The wave and current interaction is usual in a river mouth or in a gulf. The wave height becomes more flat in a rising tide as the wave travel in the same direction with current. The wave height becomes steep in the ebb tide as the wave travel in the opposite direction with current. Sometimes the wave will become breaking wave if its wave height is too steep. Therefore, during ebb tides where waves are steepened by the opposing current, entrance of some harbours may be hazardous for small boats.
Due to the interactions between waves and currents, the wave characteristics might change and this effect will alter the loading on structures. The influence of wave and current interaction to marine structure might not be negligible. Therefore the interaction between wave and current is one of the key problems needs to be researched for developing resources in the sea.
On the other hand, to model a body moving with constant speed in the wave using CFD method, as the computational domain is limited, a stationary body in a wave field with a relative velocity of the body will normally be selected. Because of the interaction between wave and current, the result might be different from the real scenario of body-wave encounter.
With significantly development of computer in these decades, design tools such as computational Fluid dynamics(CFD) become more and more useful. In addition, a deep understanding of this wave and current interaction can benefit the CFD simulation of strucutre travelling in waves. In this project, numerical wave tank is performed to simulate generation of incident wave and interaction between waves and structures instead of the test in real wave tank.
Aim and objectives
The overall aim of this Msc project is to develop necessary techniques and methodology using commercial CFD software for the prediction of wave and current interactions. This study is to find the influence of wave and current interaction by comparing theoretical analysis with computational simulation. 2D model will be used initially before 3D modelling. CFD package STAR-CCM+ Version 4.06.011 is used for this project.
The objectives can be summarised as below:
Complete literature review for numerical wave tank and wave-current interaction. Enhance the understanding of wave and current interaction and their effect on structures.
Use STAR-CCM+ to model numerical wave tank to simulate the linear gravity wave based on linear wave theory.
Simulate and investigate wave-current interaction. Compare the numerical results with theoretical solution of wave-current interaction.
Investigate coordinate transformation. Simulate a moving cylinder with constant speed in linear gravity wave and simulate a stationary cylinder in wave with relative speed current. Compare the difference between these two cases.
The schematic of this project is shown as Figure 1.
Figure 1: Project process and tasks
Research challenges
The first challenge is to confirm that the free surface behaviour of flooding can be modelled by computational fluid dynamics (CFD).
The second challenge is to model wave travels with current based on the gravity wave have been successfully modelled. The behaviour of wave travels with current will be validated with theoretical calculation.
The third challenge is to model a structure moving towards the linear gravity wave travelling and shown the pressure loading on the structure to compare with the stationary structure towards the wave and current interaction and see their difference. For the structure moving case, it might require moving mesh which is difficult to set up and get converged result.
Literature review
Wave-current interaction
Unna (1942) researched the influence of stream on the wave and concluded that steepness of wind-formed waves is not affected by the tidal stream. Longuet-Higgins and Stewart (1960,1961) introduced the concept of radiation stress. It stated that energy of waves over variable current is not conserved but is created or destroyed because of the rate of work done by the radiation stress and the current strain. Longuet-Higgins and Stewart (1960) showed that crests of longer wave become shorter and steeper while troughs becomes longer and lower when longer wave interacted with short wave which is a little different with Unna. Longuet-Higgins and Stewart (1961) concluded that amplification of the waves has some bearing on the theoretical efficiency of hydraulic and pneumatic breakwaters are affected by non-linear interactions. Consequently, Variation of amplitude of waves propagating on a non-uniform stream was derived by using the equations of conservation of mass, momentum and energy in Whitham (1962). More research related had been done by Phillips (1966).
Bretherton and Garrett (1968) stated wave energy divided by the intrinsic frequency called as conservation of wave action could be used to compute amplitude change for a board range of conservative systems. The break behavior of waves encountering an opposing current is described by a model based on wave action conservation.
Based on irrotational flow and second order stokes wave motion, Jonsson et al.(1970) carried on computation of the wave length in a current field by concept of the mean energy level and a set of conservation equations were used to calculate the current wave set down. Conservation of the wave action was discussed by Jonsson (1978) to calculate the depth of mean water surface and the mean Eulerian current velocity. The reason of a non-zero mass transport in a pure wave motion was also reviewed.
Nonlinear evolution of a wave packet moving over a variable depth was studied by Djordjevic and Redekopp (1978).
Review of work had been made by Peregrine (1976) and Peregrine & Jonsson (1983). Peregrine, Jonsson & Galvin (1983) gave an annotated bibliography of papers on wave-current interactions. Dalrymple (1973) describes various methods for calculating wave-current interaction. Beiboer(1984) has discussed wave-current interactions in relation to engineering design applications. He notes the need for a better statistical description of the joint probability of occurrence of extreme waves and currents. Srokosz (1985) reviewed the method for calculating the interaction of waves with a vertically varying current with which is steady and uniform in the horizontal plane.
Wave-current interaction in shallow water was studied by Yoon et al.(1989) on the basis of the classical Boussinesq theory. A set of Boussinesq-type equations were derived from the continuity equation and the Euler equations of motion while considering the relative magnitude of wave and current velocity components.
Based on estimating current-free plane surface wave and a uniform wave-free current which is normal to wave crest, Baddour and Song(1990a) calculated numerically wave height, wave length and water depth by solving a set of nonlinear equations to describe the interaction of wave and current. Baddour and Song(1990b) developed the velocity potential, dispersion relation, the particle kinematics and pressure distribution up to the third order in wave amplitude.
Numerical wave tank
Schäffer (1996) derived a complete mathematical model for position-controlled wavemakers including the sub- and superharmonic effects arising at second-order. Schäffer and Jakobsen (2003) derived a new theory of non-linear wave generation in position-control with active absorption.
Armenio (1998) employed SIMAC method to study the wave generation and propagation into a numerical wave tank. A pneumatic wave-maker at the left-hand side of the tank was implemented by the use of a pressure perturbation at the free surface. The dynamic loads over submerged square and rectangular cylinders were evaluated.
Chen, et al. (1998) derived Boussinesq-type equations for the combined motion of waves and currents in shallow water areas Based on the hypothesis that the current have uniform depth and have a magnitude as large as the shallow water wave celerity. Four scales including the particle velocity, the wave-nonlinearity, the surface elevation of the total wave-current motion, and the wave-dispersion are used to derive Boussinesq-type equations for the combined wave-current motion.
Dalzell (1999) extended existing deep-water wave-wave interaction theory to the second-order wave-wave interaction coefficients for finite depth.
Jensen and Grue (2002) found that the wave speed in the physical wave tank is slightly less than in the computations by comparing the result from a physical wave tank and inviscid linear computations.
The linear and nonlinear irregular waves are simulated in numerical wave tank by Boo (2002). The kinematic and dynamic free surface conditions were integrated in time using a multi-step method. The simulations of linear wave were verified by comparing the simulation results of linear regular and irregular waves with the theoretical values. It proven that the open boundary model can be applied to multi-component wave simulation.
Westphalen et al.(2007) successfully used two commercial softwares CFX and STAR CCM+ to model free surface waves. Regular waves are simulated by defining velocity components along vertical boundary as wave maker. 3 dimensional rectangular domain is used to represent a wave tank with 5 types of meshes are tried. Westphalen et al.(2008) studied extreme free surface wave has been simulated in the numerical tank by CFX and STAR-CCM+.
A mathematical model which is second order for the operation of wave makes using force-fedback control has been described by Spinneken and Swan (2009). Numerical tests shown the force-control is better than position-controlled.
Wave theory
Linearised theory of gravity waves
Stoker (1957) described the simple harmonic oscillated water wave of constant depth. Bishop and Price (1979) gave the linearised theory of gravity waves based on the assumption that the maximum wave amplitude is much smaller than the wave length. Therefore, the slope of the wave profile is small everywhere.
The velocity potential Φ satisfy the Laplace equation as equation 2.1.
^2 Φ=0 2.1
The linearised free surface condition is as equation 2.2.
∂Φ/∂t-gζ=0 2.2
Thus,
(∂^2 Φ)/ã€-∂tã€-^2 +g ∂Φ/∂Z=0 on Z= ζ(X,Y,t) 2.3
Therefore
∂Φ/∂Z=0 on Z= -d(X,Y) 2.4
Linear gravity wave elevation was given as equation 2.5.
η=acosâ¡(kx-ωt) 2.5
where a is amplitude, k is wave number, ω the angular frequency.
The relationship between wavelength λ and wave number k is given as equation 2.6.
kλ=2π 2.6
Where λ is wavelength. Wavelength is the distance over which the wave's shape repeats.
The relationship between angular frequency ω and period T is as equation 2.7.
Tω=2π 2.7
Wave speed c and wave number k have the relationship as equation 2.8.
c=L/T=ω/k 2.8
The dispersion relation was determined as equation 2.9.
ω^2=gktanhâ¡(kd) 2.9
Faltinsen (1999) gave the velocity potential of water wave as equation 2.10.
Φ=ga/ω coshk(y+d)/coshâ¡(kd) sinâ¡(kx-ωt) 2.10
The x-component and y-component velocities of fluid particles in linear gravity wave were provided as equations 2.11 and equation 2.12.
u=Aω (coshk(y+d))/(sinhâ¡(kd)) sin(kx-ωt) 2.11
v=Aω (sinhkâ¡(y+d))/(sinhâ¡(kd)) cos(kx-ωt) 2.12
where u is x component velocity and v is y component velocity. A is the wave amplitude, g is the gravitational constant, k is the wave number, d is the mean water level and ω the angular frequency.
Wave and current interaction
The following theoretical solutions which are given in Zou (2005) are based on potential theory. It includes three assumptions, firstly viscosity is ignored; secondly the interaction between water and the air above is neglected; thirdly, surface tension is ignored. Dispersion relationship for wave and current interaction was expressed as equation 2.9
ã€-(ω-kU)ã€-^2=gktanh(kh) 2.9
For the deep water condition, wave speed relationship were given as equation 2.10
C_r/C_0 =1/2(1+√(1+4 U/c_0 )) 2.10
Where C_0is the original speed of non-current wave. C_r is the speed of wave-current interaction. U is the current speed.
Wave length variation was given as equation
L/L_0 =1/2 ã€-(1+√(1+4 U/c_0 ) )ã€-^2 2.11
Where L_0 is the original wave length in non current existed case; L is the new length which combined by wave and current.
The amplitudes variation were described as follow equation 2.12
A/A_0 =2/(√(1+4 U/c_0 +√(1+4 U/c_0 )) √(1+√(1+4 U/c)) ) 2.12
Where A_0is amplitude in Non-current wave case and A is the amplitude in wave-current interaction case.
 
Methodology
This chapter review computational fluid dynamics and STAR-CCM+ for the purpose of build numerical wave tank.
Computational Fluid Dynamics (CFD)
For this project, numerical towing tank could be modelled by CFD. Recently years CFD is more and more popular due to two advantages of CFD analysis.
Firstly, more and more complex problems are possible to be solved numerically contributed by historically significant growth of computational power. Moreover, the physical limitations existed in the real towing tank as tank sizes, flow speed could easily be achieved by using CFD analysis. All the variable of specific case could by fully controlled by numerical experimentalist such as domain size, boundary conditions, flow types etc. However there are still some inherent limitations of CFD, therefore using result from towing tank test to validate CFD result is recommended.
Computational Fluid Dynamics uses numerical techniques to solve the equations defining fluid flow around bodies. Numerical approximations to mathematical models describing the physics of the actual fluid flow are solved by equations. The solution is defined at discrete positions in time and space. The spatial and temporal resolution of the solution will affect the accuracy of the result. However, higher resolution will costs significantly data storage and computational power. (Lecture Notes, 2009).
CFD user procedure
CFD analysis normally includes following four processes:
Meshing. Flow domains are split into smaller sub-domains which are called grids or meshes to analyze fluid flows due to the fluid flow governing partial differential equations could be applicable in simple cases. Sufficient mesh quality in important area is required for ensure the computational convergence while grids should be minimised in unnecessary region to reduce computational cost.
Pre-processing. Numerical schemes, physical models, initial conditions and boundary conditions need to be specified. It should pay attention to choose appropriate model to represent physics and get a stable converged solution with acceptable computational cost.
Solve. Governing equations are solved by the specified setting in pre-processing and fluid flow behaviour is determined.
Post-processing. Solver results are collected and analysed by purpose. Pressure, velocity etc might be plotted or output as the analysts needed.
Mathmatical Fluid model
Navier-Stokes equations is the fundamental basis of CFD problems. Mass, momentum and energy conservation governs the behaviour of fluid flow. Moreover, relationships of physical properties of the fluid medium density and viscosity have to be built. A mathematical formulation which allows viscous and rotational flow features to be included can be derived by the physical properties and laws of fluid motion. (Anderson,1995)
Continuity equation is based Conservation of mass applied to fluid on as equation 3.1
∂Ï/∂t+
(ÏV)=0 3.1
Momentum equation of an incompressible Newtonian fluid with constant viscosity is from applying Navier-Stokes equations for the conservation of momentum as equation 3.2
Ï DV/Dt=Ï(∂V/∂t+(
∙V)V)=Ïf-
p+μ
^2 V 3.2
Numerical parameters
Discretization. An approximation could be obtained by using a discretization method to generate a set of algebraic equations which could be solved by a computer if we want to numerically solve the flow's partial differential governing equations. There are three main approaches to discretisation of the governing equations: finite difference, finite volume and finite element methods.
Numerical grid. There are three types of grid: Structured grids, Unstructured grids and Hybrid grids. The quality of grid generation will be critical to the accuracy of numerical solution. The grid generation usually consumes the lot of user time when using numerical analysis methods to solve practical problems.
Convergence criteria. Residual behaviour in simulation is a good indication of error reduction within the simulation. The simulation is likely to convergence if the residuals are decreasing to a stable level. Residual level approach 10-3 is normally acceptable and qualitative engineering convergence solution.
STAR-CCM+
STAR-CCM+ is a commercial CFD software launched by CD-adapco company. STAR-CCM+ includes the whole engineering simulation process includes CAD, meshing, model set-up and iterative design studies. Compare to other CFD software, STAR-CCM+ has the unique advantage of automated meshing such as creating polyhedral or hexahedral volume meshes.
VOF method
Volume Of Fluid (VOF) method which is a simple multi-phase model (Hirt and Nichols, 1981) will be used to model the free surface wave in this project. It is well suited to simulate flows of several immiscible fluids on numerical grids capable of resolving the interface between the mixture's phases, as shown in Figure 2. Volume fraction of a phase is the ratio of the volume occupied by the phase over the computational cell volume.
Figure 2: Illustration of grids that are unsuitable (left) and suitable (right) for two-phase flows using the VOF model (Taken from CD-Adapco (2009))
All immiscible fluid phases present in a control volume are assumed share the same velocity, pressure and temperature fields. As a consequence, the same set of basic governing equations describing momentum, mass and energy transport in a single-phase flow is solved for an equivalent fluid whose physical properties are calculated as functions of the physical properties of its constituent phases and their volume fractions (CD-Adapco,2009).
Ï=∑_iâ-’ã€-Ï_i α_i ã€- 3.1
μ=∑_iâ-’ã€-μ_i α_i ã€- 3.2
C_p=∑_iâ-’ã€-(ã€-ã€-(Cã€-_p)ã€-_i Ï_i)/Ï_i α_i ã€- 3.3
Where
α_i=V_i/V is the volume fraction and Ï_i , μ_iand ã€-ã€-(Cã€-_p)ã€-_iare the density, molecular viscosity and specific heat of the i th phase.
The transport of volume fractions α_iis described by the following conservation equation:
d/dt ∫_Vâ-’α_i dV+∫_Sâ-’α_i (v-v_g )da=∫_Vâ-’s_(α_i ) dV 3.4
Where s_(α_i ) is the source or sink of the i th phase (CD-Adapco,2009)
Rigid body motion model
In the case of moving cylinder against wave, rigid body motion model is applied in STAR-CCM+ simulation. Rigid body motion model is used for unsteady simulations in which rigid mesh motions are specified. The mesh motions typically involve sliding meshes and repeating interfaces. A moving reference frame is added to all regions in which the moving reference model is activated. The moving reference frame model is used to simulate the effect of motion on a stationary mesh. The rigid body motion model uses the motion defined by the moving reference frame to recalculate the position of vertices on each face at each time step. This calculation will provide the grid flux for the momentum equation. (CD-Adapco,2009)
Multiphase flows methods
Flow and interaction of several phases within the same system where distinct interfaces exist between the phases is called multiphase flow. A phase can be defined as a quantity of matter within a system that has its own physical properties to distinguish it from other phases within the system.
In STAR-CCM+, there are three distinct approaches to modelling multiphase flow:
Firstly, the Lagrangian Multiphase model is used for systems that consist mainly of a single continuous phase carrying a relatively small volume of discrete particles, droplets or bubbles. It calculates the trajectories of representative parcels of the discrete phase as they pass through the system.
Secondly, the Volume of Fluid model is used for systems containing two or more immiscible fluid phases, where each phase constitutes a large structure within the system. This approach captures the movement of the interface between the fluid phases, and is often used for marine applications. This model is typically used for free surface flows.
Thirdly, the Multiphase Segregated Flow model is used for systems containing two or more generalized phases. It solves distinct conservation equations for each phase variable throughout the system. Phase interaction models are provided to define the influence exerted by one phase upon another across the interfacial area between them. It is also known as Eulerian Multiphase modelling. (CD-Adapco,2009)
User filed function
Field functions provide a mechanism by which fields (raw data from the simulation stored in the cells or on the boundaries may be viewed and defined in STAR-CCM+. User field functions are field functions that user create to access field data, and share a particular set of properties. They maybe of a scalar, vector, array or position type. They are created manually and are defined in terms of other field functions. The formulation of these field functions uses special STAR-CCM+ field function syntax. (CD-Adapco,2009)
