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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering – Ghent University
Peter TROCHGhent University - Belgium
Wave generationusing the linear wavemaker theory
the Biésel Transfer Function
Lecture notes from
Dept. of Civil Engineering – Faculty of Engineering
overview of presentation
• used material for lecture notes
• introduction‣ a typical wave flume test set-up
‣ importance and development of wavemakers
• simplified theory for plane wavemakers in shallow water
• complete wavemaker theory for plane wavemakers‣ the boundary value problem with linearized boundary conditions
‣ the Biésel transfer function‣ performance graph of a wavemaker‣ preparation of input signal
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
used material for lecture notes
• Lecture notes Ph.D. course “Experimental and numerical wave generation and analysis”, Dr. Peter Frigaard, Hydraulics and CoastalEngineering Laboratory, Aalborg University, Denmark
• Dean R.G., Dalrymple R.A., 1991. Water wave mechanics for engineers and scientists. Advanced series on ocean engineering - Vol. 2. World Scientific Publishing Co., Singapore. ISBN 981-02-0421-3
• Hughes S.A., 1993. Physical models and laboratory techniques in coastal engineering. Advanced series on ocean engineering - Vol. 7. World Scientific Publishing Co., Singapore. ISBN 981-02-1541-X
Dept. of Civil Engineering – Faculty of Engineering
a typical wave flume test set-up
wave flumespending beach foreshore
wave paddlebreakwater model wave gauges
active wave absorption
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
wavemakers in 2D wave flumes and 3D tanks
2D wave flume with piston-type wave paddle for wave generation
3D wave basin with multi-segmented wavemaker for wave generation
Dept. of Civil Engineering – Faculty of Engineering
importance of wavemakers ?
• use of physical models in coastal engineering is based on the capability to create waves in small scale models
• those waves exhibit many of the characteristics of waves in nature
• waves in nature are generated by wind
• waves in the physical wave flume are generally not generated using wind, but using mechanical wave generation where a movable wave paddle (a “wavemaker”) is placed in the flume
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
development of wavemakers ?
• the earliest wavemakers generated uniform waves by moving the wave paddle in a sinusoidal motion with a given amplitude e and period T
‣ a very simplified approximation of waves in nature
‣ reasonable agreement to linear wave theory
‣ pioneering research using limited capabilities but making great strides in coastal engineering
Dept. of Civil Engineering – Faculty of Engineering
development of wavemakers ?
• development of technology (hydraulic servo-systems, computer, …) provided more control over the wave paddle motion resulting in “better” waves
‣ irregular waves in the flume‣ non-linear waves (Stokes, cnoidal, solitary waves) in the flume
‣ directional irregular waves in wave basin using multi-segmented wavemaker‣ 2D and 3D active wave absorption
‣ hybrid modelling: coupling between fysical and numerical flumes‣ etc…
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
pioneering paper
• first description by Biésel and Suquet (1951) in a series of French papers in “La Houille Blanche”, entitled “les appareils generateurs de houles en laboratoire” of:
‣ analytical solution of the theoretical problem – first order wavemaker theory
‣ for piston-type and flap-type wavemakers
‣ practical aspects
• and considered as the basis for today’s wave generation technology in hydraulic laboratories
Dept. of Civil Engineering – Faculty of Engineering
simplified theory for plane wavemakers in shallow water
• proposed by Galvin (1964)
• piston wavemaker with stroke which is constant over water depth
• shallow water region
0 2S e=h
10kh π≤
from: Dean & Dalrymple, Water Wave Mechanics for Engineers and ScientistsAdvanced Series on Ocean Engineering, Vol. 2, World Scientific
0
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
simplified theory for plane wavemakers in shallow water
• assumption:volume displaced
by wavemakerover stroke
crest volume ofpropagatingwave form
=
( )/ 2
0 2
0
sinL
HS h kx dx= ∫
0S
0
Dept. of Civil Engineering – Faculty of Engineering
simplified theory for plane wavemakers in shallow water
• calculations:
• defining as the height-to-stroke ratio, we get
• and for flap type wavemaker, water volume displaced is
( ) ( )/ 2 / 2
0 2 2
0 0
sin sin ( ) 22
L L
H Hk
H HS h kx dx kx d kx
k k= = = =∫ ∫
0fK H S=
0
pistonf
HK kh
S= =
12
0
flapf
HK kh
S= =
102 S h
0S
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
simplified theory for plane wavemakers in shallow water
0fK H S=
from: Dean & Dalrymple, Water Wave Mechanics for Engineers and ScientistsAdvanced Series on Ocean Engineering, Vol. 2, World Scientific
kh
Dept. of Civil Engineering – Faculty of Engineering
complete wavemaker theory for planewavemakers
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
complete wavemaker theory for planewavemakers
• assuming‣ inviscid incompressible fluid‣ irrotational flow field
• a velocity potential exists and the velocity field is
• governing equations for potential flow are continuity equation and momentum equation
• boundary value problem similar to linear wave theory
0ν =0v∇ × =
�
constρ =
( , , )x z tϕ
v gradϕ ϕ= ∇ =�
Dept. of Civil Engineering – Faculty of Engineering
complete wavemaker theory for planewavemakers
• the continuity equation for incompressible flow
• combined with the definition yields the well-known
Laplace equation:
• the Laplace equation is a linear partial differential equation(PDE) in and is solved analytically for a specified set of linearized boundary conditions (BC) including paddlemovement
v ϕ= ∇�
0v∇⋅ =�
( ) 0ϕ∇⋅ ∇ = 2 0ϕ∇ =2 2
2 20
x z
ϕ ϕ∂ ∂+ =∂ ∂
( , , )x z tϕ
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
complete wavemaker theory for planewavemakers
Dept. of Civil Engineering – Faculty of Engineering
complete wavemaker theory for planewavemakers
• details of solution procedure e.g. in Dean & Dalrymple, or in Hughes
•
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
complete wavemaker theory for planewavemakers
one progressive wave with k0
series of standing waves with k1, k2, …
Dept. of Civil Engineering – Faculty of Engineering
the Biésel transfer function
far field solution• generated progressive wave
• wave amplitude doesn’t changewith location
• phase shift π / 2 relative topaddle displacement e given by
near field solution• series of standing waves
• exponential decay of wave amplitude with distance
• “disturbance” exists only near wave paddle for x < 2L
• take into account using active wave absorption at paddle
• incorporates difference between velocity profile generated by paddle and actual waves
nk xe− ⋅
( )( )( , ) sin
2S z
e z t tω=
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
the Biésel transfer function
Dept. of Civil Engineering – Faculty of Engineering
the Biésel transfer function
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
the Biésel transfer function
0fK H S=
from: Dean & Dalrymple, Water Wave Mechanics for Engineers and ScientistsAdvanced Series on Ocean Engineering, Vol. 2, World Scientific
kh
Dept. of Civil Engineering – Faculty of Engineering
the Biésel transfer function
• observations‣ piston-type wavemaker
• gradually increases from 0 to constant factor 2 for increasing frequencies
• increase is slower for smaller water depths, so decreasing wave generating capability for decreasing water depths• factor 2 is asymptotic, usually in increasing part (0.5 – 1 Hz)
• in shallow water, linear approximation by Galvin• for Kf � 0, paddle amplitude e � infinity: long wave compensation problematic
• for shallow water waves
0fK H S=
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
the Biésel transfer function
• observations‣ flap-type wavemaker
• more difficult to build and operate due to hinges
• less “value for money” (smaller wave heights for same paddle displacement)• even more difficult for long wave compensation• for deeper water waves
Dept. of Civil Engineering – Faculty of Engineering
performance graph of a wavemaker
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Wave generation using linear wavemaker theoryDr. Peter TROCH – Ghent University
Dept. of Civil Engineering – Faculty of Engineering
performance graph of a wavemaker
Dept. of Civil Engineering – Faculty of Engineering
preparation of input signal