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Simple absorbing layer conditions for shallow wave simulationswith Smoothed Particle Hydrodynamics
Diego Molteni a,n, Rosario Grammauta a, Enrico Vitanza b
a Dipartimento di Fisica, Viale delle Scienze, Universit !a di Palermo, Italyb Dipartimento di Ingegneria Civile, Ambientale e Aerospaziale, Universit !a di Palermo, Italy
a r t i c l e i n f o
Article history:Received 27 February 2012Accepted 23 December 2012
Keywords:Fluid mechanicsBoundary conditionAbsorbing layerLagrangian numerical methodSPHShallow water model
a b s t r a c t
We study and implement a simple method, based on the Perfectly Matched Layer approach, to treat nonreflecting boundary conditions with the Smoothed Particles Hydrodynamics numerical algorithm. Themethod is based on the concept of physical damping operating on a fictitious layer added to thecomputational domain. The method works for both 1D and 2D cases, but here we illustrate it in the caseof 1D and 2D time dependent shallow waves propagating in a finite domain.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
The problem of non reflecting boundary conditions is an oldsubject of the study of wave propagation in limited domains. Theso called radiation boundary conditions at infinity have beenstudied since 1912 by Sommerfeld, but its practical implementa-tion in computational solutions of electromagnetic field propaga-tion can be referred to Engquist and Majda (1977). It is obviousthat the occurrence of boundaries affects the evolution of aphysical event that would otherwise propagate into open space.Many different strategies have been adopted to circumvent theproblem. Among numerous approaches the method of character-istics is well exploited in the fixed grid numerical methods(Poinsot and Lele, 1992). The perfectly matched layer (PML)approach, i.e. the use of an artificial absorbing layer, was devisedby Berenger (1994) for simulations of electromagnetic waves andsuccessively adopted in many wave field simulations: acoustics,seismic vibrations and fluids. The general idea of the PMLapproach is very simple. An absorbing layer is added to thephysical domain. In this layer, sink or source terms are activated,multiplied by a coefficient varying from zero, inside the physicaldomain, to a maximum at the outer edge of the layer zone. Themathematical properties to be attributed to this zone can reachgreat accuracy and complexity, as shown in the paper by Lin et al.(2011) on recent advancements for non-linear regime of the Euler
equations to be adopted in the layer. Recently Modave et al.(2010) set up a simple and accurate PML method that is useful forlinear and non-linear shallow water simulations. We essentiallyadopt this simpler approach.
2. The absorbing layer method
In general, the model equations governing the fluid dynamicsare rewritten adding a sink or source term to the originalequations, as follows:
@A@t! f A,
@A@x
,x
! ""s#A"Aout$ #0:1$
where A is a generic fluid variable, "s(A"Aout) is the correspond-ing sink or source term, Aout is the external boundary value, s isthe damping coefficient different from zero only in the dampingregion. With an appropriate choice of the s spatial function thisprocedure produces extremely small reflection waves.
All these techniques are used for fixed grids discretization ofthe equations. In the Lagrangian approach the characteristiclines method has been suggested by Lastiwka et al. (2009) andVacondio et al. (2012) which uses a simplified version of thatprocedure. Instead the PML approach is by far simpler, but,as far as we know, it has not been studied in the context of aLagrangian approach. We adopted this strategy for the LagrangianSmoothed Particle Hydrodynamics scheme and tested it in thecase of waves propagating in a finite tank. We show that theresults are fairly good.
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journal homepage: www.elsevier.com/locate/oceaneng
Ocean Engineering
0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.oceaneng.2012.12.048
n Corresponding author. Tel.: %39 091 6615055; fax: %39 091 6615069.E-mail addresses: [email protected],
[email protected] (D. Molteni).
Ocean Engineering 62 (2013) 78–90
The Smoothed Hydrodynamics method (SPH) is a Lagrangianmesh-free method based on a single basic interpolating functionassociated with each node of the moving mesh.
Here we give the basic ideas. For an up to date detailedpresentation of the SPH method see Colagrossi and Landrini(2003). A function f is interpolated from its known values atpoints k, by the approximation of the Dirac function integral:
f #x$ !Z
f #y$d#x"x0$dy) ~f #x$ !X
k
f kW#x,x0k$Dx0k #0:2$
where W#x,x0k$ is the interpolating function, named kernel, cen-tered in the x0k point. This interpolating function has a scale factorh and must have the properties to mimic the Dirac function,therefore
Z %1
"1W#x=h$dx! 1 and lim
h-0W#x=h$ ! d#x$
and ~f #x$ is the approximated function. Exploiting the mass density r,we can attribute to each moving node a mass mk ! rkDx0k and
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0 200 400 600 800 1000
Vx
X
Initial conditionTotal reflection
Infinity
Fig. 1. Initial pulse profile chosen H0!1, xc!3/4X and A!9h solid line; the perfectly reflected pulse: dashed line and the infinity case: dotted line.
350 400 450 500 550 6000.998
1
1.002
1.004
1.006
1.008
1.01
1.012
1.014
X
Rho
Fig. 2. Time evolution of the wave height at intervals of 7.4 s; the bold solid line represents the initial configuration.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–90 79
therefore the approximated function is given by
~f #x$ !X
k
mkf k
rkW#x,x0k$ #0:3$
Consequently the space derivative can be approximated as
@f@x&Z@f #x0$@x0
W#x"x0$dx0 &X
k
mk
rk
f k@Wik
@xi#0:4$
Many details on the SPH approach can be also found in thereview of Monaghan (2005). We give here the final formulae.
The continuity equation is given by
drdt!"rr v
!)dri
dt!X
k
mk# v!
i" v!
k$UriW# r!
i, r!
k$ #0:5$
The momentum equation is
d v!
dt!"
1rrP)
d v!
i
dt!"
X
k
mkPi
r2i
%Pk
r2k
%Pik
!riW# r
!i, r!
k$
#0:6$
P is the pressure to be given by an equation of state specific tothe problem to be studied, that will be specified in the subsec-tions. Pik is an artificial viscosity term needed to stabilize theequations (cfr. Monaghan, 2005) and W# r
!i, r!$ is the interpolat-
ing function, named kernel, centered in the r!
i point. Thisinterpolating function has a scale factor h and must have theproperties to mimic the Dirac function, thereforeZ %1
"1W#r=h$dnx! 1 and lim
h-0W#r=h$ ! d#r$
The kernel used for our 2D simulation is the Wendland kernelfunction, where n is the spatial dimension
W#r,h$ !7
4phn
1" r2h
# $41%2 r
h
# $if r
h r2
0 if rh 40
(#0:7$
The integration in time is carried out by the predictor correctoralgorithm for low Mach number flows, accurate to second order inthe time step, proposed by Monaghan (2006). The time step islimited by the usual Courant condition.
3. Lagrangian formulation of shallow water equations
We focus our attention to the open boundaries problem for theshallow water waves in the SPH framework. The general gravitywave case has been accurately studied, but not in the context ofboundary problems, by Antuono et al. (2011). The governingequations, in conservative Eulerian form, for shallow waterwaves, derived under the usual approximations of wave elevationmuch smaller than the full water depth and for constant bottomelevation, are well known:
@H@t%div#H v
!$ ! 0 #0:8$
@H v!
@t%div H v
!' v!%g
H2
2I
!! 0 #0:9$
where H is the full height of the water level, g is the gravitationalacceleration, and v
!is the fluid velocity.
De Leffe et al. (2010) also derived an SPH formulation slightlydifferent from the one presented in this paper.
3.1. The 1D shallow water case
To outline important physical elements in the absorbing zonewe begin to analyze the simple 1D case. When written in theLagrangian form, we have for the wave height equation
dHdt!"H
@v@x
#0:10$
where dy/dt is the comoving derivative. The equation of motion is
dvdt!"g
@H@x
#0:11$
These equations can be formally satisfied by a fictitious fluidhaving a density r!H and an equation of state for the pressureP!(1/2)gr2, so that the shallow water equations are fulfilled bythis special fluid and therefore can be immediately approximatedby the standard SPH formulae. Then the SPH shallow wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140 160 180 200
Ref
lect
ion
Rat
io
Absorbing Layer
Gaussian Aplitude 9h
Hyperm=1m=2m=3m=4
Fig. 3. Reflection ratio versus the thickness of the absorbing layer for s function, hyperbolic and power law function with different values of the exponent m. Theparameters are identical to those used in Fig. 1.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–9080
equations are
dHdt!"Hr v
!)dHi
dt!X
k
mk# v!
i" v!
k$UriW# r!
i, r!
k$ #0:12$
where mk!HDxk. For the momentum equation we adopted a slightlydifferent formulation that produces more accurate results due to itshigher sensitivity to the pressure gradient for this peculiar equationof state. This equation has been used for instance by Riadh andAzzedine (2005) in their study of shallow water flows with SPH:
d v!
dt!"
1rrP)
d v!
i
dt!"
X
k
mkPi%Pj
rirj
%Pik
!riW# r
!i, r!
k$
Since P!(1/2)gr2 and r(H, then for shallow water momen-tum equation we have
d v!
i
dt!"
X
k
mk12
gH2
i %H2j
HiHj%Pik
!riW# r
!i, r!
k$ #0:13$
3.2. Test description and damping technique
In a domain of amplitude X!500 we produce a Gaussian pulsein the density profile and a corresponding fluid speed according tothe following prescription:
H#x$ !H0 1%0:01exp "#x"xc$2
A2
! !
#0:14$
v#x$ ! #H#x$"H0$%%%%%%%%%gH0
p#0:15$
So we have a 1D soliton traveling towards the right side ofthe domain. The following pulse parameters have been chosenH0!1, xc!3/4X and A!9h. The interpolating particle size ish!2, and the particle spacing is Dx!1. All quantities are inSI units.
A physical damping can, obviously, produces attenuation ofoutgoing signals; however, Fourier analysis shows that not all theharmonics belonging to a signal are attenuated and a specialshape of the damping is needed, as developed for example byModave et al. (2010) who made a good mathematical analysisthat can be assumed to be valid also for the Lagrangian SPHmethod at least in the shallow waves case, since the fluid speed isvery small (essentially no mass transport) compared to thewave speed.
To damp appropriately the waves in proximity of the domainedge an extra spatial layer is added to the domain and theequations in this damping layer are
dHdt!"H
@v@x"s#x$#H"H0$ #0:16$
for the water level, and
dvdt!"g
@H@x"s#x$#v"v0$ #0:17$
for the speed of the fluid,
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140 160 180 200
Ref
lect
ion
ratio
Absorbing Layer
Gaussian Amplitude 9 h m=1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140 160 180 200
Ref
lect
ion
ratio
Absorbing Layer
Gaussian Amplitude 9 h Hyperbolic
Fig. 4. Reflection versus the thickness of the absorbing layer for two different absorbing coefficient functions (left panel: m!1, right panel: hyperbolic). The solid lineidentifies the case with the use of only the clean function, and the dashed line identifies clean function multiplied by f1. The dotted line identifies the clean functionmultiplied by f2. The parameters are identical to those used in Fig. 1.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–90 81
where s(x) is the damping coefficient, which is a functionof the position in the layer, having an appropriate spatialdependence (discussed below); and v0 is the outflow speed. Inour study we impose v0!0 since in our case waves are propagat-ing in a closed water tank.
To produce a damping layer we add the following terms:
1. S!"s(x)(H"H0) to the density equation.2. Q!"s(x)(v"v0) to the momentum equation.
We tested for the coefficient s the following functionssuggested by Modave et al. (2010): s!s0[(x"x0)/L]m wherem is a positive integer, and s!s0[(x"x0)/((x0%L)"x)] wherex0 is the starting point of the absorbing zone. L is the thicknessof the zone and m is an exponent to be tuned.
Furthermore we tested also two ad hoc treatments, we callthem switches, based on physical intuition:
a. decrease the horizontal pressure force, only in the dampinglayer, with particular functions f1 or f2.
f1#x$ !" x"#x0% L$
L x0oxox0%L
1 xrx0
(
f2#x$ !L2"#x"x0$2
L2 x0oxox0%L
1 xrx0
(
f1 is a linear function, f2 is a parabolic function with itsmaximum at x0. The forces on the particles are simply multi-plied by these functions. We call them, cutting functions sincethey reduce to zero the horizontal force acting on particlesclose to the end of the damping layer. In this case the basicphysical idea is to decrease smoothly the pressure gradientnear the boundary.
b. Use the damping friction s40 only if vxo0. In this case theidea is to damp preferentially fluid motion directed into thedomain.
Fig. 1 shows the initial analytical configuration for the fluidspeed Vx (not the wave speed) and both the totally reflectedGaussian and the undisturbed propagated wave (we call it infinitycase) both computed at the time t! ##X=2$=#
%%%%%%%%%gH0
p$$. By ‘‘infinity’’
we mean the remnant of the numerical solution in the computa-tional domain when the corresponding peak of the analyticalsolution went out of the integration domain. It is obtained bysimply integrating the numerical solution into a much largernumerical domain, obviously consuming more CPU time.
The total reflection profile is due to the wave coming back atthe perfectly rigid boundary located at the right domain edgeX!500 m. The reflection ratio R for this set of 1D simulations is
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140 160 180 200
Ref
lect
ion
ratio
Absorbing Layer
Gaussian Amplitude 9 h m=1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140 160 180 200
Ref
lect
ion
ratio
Absorbing Layer
Gaussian Amplitude 9 h Hyperbolic
Fig. 5. Reflection ratio versus the thickness of the absorbing layer for two different distributions of the absorption coefficient (left panel: m!1, right panel: hyperbolic).The solid line identifies the case with the use of the clean function, and the dashed line identifies the same case with the attenuation of the force added with a linearfunction f1 within the layer zone. The dotted line identifies the previous case with a further switch on the speed. The parameters are identical to those used in Fig. 1.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–9082
computed following the Modave et al. (2010) formula, i.e. theratio of the errors
R!
%%%%%%%%%%%%%Elay,1
Eref l,1
s
#0:18$
given by
Eref l,1#t$ !12
gZ#Href l"H1$2dx%
12
HZ#vref l"v1$2dx #0:19$
where the label N identifies the values obtained with anextremely far right edge, i.e. no boundary condition (BC), thelabel lay refers to the quantities evaluated with a specificabsorbing layer, the label refl refers to the quantities evaluatedwith a totally reflecting BC. Obviously the integrals have beenreplaced by a sum over the particles. Essentially we aremeasuring the differences of the flow variables and then wecompute the relative energy, i.e. we are not computing thedifferences of the energies contained in the integration domain.1
The time evolution of the wave height, for a 100 m amplitudelayer with a hyperbolic damping function (Section 3.3.), typicallyappears as depicted in Fig. 2, where each profile is taken at timeintervals of Dt!7.4 s.
3.3. 1D simulation results
Fig. 3 shows the reflection ratio for the various damping functionsobtained with different values of the exponent m and the hyperbolicfunction. For small layers the best performances are obtained for thelinear and the hyperbolic functions. However we focused our studyon the hyperbolic function since it shows better results when we addthe ad hoc physical switches.
Fig. 4 shows the reflection ratio as a function of the thicknessof the absorbing layer for two different functions of the absorp-tion (left panel: m!1, right panel: hyperbolic), i.e. we compare
functions which are more efficient when it is the small absorbinglayer case. The solid line identifies the case with the use of thefunction without any switch, and we call it the clean function; thedashed line identifies the results obtained with the same cleanfunction with the addition of a further tool: attenuation of thepressure force with a linear function f1. The dotted line identifiesthe case of attenuation of the pressure force with a quadraticfunction f2. The best results are obtained with the dashed line, i.e.linear cutting function.
Fig. 5 shows the reflection ratio results obtained by adding thevelocity switch. The best results are displayed with a dotted line,corresponding to a damping with the use of the cutting functionf1 and with the simultaneous use of unidirectional friction, i.e. usethe damping friction s only if vxo0, so that the damping actsonly if the speed of the particles (not of the wave) is negative.
Fig. 6 compares the best results obtained with the m!1 andthe hyperbolic damping function using both the switches f1 andsa0 if vxo0. The hyperbolic function works only moderatelybetter, but in the 2D case we find a much better performance andtherefore we focus on that function.
3.4. 1D simulation conclusion
Resuming the 1D case, we may say that the best results havebeen obtained with the hyperbolic damping function s0[(x"x0)/((x0%L)"x)], plus two further treatments: the decrease of thehorizontal pressure force with a linear function f1 only in thedamping layer and the use of damping only if the speed of theparticles in the absorbing layer is negative vxo0.
We have to comment that, since we are using a Lagrangianapproach (the particles are free to move), we added in thedenominator an extra softening term 0.5h to avoid division byzero if a particle reaches the left edge s!s0[(x"x0)/((x0%L)"x%0.5h)]. So hereafter we report only the resultsobtained with the hyperbolic damping function.
Finally Fig. 7 shows the values of the reflection coefficientversus the amplitude of the damping layer for different widths of
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140 160 180 200
Ref
lect
ion
Rat
io
Absorbing Layer
Gaussian Aplitude 9h damp*f1 and if vx<0
m=1Hyperbolic
Fig. 6. Reflection ratio versus the thickness of the absorbing layer for two different distributions of the absorption coefficient (solid line m!1 and dashed line hyperbolic).With attenuation of the horizontal pressure force with a linear function f1 within the damping zone and use of damping only if the speed of the particles (not of the wave)is negative vxo0 in the absorbing layer. The parameters are identical to those used in Fig. 1.
1 That would be #1=2$gR#H2
l "H21$dx%#1=2$H
R#v2
l "v21$dx.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–90 83
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200
Ref
lect
ion
ratio
Absorbing Layer
Gaussian Amplitude 3h
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200
Ref
lect
ion
ratio
Absorbing Layer
Gaussian Amplitude 9 h
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200
Ref
lect
ion
ratio
Absorbing Layer
Gaussian Amplitude 18 h
Fig. 7. Reflection coefficients for increasing length of the damping layer. The results displayed with the solid, dashed and dotted lines are obtained with the cleanhyperbolic damping function, f1 pressure factor, f1 and speed switch respectively.
-1
0
1
2
3
4
5
-1 0 1 2 3 4 5 6 7
Palette
Mirror particles
Check points
Fig. 8. Tank with wavemaker and mirror particles, with no layer added.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–9084
the Gaussian pulse. It shows the predictable result that theincrease of the amplitude requires a larger damping layer toobtain the same reflection ratio.
4. Waves in tank: 2D case
In this case we study the wavy motion produced by a wavemaker palette in a water tank. The dynamics is truly twodimensional. To simulate incompressible water waves we usethe weakly compressible approximation (Monaghan, 2005),which consists essentially in the use of a sound speed an orderof magnitude larger than the maximum typical speed of thewater, and we chose cs!20vtyp, where vtyp !
%%%%%%%gH
p. Note that, as
suggested by Madsen and Shaffer (2006), here we use as typicalspeed the wave speed and not the very small fluid speed. Thegoverning equations are the previous ones, but with the equation
of state given by the Tait equation:
P!r0c2
s
grr0
! "g"1
& '#0:20$
with g!7.The wavemaker is placed at the left side of the rectangular
tank. A damping layer is added in the right side. Check points ofthe water level are defined at regular space intervals. The setup isshown in Fig. 8.
The aim is to produce, in the finite tank of length X, a motion,unaffected by the right boundary, i.e. equal to the one obtained inthe same zone but in an infinite tank.
We added the same source terms used for the 1D case to theequations of motion and continuity, taking into account thedimensionality of the problem, and we added a term Qa !"sa#va"v0a $ to each component of the momentum equation. Inthis study v0a ! 0.
-0.5
0
0.5
1
1.5
-1 0 1 2 3 4 5 6 7
Z
X
Fig. 10. Velocity field of the water in the resonant case, with no damping layer.
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
0 5 10 15 20 25
leve
ls
time
Fig. 9. Levels at x!1, 2, 3, 4, 5 m in an ‘‘infinite tank’’.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–90 85
Also we explored some ‘‘ad hoc’’ terms, guided by the 1Dexperience and physical intuition:
– a switch on the damping triggered by the speeds vx and/or vy,– the use of cutting functions to reduce the horizontal compo-
nent of the force due to the pressure.
We examined the case of a continuous periodic wave and thatof a wave generated by a single sinusoidal oscillation.
4.1. Continuous periodic wave
We made a tank of length X!6.061 meters and height 1 m.The particles have an intrinsic width h!0.1 and are placed at aregular spacing Dl!0.05 in X and Z. The number of particles in thetank is N!2570 (wavemaker included). The boundaries are madewith mirror particles procedure. The wavemaker oscillates with aperiod P!2.23 s and with an angle amplitude of 51. With these
values the water in the tanks enters in a resonant state. Fig. 9reports the levels of the water column, measured at five differentpositions (at x!1, 2, 3, 4, 5 m) along the tank, versus elapsedtime, in the case of a very long ‘‘infinite’’ tank.
The levels are vertically shifted for clarity. It is clear that thewaves propagate without disturbances.
Fig. 10 shows the velocity field of the tank in resonantcondition, with the wavemaker and the mirror points.
Fig. 11 shows the levels for the resonant tank. The levels areshifted in the Z coordinate by a small amount dz!0.01 for clarity.
It is clear that the oscillations are larger and increasing with time.In Fig. 12 we show the resulting levels when a damping
layer of extension L!6.061 m is added. That is, we are using afull simulation domain X!12.122 m. In all these tests thehyperbolic function s!s0[(x"x0)/((x0%L)"x%0.5h)] isadopted. The coefficient s0 has the dimension of 1/time andwe chose its value as s0"vref/L. For this study the referencespeed has been chosen equal to the sound speed: vref!csound
(case with the label M0).
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
0 5 10 15 20 25
leve
ls
time
Fig. 11. Water levels in the resonant case.
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
0 5 10 15 20 25
leve
ls
time
Fig. 12. Water levels for the case M0: plain hyperbolic absorption function.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–9086
0
5
10
15
20
25
30
35
5 10 15 20
Kin.
Ene
rgy
time
InfinityVx Vy switch
M0Vx switch
Fx & Vx switchResonance
Fig. 13. Kinetic energy versus time for resonance and for boundary layer with switches.
5
5.2
5.4
5.6
5.8
6
6.2
12 12.5 13 13.5 14
Kin.
Ene
rgy
time
InfinityVx Vy switch
M0Vx switch
Fx & Vx switch
Fig. 14. Zoom on the kinetic energy versus time.
Fig. 15. The particles configuration and their speed (enlarged by a factor 2.5) for the water in a reflecting tank.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–90 87
It is clear that the wave profiles are very similar to the ones ofthe infinity case.
The kinetic energy content can be used as an indicator of thesimilarity of the flows. Fig. 13 shows the kinetic energy of thewater (computed excluding the damping layer contribution)versus time for various cases. The increase of the energy in theresonant condition, the steady oscillating energy for the infinitycase, together with the very close values obtained with differentdamping layer cases are clearly shown.
If we zoom Fig. 13, on the values of the kinetic energy, weobtain Fig. 14 and find some discrepancies between the infinitecase and the M0 one.
We made the same simulation to test the following set ofswitches:
‘‘Vx Vy’’: the damping function works only if vxo0 and vyo0,
‘‘Vx’’: the damping function works only if vxo0,‘‘Fx and Vx’’: the damping function works only if fxo0 andvxo0.
If we look at the kinetic energy we find that performances betterthan M0 are obtained with each of the switch options mentioned,since for M0 the kinetic energy has oscillations larger than the onesof the infinity case. Qualitatively, the best result seems to be obtainedwith the simple ‘‘Vx’’ switch. However further investigations shouldbe carried out to produce a numerical estimate.
4.2. Sinusoidal single impulse wave
We made also simulation of a single impulsive sinusoidalwave. In the same tank, the wavemaker makes a single oscillationwith the same angular amplitude and period of the previoussimulation. In this case it is easy to see the effects of the reflected
Fig. 16. Ten levels for reflecting BC tank compared to the levels of the infinity case.
Fig. 17. Comparison of levels obtained with the M0 prescription and the infinity case.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–9088
wave. To have a detailed information, in this case we chose tenelevation level points located at intervals of 0.6 m, starting from0.3 m from the left side. Fig. 15 shows the particles and theirvelocity field for the water in the tank at time t!25 s; this is theconfiguration without any damping layer, i.e. clean reflectionconditions at the right side. The vertical lines identify the pointsof level measurement. The particles distribution and the speedarrows show that the water level is still oscillating.
Fig. 16 shows in the same panel the levels when the tank has asimple reflecting boundary on the right side; the levels obtainedfor the infinite tank (the thicker and straighter horizontal lines)are also plotted. It is obviously clear that oscillations are presenteven after the long time the pulse had to be outside the tank.
In Fig. 17 are shown the levels obtained with the M0 prescrip-tion compared with the infinity case.
In Fig. 18 the M0 levels are plotted together with the onesobtained with the ‘‘Vx Vy’’ switch. The M0 lines show a small bumparound time t!5–6 s, while the thick lines are more straight; theycorrespond to the use of the ‘‘Vx Vy’’ switch.
We studied also the same two problems using smaller damp-ing layers. The results are similar to the one presented here withthe obvious difference that the damping effects diminish as the
layer thickness decreases. We show the results of the singlesinusoidal impulse. Fig. 19 shows the kinetic energy of the tankversus time when the length of the damping zone is L!6.06.
Fig. 20 shows the kinetic energy of the water in the tank for ashorter damping layer L!3.03 m. We also tested the use ofcutting functions, but the improvements are very small to beappreciated in the figure. From Fig. 20 it is clear that the reducingaction of the residual oscillations is due to the velocity switchwhen added to the plain damping function.
Fig. 21 shows a zoom of Fig. 20, to show clearly the different effectsof the damping criteria. It shows that for the 2D problem the cuttingfunction improves the results over the plain damping, but it is notbetter than the simple velocity switch. The joint action of the velocityswitch and of the cutting function does not improve the result.
5. Conclusion
The use of a damping layer is successful in avoiding boundaryreflections into the computational domain. An obvious require-ment is that the absorbing layer must be greater than or equal tothe maximum significant wavelength produced by the physical
Fig. 18. Comparison of levels obtained with M0 and with the switches Vx and Vy (thick lines).
0
5
10
15
20
25
30
35
0 2 4 6 8 10 12 14
Kine
tic E
nerg
y
time
Layer L=6.06 m
M0Infinity
Reflecting BCVx switch
Fig. 19. Kinetic energy for different damping actions in the case of a 6.06 m large damping zone.
D. Molteni et al. / Ocean Engineering 62 (2013) 78–90 89
simulation. Both in the 1D and 2D cases the basic procedure canbe improved by the use of appropriate switches. A simple andefficient switch is the one that makes the damping operate onlyfor negative speeds, i.e. vxo0. The switch that reduces thehorizontal component of the force, Fx, is efficient in 1D, but notso much for the 2D problem we studied. Further work is inprogress to make a quantitative evaluation of the 2D simulationsand verify the affordability of the method in the case of highlycompressible fluid dynamics.
Acknowledgments
R. Grammauta acknowledges CRRSN for a six months grantexploited for this study.
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0
5
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tic E
nerg
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Fig. 20. Kinetic energy of the water in the case of damping layer L!3.03 m.
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Fig. 21. Zoom of Fig. 20 to show better results of the different damping algorithms.
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