To better understand the insight of the massive flow fields, we use enstrophy to extract attractive components and potential regions. By using properties of components to track their behavior across time, we visualized the volume of all the large components to show the trend of components’ shapes and sizes with time-varying simulation. The mutual information visualization provides the correlation between combustion velocity field and the enstrophy of the combustion turbulence.

### 5.1 Enstrophy analysis

By visualizing the enstrophy at different value, we analyzed a large turbulent combustion simulation, and discussed the evolution of structures and statistics of turbulent fluctuations. As shown in Fig. 1, we visualized the components with high enstrophy over time (color highlight parts), and we can see that during the first time step, small components form up a circle at the front, then it dissipates and disappears in the following process. The high enstrophy part begins to aggregate in the middle region of the turbulent flow over time. The figures (outside the color highlight parts) also refer to components with low enstrophy value, where the components connect together and form a larger shape. For example, they have a complete circular ring in the head of the jet combustion. At the very beginning, the scattered components form a donut winding in the middle region, and then they move backward and amalgamate with posterior components. We can see that in the middle area, some small components grow larger and become one larger component. In the rear area, we can observe that there are some large flat structures at the beginning, but they disappear at the end time, and the components also move to the middle area and amalgamate into fusiform structure.

### 5.2 Components trace

Other than visualization of the contouring of the components, we wanted to quantify the property of the components. Thus, we calculated the volume of the components for high enstrophy value and low enstrophy value. Then we visualize them as frequency histogram. As the volume distribution ranges of low and high enstrophy components are different, we define the total number of bins of the histogram to be 64, which is larger than the actual number. As we can see in Fig. 2, at all time, high enstrophy components tend to have smaller volume, since the distribution of volume at high enstrophy aggregates at low values. However, the volume of low enstrophy components is relatively large. The maximum volume of the high enstrophy components is about 7 cells, while the low enstrophy components can reach 13 cells.

At the start time of the combustion process, high enstrophy components tend to have smaller volume, and over time, the percentage of small volume components decreases. But for low enstrophy components, this kind of trend doesn’t appear.

We also performed statistical analysis on the components. For example, in Fig. 3, we calculated the total volume, the max volume for the components at high enstrophy value and low enstrophy value over time separately. After visualization, we can see that the total volume of high and low enstrophy components tends to increase over time and then decrease at the end of the combustion. And the volume of low enstrophy components is larger than the high enstrophy components. For the max volume of the components, the low enstrophy components have smaller fluctuation range, but we can still observe that it first rises to a high level and then falls. For high enstrophy value, the trend is more obvious.

Then we isolated high enstrophy components and tracked them over time. The number of components is enormous, which makes it hard to isolate them from each other and classify the components. Thus, we filtered the components with a total volume below a fixed value. In this manner, we reduce the number of components and focus more on the salient, connected structures, which makes it easier to track them. We determine a center coordinate of each component based on the coordinates of its boundary points. At every two adjacent time steps, we treat the two components whose center coordinates are closest to each other as the same component. For each component tracked, we set a distance range, and if there are no new components in the range at the next time step, the evolution of the component is considered finished.

To track the evolution of the component in the turbulent combustion process, we use the turbulence generated by the unstable spherical burst simulation process to track the high enstrophy components. Through the above component tracking method, we visualized the visualization results of the high components of the surface at several consecutive time steps when turbulence was generated, as shown in the figure. As shown in Fig. 4, yellow, blue, white, and red are the evolution of the high component of turbulence in 4 consecutive periods. It can be seen that it gradually decreases and dissipates in the process of outward diffusion. Among them, the ring structure of the high component of turbulence at the fourth time step (red) has begun to dissipate.

We then visualized the volume at every time step for each component. In Fig. 5, we can observe that the volume of components ranges from 17 cells to 35 cells, but mostly from 20 cells to 30 cells. The change in volume from time to time follows consistent trends over the whole time, which makes the track process more valid. We can see a global trend that the volume for most components increases initially and decreases afterwards. The volume reaches its peak at middle time of the simulation of combustion.

### 5.3 Mutual information

To further illustrate the effectiveness of our visual analysis system, we evaluate the results of visualization of enstrophy by calculating the mutual information between enstrophy and velocity field and performing normalization calculations. As shown in Fig. 6, the first and second rows of the figure respectively visualize the distribution of the data velocity field and the corresponding enstrophy distribution. The value of mutual information is 0.999 throughout the simulation process. It can be seen that the enstrophy distribution and the velocity field distribution have some similarities in the visualization results. The calculation results of mutual information show that enstrophy and the combustion velocity field have a very high correlation, which can better deal with the analysis and visualization of combustion turbulence.