Numerical modelling of wave reflection and transmission in vertical porous structures
Jeroen van den Bos 1,2, Henk Jan Verhagen1 and Coen Kuiper1,3
1Delft University of Technology, Civil Engineering and Geosciences, section Coastal Engineering, PO Box 5048, 2600 GA Delft, The Netherlands. Corresponding author: j.p.vandenbos@tudelft.nl
2Royal Boskalis Westminster NV, Hydronamic Engineering Department, PO Box 43, 3350 AA Papendrecht, The Netherlands
3 Witteveen+Bos consulting engineers, PO Box 233, 7400 AE Deventer, The Netherlands
ABSTRACT
This paper presents the results of a systematic validation of numerical simulation of reflection and transmission of rubble-mound breakwater structures. The aim of the study is to provide a body of validation measurements for increasingly complex situations, starting simple and building up the complexity in terms of breakwater geometry, wave conditions and numerical simulation models. The predictions from a set of three different numerical models were compared against measurements from small scale physical model tests. The results of the comparison and the achieved level of accuracy are reported. Finally, a hypothetical application for the design of rubble-mound breakwaters is presented.
INTRODUCTION
Numerical simulation models are becoming an increasingly powerful tool to study wave-structure interaction for design of rubble-mound breakwaters. In an idealized situation (see Figure 1), a designer can draw from rules of thumb or design formulas, physical model tests and numerical modelling tools to make a design.
The way the designer chooses to combine these tools will depend e.g. on the design phase, the complexity of the project and the time and budget available. The design formulas in turn are usually (semi-)empirical methods that are themselves based on model tests or field measurements. Perhaps in contrast to other fields of civil engineering, the right-hand side of Figure 1 is as yet not fully developed when it comes to design of rubble-mound breakwaters. The aim of the present study is to make a contribution towards improving that situation, by presenting a systematic validation of several numerical models on simplified structures and basic hydraulic parameters: reflection and transmission.
The basic philosophy behind the present validation is based on three principles.
Increasing complexity (structures). The modelled situations start at the most
simplified geometry of a rubble mound breakwater possible, i.e. a vertical homogeneous rock structure, loaded by regular harmonic waves in shallow water with constant water depth. Step by step, complexity is increased representing more real-life breakwater geometries such as multiple layers, irregular waves, closed cores and sloping structures. In this way, the degree in which each numerical model is able to cope with ever more complex structures can be analyzed.
Increasing complexity (models). The same principle applies for the numerical
models that have been used in the validation. A representative model from a series of “families” of models of increasing complexity were used, starting at a strongly simplified analytical model, followed by a non-hydrostatic wave-flow model (SWASH), and finally with a full RANS-VOF model (IH2-VOF).
Known porous flow parameters. All selected numerical models rely on the
Forcheimer friction model for flow in a porous medium. This requires knowledge of the associated model parameters in order to find a solution. These parameters are usually not known and need to be estimated. In the present study we have performed the physical model tests using pre-fabricated blocks of stone material, bound together using Elastocoast, resulting in a fixed porosity and layer thickness. Six of these
blocks were prepared earlier for this purpose (Zeelenberg and Koote 2012) and their hydraulic properties (porosity, permeability, grain diameter) were measured. Since the stones in the Elastocoastblocks are bound together they form a rigid block and the properties measured by Zeelenberg and Koote remain fixed throughout the tests. This avoids the need to estimate the porous friction parameters and makes the use of these blocks very practical for model validation purposes.
Table 1. Hydraulic properties of Elastocoast blocks Block # Thickness [mm] Laminar friction α0 [-] Turbulent friction β0 [-] Porosity n[-] Grain size dn50 [m] Grading width d85/d15 [-] 1 39 700 1.10 0.386 0.007 1.45 2 88 1200 1.25 0.405 0.020 1.23 3 132 1200 1.25 0.423 0.020 1.23 4 80 1900 1.70 0.410 0.039 1.18 5 160 1150 1.60 0.466 0.039 1.18 6 240 1020 1.45 0.460 0.039 1.18
Study outline. In line with the principle of increasing complexity, the present study
has focused on the hydraulic properties of reflection and transmission. These are simple and robust “bulk” properties that may not need very complex models to study them. Other, more complex and more localized hydraulic properties such as (pore) velocities, pressure distribution, wave overtopping etc. may be studied in later stages. The overall setup of the study consist of a series of physical model tests measuring reflection and transmission for an increasingly complex set of configurations, followed by numerical simulations of the same phenomena by means of three different models. The validation consists of comparison of the obtained results and presentation of the observed accuracy of the numerical prediction. These steps are described in more detail in the following sections.
After the validation, we will present a hypothetical application of the results in the field of the estimation of the notional permeability of a rubble mound breakwater.
PHYSICAL MODEL TESTS
Test data for the validation was obtained from physical model tests performed in the wave flume at the hydraulics laboratory at Delft University. The wave flume measures 38 m x 0.8 m x 1.0 m and is equipped with an Automatic Reflection Compensator (ARC) at the wave maker and second order wave steering. Mellink (2012) performed tests for single blocks. Additional tests with multiple blocks, closed boundaries, irregular waves and sloping structures were performed in the context of the present study.
All model setups with regular waves were performed for 12 wave conditions, being all combinations of four wave heights and three periods as indicated in table 2. The tests with irregular waves were performed using a JONSWAP spectrum with peak enhancement factor γ = 3.3, a significant wave height Hm0 = 0.10 m, and two different peak periods Tp = 1.5 s and Tp = 3.0 s. The water depth was constant at h0 = 0.65 m for all tests.
A total of 113 tests were performed in 12 test series. Each series used a different configuration of blocks, see table 3. In test series M1-M4, four different single vertical blocks were used with an open boundary. In series A and B a double block was used consisting of block #5 on the exposed side (representing the armor layer of a rubble mound breakwater) and block #3 on the lee side (representing the core or underlayer). In series A a gap of 18 mm was left between the blocks and an additional wave gauge was placed in the gap, in series B this gap was closed. In series C a wooden plate was placed behind block #3 representing a closed breakwater core (i.e. sand or clay). In series D-F the plate was removed and the blocks were tested with irregular waves, first for both blocks (series D) and later for block #5 only (series E and F). Test series F was a repetition of the last test condition of series E with a different wave gauge spacing, in order to test the sensitivity of the test setup (no sensitivity was observed). Finally, the wooden plate was re-introduced and the test series continued with block #5 only and a “closed core” (series G-H). For the last series (H), the block and wooden plate were placed at a 45 degree angle in order to get a rough idea of the impact of the structure slope. Of course, more extensive
datasets of wave reflection on sloping structures exist in literature but these were not analyzed in the present publication.
Reflection was measured in all tests by means of three wave gauges in front of the structure and separating the in incident and reflected wave signal by standard methods, i.e. Goda and Suzuki (1976) for regular waves and Mansard and Funke (1980) for irregular waves. The method of Goda and Suzuki is a two-gauge method; results for three gauges were obtained by applying the method for all permutations of two gauges and averaging the result. In all cases the reflection coefficient is defined as R = ar1/ai1, where ar1 and ai1 are the measured wave amplitudes of the reflected and incident signal, respectively.
Transmission was measured for all cases that did not have a closed boundary, by means of three wave gauges behind the structure. The incident and reflected signals were separated in the same manner as described above and the transmission coefficient is defined as T = ai2/ai1, where ai2 is the wave amplitude of the in incident signal for the set of wave gauges behind the structure and ai1 is the wave amplitude of the in incident signal for the set of gauges before the structure.
Table 2. Target wave conditions Period T [s] Water depth h0 [m] Local wave length L [m] kh [-] Steepness s = H/L [m] H = 0.075 m H = 0.100 m H = 0.125 m H = 0.150 m 1.5 0.65 3.06 1.335 0.025 0.033 * 0.041 0.049 2.0 0.65 4.50 0.908 0.017 0.022 0.028 0.033 3.0 0.65 7.20 0.567 0.010 0.014 * 0.017 0.021 The majority of tests were carried our using regular waves. Conditions marked with an asterisk (*)
were also performed using irregular waves, in which case T = Tp, H = Hm0 and L = Lop
Table 3. Test matrix
Series Reference Blocks Wave
type Boundary
# of
tests Remark
M1 Mellink 2012 2 Regular Open 12
M2 Mellink 2012 3 Regular Open 12
M3 Mellink 2012 5 Regular Open 12
M4 Mellink 2012 6 Regular Open 12
A This publication 3 and 5 Regular Open 12 18 mm gap between blocks
B This publication 3 and 5 Regular Open 12
C This publication 3 and 5 Regular Closed 12
D This publication 3 and 5 Irregular Open 2
E This publication 5 Irregular Open 2
F This publication 5 Irregular Open 1 Repeatability test
G This publication 5 Regular Closed 12
H This publication 5 Regular Closed 12 Sloping structure 45 deg
NUMERICAL SIMULATIONS
Numerical simulations were made for all cases using the three models described above. In all cases, the simulations were performed using the model dimensions of the physical model (no scaling). The wave conditions were re-generated in each numerical from the target wave conditions given above (so no attempt was made to reproduce the exact same boundary conditions using the steering file input from the physical model tests. Even though such methods are possible in principle, it would seem too advanced for the present purpose).
Analytical model. Madsen and White (1976) have described a mathematical model
to calculate reflection and transmission in vertical porous structures in the 1D case of regular waves in constant water depth. Their model is based on a description of the Shallow Water Equations (SWE) with the addition of a linearized friction term in the porous medium. An analytical solution to the SWE is imposed consisting of a linear superposition of two harmonic waves propagating in opposite directions, i.e. “incident” and ”reflected”, in each of the three sections (1) open water before the structure, (2) porous flow inside the structure and (3) open water behind the structure. Finally, continuity of the solution in both the horizontal velocity and the surface elevation is imposed at all interfaces between these sections. This enables the elaboration of an explicit formula for the reflection and transmission coefficients.
Porous flow is modelled by adding a friction term to the equation of motion. The friction term is based on the Forcheimer model of friction loss in a porous
medium, so it contains a laminar friction parameter a = α((1-n)3/n2)(ν/dn502) and a turbulent friction parameter b = β((1-n)/n3)(1/dn50), where n is the porosity, ν is the viscosity of water, dn50 is the characteristic grain size of the porous material, and α and β are material-specific constants. The turbulent friction term is linearized using the principle of equivalent work, to (3/8)b|U| where |U| is the velocity amplitude. This requires an iteration, as |U| is part of the solution itself. In the original description of Madsen and White this iteration was avoided by estimating |U| beforehand since at the time, a full iteration was thought to take too much computational effort. In the present application the iteration loop is included in the solution which leads to improved results (Van den Bos et al 2014). In addition, van Gent (1994) found that the turbulent friction parameter β is not constant in oscillating flow. Van Gent proposes a correction β = β0(1-7.5/KC) where β0 is the measured material constant under unidirectional flow and KC is the Keulegan-Carpenter number for porous flow, defined as KC = (|U|.T)/(n.dn50). This correction is also implemented in the friction-velocity iteration loop in the present application.
The original Madsen and White model uses only a single vertical structure. However, the method can be relatively easily extended for an arbitrary number N of blocks, by writing out the flow equations for each individual block (using the local friction parameters that may vary for each block) and imposing continuity on each interface. Instead of one single explicit formula, the end result is now a system of 2(N+1) equations and 2(N+1) boundary conditions which can be solved by matrix algebra.
When the porous structure has a closed downstream boundary, i.e. there is no transmission, the system of equations can be modified by omitting the set of equations in the open water behind the structure and introducing an appropriate extra boundary condition. In the present application a simple approach was followed in which a no-flow condition (|U| = 0) is imposed at the far boundary.
The mathematics behind the (extended) Madsen and White model (EMW) is based on the assumption of regular waves, and furthermore sloping structures cannot be modelled. For that reason, only test series M1-M4, A-C and G were simulated.
SWASH. The numerical model SWASH was developed at Delft University to
simulate the behavior of waves approaching the shoreline in shallow waters. It is based around a numerical solver for the nonlinear, non-hydrostatic Shallow Water Equations including a module for porous flow using the Forcheimer model. The equations are solved on a calculation grid in x-direction (in 1D) or both x and y directions (in 2DH). Being a 1D (or 2DH) model, sloping structures cannot be modelled accurately and as a consequence only the purely vertical test series (M1-M4, A-G) were simulated. In the present application, SWASH was used in 1D, non-hydrostatic mode with k = 2 vertical layers.
In SWASH, the user can define a “porosity grid” in x-direction with is then mapped onto the computational domain. This allows the modelling of porous structures, including several “blocks” of varying porosity. However, only the porosity n can be varied, and the other parameters that determine the porous flow (α, β0, dn50) can only have a single constant value. For the tests with single blocks this poses no problems, for the tests with double blocks a weighted average was used, i.e. α = (α1.t1
+ α2.t2)/(t1 + t2) etc where t1 and t2 are the respective thicknesses of the two blocks). The Van Gent correction on β is not implemented in SWASH, and furthermore β is bounded by a minimum β 1.8.
The grid size Δx and time step Δt were chosen to have at least 100 points per wave length, and a Courant number Cr < 0.5, see table 4.
Reflection and transmission coefficients were determined by requesting output for the free surface elevation at the exact same x-coordinates of the wave gauges used in the physical models and then applying the decomposition routines for in incident and reflected wave height as before.
SWASH does not offer a validated solution to model a closed boundary. In the present study, a closed boundary was modelled by setting the porosity in the area behind the interface to n = 0.1.
IH-2VOF. One step further up the chain of model complexity we have used a
RANS-VOF model which solves the full (Volume Averaged, Reynolds Averaged) Navier-Stokes equations on a 2DV rectangular grid, with a Volume of Fluid (VOF) routine to track the free surface. Several such models exist, but the IH-2VOF model developed by the Hydraulic Institute of Cantabria (Lara et al 2005) is readily available at Delft University and we have selected this model for the present purpose.
The model allows for a direct one-on-one modelling of the geometry of the porous structures, including all details such as slopes and different Forcheimer parameters for all layers. Therefore none of the simplifications described above had to be applied, and all block configurations tested in the physical model could be simulated.
It was found in earlier experiments that at least 150 grid points per wave length are needed (Van den Bos et al 2014), in addition to the recommendation to have an aspect ratio Δx/Δy = 2.5 approximately. For reasons of computational efficiency the calculation grid was set “as large as possible” for each of the three tested wave periods, see table 5. The time step Δt is set automatically by the model to satisfy internal criteria for the Courant number. Reflection and transmission were analyzed in the same way as for SWASH, using the free surface output at the specified wave gauge locations.
Table 4. Grid dimensions SWASH modelling Wave period T [s] Local wave length L [m] Wave celerity c [m/s]
Grid size Ratio
Δx [m] Δt [s] L/Δx [-] Cr [-]
1.5 3.06 2.04 0.030 0.005 102 0.34
2.0 4.50 2.25 0.030 0.005 150 0.38
3.0 7.20 2.40 0.030 0.005 240 0.42
Table 5. Grid dimensions IH-2VOF modelling Wave period
T [s]
Local wave length L [m]
Grid size Ratio
Δx [m] Δy [m] L/Δx [-] Δx/Δy [-] 1.5 3.06 0.020 0.008 153 2.5 2.0 4.50 0.030 0.012 150 2.5 3.0 7.20 0.030 0.012 240 2.5
VALIDATION
Results. The simulated reflection and transmission coefficients R and T were
compared to the measured result in the physical model tests. The result is given in Figure 4 as a function of the complexity of the configuration. These results show that the results for the EMW model with the closed boundary are far off the expected values. We conclude that the method of introducing the closed boundary condition (i.e. setting u = 0 across the boundary) is likely to be oversimplified. Also, we found that the SWASH runs for simulations with a closed boundary gave no results. The VOF model experienced no problems while simulating situations with a closed boundary. In the context of this study, we conclude that the first two models are not applicable in situations with a closed boundary and we omit the simulation results (if any) from the subsequent analysis.
Accuracy. We define the prediction error as ER = (1/N).{|Rmeasured – Rsimulated| / Rmeasured}, where N is the number of tests in the dataset, and with a similar expression for the prediction error in T. These results are given in tables 6 and 7 below. Because at least some of the numerical models are based on the assumption of shallow water waves, their accuracy could be a function of the validity of the relative water depth, or kh-number. In order to analyze that possibility we have split the results for (1) all tests in the data set (so T 1.5 s or kh 1.34), (2) only tests with T 2.0 s (kh 0.91) and (3) only tests with T 3.0 s (kh 0.57).
Table 6. Mean prediction error - Reflection Situation
Tests kh 1.34 Tests kh 0.91 Tests kh 0.57
EMW SWA SH IH2-VOF EMW SWA SH IH-2VOF EMW SWA SH IH-2VOF (1) Single layer 0.08 0.05 0.08 0.05 0.04 0.07 0.05 0.05 0.06 (2) (1) + double layer 0.08 0.08 0.06 0.05 0.08 0.06 0.05 0.10 0.07 (3) (2) + irregular waves - 0.08 0.06 - 0.08 0.06 - 0.10 0.06 (4) (2) + closed boundary - - 0.06 - - 0.06 - - 0.06 (5) (2) + irr waves + cl. boundary - - 0.06 - - 0.06 - - 0.06 (6) (5) + sloping structure - - 0.07 - - 0.06 - - 0.07
Table 7. Mean prediction error - Transmission Situation
Tests kh 1.34 Tests kh 0.91 Tests kh 0.57
EMW SWA SH IH2-VOF EMW SWA SH IH-2VOF EMW SWA SH IH-2VOF (1) Single layer 0.23 0.24 0.13 0.16 0.12 0.09 0.15 0.07 0.06 (2) (1) + double layer 0.20 0.31 0.17 0.15 0.20 0.12 0.14 0.13 0.08 (3) (2) + irregular waves - 0.34 0.17 - 0.22 0.13 - 0.17 0.10 (4) (2) + sloping structure - - 0.17 - - 0.13 - - 0.10 CONCLUSIONS
This paper describes the research work done to investigate the suitability of different numerical models to calculate the reflection and transmission. The calculations results have been compared to physical model tests results. Overall, the
performance of the models is good. The achieved prediction errors are given in tables 6 and 7.
For reflection, the prediction error of the models is in the order of 5-10 percent. We conclude that the difference between model performance is mainly a function of complexity of the structure (not all numerical models are able to simulate the structures with increasing complexity). In the simplest geometries, a simpler model can be used without significant loss of accuracy, while the simulation time is considerably faster. There does not seem to be a large influence of the dimensionless water depth (kh number).
For transmission the situation is slightly different in the sense that there is some difference between the models, even in the simplest geometries. In general, the IH-2VOF model is able to predict transmission more accurately, with a prediction error in the order 10-15 percent. Also, there appears to be some dependence on the kh number, with longer waves (lower kh numbers) resulting in more accurate results. This only applies to the models based on the Shallow Water Equations, i.e. the EMW model and SWASH.
The selection of a model for a given application will also depend on available time and computational resources. In this study, the calculations with the EMW model were immediate, SWASH simulations took in the order of ten minutes and IH-2VOF in the order of hours for a single simulation. The total file size, for all tests, is in the order of 100 KB for the EMW model, 100 MB for SWASH and 10 GB for the IH-2VOF model.
This type of validation studies may enable the direct application of numerical models in a rubble-mound breakwater design. A hypothetical example of such an application is given in Appendix A.
REFERENCES
Goda, Y and Suzuki, Y (1976) Estimation of incident and reflected waves in random wave experiments, Proceedings of 15th International Conference on Coastal Engineering, New York, United States of America
Lara, J (2005) A numerical wave flume to study the functionality and stability of coastal structures, PIANC magazine, nr 121
Mansard, E P D and Funke, E R (1980) The measurement of incident and reflected spectra, using least squares method, Proceedings of 17th International Conference on Coastal Engineering, Sydney, Australia
Mellink, B (2012) Numerical and experimental research of wave interaction with a porous breakwater, MSc thesis Delft University of Technology, Delft, The Netherlands
Madsen, O and White, S (1976) Reflection and transmission characteristics of porous rubble-mound breakwaters, Technical Report 76-5, US Army Corps of
Engineers, Coastal Engineering Research Center, Fort Balvoir, USA. Postma, G M (1989) Wave reflection from rock slopes under random wave attack,
MSc thesis Delft University of Technology, Delft, The Netherlands Van den Bos, J P, Verhagen, H J, Zijlema, M and Mellink, B (2014), Towards a
interaction: an initial validation, Proceedings of 34th International Conference on Coastal Engineering, Seoul, Korea
Van der Meer, J (1988) Rock slopes and gravel beaches under wave attack, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands
Van Gent, M (1995) Wave Interaction with Permeable Coastal Structures, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands
Zeelenberg, W and Koote, M (2012) The use of Elastocoast in breakwater research, Technical report, Delft University of Technology, Delft, The Netherlands Zijlema M, Stelling, G and Smit, P (2011) SWASH: an operational public domain
code for simulating wave fields and rapidly varied flows in coastal waters, Coastal Engineering (58) pp 992-1012.
Acknowledgements
Elastocoast is a registered trademark of Elastogran GmbH, Lemförde, Germany (subsidiary of BASF)
The authors would like to thank prof J. van der Meer for providing additional information related to the HADEER tests (see appendix) and mr Gerben Jan Vos for his assistance during the physical model tests.
APPENDIX: A POTENTIAL APPLICATION
In the following we will elaborate a potential application of this type of numerical analysis in a design context. The example that we have taken is the notional permeability of rubble-mound breakwaters. The method that we are describing is hypothetical at this stage, additional research is being carried out at Delft University at present to validate this approach against further measurements.
Introduction. In the context of this example, we will loosely define notional
permeability as the degree in which the core of a rubble-mound breakwater affects the stability of the armor rocks. When the core has a relatively open structure, for instance when it is made of coarse material, the wave energy can dissipate in the core and less energy is available for moving the armor rocks, and vice versa for a relative closed core (e.g made of sand). In design practice, this effect can be allowed for by selecting an appropriate value of the parameter P in the design formula (Van der Meer 1988). This parameter P is a function of the cross section of the breakwater. For four well-defined cases, P has a known value, see Figure A1.. The problem emerges when an structure is being designed that is not exactly one of these four cases, for instance when a sand core is covered by a relatively thick filter layer. At present no commonly accepted method exists to estimate or determine P in these cases.
Link to flow into the core. In his original publication on the subject, Van der Meer
(1988) hypothesizes that notional permeability may be related to the flow rate of water into the core. In other words, if the flow rate can be determined for the four cases and plotted on a “flow rate Q vs P diagram”, the flow rate for an intermediate structure can also be found and plotted on the diagram, after which the corresponding value of P can be found by interpolation.
Van der Meer describes a hypothetical experiment on a rubble-mound breakwater with a single layer of armor rock overlying a core. If the wave conditions and the diameter of the armor rock are fixed, but the diameter of the core rock is changed, the flow rate into the core should increase with increasing core rock size. Van der Meer simulated this behavior by means of the then-current (groundwater) flow model HADEER using regular waves and three different wave periods, and indeed found the expected behavior. His results, in terms of flow rate per wave have been plotted against the core rock diameter, is reproduced below in Figure A4 (right hand pane).
These results can be converted to a “Q-P” diagram. The largest core stones in the experiment have the same size as the armor rocks. The structure corresponds to the “homogenous” standard structure, so the flow rate found for this diameter corresponds to P = 0.6. The flow rate for the ratio D50armr/D50core = 3.2 corresponds to the standard “permeable” case with P = 0.5. Finally, a zero flow rate corresponds to the standard “impermeable” case with P = 0.1. Van der Meer then normalizes the flow rate with the highest value (so all rates have a value between 0 at P = 0.1, and 1 at P = 0.6) and draws the Q-P diagram. As an example, Van der Meer interpolates the plot for a core diameter of Dcore = 0.05 m and finds a value of roughly P = 0.4 (depending on the wave period). His result, for each of the three wave periods, is reproduced in Figure A5 (right hand pane).
Figure A1: Notional permeability cases (from Van der Meer 1988)
h = 5.0 m 3.5 m Armour d50 = 0.25 m Thickness 0.50 m 8.0 m 1:3 1:1.5 Core d50 varies
Reproduction of results. The IH-2VOF model was used to reproduce these
experiments. The same breakwater dimensions, rock sizes and regular wave conditions are used as those described by Van der Meer. The porous flow parameters (α, β0 and n) were unknown and had to be estimated. A total of 15 simulations were performed, for three different wave conditions and five core rock diameters. The simulated velocity field inside the breakwater was retrieved from the model and some extensive post-processing was required to obtain the velocity vectors on the interface armor-core and finally, by integration, the flow rate into the core. A full description of these simulations is outside the scope of the present paper.
Figure A3: Snapshot instantaneous flow into the core
The results from this analysis are given in figures A4 and A5, left hand pane, juxtaposed against the original results of Van der Meer (1988). As can be seen, the flow rates can be reproduced to quite a reasonable degree. The difference is roughly a factor of 2, which seems reasonable given the difference between the two models and the large number of assumptions and estimates that had to be made.
Interestingly, despite these differences, the interpolation on the Q-P diagram yields very closely the same results. It is striking that the shape of the lines is different, and the lines for different wave periods are reversed in order, but the end result in terms of an interpolated P value is almost the same.
This methods looks promising, but more research and validation is required before application in a real design case can be supported. In addition, the computational cost is significant (our simulations ran over the course of a weekend and required more than 250 GB of disk space) and complicated post-processing is required.
Figure A4: Flow into the core (left: reproduced result. Right: original result (Van der Meer 1988)
Figure A5: Interpolated notional permeability (left: reproduced result. Right: original result (Van der Meer 1988)
Link to reflection. Finally, we return to the topic of the present paper which is
reflection. This can be directly linked to notional permeability, in the sense that a less permeable structure will also be more reflective. Postma (1989) analyzed the reflection measured in all model tests by Van der Meer (1988) and presented a prediction formula for the reflection coefficient R that shows a weak dependency on P. If this formula would be reversed, P could be calculated if R was known. A hypothetical design method could then be devised in which the reflection coefficient for a given structure is simulated by any of the models described in this paper, and the corresponding value of P is found by entering this value in a “reversed Postma formula”. This relationship is too weak however to be reversed: the scatter in Postma’s curve fitting is too large to allow such a hypothetical method within any practical margin of accuracy.
However, where a direct method cannot be applied, an indirect method may be possible. We hypothesize here that a similar method can be used as proposed by Van der Meer, but not interpolating the structures based on the flow into the core but by their reflection. In such a method, the structure with unknown P would be simulated in any suitable numerical model to find its reflection coefficient, along with “equivalent cases with known P” being rubble mound breakwaters with the same armor rock size, slope and design wave conditions, but different core structures. The results can then be plotted on a “R vs P” diagram and the required value of P for the
unknown structure can then be found by interpolation. Such a method would be easier to calculate and would require less disk space than the method based on flow into the core.
As a proof of concept, we have calculated the reflection coefficients for our IH-2VOF simulations used for the reproduction of Van der Meer’s “flow into the core” tests. Figure A6 shows the resulting “R-P” curve and the interpolation for D50core = 0.05. Interestingly, this interpolation yields the same results (P = 0.4 approximately) as the previous method. This puts some confidence in the hypothesis. Additional research work is ongoing at Delft University to elaborate this idea and support it with validation tests.
