Study of wave wind interaction at a seawall using a numerical wave channel

Study of wave wind interaction at a seawall using a numerical wave channel tài liệu, giáo án, bài giảng , luận văn, luận...

To appear in: Appl Math Modelling

Received Date: 28 November 2012

Revised Date: 13 January 2014

Accepted Date: 15 April 2014

Please cite this article as: P.D Hieu, P.N Vinh, D Van Toan, N.T Son, Study of Wave – Wind Interaction at a

Seawall Using a Numerical Wave Channel, Appl Math Modelling (2014), doi: http://dx.doi.org/10.1016/j.apm 2014.04.038

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Study of Wave – Wind Interaction at a Seawall Using a

Numerical Wave Channel

Phung Dang Hieu 1* , Phan Ngoc Vinh2, Du Van Toan1, Nguyen Thanh Son3

Faculty of Hydro-Meteorology and Oceanography, Hanoi University of Science, 334 Nguyen Trai

Str., Thanh Xuan, Hanoi, Vietnam

*

Corresponding author: Fax:+84-4-3259-5429

a specific value of about 1.25

Keywords: Wave overtopping; Two-phase model; Overtopping rate; Seawall; Wind effect

1 Introduction

Accurate assessment of wave overtopping at seawall or coastal structures is a key requirement for effective design of coastal defenses Wave overtopping has been studied extensively over the last 30 years (Besley et al [1], Goda et al [2], Owen [11], Van de Meer and Waal [15]) The knowledge on the wave overtopping of seawalls has contributed significantly to the practical industry and published in some distinct guide books so far Among those, it could be mentioned such as “EuroTop Wave Overtopping of Sea Defenses and Related Structures” [10], TAW (2002) [14] and so on However, strong waves dangerous for coastal structures are mostly appeared during the storm weather with the presence of strong wind Whereas, formulae for the estimation of wave overtopping have been empirically formed using experimental data which were measured indoor from experiments done in wave flumes without presence of wind Therefore, wave overtopping in practice may

be significantly affected by wind and different from which estimated by using those empirical formulae

In the guide book [10], it is pointed out that wind may affect overtopping processes and thus discharges by: changing the shape of the incident wave crest at the structure resulting in

a possible modification of the dominant regime of wave interaction with the wall; blowing up-rushing water over the crest of the structure (for onshore wind) resulting in possible modification of mean overtopping discharge and wave-by-wave overtopping volumes; modifying the physical form of the overtopping volume or jet, especially in terms of its aeration and break-up resulting in possible modification to post-overtopping characteristics such as throw speed, landward distribution of discharge and any resulting post-overtopping loadings [10] However, very few experimental studies of wind effect on wave overtopping have been found so far Iwagaki et al [6] studied the wind effect on wave overtopping of

vertical seawalls by doing experiments in a small scale wave flume with a wind tunnel Their results confirmed that the wind effects on the rate of wave overtopping on vertical seawalls were very important However, they commented that their results were of only some cases for vertical seawalls and not sufficient to apply to practical purposes Ward et al [16, 17] carried out experiments using a physical wave flume with wind facilities to study the effects

of strong onshore winds on run-up and overtopping of coastal structures Although, it is widely assumed that onshore winds significantly result in increasing run-up and overtopping, very few formulae and experimental data estimating the wind effects on run-up and overtopping have been published [8] Thus, further studies with more systematic investigations need to be carried out to disclose the mechanism of wave overtopping

Numerical simulation of wave overtopping is very difficult due to the complex process

of the wave overtopping itself and in the treatment of the overturning free surface in a numerical model [5] For a decade, the numerical model based on the Navier-Stokes equations together with the volume of fluid (VOF) method has been known as a potential tool for the simulation of wave breaking and wave overtopping However, the simulation of wave overtopping with wind effect is still limited Recently, Li and He [7] have studied the wind effects on wave overtopping by using a two-phase solver Their results showed the capability

of the two-phase model in simulation of wind and wave movement and therefore showed the wind effects on wave overtopping of a structure

Hieu et al [3], Hieu and Tanimoto [4], Hieu and Vinh [5] proposed a numerical based two-phase flow model for wave breaking, wave-structure interaction and wave overtopping of seawall supported by porous structures Their studies on verification of the model for wave breaking, wave structure interaction and wave overtopping showed that the model has good capability in making numerical experiments on wave motion and wave structure interaction including wave breaking and overtopping In this study, the proposed

model [4, 5] is used as the core of a numerical wave channel for carrying out numerical experiments on wind-wave interaction and studying wind effects on wave overtopping of a slopping seawall Firstly, the numerical wave channel is used to carry out an experiment in the condition similar to the experiment done in a laboratory wave flume in order to verify the numerical wave channel Then, a series of numerical experiments are carried out for the investigation of wind effects on wave overtopping and wave quantities as well as for the study of wind modification by wave motion

2 Numerical wave channel and experiment setup

2.1 Numerical wave channel

The numerical model proposed by Hieu and Tanimoto [4] and Hieu and Vinh [5], which was based on the Navier-Stokes equations extended to porous media (Sakakiyama and Kajima, [12]) and the Smagorinsky turbulence model [13], was applied as the core of the numerical wave channel for conducting numerical experiments The numerical wave channel used a source wave maker method in order to minimize the reflection of waves at the wave maker boundary The source wave maker consists of two parts the source function and the damping zone The source function is added to the mass conservation equation in order to generate the desired incident waves While the damping zone works as an energy dissipation one by adding a resistance force proportional to the flow velocity to the momentum equations (refer to [4], [9] for more detail) Fig 1 presents a schematic view of the numerical wave channel

2.1.1 Governing equations

The governing equations for the numerical wave channel in a 2-dimensional model are briefly written as follows:

Continuity equation:

v m z x

q z

w x

u

γγ

γ

=

∂

∂+

z e

x v

z x

x

w z

u z

x

u x

x

p z

wu x

uu t

∂

∂

∂

∂ +

∂

∂ +

∂

∂

ν γ ν

γ ρ

γ λ

λ

w v z z e

z e

x v

z x

z

w z

z

u x

w x

z

p z

ww x

uw t

∂

∂

∂

∂ +

∂

∂ +

∂

∂

γ ν

γ ν

γ ρ

γ λ

λ

Advection equation for the volume of fluid fractional function:

qF z

F w x

F u t

∂

∂+

where t: time, x and z are the horizontal and vertical coordinates, u, w: horizontal and vertical

velocity component respectively, ρ: density of the fluid, ρa: air density, ρw: water density,

a

ν and νw are molecular kinematic viscosity of air and water, respectively, p: pressure, νe: kinematic viscosity (summation of molecular kinematic viscosity and eddy kinematic

viscosity), g: gravitational acceleration, γv: porosity, γx, γz : areal porosities in the x and z

projections, q m is the source of mass for wave generation q u , q w is the momentum source in

x and z direction F is the volume of fluid fractional function; qF is the source of F due to

the wave maker source method

−+

=

−+

=

M z z

z

M x x

x

M v v

v

C C C

γγ

λ

γγ

λ

γγ

λ

11

1

(7)

where C M is the inertia coefficient

The resistance force R x and R z are described by the following equations

12

1

w u u x

1

w u w z

locationsource

at the

s m

q

The momentum source in x and z direction (here we neglect the momentum source

contributed by the viscous terms) is respectively given as q u =uq m, q w =wq m

2.1.2 Initial and boundary conditions

At the initial time, still water is assumed inside the computation domain There are two kinds

of boundaries, namely, interface boundary and domain boundary The interface boundary

For the domain boundary, at the computational cell adjacent to the solid cell, the no-slip boundary condition is adopted At the top boundary, where the computational domain is connected to the open air above, the continuative conditions are applied for velocity and pressure These conditions mean that the velocity components fully satisfy the continuity equation and the gradient of pressure at the boundary set equal to the hydrostatic pressure gradient

2.1.3 Solution method

The governing equations are discretized by a finite difference scheme on a staggered grid mesh The velocity components are evaluated at cell sides while scalar quantities are evaluated at the cell center The SMAC method (Simplified Marker and Cell Method) is used

to get the time evolution solution of the governing equations The resultant Poisson equation

of pressure correction due to the SMAC method is solved using a Bi-conjugate gradient method Here the brief explanation is given as follows (for more detail, see [3, 4]):

(a) Give initial values for all variables;

(b) Give boundary conditions for all variables;

(c) Solve explicitly the momentum equations for the predicted velocities;

(d) Solve the Poisson equation for the pressure corrections;

(e) Adjust the pressure and velocity;

(f) Solve the advection equation of VOF function using the PLIC algorithm for tracking the free surface

(g) Calculate the new density and kinematic viscosity based on the VOF values

(h) Calculate the turbulence eddy viscosity

Return to step (b) and repeat for next time step until the end of specified time

by calculating the total water contained behind the vertical wall Therefore, the accumulated overtopping water and averaged overtopping rate can be obtained

The numerical experimental wind speeds, incident wave heights and wave periods are presented in Table 1

In order to validate the numerical wave channel in conducting the experiments on wave overtopping, an experiment with the conditions similar to the experiment N1 was carried out

by using the physical wave flume at Department of Coastal Engineering, Water Resource University, Hanoi, Vietnam The wave flume equipped by Delft, The Netherlands was 40m

long, 1.2 m high and 0.8m wide The slope and vertical wall of the seawall was made of wood The stones with size of 3cm x 4cm were used to build up the porous structure at the toe of the slope The porosity of the porous structure was 0.42 The water surface elevation

at some locations was measured by using the capacitance wave gauges The overtopping water volume was measured wave by wave by capturing the overtopped water in each container for each wave The formula for calculation of overtopping rate is commonly as

follows:

bT

V

q= w where, q is the overtopping rate, V w is the captured water volume, T is the

wave period and b is the width of the leading water channel (a small channel attached at the top of the seawall to lead the overtopping water to the capturing container) The experiment was done three times with the same experimental conditions in order to ensure the consistency of the experiments There were nine capacitance wave gauges set in the flume for measuring the water surface elevation Table 2 shows the relative positions of the wave gauges to the reference point

Regular waves with a period T =1.6s and H I = 17.3cm are generated in the wave flume

by a piston type wave maker system

3 Results and discussion

3.1 Comparison between simulated results and laboratory data

For verification of wave overtopping of the seawall between simulated results and laboratory data, it is necessary to compare the input wave motion between numerical and laboratory experiments to ensure similar input wave conditions The water surface elevations

at the same location in the numerical wave channel and physical wave flume were used for the comparison

Fig 3 shows the comparison of water surface elevations between simulated results and laboratory data at the wave gauges G1 and G4 After about seven wave periods, good

agreements are observed both at the gauge G1 for the incident waves (Fig 3a) and at the gauge G4 for the wave motion nearby the wave breaking location (Fig 3b) However, at the initial time, there are discrepancies in the water surface elevation between the simulated results and experimental data due to the difference in the slow start of wave maker method between the laboratory experiment and numerical experiment After about seven wave periods, the incident wave motions of both experiments become similar (Fig 3a) That gives

a confidence in further comparisons of wave quantities It is also seen that at the initial stage, the experimental data show the move up of the still water surface elevation from the initial still water level That could be due to the forward movement of physical wave paddle in the laboratory wave flume, which creates a long wave and results in that phenomenon

Fig 4 shows the simulated results and experimental data of the cross shore distribution

of wave quantities nearby the seawall The results show good agreements for wave crests, wave troughs and wave heights at all measured locations However, simulated wave troughs tend to be a bit smaller than the measured data

Fig 5 presents the comparison of accumulated water volume of wave overtopping between simulated results and three sets of measured data In general, the simulated results agree with the experimental data and are a little higher than the averaged values of the three experimental data

The results of the above comparisons give a great confidence in carrying out further numerical experiments using the numerical wave channel for the study of wave and wind interactions at the seawall including the overtopping processes

3.2 Wind effects on wave overtopping rates and wave crests

The numerical experiments using the numerical wave channel were carried out for 121 cases with the experimental conditions as mentioned in Table 1 For each experiment, the

variation of normalized overtopping rate versus the normalized wind velocity The wind speeds were normalized by the local wave speed at the breaking location and the overtopping rates were normalized by the quantity gH I3 In the figure, q w denotes the overtopping rate with wind effects It is seen that the influence of wind on overtopping rates is very complicated For a specific incident wave condition, it seems to have its own wind effect on wave overtopping rate Generally, the wind effect on overtopping rate becomes significant

when the relative wind speed

b

gh

W

is greater than 2 It means that, the wind effect becomes

significant when the wind speed is two times greater than the local wave speed It is very interesting that the overtopping rate does not monotonically increase with wind speed but it seems to have an effective range of the relative wind speed that gives significant effects on overtopping rates and this effective range is estimated from 2 to 7 This phenomenon may be due to the effect of wind on the wave breaking condition resulting the modification of the condition of wave energy dissipation in the surf zone Therefore, it creates the complicated effects of wind on the overtopping rates It is also seen that the wave overtopping rate has its own maximal value for each wave condition

It is very useful if the maximum effect of wind on overtopping rates can be quantified in

a certain manner Based on the simulated results, the wind adjustment coefficient is determined as a ratio of the overtopping rate with wind and the overtopping rate without wind effects, called f w Fig 7 shows the variation of wind adjustment coefficient versus the averaged overtopping rate without wind effects

It is observed that when the mean overtopping rate is smaller than 5×10−4m3/s/m, the maximum of the wind adjustment coefficient varies in a complicated manner but has a maximum of about 1.6 (Fig 7) However, when the mean overtopping rate is over the value

3.3 Wind modified by waves

Not only wave quantities are affected by wind but also the wind field on the water surface is modified by the motion of water waves

Fig 9 presents the vertical distributions of maximum horizontal wind speeds from the still water surface at three locations in the case for wind speed of 4 m/s Wind input is uniform and constant of 4 m/s, but it is seen that the vertical profiles of maximum values of horizontal wind speed are not uniform and greater than 4 m/s at some levels That is due to the reverse influence of waves on wind The vertical profiles at all locations agree fairly well with the logarithmic form At the third location just at the vertical wall (Fig 9c), the maximum horizontal wind speed has the highest value in comparison with that at two other locations (Fig 9a, 9b) That appears due to the contribution of water flow during the wave overtopping process

The wind modification by waves is also observed in Fig 10 In the figure, the time profiles of wind velocity components at the level of 40cm above the still water level are plotted for three locations Note that the level of 40cm high above the still water level in the experiment is similar to 10m above the sea surface level in practice, as the small scale is 1:25 In general, it is seen that at all locations, the wind velocity components are fluctuated in some periods similar to the incident wave period At the location near the wind source and far

and not so complicated, and has the major period similar to the wave period At the location far from the wind source and nearby the wave breaking location (Fig 10b), the wind disturbance becomes very complicated and stronger The wind disturbance has not only the period of the incident wave but also some smaller periods, which are due to the effects of waves in surf zone The similar phenomenon is also observed at the last location (Fig 10c)

At the last location, the vertical movement of wave overtopping contributes much to the fluctuation of wind In general, at all locations the fluctuation amplitudes of the vertical wind component are larger than those of the horizontal wind component This means that the waves have significant effects on bending the air flow Therefore, event a constant and uniform wind flow blows on a wavy water surface, the wind flow becomes fluctuated

The wind modification by waves is also clearly observed in Fig 11 The water waves together with the wind field are plotted in the figure for the case with H I =15cm, T =1.6s and wind speed W = 4m/s The curved line represents the interface of the water zone (below

the curve) and the air zone (above the curve) The velocity is plotted in every eight cells in x direction and every six cells in z direction It is seen that the wind field is significantly

modified by wave crests

When a wave overtops the vertical wall, it intensifies the vortex behind the wall and the vertical component of wind velocity also increases There is a clockwise wind vortex that appears at the wave trough Wave crests coming on the slope periodically result in the fluctuation of the wind flow Thus, an input constant wind can become fluctuated under the action of surface waves and inversely, summation of periodical wind flows could give more effects on wave motion

4 Conclusions

Wind and waves interaction experiments were conducted for wave overtopping of a ¼ sloping seawall using a numerical wave channel for several conditions of incident waves and wind speeds The numerical results were verified against laboratory data in a case for wave overtopping without wind effects

The comparison results showed satisfactory agreements between the laboratory measured data and simulated results for wave quantities and overtopping rates

The interaction of waves and wind was analyzed in term of mean wave quantities, overtopping rate and variation of wind velocity at some selected locations The results showed that the overtopping rate was strongly affected by wind and the wind field was also significantly modified by waves There exists an effective range of relative wind speed

The work in this study is the first approach to quantify wind effect on wave overtopping The wind input was only of “academic” case (linear wind profile and only 1m long fetch) and not of realistic one However, this study pointed out that wind effect is significant Therefore, future works with more realistic wind input (a longer fetch and a logarithmic wind profile) are necessary In addition, consideration on experiments for irregular waves is important

Acknowledgments

This work has been done under the financial support of Vietnam's National Foundation for Science and Technology Development (NAFOSTED) The financial support from NASFOSTED is gratefully acknowledged The first author wishes to bring thanks to Dr Tai Van Nguyen and Mr Luong Quang Nguyen, Department of Coastal Engineering, Hanoi Water Resource University, Vietnam for carrying out the laboratory experiments and providing with the experimental data