Numerical Simulation of Nonlinear Water Wave Propagation Over Rippled Bed

2007 ◽  
Author(s):  
Gerasimos A. Kolokythas ◽  
Athanassios A. Dimas
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Water Waves ◽  
Stokes Equations ◽  
Surface Flow ◽  
Surface Boundary ◽  
Computational Domain ◽  
Time Step ◽  
Outflow Boundary ◽  
Rippled Bed

In the present study, numerical simulations of the free-surface flow, developing by the propagation of nonlinear water waves over a rippled bottom, are performed assuming that the corresponding flow is two-dimensional, incompressible and viscous. The simulations are based on the numerical solution of the Navier-Stokes equations subject to the fully-nonlinear free-surface boundary conditions and the suitable bottom, inflow and outflow boundary conditions. The equations are properly transformed so that the computational domain becomes time-independent. For the spatial discretization, a hybrid scheme with finite-differences and Chebyshev polynomials is applied, while a fractional time-step scheme is used for the temporal discretization. A wave absorption zone is placed at the outflow region in order to efficiently minimize reflection of waves by the outflow boundary. The numerical model is validated by comparison to the analytical solution for the laminar, oscillatory, current flow which develops a uniform boundary layer over a horizontal bottom. For the propagation of finite-amplitude waves over a rigid rippled bed, the case with wavelength to water depth ratio λ/d0 = 6 and wave height to wavelength ratio H0/λ = 0.05 is considered. The ripples have parabolic shape, while their dimensions — length and height — are chosen accordingly to fit laboratory and field data. Results indicate that the wall shear stress over the ripples and the form drag forces on the ripples increase with increasing ripple height, while the corresponding friction force is insensitive to this increase. Therefore, the percentage of friction in the total drag force decreases with increasing ripple height.

Large-Scale Free-Surface Flow Simulations Using FEM

2005 ◽  
Author(s):  
Shahrouz Aliabadi ◽  
Marvin Watts ◽  
Andrew Johnson
Keyword(s):  
Finite Element ◽  
Free Surface ◽  
Water Waves ◽  
Stokes Equations ◽  
Surface Flow ◽  
Rigid Body Dynamics ◽  
Element Formulation ◽  
Computational Domain ◽  
Governing Equations ◽  
Floating Objects

A parallel finite element fluid-structure interaction free-surface solver is developed for numerical simulation of water waves interacting with floating objects. In our approach, the governing equations are the Navier-Stokes equations written for two incompressible fluids. An interface function with two distinct values serves as a marker identifying the location of the interface. The numerical method is based on writing stabilized finite element formulations in an arbitrary Lagrangian-Eulerian frame. This allows us to handle the motion of the floating objects by moving the computational nodes. In the mesh-moving schemes, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements. In order to update the position of the floating object, the nonlinear rigid body dynamics equations are coupled with the governing equations of fluids and are solved simultaneously. The mooring forces are modeled using nonlinear cables and linear spring models. The finite element formulation is implemented on Cray X1.

The mechanism of vortex connection at a free surface

1999 ◽  
Vol 384 ◽  
pp. 207-241 ◽  
Author(s):  
CHIONG ZHANG ◽  
LIAN SHEN ◽  
DICK K. P. YUE
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Vortex Ring ◽  
Stokes Equations ◽  
Incidence Angle ◽  
Surface Boundary ◽  
Viscous Layer ◽  
Vertical Vorticity ◽  
Vorticity Component ◽  
Stress Boundary Conditions

Vortex connections at the surface are fundamental and prominent features in free-surface vortical flows. To understand the detailed mechanism of such connection, we consider, as a canonical problem, the laminar vortex connections at a free surface when an oblique vortex ring impinges upon that surface. We perform numerical simulations of the Navier–Stokes equations with viscous free-surface boundary conditions. It is found that the key to understanding the mechanism of vortex connection at a free surface is the surface layers: a viscous layer resulting from the dynamic zero-stress boundary conditions at the free surface, and a thicker blockage layer which is due to the kinematic boundary condition at the surface. In the blockage layer, the vertical vorticity component increases due to vortex stretching and vortex turning (from the transverse vorticity component). The vertical vorticity is then transported to the free surface through viscous diffusion and vortex stretching in the viscous layer leading to increased surface-normal vorticity. These mechanisms take place at the aft-shoulder regions of the vortex ring. Connection at the free surface is different from that at a free-slip wall owing to the generation of surface secondary vorticity. We study the components of this surface vorticity in detail and find that the presence of a free surface accelerates the connection process. We investigate the connection time scale and its dependence on initial incidence angle, Froude and Reynolds numbers. It is found that a criterion based on the streamline topology provides a precise definition for connection time, and may be preferred over existing definitions, e.g. those based on free-surface elevation or net circulation.

Time-Domain Nonlinear Wave Making Simulation of the Catamaran Based on Velocity Potential Boundary Element Method

2010 ◽  
Author(s):  
Dakui Feng ◽  
Xianzhou Wang ◽  
Zhiguo Zhang ◽  
Yanming Guan
Keyword(s):  
Free Surface ◽  
Radiation Condition ◽  
Flow Fields ◽  
Design Stage ◽  
Hydrodynamic Characteristics ◽  
Outer Side ◽  
Surface Boundary ◽  
Computational Domain ◽  
Time Step ◽  
Surface Boundary Conditions

The catamaran is composed of two monohulls, the flow fields between the inner and outer side of each monohull are different, the bodies must be considered as lifting bodies. So it is very important to know the lifting effect on hydrodynamic characteristics of catamaran hull at the preliminary design stage of its hull form. The pressure Kutta condition is imposed on the trailing-surface of the lifting body by determining the dipole distribution, which generates required circulation on the lifting part. The method is based on Green’s second theorem. Rankine Sources and dipoles are placed on boundary surfaces. Time-stepping scheme is adopted to simulate the wave generated by the catamaran with a uniform speed in deep water. The values of the potential and position of the free surface are updated by integrating the nonlinear Lagrangian free surface boundary conditions for every time. A moving computational window is used in the computations by truncating the fluid domain (the free surface) into a computational domain. The grid regeneration scheme is developed to determine the approximate position of the free surface for the next time step. An implicit implement of far field condition is enforced automatically at the truncation boundary of the computational window, Radiation condition is satisfied automatically. The influences on the wave making resistance of the distance between the twin hulls of the Wigley catamaran on the hydrodynamic characteristics are discussed. The numerical results are presented compared with the existing simulation result. The method can be used to simulate the flow fields around the foil near free surface.

scholarly journals LARGE-WAVE SIMULATION OF TURBULENT FLOW INDUCED BY WAVE PROPAGATION AND BREAKING OVER CONSTANT SLOPE BED

2012 ◽  
Vol 1 (33) ◽  
pp. 65
Author(s):  
Gerasimos Kolokythas ◽  
Aggelos Dimakopoulos ◽  
Athanassios Dimas
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Wave Breaking ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Surface Boundary ◽  
Large Wave ◽  
Wave Simulation ◽  
Constant Slope ◽  
Good Agreement

In the present study, the three-dimensional, incompressible, turbulent, free-surface flow, developing by the propagation of nonlinear breaking waves over a rigid bed of constant slope, is numerically simulated. The main objective is to investigate the process of spilling wave breaking and the characteristics of the developing undertow current employing the large-wave simulation (LWS) method. According to LWS methodology, large velocity and free-surface scales are fully resolved, and subgrid scales are treated by an eddy viscosity model, similar to large-eddy simulation (LES) methodology. The simulations are based on the numerical solution of the unsteady, three-dimensional, Navier-Stokes equations subject to the fully-nonlinear free-surface boundary conditions and the appropriate bottom, inflow and outflow boundary conditions. The case of incoming second-order Stokes waves, normal to the shore, with wavelength to inflow depth ratio λ/dΙ = 6.6, wave steepness H/λ = 0.025, bed slope tanβ = 1/35 and Reynolds number (based on inflow water depth) Red = 250,000 is investigated. The predictions of the LWS model for the incipient wave breaking parameters - breaking depth and height - are in very good agreement with published experimental measurements. Profiles of the time-averaged horizontal velocity in the surf zone are also in good agreement with the corresponding measured ones, verifying the ability of the model to capture adequately the undertow current.

A Stable Semi-Implicit Method for Free Surface Flows

2005 ◽  
Vol 73 (6) ◽  
pp. 940-947 ◽  
Author(s):  
Cassio M. Oishi ◽  
José A. Cuminato ◽  
Valdemir G. Ferreira ◽  
Murilo F. Tomé ◽  
Antonio Castelo ◽  
...  
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Free Surface Flow ◽  
Free Surface Flows ◽  
Surface Flow ◽  
Navier Stokes ◽  
Variable Order ◽  
Implicit Methods ◽  
Time Lagged

The present work is concerned with a semi-implicit modification of the GENSMAC method for solving the two-dimensional time-dependent incompressible Navier-Stokes equations in primitive variables formulation with a free surface. A projection method is employed to uncouple the velocity components and pressure, thus allowing the solution of each variable separately (a segregated approach). The viscous terms are treated by the implicit backward method in time and a centered second order method in space, and the nonlinear convection terms are explicitly approximated by the high order upwind variable-order nonoscillatory scheme method in space. The boundary conditions at the free surface couple the otherwise segregated velocity and pressure fields. The present work proposes a method that allows the segregated solution of free surface flow problems to be computed by semi-implicit schemes that preserve the stability conditions of the related coupled semi-implicit scheme. The numerical method is applied to both the simulation of free surface and to confined flows. The numerical results demonstrate that the present technique eliminates the parabolic stability restriction required by the original explicit GENSMAC method, and also found in segregated semi-implicit methods with time-lagged boundary conditions. For low Reynolds number flows, the method is robust and very efficient when compared to the original GENSMAC method.

Numerical Simulation of Two-Dimensional Free-Surface Flow and Wave Transformation Over Constant-Slope Bottom Topography

2007 ◽  
Author(s):  
Aggelos S. Dimakopoulos ◽  
Athanassios A. Dimas
Keyword(s):  
Free Surface ◽  
Numerical Solution ◽  
Euler Equations ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Wave Transformation ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Time Step ◽  
Good Agreement

A numerical model is presented for the simulation of the two-dimensional, inviscid, free-surface flow developing by the propagation and breaking of water waves over a flat bottom of steep slope. The simulation is based on the numerical solution of the unsteady, two-dimensional, Euler equations subject to the fully-nonlinear free-surface boundary conditions, the non-penetration condition at the bottom and appropriate inflow and outflow conditions. A boundary-fitted transformation, which includes both the time-dependent free surface and the arbitrary bottom shape, is applied. For the numerical solution of the Euler equations, a two-stage fractional time-step method is employed for the temporal discretization, while a hybrid scheme is used for the spatial discretization. Finite differences are used in the streamwise direction and a pseudo-spectral method in the vertical direction. An absorption zone is placed at the outflow region in order to minimize wave reflection by the outflow boundary. Wave breaking is modeled by a surface roller breaking model, which modifies the dynamic free-surface condition. The simulation results are in very good agreement with available experimental results for the wave propagation and breaking over bottom with slope 1:35. Results, from the simulations over bottom with steeper slopes of 1:15 and 1:10, which generate strong spilling and mild plunging breakers, respectively, are also in very good agreement with available predictions for the breaking depth and wave height. In all cases, a vortex is formed under the breaking wave front and convected in the surf zone.

Nonlinear Ship Motions in the Time-Domain Using a Body-Exact Strip Theory

2008 ◽  
Author(s):  
Piotr J. Bandyk ◽  
Robert F. Beck
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Time Domain ◽  
The Body ◽  
Ship Motions ◽  
Surface Boundary ◽  
Theory Approach ◽  
Offshore Structure ◽  
Time Step ◽  
Strip Theory

Modern offshore structure and ship design requires an understanding of responses in large seas. A nonlinear time-domain method may be used to perform computational analyses of these events. To be useful in preliminary design, the method must be computationally efficient and accurate. This paper presents a body-exact strip theory approach to compute wave-body interactions for large amplitude ship motions. The exact body boundary conditions and linearized free surface boundary conditions are used. At each time step, the body surface and free surface are regrided due to the changing wetted body geometry. Numerical and real hull forms are used in the computations. Validation and comparisons of hydrodynamic forces are presented. Selected results are shown illustrating the robustness and capabilities of the body-exact strip theory. Finally, an equation of motion solver is implemented to predict the motions of the vessel in a seaway.

scholarly journals Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 461
Author(s):  
Kenta Oishi ◽  
Yoshihiro Shibata
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Mhd Flow ◽  
Transmission Conditions ◽  
Well Posedness ◽  
Anisotropic Space ◽  
Free Boundary Conditions ◽  
Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

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