Modelling wave damping by fluid mud

1
Modelling wave damping by fluid mud

• Wouter Kranenburg, March 2008, NCK-days

2
Modelling wave damping by fluid mud
Presentation on the MSc Thesis

• Wouter Kranenburg
• NCK-days, March 2008

3
Project objective
The development and testing of an adaptation to
the wave model SWAN
with which it is possible
to model the decrease of energy during the
propagation of a wave field over fluid mud.
INTRODUCTION objective - problem illustration -
outline presentation - PART I WAVE
NUMBER - PART II ENERGY -
CONCLUSION
4
Illustration of the problem
damped waves
breaking waves
(Picture by G.J. de Boer)
INTRODUCTION objective - problem illustration -
outline presentation - PART I WAVE
NUMBER - PART II ENERGY -
CONCLUSION
5
Fluid mud

• (I) when tb gt tyield ? mud structure is
  disrupted
• (I) mud becomes fluid and starts to wave
• (II) energy is transferred to viscous mud layer
  and dissipated

INTRODUCTION objective - problem illustration -
outline presentation - PART I WAVE
NUMBER - PART II ENERGY -
CONCLUSION
6
Outline of the presentation
Part I wave number a single wave 2-layer
models dispersion relations calculation and
comparison Part II energy energy
dissipation energy propagation validation
? SWAN
INTRODUCTION objective - problem illustration -
outline presentation - PART I WAVE
NUMBER - PART II ENERGY -
CONCLUSION
7
Linear wave theory
Assumptions / schematization
Equations linearized B.C.
u,w,p,E Dispersion Relation
?2gk tanh(kHw0)
Assumptions / schematization
Equations linearized B.C.
u,w,p,E Dispersion Relation
4 solutions
viscosity
complex
?a exp(-kix) exp(i(krx-?t))
PART I WAVE NUMBER LinearWaveTheory
schematization - dispersion relation - conclusion

8
Schematization
Non-hydrostatic non-viscous
(Schematization of De Wit, 1995)
Sketch of two-layer fluid mud system
Quasi-hydrostatic viscous
PART I WAVE NUMBER LinearWaveTheory
schematization - dispersion relation - conclusion

9
Dispersion relation (DELFT)

• valid for all relative water depths kHw0, small
  relative mud depth kHm0
• computation with Starting Value and Iteration
  procedure

PART I WAVE NUMBER LinearWaveTheory
schematization - dispersion relation - conclusion

10
Dispersion relation (comparison)
imaginairy wave number
ki
knm
normalized mud layer thickness
DELFT dispersion relation
Hm0
v(2 ?m/?)
low kH
high kH
PART I WAVE NUMBER LinearWaveTheory
schematization - dispersion relation - conclusion

11
Dispersion relation (behaviour)
ki
kguo
Hm0
v(2 ?m/?)
PART I WAVE NUMBER LinearWaveTheory
schematization - dispersion relation - conclusion

12
Conclusions Part I wave number
A function and procedure are available to
calculate the complex wave number k in the domain
of interest
The DELFT dispersion relation shows plausible
behaviour with regard to - peak dissipation
- location of peak - limits
PART I WAVE NUMBER LinearWaveTheory
schematization - dispersion relation - conclusion

13
Introduction to part II Energy
Part II energy energy dissipation energy
propagation validation
? SWAN
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
14
The SWAN-model
SWAN Simulating Waves Nearshore
picture from L.H. Holthuijssen (2007)
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
15
Implement effect mud on dissipation
(simplification of) Main equation in SWAN
Phase averaged energy balance
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
16
Implement effect mud on dissipation
pressure on interface
displacement of interface
work on interface
period averaged work
energy dissipation Sb,mud
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
17
Implement effect mud on dissipation
p ? from DELFT dispersion equation
pressure on interface
displacement of interface
work on interface
P1(Hw0) pressure amplitude
period averaged work
b/a Amplitude ratio
f Phase difference
energy dissipation Sb,mud
? Wave frequency
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
18
Energy dissipation term
old expr
Sum
kHw0
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
19
Validation SWAN with mud dissipation
TEST 1 one-directional, mono-chromatic waves
over a flat bottom with a mud layer of constant
thickness
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
20
Validation SWAN with mud dissipation
consistent model
kHw0 1
inconsistent model
Hs
Distance x m
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
21
Validating SWAN with mud dissipation
TEST 2 a spectrum of waves over a flat bottom
with a mud layer of constant thickness
Questions Where in the spectrum is the damping
the strongest? Is this correctly represented in
the shape of the spectrum / mean frequency?
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
22
Validating SWAN with mud dissipation
Hs m
f Hz
Distance x m
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
23
Validating SWAN with mud dissipation
Hs m
f Hz
Distance x m
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
24
Implement effect mud on propagation
regular DR
k
with energy propagation velocity cg
propagation
shoaling and refraction
shoaling
refraction
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
25
Validating SWAN-mud for propagation
shoaling
1D-shallow water test case
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
26
Validating SWAN-mud for propagation
profile
Im(k)
Re(k)
___ with SWAN-mud analytical sol. for GADE
Hs
(dissipation is turned off)
Distance
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
27
Validating SWAN-mud for propagation
shoaling
shoaling and refraction
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
28
Validating SWAN-mud for propagation
profile
Im(k)
Re(k)
__ with SWAN-mud
Dir
Hs
(dissipation is turned off)
Distance
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
29
Conclusion
Project objective
The development and testing of an adaptation to
the wave model SWAN
to model the decrease of energy during the
propagation of a wave field over fluid mud.
INTRODUCTION - PART I WAVE NUMBER -
PART II ENERGY - CONCLUSION objective
- conclusions - recommendations
30
Conclusion
Part I Wave Number - Derivation of DELFT
dispersion equation - Design of a solving routine
to compute k - Comparison DELFT with other
disp.eq.
Part II Energy - Derivation of Energy
Dissipation Term - SWAN-mud (I) Dissipation (
tests) - SWAN-mud (II) Propagation (tests)
INTRODUCTION - PART I WAVE NUMBER -
PART II ENERGY - CONCLUSION objective
- conclusions - recommendations
31
Conclusion
Recommendations - Improve solving procedure -
Calibrate the model on practical case
Ready for use!
INTRODUCTION - PART I WAVE NUMBER -
PART II ENERGY - CONCLUSION objective
- conclusions - recommendations
32
Conclusion
Project objective
dispersion equation
The development and testing of an adaptation to
the wave model SWAN
complex wave number k
to model the decrease of energy during the
propagation of a wave field over fluid mud.
dissipation
propagation
tests
tests
Calibration on practical case
INTRODUCTION - PART I WAVE NUMBER -
PART II ENERGY - CONCLUSION objective
- conclusions - recommendations
33
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34
Extra sheets
35
Modelling wave damping by fluid mud
Presentation on the MSc project

• Wouter Kranenburg
• February 2008

36
Modelling wave damping by fluid mud
Presentation on the MSc project
graduation committee prof.dr.ir. G.S.
Stelling dr.ir. J.C. Winterwerp ir. G.J. de
Boer dr.ir. M. Zijlema ir. J.M. Cornellisse dr.
A. Metrikine
Wouter Kranenburg, student Delft University of
Technology Faculty of Civil Engineering and
GeoSciences Section of Environmental Fluid
Mechanics Thesis prepared at WL delft
hydraulics UFRJ March 2007 January 2008
37
Project objective
The development and testing of an adaptation to
the wave model SWAN
with which it is possible
to model the decrease of energy during the
propagation of a wave field over fluid mud.
INTRODUCTION objective - problem illustration -
outline presentation - PART I WAVE
NUMBER - PART II ENERGY -
CONCLUSION
38
Illustration of the problem
damped waves
breaking waves
(Picture by G.J. de Boer)
INTRODUCTION objective - problem illustration -
outline presentation - PART I WAVE
NUMBER - PART II ENERGY -
CONCLUSION
39
Restriction of the project
rheological model
wave model
hydraulical model
liquefaction
damping
flow transport
INTRODUCTION objective - problem illustration -
outline presentation - PART I WAVE
NUMBER - PART II ENERGY -
CONCLUSION
40
Outline of the presentation
Part I wave number a single wave 2-layer
models dispersion relations calculation and
comparison Part II energy energy
dissipation energy propagation validation
? SWAN
INTRODUCTION objective - problem illustration -
outline presentation - PART I WAVE
NUMBER - PART II ENERGY -
CONCLUSION
41
Fluid mud

• (I) when tb gt tyield ? mud structure is
  disrupted
• (I) mud becomes fluid and starts to wave
• (II) energy is transferred to viscous mud layer
  and dissipated

PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
42
Schematization
Sketch of two-layer fluid mud system
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
43
Intermezzo linear wave theory 1 layer
mono-chromatic uni-directional periodic free
surface gravity wave
simple wave
Sketch of one-layer system
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
44
Intermezzo linear wave theory 1 layer
mono-chromatic uni-directional periodic free
surface gravity wave
simple wave
Sketch of one-layer system
Assumptions / schematization
Equations linearized B.C.
u,w,p,E DR
velocity potential
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
45
Intermezzo linear wave theory 1 layer
mono-chromatic uni-directional periodic free
surface gravity wave
simple wave
Sketch of one-layer system
Assumptions / schematization
Equations linearized B.C.
u,w,p,E DR
velocity potential
deep c g/?
Dispersion relation
?2gk tanh(kHw0)
shallow c2 gHw0
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
46
Intermezzo linear wave theory 2 layers
Assumptions / schematization
Equations linearized B.C.
u,w,p,E
Dispersion relation
4 solutions
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
47
Intermezzo linear wave theory 2 layers
Assumptions / schematization
Equations linearized B.C.
u,w,p,E
Dispersion relation
4 solutions
Internal wave, figure from C.Kranenburg
External wave, figure from C.Kranenburg
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
48
Intermezzo linear wave theory 2 layers
Assumptions / schematization
Equations linearized B.C.
u,w,p,E
Dispersion relation
4 solutions
viscosity ? complex k
?a exp(-kix) exp(i(krx-?t))
Internal wave, figure from C.Kranenburg
External wave, figure from C.Kranenburg
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
49
Schematization
Schematizations - assumptions -
equations Dispersion equations - domain -
complex k - implicit / explicit
Sketch of two-layer fluid mud system
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
50
Schematization
Non-hydrostatic non-viscous
(Schematization of De Wit, 1995)
Sketch of two-layer fluid mud system
Quasi-hydrostatic viscous
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
51
Dispersion relation (DELFT)
valid for all relative water depths kHw0, small
relative mud depth kHm0
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
52
Dispersion relation (DELFT)
f(? , Hw0 , Hm0 , ?1 , ?2 , ?m , g , k)0
implicit k-? relation? iteration
? , Hw0 , Hm0 , ?1 , ?2 , ?m , g ? k
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
53
Dispersion relation (DELFT)
f(? , Hw0 , Hm0 , ?1 , ?2 , ?m , g , k)0
implicit k-? relation? iteration
Iteration procedure
Starting Value for k
k-value
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
54
Comparison dispersion relations (1)
Table Overview Dispersion Equations
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
55
Comparison dispersion relations (1)
Table Overview Dispersion Equations
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
56
Comparison dispersion relations (1)
Table Overview Dispersion Equations
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
57
Comparison dispersion relations (1)
Table Overview Dispersion Equations
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
58
Comparison dispersion relations (2)
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
59
Comparison dispersion relations (2)
kre/im / kGuo normalized real imaginary wave
number
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
60
Comparison dispersion relations (2)
v(?/2/?m) Hm0 normalized mud layer thickness
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
61
Comparison dispersion relations (2)
kH-value indication of relative water depth
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
62
Comparison dispersion relations (2)
GADE-analytical
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
63
Comparison dispersion relations (2)
DELFT
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
64
Comparison dispersion relations (2)
Dalrymple Liu
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
65
Comparison dispersion relations (2)
Iteration errors
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
66
Solving the wave number (1)

• Problems
• - solution not explicit ? iteration
• - sometimes wrong solution, sometimes no solution
  at all
• - normalization (to much parameters to get
  insight function)
• Investigation
• - starting values
• - dimensionless parameters

PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
67
Solving the wave number (2)
non-viscous shallow water layer
viscous mud layer
Assemblage of Starting Value
High kHw-value deep water limit Starting
Value regular dispersion relation
Low kHw shallow water limit Starting
value iteration GADE-analytical
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
68
Comparison dispersion relations (3)
Proper solution DELFT
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
69
Comparison dispersion relations (3)
note peak!
Proper solution DELFT
running ?, ?m, Hm0
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
70
Comparison dispersion relations (4)
ki
kguo
Hm0
v(2 ?m/?)
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
71
Conclusions Part I wave number

• Combined SV (Gade Guo) functions over entire
  domain
• DELFT shows plausible behaviour
• with regard to peak dissipation, location of
  peak, limits

So now a function and procedure are available to
calculate the wave number k
PART I WAVE NUMBER fluid mud - schematization
- LinearWaveTheory - dispersion relation -
comparison - solution - comparison (3) -
conclusion
72
Introduction to part II Energy
Part II energy energy dissipation energy
propagation validation
? SWAN
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
73
The SWAN-model
SWAN Simulating Waves Nearshore
Phase averaged energy balance
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
74
The SWAN-model
pictures from L.H. Holthuijssen (2007)
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
75
The SWAN-model
pictures from L.H. Holthuijssen (2007)
(simplification of) Main equation in SWAN
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
76
Implement effect mud on dissipation
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
77
Implement effect mud on dissipation
pressure on interface
displacement of interface
work on interface
period averaged work
energy dissipation Sb,mud
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
78
Implement effect mud on dissipation
pressure on interface
displacement of interface
work on interface
period averaged work
energy dissipation Sb,mud
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
79
Implement effect mud on dissipation
p ? from DELFT dispersion equation
pressure on interface
displacement of interface
work on interface
period averaged work
P1(Hw0) pressure amplitude
b/a Amplitude ratio
energy dissipation Sb,mud
f Phase difference
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
80
Energy dissipation term
kHw0
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
81
Energy dissipation term
kHw0
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
82
Energy dissipation term
kHw0
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
83
Comparison with literature
Extension of applicability of the model by
derivation of energy dissipation term that is
valid for the whole kHw0-range and consistent
with the applied dispersion relation.
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
84
Validation SWAN with mud dissipation
TEST 1 one-directional, mono-chromatic waves
over a flat bottom with a mud layer of constant
thickness
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
85
Validation SWAN with mud dissipation
consistent model
kHw0 1
inconsistent model
Hs
Distance x m
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
86
Validating SWAN with mud dissipation
TEST 2 a spectrum of waves over a flat bottom
with a mud layer of constant thickness
Questions Where in the spectrum is the damping
the strongest? Is this correctly represented in
the shape of the spectrum / mean frequency?
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
87
Validating SWAN with mud dissipation
Hs m
f Hz
Distance x m
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
88
Validating SWAN with mud dissipation
Hs m
f Hz
Distance x m
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
89
Validating SWAN with mud dissipation
Hs m
f Hz
Distance x m
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
90
Validating SWAN with mud dissipation
Hs m
f Hz
Distance x m
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
91
Implement effect mud on propagation
with energy propagation velocity cg
shoaling and refraction
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
92
Implement effect mud on propagation
with energy propagation velocity cg
shoaling and refraction
shoaling
refraction
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
93
Implement effect mud on propagation
k
cg ??/?k
propagation
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
94
Validating SWAN-mud for propagation
shoaling
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
95
Validating SWAN-mud for propagation
shoaling
1D-shallow water test case
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
96
Validating SWAN-mud for propagation
___ with SWAN-mud analytical sol. for GADE
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
97
Validating SWAN-mud for propagation
shoaling
shoaling and refraction
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
98
Validating SWAN-mud for propagation
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
99
Validating SWAN-mud for propagation
cg
general formula for c?
formula applied in SWAN
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation - tests
100
Validating SWAN-mud for propagation
general formula for c?
regular dispersion equation ?2 gk
tanh(kHw0)
formula applied in SWAN
PART II ENERGY introduction - SWAN -
dissipation - tests - propagation tests
101
Conclusion
Project objective
The development and testing of an adaptation to
the wave model SWAN
to model the decrease of energy during the
propagation of a wave field over fluid mud.
INTRODUCTION - PART I WAVE NUMBER -
PART II ENERGY - CONCLUSION objective
- conclusions - recommendations
102
Conclusion
Part I Wave Number - Study of various 2-layer
models - Derivation of DELFT dispersion
equation - Design of a solving routine to compute
k - Comparison DELFT with other disp.eq.
INTRODUCTION - PART I WAVE NUMBER -
PART II ENERGY - CONCLUSION objective
- conclusions - recommendations
103
Conclusion
Part II Energy - Study of previous
implementations - Derivation of Energy
Dissipation Term - SWAN-mud (I) Dissipation (
tests) - SWAN-mud (II) Propagation
(tests) shoaling (OK) refraction (not yet OK)
INTRODUCTION - PART I WAVE NUMBER -
PART II ENERGY - CONCLUSION objective
- conclusions - recommendations
104
Conclusion
Recommendations - Improve solving procedure -
Modify refraction formula (c?) - Calibrate the
model on practical case
INTRODUCTION - PART I WAVE NUMBER -
PART II ENERGY - CONCLUSION objective
- conclusions - recommendations
105
Conclusion
Project objective
dispersion equation
The development and testing of an adaptation to
the wave model SWAN
complex wave number k
to model the decrease of energy during the
propagation of a wave field over fluid mud.
dissipation
propagation
tests
tests
Vragen in het Nederlands ook van harte welkom!
Calibration on practical case
INTRODUCTION - PART I WAVE NUMBER -
PART II ENERGY - CONCLUSION objective
- conclusions - recommendations