Bingham, Harry B.

Read, Robert

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 18, 2017Linear wave-structure interaction using overset gridsRead, Robert; Bingham, Harry B.Published in:IWWWFB26Publication date:2011Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Read, R., & Bingham, H. B. (2011). Linear wave-structure interaction using overset grids. In IWWWFB26Linear wave-structure interaction using overset gridsRobert W. Read∗ & Harry B. BinghamCoastal, Maritime, and Structural Engineering SectionDepartment of Mechanical EngineeringTechnical University of DenmarkDK-2800 Kgs. LyngbyDenmarke-mail: rrea@mek.dtu.dk & hbb@mek.dtu.dkBackgroundThis abstract describes recent progress in the development of a finite-difference model for non-linear water waves and wave-structure interaction. Our central objective is to produce a com-putational tool that can accurately and efficiently simulate the interactions between waves andmaritime structures. Amongst other applications, this tool will be used to analyse the perfor-mance of ocean wave energy devices featuring complex geometries.Engsig-Karup et al. (2009) have recently developed a single-block, finite-difference potentialflow model to represent fully non-linear water wave propagation and development up to thepoint of wave breaking. A range of validation tests have confirmed the stability, accuracy,efficiency, and robustness of this model. Of particular significance are the adoption of arbitrary-order spatial finite-difference schemes to maximise accuracy, and the use of a multigrid solverto ensure that solution effort scales linearly with problem size.The focus of present work is to extend this model to support generalised curvilinear bound-aries. This is necessary to permit the representation of structures with complex geometries,floating in water of varying depth. To achieve this, the overset approach to grid generation isemployed, allowing accurate and efficient implementation of high-order algorithms on a collec-tion of simply-generated, overlapping, structured grids. This paper addresses preliminary workto model wave-structure interaction on a simple two-dimensional, two-block grid domain usingthe Overture software framework. The added mass and damping of a half-submerged, heavingcylinder is determined by evaluating the force exerted on the body in response to a prescribeddisplacement. These numerical results are compared with an analytical solution.Problem formulationA Cartesian coordinate system is adopted with the xy-plane located at the free surface and thez-axis directed upwards. The depth of the undisturbed fluid is defined as h(x), where x = (x, y)is the horizontal coordinate vector. The free-surface elevation is z = η(x, t). Assuming that theflow is inviscid and irrotational, the fluid velocity (u, v, w) = (∇φ, ∂zφ), can be expressed as thegradient of a scalar velocity potential φ(x, z, t), where ∇ = (∂x, ∂y). The temporal development∗presenting authorof the free surface is governed by kinematic and dynamic boundary conditions, here expressedin terms of the free surface variables φ˜ = φ(x, η, t) and w˜ = ∂zφ z=η:∂tη = (1 +∇η · ∇η) w˜ −∇η · ∇φ˜, (1)∂tφ˜ = −gη − 12(∇φ˜ · ∇φ˜− (1 +∇η · ∇η) w˜2), (2)In the linear case, where it is assumed that the wave elevation is small compared to the wave-length, the kinematic and dynamic boundary conditions simplify to:∂tη = w˜, (3)∂tφ˜ = −gη. (4)With a knowledge of φ˜ and η, and assuming an incompressible flow, these equations may bedeveloped in time by solving Laplace’s equation in the fluid domain subject to a zero-flowcondition at the impermeable boundaries:φ = φ˜, z = η, (5)∇2φ+ ∂zzφ = 0, −h ≤ z ≤ η, (6)(n, n3) · (∇, ∂z)φ = 0, (x, z) ∈ ∂Ω, (7)where (n, n3) is a vector pointing outwards normal to the solid boundary surface ∂Ω.Overset gridsThe governing equations stated above have been solved numerically using the overset gridmethodology. According to this approach, the fluid domain is decomposed into a series ofoverlapping, structured grid blocks (see Figure 1). Blocks with curvilinear edges or surfaces arecreated at the fluid boundaries to represent the geometrical profile of floating structures andthe bottom. Spatial derivatives in the curvilinear, physical domain can be expressed in terms ofa weighted sum of operators on a square, unit-spaced computational domain. The weightingsapplied to the operators on the computational grid depend on the physical geometry, but needonly be calculated once when solving the linear wave problem. A comparatively small number ofpoints along the block boundaries serve to interpolate the solution between grid blocks. Follow-ing this method, a linear system of equations is formed incorporating information from all of thecomponent grid blocks, and solved in the conventional manner using direct or iterative methods.The Overture software framework developed at Lawrence Livermore National Laboratories pro-vides a convenient tool for solving partial-differential equations on multiple curvilinear blocks(Henshaw 2008). This collection of high-level C++ libraries includes a powerful grid generatorand advanced parallel-processing capabilities. Using this framework it is presently possible toimplement spatial schemes to fourth-order accuracy.Heaving cylinder test caseThe governing equations described above have been solved in two dimensions using a methodof lines approach with fourth-order accuracy in space and time. Centred, five-point, finite-difference schemes are developed on the computational grid using Taylor-series expansions. WithFigure 1: Details of the simple two-block overlapping grid used to simulate a heaving cylinder.The interpolation points are identified as blue and green dots.a knowledge of the transformation function relating the physical and computational domains, thecomputational grid operators are weighted to satisfy the appropriate governing equations on thephysical grid. Laplace’s equation is satisfied at all of the internal and boundary points of the fluidwith the exception of the free surface, where a Dirichlet boundary condition is applied directly.To maintain centred schemes throughout the fluid domain, ghost points are introduced at eachblock boundary. These points may be used to enforce Neumann boundary conditions at solidsurfaces, or may provide the interpolation points required to transfer information between blocks.The resulting linear system of equations is solved directly. Having evaluated the potential field,the free-surface conditions are advanced using the classical fourth-order Runge-Kutta scheme.Following this approach, the linear fluid motions and surface waves induced by a half-submerged heaving cylinder have been simulated using a two-block model implemented withinOverture. The physical domain consists of a rectangular tank 120 m long by 6.4 m deep, with aquarter circle of radius 1 m in the top left hand corner (see Figure 1 for detail). Under the linearassumption, vertical motion of the cylinder can be represented as a non-homogeneous boundarycondition at the cylinder surface. A Gaussian range of wavelengths was introduced by imposingthe cylinder heave displacement profile shown in Figure 2. The pressure at the cylinder surfacewas evaluated and integrated to give the force acting on the cylinder in response to the displace-ment. After performing a discrete Fourier transform on the displacement and force signals, thereal and imaginary parts of the ratio of these signals describe the added mass and damping ofthe cylinder respectively. As illustrated in Figure 3, the numerical results obtained compare wellwith the analytical results of Greenhow and Ahn (1988). Note that the discrepancy observedat low values of√kR is expected as the analytical result assumes an infinite fluid depth. Thismodel is presently being extended to three dimensions.ReferencesEngsig-Karup, A. P., H. B. Bingham, and O. Lindberg (2009). An efficient flexible-ordermodel for 3D nonlinear water waves. Journal of Computational Physics 228, 2100–2118.Greenhow, M. and S. I. Ahn (1988). Added mass and damping of horizontal circular cylindersections. Ocean Engineering 15, 495–504.Henshaw, W. D. (2008). Overture documentation. https://computation.llnl.gov/casc/overture/.−10 −5 0 5 10−4−2024time (s)amplitudezF/piρR2Figure 2: The displacement and force profiles associated with the heaving cylinder simulation.0 0.3 0.6 0.9 1.2 1.500.511.52√kRF/piρR2ω2AB/ωAexact(B/ω)exactkh = 0.125 kh = pi kh = 11Figure 3: Comparison of analytical and numerical results for the added mass and damping of ahalf-submerged heaving cylinder.AcknowledgementsThe authors wish to thank the Danish Agency for Science, Technology and Innovation (grant# 09-067257) for funding, and the Danish Center for Scientific Computing for supercomputingresources.

Linear wave-structure interaction using overset grids

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Linear wave-structure interaction using overset grids

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