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SPH modeling of resonance wave pumping in
closed tank: Parametric study
Remi Carmigniani
Saint-Venant Laboratory for hydraulics
Universit´
e Paris-Est
Chatou, France
rcarmi@me.edu
Antoine Joly
EDF & Saint-Venant Laboratory for hydraulics
Universit´
e Paris-Est
Chatou, France
Agnes Leroy
EDF & Saint-Venant Laboratory for hydraulics
Universit´
e Paris-Est
Chatou, France
Damien Violeau
EDF & Saint-Venant Laboratory for hydraulics
Universit´
e Paris-Est
Chatou, France
Abstract—The Smoothed Particle Hydrodynamics (SPH)
method with the unified semi-analytical wall boundary conditions
is used to simulate a novel approach for free-surface wave
pumping inspired from Libeau impedance pump: Resonance
wave pumping. The results are compared to experimental data.
Using the rule of thumb numerical parameters for the weakly in-
compressible method (small enough dr and a speed of sound such
that c0≈10√ghmax , where hmax is the maximum water detph),
good agreement is found for both the quantitative instantaneous
flow rate and qualitative surface wave dynamics near the two
resonance frequencies of the resonance wave pump considered in
this study. It is observed that varying the speed of sound changes
significantly the mean flow for the highest resonance frequency
considered. This effect is studied using compressible potential
flow theory approach. Two effects of the weakly compressible
method are identified: shifts of the resonance frequencies and
compressible source of pumping. This approach is an interesting
way of understanding the origin of the observed effect of c0
on the pumping behaviour and more generally in free-surface
dynamics in weakly compressible SPH.
I. INTRODUCTION
Smoothed Particle Hydrodynamics (SPH) has been used to
simulate a variety of problems involving the Navier-Stokes
equations. It is a particularly promising method for free-
surface or multi-phase flow applications. In the present study,
a novel approach for free-surface wave pumping is introduced
and the SPH method is used to reproduce the experimental data
available. The system, resonance wave pump, is inspired by
the Liebau impedance pump introduced in 1954 [1]. A Liebau
pump consists of a flexible and rigid tube connected together
at each extremities to form a closed loop. The flexible tube
is pinched at an off-centred position. For certain frequencies
(resonance frequencies) a unidirectional flow rises in the
tubes. This elastic pump was studied both experimentally
[2]–[5] and numerically [6], [7]. A free-surface version of
such pump was recently investigated and similar behaviour
reported in an experimental study [8]. This near resonance
pumping behaviour results in a complex free-surface dynamic
that seems suitable for the SPH method.
In section II , the SPH method used in this study is briefly
described and the main experimental results of the resonance
wave pump are summarised. In section III, the SPH method is
used to simulate the behaviour of the pump with fairly good
agreement with the experiment when using the general strategy
used in the literature to chose the numerical parameters (small
enough resolution, speed of sound of about 10 time the wave
speed and an adiabatic index ζ= 7 in the closure state
equation P V ζ=cte). The different numerical parameters are
then varied. It is outlined that the parameter speed of sound
of the weakly-compressible method has the most influence on
the results. In the last section, the origin of this dependance
to the speed of sound is investigated using a potential flow
approach. It appears that the compressibility enhances the
pumping mechanism and shifts the resonance frequencies.
II. TH E SPH MET HO D AN D EXP ER IME NTAL RE SU LTS
A. Unified semi-analytical boundary conditions
The SPH method used throughout this paper is presented in
detail in the papers by Ferrand et al. [9] and Mayrhofer et al.
[10], [11]. In the present section a brief summary highlights
the main characteristics of the method.
The main difference from the classical SPH methods is that
the SPH approximation of a function fevaluated at a position
ais given by
[f]a=1
γa
b∈P
VbfbWab,(1)
where Pis the set of all particles, Vbdenotes the volume
of the particle b,Wab the kernel function as a function of
the distance between the particle band the position awhich,
throughout the paper, is the quintic polynomial by Wendland
[12]. Finally, γis a kernel wall renormalisation factor which
is defined as
γa=Ω
W(ra−rb)drb,(2)
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11th international SPHERIC workshop Munich, Germany, June, 14-16 2016
where Ωis the fluid domain. Note that this integral is equal to
one inside the fluid domain and its value is bounded between
0 and 1 near the vicinity of a boundary. It is possible to
analytically evaluate this integral in 2-D [9] but in the present
study the value is computed throughout a governing equation:
dγa
dt=vR
a·∇γa,(3)
where vR
ais the relative velocity to the rigid boundary and
∇γais given by a surface integral:
∇γa=∂Ω
W(ra−rb)ndrb,(4)
with nthe inward oriented unit normal. The gradient and
divergence operators are then given by:
Gγ
a(f) = 1
γab∈PVb(fa+fb)∇aWab
−1
γas∈S(fs+fa)∇γas,(5)
Dγ
a(B) = 1
γab∈PVb(Ba−Bb)∇aWab
−1
γas∈S(Bs−Ba)∇γas.(6)
The additional terms containing the ∇γas come from the
integration by parts that is used to move the differential from
the unknown function to the known kernel. The boundary in
the present method is discretised into discrete elements s∈S,
refered to segments with no associated mass. The masses are
located at the extremities of these segments where particles are
placed with a mass that depends on the opening angle of the
connected adjacent segments. In 2-D, the boundary segments
are 1-D line segments and:
∇γas =s
W(ra−rb)ndrb,(7)
i.e. the integral of the kernel on this boundary segment. Fur-
thermore second-order differential operators are approximated
using:
∇·(f∇ ⊗ B)≈
Lγ
a(f, B) = ρa
γab∈Pmbfa+fb
ρaρb
Bab
r2
ab
rab ·∇aWab
−2
γas∈Sfs(∇B)·∇γas,
(8)
where rab =rab.
The method then solves the Navier-Stokes equations for com-
pressible fluid:
d
dtρ
u=−ρ∇·u
1
ρ(g+∇·¯
¯σ),(9)
where g=−gezis the gravitational acceleration, and ¯
¯σis the
stress tensor:
¯
¯σ=−p¯
¯
I+µ∇u+∇uT.(10)
In order to close this system of equations, an equation of state
is defined to relate the pressure and the density [13]:
p=ρ0c2
0
ζρ
ρ0ζ
−1,(11)
Fig. 1. Sketch of the experimental setup and notations. The PIV windows
shows where the data are recorded in the experiment while the vertical
dashed lines labeled xi∈{1,2,3}show where the data are sampled for the
SPH simulations. The flow rate is computed by trapezoidal rule integration.
The values of the different parameters are listed in table I. For the simulations
the values are the one without the error bars.
where ρ0is the reference density, c0is the numerical speed
of sound and ζis the adiabatic index equals to 7 for water.
The suitable values for c0are discussed later in this paper but
are generally of the two order of magnitude smaller than the
physical speed of sound in water.
B. Moving wall boundary conditions
At the moving boundary (a wave paddle in the present case,
as explained later), a no-slip boundary condition is imposed
and the velocity of the wall is prescribed. The pressure of a
wall particle has to be calculated from the fluid to accurately
approximate the pressure gradient in the fluid particle in the
vicinity of the boundary. As suggested by [14], a force balance
at the wall interface gives:
dvw
dt=−∇pw
ρw
+g=aw(12)
where awis the acceleration of the wall. After a bit of algebra
it is possible to show that:
pw=b∈PpbWwb + (g−aw)·b∈Pρbrwb Wwb
b∈PWwb
.(13)
Even though in the present case it is easy to compute the
acceleration analytically, the wall acceleration is calculated
using a first order marching scheme, thus only the previous
and current prescribed time steps are required to compute
the acceleration. This enables future integration of generalised
solid-fluid interactions in the present method.
C. The resonance wave pump
In Fig. 1 a sketch of the experimental setup is shown. A
tank with a centred submerged plate of length L−2Δxand
thickness efixed at a water depth of hmin is filled with water
up to hmax. A paddle controlled in heave motion is generating
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11th international SPHERIC workshop Munich, Germany, June, 14-16 2016
Fig. 2. Experimental results: bulk and first harmonic flow rate as a function
of the forcing frequency (filled and empty circles and left and right axis
respectively) for a paddle of width 2a = 10.46 cm and a fixed stroke amplitude
of S0= 0.69 cm. The points shaded in grey indicate cases where the bulk
flow is larger than the oscillatory first harmonic flow rate. The red points
outline the frequency considered in this paper.
waves with a controlled frequency f0and S0. The width of the
paddle is 2a. Its mean position is such that the centre of the
paddle is located l1away from the wall and its mean draft is
d. The parameters used in the experiments are shown in table
I. In the experiment, the flow rate, φ, was measured under the
submerged plate using Particle Image Velocimetry (PIV) data
and a second camera was recording the entire tank to visualise
qualitatively the free surface dynamic.
Starting from a quasi at rest tank, the flow rate under the
submerged plate has a transitional stage before reaching an
asymptotic regime. The asymptotic flow rate is analysed using
a Fast Fourier Transform (FFT) and the mean flow (φ) and
first harmonic (φ1at the frequency f0) are plotted in Fig. 2
as a function of the forcing frequency, f0, on two different
vertical axes (left and right respectively).
In [8], the authors show that the first harmonic oscillatory
amplitude φ1can be found by solving a potential flow asymp-
totic problem. It was then pointed out by the authors that,
similarly to the flexible tube pump, the mean flow is maxi-
mum at forcing frequencies where resonances are predicted.
The interesting behaviour occurs near the forcing frequency
f0= 1.2and 1.6 Hz (or f0L/√ghmin = 1.585 and 2.11
respectively). In both cases the oscillatory flow of the paddle
is converted into a pulsating flow. In the first case, f0= 1.2
Hz, the oscillations are as large as the mean flow resulting in
a pulsating flow while in the second case, f0= 1.6Hz, the
oscillations are much smaller than the mean flow resulting in
a quasi continuous flow. The instantaneous response measure
experimentally are shown in Fig. 3 and Fig. 5 respectively
(gray lines). Such behaviours are particularly interesting for
future applications since the oscillatory motion of the paddle
is converted into a quasi continuous flow. The system behaves
like a rectifier: the alternative current (AC) is converted to
direct current (DC) through the system.
TABLE I
EXPERIMENTAL AND NUMERICAL PARAMETERS
Symbol Physical quantity Values
g Gravitational acceleration 9.81 m.s−2
L Tank length 77.4 ±0.1 cm
l1Paddle off-centred position 18.7 ±0.1 cm
2 a Paddle length 10.46 ±0.05 cm
d Paddle mean draft 0.07 ±0.1cm
Δx Openings’ length 3 ±0.1cm
hmax Water depth 10.82 ±0.1 cm
W Recirculation section height 6.185 ±0.05 cm
eSubmerged plate thickness 1.2 ±0.05 cm
S0Paddle stroke amplitude 0.69 ±0.05 cm
f0Forcing frequency {1.6,1.2}Hz
xiPosition of the numerical sampling {3.2,38.7,74.2}cm
III. PARAME TR IC ST UDY O F THE RE SO NA NC E WAVE PUMP
The present study focuses on the response of the system
near the fourth resonance peak where the mean flow is larger
than the oscillation resulting in a unidirectional flow under
the submerged plate in the negative direction. The response
near the third peak (f0= 1.2Hz) is also considered at some
point to verify that the SPH method works in other parts of
the resonance wave pump spectrum. In all cases, the response
is near a linear resonance as shown in section IV. Different
numerical parameters are considered in this study. The code
being weakly compressible, one needs to worry about selecting
the proper value for the speed of sound c0. In the literature, it
is usually advised to fix the speed of sound at the beginning
of the simulation so that:
c0≥10 max (|u|)t(14)
where max (|u|)tis the maximum speed in modulo of the
fluid, obtainable along the whole simulation. In these condi-
tions the Mach number M=u/c0is always less than 0.1 and
it is possible to consider the fluid as weakly compressible.
For free surface flow it is usually considered that a proper
value for the maximum value can be taken as √gh and then
the Mach number is identified to the Strouhal number. In our
case, we first select the speed of sound to be c0= 11 m.s−1.
Note that this value is about 20√ghmin and about 10√ghmax
and thus should be reasonable for this kind of simulations.
It is also possible to verify that the maximum speed in the
breaking waves induced in the shallow water is of the order
of magnitude of 1 m.s−1. The value of the spatial resolution
is selected to ensure that there are about 70 particles in the
shallow water, thus dr = 0.0005 m (or hmin/dr = 68.7). The
value of dr is also consistent with the error bars of the different
measurement. The value for the state equation adiabatic index
is ζ= 7 for water. The numerical results are compared
to experimental results. In the available experiment data the
starting process of the paddle has not been recorded precisely
so a discrepancy at the beginning of the simulation is expected.
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11th international SPHERIC workshop Munich, Germany, June, 14-16 2016
SPH with c0=11 m/s, hmin/dr =68.7
Experiment
0 5 10 15 20
25
-40
-30
-20
-10
0
10
20
t(s)
∫
u.dS (cm2.s-1)
Fig. 3. Compared experimental (gray continuous line) and numerical flow rate
using the SPH method (black continuous line) for the numerical parameters
c0= 11 m.s−1,ζ= 7,hmin /dr = 68.7,f0= 1.6Hz.
The paddle in the experiment was set-up near its top position
before starting while in the simulation it is started at its
mean position and its displacement is S(t) = S0sin (2πf0t).
In the experiment the authors were mostly interested in the
asymptotic behaviour thus this lack of care on the starting
process. For the sake of simplicity and to ease the comparison
the data of the experiment are shifted such that the first peak
matches in the flow rate measurement. In the experiment the
flow rate was measured under the submerged plate using PIV.
Since the flow is weakly compressible here to calculate the
mean flow, data are sampled at three different positions under
the submerged plate (x={3.2,38.7,74.2}cm, see vertical
gray dashed lines in Fig. 1), integrated in the entire profile
using a trapezoidal rule and averaged. Figure 3 shows the
results obtained with the parameters described in the previous
paragraph for the 25 first seconds with the forcing frequency
f0= 1.6Hz. The results are qualitatively and quantitatively
in good agreement with the experiments with these numerical
parameters. The free surface dynamic is also compared to the
experimental data for the flow around 15 seconds after the
start other a wave period. The results are shown in Fig. 4.
The wave dynamic is qualitatively correctly captured by the
SPH method. Finally, the frequency is varied to f0= 1.2Hz
and compared in a similar manner to the experimental results
to validate the method. The flow rate instantaneous response
is given in Fig. 5. There is a good agreement between the
simulation and the experiment. This validates the SPH method
with this set of parameters.
It is now interesting to see what are the effects of separately
varying the resolution and speed of sound. This part of the
study focuses on the frequency f0= 1.6Hz. First, the particles
size is varied. The resolution is reduced with hmin/dr = 34.35
(in blue) and increased hmin/dr = 137.4(in red) and compared
to the previous simulation (in black) and experiment (in gray).
Fig. 6 shows the different instantaneous responses. In all cases,
the pumping occurs. It is visible that the oscillations are
slightly better captured by the most refined simulations. This
results confirms the convergence of the method with these
Fig. 4. Compared numerical (top) and experimental (bottom) surface
dynamics at four different instances during a wave period for the numerical
parameters c0= 11 m.s−1,ζ= 7,hmin/dr = 68.7,f0= 1.6Hz. The blue
square in the top images is the paddle position in the simulations.
SPH with c0=11 m/s, hmin/dr =68.7
Experiment
0 5 10 15 20 25
-40
-30
-20
-10
0
10
20
t(s)
∫
u.dS (cm2.s-1)
Fig. 5. Compared experimental (gray continuous line) and numerical flow rate
using the SPH method (black continuous line) for the numerical parameters
c0= 11 m.s−1,ζ= 7,hmin /dr = 68.7,f0= 1.2Hz.
parameters (c0= 11 m.s−1,ζ= 7).
In the rest, the spatial resolution is now fixed to hmin/dr =
68.7and the speed of sound is varied. The speed of sound is
varied between c0= 6 and 44 m.s−1. Fig. 7 shows the flow
rate for four different values of the speed of sound for the case
f0= 1.6Hz. In all cases the 5 first seconds of simulations
are fairly identical. However the behaviour changes drastically
with the different value of c0afterwards. Reducing the speed of
sound leads to a faster asymptotic flow rate while larger speed
of sound and thus more incompressible fluid, leads to a slow
down and almost a complete stop of the pumping mechanism.
It is important to outline though than in all cases the mean
flow is negative in most part of the simulation. Also it is
important to remind the reader that up to this point the flow
rate was computed by averaging the horizontal velocity under
the submerged plate at three different locations because the
flow rate we are interested in is the incompressible one and
this way we eliminate the bias of the compressibility.
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11th international SPHERIC workshop Munich, Germany, June, 14-16 2016
SPH with c0=11 m/s, hmin/dr =34.35
SPH with c0=11 m/s, hmin/dr =68.7
SPH with c0=11 m/s, hmin/dr =137.4
Experiment
0 5 10 15 20 25
-40
-30
-20
-10
0
10
20
t(s)
∫
u.dS (cm2.s-1)
Fig. 6. Compared experimental (gray continuous line) and numerical flow
rate using the SPH method for the numerical parameters c0= 11 m.s−1,
ζ= 7,f0= 1.6Hz and different resolutions (in blue, black and red).
Identically, we varied the speed of sound for f0= 1.2Hz.
The results are shown in Fig. 8 for c0= 6 and c0= 22
m.s−1. The variation of c0in this range does not have any
significant impact on the dynamic. It is noticeable though that
the flow is slightly slower for c0= 22 m.s−1but the impact
is not as significant as for the frequency f0= 1.6Hz. In the
next section a potential approach is developed to investigate
the influence of compressibility on the resonance wave pump
with the forcing frequency f0.
IV. COM PRE SS IBL E POT ENT IA L FLOW HA RMO NI C TH EORY
To investigate the effect of the compressibility on the
simulation results and have a better understanding of the
dependance of the flow rate with c0, identically to what
was done in [8], we analyse the potential flow linearised
response of the system to an infinitesimal stroke amplitude.
Since there seems to exist an asymptotic regime, we seek
solution in the form u=U(x) + u1(x, t) + h.o., where
u1=iS0ω∇ϕ(x)eiωt. We further assume that the mean
flow is small compare to the oscillatory part. If this assumption
may seem wrong, it is important to notice that in the Fig. 2
the mean flow is generally largelly smaller than the oscillatory
part. Moreover, the mean flow was reported to vary quasi-
quadratically with the stroke amplitude and thus if the stroke
amplitude is small enough this assumption holds.
The potential scalar ψ=iS0ωϕ (x)eiωtverifies the
following set of linearized equations and boundary conditions:
∂t,tϕ−c2
0∇2ϕ+g∂zϕ= 0 ,for x∈Ω
∂nϕ= 0 ,for , x ∈Γwall
∂nϕ=ez·n,for , x ∈Γpaddle
−ω2ϕ+g∂zϕ= 0 ,for , x ∈ΓFS
(15)
where Ωis the fluid domain, Γwall the fixed wall boundaries,
Γpaddle the moving paddle boundary in contact with water,
ΓFS the free surface (see Appendix A). This problem can be
solved using the Finite Element Method (FEM). In the present
study, we used Mathematica’s FEM package to solve this set
of equations. The free surface deformation is reconstructed
using the relation η=S0∂zϕ|z=0 eiωt. The equivalent
Fig. 7. Compared experimental (gray continuous line) and numerical flow
rate using the SPH method for the numerical parameters ζ= 7,f0= 1.6
Hz, hmin/dr = 68.7and different values of the speed of sound.
problem for incompressible fluid is the limit c0→ ∞ and the
first equation of the system Eq. 15 simplifies to the Laplace
equation for ϕ.
First, we look at the amplitude of oscillations of the flow rate
as a function of frequency defined as the average potential
flow rate at the three position x1,x2and x3:
φ1=S0ω
3W
0
i∈{1,2,3}
∂xϕ(xi, z)dz
(16)
The frequency response of the system is shown for different
values of the speed of sound (or Mach numbers M=
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11th international SPHERIC workshop Munich, Germany, June, 14-16 2016
SPH with c0=6 m/s, hmin/dr =68.7
Experiment
-30
-20
-10
0
10
∫u.dS (cm2.s-1)
SPH with c0=22 m/s, hmin/dr =68.7
Experiment
0 5 10 15 20
25
-30
-20
-10
0
10
t(s)
∫u.dS (cm2.s-1)
Fig. 8. Compared experimental (gray continuous line) and numerical flow
rate using the SPH method for the numerical parameters ζ= 7,f0= 1.2
Hz, hmin/dr = 68.7and different values of the speed of sound.
√ghmax /c0) near the frequencies f0= 1.2and 1.6 Hz in
Fig. 9. In both cases, a smaller speed of sound (larger Mach
number) shifts the curves to left. The shift is more important
near f0= 1.6Hz resulting in a larger impact of the speed
of sound on the results. We see that for the case f0= 1.2
Hz the difference between the spectrum with values of the
speed of sound c0= 11 m.s−1(M= 0.09, black curve)
and the incompressible case is rather small, suggesting that
the solution is fairly converged in the incompressible sense.
However, in the case f0= 1.6Hz, the curve c0= 11
m.s−1(M= 0.09) the difference is still large compared to the
incompressible case. This might explain the drastic difference
of response between the different values of the speed of sound
investigated in the previous section. This is outlined in Fig. 10
where the value of the amplitude at the frequencies f0= 1.2
and f0= 1.6Hz are plotted as a function of the Mach number.
Furthermore, one characteristic of the potential theory for
compressible fluid is that the amplitude of oscillation of
the flow rate under the submerged plate is a function of
the position x. In particular, the amplitude of oscillation
of the flow rate between the position x1and x3might be
different. A larger amplitude of oscillation at the position
x1compared to x3results in a lower mean pressure at the
position x1that might amplify artificially the flow rate. Fig.
11 shows the theoretical amplitude of oscillation of the flow
rate under the submerged plate as a function of the position
nondimensionalised by the incompressible potential solution.
The amplitude of oscillations are larger at the position x1than
x3and thus the effect described before could rise. We see that
M=0.17
M=0.09
M=0.02
M=0
1.2 1.3 1.4 1.5 1.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
f
(
Hz
)
ϕ1
2a S0ω
Fig. 9. Oscillatory flow rate amplitude as function of frequency in the
compressible potential flow for different Mach numbers. When the Mach
number increases the curves are shifted to the left resulting in a modification
of the resonance frequencies. It appears that this effect is magnified near the
frequency f0= 1.6Hz, while near f0= 1.2Hz the black and red curves
are collapsed with the incompressible one (gray dashed line).
f0=1.2 Hz
f0=1.6 Hz
10 15 20 25 30 35 40
0.0
0.2
0.4
0.6
0.8
1.0
1/M
ϕ1comp/ϕ1incomp
Fig. 10. Ratio of the oscillatory flow rate amplitude for compressible and
incompressible potential flows as function of the Mach numbers. The ratio
goes to 1 when the Mach number goes to zero. For f0= 1.2Hz (in black)
the compressible error is smaller than in the case f0= 1.6Hz (in gray).
this could generate a source of mean flow and that this source
is less important as the speed of sound increases.
To investigate these effects in the SPH method, an FFT
analysis of the flow rate at the three different positions sampled
in the simulations is performed at the end of time series. The
results are shown for the two extreme values of the speed
of sound c0= 6 m.s−1(M= 0.17) and c0= 44 m.s−1
(M= 0.02) in Fig. 12 and Fig. 13, respectively. The data are
sampled on the last 5 seconds of the simulation for t ∈[20; 25]
s. In the case c0= 44 m.s−1(M= 0.02) the FFT mostly
consists of a mean flow and a peak near the forcing frequency
as expected. However the mean flow is smaller than observed
experimentally. For c0= 6 m.s−1(M= 0.17), there is a
larger variety of modes. Some of these modes are acoustic
waves in the recirculation section. We do observe that the
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11th international SPHERIC workshop Munich, Germany, June, 14-16 2016
c0=6 m/s
c0=11 m/s
c0=44 m/s
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.0
0.2
0.4
0.6
0.8
1.0
x
∫-H
-(h+e)ucomp (z).ⅆz
∫-H
-(h+e)uincomp (z).ⅆz
Fig. 11. Compressible potential amplitude of oscillation of the first harmonic
under the submerged plate as a function of the position for different values
of the speed of sound nondimensionalised by the incompressible potential
solution. The gray dashed vertical lines show the position x1and x3. The
amplitude are larger at the position x1than x3in all the cases. The difference
is more important for the lower speed of sound. This might amplify artificially
the mean flow.
amplitude of oscillations are larger for case c0= 44 m.s−1
(M= 0.02) at the forcing frequency f0than for c0= 6
m.s−1(M= 0.17). This is consistent with Fig. 9 where
we see that the shift increases the amplitude of oscillation
at the constant frequency f0= 1.6Hz. We see that the
oscillations are larger at the position x1than x3for the case
with c0= 6 m.s−1(M= 0.17). This might help the flow to
pump in the negative direction. Nonetheless, the effect is not
as drastic as the compressible potential theory may suggest.
The oscillations are also smaller at the center for this case. It
is however interesting to point out that the compressibility is
a source of mean flow. The global direction of this secondary
source of mean flow is however uncertain.
V. CONCLUSION
In the present study the SPH method was applied to a
resonance free-surface wave pump near two resonance fre-
quencies. Good agreement is found for a large range of
numerical parameters for both the mean flow generated by
the pump and the free surface dynamic when compared to the
experiment. Nonetheless, it is pointed out that compressibility
might generate pumping that does not occur for larger speed
of sound. A potential approach is developed to demonstrate
this behaviour. It is outlined that the observed influence to
the mean flow is nonetheless small and this effect should
be negligible. This approach is however an interesting way
of understanding the origin of the observed effect of c0on
the pumping behaviour (and more generally in free-surface
dynamics in weakly compressible SPH). The main difference
appeared to be the shift in the spectrum due to compressibility
which is more likely to explain the difference of behaviour. In
the present study the water depth was not varied in the range
of the experimental error and the particle size was not refined
for larger speed of sound.
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02468
0
1
2
3
4
5
6
7
f/f
0
|ϕ|(cm2.s-1)
Fig. 12. FFT of the simulated flow rate at the different positions for
c0= 6 m.s−1(or M= 0.17) and a forcing frequency f0= 1.6Hz. The
oscillations are larger at the position x1than x3due to the incompressibility.
This difference in amplitude results in a lower time averaged pressure at the
position x1. Acoustic waves oscillations are visible in the FFT spectrum near
the abscissa 4. The data are sampled on the last 5 seconds of the simulation
for t∈[20; 25] s.
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02468
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Fig. 13. FFT of the simulated flow rate at the different positions for c0= 44
m.s−1(or M= 0.02) and a forcing frequency f0= 1.6Hz. There is almost
no difference between the oscillations at the different position and almost no
acoustic waves. The data are sampled on the last 5 seconds of the simulation
for t∈[20; 25] s.
APPENDIX A
DER IVATIO N OF TH E POT ENT IA L EQUATI ON FO R
CO MP RES SI BL E FLUI D
One can show that the potential scalar ψ=
iS0ωϕ (x)eiωt, verifies the generalised Bernoulli
equation for compressible fluid:
∂tψ+1
2(∇ψ)2+p
pref
dp�
ρ(p�)+gz =cte (17)
106
11th international SPHERIC workshop Munich, Germany, June, 14-16 2016
where the constant is assumed independent of time. Taking
the time derivative of the Bernoulli equations and noting that
∂tp
pref
dp�
ρ(p�)=1
ρ∂tp(18)
it is easy to show that:
∂t,tψ+1
2∂tu2+1
ρ∂tp= 0.(19)
Recall the momentum equation for irrotational fluid (or
inviscid):
dtui+1
ρ∂xip+gδi3= 0 (20)
where dtis the Lagrangian derivative and ∂xiis the spatial
derivative with respect to the direction i,δij is the kronecker
delta, and using Einstein’s notation. We multiply this equation
by ui. Using the continuity equation and deriving the relation:
1
ρui∂xip=−c2
0∂xiui−1
ρ∂tp, (21)
it comes after some basic algebra:
1
2∂tu2
i+1
2ui∂xiu2
j−c2
0∂xiui−1
ρ∂tp+gu3= 0.(22)
Finally, summing Eq. 19 and Eq. 22 and using the fact that
ui=∂xiψ, we derive the potential equation for compressible
fluid:
∂t,tψ−c2
0∇2ψ+g∂zψ=−∂tu2−1
2u·∇u2.(23)
Considering now only the first order terms, it yields:
∂t,tψ−c2
0∇2ψ+g∂zψ= 0.(24)
Note that in the limit of incompressible fluid c0become infinite
and we get the Laplace equation for ψ.
ACK NOWL ED GME NT
The authors would like to thank EDF and Caltech for the
help with running the simulations and the assistance in con-
ducting the experiment. We wish to acknowledge the funding
received from EDF to conduct this research and the Gordon
and Betty Moore Foundation for their generous support for
the experiments. Our gratitude also goes to Martin Ferrand
and Alex Ghaitanellis for their support and help to run the
simulations.
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