Wave-structure interaction for long wave models in the presence of a freely moving body on the bottom Krisztián BENYÓ Laboratoire d’Hydraulique Saint-Venant July 29, 2019 Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 1 / 16
Contents Setting of the problem 1 The physical domain of the problem The governing equations Nondimensionalisation The Boussinesq regime 2 The Boussinesq system and the approximate solid equations Theoretical results Numerical simulations 3 The numerical scheme The numerical experiments Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 2 / 16
Introduction Motivation Mathematical motivation : a better understanding of the water waves problem Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 3 / 16
Introduction Motivation Mathematical motivation : a better understanding of the water waves problem Real life applications : Coastal engineering and wave energy converters (a) Wave Roller (b) Wave Carpet Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 3 / 16
Setting of the problem The physical domain of the problem The physical domain for the wave-structure interaction problem ( x , z ) ∈ R 2 : − H 0 + b ( x − X S ( t )) < z < ζ ( t , x ) � Ω t = � . Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 4 / 16
Setting of the problem The physical domain of the problem References · In the case of a predefined evolution of the bottom topography : − T. Alazard, N. Burq, and C. Zuily, On the Cauchy Problem for the water waves with surface tension ( 2011 ), − F. Hiroyasu, andT. Iguchi, A shallow water approximation for water waves over a moving bottom ( 2015 ), − B. Melinand, A mathematical study of meteo and landslide tsunamis ( 2015 ) ; · Fluid - submerged solid interaction : − G-H. Cottet, and E. Maitre, A level set method for fluid-structure interactions with immersed surfaces ( 2006 ), − P. Guyenne, and D. P. Nicholls, A high-order spectral method for nonlinear water waves over a moving bottom ( 2007 ), − S. Abadie et al., A fictious domain approach based on a viscosity penalty method to simulate wave/structure interactions ( 2017 ). Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 5 / 16
Setting of the problem The governing equations The governing equations Fluid dynamics The free surface Euler equations in Ω t ∂ t U + U · ∇ U = − ∇ P + g , ̺ ∇ · U = 0 , ∇ × U = 0 , with boundary conditions � 1 + |∇ x ζ | 2 U · n = 0 on { z = ζ ( t , x ) } , ∂ t ζ − � 1 + |∇ x b | 2 U · n = 0 on { z = − H 0 + b ( t , x ) } , ∂ t b − P = P atm on { z = ζ ( t , x ) } . Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 6 / 16
Setting of the problem The governing equations The governing equations Fluid dynamics The free surface Bernoulli equations in Ω t � ∆Φ = 0 in Ω t � 1 + | ∂ x b | 2 ∂ n Φ bott = ∂ t b . Φ | z = ζ = ψ, An evolution equation for ζ , the surface elevation. An evolution equation for ψ , the velocity potential on the free surface. Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 6 / 16
Setting of the problem The governing equations The governing equations Fluid dynamics A formulation of the water waves problem � ∂ t ζ + ∂ x ( hV ) = ∂ t b , 2 | ∂ x ψ | 2 − ( − ∂ x ( hV ) + ∂ t b + ∂ x ζ · ∂ x ψ ) 2 ∂ t ψ + g ζ + 1 = 0 , 2(1 + | ∂ x ζ | 2 ) where � ζ V = 1 ∂ x Φ( · , z ) dz . h − H 0 + b Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 6 / 16
Setting of the problem The governing equations The governing equations Fluid dynamics A formulation of the water waves problem � ∂ t ζ + ∂ x ( hV ) = ∂ t b , 2 | ∂ x ψ | 2 − ( − ∂ x ( hV ) + ∂ t b + ∂ x ζ · ∂ x ψ ) 2 ∂ t ψ + g ζ + 1 = 0 , 2(1 + | ∂ x ζ | 2 ) where � ζ V = 1 ∂ x Φ( · , z ) dz . h − H 0 + b Solid mechanics By Newton’s second law : F total = F gravity + F solid − bottom interaction + F solid − fluid interaction . Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 6 / 16
Setting of the problem The governing equations The governing equations Fluid dynamics A formulation of the water waves problem � ∂ t ζ + ∂ x ( hV ) = ∂ t b , 2 | ∂ x ψ | 2 − ( − ∂ x ( hV ) + ∂ t b + ∂ x ζ · ∂ x ψ ) 2 ∂ t ψ + g ζ + 1 = 0 , 2(1 + | ∂ x ζ | 2 ) where � ζ V = 1 ∂ x Φ( · , z ) dz . h − H 0 + b Solid mechanics The equation of motion for the solid � � � � M ¨ X S ( t ) = − c fric Mg + e tan + P bott dx P bott ∂ x b dx . I ( t ) I ( t ) Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 6 / 16
Setting of the problem Nondimensionalisation Characteristic scales of the problem · L , the characteristic horizontal scale of the wave motion, · H 0 , the base water depth, · a surf , the order of the free surface amplitude, · a bott , the characteristic height of the solid. Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 7 / 16
The Boussinesq regime The Boussinesq system and the approximate solid equations The coupled Boussinesq system With an order O ( µ 2 ) approximation, we are going to work in the so called weakly nonlinear Boussinesq regime 0 � µ � µ max ≪ 1 , ε = O ( µ ) , β = O ( µ ) . (BOUS) The coupled Boussinesq system with an object moving at the bottom writes as ∂ t ζ + ∂ x ( hV ) = β ε ∂ t b , ( − ∂ x ( hV ) + β ε ∂ t b + ∂ x ( εζ ) · ∂ x ψ ) 2 ∂ t ψ + ζ + ε 2 | ∂ x ψ | 2 − εµ = 0 , 2(1 + ε 2 µ | ∂ x ζ | 2 ) � � 1 e tan + 1 X S ( t ) = − c fric ¨ � � 1 + I ( t ) P bott dx R P bott ∂ x b dx . √ µ β ˜ ˜ M M Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 8 / 16
The Boussinesq regime The Boussinesq system and the approximate solid equations The coupled Boussinesq system With an order O ( µ 2 ) approximation, we are going to work in the so called weakly nonlinear Boussinesq regime 0 � µ � µ max ≪ 1 , ε = O ( µ ) , β = O ( µ ) . (BOUS) The coupled Boussinesq system with an object moving at the bottom writes as ∂ t ζ + ∂ x ( hV ) = β ε ∂ t b , 1 − µ ∂ t V + ∂ x ζ + ε V · ( ∂ x V ) = − µ � � 3 ∂ xx 2 ∂ x ∂ tt b , � � 1 X S ( t ) = − c fric ε e tan + ε ¨ � � β c solid + R ζ ( t , x ) ∂ x b dx , I ( t ) ζ dx √ µ β ˜ ˜ M M Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 8 / 16
The Boussinesq regime The Boussinesq system and the approximate solid equations The coupled Boussinesq system With an order O ( µ 2 ) approximation, we are going to work in the so called weakly nonlinear Boussinesq regime 0 � µ � µ max ≪ 1 , ε = O ( µ ) , β = O ( µ ) . (BOUS) The coupled Boussinesq system with an object moving at the bottom writes as ∂ t ζ + ∂ x ( hV ) = β ε ∂ t b , 1 − µ ∂ t V + ∂ x ζ + ε V · ( ∂ x V ) = − µ � � 3 ∂ xx 2 ∂ x ∂ tt b , � � 1 X S ( t ) = − c fric ε e tan + ε ¨ � � β c solid + R ζ ( t , x ) ∂ x b dx , I ( t ) ζ dx √ µ β ˜ ˜ M M Aim : Long time existence result T 0 ε timescale for Boussinesq system over flat bottom : C. Burtea, New long time existence results for a class of Boussinesq-type systems (2016), T 0 timescale for water waves over a moving bottom : B. Melinand, A mathematical study of meteo and landslide tsunamis : the Proudman resonance (2015). Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 8 / 16
The Boussinesq regime Theoretical results L 2 estimates ˜ � � � E B ( t ) = 1 ζ 2 dx + 1 2 dx + 1 µ M � 2 , 3 h ( ∂ x V ) 2 dx + � ˙ � � X S ( t ) hV 2 2 2 2 ε R R R Proposition Let µ ≪ 1 sufficiently small and let us take s 0 > 1 . Any U ∈ C 1 ([0 , T ] × R ) ∩ C 1 ([0 , T ]; H s 0 ( R )) , X S ∈ C 2 ([0 , T ]) solutions to the coupled system, with initial data U (0 , · ) = U in ∈ L 2 ( R ) and ( X S (0) , ˙ X S (0)) = (0 , v S 0 ) ∈ R × R , verify e −√ ε c 0 t E B ( t ) � � sup � 2 E B (0) + µ Tc 0 � b � H 3 , t ∈ [0 , T ] where c 0 = c ( |||U||| T , W 1 , ∞ , |||U||| T , H s 0 , � b � W 4 , ∞ ) . Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 9 / 16
The Boussinesq regime Theoretical results Long time existence for the Boussinesq system Theorem Let µ sufficiently small and ε = O ( µ ) . Let us suppose that the initial values ζ in and b satisfy the minimal water depth condition. If ζ in and V in belong to H s +1 ( R ) with s ∈ R , s > 3 / 2 , and that X S 0 , v S 0 ∈ R , then there exists a maximal time T > 0 independent of ε such that there exists a solution �� 0 , T � � ∩ C 1 �� 0 , T � � ; H s +1 ( R ) ; H s ( R ) ( ζ, V ) ∈ C √ ε √ ε , 0 , T X S ∈ C 2 �� �� √ ε of the coupled system � D µ ∂ t U + A ( U , X S ) ∂ x U + B ( U , X S ) = 0 , ¨ t , X S ( t ) , ˙ X S ( t ) = F [ U ] � X S ( t ) � . with initial data ( ζ in , V in ) and ( X S 0 , v S 0 ) . Krisztián BENYÓ (ENPC) Wave structure interaction July 29, 2019 10 / 16
Recommend
More recommend
