Empirical Drag Coefficient Research Articles

A multi-faceted methodology for calibration of coastal vegetation drag coefficient

The accurate prediction of wave height attenuation due to vegetation is crucial for designing effective and efficient natural and nature-based solutions for flood mitigation, shoreline protection, and coastal ecosystem preservation. Central to these predictions is the estimation of the vegetation drag coefficient (Cd). The present study undertakes a comprehensive evaluation of three distinct methodologies for estimating the drag coefficient: traditional manual calibration, calibration using a novel application of state-of-the-art metaheuristic optimization algorithms, and the integration of an empirical bulk drag coefficient formula (Tanino and Nepf, 2008) into the XBeach non-hydrostatic wave model. These methodologies were tested using a series of existing laboratory experiments involving nearshore vegetation on a sloping beach. A key innovation of the study is the first application of metaheuristic optimization algorithms for calibrating the drag coefficient, which enables efficient automated searches to identify optimal values aligning with measurements. We found that the optimization algorithms rapidly converge to precise drag coefficients, enhancing accuracy and overcoming limitations in manual calibration which can be laborious and inconsistent. While the integrated empirical formula also demonstrates reasonable performance, the optimization approach exemplifies the potential of computational techniques to transform traditional practices of model calibration. Comparing these strategies provides a framework to determine effective methodology based on constraints in determining the vegetation drag coefficient.

Determination Of Drag And Inertia Coefficients By An Analytical Model

Hard structures like dykes or groins have been recognized for their negative environmental impact and limited durability in the face of climate change (Sutton-Grier et al., 2015). The concept of Shore Soft Engineering (SSE) has allowed ecological considerations to be embedded in the design of coastal protection (Hartig et al., 2011). Among emerging defense structures, nature-based solutions are built with natural materials and rely on physical properties and mechanisms observed in nature. They aim to protect, manage and restore ecosystems while providing some benefit to the human-being and biodiversity (Cohen-Shacham et al., 2016); they offer an interesting alternative to hard structures. However, implementing these solutions in urbanized coastal areas where ecosystems are vulnerable can be challenging. Another alternative approach is using biomimetic solutions, which can provide the functions of natural systems like seagrass, dunes, or coral through robust human-made constructions. Natural habitats exist in a variety of more or less complex forms, ranging from rigid (mangroves, coral) to flexible (seagrass) (Mullarney and Henderson, 2018); and they drive changes in the current profile, in wave dissipation or in sediment motion. Understanding how to mimic these systems, in particular their internal geometry and hydrodynamic effects, is challenging. This complexity makes the development of biomimetic solutions difficult (Perricone et al., 2023). In this context, this study strictly focuses on wave dissipation by flexible systems. Wave dissipation by natural habitats has been extensively studied through laboratory experiments (Houser et al., 2015) and in-situ measurements (Bradley and Houser, 2009). The pioneer analytical model (Dalrymple et al., 1984) depicted aquatic vegetation as rigid cylinders; however, the rigidity assumption fails to capture the inherent flexibility of plants. Alternative strategies emerged to account for this flexibility, such as using an empirical drag coefficient (Mendez and Losada, 2004) or introducing an effective length, representing the length that a rigid cylinder would have to dissipate the same wave height as flexible cylinders (Luhar and Nepf, 2016). However, to define this effective length or an empirical drag coefficient, it is necessary to carry out in-situ measurements. Numerous analytical models based on force balance (Luhar and Nepf, 2016; Leclercq and de Langre, 2018) have been developed to represent the movement of flexible vegetation like seagrass. These models depict the flexible vegetation as a series of segmented rigid stems attached to each other and subjected to oscillating flows. Some models attempt to link stem motion with wave dissipation to integrate this coupling into numerical models (Yin et al., 2022). However, these models still require unknown quantities such as drag and inertial coefficients. The present study aims to develop an analytical model for determining drag and inertial coefficients based solely on the geometry, the structure flexibility and wave forcing. At last, the method could help better represent the dissipation into numerical simulations without the need for prior parametrization.

Drag coefficients of double unequal-diameter flexible cylinders in tandem undergoing vortex/wake-induced vibrations

A reliable load-to-response prediction for flexible risers under drag forces can protect the multiple-riser system from collision and optimize its spatial arrangement. To investigate the drag coefficients for double risers in tandem, a series of experiments for two unequal-diameter flexible cylinders in different initial gap spacings was performed under uniform flows with Reynolds numbers up to 11,328. Based on the reconstructed mean drag displacement from measured strains, a distributed drag force amplified by vortex-induced vibrations can be identified reversely. Meanwhile, a modified wake model was developed to estimate the mean wake velocity for the downstream cylinder. With this mean wake velocity, the drag coefficients of the downstream cylinder were obtained. The features of drag coefficients and vortex/wake-induced vibrations were found to be closer to the corresponding features of an isolated cylinder. Furthermore, an empirical drag coefficient model for double unequal-diameter cylinders in tandem was proposed, which accounts for the drag amplification effect by vibrations, tandem spacing, upstream-to-downstream cylinder diameter ratio, mean wake velocity, etc. The results show that this empirical drag coefficient model can obtain the prediction results with relatively higher precision.

Modelling wave attenuation through submerged vegetation canopies using a subgrid canopy flow model

Coastal canopies formed by aquatic vegetation can significantly modify nearshore wave processes, which has motivated numerous laboratory and field studies investigating how wave attenuation occurs for a wide range of canopies as a function of wave conditions and canopy properties. To predict wave dissipation by canopies, numerous formulations have been derived to describe canopy drag through use of empirical drag coefficients. In many cases, drag coefficients are used as a tuning parameter to account for complex processes that govern how wave-driven flows interact with individual plants within canopies. This approach, however, can greatly limit the predictive capabilities of both empirical and numerical models. In this study, we extend a non-hydrostatic (phase-resolving) depth-integrated wave model (XBeach non-hydrostatic) with an efficient and robust subgrid canopy flow model to account for the important properties of submerged canopy flows that govern canopy drag forces and wave dissipation. A combination of wave heights, velocities and direct canopy force measurements, as well as data obtained from literature, were used to validate the model, showing the model has good skill in capturing the governing flow dynamics for both rigid and flexible vegetation canopies. By incorporating the canopy flow dynamics, the extended model allows for application of well-established drag coefficients for individual canopy elements to accurately predict wave attenuation over submerged vegetation canopies, rather than relying on empirical formulations or calibration. With the current extension, the model remains computationally efficient, allowing for use on coastal (∼km) scales. • A canopy flow subgrid model was implemented in the open-source XBeach model. • The model was verified using observed wave heights, velocities and canopy forces. • It can accurately predict wave evolution over rigid and flexible submerged canopies. • The model is robust and computationally efficient for use in large scale applications. • Not accounting for canopy flow can lead to substantial errors in wave predictions.

Pressure- and Size-Dependent Aerodynamic Drag Effects on Mach 0.3–2.2 Microspheres for High-Precision Micro-Ballistic Characterization

The acceleration of microparticles to supersonic velocities is required for microscopic ballistic testing, a method for understanding material characteristics under extreme dynamic conditions, and for projectile gene and drug delivery, a needle-free administration technique. However, precise aerodynamic effects upon supersonic microsphere motion at sub-300 Reynolds numbers have not been quantified. We derive drag coefficients for microspheres traveling in air at subsonic, transonic, and supersonic velocities from the measured trajectories of microspheres launched by laser-induced projectile acceleration. Moreover, the observed drag effects on microspheres in atmospheric (760 Torr) and reduced pressure (76 Torr) are compared with existing empirical data and drag coefficient models. We find that the existing models adequately predict the drag coefficient for subsonic microspheres, while rarefaction effects cause a discrepancy between the model and empirical data in the supersonic regime. These results will improve microsphere flight modeling for high-precision microscopic ballistic testing and projectile gene and drug delivery.

Evaluation of Morison approach with CFD modelling on a surface-piercing cylinder towards the investigation of FOWT Hydrodynamics

To predict hydrodynamic forces on slender cylinders, Morison formula is commonly used. The expression is composed of two terms including an empirical drag and inertia coefficient. In the present study, a constrained cylinder subjected to regular waves is modelled using Computational Fluid Dynamics (CFD) and Morison coefficients are derived from the CFD loads.A numerical wave tank, including a surface-piercing cylinder, is implemented to generate, propagate and absorb waves in a controlled way. The cylinder is discretised into slices for each of which the Morison formulation is fitted to the CFD loads to determine Morison coefficients.Morison formulation is in very good agreement with the numerical load over the entire length of the cylinder, except for the slices near the free surface which are outside the applicability range of Morison formula. The loads near the free surface being negligible compared to the total load, the total theoretical load obtained reproduces well the experimental and the CFD loads.The CFD simulations proved to be a powerful tool to predict the wave loads and calibrate Morison coefficients. On this basis, the methodology will be extended to floating offshore wind platforms, combining several cylinders with different inclinations, sizes and positions.

Assessment of behavioral modification techniques through immersed boundary method simulation of binary particle interactions in isotropic turbulence

Behavioral modification effects for particle-laden turbulent flows are developed and assessed through high-fidelity modeling using an implementation of the mirroring ghost-cell based immersed boundary method in conjunction with direct numerical simulation. The continuous phase uses the open-source spectral element method-based solver, Nek5000. A dynamic form of the mirroring immersed boundary method is described that also solves for interparticle attraction and repulsion forces allowing for nontrivial collision outcomes such as agglomeration. The solid-phase solver is validated against empirical drag coefficient data as well as spherical bouncing experiments with excellent agreement obtained at low particle Reynolds numbers. Periodic boxes of homogeneous isotropic turbulence are generated using the linear forcing method at Reλ=29, 51, and 120. Ensembles of structure-resolved binary particle collisions are then studied within these boxes, considering the variation of six key mechanical and chemical parameters. These are the coefficient of restitution, Hamaker constant, surface charge potential, inverse Debye length, temperature, and Reynolds number. It is established that the coefficient of restitution, inverse Debye length, and Reynolds number have the greatest impact on the resulting particle motion and interaction by considering probability density functions of intersurfacial distance and relative particle velocities. Suggestions for real-world procedures that modify these parameters in order to either encourage or discourage particle interaction and potential agglomeration are discussed.

Distribution of drag coefficients along a flexible pipe with helical strakes in uniform flow

The influences of helical strake geometry and vortex-induced vibration (VIV) on the drag coefficients of a long flexible pipe with helical strakes are investigated through uniform flow experiments with the Reynolds (Re) number ranging from 1.1 × 104 to 9.5 × 104. The strains caused by drag forces of a bare pipe and six different straked pipes are measured by fiber bragg grating (FBG) sensors. According to the inverse analysis method and Morison formula, the drag forces and coefficients at each section of the pipe models are identified. Then, the mean drag coefficients of the bare and straked pipes are compared. The results show that helical strakes can effectively stabilize the fluctuation of drag coefficients with Re numbers. An unexpected phenomenon is that helical strakes with pitches of 5 D can reduce the mean drag coefficients. Moreover, the mean drag coefficients of straked pipes are less sensitive to VIV response. Furthermore, an empirical mean drag coefficient prediction formula for straked pipes is proposed, which accounts for the VIV response and geometry of helical strakes. Its applicability is well verified via comparison between predicted and experimental results.

Discussion of “Field-based numerical model investigation of wave propagation across marshes in the Chesapeake Bay under storm conditions” by Juan L. Garzon, Tyler Miesse and Celso M. Ferreira

In the recent paper by Garzon, Miesse, and Ferreira (Coastal Engineering, DOI: https://doi.org/10.1016/j.coastaleng.2018.11.001), the authors collected field data from Chesapeake Bay, and numerically simulated the wave attenuation by vegetation using five different empirical drag coefficient formulas. The implementation of the formula from Jadhav (2012) was incorrect by Garzon et al. (2018), resulting in considerable under-prediction of wave height in comparison of the field data. This discussion aims to present the correct way to use the drag coefficient formula from Jadhav (2012). Our numerical results show that the drag coefficient formulas of Jadhav (2012) and Jadhav and Chen (2012) are suitable for modeling wave attenuation by salt marshes in Chesapeake Bay.

Eulerian–Lagrangian flow-vegetation interaction model using immersed boundary method and OpenFOAM

This paper presents a novel coupled flow-vegetation interaction model capable to resolve the flow and motion of flexible vegetation with large deflections simultaneously on a hybrid Eulerian–Lagrangian grid. The hydrodynamics model is based on a Navier–Stokes flow solver with the Volume of Fluid surface capturing method in OpenFOAM. The deforming and moving vegetation are tracked through a series of Lagrangian grids attached to the vegetation and embedded in the computational domain of OpenFOAM. The governing equations for the flexible vegetation motion are based on a slender rod theory and are solved by a Finite Element Method on a vegetation-following Lagrangian grid. The hydrodynamics and vegetation models are coupled through the vegetation-induced hydrodynamic forces using a novel diffused immersed boundary method to avoid excessive fluid grid refinements around the vegetation. The standard two-equation k − ε turbulence model is extended for vegetated flow by incorporating additional closure terms of vegetation effects. The newly developed coupled flow-vegetation model is validated against experiments for both individual single-stem vegetation and a vegetation patch in a large-scale wave flume. Model results of asymmetric displacement of the vegetation motion, wave height decay, and wave kinematics within and outside the vegetation patch are in good agreement with measurements. The drastical change in wave radiation stress by the presence of vegetation patch leads to a strong current jet near the top of vegetation patch, which in turn drives a local circulation pattern around the vegetation patch. The effect of vegetation bending stiffness on the model results and empirical drag and inertia coefficients for calculating the hydrodynamic forces are examined in detail. Maximum turbulence energy is observed close to the bed and the top of vegetation patch just before and after the wave crest/trough, where the relative flow velocity to vegetation is large.

Hybrid modeling of flows over submerged prismatic vegetation with different areal densities

In the modeling of vegetated flows an efficient approach is to solve the Double Averaged Navier Stokes (DANS) equations, which are obtained from the spatial and temporal averaging of the Navier Stokes equations. The resistance effects of vegetation are modeled by a drag force density term with an empirical bulk drag coefficient, as well as by turbulence terms characterized by a length scale parameter. These empirical parameters are dependent on the areal density of vegetation and the spatial distribution pattern of vegetation elements and have seldom been studied. In this work the effect of spatial distribution of vegetation elements on the bulk drag coefficient is investigated by computing explicitly the flows around the vegetation elements. The results show that the bulk drag coefficient increases with the longitudinal vegetation element spacing, decreases with the lateral vegetation element spacing and can have multiple values for a given areal density of vegetation. In the DANS model the vegetation induced turbulence is simulated by a novel k-ε type model embracing an empirical length scale parameter. The length scale parameters are calibrated against previous experiments and a new set of experiments with high areal density of vegetation. The DANS model is subsequently verified by two cases with the same vegetation density and different distribution patterns.

Sensitivity of surface drag predictions using Monin–Obukhov similarity theory in non-ideal flow conditions

Turbulent flux measurements in the atmospheric surface layer (ASL), acquired from a vertical tower of sonic anemometers, are used to investigate the transient behavior of the drag coefficient over a relatively flat, homogeneous, desert playa surface. The goal of the study is to quantify the extent to which Monin–Obukhov similarity theory (MOST) successfully predicts the empirical drag coefficient when the data are not necessarily restricted to idealized conditions. Here, “idealized” refers to the conditions under which MOST is assumed to be valid, namely that the ASL exhibits statistically stationary behavior and stresses are constant with height. In order to quantify the agreement between the empirical and theoretical drag coefficients, uncertainty bands as a function of time of day were calculated for each. These uncertainty bands account for random errors and nonstationarity effects in the momentum flux, horizontal wind velocity, and stability parameter, as well as uncertainties associated with the model parameters appearing in MOST, i.e., the von Karman constant and aerodynamic roughness length. Results show that the average uncertainties in the empirical and theoretical drag coefficients are about $$\pm \,30\%$$ and $$\pm \,20\%$$ , respectively. In the case of the former, the major contribution lies from uncertainty in the momentum flux measurements; while, the latter is dominated by uncertainty in the von Karman constant. Joint probability density functions reveal that large differences between the empirical and theoretical drag coefficient have a high probability of occurrence when the ASL is nonstationary and/or exhibits a non-constant stress layer.

Electrohydroelastic Euler–Bernoulli–Morison model for underwater resonant actuation of macro-fiber composite piezoelectric cantilevers

Bio-inspired hydrodynamic thrust generation using smart materials has received growing attention over the past few years to enable improved maneuverability and agility, small form factor, reduced power consumption, and ease of fabrication in next-generation aquatic swimmers. In order to develop a high-fidelity model to predict the electrohydroelastic dynamics of macro-fiber composite (MFC) piezoelectric structures, in this work, mixing rules-based (i.e. rule of mixtures) electroelastic mechanics formulation is coupled with the global electroelastic dynamics based on the Euler–Bernoulli kinematics and nonlinear fluid loading based on Morison’s semi-empirical model. The focus is placed on the dynamic actuation problem for the first two bending vibration modes under geometrically and materially linear, hydrodynamically nonlinear behavior. The electroelastic and dielectric properties of a representative volume element (piezoelectric fiber and epoxy matrix) between two subsequent interdigitated electrodes are correlated to homogenized parameters of MFC bimorphs and validated for a set of MFCs that have the same overhang length but different widths. Following this process of electroelastic model development and validation, underwater actuation experiments are conducted for different length-to-width aspect ratios (L/b) in quiescent water, and the empirical drag and inertia coefficients are extracted from Morison’s equation to establish the electrohydroelastic model. The repeatability of these empirical coefficients is demonstrated for experiments conducted using aluminum cantilevers of different aspect ratios with a focus on the first two bending modes. The convergence of the nonlinear electrohydroelastic Euler–Bernoulli–Morison model to its hydrodynamically linear counterpart for increased L/b values is also reported. The proposed model, its harmonic balance analysis, and experimental results can be used not only for underwater piezoelectric actuation, but also for sensing and energy harvesting problems.

Distribution of drag force coefficient along a flexible riser undergoing VIV in sheared flow

The drag force coefficients of a flexible riser undergoing vortex-induced vibration (VIV) in sheared flow are investigated for Reynolds numbers (Re) up to 1.2×105. Based on the drag forces theoretically calculated by the beam theory using the strains measured in a scale model test, the properties and distribution of the drag coefficients are investigated, and a new empirical model for estimating the drag coefficient on a flexible riser undergoing VIV is proposed. The results show that VIV leads to non-uniform distribution of the drag coefficient and amplifies the drag coefficient, and the local drag coefficient can reach up to 3.2. For Re values from 1.0×104–1.2×105, the mean drag coefficient is between 1.3 and 2.0 and decreases as Re increases. Furthermore, the empirical drag coefficient prediction model obtained from experiments under low Re is not suitable for high Re. The corrected empirical prediction model, which accounts for the effect of the flow velocity, the VIV dominant mode number and the dominant frequency, can be used to predict riser drag coefficients under VIV more accurately at high Re values up to 1.2×105.

Experimental study on the drag coefficient of single bubbles rising in static non-Newtonian fluids in wellbore

Using an empirical drag coefficient model to investigate the laws of bubble rising in non-Newtonian fluids is important for calculating the safe cycle time in avoiding typhoon, tripping operation in offshore drilling engineering. Herein, the drag coefficient for a single bubble rising steadily in a static non-Newtonian fluid within a wellbore was experimentally investigated using a vertical, cylindrical wellbore mimic. The effects of viscosity, density and surface tension of the fluids, as well as bubble diameters on the drag coefficient were studied. The experimental results indicate that the drag coefficient increases with the increase of the fluid viscosity, surface tension and bubble diameters, while the solution density has little effect on the drag coefficient. Meanwhile, the relationship between Reynolds number (Re) and drag coefficient and that between Re and Eötvös numbers (Eo) were analyzed, respectively. The result suggests that surface motion leads to a transition of drag coefficient. A new correctional model for the drag coefficient of single bubbles rising in static non-Newtonian fluids in a wellbore was obtained, which showed a good agreement with the experimental data.

Drag Forces on Large Cylinders

AbstractIntroducing large woody debris into streams is a common practice in restoration projects. Beyond the complexity of flow patterns and sediment movements in streams where woody debris are found or placed, it seems that our understanding of the basic hydraulics of large roughness elements in small channels remains limited. Underestimating the drag force affecting large roughness elements can compromise the success of stream restoration projects. Results from a simple experimental setting confirm that drag force estimates based on approaches developed for small cylinders are not valid when applied to large cylinders. Indeed, the classic drag force equation that uses an empirical drag coefficient is found to significantly underestimate measured drag forces, even when corrected for the ‘blockage ratio’. In contrast, application of specific momentum can yield good estimates of the drag force. A dimensionless depth is defined in a 1D context as a function of the flow depth, critical flow depth and cylinder diameter. A cylinder is considered to be ‘large’ when this dimensionless depth is smaller than 2. In this instance, a relationship is established to estimate the upstream flow depth and the drag force acting on the cylinder. This research bridges the small roughness element theory widely recognized in hydraulic engineering with the theory applicable to large, flow controlling structures such as weirs. From a practical perspective, this research can be used to assist in the design of engineered large woody debris structures. Copyright © 2015 John Wiley & Sons, Ltd.

Comparison of Model-Scale and Full-scale Deceleration of a Fully Skirted Air Cushion Vehicle in Sideslip

This article provides experimental results and shows comparisons of 90 sideslip performance of a fully skirted air cushion vehicle using theoretical predictions, 1/12th model scale data, and full-scale data. The goal is to establish a relation among the three data sets and draw conclusions for use in future predictions. First, this article presents results and analysis of tow tank data obtained in late 2008. Then, the Froude scaled data are used to obtain an empirical drag coefficient. A comparison is made between this approach and a theoretical prediction proposed in previous work. An iterative, one dimensional deceleration algorithm is then constructed using the coefficients to predict the deceleration of a full-scale craft having similar skirt characteristics. The predictions are performed for three craft weights and the resulting deceleration rates as a function of Froude number are presented. Data obtained from full-scale testing are then compared with the results of both the model-based algorithm and the theoretical prediction. In general, the model-based simulation overpredicts the deceleration rate for a full-scale craft, whereas the theoretical prediction is more accurate. The model simulation is recomputed using a developed correction factor and is plotted against the theoretical and full-scale deceleration, revealing favorable results. Lastly, a review of the technique is described and recommendations for improvements and following work are provided.

Comparison of Model-Scale and Full-scale Deceleration of a Fully Skirted Air Cushion Vehicle in Sideslip

This article provides experimental results and shows comparisons of 90° sideslip performance of a fully skirted air cushion vehicle using theoretical predictions, 1/12th model-scale data, and full-scale data. The goal is to establish a relation among the three data sets and draw conclusions for use in future predictions. First, this article presents results and analysis of tow tank data obtained in late 2008. Then, the Froude scaled data are used to obtain an empirical drag coefficient. A comparison is made between this approach and a theoretical prediction proposed in previous work. An iterative, one-dimensional deceleration algorithm is then constructed using the coefficients to predict the deceleration of a full-scale craft having similar skirt characteristics. The predictions are performed for three craft weights and the resulting deceleration rates as a function of Froude number are presented. Data obtained from full-scale testing are then compared with the results of both the model-based algorithm and the theoretical prediction. In general, the model-based simulation overpredicts the deceleration rate for a full-scale craft, whereas the theoretical prediction is more accurate. The model simulation is recomputed using a developed correction factor and is plotted against the theoretical and full-scale deceleration, revealing favorable results. Lastly, a review of the technique is described and recommendations for improvements and following work are provided.

Breaking wave loads on monopiles for offshore wind turbines and estimation of extreme overturning moment

Measurements of breaking wave forces on vertical circular columns have been re-analysed, where substantial magnification over forces because of non-breaking non-linear waves may occur, by a factor of up to 2.8. The analysis shows that this factor increases as the depth parameter kd decreases, being close to unity for kd > 1.5 (k is the wave number and d is depth). Non-breaking wave forces are predicted reasonably by non-linear stream function wave theory using Morison's equation with empirical drag and inertia coefficients. A study was then made of the magnitude of wave overturning moment in relation to hub moment due to wind on standard 2 and 5 MW turbines with a 6 m diameter column. This showed that the wave moment in extreme conditions is greater than wind ‘hub’ moment for depths greater than about 7 and 13 m for the 2 and 5 MW turbines, respectively. Monopiles are normally used in depths below about 30 m and extreme moments due to waves occur when waves are depth-limited and defined by the Miche criterion, well below the limiting value of kd for the onset of depth-induced breaking. The maximum moment for these depths is expected to be predicted reasonably.

Wave-induced drag force on vegetation under shoaling random waves

This paper provides a practical method for estimating the drag force on a vegetation field in shoaling conditions beneath non-breaking and breaking random waves. This is achieved by using a simple drag formula based on two empirical drag coefficients given by Méndez et al. (1999) and Méndez and Losada (2004), respectively, in conjunction with a stochastic approach. Here the waves are assumed to be a stationary narrow-band random process and propagating in shallow waters. The effects of shoaling and breaking waves are included by adopting the Méndez et al. (2004) wave height distribution. Results are presented and discussed for different slopes, and an example of calculation is also provided to demonstrate the application of the method.