Surface Boundary Conditions Research Articles

Design and Preparation of Nematic Colloidal Particles.

Colloidal systems are abundant in technology, in biomedical settings, and in our daily life. The so-called "colloidal atoms" paradigm exploits interparticle interactions to self-assemble colloidal analogs of atomic and molecular crystals, liquid crystal glasses, and other types of condensed matter from nanometer- or micrometer-sized colloidal building blocks. Nematic colloids, which comprise colloidal particles dispersed within an anisotropic nematic fluid host medium, provide a particularly rich variety of physical behaviors at the mesoscale, not only matching but even exceeding the diversity of structural and phase behavior in conventional atomic and molecular systems. This feature article, using primarily examples of works from our own group, highlights recent developments in the design, fabrication, and self-assembly of nematic colloidal particles, including the capabilities of preprogramming their behavior by controlling the particle's surface boundary conditions for liquid crystal molecules at the colloidal surfaces as well as by defining the shape and topology of the colloidal particles. Recent progress in defining particle-induced defects, elastic multipoles, self-assembly, and dynamics is discussed along with open issues and challenges within this research field.

Comparison of wave diffraction forces on a surface-piercing body for various free-surface grid update schemes

This study analyzed the difference in wave diffraction force of a surface piercing body according to the grid update scheme of free surface nodes in the time integration process. A two-dimensional fully nonlinear numerical wave tank based on potential flow theory and boundary element method was used for numerical analysis. The MEL (mixed Eulerian–Lagrangian) method was used for free surface node treatment. This method was divided into a semi-Lagrangian approach and a full Lagrangian approach (material node approach).The Runge–Kutta 4th-order integration method was used to integrate the free surface boundary condition temporally. In the internal sub-time step of the time integration process, the frozen coefficient scheme and the full update scheme were used according to the node update scheme, respectively. The vertical diffraction forces of each frequency component for node update schemes were compared. The difference in the 2nd-order vertical forces according to wave frequency, wave steepness, time step interval, and node spacing was investigated. A more precise time-domain numerical analysis method was proposed for high-order external forces in the wave-floating body interaction.

The Influence of Characteristic Sea State Parameters on the Accuracy of Irregular Wave Field Simulations of Different Complexity

The accurate description of the complex genesis and evolution of ocean waves, as well as the associated kinematics and dynamics is indispensable for the design of offshore structures and the assessment of marine operations. In the majority of cases, the water-wave problem is reduced to potential flow theory on a somehow simplified level. However, the nonlinear terms in the surface boundary conditions and the fact that they must be fulfilled on the unknown water surface make the boundary value problem considerably complex. Hereby, the contrary objectives with respect to a very accurate representation of reality and numerical efficiency must be balanced wisely. This paper investigates the influence of characteristic sea state parameters on the accuracy of irregular wave field simulations of different complexity. For this purpose, the high-order spectral method was applied and the underlying Taylor series expansion was truncated at different orders so that numerical simulations of different complexity can be investigated. It is shown that, for specific characteristic sea state parameters, the boundary value problem can be significantly reduced while providing sufficient accuracy.

Hydroelastic potential flow solver suited for nonlinear wave dynamics in ice-covered waters

The aim of this paper is to introduce a fully nonlinear numerical finite element solver for the simulation of nonlinear wave processes in the presence of a solid ice sheet. In this study, solid ice cover referred to the size of the ice sheet and denoted that the length of the ice sheet was many times larger than the longest relevant wavelength. The complexity of the ice sheet characteristics was assumed to be homogeneous, isotropic and in the linear elastic plate regime so that the deformation process could be modeled by the Kirchhoff–Love plate ansatz. The method presented was verified and validated for different ice dimensions and wave scenarios. At the beginning, the implementation of the flexural rigidity to the free surface boundary condition was verified by comparing the analytical wave-ice dispersion relation to simulation results with small amplitude regular waves and varying ice dimensions. Afterwards, the general applicability was validated by means of wave-ice experiments. The experiments were performed in the ice tank at Hamburg Ship Model Basin HSVA comprising regular waves and transient wave groups.

Towards automatic finite-element methods for geodynamics via Firedrake

Abstract. Firedrake is an automated system for solving partial differential equations using the finite-element method. By applying sophisticated performance optimisations through automatic code-generation techniques, it provides a means of creating accurate, efficient, flexible, easily extensible, scalable, transparent and reproducible research software that is ideally suited to simulating a wide range of problems in geophysical fluid dynamics. Here, we demonstrate the applicability of Firedrake for geodynamical simulation, with a focus on mantle dynamics. The accuracy and efficiency of the approach are confirmed via comparisons against a suite of analytical and benchmark cases of systematically increasing complexity, whilst parallel scalability is demonstrated up to 12 288 compute cores, where the problem size and the number of processing cores are simultaneously increased. In addition, Firedrake's flexibility is highlighted via straightforward application to different physical (e.g. complex non-linear rheologies, compressibility) and geometrical (2-D and 3-D Cartesian and spherical domains) scenarios. Finally, a representative simulation of global mantle convection is examined, which incorporates 230 Myr of plate motion history as a kinematic surface boundary condition, confirming Firedrake's suitability for addressing research problems at the frontiers of global mantle dynamics research.

Application of normalized SPH method to inviscid fluid flows with large deformation of free surface

A new normalized smoothed particle hydrodynamics (SPH) method has been presented recently and applied to transient solid mechanics problems of small deformation. The method can overcome boundary deficiency of the conventional SPH method without troublesome inversion of corrective matrix and provide convenience for treating boundary conditions. In this paper, a formulation of the normalized SPH method is derived for modeling inviscid fluid flows with large deformation of free surface. The treatment of dynamic free surface boundary conditions in the present method is addressed and theoretically compared with that in the conventional SPH method. A method based on kernel estimation is proposed for calculation of boundary summation and compared with the method based on α-shape. The normalized SPH method is applied to modeling inviscid free surface flows. Numerical results demonstrate that the present method is capable of modeling large deformation of free surface and can produce higher accuracy than the conventional SPH method, especially at boundary regions. The newly developed kernel estimation method for the calculation of boundary summation is more efficient than the α-shape-based method, and makes the computational efficiency of the normalized SPH method close to the conventional SPH method. To simulate violent free surface flows, a particle shifting technique is introduced in the present method. With this technique, the method is applied to simulating extremely large deformation, fragmentation and reconnection of free surface.

Development of Analytical Solutions on Velocities of Regular Waves Generated by Bottom Wave Makers in a Flume

Analytical solutions for two-dimensional velocities of regular waves generated by bottom wave makers in a flume were derived in this study. Triangular and rectangular bottom wave makers were adopted. The velocity potential was derived based on the linear wave theory with the bottom moving boundary condition, kinematic and dynamic free surface boundary conditions. Then, analytical solutions of two-dimensional particle velocities were derived from the velocity potential. The velocity potential and two-dimensional particle velocities which were derived as complex integral equations were numerically calculated. The solutions showed physically valid results as velocities of regular waves generated by bottom wave makers in a flume.

Dynamic frictional contact mechanics between a functionally graded orthotropic medium and a moving flat punch

The present study establishes a general theory for dynamic frictional contact mechanics between an orthotropic graded half-plane and a moving rigid punch of a flat profile. The rigid punch moves over the orthotropic graded half-plane at a constant subsonic speed. Analytical method is developed for the contact mechanics problem based on two-dimensional elastodynamics. Stresses in the orthotropic graded medium are determined considering plane elasticity. Governing partial differential equations are solved analytically by the use of Galilean and Fourier transformations. The unknown functions appearing in the displacement components are determined considering the surface boundary conditions. The mixed boundary value problem is then reduced to a singular integral equation of the second kind which is solved numerically using an appropriate expansion-collocation technique. The contact stress results of the present analytical study are compared to those generated through a finite element analysis regarding the elastostatic case. Moreover, the contact stress results obtained in the elastodynamic case are also verified using some analytical results available in the literature. Successful agreement is attained in all the comparative analyses. Normal contact stress, lateral contact stress and punch stress intensity factors are calculated for various values of punch speed, coefficient of friction, in-homogeneity constant and orthotropic elastic constants. Results of the present study clearly indicates that the utilization of elastodynamic theory is crucial to get more realizable and accurate estimations of contact stresses especially in high speed sliding conditions.

Simultaneous Identification of Fundamental Frequency, Mode Shape, and Damping of a Bridge by a Passing Vehicle-Trailer System

A theoretical framework is proposed to identify fundamental frequency, mode shape, and damping of a bridge simultaneously by a passing vehicle-trailers system with only accelerometers. The framework is the first attempt to use indirect method to identify all fundamental modal properties of a bridge simultaneously without shaker, laser sensors and pre-known information. The vehicle performs as both “tractor” and “excitor” and the trailers perform as “message carriers”. The residual response defined as the response difference of one trailer to another one was usually adopted in the previous studies to remove the adverse effect of road roughness; however, it lacks theoretical support. Instead, the residual response of one trailer–bridge contact point to another one is used in this study, and theoretical analysis proves that it can effectively eliminate the adverse effect of road roughness. Different from previous studies, the residual response of contact point in this study is obtained in the frequency domain rather than the time domain. Frequency and damping can be evaluated by applying single degree-of-freedom (SDOF) model-based curve fitting technique to several discrete values from the Fourier transform of the residual response. Mode shapes can be reconstructed by dividing the Fourier transform of one residual response at the natural frequency by another one without time lag. The proposed framework is validated by independent finite element (FE) analysis and it shows great performance. Parametric study including vehicle speed, measurement noise, calibration error, road surface roughness and boundary condition has also been conducted to check its feasibility in various scenarios.

Diagnosing Discrepancies between Observations and Models of Surface Energy Fluxes in a Midlatitude Lake

Abstract The physical processes of heat exchange between lakes and the surrounding atmosphere are important in simulating and predicting terrestrial surface energy balance. Latent and sensible heat fluxes are the dominant physical process controlling ice growth and decay on the lake surface, as well as having influence on regional climate. While one-dimensional lake models have been used in simulating environmental changes in ice dynamics and water temperature, understanding the seasonal to daily cycles of lake surface energy balance and its relationship to lake thermal properties, atmospheric conditions, and how those are represented in models is still an open area of research. We evaluated a pair of one-dimensional lake models, Freshwater Lake (FLake) and the General Lake Model (GLM), to compare modeled latent and sensible heat fluxes against observed data collected by an eddy covariance tower during a 1-yr period in 2017, using Lake Mendota in Madison, Wisconsin, as our study site. We hypothesized transitional periods of ice cover as a leading source of model uncertainty, and we instead found that the models failed to simulate accurate values for large positive heat fluxes that occurred from late August into late December. Our results ultimately showed that one-dimensional models are effective in simulating sensible heat fluxes but are considerably less sensitive to latent heat fluxes than the observed relationships of latent heat flux to environmental drivers. These results can be used to focus future improvement of these lake models especially if they are to be used for surface boundary conditions in regional numerical weather models. Significance Statement While lakes consist of a small amount of Earth’s surface, they have a large impact on local climate and weather. A large amount of energy is stored in lakes during the spring and summer, and then removed from lakes before winter. The effect is particularly noticeable in high latitudes, when the seasonal temperature difference is larger. Modeling this lake energy exchange is important for weather models and measuring this energy exchange is challenging. Here we compare modeled and observed energy exchange, and we show there are large amounts of energy exchange happening in the fall, which models struggle to capture well. During periods of partial ice coverage in early winter, lake behavior can change rapidly.

Added value of assimilating springtime Arctic sea ice concentration in summer-fall climate predictions

Prediction skill of continental climate in the Northern Hemisphere (NH) midlatitudes is generally limited throughout the year in dynamical seasonal forecast systems. Such limitations narrow the range of possible applications by different stakeholders. Improving the predictive capacity in these regions has been a challenging task. Sea ice is a central component of the Arctic climate system and a local source of climate predictability, yet its state is often not fully constrained in dynamical forecast systems. Using the EC-Earth3 climate model, we study the added value of assimilating observed Arctic sea ice concentration on the NH extratropical climate in retrospective forecasts of summer and fall, initialized every spring over 1992–2019. Predictions in the North Atlantic and Eurasia benefit from better initialization of sea ice in the Atlantic sector of the Arctic in a two-step mechanism. Initially, sea ice influences the central North Atlantic Ocean through an atmospheric bridge that develops in the first forecast weeks, subsequently leading to preserved skill in the sea surface temperatures (SSTs) throughout summer and early fall. Secondly, these long-lasting SST improvements provide better surface boundary conditions for the atmosphere and lead to more skillful predictions of circulation and surface climate in the Euro-Atlantic and Asian regions. In addition, our findings suggest that fully coupled ocean-atmosphere-sea ice models are likely necessary to study linkages between Arctic sea ice and midlatitudes, by better representing the interactions and feedbacks between the different components of the climate system.

Experimental and theoretical analysis of sub-ambient cooling with longwave radiative coating

With radiative cooling coating it is possible to keep the temperature of a surface around or even below air temperature under solar radiation exposure. This study conducted experimental and theoretical analysis of its effectiveness. Setups with different surface coatings and boundary conditions were placed on a roof of a building in Chongqing, China. The 24-h measurements on a representative summer day showed that different boundary conditions at the bottom of the coating led to varied cooling effects, but the surface temperature of the radiative cooling coating stayed below air temperature most of the day when the incident solar radiation was below 800 W m−2. Theoretical models were also established to investigate the heat transfer process. Analysis showed that the longwave radiation absorption amounted to 89.0%–91.9% of the total daily heat gain of the cooling coating. The longwave radiative emission was the sole and significant heat loss pathway for the cooling coating, making the net longwave radiative heat transfer a pathway for heat dissipation. Sensitivity analysis further demonstrated that on a typical summer day in Chongqing, shortwave reflectivity and incident solar radiation were the dominant factors impacting the diurnal cooling effect, while wind velocity was the most important parameter for nocturnal cooling.

Analytical Solution for Bichromatic Waves on Linearly Varying Currents

It is well-known that the bound long waves play an important role on beach morphology. The existence of the current will modify the intensity of the waves. To examine the more realistic problem where the current is non-uniform, a third-order analytical solution for bichromatic waves on currents with constant vorticity is derived using a perturbation method. Earlier derivations of theories for interactions between waves and currents have been limited to cases of bichromatic waves on depth-uniform currents and cases of monochromatic waves on linear shear currents. Unlike the derivations of monochromatic waves, the moving-frame method cannot be used in the case of bichromatic waves because there are multiple waves with different celerities, and the flow can in no way be treated as steady-state. Moreover, for shear currents where the flows are rotational, the velocity potential cannot be simply defined. For these reasons, it is difficult to carry out mathematical operations when the dynamic free surface boundary condition is applied. However, with the consideration of the wave part of the fluid motion remaining irrotational, some of the terms in the expanded boundary conditions can be ignored; thus, the derivations can be further processed. As a result, the third-order explicit expressions for the stream function, the velocity potential, and the surface elevation can be solved. The nonlinear dispersion relation is also derived to account for interacting wave components with different frequencies and amplitudes which can be used to solved for wavenumbers of both wave trains. The obtained solutions are verified by reducing to those of previous results for monochromatic waves and uniform currents. Furthermore, the influence of constant vorticity on the wave kinematics is illustrated. A comparison between following/opposing currents is also carried out. Finally, the effects of the shear current on the strength of the bound long waves are also illustrated.

Similarity solution for boundary layer flow near a moving vertical porous plate with combined effects of nonlinear thermal radiation and suction/injection having convective surface boundary condition

This piece of work explored the impact of nonlinear thermal radiation on natural convection flow in the vicinity of a constant moving vertical porous plate with suction/injection. The moving vertical porous plate is considered to be either move in the same direction with fluid in the free stream or move in the negative x-direction while the fluid move in the positive x-direction. The partial differential equations (PDEs) associated with the flow properties are simplified to ordinary differential equations (ODEs) using similarity variables. Then, the resulting ODEs and the boundary condition are translated into the IVP through the shooting method. Furthermore, the final IVP and the reduced boundary conditions are solved numerically through the Runge–Kutta method. The influence of the emerging dimensionless quantities is analyzed through tables and line graphs. The numerical results obtained revealed that with suction/injection, the shear stress could be enhanced by propagating the local convective heat transfer parameter Bix, nonlinear thermal radiation R and the local Grashof number Grx for the two cases of plate motion considered. The result also revealed that in the presence of suction/injection, the heat transfer rate decreases with Bix and Grx, whereas enhanced with R augment for the two plate motions considered. Furthermore, for a constant moving vertical porous plate and constant moving freestream velocity, the convective heat transfer parameter at any point x is proportional to the corresponding heat transfer coefficient related to hot fluid. The nonlinear thermal radiation enhances the velocity and temperature profiles.

Modeling Graphene-Based THz Devices With an Improved Surface Boundary Condition in Leapfrog FDTD Scheme

Finite-difference time-domain (FDTD) method is an efficient full-wave numerical method for simulating graphene-based optoelectronic devices. The previous investigation on modeling graphene in leapfrog FDTD methods shows that dispersive errors increase with tuning up the chemical potential of graphene. In this article, a new implementation of graphene’s surface boundary condition (SBC) is proposed to reduce the dispersive errors in the leapfrog FDTD update scheme. To achieve lower dispersive errors, the graphene layer is positioned exactly on the grid lattice of transverse magnetic fields, and each of the transverse magnetic fields is split into two fields on the two sides of graphene. The split magnetic fields are updated using forward and backward differences in the electric fields on the normal direction of graphene plane. Numerical analysis shows that the maximum dispersive error of the proposed graphene model is reduced to only 37.5% of the other publicly reported graphene models in the leapfrog FDTD update scheme. However, the CPU time cost keeps almost the same as the other graphene models. The proposed implementation of the graphene SBC model also shows less CPU time consumption compared with the graphene SBC model which requires to solve a linear system and breaks the leapfrog FDTD update scheme, while the maximum numerical error is the same. The proposed graphene model is applied to simulate graphene terahertz (THz) surface plasmon resonances and modulators to demonstrate the applications in accurately modeling and analyzing graphene-based THz devices.

Investigation of nanofluid flow in a vertical channel considering polynomial boundary conditions by Akbari-Ganji's method

In this research, a vertical channel containing a laminar and fully developed nanofluid flow is investigated. The channel surface's boundary conditions for temperature and volume fraction functions are considered qth-order polynomials. The equations related to this problem have been extracted and then solved by the AGM and validated through the Runge-Kutta numerical method and another similar study. In the study, the effect of parameters, including Grashof number, Brownian motion parameter, etc., on the motion, velocity, temperature, and volume fraction of nanofluids have been analyzed. The results demonstrate that increasing the Gr number by 100% will increase the velocity profile function by 78% and decrease the temperature and fraction profiles by 20.87% and 120.75%. Moreover, rising the Brownian motion parameter in five different sizes (0.1, 0.2, 0.3, 0.4, and 0.5) causes lesser velocity, about 24.3% at first and 4.35% at the last level, and a maximum 52.86% increase for temperature and a 24.32% rise for Ψ occurs when Nb rises from 0.1 to 0.2. For all Nt values, at least 55.44%, 18.69%, for F(η), and Ω(η), and 20.23% rise for Ψ(η) function is observed. Furthermore, enlarging the Nr parameter from 0.25 to 0.1 leads F(η) to rise by 199.7%, fluid dimensionless temperature, and dimensional volume fraction to decrease by 18% and 92.3%. In the end, a greater value of q means a more powerful energy source, amplifying all velocity, temperature, and volume fraction functions. The main novelty of this research is the combined convection qth-order polynomials boundary condition applied to the channel walls. Moreover, The AMG semi-analytical method is used as a novel method to solve the governing equations.

Wind Farm Inflow Wind Simulation based on Mesoscale and Microscale Coupling

Inflow wind simulation is a critical issue that dominates the wind farm design regarding the actual wind environment. Different from traditional real measurement based wind simulation, this paper proposed a mesoscale and microscale coupling strategy, which applies the forecasting information from WRF model to the LES model based SOWFA. Firstly, an offline coupling strategy is implemented with the modular software interface between WRF and SOWFA. The wind speed, potential temperature and pressure data are converted from Geographic Coordinate to Cartesian coordinate that is a readable format to SOWFA. Then, the simulation domain is selected in daytime for neutral ABL condition at a 1km×1km region where the wind information from WRF is interpolated and averaged at center point with 100m height. Time-series ABL conditions are extracted from center point and force the SOWFA internal solver to simulate the same environment with predictive data. The mesoscale lacked information, turbulence, is generated by periodically running precursor with the surface roughness and boundary conditions. Finally, the comparison between WRF exacted data and SOWFA output verifies the coupling strategy. The result shows that the mesoscale and microscale coupling has high fidelity and accuracy simulation at stable ABL conditions and slow-changing wind environments. This work provides a low cost and reliable data source which allows the inflow wind simulation to have the predictive ability for actual wind.

Medium-Scale Methanol Pool Fire Model Validation

Abstract Medium scale (30 cm diameter) methanol pool fires were simulated using the latest fire modeling suite implemented in Sierra/Fuego, a low Mach number multiphysics reacting flow code. The sensitivity of model outputs to various model parameters was studied with the objective of providing model validation. This work also assesses model performance relative to other recently published large eddy simulations (LES) of the same validation case. Two pool surface boundary conditions were simulated. The first was a prescribed fuel mass flux and the second used an algorithm to predict mass flux based on a mass and energy balance at the fuel surface. Gray gas radiation model parameters (absorption coefficients and gas radiation sources) were varied to assess radiant heat losses to the surroundings and pool surface. The radiation model was calibrated by comparing the simulated radiant fraction of the plume to experimental data. The effects of mesh resolution were also quantified starting with a grid resolution representative of engineering type fire calculations and then uniformly refining that mesh in the plume region. Simulation data were compared to experimental data collected at the University of Waterloo and the National Institute of Standards and Technology (NIST). Validation data included plume temperature, radial and axial velocities, velocity temperature turbulent correlations, velocity velocity turbulent correlations, radiant and convective heat fluxes to the pool surface, and plume radiant fraction. Additional analyses were performed in the pool boundary layer to assess simulated flame anchoring and the effect on convective heat fluxes. This work assesses the capability of the latest Fuego physics and chemistry model suite and provides additional insight into pool fire modeling for nonluminous, nonsooting flames.

Examining Parameterizations of Potential Temperature Variance Across Varied Landscapes for Use in Earth System Models

AbstractEarth system models (ESMs) and mesoscale models have come to employ increasingly complex parameterization schemes for the atmospheric boundary layer, requiring surface boundary conditions for numerous higher order turbulence statistics. Of particular interest is the potential temperature variance (PTV), which is used not only as a boundary condition itself but also to close boundary conditions of other statistics. The existing schemes in ESMs largely rely on the assumptions of Monin‐Obukhov similarity theory (MOST) and are not necessarily applicable over complex and heterogeneous surfaces where large scale circulations and roughness sub‐layer effects may cause deviations from MOST. The National Ecological Observation Network is used here to evaluate existing parameterizations for the surface boundary of PTV, note key deficiencies, and explore possible remedies. The results indicate that existing schemes are acceptable over a variety of surface conditions provided the analysis of a priori filters out low frequency variability not associated with turbulent time scales. There was, however, significant inter‐site variability in observed similarity constants and a significant bias when compared to the textbook values of these parameters. Existing models displayed the poorest performance over heterogeneous sites and rough landscapes. Attempts to use canopy structure and surface roughness characteristics to improve the results confirmed a relation between these variables and PTV but failed to significantly improve the predictive power of the models. The results did not find strong evidence indicating that large scale circulations caused substantial deviations from textbook models, although additional analysis is required to assess their full impacts.

A Fourier Series Approximation for Deep-water Waves

Dean (1965) proposed the use of the root mean square error (RMSE) in the dynamic free surface boundary condition (DFSBC) and kinematic free-surface boundary condition (KFSBC) as an error evaluation criterion for wave theories. There are well known wave theories with RMSE more than 1%, such as Airy theory, Stokes theory, Dean’s stream function theory, Fenton’s theory, and trochodial theory for deep-water waves. However, none of them can be applied for deep-water breaking waves. The purpose of this study is to provide a closed-form solution for deep-water waves with RMSE less than 1% even for breaking waves. This study is based on a previous study (Shin, 2016), and all flow fields were simplified for deep-water waves. For a closed-form solution, all Fourier series coefficients and all related parameters are presented with Newton’s polynomials, which were determined by curve fitting data (Shin, 2016). For verification, a wave in Miche’s limit was calculated, and, the profiles, velocities, and the accelerations were compared with those of 5th-order Stokes theory. The results give greater velocities and acceleration than 5th-order Stokes theory, and the wavelength depends on the wave height. The results satisfy the Laplace equation, bottom boundary condition (BBC), and KFSBC, while Stokes theory satisfies only the Laplace equation and BBC. RMSE in DFSBC less than 7.25×10-2% was obtained. The series order of the proposed method is three, but the series order of 5th-order Stokes theory is five. Nevertheless, this study provides less RMSE than 5th-order Stokes theory. As a result, the method is suitable for offshore structural design.