Free Surface Boundary Research Articles

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.

Solutions for Groundwater Flow to a Pumping Well with an Exponentially Decreasing Variable-Rate in a Sloping Aquifer.

During aquifer pumping tests in the field, the attenuation process of the pumping rate is generally monitored, and the aquifer may be slanted, especially in riverbank terraces, piedmont plain and other similar terrains. The combination of these two factors (attenuation process and aquifer slope) has not received enough academic attention, therefore, a two-dimensional groundwater flow model is developed for an exponentially decreasing variable rate pumping well in a sloping aquifer in this study. The new semi-analytical solution (based on a linearized free-surface boundary) is obtained using the Laplace transform and finite Fourier cosine transform. Meanwhile, to depict the linearized boundary precisely with the first-order free surface equation, a numerical model that considers the decline of the water table is also developed. Results show that under the effect of the variable rate, the time-drawdown curve presents a downward trend at intermediate times. Additionally, the error caused by the linearization of the free-surface equation on the analytical model is found to be acceptable except when the sloping angle is larger than π/6. Furthermore, since this study extends the work of Huang et al. (2014), who considered a constant pumping rate, the new solution was found better to interpret real pumping test data reported by Antonio and Pacheco (2002), and the laboratory experiment results. Overall, the new solution proves to be useful for parameter inversion accurately as the average pumping rate tended to underestimate hydraulic parameters when fitting short-term pumping data.

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.

Acoustic radiation from a sphere pulsating near an impedance plane using a boundary integral equation method

In this study, a boundary integral equation method is proposed for investigating acoustic pressure radiation from a sphere pulsating near a free surface or an impedance plane. The half-space and free-space problems are investigated for the acoustic radiation of pulsating sphere. The effects of free surface and impedance boundaries are introduced into the mathematical model by employing three different half-space Green’s functions, respectively. These Green’s functions are derived, respectively, using the single image-source method, multiple equivalent-source method, and complex equivalent-source method. Green’s functions are implemented into the boundary element (BE) formulation. The surface of the pulsating sphere is discretized with linear and quadratic BEs, and the Combined Helmholtz Integral Equation Formulation (CHIEF) is employed to overcome the non-uniqueness problem. Four different case studies are considered for the sphere pulsating near a free surface or an impedance plane. The first case study involves the sphere pulsating near a free surface (perfectly reflective) and the single image-source method is used in the boundary element method (BEM) formulation. In the second case study, the sphere is assumed as pulsating near a perfectly reflecting and perfectly absorbing impedance planes, respectively. The multiple equivalent-source method is employed for the perfectly reflecting plane, but the multiple equivalent-source method and complex equivalent-source methods for the perfectly absorbing plane. The third case study involves a general impedance plane, and all the methods are employed, respectively, in the BE formulation. The final case study assumes a general impedance plane forming a perpendicular incidence and the complex equivalent-source method is used in this particular case. It is observed that there is a very good comparison between the results obtained from all these methods.

Numerical and experimental study on free-surface wave generation by a submerged moving body in a towing tank

Free surface waves were generated by a submerged moving body in a towing tank using a numerical and a physical towing tank. Unlike a circulating water channel, a towing tank allows easy to control of acceleration and deacceleration range of the body motion. Numerical and experimental analysis can make the same moving conditions of the submerged body. Using a Numerical Towing Tank (NTT), free surface elevation was calculated according to body velocities, submerged depths and body volumes. The measured experimental data agreed well with the results of a fully nonlinear simulation and the data was also compared with the Russell comprehensive error method. The nonlinearity of the free surface waves increased as the submerged depth of the body became shallower and the body velocity became faster. The increase in wave height according to the increase of the body velocity or the increase in the wave height according to the decrease in the submerged depth of the body generally showed a proportional relationship. Overall, the nonlinear free surface boundary condition needs to be applied when calculating the free surface wave due to the motion of a submerged body. The ratio of wave height difference according to body displacement was similar to the ratio (percentage) of increase in body displacement to NACA shape displacement as body velocity becomes faster.

An isogeometric FE-BE method to investigate fluid–structure interaction effects for an elastic cylindrical shell vibrating near a free surface

In this study, an isogeometric finite element-boundary element (FE-BE) framework is proposed to investigate frequency-dependent hydrodynamic characteristics of elastic structures vibrating at free surface or in the vicinity of free surface. The overall numerical framework consists of two parts. In the first part, in vacuo dynamic characteristics are obtained by an isogeometric finite element method (IGAFEM) formulation. Then, in the second part, frequency-dependent generalized added mass and generalized hydrodynamic damping coefficients are obtained by an isogeometric boundary element method (IGABEM) formulation, assuming that the surrounding fluid is ideal, i.e., inviscid, incompressible and its motion is irrotational. To include free surface effects, the free surface boundary conditions are satisfied with an appropriate Green’s function in the mathematical formulation. To show the applicability of the presented framework, an elastic cylindrical shell which was studied previously in the literature is investigated considering that the structure is partially or fully submerged into fluid of infinite depth. It is shown that the results obtained in this study compare very well with the results available in literature.

NUMERICAL MODELLING OF THE JET FLOW

In this work, plane turbulent jets are simulated numerically using a three-dimensional computational model. The simulated jets exclude the influence of the free surface and solid boundaries on the jets. The accuracy of turbulent schemes is assessed by analyzing the decay of the average velocity, the rate of growth of the jet, and the similarity of the profiles of the longitudinal and vertical velocities. The results of the two turbulent closure schemes are compared with experimental and theoretical studies to determine their accuracy.

A flexible fully nonlinear potential flow model for wave propagation over the complex topography of the Norwegian coast

Coastal wave propagation and transformation are complicated due to the significant variations of water depth and irregular coastlines, which are typically present at the Norwegian fjords. A potential flow model provides phase-resolved solutions with low demands on computational resources. Many potential flow models are developed for offshore waves and lack of numerical treatments of coastal conditions. In the presented work, several modifications are introduced to a fully nonlinear potential flow model with a σ-grid for the purpose of coastal wave modelling: Shallow water breaking criteria are included in addition to deepwater breaking algorithms to approximate breaking waves over a complete range of water depth. A new coastline algorithm is introduced to detect complex coastlines and ensure robust simulations near the coast. The algorithm is compatible with structured grid arrangement in the horizontal plane and allows for high-order discretisation schemes for the free surface boundary conditions for an accurate representation of complex free surfaces. A parallelised solver for the Laplace equation is utilised to ensure fast simulations for large domains with multi-core infrastructures.The proposed model is validated against theories and experiments for various two- and three-dimensional nonlinear wave propagation and transformation cases that represent typical coastal conditions. The simulations show a good representation of nonlinear waves, and the results compare well with experiments. Furthermore, two large-scale engineering scenarios are simulated, where the applicability of the coastline algorithm and the parallel commutation capability of the model are demonstrated.

An alternative linear flow model and boundary integral flow representation for ship motions in regular waves

The basic issue of formulating a linear flow model for the analysis of potential flow around a ship advancing at a constant speed in regular waves is considered. Specifically, the ‘rigid-waterplane flow model’, previously considered for diffraction–radiation of regular waves by offshore structures, is applied to analyze flows around ships advancing in regular waves. A straightforward application of Green’s fundamental identity to the rigid-waterplane flow model yields a remarkable new boundary integral flow representation. Indeed, this flow representation does not involve a line integral around the mean waterline of the ship hull surface, is weakly singular, does not involve a distribution of dipoles over the ship-hull surface, and is free from irregular frequencies. In the particular case of flow around a ship steadily advancing in calm water, the boundary integral flow representation obtained in the study via an analysis based on the rigid-waterplane flow model is identical to the flow representation associated with the Neumann–Michell theory, which is based on a different linear flow model that only considers the flow region strictly outside the ship but consistently accounts for the linear contribution of the thin band of water between the wave profile along the ship hull surface and the undisturbed free surface. This result strongly suggests that the classical application of Green’s basic identity to analyze linear potential flow around a ship advancing in regular waves or in calm water, which yields a notoriously troublesome line integral around the ship waterline, may be questionable due to possible incompatibilities between the Neumann boundary condition at the ship hull surface and the linear free surface boundary condition associated with flows around ships advancing in calm water or in regular waves. This fundamental difficulty may be alleviated in the rigid-waterplane flow model, which imposes a compatibility condition between the Neumann boundary condition at the rigid lid that artificially closes the open ship hull surface and the linear boundary condition at the free surface above the waterplane-lid. The boundary integral flow representation given in the study also holds for diffraction–radiation of regular waves by offshore structures. Thus, this new flow representation provides a common mathematical basis, which fundamentally differs from the basis steadfastly adopted over the past fifty years, for the analysis of three basic classes of flows in ship and offshore hydrodynamics, and indeed can be applied more generally to other boundary-value problems such as those associated with flexural-gravity waves for ice sheets or very large floating structures.

Hydrodynamic Coupling of Viscous and Nonviscous Numerical Wave Solutions Within the Open-Source Hydrodynamics Framework reef3d

Abstract A comprehensive understanding of the marine environment in the offshore area requires phase-resolved wave information. For far-field wave propagation, computational efficiency is crucial, as large spatial and temporal scales are involved. For the near-field extreme wave events and wave impacts, high resolution is required to resolve the flow details and turbulence. The combined use of a computationally efficient large-scale model and a high-resolution local-scale solver provides a solution that combines accuracy and efficiency. This article introduces a coupling strategy between the efficient fully nonlinear potential flow (FNPF) solver REEF3D::FNPF and the high-fidelity computational fluid dynamics (CFD) model REEF3D::CFD within the open-source hydrodynamics framework REEF3D. REEF3D::FNPF solves the Laplace equation together with the boundary conditions on a sigma-coordinate. The free surface boundary conditions are discretized using high-order finite difference methods. The Laplace equation for the velocity potential is solved with a conjugated gradient solver preconditioned with a geometric multigrid provided by the open-source library Hypre. The model is fully parallelized following the domain decomposition strategy and the message passing interface protocol. The waves calculated with the FNPF solver are used as wave generation boundary conditions for the CFD-based numerical wave tank REEF3D::CFD. The CFD model employs an interface capturing two-phase flow approach that can resolve complex wave structure interaction, including breaking wave kinematics and turbulent effects. The presented hydrodynamic coupling strategy is tested for various wave conditions and the accuracy is fully assessed.

Numerical simulation of the free surface flow passing two hydrofoils

In this paper, a numerical method based on potential flow theory is proposed to investigate the hydrodynamic interaction between two foils with free surface effect. The integral equation is solved using boundary element method. On foil surfaces, the impermeable boundary condition is satisfied. On free surface, nonlinear kinematic and dynamic boundary conditions are satisfied. A robust numerical model is proposed to avoid the complicated vortical interference between the two foils. A fully nonlinear scheme is developed for the numerical simulation. The proposed method is validated by extensive convergence studies. The hydrodynamic loadings on the foils are investigated in time domain. Two major quantitative findings are discussed. Firstly, the free surface elevation could be weakened significantly if the two foils are appropriately arranged. Secondly, the lifting force of the following foil varies periodically with the increase of the horizontal distance between the foils.

The instability of a helical vortex filament under a free surface

We perform a theoretical investigation of the instability of a helical vortex filament beneath a free surface in a semi-infinite ideal fluid. The focus is on the leading-order free-surface boundary effect upon the equilibrium form and instability of the vortex. This effect is characterised by the Froude number $F_r = U(gh^*)^{-{1}/{2}}$ where $g$ is gravity, and $U = \varGamma /(2{\rm \pi} b^*)$ with $\varGamma$ being the strength, $2{\rm \pi} b^*$ the pitch and $h^*$ the centre submergence of the helical vortex. In the case of $F_r \rightarrow 0$ corresponding to the presence of a rigid boundary, a new approximate equilibrium form is found if the vortex possesses a non-zero rotational velocity. Compared with the infinite fluid case (Widnall, J. Fluid Mech., vol. 54, no. 4, 1972, pp. 641–663), the vortex is destabilised (or stabilised) to relatively short- (or long-)wavelength sub-harmonic perturbations, but remains stable to super-harmonic perturbations. The wall-boundary effect becomes stronger for smaller helix angle and could dominate over the self-induced flow effect depending on the submergence. In the case of $F_r > 0$ , we obtain the surface wave solution induced by the vortex in the context of linearised potential-flow theory. The wave elevation is unbounded when the $m$ th wave mode becomes resonant as $F_r$ approaches the critical Froude numbers ${\mathcal {F}} (m) = (C_0^*/U)^{-1} (mh^*/b^*)^{-{1}/{2}}$ , $m=1, 2, \ldots,$ where $C_0^*$ is the induced wave speed. We find that the new approximate equilibrium of the vortex exists if and only if $F_r < {\mathcal {F}}(1)$ . Compared with the infinite fluid and $F_r \rightarrow 0$ cases, the wave effect causes the vortex to be destabilised to super-harmonic and long-wavelength sub-harmonic perturbations with generally faster growth rate for greater $F_r$ and smaller helix angle.

An implicit method for slow-moving free-surface boundary of open channel flows

This paper outlines the implementation of an efficient implicit method for a free surface boundary for solving free surface flows with larger time steps. In this method, the governing equations are solved with dynamic conditions applied to the free surface boundary. The position and shape of the free surface are represented with adaptive boundary grid points. Unlike existing explicit methods, the proposed method solves the space conservation equation implicitly, making the calculations more stable and at a lower computational cost. The approach was implemented numerically by developing a new boundary field in OpenFOAM. Several test cases, including propagations of tidal wave, channel surge and solitary wave, were simulated to demonstrate the method’s accuracy. The proposed method accurately predicted the first two cases using larger time steps. The Courant number of the solitary wave case should be less than 1 to reduce numerical errors, which would damp the wave amplitude.

A free boundary isometric embedding problem in the unit ball

We study a free boundary isometric embedding problem for abstract Riemannian two-manifolds with the topology of the disc. Under the assumption of positive Gauss curvature and geodesic curvature of the boundary being equal to one, we show that every such disc may be isometrically embedded into the Euclidean three-space mathbb {R}^3 such that the image of the boundary meets the unit sphere mathbb {S}^2 orthogonally. We also show that the embedding is unique up to rotations and reflections through planes containing the origin. Finally, we define a new Brown-York type quasi-local mass for certain free boundary surfaces and discuss its positivity.

On Particle Shifting Techniques (PSTs): Analysis of existing laws and proposition of a convergent and multi-invariant law

This paper addresses the Particle Shifting Technique (PST) in the SPH schemes. Improving the accuracy of SPH schemes leads to particle clustering along the flow streamlines which turns to be detrimental for the simulations. PSTs aim at avoiding this adverse effect by slightly disordering the particles, allowing to retrieve a regular particle distribution within the kernel interpolation support. The gain in accuracy is such that this technique is now commonly adopted by the SPH practitioners, however the conditions that should be respected by a PST are not clearly discussed in the literature. In this paper, such conditions are exposed and their fulfillment by the main existing PSTs of the literature is analyzed. None of these existing PSTs fully satisfying these conditions, a novel PST is introduced. The proposed PST is validated for three different SPH schemes on 2D and 3D test cases, in presence of free-surface and solid boundaries.

A nonlinear time-domain BEM for the performance of 3D flapping-wing thrusters in directional waves

Recent studies indicate that flapping-foil biomimetic systems can significantly improve ship propulsion in waves. In most cases horizontal wing(s) are used performing vertical oscillatory motion induced by ship heaving and pitching responses in waves, while foil's rotational motion is actively controlled. Aiming to the detailed investigation of the free-surface effects, a GPU-accelerated Boundary Element Method (GPU-BEM) has been developed for the hydrodynamic analysis of 3D lifting bodies in non-linear waves with an angle of incidence and motions. The nonlinear free-surface conditions are satisfied at the exact location of the free-surface boundary. A nonlinear pressure-type Kutta condition is applied at the trailing edge and the dynamic evolution of the free vortex wake is obtained by a time-stepping method. The free-surface boundary conditions and Kutta condition are used to set up the nonlinear and nonlocal evolution equations. For the treatment of the dynamical system, the present BEM is coupled to the dynamical equations and it is integrated in time domain with an iterative scheme. Results are presented for a few subproblems of special interest, such as the wave resistance and the enforced radiation problems of smooth immersed bodies and compared with other methods for verification. Subsequently, the unsteady biomimetic thruster is examined in unbounded domain, beneath the free surface in the absence of incident waves, and in the presence of obliquely incident waves, illustrating the performance of the system. The present method can be exploited for the systematic investigation and the design of flapping-foil thrusters for augmenting ship propulsion in waves.

An upright bottomless vertical cylinder with baffles floating in waves

Damping of the surge and pitch motions, as well as the first lateral sloshing mode in a rigid free-floating upright circular dock with bilge boxes and open bottom is investigated. Model tests are carried out on a 0.80 m diameter model in regular waves with wave periods near the highest natural sloshing period, and the internal free-surface elevation and model’s rigid body motions are measured. Perforated and solid annular baffles of relatively small widths are also installed inside the dock at various submergences. The experimental results are compared to a semi-analytical approach, where a three-dimensional domain decomposition method based on linear potential flow theory is adopted to calculate the hydrodynamic coefficients and exciting forces in heave, surge and pitch. A reduced natural sloshing frequency, as well as a damping ratio estimated from the energy dissipated due to flow separation from the baffles, are introduced in the free-surface boundary conditions to model the effects of the baffle. It shows good agreement with experimental data when the ratio between the draft of the baffle and the internal radius of the cylinder is dB/a=0.27, and tends to under-predict the damping ratio for shallower drafts, most likely due to free-surface interactions. The solid baffle damps the sloshing response most efficiently, reducing the amplitude at the resonant peak by more than 56%.

Fatigue crack growth simulation of multiple surface cracks using dislocation dynamics

This paper presents a numerical analysis method for surface crack problems based on the discrete dislocation dynamics (DDD) method. To satisfy the free surface boundary condition, the superposition principle is utilized. Virtual dislocation loops are placed at the outside of the body of interest to remove the dislocation singularity at the free surface. In this study, the DDD method for surface cracks was applied to the fatigue crack propagation problem for double surface cracks. First, fatigue crack growth analysis was performed for one surface crack, and the results were compared with the Newman-Raju solution to verify the validity of the numerical analysis method using the DDD method. Next, fatigue crack growth analysis was performed for double surface cracks and the results were compared with FEM.

On the free surface boundary of moving particle semi-implicit method for thermocapillary flow

The moving particle semi-implicit (MPS) method has great potential in dealing with free surface flow due to its Lagrangian nature. In most cases, the free surface boundary is simply served as the pressure boundary condition. In this paper, an improved MPS method is presented for thermocapillary driven free surface flow. A series of surface nodes explicitly represent the free surface boundary. The normal stress on the free surface provides the Dirichlet pressure boundary condition, while the velocity boundary condition, i.e., Marangoni stress, is enforced through the Taylor series expansion and least squares method. Meanwhile, a quasi-Lagrangian formulation is introduced to avoid particle clustering and the corresponding numerical instability by slightly modifying the advection velocity. The upwind scheme is employed for the convection term to obtain accurate and stable results. A novel constraint scheme with the divergence of provisional velocity is developed for the pressure gradient to enhance stability further. The consistency of the derived generalized boundary condition is firstly verified with a simple convergence test. Then, several numerical tests, including square patch rotation, lid-driven and square droplet oscillation, are simulated to show the improvements. Finally, thermocapillary driven flows in an open cavity without and with buoyancy effect are studied. Good agreements are obtained by comparing with reference simulations taken from literature. Heat transfer characteristics are further investigated for different dimensionless numbers, including the Rayleigh number and Marangoni number.<PE-FRONTEND>

An enhanced semi-implicit particle method for simulating the flow of droplets with free surfaces

The accurate modeling of the surface-tension on free surfaces remains a challenging problem in meshfree particle methods. Recently, Matsunaga et al. (2020) have made an important step forward toward addressing this issue by introducing a moving surface mesh to represent the free-surface boundary in 2D. However, the surface mesh makes it difficult to extend this methodology to 3D and to treat topological changes in free surfaces. This study aims to extend this methodology to 3D and restore the capability of the model to simulate topological changes by carefully detecting the free-surface particles. In this situation, the problems regarding fluctuated free-surface boundaries and the dynamic intersection of a free surface with a wall boundary arise. To resolve these problems, a new particle shifting method in the surface normal direction is proposed for the free-surface particles to reduce fluctuations. A new contact angle model is proposed to compute the intersection point of a free surface and a wall boundary. In this manner, an enhanced particle method suitable for simulating the dynamics of liquid droplets with possible topological changes in 3D is developed. Eight numerical examples verified the developed method and demonstrated the enhancement of the new particle shifting technique.