Revetment Rock Armour Stability Under Depth-Limited Breaking Waves

Abstract

This article presents a rock armour stability formula for coastal revetments under depth-limited breaking waves that defines requisite armour mass as a function of incident wave energy. Parameters include wave height, wave period, toe depth, revetment slope, specific gravity of armour and water, percentage damage and the number of waves. The formula has been calibrated empirically based on university research flume test data. It departs from existing approaches by using wave energy in lieu of wave height as the disturbing parameter, but adopts other parameters developed by previous researchers. Results are compared with established formulae and display better coherence with the flume data. Testing constraints including possible scale effects are highlighted. Recommendations are made for further testing including the effects of seabed slope.

1. Introduction

1.1. Background

Typically, coastal revetments are constructed to prevent wave erosion of dunes and foreshores. Revetments may be subjected to waves that shoal and dissipate on the revetment slope, the revetment toe being in relatively deep water, to waves that shoal on the seabed immediately in front of the revetment, breaking directly onto it, and to waves that have shoaled and broken on the seabed prior to reaching the revetment. The latter conditions have revetment toes in relatively very shallow water, being the subject of this research.

Commonly, revetment armour is underlain by a separating filter layer overlying fine-grained sediment. While allowing for groundwater seepage, these underlying materials are impermeable to incident wave energy, which is reflected onto the revetment armour. Therefore, the stability of revetment armour is somewhat compromised compared with breakwater armouring that, commonly, overlies a permeable core.

Armour units for revetments subjected to wave impact are designed with formulae developed and calibrated empirically using hydraulic scale models [1,2,3]. Early work defined the median requisite armour mass, M₅₀, as a function of wave height, rock density, water density and revetment slope following the passage of some 1000 regular waves, being the Hudson formula [1]. These tests were undertaken on sloping structures for permeable breakwaters under regular non-breaking waves. Empirical coefficients for a range of various concrete armour units and rock types for both non-breaking and breaking waves have been developed subsequently. The Hudson formula is:

$$M_{50}={\displaystyle \frac{{\rho }_a{H}_s^3}{K_D{∆}^{3}cot\alpha }}$$

(1)

where M₅₀ is the median armour mass (kg), ρ_a is the armour density (kg/m³), H_s is the incident significant wave height (m), Δ is the relative density of the armour to water ((ρ_a − ρ_w)/ρ_w) (-), K_D is a damage coefficient derived empirically (-) and cotα is the revetment armour slope (-).

Testing of rock armour and various concrete armour units under breaking waves was undertaken in a wave flume at the University of New South Wales Water Research Laboratory (UNSW WRL) [4,5]. The testing under regular waves found that K_D depended on the percentage damage, D, thus [4], [5] (p. 28):

$$K_D=a{D}^n$$

(2)

where (a, n) = (1.2, 0.51) for two layers of random rock on an impermeable core and D is percentage damage (0 < D < 20) (-). This form of expression for the Hudson damage coefficient has been adopted for subsequent modifications to the Hudson formula by others [2,3].

Later research included formulae calibrated empirically for both surging and plunging random waves on sloping revetments and breakwaters. The van der Meer formulae [2] encompassed a larger range of relevant parameters including the Iribarren number (wave period), core permeability, degree of damage and the number of zero-crossing waves. Most of the tests comprised relatively deep water at the structure toe (H_s/h_toe < 0.3). The requisite armour mass was formulated as a function of wave height cubed for plunging conditions:

$$M_{50}={\displaystyle \frac{{\rho }_a{\epsilon }_m^{1.5}{H}_s^3}{c_{pl}^3{{\left(\frac{S}{\sqrt{N}}\right)}^{0.6}{P}^{0.54}∆}^3}}$$

(3a)

and for surging conditions:

$$M_{50}={\displaystyle \frac{{\rho }_a{P^{0.39}H}_{\mathrm{s}}^3}{c_s^3{{\left(\frac{S}{\sqrt{N}}\right)}^{0.6}{\epsilon }_m^{3P}∆}^3{cot\alpha }^{1.5}}}$$

(3b)

where P is a permeability parameter of the armour underlayers (-), ε_m is the Iribarren number using the mean wave period (-), $S=A_E/D_{n50}^2$ (-), A_E is the cross-sectional eroded area (m²), D_n50 is the nominal median diameter of the armour stones (m), N is the number of zero-crossing waves (-) with calibration coefficients C_pl = 6.2 (-) and C_s = 1.0 (-). As $M_{50}={\rho }_a{D}_{n50}^3$ and $S=A_E/D_{n50}^2$, D_n50 is included both in the dependent and independent variables.

The most recent development of these formulae, being that of van Gent [3,6], included testing of two sloping structures with impermeable underlayers. The van Gent formula can be expressed as:

$$M_{50}={\displaystyle \frac{{\rho }_a{H}_s^3}{5.4{{\left(\frac{S}{\sqrt{N}}\right)}^{0.6}\left(1+\raisebox{1ex}{$D_{n50core}$}\!\left/ \!\raisebox{-1ex}{$D_{n50}$}\right.\right)}^3{∆}^3{cot\alpha }^{1.5}}}$$

(4)

with D_n50 included in both the dependent and independent variables.

With this development, the influence of wave period was considered small compared with the amount of scatter in the data due to other reasons [6]. Therefore, the wave period was not used in this formula and there was no separation between “plunging” conditions and “surging” conditions [6].

The Formulae (3a) and (3b) have been revised recently with changes made to the calibration coefficients and to the wave period parameters to be used [7,8]. However, their basic structures have remained.

Typical conditions studied by van der Meer [2] had waves shoaling and dissipating on the revetment slope, whereas Gordon [5] and van Gent et al. [3] considered waves breaking onto the revetment armour directly from the fronting seabed.

More recently, numerical models such as SPH (Smoothed Particle Hydrodynamics), IHFOAM, waves2Foam, and DualSPHysics have been developed for overtopping and armour stability research [9,10]. These models can be expensive computationally, particularly when applied to solve large domains and long duration wave trains, which may be required for wave-structure interaction, although, significant progress has been made on wave interaction with porous structures with these models [11] (p. 80).

As many numerical models rely on physical modelling for validation, the role of physical hydraulic scale modelling of coastal protection structures remains relevant [12]. The empirical approaches based on physical model studies defined the median requisite armour mass as a function of, inter alia, incident wave height cubed [1,2,3]. No explanation for this has been proffered other than it would appear to have been necessary to ensure dimensional integrity of the formulae [1] (p. 612), [2] (p. 115). However, other researchers have proposed sediment transport and revetment toe scour as a function of incident wave energy [13] (p. 27), [14].

1.2. Thesis

It is the energy of breaking waves, that is, the breaking velocity of the fluid mass, that applies the disturbing forces of lift and drag to revetment armour that may cause it to shift, and it is the mass of that armour under the restoring force of gravity that secures or otherwise its stability. This article examines revetment rock armour stability as a function of incident wave energy, E, which is a function of the product of wave height squared and wavelength [15] (p. 2-26) (Equation 2-38) thus:

$$E={\displaystyle \frac{{\rho }_{w}g{H}_s^{2}L}{8}}$$

(5)

where g is gravitational acceleration (m/s²) and L is the nearshore wavelength (m). In shallow water, where the depth is less than 0.04L (0.016T²) [15] (p. 2-9), the nearshore wavelength can be approximated by [15] (p. 2-25) (Equation 2-37), [16]:

$$L=T\sqrt{g{h}_{toe}}$$

(6)

where h_toe is the nearshore depth at the revetment (m).

The following wave energy formula is proposed for the requisite rock armour mass on an impermeable core revetment, adopting the dependence on armour slope from van der Meer [2] and incorporating nearshore depth and wave period through wavelength, preserving the dimensional integrity of the formula:

$$M_{50}={\displaystyle \frac{{\rho }_a{H}_s^{2}T\sqrt{g{h}_{toe}}}{{K^{\prime }∆}^3{cot\alpha }^{1.5}}}$$

(7)

The damage coefficient, $K^{\prime }=f\left(D/\sqrt{N}\right)$, is calibrated empirically herein in the form of Equation (2) for double layered rough angular rock [4,5,6]. While the nearshore wave height also is a function of depth, period and beach slope [17], [18] (p. 81), paucity of data precluded considering the influence of seabed slope on the rock armour formula proposed.

2. Method

2.1. Data

Data for this research comprised:

• Research from a Master degree thesis for amour rock stability under breaking waves undertaken in a flume at the University of New South Wales Water Research Laboratory (UNSW WRL) [4,5].
• Research for amour rock stability under breaking waves generated in flumes at Delft Hydraulics [2,3,6].

2.2. Physical Modelling

2.2.1. Water Research Laboratory

Testing of rough angular rock armour on impermeable core slopes of cotα = 1.25, 1.5 and 2.0 with regular waves was undertaken in a flume 1 m wide, 1.5 m deep and 30 m long [4,5]. The flume bed for 8 m fronting the structure had a slope m = 0.033, beyond which the bed was flat. The limestone rock armour had a specific gravity of 2.7, was graded with a ratio of maximum to minimum mass of approximately 2.0, a median mass of M₅₀ = 0.064 kg, an aspect ratio not exceeding 2.0 with edges that were moderately sharp and was placed randomly in two layers.

For a given toe depth, the wave height and period were adjusted until the wave formed a plunging break immediately onto the structure. The toe depth was increased by small increments allowing larger waves to break onto the structure. Damage was assessed as the percentage of rock dislodged over the revetment area extending from one wave height below still water level (SWL) to the limit of wave uprush. The testing regime was set up to enable the effects of up to 10,000 regular waves to be observed for each water level and period combination. Under the limiting conditions, progressive damage to the armour was observed up to failure, which occurred generally after around 2500 waves [4,5]. Progressive damage to failure developed once damage had reached around 20%. For wave attack producing lesser damage levels the structures adopted a stable configuration once the initial damage had occurred. The test data and reduced test data are presented in

Table 1. 1973 WRL Flume test and reduced data of damage from regular waves shoaling on a seabed slope m = 0.033 and breaking onto double layered rock armoured revetments of various slopes with impermeable underlayers [5].

Test Data						Reduced Data
cotα	T	hb	Hb	D	N	L	(1)  Hs	(2)  N0x	$\mathit{D}/\sqrt{{\mathit{N}}_{0\mathit{x}}}$	LHs2/((ΔDn50)3cotα1.5)
(-)	(s)	(m)	(m)	(%)	(-)	(m)	(m)	(-)	(-)	(-)
2	1.37	0.095	0.085	3.0	2500	1.33	0.061	125,000	8.49 × 10−5	14.8
2	1.35	0.100	0.090	6.0	3000	1.34	0.064	150,000	1.55 × 10−4	16.9
2	1.35	0.115	0.104	18.0	3000	1.43	0.075	150,000	4.65 × 10−4	24.2
1.5	1.31	0.075	0.067	2.2	3000	1.13	0.048	150,000	5.68 × 10−5	12.0
1.5	1.37	0.084	0.077	3.4	2000	1.24	0.055	100,000	1.08 × 10−4	17.6
1.5	1.35	0.094	0.084	8.3	4000	1.29	0.060	200,000	1.86 × 10−4	21.9
1.25	1.34	0.067	0.060	6.5	5000	1.09	0.043	250,000	1.30 × 10−4	12.3
1.25	1.35	0.077	0.069	11.6	5000	1.17	0.050	250,000	2.32 × 10−4	17.7

Note: ⁽¹⁾ H_b = H_2%; H_s = H_2%/1.4 [2] (p. 81) ⁽²⁾ N_0x = N/2% denotes the number of zero-crossing waves.

.

2.2.2. Delft Hydraulics

Data were derived from tests undertaken in the 1 m wide flume of Delft Hydraulics in 1983–1986 [2] and in 2003 [3,6]. The former are documented comprehensively in the thesis of professor Jentsje van der Meer, which is publicly and readily available. Included were some 132 tests with rough angular armourstone of D_n50 = 0.036 m on impermeable underlayers of slopes cotα = 2, 3, 4, and 6, T_m varying from 1.3 s to 3.3 s, damage recorded after 1000 and 3000 waves with damage after 3000 waves varying from S = 0.6 to 33.0. As H_s/h_toe varied from 0.06 to 0.32 with an average value of 0.16, the revetment toe was in relatively deep water (h_toe > 0.04L) with waves collapsing and dissipating on the revetment slope, rather than shoaling and breaking onto it from the fronting seabed. These data were not considered further in this analysis.

For the tests in 2003 [3], the armour comprised well sorted “standard rough angular rock” of specific gravity 2.75 with D_n50 = 0.026 m [6]. The revetments were on impermeable cores at slopes of cotα = 2 for Structure No. 6 and cotα = 4 for Structure No. 7. The seabed slope was m = 0.033. Considered were results with damage 1.6 < S < 17 (2% < D < 20%) for severe wave breaking, defined herein as H_s/h_toe > 0.5 for this moderate seabed slope (m = 0.033). The flume test data used and reduced data are in

Table 2. 2003 Delft Hydraulics flume test and reduced data of damage from waves shoaling on a seabed slope m = 0.033 and breaking onto double layered rock armoured revetments of various slopes with impermeable underlayers [3].

Test Data							Reduced Data
Test	Hm0toe	htoe	Hm0toe/htoe	N	Tp	Smeasured	$\mathit{D}/\sqrt{\mathit{N}}$	LH2/((ΔDn50)3cotα1.5)
No.	(m)	(m)	(-)	(-)	(s)	(-)	(-)	(-)
Structure 6 cotα = 2
16	0.076	0.150	0.51	1193	1.981	11.97	4.3 × 10−3	33.5
17	0.075	0.150	0.50	1216	1.863	10.13	3.6 × 10−3	52.1
18	0.063	0.125	0.50	1039	1.589	7.09	2.7 × 10−3	31.6
19	0.070	0.125	0.56	1286	2.471	10.34	3.6 × 10−3	33.2
Structure 7 cotα = 4
8	0.106	0.200	0.53	1032	1.888	6.96	2.7 × 10−3	40.5
9	0.111	0.200	0.56	1333	3.000	12.20	4.2 × 10−3	70.5
10	0.081	0.150	0.54	1017	2.584	1.72	6.7 × 10−4	28.0
11	0.088	0.150	0.59	1068	3.038	3.55	1.4 × 10−3	38.9

.

3. Results

The data were used to quantify the calibration coefficient K′ in Equation (7). The formula was re-written as follows to make the calibration coefficient, K′, the dependent variable:

$$K^{\prime }={\displaystyle \frac{H_s^2{T}_p\sqrt{g{h}_{toe}}}{D_{n50}^3{∆}^3{cot\alpha }^{1.5}}}=f\left({\displaystyle \frac{D}{\sqrt{N}}}\right)=a{\left({\displaystyle \frac{D}{\sqrt{N}}}\right)}^n$$

(8)

Plotting ${\displaystyle \frac{H_s^2{T}_p\sqrt{g{h}_{toe}}}{D_{n50}^3{∆}^3{cot\alpha }^{1.5}}}$ against $\left({\displaystyle \frac{D}{\sqrt{N}}}\right)$, as done in Figure 1, gave the values for the factor a and exponent n in Equation (8) by fitting a power curve. Essentially, the relationship becomes:

$$k^{\prime }=f\left({\displaystyle \frac{Wave Energy}{bouyant mass \times revetment slope}}\right)$$

(9)

The resulting calibrated wave energy formula is:

$$M_{50}={\displaystyle \frac{{\rho }_a{H}_s^2{T}_p\sqrt{g{h}_{toe}}}{310{\left(\frac{D}{\sqrt{N}}\right)}^{0.33}{∆}^3{cot\alpha }^{1.5}}}$$

(10)

4. Discussion

Data presented herein have been drawn from university research projects undertaken in 1973 [4,5] and 2003 [3,6]. No other data for very shallow waves breaking onto impermeable core rock armoured revetments fronted by a nearshore seabed slope of m = 0.033 (1V:30H) were found. The results are qualified as follows.

4.1. Comparison with Established Formulae

To compare the wave energy formula with the Hudson formula, in Figure 2 the very shallow water flume data [3,5] were plotted against normalised damage, $D/\sqrt{N}$, with the wave energy damage coefficient, $K^{\prime }=L{H}_s^2/{cot\alpha }^{1.5}{\left(∆D_{n50}\right)}^3$ and a modified Hudson coefficient $K_D=H_s^3/{cot\alpha }^{1.5}{\left(∆D_{n50}\right)}^3$. The Coefficients of Determination from the “best fit” trend lines (Figure 2) indicated that the wave energy formula using LH² (R² = 0.84) was significantly more coherent than the Hudson approach using H³ (R² = 0.39).

Intuitively, the greater the breaking wave energy the larger would be the damage to rock armour. Hence, waves of longer period would require larger armour rock mass for stability. However, neither the Hudson [1] nor the van Gent [3,6] formulae include a wave period parameter.

Calibrating rock armour damage to the energy of the depth-limited breaking waves introduces both toe depth and wave period to the design formula. As shown in Figure 3, both the wave energy formula and that of van der Meer [7,8] require larger requisite mass for waves with longer periods. The wave energy formula predicts a lesser armour mass than does the Hudson [1] formula but over-estimates the results the van der Meer [7,8] formulae for longer wave periods, under-estimates the van der Meer formula for shorter periods but is more conservative for greater damage. For initial damage, the formula over-estimates and under-estimates results from the van Gent formula [3,6] for longer and shorter wave periods respectively, as expected, as the van Gent formula does not include wave period.

4.2. Storm Duration

Flume testing has found that rip-rap damage may not have stabilised after more than 5000 to 20,000 zero-crossing waves (Figures 7, 16, 28–29, 31, 33–34 in [19]). For many of the tests undertaken herein the duration was some 1000 to 5000 zero-crossing waves, which may not have been enough to reach stabilisation of damage.

The number of waves matters. Should a design criterion comprise 2% damage, that is, virtually no damage after a storm of 1000 waves, then failure of the rock armour protection, assumed to be at 15% damage, is predicted not to occur until the passage of a further 55,000 design waves (Figure 4). Further, however, the predictor indicates that should a concept design comprise 5% damage under 1000 waves, then failure could occur following the passage of an additional 8000 design waves (Figure 4). For 10% damage/1000 waves design criteria, failure is predicted following a further 1300 design waves (Figure 4). Such considerations could inform an economic analysis of capital cost versus delayed maintenance costs.

4.3. Seabed Slope

The ratio of depth-limited incident breaking wave height to depth is dependent upon revetment toe depth, wave period and seabed slope (Figure 5). The ratio increases with longer wave periods and steeper seabed slopes. However, Figure 5 over-predicts the breaking wave height on flat beds [20,21] and has not been validated for slopes steeper than m = 0.1 [17]. Insufficient data were available to investigate the impact of seabed slope on the stability of armour rock.

4.4. Limits of Application

Ranges in the values of pertinent parameters of the model studies adopted herein are presented in

Table 3. Limits of application of the adopted flume data.

Parameter	Symbol	WRL 1973	Delft 2003
Revetment slope	cotα	1.25, 1.5, 2.0	2.0, 4.0
Median rock mass	M50	0.064 kg	0.047 kg
Rock Grading	Dn85/Dn15	1.3	1.4
Seabed slope	m	0.033	0.033
Wave Conditions		Regular	Random
Wave period	Tp	1.3–1.4	1.5–3.0 s
Wave height/depth	Hstoe/htoe	0.64–0.65	0.50–0.59
Number of waves	N	5000	1100–1300

.

Depending on the scale applied for design, prototype conditions would lay beyond some of these limits and extrapolating results from the predictor proposed should consider experimental constraints and possible scale effects. Experimental constraints include:

• Data are available for only one fronting seabed slope of m = 0.033. A likely range of nearshore seabed slopes could be 0.025 < m < 0.050. Test data for these slopes are not available but may result in a modified formula.
• The predictor is based on a paucity of data, with only 16 suitable test results available. A much larger data set is warranted.
• The shortest wave period used in the models was 1.3 s, which limits a small 1:50 Froude scale model to schematise only prototype wave periods shorter than 8 s. For larger scale models the maximum prototype period schematised could be far shorter. Prototype wave periods may exceed 15 s.
• The smallest significant wave height modelled was 0.04 m, which limits a small 1:50 Froude scale model to schematise wave heights only up to H_s = 2 m. For larger scale models the maximum prototype significant wave height schematised could be far smaller. Depth-limited prototype waves incident upon coastal revetments can exceed H_s = 2 m.
• The lightest model armour rock was M₅₀ = 0.047 kg, which limits a small 1:50 Froude scale model to schematise armour rock only up to M₅₀ = 6 t. For larger scale models the maximum mass of prototype armour rock schematised could be far smaller.
• Much of the data is based on around 1000 zero-crossing waves, which limits a small 1:50 Froude scale model to schematise a storm duration of up to 2 h. For larger scale models the maximum prototype storm duration schematised could be far shorter. Revetments in depth-limited situations are likely to be subjected to much longer storms and a much larger number of waves.
• The predictor is based on results of 2-dimensional flume studies, which do not replicate any 3-dimensional processes that occur in the nearshore surf zone fronting revetments.
• The predictor is based on flume studies that did not consider wave overtopping.

Reynolds number scale effects are possible. For example, if the armour rock was assumed to be spherical, to ensure similitude for drag and lift coefficients on a spherical object, the Reynolds number 2 × 10³ < Re < 2 × 10⁵ for both model and prototype (Figure 6). For Re < 2 × 10³ the drag and lift coefficient for spheres increases whereas the coefficient decreases for Re > 2 × 10⁵ (Figure 6).

Reynolds Number is Re = ρVL/μ or VL/υ where ρ is fluid density (kg/m³), μ is dynamic viscosity (N s/m²), υ is kinematic viscosity (m²/s), the wave breaking velocity jet being $V=\sqrt{g{h}_{toe}}$ (m/s) and the characteristic length $L=D_{n50}$ (m). Should the model data be applied at normal scales for prototype concept design, the likely prototype ranges of Reynolds number would be Re > 2 × 10⁵ as indicated in

Table 4. The range of model Reynolds numbers for the flume data presented herein and the range of likely prototype Reynolds numbers for the model data applied at two different scales.

	Dn50		htoe		Re
	Min.	Max.	Min.	Max.	Min.	Max.
	(m)	(m)	(m)	(m)	(-)	(-)
Model	0.026	0.029	0.060	0.200	2.0 × 104	4.1 × 104
Prototype for Scale 1:10	0.26	0.29	0.60	2.00	6.3 × 105	1.3 × 106
Prototype for Scale 1:50	1.3	1.5	3.0	10.0	7.1 × 106	1.5 × 107

and Figure 6. Values of Re > 2 × 10⁵ in a prototype with spherical armour units, such as armour rock, may cause the drag and lift forces of the flow being relatively lower than that in the model, leading to conservative estimates for requisite armour mass. However, Figure 6 applies to spheres that are submerged in flow, such as under non-breaking waves, but its applicability to armour rock in aerated flows of breaking waves is not known. Nevertheless, similar scale effects occur with certain types of concrete armour units [22,23].

Figure 6. The variation with Reynolds number of the drag coefficient for spheres submerged in uniform flow [24]. The range of model values and likely prototype values is indicated.

Figure 6. The variation with Reynolds number of the drag coefficient for spheres submerged in uniform flow [24]. The range of model values and likely prototype values is indicated.

5. Conclusions

A formula has been presented for coastal revetments located in very shallow water that relates the requisite stable mass of armour rock to the energy of the incident breaking waves. This formula includes parameters for significant wave height, wave period, toe depth, degree of damage, the number of waves, revetment slope, and armour and water densities. The formula has been calibrated using flume data generated by research at University hydraulic laboratories in Australia and the Netherlands. The formula is compared with those of Hudson and those as modified by van der Meer and van Gent, and shows more coherence with the available flume data than those of the (modified) Hudson formulae. The formula allows for the investigation of capital expenditure versus maintenance for the design of rock armoured revetments. Scale effects should be considered for design as they may produce very conservative results. This new development would benefit from much more laboratory data, which is recommended as it is required also to investigate the impact of seabed slope as well as possible Reynolds number scale effects.