From the surface to the sea bed:
Recent advances in computational
coastal dynamics
David R. Fuhrman, PhD
Associate Professor
Department of Mechanical Engineering
Technical University of Denmark
Inspire ME seminar, Boise State University, October 9, 2015
Lower Red River Meadow, Idaho
UNESCO-IHE Delft, The Netherlands
Technical University of Denmark
Wave-induced turbulence near a
change in bottom roughness
Fuhrman, D.R., Sumer, B.M. & Fredsøe, J. 2011 Roughness-induced streaming in turbulent
wave boundary layers. J. Geophys. Res. 116, article no. C10002.
Sediment transport around a
monopile wind turbine foundation
Crescent waves
Fuhrman, D.R., Madsen, P.A. & Bingham 2004 A numerical study of crescent waves.
J. Fluid Mech. 513, 309—341.
Photograph from
Su (1982).
Wave modelling
Numerical model
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Boussinesq-type formulation of Madsen,
Fuhrman & Wang (2006)
Finite difference solutions of Fuhrman &
Bingham (2004)
Capable of accurately simulating:
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Free surfaces for roughly kh<30 (h/L<5)
Velocity kinematics for roughly kh<12 (h/L<2)
Note that “shallow water” corresponds to kh<0.3
(h/L<0.05) and “deep water” to kh>3 (h/L>0.5)
Short-crested wave instabilities
Fuhrman, D.R., Madsen, P.A. & Bingham, H.B. 2006 Numerical simulation of
lowest-order short-crested wave instabilities. J. Fluid Mech. 563, 415—441.
Class Ia:
Class Ib:
”Freak” or ”Rogue” wave
Focusing of a JONSWAP spectrum
(Tp = 13.5 s, s = 7,  = 1.7)
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After Bateman, Swan &
Taylor (2001)
Nonlinear
Linear
Extreme waves
Fuhrman, D.R. & Madsen, P.A. 2006 Numerical simulation of extreme events from
focussed directionally spread wavefields. In: Proc. 30th Int. Conf. Coast. Eng., San Diego.
Following: Johannessen & Swan (2001)
Bateman, Swan & Taylor (2001, 2003)
Tsunami wave run-up on a conical island
Fuhrman, D.R. & Madsen, P.A. 2008 Simulation of nonlinear wave run-up with a high-order
Boussinesq model. Coast. Eng. 55, 139—154.
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Experiments of
Briggs et al.
(1995)
s = 1/4
H/h = 0.18
Impulsive bed upthrust (simulation based
on experiments from the 1970s)
Fuhrman, D.R. & Madsen, P.A. 2009 Tsunami generation, propagation, and run-up with a
high-order Boussinesq model. Coast. Eng. 56, 747—758.
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Depth: h=1 m
Upthrust a=0.1 m
Width b=12.2 m
c  gh
Solitary wave evolution
(after long evolution time)
Note that the ”nonlinearity”
a/h=0.1 far exceeds that for
tsunamis in the open ocean
where a/h=O(0.001)!
Solitary waves propagate
with constant form in
constant depth.
Formation requires a
balance between
nonlinearity and dispersion!
t
g
 2750
h
Background
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A positive initial disturbance will eventually
result in leading solitary waves
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Shown theoretically by Hammack and Segur (1970s)
Frequently used as justification for use of solitary waves as
model tsunamis
This ”solitary wave paradigm” has dominated experimental
and mathematical research on tsunamis since the 1970s
Link to geophysical scales never established!
Questions:
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Will this really occur at geophysical scales?
Even if it does, is a solitary wave a good model for the
”bulk” tsunami?
Tsunami life cycle (prior to run-up)
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Propagation across deep ocean (h≈4000 m)
”Shoaling” up the continental rise/slope
Propagation across continental shelf
(Nonlinearity
a/h0=0.0005)
c  450 mph
b=200 km
a=2 m
h0=4000 m
Tsunami propagation and shoaling
Tsunami life cycle (prior to run-up)
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Propagation across deep ocean (h≈4000 m)
”Shoaling” up the continental rise/slope
Propagation across continental shelf
(Nonlinearity
a/h0=0.0005)
c  450 mph
b=200 km
a=2 m
h0=4000 m
Propagation across deep ocean
(depth h0=4000 m)
Madsen, P.A., Fuhrman, D.R. & Schaffer 2008 On the solitary wave paradigm for tsunamis.
J. Geophys. Res. 113, article no. C12012.
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Allowed to propagate for 20,000 km (i.e. across
the entire Pacific ocean)
No solitary wave formation!
Results are dispersive but linear
Tsunami life cycle (prior to run-up)
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Propagation across deep ocean (h≈4000 m)
”Shoaling” up the continental rise/slope
Propagation across continental shelf
(Nonlinearity
a/h0=0.0005)
c  450 mph
b=200 km
a=2 m
h0=4000 m
Shoaling tsunami to continental shelf
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1/200 slope
No sign of
disintegration into
shorter waves
Nonlinear effects are
important, but
dispersion is not
h  14 m
h  40 m
h  4000 m
h  400 m
Tsunami life cycle (prior to run-up)
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Propagation across deep ocean (h≈4000 m)
”Shoaling” up the continental rise/slope
Propagation across continental shelf
(Nonlinearity
a/h0=0.0005)
c  450 mph
b=200 km
a=2 m
h0=4000 m
Tsunami propagation across shelf
(h=20 m)
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Due to nonlinearity, the
front face of the wave
steepens
At this point dispersion
becomes important
Undular bore develops
(will eventually shed
solitary waves)
These do not represent
the bulk tsunami!
Tsunami propagation across shelf
(h=20 m)
3D perspective:
2D perspective:
Visual comparison with the 2004
Indian Ocean tsunami
3D perspective:
The 2004 Indian Ocean tsunami:
The Sumatra 2004 tsunami reaching the island Koh Jum off
the coast of Thailand.
Conclusions
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Use of solitary waves to
represent tsunamis must be
done with great care
This can result in durational
errors that are several orders
of magnitude
We must not be afraid to
challenge traditional or
widely accepted practices!
From: Schimmels et al. (2014) ”On the
generation of a tsunami in a large scale
wave flume” ICCE 2014.
Sediment transport
”Vortex ripples”
Liseleje, Denmark
”Fully-coupled” model
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OpenFOAM ”sediMorph” model
Hydrodynamics: Reynolds-averaged NavierStokes (RANS) equations
Turbulence: Two-equation k- turbulence closure
(Wilcox 2006)
Sediment transport model
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Bedload transport: Engelund-Fredsøe method
Suspended sediment: Turbulent-diffusion equation
Bed morphology: Based on sediment continuity
Simulated flow and scour around a
monopile foundation
Pure-wave scour (KC = 30)
Fuhrman, D.R., Baykal, C., Sumer, B.M., Jacobsen, N.G. & Fredsøe, J. 2014 Numerical simulation
of scour and backfilling processes beneath submarine pipelines. Coast. Eng. 94, 10—22.
S eq
D
 0.1 KC
KC 
U wTw
D
Combined wave-plus-current
scour beneath pipelines
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Practical ”engineering” models of scour
require simple expressions for:
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Equilibrium scour depth
Time scale
Both are established for pure waves and pure
currents
No expression exists for the time scale in
combined wave-plus-current environments!
Computational domain
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Model domain 40D x 10D
Flow introduced at left-hand boundary
Validation: pure-current scour
Combined wave-current scour
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Approx. 80 runs
Keulegan-Carpenter number:
5<KC<50
KC 
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Current-strength parameter:
0<m<1
U
m
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U wTw
D
c
Uc Uw
Shields parameter:

b
g s  1d
Computed
equilibrium scour
depth
(combined waves
and currents)
Background
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For both pure current and pure waves, the
time scale is given by (Fredsøe et al. 1992):
T 
*
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g s  1d 3
1 5 3
T

 max
2
50
D
The scour versus time is assumed to
approximately follow:

 t 
S t   S eq 1  exp   
 T 

Preliminary comparison
Preliminary conclusions
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For a given max. Shields parameter, the
scour time scale is increased for combined
waves and currents
The time scale seems to follow the same
scaling with respect to the Shields parameter
The lead coefficient is clearly not constant for
combined waves and currents!
We therefore propose the generalization:
T* 
1 5 3
 max
50
5 3
 T *  m  cw
Final results
m  
T  m 
*

2
2
1
 0.04 e 170m 0.5   e  20m 0.5 
50
5 3
cw

T*
5 3
  cw
m 

Conclusions
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Through combined simulation and physical
insight, we have unified the time scale of
scour for general wave-plus-current flows!
Matches previous research at both pure wave
and current limits
Simple expression, appropriate for
engineering use
From the surface to the sea bed:
Recent advances in computational
coastal dynamics
David R. Fuhrman, PhD
Associate Professor
Department of Mechanical Engineering
Technical University of Denmark
Inspire ME seminar, Boise State University, October 7, 2015