particle image velocimetry applied to water sloshing due...

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International Symposium on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2014

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Particle Image Velocimetry applied to water sloshing due to a harmonic external excitation.

Alessia Simonini1,*, Maria Rosaria Vetrano1, Pierre Colinet2, Patrick Rambaud1

1: Environmental & Applied Fluid Dynamics Department, von Karman Institute, Rhode-Saint-Genèse, Belgium

2: TIPs – Fluid Physics, Université Libre de Bruxelles, Brussels, Belgium

* correspondent author: [email protected]

Abstract In this paper the asymmetric water sloshing in a partially filled circular cylindrical container subjected to

horizontal harmonic excitations is investigated. The understanding of this phenomenon is fundamental for the design of

propellant management devices; moreover, the dynamic response of the moving liquid inside the tank can dramatically

influence the trajectory of vehicles. The asymmetric sloshing is controlled by the liquid depth, forcing frequency and

amplitude. Above all, the behavior of the free liquid surface is of major interest. The analytical and equivalent

mechanical models of the simplified undamped problem are here introduced. Fluorescent Particle Image Velocimetry is

used to investigate the velocity field in the liquid phase. Phase locked PIV with a selected phase of the external

excitation is used in order to provide statistical data. An image processing algorithm is developed to detect the free

surface and to properly define the fluid field of interests. The commercial algorithms of PIV and the combined

PIV+PTV are compared. Selected examples of information that can be extracted from the PIV phase locked results for

the sloshing phenomenon are presented.

1. Introduction The motion of the free liquid surface inside its container is called ‘sloshing‘. It is strongly affected by

external excitations applied to the partially filled container. The understanding and the prediction of this

particular motion is of special importance in many fields. Indeed, the motion of liquids inside containers can

refer to embarked containers filled of fluid, motion of cooling liquids in systems subjected to earthquake or

motion of propellant inside the tank. The latter can influence the dynamic stability and performance of the

vehicle, be a ship, a missile, a rocket or a satellite. The motion of a liquid inside its container can have an

infinite number of natural frequencies, but the lowest few modes are most likely to be excited from the

vehicle motion (Ibrahim, 2005). An example of liquid sloshing in a circular upright cylinder subjected to

lateral excitation is shown in Fig. 1.

Fig. 1 Example of liquid sloshing in a circular upright cylinder subjected to lateral excitation.

The prediction and control of this motion is far to be understood due to the different parameters that play a

role in the dynamic system such as the geometry of the container, the type of external excitation (shape,

frequency content and amplitude) and finally the level of the liquid. In this context, the creation of a reliable

and consistent experimental database is crucial for assessing the accuracy and the range of validity of

existing numerical models. The motivation behind this paper is the characterization of the sloshing motion

inside a partially filled cylinder in order to build a test case and provide experimental data for calibrating and

validating numerical codes. The velocity in the liquid phase and the behavior of the free surface will be

studied. The geometry chosen is a circular cylinder: this is a common shape for liquid storage due to

pressurization issues. For this geometry the analytical results are well known. In this work the experiments

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were performed with water in a partially filled circular cylinder subjected to sinusoidal lateral excitations.

Water was chosen in order to figure out the experimental procedures and the post processing algorithm with

a well-known liquid. Fluorescent 2D Particle Image Velocimetry (f-PIV) has been applied to characterize the

velocity field in the plane in which is applied the external excitation. The images have been processed with a

standard PIV cross-correlation algorithm with a structured grid. A comparison with the results obtained with

Hyper Resolution PIV algorithm is shown.

2. Sloshing theory The theory of liquid sloshing dynamics in partially filled containers consists in developing the fluid field

equations. The Navier-Stokes equations are integrated taking into accounts different hypotheses and

boundary conditions relative to the particular problem. For an open surface the boundary condition specifies

either the values of the field at every point of the surface, the normal gradient to the container walls or both.

Explicit solutions are possible only for few cases such as the upright partially filled cylinder. The boundary

value problem is usually solved for modal analysis (free sloshing) and for the dynamic response

characteristics to external excitations (forced oscillations). The fluid free-surface motion and the resulting

hydrodynamic forces and moments can be estimated from the system of fluid field equations. Mechanical

analogies can be found in literature beside the analytical solutions (Ibrahim, 2005; Abramson, 1966; Dodge,

2000).

2.1 Free oscillations The container-fluid system investigated is shown in Fig. 2.

Fig. 2 Container-fluid system: a circular cylindrical tank of radius R, partially filled to an arbitrary depth h

As a first approximation, the analysis of the fluid dynamic problem is generally simplified with the following

assumptions:

rigid and impermeable container;

inviscid, incompressible and irrotational fluid;

if the system is in normal gravity environment, the effect of surface tension can be neglected.

It is of interest to determine the regime of validity of these simplifications. The dimensionless parameters

controlling the validity of the inviscid and negligible surface tension assumptions are the Reynolds number

and the Bond number, respectively. The Reynolds number in the sloshing field is often express in terms of

importance of gravity forces with respect to viscous forces, the so called Galilei number (eq.1)(Arndt,

2008),(Hopfinger, 2009). The Bond number (eq.2) gives information on the relative importance of gravity

forces with respect to surface tension ones (capillary forces). Physically the Bond number can give a

prediction on the shape of the interface: for Bo>>1, gravitational forces are dominant and the free surface is

flat (except right at the wall where the contact angle condition must be satisfied) while for Bo<<1, capillary

forces are dominant and the free surface will be highly curved.

(1)

(2)

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From the hypothesis stated here-above, the field equation can be written as:

in

(3)

The non permeability conditions at the rigid container are:

at r = R (4)

at z = -h (5)

Calling (r, θ, t) (Fig. 3) the vertical displacement of the free surface above the undisturbed free surface and,

for simplicity, setting the value of the pressure on the free surface to zero, the dynamic condition at the free

surface can be derived.

A second condition has to be expressed on the kinematic problem: the vertical velocity of a fluid particle

located on the free surface is equal to the vertical velocity of the free surface itself. Assuming

amplitude excitations and fluid responses to be small, the problem can be linearized and the two conditions

are expressed in equations 6 and 7.

Fig. 3 Free surface displacement in a cylindrical tank.

at (6)

at (7)

Combining equations 6 and 7 we obtain the linearized condition at the free surface:

at (8)

Taking equation 8 and substituting a possible solution of the Laplace equation 3, which satisfies conditions 4

and 5, the natural frequencies of the liquid free-surface can be calculated from the following expression:

(9)

where m refers to the diametral nodes and n to the circumferential nodes for the different kind of vibration;

are the roots of = 0, where J1 is the Bessel function of first order and first

kind(Abramson, 1966). The asymmetric modes are of major concern since they are associated to a horizontal

oscillation of the center of mass (Fig. 4). Symmetrical modes about the vertical axis are possible but not

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excited so often(Institute, 2007). In the case of a lateral excitation, the first asymmetrical mode (m=1, n=1) is

predominant(Lance, 1966) and it is related to the lowest natural frequency of the system. For the first mode,

m=1, we have a unique nodal diameter perpendicular to the direction of motion.

Fig. 4 Scheme of first asymmetrical modes (Dodge, 2000).

2.2 Small amplitude forced oscillations It is possible to determine the linear response to a sinusoidal external excitation of the liquid contained inside

the cylindrical tank of Fig. 2. We assume that the excitation is only acting in the x-direction and it has the

form:

(10)

The linearized fluid field equation is the equation 3, while the linearized free surface condition (eq. 8)

becomes:

at (11)

Substituting the general solution of the potential function it is possible to determine the surface wave

height function:

(12)

with

where only the first asymmetric mode is considered (m=1). It is obvious from equation 12 that how the

applied external pulsation, Ω, approaches a natural frequency, , the solution becomes unbounded due to

the lost of validity of the linearized theory.

2.3 Equivalent mechanical model An equivalent mechanical model of the linear sloshing phenomenon can be studied. The liquid motion inside

the container can be distinguished in two different components: one part which is moving in unison with the

rigid tank and a second part which experiences the sloshing motion. Experiments have shown that the fluid

velocity decreases exponentially with depth (Ibrahim, 2005). For this reason, the fluid dynamics of an

undamped fluid inside a container in response to a lateral excitation might be represented by a non-sloshing

mass and an infinite series of oscillating spring masses as shown in Fig. 5 (Lance, 1966). Each spring-mass

element represents a single slosh mode.

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Fig. 5 Equivalent spring-mass system for undamped propellant slosh in cylindrical tank.

For the pure translational excitation of eq. 10 and with the hypothesis of zero damping, the sloshing equation

for each mass n takes the form:

(13)

Comparing the results from the analytical solution with those of the mechanical model and considering only

the fundamental mode, we obtain the following equivalent parameters(Ibrahim, 2005):

(14)

(15)

(16)

(17)

(18)

where is the non-sloshing mass, is the total fluid mass, and are the mass and the spring

constant concerning the first sloshing mode. An interesting result from this mechanical model is the

dependence of the non-sloshing mass ration, / , and the modal slosh mass ratio, / , on the liquid

depth ratio, h/R, for the firsts three modes, as shown in Fig. 6. It can be observed that the size of the slosh

mass rapidly decreases for all modes exceeding the first.

Fig. 6 Ratio of mechanical model masses and fixed mass to total fluid mass for a circular cylindrical tank (Ibrahim,

2005).

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3. Experimental methodology This chapter is dedicated to the experimental methodology: the test matrix chosen will be stated and the

facility to perform f-PIV during sloshing will be presented with all the sensors necessary to track the external

excitation. One paragraph is dedicated to the choice of the fluorescent particles with particular attention to

their physical properties and their behavior inside the cell when no excitation is present. Finally the pre-

processing related to the detection of the interface and its importance will be presented together with the

chosen settings of the cross-correlation PIV algorithm used to process the images.

3.1 Facility A sketch of the experimental facility developed to characterize the behavior of a fluid contained in a circular

cylindrical cell subjected to sinusoidal excitation is shown in Fig. 7.

Fig. 7 General sketch of the sloshing facility.

The principal element of the experimental setup consists in a partially filled circular cylindrical cell. Two

different cell diameters R = 40mm and R = 20mm are used which, from now on, are respectively called big

and small cell. Both cylindrical cells are designed in order to minimize image distortion due to the curved

walls by means of refractive index matching. Both the cells are supposed to be filled with water, which is the

testing fluid, and which possesses a refractive index nw=1.333, and are manufactured in Plexiglas with

refractive index npx=1.489. In the case of the small cell, the external faces of the vertical walls are flat in

order to prevent major distortion, while the big cell was designed with a double layer containing water. The

drawings of the cells are shown in Fig. 8.The cell is bolted to a table that can generate harmonic oscillations.

From now on this table is here called sloshing table. The latter is able to produce a sinusoidal motion in one

direction with amplitude in the range [1-10] mm and with a frequency in the range [0.1-6.6] Hz. The

accuracy on the selected frequency is 0.02Hz. The position of the table is recorded online by an Optical

Displacement Sensor (ODS) which has an accuracy of 18µm (Confidence level C.L.=95%). An

accelerometer can be mounted on the cell to record the accelerations experienced by the cell itself. The

uncertainty on the acceleration is 0.01mm/s2 (C.L.=95%). The signal of the ODS can be used as input for a

synchronizer in order to perform phase locked image acquisitions with a selected phase of the sinusoidal

motion. In this setup it is possible to study the behavior of the fluid during sloshing by means of backlighting

high speed visualizations and Particle Image Velocimetry phase locked measurements. The optical

arrangement for f-PIV is shown in Fig. 9, where in addition to the standard PIV optics, another filter (high

pass, cut off=580nm) were added in front of the camera in order to see only the fluorescent light and discard

the laser light.

The PIV system used is the LaVision FlowMaster 2D PIV. It is composed by:

A double pulsed Nd:Yag laser (EverGreen) with wavelength 532nm and with a maximum energy of

250mJ

An Imager ProSX Camera with a resolution of 2500x1700 (pixels)2 and a mounted objective of

35mm (Nikkor 35mm F/1.8)

A PTU synchronizer which is able to be triggered externally with a specified event.

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Cell R=20mm Cell R=40mm

Fig. 8 Drawing of the sloshing cells

Fig. 9 Optical arrangement for f-PIV

The signal from the ODS is used as event to trigger the acquisition. Indeed, based on a given threshold on the

ODS voltage, a signal is given as input to a signal generator which sends a trigger signal to the PIV system.

This chain permits to acquire images phased locked in a selected phase of the sloshing table motion. As far

as the spatial resolution is concerned, the resolution of the images recorded using the small cell (R=20mm) is

of 0.0546mm/pixel while for the big cell (R=40mm) is 0.109mm/pixel.

3.2 Seeding and particle motion Fluorescent particles (brand Cospheric) were used. The properties of the particles are summarized in Table 1

in which also the order of magnitude of the Stokes number for this application is shown. The density of the

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particles is very close to the one of water. For this reason some particles can float on the free surface and this

it helps to detect the free surface in the images as it will be explained in a next paragraph.

Table 1 Properties of fluorescent particles

Diameter [µm]

Density [kg m-3]

Stokes number [-]

Absorption band [nm]

Emission peak [nm]

10-45 998 ≈10-4 460-650 605

In order to assess the background motion of the particles due to the density difference, a preliminary PIV

experiment without external excitation has been done. The cell used for this test is the small cell. The delay

between frames is 90ms and the images have been processed with the same parameters used for the PIV

sloshing results. The result of this preliminary test is showing that there is a really slow motion of the

particles when the external excitation is not present (maximum values ~ 1mm/s). For this reason before

starting the acquisition of the experiment, the cell has been always sloshed for about 1000 periods of the

external excitation. Moreover, the external excitation has been kept above 0.6 times the natural frequency fn

in order to be able to neglect the contribution of the particle motion respect to the sloshing motion (1%

maximum).

3.3 Conditions The sloshing motion has been characterize below the first natural frequency related to both geometries. The

natural frequency for the small cell is fn=4.78 Hz while for the big one is fn=3.38Hz. The acquisitions were

carried out phase locked with the external excitation and data were acquired for 10 different phases of the

motion. The experimental test matrix of the experiment is presented in Table 2. As already mentioned in the

previous paragraph, at least 1000 periods lasted before starting the acquisition in order to have a well-

established sloshing motion. The external excitation is a sinusoidal displacement of frequency F and of

amplitude X0=1.2mm.

Table 2 Test matrix

R [mm] F/fn h/R Time discretization

(points)

20 0.6 2.4/1.8 10

0.8 2.4 10

40 0.6 2.4 10

0.9 2.4 10

3.4 Data Processing

Masking One of the major problems, encountered performing PIV on sloshing flows, is the presence of a liquid gas

interface that has a complex shape evolving in time. Hereafter it will be shown that an appropriate post

processing can be done to overcome this problem and consequently have a reliable vector field close to the

liquid/gas interface. In this work PIV fluorescent particles with density close to the one of the liquid

investigated were used. For this reason the particles are homogeneously distributed in the field, can float on

the free surface helping in the detection of the interface position. In Fig. 10 the free surface of a typical PIV

image acquired during the experimental campaign is shown for two different external excitations. From these

images it can be noticed that there are two topology of intensity images: in the case F/fn=0.6, the external

motion is such that a lot of particles float on the free surface; in the case F/fn=0.9, the external motion starts

to be too important and due to a higher mixing the particles are trapped below the interface. In both images,

the presence of “ghost particle” (Berger, 2010), (Sanchis, 2011) as reflection of the real particles on the

interface is evident due to the symmetry of their placement.

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The RMS of the light intensity over 225 frames is shown for the same zoom of Fig. 10. It presents a

maximum on the interface for the case with F/fn=0.6, while it shows a local minimum for the one with

F/fn=0.9. Taking into account all these observations an algorithm for the automatic detection of the interface

and the consequent masking of the PIV images has been written with the commercial code MATLAB.

F/fn =0.6

F/fn =0.9

Fig. 10 PIV images for the case R=40mm, zoom on the interface

F/fn = 0.6

F/fn = 0.9

Fig. 11 RMS of light intensity over 225 frames.

The detection algorithm is based on several successive steps:

1. After applying a threshold to remove the background, the edge function is applied for both frames of

the pairs PIV images in order to detect the contour presents, either the particles or the box contour.

2. The detected edges for each frame are summed in order to detect the full fluid domain. At this step

also the ghost particles are considered part of the fluid domain. The output of this process is

converted to a binary image.

3. The vertical derivative for each pixel column is calculated and the maximum location, where the

fluid domain is identified, is detected. In this way a vector of maxima location is created.

4. The vector calculated at the step 3 is fitted by a linear, second order function or a sum of sinusoids. It

has to be underlined that the real interface will be most of the time in a lower position compared to

the 1st fitting. These firsts four steps are used to delimit the domain.

5. The light RMS of the set of images (see Fig. 11) is processed to identify, depending on the topology

of the RMS, the local maximum or the local minimum. The detection is performed in the domain

delimited by the firsts four step.

6. The detected profile in the light RMS images is fitted with a sum of sinusoids. This fitting was

selected due to the fact that the free surface is expected from the theory to be a sum of sinusoids (see

chapter 2), as consequence the fitting is considered to be the free surface. The error on this detection

is estimated to be of maximum 8 pixels.

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7. Finally, a mask, which excludes all the part external to the walls and above the detected free surface,

is created (see Fig. 13).

This processing is performed for each frame of the pair of PIV images and finally a mask for each frame is

created. Since the commercial code PIV used for PIV processing was not able to apply separated mask for

the first and second frame of a pair of image, the union of the masks was used as final mask for the PIV

processing. In Fig. 13 the result of the masking algorithm is shown.

PIV image

Edge of a single PIV image

Sum of single edge images

1st detection of the “fluid domain”

2

nd detection of the fluid domain

Fitting of the 2

nd detection – free surface detection

Fig. 12 Steps for the detection of the free surface

Frame a Frame b Union

Fig. 13 Result of the masking.

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The developed algorithm shows how using appropriate particles (e.g. fluorescent and of density close to that

one of the liquid) it is possible to perform a good free surface detection for sloshing experiments.

PIV processing The PIV images were processed using the commercial code DaVis8 from the software house LaVision. After

applying the mask the double frame were processed with the multipass correlation algorithm with initial

window size 128x128 (pixels)2 and final window size 48x48 (pixels)

2 and 75% of overlap, obtaining a final

resolution of 12x12 (pixel)2. The final results were filtered eliminated all the vector coming from a window

with SN<1.5.

PIV+PTV processing The PIV+PTV algorithm is a combined algorithm implemented in the commercial code DaVis8 which give

an integrated method PIV+PTV. A preliminary guess of the velocity is given with the standard PIV

evaluation to determine a velocity field. This can be used to get a robust estimator of the local velocity field.

This step was done in window of 64x64 pixels2, 75% of overlap and filtering the results with a SN<1.5. Then

the resolution of the vector field is further refined by the detection and tracking of individual particles (PTV).

The particle detection is looking for the position of the highest intensity and then it fixes an interrogation

window of the size of the particle diameter around the position of the maximum intensity. The particle size

diameter was set to be in the range 4-7 pixels. The interrogation window in the second frame is shifted by the

vector found in the PIV processing result. Finally an individual particle correlation will correlate the

intensity pattern of both windows to determine the exact shift. The final correlation window size was chosen

to be 32x32 pixels. The result of this processing gives a vector for each particle detected. In order to compare

the result of the PIV algorithm with that one of PIV+PTV, the final results of the PTV algorithms has been

converted to a Cartesian grid of size 12x12(pixel)2 by means of a linear interpolation.

4. Results In this section some of the results obtained with the test matrix of Table 2 will be presented. First, a

comparison between the results from PIV and PIV+PTV processing algorithm will be shown. Then

horizontal profiles for both component and different synchronization and L/D will be presented. Finally the

evolution of a local velocity during a typical period is shown. The synchronization of the phase locked

acquisition respect to a period of the external motion is shown in Fig. 14, where the range T#N represents an

indication of the phase of the recording respect to the period of the external motion.

External displacement applied External velocity adimensionalized with the maximum

velocity applied

Fig. 14 Synchronization for the phase locked measurements.

Since the synchronization depends on a voltage threshold over a signal which can be affected by electrical

noise, the trigger has been acquired for each test. This permits to know the exact synchronization for each

phase locked acquisition, even if there is a small phase delay.

4.1 Comparison PIV and PIV+PTV algorithm The two algorithm of the commercial code DaVis8, PIV and PIV+PTV has been tested with the parameters

introduced in the paragraph 3.4. The average velocity magnitude obtained by the PIV+PTV algorithm before

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the reduction to a Cartesian grid is shown on the left of Fig. 15 for the case R=20mm, F/fn=0.8 and h/R=2.4.

It can be noticed that at the interface and at the solid walls, the algorithm is not able to detect properly the

particles. On the interface this is due to the presence of the floating particles which illuminate all the area

around, as it can be seen from Fig. 11. Reducing the field to a Cartesian grid of the same size as the final

window size of the PIV processing, we obtain the result shown on the right of Fig. 15.

PIV+PTV average velocity PIV+PTV on Cartesian grid

Fig. 15 PIV+PTV result for R=20mm, F/fn=0.8, h/R=2.4 and phase locked T3.

Processing the same test with the PIV algorithm with parameters introduced in the paragraph “PIV”, the

average velocity field obtained is shown in Fig. 16 on the left. From the comparison of both algorithms it can

be noticed that the average velocity field has similar typology in both results, while the PIV+PTV results

lose some information on the solid walls and in the vicinity of the interface. Since no improvement can be

seen from PIV to PIV+PTV in the region of most interest, i.e. the free surface, PIV has been chosen to

process the results.

PIV – average velocity PIV – rms of the velocity

Fig. 16 PIV result for R=20mm, F/fn=0.8, h/R=2.4 and phase locked T3.

4.2 Sloshing reconstruction Hereafter some results that can be extracted from f-PIV in sloshing are shown as example. The average

velocity component profiles, Vx and Vz respectively, for R-20mm, F/fn=0.6 and z=1.8h are shown in Fig. 17

and Fig. 18. Considering the synchronization of Fig. 14, it is evident that the horizontal profile is in phase

with external excitation since it present a maximum negative value for T1 and a maximum positive value for

T3.

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Fig. 17 Horizontal profile of the horizontal average velocity component, Vx, for the test with R=20mm, F/fn=0.6

extracted at z=0.9h

From the boundary conditions (7) and the analytical relation for the displacement of the free surface, eq.

(12), a sinusoidal shape is expected for the vertical velocity component. Moreover since the external

excitation is below and far from the natural frequency, the horizontal profile of the vertical velocity

component is to expect to be a sinusoid of 2 diameters wavelength. This is confirmed from Fig. 18, taking

into account the presence of the boundary layer at the wall which slows down the free surface motion.

Indeed in Fig. 18 a reduction of the velocity is observed at the wall.

Fig. 18 Horizontal profile of the vertical average velocity component, Vz, for the test with R=20mm, F/fn=0.6 extracted

at z=0.9h

Finally, an example of the results that can be extracted performing the phase locked average in several

phases of the external motion period is shown in Fig. 19. This figure is obtained extracting the vertical

average velocity component, Vz, for the radial position at -0.63R and the vertical position corresponded to

the first PIV window below the free surface. This position will be called with the greek letter Σ. From Fig. 19

it can be noticed that the order of magnitude of the vertical average velocity component Vz is independent

from the size of the cell, while it is strongly dependent on the ratio F/fn.

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Fig. 19 Vertical average velocity component, Vz, as function of phase in a period extracted for (R, z)= (-0.63R, Σ). Σ

refers to the first PIV window at the interface.

5. Conclusions Fluorescent Particle Image Velocimetry in the form of phase locked measurements has been found a reliable

tool for the characterization of the sloshing phenomenon. A further step has been done for the detection of

the free surface by image processing of the same f-PIV images, for the particular application to the sloshing

phenomenon. The selection of the density of the fluorescent particles in order to apply the explained

detection algorithm is fundamental. Two different processing to retrieve the velocity from double PIV

images has been tested: standard PIV correlation algorithm and a combined approach of PIV with Particle

Tracking Velocimetry algorithm. The latter cannot distinguish the particles at the free surface and at the

solid walls due to their high concentration in those areas. The standard PIV algorithm for the phase locked

acquisition has been evaluated over 225 images and the maximum variations from the average velocity has

been found of the order of 2%. Some results are reported as example of the information that can be extracted

from PIV applied to the sloshing phenomenon. A further step has to be done to retrieve the velocity at the

free surface: once the free surface has been detected thanks to the algorithm explained in the paragraph 3.4:

the PIV grid should be an adaptive grid in order to be locally perpendicular to the interface and don’t lose

any information at the interface (Theunissen, 2010).

Acknowledgments The financial support of the European Space Agency is gratefully acknowledged.

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Arndt, T. a. (2008). Damping behavior of sloshing liquid in laterally excited cylindrical propellant vessels.

Journal of Spacecraft and Rockets.

Berger, R. (2010). PhD Thesis: Integration et validation exprimentale de la methode VOF dans les calculs

arodynamiques automobiles : application au cas de l’entrainement d’eau dans les conduits de

climatisation. von Karman Institute.

Dodge, F. (2000). The "new" dynamic behavior of liquids in moving containers. SRI.

Hopfinger, E. a. (2009). Liquid sloshing in cylindrical fuel tanks. Progress in Propulsion Physics, 279-292.

Ibrahim, R. (2005). Liquid sloshing dynamics: theory and applications. Cambridge University Press.

Institute, T. (2007). Fluid structure interaction effects on and dynamic response of pressure vessels and

tanks subjected to dynamic loading. Berks: Health and Safety Executive.

Lance, T. (1966). Analysis of propellant slosh dynamics and generation of equivalent mechanical model for

use in preliminary voyager autopilot design studies. NASA.

Sanchis, A. a. (2011). Dynamic masking of PIV images. Experiments in Fluids, 51, 871-880.

Theunissen, R. (2010). PhD Thesis: Adaptive image interrogation for PIV - Application to compressible

flows and interfaces. von Karman Institute.

particle image velocimetry applied to water sloshing due...

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