Application of numerical models and codes Best practice report Application of numerical models and codes Task 3.4.4 of WP3 from the MERiFIC Project A report prepared as part of the MERiFIC Project "Marine Energy in Far Peripheral and Island Communities" February 2014 Written by Thomas Vyzikas ([email protected]), University of Plymouth With contributions from Deborah Greaves ([email protected]), University of Plymouth Dave Simmonds ([email protected]), University of Plymouth Christophe Maisondieu ([email protected]), Ifremer Helen Smith ([email protected]), University of Exeter Lisa Radford, ([email protected]) MERiFIC was selected under the European Cross-Border Cooperation Programme INTERREG IV A France (Channel) – England, co-funded by the ERDF. The sole responsibility for the content of this report lies with the authors. It does not represent the opinion of the European Communities. The European Commission is not responsible for any use that may be made of the information contained therein. The present report is intended for academic purposes only. It describes some of the numerical models for marine and ocean applications to the authors’ best knowledge on the date of its publication. The authors thus bear no responsibility for any inaccuracies or omissions regarding the description of the models. © University of Plymouth, February 2014 2 MERiFIC Application of numerical models and codes Contents Introduction .......................................................................................................................................... 9 The MERiFIC Project ......................................................................................................................... 9 This report ......................................................................................................................................... 9 Structure of the report...................................................................................................................... 10 Scope of the report .......................................................................................................................... 11 1. The Modelling Process ............................................................................................................. 12 1.1 Introduction to modelling ..................................................................................................... 12 1.2 Physical and Numerical modelling ...................................................................................... 12 1.2.1 SWOT analysis of physical models ................................................................................. 12 1.2.1.1 Strengths of physical models ................................................................................... 12 1.2.1.2 Weaknesses of physical models ............................................................................. 13 1.2.1.3 Opportunities for physical models ........................................................................... 13 1.2.1.4 Threats to physical models ...................................................................................... 14 1.2.2 SWOT analysis of numerical models .............................................................................. 14 1.2.2.1 Strengths of numerical models ................................................................................ 14 1.2.2.2 Weaknesses of numerical models ........................................................................... 15 1.2.2.3 Opportunities for numerical models ......................................................................... 15 1.2.2.4 Threats to numerical models ................................................................................... 15 1.2.3 Summary of SWOT analysis ........................................................................................... 16 2. Fundamentals of Fluid Mechanics ........................................................................................... 17 2.1 Governing equations of the fluid flow .................................................................................. 17 2.1.1 Mass conservation in three dimensions .......................................................................... 18 2.1.2 Momentum equation in three dimensions ....................................................................... 19 2.1.3 Energy equation in three dimensions .............................................................................. 22 2.2 Equation of state .................................................................................................................. 24 2.3 Navier-Stokes equations ..................................................................................................... 25 2.4 Summary of governing equations of fluid motion ................................................................ 28 2.5 Turbulence and its modelling............................................................................................... 28 2.5.1 Introduction to turbulence ................................................................................................ 28 2.5.2 Characteristics of turbulence ........................................................................................... 30 2.5.3 Mathematical modelling of turbulence ............................................................................. 30 3. Fundamentals of Wave Mechanics .......................................................................................... 32 3.1 Introduction .......................................................................................................................... 32 3.2 Characteristics of waves ...................................................................................................... 33 3.3 Wave theories ...................................................................................................................... 34 3.3.1 Linear wave theory (Stokes 1st) ....................................................................................... 34 3.3.1.1 The dispersion relationship ...................................................................................... 36 3.3.1.2 Deep water approximation ....................................................................................... 37 3.3.1.3 Shallow water approximation ................................................................................... 37 3.3.1.4 Phase and group velocity ........................................................................................ 38 3.3.1.5 Wave energy............................................................................................................ 39 3.3.2 Nonlinear waves .............................................................................................................. 41 3 MERiFIC Application of numerical models and codes 3.3.2.1 Introduction .............................................................................................................. 41 3.3.2.2 Stokes theory of nonlinear waves............................................................................ 41 3.3.2.3 Stream function ....................................................................................................... 42 3.3.2.4 Cnoidal and solitary wave theory............................................................................. 42 3.4 Transformation of waves in near shore environment .......................................................... 44 3.4.1 Shoaling ........................................................................................................................... 44 3.4.2 Refraction ........................................................................................................................ 44 3.4.3 Diffraction ......................................................................................................................... 45 3.4.4 Waves breaking in shallow water .................................................................................... 46 3.4.5 Wave reflection ................................................................................................................ 47 3.4.6 Wave-current interaction ................................................................................................. 48 3.5 Wave-structure interaction................................................................................................... 49 3.5.1 Wave forces ..................................................................................................................... 49 3.5.2 From potential flow to Morison equation ......................................................................... 49 3.5.3 Numerical methods for wave loading on large objects of arbitrary shapes ..................... 52 3.5.4 Spectral approach to wave force prediction .................................................................... 53 3.6 Irregular waves .................................................................................................................... 53 3.6.1 Wind generated waves .................................................................................................... 53 3.6.2 Wave statistics ................................................................................................................. 54 3.6.3 Wave spectrum ...............................................................................................................