Sloshing in a tank - Development of a water bombing system for fire-fighting


Industrial Study - ENSEEIHT 2013






November, 12th 2012 - March, 15th  2013


Margaux JALBY


Supervisor : Dominique LEGENDRE


In this first part, the context of the project and its objectives will be explained to set the study.

Context of the study

Every year, thousands of forest's hectares go up in smoke because of wildfires wether they are criminal acts or not. In the framework of firefighting, airtankers (or water bombers) play a crucial role. The airtanker can release either pure water or water completed with retardant which delays the spreading of the fire by covering the vegetation with a thin fireproof layer. When the weather is not clement (often during summertime), the water bombers can practice air suveillance, with full tanks, as a precaution. 


Illustration : Airtanker (or water bomber) (source :

Airtankers are subject to important mechanical stress :

  • they have to release the water as close as possible to the fire, so they have to fly at low altitude, with turbulences caused by the fire, and often strong wind.
  • the release of water or retardant causes strong mechanical stress on the plane.


That is the second point hereabove which is the base of this study. Indeed, airtankers are subject to flying difficulties and to important stresses. This can explain the frequency of accidents, and especially rupture in flight. That is why limiting this mechanical stress is a real stake. This limitation can be realised by several ways, and one of them is to reduce the sloshing in the tanks of water bombers and this is the object of our study.







The objectives

In this study, we will be particularly interested in the using of the cargo plane CASA 235-295 A400M of Airbus. The global objective of this work is to develop a release system for firefighting. During the previous works on this subject, the former students focused on the numerical modelling of the print of released water on the ground, and on the ejection system. This year, we will particularly focus on the sloshing in the water (or retardant) tank in order to characterise it and then reduce it.

First of all, it was important to carry out a bibliographical work, to study the articles and thesis which can help us, give us directions to progress in the work to be done. After that, we set the study from the theoretical point of view, writing the preliminary calculations governing the problem. Then, we proceeded to the numerical modelling of the situation, thanks to the software FLUENT. This numerical study is divided in several parts : 

    - Creating of the geometry and mesh representing the tank containing water (or retardant)

    - Submiting the mesh to a "reference" situation and then run several simulations, making the following parameters vary : acceleration of the plane, viscosity of the fluid, fill rate of the tank.  Getting the oscillation frequency and the dissipation time function of these parameters.

    - Running more simulation with a new geometry : a compartmentalized the tank (with inserted walls in the tank) in order to reduce sloshing.

   - Simulating a complete flight with all the phases : take off acceleration, gaining of altitude, release of water (or retardant)



Theoretical analysis

Several theoritical expressions will be established in this part :

The main parameters of the problem are presented thereafter : 

- H=1.2 m : height
​- L= [1.5 ; 3.5] m : lenght
- h=[0.2 ; 1.1] m : initial water level
- a=[1 ; 2.5] m/s2 : plane's acceleration
- g=9.81 m/s2 : gravity 


Preliminary calculation

Elevation e of the fluid in a tank subject to an acceleration a

In a first time, we theoretically calculate the level of water (or retardant) in a tank subject to an acceleration and gravity. Herebelow a diagram of the problem : 


The blue line represent the interface between air and water and we have :

\[ \vec{a}=a.\vec{x} \]

\[\vec{g}=-g.\vec{y} \]

The forces' balance gives :

\[\rho \vec{a}=-\Delta P+\rho\vec{g}\]

After projection, we have : \[\frac{\partial P}{\partial x}=- \rho a \:  and \: \frac{\partial P}{\partial y}=- \rho g \]

On the surface, $ P= constant $ so : \[dP=0= \frac{\partial P}{\partial x} dx + \frac{\partial P}{\partial y} dy \]

Therefore, \[ \rho a.dx + \rho g.dy = 0 \]

 \[ \frac{dy}{dx} |_{surface} = - \frac{a}{g} \]

\[ y = - \frac{a}{g}.x + C \] and we know that $ \: y(x=0)=h \rightarrow C= h $ so $  \: y = - \frac{a}{g}x + h $ and now we can calculate $e$, the level of water : $ e + h = \left(- \frac{a}{g} \right)  . \left(- \frac{L}{2} \right) + h $ and finaly we get :

\[e = \frac{a.L}{2.g}\]


Dispersion time T of waves

This calculation is useful to know the time which is necessary for a wave to disperse when the acceleration moves from a value $ a $  to $ 0 $ . Yet, it is important to precise that the calculation is valid only if the value of the acceleration $ a $ is not too high (not over 2.5g). Herebelow is a diagram of the dimensions of the tank we need for this calculation :


Hypothesis :

\[\omega^2 = (gk + \frac{\sigma}{\rho}k^2) \tanh (kh)\]

Two hypothesis can be made to calculate the characteristic dispersion time :

  • hypothesis of "deep water" : $ \lambda << h $,  $ \omega^2 = gk $,  $ c=\sqrt{\frac{g \lambda}{2 \pi}} $
  • hypothesis of "long waves" : $ \lambda >> h $,  $ \omega^2 = gk^2h $,  $ c=\sqrt{gh} $

In our case, we choose to work with the hypothesis of "long waves" .


Calculation :

We have : \[\frac{\partial E_c}{\partial t} = D \thicksim 2\mu S_{ij}S_{ij} \thicksim 2\mu \frac{V^2}{L^2}\]

\[E_c \thicksim \rho V^2 \rightarrow \mu \frac{V^2}{L^2} \thicksim \frac{\rho V^2}{T}\]

\[\longrightarrow T = \frac{\rho V^2 L^2}{\mu V^2} \thicksim T=\frac{\rho L^2}{\mu}\]

Here, $ L = e $ (calculated just before), so we have : \[ T \thicksim \frac{\rho}{\mu}e^2\]

Thanks to the previous calculation, we know the expression of $ e $ :

\[e = \frac{aL}{2g}\] So we can write : 

\[T = \frac{\rho a^2 L^2}{4\mu g^2}\]


Natural frequency f0

In the light of the framework of this problem, we can make the parallel with the problem of the oscillating pendulum and suppose, in a first time, that the natural frequency of a wave can be written as following : 

\[f_0 = \frac{1}{2\pi} \sqrt{\frac{g}{L}}\]

Numerical analysis

A parametric study was performed with the software Fluent ANSYS in laminar flow.

The four following parameters varied : 

- Fluid's viscosity : $\mu $ = {0.001, 0.25, 0.75, 1} Pa.s

- Tank's lenght : L = {1.5, 2, 2.5, 3, 3.5} m

- Inital water level : h = {0.2, 0.5, 0.7, 1.1} m

- Plan's acceleration : a = {1, g, 2.5g} m/s2

The elevation of water level obtained with Fluent match the theoritical expression for all the cases.

Therefore, theoretical expressions and Fluent's results of dissipation time and natural frequency will be compared in the next parts. 

Setting up of the simulations


The first step of the numerical analysis is creating the mesh which will be our base to launch simulations. To do that, we used the software ICEM CFD, and we created a rectangular mesh representing the tank of water or retardant. The study is done in 2D.       

Once the geometry created, we have to mesh it and to define  the type of boundary conditions (only walls) before exporting it to FLUENT.

Initially, we worked with a rough mesh (5000 nodes) and then we refined it (up to 200 000 nodes), but the results were pretty much the same. So, to avoid longer calculation time, we opted for the rough mesh for all the simulations.


Fluent - VOF Model

The first step was to comprehend the diphasic mode on FLUENT, given the fact that we never used that mode before. There are different options in the diphasic model : Eulerian mode, mixture mode, and Volume Of Fluid mode. We chose to work with the VOF (Volume Of Fluid) option. 

In the VOF model, the input are the following : 

  • number of phases
  • volume fraction scheme (explicit or implicit)
  • inclusion of the implicit body force formulation (optional)
  • coupled level set with VOF (optional)
  • inclusion of open channel flow
  • inclusion of open channel wave boundary conditions
  • inclusion of zonal discretization for applications such as diffused interface modeling in one zone and sharp interface modeling in another zone

Concerning our problem, we chose two phases, an explicit volume fraction scheme and the inclusion of the implicit body force formulation. 


The explicit scheme

Explicit schemes can take on different forms :

  • time-dependent with the explicit interpolation scheme : used for hybrid meshes (containing twisted hexahedral cells)
  • time-dependent with the geometric reconstruction interpolation scheme : used if there is an interest in the time-accurate transient behavior of the VOF solution.
  • time-dependent with the donor-acceptor interpolation scheme : available only for quadrilateral and hexahedral meshes.
  • CICSAM scheme : gives interface sharpness of the same level as the geometric reconstruction scheme and is particularly suitable for flows with high viscosity ratio between the phase.

In our case, we can use georeconstruct scheme or CICSAM scheme.


Including body forces

When large body forces (like gravity or surface tension forces) exist in multiphase flows, the body force and pressure gradient terms in the momentum equation are almoste in equilibrium, with the contributions of convective and viscous terms small in comparison. Segragated algorithms converge poorly unless partial equilibrium of pressure gradient and body forces is taken into account. That is why the optional "implicit body force" treatment has its interest in FLUENT. Indeed, this option account this effect and in this way make the solution more robust by achieving a realistic pressure field very early in the iterative process.

Then, after choosing the model and the options, we defined the phases : their material properties, interaction between them (here surface tension). In the VOF model, you can specify the primary and secondary phases as you prefer. In our case, we have air and water and it is preferable to specify the compressible ideal gas as the primary phase to improve solution stability. So the primary phase is air and the secondary is water.

If you have a difference between the densities of the two phases around 1000 or more of that, you have to choose, in the boundary conditions>operating conditions an operating density equel to 0 kg/m3. This choice allow users to have a better convergence.



  • Initialize all the zone with a water volume fraction equal to 0
  • Define a region of water (specifying the level of water) : >Adapt>Region and click on Mark
  • Patch the water volume : Initialization>Patch. Choose, water volume function, value equals to 1, hewahedron-0. Click on Patch
  • Check that the two phases are well defined : >Display>Contours>Phases>Water volum fraction (options : click on filled).










Recap chart of the different cases

Several parameters have influence on the results of the simulations and can be adjusted :

  • the accelereration of the plane $ a [m/s²] $
  • the viscosity of the fluid (water or retaradant) $ \mu [Pa.s] $
  • the length of the tank $ L [m] $
  • the fill rate $ N=\frac{h}{L} $, or water level $ h [m] $ in the tank

By making vary all these parameters, we tested different cases ; herebelow is a recap chart of the different simulations :

Dissipation time

The theoretical expression of waves' dissipation time is $ \displaystyle T = \frac{\rho a^2 L^2}{4 \mu g^2} $.

The influence of the parameters (viscosity $\mu$, tank's length L, initial water level h, and plan's acceleration a) will be studied to obtain the limits of this expression and to validate the numerical model. 

The following drawing remind the configuration of the problem :



Influence of viscosity

For the first simulations, the liquid phase is water but then we tested a liquid phase which comes up to a retardant substance. We take as hypothesis that the viscosity of a retardant is  between the water viscosity and a thousand times the water viscosity. So we run four simulations for the four following cases with the same acceleration (a=1m/s²):


Water level on the left wall function of time for the four viscosities 

For the four cases, we can see the oscillations obtained with Fluent and the theoretical envelope :  $ y=A.e^{\frac{t}{T}} $


  • $ \mu = 1000 \mu_{water} $ 


  • $ \mu = 750 \mu_{water} $ 


  • $ \mu = 250 \mu_{water} $ 



  • $ \mu = \mu_{water} $ 

The results fit the theoretical dissipation time for the four viscosities


Waves dissipation for the four viscosities (exponential envelopes)

The more retardant there is, the shorter is the dissipation time  and the briefer the sloshing is.

Influence of tank's length

Water level on the left wall for several tank's length 


- L=1.5 m                                                                                                 - L=2m


- L=2.5 m                                                                                                 - L=3m


- L=3.5 m  


The results fit the theretical dissipation time for most of tank's lengths. However, for L=1.5m, Fluent results does not match the theory. For this case the length (L) and the height (H) are almost the same so the long waves hypothesis is no longer valid.


Waves dissipation for several tank's length (exponential envelopes)  

Smaller is the tank,  briefer is the sloshing.

Influence of initial level

Another parameter to take into account is the water/retardant intial level (or fill rate) in the tank.


The three cases in the middle of the graph (green, magenta and blue) have the same dissipation time. For the others two cases, the sloshing is briefer. During these two cases, there is a wetting, that is to say that the free surface touch the bottom. 

So, without wetting, the dissipation time does not depend on the intial water level, and with wetting, the sloshing is brief and theoretical dissipation time is wrong.

Influence of the acceleration

A really important factor in this study is the acceleration of the plane. This is the more complicated factor to study because of high accelerations, with which wetting happens.

Firstly, the theoretical dissipation time is proportional to the squared acceleration over the squared gravity : $ T^2 \propto \frac{a^2}{g^2} $.  From the plot, the dissipation time is larger for the high acceleration (blue curve). There is a contradiction with the theory.

In fact, $ T_{blue curve_ {th}} \approx 10^{4} s $ and $ T_{blue curve_{plot}} \approx 50 s $

But in this case (high acceleration), wetting happens. The following drawing shows the free surface form : 

We try to imagine a new case : we take the same tank turned of 90 degrees and we reverse the acceleration and the gravity : 

We obtain a new dissipation time : $ T'_{blue curve_{th}} = 500 s $. This result is closer but it is not exact. 


$\Rightarrow$ To conclude this part about dissipation time, its theoretical expression is correct without wetting.

Natural frequency

Thanks to a parallel with the oscillating pendulum problem, we suppose that the waves' natural frequency has the following expression : $ f_0= \frac{1}{2\pi} \sqrt{\frac{g}{L}} $.

In a first time, we will check that the fluid's viscosity does not influence the natural frequency and we will verify the theoretical expression. 

Influence of fluid's viscosity

The natural frequency is the same for the four viscosities so the viscosity does not have any influence on the natural frequency.

The theoretical expression gives a natural frequency equal to 0,27 Hz but with Fluent results (plot) we obtain a natural frequency equal to 0,35 Hz. The expression is not right.

The expression of the natural frequency is modified supposing that it depends on all the dimensions of the problem. The expression becomes : $ \displaystyle f_0 = \frac{k}{2 \pi} \sqrt{\frac{g}{L*}} $ with L*=L*(L,h) a characteristic length and k a constant. We will find this characteristic length studying the influence of the initial level (h) and of the tank's length (L) on the natural frequency (f0).

Influence of initial level

The Fast Fourier Transform of initial water level for five various initial water levels (first plot) allows to know that the initial water level has an influence on the natural frequency. Thanks to these Fast Fourier Transforms, the natural frequency can be extracted for all the cases and we obtain the second plot.





This plot represents the natural frequency (obtained after a fast fourier transform) for four initial water levels. Blue points are the Fluent results. A curve fitting returned a square root function (red points). So the natural frequency is proportional to the squared root of the initial water level : $f_0 \propto \sqrt{h} $. Therefore the characteristic length is proportional to the inverse of the initial level : $ L^* \propto \frac{1}{h} $. We can suppose that the characteristic length is proportional to the squared of tank's lenght over the intial water level : $ L^* \propto \frac{L^2}{h} $. We will verify this expression studying the influence of tank's lenght on the natural frequency.

Influence of tank's length

The Fast Fourier Transform of water level for five various tank's length (first plot) allows to know that the length have an influence on the natural frequency. Thanks to these Fast Fourier Transforms, the natural frequency can be extracted for all the cases and we obtain the second plot.


This plot represents the natural frequency (obtained after a fast fourier transform) for five tank's length. Blue points are the Fluent results. A curve fitting returned an inverse function (red points). So the natural frequency is proportional to the inverse of the tank's length : $ f_0 \propto \frac{1}{L} $. Therefore the characteristic length is proportional to the squared of tank's length : $ L^* \propto  L^2 $. 

Thus the characteristic length is proportional to the squared of tank's lenght over the intial water level : 

$ \Rightarrow \displaystyle L^* \propto \frac{L^2}{h} $  

All cases will be ploted in function of this characteristic length in the next part.

Influence of the characteristic length

Thanks to the previous studies, the expression of the characteristic length is known as $ \displaystyle L^* = \frac{L^2}{h} $. All the studied cases are gathered on the following plot : 

Blue points are Fluent Results. Red points are the mathematical approximation :

$ \displaystyle f_0 = \frac{2.73}{2 \pi} \sqrt{\frac{g}{L*}} $

Results and approximation match for almost all the cases. For the first one, there is a difference which can be explained by the non-validity of the long waves hypothesis for this case (L=1.5m).

The characteristic length and so the expression of the natural frequency were groped for but we find the waves celerity. In fact, with the long waves hypothesis, the waves' celerity is equal to : $ c = \sqrt{gh} $

So the new expression of the natural frequency is :

$ \displaystyle f_0 = 0.53 \frac{c}{L} $

$\Rightarrow$ To conclude this part about the natural frequency, we found an expression function of a characteristic length and it does not depend on the fluid's viscosity.

Split tank

A solution to reduce sloshing in the tank is to compartmentalize the latter. Indeed, inserting walls into the tank can be a way to counter the sloshing of water or retardant, and consequently it can help to reduce the mechanical stress exerted on the plane. To realize that part of the study, we create a new geometry, including the new walls. Herebelow is a diagram of this new geometry in which we inserted four walls :


Compartmentalized tank with four walls (same dimensions of the tank as before)


Herebelow are two plots : water level function of time on the left wall and on the fourth inside wall, and fast fourier transform of water level for the same two walls :

We can notice that there is the same frequency on the two walls, but a phase difference appears. Plus, if we compare Fluent's results with the theory, we have :

Analysing this values, we see that the water level elevation is the same with Fluent and in theory. However, it is not the case for frequency and dissipation time. Indeed, Fluent results differs from theory for frequency and dissipation time. This can be due to the fact that the "long waves" hypothesis is no longer valid or to the fact that each part of the tank communicates with the others at the bottom.

Apart from that, the compartmentalized tank solution obviously helps to reduce the sloshing (elevation of water level much less important than before).



Finally, assuming that the long waves hypothesis is valid, we characterized the sloshing in the tank through different parameters which are :

Plus, we also made sure that the compartmentalized solution really helps to reduce the sloshing in the tank.

Then, after making sure that the compartmentalized tank solution helps to reduce the sloshing, we tested it on a simulation of a complete flight. Indeed, we simulated all the phases of a water bomber flight : take off acceleration phase, climbing phase, constant velocity phase (plane approaching the release zone). This last simulation showed that, just before releasing, the sloshing is stopped in the tank, as expected.

To finish, we could say that the study could be continued by :

Furthermore, it was, from several points of view, an enriching study. Indeed, the complexity and diversity of this industrial project had several interests. First of all, new functionnalities of the CFD-Software Fluent were mastered. This project also helped developing communicational and organizational skills. Finally, it represented the opportunity to carry out a project in an imposed period of time, respecting expectations.


Throughout our work we refered to several websites and documents (articles, thesis) of which the references are herebelow :


- ROYON-LEBEAUD Aude, " Ballottement des liquides dans les réservoirs cylindriques soumis à une oscillation harmonique ", Mars 2005 (thesis)

- —EL-KAMALI Mahdi, « Ballottement des Liquides avec Tension Superficielle : Etudes Statique et Dynamique », March 2010 (thesis)

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Photographies :