university of twente faculty of engineering technology (CTW) department of engineering fluid dynamics Numerical modelling of non-linear shallow water waves Supervisor Author Prof. Dr. Ir. H.W.M. L.H. Lei BSc Hoeijmakers August 22, 2017 Preface An important part of the Master Mechanical Engineering at the University of Twente is an internship. This is a nice opportunity to obtain work experience and use the gained knowledge learned during all courses. For me this was an unique chance to go abroad as well. During my search for a challenging internship, I spoke to prof. Hoeijmakers to discuss the possibilities to go abroad. I was especially attracted by Scandinavia. Prof. Hoeijmakers introduced me to Mr. Johansen at SINTEF. I would like to thank both prof. Hoeijmakers and Mr. Johansen for making this internship possible. SINTEF, headquartered in Trondheim, is the largest independent research organisation in Scandinavia. It consists of several institutes. My internship was in the SINTEF Materials and Chemistry group, where I worked in the Flow Technology department. This department has a strong competence on multiphase flow modelling of industrial processes. It focusses on flow assurance market, multiphase reactors and generic flow modelling. During my internship I worked on the SprayIce project. In this project SINTEF develops a model for generation of droplets sprays due to wave impact on marine structures. The application is marine icing. I focussed on understanding of wave propagation. I would to thank all people of the Flow Technology department. They gave me a warm welcome and involved me in all social activities, like monthly seminars during lunch and the annual whale grilling. In special I would like to thank Sverre Gullikstad Johnson and Stein Tore Johansen for the daily supervision, willingness to answer questions and discussing progress and results! My internship assignment was really interesting and challenging. These three months gave me a nice insight in how it is to work in fundamental research and especially in the field of numerical modelling. Beside daily work, I attended the international conference on computa- tional fluid dynamics organized in Trondheim, which showed me the wide range of applications in computational fluid dynamics. All of this made me have an awesome time in Norway! 3 Summary Marine icing is ice accretion on offshore structures and vessels. Heavy ice accretion can be a severe threat, since it leads to safety problems or damaging of equipment. It can even lead to instability with all its fatal consequences. In SINTEF, fundamental research is done to develop models for forecasting marine icing. Marine icing is mainly caused by sea spray, which is induced by water waves impacting on vessels or offshore structures. In this internship a numerical study is done to verify the possibility of modelling of non-linear water waves using ANSYS Fluent. Before a numerical study was done in ANSYS Fluent, a 1D Boussinesq model had been elab- orated. Boussinesq type of systems are mathematical models to describe shallow water waves. One type of shallow water waves are solitary waves, which are characterized by maintaining its shape over a long distance, while it propagates with constant velocity. Solitary waves will never merge. The 1D Boussinesq model was used to gain insight in (numerical) behaviour of these waves, which could be used as a benchmark for the numerical study in ANSYS FLe- unt. The Boussinesq model was discretised on a staggered grid using upwinding schemes. The initialization was based on the analytical expression of solitary waves. Results were validated with experiments of solitary wave interactions. In these experiments two cases were considered; head-on and overtaking collisions. For both cases, the 1D Boussinesq model was capable in reproducing the wave interaction with a relatively small error. For overtaking collisions friction was added to come to a better fit. In ANSYS Fluent a 2D geometry was built, which was meshed on a uniform grid. Volume of fluid method was used to model the multi phases of water and air. Initial conditions were set with a user-defined functions. This user-defined function enhanced the standard the features of ANSYS Fluent. The volume fractions, velocity field and pressure distribution were initial- ized using this function. The results were again validated with experimental data. The data was really well presented by the 2D model. However, numerical dissipation was quite significant. It can be concluded that ANSYS Fluent is a suitable CFD method to handle wave propa- gation. In the continuation of the project, geometry and user defined functions can be adjusted to make it applicable for this marine icing project at SINTEF. 4 Contents 1 Introduction 5 2 Wave Theory 7 2.1 Linear wave theory . .7 2.2 Non-linear wave theory . .9 3 1D Boussinesq model 12 3.1 Discretization Boussinesq model . 12 3.1.1 Mass conservation . 13 3.1.2 Momentum conservation . 14 3.1.3 Initial and boundary conditions . 15 3.2 Single wave propagation . 16 3.3 Experimental and numerical comparisons . 17 3.3.1 Head-on collisions . 19 3.3.2 Overtaking collisions . 19 3.3.3 Accuracy study . 25 4 2D Naviers-Stokes model 26 4.1 Initial conditions . 26 4.1.1 Initialization volume fraction . 27 4.1.2 Initialization velocity . 27 4.1.3 Initialization pressure . 28 4.2 Solution methods . 29 4.3 Single wave propagation . 29 4.4 Experimental and numerical comparisons . 30 4.4.1 Head-on collisions . 30 4.4.2 Overtaking collisions . 30 4.4.3 Accuracy study . 36 5 Conclusion 37 6 Recommendations 38 7 Bibliography 39 8 Appendix A1 5 1. Introduction Marine icing is ice accretion on offshore structures and can occur in sufficiently cold and windy conditions. It can lead to severe safety problems, such as slippery decks, ladders and handrails. Equipment like valves and winches can be become useless, causing delays in operation. Radar antennas can be damaged by icing. Heavy ice accretion can increase the size of structural members, which can lead to higher wind forces. It also increases the total weight and raises the center of gravity, which can lead to instability. In figure 1.1 examples of marine icing are shown. It clearly visualises the enormous impact of marine icing. (a) (b) Figure 1.1: Examples of marine icing Marine icing is mainly caused by sea spray. It can be induced by wave impacts on vessels or offshore structures. Due to wave impact, a spray of droplets can be generated, which is moved and cooled by ambient air. If the ambient air temperature is below the freezing point of the seawater, these small supercooled water droplets impacts on vessels or offshore structures and freeze. The process of spray formation is related to different complicated phenomena, including propagation of the free water surface, wave slamming, spray formation after impact and droplet break up. These droplets will eventually lead to ice formation on a surface. Figure 1.2 presents an overview of these phenomena. In the SprayIce project, fundamental research is done to develop models for spray generation, which contain all these phenomena. This gives insight in the droplet distribution and where and in which amount marice icing occurs. 6 CHAPTER 1. INTRODUCTION Figure 1.2: Formation of sea spray This internship assignment is part of the SprayIce project. It focusses on numerical mod- elling of wave propagation. A method is needed which generates oncoming waves to an object. It must be verified that the wave structure is not destroyed by numerical dissipation before impact. In chapter 2, an introduction to wave theory is given. Starting from the Navier-Stokes equations the linear wave theory is derived, which is expanded to the non-linear wave theory ending up with the 1D Boussinesq model. In chapter 3, the 1D Boussinesq model has been elaborated in more detail. The discretized model is given, together with initial and boundary conditions. Results based on this model are presented and compared with experimental data. A grid refinement study has been performed to check the accuracy of different grid sizes and time steps. These observations can be used as a benchmark for 2D simulations. The 2D simula- tions have been perfomed in commercial CFD software, which are discussed in chapter 4. The initialization of different flow variables is explained and similar results as for the 1D model are presented. Conclusions can be found in chapter 5. Recommendations for future work will be given in chapter 6. 7 2. Wave Theory Surface waves are disturbances on the interface between two fluids. In this report there will be focussed on surface waves between water and air in oceans. Ocean waves can be mainly divided in two types: gravitational and capillary waves. Capillary waves can be observed as small ripples on even flat sea surfaces. This report focusses on gravitational waves. These are sustained by the gravitational force after an initial disturbance on the surface. This can occur due to tide, wind, currents, earthquakes, ships, etc. In figure 2.1 examples of both types of waves are shown. (a) Capillary waves (b) Gravitational waves Figure 2.1: Examples of different type of water waves 2.1 Linear wave theory The simplest way to describe propagation of water waves is the linear wave theory. The math- ematical theory will be reviewed, which leads to expressions for the shape of the free surface and velocity field of the wave [1]. It is assumed that the ocean is incompressible and inviscid. The equations for respectively conservation of mass and momentum are then given as r · u = 0 (2.1) @u 1 + u · ru = − rp + g (2.2) @t ρ The ocean can also be considered as irrotational, and hence the velocity field u can be written as the gradient of a potential function, u = rφ.