Biesel Transfer Function

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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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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


a typical wave flume test set-up

wave flumespending beach foreshore

wave paddlebreakwater model wave gauges

active wave absorption

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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


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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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


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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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


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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• assumption:volume displaced

by wavemakerover stroke

crest volume ofpropagatingwave form

=

( )/ 2

0 2

0

sinL

HS h kx dx= ∫

0S

0



• 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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0fK H S=


kh


complete wavemaker theory for planewavemakers

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• 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= ∇ =�



• 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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• details of solution procedure e.g. in Dean & Dalrymple, or in Hughes

•

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one progressive wave with k0

series of standing waves with k1, k2, …


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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12




0fK H S=


kh



• 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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• 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


performance graph of a wavemaker

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performance graph of a wavemaker


preparation of input signal

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Biesel Transfer Function