# Central concepts and methods

We focus on surface gravity wave modelling on the time scale of a few wave periods and the length scale of a few wave lengths. On these scales both the water and the air phase above are well described by the incompressible Navier-Stokes equations with gravity as a source term. These equations relate the velocity field and pressure field, together defining the state of each fluid phase for given instantaneous water surface shape.

The most common approach in CFD for solving these equations is the Finite Volume Method (FVM) [3]. Here the physical domain of interest is first divided into many small computational cells. The governing equations are then formally integrated over the volume of each cell and over a small time interval. The different terms in the equations are then approximated with numerical schemes yielding linear relations between the former and present values of the fluid state variables in adjacent cells. With up to billions of cells in today’s state-of-the-art calculations there are also billions of coupled equations to be solved by the computer.

Free surface simulations are particularly challenging because the position of one of the fluid boundaries – the free surface – must be calculated as part of the solution. During the past half-century a myriad of methods for solving this problem have emerged [4]. To date the most successful method for ocean wave applications is the so-called Volume-of-Fluid (VOF) method [5].

The VOF method exploits that the density of a fluid particle is constant; ρ_water≈ 1000 kg/m3 if it is a water particle and ρ_air≈ 1.2 kg/m3 if it consists of air. The density field is thus a passive tracer field. By a constant shift and rescaling, this field is converted into a dimensionless fluid marker field taking the value 1 in water and 0 in air. The equation of motion for this field is the passive advection equation. In the spirit of FVM, this equation is formally integrated over each cell to obtain an evolution equation for the cell water content; a quantity usually denoted as the volume fraction field. The constructed equation is essentially an accountantship with the volume fraction in terms of the flow of water to and from neighbouring cells. This flow depends on the instantaneous velocity and on the internal distribution of water in the cells. Unfortunately, this distribution is not available and must therefore be approximated from the available information, to obtain a closed system of equations for the velocity, volume fraction and pressure field. VOF methods differ in the numerical schemes used in this approximation procedure. The aim of the present project is to develop a new VOF method to circumvent the problems and limitations of existing schemes.

A convenient quality of the VOF approach is its inherent ability to conserve the amount of water and air present in the domain both locally and globally as the volume fraction field is advanced in time. The main drawback of the VOF method is that it operates with a discontinuous field on a discrete mesh. As the initially sharp jump in density is propagated across the mesh it will diffuse, eventually extending over many cells. The challenge is to construct a VOF scheme that avoids or repairs this diffusion, thus keeping the surface sharp and well defined at all times.

# Project foci

Free surface kinematics. A particular challenge when modelling air-water interfaces is the almost 3 orders of magnitude in density difference. This means that the amount of momentum in air and water moving with the same speed will also differ by 3 orders of magnitude. The VOF method is therefore highly sensitive to any mismatch in the numerical treatment of mass and momentum advection. In practical calculations this often leads to high artificial air currents generated along the water surface [6]. In the worst cases this destabilizes the water surface causing a nonsense solution or simulation crash. Even in simulations that look superficially reasonable, large overshoots in the water velocity near the free surface are often observed. Some codes circumvent this problem by simply omitting the air phase in the calculations [7]. Unfortunately, this strategy prevents proper modelling of situations such as wave-in-deck and slamming load calculations, where air entrapment may have large effects. In this project we will therefore retain the air phase and instead work on obtaining a proper detailed momentum balance near the free surface with the developed numerical schemes. The hypothesis is that this can only be obtained by a more integrated approach to the scheme choices in the volume fraction equation and the momentum equation.

Unstructured meshes. The FVM method for single phase flows with properly chosen schemes is generally second order accurate with respect to mesh refinement. The additional challenges encountered in two-phase flows, due to the presence of the free surface, significantly lowers the order of convergence for virtually any existing methods including the VOF approach. This means that very fine meshes must be used to obtain a desired resolution leading to excessive computation times and unfeasibility of CFD for some applications. Recently developed methods have shown better convergence properties, but only on simple Cartesian meshes [8] [9]. For the methods to be useful in engineering applications with complicated structures present, more general higher order methods must be developed also for unstructured meshes. This will be the second project focus.

Active wave absorption. When simulating waves in CFD, boundary conditions for the fluid state variables are an important part of the problem setup. A wave travelling through the computational domain and hitting a domain wall is reflected back into the domain if conventional slip or no-slip conditions are imposed. This is undesirable when modelling a section of the open sea, e.g. around an offshore wind turbine. Conventional methods to remedy this problem involve extending the domain with large wave damping or relaxation zones[10]. This significantly increases the required computational costs, often to an impractical level. Alternatively, active wave absorption boundary conditions can be developed where the field values on the boundary are set up to model full transmission of the wave through the boundary[11]. This is already used in DHI’s physical model test facilities, where the reflection problem also occurs due to the finite size of wave tanks[12]. Here, feedback control systems make the wave generating paddles on the sides of the basin take into account any incoming wave absorbing its energy as efficiently as possible, whilst simultaneously generating the desired input wave signal. In physical tests, the success of the active absorption strategy is limited by restrictions in paddle motion and by response time delays. Information about the wave field in front of the wave maker is typically also rather sparse. None of these limitations are present in CFD. Therefore the potential for efficient active absorption is large if the flexibility and available data is exploited cleverly. This is the third focus of the project.

# Code development

The coding framework of choice is OpenFOAM®; an extensive, object-oriented, parallelised C++ library for computational continuum mechanics. It is open source and has gained much momentum during the past decade with a large international community contributing to its development and maintenance. It includes all essential CFD core functionality allowing us to focus on the aspects of relevance to free surface calculations.

In line with the Open Access policy of Danish funding agencies the developed code will be published as open source.

This choice of framework increases the chance of the developed code “staying alive” in the community after project termination.

# Literature

[1]M. Perlin, W. Choi, and Z. Tian, ‘Breaking Waves in Deep and Intermediate Waters’, Annual Review of Fluid Mechanics, vol. 45, no. 1, pp. 115–145, 2013.

[2]DNV, “Environmental Conditions and Environmental Loads”, DNV RP-C205, October 2010.

[3]J. H. Ferziger and M. Perić, Computational methods for fluid dynamics. Springer London, Limited, 2002.

[4]R. Scardovelli and S. Zaleski, ‘DIRECT NUMERICAL SIMULATION OF FREE-SURFACE AND INTERFACIAL FLOW’, Annual Review of Fluid Mechanics, vol. 31, pp. 567–603, Jan. 1999.

[5]C. . Hirt and B. . Nichols, ‘Volume of fluid (VOF) method for the dynamics of free boundaries’, Journal of Computational Physics, vol. 39, no. 1, pp. 201–225, Jan. 1981.

[6]J. Mencinger and I. Žun, ‘On the finite volume discretization of discontinuous body force field on collocated grid: Application to VOF method’, Journal of Computational Physics, vol. 221, no. 2, pp. 524–538, Feb. 2007.

[7]E. D. Christensen, ‘Large eddy simulation of spilling and plunging breakers’, Coastal Engineering, vol. 53, no. 5–6, pp. 463–485, Apr. 2006.

[8]J. E. Pilliod Jr. and E. G. Puckett, ‘Second-order accurate volume-of-fluid algorithms for tracking material interfaces’, Journal of Computational Physics, vol. 199, no. 2, pp. 465–502, Sep. 2004.

[9]G. D. Weymouth and D. K.-P. Yue, ‘Conservative Volume-of-Fluid method for free-surface simulations on Cartesian-grids’, Journal of Computational Physics, vol. 229, no. 8, pp. 2853–2865, Apr. 2010.

[10]N. G. Jacobsen, D. R. Fuhrman, and J. Fredsøe, ‘A wave generation toolbox for the open-source CFD library: OpenFoam®’, International Journal for Numerical Methods in Fluids, vol. 70, no. 9, pp. 1073–1088, 2012.

[11]P. Higuera, J. L. Lara, and I. J. Losada, ‘Realistic wave generation and active wave absorption for Navier–Stokes models: Application to OpenFOAM®’, Coastal Engineering, vol. 71, pp. 102–118, Jan. 2013.

[12]H. Schäffer and G. Klopman, ‘Review of Multidirectional Active Wave Absorption Methods’, Journal of Waterway, Port, Coastal, and Ocean Engineering, vol. 126, no. 2, pp. 88–97, 2000.