A theoretical approach for the notional permeability factor P PIANC Article Ir. H.D. JUMELET De Vries & van de Wiel Kust‐ en Oeverwerken, Schiedam, The Netherlands ABSTRACT This article describes a theoretical approach for a physical description of the notional permeability factor P in the stability formu‐
las 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 determination of the value of this factor. To give this factor a physical description a volume exchange model was introduced to express the effect of core permeability on the external wave run‐up process. This volume exchange model couples the external with the internal process. The external process is described by a wave run‐up model. In this model the wave run‐up wedge approach of HUGHES [2004] is linked to the wave kinematics in front of the structure. The internal process is described by the ‘Forchheimer’ equation for the water flow through a porous medium. In this study it is stated that the notional permeability factor P is highly related to this volume exchange model. With this correlation it is possible to choose a value of the notional permeability factor that is based on a physical description. Besides the volume exchange model shows that the P‐factor is not only related to the structural parameters, as stated by Van der Meer [1988], but also on the hydraulic parameters. Key words: notional permeability, run‐up reduction coefficient and volume exchange model. types. For that reason, the notional permeability is only de‐
fined for three structure types, a fourth (P=0.4) has been as‐
sumed. The notional permeability coefficient has no physical meaning, but was introduced to ensure that the permeability of the structure is taken into account. 1. INTRODUCTION The interaction of a wave with rubble mound breakwater results in a complex flow, which is both non‐linear and turbu‐
lent, this results in a complex process. It is in present time not possible to give an accurate description of the wave‐structure interaction during this complex process of wave dissipation. Therefore, the design of such structures is often based on em‐
pirical relationships, scale tests in research laboratories and a synthesis of knowledge from different disciplines. The most important part of the breakwater design is the prediction of the rock‐size of the armour units (Dn50), to with‐
stand the wave attack. By reason of the wide applicability the stability formulae of VAN DER MEER [1988] are often used as the design formulae. These design formulae can be described as follows: .
.
∆
.
√
Figure 1.1 ‐ Notional permeability P according to VAN DER MEER [1988] for plunging waves (1.1) .
.
∆
√
√
Caused by the absence of a physical description it is not possible to determine the notional permeability factor for dif‐
ferent types of structure. In practice this leads to ambiguities in determination of the value of this factor, this results in a over‐
designed size of the Dn50. Figure 1.2 shows the influence of this factor on the Dn50 of the armour layer. for surging waves (1.2) Besides the significant wave height ( ), damage level ( ), relative mass density (Δ), Iribarren number ( ), number of waves (N) and the slope steepness (α), the stability is strongly related to the notional permeability factor (P). The value of this factor is based on curve fitting results of the test program of VAN DER MEER [1988]. This test program includes three structure 1 internal water movement. The internal water movement, in this study, is defined as the water movement that takes place in the construction (filter and core) for an imposed external water movement. In the next two paragraphs these two processes will be discussed. At first both processes are consi‐
dered to be independent of each other, meaning an imperme‐
able structure slope has been assumed. The last paragraph of this chapter describes some methods to couple both processes. Moreover, a model has been chosen which will elaborated further. This choice depends mainly on whether the model can be developed analytically. 2.2 External process The external process can be described in several ways. The first way is a velocity description of the wave on the structure slope. Analytical approaches are the method of HUGHES ET AL. [1995] and the energy balance by VAN DER MEER [1990]. These approaches are very conservative and therefore not useful. In the last decades a lot of numerical velocity approaches have been introduced. These models are based on the shallow water equations, Boussinesq equations or the Navier Stokes equa‐
tions. Disadvantage of these models is the time it takes time to develop a useful and accurate operational model. Figure 1.2 ‐ Example of the influence of notional permeability factor P on the Dn50 for plunging and surging waves By means of these considerations it is clear that the influence of the core permeability on armour layer stability is of both academic and practical importance. The aim of this study is to investigate whether more physical background can be given to the influence of core permeability on the armour layer stability. A more physical background in this field can result in a more precise choice of the notional permeability coefficient P in the calculation of the breakwater design. Since the permeability of the structure primarily affects the physical process in the case when the wave does not break, this study only considered surging waves. 2. BACKGROUND LITERATURE Another way to describe the external process is a description of the water elevation. This description can be used as a vo‐
lume change approach instead of the velocity approach men‐
tioned above. An analytical method based on the solution of the classical shallow water equations is given in CARRIER AND GREENSPAN [1958]. A non‐linear solution for the classical shal‐
low water equations, which describes the wave characteristics on a slope, is obtained by LI [2000]. Another external wedge approach is introduced by HUGHES [2004]. This method assumes a triangular wave run‐up and the run‐up area can be calculated with the following equation: 2.1 General In the past VAN DER MEER [1988] has tried to give the notional permeability coefficient a physical background using the model HADEER. This model can calculate the discharge dissipation in the core. With the use of the ODIFLOCS‐model of VAN GENT [1995] the same principle predictions are also tried by DE HEIJ [2001]. He considered the discharge, extreme velocities and U % . However, these methods are not very accurate. This study has the aim to develop a practical application for determining the notional permeability. Therefore not only numerical approaches are considered, but also analytical op‐
tions have been investigated. For developing a notional per‐
meability approach it is necessary to have insight in the wave‐structure interaction process. With a description of this process it is possible to determine the influence of core per‐
meability on the external process. Area ABD
·
R
α
·
α
1 (2.1) The only unknown variable in this method is the angle θ.This variable is the angle between still water level and run‐up water surface (which is assumed to be a straight line). Earlier studies have shown that a detailed description of the full wave‐structure interaction is complex. For that reason, the full wave‐structure interaction is divided into two separate processes. The first process can be described as the external water flow in and on the armour layer that is influenced by the presence of the structure. The other process is defined as the 2 The Forchheimer equation (Eq. 2.3) is valid for a stationary flow. However, the porous flow in the core of a rubble break‐
water is usually not permanent, as a result of the dynamic wave effect. The water flow speeds up and slows down in al‐
ternating directions within a full wave period. In the case of non‐continuous flow in a grainy porous medium the inertia must be taken into account. POLUBARINOVA‐KOCHINA [1952] (from VAN GENT [1995]) added therefore a time‐dependent term. This type of formula for unsteady porous flow is referred to as the extended Forchheimer equation: Figure 2.1 – Triangular wedge approach based on the theory of HUGHES [2004] The last way to describe the external process is by using an energy consideration. This can be done by separating the energy of the incoming wave in case of a wave‐structure inte‐
raction in two different components: (1) dissipation on the slope, and (2) the residual wave energy (reflective wave). The total energy of a wave can be expressed as sum of the kinetic energy density E and the potential energy density E : I
E
E
E
ρg
Inertia term: (2.3) VAN GENT [1995]: “The first term can be seen as the laminar contribution and the second term can be seen as the contribu‐
tion of turbulence. For turbulent porous media flow, and in the transition between laminar and turbulent flow, this equation can be used.” The friction coefficients (s/m) and (s2/m2) are dimensional and contain several parameters. Turbulent term: (2.7) With this consideration only a volume exchange approach, by coupling the internal and external water elevation, remains. The analytical approach of HUGHES [2004] can be used as an external water description. With the use of a similar internal water level fluctuations model, the internal and external wa‐
terline can be coupled. The main problem is that the method of An analytical way to couple the external and internal velocity approach is in present time not available. In order to link the external and internal velocity approaches, a numerical model is needed. Considering the practical design formulae by VAN DER MEER [1988] a numerical modification is not the ideal solution. Also a few recently released articles show the work intensive‐
ness and their disadvantages. Therefore in this research is tried to find an analytical way. For review of numerical modeling before the year 1992 is referred to the paper of HALL AND HET‐
TIARACHCHI [1992]. The latest review is given by LIN [2008]. In this book a reference is made to numerous widely used pack‐
ages. υ
c
D
(2.6) According to the results in study the of VAN GENT [1995], the influence of the turbulent term is the highest and is in the or‐
der of 90%. 2.4 Coupling of the internal and external process The possibilities to describe the internal process accurately are limited; therefore the way of coupling is determined by the internal process. The previous sections clearly indicate that a link between the two processes can be realized in two ways. The first way is to link an internal and external velocity ap‐
proach. The second way is to couple the internal and external water elevation by using a volume exchange principle. With help of a reflective wave approach the energy dissipation on the slope can be calculated. 2.3 Internal process Due to the complexity of the internal process it is difficult to give an accurate description of this process. A straightforward way is to describe the water flow through a porous medium. It has been assumed that the flow in the structures, studied in this research, is a ‘fully’ turbulent or a ‘Forchheimer’ flow. Therefore, Darcy is not valid. The gradient (or the reistance over covered length) of this water flow through a porous me‐
dium of coarse granular material can be reasonable well ex‐
pressed by a term that is linear with the flow velocity a · u and a term that is quadratic with the flow velocity b · |u|u . Where and are both dimensional coefficients. Such a relation was proposed by FORCHHEIMER [1901]. α
Laminar term: c
(2.2) | | bu|u|
Where is the inertia term (m2/s) which take the added mass into account. The expression for the inertia term is: H
au
(2.4) (2.5) 3 d and R can be calculated, which leads to the following expressions: HUGHES [2004] still contains an unknown variable (angle θ). The next chapter deals with this problem and gives a derivation of the coupling between the external and internal water level fluctuations. 3. ANALYTICAL MODEL 3.1 Deriving a run‐up volume approach This model used the principle of a triangular wave run‐up shape introduced by HUGHES [2004]. As mentioned before in the theory of HUGHES [2004] an unknown variable (angle θ between still water level and run‐up water surface) is included, thus this description is not directly applicable. A more com‐
plete description is achieved by linking the run‐up wedge ap‐
proach of HUGHES [2004] to the wave kinematics in front of the structure. The basic principle behind the ‘new’ run‐up volume description is that the incoming ‘sinus’ wave crest has the same volume as the run‐up volume. If the incoming wave is assumed as a sine wave, the crest volume (half of the wave length) can then be described as follows (see Figure 3.1): R
(3.6) Figure 3.1 ‐ Schematization of the wave run‐up model for a frictionless slope. With the use of these equations the run‐up volume (eq. 3.2) can now be expressed as: VR
·
L0
·R π2
(3.2) According to this theory the run‐up for non‐breaking waves is only related to the wave height. This theory is confirmed by the following quote of HUGHES [2004]: “run‐up data for non‐breaking waves that surge up steeper slopes does not correlate as well to the Iribarren number, and instead run‐up appears in this case to be directly related to wave height.” (3.1) With a triangular wedge approach the total wave run‐up vo‐
lume can be expressed as: (3.2) In this approach is assumed that no energy losses due to bottom friction (foreshore and structure slope) and viscous effects on the free surface will occur. With this assumption it can be stated that the energy of the incoming wave crest is equal to the total energy of the run‐up volume. In the previous chapter is given that the total wave energy density of an inci‐
dent wave can be expressed as sum of the kinetic energy den‐
sity E and the potential energy density E . For a half wave‐
length, the expression of the total energy is: Reliability test The assumption that the run‐up volume has a triangular shape will lead to an underestimation of the actual run‐up in case of non‐breaking waves because non‐breaking waves will have more concave shaped sea surface elevation (see also HUGHES [2004]). However, it is expected that the simplifications used in the ‘new’ run‐up wedge approach are not very unrea‐
listic. By testing the reliability an indication of the inaccuracy can be given. This is done by comparing this approach with the run‐up approach described in the CUR/CIRIA [2007] However, it should be remembered that also this approach is based on a simplification of the reality. This approach can be expressed as: ρgH
·L (3.3) When the wave reaches the maximum run‐up position, the potential energy reaches a maximum. The kinetic energy at this position is very small. This small amount of energy may be associated with the mild reflected wave from the slope or the small and negligible water particle velocity associated with the run‐up tongue. This small amount of kinetic energy will be neglected here. With these assumptions the energy of the wave run‐up triangle is: R
H
E
ρgR d %
Aξ
B (3.7) For this reliability test the coefficients of ALLSOP ET AL. [1985], described in the CUR/CIRIA [2007], are used (which do not include safety margins): A
0.21 and B 3.39. In this example the Iribarren breaker parameter is between 3.5 and 6 and the significante wave height H is 5. This leads to the following results: E
(3.5) E
L
d
πH (3.4) Using the equations and assumption from above the unknown 4 Run‐up according to ALLSOP ET AL. [1985]: R
13.3
10.7
Run‐up according to the ‘new’ run‐up wedge approach: 11.7 m Without drawing conclusions, this comparison does show that the run‐up using the ‘new’ run‐up wedge approach gives a realistic value. It can be concluded that this external volume approach is a realistic basis for the exchange volume model, described in the next section. V (3.8) With the assumption that the internal water volume has also a triangular shape, the total internal water volume for a smooth vertical transition can be calculated with the following formula (see figure 3.2), where n is the porosity: n · 0.5 · R
·
I
VR
,
VR
L0
·R , π2
(3.11) V (3.12) ,
3.3 Internal water level gradient for a vertical transition In this study the internal water volume is determined, for a given maximum wave run‐up, on the basis of a previously es‐
timated internal water gradient. With a previously estimated gradient the (internal) velocity in a porous medium can be determined. Initially, only the turbulent term (b · |u|u) is taken into account. The turbulent friction term can be expressed with Eq. 2.5. As already mentioned the influence of the turbu‐
lent term is in the order of 90%. It is also assumed that the porosity has one value only (core and filter layer(s)). In a later case these two assumptions can still be refined. The volume inflow (in the same period) should be equal to the water vo‐
lume that the gradient 'prescribes'. If this is not the case, the gradient should be modified until both volumes are the same (iteration). V
·
VR
,
The only unknown variable in the exchange volume method (Eq. 3.12) is the internal water level gradient (I) required for calcu‐
lating the body volume V . A determination of this gradient will be described in the next section. 3.2 Deriving the volume exchange model In this study the permeability of the structure is expressed as a reduction of the external wave run‐up volume. This reduction is caused by the inflowing water volume V that prevails dur‐
ing the maximum wave run‐up. The volumes of water in and on the structure can be divided into three volumes: the run‐up volume VR , the reduced run‐up volume VR , and the vo‐
lume in the body of the structure V . In formula form this is: ,
VR
VR
(3.10) Also for determination of the run‐up volume (3.2) and the reduced run‐up volume (3.8) this friction factor should be in‐
cluded. This leads to the following equations: R
γ ·R ,
(3.9) When only the turbulent term is taken into account, the flow velocity in the porous medium can be expressed as the relation between the water level gradient and turbulent term: u
I
(3.13) For determining the volume inflow a sinusoidal wave run‐up is considered with a maximum wave run‐up height in a time span of one quarter of the wave period (
0,25 . This time span is based on test results of MUTTRAY [2001]. The pe‐
riod represents the up‐rush time of the wave run‐up from SWL till the maximum wave run‐up height. The volume inflow for a rough sloped transition can now be expressed as: Figure 3.2 – Schematization of the volume inflow for a homogeneous structure Initially, the slope roughness is not included. This term can easily be taken into account, by multiplying the run‐up R with a roughness reduction factor. In this study this friction factor has an assumed value of 0.75; this value is based on VAN DER MEER [2002]. Further research should be done to be able to express the porosity and roughness separately. The reduced run‐up due to slope roughness can be described as follows:
V
⁄ ·
⁄ ·
,
,
· 1
·
·
(3.14) To determine the occurring water level gradient (I), the volume inflow should be equal to the water volume that the gradient 5 prescribed. Therefore, the following equation must be satis‐
fied. ⁄ ·
,
· 1
·
n
,
I
(3.15) 3.3 Internal water level gradient for a sloped and mul‐
ti‐layered transition For a sloped transition, the volume inflow remains in princip‐
al the same. However, for a sloped transition the waterline hangs under the sloped transition, wherefore the water inflow is limited. In such case it is difficult to determine the water inflow, because the water flow is not exactly horizontal (land‐
wards) but mainly downwards. VAN GENT [1994]: “The down‐
ward vertical velocity of the phreatic surface has a maximum. This is the result of the equilibrium of gravity and friction. If this maximum would be exceeded, the gradient in the pressures would be larger than one. This means that the water would flow quicker than the free seepage velocity which is not possi‐
ble”. The maximum gradient for the vertical velocity is one. Figure 3.3 ‐ Schematization of the volume inflow for a double layered sloped structure Note: The body volume consists only of the water volume in filter and core, because the notional permeability coefficient reflects the influence of the core (and filter) permeability and not the permeability of the complete structure. 3.4 Determination of the run‐up reduction factor With an iteration of the internal water level gradient it is possible to calculate the total internal water volume for an imposed wave run‐up. It is assumed that the wave run‐up tri‐
angle for permeable structure slopes is proportional to the wave run‐up triangle for impermeable slopes. The reduced run‐up is then the volume ratio of the two run‐up wedges mul‐
tiplied by the run‐up for a rough impermeable slope:
A non‐horizontal inflow means that the flow velocities can‐
not be calculated on the same principle as for a vertical transi‐
tion (Eq. 3.15): the water level gradients should be limited to a maximum value corresponding to the geometric properties of the structure (slope angle α, porosity and Dn50). If the gradient from the water level iteration surpasses this maximum prede‐
fined gradient value, then this maximum value should be used in the calculation of the volume inflow. The gradient iteration for a rough sloped transition can be expressed similarly as for a vertical transition but with a different distance: R
,
· 1
·
n
I
,
VR , V
VR ,
,
(3.17) However, this reduced volume VR , will be smaller in reality, because the incoming volume is determined for a run‐up height with only a reduction for the slope friction. In reality, inflow will only take place over a run‐up height that is reduced by friction and permeability. By a new iteration of the water level gradient (Eq. 3.15) using the corrected R , in‐
stead of the original R , (Eq. 3.16), the inflowing volume is determined for a reduced run‐up with slope friction and per‐
meability included. ⁄ ·
,
(3.16) For a homogeneous structure it is assumed that the above method works well. However, in case of a layered structure, this method is no longer applicable as the water levels in the different layers will be subject to phase differences. By assum‐
ing the predefined maximum gradient is the same as the max‐
imum gradient for vertical velocity 1 , the inflowing body volume for a layered structure can be determined. The im‐
posed run‐up in the core is assumed to be 50% of the maxi‐
mum external run‐up (
0,5). This assumption is based on MUTTRAY [2001], in which it is assumed that the seaward gra‐
dient is not affected by the landward gradient. For an illustra‐
tion of this volume exchange model see figure 4.3. By introducing the so‐called run‐up reduction factor the influence of the structure permeability on the external run‐up process is given. This reduction factor is the relation between the run‐up according to the ‘new’ run‐up wedge approach and the reduced run‐up according to the exchange volume ap‐
proach. The run‐up reduction coefficient can be expressed as: c
6 R ,
R ,
(3.18) sumption is made for the grading values. Since this structure type is a typical practical example, were the core is mostly built of quarry run, a large grading is selected. This results in a low porosity. Table 1 shows an overview of the concerning values. 4. DETERMINING THE INFLUENCE OF THE CORE PERMEABILITY WITH THE EXCHANGE VOLUME MODEL 4.1 General With the exchange volume model a description of the influ‐
ence of the permeability on the run‐up process can be given. This run‐up process is strongly correlated to the destabilization process of the armour units. As already mentioned this stabili‐
zation process is empirical expressed in the stability formulae of VAN DER MEER [1988]. With the correlation it is possible to use the volume exchange model for a determination of the notional permeability factor. It should be noted that the no‐
tional permeability coefficient does not describe the permea‐
bility of the structure, but expresses the correlation of the permeability of the structure with the stability of the armour layer units. By assuming a fixed shape factor of β = 3.6 and the above values it is possible to determine for each individual layer the turbulent friction term b by using Eq. 2.5. Using an iteration of the internal water level gradient the volume inflow and thus the run‐up reduction coefficient for the four structure types can be calculated. First, this is done for a vertical transition. Note: In reality, the turbulent friction term is not fixed and should in principle be treated as instantaneous values, see VAN GENT [1993]. Example Calculation The run‐up volume Vru can be calculated with Equation 3.2. The internal water level gradients for each layer can be iterated with Equation 3.15. The water level gradient depends on the imposed water level and geometric properties of each layer (porosity and Dn50). With the use of the calculated water level gradients the body volume can be calculated and therewith the reduced run‐up volume (Eq. 3.12). With the reduced run‐up volume the reduced run‐up and run‐up reduction coefficient can be calculated (resp. Eq. 3.16 and Eq. 3.23). With these equations and the geometric properties in Table 1 the volume exchange model can be applied for the defined ‘notional permeability structures’. This is done by calculating for a given wave steepness s the run‐up reduction coefficient for each structure type. A link between the exchange volume model and the notional permeability coefficient is achieved by determining for all four notional permeability factors the run‐up reduction factor. Sen‐
sitivities in this provision are the value of the turbulent friction term (b) and thus the porosity (n), Dn50 and the shape factor of the granular medium (β) for determining the internal water level gradient. To couple the exchange volume model with the notional permeability coefficient P correctly it is important to use the normative variables on basis of the tested variables by VAN DER MEER [1988]. The ratio between the layers for different struc‐
tures are given in Figure 1.1, in which the notional permeability is defined. ⁄
Notional permeability P=0.6 1.25 P=0.5 1.25 P=0.4 1.25 P=0.1 1.25 0.4 0.4 0.4 0.4 Armour layer Porosity n 2 4.5 Filyer ⁄
2.25 2.25 layer Porosity n 0.38 0.38 ⁄
⁄
Core 3.2 8 1.5 4 Porosity n 0.4 0.3 ⁄
2.91 2.91
2.91
2.91 2.00
3.00
4.00 5.00 Steepness 1.30% 1.10%
0.95%
0.83% 0.65%
0.98%
1.31% 1.64% 12 13
14
15 14
14
14 14 P=0.6 0.967 0.970 0.972 0.975 0.976 0.972 0.969 0.966 P=0.5 0.977 0.979 0.981 0.983 0.984 0.981 0.979 0.977 P=0.4 0.984 0.986
0.987
0.988 0.988
0.987
0.986 0.985 P=0.1 0.997 0.998
0.998
0.998 0.998
0.998
0.998 0.998 Table 4.1 – Values of Table 1 ‐ Filter laws, grading values and porosity values of the four defined layer composition by VAN DER MEER [1988] Note: The porosity values (blue) are based on values of prototype breakwater in Zeebrugge (reference: TROCH [2000], the grading values (red) are assumed values and the other values (black) are taken from the tested structures by VAN DER MEER [1988] for values of P and various waves conditions [JUMELET 2010] Regression analyses (VERHAGEN ET AL. [2011]) shows the follow‐
ing relationship: 0.72
The porosity of the concerned layers is based on a prototype breakwater built in Zeebrugge, reference TROCH [2000]. The grading values for the tested structures are given in the VAN DER MEER [1988]. For the non‐tested structure type (P=0.4) an as‐
1
.
2.5
.
(4.1) Note: The body volume consists only of the water volume in filter and core, because the notional permeability coefficient reflects the influence of the core (and filter) permeability and 7 not the permeability of the complete structure. To calculated , equation (3.11) can be used, using , instead of , . The imposed run‐up at the breakwater core can be calculated with the following equation: In VILAPLANA [2010] is shown that a fit factor β of 4.2 in the friction coefficient b, see equation 2.5, lead on average to a good relation between the suggested values of VAN DER MEER [1988]. See figure 4.2 ·
,
, (5.5) On base of these improvements of the VE‐model, VERHAGEN ET AL [2011] introduces by using regression analysis the following relationship: 0.5
,
(5.6) 6. CONCLUSION From this research follows that the volume exchange model can be used to provide a physical background of the notional permeability coefficient, contributing to a more accurate value determination of this coefficient. By using equation (5.6) the notional permeability factor can be calculated for structures that are not defined by VAN DER MEER [1988]. Figure 4.1 – Computed values of P with eq. 9 compared with the suggested value of P for this condition by Van der Meer (P=0.5) for a variation coefficient β = 4.2. 5. FURTHER RESEARCH After the introduction of the volume exchange model by JUME‐
LET [2010], further research is done to improve the model. Weak points in the VE‐model of JUMELET [2010] are the as‐
The actual used values for the existing notional permeability coefficient P are questionable and therefore in this study it is pretended that the exchange model currently can be used as a physical background of the notional permeability coefficient, but eventually must lead to a separate permeability descrip‐
tion. sumptions regarding the value of γf and γRu. Therefore further research has been carried out in order to separate the influ‐
ence of friction and infiltration on the total run‐up on a break‐
water. Furthermore, the description of the notional permeability values (Figure 1.1) implies that the permeability of the struc‐
ture depends only on the geometric properties of the break‐
water. This study shows that the permeability of the structure also depends on the hydraulic parameters (Hs and T0). This explains also the dual permeability notation in the stability formula of VAN DER MEER [1988] for surging waves. From the test results in VAN BROEKHOVEN [2011] followed that the reduction factor γRu was a function of the Iribarren number: 1.0 tanh 0.31 (5.1) The tests also showed that the infiltration period is somewhat shorter than assumed in eq. (3.14). The experiments showed that: REFERENCE CARRIER, G.F. AND GREENSPAN, H.P. [1958] Water waves on finite amplitude on a sloping beach. Journal of Fluid Mechanics, Vol.4, 97‐109, Pierre Hall, Harvard University CIRIA, CUR, CETMEF [2007] The rock manual, The use of rock in hydraulic engineering, C683 CIRIA, London HALL K.R. AND HETTIARACHCHI S. [1992] Mathematical modeling of interaction with rubble mound breakwaters, Coastal struc‐
tures and breakwaters. Thomas Telford, London, pp.123‐147 HEIJ, J. DE [2001] The influence of structural permeability on armour stability of rubble mound breakwaters, MSc‐thesis, Delft University of Technology, Delft, The Netherlands HUGHES, S.A. AND FOWLER, J.E. [1995] Estimating wave‐induced kinematics at sloping structures, J. of Waterway, Port, Coast‐
al and Ocean Engineering, ASCE, vol. 121, no. 4, p. 209‐215 Li, Y., [2000] Tsunamis: non‐breaking and breaking solitary wave run‐up, Q Report No. KH‐R‐60,W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of γinf ≈0.15. In the test results of VAN BROEKHOVEN [2011] also a difference was observed between the calculated run‐up at the core and the observed run‐up at the core. Therefore an empirical cor‐
rection factor was introduced: 1.0 tanh 0.31 (5.2) The reduced core run‐up in the adjusted volume exchange model is now given by: ,
,
,
,
(5.3) The run‐up reduction coefficient becomes: ,
,
(5.4) 8 Dimensionless friction coefficients in the Forch‐ heimer equation [m2/s] Run‐up reduction coefficient [‐] , /
, Run‐up reduction coefficient /
[‐] ,
,
d Base of the run‐up triangle [m] Energy of the incoming wave [J] Kinetic energy [J] Potential energy [J] Gravitational acceleration [m/s2] Significant wave height, HS H / [m] Hydraulic gradient [‐] Wave number, k 2π/L [m‐1] Deep water wave length, L
gT /2π [m] Number of waves in the Van der Meer formula [‐] Notional permeability factor [‐] Run‐up level, relative to SWL [m] Imposed run‐up at the breakwater core, relative to ,
SWL [m] run‐up, run‐up level exceeded by only 2% of %
run‐up tongues [m] 1. Slope(gradient), 1/
[‐] 2. Wave steepness, s H/L [‐] Damage level in the Van der Meer formula [‐] Wave period [s] % Average of the highest or lowest 2% of velocities on the slope [s] Volume that flows into the structure [m3] Volume of the run‐down [m3] Volume of the run‐up [m3] Weight of water per unit crest width in area ABC [] 1. Angle of slope of breakwater [°] 2. Shape factor in the friction coefficient a in the Forchheimer equation [‐] Shape factor in the friction coefficient b in the Forchheimer equation [‐] Shape factor in the friction coefficient c in the Forchheimer equation [‐] correction factor for observed/calculated run‐up at the core [‐] Run‐up reduction coefficient [‐] , /
, Reduction for the time the water is infiltrating into the core [‐] Run‐up reduction coefficient for infiltration [‐] , /
, Unknown angle between still water level and run‐up water surface [°] Iribarren number [‐] Angular frequency (2π/T) [s‐1] Technology, Pasadena, California. LIN P. [2008] Numerical modeling of water waves, London, England MUTTRAY, M. [2000] Wellenbewegung in einem geschütteten Wellenbrecher, Ph.D.‐Thesis, Technical University Braunsch‐
weig, Braunschweig, Germany TROCH, P. [2000] Experimentele studie en numerieke modelle‐
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schrift, Gent University, Gent, Belgium
VAN DER MEER, J.W. [1988] Rock slopes and gravel beaches under Wave attack, Ph.D.‐Thesis, Delft University of Technology, Delft, The Netherlands VAN DER MEER, J.W. [1990] Measurement and computation of wave induced velocities on smooth slope, proceedings 22nd ICCE, Delft, p. 191‐204 VAN DER MEER, J.W. [2002] Invloedsfactoren voor de ruwheid van toplagen bij golfoploop en overslag, bijlage bij het Technisch Rapport Golfoploop en Golfoverslag bij dijken, Rijkswater‐
staat, Dienst Weg‐ en Waterbouwkunde, publicatienummer DWW‐2002‐112 VAN GENT M.R.A. [1993] Stationary And Oscillatory Flow Through Coarse Porous Media, Communications on hydraulic and geotechnical engineering, Report No. 93‐9. Delft Univer‐
sity of Technology, Delft. VAN GENT M.R.A., TÖNJES P., PETIT H.A.H., VAN DEN BOSCH P., [1994]. Wave action on and in permeable coastal structures. In: Proceedings 24th International Conference on Coastal Engi‐
neering, Kobe, Japan, Vol.2, pp.1739‐1753. VAN GENT M.R.A., [1995]. Wave interaction with permeable coastal structures, Ph.D.‐Thesis TU Delft, Nederland. ISBN 90‐407‐1182‐8. VAN BROEKHOVEN, P.J.M., [2011]. The influence of armour layer and core permeability on the wave runup. MSc thesis, Delft University of Technology
VERHAGEN H.J., JUMELET H.D., VILAPLANA DOMINGO A.M., VAN BROEK‐
HOVEN P.J.M., [2011]. Method to quantify the notional per‐
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lume‐exchange model with Van der Meer laboratory test re‐
sults. Additonal MSc thesis, Delft University of Technology NOTATION 1. Amplitude of incident regular wave [m] 2. Porous friction coefficient in the Forchheimer equation [s/m] Fitting coefficients in the run‐up formula [‐] Turbulent friction coefficients in the Forchheimer equation [s2/m2] Fitting coefficients in the run‐up formula [‐] 9