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Modelling Viscous Free Surface Flow

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Modelling Viscous Free Surface Flow Chapter 8 Modelling Viscous Free Surface Flow Free surface flows, where a boundary of a fluid body is free to move constrained only by forces across the surface, are possibly the most commonly observed flow phenomenon, with the motion of the free surface readily allowing the observation of flow of the fluid. Flows that are commonly encountered by the layperson include the motion of the surface of a river, the waves on the surface of the ocean, and the more personal flow of that in a cup of tea. From an engineering perspective, flows of interest include the open channel flow of rivers and canals, the erosive forces of waves on the shoreline, and the wakes generated by ships when under way. When modelling these flows, the problems associated with the solution of the Navier–Stokes equations are compounded by the free motion of the surface of the fluid, with the boundaries of the flow domain being a function of the flow structure, and thus an unknown which must be calculated along with the flow field. The movement of the free surface therefore both aids the observation of the flow, and hinders it’s modelling. In this chapter an attempt is made to model the steady flow around a ships hull. The modelling of such a flow is of great interest to Naval Architects, with the ability to predict the resistance of ships allowing the design of more efficient hull forms, whilst the accurate modelling of the ships wake allows the design of hulls that create a smaller disturbance in confined waterways. A brief overview of the modelling of ship flows is given, with emphasis being given to the use of CFD, being followed by the derivation of the numerical method used in this study. The implemented code is then used to model two test problems, open channel flow and flow in a free surface driven cavity, before being applied to the flow around a ships hull. 8.1 The Modelling of Flow About Ships The ability to predict the resistance of ships hulls is of interest to the ship building industry. With the ability to predict ships speeds the designers can hopefully design a ship that meets the performance specifications of the clients, whilst an understanding of ship resistance enables the designers to pro- duce a more efficient hull form or to reduce the wave generation, an issue of interest when running ships in confined waterways. Three methods are commonly used to predict the resistance of a ship. The earliest technique, as used by Froude, uses empirical data gained from testing a model in a towing tank. This process is not as straight-forward as modelling vehicles that are wholly immersed in a fluid, such as the case of aircraft models tested in a wind tunnel, since there are two length scales of interest in a ships flow, the Froude number ¢¡ , which is the ratio of the characteristic velocity and the wave speed for the flow, and the 195 CHAPTER 8. VISCOUS FREE SURFACE FLOW 196 Reynolds number £¥¤ , the ratio of the dynamic and viscous forces. These dimensionless groups are defined as1 ¦¡¨§ © (8.3) § © £¥¤ (8.4) When testing a model it is practically impossible to match both the Froude number and the Reynolds number of the full scale flow. The resistance of the full scale ship is therefore estimated by assuming that the total resistance of the ship can be broken down into two independent components, the wave drag, due to disturbances to the free surface and characterised by the Froude number, and the frictional resistance, due to skin friction and characterised by the Reynolds number. The frictional resistance of the model is traditionally approximated by the drag of a flat plate of the same cross section as the ship model, and so the wave drag of the model can be estimated as the total resistance of the model less this estimated frictional resistance. The resistance of the full scale ship can then be calculated as the sum of the scaled wave resistance of the model and the frictional resistance calculated for the Reynolds number of the full scale ship. A second class of techniques models the flow about the ships hull using a distribution of source singularities to perturb the free stream flow. The initial method for such a flow was developed by Mitchell[112]2, a summary of the methods being given by Wehausen[177]. The neglect of viscous effects from the models however becomes problematical when modelling ships with blunt sterns, propeller wakes, and vessels at an angle of attack to the free stream flow, where viscosity can effect the flow topology by causing and controlling flow separation. In more recent years CFD codes have been applied to modelling ship flows. These methods overcome some of the problems of the previous two techniques, but suffer from deficiencies of their own. Due to the large range of scales, varying from the length of the ships wave train to the smallest turbulent eddy, and the slow speed of the techniques, the resolution of the model used is a compromise between the physics of the problem and the computer resources and time available. Typically some form of turbulence model must be relied upon to model the finer structures of the flow. In addition, the discretisation schemes are dispersive and dissipative to varying degrees, affecting the accuracy of the solution. Worst of all, the methods are slow. Nevertheless, there is some hope for these techniques. With much research being performed into turbulence modelling and discretisation techniques it is hoped that the overall accuracy of the methods will improve. In addition, the rise in computer speeds and the use of parallelisation methods reduces the overall time taken for a computation and allows for a refinement in the mesh used to model the flow. 8.1.1 A Review of CFD Schemes Described in the Literature In the earliest paper to apply CFD techniques to the modelling of ship flows, Miyata and Inui[113] described a three dimensional finite difference model of the flow past a ship’s bow, their TUMMAC code. A laminar Navier–Stokes code using primitive variables, the differencing was done on a regular 1There is some variation in the definition of the Froude number. Most fluids texts define it as (8.1) However, some Authors (such as Lighthill[102]) use "! (8.2) as their definition. White[179] equivocates and gives both forms in different chapters of his book, without drawing attention to the inconsistency. 2An earlier model of the far field wake of a ship in deep water was developed by Lord Kelvin[171]. CHAPTER 8. VISCOUS FREE SURFACE FLOW 197 Cartesian mesh, with the bow of the ship being modelled by blocking out mesh nodes that lay within the ships volume. The surface of the flow was modelled using a “Marker and Cell” approach as developed by Harlow and Welch[57, 178], using a particle tracking approach to track the free surface, using the assumption that particles on the free surface remain on the surface. To model the steady state wave patterns, the solution is advanced using pseudo time stepping, with the positions of the particles on the surface being extrapolated at each time step to get the new position of the free surface using a first order explicit scheme. The vertical spacing of the mesh points at the free surface is irregular, with points being added and removed from the domain as the free surface rises and falls. Later developments to the TUMMAC code are described by Miyata, Nishimura and Masuko[116, 115], with the code being used to model a full ships geometry. For some calculations the viscosity was set to zero, reducing the Navier–Stokes equations to the Euler equations. A later paper applied the scheme to the modelling of a ship advancing into head seas[114]. To overcome problems with the original model, Miyata, Sato and Baba[117] developed the WISDAM code, using a non-orthogonal curvilinear mesh that fitted both the ship geometry and the free surface. At each time step a new wave pattern was calculated, and the mesh was regenerated, the equations being formulated to account for mesh movement from the remeshing. The Sub Grid Scale (SGS) turbulence model was incorporated into the momentum equations, which were applied in Reynolds averaged form. Further development by Zhu, Miyata, Kajitani and Watanabe[186, 118], WISDAM-V , reworked the code using a staggered grid finite volume discretisation. Initially the method did not account for movement in the free surface[186], but this was later implemented into the code[118]. Hino[61] developed a finite difference method that used a body fitted mesh that was fitted to the hulls surface. The method did not align with the free surface however, and a marker and cell scheme was used with mesh points being added and subtracted as the free surface rises and falls, in a manner similar to the original method of Miyata[113]. Later the author[62] switched to using a finite volume artificial compressibility method coupled with a Baldwin–Lowmax[6] turbulence model and a free surface fitted mesh. Kodama used an Implicit Approximation Factorisation scheme (IAF, a pseudo-compressibility method) to model the flow around a Wigley hull at zero Froude number[80], using a Baldwin–Lowmax turbu- lence model. A later paper discusses the details of the mesh generation used[81]. The method was then reworked to use a finite-volume scheme, and was applied to modelling double body flows around a flat plate and a series 60 ship hull[82].
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