Numerical simulation of dispersive waves

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Numerical simulation of dispersive waves DENYS DUTYKH 1 Chargé de Recherche CNRS 1 LAMA, Université de Savoie 73376 Le Bourget-du-Lac, France Colloque EDP-Normandie DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 1 / 25

Acknowledgements Special thanks to: Frédéric Dias: Professor, ENS Cachan on leave to: University College Dublin Collaborators: Dimitrios Mitsotakis: Post-Doc, Paris-Sud, Orsay Theodoros Katsaounis: Assistant professor, Crete DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 2 / 25

Coastal hydrodynamics Physical motivation for this study Tsunami hazard estimation Harbour protection Ship generated waves Beach erosion by wave generated currents DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 3 / 25

Water wave problem - I Physical setting and assumptions z h() (, t) H(, t) O Physical assumptions: Fluid is perfect: Re = Flow is incompressible: u =... and irrotational: ω = u = Free surface is a graph z = (, t) Above free surface there is void DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 4 / 25

Water wave problem - II Mathematical setting the governing equations Continuity equation: u = ( φ) = φ = 2 φ+ 2 zz φ =, (, z) Ω [ h,] Free surface kinematic condition: t + φ z φ =, z = (, t) Free surface dynamic condition (free surface isobarity): t φ+ 1 2 φ 2 + 1 2 ( zφ) 2 + g =, z = (, t) Bottom kinematic condition: φ h+ z φ =, z = h( ) DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 5 / 25

Dispersive long wave models Boussinesq-type equations Long wave (Boussinesq) scaling: Nonlinearity: ε := a h 1 Dispersion: µ 2 := ( h l ) 2 1 Stokes-Ursell number: S := ε µ 2 1 BCS (22); DL et al. (25): T = O( 1 ε ) t + u +ε(u) +ε[au b t ] = u t + +εuu +ε[c du t ] = DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 6 / 25

Conservative form of governing equations Flat bottom 1D case Boussinesq equations: (I D)v t +[F(v)] +[G(v )] = Conservative variables: v := (, u) Advective flu: F(v) = ( (1+)u, + 1 2 u2) Elliptic operator: I D = diag(1 b 2, 1 d 2 ) Dispersion: G(v ) = ( au, c ) Reference: D. Dutykh, T. Katsaounis, D. Mitsotakis. Finite volume schemes for dispersive wave propagation and runup. Submitted. 21 http://hal.archives-ouvertes.fr/hal-472431/ DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 7 / 25

Finite volume schemes for dispersive waves Ref: Dutykh, Katsaounis, Mitsotakis (hal-472431) (I D)v t +[F(v)] +[G(v )] = Advective flues: m-flu, KT, FVCF Reconstructions: TVD MUSCL2, UNO2, WENO3 Elliptic operator: 2nd or 4th order FD: (I D)v v i 1 + 1v i + v i+1 (b, d) v i+1 2v i + v i 1 12 2 Dispersion: Centered flu: G m i+ 1 2 = (a, c) v(2) i + v (2) i+1, G lm 2 i+ 1 2 Time stepping: Eplicit SSP-RK2, SSP-RK3 v (2),L + v (2),R i+ = (a, c) 1 i+ 1 2 2 2 DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 8 / 25

Solitary wave propagation Convergence tests for Bona-Smith system Bona-Smith system (we take θ 2 = 8 1 ): t + u +(u) 3θ2 1 6 t =, u t + + uu + 2 3θ2 3 3θ2 1 6 u t =. Eact solitary wave solution: (ξ) = sech 2 (λξ), u(ξ) = B(ξ), ξ := c s t = 9 2 θ2 7/9 1 θ 2, c s = 4(θ 2 2/3) 2(1 θ 2 )(θ 2 1/3) Ivariants: mass conservation I = (, t) d R ( I 1 = 2 (, t)+(1+(, t))u 2 (, t) c(, 2 t) au 2 (, t) ) d R DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 9 / 25

Convergence test-cases - I Numerical results on the solitary wave propagation ( ) 1/2 Eh 2 (k) = Uk h / U h, U k h = Ui k 2, Eh (k) = U k h, / U h,, U k h, = ma U k i i, i Rate(Eh 2) Rate(E h ).5 1.91 1.978.25 1.91 1.954.125 1.923 1.937.625 1.936 1.941.3125 1.946 1.948 Rate(Eh 2) Rate(E h ).5 2.42 2.32.25 2.33 2.29.125 2.26 2.23.625 2.21 2.19.3125 2.17 2.16 Table: m-flu Table: MUSCL2 MinMod DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 1 / 25

Convergence test-cases - II Invariants preservation Numerical parameters: [ 5, 5], T = 2, =.1, CFL =.5; = 1 2 Mass conservation: I h = 1.932183566158.51 1.5.5 1.45.49 1.4 U h,.48.47 I h 1 1.35 1.3.46 1.25.45 CF UNO2 CF TVD2 average.44 2 4 6 8 1 12 14 16 18 2 t (a) -amplitude 1.2 CF UNO2 CF TVD2 average 1.15 2 4 6 8 1 12 14 16 18 2 t (b) I 1 invariant DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 11 / 25

Head-on collision of solitary waves Comparison with eperimental data Comparison among following models: Eperimental data BBM-BBM system Bona-Smith system (θ 2 = 9 11 ) Numerical scheme: FVCF + UNO2 + SSP-RK3 Numerical parameters: [ 3, 3], =.5, CFL =.2 Reference: W. Craig, Ph. Guyenne, J. Hammack, D. Henderson, C. Sulem. Solitary water wave interactions. Phys. Fluids, 26 DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 12 / 25

Head-on collision eperiment Comparison with eperimental data by D. Henderson 3 2.5 Bona Smith θ 2 =9/11 BBM BBM Eperimental data 2 1.5 1.5.5 3 2 1 1 2 3 Figure: t = 18.29993 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 13 / 25

Head-on collision eperiment Comparison with eperimental data by D. Henderson 3 2.5 Bona Smith θ 2 =9/11 BBM BBM Eperimental data 2 1.5 1.5.5 3 2 1 1 2 3 Figure: t = 18.867 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 13 / 25

Head-on collision eperiment Comparison with eperimental data by D. Henderson 3 2.5 Bona Smith θ 2 =9/11 BBM BBM Eperimental data 2 1.5 1.5.5 3 2 1 1 2 3 Figure: t = 19.956 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 13 / 25

Head-on collision eperiment Comparison with eperimental data by D. Henderson 3 2.5 Bona Smith θ 2 =9/11 BBM BBM Eperimental data 2 1.5 1.5.5 3 2 1 1 2 3 Figure: t = 19.1587 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 13 / 25

Head-on collision eperiment Comparison with eperimental data by D. Henderson 3 2.5 Bona Smith θ 2 =9/11 BBM BBM Eperimental data 2 1.5 1.5.5 3 2 1 1 2 3 Figure: t = 19.19388 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 13 / 25

Head-on collision eperiment Comparison with eperimental data by D. Henderson 3 2.5 Bona Smith θ 2 =9/11 BBM BBM Eperimental data 2 1.5 1.5.5 3 2 1 1 2 3 Figure: t = 19.3294 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 13 / 25

Head-on collision eperiment Comparison with eperimental data by D. Henderson 3 2.5 Bona Smith θ 2 =9/11 BBM BBM Eperimental data 2 1.5 1.5.5 3 2 1 1 2 3 Figure: t = 19.84514 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 13 / 25

Head-on collision eperiment Comparison with eperimental data by D. Henderson 3 2.5 Bona Smith θ 2 =9/11 BBM BBM Eperimental data 2 1.5 1.5.5 3 2 1 1 2 3 Figure: t = 2.49949 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 13 / 25

Further numerical eperiments Additional test-cases: Overtaking solitary waves collision Dispersive shock wave formation (conservation of invariants, etc.).9.8.7.6.5.4.3.2.1.1 4 3 2 1 1 2 3 4 t= t=2 Reference: D. Dutykh, T. Katsaounis, D. Mitsotakis. Finite volume schemes for dispersive wave propagation and runup. Submitted. 21 http://hal.archives-ouvertes.fr/hal-472431/ DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 14 / 25

Dispersive wave runup computation Numerically stiff and tricky problem P. Madsen et al. (1997): However, to make this technique [slot technique] operational in connection with Boussinesq type models a couple of problems call for special attention. [...] Firstly the Boussinesq terms are switched off at the still water shoreline, where their relative importance is etremely small anyway. Hence in this region the equations simplify to the nonlinear shallow water equations. G. Belotti & M. Brocchini (22): In our attempt to use these equations from intermediate waters up to the shoreline we run into numerical troubles when reaching the run-up region, i.e. >. These problems were essentially related to numerical instabilities due to the uncontrolled growth of the dispersive contributions (i.e. O(µ 2 )-terms). DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 15 / 25

Boussinesq-type equations for non-flat bottom The celebrated Peregrine system (1967) Peregrine system in its original form: t + ( (h+)u ) =, u t + uu + g h 2 (hu) t + h2 6 u t = Reference: D.H. Peregrine. Long waves on a beach. JFM, 1967 Vertical translations (group G 5, Benjamin & Olver (1982)): z z+d, d, h h+d, u u, This symmetry is broken, only H = h+ remains invariant DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 16 / 25

Peregrine system invariantization Invariantization under vertical translations + conservative form of equations Pre-conservative form: H t +(Hu) =, (Hu) t + ( εhu 2 + 1 2ε H2) µ2( Hh 2 (hu) t Hh2 ) 6 u t Using h = H +O(ε), H t = O(ε), Q = Hu: = 1 ε Hh H t + Q = ( (1+ 1 3 H2 1 6 HH ) 1 Q 3 H2 Q 1 ) ( Q 2 3 HH Q + t H +g 2 H2) = ghh New system is asymptotically equivalent to the original one Symmetry G 5 is recovered Dispersive terms naturally near the shoreline DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 17 / 25

m-peregrine system discretization With finite volumes method [D(v)] t +[F(v)] = S(v) Advection: KT, FVCF + MUSCL2, UNO2 Dispersion: 2nd order FD ( ) H D(v) = (1+ 1 3 H2 1 6 HH )Q 1 3 HH Q H2 3 Q Source terms: Well-balanced hydrostatic reconstruction Time stepping: Eplicit SSP-RK3 Reference: D. Dutykh, T. Katsaounis, D. Mitsotakis. Finite volume schemes for dispersive wave propagation and runup. Submitted. 21 http://hal.archives-ouvertes.fr/hal-472431/ DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 18 / 25

W.M. Keck Laboratory of Hydraulics, Caltech PhD thesis of Costas Synolakis DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 19 / 25

Solitary wave runup on a plane beach Eperiments by C. Synolakis (1987), Caltech D H β Bottom shape: Initial condition: h() = { tanβ, cotβ, 1, > cotβ, () = A s sech 2 () (λ( X )), u () = c s D + () DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 2 / 25

Solitary wave runup: A s =.4 Data from C. Synolakis (1987), β = 2.88.3.25.2 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.15.1.5.5.1.15.2 1 5 5 1 15 2 Figure: t = 2 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 21 / 25

Solitary wave runup: A s =.4 Data from C. Synolakis (1987), β = 2.88.3.25.2 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.15.1.5.5.1.15.2 1 5 5 1 15 2 Figure: t = 26 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 21 / 25

Solitary wave runup: A s =.4 Data from C. Synolakis (1987), β = 2.88.3.25.2 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.15.1.5.5.1.15.2 1 5 5 1 15 2 Figure: t = 32 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 21 / 25

Solitary wave runup: A s =.4 Data from C. Synolakis (1987), β = 2.88.3.25.2 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.15.1.5.5.1.15.2 1 5 5 1 15 2 Figure: t = 38 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 21 / 25

Solitary wave runup: A s =.4 Data from C. Synolakis (1987), β = 2.88.3.25.2 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.15.1.5.5.1.15.2 1 5 5 1 15 2 Figure: t = 44 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 21 / 25

Solitary wave runup: A s =.4 Data from C. Synolakis (1987), β = 2.88.3.25.2 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.15.1.5.5.1.15.2 1 5 5 1 15 2 Figure: t = 5 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 21 / 25

Solitary wave runup: A s =.4 Data from C. Synolakis (1987), β = 2.88.3.25.2 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.15.1.5.5.1.15.2 1 5 5 1 15 2 Figure: t = 56 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 21 / 25

Solitary wave runup: A s =.4 Data from C. Synolakis (1987), β = 2.88.3.25.2 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.15.1.5.5.1.15.2 1 5 5 1 15 2 Figure: t = 62 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 21 / 25

Solitary wave runup: A s =.28 DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 22 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 1 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 15 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 2 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 25 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 3 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 45 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 55 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 6 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 7 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Solitary wave runup: A s =.28 Data from C. Synolakis (1987), β = 2.88 1.5 1 Boussinesq (CF UNO2) Boussinesq (KT UNO2) Shallow water (CF UNO2) Eperimental data.5.5 1 1 5 5 1 15 2 Figure: t = 8 s DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 23 / 25

Conclusions Several schemes to discretize dispersive wave equations have been proposed FVs are sufficiently accurate for solitons dynamics Dispersive effects are beneficial for breaking wave description DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 24 / 25

Thank you for your attention! http://www.lama.univ-savoie.fr/ dutykh/ DENYS DUTYKH (CNRS LAMA) Dispersive waves EDP Normandie (Caen) 25 / 25

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