1 

Evaluation of the volumeexchange model with Van der Meer laboratory tests results
This study extends the theoretical approach developed by JUMELET [2010] to acquire a physical description of the notional permeability coefficient applied in the stability formulae of VAN DER MEER [1988]. Van der Meer introduced this coefficient to ensure that the permeability of the structure is taken into account, however due to the empirical character of Van der Meer equations and, until Jumelet.s research, there was not an available physical description of the notional permeability factor; hence, the determination of this factor was rather vague. Because of the fact that the stability relationship includes the Pcoefficient, it has to be estimated somehow and, therefore, the research carried out by JUMELET [2010] is, to some extent, the starting point to achieve the required physical description of the notional permeability coefficient.
In order to get this physical description, it is introduced the volumeexchange model where the external and internal processes that take place within a breakwater are coupled. The external process is described by a wave runup model while the internal process is described by the .Forchheimer. equation for the water flow through a porous medium. According to JUMELET [2010], the notional permeability parameter P is highly related to the runup reduction coefficient from the volumeexchange model, and so that Jumelet defines an expression for this coefficient by means of coupling the notional permeability factor with the volumeexchange model. Because of the simplicity of the notional permeability coefficient formula developed by JUMELET [2010] further research is required so as to analyze the actual correlation between the notional permeability factor and the socalled runup reduction coefficient (obtained from the volumeexchange model).
This study focuses in developing a general formula for the notional permeability coefficient based on JUMELET [2010] and analyzing the real influence of the hydraulic parameters and structural properties on the Pfactor. As stated by JUMELET [2010], the permeability of the structure not only depends on the structural properties but also on the hydraulic parameters. In this way, a physical description of the notional permeability coefficient is given and is ready to be applied in Van der Meer stability equations to design breakwaters. In addition, the combined method of Jumelet.s model, the generalized formula for notional permeability coefficient and Van der Meer stability equations should be introduced as a tool to determine the maintenance policies in breakwaters by taking into account the damage that waves causes on them.

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2 

The notional permeability of breakwaters: experimental research on the permeability factor P
The stability formula developed by Van der Meer is used for the design of different kind of rock slopes. In the formula is among a number of other parameters also the permeability of the structure represented. A more permeable structure has the ability to dissipate more water and therefore more energy, this results into a lower required weight of the armour layer.
This coefficient, described as the Notional Permeability P, has been determined for three different types of structures. A homogeneous structure, a structure with a permeable core and an impermeable structure. In practice structures are being build who deviate from these standard situations. Therefore there is a demand for values of P about structures other than the known standard situations. In this thesis P values are found by means of physical scale model tests.
First of all two reference structures were tested. The permeable and the impermeable structure with known values of P= 0.5 and respectively P=0.1. The values found in this study are almost equal to the values above.
The new structure has an impermeable core covered with a thick filter layer. On top of that an under layer is placed and finally there is a double armour layer.

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3 

Physical model tests of the notional permeability on breakwaters
Breakwaters are important objects to protect coastal and harbour areas. To minimalize the probability of failure of breakwaters, a lot of research has been conducted concerning the stability of breakwaters. After Iribarren and Hudson, an influential research is conducted by Van der Meer. The literature research of this report will provide more background information concerning their researches on the stability of breakwaters.
Van der Meer tested three sorts of breakwater constructions. The first breakwater structure contained a homogeneous construction (P=0.6) The second and third structure consisted of respectively a construction with impermeable core (P=0.1) and a structure with a filter layer and a permeable core (P=0.5). These variants of breakwaters were constructed with different slopes angles to require as much information possible concerning the stability of breakwaters.
Van der Meer discovered two formulas for the stability of breakwaters. The first formula is used for plunging waves while the second formula is used for surging waves.Within these formulas, important factors as damage, wave height and notional permeability are included. The most important parameter of the formulas of Van der Meer is the notional permeability factor P.
Van der Meer conducted his research on three different constructions and has designed a fourth construction based on the stability curves. This fourth construction has a value of permeability of 0.4. This value is estimated based on curve fitting.
Following the research done by Van der Meer, Kik has subsequently researched the notional permeability of three breakwater constructions. Firstly, Kik repeated the test with a construction of impermeable core (model 1/P=0.08) and the test with the construction of filter layer and permeable core (model 2/ P=0.05) of Van der Meer. Lastly, Kik did a third test existing of a variant of the design of the fourth construction of Van der Meer (model 3 / P=0.35). Concluding from his research, Kik stated that the ‘Root mean square equation’ is a reliable method to determine the notional permeability P.
During this research the influence of the thickness of the filter layer on the notional permeability P is studied. This research will also try to answer the question whether other relevant aspects might influence the notional permeability as well. The elaboration of this research is performed in a practical way in a wave flume in the water laboratory of the faculty of civil engineering of the TU Delft. Scale models of the breakwaters were constructed to test the notional permeability of the breakwaters.
In the water laboratory three models were tested. Firstly, model 3 of Kik is repeated as model 3A, with a calculated value of notional permeability P 0.38. The construction of model 3A is build with a top layer, filter layer 1, filter layer 2 and a impermeable core.
Second, another variant of model 3 of Kik is designed and tested (model 4). However, the measured damage figures were too low and therefore they could not be used to calculate a value for the notional permeability P. The construction of model four is build with a top layer, filter layer 1, filter layer 2 which is thicker as model 3A and an impermeable core.
Finally, model 5 is tested with a calculated value of notional permeability of P 0.45. This model is designed from the fourth construction of Van der Meer. The construction of model 5 is build with a top layer, filter layer 1 and a permeable core with the same material of filter layer 2 of model 3A and model 4.
The results of this research show that the influences of the notional permeability P exists of the ratio of the armour layer thickness and the thickness of the second filter layer. If the layer thicknesses are equal the value for notional permeability P is 0.38, which follows from model 3A. If the second layer has an infinite thickness (permeable core), the value for notional permeability P is 0.45, which follows from model 5.
The value of the notional permeability P of model 5 corresponds to the design calculations of the computer model HADEER. Van der Meer discovered using this computer model that the ratio of dn50a/ dn50f = 5 has a value on the notional permeability P of 0.43 –0.44. During this research, while using two different methods, a value of the notional permeability P of 0.45 was calculated.

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4 

The use of Elastocoast in breakwater research
In this report will be explained how breakwaters can be made with the use of Elastocoast, a sort of glue. This makes it possible to fix the individual rocks, and allows repetitive tests possible with exactly the same layer properties. We made six samples of breakwater rock layers, made with Elastocoast and stones, which can be placed and tested in the wave flume.
For doing tests it is important to know the properties of the breakwater, such as the grain size distribution, the porosity and the permeability.
The permeability and porosity tests were performed on smaller parts than the slabs to be used in the model breakwater. After making the little samples we made the large ones, on the same way, for use in the wave flume. For the testing of the permeability we used a construction in which we could let water flow through the samples.
This report shows the results of our tests, so these results can be used for further purposes, when other people use these breakwater samples.

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5 

Analysis of the notional permeability of rubble mound breakwaters by means of a VOF model
When designing a rubble mound breakwater, one would like to predict the stability of the armour layer of the breakwater. For this purpose several armour layer stability formula are developed. The most reliable is considered to be the stability formula of Van der Meer (1988). With this method, the armour layer stability is predicted based on several parameters. One of these parameters is the socalled notional permeability, P.
The definition of the word notional is important to keep in mind; existing as or based on a suggestion, estimate, or theory; not existing in reality. Or, in other words, the notional permeability is not a physical description of the real permeability of a breakwater. Furthermore, there are no methods to calculate the notional permeability. This makes it difficult to estimate a safe and reliable Pvalue for an arbitrary structure.
The definition of the notional permeability suggests that a single Pvalue can be attributed to a structure. However, previous research has suggested that the notional permeability is not only dependent on structural parameters, but also on hydraulic parameters. This makes the assumption of a single Pvalue for a particular structure under arbitrary hydraulic conditions invalid. The Pvalue should therefore vary under varying hydraulic conditions.
This thesis aims for a better understanding of the physical processes the notional permeability. Some of the original physical model tests of Van der Meer (1988) are selected for analysis to achieve this goal. As a tool the numerical model IH2VOF is used. This 2D vertical model is able to simulate flow interaction with porous media. And makes it possible to measure pressures and flow velocities at any point within a breakwater. In this way physical processes can be described as functions of pressures and flow velocities.
The model is able to simulate porous flows by Volume Averaging the Reynolds Averaged Navier Stokes equations. This Volume Averaging introduces a porosity into the equations. Furthermore, the extended Forchheimer equation is needed to close the equations. This additionally introduces the laminar and turbulent Forchheimer coefficients and added mass coefficients to the equations. These four variables describe the porous media and are required as input to the model.
Based on a literature study a hypothesis is made about the variables on which the notional permeability is dependent. The Buckingham PI theorem is applied to these variables, resulting in four dimensionless PI terms. Eventually it is concluded that the PI terms are best measured 0.5 times the significant wave height below the initial water level, inside the armour layer and in a normal direction to the front slope of the structure. With this approach, all four PI terms show a positive correlation with P.

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6 

Notional Permeability
Different layer design of a rock slope and under layers has a large effect on the strengths on the rock slope itself. In the stability formula developed of VAN DER MEER [1988] this effect is represented by the term Notional Permeability with symbol P. A more open, or permeable, structure underneath the armour layer has the ability to dissipate more wave energy and therefore requires less weight of the armour layer. The influence of this parameter is thus very important in economic sense. Up until now only three configurations have been tested. In practice often intermediate structures were designed which do not correspond to the standard situations. Pvalues then have to be estimated in comparison with the known structures, which gives some uncertainty around the Pvalue. Therefore there is the demand for more validated values of the notional permeability representing other structures. During this study physical scale modelling is used to produce a value of P for a new structure.

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7 

Evaluation of the VolumeExchange Model using Van der Meer Laboratory Tests Results
This study extends the theoretical approach developed by JUMELET [2010] to acquire a physical description of the notional permeability coefficient applied in the VAN DER MEER stability formulae [1988]. Van der Meer introduced this coefficient to ensure that the permeability of the structure is taken into account, however due to the empirical character of Van der Meer equations and because prior to Jumelet's research there was not an available physical description of the notional permeability factor, the determination of this factor was rather vague. Because of the fact that the stability relationship includes the Pcoefficient, it has to be estimated somehow and, therefore, the research carried out by JUMELET [2010] is, to some extent, the starting point to achieve the required physical description of the notional permeability coefficient.
To obtain this physical description, the volumeexchange model is introduced, in which the external and internal processes that take place within a breakwater are coupled. The external process is described by a wave runup model while the internal process is described by the „Forchheimer‟ equation for the water flow through a porous medium. According to JUMELET [2010], the notional permeability parameter P is highly related to the runup reduction coefficient from the volumeexchange model, and thus Jumelet defines an expression for this coefficient by means of coupling the notional permeability factor with the volumeexchange model. Because of the simplicity of the notional permeability coefficient formula developed by JUMELET [2010], further research is required to analyze the actual correlation between the notional permeability factor and the socalled runup reduction coefficient (obtained from the volumeexchange model).
This study focuses on developing a general formula for the notional permeability coefficient based on JUMELET [2010] and analyzing the real influence of the hydraulic parameters and structural properties on the Pfactor. As stated by JUMELET [2010], the permeability of the structure depends not only on the structural properties but also on the hydraulic parameters. In this way, a physical description of the notional permeability coefficient is given and can be applied in Van der Meer stability equations to design breakwaters. Moreover, a damage level analysis has been performed to compare the observed damage by VAN DER MEER [1988] with the estimated damage through the combined method of Jumelet's model, the generalized formula for the notional permeability coefficient and Van der Meer stability equations, which leads to introducing the combined method as a tool to determine the maintenance policies in breakwaters by taking into account the damage that waves causes on them.

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8 

The influence of core permeability on armour layer stability
This study describes a theoretical approach of a physical description of the notional permeability factor in the stability formulae of Van der Meer [1988]. Caused by the empirical character of these stability formulae a physical description is not available for the notional permeability factor. In practice this leads to ambiguities in determining the value of this factor. To give this factor a physical description a volumeexchangemodel was introduced to express the effect of core permeability on the external wave runup process.

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9 

The influence of armour layer and core permeability on the wave runup
In JUMELET [2010] a method with a physical basis (so called “Volume Exchange model”) to determine the ‘notional’ permeability coefficient P was developed. The ‘notional’ permeability coefficient was previously introduced in the stability formula of the armour layer; see VAN DER MEER [1988]. In this latter study this coefficient was empirically based for three different structures. Due to the limited validity it is difficult to apply a coefficient for different breakwater configurations. The Volume Exchange model determines the influence of the core permeability by computing the difference between the surface wave runup on an impermeable core and a permeable core. The volume of water that flows into the core causes a reduction of the wave runup. Reduction of the wave runup is not only caused by infiltration, but also by the slope surface roughness and energy dissipation inside the pores of the armour layer.
To investigate the influence of the above three mentioned factors physical model tests have been conducted. The tests were carried out in the wave flume in the water laboratory at Delft University of Technology. On four different configurations (smooth impermeable slopes, rough impermeable slope, armour layer on an impermeable core and permeable core) tests were conducted.
In the analysis of the results the influence of the surface roughness, energy dissipation in the pores of the armour layer and the reduction of the surface wave runup due to the inflow into the core could be determined. Besides, the surface wave runup also the wave runup on the core is measured. The results showed that the slope surface roughness has no influence on the wave runup, when the waves are of the surging breaker type. Also, the surface wave runup is not reduced by a permeable core. Wave runup measurements showed the same wave runup height for armour layers on an impermeable and a permeable core. Wave runup on the core showed a considerable difference between runup on an impermeable core and a permeable core. Therefore, in the Volume Exchange model the wave runup on the core should be considered. The adjusted Volume Exchange model is used to determine a formula for the permeability coefficient. This has led to the conclusion that the permeability coefficient is dependent on the Iribarren number and the structural configurations and /or properties.

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10 

Numerical and experimental research of wave interaction with a porous breakwater
The design formula for rubble mound breakwaters by Van der Meer has an unclear Notional Permeability term. This term causes a lot of confusion for designers. In the past many people have tried to derive a better formulation for that term by experimental and analytical research. The goal of this study was to obtain a better formulation along a numerical way. This study explores the numerical possibilities and tries to define which direction has to be taken in future research.
As a first step, a very simplified case is taken with a vertical homogeneous breakwater which interact with monochromatic waves. In total six different blocks were made of epoxy and elastocoast. Only 4 out of the 6 blocks were tested. Also the porosity (n), laminar friction (α) and turbulent friction constant (β) of the blocks were determined experimentally. This way the experimental results could be compared with computations.
These experiments have been done in the large flume of the Environmental Fluid Mechanics Laboratory of the TU Delft. Two types of data were collected: pore pressures and water levels in front and behind the block. The water levels seemed to be the most reliable data. The main deficit of the setup was the wave absorber at the end of the flume. The wave absorber is not able to sufficiently absorb long waves. So the dataset had to be corrected for that effect. The created dataset was in line with results from earlier experiments.
Results were compared with an analytical solution and the numerical SWASH model. Comparisons with the analytical solution showed a reasonable fit without any calibration. The SWASH model showed in first instance large deviations using the same dataset. By calibrating the turbulent flow resistance β, it was possible to generate a decent fit. However, the used β constants are 610 times higher than the measured β constants. This is physically unrealistic high. Therefore the most likely explanation is an error in the transition between the water and the porous medium. During the experiment discontinuities can occur on this transition while SWASH uses an continuity requirement.
Numerical tests were performed on some multilayered combinations of the different blocks in order to derive a "Vertical P" value in a similar way as Van der Meer determined his P=0.4 structure. The results showed, nevertheless, quite some different patterns as the computations done by Van der Meer. However, taking into account all the problems with calibrating the SWASH model the results for the notional permeability seemed very promising. This numerical method shows the possibility of numerically calculating a notional permeability and should be investigated further in the future.

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