Modelling and simulation of a wave energy converter E. BOCCHI , J. - PowerPoint PPT Presentation

Modelling and simulation of a wave energy converter E. BOCCHI ∗ , J. HE † and G. VERGARA-HERMOSILLA ∗ ∗ Institut de Math´ ematiques de Bordeaux † Institut Camille Jordan Supervisor: D. LANNES Luminy, 21 August, 2019 Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 1 / 25

Overviews Introduction 1 Motivation Wave energy converter 2 Derivation of the model Discretization Numerical results 3 Wave energy converter Absorbed power of the device Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 2 / 25

Introduction: Oscillating water column (OWC) Closed chamber submerged with an opening below the free surface towards the incident wave Due to the waves motion, the water column acts as a piston compressing the air trapped inside the chamber. Pressurized air activates a turbine that is attached to the energy generator. Some Advantages Easy maintenance There are no machine components in the water Taken from Falcao, Henriques, Renewable Energy, 2015. Efficient use of the marine space and is environment friendly Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 3 / 25

Motivation: some experiences All these pictures are taken from Falcao, Henriques, Renewable Energy, 2015. Offshore OWC installed in Ireland, about 2008. Offshore OWC installed in Australia, about 2005. Onshore OWC installed in 1990 at Trivandrum, India. Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 4 / 25

Wave energy converter: configuration Figure: Configuration. Notations ζ is the surface elevation around the rest state, h is the fluid height, q is the horizontal discharge, P is the surface pressure. Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 5 / 25

Wave energy converter: constrains Exterior domain E , P e = P atm and ζ e is unknown. Interior domain I , P i is unknown and ζ i = ζ w . where � f � denotes the difference of f on the two side-walls of the solid, namely � f � = f ( l 0 + r ) − f ( l 0 − r ) . Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 6 / 25

General Settings The motion of the fluid is governed by the following nonlinear shallow water equations (NSW):  ∂ t ζ + ∂ x q = 0   � q 2 � x ∈ ( −∞ , l 0 − r ) ∪ ( l 0 + r , l 1 ) ∂ t q + ∂ x + gh ∂ x ζ = 0   h The wave-structure interaction is described by the following two transmission conditions : � q � = 0 , � q 2 � = − 2 r dq i 2 h 2 + g ζ dt . h w Initial conditions : q ( t = 0 , x ) = q 0 ( x ); ζ ( t = 0 , x ) = ζ 0 ( x ) . Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 7 / 25

Derivation of the Model Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 8 / 25

Derivation of the Model Step 1 : Reduce the problem The motion of wave is described by the 1D shallow water equations :  ∂ t ζ + ∂ x q = 0   � q 2 � In the exterior domain E : + gh ∂ x ζ = − 1 ∂ t q + ∂ x ρ h ∂ x P atm = 0   h  ∂ t ζ i + ∂ x q i = 0   � q 2 � In the interior domain I : + gh i ∂ x ζ i = − 1 i ∂ t q i + ∂ x ρ h i ∂ x P i   h i Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 9 / 25

Derivation of the Model Step 1 : Reduce the problem The motion of wave is described by the 1D shallow water equations :  ∂ t ζ + ∂ x q = 0   � q 2 � In the exterior domain E : + gh ∂ x ζ = − 1 ∂ t q + ∂ x ρ h ∂ x P atm = 0   h  ∂ t ζ i + ∂ x q i = 0   � q 2 � In the interior domain I : + gh i ∂ x ζ i = − 1 i ∂ t q i + ∂ x ρ h i ∂ x P i   h i Coupling conditions : q ( t , l 0 ± r ) = q i ( t , l 0 ± r ) ∂ t ζ i = 0 ❀ q i ( t , x ) = q i ( t ) Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 9 / 25

Derivation of the Model Step 1 : Reduce the problem The motion of wave is described by the 1D shallow water equations :  ∂ t ζ + ∂ x q = 0   � q 2 � In the exterior domain E : + gh ∂ x ζ = − 1 ∂ t q + ∂ x ρ h ∂ x P atm = 0   h  ∂ t ζ i + ∂ x q i = 0   � q 2 � In the interior domain I : + gh i ∂ x ζ i = − 1 i ∂ t q i + ∂ x ρ h i ∂ x P i   h i Coupling conditions : q ( t , l 0 ± r ) = q i ( t , l 0 ± r ) first transmission condition : ❀ ∂ t ζ i = 0 ❀ q i ( t , x ) = q i ( t ) � q � = 0 Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 9 / 25

Step 1 : Reduce the problem  ∂ t ζ + ∂ x q = 0   � q 2 � In the exterior domain E : ∂ t q + ∂ x + gh ∂ x ζ = 0   h In the interior domain I : ∂ t q i = − 1 ρ h i ∂ x P i Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 10 / 25

Step 1 : Reduce the problem  ∂ t ζ + ∂ x q = 0   � q 2 � In the exterior domain E : ∂ t q + ∂ x + gh ∂ x ζ = 0   h In the interior domain I : ∂ t q i = − 1 ❀ − 2 r ρ ρ h i ∂ x P i ∂ t q i = � P i � h w Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 10 / 25

Step 1 : Reduce the problem  ∂ t ζ + ∂ x q = 0   � q 2 � In the exterior domain E : ∂ t q + ∂ x + gh ∂ x ζ = 0   h In the interior domain I : ∂ t q i = − 1 ❀ − 2 r ρ ρ h i ∂ x P i ∂ t q i = � P i � h w Remark : Free surface, constrained pressure in the exterior domain : ζ , P atm Constrained surface, free pressure in the interior domain : ζ w = ζ i , P i Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 10 / 25

Step 1 : Reduce the problem  ∂ t ζ + ∂ x q = 0   � q 2 � In the exterior domain E : ∂ t q + ∂ x + gh ∂ x ζ = 0   h In the interior domain I : ∂ t q i = − 1 ❀ − 2 r ρ ρ h i ∂ x P i ∂ t q i = � P i � h w Remark : Free surface, constrained pressure in the exterior domain : ζ , P atm Constrained surface, free pressure in the interior domain : ζ w = ζ i , P i Goal : find the evolution equation for q i ! Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 10 / 25

Step 2: Derive the transmission condition  ∂ t ζ + ∂ x q = 0   � q 2 � + gh ∂ x ζ = − 1 ∂ t q + ∂ x ρ h ∂ x P   h Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 11 / 25

Step 2: Derive the transmission condition  ∂ t ζ + ∂ x q = 0   � q 2 � + gh ∂ x ζ = − 1 ∂ t q + ∂ x ρ h ∂ x P   h Local energy conservation in the exterior region. ∂ t e ext + ∂ x f ext = 0 . with e ext = q 2 2 h + g ζ 2 f ext = q 3 and 2 h 2 + g ζ q , 2 Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 11 / 25

Step 2: Derive the transmission condition  ∂ t ζ + ∂ x q = 0   � q 2 � + gh ∂ x ζ = − 1 ∂ t q + ∂ x ρ h ∂ x P   h Local energy conservation in the exterior region. ∂ t e ext + ∂ x f ext = 0 . with e ext = q 2 2 h + g ζ 2 f ext = q 3 and 2 h 2 + g ζ q , 2 � � � q 2 ρ � i + g ζ 2 Total energy : E tot = e ext + w 2 h w E I Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 11 / 25

Step 2: Derive the transmission condition  ∂ t ζ + ∂ x q = 0   � q 2 � + gh ∂ x ζ = − 1 ∂ t q + ∂ x ρ h ∂ x P   h Local energy conservation in the exterior region. ∂ t e ext + ∂ x f ext = 0 . with e ext = q 2 2 h + g ζ 2 f ext = q 3 and 2 h 2 + g ζ q , 2 � � � q 2 ρ � i + g ζ 2 Total energy : E tot = e ext + w 2 h w E I Energy conservation : 0 = � f ext � + 2 r ρ d q i dt q i h w Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 11 / 25

Step 2: Derive the transmission condition  ∂ t ζ + ∂ x q = 0   � q 2 � + gh ∂ x ζ = − 1 ∂ t q + ∂ x ρ h ∂ x P   h Local energy conservation in the exterior region. ∂ t e ext + ∂ x f ext = 0 . with e ext = q 2 2 h + g ζ 2 f ext = q 3 and 2 h 2 + g ζ q , 2 � � � q 2 ρ � i + g ζ 2 Total energy : E tot = e ext + w 2 h w E I Energy conservation : � q 2 � 0 = � f ext � + 2 r ρ = − 2 r ρ d d q i dt q i 2 h 2 + g ζ dt q i ❀ h w h w Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 11 / 25

Wave-structure interaction The original problem can be reduced to a transmission problem :  ∂ t ζ + ∂ x q = 0   � q 2 � x ∈ E (1) ∂ t q + ∂ x + gh ∂ x ζ = 0   h with transmission conditions provided at the contact points x = l 0 ± r : � q � = 0 , (2) � q 2 � = − 2 r ρ dq i dt = − α dq i 2 h 2 + g ζ dt . (3) h w Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 12 / 25

Discretization Bocchi, He, Vergara-Hermosilla Modelling and simulation of a WEC Luminy, 21 August, 2019 13 / 25