Mixed Eulerian-Lagrangian Method Research Articles

Fully nonlinear wave interaction with 2D floating body including nonlinear sloshing tank

Two-dimensional fully-nonlinear numerical sloshing tank (FN-NST) simulation program is developed based on boundary element method (BEM), potential theory with artificial free-surface damping, and space-fixed (global) coordinate system. The free surface in sloshing tank is updated by the mixed Eulerian-Lagrangian (MEL) method with material node approach (Full Lagrangian approach). The developed potential-based nonlinear model was validated through comparisons with available experimental and CFD (computational fluid dynamics) results and they show good agreements. The nonlinear sloshing model is then further implemented in the fully nonlinear wave and floating-body simulation program to observe the effects of liquid sloshing motion. In the fully nonlinear program, the entire nonlinear fluid pressure is integrated over the instantaneous wetted body surface and position, which is subsequently compared with the corresponding linear theory and results. The second-order effects are also identified from the fully-nonlinear simulation. The floater Response Amplitude Operators (RAOs) for frozen- and liquid-cargo cases are also compared for various wave frequencies and amplitudes to better identify the role of liquid cargo in the stability of rotational motions. The effects of the vertical location of liquid tank and artificial free-surface damping are also investigated.

Non-linear slosh dynamics of sloped wall tank with bottom-mounted object under seismic excitation

The present study deals with the investigation of the non-linear slosh dynamics of a sloped wall tank with bottom-mounted object under seismic excitation. The potential flow theory has been adopted for modeling the liquid domain with a mixed Eulerian-Lagrangian method. The nonlinear seismic response of a sloped wall tank with bottom-mounted rigid internal object is suitably investigated under six different earthquakes. The selected earthquakes are classified based on their PGA/PGV ratios. The base shear force and hydrodynamic pressure are successfully quantified along with the impulsive and convective components. In order to study the influence of the internal object on the hydrodynamic behavior of the liquid tank, a parametric investigation is performed by altering the object's height. The necessity of non-linear analysis is reasonably justified by comparison of the results with those obtained from linear analysis, where a significant difference is observed between the linear and non-linear hydrodynamic responses. The developed non-linear finite element model in the current study is found to be more effective than the linear model and may be employed in designing structure-coupled sloped wall TLDs.

Free surface and internal periodical waves generated by an oscillating floater in two-layer fluids using fully nonlinear numerical wave tank

This study simulated free-surface and internal periodical waves generated by an oscillating floater in two-layer fluids using a numerical wave tank (NWT) with dual-fluid domains. Each domain was based on the potential flow and boundary element method (BEM). To validate the present NWT, the simulated results were compared with Kashiwagi's theoretical and experimental data, which results in good agreement. The fully nonlinear NWT results were compared systematically with the linear results to quantify their differences through higher-order components in barotropic (or surface wave) and baroclinic (or internal wave) modes both in the surface and internal waves. Free-surface and internal waves were analyzed systematically in terms of the two wave modes, respective-order components, several density ratios, and body-oscillation amplitudes and frequencies. For the present NWT(FN-NWT) with dual fluid domains, three different approaches for updating the instantaneous free surface and interface in the mixed Eulerian-Lagrangian (MEL) method were compared to cross-check the robustness of the developed schemes. When the forced oscillation amplitude and density difference are large, both surface and internal waves have strong nonlinearity, especially in the low-frequency region mainly due to the increased effects of the baroclinic mode. On the other hand, for the surface wave, the increased effects of primary and higher-order components from the barotropic mode are observed, particularly in high frequencies.

A fully nonlinear BEM-beam coupled solver for fluid–structure interactions of flexible ships in waves

A fully nonlinear hydroelastic numerical method is proposed for the fluid–structure interactions of elastic ships. The boundary value problem is solved by the three-dimensional boundary element method. To simulate the instantaneous evolution on the free surface, an effective mesh optimization solution consisting of the local coordinate system and an elastic mesh technique has been developed. The mixed Eulerian–Lagrangian method is implemented to update the fully nonlinear free surface conditions in the local coordinate system. The hull structure is simplified as a non-uniform Timoshenko beam, and the modal superposition approach is adopted to analyse the symmetric responses in the head sea. Convergence of the model is evaluated for mesh resolutions and the time step size, and then comprehensive investigations of vertical motions as well as vertical bending moment are carried out for an advancing ship in a range of wave frequencies. The numerical solutions are compared with the experimental data and a satisfactory agreement is demonstrated. The distribution of high-order harmonics and whipping responses are predicted by the amplitude spectrum, in which the water entry and water exit are considered. The high-frequency vertical bending moments are investigated, in which the effects of higher harmonics and whipping responses are addressed.

Adaptive fully nonlinear potential model for the free surface under compressible air pressure of oscillating water column devices

A two dimensional fully nonlinear numerical wave flume is extended with a pneumatic model to compute the free surface flow inside the chamber of land-based Oscillating Water Column (OWC) devices. Potential theory and the Mixed Eulerian-Lagrangian method are employed to compute the fluid flow using the direct Boundary Element Method (BEM) enhanced by a non-uniform rational B-Spline function. Meanwhile, a pneumatic model is manipulated to obtain the instantaneous air pressure on the chamber's free surface. Spring-like behavior of a perfect gas through an adiabatic process and a linear power take-off mechanism are used in the pneumatic model that makes time derivative of the air pressure turns to a function of reciprocating air and fluid flows in the chamber. Therefore, temporary air pressure is taken into account on the fully nonlinear free surface boundary conditions implicitly along with the time marching algorithm. The hydrodynamic efficiency of an OWC interacting with second-order Stokes waves is compared with former studies to verify the present model. To assess the present model for simulating the air compressibility in a large-scale case study, the present solution is compared with experimental and mathematical results under an irregular wave. Afterwards, an OWC device is exposed to fifth-order Stokes waves in the flume to examine the numerical stability of computing the free surface flow outside and inside of the chamber.

Slow Rhythmical Brain Activity During Coordination of Cognitive Processes

A two dimensional fully nonlinear numerical wave flume is extended with a pneumatic model to compute the free surface flow inside the chamber of land-based Oscillating Water Column (OWC) devices. Potential theory and the Mixed Eulerian-Lagrangian method are employed to compute the fluid flow using the direct Boundary Element Method (BEM) enhanced by a non-uniform rational B-Spline function. Meanwhile, a pneumatic model is manipulated to obtain the instantaneous air pressure on the chamber's free surface. Spring-like behavior of a perfect gas through an adiabatic process and a linear power take-off mechanism are used in the pneumatic model that makes time derivative of the air pressure turns to a function of reciprocating air and fluid flows in the chamber. Therefore, temporary air pressure is taken into account on the fully nonlinear free surface boundary conditions implicitly along with the time marching algorithm. The hydrodynamic efficiency of an OWC interacting with second-order Stokes waves is compared with former studies to verify the present model. To assess the present model for simulating the air compressibility in a large-scale case study, the present solution is compared with experimental and mathematical results under an irregular wave. Afterwards, an OWC device is exposed to fifth-order Stokes waves in the flume to examine the numerical stability of computing the free surface flow outside and inside of the chamber.

Numerical investigation on a container ship navigating in irregular waves by a fully nonlinear time domain method

A three dimensional fully nonlinear time domain method is introduced to simulate ship advancing in regular and irregular waves with forward speed. The mixed Eulerian-Lagrangian (MEL) method and boundary element method are adopted to solve the boundary value problem. Spring analogy method is utilized to guarantee the optimization mesh of the transient free surface and wetted hull body. A local coordinate system is introduced, and the fully nonlinear disturbed wave is separated from the total wave. Auxiliary functions are deduced to decouple the strongly nonlinear hydrodynamic forces and ship motion. Then the motion and velocity potential can be updated with the fourth-order Runge Kutta method. A 13500 TEU container ship is chosen to verify the numerical codes, the numerical results in regular waves and irregular waves are both in good agreement with experimental data. The instantaneous pressure distribution around wetted body surface and wave elevation on the free surface of dangerous moments are presented and the strongly nonlinear states in the harsh irregular waves are analyzed.

A mixed Eulerian-Lagrangian simulation of nonlinear wave interaction with a fluid-filled membrane breakwater

This paper introduces a time-domain potential flow method for simulating fully nonlinear wave interaction with a fluid-filled membrane mounted on the seabed. The numerical method is based on a mixed Eulerian-Lagrangian method, which tracks the free water surface and the membrane using a Lagrangian method and solves the flow fields inside and outside the membrane using an Eulerian method. To remove the numerical instability, a method that satisfies the no-flux condition on the membrane at any location and any time was developed to calculate the radial displacement of the membrane at every time step and a Fourier series expression for the radial displacement of the membrane was used to calculate the local curvature of the membrane accurately. Experimental results from an existing experimental study were used for model validation and verification. The response of the membrane and the nonlinear wave scattering were simulated. The simulation results show that the transmission coefficients of the fundamental waves, second harmonic free and locked waves approach zero when a resonant response of the membrane to waves occurs. Sample results of the simulated tension in the membrane and interior pressure are also included in the discussion.

Irregular wave transmission on bottom bumps using fully nonlinear NURBS numerical wave tank

In this study, a fully nonlinear three-dimensional numerical wave tank (NWT) is developed to simulate propagation of nonlinear random sea wave over the bottom ripple patches. Evolution of the free surface is performed by the non-uniform rational B-spline formulation (NURBS) and the mixed Eulerian–Lagrangian method (MEL). A high-order boundary integral equation is used to solve the Laplace equation in the Eulerian frame. The free surface is updated by the material node method and the fourth-order Runge–Kutta time integration scheme. Appropriate numerical solutions are obtained by deploying damping zones at the both tank end walls. Also, the NURBS approximation is applied to compute the kinematics of the free surface particles. Propagation of the irregular waves in an NWT is conducted and compared with the available experimental and numerical data. Bragg reflection of the random sea wave due to bottom ripple patches are compared with the prior numerical studies. The fully nonlinear free surface evolution of transmitted spectrum due to presence of the hemispherical bumps is modeled successfully. Also fifth-order Stokes wave is applied as a strong nonlinear incident wave to interact with the hemispherical bottom bumps.

Numerical investigation of solitary wave action on two rectangular boxes with a narrow gap

Based on the time-domain higher-order boundary element method (HOBEM), a two-dimensional numerical wave flume is developed to investigate solitary wave interaction with two rectangular boxes with a narrow gap. In the numerical model, the fully nonlinear boundary conditions are satisfied on the free surface, the mixed Eulerian-Lagrangian method is adopted to track the transient water surface and the fourth-order Runga-Kutta method is used to predict the velocity potential and wave elevation on the free surface. The acceleration potential technique is used to compute the transient wave forces along the wetted object surface. A piston-type wavemaker is used to generate solitary waves. The proposed model is validated by comparing the simulated wave run-up and the wave loads with the published experimental and numerical results of the reflection of a solitary wave from a vertical wall. Then, numerical experiments are performed to study the effects of the narrow gap and the size of each box on the wave run-ups at the two sides of the two-box system and in the narrow gap between two boxes, and the wave loads on the two boxes. The interaction between double solitary waves with a time interval between them with a two-box system is also investigated.

Nonlinear and unsteady waves generated by a traveling pressure distribution and the associated waveless shapes

The wave elevation and flow near a moving pressure distribution of characteristic intensity $$p_0$$ and length $$2a$$ are investigated using a fully nonlinear and unsteady mixed Eulerian–Lagrangian method. Special attention is paid to the zero-wave-resistance (ZWR) conditions and the associated waveless free-surface profiles. Large-time solutions for free-surface elevation are in good agreement with linear steady-state solutions for small applied pressure parameter $$p_0/(\rho ga)$$ , with $$\rho g$$ being the weight density of the fluid. The ZWR phenomenon predicted by the linear wave theory at certain critical speeds is recovered from these nonlinear solutions. As $$p_0/(\rho ga)$$ increases, large-time nonlinear solutions indicate that trailing waves of near-zero amplitude can still be achieved at certain adjusted critical speeds. The time evolution of the trailing waves is presented and discussed.

Wave propagation in a 3D fully nonlinear NWT based on MTF coupled with DZ method for the downstream boundary

Wave propagation in a three-dimensional (3D) fully nonlinear numerical wave tank (NWT) is studied based on velocity potential theory. The governing Laplace equation with fully nonlinear boundary conditions on the moving free surface is solved using the indirect desingularized boundary integral equation method (DBIEM). The fourth-order predictor-corrector Adams-Bashforth-Moulton scheme (ABM4) and mixed Eulerian-Lagrangian (MEL) method are used for the time-stepping integration of the free surface boundary conditions. A smoothing algorithm, B-spline, is applied to eliminate the possible saw-tooth instabilities. The artificial wave speed employed in MTF (multi-transmitting formula) approach is investigated for fully nonlinear wave problem. The numerical results from incorporating the damping zone (DZ), MTF and MTF coupled DZ (MTF+DZ) methods as radiation condition are compared with analytical solution. An effective MTF+DZ method is finally adopted to simulate the 3D linear wave, second-order wave and irregular wave propagation. It is shown that the MTF+DZ method can be used for simulating fully nonlinear wave propagation very efficiently.

Simulation of irregular waves over submerged obstacle on a NURBS potential numerical wave tank

In this paper, a fully non-linear three-dimensional Numerical Wave Tank (NWT) is developed for studying propagation and scattering of non-linear random sea wave over bottom submerged bars. The simulation of fully non-linear free-surface is based on Non-Uniform Rational B-Spline formulation (NURBS) as a novel approach and Mixed Eulerian-Lagrangian method (MEL). High-order boundary integral equation is used to solve the Laplace equation in the Eulerian frame. To update the free-surface, time marching approach including material node method and fourth order Runge-Kutta time integration scheme is used. To obtain appropriate numerical solutions for wave propagation problem, damping zone is set at the downstream. Also, the NURBS approximation is employed to evaluate the velocity of the free-surface particles. Propagation of regular and irregular waves in a NWT is investigated and compared with the available experimental and numerical data. Transmission of the random sea wave over submerged bars is also compared with the experimental and prior numerical studies.

Numerical simulation of motion and deformation of ring bubble along body surface

Numerical simulation for fluid flow over an attached rigid body with a deformable ring bubble is analyzed based on the velocity potential theory together with the boundary element method (BEM). The analysis is focused on the axisymmetric case. The bubble surface is treated as a well defined air-liquid interface and is tracked by a mixed Eulerian-Lagrangian method. The points of intersection between the bubble and body are treated, specially in the numerical procedure. The auxiliary function method is adopted to calculate the pressure on the body surface and in the flow field. The convergence study is undertaken to assess the developed numerical method and the computation code. Some case studies are undertaken in which the interactions between the bubble/body and the incoming flow field are simulated. The effects of various physical parameters on the interactions are investigated.

A heterogeneous flow model based on DD method for free surface fluid–structure interaction problems

SUMMARYAn efficient heterogeneous flow model that combines incompressible viscous flow and potential flow(PF)(both) with free‐surface boundaries for fluid–structure interaction(FSI) problems was developed and solved numerically. In the model, the near field(viscous) fluid subdomain of the FSI simulation is described by the Navier–Stokes equations(NSE), whereas the far field(inviscid) subdomain is described by PF. This method is particularly suitable for modeling a large fluid domain while avoid solving the NSE at the far field, where the inviscid flow assumption is appropriate and adopting a simplified(e.g., PF) flow description reduces computing cost. Numerically, the NSE are solved using a FEM, and the PF is solved using a BEM. Free surface tracking is achieved through a hybrid level set method for the NSE and the mixed Eulerian–Lagrangian method for the PF. The coupling is based on a non‐overlapping DD method similar to the Dirichlet–Neumann method. At the interface between the two flow model subdomains, velocity and pressure are matched to ensure continuity of the corresponding variables at the common boundary. In this framework of the heterogeneous DD method, an explicit scheme is developed and implemented. The relation between this scheme and the classical Dirichlet–Neumann method for a homogeneous problem is revealed. Numerical examples demonstrating the capability of the method to model nonlinear wave phenomena and FSI problems with flexible structures are presented. Copyright © 2013 John Wiley & Sons, Ltd.

Influences of different forces on the bubble entrainment into a stationary Gaussian vortex

Simulations of bubble entrainment into a stationary Gaussian vortex are performed by using the combined particle tracking method (PTM) and boundary element method (BEM). Before the bubble is captured by the vortex core, oscillation and migration of the quasi-spherical nucleus are solved by using improved RP equation and the momentum theorem in the Lagrangian reference frame simultaneously, and the trajectory of the nucleus presents a kind of reduced helix shape. After captured by the vortex core, the bubble grows immediately and moves and deforms along the vortex core axis. The non-spherical evolution and deformation of the bubble is simulated by adopting a mixed Eulerian-Lagrangian method. The output of quasi-spherical stage is taken as the input of non-spherical stage, and all the behaviors of the entrained bubble can be simulated such as inception, motion, deformation and split. Numerical results agree well with published experimental data. On this basis, the influences of various factors such as viscosity, surface tension, buoyancy are studied systemically. Hopefully the results from this paper would provide some insight into the control on vortex bubble entrainment.

Quantification of nonlinear seismic response of rectangular liquid tank

Seismic response of two dimensional liquid tanks is numerically simulated using fully nonlinear velocity potential theory. Galerkin-weighted-residual based finite element method is used for solving the governing Laplace equation with fully nonlinear free surface boundary conditions and also for velocity recovery. Based on mixed Eulerian-Lagrangian (MEL) method, fourth order explicit Runge-Kutta scheme is used for time integration of free surface boundary conditions. A cubic-spline fitted regridding technique is used at every time step to eliminate possible numerical instabilities on account of Lagrangian node induced mesh distortion. An artificial surface damping term is used which mimics the viscosity induced damping and brings in numerical stability. Four earthquake motions have been suitably selected to study the effect of frequency content on the dynamic response of tank-liquid system. The nonlinear seismic response vis-a-vis linear response of rectangular liquid tank has been studied. The impulsive and convective components of hydrodynamic forces, e.g., base shear, overturning base moment and pressure distribution on tank-wall are quantified. It is observed that the convective response of tank-liquid system is very much sensitive to the frequency content of the ground motion. Such sensitivity is more pronounced in shallow tanks.

A numerical model of an air pocket impact during sloshing

The present work concerns a nonlinear numerical model of a sloshing experiment which consists of a gravity water wave inside a rectangular tank. At a certain time instant the free surface is shaped such that it entraps an air pocket close to the upper left corner of the tank. The air pocket is compressed and starts to oscillate. The pressure oscillations in the air pocket resemble the free oscillations of a single degree of freedom under-damped mass-spring system. The time before the wave touches the roof is modeled by a new numerical method called the boundary-element finite-difference method. This is a two-phase numerical method, designed to model the interaction effect between the escaping air and the water prior to impact. Both the water and the air are assumed incompressible and inviscid. The water flow is assumed irrotational and two dimensional and the air flow is assumed quasi-one-dimensional. Through different test cases the results from the new numerical method is compared with a linear analytical solution, a nonlinear multimodal solution and experimental results. The second part of the experiment, which is after the wave touches the roof and the air pocket is entrapped, is modeled using a mixed Eulerian–Lagrangian method. The air inside the air pocket can be assumed uniform in space. The air pocket is then modeled using a polytropic gas law. The water flow is described by potential flow theory. The results from the numerical model of the air pocket oscillations are compared with experiments. The comparison shows that the numerical model overestimates the maximum pressure inside the air pocket by 17%.

Numerical analysis of the hydroelastic behavior of a vertical plate due to solitary waves

Interactions of a vertical elastic plate with fully nonlinear water waves were simulated. Utilizing the mixed Eulerian Lagrangian method for the free-surface flow and the finite element method for the deflection of an elastic plate, a fully coupled scheme for accurately determining fluid–plate motions was developed. Using this scheme, some modifications to the solvers for both fluid and plate were made. A hybrid wave-absorbing beach was installed to prevent wave reflection from the end of the wave tank. A fourth-order Runge–Kutta time-marching scheme with a uniform time step was applied to achieve numerical stability. The method was validated by simulating the wave generated by the initial deformation of a vertical plate and comparing the result with the corresponding analytical solution. For further validation, the hydroelastic behavior of a vertical plate induced by a pulse-type wave (where the initial pulse-type elevation of the free surface is specified) was computed, and the result was compared with another numerical result from a mode-expansion method. The interaction of a surface-piercing plate with nonzero initial free surface was then simulated, and the result was compared with the corresponding linear analytical solution. Finally, the hydroelastic response of a surface-piercing vertical plate due to a solitary wave (generated by actuating the vertical plate at the right end of the tank only at the beginning) was computed and investigated systematically.

Numerical simulation of fully nonlinear sloshing waves in three-dimensional tank under random excitation

Based on the fully nonlinear velocity potential theory, the liquid sloshing in a three dimensional tank under random excitation is studied. The governing Laplace equation with fully nonlinear boundary conditions on the moving free surface is solved using the indirect desingularized boundary integral equation method (DBIEM). The fourth-order predictor-corrector Adams-Bashforth-Moulton scheme (ABM4) and mixed Eulerian-Lagrangian (MEL) method are used for the time-stepping integration of the free surface boundary conditions. A smoothing scheme, B-spline curve, is applied to both the longitudinal and transverse directions of the tank to eliminate the possible saw-tooth instabilities. When the tank is undergoing one dimensional regular motion of small amplitude, the calculated results are found to be in very good agreement with linear analytical solution. In the simulation, the normal standing waves, travelling waves and bores are observed. The extensive calculation has been made for the tank undergoing specified random oscillation. The nonlinear effect of random sloshing wave is studied and the effect of peak frequency used for the generation of random oscillation is investigated. It is found that, even as the peak value of spectrum for oscillation becomes smaller, the maximum wave elevation on the side wall becomes bigger when the peak frequency is closer to the natural frequency.