Amini Afshar, Mostafa

Bingham, Harry B.

Read, Robert

During recent years a computational strategy has been developed at the Technical University of Denmark for numerical simulation of water wave problems based on the high-order nite-dierence method, [2],[4]. These methods exhibit a linear scaling of the computational eort as the number of grid points increases. This understanding is being applied to develop a tool for predicting the added resistance (drift force) of ships in ocean waves. We expect that the optimal scaling properties of this solver will allow us to make a convincing demonstration of convergence of the added resistance calculations based on both near-eld and far-eld methods. The solver has been written inside a C++ library known as Overture [3], which can be used to solve partial dierential equations on overlapping grids based on the high-order nite-dierence method. The resulting code is able to solve, in the time domain, the linearised potential ow forward-speed hydrodynamic problems; namely the steady, radiation and diraction problems. The near-eld formulation of the wave drift force has also been implemented, and development is under way to include far-eld methods. This paper presents validation results based on analytical solutions for exact geometries

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Downloaded from orbit.dtu.dk on: Dec 20, 2017A high-order finite-difference linear seakeeping solver tool for calculation of addedresistance in wavesAmini Afshar, Mostafa; Bingham, Harry B.; Read, RobertPublication date:2015Link back to DTU OrbitCitation (APA):Amini Afshar, M., Bingham, H. B., & Read, R. (2015). A high-order finite-difference linear seakeeping solver toolfor calculation of added resistance in waves. Abstract from 30th International Workshop on Water Waves andFloating Bodies, Bristol, United Kingdom.(Abstract submitted to the 30th IWWWFB workshop, Bristol, UK, 2015)A high-order finite-difference linear seakeeping solver tool forcalculation of added resistance in waves ∗Mostafa Amini Afshar, Harry B. Bingham, and Robert ReadDepartment of Mechanical Engineering, Technical University of DenmarkE-mail: maaf@dtu.dk, hbb@dtu.dk, rrea@mek.du.dk1 IntroductionDuring recent years a computational strategy has been developed at the Technical University ofDenmark for numerical simulation of water wave problems based on the high-order finite-differencemethod, [2],[4]. These methods exhibit a linear scaling of the computational effort as the number ofgrid points increases. This understanding is being applied to develop a tool for predicting the addedresistance (drift force) of ships in ocean waves. We expect that the optimal scaling properties ofthis solver will allow us to make a convincing demonstration of convergence of the added resistancecalculations based on both near-field and far-field methods. The solver has been written inside aC++ library known as Overture [3], which can be used to solve partial differential equations onoverlapping grids based on the high-order finite-difference method. The resulting code is able to solve,in the time domain, the linearised potential flow forward-speed hydrodynamic problems; namely thesteady, radiation and diffraction problems. The near-field formulation of the wave drift force has alsobeen implemented, and development is under way to include far-field methods. This paper presentsvalidation results based on analytical solutions for exact geometries.2 Mathematical formulationA moving Cartesian coordinate system O-xyz is adopted, which is in steady translation with thebody’s forward speed U. The origin of the coordinate system is at the mean free surface position, andz is vertically upward. The body is under the action of incoming waves, and is free to oscillate in 6degrees of freedom. Assuming a potential flow model, the governing equation is:∇2φ = ∂2φ∂x2+∂2φ∂y2+∂2φ∂z2= 0, where φ = −Ux + φb + φu and φu = φ0 + φs +6∑k=1φk.−Ux + φb arises due to the forward speed of the body, and represents the solution of the steadywave resistance problem. For the Neumann-Kelvin linearisation (φb = 0), and for the double-bodylinearisation, the following boundary value problem is solved to obtain the base flow, φb = φdb.∇2φdb = 0, ∂φdb∂n= ~W · n on s0, ∂φdb∂z= 0 on z = 0, ∇φdb → 0 in the far field,where and ~W = (U, 0, 0). Moreover φ0 and φs are the velocity potentials of the incident waves andthe scattered waves respectively, while the body is assumed to be fixed (the diffraction problem). Thevelocity potentials due to oscillatory motions of the body in the kth direction (the radiation problem)are given by φk. The decomposed velocity potentials mentioned above are substituted into the non-linear boundary conditions. The linearised conditions are then obtained by Taylor-series expansionaround the mean water level z = 0 and mean body position s0 respectively:∂φu∂t= −gηu + U ∂φu∂x− ∇φb · ∇φu − 12∇φb · ∇φb + U ∂φb∂x− gηb = 0, (1)∂ηu∂t=∂φu∂z+ U∂ηu∂x− ∇φb · ∇ηu + ηu∂2φb∂z2− ∇φb · ∇ηb + U ∂ηb∂x+ ηb∂2φb∂z2= 0. (2)and∂φu∂n=6∑k=1(ξ˙k · nk + ξk ·mk), and ∂φs∂n= −∇φ0(r, t) · n, on z = s0.∗The authors wish to thank the Danish Maritime Fund for supporting this work.Here, ξk is the translation or rotation of the body, and ξ˙k is the corresponding velocity. The normalvectors nk and the m terms are defined as follows:(n1, n2, n3) = n, (m1,m2,m3) = (n · ∇)( ~W −∇φb),(n4, n5, n6) = (r× n), (m4,m5,m6) = (n · ∇)(r× ( ~W −∇φb)),where r is the position vector. The above mentioned linear initial boundary value problem is solvedin the time domain. Instead of finding the radiation and diffraction impulse response functions, theresponse of the body is calculated when exposed to a pseudo-impulsive Gaussian forcing given byξk(t) = ζ0(t) = e−2pi2s2 t2 , where s is a parameter to control the shape of the pseudo impulse. ζ0 isthe amplitude of the incident waves for the diffraction problem. Having solved for φu, the first-orderforce on the body can also be obtained from the linearised Bernoulli equation:p = −ρ[(∂∂t− ~W · ∇)φu − ~W · ∇φb +∇φb · ∇φu + 12∇2φb]. (3)The frequency-domain added mass and damping coefficients ajk, bjk, and the wave exciting forcecoefficient Xj are then obtained by a Fourier transform of the time-domain data. The Froude-Krylovpart of the wave excitation force is however, computed from the closed form expression in the frequencydomain. Finally the body motion ξˆk in the frequency domain is found by solving the equations ofmotion:6∑k=1ξˆk[−ω2(Mjk + ajk) + iωbjk + cjk] = AXj , j = 1, 2, . . . , 6 (4)where Mjk is the inertial mass, cjk is the hydrostatic coefficient matrix, and A is the incident waveamplitude. The wave drift force is calculated in the frequency domain by considering the second-orderpressure terms in the Bernoulli equation. All the terms in these expressions are obtained as describedabove, via Fourier transform of the corresponding pseudo-impulsive quantities [1].3 The numerical methodThe entire physical domain is descretised by overlapping structured and body-fitted grids using theOgen grid generator. There are three sets of grid points including discretisation, interpolation andhole points. Each grid is mapped to a uniform Cartesian computational grid, where discretisationof the continuous derivatives takes place using fourth-order finite-difference schemes. There are twolayers of ghost points generated to handle the derivatives at the boundaries. The continuity equationis descretised by a fourth-order centered scheme. A non-homogeneous Neumann boundary condition isapplied at the body to satisfy the body boundary condition. The Dirichlet boundary condition is usedto specify the velocity potential at the free surface. The resulting system of equations including theinterpolation equations, and the right hand side vector b is constructed as [A][φ] = [b], and solved by adirect LU factorisation method. The free-surface conditions, (1) and (2) are integrated in time usingthe fourth-order Runge-Kutta method, and φ and η at the free surface are updated at each time step.As the free-surface conditions are in fact a system of hyperbolic equations, a special care is requiredto set the ghost points while evaluating the convective derivatives at the boundaries. In the regionwhere ~W · n < 0, extrapolation from the internal points is used to populate the ghost layers, whichturns the centered scheme at the boundary into an upwind scheme. For the region where ~W · n > 0,a Neumann condition is used to set the ghost points at the boundaries. The dynamic free-surfaceboundary condition and the known Neumann condition for φ on the body boundary can be used toderive a corresponding Neumann condition for η:∇η · n = −1g(∂∂t(∇φ · n)− U ∇(∂φ∂x)· n)(5)where this expression corresponds to the Neumann-Kelvin linearisation. A similar relation can beobtained for the double-body flow linearisation. The above treatments are necessary for ensuring astable numerical scheme and for preventing spurious reflections at the boundaries. Moreover in thecase of grid stretching, the centered scheme is not suitable for evaluation of the convective derivatives.The unequal weighting of the neighboring points in the convective derivative on a stretched grid willalways produce an effective downwinding in some areas of the free-surface, which leads to instability.Two remedies are available: either a centered scheme is used and at the same time a Savitzky-Golaytype filter is applied to the solution at each time step, or an upwind-biased scheme is used for evaluationof the convective derivatives.4 ResultsThe radiation problem for surge and heave body motion has been solved for a floating cylinder withFr = U/√ga = 0.03, where a is radius of the cylinder. The radiation problem for sway and heavemotion has been solved for a submerged sphere with Fr = 0.40 with a submergence depth of h = 2a.The diffraction problem in the case of the head waves for the submerged hemisphere with Fr = 0.40 hasalso been solved. Just the imaginary part of the complex force is shown here, and ν = ω2e/g, ν0 = ω/g.The numerical results show very good agreement with the analytical solutions from [5] and [6]. Thedeep-water limit has also been shown in the figure, as the analytical results are for the deep watercondition. The wave drift force along the x coordinate and the body motions have been calculated forthe case of a floating hemisphere with Fr = 0, and results have been compared with those calculatedusing WAMIT. Contributions from both the water line integral and the body surface integral in thenear-field method are shown in the figure, and d is the diameter of the hemisphere. The horizontalline in the drift force figure shows the analytical asymptotic value −13 for the short waves. The resultsof the developed solver in the figure are shown with the name OceanWave3D.5 ConclusionsA seakeeping solver has been developed based on the high-order finite-difference method applied onoverlapping grids. A stable numerical scheme has been achieved either by using a centered schemeand applying a mild filter to the solution, or by using the upwind-biased difference scheme for theconvective derivatives in the free-surface boundary conditions. The solver has been validated for exactgeometries, and work is under way to validate the code for real ship geometries. The solver is beingdeveloped to include more models for the wave drift force calculation, and to run in parallel in orderto demonstrate convergence of the added resistance based on both far-field and near-field methods.References[1] Amini Afshar, M. Towards Predicting the Added Resistance of Slow Ships in Waves. PhD thesis,DTU Mechanical Engineering, 2015.[2] Bingham, H. B. and H. Zhang. On the accuracy of finite-difference solutions for nonlinear waterwaves. Journal of Engineering Mathematics, 58(1-4):211–228, 2007.[3] Brown, D. L., W. D. Henshaw, and D. J. Quinlan. Overture: An object-oriented framework forsolving partial differential equations on overlapping grids. Object Oriented Methods for Interoper-able Scientific and Engineering Computing, SIAM, pages 245–255, 1999.[4] Engsig-Karup, A. P., H. B. Bingham, and O. Lindberg. An efficient flexible-order model for 3dnonlinear water waves. Journal of computational physics, 228(6):2100–2118, 2009.[5] Wu, G. and R. E. Taylor. Radiation and diffraction of water waves by a submerged sphere atforward speed. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences,417(1853):433–461, 1988.[6] Wu, G. and R. E. Taylor. The hydrodynamic force on an oscillating ship with low forward speed.Journal of Fluid Mechanics, 211:333–353, 1990.0 1 2 3 4 5 6−0.500.511.52kaaijρpia2ωU g  Wu and Taylor (1990)OceanWave3Da31a13Floating cylinder, a13 and a310 1 2 3 4 5 6 70.511.522.533.544.555.5νaa22ρpia3  Wu and Taylor (1998)OceanWave3DSubmerged sphere, a220 0.5 1 1.5 2 2.5−0.00500.0050.010.0150.020.0250.030.035ν0aIm{X3}ρgpia3ν  Wu and Taylor (1998)OceanWave3Dν0h = piSubmerged sphere, heave force imaginary0 1 2 3 4 5 6 70.511.522.533.544.555.5νaa33ρpia3  Wu and Taylor (1998)OceanWave3DSubmerged sphere, a330 2 4 6 8 10 12 14 16 1800.20.40.60.811.21.41.61.82ω2 dgξ3 A  WAMITOceanWave3DFloating hemisphere, RAO0 0.5 1 1.5 2 2.5 3−1.2−1−0.8−0.6−0.4−0.200.20.40.60.8kaF(2)1ρgA2L  OceanWave3D (total)OceanWave3D (body)OceanWave3D (waterline)WAMITFloating hemisphere, wave drift forceFigure 1: Validation results for various quantities using a floating two-dimensional cylinder, a sum-berged sphere, and a floating hemispere.

A high-order finite-difference linear seakeeping solver tool for calculation of added resistance in waves

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A high-order finite-difference linear seakeeping solver tool for calculation of added resistance in waves

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