Resonant wave run-up on sloping beaches and vertical walls

Transcription

1 Resonant wave run-up on sloping beaches and vertical walls DENYS DUTYKH 1 Chargé de Recherche CNRS 1 Université de Savoie Mont Blanc Laboratoire de Mathématiques (LAMA) Le Bourget-du-Lac France Seminar: Conservation Laws & Invariants for PDEs Institute of Computational Technologies, SB RAS October 24, 2014 DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

2 Acknowledgements Collaborators: Themistoklis Stefanakis: formely a PhD UCD & ENS de Cachan Francesco Carbone: University College Dublin Frédéric Dias: Professor, University College Dublin (UCD) (on leave from ENS de Cachan) DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

3 Outline 1 Sloping beach Plane beach Numerical simulations Multiple slopes 2 Vertical wall Sinusoidal waves Cnoidal waves Forces estimation Robustness DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

4 Outline 1 Sloping beach Plane beach Numerical simulations Multiple slopes 2 Vertical wall Sinusoidal waves Cnoidal waves Forces estimation Robustness DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

5 Wave run-up Definition of the wave run-up Figure : SWL indicates Still Water Level Definition (Sorensen [1]): Wave runup is the maximum vertical extent of wave uprush on a beach or a structure above the still water level DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

6 July 17, 2006 Java Tsunami By courtesy of Dr. Widjo Kongko (FI-LUH, Hannover) How to explain extreme runup values? DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

7 Governing equations The classical nonlinear shallow water (Saint-Venant) model [3] Nonlinear Shallow Water Equations (NSWE) (without friction): H t +(Hu) x = 0, ( (Hu) t + Hu 2 + g 2 H2) = ghd x x Solver validation [2]: Dutykh, D., Katsaounis, T., & Mitsotakis, D. (2011). Finite volume schemes for dispersive wave propagation and runup. J. Comput. Phys, 230(8), Remark: The same effects can be observed in a dispersive model DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

8 Simple academic test-case Monochromatic wave runup Left boundary condition: H 0 (t) = d 0 + a 0 sin(ωt) η (x, t) θ z x Incoming periodic monochromatic wave u (x, t) Reference: I. Didenkulova, E. Pelinovsky. Run-up of long waves on a beach: the influence of the incident wave form. Oceanology, 48, 2008 Linear theory was shown to predict correctly at least the maximal run-up Linear prediction: R max ω (in certain range of validity) We compute numerically the R max for various values of ω DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

9 Wave run-up amplification phenomenon Constant sloping beach case Rmax /η0 Rmax /η Runup Amplification vs Angular F requency ω/ g tan (θ)/l Runup Amplification vs Wavelength tan(θ) = 0.13 ; L = 12.5 tan(θ) = 0.26 ; L = 12.5 tan(θ) = 0.30 ; L = 12.5 Disp. tan(θ) = 0.13 ; L = 12.5 tan(θ) = 0.13 ; L = λ 0 /L tan(θ) = 0.13 ; L = 12.5 tan(θ) = 0.26 ; L=12.5 tan(θ) = 0.30 ; L=12.5 Disp. tan(θ) = 0.13 ; L = 12.5 linear theory tan(θ) = 0.13 ; L = 4000 DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

10 Runup amplification: non-resonant interaction Numerical illustration of a non-resonant case 3 2 Free surface elevation at t = s η(x,t) x, m Shoreline Elevation. Non resonant case R(t)/a t, s Reference [4]: T. Stefanakis, F. Dias, D. Dutykh. Local Runup Amplification by Resonant Wave Interactions. Phys. Rev. Lett. 107, (2011) DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

11 Runup amplification: resonant interaction Numerical illustration of a resonant case 3 2 Free surface elevation at t = s η(x,t) x, m Shoreline Elevation. Non resonant case R(t)/a t, s Reference [4]: T. Stefanakis, F. Dias, D. Dutykh. Local Runup Amplification by Resonant Wave Interactions. Phys. Rev. Lett. 107, (2011) DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

12 Fluid volume evelotion inside the domain Wave tank is not closed! V /Vi t gtanθ/l DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

13 Bi-chromatic wave run-up Result of extensive numerical simulations Rmax/η ω 2(L/gtanθ) 1/ ω 1 (L/gtanθ) 1/2 4 DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

14 Physical simulation of wave run-up Experimental validation of the resonance phenomenon Reference [5]: Ezersky, A., Abcha, N., & Pelinovsky, E. (2013). Physical simulation of resonant wave run-up on a beach. Nonlin. Processes Geophys., 20, DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

15 Physical simulation of wave run-up Experimental validation of the resonance phenomenon In Discussion section: Experimental values of frequencies f 3 4 practically coincide with frequencies of modes having nodes near the wave maker; numerical values (Stefanakis et al., 2011) exceed this frequency by 2.5% for all bottom inclinations. The reason of such differences is not clear yet. Reference [5]: Ezersky, A., Abcha, N., & Pelinovsky, E. (2013). Physical simulation of resonant wave run-up on a beach. Nonlin. Processes Geophys., 20, DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

16 A simple theoretical explanation Based on a linearized NSWE equations LSWE An elementary bounded solution to LSWE on a sloping beach: η (x, t) θ z x ( 4ω η(x, t) = AJ 2 x ) 0 cos(ωt) g tanδ u (x, t) Incoming monochromatic wave at x = l: η( l, t) = η 0 cos(ωt) Solution to the BVP: ( 4ω η(x, t) = J 2 x ) ( 4ω 0 /J 2 l ) 0 cos(ωt) g tanδ g tanδ DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

17 What happens if we have two slopes? A natural generalization of the previous situation The resonance condition this time reads: J 0 (σ 1 ) J 0 (σ 2 ) Y 0 (σ 2 ) J 1 (σ 1 ) J 1 (σ 2 ) Y 1 (σ 2 ) 0 J 0 (σ 3 ) Y 0 (σ 3 ) = 0. DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

18 What happens if we have two slopes? A natural generalization of the previous situation The resonance condition this time reads: J 0 (σ 1 ) J 0 (σ 2 ) Y 0 (σ 2 ) J 1 (σ 1 ) J 1 (σ 2 ) Y 1 (σ 2 ) 0 J 0 (σ 3 ) Y 0 (σ 3 ) = 0. DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

19 Almost conclusions... Question: Why previous investigations did not unveil this phenomenon? One has to consider a BVP instead of the IVP! Literal descriptions existed however [6]: [Resonant run-up] occurs when run-down is in a low position and wave breaking takes place simultaneously and repeatedly close to that location. Alternative view of this work: We proposed a hydrodynamic method to compute zeros of the Bessel function J 0 (ξ) = 0 DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

20 Outline 1 Sloping beach Plane beach Numerical simulations Multiple slopes 2 Vertical wall Sinusoidal waves Cnoidal waves Forces estimation Robustness DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

21 Ultimate engineering question: What are the forces exerted on structures? Dynamic pressure on the wall: Minikin s equation (1963) [7] Goda s formula (1974) [8] Other empirical approaches... Design wave: Significant wave height: H 1/3, H s Average of highest 1% of all waves: H 1 = 1.67 H 1/3 Goda is more cautious: H 1 = 1.8 H 1/3 DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

22 Ultimate engineering question: What are the forces exerted on structures? Dynamic pressure on the wall: Minikin s equation (1963) [7] Goda s formula (1974) [8] Other empirical approaches... DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

23 Theoretical predictions Asymptotic analysis in 2D: dimensionless amplitude α := a/d Linear theory [9]: R max /d = 2α Nonlinear shallow water equations [10]: R max /d = 4 ( 1+α 1+α ) = 2α+1/2α 2 1/4α 3 +O(α 4 ) Third order theory [11]: R max /d = 2α+1/2α 2 + 3/4α 3 +O(α 4 ) Preliminary conclusion: Maximal run-up R max 2α + higher order corrections DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

24 Recent experimental study Reference [12]: W. Li, H. Yeh & Y. Kodama, JFM, 2011 Mach reflection of an obliquely incident solitary wave Mechanism is substantially 3D DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

25 Recent experimental study Reference [12]: W. Li, H. Yeh & Y. Kodama, JFM, 2011 Mach reflection of an obliquely incident solitary wave Mechanism is substantially 3D DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

26 Mathematical model R-S-SG-GN-PZh equations The governing equations [13]: h t +(hu) x = 0, u t + ( 1 2 u2 + gh ) x = 1 3 h 1[ h 3( u xt + uu xx u 2 x) ] x, Credits: John William Strutt (Lord Rayleigh) (1876) F. Serre (1953) C. Su & C. Gardner (1969) A. Green & P. Naghdi (1976) E. Pelinovsky & M. Zheleznyak (1985) Some properties: A long wave model (weak dispersive effects) Fully nonlinear equations Does possess several conservative quantities DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

27 Numerical set-up I The idealized situation 2D uniform channel of constant depth Left: wavemaker, Right: vertical wall Quantity of interest: run-up on the wall R(t)/a 0 All simulations start from the rest state: η 0, u 0 Wavemaker motion: η(x = 0, t) = a 0 sin(ωt)h(nt t) Figure : A schematic view of the numerical experiments. DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

28 Numerical results Maximum measured wave run-up on the wall D ENYS D UTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

29 Numerical results Maximum measured wave run-up on the wall D ENYS D UTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

30 Numerical results Maximum measured wave run-up on the wall D ENYS D UTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

31 Numerical results Maximum measured wave run-up on the wall D ENYS D UTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

32 Space-time dynamics Run-up time series on the vertical wall Time ω = x ω = x ω = x R L (t) / a ω = 0.01 ω = ω = Time Time Time DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

33 Numerical set-up II The idealized situation 2D uniform channel of constant depth Left: wavemaker, Right: vertical wall Quantity of interest: run-up on the wall R(t)/a 0 All simulations start from the rest state: η 0, u 0 Wavemaker motion: η(x = 0, t) = a 0 sn(ωt, m)h(nt t) Figure : A schematic view of the numerical experiments. DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

34 Maximum run-up on the wall Dependence on parameters ω and m Unperturbed m ω DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

35 Maximum run-up on the wall Dependence on parameters ω and m Maximum Elevation m DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

36 Estimation of forces In the framework of the Serre-Green-Naghdi equations Pressure distribution in the bulk of the fluid: [ ( P(x, y, t) = η y + 1 ) ] h 2 ( 1+ y ) 2 ρgd d 2 d d Depth-averaged force: F(x, t) ρgd 2 = η d P ρgd 2 dy = Tilting moment (w.r.t. bottom): M(x, t) ρgd 3 = η d P (y + d) dy = ρgd 3 ( γ )( ) h 2. 3 g d γ d g h, ( γ )( ) h 3. 8 g d where γ is the vertical acceleration on the free surface: { γ ṽ t + ū ṽ = h ( ū) 2 ū t ū [ ū] }, DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

37 Depth averaged force Dependence on parameters ω and m m ω DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

38 Numerical set-up III The idealized situation robustness test 2D uniform channel of constant depth Left: wavemaker, Right: vertical wall Quantity of interest: run-up on the wall R(t)/a 0 All simulations start from the rest state: η 0, u 0 Wavemaker motion: η(x = 0, t) = ( a 0 sn(ωt, m)+εξ(t) ) H(nT t) Figure : A schematic view of the numerical experiments. DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

39 Maximum run-up on the wall (perturbed case) Dependence on parameters ω and m Perturbation 25% m ω DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

40 Maximum run-up on the wall (perturbed case) Dependence on parameters ω and m 6.4 Perturbation 25% Maximum Elevation m DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

41 Conclusions & Perspectives Conclusions: Extreme run-up on the wall was highlighted Design wave definition to be revisited H s 3H 1/3 or even H s 3H 1 Perspectives: Validation by the full Euler / laboratory experiments Investigation of 3D focussing mechanisms Important remark: Waves never come isolated. Wave groups have to be considered. DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

42 Thank you for your attention! DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

43 References I R. M. Sorensen. Basic coastal engineering. Springer, D. Dutykh, T. Katsaounis, and D. Mitsotakis. Finite volume schemes for dispersive wave propagation and runup. J. Comput. Phys, 230(8): , April A. J. C. de Saint-Venant. Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l introduction des marées dans leur lit. C. R. Acad. Sc. Paris, 73: , T. Stefanakis, F. Dias, and D. Dutykh. Local Runup Amplification by Resonant Wave Interactions. Phys. Rev. Lett., 107:124502, DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

44 References II A. Ezersky, N. Abcha, and E. Pelinovsky. Physical simulation of resonant wave run-up on a beach. Nonlin. Processes Geophys., 20:35 40, July P. Bruun and A. R. Gunbäk. Stability of sloping structures in relation to \xi = \tan \alpha/\sqrt{h/l 0} risk criteria in design. Coastal Engineering, 1: , R. R. Minikin. Winds, Waves and Maritime Structures. Arnold, London, 2nd revise edition, Y. Goda. New wave pressure formulae for composite breakers. In Proc. 14th Int. Conf. Coastal Eng., pages , DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

45 References III C. C. Mei. The applied dynamics of water waves. World Scientific, N. R. Mirchina and E. Pelinovsky. Increase in the amplitude of a long wave near a vertical wall. Izvestiya, Atmospheric and Oceanic Physics, 20(3): , C. H. Su and R. M. Mirie. On head-on collisions between two solitary waves. J. Fluid Mech., 98: , W. Li, H. Yeh, and Y. Kodama. On the Mach reflection of a solitary wave: revisited. J. Fluid Mech., 672: , DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37

46 References IV F. Serre. Contribution à l étude des écoulements permanents et variables dans les canaux. La Houille blanche, 8: , DENYS DUTYKH (CNRS LAMA) Resonant wave run-up Novosibirsk, October 24, / 37