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71st International Astronautical Congress (IAC) – The CyberSpace Edition, 12-14 October 2020.
Copyright 2020 by Mr. James E. McKevitt. Published by the IAF, with permission and released to the IAF to publish in all forms.
IAC-20-E2.1.9 Page 1 of 12
IAC-20-E2.1.9
ASTrAEUS: An Aerial-Aquatic Titan Mission Profile
James E. McKevittab*
a Department of Aeronautical and Automotive Engineering, Loughborough University, Epinal Way, Loughborough
LE11 3TU, j.e.mckevitt-17@lboro.ac.uk
b Institute for Astrophysics, University of Vienna, Türkenschanzstraße 17, 1180 Wien, Austria,
jamesm20@univie.ac.at
* Corresponding Author
Abstract
Key questions surrounding the origin and evolution of Titan and the Saturnian system in which it resides remain
following the Cassini-Huygens mission. In-situ measurements performed at key locations on the body are a highly
effective way to address these questions, and the aerial-aquatic platform proposed in this report serves to deliver
unprecedented access to Titan's northern surface lakes, allowing an understanding of the hydrocarbon cycle, the
potential for habitability in the environment and the chemical processes that occur at the surface. The proposed heavier-
than-air flight and plunge-diving aquatic landing spacecraft, ASTrAEUS, is supported by the modelling of the
conditions which can be expected on Titan's surface lakes using multiphysics fluid-structure interaction (FSI) CFD
simulations with a coupled meshfree smoothed-particle hydrodynamics (SPH) and finite element method (FEM)
approach in LS-DYNA.
Keywords: Titan, Bioinspiration, CFD, FSI, SPH
1. Introduction
The Cassini-Huygens mission gave rise to a much
more comprehensive understanding of the Saturnian
system [1,2], confirming Titan, Saturn's largest moon, as
unique in the Solar System for sustaining a nitrogen-
based organically rich atmosphere [3,4], a highly active
surface at which complex geological processes occur [5]
and a subsurface ocean [6,7].
A multi-phase alcanological* cycle is present, active
across the surface of the body, which includes
hydrocarbon lakes and seas primarily clustered around
Titan's northern pole [8,9], as seen in Fig. 1. This makes
Titan the only location in the Solar System other than
Earth to have bodies of liquid on the surface and when
considered alongside the presence of organically-rich
dunes [10], aeolian activity [11], fluvial features [5] and
cryovolcanic activity [12], Titan can be seen as
analogous to the early Earth, and so is naturally of interest
to abiogenetic studies.
The Cassini-Huygens mission proved a great success,
ending with the spacecraft's intentional destruction in
Saturn's atmosphere in a Grand Finale on the 15th of
September, 2017. Since then, a great number of proposals
for aerobots, balloons, atmospheric probes, lake probes
and orbiters have been presented. These are explored
further in Section 2.
This report proposes an aerial-aquatic spacecraft for
use in the further exploration of Titan and as similar
vehicles have been proven viable on Earth [13–15] but
* This can be seen as analogous to Earth's
hydrological cycle, but is instead driven by methane
never been studied in this context, attempts to perform
the required analysis to assess the concept’s feasibility.
This report, therefore, aims to present a clear case for
the further exploration of Titan by such an aerial-aquatic
spacecraft, explain the numerical modelling techniques
used and justify their selection, present the results of this
modelling and then discuss the implications.
Fig. 1. Mercator map of Titan’s major geomorphological
units. Reproduced from Lopes et al. [9].
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IAC-20-E2.1.9 Page 2 of 12
2. Mission Analysis
Previous mission proposals for Titan show a keen
interest of the community to both fund and develop in
great detail innovative engineering approaches to further
explore the body. Each proposal offers excellent access
to one of two mediums at Titan, either surface liquid or
solid surface. Titan Mare Explorer (TiME) [16] offers the
opportunity for limited atmospheric analysis during
descent in a similar fashion to Huygens, and only offers
access to the surface of one of Titan’s lakes. NASA’s
selected Dragonfly mission offers the same scientific
opportunities as a traditional rover, with a new relocation
mechanism. However, following its mission, there will
be adequate scope for follow up missions to explore the
still unvisited lakes, and the Titan Saturn System Mission
(TSSM) [17] provides the perfect accommodation for a
small and innovative lake lander.
2.1 ASTrAEUS
With a unifying aspect of current proposals being
their specificity, there is space for a concept which could
allow access to multiple mediums with a single platform.
The ASTrAEUS (Aerial Surveyor for Titan with
Aquatic Operation for Extended Usability) spacecraft
(Fig. 2) provides an aerial-aquatic platform inspired by
the flight and ‘plunge-diving’ landing of the gannet sea
bird within a field termed bioinspiration and biomimetics.
Fig. 2. Impression of a 'plunge-diving' manoeuvre by an
aerial-aquatic vehicle inspired by the gannet seabird
(inset). Inset adapted from [14].
2.1.1 Bioinspiration and biomimetics
The area of bioinspiration and biomimetics is one
which studies the natural world using evolution as a
guide for the design of new systems to complete
processes, in this case, the traversal of two mediums with
a single vehicle.
The field is one which has gathered a much-increased
following in recent years due to advancements in
microrobotics, enabling the replication of the processes
which have been observed in nature for many years.
Although aerial-aquatics (the study of biological
organisms which traverse atmospheric and liquid
mediums) has been given more attention recently, to the
knowledge of the author it has never before been
considered within the context of space exploration.
2.1.2 Aerial-aquatic operation at Titan
A key and unique benefit of the ASTrAEUS proposal
is the ability to make in-situ measurements in various
separate bodies of surface liquid on Titan, allowing
characterisation of these areas to a level currently
unavailable with any planned or proposed mission.
Kraken Mare is selected as an initial landing site due to
it being Titan’s largest lake, and due to its proximity to a
number of other lakes (see Fig. 3) of key interest to the
scientific community.
Fig. 3. Cassini SAR mosaic images of the north polar
region showing Kraken, Ligeia and Punga Maria.
Reproduced from Mitri et al. [18].
Just as the Huygens probe provided a view to the
surface, which was obscured to Cassini orbiter, the
ASTrAEUS aerobot will also give wide-ranging access to
the surface and near-surface atmosphere of Titan.
2.1.3 Science Case
The opportunities for in-situ study of surface lakes
and the near-surface atmosphere above them, given by
this vehicle, are unprecedented. Of particular interest in
these locations and deducible by measurements made
here will not only be the composition of Titan's lakes and
near-surface atmosphere, but the methane flux from
surface liquid and terrain. It can be hoped that this will
aid the development of current understanding, or lack
thereof, about the presence of methane at Titan and any
sources of resupply – a topic relevant to the discussion
about the potential presence of life at the Saturnian body.
Key science objectives of the vehicle would include:
1. Characterise Titan’s lakes and determine
their impact on the hydrological cycle
2. Find the source of Titan’s methane
3. Characterise Titan’s near-surface
atmosphere
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IAC-20-E2.1.9 Page 3 of 12
3. Numerical Modelling
This numerical study involves the CFD simulation of
a rigid body entering a finite domain free-surface †
without heat transfer.
A two-stage approach was taken to first validate
modelling techniques, for example, non-reflecting
boundaries and equation of state selection – more
information on which is detailed in Section 4. This
involved using a documented test rig [19,20], which has
been refined with experimental data [21], for testing
equation of state and other material property changes for
the verification of expected behaviour.
Further improvement of fluid behaviour with, for
example, bulk viscosity controls can then be put in place
in an environment where behaviour is easier to refine.
The test rig was chosen as it not only clearly displays
sloshing behaviour, useful for verifying the effects of
relative viscosity changes, but it also includes an FEM
mesh for FSI modelling. This is useful in refining contact
behaviour between materials, as is detailed in Section 4.
The test rig was first used to simulate liquid water, and
then adapted to the Nitrogen-Ethane-Methane mix found
in Kraken Mare, this having density
= 664 kg/m3 and
dynamic viscosity of
= 1014 µPas [22].
For the surface liquid impact, a Computer Aided
Design (CAD) model of the ASTrAEUS spacecraft was
greatly simplified to consist solely of a quasi-ellipsoid
fuselage, modelled by a rotated symmetrical NACA-
0010 aerofoil. An atmosphere can be neglected by
beginning simulations immediately before liquid impact,
meaning any drag effects on the projectile would be
trivial. SPH particles are also deactivated once they are
thrown up just beyond the surface as they cease to be of
importance, and so atmospheric interactions with these
are irrelevant.
4. Methodology
4.1 Wave-structure interaction
4.1.1 Smoothed-particle hydrodynamics
SPH is a pure Lagrangian method using meshfree
particles, as opposed to an Eulerian fixed mesh method.
SPH was initially used for astrophysical problems, and
now serves to efficiently model situations where large
boundary deformations are present (like in this case), or
when a free-surface flow needs to be defined. This
method serves as the most appropriate for free-surface
flow and hydrodynamics problems due to the versatility
and simple method of numerical analysis and uses
interpolation to compute smooth field variables.
It is widely accepted that for problems with large
deflections or boundary deformations, an SPH method is
more appropriate than the alternative meshed method as
it avoids problems such as mesh distortion, mesh tangling
† An undisturbed liquid surface at rest which is
subjected to no parallel shear stress
and inaccurate modelling due to unrealistic mesh
interdependence.
An SPH method was more readily available to the
author at the time of writing, and when beginning the
project, large mesh deformations were very possible. It
was unknown to what depth the projectile would
penetrate the liquid, and so it proved a safe starting point,
with FEM always available for adoption later.
The properties of each individual particle are
determined by integrating the discretized Navier-Stokes
equations according to the physical properties of
neighbouring particles which fall within a given range,
called the ‘smoothing length’. A distance-based function,
acting within the smoothing length, weights particles
with a closer proximity more heavily to accurately
estimate a local density. This works in a similar way to
the traditional method of estimating density, that being to
divide the total mass inside a given sampling volume by
the volume itself. Here, however, to address clustered
and sparse regions giving erroneous results, a weighting
based on proximity to the sampling volume's centroid
helps to smooth the local density estimates.
LS-DYNA requires the definition of an initial number
of particles to interact with for each individual particle,
so that the smoothing length can be defined according to
user requirements. This value was adjusted to improve
the accuracy of the simulation, at the obvious cost of
computational performance. The accuracy was attained
by comparison with the empirical data of Gomez-
Gesteira et al. [21], and a final value of 500 particles was
defined.
Fig. 4. Set of neighbouring particles in 2D. The possible
neighbours of a fluid particle are in the adjacent cells but
the particle only interacts with particles marked by black
dots. Reproduced from Gomez-Gesteira et al. [21].
4.1.2 Particle approximation theory
A number of equations are used to define the
behaviour of SPH particles. Most notable of these are the
equations for energy and momentum, Equ. 1 and Equ. 3
respectively.
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IAC-20-E2.1.9 Page 4 of 12
()=,()
,
(1)
()=
()
(2)
For deployment in LS-DYNA, Equ. (1) can be
adapted in this instance to better represent the behaviour
of two mediums of significantly different densities. This
adaption is seen in Equ. 3.
()=,()
,
(3)
This is referred to as the fluid formulation of the
momentum equation. Computational simulations were
performed using both Equ. 1 and Equ. 3. As it is
convention to use the latter for fluid modelling, this was
always preferred, and as only a very minor increase in
simulation time accompanied this, it was selected.
4.1.3 Equation of state
When using an SPH method to simulate liquid
behaviour, an equation of state (EOS) is required. This is
used to define the hydrostatic or bulk behaviour by
calculating pressure as a function of other material
properties such as density, energy, and temperature.
While previously it was common to use either a linear
polynomial or Gruneisen EOS to model internal liquid
behaviour, recent developments in LS-DYNA have
allowed a new approach using the Murnaghan EOS [23].
This can be used when assuming little to no
compressibility in the subject fluid - sometimes referred
to in relevant reading as a quasi-incompressible fluid.
The Murnaghan equation of state defines pressure at
a point in the fluid as:
=
1 (4)
while satisfies:
=
10max (5)
where vmax is the maximum expected flow velocity.
is typically chosen to be around 7 and when the
above conditions are satisfied, a low compressibility is
maintained while relatively large time steps are allowed.
J. J. Monaghan and A. Kos [24] present an equation
to determine the coefficient k0 as
=
100
(6)
where g is the acceleration due to gravity and H is a
specific height taking the depth of the liquid. Taking the
gravitational acceleration at Titan's surface as 1.35 m/s,
the depth of the test tank as 1 m and the liquid density as
664 kg/m3, k0 takes the value of approximately 1.2 × 10.
This compares to a value of approximately 1.5 × 10 for
water in the same conditions.
When performing the verification for stability with
Equation 5, it is found that the permitted max is around
1.2 m/s, as opposed to around 3.2 m/s for a similar water
case. Under these conditions, however, the criterion is
satisfied for the proven value of 7.
When defining the material properties, a density and
viscosity is described, which is then related to pressure
with the above relationship.
Given the simplicity with which Titan’s liquid can be
modelled using this EOS, it was used in all simulations.
4.1.4 Further viscosity control
The default viscosity parameters assumed by LS-
DYNA are typically overly dissipative when a low-
viscosity fluid is considered. The dynamic viscosity of
the subject liquid is approximately 1 × 10 Pa s,
comfortably qualifying for this designation. There are
two main approaches to correct for this, both of which are
discussed below.
The bulk viscosity of the fluid can be constrained,
applying across the whole domain, and requires the
definition of two additional coefficients. These are the
quadratic bulk viscosity coefficient and the linear bulk
viscosity coefficient, denoted as and and taking
the recommended values of 0.01 - 0.1 and approximately
0 respectively.
An additional constant can be introduced, called
the hourglass coefficient. This allows control of the
hourglass modes within the fluid which are non-physical,
zero-energy modes of deformation resulting in no stress
or strain. Hourglass energy is expended internally within
the fluid to resist these modes and so is a way the fluid
dissipates its energy.
These modes occur when performing one-point
integration and as their motion is orthogonal to the plane
in which strain calculations are performed, work done by
their resistance is not accounted for in the energy
equation. When is combined with and ,
hourglass modes can be controlled.
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For the cases considered in this study, bulk viscosity
control is sufficient as no significant hourglass nodes are
excited in the fluid, given they are primarily present in
FEM simulations with high deformation.
4.1.5 Contact modelling
Contact modelling is a key element of accurate
simulations, and as the boundary deformation of a
problem increases, so does the importance of the contact
methodology selected.
In LS-DYNA, contacts are modelled in pairs, with
nodes belonging to one part acting in a slave role and the
nodes belonging to the other in a master role. Each
timestep, a search is made for penetrations of the two
selected node sets and a force, proportional to the
penetration depth, is applied to resist the penetration and
move nodes to eliminate the penetration. Called penalty-
based contact, restoring forces are calculated as a
function of the larger elemental material properties. An
alternative methodology called constraint-based contact
can also be considered, where the restoring force is
calculated as a function of individual nodal mass
performing the penetration. These are compared in Equ.
7 and 8 respectively.
Fig. 5. Slave node-set penetrating master surface,
prompting a restoring force
A contact type is now selected. A vast number of
options are available in LS-DYNA, the majority of which
are not of concern for this problem. For example,
surfaces can be specified to react only to specific relative
degrees of freedom, allowing sliding interfaces to be
defined, or surfaces can be tied together until a specified
terminal stress, at which point a fracture forms.
A group of contact types, classified as ‘automatic’,
are a more recently introduced option in LS-DYNA.
They are almost always selected as they are capable of
handling contacts from any number of directions, making
them ideal for handling highly distorted meshes, or as in
this case, for handling a large number of individual SPH
particles.
For this group of contact types, and many others
available, a maximum penetration depth is defined before
the slave node's elimination. This acts to ensure restoring
forces do not become disproportionate to those
experienced in the rest of the model, with the aim of
maintaining realistic behaviour.
A different grouping of contact types also separates
those which are one-way and those which are two-way.
One-way methods identify penetrations of the slave node
set within the boundary of the master node set (called the
master segment) and apply appropriate restoring forces.
Two-way methods, however, identify relevant nodes and
apply restoring forces to both node sets, meaning the
label of slave and master applied to the node sets in this
instance is arbitrary.
As aforementioned, a penalty-based approach to
contact modelling has been taken, where a restoring force
is a function of penetration depth. This is achieved by the
modelling of a spring force, acting between the
penetrating node and nearest master segment, with
stiffness
=
××
(7)
where is the penalty scale factor and and are
the area and bulk modulus of the contact segment
respectively. It should be noted that this equation is only
valid for solid elements. An alternative equation is
required when shells are considered.
However, an alternative method to calculate can be
to only consider the properties of the individual node. In
this case, the contact stiffness is found as a function of
the nodal mass where
=
(8)
where is the nodal mass and is the timestep.
This approach is recommended for impact analysis of
dissimilar materials, and after an empirical comparison
of both methods, it was confirmed that using Equ. 8
produced the more accurate contact behaviour.
An additional option is available in the software,
where further work is done to smooth the contact
modelling by interpolating the subject surface. This
means considerations like the mesh resolution become
less important for low-interest areas, meaning a lower
mesh resolution can be used. While it is not
recommended to use the method on strongly undulating
surfaces, on gentle curves with a uniform structure, this
option can be of great help in reducing computational
load. While this was considered for application to this
problem, access to multiple processors is required if the
smoothing is to be used with a two-way contact definition,
and this was not available to the author at the time of
writing. It can, however, be hoped that this option is
available in the future to continue this investigation.
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4.1.6 Material
For use with the equation of state that has been
defined, a further definition of material properties is
needed. Fortunately, this proves relatively simple and
only the density and dynamic viscosity of the liquid need
specifying for use with the Murnaghan EOS. As
mentioned above, Titan’s liquid has density
= 664
kg/m3 and dynamic viscosity of
= 1014 µPas, using the
assessment in Hartwig et al. [22], that Titan’s largest lake,
Kraken Mare consists of a Nitrogen-Ethane-Methane mix.
4.1.7 Gravity
The gravitational acceleration acting on SPH particles
and the rigid body is set to 1.35 m/s2 [25].
4.1.8 Infinite domain modelling
An intrinsic and highly problematic feature of
modelling an infinite domain such as a lake is that in
reality, shockwaves caused by a projectile entry will
propagate without reflection for a long period, and any
reflections that do occur will be weakened to such an
extent that by the time of their return, their effect is
negligible when compared to other entry effects.
LS-DYNA has recently introduced a new technique
to deal with these problems, allowing infinite domains to
be modelled in a finite volume, by attempting to
eliminate reflections from incident SPH particles. The
elimination is achieved by accelerating incident particles
parallel to the wall so as to remove them from the path of
further incoming particles, without translating the
resisting force back into the area of interest. Given the
combination of contact models, EOS and material
properties, infinite domain modelling attempts resulted in
high simulation instability, and non-physical SPH
particle velocities. They, therefore, were not
implemented into solutions, meaning shockwave
reflection time heavily limited results, seen in Section 5.
(a) Instant after boundary
interaction at time = 0 s
(b) Boundary instabilities
beginning at = 0.02 s
(c) Boundary instabilities fully established at =
0.125 s. max >100 m/s
Fig. 6. Instability caused by the inclusion of non-
reflecting boundaries
4.2 Projectile impact
4.2.1 Resolution and domain size
The mesh used was generated using LS-PrePost solid
meshing using a rotated NACA-0010 aerofoil, and is
staged so as to provide a higher resolution at the nose of
the structure. The intrinsic problem limited
computational power presents to CFD simulations was,
as ever, present in this study. However, a condition of the
LS-DYNA licence used also meant node-count was
limited below 10,000. As it turned out, the requirements
of the RAM of the host computer were the limiting factor,
and both the resolution of the solid mesh and SPH
particle field were responsible for contributing to this.
Fig.7. Rotated NACA-0010 trimmed and meshed
projectile
Through a mesh sensitivity study, it was seen that a
high resolution at the nose was essential. Fig. 8 best
represents the problems that are caused by a low
resolution at the tip of the projectile. As there are many
SPH particles per mesh face, they interact with singular
faces in a planar fashion and so produce the non-physical
behaviour that can be clearly seen, taking the form of
shockwaves propagating not in the form of a uniform
wave, but the form of individual particle streams. This
effect, however, is only present at the very tip of the nose
and is greatly reduced from when an initial global
meshing resolution was applied across the whole
projectile. Given increased computational power, it
would be possible to increase this resolution further and
allow for the region of accurate simulation to move
forward and closer to the nose. However, that was outside
the scope of this study.
Fig. 8. High SPH resolution, double-precision solved
solution with nodes travelling over a specific velocity
shown and with attached velocity vectors
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Alternatively, a smaller refinement region could be
added, in addition to the three already present. However,
as can be most clearly seen in Fig. 7, a discontinuity in
the mesh is present at boundaries between vastly
changing mesh resolutions. This is undesirable at any
location, but specifically at a region of high interest. It
was concluded through empirical testing that this was
unfeasible, and that to eliminate non-physical behaviour
caused by mesh resolution, a computer with higher
computational power is required.
An alternative method for reducing this effect is to
reduce the resolution of the SPH particles, meaning that
they react according to multiple FEM faces, as opposed
to a single one as is present below around the projectile
tip. There are also further reasons for doing this. The
simulation below, running on a single PC core (another
condition of the software licence available) and with
2133 MHz RAM took 36 hours to progress this small
distance into the liquid. The timestep (which is also
discussed in more detail below) required for simulation
stability for such a high SPH resolution was
approximately 1 × 10s, and as data was recorded at
each timestep to ensure the capture of all simulation
detail, produced over 600GB of data. The scale of this
simulation, while perhaps expected in industrial
applications, is far out of the hardware range of the author,
and so the SPH resolution was reduced for the final
simulations.
A double-precision solver was also used in this case,
with further identical simulations carried out with a
single-precision solver. This is recommended practice for
initial simulation runs so that results can be compared,
and the use of the single-precision solver over the double
(the latter having an approximately 30% longer
computation time) can be justified. It is also common
practice to use the double-precision solver for explosion
modelling, and therefore necessary to test it with high-
velocity impact studies. Given that the timestep used was
small enough, the single and double-precision solvers
produced identical results, and so the single-precision
solver was used from this point on.
Another problem associated with RAM is not just its
speed but its size. The RAM size required to initialise a
simulation with SPH particles scales with the domain size
and number of particles. Therefore, for an increased
domain size, the number of particles must be reduced. A
compromise was found between these two competing
requirements, whereby the domain size meant the
projectile could traverse to a depth sufficient to see a
divergence between its deceleration in water and
nitrogen-ethane-methane liquid (as seen in results in
Section 5), and the SPH resolution meant shockwave
propagation was effectively modelled (this being defined
by the shockwave travelling at the speed seen in the high-
resolution, double-precision simulation).
4.2.2 Equation of state
Simply by maintaining the same characteristic length
from the wave-structure interaction simulation in the
impact simulation, the EOS can remain unchanged.
Therefore, the modelled SPH domain is constructed with
the same reference dimension. As before, the Murnaghan
EOS is used and the parameter is kept the same.
4.2.3 Timestep resolution
As previously mentioned, the timestep required to
ensure stability in the high resolution, double-precision
simulation of Fig. 8 was very small relative to the period
of interest for the simulation. This timestep can be
specified by adding a scale factor to the calculated time
step to ensure a resolution high enough to capture all
features. This is however a trial-and-error, empirical
process and can be time-consuming. It is also a ‘fudge
factor’ method. A neater approach is to correctly specify
contact penalties (as discussed above in Section 4.1.5),
which relate to the relative velocities which are seen
between our two mediums.
For this simulation, the LS-DYNA computed
timestep was around 1 × 10s, and as a timestep of
1 × 10s was required, a very significant fudge factor
would have been required – a particularly bad approach
as LS-DYNA’s computed timestep changes through the
course of the simulation, meaning and small changes are
largely amplified, effectively invalidating the simulation.
Following the specification of contact velocities, a
relatively small timestep scale factor of 0.6 was required,
just below the recommended value for explosive analysis.
5. Results and discussion
5.1 Wave-structure interaction
5.1.1 Sloshing
Fig. 9 shows the simulation of Titan’s surface liquid,
with identical conditions present for liquid water at
identical time steps for reference. While a nitrogen-
ethane-methane mix is more viscous than liquid water,
this difference is relatively small at = 1.24 ×10.
This represents an approximately 15% increase from
liquid water. The largest relative change between
materials is the density, as the 664 kg/m3 density of
N/C2H6/CH4 represents an approximately 35% decrease
from that of water.
Both of these key material changes would bring about
the behaviour observed in Fig. 9, with viscosity affecting
the speed at which the fluid progresses along the
container (which can be observed to be much slower),
and density affecting the manner of its impact with the
structure (as can be seen by the magnitude of the
interaction).
An observation, useful for projectile simulations later,
about the relative speed and magnitude of energy transfer
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by the fluids can be made. It can be seen that due to the
different behaviours, energy is not only transferred to the
terminating wall at a slower rate, a smaller magnitude of
it is transferred. This is clearly demonstrated in Fig. 12
and is discussed in Section 5.1.3.
5.1.2 Fluid-structure interaction
Further interaction with the FEM structure can be
seen from above in Fig. 10. In Fig. 10(a), the lower
velocity at which the nitrogen-ethane-methane mixture
passes the structure means the beginning of a von
Kármán vortex street can be seen due to the vortex
shedding by the structure. This simulation was not run
with the observation of this phenomenon in mind, nor is
it the purpose of this study, meaning the container does
not provide sufficient space or time for the effect to
become fully established. Therefore, while a frequency
of oscillation cannot be obtained, it is nonetheless an
interesting effect to observe.
It serves as a particularly useful way to observe the
changes a different viscosity, density and the subsequent
velocity change have on fluid behaviour.
(a) N/C2H6/CH4 FSI wake
(b) H2O FSI wake
Fig. 10. Wave-structure wake formation of Kraken Mare
liquid and Earth water with velocity displayed
It is known that turbulence is related to Reynolds
number Re, and below a critical value of this, turbulence
begins. While this problem is three-dimensional and not
concerning a constant flow, this theory can be used to
verify the viscosity change.
=
(9)
where v is the fluid velocity, d is the characteristic length
and is the kinematic viscosity. With a reduced density,
velocity and increased kinematic viscosity, all changes
act to reduce the Reynolds number, scaling it to give
approximately titan = 0.17earth , where titan is the
Reynolds number of the N/C2H6/CH4 flow and earth is
the Reynolds number of the liquid water flow.
Fig. 11 shows this effect in greater clarity and
attaches relevant streamlines.
Fig. 11. Vortex shedding for N/C2H6/CH4 liquid with
streamlines on particles in area of interest
5.1.3 Fluid force
Fig. 12 shows the force on the terminating wall by the
respective fluids. As was evident in Fig. 9 the peak force
of the water occurs much earlier than the nitrogen-
ethane-methane mix. What however was less clear was
the relative sharpness of each peak, or in this case the
peak of water, as the nitrogen-ethane-methane mix does
not appear to demonstrate a peak. It has already been
(a) N/C2H6/CH4 at t = 0.3 s (b) N/C2H6/CH4 at t = 0.6 s (c) N/C2H6/CH4 at t = 0.8 s (d) N/C2H6/CH4 at t = 1.6 s
(e) H2O at t = 0.3 s (f) H2O at t = 0.6 s (g) H2O at t = 0.8 s (h) H2O at t = 1.6 s
Fig. 9. Wave development and FSI of Kraken Mare liquid and Earth water with velocity displayed. Red corresponds
to a higher relative velocity.
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observed that the rate and magnitude of energy transfer
across the container was higher for water, but this
magnitude can be much more clearly seen in Fig. 12.
5.2 Projectile impact
5.2.1 Shockwave propagation
One main problem with modelling a quasi-infinite
domain in a finite domain is that the solution becomes
much less accurate, perhaps to the point of being invalid,
once shockwaves caused by the impact are reflected back
to the impacting body.
For this simulation, this phenomenon can be
visualised by showing the acceleration, within a fine
tolerance, of the free-surface liquid. This has been done
in Fig. 13, and as expected, the propagating shockwave
is generally uniform. The time at which this figure was
produced has been chosen to show that the momentum of
the incoming projectile pushes the centre of shockwave
propagation below the free-surface, as can be seen by the
exposed SPH particle wall.
Fig. 13. Shockwave propagation in N/C2H6/CH4 mix
with nodal acceleration displayed
The exact mechanics of shockwave reflection can be
more clearly seen in Fig. 14, with the forces on the
projectile (causing the plotted deceleration) clearly
affected at around 5.75 ×10 s, the point at which the
shockwave returns to the body. Therefore, the figure
limits data to that gathered before the shockwave
reflection, for the discussion of projectile behaviour.
While the figure is cropped to the resultant acceleration
values of interest, it should be noted that they exceeded 8
times their peak value of the valid period. Therefore,
even if local particles return to enacting accurate physical
behaviour after the peak of shockwave reflection, an
offset would be needed for the data.
5.2.2 Projectile behaviour
The projectile was impacted into the fluid with a
velocity of 25 m/s and at an angle of 75° to the horizontal.
The object moved under 6 degrees of freedom, and its
body was parallel with its initial direction of motion.
The simulation was completed with timesteps of
approximately 1 × 10 s, with data taken at each
timestep. The results presented represent a 75-point
moving average plot, meaning that data points are plotted
at approximately 7.5 × 10 s intervals. This was done
to reduce noise which made the data unintelligible.
However, the oscillating forces that the body experienced
are still clearly seen. When first observed, these were
much larger, and bulk viscosity controls were tightened,
in line with the procedure discussed previously. However,
it is now understood that constraining hourglass modes
would also contribute to damping the oscillations,
presenting an area for future work. The oscillations are
not discussed further as their presence is erroneous,
although it is appreciated that their presence is a cause for
further investigation. As discussed above, the reflected
shockwave can be seen and so data until this point is what
is of interest.
The data shows a similar behaviour of water and the
nitrogen-ethane-methane mix initially. The initial
increase in deceleration magnitude and its subsequent
decline can be attributed to the initial impact on a non-
moving body and its attenuation is as expected.
Beginning at approximately 2 × 10 s, the
behaviour of the projectile in the two mediums begins to
Fig. 12. Force caused by sloshing fluid on the terminating wall for water and N/C2H6/CH4 mix
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diverge. The resisting force experienced by the projectile
in liquid water begins to become larger than that
experienced in the nitrogen-ethane-methane medium.
Viscosity is a measure of a fluid’s resistance to
deformation. Therefore, given the two medium’s relative
viscosities (discussed previously), it could have been
expected that nitrogen-ethane-methane would have
provided a larger resisting force to the projectile’s motion.
However, when the densities of the fluids are considered
relative to one another, the increased inertia of water is to
be expected (it is well known that fluid inertia is directly
proportional to density [26]), and it can be clearly seen
that this has a larger effect on the projectile’s motion than
the difference in viscosities.
An additional unexpected observation of the
projectile behaviour was made. As seen in Fig. 15, the
projectile experienced small, but well established and
uniform, oscillations during entry. These occurred with a
vector perpendicular to the direction of motion, and in the
plane of the projectiles angle to the horizontal with a
frequency of 280 Hz.
As mentioned previously, the projectile was
orientated in line with its velocity vector, and given six
degrees of freedom. However, upon entry, the x and y-
components of its velocity were attenuated at different
rates, given the geometry of the projectile. Therefore, and
oscillating stress within the projectile was excited.
It was not within the initial scope of the project to
detect these, and further work needs to be done to
properly understand their implications.
Fig. 15. Oscillation of projectile during fluid entry seen
with nodal acceleration displayed
6. Conclusions and further work
This work has successfully modelled Titan’s surface
liquid and presented the results of two key fluid-structure
interaction simulations. Using Titan’s largest lake
Kraken Mare’s composition, shown by Hartwig et al. [22]
to be a nitrogen-ethane-methane mixture and adapting
the Murnaghan equation of state and work done by J. J.
Monaghan and A. Kos [24], a model has been built.
Using the results of simulations making use of this
model, the behaviour of Titan’s surface liquid can now
be better understood, and the deceleration caused by its
liquid can be anticipated. It can also now be understood
that despite the higher viscosity, the lower density causes
smaller forces on the impact body than those of water.
Taking the simulated projectile impact case, the
deceleration of the projectile was very gradual, meaning
the projectile could be expected to penetrate the lake to a
depth equal to around 25 body lengths given the entry
velocity of 25m/s and angle of 75°. This would be ideal
given the interest in profiling the lakes of Titan, and
appropriate for Kraken Mare, given its estimated depth
of 160 m [27].
The mission concept demonstrates key strengths
against the other proposals discussed in this paper, given
the ability for the ASTrAEUS platform to perform almost
all of their individual functions, and has certainly
inspired interest for future study.
In the future, investigations would benefit greatly
from the allocation of more resources such as
computational power and licensing capabilities. Given
this, more complex geometries could be modelled in a
larger domain, increasing simulation accuracy and the
length of time for which simulations remain accurate.
Further work should also look at the ejection of the
vehicle from the surface liquid. The beginnings of this
have already been completed using classical mechanics
and hydrodynamics equations, in conjunction with this
project, by the author.
In addition to this, higher-level components of space
mission proposals should receive some future attention.
For example, an analysis of the power requirements of a
(a) Projectile entry at
time t
(b) Projectile at time +
3.6 × 10
s
Fig. 14. Projectile deceleration due to water and N/C2H6/CH4 mix
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vehicle such as this would allow the selection of an
appropriate power supply such as a Radioisotope
Thermal Generator (RTG) and the assessment of whether
this would be sufficient, or simply provide enough power
for a short release power supply to enable short flight
from one body of surface liquid to another. There are a
large number of considerations to make at this level,
however, and it is recognised by the author that a team
would need to be assembled to complete these, another
priority for the future.
Acknowledgements
This work is the result of a research project funded by
the Royal Academy of Engineering, the Royal
Astronomical Society and the British Interplanetary
Society. Research was conducted in, and with the support
of, the Blackett Laboratory at Imperial College London
and the Aeronautical Engineering department at
Loughborough University. Thanks are given to members
of these departments for their belief in this project.
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