Wave Overtopping on Coastal Structures - Essay Example

Wave-Topping On Coastal Structures: Introduction: Breakwaters are used extensively along coastlines all over the Earth and those that can allow wave overtopping without significant damage to effective structures and, thus, to properties and life they shelter are considered of eminent construction. Wave overtopping is a violent natural phenomenon that causes serious damage to protecting structures and life and property along coastlines (Shao, 2006). Traditionally, laboratory experiments and field observations have been used to study this turbulent oceanographic phenomenon and empirical formulae have been derived from these but severe limitations existed since parameters to which these derivations fitted were local (Shao, 2006). Thus, in recent time, universal derivations that can fit across a wide range of parametric requirements of structure geometry, water conditions and wave dynamics are considered essential and desirable. To this end, fluid dynamics proves a somewhat reliable model generator but traditional Eulerian approaches that discretise governing equations over a computational field divided into a grid system based on local parameters develop problems of numerical diffusion that transcends localised grid patterns and tend to encompass the entire grid so that the discretised development of the equation into an unified whole is seriously affected (Shao, 2006). More recently, to solve this diffusion effect for traditional dynamics, a particle method has been developed wherewith the discretised equation utlises individual particles in the flow as centres of development. The diffusion effect is effectively smoothed by a functional kernel that identifies and utilises the combined functions of the angular and linear momentums of each particle (Shao, 2006). The smoothing out of the diffusion effect generated at each particle location within the flow thus allows the fluid, in this case seawater in wave form, to be accounted for as an incompressible one (Shao, 2006). One such method that utilises this unique strategy is the moving particle semi-implicit method (MPS) applied somewhat successfully by Japanese scientists to wave flow patterns (Shao, 2006). The model that this paper will demonstrate is the smoothed particle hydrodynamic (SPH) method as developed and tested by Shao, 2006. The paper shall now study a little of how this manner of computational strategy developed. Smoothed Particle Hydrodynamics (SPH): The smoothed particle hydrodynamic method was one of the earliest meshfree methods applying Langrangian description of motion. It was primarily proposed by Lucy (1977) and Gingold and Monaghan (1977) (source: Zhang and Batra, 2004) for problems in astrophysics in three-dimensional space (Zhang and Batra, 2004). In the conventional smoothed particle hydrodynamic (SPH) method, for a function f at a point x within a domain , the approximate value of is given as below: = (Eqn. 2.1, p. 137, Zhang and Batra, 2004) In this equation, is the kernel or smoothing function. The approximate value of of f depends upon two parameters - the kernel W and the dilation h, the last providing support for W. It is essential that the kernel W should have the following properties - I) = 0, for , II) , III) , here is the Dirac delta function, IV) , and V) =. (Zhang and Batra, 2004) This conventional SPH method is not even zero-order consistent at the boundaries (Zhang and Batra, 2004). This forced Liu et al, 1995a,b, to introduce a corrective function that is a polynomial of the spatial coordinates, making the method order consistent (Zhang and Batra, 2004). Chen et al, (1999a,b) and Zhang and Batra, 2004, also sought to improve the conventional SPH method consistency in some manner. It is notable that the smoothed particle hydrodynamics method is a macroscopic model but it can be considered both as a continuum and particle method (Meakin et al, 2007). This is in particular context to the fact that the computational efficiency of purely particle methods is low in comparison to purely continuum ones (Meakin et al, 2007). Even for the SPH method, which can be taken as a continuum one, when computation is accepted at the particle stage, the afore-mentioned low consistencies develop at the boundaries as the no-slip principle cannot be successfully sustained. Also, the proper time scale for computing at simulated particle and molecular stages is so minuscule that low efficiencies have to be accepted in comparison to other macroscopic models such as experimental ones (Meakin et al, 2007). This is specifically why the inconsistency problem is removed to the point where the fluid can be considered incompressible and a continuum can thus be generated throughout the fluid structure. This paper will now investigate how Shao, 2006, has approached the inconsistency problem to develop a convention that can approximate an incompressible system upon waveforms in seawater. Shao, 2006: Model Application - Section 6, p. 606 Now the paper shall look into how Shao, 2006, has applied his model of the modified SPH protocol to investigate functional properties of wave breaking and overtopping at a sloped sea wall. The importance of such investigations has already been emphasised earlier in the paper. Shao, 2006, employed a wave tank to carry out his experiments. The wave tank had the following features: a sea wall 6.3m long and 1.0m high with a 1:6 seaward and a 1:3 landward slope; crest height of sea wall is 0.8m and width is 0.3m; distance between the upper offshore boundary of the sea wall and its toe was 1.0m; inlet and outlet were defined as x=0.0 and 6.3m respectively; original still water line was taken as y=0.0; the constant water depth was 0.7m; and regular waves at height 0.16m and 2.0s periods were taken; the resultant wavelength was accepted as 4.62m based on the height and period of waves (Shao, 2007). Diagram 1, Appendix, may be consulted. Time-dependent water surface elevations were measured at several locations within the experimental conditions and they were: WG4 (x=5.20m); WG3 (x=3.81m); WG2 (x=2.02m); and WG0 (x=0.0m). A particle spacing of was used and was considered fine enough to allow consideration of wave features like free surface deformations and velocity structures (Shao, 2006). This particle spacing implied that a total of N=6000 particles, including 400 constituted by the offshore wave maker, the sea bottom and the sloped sea wall, was involved on average in each wave cycle. Overtopped particles were separated and it was found that they constituted 3% of total for 10 waves. This is considered not too severe by real standards. The highly complex waveforms (Diagram 2) constituted of wave attack, run-up, run-down, breaking and overtopping were large deformations on the free surface. The corresponding velocity fields (Diagram 3) conformed to the waveform deformations, providing explanations. All diagrammatic representations were available within one wave period. Shao, 2006, notes that these were considered close enough to observed finite volume computations. In explanation, the concatenated sequence of events during the wave breaking and overtopping phenomenon as per the two diagrams - 2 & 3 - appendix, are as follows - as the first wave comes in it is observed in 2a and 3a that some particles of the previous wave continue to overtop the wall crest while the rest of the particles proceed to retreat under gravitational pull. As the incoming wave particles meet with the retreating particles at x=4.2m there is a strong backwash. There is a roller and large hydraulic jump evidenced at t=12.5s and, though much energy is dissipated by this contra-action of the particles of the previous wave, the incoming wave particles continue to move up the wall slope, though with decreased velocities, as is evidenced by the slowing wave front. Finally, the incoming wave particles manage to overtop at t=13.2s. Turbulence energy distribution during this phenomenon is as follows. The turbulence energy is normalised by the wave velocity c=. When the high turbulence areas are plotted (Diagram 4) Fig. 4a demonstrates that increases to the maximum at the point of wave breaking. The surf similarity parameter =0.958 demonstrates that the wave behaves as a surf with top curling and breaking as a surf breaker. The surf similarity parameter is based on the deep water wave and slope parameters - ; ; and . After the curling and breaking the wave continues to run upslope but the energy is much dissipated and the turbulence advected. Now the peak turbulence energy (Diagram 4b, Appendix) available for the upslope run is only 45% of that made available at the wave-breaking phenomenon. As per Diagram 4c, Appendix, the energy increases again as the wave commences to overtop and the maximum becomes manifest again at the temporal point when the wave overtops the wall crest. Fig 4c, Diagram 4, Appendix, also makes manifest the slight turbulence energies beginning to form at the next incoming wave front. Diagram 5, Appendix, makes manifest that the experimental data, Shao, 2004, when compared with the numerical data derived by SPH and from Li et al, is typically non-linear and in agreement. The nonlinearity is manifest by high and narrow wave crests and broad and shallow wave troughs. For the experimental data, only WG2 and WG3 are available but all the data are present for t=20.0-22.0s. For Li et al, data from the finest grid at 251*40 under the Smagorinsky Model (Shao, 2004) is made available as it presents the best numerical fit. AS per Diagram 5, Appendix, the points WG3 (close to breaking point), WG4 (surf zone) and WG5 (sea wall crest) the wave profile increases rapidly at first to decrease slowly next, in conformation with the nonlinear wave theory. This is as per the turbulence kinetic energies explained previously where the normalised turbulence energy is maximum at wave breaking point (the three breaking points at surf zone and wave crests on the central parts of the sea wall slope and the wave overtopping point near the crest of the sea wall). From Fig. 5a and 5b it is notable that the SPH computed lines conform more closely with the experimental data lines than the Li et al numerically derived ones. Shao, 2004, postulates that this is because of the numerical diffusion the Li et al data is subjected to and that is absent from the experimental and SPH data as smoothing function is operable in these on the particulate equations. In the contrast, the wave patterns available from Fig. 5c and 5d demonstrate that the SPH computations tend to over-emphasise the turbulence energies while underestimating the surface deformations in comparison with the Li et al numerical computations. This is so because the Li et al Smagorinsky model utilises a dynamic construct with an adjustable constant derived from water flow properties while the SPH incompressible flow characteristics allow derivation of a partly adjustable constant that is dependent upon quasi-steady flow characteristic that is not wholly conformant with transient flow characteristics available in wave features like wave breaking and overtopping. Thus, the SPH construct model with comparatively inadjustable constant coefficients is not as true to wave breaking and overtopping characteristics as the numerically derived Li et al data. Shao, 2004, concludes, as have other researchers before him, that the SPH time-averaged model tends to distribute turbulence energies uniformly across the wave period while the real time model may provide skewed distribution characteristics with higher turbulence energies at points immediately preceding wave breaking and overtopping as per the previously studied experimental data. Otherwise, as per Diagram 5, Appendix, the three data sets produce lines that conform closely as per wave phase and shape. Now, as per Diagram 6, Appendix, the SPH computations produce finite volume wave characteristics that conform across wave phases closely enough. The volume difference due to each phase difference is very small and the periodic overtopping features are faithfully reproduced. From Diagram 6, Appendix, it is observable that the mean overtopping volume per wave is 0.01 and the standard deviation is 0.0005. From these it is computable that the mean overtopping rate is 0.005 at T=2.0s. This is conformant with relevant coastal manuals that predict similar moderate overtopping events with similar characteristics. Such events may lead to structural and functional damage to protection structures available at coastlines. Overtopping Discharge: The paper shall now examine overtopping discharge calculated from the parametric values available from Shao, 2006, using several methods to compare and determine which of these methods produce the most approximate values. Weggel (1976): Weggel (1976) analysed the scaling models for wave overtopping by Saville en Caldwell (1953). He developed the following overtopping volume formula using the following parameters (Cui, 2004): V = volume of water in a wave above the still water level () = altering of the water surface (m) H = wave height (m) L = wave length (m) (Cui, 2004) He developed the following formula for calculating the overtopping volume: Q = = (Cui, Eqn. 2.4, p. 3, 2004) Where, T = wave period (s) Data from Shao, 2004, gives, H = 0.16m, L = 4.62m and T = 2s Therefore, = = 0.058 (Volume of water above the stillwater mark). Note: The above overtopping volume as per Weggel (1976) (Source: Cui, 2004) is not distributed across the length of the seawall that is 6.3m as per Shao, 2006. Therefore mean overtopping volume V: V = , where is length of seawall. Therefore, V = .0092/ Battjes Overtopping Volume: According to Battjes, Cui, 2004, the dimensionless wave overtopping rate 'b': b = 0.1() where, B = wave overtopping discharge volume per wave per crest length ; = deepwater wave length = 6.245m; = slope = 1:6; and H = wave height = 0.16m. Here B is taken as derived from Shao, 2006, and is 0.005/. Other parameters are as per Shao, 2006. 0.1 is a coefficient that has been developed by Cui, 2004, after comparison of the test results of Saville & Caldwell (1953). b = 0.1() = 0.0093 Daemrich et al, 2006: According to Daemrich et al, 2006, the derivation of the dimensionless overtopping volume Q is as follows: Q = , where = 0.0167 (Values used are from Shao, 2006, where is deepwater wave height at 0.189m; is deepwater wave length at 6.245m; is slope of seawall at 1:6. , as per Daemrich et al, 2006, is the breaker parameter. Therefore, Q = = .0086, where q is mean overtopping volume at .01 per metre of seawall and H, wave height, at 0.16m as per Shao, 2006. HR Wallingford Ltd., 1999: The HR Wallingford Ltd. 1999, R & D Technical Report W178 on design and assessment of wave overtopping of seawalls proves to be a singular document with valid instructions for calculating the maximum individual overtopping volume for a series of waves with particular parametric characteristics. The maximum individual overtopping volume per metre run of seawall for a sequence of waves is given by: = a (Eqn. 46, Box 4.5, p.31, Wallingford, 1999), where a, b are empirical coefficients and is the number of overtopping waves in the sequence . For sloped seawalls and for = 0.02 (= wave steepness) a = 0.85 and b = 0.76 (Box 4.5, p.31, Wallingford, 1999),. Using parameters from Shao, 2006, =8; and =0.005 (=mean individual overtopping discharge volume). a = 0.85= 0.85 0.005 = .00425 It is assumed that, as per Shao, 2006, the waves with height 0.16m and length 4.62m have steepness >.02 to meet criteria (Wallingford, 1999) for values of parameters a & b. For mean overtopping volume (= .005; Shao, 2006) lying between .0008 to .01: (Eqn. 37, p.27, Wallingford, 1999) Therefore, = = 6.97 (As per Wallingford, 1999, Box 4.5, p. 31, the value of 5 allows the Eqn. 46 to be used in this context). Therefore, for: = a= .00425= 0.0101 Results: Table 1: Method Weggel (1976) Battjes (Source: Cui, 2004) Daemrich et al, 2006 Wallingford, 1999 Formula = b = 0.1() Q = , = a Value (Overtopping Volume) .0092/ 0.0093 (dimensionless) .0086(dimensionless) 0.0101(dimensionless) Extra Parameters 0.1 a, b Notes: 1. For Weggel (1976) the overtopping volume formula involves as the parametric adjustment for the integration of (altering of the water surface) for the interval L(wavelength) to L. Also, the formula does not adjust for crest length of seawall and this is done separately. 2. For Battjes, (Source: Cui, 2004) the parametric adjustment is 0.1, a coefficient developed by Cui, 2004, as per test results of Saville & Caldwell (1953). 3. For Daemrich et al, 2006, the parametric adjustment, the breaker parameter similar to Shao, 2006, surf similarity parameter, is as in the table above. 4. For Wallingford, 1999, the parametric adjustments pertain to wave sequences where overtopping waves exceed the value 5. Discussion: It is noted here that the last three results are dimensionless while the first one, Weggel (1976), is in /. This is so because, for Weggel (1976), the formula is more universal and the mean overtopping discharge volume derived by Shao, 2006, at .005/s//s has not been an essential input. Actually, it is noted that Wallingford requires the mean volume per wave as input that is as per Shao, 2006, 0.01/. Nevertheless, the input value is taken as .005/ as the minimum volume per wave. Since the parameters have been chosen for individual overtopping volume at >0.01 the equation is considered valid. The rest of the three results are dimensionless and all of them require input from Shao, 2006, mean overtopping volumes. For Battjes and Wallingford, 1999, the value input is the averaged overtopping discharge volume at .005/s/ while for Daemrich et al, 2006, it is the individual overtopping volume at 0.01/ unaffected by the wave time period. It is thus noted that the Battjes and Wallingford, 1999, formulae coefficients at 0.1 and a and b respectively adjust for the wave time period which the Weggel (1976) and Daemrich et al, 2006, formulae parametric adjustment factors do not. It is notable that the HR Wallingford Ltd, 1999, formula extracts the most precise results for the individual dimensionless volume 0.0101 very near to Shao, 2006 0.01/. Nevertheless, the Wallingford coefficients a and b are fixed for a range of wave parameters and thus, there is little scope for diversity in parameters across this range. This is also true for Weggel (1976) where the parametric adjustment is relatively fixed at for an undefined range of wave parameters and offers less scope for more minute parametric adjustment. In that sense the Battjes (Source: Cui, 2004) and Daemrich et al, 2006 formulae are more flexible. There are the parametric adjustments very similar to the Shao, 2006 surf similarity parameter and though the dimensionless volumes, at .0093 (Battjes) and .0086 (Daemrich et al, 2006), are not as accurately aligned to the Shao, 2006 individual volume derivation at .01/ there is more scope for parametric adjustment across all ranges of wave parameters. In lieu of parameters taken into account the both the Battjes and Daemrich et al, 2006, formulae, both being more flexible than the others, take the tangent of the slope for their parametric adjustments and are more precise in this sense than Shao, 2006, who does not do this. Nevertheless, the Shao, 2006, SHP incompressible model is based on more parametric adjustments. The turbulence factor is regarded and turbulence is somewhat offset by wave celerity. Also, eddy viscosity is accounted for. Another major factor is the roughness of seawalls. Shao, 2006, mentions these last two. Though the Wallingford model also accounts for impacting and reflecting waves the Shao, 2006 turbulence k-c model is capable of more complex operations with possibilities of turbulence decrease with reverse energy from reflected waves as the new waves come in. There is some anomaly in units as the Shao, 2006, model provides for individual volumes in units while rest require volume in . This is because the Shao, 2006, model is an incompressible two-dimensional particle one. Nevertheless, the dimensionless results achieved have not been affected by this anomaly. It is also noted that the Wallingford result is for the maximum overtopping volume per wave as per Shao, 2006, parameters. The result achieved thus coincides spectacularly with that of Shao, 2006, for mean individual overtopping volume. Conclusion: The paper believes that the Shao, 2006, k-c turbulence SPH partly incompressible model is much more amenable to parametric adjustments than the others in study here and that it is also more flexible and accounts for more parameters than the others. Appendix: Diagram 1: Numerical wave flume and sloping sea wall for wave overtopping Source: Shao, 2006, Fig. 1, p. 607. Diagram 2: Wave breaking and wave overtopping Source: Shao, 2006, Fig. 2, p. 608. Diagram 3: Velocity fields during wave breaking and overtopping Source: Shao, 2006, Fig. 3, p. 609. Diagram 4: Turbulence kinetic energy distribution during wave breaking and overtopping Source: Shao, 2006, Fig. 4, p. 610. Diagram 5: Comparisons of computed water surface elevations by SPH (solid lines) with experimental (circles) and numerical (dotted lines) data of Li et al. Source: Shao, 2006, Fig. 5, p. 611. Diagram 6: Computed Overtopping Volume for each Wave Source: Shao, 2006, Fig. 7, p. 614. References: Daemrich, Karl-Friedrich, et al, Irregular Wave Overtopping Based on Regular Wave Tests, November, 2006. Accessed on 30th March, 2008, at: http://www.comc.ncku.edu.tw/joint/joint2006/pdf/47%20COS%2008.pdf Meakin, Paul, et al, Particle methods for simulation of multiphase fluid flow and biogeological processes, 2007. Accessed on 16th March, 2008, at: http://www.iop.org/EJ/article/1742-6596/78/1/012047/jpconf7_78_012047.pdfrequest-id=7dca4705-a112-4717-84aa-e3e14427b5ca Shao, Songdong, Incompressible SPH simulation of wave breaking and overtopping with turbulence modeling, International Journal for Numerical Methods in Fluids, 2006; 50; 597-621. Wallingford, HR, Ltd., Wave Overtopping of Seawalls, R & D Technical Report W178, 1999. Accessed on 29th March, 2006, at: http://ftp.hrwallingford.co.uk/downloads/projects/overtopping/overtopping_manual.pdf Zhang, G.M., and Batra, R.C., Modified smoothed particle hydrodynamic method and its application to transient problems, Computational Mechanics 34(2004); 137-146. Read More