SURFZONE WAVES II (PHASERESOLVING)

Surf-Zone Wave Models II Phase-Resolving Models Waves in Shallow Water Airy 1845: 0     conservation of mass momentum equation... P   Mass conservation: Momentum conservation: Ma

Surf-Zone Wave Models II

Phase-Resolving Models

Waves in Shallow Water

Airy (1845):

0

 

 

conservation of mass

momentum equation

 

3

2

0 3

 

 

original, 1D form of Boussinesq equations

(linear variation in vertical velocity,

pressure not hydrostatic)

Scaling Parameters and Limits

(long-wave theory)

h h L



 

 

Non-linear effects

Dispersive effects

Non-dimensional equations:

2

'

(1 ') ' 0

u

u

 

Different forms of the Boussinesq Equations

• integration procedures

• dependent variables

Causes for differences:

Abbott (1979):

3

3

3

2

3

I

II

III

Peregrine (1967):

3

(different integration procedure)

Other Velocity Variables

Nwogu (1993):

2

2

0

u





1

0

0

2

P

gh

 

 

     

P is the depth-averaged volume flux

(B is a linear dispersion coefficient)

Applications for the Boussinesq Equations

• wave-current interaction

• nearshore currents

• interaction waves/structures

Application of Boussinesq Equation to

Simulate Waves in the Surf Zone

P

 

Mass conservation:

Momentum conservation:

(Madsen et el 1991)

2

, 2

1 3 1

 

Momentum Loss due to Wave Breaking

,

ˆ

g Q Q





Watanabe and Dibajnia (1988):

0.4 0.57 5.3 tan

s

Wave-induced flow:

Flow in reformed waves:

3

0.135

r

D=2.5

Sato et al (1992):

2

D b b

Q

M

x



 



2

ˆ

tan

b

 

1/ 2 tan ( ) tan (tan tan ) exp ln 2 B

t

t

: angle defining roller (b indicates breaking

and o terminal conditions)

Evolution of roller (geometry):

D b

P



       

      

Momentum loss:

Date Profile Type H mo (m) T p(s) S 0 S br  0  br  0(10 -3

)

Sept 5, 1985, 09.55 Shelf 0.61 11.4 0.81 1.12 0.16 0.70 1.68

Sept 5, 1985, 10.15 Shelf 0.42 13.1 0.93 1.29 0.11 0.66 1.27

Sept 5, 1985, 13.52 Shelf 0.64 10.9 0.77 1.00 0.17 0.68 1.88

Sept 5, 1985, 15.25 Shelf 0.53 11.1 0.83 1.13 0.12 0.68 1.61

Sept 15, 1986,14.45 Shelf 1.07 10.1 0.62 0.72 0.15 0.48 2.84

Sept 19,1986, 11.00 Bar 0.70 11.2 0.79 1.00 0.10 0.47 2.5

Model Validation: Duck Data

(Hrms) (H1/3) (H1/10)

Date Roller Mod W-D Mod Roller Mod W-D Mod Roller Mod W-D Mod

Sept 5, 1985, 09.55 19.0 25.0 18.0 23.0 18.0 24.0

Sept 5, 1985, 10.15 16.8 27.4 27.0 21.0 30.0 23.0

Sept 5, 1985, 13.52 26.0 28.0 27.0 28.0 24.0 28.0

Sept 5, 1985, 15.25 17.0 23.0 18.0 19.0 12.0 13.0

Sept.15, 1986, 14.45 12.5 15.0 13.0 16.0 16.7 22.3

Sept.19, 1986, 11.00 8.8 13.7 6.3 15.1 13.8 19.4

Duck85 Beach Profile Shape

shelf-type profile

SuperDuck Beach Profile Shape

1985-09-05 09:55

Calculated and Measured Water Surface Elevation

1986-09-19 11:00

Measured and Computed Skewness and Kurtosis

Shelf

profile