Next, the natural HHD is used to decompose a number of different simulated flows reflecting a variety of physical phenomena. In particular, the results presented in this section show how physical flows do not conform with traditional boundary conditions, and, thus, existing techniques are not applicable.
3.3.2.1 Flow beh in d th e cylin d er and cu b o id . The data shown in Figure 3.6 is a single time-slice of the time-varying simulated flow behind a cylinder. In this setup, the flow is injected from the left boundary, and its behavior is observed behind the cylinder, which is located at the left boundary of the domain (see Data B.1.3). As seen in the figure, the natural HHD extracts the harmonic components completely, and r* reveals the vortices. d* is zero since the original flow is incompressible. Note how r* is not aligned with boundary— a result not feasible with traditional techniques. Especially, note that the vortices in this data move left to right as time progresses. Therefore, depending upon the current value of time, the alignment of the flow inside the rightmost vortex changes with the right boundary of the domain, thus creating serious time-dependent artifacts. The computation of the natural HHD for this [400 x 50] data took about 1.06 seconds.
Figure 3.7 shows the results of the natural HHD on a similar 3D flow. The data represents the flow behind a cuboid, where the flow is injected from one direction, and its behavior is observed behind the obstacle. Notice that r* reveals the vortical structures, and is not
Figure 3.6. The natural HHD of the flow behind a cylinder (top). r* reveals the von Karman vortex street, and is not tangential with the boundary (middle). h* represents the background flow (bottom).
compatible with the standard boundary condition, whereas h* captures the background flow. The decomposition of this [101 x 101 x 101] flow took about 120 seconds with 144 processes in parallel.
3.3.2.2 Lifted ethylene je t flame. The second example is a direct numerical simulation of a turbulent lifted ethylene jet flame [222] (see Data B.3.1). Unlike the previous dataset, this is a compressible and highly turbulent flow. The fuel is injected on the bottom of the domain, creating a strong harmonic flow towards the top. Figure 3.8 shows one snapshot of a 2D slice from the center of the 3D flow. Both r* (Figure 3.8(b)) and d* (Figure 3.8(c)) are highly complex, not aligned with the boundary, and show some surprising structures. In particular, r* shows two global counter-rotating vortices rather than a streak of smaller vortices one may have expected. Finally, h* (Figure 3.8(d)) reveals an elliptical shape reflecting the nonconstant velocity profile imposed by the simulation. The decomposition of this [800 x 2025] data took about 290 seconds with 144 processes in parallel.
3.3.2.3 Jet in cross-flow . The next dataset represents a simulation of a jet in cross flow, which is a fundamental flow phenomenon in many engineering applications [60, 73], for
(c)
Figure 3.7. The natural HHD of the flow behind a cuboid (a). The figure shows streamlines in the interesting regions. (b) r* reveals the vortical structures, and (c) h* shows the translation present in the flow (right).
example, film cooling of turbines, fuel injections, and dilution jets in gas-turbine combustors. The experimental setup contains injection of flow through a jet at the bottom in the presence of a strong background transverse flow, the cross-flow (see Data B.3.2).
To simplify the illustration, a 2D slice is taken through the center of the 3D flow, such that the cross-flow is directed from left to right, and the jet appears at the bottom. Figure 3.9 compares the rotational fields obtained through HHDnp and the natural HHD. The topological skeletons of these flows are computed by decomposing the domain into
(a (b) (c (d)
Figure 3.8. The natural HHD of the flow at the center of a lifted ethylene jet flame. (a) Flow at the center of a lifted ethylene jet flame [222]; (b) r* showing two global counter-rotating vortices encapsulating smaller structures; (c) d* showing a large number of sources and sinks related to volume changes; and (d) h* computed as the residual showing an elliptical flow structure around the flame center.
nonoverlapping regions using the critical points and the saddle separatrices of the flow. Since the topological skeletons of the two rotational fields are significantly different, the im portant question is which of these flows corresponds to the expected combustion phenomena. Through scientific insights of domain experts, it is known that the small vortices near the top-left and bottom-left corners of fNp are unphysical, since the underlying physical model expects vortices to be generated behind the jet only. Furthermore, note how r* successfully captures the behavior of jet (green color), which is expected to rise up and go left to right (compare with the illustration shown in Appendix B). On the other hand, fNp shows the jet (brown color) to follow a circular path that does not leave the domain from the right side, but comes back to the jet’s point of entry. This behavior is also deemed unphysical by domain scientists. Thus, imposing parallel flow on the boundary produces false features (vortices) in the analysis.
These results demonstrate that the flow obtained through the natural HHD captures the physical model successfully, and it becomes possible to make enquiries about the accuracy or uncertainty of the computation. However, the flow obtained by using the boundary
Figure 3.9. Comparison of the natural HHD and the HHD with NP boundary conditions on a 2D slice of the jet in cross-flow. The rotational fields rNp and r* are shown in the top and the bottom, respectively. Unphysical rotational structures are observed at the bottom-left and top-left corners in tnp caused due to imposing parallel flow.
conditions does not respect the underlying physical model, and cannot be used to derive insights. Furthermore, the concerns of accuracy and/or uncertainty do not even make any sense in this case since the flow represents a different physical phenomena.
3.3.2.4 O ceanic currents. The final example is the surface flow taken from a 3D simulation of global ocean currents [133] (see Data B.3.3), as shown in Figures 3.10, with the focus on a Qi = [200 x 200] region in the South Pacific ocean. To demonstrate the practical utility of the local computation and the local approximation of the natural HHD, the decomposition for Q1 is computed using the data from concentric grids Q = Gn. Figure 3.11
(c) (d)
F igure 3.10. Local approximation of the natural incompressible component of the global oceanic flow, computed for the innermost tile, G200, shown in (a) by separately computing over three concentric tiles: (b) G200, (c) G400, and (d) G600.
shows a quantitative comparison of the results when N is increased from 200 to 600. It should be noted that whereas the time needed for the computation grows quadratically, the L2 and norms, and hence the decomposition, already seem to have converged. As a result, in order to analyze the flow in Q i, the natural HHD provides a local and faster technique, instead of using the existing global solutions, which would require the computation for the entire domain with over 5.3 million vertices. Figure 3.10 shows the rotational flows obtained for Q = G200, Q = G400, and Q = G600.
Size
Figure 3.11. Quantitative evaluation of local approximation of the natural incompressible component of the global oceanic flow. Local computation for Q1 = G200 using the data from Q = Gn with increasing N requires a quadratically increasing time (scaled to the right axis). On the other hand, the and the normalized L 2 norms (log-scaled to the left axis) of |r| and V x r seem converged.