A simple and accurate nonlinear method for recovering the surface wave elevation from pressure measurements

Transcription

1 ICCE 2018 Baltimore July 30-August 3, 2018 A simple and accurate nonlinear method for recovering the surface wave elevation from pressure measurements Bonneton P. 1, Mouragues A. 1, Lannes D. 1, Martins K. 2, Michallet H. 3 1 Bordeaux Univ., 2 Bath Univ., 3 Grenoble Univ. ζ NNNN = ζ LL 1 gg tt(ζ LL tt ζ LL ) Garonne - Bonneton et al., 2015 UHAINA model

2 Introduction Wave elevation measurement Accurate measurements of surface wave elevation ζ are crucial for many coastal applications: wave overtopping and submersion navigation and platform safety wave-induced sediment transport

3 Introduction Wave elevation measurement Direct measurement of the surface elevation highly accurate measurement Acoustic surface tracking AST Lidar scanning expensive difficult to deploy and fragile AST sensitive to air bubbles and turbidity lidar requires the presence of nearshore structures Nortek Martins et al 2018

4 Introduction Wave elevation measurement Pressure sensors are still a very useful tool for measuring waves Pressure sensor z z = ζ(x 0,t) cheap easy to deploy robust (storms, bottom trawling, ) Ocean Sensor Systems δ m P m (x 0,t) x 0 x not sensitive to air bubbles and turbidity not a direct measurement of ζ methods for recovering ζ

5 Introduction Wave elevation measurement Commonly used reconstruction methods and their limitations Pressure sensor z z = ζ(x 0,t) Ocean Sensor Systems δ m P m (x 0,t) x 0 x

6 Introduction Commonly used reconstruction methods long waves: tsunamis, tides, hydrostatic reconstruction z z = ζ(x 0,t) h 0 δ m x 0 P m (x 0,t) x = ρ 0gg h HH (xx 0, tt) = P m P a ρ 0 gg + δ mm ζ HH = P m P a ρ 0 gg + δ mm h 0

7 Introduction Commonly used reconstruction methods swell, wind-generated waves non-hydrostatic reconstruction

8 Introduction Commonly used reconstruction methods swell, wind-generated waves non-hydrostatic reconstruction recover the wave field by means of a transfer function based on linear theory transfer function method z z = ζ(x 0,t) h 0 δ m x 0 P m (x 0,t) x pressure response factor ζ LL ω = cosh(h 0 kk ) cosh δ mm kk ζ HH ω ζ HH = P m P a ρ 0 gg + δ mm h 0 ω 2 = gg kk tanh h 0 kk ω

9 Introduction Commonly used reconstruction methods swell, wind-generated waves non-hydrostatic reconstruction transfer function method gives reasonable estimates for bulk wave parameters such as H s Guza and Thornton 1980, Bishop and Donelan 1987, Tsai et al. 2005, fails to describe the shape of nonlinear nearshore waves lidar scanner ζ linear reconstruction ζ L ζ Martins et al. 2017, BARDEX II, h 0 =1.17 m, T p =12.1 s provides a poor description of the peaky and skewed shape of nonlinear waves underestimates the individual wave height by up to 30 %

10 Introduction Nonlinear reconstruction methods There is a critical need for nonlinear reconstruction methods Constantin 2012, Deconinck et al. 2012, Olivears et al. 2012, Clamond, 2013 periodic waves of permanent form Oliveras et al heuristic approximation for irregular waves

11 Introduction Nonlinear reconstruction methods Two nonlinear reconstruction methods for irregular waves in the field Bonneton, Lannes, Martins, Michallet, Coastal Eng weakly dispersive Bonneton and Lannes, JFM 2017 fully dispersive

12 Nonlinear reconstruction formulas Scaling analysis z z = ζ(x,t) c a 0 h 0 P b (x 0,t) x 0 λ 0 =2πL 0 x ε = aa 0 = 0(1) µ = h 0 h 0 LL 0 2 < 1 σ = aa 0 LL 0 = ε µ 1

13 Nonlinear reconstruction formulas fully dispersive reconstruction Asymptotic expansion of the Euler equations in terms of σ φ = φ 0 + σφ 1 + OO(σ 2 ) ζ NNNN = ζ LL µσ tt (ζ LL tt ζ LL ) Bonneton and Lannes (JFM 2017) ζ LL (kk) = cosh µ kk ζ HH (kk) In variables with dimension ζ NNNN = ζ LL 1 gg tt (ζ LL tt ζ LL ) ζ LL (kk) = cosh h 0 kk ζ HH (kk)

14 Nonlinear reconstruction formulas weakly dispersive reconstruction Waves in shallow water (µ 1) Asymptotic expansion of the Euler equations in terms of µ φ = φ 0 + µφ 1 + OO(µ 2 ) ζ SSNNNN = ζ SSLL 1 gg tt (ζ SSLL tt ζ SSLL ) Bonneton et al., Coastal Eng ζ SSLL = ζ HH h 0 2gg tt 2 ζ HH

15 Applications to laboratory and field data Laboratory experiments Field measurements LEGI wave flume

16 Applications to laboratory and field data Fully dispersive reconstruction LEGI wave flume experiments, Michallet et al bichromatic waves propagating over a gently sloping (1/20) movable bed LEGI wave flume 36 m long, 0.55 m wide ζ and P m were synchronously measured in the shoaling zone, at 18.5m from the wave maker; h 0 =0.326 m, T m =1.7 s, µ=0.53

17 Applications to laboratory and field data Fully dispersive reconstruction bichromatic waves propagating over a gently sloping movable bed f 1 f 2 Energy density spectra

18 Applications to laboratory and field data Fully dispersive reconstruction ζ LL (ω) = cosh h 0 kk(ω) ζ HH (ω) linear reconstruction ω 2 = gg kk tanh(h 0 kk ω ) f c

19 Applications to laboratory and field data Fully dispersive reconstruction ζ LL (ω) = cosh h 0 kk(ω) ζ HH (ω) linear reconstruction ω 2 = gg kk tanh(h 0 kk ω ) f > f c ζ LL (ω) = ζ HH (ω) f c

20 Applications to laboratory and field data Fully dispersive reconstruction ζ LL (ω) = cosh h 0 kk(ω) ζ HH (ω) linear reconstruction ω 2 = gg kk tanh(h 0 kk ω )

21 Applications to laboratory and field data Fully dispersive reconstruction ζ NNNN = ζ LL 1 gg tt ζ LL tt ζ LL non-linear reconstruction

22 Applications to laboratory and field data Fully dispersive reconstruction ζ NNNN = ζ LL 1 gg tt ζ LL tt ζ LL non-linear reconstruction ζ direct measurement ζ L ζ NL S K S K error 25% 3%

23 Applications to laboratory and field data Fully dispersive reconstruction ζ NNNN = ζ LL 1 gg tt ζ LL tt ζ LL non-linear reconstruction f c

24 Applications to laboratory and field data Highest wave nonlinearities generally occur in shallow water close to the onset of breaking high nonlinearities nonlinear weakly dispersive reconstruction ζ SSNNNN = ζ SSLL 1 gg tt (ζ SSLL tt ζ SSLL ) local in time no frequency cut-off f c reconstruction of the whole elevation density spectrum µ < 0.3 ζ SSLL = ζ HH h 0 2gg tt 2 ζ HH

25 Applications to laboratory and field data Weakly dispersive reconstruction Field campaign (April 13-14, 2017) La Salie beach, southern part of the French Atlantic coast Instruments were deployed at low tide: pressure transducers (Ocean Sensor Systems) Nortek ADCP, Signature 1000 direct measurement of ζ from the vertical beam of the ADCP Nortek

26 Applications to laboratory and field data Weakly dispersive reconstruction Field campaign nonlinear waves in the shoaling zone just prior to breaking H max =1.4 m h 0 =2.25 m, H s =0.70 m, T p =11.1 s µ=0.075 weakly dispersive waves

27 Applications to laboratory and field data Weakly dispersive reconstruction Field campaign

28 Applications to laboratory and field data Weakly dispersive reconstruction Field campaign ζ SSNNNN = ζ SSLL 1 gg tt ζ SSLL tt ζ SSLL ζ SSLL = ζ HH h 0 2gg tt 2 ζ HH

29 Applications to laboratory and field data Weakly dispersive reconstruction Field campaign

30 Applications to laboratory and field data Weakly dispersive reconstruction Field campaign wave by wave analysis 40 % ζ c

31 Applications to laboratory and field data Weakly dispersive reconstruction Field campaign wave by wave analysis ζ c

32 Conclusion Two novel nonlinear methods for recovering the surface wave elevation from pressure measurements general fully dispersive method weakly dispersive method, µ 0.3 ζ NNNN = ζ LL 1 gg tt ζ LL tt ζ LL ζ SSNNNN = ζ SSLL 1 gg tt ζ SSLL tt ζ SSLL µ=0.53 ζ SSLL = ζ HH h 0 2gg tt 2 ζ HH µ=0.28 f c

33 Conclusion Two novel nonlinear methods for recovering the surface wave elevation from pressure measurements provide much better results compared to the transfer function method commonly used in coastal applications are very simple and easy to use represent an economic alternative to direct wave elevation measurement methods (AST and Lidar) are a valuable tool for accurately characterizing extreme waves in shallow and intermediate water depths

34 Thank you for your attention