Subsidence of the Ekofisk Platforms: Wave in Deck Impact Study — Various Wave Models and Computational Methods

2002 ◽  
Author(s):  
Bogdan Iwanowski ◽  
Henrik Grigorian ◽  
Ingar Scherf
Keyword(s):  
Computational Methods ◽  
Stokes Equations ◽  
Type Equation ◽  
Navier Stokes ◽  
Stokes Wave ◽  
Wave Loads ◽  
Wave Impact ◽  
Navier Stokes Equations ◽  
Wave Models ◽  
Displacement Approach

Subsidence of the Ekofisk platforms creates several operational challenges. For safety of the platforms, it is of great importance to find the wave impact loads acting on the platforms’ decks. The paper describes how such loads can be computed. Three theoretical wave models are discussed in the paper: the Airy wave, Airy wave modified through Wheeler stretching and the 5th order non-linear Stokes wave. The wave loads for these wave models are computed by various methods. The method based on momentum displacement approach and Morison-type equation developed by Dr. Kaplan is used as a reference point. The loads are also computed through a solution of complete Navier-Stokes equations, with the Volume of Fluid (VOF) method used to trace motion of the fluid’s free surface. Results of different wave models and different computational methods are compared and discussed.

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.

scholarly journals Fluid-kinetic modelling for respiratory aerosols with variable size and temperature

2020 ◽  
Vol 67 ◽  
pp. 100-119 ◽  
Author(s):  
Laurent Boudin ◽  
Céline Grandmont ◽  
Bérénice Grec ◽  
Sébastien Martin ◽  
Amina Mecherbet ◽  
...  
Keyword(s):  
Stokes Equations ◽  
Kinetic Modelling ◽  
Type Equation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Aerosol Dynamics ◽  
Convection Diffusion Equation ◽  
Time Marching ◽  
Branched Structure ◽  
The Impact

In this paper, we propose a coupled fluid-kinetic model taking into account the radius growth of aerosol particles due to humidity in the respiratory system. We aim to numerically investigate the impact of hygroscopic effects on the particle behaviour. The air flow is described by the incompressible Navier-Stokes equations, and the aerosol by a Vlasov-type equation involving the air humidity and temperature, both quantities satisfying a convection-diffusion equation with a source term. Conservations properties are checked and an explicit time-marching scheme is proposed. Twodimensional numerical simulations in a branched structure show the influence of the particle size variations on the aerosol dynamics.

scholarly journals Development of An Integrated Numerical Model for Simulating Wave Interaction with Permeable Submerged Breakwaters Using Extended Navier–Stokes Equations

2020 ◽  
Vol 8 (2) ◽  
pp. 87 ◽  
Author(s):  
Paran Pourteimouri ◽  
Kourosh Hejazi
Keyword(s):  
Porous Medium ◽  
Wave Propagation ◽  
Numerical Model ◽  
Analytical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Stokes Wave ◽  
Navier Stokes Equations ◽  
Numerical Predictions ◽  
The Impact

An integrated two-dimensional vertical (2DV) model was developed to investigate wave interactions with permeable submerged breakwaters. The integrated model is capable of predicting the flow field in both surface water and porous media on the basis of the extended volume-averaged Reynolds-averaged Navier–Stokes equations (VARANS). The impact of porous medium was considered by the inclusion of the additional terms of drag and inertia forces into conventional Navier–Stokes equations. Finite volume method (FVM) in an arbitrary Lagrangian–Eulerian (ALE) formulation was adopted for discretization of the governing equations. Projection method was utilized to solve the unsteady incompressible extended Navier–Stokes equations. The time-dependent volume and surface porosities were calculated at each time step using the fraction of a grid open to water and the total porosity of porous medium. The numerical model was first verified against analytical solutions of small amplitude progressive Stokes wave and solitary wave propagation in the absence of a bottom-mounted barrier. Comparisons showed pleasing agreements between the numerical predictions and analytical solutions. The model was then further validated by comparing the numerical model results with the experimental measurements of wave propagation over a permeable submerged breakwater reported in the literature. Good agreements were obtained for the free surface elevations at various spatial and temporal scales, velocity fields around and inside the obstacle, as well as the velocity profiles.

Numerical Prediction of Impact-Related Wave Loads on Ships

2006 ◽  
Vol 129 (1) ◽  
pp. 39-47 ◽  
Author(s):  
Thomas E. Schellin ◽  
Ould el Moctar
Keyword(s):  
Stokes Equations ◽  
Numerical Procedure ◽  
Navier Stokes ◽  
Ship Motions ◽  
Normal Velocity ◽  
Wave Loads ◽  
Navier Stokes Equations ◽  
Critical Areas ◽  
Strip Theory ◽  
A Chain

We present a numerical procedure to predict impact-related wave-induced (slamming) loads on ships. The procedure was applied to predict slamming loads on two ships that feature a flared bow with a pronounced bulb, hull shapes typical of modern offshore supply vessels. The procedure used a chain of seakeeping codes. First, a linear Green function panel code computed ship responses in unit amplitude regular waves. Ship speed, wave frequency, and wave heading were systematically varied to cover all possible combinations likely to cause slamming. Regular design waves were selected on the basis of maximum magnitudes of relative normal velocity between ship critical areas and wave, averaged over the critical areas. Second, a nonlinear strip theory seakeeping code determined ship motions under design wave conditions, thereby accounting for the nonlinear pressure distribution up to the wave contour and the frequency dependence of the radiation forces (memory effect). Third, these nonlinearly computed ship motions constituted part of the input for a Reynolds-averaged Navier–Stokes equations code that was used to obtain slamming loads. Favorable comparison with available model test data validated the procedure and demonstrated its capability to predict slamming loads suitable for design of ship structures.

Numerical Solution of 3-D Water Entry Problems With a Constrained Interpolation Profile Method

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Qingyong Yang ◽  
Wei Qiu
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
Water Entry ◽  
Navier Stokes ◽  
Time Step ◽  
Constrained Interpolation ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
Calm Water

This paper presents the numerical solutions of slamming problems for 3D bodies entering calm water with vertical and oblique velocities. The highly nonlinear water entry problems are governed by the Navier-Stokes equations and were solved by a constrained interpolation profile (CIP)-based finite difference method on a fixed Cartesian grid. In the computation, the 3D CIP method was employed for the advection calculations and a pressure-based algorithm was applied for the nonadvection calculations. The solid body and the free surface interfaces were captured by density functions. For the pressure computation, a Poisson-type equation was solved at each time step by using the conjugate gradient iterative method. Validation studies were carried out for a 3D wedge, a cusped body vertically entering calm water, and the oblique entry of a sphere into calm water. The predicted hydrodynamic forces on the wedge, the cusped body, and the sphere were compared with experimental data.

scholarly journals MODELLING OF BREAKING WAVES IN TSUNAMI AND SLOSHING WAVES BY A NEW PARTICLE METHOD

2014 ◽  
Vol 34 ◽  
pp. 1460375 ◽  
Author(s):  
Mimi Gao ◽  
Chan Ghee Koh ◽  
Min Luo ◽  
Wei Bai
Keyword(s):  
Large Amplitude ◽  
Stokes Equations ◽  
Hydrodynamic Pressure ◽  
Particle Method ◽  
Breaking Waves ◽  
Particle Methods ◽  
Navier Stokes ◽  
Implicit Time ◽  
Wave Impact ◽  
Navier Stokes Equations

The recently developed Consistent Particle Method (CPM) is used to model breaking waves in tsunami and violent sloshing waves in a moving tank. Solving the Navier-Stokes equations in a semi-implicit time stepping scheme, the CPM eliminates the use of kernel function which is somewhat arbitrarily defined and used in other particle methods. It is demonstrated that the method is applicable to large amplitude free surface wave problems that involve breaking phenomenon. Tsunami wave impact on a fixed structure is modeled using CPM. The simulated results show fairly good agreement to the actual nonlinear wave motions including overturning and breaking of waves. Large amplitude sloshing waves in a moving tank are investigated with CPM. Experiment was conducted in the laboratory to verify the CPM solutions. The hydrodynamic pressure computed by the CPM agrees well with the experimental results.

scholarly journals Spatiotemporal dynamics in two-dimensional Kolmogorov flow over large domains

2014 ◽  
Vol 750 ◽  
pp. 518-554 ◽  
Author(s):  
Dan Lucas ◽  
Rich Kerswell
Keyword(s):  
Stokes Equations ◽  
Type Equation ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Multiple Attractors ◽  
Navier Stokes Equations ◽  
Kolmogorov Flow ◽  
Long Wavelength ◽  
Collisional Dynamics

AbstractKolmogorov flow in two dimensions – the two-dimensional (2D) Navier–Stokes equations with a sinusoidal body force – is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimics the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn–Hilliard-type equation and as a result coarsening dynamics is observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) partial differential equations (PDEs) based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions – a kink and antikink – which connect two steady one-dimensional (1D) flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour.

On the Mechanics of Non-Newtonian Media

1965 ◽  
Vol 87 (3) ◽  
pp. 689-693
Author(s):  
N. Tipei
Keyword(s):  
Stokes Equations ◽  
Practical Importance ◽  
Type Equation ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Navier Stokes Equations ◽  
Parallel Surfaces ◽  
Steady Motions ◽  
Isotropic Bodies ◽  
Good Agreement

The extension of Newton’s shearing law is aimed at the tensions being expressed by means of a general type equation. The case of the Bingham type media is considered and the components of the tensions tensor for homogeneous isotropic bodies are obtained in an orthogonal system of coordinates. The notion of viscosity is also extended by the introduction of viscosities of any order n, having the dimensions (M/L)Tn−2. The equation of motion upon any direction xi is then derived, extending thus the Navier-Stokes equations. Further the particular cases of incompressible fluids and steady motions are considered. Applications to simple cases are performed: The motion in tubes, coaxial cylinders, or between solid parallel surfaces. These applications lead, as particular forms of the general formulas obtained, to result in good agreement with those found by other authors (Paslay and Slibar, Milne, and so on). The flow between parallel plates is studied too, for different shearing laws. Finally, a more general form of the shearing stresses is considered and by the proposed generalization, the possibility of a direct unitary study of various continuous bodies of a great practical importance is obtained.

Computation of Slamming Forces on 3-D Bodies With a CIP Method

2010 ◽  
Author(s):  
Qingyong Yang ◽  
Wei Qiu
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
Navier Stokes ◽  
Cip Method ◽  
Time Step ◽  
Constrained Interpolation ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
Calm Water

This paper presents the numerical solutions of slamming problems for 3D bodies entering calm water. The highly nonlinear water entry problems are governed by the Navier-Stokes equations and were solved by a Constrained Interpolation Profile (CIP)-based finite difference method on a fixed Cartesian grid. In the computation, the 3D CIP method was employed for the advection calculations and a pressured-based algorithm was applied for non-advection calculations. The solid body and the free surface interfaces were captured by density functions. For the pressure computation, a Poisson-type equation was solved at each time step by the Conjugate Gradient iterative method. Validation studies were carried out for a 3D wedge entering calm water and the entry of a sphere into calm water at both vertical and horizontal velocities. The predicted hydrodynamic forces on the wedge and the sphere were compared with experimental data.

The Evolution of Computational Methods in Aerodynamics

1983 ◽  
Vol 50 (4b) ◽  
pp. 1052-1070 ◽  
Author(s):  
A. Jameson
Keyword(s):  
Euler Equations ◽  
Computational Methods ◽  
High Speed ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Flow Equation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Electronic Computers ◽  
Reynolds Averaged Navier Stokes

This paper surveys the evolution of computational methods in aerodynamics. Improvements in high-speed electronic computers have made it feasible to attempt numerical calculations of progressively more complex mathematical models of aerodynamic flows. Numerical approximation methods for a hierarchy of models are examined in ascending order of complexity, ranging from the linearized potential flow equation to the Reynolds averaged Navier Stokes equations, with the inclusion of some previously unpublished material on implicit and multigrid methods for the Euler equations. It is concluded that the solution to the Euler equations for inviscid flow past a complete aircraft is a presently attainable objective, while the solution to the Reynolds averaged Navier Stokes equations is a possibility clearly visible on the horizon.

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