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Variational modelling of wave-structure interactions with an offshore wind-turbine mast Tomasz Salwa, Onno Bokhove, Mark A. Kelmanson [email protected], [email protected], [email protected] 1. Introduction We present a mathematical model of water waves inter- acting with the mast of an offshore wind turbine. A vari- ational approach is used for which the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with non- linear elasticity, and the coupling between them. We de- velop a linearized model of the fluid-structure or wave-mast coupling, which is a linearization of the variational princi- ple for the fully coupled nonlinear model. Our numerical results in Firedrake for the linear case indicate that our variational approach yields a stable numerical discretiza- tion of a fully coupled model of water waves interacting with an elastic beam. 2. Problem formulation The problem is formulated with the linearized version of the fully nonlinear functional from [1]: Linearized variational principle 0=δ Z T 0 ZZ ∂ Ω f ρ∂ t ηφ f - 1 2 ρgη 2 dz dy - ZZZ Ω 1 2 ρ|∇φ| 2 dz dy dx + ZZ ∂ Ω s ρn · ∂ t X s φ s dz dy + ZZZ Ω 0 ρ 0 ∂ t X · U - 1 2 ρ 0 |U| 2 - 1 2 λe ii e jj - μe 2 ij dz dy dx dt (1) in which the variables and parameters are as follows: φ is the flow velocity potential, η , free surface deviation, g, gravita- tional acceleration, ρ, ρ 0 , fluid and structure densities, λ, μ, first and second Lamé constants, X, structure displacement, U, structure velocity and the stress tensor is e jk = 1 2 ∂X j ∂x k + ∂X k ∂x j . The fluid domain is denoted by Ω and the structural one by Ω 0 : in the linear approximation both domains are fixed. The free surface is denoted with index f and the common fluid-structure boundary with s. Evaluation of individual variations yields the equations of motion. 3. Discretization scheme P n , Q n φ n h ,η n h ,X n h ,P n h L = P dQ dt - H (P, Q) Eliminate internal φ Temporal discretization Recover internal φ Spatial discretization Find X-conjugate momentum P φ n h ,η n h ,X n h ,U n h Recover U φ, η, X, U φ h ,η h ,X h ,U h φ h ,η h ,X h ,P h Transform to Hamiltonian form Figure 1: Solution procedure for the discretization. The discretization procedure, as depicted in Fig. 1, reduces to the transformation of the coupled system into the ab- stract Hamiltonian form. This is performed first by spatial discretization with the Finite Element Method. The mesh with test functions from linear continuous Galerkin space, localized at each node, is introduced. Test functions are space-dependent only. Time-dependence of the solution is contained within the coefficients of the discrete expansion of the numerically computed free-surface height, denoted by h: φ h (~x,t)= φ i (t)˜ ϕ i (~x) φ fh (x, y, t)= φ α (t)ϕ α (x, y ) η h (x, y, t)= η α (t)ϕ α (x, y ) X a h (~x,t)= X a k (t) ˜ X a k (~x) U a h (~x,t)= U a k (t) ˜ X a k (~x). These expressions can be plugged directly into the vari- ational principle (1). Then, through finding the X- conjugate momentum and expressing the interior φ in terms of its value at the free surface φ f , after some al- gebra one ends with the system in Hamiltonian form. In this form an existing time discretization scheme can be applied, e.g.,1 st -order symplectic Euler or 2 nd -order Störmer-Verlet, which is stable by construction. In the end we have to return to original variables. 4. Firedrake implementation Firedrake accepts equations in space-continuous form. Since the space-discrete form was used to obtain time discretization, one has to return to time-discrete, space-continuous equations. Nondi- mensionalized final equations with the symplectic Euler scheme are shown on the right, together with a code excerpt of the actual im- plementation below. The F,S,f,s indices respectively denote inte- gration over fluid, structure, free surface and fluid-structure interface. The subdomain functionality was used to mark fluid and structure regions in a common mesh. Final equations Z vφ n+1 dS f = Z v (φ n - Δtη n )dS f Z ρ 0 v · U n+1 dV S + Z n · v φ n+1 dS s = ρ 0 Z v · U n dV S - Δt Z (λ∇· v∇· X n + μ∂ a X n b (∂ a v b + ∂ b v a )) dV S + Z n · v φ n dS s Z ∇v ·∇φ n+1 dV F - Z v n · U n+1 dS s =0 Z vη n+1 dS f = Z vη n dS f +Δt Z ∇v ·∇φ n+1 dV F -Δt Z v n · U n+1 dS s Z v · X n+1 dV S = Z v · (X n +ΔtU n+1 )dV S ... a_phi_s = trial * v * ds (top_id) L_phi_s = (phi_s - dt * eta) * v * ds (top_id) LVP_phi_s= LinearVariationalProblem (a_phi_s,L_phi_s,phi_s,bcs=exclude_beyond_surface) LVS_phi_s = LinearVariationalSolver (LVP_phi_s) ... 5. Results Firedrake results are computed for the parameter values from the table below. The initial con- dition consists of the first mode of the analytical solution for the free surface deviation without the beam and with no flow. The beam is initially undeformed, see Fig. 2 (top subfigure). Nu- merical results confirm the stability of the scheme, as predicted by construction. The energy of the system during the time evolution is conserved up to bounded oscillations that decrease by a factor of four with halved timestep for the Störmer-Verlet scheme, as indicated in Fig. 3 (right subfigure), thus confirming its 2 nd -order convergence in time. Parameter Value Comment g 9.8 m/s 2 gravitational acceleration L x × L y × H 0 10 m × 2.5 m × 4 m water domain R i 0.6 m beam inner radius R o 0.8 m beam outer radius H 12 m beam height ρ 1000 kg/m 3 water density ρ 0 7700 kg/m 3 beam density (steel) λ 1 × 10 7 N/m 2 first Lamé constant μ 1 × 10 7 N/m 2 second Lamé constant Figure 2: Initial geome- try (top) and at 5.9s (bot- tom). Figure 3: Energy partitioning in the system (left), detail of energy partition at the beam (middle) and numerical accuracy of energy conservation with halving timestep for Störmer-Verlet scheme (right). 6. Conclusions The proposed variational method yields stable, structure- preserving schemes for the linear fluid-structure interac- tion problem with a free fluid surface. The energy ex- change between the subsystems is seen to be in balance, yielding a total energy that shows only small and bounded oscillations whose amplitude tends to zero with 2 nd -order convergence as the timestep goes to zero. Similar 2 nd -order convergence is observed for spatial mesh refinement. The implementation of the nonlinear model extending [1] is in progress. 7. References [1] T. Salwa, O. Bokhove, and M. Kelmanson. Variational modelling of wave-structure interactions with an offshore wind-turbine mast. J. Eng. Math., 2017. Submitted. [2] J. C. Luke. A variational principle for a fluid with a free surface. J. Fluid Mech., 27:395–397, 1967. [3] J. W. Miles. On Hamilton’s principle for surface waves. J. Fluid Mech., 83:153–158, 1977.
