withano shorewind-turbinemast · Variationalmodellingofwave-structureinteractions withano...
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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 φ η ρ, ρ 0 λ, μ e jk = 1 2 ∂X j ∂x k + ∂X k ∂x j Ω Ω 0 f s 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 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). X φ φ f F,S,f,s 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) ... g 9.8 2 L x × L y × H 0 10 × 2.5 × 4 R i 0.6 R o 0.8 H 12 ρ 1000 3 ρ 0 7700 3 λ 1 × 10 7 2 μ 1 × 10 7 2
Transcript of withano shorewind-turbinemast · Variationalmodellingofwave-structureinteractions withano...
Variationalmodellingofwave-structure interactionswithanoshorewind-turbinemast
Tomasz Salwa, Onno Bokhove, Mark A. [email protected], [email protected], [email protected]
1. Introduction
We present a mathematical model of water waves inter-acting with the mast of an oshore wind turbine. A vari-ational approach is used for which the starting point is anaction functional describing a dual system comprising apotential-ow uid, a solid structure modelled with non-linear elasticity, and the coupling between them. We de-velop a linearized model of the uid-structure or wave-mastcoupling, which is a linearization of the variational princi-ple for the fully coupled nonlinear model. Our numericalresults in Firedrake for the linear case indicate that ourvariational approach yields a stable numerical discretiza-tion of a fully coupled model of water waves interactingwith 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
∫ T
0
∫∫∂Ωf
ρ∂tηφf − 1
2ρgη2dz dy −
∫∫∫Ω
1
2ρ|∇φ|2 dz dy dx+
∫∫∂Ωs
ρn · ∂tXsφs dz dy
+
∫∫∫Ω0
ρ0∂tX ·U− 1
2ρ0|U|2 − 1
2λeiiejj − µe2ij dz dy dxdt (1)
in which the variables and parameters are as follows: φ is the ow velocity potential, η, free surface deviation, g, gravita-tional acceleration, ρ, ρ0, uid and structure densities, λ, µ, rst and second Lamé constants, X, structure displacement,
U, structure velocity and the stress tensor is ejk = 12
(∂Xj
∂xk+ ∂Xk
∂xj
). The uid domain is denoted by Ω and the structural
one by Ω0: in the linear approximation both domains are xed. The free surface is denoted with index f and the commonuid-structure boundary with s. Evaluation of individual variations yields the equations of motion.
3. Discretization scheme
Pn,Qn
φnh, ηnh, X
nh , P
nh
L = PdQ
dt−H(P,Q)
Eliminate internal φ
Temporal discretization
Recover internal φ
Spatial discretization
Find X-conjugate momentum P
φnh, ηnh, X
nh , U
nh
Recover U
φ, η,X, U
φh, ηh, Xh, Uh
φh, ηh, Xh, Ph
Transform to Hamiltonian form
Figure 1: Solution procedure for the discretization.
The discretization procedure, as depicted in Fig. 1, reducesto the transformation of the coupled system into the ab-stract Hamiltonian form. This is performed rst by spatialdiscretization with the Finite Element Method. The meshwith test functions from linear continuous Galerkin space,localized at each node, is introduced. Test functions arespace-dependent only. Time-dependence of the solution iscontained within the coecients of the discrete expansionof the numerically computed free-surface height, denotedby h:
φh(~x, t) = φi(t)ϕi(~x)
φfh(x, y, t) = φα(t)ϕα(x, y)
ηh(x, y, t) = ηα(t)ϕα(x, y)
Xah(~x, t) = Xa
k (t)Xak (~x)
Uah (~x, t) = Ua
k (t)Xak (~x).
These expressions can be plugged directly into thevariational principle (1). Then, through nding theX-conjugate momentum and expressing the interior φ interms of its value at the free surface φf , after some al-gebra one ends with the system in Hamiltonian form.In this form an existing time discretization scheme canbe applied, e.g., 1st-order symplectic Euler or 2nd-orderStörmer-Verlet, which is stable by construction. In theend we have to return to original variables.
4. Firedrake implementation
Firedrake accepts equations inspace-continuous form. Since thespace-discrete form was used toobtain time discretization, onehas to return to time-discrete,space-continuous equations. Nondi-mensionalized nal equations withthe symplectic Euler scheme areshown on the right, together witha code excerpt of the actual im-plementation below. The F, S, f, sindices respectively denote inte-gration over uid, structure, freesurface and uid-structure interface.The subdomain functionality wasused to mark uid and structureregions in a common mesh.
Final equations∫vφn+1 dSf =
∫v(φn −∆tηn) dSf∫
ρ0v ·Un+1 dVS+
∫n · v φn+1 dSs = ρ0
∫v ·Un dVS
−∆t
∫(λ∇ · v∇ ·Xn + µ∂aX
nb (∂avb + ∂bva)) dVS+
∫n · v φn dSs∫
∇v · ∇φn+1 dVF−∫
vn ·Un+1 dSs = 0∫vηn+1 dSf =
∫vηn dSf +∆t
∫∇v · ∇φn+1 dVF−∆t
∫vn ·Un+1 dSs∫
v ·Xn+1 dVS =
∫v · (Xn +∆tUn+1) dVS
...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 rst mode of the analytical solution for the free surface deviation withoutthe beam and with no ow. The beam is initially undeformed, see Fig. 2 (top subgure). Nu-merical results conrm the stability of the scheme, as predicted by construction. The energy ofthe system during the time evolution is conserved up to bounded oscillations that decrease bya factor of four with halved timestep for the Störmer-Verlet scheme, as indicated in Fig. 3 (rightsubgure), thus conrming its 2nd-order convergence in time.
Parameter Value Comment
g 9.8m/s2
gravitational accelerationLx × Ly ×H0 10m× 2.5m× 4m water domain
Ri 0.6m beam inner radiusRo 0.8m beam outer radiusH 12m beam height
ρ 1000 kg/m3
water density
ρ0 7700 kg/m3
beam density (steel)
λ 1× 107N/m2
rst Lamé constant
µ 1× 107N/m2
second Lamé constant
Figure 2: Initial geome-try (top) and at 5.9s (bot-tom).
0 2 4 6 8 10time [s]
0
20000
40000
60000
80000
100000
120000
140000
Ener
gy [J
]
Energy(time)EpwEkwEtwEpbEkbEtbEt
0 2 4 6 8 10time [s]
0
200
400
600
800
1000
1200
1400
1600
Ener
gy [J
]
Beam energy(time)potentialkinetictotal
0 2 4 6 8 10t[s]
0
1
2
3
4
5
6
7
|E(t
)−E
(0)|/E
(0)
1e 7 |E(t)−E(0)|/E(0) as ∆t→∆t/2
∆t
∆t/2
Figure 3: Energy partitioning in the system (left), detail of energy partition at the beam (middle) and numerical accuracy ofenergy conservation with halving timestep for Störmer-Verlet scheme (right).
6. Conclusions
The proposed variational method yields stable, structure-preserving schemes for the linear uid-structure interac-tion problem with a free uid surface. The energy ex-change between the subsystems is seen to be in balance,yielding a total energy that shows only small and boundedoscillations whose amplitude tends to zero with 2nd-orderconvergence as the timestep goes to zero. Similar 2nd-orderconvergence is observed for spatial mesh renement. Theimplementation of the nonlinear model extending [1] is inprogress.
7. References
[1] T. Salwa, O. Bokhove, and M. Kelmanson. Variational modelling of wave-structure interactions with an oshore wind-turbine mast. J.
Eng. Math., 2017. Submitted.
[2] J. C. Luke. A variational principle for a uid with a free surface. J. Fluid Mech., 27:395397, 1967.
[3] J. W. Miles. On Hamilton's principle for surface waves. J. Fluid Mech., 83:153158, 1977.
