WP-1 Hydrodynamics: Background and Strategy

Transcription

1 WP-1 Hydrodynamics: Background and Strategy Harry B. Bingham Robert Read (Post-doc) Torben Christiansen (PhD) Mech. Engineering Tech. Univ. of Denmark Lygnby, Denmark Eric D. Christensen Hans Hansen DHI Water and Environment Hørsholm, Denmark SDWED 1st Symposium, DHI, Hørsholm, Denmark August 30-31, 2010

2 Schematic of the hydrodynamic problem S Deep water waves f Wave forcing at the device Refraction from the bottom Triad interaction S f Wave farm Diffraction from coastal boundaries

3 S S f f 10 50m > 100m

4 WP-1 Scope 1. Predicting the Incident Wave climate at the device location (10-50m water depth) Wave refraction and diffraction plus nonlinearity are all important No simple description of the waves exists Efficient potential flow solvers can do the job (OceanWave3D - DTU) 2. Mildly-nonlinear wave-structure interaction - Optimization and long-term loading Linear hydrodynamic coefficients Nonlinear hydrostatics, mooring system, and PTO forces included in the equations of motion (WAMSIM - DHI/DTU) Model under development to compute linear hydrodynamics on a variable depth, possibly semi-enclosed domain (Rob Read - DTU) 3. Strongly nonlinear wave-structure interaction - Final design analysis Fully nonlinear inviscid flow solver (PhD - DTU) Viscous flow solver (DHI)

5 The incident waves - OceanWave3D 3D potential flow Solve using high-order finite-differences on overlapping, boundary-fitted blocks Numerical order of accuracy is flexible Two paths to convergence h-type and p-type

6 The nonlinear potential flow wave problem ζ(x, t) z L ( φ, w) H x h(x) φ(x, z, t) (u, v, w) = ( φ, φ z ) ζ t φ t = ζ φ + w(1 + ζ ζ) = g ζ 1 2 φ φ w 2 (1 + ζ ζ)

7 Laplace Problem for w 1 2 φ + φ zz = 0, h < z < ζ φ z + h φ = 0, z = h ζ h z x 1 Bingham & Zhang (2007) J. Eng. Math. 58

8 Laplace Problem for w 1 2 φ + φ zz = 0, h < z < ζ φ z + h φ = 0, z = h ζ h z x Sigma transform the vertical coordinate: σ(x, z, t) = z + h(x) ζ(x, t) + h(x) 1 Bingham & Zhang (2007) J. Eng. Math. 58

9 Laplace Problem for w 1 2 φ + φ zz = 0, h < z < ζ φ z + h φ = 0, z = h ζ h z x Sigma transform the vertical coordinate: σ(x, z, t) = z + h(x) ζ(x, t) + h(x) Φ = e φ, σ = 1 2 Φ + 2 σ( σφ) + 2 σ. ( σφ) + σ. σ + σ 2 z ( σσφ) = 0, 0 σ < 1 ( zσ + h. σ) ( σφ) + h. Φ = 0, σ = 0 σ σ = 1 with Φ(x, σ, t) = φ(x, z, t) Gives a fixed computational geometry, no need to re-grid 1 Bingham & Zhang (2007) J. Eng. Math. 58

10 Solution by Arbitrary-Order Finite Differences 2 Structured, but non-uniform grid. Choose r neighbors to develop 1D, r 1 order FD schemes. Leads to a linear system Ax = b A is sparse with at most r d, non-zeros, in d = 2, 3 dimensions. GMRES iterative solution preconditioned by the linearized, 2nd-order version of the matrix A 2 : A 1 2 {A(t) x = b} Multigrid solution of the preconditioning step. Solution in O(10) iterations, independent of physics and N. Time stepping by the classical 4th-order Runge-Kutta scheme. 2 Engsig-Karup, Bingham & Lindberg (2009) J. Comp. Phys. 228

11 Solution by Arbitrary-Order Finite Differences 2 Structured, but non-uniform grid. Choose r neighbors to develop 1D, r 1 order FD schemes. Leads to a linear system Ax = b A is sparse with at most r d, non-zeros, in d = 2, 3 dimensions. GMRES iterative solution preconditioned by the linearized, 2nd-order version of the matrix A 2 : A 1 2 {A(t) x = b} Multigrid solution of the preconditioning step. Solution in O(10) iterations, independent of physics and N. Time stepping by the classical 4th-order Runge-Kutta scheme. 2 Engsig-Karup, Bingham & Lindberg (2009) J. Comp. Phys. 228

12 Scaling of the solution in 3D Nonlinear test case, 6 th -order accurate operators CPU/Iter [s] O(n) scaling GMRES+LU GMRES+MG 1F(1,1) n CPU time Memory [MB] O(n) scaling GMRES+LU GMRES+MG 1F(1,1) n RAM memory use

13 Error in linear dispersion at different resolutions Relative dispersion error Relative dispersion error r=3 uniform grid, N x =50 =25 =50 =100 = kh r=5 clustered grid, N x =15 =9 =12 =17 = kh Relative dispersion error Relative dispersion error r=3 uniform grid, N x =150 =75 =150 =300 = kh r=7 clustered grid, N x =9 =7 =9 =12 = kh

14 Highly nonlinear waves in 3D, kh = 2π, H/L = 90% 30 x 7.5 wavelengths, N = = , C r = 0.5. (6 th -order. Approx. 5 min. CPU/wave period.) Energy/ρ Total Potential Kinetic t/t Error, ε 8 x x/l Error, ε x y/l

15 Extensions in progress Parallelization of the code Introduction of a wave-breaking model Introduction of a shoreline run-up model

16 Mildly nonlinear wave-structure interaction - WAMSIM 3 Assumptions Small wave steepness at the structure (linear hydrodynamics). Mooring/PTO adds: Nonlinearity, and resonant horizontal modes (models provided by other WPs). Body motions are still small. Incident wave and motions problems are de-coupled. 1. Wave forcing: OceanWave3D + Haskind relations. 2. Wave-body interaction: Panel Method (WAMIT), OceanWave3D (under development) 3. Ship response: Time-domain equations of motion. 3 H.B. Bingham. Coastal Engineering 40 (2000)

17 equations of motion for the structure: 6 k=1 t (M jk + a jk ) ẍ k (t) + K jk (t τ) ẋ k (τ)dτ + C jk x k (t) = 0 F jd (t) + F jnl (t); j = 1, 2,..., 6 M jk, C jk : K jk, a jk : F jd : F jnl : 1 st order inertia and hydrostatics 1 st order wave radiation impulse response function 1 st order wave diffraction exciting force Nonlinear forcing. Mooring lines, fender friction, viscous effects,...

18 Linear wave-structure interaction with OceanWave3D ζ(x, t) z L ( φ, w) H x h(x) φ(x, z, t) (u, v, w) = ( φ, φ z ) ζ t φ t = ζ φ + w(1 + ζ ζ) = g ζ 1 2 φ φ w 2 (1 + ζ ζ) At any given time step, the 3D solution is obtained from: 2 φ + zz φ = 0, h z < ζ, (n, n z ) ( ; z )φ = 0, (x, z) Ω

19 Multi-block boundary-fitted discretization of the Laplace problem y x

20 Each block is mapped to the unit-spaced rectangle

21 Each block is mapped to the unit-spaced rectangle Physical and computational grids defined by the point mappings: [ξ(x, y), η(x, y)], and [x(ξ, η), y(ξ, η)] Partial derivatives in the two domains are related by: x = ξ x ξ + η x η, y = ξ y ξ + η y η, etc. Express the grid transformations in computational space via» ξx η x ξ y η y» xξ y ξ x η y η =» ,» ξx η x ξ y η y» xξ y = ξ x η y η 1 Derivatives via operations entirely in the computational space: x = 1 J (yη ξ y ξ η), J = (x ξ y η x ηy ξ ), etc.

22 A heaving horizontal axis cylinder (linear) Pseudo-impulsive motion (Gaussian in time) z/r x 3 F 33 /ρ π R x/r t/t min

23 Fourier transform gives the added mass and damping A B/ω A Exact B/ω Exact F/ρ π R 2 ω kr

24 Fully nonlinear analysis - Inviscid flow Torben Christiansens PhD project. Starts Oct. 1, 2010 Apply the OceanWave3D strategy to the fully nonlinear problem

25 Fully nonlinear analysis - CFD

26 Preliminary Work Plan 1. The incident waves Choose a set of test case conditions for the near-shore geometry/bathymetry and the offshore wave conditions Run OceanWave3D for these conditions, store data 2. Mildly nonlinear analysis Complete the OceanWave3D linear hydrodynamics solver Compute IRF s for a test concept WEC Use WAMSIM to optimize the concept 3. Fully nonlinear analysis Build the nonlinear inviscid solver Apply it to the final concept design for extreme loading and performance analysis