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REGULAR PAPER
Moritz Heinemann •Steffen Frey •Gleb Tkachev •Alexander Straub •Filip Sadlo •
Thomas Ertl
Visual analysis of droplet dynamics in large-scale
multiphase spray simulations
Received: 23 December 2020 / Revised: 4 February 2021 / Accepted: 28 February 2021
ÓThe Author(s) 2021
Abstract We present a data-driven visual analysis approach for the in-depth exploration of large numbers
of droplets. Understanding droplet dynamics in sprays is of interest across many scientific fields for both
simulation scientists and engineers. In this paper, we analyze large-scale direct numerical simulation
datasets of the two-phase flow of non-Newtonian jets. Our interactive visual analysis approach comprises
various dedicated exploration modalities that are supplemented by directly linking to ParaView. This hybrid
setup supports a detailed investigation of droplets, both in the spatial domain and in terms of physical
quantities . Considering a large variety of extracted physical quantities for each droplet enables investigating
different aspects of interest in our data. To get an overview of different types of characteristic behaviors, we
cluster massive numbers of droplets to analyze different types of occurring behaviors via domain-specific
pre-aggregation, as well as different methods and parameters. Extraordinary temporal patterns are of high
interest, especially to investigate edge cases and detect potential simulation issues. For this, we use a neural
network-based approach to predict the development of these physical quantities and identify irregularly
advected droplets.
Keywords Flow visualization Time-varying data Visualization in physical sciences and engineering.
1 Introduction
Flow visualization has mainly been concerned with the analysis of single-phase flow, i.e., flows where a
single type of fluid is involved (e.g., airflow around objects or liquid flow through machinery). However, in
many science and engineering problems, two or even more phases are involved, such as water flowing in a
domain containing air or in the dynamics of immiscible liquids. A major difficulty with the analysis of such
multiphase flow is, however, its various degrees of complexity. On the one hand, it inherits all complexity of
single-phase flow, whose visualization is subject to ongoing research. On the other hand, the dynamics and
physics of the interface between the different phases are closely interrelated with the flow, as well as
Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s12650-
021-00750-6.
M. Heinemann (&)G. Tkachev A. Straub T. Ertl
VISUS, University of Stuttgart, Allmandring 19, 70569 Stuttgart, Germany
E-mail: moritz.heinemann@visus.uni-stuttgart.de
S. Frey
Bernoulli Institute, University of Groningen, Nijenborgh 9 (Bernoulliborg), 9747 AG Groningen, The Netherlands
F. Sadlo
IWR, Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
J Vis
https://doi.org/10.1007/s12650-021-00750-6
phenomena from solid mechanics such as collision, deformation, and adhesion. Furthermore, the volume of
fluid method (VOF) (Hirt and Nichols 1981), which is typically used for simulating multiphase flow, further
complicates analysis and interpretation. One reason is that the interface between the phases is not tracked
but reconstructed at each time step during simulation, leading to inconsistencies between the flow in the
vector field and the motion of the interface.
A phenomenon in two-phase flow with particularly high complexity is the formation of sprays. Sprays
play an essential role in a wide range of natural phenomena and production, including precipitation,
combustion, food processing, production and application of drugs, and cooling. In technical applications,
sprays are typically generated by guiding the liquid through a spray nozzle, which produces an unstable jet
that eventually breaks up into droplets. The dynamics of this breakup is highly complex, with primary
breakup producing elongated components called ligaments, followed by a secondary breakup of ligaments
into droplets.
All these processes and complexities make the resulting data extremely hard to navigate and analyze due
to their spatiotemporal nature. While traditional visualization approaches are applicable to subsets and
partial aspects of the data, they cannot provide an effective means for hypothesis forming and hypothesis
testing for the entire data, in particular because the importance and interrelations of specific quantities and
processes are buried in the discussed degrees of complexity. Another difficult aspect is that in order to be
able to resolve the complex dynamics involved at small scales, high temporal and spatial resolution of the
data is required. This results in data sizes quickly reaching terabytes, impeding the direct application of
many types of advanced analysis procedures. While interactive exploration is required to analyze the
complex, small-scale effects, this is challenging with the large data sizes and large numbers of droplets.
In this work, we aim to analyze the droplets in a two-phase DNS flow simulation of the breakup of a
liquid jet in air (cf. Fig. 1). This simulation was conducted by Ertl (Ertl and Weigand 2017; Ertl 2019) using
FS3D (Eisenschmidt et al. 2016) and considered water with 0.3% flocculant, leading to non-Newtonian fluid
behavior. In this case, the simulation scientists’ interest focused on the fully converged phase of the jet.
Accordingly, we mainly consider a respective subsequence of 101 time steps covering 0.679 ms of the
entire simulation (except for the collision and separation counters, which cover the complete 623 time
steps). The total data size is around 7 TB (i.e., 1 TB for the focused subset of 101 time steps).
Our dedicated visual analysis approach provides different hypothesis forming tools to analyze droplet
behavior in the data. Our goal is (i) to allow simulation and domain scientists to analyze what common types
of droplets and their behaviors are, and (ii) to identify and study anomalous—i.e., uncommon—cases in
depth. Below, after discussing related work (Sect. 2), we describe our workflow enabling the interactive
visual analysis of large sets of droplets based on the extraction of meaningful quantities and advanced
further automated analysis, including clustering and machine learning-based anomaly detection (Sect. 3).
Our visual analytics system then provides different perspectives on abstract and spatial quantities and allows
for detailed flow analysis with the original raw data (Sect. 4). Finally, we present insights gained from
visual analysis (Sect. 5) before concluding (Sect. 6).
We present the first visual analysis approach for complex time-dependent multiphase flow data. By
combining domain-specific clustering techniques with an artificial neural network for learning droplet
Fig. 1 Interface between liquid and air in the Jet Simulation dataset after the formation of a stable jet. Each time step features a
rectilinear grid with a resolution of 1 536 512 512 cells covering a domain of 12 cm 4cm4 cm; 623 time steps
represent a time span of 5.567 ms. The result is a volume of fluid field fand a velocity field u. Both are given in cell-based
representation
M. Heinemann et al.
anomaly behavior, we developed a highly interactive framework to support flow scientists in the compli-
cated tasks of analyzing terabytes of simulation data.
2 Related work
Our data stems from a CFD solver for multiphase flow simulation of incompressible fluids. This solver
(Eisenschmidt et al. 2016) uses the VOF method (Hirt and Nichols 1981), in combination with Piecewise
Linear Interface Calculation (PLIC) as interface reconstruction algorithm, which we also use in this work to
analyze droplets (cf. Karch et al. (2013) for a discussion of PLIC in visualization).
The visual analysis of time-dependent flow fields is challenging. Aigner et al. (2007) discuss the con-
sideration of time as an additional dimension in visual analytics. Bu
¨rger et al. (2007) integrate local feature
detectors in the visual analysis of time-dependent flow simulations. Another line of work concerned itself
with the interactive exploration of complex time-dependent flow simulations and real-world data (Doleisch
et al. 2003,2004a,b. Shi et al. (2009) introduced an approach to visually analyze time-dependent flow fields
using pathlines in particular. An overview of different feature tracking techniques for flow visualization was
presented by Post et al. (2003). Further Theisel and Seidel (2003) show a streamline-integration-based
method for feature tracking in instationary vector fields. Garth et al. (2004) discuss an approach for tracking
singularities within a vector field. In contrast to these previous works, while we also employ techniques from
flow visualization for the detailed investigation of individual droplets, our specific focus on large numbers of
droplet data features unique challenges that we aim to address in this work. Further previous work already
focused on feature tracking in the context of large-scale datasets (Dutta and Shen 2016) and calculating
tracking information in situ (Biswas et al. 2020).
Combining flow visualization and machine learning, Tzeng and Ma (2005) used neural networks to
generate adaptive transfer functions based on user input. Bai et al. (2017) applied linear discriminant
analysis to classify experimental images into actuated and unactuated flow. In this paper, we also apply
neural networks to support flow visualization but use them to generate visual features that guide the analysis.
Tkachev et al. (2019) trained neural networks on spatiotemporal volumes to detect irregular behavior. We
use a similar idea for our anomaly detection in this work, but we apply our model to time series of extracted
droplet quantities. Machine learning has also been applied in fluid simulations. Artificial neural networks are
even used within solvers of the Navier–Stokes equation (Tompson et al. 2017) for accelerating the com-
putation. In visualization in general, machine learning methods are particularly popular in visual analytics
approaches (Endert et al. 2017).
This work started in the context of a master thesis (Heinemann 2018) which was reported in a very early
version in a project report (Straub et al. 2019) and within a book chapter (Straub and Ertl 2020). The idea of
using ML for prediction on droplet quantity time series is already mentioned there, but due to its overview
character, no details of our technique were presented. Further, an early version of the surface view of our
prototype was depicted there. This article introduces our framework and elaborates on new features like
clustering, improvements to the 3D surface view, and extensions to the complete system, including the
quantity and flow view and an analysis and discussion of the used dataset.
3 Preprocessing: extraction, clustering and anomaly detection
To make the data (interactively) explorable while still capturing the various degrees of complexity of droplet
behavior, we first extract different quantities based on individual droplet instances (Sect. 3.1). Next, we
establish temporal relationships by connecting individual droplet instances between the time steps to time-
continuous droplet traces (Sect. 3.2). To get an overview of the droplet quantities, we use trace-based
hierarchical clustering (Sect. 3.3). Also, we train a regression model to capture typical temporal patterns in
droplets’ quantities and then compute the deviations from this model to guide the researcher to anomalous
cases in the sense of being uncommon. Akin to previous work (Tkachev et al. 2019), we chose artificial
neural networks (ANNs) due to their generality, performance efficiency on large data (compared to, e.g.,
non-parametric models), and their successful applications across many diverse tasks (Sect. 3.4). Below, we
mainly focus on applying and adapting the different methods involved to enable the interactive exploration
of large numbers of droplets (Sect. 4). An overview of the order and dependencies of our processing steps
can be seen in Fig. 2.
Visual analysis of droplet dynamics in large-scale multiphase spray simulations
3.1 Extraction of quantities
We extract different instantaneous quantities for each droplet instance, which capture the various degrees of
complexity of droplet behavior. In collaboration with domain experts, the following set of instantaneous
quantities has been determined, ranging from purely geometric to purely physical.
Data representation and segmentation. The first step consists in the identification of the individual
droplets. The employed VOF approach in our two-phase flow simulation maintains a scalar field fðxÞ2R
during the simulation. This field stores conceptually for each point x2R3the volume fraction of the liquid,
i.e., f¼0 representing only gas, f¼1 representing only liquid, and 0\f\1 mixtures of both. Here, fðxÞis
defined in a cell-based manner, i.e., the field stores this total fraction for each cell iindividually, in
piecewise constant representation. Thus, firepresents the value of the f-field for cell i. Similarly, the
velocity field uðxÞis given in cell-based representation ui. In the domain, a droplet is commonly defined as
the face-connected component of cells that exhibit fi[0. In practice, due to the numerical limitations of the
simulation, we use the slightly modified definition fi[sf, with sf¼106. This value is defined by our
domain experts, who run the simulation. Thus, we obtain the droplets by connected component labeling
using region growing, resulting in a cell-based label field lðxÞ, where listores the droplet identifier iof the
droplet that cell ibelongs to.
A further peculiarity of two-phase flow simulation represents the interface between the two phases via
PLIC (Youngs 1984). During the simulation, the interface is represented in a piecewise planar manner, i.e.,
within each cell with 0\f\1, a planar patch represents the interface. It is determined by using the negative
gradient of the f-field as a plane normal and choosing the position such that the volume below the patch is
precisely the volume fraction represented by fi. This piecewise linear representation per cell results in a
discontinuous global interface containing gaps (see, e.g., Fig. 13). Using isosurface extraction for interface
reconstruction would violate both the connected component definition of droplets and interface represen-
tation. Thus, it is dissuaded by domain experts. Therefore, we use the PLIC patches as geometric interface
representation for both the computation of derived quantities and rendering, similar to Karch et al. (2013), at
the cost of accepting the discontinuities.
Main instantaneous quantities. We want the extracted quantities to capture droplet behavior reasonably
accurate, but at the same time, we are limited to quantities that can be calculated purely from the VOF and
velocity field. Furthermore, we need a minimum of numeric stability in calculating them. While the overall
droplet quantities (Sec. 3.1)
droplet traces (Sec. 3.2)
clustering (Sec. 3.3) prediction-based anomaly
detection (Sec. 3.4)
raw spatial simulation data
droplet surface view
(Sec. 4)
quantity relationship view
(Sec. 4)
filter & highlight
brushing & linking
gn
is
s
ecor
perp
noitarolpxe
flow view (Sec. 4)
droplet selection
preprocessed data
Fig. 2 Overview of the components of our visual analysis approach. This represents the execution order of our processing
pipeline and shows data dependency of all steps described within Sect. 3and Sect. 4. Each of the black arrows represents data
usage from the previous step
M. Heinemann et al.
jet simulation is high resolution, single droplets within this dataset may only be resolved by few cells. Due
to this, we have observed instabilities, mainly when calculating derivatives on the velocity fields. We
therefore focus on derivative-free quantities and compute a total of 11 scalar quantities for each temporal
instance of droplet i: volume li, area Ai, area-to-volume ratio ai, velocity kuik, momentum kpik, angular
velocity kxik, angular momentum kLik, total energy Ei, kinetic energy Eu
i, rotational energy Ex
i, and
residual energy Ed
i. The formal definition of these quantities is provided within the appendix of this
paper (Appendix A). Note that magnitudes are used in the case of vector-valued quantities for ensuring
rotational invariance.
Additional quantities for evaluation and visualization. In addition to the main quantities used for the
further preprocessing steps, we use additional quantities for visualization and evaluation. The polygons of
the PLIC surface are stored to visualize droplet surfaces later. While we deemed derivation-based quantities
to be not numerically stable enough to be used in automatic analysis, we still consider them useful when
carefully employed in supplementing the analysis. To discern between droplets and ligaments, we compute
spherical anisotropy measure from our segmented droplets. We achieve this by computing fractional
anisotropy (Rosenberger et al. 2012; Basser and Pierpaoli 2011) of the covariance matrix of all cell centers
comprising a droplet component i, which we denote cs, with cs¼0 indicating linear or planar shape, and
cs¼1 for a perfectly spherical shape of a droplet. Furthermore, we compute the radius from the smallest
surrounding sphere around the center of mass. While our main quantities above are designed to include
droplet-local quantities, we include location-based quantities to refer to a position within the jet for analysis.
Next to the absolute center of mass position, we use the distance of a droplet’s center of mass to the jet’s
base axis and call it radius. In contrast, we do not include different velocity components, i.e., axial and radial
velocity of a droplet, because the axial velocity is expected to approximate the overall velocity and the radial
velocity is assumed to be a relatively small constant only depending on the position within the jet, as
described by domain experts. The number of cells of droplet component iis provided as a discretized
alternative to liwith two variants: (1) including the number of all cells contributing to a droplet and (2) the
number of cells that are at least filled by 50% with the liquid phase. Additionally, we calculate the vortex
core lines of each droplet according to Sujudi and Haimes (1995) and count the number of line segments to
quantify the presence of a vortex. Furthermore, we save error flags during the computation process, i.e., if
droplets are too small or the trace of a single droplet cannot be determined. These error flags have proven
useful for filtering later on.
3.2 Trace generation
From the segmented droplet instances, we now establish a space-time graph depicting their temporal
correspondence (i.e., a node is a droplet instance, and each edge a temporal relationship). Initially, the graph
consists only of nodes and is extended by adding edges if we find correspondence between droplet instances
in neighboring time steps. This is achieved by tracing imaginary particles from the center of each cell to the
next time step (Karch et al. 2018). Unfortunately, using higher-order integration schemes for this, e.g.,
Runge–Kutta, would require interpolation in space and time, which is very likely to sample data from the
gaseous phase and lead to erroneous results as pointed out by Karch et al. (2018). Therefore, we are using a
forward Euler step:
xðtþDtÞ:¼xiðtÞþDtuiðtÞ:ð1Þ
An edge is added if xðtþDtÞbelongs to a droplet, i.e., xðtþDtÞis located at time tþDtin a cell jwith a
valid droplet label. We also do a backward Euler step from the center ukof each cell kthat is part of a
droplet at time tþDt:
xðtÞ:¼xkðtþDtÞDtukðtþDtÞ:ð2Þ
If xðtÞbelongs to a droplet at time t, i.e., if xðtÞis located in a cell lwith a valid droplet label at t, we add the
respective edge (if not already present). If a node njhas degree d[2 at time t, and only one connected
neighbor at time tDt, a breakup event has happened at node njat time t. If there is more than one
connected neighbor at time tDt, coalescence is involved. Our analysis focuses on the dynamics of single
droplets (not considering splits and merges; e.g., cf. Karch et al. (2018) for an analysis of these).
Accordingly, we split the graph at nodes where more than two edges meet—i.e., coalescence and breakup
events—, and base our approach on the isolated linear subgraphs, each representing the trace of a single
Visual analysis of droplet dynamics in large-scale multiphase spray simulations
droplet over time. A trace contains droplet quantities as described above and serves as input to our
regression model (Sect. 3.4).
3.3 Clustering
We want to cluster the droplets to reveal their different types of behavior in the simulation. With our
distribution of droplet quantities, no meaningful results could be obtained with density-based clustering
algorithms such as DBSCAN (Ester et al. 1996) (yielding just one or two major clusters and a large number
of outliers). In contrast, hierarchical clustering according to Ward’s method (Johnson 1967) has generated
more expressive clusters in our experiments. However, hierarchical clustering exhibits quadratic memory
complexity. The significant part is a distance matrix requiring ðn2nÞ=2 values to store. Handling our
1 000 000 droplet instances (10 000 droplets over 100 time steps) would require at least 3.6 TB of
memory and is therefore not possible on our machine. Instead, we aim to reduce the number of data points
significantly. To achieve this, we limit ourselves for this modality to just one aggregate of each trace. Albeit
at the cost of omitting temporal information, this can still serve the original purpose of identifying different
droplet types (cf. Sect. 5). Note that our learning-based method presented below (Sect. 3.4) does not require
this pre-aggregation.
3.4 Learning-based droplet anomaly detection
In our droplet anomaly approach, we first train a basic regression model on traces, using the surrogate task of
predicting future values from a preceding time window. Assuming that the model does not overfit the
training data (which we verify with a hold-out validation), it captures the most common and pre-
dictable patterns in droplet behavior. Then, we can quickly check droplets against the model to find ones that
deviate from the typical behavior. In this work, we employ ANNs for their ability to handle a large number
of elements and learn a useful data representation (Bengio et al. 2013).
As a surrogate task for training, we define a fixed-sized input window and slide it along the trace,
applying the model at every window position to predict the next value in that trace. That is, given a trace of
length nt, and a fixed window size w, we obtain ntwþ1subtraces, and for each subtrace, the model
takes the first w1 time steps as input and predicts the droplet quantities at the last (w-th) time step in the
subtrace. This time-window approach allows for comparably simple models to be used (lowering costs) and
reduces the risk of overfitting because shorter subtraces are generally ambiguous and thus are more difficult
to fit during the training process. We train a separate prediction model for each quantity to affirm that each
quantity is given the same importance, avoiding compromises regarding accuracy as would be the case with
multiple output variables. We split 20% of the data into a held-out validation dataset, using the rest for
training. Of course, we want the training set to include as much data as possible, but the validation set needs
to be a reasonably sized sample of the whole dataset. Due to containing a high amount of relatively similar
droplets and a low amount of relatively widely spread outliers, a validation size smaller than 20% may
introduce sampling bias. Beforehand, we normalize each quantity to have zero mean and a standard
deviation of one, using the mean and standard deviation estimated on the training set. Each model is a fully-
connected neural network with two hidden layers of 64 neurons using Rectified Linear Unit (ReLU) (Glorot
et al. 2011) activation and a single linear unit in the output layer. The models are trained using the Adam
(a) angular mommentum (b) residual energy
Fig. 3 An example of training and validation loss for the ANNs of angular momentum and residual energy. The curves for the
remaining networks have a very similar shape
M. Heinemann et al.
optimizer (Kingma and Ba 2014), with a learning rate 105, and a batch size of 32 throughout 100 epochs
until convergence of the validation MSE loss (Fig. 3). Our setup was chosen empirically by relying mostly
on common values for most ML parameters, as our focus is more on the overall framework. We think there
might be potential for further optimization, but we leave it for future work.
Once the models have been trained, we use them to perform predictions on all available subtraces. The
first w1 time steps of a trace cannot be predicted by design and have to be omitted in the following. We
then determine the difference between the actual and the predicted value of each quantity. In total, this
results in an 11-dimensional vector of deviations, whose Euclidean norm finally yields total error e—our
measure of estimated droplet anomaly.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(a)
(x)
(xi)
(xii)
(b) (c)
Fig. 4 System overview: a3D surface view for showing droplet surfaces, consisting of filter panel (i), coloring tool (ii), bar
chart (iii), main 3D view (iv) (here color mapped to anomaly measure e), time view (v) for selected droplet (crosshair in (iv))
with temporal scrolling. The similarity search (vi) provides cases similar to the selection, based on feature vector distance. In
addition, we depict droplet instance information (vii), droplet trace similarities (viii) for (vi), and spider chart as a complement
for (iii). bQuantity relation view: Input parallel coordinates plot for data filtering (x), scatterplot matrix (xi), and second
parallel coordinates plot (xii) allows analysis of data. Moreover, any quantity can be mapped to color. cA ParaView instance is
integrated within the system for analysis of the raw flow field for a single selected droplet and advanced flow feature extraction.
A droplet selected in the 3D surface view can be automatically loaded into ParaView, including a useful default filter pipeline
as shown within the figure
Visual analysis of droplet dynamics in large-scale multiphase spray simulations
4 Visual analytics system
Our visual analysis system has three different views: the abstract quantity relationship view, the droplet
surface view, and the flow view. These components and their interplay enable the detailed analysis of a large
droplet collection by supporting various kinds of investigation. The droplet surface view displays a set of
droplets in the 3D domain, allowing interactive exploration within the spatial and temporal context for
selected droplet instances. The quantity relationship view provides an overview of extracted droplet char-
acteristics and supports analyzing the interdependencies between them. Finally, the flow view can be used
for the detailed investigation of a single selected droplet using complete raw input data. Different views are
linked to supplement each other efficiently. An overview of the system can be seen in Fig. 4, and the
different views are described in detail below.
Droplet surface view. The 3D droplet surface view allows a user to explore the droplet dataset spatially
(Fig. 4a(iv)). Droplets of interest can be selected directly via picking (indicated with a crosshair). As
showing all droplets would lead to significant occlusion and visual clutter, filtering the data is cru-
cial (Fig. 4a(i)). We support filtering w.r.t. arbitrary physical or geometrical quantities, clusters, and pre-
diction errors (and combinations thereof). Color mapping can flexibly depict chosen quantities, with the
color-coded anomaly measure ebeing the default choice (Fig. 4a(ii)). A user can navigate within the full
trace of a selected droplet and explore its temporal evolution (Fig. 4a(v)). Below, similar traces are shown
that were identified via feature vector distance, i.e., by Euclidean distance between droplet instances in the
11-dimensional quantity space. This helps, on the one hand, to assess the uniqueness of a droplet evolution,
and on the other hand, the comparison to similar droplets can help to gain further insights into which
commonalities or differences have led to certain behavior patterns (Fig. 4a(vi)). Additionally, we also
provide the values for the selected droplet in our time view with a bar chart (Fig. 4a(iii)) and a spider
chart (Fig. 4a(ix)), which can present its prediction error for different physical quantities. Detailed droplet
instance information (Fig. 4a(vii)) and droplet trace similarities (Fig. 4a(viii)) are provided as well.
Quantity relationship view. The quantity relationship view is focused on displaying the different (abstract)
droplet quantities in context with each other. Here, we use classical information visualization methods, namely
parallel coordinates plots (PCP) and a scatterplot matrix (SPLOM), to present an overview of the physical
quantities in the dataset as well as all derived quantities (e.g., clusters and prediction errors). In detail, this view
consists of three separate components as shown in Fig. 4b. On top, we see the quantities of all droplets, shown
within an interactive PCP (Fig. 4b(x)), where sliders on each axis can be used to filter the data. Positioned
below are a scatterplot matrix (Fig. 4b(xi)) and a second PCP (Fig. 4b(xii)), in which we can analyze the
filtered data. We provide both SPLOM and PCP to make use of the strengths of both visualization techniques.
The PCP is ideal for getting an overview of the data and locating single data points with the overall value range
context, while the SPLOM can show pairwise relations and is well-suited to identify patterns and relations
within the data. Naturally, highlighting data in one view also will highlight the data in the other view. They are
also linked to the droplet surface view (Fig. 4a), i.e., brushing within the SPLOM or PCP can be used to filter in
the 3D surface view. This can help to obtain spatial context regarding location and surface shape to the abstract
quantity data points. This component is implemented on top of MegaMol (Gralka et al. 2019), employing
OpenGL to render millions of points and lines at highly interactive frame rates.
Flow view. For an in-depth exploration of the underlying flow field, e.g., to analyze the reason for a high
anomaly, we further incorporate various classic flow visualization techniques by directly integrating
ParaView (Ayachit 2015) into our system (Fig. 4c). ParaView is controlled from our application by loading
the droplet data of the currently selected droplet within our 3D droplet surface view and automatically setting
up the ParaView visualization pipeline. Not only the droplet itself is exported, but its full trace (in a droplet-
local coordinate system for convenience), allowing animation. Moreover, we provide the precomputed surface
and all other quantities. As frontend, the user has the classical fully functional ParaView interface.
5 Results
We will now analyze the Jet Simulation (cf. Sect. 1) in more detail to gain some insights and demonstrate
our methods and system in practice. From the dataset, we extracted 1 043 168 droplet instances. 273 928
were omitted due to insufficient size (around two-thirds of them are artifacts at the simulation boundary),
some more due to not being part of a trace with a minimum length of six. This results in 575 833 droplet
M. Heinemann et al.
instances, which we use in our analysis. They are organized within 23 738 unique traces of sufficient length.
These traces were split into 457 143 subtraces for training and validation data.
We used a machine featuring an Intel Core i7-8700K, 64 GB RAM, and an NVIDIA RTX Titan. The
data had to be stored on an HDD due to its size. Therefore, disk I/O is the bottleneck for many of the tasks.
The computation took 11 hours for segmentation (Sect. 3.1, CPU) and 25 hours for tracking (Sect. 3.2,
CPU). Computation of the droplet quantities took 9.5 hours (CPU). Trace generation and additional data
handling completed in 2 hours. The machine learning (Sect. 3.4, GPU) took 4 hours for the 11 ANNs,
while clustering was finished in only 5 minutes (CPU). Our quantity view achieves more than 60 frames
per second for a single time step and drops to around 5 frames per second when looking at the whole data at
once (GPU).
5.1 Quantity relationships
We first consider a single time step with the quantity relationship view (Fig. 4b). As the jet we are looking
at is fully converged, all time steps are quite similar regarding general structure. First, we investigate the
filter PCP at the top to get an overview of the different value ranges. We notice the wide value range of the
droplet quantities due to single outliers, which leads to the majority of droplets being squashed together for
some of the quantities, e.g., this becomes quite apparent at the volume axis. We use the filter markers on the
axis to remove the single droplet with very high volume, the jet base, and also omit droplets with an error
flag for being too small—our expert specified that droplets below a threshold of 15 cells show unphysical
behavior—or not being part of a trace. With the filtered data being rescaled on each axis within the SPLOM
and second PCP, we can now see details and structures lost in the first overview PCP.
The first noticeable correlation we see is within the distance between droplet and jet center to velocity
scatterplot (Fig. 5). In particular, we identified two main clusters, which are mostly symmetric to the
diagonal: one cluster with high distance and high velocity (Fig. 5a) and a second cluster with small distance
and small velocity (Fig. 6b). Furthermore, there are a few more outliers (Fig. 5c). With the first two main
clusters, we conclude that the droplets in the center of the jet are slower than the droplets farther away from
the center. We compare the scatterplots with the corresponding droplets in the 3D surface view to provide
spatial context (Fig. 5d–f). In addition, the 3D droplets are colored by velocity using the Viridis colormap.
The reason for these two clusters is that our dataset has a two-nozzle setup where we have an inner and outer
zone with a different velocity at the boundary of the simulation domain. This can be easily seen by looking
Radius [cm]
Velocity [cm/s]
(a)
Radius [cm]
Velocity [cm/s]
(b)
Radius [cm]
Velocity [cm/s]
(c)
(d) (e) (f)
Fig. 5 Our analysis of the distance between droplet and jet center (here called radius) to velocity relationship. The top row
shows the scatterplot with different selected data points (red). The linked 3D view below provides a spatial context for selection
Visual analysis of droplet dynamics in large-scale multiphase spray simulations
at the velocity magnitudes and the surface of the jet base in the first time step of the simulation (Fig. 6).
However, it has previously been unclear how this setup affects droplet velocities further within the simu-
lation domain.
Next, we look at the correlation between angular velocity and the number of vortex coreline segments
(Fig. 7). Originally, our hypothesis was that a high angular velocity in the sense of rigid body rotation could
be seen as a vortex, but in fact the plot shows exactly the opposite. We see that only droplets with a
relatively small angular velocity have a high number of vortex core line segments, while droplets that have
Velocity Magnitude [cm/s]
0 13706.9
(a) (b)
Fig. 6 a Inflow velocity magnitude profile at jet nozzle depicting the two-nozzle setup. bIsosurface of the jet base in the first
simulation time step, reflecting the influence of the two-nozzle setup on the jet
Angular Velocity
[
rad
/
s
]
#Vortex core lines
Fig. 7 A comparison of angular velocity to the number of vortex coreline segments within a droplet. These two quantities
behave inversely, i.e., higher velocity yields a lower number of coreline segments
Radius
[
cm
]
#Collisions in history
Fig. 8 A comparison of the radius of the droplet position around the jet center to the number of collision events in the history
of a single droplet. Highlighted (red) is the radius range with a peak in the number of collision events
M. Heinemann et al.
at least one vortex core line segment have a relatively low angular velocity. We consider this to be an
interesting finding, but were not yet able to confirm a physical explanation for this phenomenon.
The scatterplot of the distance between droplet and jet center compared to the number of merge events in
the history of the jet also looks interesting (Fig. 8). It exhibits a triangular shape, except for a few outliers.
The most inner and outer droplets seem to have only few merge events, while there is a ring of droplets with
average distance, that appears to have a very high number of such occurrences. We attribute this to the dual
nozzle injection, where slower droplets in the inner and faster droplets in the outer ring collide in a transition
area. The peak in the number of merge events accordingly indicates this contact area.
5.2 Clustering
The clustering results depict different characteristics (cf. Fig. 9). We notice that the volume is one of the
main factors, which separates these clusters. It further shows in Fig. 9b that the velocity is a factor
orthogonal to the volume. In Fig. 9c, we can see the contour of a cluster with respect to the spherical
anisotropy. This is especially remarkable because this value was not used for the clustering. Accordingly,
we assume that this is an indicator for the correlation of quantities. A limitation may be that there is no direct
physical interpretation of these clusters. While it would be a great result if we had found one, our clustering
method is based on the trace average of the extracted quantities. This is a very simplified projection of a
droplet, probably not covering all physical laws and may not be entirely based on intrinsic physical
quantities. Therefore, the clusters could only have a phenomenological interpretation by looking at the
quantity distribution within the SPLOM view as sketched by Fig. 9. Nevertheless, we think these are still
structural and dynamical relevant clusters representing groups of similar droplet behavior within the abstract
quantity space.
Looking at the 3D surface view, we find that each cluster has similar droplets in reference to its shape
and size (Fig. 10). Overall this clustering gives us an overview of the different types of droplets within this
dataset. It provides additional information to statistical quantity value distribution, for example, within the
PCP view, by addressing higher-order connection in the high-dimensional space of the quantity values.
However, remember that we neglect temporal effects as we averaged traces before clustering (cf. Sect. 3.3).
5.3 Droplet anomaly-guided analysis
Finally, we analyze the results based on prediction anomaly. In Fig. 11a, we see a correlation between the
angular velocity and the momentum, with most data points being close to the axis. That means a droplet
generally exhibits high velocity or high momentum, but not both. By coloring the data points by the trace
prediction error, we see that the further droplets are away from this trend along the axis, the higher is the
prediction error. In the bottom left, we see a dense region with mostly low anomaly droplets. Zooming
reveals a few droplets with a high anomaly, which attract our attention (Fig. 11b). Using brushing on the
highest anomaly data point in this region (marked red) and observing it in the linked 3D surface and flow
view, we see that this droplet is located at the boundary of the domain. Identifying such problematic cases is
important for studying edge cases in the simulation and discarding them from further consideration.
X-Position [cm]
Volume [mm3]
(a)
Velocity [cm/s]
Surface area [cm2]
(b)
Velocity [cm/s]
cs
(c)
Fig. 9 Data points are colored by cluster ID from the eight-class hierarchical ward clustering. aPosition in x-direction (flow
direction of the jet) compared to volume. Notice the relation of the clusters for higher volumes. For lower volumes, clusters are
not distinguishable. bComparing velocity to surface, we can see separation of clusters in both dimensions. cSame holds for
comparing velocity to spherical anisotropy (anisotropy is not considered during clustering)
Visual analysis of droplet dynamics in large-scale multiphase spray simulations
Figure 4a shows the 3D surface view depicting time step 410. Color depicts our anomaly estimation e,
with higher anomaly indicated by reddish colors and lower anomaly by yellowish ones. Gray indicates
droplet instances that are within the w1 first time steps of a trace and thus do not provide prediction nor
anomaly information e(this always applies to the jet itself, it is always at the beginning of the main trace
due to droplets splitting from it continuously). To reduce occlusion and clutter, a typical first step is to omit
those structures, which can be accomplished by requiring e0, because eis set to a negative constant in our
implementation if no evalue can be computed (cf. Fig. 12a). Here, we observe that ligaments, i.e., the
elongated droplets that typically break up into smaller droplets, later on, exhibit very high e, i.e., they show
temporal behavior that strongly deviates from the trends of the majority of the droplets. We also note that
the structures with the highest eare typically tiny droplets (Fig. 13), which led us to the hypothesis that the
quantities that we computed suffered from discretization artifacts for small droplets, even if they are still
larger than the required minimum size initially indicated by our domain scientists. In particular, such
droplets tend to exhibit alternating e, switching between high and low values at a high temporal frequency,
which supports our hypothesis of discretization issues due to insufficient resolution. We also observed such
alternating behavior of efor very thin ligaments. As a result, we do not consider very small droplets nor
ligaments with this analysis component in the following. To accomplish this, we filter by spherical
anisotropy cs, requiring cs[0:4 in this case. Additionally, we suppress all droplets with low anomaly by
adjusting our filter to e0:15. To obtain a better space-time overview, we now enable the simultaneous
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Fig. 10 Same clusters as in Fig. 10, but within the 3D surface view. aAll clusters in one view. b–iEach cluster on its own.
Note the similarity of droplet surface shape and size within clusters
Angular Velocity [rad/s]
Momentum
[0.01Ns]
(a) (b) (c)
Fig. 11 a The comparison of angular velocity to momentum shows a trend of data points aligning along the axis. Coloring by
prediction anomaly seems to correlate with outliers of this trend. bZoom to bottom left of a. Selected (red) is the data point
with the highest anomaly in this range. c3D surface view of the selected droplet in b. We can see the domain boundary cuts
this droplet. This implies to be the source for this droplet being an anomaly
M. Heinemann et al.
display of all time steps with the chosen filtering criteria, leading to Fig. 12b. This way, we observe many
long traces in the 3D surface view, which provides the point of origin for our further investigations.
We now pick a trace with exceptionally high anomaly estimation (selection indicated by a black cross in
Fig. 12a). A closer inspection of that trace in the time-view (Fig. 4a(v)) reveals that the droplet becomes
rounder over time and that the anomaly indicator estays almost constant over time. This observation turned
out to be quite rare, since typically, the anomaly measure reduces quite quickly, particularly if the respective
droplet becomes roundish. A thorough investigation of the respective plots of the original traces and the
individual deviations of the predicted quantities from the original ones did not provide insights on the causes
of this behavior. We started to hypothesize that the internal flow within the droplet might provide insights.
We thus initiated flow visualization of the liquid phase of the droplet (Fig. 14) (left). Note that we use a
linear mapping of velocity magnitude to glyph color, but a logarithmic mapping of velocity magnitude to
glyph size within Fig. 14 and following. Interestingly, we observed a distinguished and strong saddle-type
flow pattern in the internal flow, in the frame of reference moving with the velocity of the center of mass of
the droplet. We investigated other cases with high anomaly measure e, either by direct selection in the 3D
Fig. 12 Exploration of the dataset within the 3D surface view (cf. Figure 4a(iv)). After loading the dataset, the time filter is
set to show only a single time step and a second filter omits all droplet instances that lack total error (our anomaly indicator) e,
leading to the view in a. The user now selects a droplet of interest for further analysis (black cross). Here, the droplet shown in
Figure 14 (left) is selected. bFurther example. Now the time filter is set to aggregate all time steps of the dataset, leading to
dotted line-like structures showing the same droplet over time. To further reduce visual clutter, filters are used to show only
droplets with anomaly measure e[0:15 and for suppressing ligaments by requiring spherical anisotropy cs[0:4. Here, the
droplet in Fig. 15 is selected
Fig. 13 Droplet exhibiting the highest anomaly. Structures with the highest anomaly measure eare tiny droplets whose
quantity computation suffers from discretization issues. In bthe collision with a smaller droplet artifact can be observed, which
was missed by the trace generation (cf. Sect. 3.2). Further, due to the nature of PLIC, the strongly discontinuous surface
reconstruction can be seen in the form of gaps (cf. Sect. 3.1). Successive time steps a–eexhibit alternating behavior of e,
supporting the hypothesis of insufficient resolution of the simulation grid
Visual analysis of droplet dynamics in large-scale multiphase spray simulations
view or using our similarity search, as provided in the lower half of our tool. Interestingly, most cases of
non-ligament (more or less roundish) droplets with high eturned out to exhibit such saddle-type flow
patterns in their interior, including the case shown in Fig. 14 (right), found as most similar to that droplet by
similarity search.
This motivated us to investigate droplets with different anomaly behavior. While decaying eis quite
common, we wanted to look at droplets for which eincreases over time. The hypothesis behind this
reasoning is that a collision of droplets often causes anomalous droplet behavior, but while the products of
the collision move over time, they tend to ‘‘calm down’’ and become better predictable by our regression
approach, and thus edecays. In Fig. 12b, we were able to identify such a droplet, select it by picking, and
investigate its temporal behavior in the time view. Detailed flow analysis is shown in Fig. 15. Interestingly,
this droplet has been the only one we could find with a strong vortex in its interior.
Finally, for comparison and validation of the utility of our anomaly estimation, we investigated droplets
with low anomaly measure e. We selected them either by direct picking or by similarity search. The majority
of droplets with low eexhibit shear flow in its interior, as shown in Fig. 16 (left). We were able to find only
one case with deviating flow pattern. This droplet (Fig. 16) (right) exhibits a ‘‘half’’ saddle-type flow pattern
(imagine splitting a 3D saddle along its 2D manifold) but with low flow velocity in the frame of reference
moving with the observer.
6 Discussion and future work
We developed a visual analysis approach for investigating droplet characteristics and behaviors occurring in
large ensembles such as sprays. It allows exploring what initially are terabytes of simulation data depicting
complex small-scale effects with a multi-stage workflow. On the one hand, we reduced the data by
(a) (b) (c) (d) (e)
(f) (g) (h) (i) (j)
High Anomaly Droplet case.
(k) (l) (m) (n) (o)
(p) (q) (r) (s) (t)
High Anomaly Droplet II case.
Fig. 14 Left: Selected droplet from Fig. 12a. We found that droplets with low temporal decay of eexhibit distinguished
saddle-type flow patterns in their interior flow, in the frame of reference moving with the center of mass of the droplet. Right:
This is the droplet found most similar by similarity search in Fig. 4a(vi)
Fig. 15 An increasing anomaly droplet case. The scarce case of temporally increasing ebrought our attention to this case,
which, in the frame of reference moving with the center of mass of the droplet, exhibits a strong vortex in its interior flow. This
droplet has been the only one we could identify to contain a vortex
M. Heinemann et al.
extracting droplet surfaces and physical quantities. This allowed us to give an overview and investigate large
numbers of droplets at once during interactive exploration. The physical quantities were further used in
advanced analysis steps to (1) get an overview of different kinds of droplets via clustering (where we
introduced a domain-specific optimization to be able to handle a large number of droplets), and (2) identify
droplets with irregular temporal behavior by adapting a recent machine learning-based method. On the other
hand, the complete original data were still available for the detailed investigation of selected cases. We
integrated different analysis components to be able to provide the user with the full breadth of required
interactive analysis functionality from investigating relationships of extracted physical quantities, droplet
traces, and results from clustering and ML-based analysis to a spatial overview, and finally, a direct link to
ParaView supporting advanced flow analysis for the detailed investigation of selected droplets.
So far, we have applied our tool to gain insights from a single large-scale direct numerical simulation
datasets of two-phase flow of non-Newtonian jets. However, we believe our approach directly generalizes to
various simulations of this kind. We have chosen basic physical quantities based on raw simulation data
from the VOF method (and accordingly will be available from nearly all VOF-based simulations).
Depending on the simulation results, the list of properties could be extended, e.g., if pressure is available, or
a higher resolution allows for a precise calculation of derivation-based quantities. To test the stability of our
quantity calculation, we have downsampled the simulation grid of each droplet by merging a cube of
222 cells into a single cell and repeated the quantity calculation thereon. As expected, the calculations
are quite stable for large droplets containing many grid cells and less stable for smaller droplets consisting of
fewer cells. An example for this behavior in the form of the velocity value is shown in Fig. 17a. Further, we
can look at the temporal development of droplet quantities. Due to the relatively high temporal resolution of
our dataset, we expect smooth changes in the quantities. Figure 17b shows that this only holds for the high-
resolution grid, while the downsampled grid introduces numerical errors due to the coarse grid.
We would generally expect the ANN-based anomaly detection to work likewise for other data. However,
as our prediction is based on traces, we need a minimum temporal resolution, especially some highly chaotic
(a) (b) (c) (d) (e)
(f) (g) (h) (i) (j)
Low Anomaly Shear case.
(k) (l) (m) (n) (o)
(p) (q) (r) (s) (t)
Low Anomaly Half-Saddle case.
Fig. 16 Left: Droplets with low anomaly measure egenerally exhibit shear flow patterns in their interior, in frame of reference
moving with the center of mass of the droplet. Right: The only exception to our hypothesis, that low anomaly droplets exhibit
shear flow patterns in the interior. This droplet exhibits a weak ‘‘half-saddle’’ pattern, i.e., half of a 3D saddle, similar to a
detachment point in its interior, in the frame of reference moving with the droplet
#Cells/Droplet
Velocity
orig./down.
(a) (b)
Fig. 17 An analysis of the quantity calculation on the original grid vs. downsampled grid. aQuotient of velocity on the
original and downsampled grid, plotted over the number of cells per droplet. Note that for a large number of cells per droplet,
this quotient is near one, indicating the quantity calculation is stable over different grid resolutions, while for a low number of
cells, we see diverging calculation results. bSurface area over time for the droplet in Fig. 14. The number of cells for this
droplet on the original grid varies between 811 and 862 cells. The surface area changes are smooth on the original grid but less
stable on the downsampled grid
Visual analysis of droplet dynamics in large-scale multiphase spray simulations
datasets might be problematic, where splits and merges happen in nearly every time step for each droplet.
That being said, we generally consider jets to be already among the more difficult real-world examples.
However, the trace aggregation approach we use for hierarchical clustering might not adequately transfer to
other examples. While the representation of traces through aggregates proved useful in our analysis sce-
nario, more elaborate, adaptive approaches are potentially required in other contexts.
In future work, we aim to study additional variants of jets, and beyond this, we will explore possibilities
to generalize our technique for types of simulation data in which large amounts of small entities need to be
investigated. We plan to incorporate further split and merge events to understand the atomization process
better. While we already collect the number of these events in the entire history of a droplet, we also aim to
integrate a graph view in our tool (enriched with our extracted quantities) and extend our ML-based
anomaly detection to consider split events and merge events explicitly. Finally, the main goal would be to
allow a user to directly influence our clustering and machine learning component during interactive analysis
for specified subsets or partitions of the data.
Acknowledgements We acknowledge the continuous support of our colleagues and domain scientists from the Institute of
Aerospace Thermodynamics (ITLR) of the University of Stuttgart and the collaborative research center on droplet dynamics
under extreme conditions. We kindly acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation)—Project SFB-TRR 75, Project number 84292822 and EXC2075, Project number 390740016,
under Germany’s Excellence Strategy.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use,
sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the
original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The
images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your
intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly
from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funding Open Access funding enabled and organized by Projekt DEAL.
Appendix A
A.1 Definition of Physically Motivated Quantities
Droplet volume: We compute the volume of each droplet by integrating the f-field. That is, the volume liof
the droplet with identifier iis obtained by integrating the product of the f-field and the cell volume for all
cells iwhere li¼i:
li:¼X
i2Ci
fili;ð3Þ
with Ci:¼fjjlj¼ig, and libeing the volume of cell i.
Droplet area-to-volume ratio: As motivated above, we compute the area Aiof droplet iby integrating
the area of the respective PLIC patches, i.e.,
Ai:¼X
i2Ci
Ai;ð4Þ
with Aibeing the area of the PLIC patch in cell i, i.e., Ai¼0iffi¼0orfi¼1. From that, we compute the
area-to-volume ratio ai(Karch et al. 2018) of the droplet as
ai:¼Ai
li
rs
3;ð5Þ
with rs:¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð3liÞ=ð4pÞ
3
pbeing the radius of a sphere with volume li.
Droplet velocity and momentum: The velocity of the center of mass of droplet ifollows to be
M. Heinemann et al.
ui:¼1
liX
i2Ci
miui;ð6Þ
with cell-based flow velocity uiin cell i, and mi:¼fili. From that, the droplet’s momentum computes
pi:¼liui:ð7Þ
Notice that we assume the density of the liquid to be one because the density is often not provided explicitly
for the liquid phase in two-phase simulations (as in our case), and since density represents only a scaling
factor (note that liquids are generally treated as being incompressible). As a consequence, the total mass of
droplet iequals its volume li.
Auxiliary measures: A common measure in astrophysics and particle systems is the center of mass. It
provides a frame of reference and enables the computation of derived quantities. The center of mass xiof
droplet icomputes
xi:¼1
liX
i2Ci
mixi;ð8Þ
with xibeing the center of cell i.
From that, it is a common step to compute the total angular momentum for a set of particles, resulting in
our case in the angular momentum Liof the droplet:
Li:¼X
i2Ci
^
ximiui;ð9Þ
with ^
xi:¼xixibeing the center of cell irelative to xi. Notice that, for the example of particle systems,
this total angular momentum describes the rotational motion of the ‘‘rigid-body aspect’’ of the particle set, or
in other words, the rotational motion of the entirety of the particles. Thus, it represents the ‘‘rigid-body
rotation component’’ of the droplet in terms of the flow of the liquid in our context.
In analogy, the droplet’s inertia tensor Hicomputes
Hi:¼Pimið^
y2
iþ^z2
iÞ
Pimi^
yi^
xiPimi^zi^
xi
Pimi^
xi^
yiPimið^
x2
iþ^z2
iÞ
Pimi^zi^
yi
Pimi^
xi^ziPimi^
yi^ziPimið^
x2
iþ^
y2
iÞ
0
@1
A;ð10Þ
with ^
xi¼:ð^
xi;^
yi;^ziÞ>and i2C
i. From that, the angular velocity xiof droplet iis obtained by
xi:¼H1
iLi:ð11Þ
Notice that Hican become singular for very small droplets, which is one reason we exclude such very small
droplets from our analysis.
Total droplet energy: The total energy Eiof droplet i(notice that the addressed simulations exclude
thermodynamical and chemical processes) computes
Ei:¼X
i2Ci
1
2miu2
i:ð12Þ
This total energy can be decomposed into kinetic Eu
i, rotational Ex
i, and residual energy Ed
i:
Ei:¼Eu
iþEx
iþEd
i;ð13Þ
as explained next.
Kinetic droplet energy: The kinetic energy Eu
iof droplet iwith respect to its velocity uiis
Eu
i:¼1
2liu2
i:ð14Þ
This measure represents the kinetic energy of the rigid-body property of the liquid component representing a
droplet.
Rotational droplet energy: The rotational energy Ex
iof droplet icomputes
Visual analysis of droplet dynamics in large-scale multiphase spray simulations
Ex
i:¼1
2x>
iHixi:ð15Þ
This rotational energy captures the rigid-body part of the liquid dynamics of the droplet with respect to
rotational motion.
Residual droplet energy: The residual energy Ed
iof droplet ifinally computes
Ed
i:¼EiEu
iEx
i;ð16Þ
and includes deformation of the droplet, as well as the non-rigid flow in its interior. In that sense, Ed
i
captures the deviations of a droplet’s flow from rigid-body dynamics. Due to limited numerical accuracy, Ed
i
can turn out to be slightly negative, in which case we clamp it to zero and set Ex
i:¼EiEu
i.
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Visual analysis of droplet dynamics in large-scale multiphase spray simulations