CiteSeerX — Robustness

Resonant wave run-up

on sloping beaches and vertical walls

DENYSDUTYKH¹ Charg ´e de Recherche CNRS

1Universit ´e de Savoie Mont Blanc Laboratoire de Math ´ematiques (LAMA)

73376 Le Bourget-du-Lac France

Seminar: Conservation Laws & Invariants for PDEs Institute of Computational Technologies, SB RAS

October 24, 2014

Acknowledgements

Collaborators:

Themistoklis Stefanakis: formely a PhD student @ UCD & ENS de Cachan

Francesco Carbone: formely @ University College Dublin Fr ´ed ´eric Dias: Professor, University College Dublin (UCD) (on leave from ENS de Cachan)

Outline

1 Sloping beach Plane beach

Numerical simulations Multiple slopes

2 Vertical wall

Sinusoidal waves Cnoidal waves Forces estimation Robustness

Outline

1 Sloping beach Plane beach

Numerical simulations Multiple slopes

2 Vertical wall

Sinusoidal waves Cnoidal waves Forces estimation Robustness

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

July 17, 2006 Java Tsunami

By courtesy of Dr. Widjo Kongko (FI-LUH, Hannover)

How to explainextremerunup values?

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 +

Hu²+g 2H²

x = gHd_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), 3035-3061.

Remark:

X The same effects can be observed in a dispersive model,

Simple academic test-case

Monochromatic wave runup

Left boundary condition:

H₀(t) =d₀+a₀sin(ωt)

Incoming periodic monochromatic wave

z

x θ

u (x, t) η (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 computenumericallythe R_maxfor various values ofω

Wave run-up amplification phenomenon

Constant sloping beach case

0 1 2 3 4 5 6 7

0 10 20 30 40 50 60

ω/pg tan (θ)/L Rmax/η0

Runup Amplif ication vs Angular F requency

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 = 4000

10⁰ 10¹ 10²

0 10 20 30 40 50 60

λ0/L Rmax/η0

Runup Amplif ication vs W avelength

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

Runup amplification: non-resonant interaction

Numerical illustration of a non-resonant case

−10 −5 0 5 10 15 20 25 30

−2

−1 0 1 2 3

x, m

η(x,t)

Free surface elevation at t = 12.00 s

0 20 40 60 80 100 120

−5 0 5

t, s R(t)/a0

Shoreline Elevation. Non−resonant case

Reference [4]:

T. Stefanakis, F. Dias, D. Dutykh. Local Runup Amplification by Resonant Wave Interactions. Phys. Rev. Lett. 107, 124502 (2011)

Runup amplification: resonant interaction

Numerical illustration of a resonant case

−10 −5 0 5 10 15 20 25 30

−2

−1 0 1 2 3

x, m

η(x,t)

Free surface elevation at t = 12.00 s

0 20 40 60 80 100 120

−5 0 5

t, s R(t)/a0

Shoreline Elevation. Non−resonant case

Reference [4]:

T. Stefanakis, F. Dias, D. Dutykh. Local Runup Amplification by Resonant Wave Interactions. Phys. Rev. Lett. 107, 124502 (2011)

Fluid volume evelotion inside the domain

Wave tank is not closed!

0 5 10 15 20 25 30 35 40 45

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05

V/Vi

tpg tan θ / L

Bi-chromatic wave run-up

Result of extensive numerical simulations

0 1

2 3

4

0 1 2 3 4 0 0.5 1 1.5 2 2.5

ω1(L/g tan θ)^1/2 ω2(L/g tan θ)^1/2

Rmax/η0

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, 35–40.

Physical simulation of wave run-up

Experimental validation of the resonance phenomenon

In Discussion section:

Experimental values of frequencies f3−4practically 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, 35–40.

A simple theoretical explanation

Based on a linearized NSWE equations⇒LSWE

An elementary bounded solution to LSWE on a sloping beach:

η(x,t) =AJ0

 s

4ω²|x| g tanδ



cos(ωt)

z

x θ

u (x, t) η (x, t)

Incoming monochromatic wave at x = −ℓ: η(−ℓ,t) = η₀cos(ωt) Solution to the BVP:

η(x,t) =J₀ s

4ω²|x| g tanδ

/J₀ s

4ω²ℓ g tanδ

cos(ωt)

What happens if we have two slopes?

A natural generalization of the previous situation

Theresonance conditionthis time reads:

J₀(σ1) −J₀(σ2) −Y₀(σ2) J1(σ1) −J1(σ2) −Y1(σ2) 0 J₀(σ₃) Y₀(σ₃)

=0.

What happens if we have two slopes?

A natural generalization of the previous situation

Theresonance conditionthis time reads:

J₀(σ₁) −J₀(σ₂) −Y₀(σ₂) J₁(σ1) −J₁(σ2) −Y₁(σ2) 0 J₀(σ₃) Y₀(σ₃)

=0.

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 J₀(ξ) =0

Outline

1 Sloping beach Plane beach

Numerical simulations Multiple slopes

2 Vertical wall

Sinusoidal waves Cnoidal waves Forces estimation Robustness

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, Hs

Average of highest 1% of all waves:

H₁=1.67×H_1/3 Goda is more cautious:

H₁=1.8×H_1/3

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. . .

Theoretical predictions

Asymptotic analysis in 2D: dimensionless amplitudeα :=a/d

Linear theory [9]:

R_max/d =2α

Nonlinear shallow water equations [10]:

R_max/d =4 1+ α −√

1+ α
=2α+1/2α²−1/4α³+ O(α⁴)

Third order theory [11]:

R_max/d =2α+1/2α²+3/4α³+ O(α⁴) Preliminary conclusion:

Maximal run-upR_max≡2α+ higher order corrections

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

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

Mathematical model

R-S-SG-GN-PZh equations

The governing equations [13]:

ht + (hu)x = 0, ut + ¹₂u²+gh

x = ¹₃h⁻¹h

h³ uxt +uuxx−u²_x
i

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

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 wallR(t)/a₀ All simulations start from the rest state: η ≡0, u≡0 Wavemaker motion:η(x =0,t) =a0sin(ωt)H(nT −t)

Figure :A schematic view of the numerical experiments.

Numerical results

Maximum measured wave run-up on the wall

Numerical results

Maximum measured wave run-up on the wall

Numerical results

Maximum measured wave run-up on the wall

Numerical results

Maximum measured wave run-up on the wall

Space-time dynamics

Run-up time series on the vertical wall

x

Time

0 500 1000 1500 0

500 1000 1500 2000 2500 3000 3500 4000

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5 3

ω = 0.01

x

0 200 400 600

0

200 400

600

800

1000

1200

170 210

200

1200

−1 0 1 2 3 4 5

ω = 0.0315

x

0 50 100 150

0 50 100 150 200 250 300 350

−1 0 1 2 3 4

ω = 0.11

1000 2000 3000 4000

−2 0 2 4

Time RL(t) / a0

600 800 1000 1200

−2 0 2 4 6

Time

100 200 300

−2 0 2 4

Time

ω = 0.01 ω = 0.0315 ω = 0.11

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 wallR(t)/a₀ All simulations start from the rest state: η ≡0, u≡0 Wavemaker motion:η(x =0,t) =a0sn(ωt,m)H(nT −t)

Figure :A schematic view of the numerical experiments.

Maximum run-up on the wall

Dependence on parametersωand m

ω Unperturbed

m

0.05 0.1 0.15 0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2.5 3 3.5 4 4.5 5 5.5

Maximum run-up on the wall

Dependence on parametersωand m

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

5.52 5.54 5.56 5.58 5.6 5.62 5.64 5.66 5.68 5.7

m

Maximum Elevation

Estimation of forces

In the framework of the Serre-Green-Naghdi equations

Pressure distribution in the bulk of the fluid:

P(x,y,t)

ρgd = η −y

d + 1

2

"

h d

2

−  1+y

d

2#

˜ γd g h,

Depth-averaged force:

F(x,t) ρgd² =

Z _η

−d

P

ρgd²dy =

1 2 + γ˜

3 g

h d

2

.

Tilting moment (w.r.t. bottom):

M(x,t) ρgd³ =

Z η

−d

P

ρgd³(y+d) dy =

1 6+ γ˜

8 g

h d

3

.

whereγ˜is the vertical acceleration on the free surface:

˜γ ≡ ˜vt + ¯u· ∇v˜ = h n

(∇ · ¯u)² − ∇ · ¯ut − ¯u· ∇[ ∇ · ¯u]o ,

Depth averaged force

Dependence on parametersωand m

ω

m

0.05 0.1 0.15 0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76

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 wallR(t)/a0

All simulations start from the rest state: η ≡0, u≡0

Wavemaker motion:η(x =0,t) = a₀sn(ωt,m) +εξ(t)
H(nT −t)

Figure :A schematic view of the numerical experiments.

Maximum run-up on the wall (perturbed case)

Dependence on parametersωand m

ω Perturbation 25%

m

0.05 0.1 0.15 0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2.5 3 3.5 4 4.5 5 5.5 6

Maximum run-up on the wall (perturbed case)

Dependence on parametersωand m

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

5.7 5.8 5.9 6 6.1 6.2 6.3 6.4

Perturbation 25%

m

Maximum Elevation

Conclusions & Perspectives

Conclusions:

Extreme run-up on the wall was highlighted

Design wave definition to be revisited Hs ∼3H1/3or even Hs ∼3H1

Perspectives:

Validation by the full Euler / laboratory experiments X

Investigation of 3D focussing mechanisms

Important remark:

Waves never come isolated. Wave groups have to be considered.

Thank you for your attention!

http://www.denys-dutykh.com/

References I

R. M. Sorensen.

Basic coastal engineering.

Springer, 1997.

D. Dutykh, T. Katsaounis, and D. Mitsotakis.

Finite volume schemes for dispersive wave propagation and runup.

J. Comput. Phys, 230(8):3035–3061, April 2011.

A. J. C. de Saint-Venant.

Th ´eorie du mouvement non-permanent des eaux, avec application aux crues des rivi `eres et `a l’introduction des mar ´ees dans leur lit.

C. R. Acad. Sc. Paris, 73:147–154, 1871.

T. Stefanakis, F. Dias, and D. Dutykh.

Local Runup Amplification by Resonant Wave Interactions.

Phys. Rev. Lett., 107:124502, 2011.

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 2013.

P. Bruun and A. R. Gunb ¨ak.

Stability of sloping structures in relation to\xi =\tan\alpha/\sqrt{H/L 0} risk criteria in design.

Coastal Engineering, 1:287–322, 1977.

R. R. Minikin.

Winds, Waves and Maritime Structures.

Arnold, London, 2nd revise edition, 1963.

Y. Goda.

New wave pressure formulae for composite breakers.

In Proc. 14th Int. Conf. Coastal Eng., pages 1702–1720, 1974.

References III

C. C. Mei.

The applied dynamics of water waves.

World Scientific, 1989.

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):252–253, 1984.

C. H. Su and R. M. Mirie.

On head-on collisions between two solitary waves.

J. Fluid Mech., 98:509–525, 1980.

W. Li, H. Yeh, and Y. Kodama.

On the Mach reflection of a solitary wave: revisited.

J. Fluid Mech., 672:326–357, 2011.

References IV

F. Serre.

Contribution `a l’ ´etude des ´ecoulements permanents et variables dans les canaux.

La Houille blanche, 8:830–872, 1953.