3D Simulation of Dam-break effect on a Solid Wall using
Smoothed Particle Hydrodynamic

Suprijadia,b, F. Faizalb, C.F. Naaa and A.Trisnawanb

aDepartment of Physics, bDepartment of Computational Sciences

Faculty of Mathematics and Natural Science, Institut Teknologi Bandung

Jl. Ganesha No.10, Bandung 40132, Indonesia

email<EMAIL_ADDRESS>

Abstract. Dam is built for water supply, water flow or flooding control and
electricity energy storage, but in other hand, dam is one of the most
dangerous natural disaster in many countries including in Indonesia. The
impact of dam break in neighbour area and is huge and many flooding in remote
area, as happen in Dam Situ Gintung in Tangerang (close to Jakarta) in 2009.
Smoothed Particle Hydrodynamics (SPH), is one of numerical method based on
Lagrangian grid which is applied in astrophysical simulation may be used to
solve the simulation on dam break effect. The development of SPH methods
become alternative methods to solving Navier Stokes equation, which is main
key in fluid dynamic simulation. In this paper, SPH is developed for
supporting solid particles in use for 3D dam break effect (3D-DBE) simulation.
Solid particle have been treated same as fluid particles with additional
calculation for converting gained position became translation and rotation of
solid object in a whole body. With this capability, the result of 3D-DBE
simulation has been varies and interesting. The goals of this simulation is
for analyse fluid and solid particle interaction by using two different
scenario. The first scenario relation between height of fluid to a solid wall
barrier and second scenario, is to study relation between solid wall and its
collapse time by dam break. The results show sliding distance of a solid wall
is depend on a fluid height, and relation between fluid heght and wall
dimension will be discussed too in this paper.
Keywords: smoothed particle hydrodynamic, dam break, solid-fluid interaction

##  1 Introduction

Dam is built for water supply, water flow or flooding control and electricity
energy storage, but in other hand, dam is one of the most dangerous natural
disaster in many countries including in Indonesia. Because of the age, there
are some evidence of natural disaster by a dam failure and break were reported
for example in Situgintung Tangerang, close to Jakarta was happen in 2009 [1].
The impact of dam break in neighbour area and is huge and many flooding in
remote area, the computer calculation to predict effect of the dam broken is
important including its vast impact[2].

Smoothed Particle Hydrodynamics (SPH), is one of numerical method based on
Lagrangian grid which is applied in astrophysical simulation may be used to
solve the simulation on dam break effect. The development of SPH methods
become alternative methods to solving Navier-Stokes equation, which is main
key in fluid dynamic simulation. As a standard test case in computational
fluid dynamic the dam break simulation were often reported in many cases,
including using SPH methods [5]. The simulation using SPH became emerge
whether using two or three dimension. In other hand, simulation of solid
particle is many reported direct numerical simulation [6]. Interaction between
fluid and solid particle is still open ended to studying. Including simulation
base on SPH method.

In this paper, SPH is developed for supporting solid particles in use for 3D
dam break effect (3D-DBE) simulation. Solid particle have been treated same as
fluid particles with additional calculation for converting gained position
became translation and rotation of solid object in a whole body. With this
capability, the result of 3D-DBE simulation has been varies and interesting.
The goals of this simulation is for analyse fluid and solid particle
interaction by using two different scenario. The first scenario relation
between height of fluid to a solid wall barrier and second scenario, is to
study relation between solid wall and its collapse time by dam break. The
results show sliding distance of a solid wall is depend on a fluid height, and
relation between fluid height and wall dimension will be discussed too.

##  2 Smoothed Particle Hydrodynamics

The governing equation for fluid dynamics is Navier-Stokes equations consist
of conservation of momentum, conservation of mass and conservation of energy.
Assume the system of dam is in isothermal state, then we can neglect the
calculation of fluid’s thermal energy. The momentum equation has been written
as follows:

$\rho\,\frac{\partial\mathbf{u}}{\partial
t}+\rho\,\left(\mathbf{u}\cdot\nabla\,\mathbf{u}\right)=-\nabla
p+\mu\nabla^{2}\mathbf{v}+\rho\mathbf{g}$ (1)

with $\rho$ is fluid density, t is time, v is fluid velocity, p is pressure,
$\mu$ is viscosity, and f is external force. The fluid motion is also governed
by the conservation of mass, which stated by

$\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\mathbf{u}\right)=0$ (2)

Equations (1) and (2) will be solved by numerical approximation of SPH method.
The method simply described by discreet collection of particles or balls for
approximating fluid body. Every particle mathematically influence the
neighbouring particles via kernel function.

$W\left(r,l\right)=\frac{1}{l\sqrt{\pi}}\exp{\left(\frac{-r^{2}}{l^{2}}\right)}$
(3)

Here $W$ is kernel function, $x$ is particle distance and $l$ is smoothing
factor. The kernel function stated in equation (3) now can be use as weighting
function to calculate all physical quantity, gradient and Laplacian attached
to the particle, here we calculate the solution of equation (1) and (2) using
weighted interpolation concept and solved them numerically. The detail of this
method described clearly in [4].

Figure 1: Kernel radius in SPH simulation

Figure 1 illustrated the approximation of fluid body by collection of moveable
particle interacted along kernel radius $r$. The numerical SPH for momentum
and mass conservation int total derivative form respectively are [3]:

$\frac{D\mathbf{u}_{a}}{Dt}=\sum_{b}m_{b}\left(\frac{p_{b}}{{\rho_{b}}^{2}}+\frac{p_{a}}{{\rho_{a}}^{2}}+\Pi_{ab}\right)\nabla_{a}W_{ab}+\mathbf{g}$
(4)

and

$\frac{D{\rho}_{a}}{Dt}=\sum_{b}m_{b}\,\left(\mathbf{u}_{b}-\mathbf{u}_{a}\right)\cdot\nabla_{a}W_{ab}.$
(5)

With index $a$ and $b$ denoted particle index, $\Pi_{ab}$ is numerical
viscosity described in[3]. Here the kernel function $W_{ab}$ and it’s
derivative solved analytically. Equation (4) and (5) can be solved using
direct numerical integration such as Euler method. Another well-known methods
is Runge-Kuta method, Leap frog method and Verlet method.

##  3 Solid - Fluid Interaction

In DBE simulation, one of the important aspect is solid motion exerted by
fluid energy. Here we use solid body dynamic to simulate the motion of fluid.
Once the total force and total torsion exerted by fluid to surface of solid
already known, so do translational and rotational motion of solid can be
calculated.

Figure 2: Force diagram and solid wall motion

Figure 2 show the basic physical scheme of the solid fluid interaction with
strong assumption that no flow across the surface of fluid particle. Some
important aspect such as floor friction and center mass of solid body is also
taking into account.

Figure 3: Solid particle motion schematic

To simplified the continuum aspect of solid body, solid object will be
approximated also by collection rigid particles or balls with certain physical
quantity governed by newton equation of translation and rotation. The center
of mass of such discreet particle are also easy to calculate. Figure 3 showed
the step process of solid particle position update in rotational motion.

##  4 Results and Discussion

The scenario of DBE simulation in three dimension described as follow.
Initially the fluid position in in cubic shape dam for simplification and
without initial velocity of fluid assuming the fluid is at rest. At a certain
distance lied solid wall to block flow of the fluid. Another assumption were
made that there no crack or fracture in solid wall.

Figure 4: Simulation scenario diagram, $h_{2}$ is initial fluid height,
$h_{1}$ and $w_{1}$ is height and width of solid

Figure 4 shown the description of initial arrangement of solid and fluid
particles. As described earlier the arrangement of fluid particle in box shape
dam and initially particle at rest. As the time step moves the gravity force
work on the fluid particles and they move with the same for as if happened in
the dam failure suddenly.

Some parameter involved in 3D simulation in Figure 4 is initial fluid height,
solid wall dimension. Later in this section will be shown some result in 3D
DBE simulation with some variation in simulation parameter.

Figure 5: 3D simulation of DBE a $t_{1}=0s$, $t_{2}=0.228s$, $t_{3}=0.543s$
and $t_{4}=1.068s$, $h_{1}=55m$ and $h_{2}=40m$ and $w_{1}=40m$

Figure 5 show general result of DBE simulation wall with height $55m$ and
fluid dam with $40m$ in height at time $t_{1}=0s$, $t_{2}=0.228s$,
$t_{3}=0.543s$ and $t_{4}=1.068s$. The frame shows the fall process of the
wall as the effect of fluid flow.

Figure 6: Wall height effect on DBE simulation at $t=0.54s$. Left $h_{1}=55m$,
right $h_{1}=60m$

The variation in solid wall height is made to test the stability of moveable
wall under the same energy of fluid flow exerted on the wall surface. It is
done by setting the fluid height at $40m$. Figure 6 shows the effect of wall
height to the fall process under the influence of flow caused by dam break
event. Here the frame plot the position of particles at the same time step
$0.54s$ and the same initial height of fluid $h_{2}=40m$ but in different wall
height. The frames show different response of falling time. Higher wall fall
faster than the lower one at other parameter were keep constant. Here we can
say that the dimension of the wall influence the stability of the wall object
under the same outer forces. If the width of the wall has been increasing the
high wall also become relative stable.

In another scenario the variation of fluid height were tested to the
relatively stable wall object and observe the wall sliding distance.

Figure 7: Fluid height effect to the sliding distance on DBE simulation

Figure 7 shows sliding distance plot versus initial fluid height. The trend of
the data seems to be non linear as the height of fluid increase over 55 m,
this is make sense because of volume of the fluid increase by the power of
three to the height measure and linearly increase the potential energy of
fluid.

##  5 Conclusion

We successfully simulate 3D-DBE simulation using SPH methods, and fluid-solid
interaction capability was applied to study the dam break effect on a solid
wall object, the SPH can be used to simulate a solid particle simulation. Some
simplified assumption that no crack or fracture happen in the solid wall
object thus no separation of solid body as the impact of the fluid momentum.
The simulation results showed that the fluid height and solid wall thickness
and height play a significant effect on solid wall translational and
rotational motion.

For the future work it is also challenging to study the fracture event,
crumbling process or separation of solid body by the effect of momentum from
the fluid flow and made realistic devastation process from dam failure. This
features will increase the computational cost and difficulties in algorithm.

## Acknowledgement

A part of this research was funding by Research and Innovation Program of
Research Group ITB, grant No:243/I.1.C01/PL/2013.

## References

  * [1] BBC (2009). Indonesian dam burst toll rises. BBC News Asia-Pacific
  * [2] D. Munger (2009). Economic Consequences Methodology for Dam Failure Scenarios. Reclamation Managing in the West,Technical Memorandum Number EC-2009-01 T. U.S. Department of the Interior Bureau of Reclamation Technical Service Center Security, Safety and Law Enforcement
  * [3] J.J. Monaghan (1992). Smoothed Particle Hydrodynamic. Annu. Rev. Astron. Astrophys, 30, 543 - 74.
  * [4] G.R. Liu and M.B. Liu (2007). Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific Publishing, Singapore
  * [5] M. Kelager (2006). Lagrangian Fluid Dynamics using Smoothed Particle Hydrodynamics. Master Thesis. Univiersity of Copenhagen.
  * [6] P.Y. Huang, J. Feng, H.H. Hu and D.D Joseph (1997). Direct simulation of the motion of solid particles in Couette and Poiseuille flows of viscoelastic fluids. J. Fluid Mech., 343, 73 - 94.


[a]Salvador Urrea222Work done in collaboration with: Pilar Coloma, Jacobo
López-Pavón and Salvador Rosauro-Alcaraz

# Non-unitarity and sterile neutrinos at the DUNE near detector

###### Abstract

We study the capabilities of the DUNE near detector to probe the 3+1 sterile
formalism and the non-unitarity of the leptonic sector. We add to the current
analyses in the literature, the use of the charged current events for the
$\nu_{\tau}$ appearance channel and the consideration of the energy spectral
uncertainty (shape uncertainty) for the different channels which plays an
important role, especially at the near detector, and has been usually
overlooked in the literature. We find that even with this more conservative
and realistic approach, we still obtain an improvement in the sensitivity with
respect to the current bounds.

##  1 Introduction

The new generation of long-baseline neutrino oscillation experiments such as
DUNE are expected to probe the robustness of the three neutrino family
oscillation picture to a high level of accuracy. They will also give stronger
bounds and/or possibly detect non-standard deviations from the standard
picture.

The ability to probe new physics in oscillations is going to be heavily
affected by systematic uncertainties coming from our lack of knowledge, both
theoretical and experimental, of the cross sections and fluxes involved. In
the case of the near detector (ND), the lack of another detector which would
reduce uncertainties, results in larger systematics. We will see that the
uncertainties in the energy spectra will play the main role in these searches.

##  2 Theoretical framework and notation

In this section, we will briefly introduce the two theoretical scenarios that
we will consider in our sensitivity analysis: non-unitarity and sterile
neutrinos. For a more detailed discussion see [1].

A simple way to account for light neutrino masses is to add singlet fermions
to the SM field content. The matrix mixing between the mass and flavor basis
can be expressed as:

$\mathcal{U}=\begin{pmatrix}N&\Theta\\\ R&S\\\ \end{pmatrix}\,,$ (1)

Here $N$ is a $3\times 3$ non-unitary matrix corresponding to the PMNS active-
light sub-block, and $\Theta$ is the $3\times n$ sub-block parameterizing the
mixing between active and heavy neutrinos, with $n$ being the number of new
states. The $R$ ($S$) sub-block gives the mixing between the sterile and light
(heavy) states and does not play any role in neutrino oscillations.

The value of the mass of the new states will determine whether they can be
produced in the neutrino beam or not, changing the phenomenology we expect to
observe. We have two distinct cases: if their mass lies above the production
threshold of the neutrino source they cannot be produced, this scenario is
usually referred to as Non-unitarity (NU) and typically mass scales above the
electroweak scale are assumed; whereas if the new states are kinematically
accessible, they can be produced in the neutrino beam leading to new
oscillation frequencies driven by the new masses. We refer to the latter as
light sterile neutrinos.

###  2.1 Parameterization

We will parametrize the deviations from unitarity of the matrix $N$ as
follows:

$N=(I-T)U$ (2)

where $T$ is given by:

$T=\begin{pmatrix}\alpha_{ee}&0&0\\\ \alpha_{\mu e}&\alpha_{\mu\mu}&0\\\
\alpha_{\tau e}&\alpha_{\tau\mu}&\alpha_{\tau\tau}\end{pmatrix},$ (3)

and $U$ plays the role of the standard(unitary) PMNS matrix up to small
corrections encoded in $\alpha_{\gamma\beta}$.

###  2.2 Non-Unitarity from new physics above the electroweak scale

In this scenario, the heavy states are integrated out from the low energy
spectrum and are thus not kinematically accessible in the experiment. At very
short distances, as the ones in the DUNE ND, the standard oscillation do not
have time to develop, yielding the following simple expressions for the
oscillations probabilities:

$\displaystyle P_{\mu e}(L=0)$ $\displaystyle=$ $\displaystyle|\alpha_{\mu
e}|^{2},$ $\displaystyle P_{\mu\tau}(L=0)$ $\displaystyle=$
$\displaystyle|\alpha_{\tau\mu}|^{2}.$ (4)

###  2.3 Sterile neutrinos & Non-Unitarity from new physics at low scales

As we mentioned, sterile neutrinos are kinematically accesible and we expect a
new oscillation frequency. For simplicity, we consider the $3+1$ scenario in
which only one new neutrino is introduced.

At very short baselines, we find the following simple expressions for the
oscillation probabilities:

$\displaystyle P_{\gamma\beta}=$
$\displaystyle\sin^{2}2\vartheta_{\gamma\beta}\sin^{2}\left(\dfrac{\Delta
m^{2}_{41}L}{4E}\right),\quad$
$\displaystyle\rm{with}\;\;\sin^{2}2\vartheta_{\gamma\beta}\equiv
4\,|\mathcal{U}_{\beta 4}|^{2}|\mathcal{U}_{\gamma 4}|^{2},$ $\displaystyle
P_{\beta\beta}=$ $\displaystyle
1-\sin^{2}2\vartheta_{\beta\beta}\sin^{2}\left(\dfrac{\Delta
m^{2}_{41}L}{4E}\right),\quad$
$\displaystyle\rm{with}\;\;\sin^{2}\vartheta_{\beta\beta}\equiv|\mathcal{U}_{\beta
4}|^{2}\,.$ (5)

In the averaged-out regime ($\Delta m^{2}_{41}L/E\gg 1$) the
$\sin^{2}\left(\dfrac{\Delta m^{2}_{41}L}{4E}\right)$ average out to 1/2
leading to following energy independent expressions:

$\displaystyle P_{\mu e}=2\,|\mathcal{U}_{\mu
4}|^{2}|\mathcal{U}_{e4}|^{2}=2|\alpha_{\mu e}|^{2},P_{\mu\tau}$
$\displaystyle=$ $\displaystyle 2\,|\mathcal{U}_{\mu 4}|^{2}|\mathcal{U}_{\tau
4}|^{2}=2|\alpha_{\tau\mu}|^{2}$ (6)

Notice that, barring the factor 2, the new physics effects here are identical
to those given in Eq. 2.2. Thus we can consider this regime as a low scale
source of non-unitarity

##  3 Results

In this section, we will present our main results of our paper, for the rest
of the results including the sensitivity analysis for NSI in detection and
production check [1].

Before going to the results, we will comment on what type of systematics will
be important to have sensitivity to the different scenarios we have just
described. At the DUNE ND the sensitivity will be dominated by the spectral
information, since even for a value of $\alpha_{\gamma\beta}$ that saturates
the present bound the signal is much smaller than the background. The
sensitivity comes mainly from the differences in energy shape between the
background and signal. Shape uncertainties are generally overlooked in the
literature.

###  3.1 Non-unitarity and sterile averaged-out regime

Comparing Eq.(2.2) with Eq.(6), we see that in the appearance channels there
is only a factor 2 difference, and therefore, the results can be rescaled from
one scenario to the other. Even though, in oscillations they only differ by
this factor 2, the bounds that apply for both scenarios are different. In the
Non-unitarity scenario we have very strong bounds from high-precision
measurements of electroweak processes which are not expected to be improved by
near future oscillation experiments[2]. However, these constraints do not
apply when the sterile neutrinos are kinematically accessible. Indeed, the
bounds we show in Fig 1, which come from oscillation experiments are
considerably weaker and apply to the average-out regime of the steriles. See
[1] for details.

Therefore, our results shown in Fig. 1 for $\alpha_{\mu e}$ (left panel) and
$\alpha_{\tau\mu}$ (right panel), respectively, correspond to the averaged-out
regime of the steriles. The different lines show the results for different
choices of systematic uncertainties as indicated by the labels, as a function
of running time.

Figure 1:  Sensitivity to the off-diagonal NU parameters $\alpha_{\mu e}$
(left panel) and $\alpha_{\tau\mu}$ (right panel). The lines show the
sensitivity at 90% CL for 1 degree of freedom (d.o.f.) as a function of the
running time $\Delta t$, for different choices of the systematic
uncertainties. The vertical lines indicate the changes between neutrino and
antineutrino running modes (in the nominal beam scenario) as well as the
change to the high energy beam. For reference, the shaded gray areas show the
region disfavored at 90% CL by current constraints. For $\alpha_{\mu e}$, the
dot-dashed red line indicates the expected sensitivity using the far detector
data (taken from Ref. [2]), while for $\alpha_{\tau\mu}$ the expected
sensitivity would be worse than the range shown in the figure and is therefore
not shown.

###  3.2 Sterile neutrino oscillations

In the sterile neutrino scenario, the DUNE ND will be sensitive to different
oscillation channels. Each of them will be sensitive to a different set of
mixing matrix elements as described in Sec. 2, see Eq. (2.3).

Figure 2:  Expected sensitivity to the sterile neutrino scenario, for
oscillations in the $P_{\mu\mu}$ and $P_{\mu\tau}$ channels for the top and
lower panels, respectively. The shaded regions show current constraints from
other experiments, while the colored lines indicate the sensitivity for the
DUNE ND, for different choices of systematic uncertainties as indicated by the
legend. Finally, the colored blue bands indicate the increase in sensitivity
due to the addition of 3.5 years of data taken in the high energy mode.

In Fig. 2 we present the results obtained for the sensitivity for $\nu_{\mu}$
dissapearance and $\nu_{\tau}$ appearance channels. Shape uncertainties have a
large impact on the results in all channels. It is worth mentioning that the
most dramatic effect occurs for the sensitivity via $P_{\mu\mu}$, where shape
uncertainties at the level of 2% lead to a decrease in sensitivity of over an
order of magnitude with respect to the scenario where only normalization
uncertainties are considered. This is expected, since the $\nu_{\mu}$ flux
constitutes the largest component of the incoming flux and the same relative
error results in a larger absolute error.

Figure 3, on the other hand, shows the DUNE ND sensitivity to combination of
mixing matrix elements that appear in $P_{\mu e}$. Looking at Eq. 2.3 we can
see that the same parameters are accessible by the combination of $P_{ee}$ and
$P_{\mu\mu}$. Therefore, the three channels together will give us the best
sensitivity. From these results, we see that for DUNE ND is expected to probe
the regions favoured by the LSDN and MiniBooNE results.

Figure 3:  Expected sensitivity to the sterile neutrino scenario, for
oscillations in the $P_{\mu e}$ channel. The colored lines indicate the
sensitivity for the DUNE ND, for different choices of systematic
uncertainties. In each case, the region to the right of the lines would be
disfavored at 90% CL (2 d.o.f.). Finally, the shaded band to the left of the
dashed lines indicate the increase in sensitivity due to the addition of 3.5
years of data taken in the HE mode.

##  4 Conclusions

In this work we have studied the sensitivity to new physics affecting neutrino
oscillations in the DUNE ND. We have presented a conservative but realistic
approach, including shape uncertainties, for different scenarios and channels.
Even though, the expected sensitivity is, therefore, reduced we generally
expect an improvement over present bounds. With this work we want to stress
the importance of the reduction of these type of systematics from an
experimental and also theoretical point of view.

Acknowledgments: I acknowledge support from Generalitat Valenciana through the
plan GenT program (CIDEGENT/2018/019) and from the Spanish MINECO under Grant
FPA2017-85985-P.

## References

  * [1] P. Coloma, J. López-Pavón, S. Rosauro-Alcaraz and S. Urrea, _New physics from oscillations at the dune near detector, and the role of systematic uncertainties_ , _Journal of High Energy Physics_ 2021 (2021) .
  * [2] M. Blennow, P. Coloma, E. Fernandez-Martinez, J. Hernandez-Garcia and J. Lopez-Pavon, _Non-Unitarity, sterile neutrinos, and Non-Standard neutrino Interactions_ , _JHEP_ 04 (2017) 153 [1609.08637].