A finite element model of interaction between viscous free surface waves and submerged cylinders
Introduction
Free surface flows are encountered in a wide variety of hydraulic engineering situations. Sloshing in containers may result from transportation of liquids, earthquake-induced base-excitation, wave-induced oscillations on a floating offshore platform, or so-called green water flow on ship decks (Jones and Hulme, 1987). Wind-induced surface stresses cause set-up and waves to form in the open sea and lakes. Steep waves may induce ringing of offshore structures (Chaplin et al., 1997). In the recent past, Computational Fluid Dynamics (CFD) models of free surface flows have tended to assume the liquid is inviscid (the majority of sloshing, solitary and progressive wave simulations) or to undertake depth-integration and assume hydrostatic pressure distributions (shallow water equations). For example, Wu et al. (1998) use a finite element model to simulate sloshing of inviscid liquid in a 3D rectangular tank, and are able to produce extreme free surface motions including travelling waves and bores. Chern et al. (1999) consider non-linear inviscid waves using a pseudospectral matrix element method with a σ-transformation between the free surface and bed. Although such models give convincingly accurate simulations when applied appropriately, they are unsuitable for cases where the free surface is changing rapidly and viscosity is important (e.g. the separated flow past a submerged obstacle under waves). With the availability of vastly improved computer resources, modelling of viscous free surface flows is becoming feasible (e.g. Floryan and Rasmussen, 1989; Yeung and Yu, 1995; Gentz et al., 1996; Ushijima, 1998).
As discussed by Ferziger and Peric (1999), two main approaches are used for free surface flow modelling; namely, interface-tracking and interface-capturing methods. Interface-tracking involves adaptive fitting of the computational grid to the moving free surface. Typically, the kinematic free surface boundary condition is used to provide a height function which defines the free surface whose velocity is determined by extrapolation or from the dynamic free surface boundary condition. Several researchers, including Farmer et al. (1994), Thé et al. (1994), Lilek (1995), Mayer et al. (1997) and Cobbin et al. (1998), have applied this method to a variety of free surface flow problems, including flow around a Wigley ship, inviscid/viscous sloshing in a tank, and shoaling waves over a submerged bar. Interface-capturing is undertaken on a fixed grid which encloses the liquid domain and contains partially filled cells at the free surface interface. Of this latter method, there are two sub-methods commonly employed, the marker-and-cell method of Harlow and Welch (1965) or the volume-of-fluid (VOF) scheme due to Hirt and Nichols (1981). Examples include the work of Lafourie et al. (1994), Armenio (1997) and Muzaferija and Peric (1998) who simulated a collapsing water column. Rogers and Szymczak (1997) have presented a generalised algorithm which may be viewed as an extension of the VOF technique, and applied it to underwater explosions. In general, interface-capturing methods permit simulations of extremely complicated violent surface motions, but lack exact definition of the free surface. Interface-tracking methods give a better approximation to the free surface, but depend on interpolation/extrapolation or mappings and may be more demanding computationally.
This paper gives brief details of an interface-tracking Navier–Stokes solver for unsteady free surface viscous flows which utilises a triangular boundary-fitted Q-tree mesh for adaptation to the moving free surface boundary. The model is used to simulate standing and progressive free surface waves over submerged cylinders.
Section snippets
Q-tree-based finite element mesh generation
A hierarchical quadtree (Q-tree) grid generator is first used to create an underlying tiled rectangular panel grid about seeding points distributed along free surface and solid wall boundaries. The Q-tree grid commences from an initial square into which the normalised flow domain is inserted. The root square panel is divided into four quadrant panels which are then recursively subdivided into further sub-quadrants according to whether or not a prescribed number of seeding points are present in
Mathematical formulation
Consider an incompressible Newtonian liquid in two spatial dimensions, vertical and horizontal. The Navier–Stokes equations may be non-dimensionalised using a characteristic length, Lc=2h, and characteristic velocity, , where h is the still water depth, g the acceleration due to gravity and A the initial wave amplitude. Defining the non-dimensional total pressure where p* is the dynamic pressure, is atmospheric pressure, and ρ is liquid density, the non-dimensional
Finite element formulation
Within any arbitrary element, the velocity components and pressure are given bywhere ϕi is the velocity interpolation shape function corresponding to the ith velocity node, ψi is the pressure interpolation shape function corresponding to the ith pressure node, Nv and Np are the total numbers of velocity and pressure nodes in the element.
Using the shape functions as weightings, and integrating over the liquid
Results
All computations were performed on a Compaq Alphaserver 8400 computer with six 300 MHz Alpha EV5 processors and 6 Gb of memory, located at the Rutherford Appleton Laboratory.
Conclusions
Adaptive finite element meshes have been produced based on an underlying Q-tree rectangular grid which is fitted to moving free surface and solid wall boundaries. Unsteady liquid motions are modelled using the Navier–Stokes equations with fully non-linear free surface boundary conditions, including viscous and surface tension terms. A streamline upwind Petrov–Galerkin scheme has been utilised to discretise the momentum equations. The Poisson pressure equation is solved directly using Crout's
Acknowledgements
This work was supported as part of the MTD Managed Programme in Marine Computational Fluid Dynamics through EPSRC grant GR/L24090. The project involved collaboration with Dr G. X. Wu and Dr D. M. Greaves (University College London) and Professor M. Giles (Oxford University). The authors received helpful advice on grid generation from Dr J. Józsa (Technical University of Budapest) through the auspices of the Hungarian–British Intergovernmental Science and Technology Co-operation programme
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