Visual analysis of droplet dynamics in large-scale multiphase spray simulations

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 \(f(\mathbf {x}) \in \mathbb {R}\) during the simulation. This field stores conceptually for each point \(\mathbf {x} \in \mathbb {R}^3\) the volume fraction of the liquid, i.e., \(f=0\) representing only gas, \(f=1\) representing only liquid, and \(0< f < 1\) mixtures of both. Here, \(f(\mathbf {x})\) is defined in a cell-based manner, i.e., the field stores this total fraction for each cell i individually, in piecewise constant representation. Thus, \(f_i\) represents the value of the \(f\)-field for cell i. Similarly, the velocity field \(\mathbf {u}({\mathbf {x})}\) is given in cell-based representation \(\mathbf {u}_i\). In the domain, a droplet is commonly defined as the face-connected component of cells that exhibit \(f_i > 0\). In practice, due to the numerical limitations of the simulation, we use the slightly modified definition \(f_i > \tau _f\), with \(\tau _f= 10^{-6}\). 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 \(l(\mathbf {x})\), where \(l_i\) stores the droplet identifier \(\iota\) of the droplet that cell i belongs 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 \(f_i\). 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 representation. 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 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 \(\iota\): volume \(\overline{\mu }_{\iota }\), area \(\overline{A}_{\iota }\), area-to-volume ratio \(\overline{\alpha }_{\iota }\), velocity \(\Vert \overline{\mathbf {u}}_{\iota } \Vert\), momentum \(\Vert \overline{\mathbf {p}}_{\iota } \Vert\), angular velocity \(\Vert \overline{\pmb {\omega }}_{\iota } \Vert\), angular momentum \(\Vert \smash {\overline{\mathbf {L}}}_{\iota } \Vert\), total energy \(E_{\iota }\), kinetic energy \(\smash {E_{\iota }^{\mathbf {u}}}\), rotational energy \(\smash {E_{\iota }^{\pmb {\omega }}}\), and residual energy \(\smash {E_{\iota }^{\delta }}\). 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 \(\iota\), which we denote \(c_s\), with \(c_s=0\) indicating linear or planar shape, and \(c_s=1\) 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 \(\iota\) is provided as a discretized alternative to \(\overline{\mu }_{\iota }\) with 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.

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:An edge is added if \(\mathbf {x}(t+\varDelta t)\) belongs to a droplet, i.e., \(\mathbf {x}(t+\varDelta t)\) is located at time \(t+\varDelta t\) in a cell j with a valid droplet label. We also do a backward Euler step from the center \(\mathbf {u}_k\) of each cell k that is part of a droplet at time \(t+\varDelta t\):If \(\mathbf {x}(t)\) belongs to a droplet at time t, i.e., if \(\mathbf {x}(t)\) is located in a cell l with a valid droplet label at t, we add the respective edge (if not already present). If a node \(n_j\) has degree \(d> 2\) at time t, and only one connected neighbor at time \(t-\varDelta t\), a breakup event has happened at node \(n_j\) at time t. If there is more than one connected neighbor at time \(t-\varDelta t\), 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 droplet over time. A trace contains droplet quantities as described above and serves as input to our regression model (Sect. 3.4).

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 \((n^2 - n)/2\) values to store. Handling our \(1\,000\,000\) droplet instances (\(\approx 10\,000\) 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.

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 predictable 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 \(n_t\), and a fixed window size \(w\), we obtain \(n_t- w+ 1\) subtraces, and for each subtrace, the model takes the first \(w- 1\) 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 optimizer (Kingma and Ba 2014), with a learning rate \(10^{-5}\), 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 \(w-1\) 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 \(\smash {\overline{\varepsilon }}\)—our measure of estimated droplet anomaly.

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 \(\approx\)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 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 simulation 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 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.

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).

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.

Figure 4a shows the 3D surface view depicting time step 410. Color depicts our anomaly estimation \(\smash {\overline{\varepsilon }}\), with higher anomaly indicated by reddish colors and lower anomaly by yellowish ones. Gray indicates droplet instances that are within the \(w- 1\) first time steps of a trace and thus do not provide prediction nor anomaly information \(\smash {\overline{\varepsilon }}\) (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 \(\smash {\overline{\varepsilon }}\ge 0\), because \(\smash {\overline{\varepsilon }}\) is set to a negative constant in our implementation if no \(\smash {\overline{\varepsilon }}\) value 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 \(\smash {\overline{\varepsilon }}\), 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 \(\smash {\overline{\varepsilon }}\) are 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 \(\smash {\overline{\varepsilon }}\), 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 \(\smash {\overline{\varepsilon }}\) for 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 \(c_s\), requiring \(c_s> 0.4\) in this case. Additionally, we suppress all droplets with low anomaly by adjusting our filter to \(\smash {\overline{\varepsilon }}\ge 0.15\). To obtain a better space-time overview, we now enable the simultaneous 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 \(\smash {\overline{\varepsilon }}\) stays 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 \(\smash {\overline{\varepsilon }}\), either by direct selection in the 3D 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 \(\smash {\overline{\varepsilon }}\) turned 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 \(\smash {\overline{\varepsilon }}\) is quite common, we wanted to look at droplets for which \(\smash {\overline{\varepsilon }}\) increases 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 \(\smash {\overline{\varepsilon }}\) decays. 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 \(\smash {\overline{\varepsilon }}\). We selected them either by direct picking or by similarity search. The majority of droplets with low \(\smash {\overline{\varepsilon }}\) exhibit 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.