The feasibility of implementing radiating boundary conditions in Triwaq

T R I - F L O W COMP. INC. 4818 KE Road P.O. BOX 122 MESA CO 81643 Tel: 970-268-5478 Fax: 970-268-5262 T H E FEASIBILITY OF IMPLEMENTING

RADIATING BOUNDARY CONDITIONS

IN TRIWAQ

Jan J. Leendertse

PREFACE

This study was made for the National Institute for Coastal and Marine Management of the Department for Public Works and Watermanagement in The Netherlands (Order number 22971233, 2 9 / 0 4 / 9 7 ) .

SUMMARY

This report describes a method for radiating disturbances generated in the computational field through the radiating boundaries. First a simulation is made without disturbances (such as a storm surge) subsequently a second simulation is made with the radiating boundary conditions activated, and the disturbance is able to pass through the boundary as if no boundary is present. The developed boundary condition permits nearly complete transmittal with reflections of only a few percent.

By making a large number of experiment, the effectiveness of the method is illustrated. It appears also possible in principle, to modify the radiation boundary during the second simulation, but this needs further study.

It has been concluded that radiating boundary conditions can be implemented in T R I W A Q .

CONTENTS

PREFACE ' SUMMARY ' Section

I. INTRODUCTION

II. T H E RADIATION BOUNDARY

III. BASIC DIFFERENCE APPROXIMATION

IV. EXPERIMENTS W I T H A RADIATING BOUNDARY V. ABSTRACTING DATA FROM A SIMULATION

VI EXPERIMENTS W I T H RADIATING DISTURBANCES IN A FORCED SYSTEM

VII. MODEL W I T H B O T T O M STRESS

MX. EXPERIMENTS W I T H BOTTOMSTRESS IX. ADJUSTED RADIATIVE BOUNDARIES

X. INTRODUCTION OF A DAMPING FUNCTION XI. IMPLEMENTATION IN TRIWAQ

XII. DISCUSSION XIII. CONCLUSION REFERENCES

I. INTRODUCTION.

Models o f estuaries and coastal seas have so-called open boundaries. At the location of these boundaries, the investigator has to prescribe numerically certain characteristics of the flow, such as the waterlevels, the horizontal currents or velocities. These model inputs are generally the major driving forces of the water movements.

In model investigations, the aim is generally to find how the physical system will behave for conditions not yet encountered. This poses particular problems as for those conditions we need to have boundary conditions which matches the changed system or the new driving forces. For example, by making a storm surge simulation of a continental shelf, waterlevels at the open boundary are norma ly used for generating the tidal motions in the system. Unfortunately the wind and pressure in the system will modify these boundary conditions. For systems where we have sufficient data, methods have been developed to adjust these boundaries, but this approach works only for model simulations using observed data, thus for so-called "hindcasts".

In a 1986 RAND Working Note^ for the Rijkswaterstaat, a radiating boundary condition is described for a three-dimensional model. In that Note is was shown how a wave which it generated in the model area can run out of the niodel as if no boundary existed. It did not describe how such a wave did run out of the model when the model is also driven by tides at the boundary.

In this study, we investigate the feasibility of designing boundary conditions for the three-dimensional model ( T R I W A Q ) which allows disturbances which are generated in the model area, to run across the boundary, while at the same time the normal inputs at this boundary are retained. The above mentioned Working

Note indicated only the principle we could use, but not the approximations to be added to the boundaries as the model of the Working Note is based on other partial differential equations than TRIWAQ.

In formulating these boundaries, we worked with a simple one-dimensional model, in which the approximations and the method of time-stepping is essentially the same as in T R I W A Q . In other words the experiments were made with a system which has very similar numerical characteristics.

A very large number of experiments were made while we tested our approach. We found that of the many difference approximations we tested the ones we have presented are working the best. We have presented only test resuits with these finite difference approximations.

In presenting the results of our investigation, we show approximations and model results with increasing complexity, in order to give the reader a good understanding of causes and effects. In one of the last Chapters of the report, it will be indicated how the implementation of this approach in TRIWAQ can be accomplished.

II. T H E RADIATION BOUNDARY.

The Sommerfeld2 radiation condition for a wave running out of an one dimensional system can be written as:

K + C | = 0 (2.1)

where ( = waterlevel above the reference level

C = speed of the wave

Thus for an outgoing wave, the increase of the waterlevel with time is determined by the slope of the waterlevel at the boundary times the wave speed. This boundary condition can be implemented in a numerical model by replacing the partial differentials by a stable finite difference approximation as will be shown in Chapter I I I .

For practical applications this boundary condition is not very usefull as it enables only a wave to run out of the model area. An boundary condition which is very usefull is the condition which allows an disturbance t o run out of the system but keep the basic water and wave movements as prescribed on these boundaries. This can be accomplished by the following procedure.

We assume that we have two wave conditions in the model area, one is the basic condition and the second one is caused by the disturbance. We assume t h a t :

( = h i + hs

where hi = wave created by the disturbance

h2 = basic wave present in the system

Consequently we can write for eq. (2.1)

dhi I r , dh2 , r 5h2 _ A fo o)

or

| h i + c | i + f ( t ) = 0 (2.3)

where

| ^ + C | ^ = f ( t ) (2.4)

To apply this approach, first a simulation is made with the basic system without the disturbance. From this simulation we abstract f ( t ) and subsequently we make a simulation with the system which contains the disturbance and use eq. (2.3) as the boundary condition.

III. BASIC FINITE DIFFERENCE APPROXIMATION.

TRIWAQ3 uses a two stage approximation of the partial differential equations which describe the three-dimensional flow in the model area. Two different approximations are used in succession in each step of the model. Each approximation is of a first order, but their combined use results in a second order scheme which is very stable and only slightly dissipative in the approximation of the advection terms.

Computations are made on a grid system, but the variables are not computed at the same locations of the grid. The velocities are computed at points between the waterlevel points. This means that if we are using waterlevel as boundary inputs, the velocities immediately adjacent to the boundary can be considered as one-dimensional flows, with some extra terms which contain contributions of the flow in the direction of the boundary. Thus, if we make an investigation with an one dimensional model, we are making an investigation which contains the major behaviour of the the boundary condition. T o be effective in this study, we have to make the approximations of the one dimensional model very similar to the one in TRIWAQ.

The two steps in TRIWAQ are an explicit step and an implicit step. In the first step, the momentum is computed directly from the waterlevels:

whe,e = + « „ ^ 1 +

if the lower boundary is a closed boundary an the radiating boundary is at M , the computation is made from 2 to M - 1 .

For the continuity equation we have also an explicit computation:

(<m - U'^' + ( ^ m - ^ m - l ) / ^ ^ = ° ^ = ^ " " ^ " ^ ^^'^^

In the second step we use

( U ; ; - U ^ ) / A t + g H ; ; ( C ^ ^ ^ - C ^ ) / A X = O for m = 2 , . . . M - 1 (3.3)

where

H " = temporal depth obtained by iteration

and for the continuity equation

( ( ; ; - C ^ ) / A t + ( U ^ - U ^ _ i ) / A x = 0 for m = 2 , . . . M - 1 (3.4)

for the boundary condition we take

where

c = [g ( h M - i + ( C M + < M + C^^-i + C M _ I ) / 4 ) 1 ' / ^

+ ( ' J < f l - i + " M - I ) / ( " M - I + " M - I ) (3-^) The finite difference approximations of Eq. 2,1) are extensively studied by Stelling^. For this boundary condition represented by Eq. (3.5) we have a first order upwind approxinnation in space. Note that the boundary condition contains information at time level t and timelevel t + 1 . The wave speed is determined centrally in the time step 2At,

The second step is solved implicitly by eliminating out of eq,(3.4) by use of eq.(3.3). We obtain a tri-diagonal system of linear equations:

where A C" m^m-_1 + B m^m m ^ m + 1 D _m (3.7) Am (3.8) ^ m

=

=

=

boundary point M we obtain:

At r _(3.12)

=

0 (3.14)

=

This system of linear equations can be solved by recursion. Once we obtain the water levels, the new transports U" can be computed by means of eq. (3.3). Note that on the boundary (at M) information of the whole

IV. EXPERIMENTS WITH A RADIATING BOUNDARY.

To test the finite difference equations for the radiating condition one dimensional experiments were made. A wave was generated near the closed end of a canal with a depth of 10 meter. The canal was 20 km long which was represented by 200 points with a distance between the points of 100 m.

The half time step was 40 seconds, thus the time step was nearly eight times the Courant speed.

The initial disturbance was sinusoidal with a length of 4 km, thus the starting condition was distributed over 40 points as shown in Fig. 4 . 1 . The wave

height was 0.25 m. In the figure we show the three water levels of step 1. The computed water level at the end of the explicit half step coincides with the starting condition as the starting velocities are zero (Eq. 3.2). As the time step is very large, we see a drop in the wave height at the end of the second implicit half time step.

In Fig. 4.2 the computed wave is shown at time step 11. The disturbance has moved over about eighty grid points as was to be expected. As we have not incorporated any bottom stress term or a viscosity term the wave height at the beginning and end of the time step are the same. At the end of the explicit step the wave height is slightly larger than at the beginning of this time step, as was to be expected by the fact that the amplification of this half time step is larger than unity. In the second half time step this wave height is reduced again. Note that a trailing wave is being generated behind the disturbance.

At time step 21 (Fig. 4.3 the front of the wave has reached the radiation boundary. The maximum wave height at this time step is essentially the same as at time step 11. The trailing wave has increased somewhat in amplitude.

In Fig. 4.4 the computed wave is shown at time step 3 1 . The disturbance has passed through the boundary and only the trailing wave is still in the system. At the left side of the figure the water levels are very close to zero.

At time step 41 (Fig. 4.5) the trailing wave has passed also through the boundary. In this figure it can be seen that the radiation through the boundary has not been complete, a very small part of the disturbance has been reflected back. The maximum of this reflection is visible at point 80. This is in agreement with the computed wave speed. At step 41 the disturbance has traveled a little less than 41 x 8 grid points. As the reflection occurs at grid point 200, the maximum reflected wave should be at about point 90.

Computing the content of the water in the canal at the beginning of the experiment and at time step 4 1 , it appears that the reflection is less than 1 percent. Using other approximation than those described in Chapter III, did result in more reflection. We found also that the reflection depends on the time step.

In this experiment, the water levels and velocities are in phase with each other as shown in Fig, 4,6 for time step 21, Note that water velocity is about 1,2 m/sec, thus contributes considerably to the value of C in Eq, 3,6,

6

To see if the disturbance runs connpletely out of the system, the computation was extended to 400 timesteps. The variations in the waterlevel were of the order of the order of 5 x lO-^. However, the mean water level at that time was about 0.00180 m. The computation gives apparently a bias, but this bias does not increase with time.

To investigate this further, the simulation was made with a half timestep of 20 sec. The mean waterlevel after 800 steps was now 0.00090 m. By reducing the timestep to 10 sec, the mean water level bias was now about 0,00042 m over the same time period. In an experiment with a timestep of 5 sec (which is about the Courant speed), the waterlevel bias is now 0.00019 m. It suggests that an approximation error is present which is linear with the timestep.

The experiments were made with a positive wave running through the channel. The question arises if this positive wave is the cause of the positive bias in the water level after the wave has passed. To investigate this we let a negative wave run through the channel, again with a half time step of 40 seconds for 400 time steps. The remainder variations in the water level were again of the order of 5 X 10-5 m. and the mean waterlevel was about 0.00190 m above the starting level of the experiment, thus comparable with the positive wave. The error is always positive.

If we make the wave smaller, reducing the height from 0.25 m to 0.125 m the bias reduces to 0,00045 m, thus the bias is a quadrastic function of the height of the disturbance.

The starting disturbance shown in Fig, 4,1 is 40 gridpoints long, if the length is increased to 80 points and the wave height is now 0.125 m the bias reduces to 0,00023 m.

As the bias diminishes with the timestep, it could be that the first order approximation of the partial differential equation ( E q , 2 , l ) couid be the source. We replaced Eq, (3.5) by

( C M - ^ M + < M - r < M - l ) / ( 4 ^ ^ ) + C ( ^ M + - ^ M - 1 " C M - I ) / ( 2 ^ > < ) = ' ( ' ' ' ^

V. ABSTRACTING DATA FROM A SIMULATION.

It is the aim of the introduction of the radiating boundary condition to let a disturbance which generated inside the model area to radiate out. Consequently, we need to be able to abstract from the basic run that data which is input to the boundary for the simulation which contains the disturbance. This is the function f ( t ) of Eq. (2.4).

In Chapter II the function f f t ) was presented in partial differential form, but as we are working with finite differences in our model, we need to compute this function in finite difference form. To be consistent with the radiation boundary described in Chapter III, we take

F«)

+ c

(^-i)

where

c = lg Hii^ + ( M + <M-i + C M - I ) / " ) ! ' ' '

+ ( U M - I + " M - I ) / ( " M - I + " M - I ) ( 5 ' 2 )

The function F(t) which is abstracted from the basic computation and which will be the boundary input of the simulation with the disturbance, it of a different nature than the other boundary conditions in TRIWAQ. The more normal boundary conditions such as water level and current velocity are values occurring at the boundary at a particular time. With the radiation boundary we insert a difference in waterlevel over the timestep and the waterlevels in the simulation with the disturbance contain the cumulative effect of boundary inputs up to the time that the timestep is computed. This means that data abstraction of the basic simulation and the data insertion of the simulation with the disturbance need to be made with high accuracy.

To make the simulation with the disturbance, F(t) becomes a forcing function and Eqs. (3.6 through 3.14) are still valid. Only Eq. (3.15) becomes

{

VI EXPERIMENTS W I T H RADIATING DISTURBANCES IN A FORCED SYSTEM.

In the first experiment with a radiating disturbance in a forced system, we want to let the wave of the experiment described in Chapter IV run out of the

right boundary of the channel, while a wave is entering the channel of that side. First a simulation was made of this wave with the model with a prescribed simussoidal water level boundary. A very low frequency wave was used as input. Slowly a standing wave is being generated, and we show the waterlevels in the channel with intervals of 10 timesteps in Figs. 6.1 through 6.5. At timestep 21 we show in addition to the waterlevel at the end of this time step, the velocity (Fig. 6.6). At this timestep the flow is negative, filling the basin.

From this simulation we abstracted F f t ) for every timestep to be used as the forcing function of the disturbed system. Subsequently, the forcing function was inserted in the model with the radiating boundary described by Eqs. (3.1 through 3.6) with the forcing function in Eq. (5.3) in the computation.

As no disturbance was inserted in the waterlevel, the same results were obtamed as shown in Figs. 6.1 through 6.6. The differences were smaller than IQ-e, thus the accuracy of the computation. This experiment verified that the boundary mput procedures are correct.

In the next experiment we repeated the simulation, but now we inserted the disturbance at the beginning of the simulation. At the end of the first timestep the water level is shown in Fig. 6.7. At timestep 11, the disturbance has moved to the right and the low frequency wave is entering the channel from the right. In the figure the periodic wave of the basic computation (Figs. 6.1 through 6.6) js also shown. By subtracting the waterlevels of the periodic wave from the result with the radiating boundary, we obtain the disturbance. This is comparable with the disturbance of the first experiment (Fig.1.2). For the following timesteps the results are presented in the same manner. At step 41 the disturbance has moved out of the channel nearly completely. Fig. 6.12 shows the waterlevel and the velocity at step 2 1 . Left of the disturbance the velocities are negative but perturbed by the trailing waves of the disturbance which moves to the right. At the location of the disturbance, the velocity of the disturbance is larger than the

negative one of the periodic wave.

In the experiments described in Chapter 4 we found a small bias in the water level after the wave had run out of the model area. Here again we found the same effect with about the same magnitude.

In the above described test a low frequency wave entered the channel from the right. In the following test we tried to simulate a condition by which the low-frequency wave starts moving out of the channel. The initial waterlevel was set at 0.05 m above the reference level and we lowered the initial waterlevel with the same frequency as in the previous test. Thus first we obtained the radiation boundary function F(t) and subsequently used the boundary function at the right boundary of the channel and started the simulation again with the disturbance on the left closed boundary. Figs. 6.13 through 6.18 show the result of the combined effect. Here again the disturbance runs effectively through the right boundary with only a very small reflection.

In the above tests, we had a short disturbance running through a forced boundary which had a low frequency variation. The question which can be raised is what will happen if the waterlevel variations at the boundary have frequencies in the same band of the disturbance. In the experiment which we made for this case, we generated waves with a wavelength of about 120 grid points as shown in Figs. 6.19 through 6.24. It will be noted that particularly in the compute water level at the end of the explicit step, a small high frequency wave is present. The computation method uses an explicit and an implicit step, each of these is first order accurate. Thus if we assign a boundary value at each half timestep which is part of a sinus curve, the first order approximation introduce a high frequency wave. This appear unimportant in testing the radiation boundary.

In Figs. 6.25 through 6.30 the combined results are shown. The wave in the channel is complex, but again the radiating boundary conditions performs very well. As shown in step 41 (Fig. 6.29) the wave has radiated effectively through the boundary.

VII. MODEL WITH B O T T O M STRESS

In the previous chapters we made our investigations with a very simple model. In TRIWAQ more terms are included in the hydrodynamic equations and it can be expected that these influence the radiation of a disturbance over the boundary.

A possibility would be to work in TRIWAQ with only the basic terms near the boundary, thus linearize the model locally. This would simplify our precedures considerably, but would cause problems in practical application, as the radiating boundary condition is intended to be used for models which already exist and are adjusted with the terms included near the boundary.

To investigate the effect of the other terms in the hydrodynamic equations, we included the bottomstress in our model. Bottomstress is generally one of the major terms influencing the outcome of an simulation. Consequently we added these stress terms to the momentum equation described in Chapter 3

(7.1)

" m = + « m + l + <J"

and C^ = Chezy value In the second step we use

( u ; -

g

(7.2) where

H " = temporal depth obtained by iteration

= ^ +

(C+i

+

The value for U " in the bottom stress term is obtained by iteration m

The continuity equation and the boundary representation (Eq. 3.5) is not changed.

The second step is solved implicitly by eliminating out of eq.(3.4) by use of eq.(7.2). We obtain a tri-diagonal system of linear equations:

A C" . + B + C ( " , , = D (7.3)

m ^ m - 1 m^m m^m + 1 m ^ ^

11 where Am Cm - Ó ' t " m (7.4) (7.5) B m

=

=

=

For the boundary point M we retain Eqs. (3.12) through (3.15). Again the tri-diagonal system is solved by recursion.

NX, EXPERIMENTS W I T H BOTTOMSTRESS

The first experiment of this serie is again a disturbance running through a channel with a half time step of 40 seconds like the first experiment described in Chapter IV. The Chezy value was taken as 40 m^^'^/sec.

Figs. 8.1 through 8.5 shows the wave traveling in the channel. In Fig. 8.5 a reflected wave with its crest around gridpoint 80 is barely perceptable, thus the radiation through the boundary is nearly complete.

Comparing Fig. 4.3 with Fig. 8.3 it appears that the wave height has decreased by the bottom stress as was to be expected. As our computation method is completely conservative, the decrease in waveheight resulted in a small water level rise behind the disturbance. This effect is retained long time after the crest has past the boundary. At step 41 (Fig.8.5) the waterlevels in the channel are every where positive. In Fig. 8.6 the waveheights versus time are shown for five stations in the channel. A small decrease in waveheight is shown for the stations near the end of the channel.

The influence of the Chezy value on the condition of the channel were also investigated. It was found that decreasing the Chezy value did not have a noticable effect on the radiation through the boundary, only the wave height decreased more as the wave travels through the channel and behind the crest of the wave the waterlevel rises more. In Fig. 8.7 is shown the situation at timestep 2 1 . In this test the Chezy value was 20 m^/'^/sec. Compared to Fig. 8.3 the waveheight has decreased from 0.12 m to 0.10 m, with a noticable rise in water level behind the crest. In Fig. 8.8 the waterlevels are shown as a function of time at five stations in the channel. The stations are 40 gridpoint apart and the figure shows well the steady decrease in wave height. With the low Chezy value the water movements in the channeld have essentially stopped at 100 timesteps.

In the next experiment we let the disturbance run toward the right and hace an incoming wave. The Chezy value is set at 20 m^/^/sec. Figs. 8.9 through 8.12 show the progression of the water levels in the channel. At step 11 (Fig. 8,9) the disturbance and the incoming progressive wave are just starting to interact. At step 21 (Fig. 8.10) the disturbance is riding on the incoming wave. In the figure we show what is called the original disturbance, which is the disturbance in a channel without the incoming wave, but with the bottom stress. The periodic wave represents the wave from which we obtained the radiation function F ( t ) . The combined effect is the result of the simulation, thus the disturbance riding on the incoming periodic wave. Finally we havem shown the disturbance on the periodic wave. This time serie is obtained by subtracting from the combined effect, the periodic wave. This disturbance of wave is slightly different from the original disturbance, as the bottom stress term in non-linear. When the disturbance progresses toward the right, we have a positive velocity, but the incoming wave has a negative velocity, thus reducing the bottomstress effect. The wave height decreases somewhat less as is shown in Fig. 8.10 and the water level behind the disturbance is less than in the original disturbance. In any case the radiation through the boundary is very high.

IX. ADJUSTED RADIATIVE BOUNDARIES.

Up to this point the investigation has been directed toward finding a a boundary formulation which allows waves generated in the field to run out of the system as if no boundary existed. As the tests in previous chapters show, this goal has been archieved.

In practical applications, for example when using data assimilation techniques, this radiative boundary need to be adjusted while retaining is original radiative properties. In this chapter we will focus on the feasibility of developing this practical application.

The process, as developed in this report requires that we first make a simulation from which we capture the boundary condition for every timestep as a function F ( t ) . This function has the dimension of length and contains the differences in waterlevels occurring over the timestep and also the local waterlevel gradient at the boundary. Thus is we want to change the boundary condition at a particular time, we need to change this function.

In the experiment we are again working with the channel represented by 200 points. The left side is closed off and at the right side ( at point 200)

is a radiating boundary condition. The function F ( t ) is zero for the first five timesteps and then we set F ( t ) = 0 . 0 1 for five steps and for the rest of the simulation again at zero. The half timestep is 40 seconds and the Chezy value is 20 m'^'/^/sec. Figs. 9.1 through 9.5 show the progress of the disturbance generated at the boundary. Note that we get an increase in waterlevel at the radiation boundary and the front moves first to the left until it is reflected at the closed off section of the channel and returns moving to the right at about double the original height. Fig. 9.6 shows waterlevels versus time for five station in the channel. It will be noted that the waterlevel stabilises with time at a constant level. The returning wave has been transmitted through the radiation boundary and only some very small oscillations are still evident.

The bottom stress has influence on the waterlevels, but how this works is 1/2

not immediately clear. By setting the Chezy value at 10 m ' /sec, we see in Fig.9.7 that the wave height decreases when the wave front moves to the left, but is stays fairly constant thereafter, as due to the reflection the velocities in the channel become practically zero.

When we make the Chezy value very large, thus effectively eliminating bottom stress, the waterlevel is exactly twice the waterlevel at the radiation boundary when we started.(Fig. 9.8)

In the above described experiments, the function was set over five steps by 0.01 m. Fig. 9.9 shows what is happening if we set F ( t ) = 0 . 0 5 at timestep seven. The system behaves the same, only short oscillations are present. The same is true when we set the Chezy value at 40 (Fig. 9.10). We might conclude from these experiments that when the waterlevel at the boundary has to be increased, it is advantageous to do that over several steps.

14

With these experiments it is shown that the radiation boundary can be modified during computation, for example as may be required when using data assimilation. To incorporate these procedures we need to express the required modification in terms of the function F ( t ) , which contains not only an increase in the waterlevel at the boundary but also the local waterlevel gradient (Eq.5.1). Clearly more research and experimentation is required.

X, INTRODUCTION OF A DAMPING FUNCTION.

The introduction of bias in the solution by a wave generated in the model field can be prevented by the introduction of a dampting function. This function is formulated in such a way that after the wave has passed the residual deviation or bias is slowly approaching zero. Many such function can be designed, we use one here wich is easy to apply.

To obtain the convergence, not only F(t) has to supplied into the model with the basic wave, but also a waterlevel. In the computation only the term D j ^ of Eq.(3.15) need to be modified. We obtain

D M = ^ M - ^ C ( C M - ^ M - I ) - ^ ( C M - < M ) (lo-i) where

0

^M ~ Water level at the beginning of the time step in the basic computation

tp = damping function

To test this we take one of the experiments with bottom stress described in Chapter MX. The original disturbance is followed over 100 time steps as shown in Fig. 8.8. Due to the relatively small Chezy value, the bottomstress had considerable influence on the disturbance. The height decreased and behind the disturbance the water level did rise.

When making the same experiment but now with the damping function and a value of 0.01 for its coefficient, the mean water level is much lower after the wave has passed and at the end of the simulation the waterlevel approaches zero as shown in Fig. 10.1.

Making the damping coefficient 0.02, the waterlevels in the channel become increasing lower and even negative as shown in Fig. 10.2. But later in the simulation the waterlevel is essentially zero (Fig. 10.3). With the smaller coefficient the waterlevel approaches zero without a long period oscillation (Fig. 10.4).

The damping coefficient has considerable influence on the transmission of the disturbance through the barrier. Setting the coefficient tp = 0.5, then about half of the disturbance is reflected as shown in Fig. 10.5. The relected wave becomes negative being partially clamped.

XI. IMPLEMENTATION IN TRIWAQ.

Prior to a simulation is made with a radiation boundary, data need to be abstracted from a regular simulation. Thus we need to define the location where the data need to be abstracted. We assume for the moment that the basic model has a water level boundary. Consequently we need to abstract F(t) as described by Eq. (5.1) and (5.2). This can best be done at the end of the explicit+implicit cycle. Both values of the water level which are two half time steps apart should be available, which means that after we have computed F ( t ) we store in an array ( " at the boundary and ( " at the point one grid distance away in the field. These values of are to be used as values for ( in Eq. (5.1) in the computations for the next data abstraction. The function F ( t ) , as well ( (to be used for the dampening) need to to written in a file with high accuracy. In implementing the procedures, it appears most expedient to file the functions in a special file, before incorporating these series in the SIMONA files.

For the model to run with the functions ( F ( t ) and ( ) as inputs, the boundary needs to defined for which the procedures o f the water level boundary can be used. The time series for ( can be inserted through the regular time varying input files like the water level boundary time series file. We need an extra input for F(t).

Besides the more administrative work for describing the location of the radiation boundary and the type of the boundary, the most additional work will be in subroutine trsrvw. But before the actual programming can begin, we have to translate the ideas presented in Chapter III, into a three-dimensional formulation. Note that in the one dimensional model we work with the momentum U, in TRIWAQ this value is obtained by integrating the layers over the vertical. In essence Eq. (3.7) is the simplified form of Eq. (3.1.10) of Ref 3. In the latter integration over the layers is included and also horizontal coordinate transformation. For the right boundary condition we need to insert Eqs. (3.12) through (3.14) and (10.1).

For the loop over all computational rows, we have to add the boundary condition for the radiating boundary at their ends. As we have already 6 possible boundaries, we have to add at the end of this list a section for number 7. Note that we can treat the radiation boundary condition in the same manner as the water level boundary. The first momentum equation near the end or beginning of each row or column is included in the normal manner, we need only to include the actual radiation condition which only contains relations between water levels.

The main programming effort will be administrative, defining locations, accumulating arrays, the additions in the computational subroutines will be limited.

We have indicated that the radiative boundary has much in common with the water level boundary. For the transport equations we can make the radiative condition the same as for the water level boundary.

XII. DISCUSSION.

The new approach to the radiating boundary problem which is presented in this report is that a separation is made between basic simulation and simulations which contain also disturbances which are generated in the computation field and which we would like to move across the boundary as if no boundary is present.

The computation of the wave motion near the new boundary can best be described as having retained all the conditions of the basic model, and in addition we let disturbances pass through nearly completely without a noticable reflection. However, a small error is introducted during the transmission, resulting in a small rise of the waterlevel in the system after the disturbance has passed the boundary. To eliminate this effect we introduced a damping function which reduces this error with time.

The approach designed here has some in common with the boundary condition designed by Blumberg and Kantha^ for their three-dimensional model. Also the Sommerfeld radiation condition was used, together with a damping function, but here the damping function acts on the difference between the solution and the prescribed water level. In their report it is stated that a damping function is required. Without the damping function their basin continiously empties after a disturbance is applied. In our experiments we get only an increase in water level during the passage of the disturbance through the boundary, there after that level remains constant. This difference has probably its origin in the difference in the approximations of Eq. (2.1). In the most simple form of Blumbergs approach, the boundaries are linearized. In the model boundaries presented here, the full approximations for the basic flow near the boundary are retained, and the disturbance is moved linearly across the boundary.

It has been found that the radiating boundary is working very well, hardly any reflection has been noted in the experiments, and it can be expected when this method is applied to the three-dimensional it would work also very well. It is noted here that at the boundary the radiation is treated as an one dimensional problem and we remove the disturbance only in the direction perpendicular to the boundary. As the transmission is nearly complete, we can still expect nearly complete removal through the boundary. From the experiments we have made, it is estimated that reflection is less than 5%, in most cases it is only one or two percent.

With the introduction of the function F ( t ) for every boundary point, it is not required that the basic model has the same aimension as the model which is used for the radiating condition, only the grid which is used for both models need to be the same. As we abstract only the function F(t) and the water level the basic model can have any type of boundary as long as we can compute F ( t ) . The only requirement would be that the model with the radiation boundary could be driven by water level boundaries alone. With water level boundaries as presently implemented in TRIWAQ, no function of the velocity distribution is prescribed, this distribution is computed from the momentum equation at the point just inside the boundary.

What in essence is being proposed in the above paragraph is nesting van models with the boundary transfer accomplished by the function F ( t ) and the waterlevel used for the damping term.

18

At this stage of the research on radiating boundaries it is not clear how we apply in practical application the adjusted radiative boundary. We can see in the experiments described in chapter IX, that modification of the Function F(t) increases the water level at that boundary. The function contains several water level values on the beginning of the time stap and at the end of the timestap, thus it is not easy to determine before hand the relation of water level rize and increase of the function. At this time we have established at least a tool to work with.

The damping factor is presently introduced as a numerical value in the computation. Thus if we change the timestep, the numerical value need to be changed also. It seems better to introduce the damping as a function of time. In studies of decaying matter, a convenient measure is T 9 0 . This is the time that the decay is 90%, thus the time that the original value is reduced to one-tenth. This would give for the modeller a value which has meaning.

At present the damping function acts on the difference between the water level of the basic flow computation at the beginning of the time step and the value in the radiating model also at the beginning of the time step (Eq. 10.1). It seems to be better to take these values at the end of the timstep.

During the simulation with the basic model, the function F(t) is abstracted. This function makes use of the wave speed C. When we compute the boundary condition for the simulation with the disturbance, the wave speed is computed again and used in the boundary determination. This wave speed is a function of temporal depth and the local velocity. As we add a disturbance in the computation local water levels and velocities change, thus influencing the wave speed. It is possible that we not accurately compute the basic wave conditions, thus it would be more accurate to abstract also the wave speed from the the basic computation. Errors would then only show up in the radiation characteristics of transmission. In our experiments we did not abstract the wave speed and used in the second simulation, we have not noticed any effects of significance.

In order to get more experience with the radiating boundaries, it is now the time to start experimentation in a three-dimensional model. As recommended in the previous Chapter, data extraction and supply should be programmed in TRIWAQ, outside the regular input and output structure of SIMONA. Once all data need is firmly established, implementation can proceed within the formal structure of SIMONA.

XIII. CONCLUSION.

A new method has been developed for the incorporation of radiation boundary conditions in a model. In most instances radiative boundaries are required to radiate disturbances which are generated in the computation field out of the model area. It is proposed to make two simulations. One with existing boundaries in the presently usual practice of modeling. From this simulation time serie are abstracted of a function and a water level. A second simulation is new made with these abstracted time series as inputs and with a Sommerfeld type boundary and which simulation contains the disturbances. In the second simulation the disturbances will run across the boundary as if no boundary is present.

The experiments show that with this new approach the transmission through the boundary is very effective, generally only a few percent of the height of the disturbances is reflected. The method contains a dampening function with a coefficient to be determined by experiments.

Experimentally we found that boundary water levels can be modified during a simulation with the radiation boundary. This needs further investigation.

It is feasible to incorporate radiating boundary conditions in TRIWAQ.

R E F E R E N C E S

1. L i u , Shiao-Kung and J . J . Leendertse, Radiation Boundary Conditions for the RAND Three-Dimensional hydrodynamic Model, Working Draft WD-3011-Neth, April 1986.

5. Sommerfeld, A., Partial Differential Equations in Physics, E . G . Strauss (translator), Academic Press, 1949, 335 pgs.

3. Lander, J . W . M . , P . A . Blokland and J . M . de Kok, The

three-dimensional shallow water model TRII/iQ with a flexible vertical grid definition, Ministry of Transport and

Watermanagement, R I K Z / 9 4 - 0 , November 1994.

4. Stelling, G . S . , On the construction of computational method for shallow water flow problems, Thesis Delft University of Technology, 1983.

5. Blumberg, Alan F . and Lakshmi H . Kantha, Open Boundary Condition for Circulation Models, Journal of Hydraulic Engineering, Vol. I l l , No. 2, February, 1985.

(

S T E P 1 (STARTING CONDITION) 0.3 0.2 0.1 Wave height (m) -0.1 E n d of Implicit s t e p — E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of Implicit s t e p — E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 1 20 40 60 80 100 120 140

Distance (in grid intervals) Fig, 4.1—Starting condition of wave

traveling in a channel. 160 180 200 S T E P 11 Wave height (m) E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 20 40 60 80 100 120 140 Distance (in grid intervals) Fig. 4.2—Wave at timestep 11 traveling in a channel with a radiative boundary at the right.

160 180 200

S T E P 21 Wave height (m) 0.1 0 — 1 1 1 1 E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 1 1 20 40 60 80 100 120 140 160 180 200 Distance (In grid intervals)

Fig. 4.3—Wave at timestep 21 traveling in a channel with a radiative boundary at the right.

S T E P 31 Wave height (m) 0.1 0 1 1 1 I E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 1 1 1 I E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p I 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 4.4—Wave at timestep 31 traveling in a channel with a radiative boundary at the right.

S T E P 41 Wave height (m) 0.3 0.2 0.1 0 l l l l E n d of implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 4.5—Wave at timestep 41 traveling in a channel with a radiative boundary at the right.

S T E P 21

Wave height (m) _{Velocity ( m / s e c )}

0.3 0.2 0.1 •0.1

-

-

Fig. 4.6—Waveheights and velocities at timestep 21.

23

- 0 . 4 160 180 200

S T E P 1 Wave height (m) 0.3 0.1 0 l l l l E n d of I m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of I m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6.1~Periodic wave (at timestep 1) being generated at the right boundary of

the channel. S T E P 11 Wave height (m) 0.3 0.1 0 l i l l E n d of I m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of I m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6.2—Periodic wave (at timestep 11) being generated at the right boundary of

the channel. 24

S T E P 21 E n d ot Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0.3 0.2 0.1 Wave height (m) -0.1 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6,3—Periodic wave (at timestep 21) being generated at the right boundary of

the channel. S T E P 31 Wave height (m) 0.3 0.1 0 l l l l E n d of implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0 20 40 60 80 100 120 140 160 180 200 Distance (In grid intervals)

Fig. 6.4—Periodic wave (at timestep 31) beiijg^generated at the right boundary of

the channel. 25

S T E P 41 Wave height (m) 0.3 0.1 0 l l l l E n d of i m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of i m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6.5—Periodic wave (at timestep 41) being generated at the right boundary of

the channel.

S T E P 21

Wave height (m) Velocity ( m / s e c )

0.3 0.2 0.1 -0.1

Distance (in grid intervals) Fig. 6.6— Waveheights and velocities at timestep 21.

160 180 200

S T E P 1 (STARTING CONDITION) 0.3 0.2 0.1 Wave height (m) D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T • U . l 20 40 60 80 100 120 140 Distance (in grid intervals) Fig. 6.7—Combined effect ( timestep 1)

of outgoing disturbance and incoming periodic wave 160 180 200 S T E P 11 0.3 0.2 0.1 Wave height (m) D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T _ Q . j I . I , _ 1 I I I I , 1 _ I 1 1 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid Intervals)

Fig. 6.8-- Combined effect ( timestep 11) of outgoing disturbance and incoming

periodic wave 27

S T E P 21 Wave height (m) 0.3 0.2 0.1 0 l l l l D I S T U R B A N C E — P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E — P E R I O D I C WAVE C O M B I N E D E F F E C T 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig, 6 . 9 ~ C o m b i n e d effect ( timestep 21) of outgoing disturbance and incoming

periodic wave S T E P 31 Wave height (m) 0.3 0.1 0 I ' l I. 1 D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

F.ig._6.10—Combined effect ( timestep 31) of outgoing disturbance and incoming

periodic wave 28

S T E P 41 Wave height (m) 0.3 0.2 0.1 0 1 1 i . .! D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T 0 20 40 60 80 100 120 140 160 180 2 0 0 Distance (in grid intervals)

Fig, 6.11—Combined effect ( timestep 41) of outgoing disturbance and incoming

periodic wave

(

S T E P 21

Wave height (m) Velocity ( m / s e c )

0.3

0.2

0.1

0

0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6.12—Waveheights and velocities at timestep 21.

S T E P 1 (STARTING CONDITION) Wave height (m) 0.3 0.2 0.1 0 I I I ! D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6.13—Combined effect ( timestep 1) of an outgoing disturbance and an

outgoing periodic wave

S T E P 11 Wave height (m) 0.3 0.2 0.1 0 1 t .1 i D I S T U R B A N C E - P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E - P E R I O D I C WAVE C O M B I N E D E F F E C T

/

Fig. 6.14—Combined effect { timestep 11) of an outgoing disturbance and an

S T E P 21 0.3 0.2 0,1 Wave height (m) -0.1 D I S T U R B A N C E - - P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E - - P E R I O D I C WAVE C O M B I N E D E F F E C T

X f /

of an outgoing disturbance and an outgoing periodic wave

160 180 200 S T E P 31 0.3 0.2 Wave height (m) D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T 0.1 •0.1 20 40 60 80 Distance 100 120 140 (in grid intervals)

160 180 200

Fig. 6.16—Combined effect ( timestep 31 of an outgoing disturbance and an

outgoing periodic wave

S T E P 41 0.3 0.2 0.1 Wave height (m) ! 1 1 i D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T 0.1 20 40 60 80 100 120 140 Distance (in grid intervals) Fig. 6.17--Combined effect (timestep 41)

of an outgoing disturbance and an outgoing periodic wave

160 180 200

S T E P 21

0.4

0.3

Wave height (m) Velocity ( m / s e c )

- 0 . 4 60 80 100 120 140 160 180 200

Distance (in grid intervals) Fig. 6.18--Waveheights and velocities at timestep 21.

S T E P 1 (STARTING CONDITION) Wave height (m) 0.3 l i l l E n d of implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig, 6.19~Periodic wave (timestep 1) being generated at the right boundary of

the channel. S T E P 11 Wave height (m) 0.1 0 1 1 1 1 E n d of Implicit s t e p E n d ot e x p l i c i t s t e p B e g i n n i n g of s t e p 1 1 1 1 E n d of Implicit s t e p E n d ot e x p l i c i t s t e p B e g i n n i n g of s t e p 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig, 6.20—Periodic wave (at timestep 11) being generated at the right boundary of

the channel.

S T E P 21 Wave height (m) 0.3 0.2 0.1 0 l l l l E n d of I m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of I m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6.21—Periodic wave (at timestep 21) being generated at the right boundary of

the channel. S T E P 31 Wave height (m) 0.3 0.2 0.1 0 l l l l E n d of I m p l i c i t s t e p E n d ot e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of I m p l i c i t s t e p E n d ot e x p l i c i t s t e p B e g i n n i n g of s t e p

J

Fig. 6.22—Periodic wave (at timestep 31) being generated at the right boundary of

the channel.

S T E P 41 0.3 0.2 0.1 Wave height (m) -0.1 I i l i E n d of i m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of i m p l i c i t s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 20 40 _{60 80 100 120 140 160 180 200} Distance (in grid intervals)

Fig. 6 . 2 3 ~ P e r l o d i c wave (at timestep 41) being generated at the right boundary of

the channel. S T E P 21 0.4 0.3 0.2 0.1

Wave height (m) Velocity ( m / s e c )

0.1 0.2 V E L O C I T Y W A V E H E I G H T V E L O C I T Y W A V E H E I G H T 20 40 60 80 100 120 140 Distance (in grid Intervals) Fig. 6.24~Waveheights and velocities at timestep 21. 1.6 1.2 0.8 0.4 - 0 . 4 - 0 . 8 160 180 200 3 5

S T E P 1 (STARTING CONDITION) Wave height (m) 0.3 0.1 0 I I I ! D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6.25~Combined effect ( timestep 1) of an outgoing disturbance and an

outgoing periodic wave

S T E P 11 Wave height (m) 0.3 0.1 0 1 1 1 1 D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6.26—Combined effect ( timestep 11) of an outgoing disturbance and an

outgoing periodic wave

S T E P 21 Wave height (m) 0.3 0.1 0 l l l l D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T

/

Fig. 6.27—Combined effect ( timestep 21) of an outgoing disturbance and an

outgoing periodic wave

S T E P 31 Wave height (m) 0.3 0.2 0.1 0 l l l l D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 6.28—Combined effect ( timestep 31) of an outgoing disturbance and an

outgoing periodic wave

S T E P 41 0.3 0.2 0.1 Wave height (m) -0.1 l i l ! D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T 20 40 _{60 80 100 120 140 160 180 2 0 0} Distance (in grid intervals)

Fig. 6.29—Combined effect ( timestep 41) of an outgoing disturbance and an

outgoing periodic wave

S T E P 21

Wave height (m) Velocity ( m / s e c )

- 0 . 4

40

- 0 . 8 60 80 100 120 140 160 180 200

Distance (in grid intervals) Fig. 6.30—Waveheights and velocities at timestep 21.

S T E P 1 (STARTING CONDITION) Wave height (m)

•0.1

20 40 60 80 100 120 140 Distance (in grid intervals) Fig. 8.1~Wave traveling in a channel

with a radiative boundary at right. Half time step • 40 s e c , Chezy • 40

160 180 200 S T E P 11 Wave height (m) 0.1 0 1 i E n d of Implicit s t e p E n d ot e x p l i c i t s t e p • B e g i n n i n g of s t e p 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 8.2~Wave traveling in a channel with a radiative boundary at right. Half time step • 40 s e c , Chezy • 40

S T E P 21 Wave height (m) 0.2 0.1 0 I i i l E n d o( Implicit s t e p E n d o( e x p l i c i t s t e p B e g i n n i n g of s t e p E n d o( Implicit s t e p E n d o( e x p l i c i t s t e p B e g i n n i n g of s t e p 1 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 8.3~Wave traveling in a channel with a radiative boundary at right. Half time step • 40 s e c , Chezy • 40

S T E P 31 0.3 0.2 Wave height (m) E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0.1 •0.1 20 40 60 80 100 120 140 Distance (in grid intervals) Fig. 8.4~Wave traveling in a channel

with a radiative boundary at right. Half time step • 40 s e o , Chezy • 40

4 0

S T E P 41 Wave height (m) 0.2 0.1 0 E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 8.5~Wave traveling in a channel with a radiative boundary at right. Half time step • 40 s e c , Chezy • 40

WAVE HEIGHT VS TIME 0,15 0.1 0.05 0 s t a t i o n 3 9

\ A

\ A

i w W

i 1/

Fig. 8.6 Wave heights caused by adjus radiative boundary at five stations Half time step • 40 s e c , Chezy • 40.

S T E P 21 Wave height (m) 0.3 0.1 0 i l l ' E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p E n d of Implicit s t e p E n d of e x p l i c i t s t e p B e g i n n i n g of s t e p i 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 8.7 Wave traveling in a channel with radiative boundary at right. Half time step • 4 0 s e c , Chezy « 20

- 0 . 0 5 ' ' ' 1 ^ i > 1 1 i 1 0 10 20 30 40 50 60 70 80 90 100

TIME (in timesteps) Fig. 8.8 Wave heights at five stations

in the channel

Half time step • 40 s e c , Chezy • 20.

- 0 . 0 5 ' ' ' ^ 1 1 i ^ i I 0 20 40 60 60 100 120 140 160 180 200

Distance (in grid intervals) Fig. 8.9~Combined effect of outgoing disturbance and incoming periodic wave

with bottomstress (C » 20)

_0.05 ' ' ' ' ^ 1 ' ^ 1 1 1 0 20 40 60 80 100 120 140 160 180 200

Distance (in grid intervals) Fig. 8.10—Combined effect of outgoing disturbance and incoming periodic wave

with bottomstress (C • 20)

S T E P 31 0.15 0.1 0.05 Wave height (m) O R I G I N A L D I S T U R B A N C E - P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E ON WAVE - 0 . 0 5 ' ' ' ' ' 1 ^ ' 1 i 1 0 20 40 60 80 100 120 140 160 180 200

Distance (in grid intervals) Fig. 8.11~Combined effect of outgoing disturbance and incoming periodic wave

with bottomstress (C • 20) S T E P 41 Wave height (m) 0.1 0.05 0 1 1 1 1 1 O R I G I N A L D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E ON WAVE O R I G I N A L D I S T U R B A N C E P E R I O D I C WAVE C O M B I N E D E F F E C T D I S T U R B A N C E ON WAVE 0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 8.12~Combined effect of outgoing disturbance and incoming periodic wave

with bottomstress (C « 20)

S T E P 11

Wave height (m) Velocity ( m / s e c )

1 ! 1

-

Distance (in grid Intervals) Fig. 9.1 —Waveheights and velocities caused by adjusted

radiative boundary (C"20) 180 200 S T E P 21 0.1 0.075 0.05 0.025 0 •0.025 - 0 . 0 5 •0.075 -0.1

Wave height (m) Velocity ( m / s e c ) i \ i

V E L O C I T Y • WAVE H E I G H T

20 40 60 80 100 120 140 Distance (in grid intervals) Fig. 9.2 —Waveheights and velocities c a u s e d by adjusted radiative boundary (C-20) 4 5 0.4 0.3 0.2 0.1 0 -0.1 - 0 . 2 - 0 . 3 - 0 . 4 160 180 200

S T E P 31

Wave height (m) Velocity ( m / s e c )

i 1 i

-

Distance (in grid intervals) Fig. 9.3 —Waveheights and velocities caused by adjusted

radiative boundary (C-20) 180 200 S T E P 41 0.1 0.075 0.05 0.025 0 •0.025 - 0 . 0 5 -0.075 -0.1

Wave height (m) Velocity ( m / s e c ) 1 I 1 V E L O C I T Y • WAVE H E I G H T 0.4 0.3 0.2 0.1 0 -0.1 - 0 . 2 - 0 . 3 - 0 . 4 20 40 60 80 100 120 140

Distance (in grid intervals) Fig. 9.4 —Waveheights and velocities c a u s e d by adjusted

radiative boundary (C-20)

160 180 200

S T E P 51 0.1 0.075 0.05 0.025 O -0.025 - 0 . 0 5 •0.075 -0.1

Wave height (m) _{Velocity ( m / s e c )}

V E L O C I T Y W A V E H E I G H T

0 20 40 60 80 100 120 140 160 180 200 Distance (in grid intervals)

Fig. 9.5 —Waveheights and velocities caused by adjusted

radiative boundary (C-20) 0.4 0.3 0.2 0.1 0 i - 0 . 1 0.2 0.3 0.4 WAVEHEIGHT VS TIME 0.15 0.1 0.05 0 - 0 . 0 5 ! ! ! S t a t i o n 3 9 S t a t i o n 7 9 ' S t a t i o n 119 S t a t i o n 159 S t a t i o n 199 ! ! ! S t a t i o n 3 9 S t a t i o n 7 9 ' S t a t i o n 119 S t a t i o n 159 S t a t i o n 199 / V / V

/ y

Fig. 9.6 Wave heights c a u s e d by adjusted radiative boundary at five stations. Half time step • 40 s e c , Chezy • 20.

WAVEHEIGHT V S TIME 0.15 0.05 O i 1 i s t a t i o n 3 9 S t a t i o n 7 9 S t a t i o n 119 - - - - S t a t i o n 159 S t a t i o n 199 i 1 i s t a t i o n 3 9 S t a t i o n 7 9 S t a t i o n 119 - - - - S t a t i o n 159 S t a t i o n 199 / 0 10 20 30 40 50 60 70 80 90 100 TIME (in timesteps)

Fig. 9.7 Wave heiglits c a u s e d by adjusted radiative boundary at five stations Half time step • 40 s e c , Chezy • 10,

WAVEHEIGHT V S TIME 0.15 0.1 0.05 1 1 i S t a t i o n 3 9 S t a t i o n 7 9 S t a t i o n 119 - - - - S t a t i o n 159 S t a t i o n 199 • - - - ^ 1 1 i S t a t i o n 3 9 S t a t i o n 7 9 S t a t i o n 119 - - - - S t a t i o n 159 S t a t i o n 199 • - - - ^ - /•- / / /

/

77

Fig. 9.8 Wave heights c a u s e d by adjusted radiative boundary at five stations Half time step • 40 s e c , Chezy - 999.

WAVEHEIGHT VS TIME 0.15

0.05

- 0 . 0 5

60 80 100 120 140 TIME (in timesteps)

Fig. 9.9 Wave heigtits caused by adjusted radiative boundary at five stations Half time step • 20 s e c , Ctiezy » 999.

160 180 2 0 0 WAVEHEIGHT VS TIME 0.15 0.1 0.05 •0.05 S t a t i o n 3 9 S t a t i o n 7 9 S t a t i o n 119 S t a t i o n 159 S t a t i o n 199 20 40 60 80 100 120 140 TIME (in timesteps)

160 180 200

Fig.9.10-Wave lieigtits caused by adjusted radiative boundary at five stations Half time step » 20 s e c , Ctiezy • 40.