in the Morison equation, which
makes it very suitable as a base for quantification. They
should be composed with measurable quantities, e.g.
relative submergence depth, spacing between obstacles,
arrangement, size and shape of the cross-section of
roughness elements and rigidity. Drag induced vortices to
the flow should be considered as well.
At the roughness elements, forces and momentums
should be calculated to be compared with experiments,
leading to a validation of the numerical model.
The following section outlines a detailed plan for
numerical and physical experiments which will be applied
in this study.
II. EXPERIMENTAL PROGRAM
The underlying assumption for this program is that the
experimental area is or contains one cell of a large scale
nearshore model, through which a solitary wave
propagates. The large scale model, such as NLSW or
Boussinesq type models, is considered as a set of
equations which act on a relatively coarse grid without
having the possibility of accounting for structures which
are so small that they are not represented in the
bathymetry. This cell lies offshore to assess the influences
of submerged or partially emerged structures.
Navier-Stokes solvers have often been applied to a wide
range of detailed simulation tasks. They have been
compared successfully with physical experiments of flow
around obstacles, as in [31]. This model type will be used
in the present study. Since it just approximates real
physics, its validation by physical experiments is required.
The previously defined losses due to inertia, drag and
vortices will be determined through the analysis of the
differences in forces and momentums in the near field of
the roughness elements as well as at the four boundaries
of the assumed large scale model cell, lying in the domain.
Wave and roughness element parameters will be varied
within each test. First tests on single elements will be
performed, followed by tests comprising groups of
uniformly shaped elements.
Single roughness elements
In case of a single large roughness element varied
parameters are
1) Shape and orientation: circular shapes, quadratic
shapes with wave facing side and quadratic shapes with
wave facing edge. They are presented in figure 1.
2) Width of the roughness element DB
3) Height of the roughness element RC=dB-d (freeboard
RC>0 and submergence depth RC<0)
Fig. 1. a) circular shape, b) quadratic shape with wave facing side, c)
quadratic shape with wave facing edge.
Groups of roughness elements
4) Basic arrangement of the roughness elements B:
side by side, tandem, 2 x staggered, as given in figure 2.
6th Annual International Workshop & Expo on Sumatra Tsunami Disaster & Recovery 2011
in Conjunction with
4th South China Sea Tsunami Workshop
use of the k- model in the boundary layers and the k-
model in the free flow region [34].
Numerical domain
An example of a numerical domain is presented in
figure 4.
Fig. 2. Basic arrangements of roughness elements.
5) Groups of roughness elements of uniform shape:
circular shape, quadratic shape with wave facing side,
quadratic shape with wave facing edge, as shown in figure
3.
The configurations can be expressed with the relation
between the diameter/width DB and the distances between
the roughness elements lB-B,x and lB-B,y. All roughness
elements will be emerged.
Fig. 3. Configuration of groups of roughness elements: a) circular shape,
b) quadratic shape with wave facing side, c) quadratic shape with wave
facing edge.
Mixed roughness elements
6) Depending on the configurations of available
experimental data, various roughness elements will be
applied in this test. This test should prove found relations
in the test program.
The following section introduces the methods applied in
the numerical tests.
III. NUMERICAL METHODS
Numerical model
The open source software package OpenFOAM will be
used to perform numerical simulations in 2D and 3D. It
consists of different solvers, pre- and post-processing
tools. interFoam is the solver for multiphase
incompressible flow, solving the Reynolds-averaged
Navier-Stokes equations for velocities and pressure for
two incompressible, isothermal, immiscible fluids using
an interface capturing approach based on the volume of
fluid (VOF) phase fraction. It uses the multidimensional
universal limited for explicit solution (MULES-VOF)
method to maintain boundedness of the phase fraction
independent of the underlying numerical scheme, mesh
structure, etc. [32; 33]. From available turbulence models,
the k- SST model has been applied here, which makes
Fig. 4. Horizontal dimensions of the numerical domain.
The mesh has to covers the area of interest, representing
one cell of a large scale model, and a surrounding area, in
which the wave can propagate further. This is required
because no outlet boundary condition is available, which
is free of reflections. Thus, size of the surrounding area
should be selected in such a way that it is large enough, so
that reflections from the boundaries cannot enter the area
of interest during the simulation time in which the
incoming wave can completely pass the area of interest
and that it lets waves, reflected on the roughness
parameters, completely leave the area of interest.
Initial and boundary conditions
The velocity components ux, uy and uz and the dynamic
pressure pdyn, the turbulent kinetic energy
k  0.50.0025c 
2
(1)
with the wave celerity c [35], the turbulent kinetic
dissipation rate
  C k 2 3 l t
(2)
with Cµ=1.44 [34] and the turbulent length scale lt being
20% of the water depth [36] and specific dissipation rate
   C  k 
(3)
are set as initial conditions.
Boundary conditions at the inlet follow the analytical
solution of the solitary wave for water level, ux and uz
[37], while uy=0. This has been done with the help of
groovy boundary condition [38]. All walls are treated with
the help of wall functions, which are available for k,  and
. For the omega wall function, the value has been
determined using
  2500 k St 2
(4)
[34]. The surface roughness of the wall boundaries
flume walls and roughness elements is considered with the
help of a wall function for the turbulent viscosity νT,
which incorporates the absolute roughness height kSt. Zero
gradient dynamic pressure condition has been used for all
boundaries except the outlet, where it has been fixed to be
6th Annual International Workshop & Expo on Sumatra Tsunami Disaster & Recovery 2011
in Conjunction with
4th South China Sea Tsunami Workshop
zero. This is required to keep pressure free from numerical
oscillations, but it is also wrong in case that the fluid is
moving. However, its influence is negligible, because the
outflow boundary is more than a wave length away from
the area of interest due to the requirements on the
surrounding area, exemplified above.
This method has been applied to first 2D and 3D
numerical simulations. The results are presented in the
next section.
Before the experimental program has been started, 2D
numerical simulations have been performed in order to
evaluate boundary conditions and turbulence model with
the help of physical experiments. Furthermore, the
interaction between a cylinder and a solitary wave has
been assessed through a 3D simulation. Momentums at the
cylinder base, caused by the solitary wave passing it have
been compared with the empirical solution, found in
literature. For both simulations, numerical properties like
mesh element sizes at the wall boundaries have been
estimated with the help of physical relations, which is
described in detail in the following sections.
Comparison of 2D numerical results with physical
experiments on a submerged reef
Physical experiments on the interaction between
solitary waves and a submerged artificial reef have been
carried out in the wave flume of Leichtweiss-Institute for
Hydraulic Engineering and Water Resources (LWI) at TU
Braunschweig. The wave flume is approximately 90m
long, 2m wide and 1.2m high. Waves are generated with a
Piston type wave maker. A detailed description of the
experiments is given in [39]. For this comparison, a
solitary wave of the height H=0.22m and the water depth
of d=0.6m has been selected. The dimensions of the wave
flume, the reef and positions of the wave gauges are
shown in figure 5.
Fig. 5. Model set-up in the wave flume at LWI [39].
The numerical setup generally follows the criteria
outlined above. Element sizes at the walls have been
calculated to fulfill the y+=50 [34], which leads to the
element height at the wall of
.
ub    ,
(6)
[34], in which the wall shear stress
  FF A .
(7)
A is the wetted area. The friction force FF can be
determined using
FF  gz .
(8)
Here, ρ is the fluid density, g is the acceleration due to
gravity and
IV. FIRST RESULTS
y  2 y  ub
In this equation, ν is the kinematic viscosity and the
friction velocity is given by
(5)
z  l E
(9)
is the loss height over the whole length of the wave
flume l. The energy gradient E can be derived from the
Manning/Strickler equation
u 0  k St E rH
23
,
(10)
in which u0 is the free stream velocity outside the
boundary layer. A roughness height of kSt=0.001m has
been used [40]. It is mentioned here, that the
Manning/Strickler equation is only valid for steady state
flow conditions, which is not fulfilled in the presence of
waves. However, aiming for the maximum energy loss
due to friction on the wave flume walls, the mean velocity
under the wave crest has been used instead of the free
stream velocity. Derived quantities have been applied to
the relation
u   u 0 ub ,
(11)
which leads to u+=0.0525 for comparisons of numerical
results of the given solitary wave with physical
experiments in the wave flume at LWI. The estimated
value corresponds to typical values for u+ for laboratory
flumes, which is approximately 0.05 [41].
The structured mesh comprises of hexahedral elements.
Their sizes in x direction range from 0.025m at the inlet to
0.005m at the reef to 1.05m at the end of the wave flume.
In z direction, element heights take values of 0.001m at
the bottom and 0.00015m above the reef. They increase
linearly to max. 0.01m at the still water depth. The wave
height has been resolved using constantly 0.01m high
elements.
The numerical and experimental time scales have been
adjusted, using data of wave gauge 1. The numerical time
scale has been shifted by adding 86.43s.
The water surface elevations of the simulations have
been compared with experimental data at the positions of
gauges 1, 2, 7, 8, 15 and 19. The time series are given in
figure 6.
6th Annual International Workshop & Expo on Sumatra Tsunami Disaster & Recovery 2011
in Conjunction with
4th South China Sea Tsunami Workshop
It is only mentioned that wave breaking has been observed
during the physical experiments (Strusinska, 2010) as well
as it has been found in the simulation results. Behind the
reef, good agreement has still been achieved comparing
the leading wave heights of simulation and experiment. It
can be concluded, that the numerical simulation
overestimated slightly the reflection and underestimated
the transmission of wave energy. The deviance is less than
10% of the measured wave height.
Fig. 6. Comparisons of water levels at wave gauges 1 (x=5m), 2
(x=10m), 7 (x=36.09m), 8 (x=6.72m), 15 (x=40.79m) and 19
(x=49.79m).
Table 1 gives an overview of the differences between
numerical simulation and physical experiment.
TABLE I
WAVE HEIGHTS OF THE NUMERICAL SIMULATION COMPARED TO THE
PHYSICAL EXPERIMENT
in front of reef
above reef
behind reef
1
2
7
8
15
19
0.73 % 2.63 % -4.23 % -2.13 % -7.97 % -4.86 %
gauge
height of
1st wave
arrival time 0.00 % 6.12 % 5.12 % 4.50 % 5.33 % 4.75 %
of first
wave
height of
14.7 % 12.2 %
-8.3 % -19.2 %
reflected
wave*
arrival time 3.57 % 2.55 % 1.10 % 1.99 %
of reflected
wave*
*gauge 1 and 2: max. water surface amplitude of the reflection from
the seaward reef slope; gauge 7 and 8: max. water surface amplitude of
the wave train evolved from wave reflected andward reef face at
transition water depth
Water surface amplitudes have been related to the still
water depth for the first wave. The heights of the reflected
waves have been defined to be the difference between
minimum and maximum water surface amplitude. In front
of the reef (gauges 1 and 2), the representation of the
wave height is good. The reflection from the seaward reef
slope is considerably overestimated. The differences in
wave height are in a range of approximately 8% of the
initial wave. Above the reef, the transmitted first wave is
still good represented. The wave train induced by wave
breaking and increasing water depth after the reef is
qualitatively well represented, but the amplitudes are
underestimated considerably. The reason for it is not been
clarified here, because the breaking of the wave plays an
important role. This lies outside of the focus of this study.
Comparison of 3D numerical results with empirical
solution of the interaction of a solitary wave with an
emerged cylinder
The area of interest, representing a computational cell
of a large scale wave model, has been selected to be
5.5x5.5m². The diameter of the emerged cylinder has been
DB=0.24m, the nominal wave height has been H=0.21m
and the water depth has been d=0.7m.
For the cylinder, the roughness height kSt=0.0001m has
been applied [40].
The element sizes at the cylinder wall, where also wall
functions have been applied, have been determined with
the help of the skin friction coefficient CF. The skin
friction is caused by viscous drag in the boundary layer.
The skin friction coefficient can be derived from the
empirical 1/7 power law
C F  0.0583 Re 0.2
(12)
[42], in which Re is the Reynolds number determined
with the diameter of the roughness element and the water
particle velocity under the wave crest which has been
averaged over the water depth. The wall shear stress
  0.5C F u 0 2
(13)
has been applied to equation (6). The largest elements
in the domain should allow resolving the vortices, which
are induced and shed from the roughness elements with at
least 10 elements [43]. With the help of Strouhal number
Sr  fD B u 0
(14)
and the relation
fDB u0  0.1981  19.7 Re
(15)
(Douglas et al., 1995), the frequency of vortex shedding
f can be determined. The equation is valid for both round
and squared shaped cylinders and for 250<Re<200,000
(Pavlov et al., 2000). It is further, conservatively,
assumed, that the diameter of a vortex DV is at least
DV  u0 f .
(16)
Because under the wave, water particle oscillate no full
vortex street can develop. The approximation might
anyway serve as initial guess. Thus, the element sizes
reach from 0.0035m at cylinder and bottom to 0.12m in
the area of interest.
At the boundaries of the area of interest (far-field), the
influence of the single cylinder on the fluxes and water
surface elevations is very small. That’s why, these results
have not been represented here, but will later be used for
comparisons.
In the near field, forces on the cylinder and momentums
at the cylinder base have been determined. They are
presented in figure 7.
6th Annual International Workshop & Expo on Sumatra Tsunami Disaster & Recovery 2011
in Conjunction with
4th South China Sea Tsunami Workshop
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empirical relations, found in literature, e.g. using
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2
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poaching worsens tsunami destruction in Sri Lanka”, EOS Trans.
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