Reduced

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Stormy Maths Create Rogue & Breaking

Waves

Onno Bokhove

with Gagarina, Van der Horn, Van der Meer, Zweers & Thornton

“Mathematics of Computational Science”, University of Twente

Conclusions

School of Mathematics, University of Leeds, March 2012

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

1 Introduction

2 The Results

3 Maths of Wave Design

4 Maths-geared Lab Experiments

5 Multiphase GFD Models

6 Conclusions

1. Introduction

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Rogue waves on the oceans are sudden “anomalously”

emerging high waves, causing severe damage to ships.

Breaking storm waves at beaches drive sand erosion and

de-positioning, with associated changes of our coastlines.

Rather than trying to model these maritime and

near-shore phenomena directly, the challenges are turned

around such as to enhance our knowledge further.

Rogues Waves

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Definition, wave is rogish when:

Height of freak wave H fr over significant wave height H s :

AI = H fr

H s

> 2. (1)

Significant wave height: average wave height among 1/3 rd

of heighest waves in time series (e.g., 30 min).

Rogue Waves on the Ocean by Kharif, Pelinovsky, Slunyaev.

Conclusions

Rogues Waves

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Proposed mechanisms for rogue waves:

wave-current interactions

geometrical and spatial focussing

focussing due to dispersion: spatio-temporal focussing

focussing due to modulational instability

soliton collisions

1D versus 2D: multidirectional wave fields.

Rogue Waves: Wave Impact

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Research on wave impact at break walls:

Pressure impuls theory of Peregrine and co.

Multiphase modelling.

Wave tank Hannover (image courtesy Peregrine):

Multiphase

GFD Models

Conclusions

Storm Waves at Beaches

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Beaches abound world’s coastlines.

Understanding and prediction of beach dynamics lead to

insights: forecast/prevent dangers of flooding and erosion.

Great advances: Soulsby (1997), Calantoni et al. (2004,

2006), Garnier et al. (2006, 2008, 2010); Roelvink et al.,

McCall et al. (CE 2009/2010), Dutykh et al. (2011).

But laws of sand & sediment transport under breaking

waves relatively poorly understood.

Storm Waves at Beaches

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Is it possible to create a manageable mathematical modeling

environment of beach dynamics?

Equations water, air, particles available: DNS too costly.

Modeling environment should include hierarchy of models.

Including the range from DNS, dispersed multiphase

models to coastal depth- and wave-averaged models.

Dispersed multiphase models: GFD mixture theory.

Would such an environment also permit a suitable

laboratory experiment for validation?

Challenges

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Rather than trying to model these maritime and near-shore

phenomena directly, the challenges are turned around:

How can maths create the highest rogue wave?

How can maths create air-water-grain interactions under

breaking waves most clearly?

2. The Results: Bore-Soliton-Splash

Stormy Maths

Onno

Bokhove

Introduction

The Results

Water wave channel:

Two sluice gates; one excavator.

Uniform channel section.

Channel linearly converging at one end:

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

The Results: Bore-Soliton-Splash

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Phenomenon:

3 “Solitons” generated by removing sluice gate: 2.5m/s.

Front soliton breaks quickly into a bore.

It crashes 14s later into linearly converging closed narrows.

Bore reflects, draws a trough . . .

. . . in which unbroken second soliton crashes:

3.5-4m vertical jet at 15-17s.

Surprise: reproducible result very sensitive. Why?

Amplification Index:

AI = H fr

= 3.5 = 10. Truly rogish? (2)

H s 0.35

The Results: Hele-Shaw Beach

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Yes 2 :

Imagine a giant’s knife, make two cuts to isolate a slice of

beach . . . including sand & water, particle and wave

motion.

Place between laterally periodic boundaries, or 2 glass

plates.

Shrink latter to table-top size: Hele-Shaw beach

experiment demonstrated at Fluid Fascinations url 2010.

Qua Art & Qua Science show: tribute Howell Peregrine url .

Sketch Hele-Shaw beach

Stormy Maths

Onno

Bokhove

Introduction

The Results

z

wave maker

g

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

l

p

! w

wedge

0 x

L

w l w

x

Sketch Hele-Shaw cell: wedge, waterline, particles &

wave-maker.

B 0

free surface

H 0

particles

(Dis)Advantages Hele-Shaw Beach

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Advantages and disadvantages:

Set-up unique: focus on particle motion by breaking waves.

Immense reduction of dof’s: quasi-2D.

Innovative: allowing great visualisation fundamental

interactions; research on hierarchy of models.

Damping too severe due to proximity of glass plates?

Determine minimal gap width for which dynamics inertial!

3. Maths of Wave Design

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

3.1. Maths Origin Bore-Soliton-Splash

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Hydraulics of water/granular flows through contraction. We

average: to go from particles to continuum

find & assess closure laws for 2D shallow layer equations?

analyse to get exact solutions &

confirm these with DGFEM of 2D continuum.

Advantage: handy & faster.

Disadvantage: closures hard to formulate & validate.

Maths Origin Bore-Soliton-Splash

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Geometry of inclined chute:

1) constant width with no contraction, W c = W 0 ,

2) with localized contraction, and

3) blocked in middle, “W c = 0”.

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

u h

0 0

x

W

0

y

g t

L c

u h

0 0

W(x)

L

u h

0 0

Block

feeder with

sluice gate

z

y

x

contraction

g

g t

g n

Wc

(1) (2) (3)

inclination angle

!

Topography b = b(x) (Tata Steel). Granular jump formation.

Assumptions

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Assume incompressible continuum equations . . . .

Explore shallowness: σ zz ≈ ρgh cos θ & σ xx ≈ Kσ zz .

Use kinematic conditions & average over depth:

∂ t h + ∂ x (hū)+∂ y (h¯v) = 0

∂ t (hū)+∂ x (α 1 hū 2 + Kg n h 2 /2) + ∂ y (α 1 hū¯v) =

− g n h∂ x b + g n h(tan θ − µ(h, F )ū/|ū|)

∂ t (h¯v)+∂ x (α 1 hū¯v)+∂ y (α 1 hū 2 + Kg n h 2 /2) =

− g n hµ(h, F )¯v/|ū|

(3a)

(3b)

(3c)

with h(x, y, t)ū(x, y, t) = ∫ b+h

b

u(x, y, z, t)dz

Closures: α 1 , K ≈ 1&µ(h, F ) with F = |ū|/ √ g n h.

Closure Relation

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Experiments, DPMs:

Coulomb’s model µ = tan θ = tan δ unrealistic.

Stoppage height function of angle of θ

h stop (θ)

Ad

= tan(δ 2) − tan(θ)

tan(θ) − tan(δ 1 )

for δ 1 < θ < δ 2 (4)

δ 1,2 minimum/maximum angle steady flow; diameter d.

Froude number function of h/h stop :

F =

Combine (4,5) using µ = tan θ:

βh − γ. (5)

h stop (θ)

µ = µ(h, F ) = tan(δ 1 )+ tan(δ 2) − tan(δ 1 )

βh/(Ad(F + γ)) + 1

From DPM simulation: 17 ◦ < θ < 32 ◦ .

(6)

1D Hydraulic Analysis

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Average equations over cross section.

Find steady states & “shocks” for:

u t + uu x + h x

F0

2 = (tan θ − δb x − µ(h, F ) u

|u| )

δF0

2 (7a)

(hW ) t +(hWu) x = 0 (7b)

variable velocity u = ū(x, t), depth h = h(x, t), given

width W = W (x) & topography b = b(x).

Froude no. F 0 = u 0 / √ gh 0 , aspect ratio δ = h 0 /W 0 .

Classification

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Steady state solutions classify flow regimes (b = 0):

Flow rate Q = hWu(= 1) constant.

ODE for Froude variable F = F 0 u/ √ h = F (x):

( F 2 − 1

) dF

( F 2

F dx = +2

) d(ln W )

+

2 dx

F 2/3 W 2/3

(tan θ − µ(F )). (8)

δ(QF 0 ) 2/3

Singular at F =1: critical; F >< 1: super/subcritical.

Regularize at F =1s.t. LHS=RHS, if mayhem at nozzle:

0 dF

dx }{{}

LHS

≠ 3 α

+ 3 1 2/3

W

2 W c 2 δ(QF 0 ) 2/3

} {{ }

RHS

c (tan θ − µ(1)) . (9)

“Invisid” Classification

Stormy Maths

Onno

Bokhove

In contraction and channel when µ(F ) = tan θ:

Steady-state flow parameter plane F 0 versus B c = W c /W 0 :

Introduction

4

The Results

Maths of

Wave Design

3.5

3

i/iii

i

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

F 0

2.5

2

1.5

1

iii

0.5

ii

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

B c

h

h

Full Classification H 2 O

h

h

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

F

4

2

5

In contraction, when θ = 0 & µ = C d u 2 (Akers & B. 2008):

Parameter plane F 0 versus B c = W c /W 0 flo ws:

0

0 1 2 3 4 5

0

0 1 2 3 4 5

x

(i)

F 0

4

3.5

3

2.5

2

1.5

(b)

iii

i/iii/iv

i

iv

F

1

0.5

1.5

1

0.5

0

0 1 2 3 4 5

0

0 1 2 3 4 5

x

(ii)

1

F

4

2

0

0 1 2 3 4 5

5

(iii)

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

B c

ii

F

4

2

0

0 1 2 3 4 5

5

(iv)

0

0 1 2 3 4 5

x

0

0 1 2 3 4 5

x

Multiple Steady States

Stormy Maths

Onno

Bokhove

Transition supercritical state:

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

0.6

0.4

h

0.2

4.5

x

5

5.5

6

Oblique jumps

0.4

0.2

y

0

-0.2

-0.4

3

3.5

4

h: 0.1 0.2 0.3 0.4 0.5

Multiple Steady States

Stormy Maths

Onno

Bokhove

to

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Hydraulic jump

Multiphase

GFD Models

Conclusions

0.6

0.4

h

0.2

4.5

x

5

5.5

0.4

0.2

y

0

-0.2

-0.4

3

3.5

4

h: 0.1 0.2 0.3 0.4 0.5 0.6

Multiple Steady States: DGFEM

Stormy Maths

Onno

Bokhove

subcritical state via slurry flow (Rhebergen et al. 2009):

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

6

Hydraulic jump

Multiphase

GFD Models

Conclusions

0.6

0.4

h

0.2

4.5

x

5

5.5

0.4

0.2

y

0

-0.2

-0.4

3

3.5

4

h: 0.1 0.2 0.3 0.4 0.5 0.6

h

F

h

h

F

F

Full Classification Granular

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

1

0.8

0.6

0.4

0.2

1.5

0.5

In contraction, when µ(F ) ≠ tan θ (Tunuguntla & B.):

Parameter plane F 0 versus B c = W c /W 0 flows:

0

−4 −3 −2 −1 0 1 2 3 4

x

1

0

−4 −3 −2 −1 0 1 2 3 4

x

4

3

(ii)

(iii)

F o

5

4.5

4

3.5

3

2.5

2

1.5

iii

F o

−B c

parameter plane

i / iii

*

i

4

3

2

1

0

−4 −3 −2 −1 0 1 2 3 4

x

5

4

3

2

1

0

−4 −3 −2 −1 0 1 2 3 4

x

(i)

2

1

−4 −3 −2 −1 0 1 2 3 4

5

4

3

2

1

0

−4 −3 −2 −1 0 1 2 3 4

x

1

0.5

ii

0

*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

B c

Brute force SPH simulation slow

Stormy Maths

Onno

Bokhove

Introduction

Idea: let soliton run into linear contraction to stow H 2 O:

Brute-force SPH calculation: slow & inaccurate.

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Velocity t =1.5, 2.6, 3.6, 3.6, 4.1, 4.5s in finite 11m 2D

channel. Maximum splash ∼ 1.2m at t ≈ 3.64s.

3.2. Maths of Hele-Shaw Beach

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Specifications quasi-2D dynamics in Hele-Shaw cell:

2 vertical glass plates: 0.6 × 0.3 × 2l or 1 × 0.3 × 2lm 3

filled with water & particles d ≤ 2lmm

wavemaker: moving rod; f ∼ 1 Hz.

What should half gap width l be?

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Asymptotic Analysis: width averaging

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Determine minimum gap width for which broken wave travels

from wave maker to beach?

1. Focus on dynamics water & scale:

- 3D Navier-Stokes with length & velocity (L, l, D) &

(U, V , W ), pressure P 0 = ρ 0 U 2 /(Re ɛ 2 );

- dimensionless: Reynolds Re = UL/ν, Froude

1/Fr 2 = gD/U 2 & aspect ratios ɛ = l/L ≪ 1&δ = D/L:

∂ t w + u∂ x w + v∂ y w + w∂ z w = − 1

Re ɛ 2 δ 2 ∂ zp

− 1

Fr 2 δ 2 + 1 (

∂

2

Re x + 1 ɛ 2 ∂2 y + 1 )

δ 2 ∂2 z w (10a)

∂ x u + ∂ y v + ∂ z w =0.

(10b)

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Simplify:

∂ t w + u∂ x w + v∂ y w + w∂ z w = − 1

Re ɛ 2 δ 2 ∂ zp

− 1

Fr 2 δ 2 + 1

Re ɛ 2 ∂2 y w

(11a)

∂ x u + ∂ y v + ∂ z w =0.

(11b)

Balance between pressure gradient and viscous terms?

ũ = − 1

2ν (∂ x ⋆p⋆ /ρ 0 )(l 2 − y 2 ⋆ )

˜w = − 1

2ν (∂ z ⋆p⋆ /ρ 0 + g)(l 2 − y 2 ⋆ )?

(12a)

(12b)

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

2. Consider ratio inertia terms/pressure gradient:

3 ρ 0 ũ 2

L|∇ xz p| = l 4 |∇ xz p|

3 ρ 0 ν 2 L = l 4 g∆h

3 ν 2 L 2 (13a)

= l 4 × 10 × (4 × 10 −2 )

3 × 10 −12 × (0.5) 2 ∼ 0.1 to 10 (13b)

for l =0.75 to 2 mm. Flow is inertial.

Pohlhausen (Rosenhead 1963) suggested Ansatz:

u = 3 2ū (l 2 − y 2 )

l 2 and w = 3 2 ¯w (l 2 − y 2 )

l 2 (14)

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

3. Average across gap: obtain 2D Navier-Stokes equations

∂ t ū + γū∂ x ū + γ ¯w∂ z ū = − 1 ρ 0

∂ x P − 3νū

l 2

∂ t ¯w + γū∂ x ¯w + γ ¯w∂ z ¯w = − 1 ρ 0

∂ z P − g −

∂ x ū + ∂ z ¯w =0

3ν ¯w

l 2

(15a)

(15b)

(15c)

with y–independent pressure P = P(x, z, t), width average

ū = ∫ l

−l

u(x, y, z, t)dy/(2l) &γ =6/5.

Asymptotic Analysis: depth averaging

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

4. Between z = b(x, t) &z = b(x, t)+h(x, t).

Kinematic conditions at bottom & free surface:

∂ t (b + h)+ū∂ x (b + h) − ¯w =0

∂ t b + ū∂ x b − ¯w =0.

(16a)

(16b)

Multiphase

GFD Models

Conclusions

Use hydrostatic balance:

∂ z p/(Re ɛ 2 )+1/Fr 2 =0 or ∂ z ⋆P ⋆ /ρ 0 + g =0. (17)

Asymptotic Analysis: depth averaging

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Shallow water equations with damping:

∂ t (hū)+∂ x

(

γhū 2 + gh 2 /2 ) = − gh∂ x b − 3νhū/l 2

∂ t h + ∂ x (hū) =0

(18a)

(18b)

and hū(x, t) = ∫ h+b

b

ū(x, z, t)dz.

Represent breaking wave as shallow-water bore (γ = 1):

discontinuity at x = x b (t); speed S = dx b /dt satisfying

[h(ū − S)] = 0 [h(ū − S) 2 + gh 2 /2] = 0. (19)

. . . depth averaging

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Original design question: Determine minimum gap width

for which broken wave can travel to from wave maker to

beach? turns into simpler question:

For which gap width 2l can bore generated offshore reach

end of beach?

. . . research-led simulations for M.Sc..

. . . depth averaging

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Answer: for a beach of length ∼ 0.5m and gap width

2 l > 1.5mm generated bore reaches end of beach.

Given availability of zeolite particles with

d =1.80 ± 0.05mm; we chose 2 l = 2mm

and build 2 experimental set-ups (B. & Zweers), to test

my asymptotic calculations.

4. Maths-geared Lab Experiments

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Sketch wave tank:

Make still water levels, h 0,1 , in sluice & main channel:

y

c

g

L

Sketch: L = 43.63 ± 0.2m, c =2.7 ± 0.1m,

l =2.63 ± 0.1m, w = 2m.

h 0

z

w

x

l

x

h 1

Procedure

Stormy Maths

Onno

Bokhove

Introduction

Open sluice gate and create a main soliton.

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Classical soliton.

Procedure

Stormy Maths

Onno

Bokhove

Introduction

Main soliton breaks into bore.

Bore and second large soliton travel trough channel.

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Runs

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Case h 0 (m) h 1 (m) Comments

1 0.32 0.67 bore

2 0.38 0.74 good splash

3 0.41 0.9 Bore Soliton Splash, like cases 6 & 8

4 0.47 1.0 bore

5 0.41 1.02 low BSS

6 0.41 0.9 BSS like cases 3 & 8

7 0.45 0.8 good splash

8 0.41 0.9 BSS & (highest?) splash

9 0.43 0.9 2 solitons & tiny splash

Table: 1 to 7: 27–09–2010; 8 & 9: 30–09–2010. Levels h 0,1 .

Sensitivity

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Cases 3, 6, & 8 with highest (recorded) splash are repeats

with h 0 =0.41m and h 1 =0.9m.

Case 9 differs 0.02m: h 0 =0.43m and h 1 =0.9m resulting

in no wave breaking and big splash.

Conclusions

Sensitivity: case 8

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Case 8 with h 0 = 41cm: the bore.

Sensitivity: case 8

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Sensitivity: case 9

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Case 9 with h 0 = 43cm: only solitons.

Sensitivity: case 9

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Hele-Shaw Beach Experiments

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Quasi-2D dynamics in Hele-Shaw cell:

2 glass plates: 0.6 × 0.3 × 0.002 or 1 × 0.3 × 0.002m 3

filled with water & heavier particles d =1.8mm

wavemaker: moving welding rod 1.5mm thin; f ∼ 1 Hz

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Wave Types

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Breaker wave types (Peregrine 1983 ARFM) observed:

Spilling: white water at wave crest spills down front face

sometimes with projection of small jet

Plunging: wave’s front face overturns, prominent jet at

base wave, causing large splash

Collapsing: lower portion front face overturns, behaves like

truncated plunging breaker

Surging: significant disturbance smooth profile occurs only

near moving shoreline

. . . Shore break: whole face from trough to crest vertical

with little/no water in front.

63, 115, 133, 8s

Beach Formation

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Monochromatic wave frequency 0.6m–cell: beach creation.

Monochromatic 0.6hz wave frequency in 1m–cell; 30s

stills: formation of sand waves & beach.

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Dune Formation

Stormy Maths

Onno

Bokhove

Introduction

The Results

Alternating f =0.6 & 0.9Hz: dune creation, hysteretic

effects. Monochromatic f =1.0Hz dune (Van der Horn)

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Dune Formation: single frequency

Stormy Maths

Onno

Bokhove

Monochromatic case reproducible (Bram van der Horn):

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Modeling challenges

Stormy Maths

Onno

Bokhove

Given design calculations & experimental results, goal becomes:

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

to predict the dynamics in the Hele-Shaw cell,

with models that can be extended to yield feasible

3D-predictions.

Consider dof’s:

DNS simulations: ≥ 10 3 /particle; tank

250 × 40 × 1 d 3 = 10 4 d 3 .

Leading to 10 7 − 10 8 dof’s for Navier-Stokes and DPMs!

Perhaps feasible in quasi-2D but not in 3D.

A hierarchy of reduced & multi-phase models required.

5. Multiphase GFD Models

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Overview:

Baer-Nunziato model & simplification by Dumbser (2011).

GFD mixture theory models granular debris flows (B. &

Thornton 2012).

Clash between conservative & compatible wave modelling

and CFD/engineering multiphase modelling.

Bore-Soliton-Splash: air-water phases, minimal/no loss of

soliton wave amplitude, break-up in splash.

Hele-Shaw beach dynamics: closures grain-water, air-water

& air-water-grains in dispersed/continuum models.

Compatible discretization water waves

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Discretization Miles’ variational principle (Gagarina et al.).

Classic DGFEM for water waves (2007) did not work [well].

Variational FEM in space & DGFEM in time.

Wave amplitude “preserved”.

Symplectic mesh movement stable via time integrator:

obeys geometric conservation law.

Numerical verification: Fenton waves, MARIN test case,

wave maker movement.

GFD version based on Baer-Nunziato

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Dumbser’s pseudo two-phase model for water (air passive):

∂ t (ϕρ)+∇· (ϕρu) =0

(20a)

∂ t (ϕρu)+∇· (ϕ(ρuu + pI)) =ϕρg∇Z (20b)

∂ t ϕ + u ·∇ϕ =0

(20c)

∂ t Z =0

(20d)

p =k 0 (ρ/ρ 0 − 1) (20e)

Multiphase

GFD Models

Conclusions

Volume fraction ϕ of water phase

Tait EOS mimicking incompressible flow: k 0 = 10 7

Liquid part Ω: ϕ(x, 0) = 1 − ɛ; air: ϕ(x, 0) = ɛ ≈ 10 −3

Challenge: DGFEM with hydrostatic balance, ϕ> 0&

new slope limiters.

GFD Mixture Theory Model

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

By analogy with granular systems (B. & Thornton 2012):

Use mixture theory to average fine-scale water-air

interactions.

First air & water,

then three-phase wave-resolving mixture theory.

Advantage: feasibility in 3D. Stability.

GFD Mixture Theory: Air & Water

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Two-phase model with friction between phases.

Volume fraction ϕ liquid phase, 1 − ϕ of gas phase.

Incompressible liquid and gas: densities ρ 0 and ρ g .

Stabilizing demixing between phases: air goes up

∂ t (ρu)+∇· (ρuu) =−∇p + ρg

(21a)

∇· u + K (ρ 0 − ρ g ) 2

ρ 0 ρ g

∂ z

(

(1 − ϕ)ϕ

)

=0 (21b)

∂ t ϕ + ∇· (ϕu) − K(1 − ρ g /ρ 0 )∂ z

(

ϕ(1 − ϕ)

)

=0

ρ = ρ 0 ϕ + ρ 0g (1 − ϕ).

(21c)

(21d)

6. Conclusions

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Designed Bore-Soliton-Splash based on maths.

It yields highest Rogue Waves: AI = 10. Exact solutions?

Designed Hele-Shaw beach dynamics based on maths.

It works and is unique: breaking waves & dune formation.

It is innovative: great quasi-2D modeling environment.

Need multi-phase models made numerically compatible

with conservative limit of PDE’s.

References

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

My www: presentation, movies & eprints.

B., Van der Horn, Van der Meer, Zweers & Thornton

2012: Breaking waves on a dynamic Hele-Shaw Beach.

Paper for Shallow Flows Symposium, 10 pp.

B., Gagarina, Zweers, Thornton 2011:

Bore-Soliton-Splash: van Spektakel to Oceaangolf.

Nederlands Tijdschrift v. Natuurkunde url .

B., Zwart & Havemans 2010: Fluid Fascinations. Photo

Severn Bore by DHP; painting by zw-artprojects url :

Appendix: 2D Confirmation 1D Hydraulic Analysis

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

Some people did not believe 1D theory. Bad luck.

F 0

3.5

3

2.5

2

1.5

1

0.4 0.5 0.6 0.7 0.8 0.9 1

B c

1D theory (thin solid); 2D theory: paddle lengths L =0.305m

(thicker) and L =0.465m (thickest); and, simulations(◦/□).

Wave-Maker

Stormy Maths

Onno

Bokhove

Introduction

The Results

Wave-maker

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

GFD Mixture Theory: Air & Water

Stormy Maths

Onno

Bokhove

Introduction

The Results

Maths of

Wave Design

Maths-geared

Lab

Experiments

Multiphase

GFD Models

Conclusions

What happens in limit ρ g → 0?

∂ t (ρu)+∇· (ρuu) =−∇p + ρg (22a)

∇· (ϕu l + (1 − ϕ)u g ) =0 (22b)

∂ t ϕ + ∇· (ϕu l ) =0

(22c)

ρ = ρ 0 ϕ + ρ g (1 − ϕ)

(22d)

w l = w − g c (1 − ρ g /ρ 0 )(1 − ϕ)

(22e)

w g = w + g c (ρ 0/ρ g − 1)ϕ.

(22f)