Results
4.2 Computational framework
For both these test cases different meshes will be considered for the two different capsule designs studied. However, since the flow around the body differs depending on either the capsule’s geometry or the trajectory point being studied, not all meshes (for the same geometry) will have the same upstream boundary. Depending on whether the shock front is closer or farther from the wall, the mesh is chosen to be distant enough to capture this shock, but also not too distant as to not have unnecessary cells computed in the freestream, and thus, reducing the code’s efficiency.
As mentioned before, the chosen mesh consisted on a structured mesh of 70x60, yielding 4200 cells.
This layout allowed achieving sufficient accuracy for computing the convective wall heat fluxes. However, for the radiative study using Spark LbL, this layout is too memory intensive, which is why every solution was also computed for a 50x60 mesh with 3000 cells. This layout will only be used for computing the radiative heat flux. Before adopting this option, an exercise was made to confirm the low influence of the number of normal cells to the final result (radiative wall heat flux), using the tangent slab approach. The results showed a difference smaller than 1% for both capsules with the chemical composition B (with CH4) and around 2% for the composition A (without CH4), confirming the aforementioned assumption.
4.2.1 Computational Domain
The computational domain was extended beyond the end of the capsule’s shoulder, to account for the hypothetical location of the trim tabs (considered for Test Case 2). Even though these cells at first sight could seem not useful for Test Case 1, they still have their usefulness. Since the Ray Tracing model computes the wall heat from all the visible rays, shortening the domain to the limit of the shoulder would underestimate the radiative wall heat in these last wall points since the downstream vicinity of these points would otherwise lie outside the computational domain.
Also, even though the cornering of the shoulder is not being considered for the solutions, this should not affect the results significantly as long as the sonic line reattaches to the body before the shoulder (which is the case for both capsule configurations for the Entry TP, shown in Figure 4.1). This makes the expansion modeling not required if one only cares for the body’s front solution. This procedure avoids the huddles related to instabilities that would arise if the expansion in the shoulder was considered and computed. In this case, ignoring the strong flow expansion, the conical region may simply be extended with no impact on the forebody flowfield [53]. While doing so, one may use the same flowfield solution for the sweeping area beyond the shoulder for accounting, or not, the flap, and just ignore the part below the shoulder if the flap region is not being analyzed. However, for Aerocapture TPθ = 60◦ this sonic line attachment does not occur and the results lose their validity. More details are presented in§4.4.2.2.
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(a) Mach contour lines for Entry TPθ= 60◦.
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(b) Mach contour lines for Entry TPθ= 45◦. Figure 4.1: Sonic lines and shock line.
The domain covered by the meshes also varies for each case, since depending on the capsule geometry and the upstream conditions considered, the shock front will be located at different distances from the wall, as seen in Figure 4.2. This means that for each case, while computing the first solutions without refinement, one should overestimate the distance of the shock to the wall. After having an adequate estimation, right after the first refinement, the mesh was shortened to include less of the freestream region and thus allow for more efficient computation of the solutions.
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Stagnation region zoom
(a) Mesh for Entry TPθ= 60◦ with CH4.
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Stagnation region zoom
(b) Mesh for Entry TPθ= 45◦ with CH4.
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Stagnation region zoom
(c) Mesh for Entry TPθ= 60◦ without CH4.
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Stagnation region zoom
(d) Mesh for Entry TPθ= 45◦ without CH4.
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Stagnation region zoom
(e) Mesh for Aerocapture TPθ= 60◦ with CH4.
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Stagnation region zoom
(f) Mesh for Aerocapture TPθ= 45◦ with CH4.
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Stagnation region zoom
(g) Mesh for Aerocapture TPθ= 60◦ without CH4.
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Stagnation region zoom
(h) Mesh for Aerocapture TPθ= 45◦ without CH4. Figure 4.2: Meshes for all cases.
4.2.2 Ray Tracing Convergence
Before choosing the number of rays per hemisphere used for the radiative analysis, some conver- gence tests (global and local) were carried out for a specific case using 50, 75 and 100 rays. For the global convergence the choice arbitrarily relied on the Entry TP forθ= 60◦. As Figure 4.3 shows, there is a marginal difference when increasing the number of rays reaching a maximum error of 3.5% but hav- ing an average value below 2 %. However, this increment adds a significant additional time for the Ray Tracing procedures computation as will be shown in Table 4.4. In view of this, remaining Ray Tracing procedures were performed with 50 rays per hemisphere.
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Dist along wall from Stag point - S [cm]
1000 1500 2000 2500 3000 3500 4000
Radiative Wall heat flux [W/cm2]
50 Rays 75 Rays 100 Rays
(a) Radiative heat flux for different number of rays per hemisphere.
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Dist along wall from Stag point - S [cm]
0 0.5 1 1.5 2 2.5 3 3.5 4
Radiative Wall heat flux Error [%]
50 Rays 75 Rays 100 Rays
(b) Error in radiative hear flux for different number of rays per hemisphere .
Figure 4.3: Convergence study on number of rays per hemisphere for Radiative Study - Entry TP for θ= 60◦ with CH4.
Using another case that requires less computational times (Aerocapture TPθ= 45◦ without CH4, for example), the convergence study was also performed locally, only analyzing the Radiative Heat in the Stagnation Point for a greater number of points. The results are presented in Table 4.3.
Table 4.3: Convergence Study for Ray Tracing for Stagnation Line - Error compared to 1500 rays case.
Number of Rays 50 75 100 150 200 250 500 750 1000 1500
Error [%] 2.36 3.31 -0.66 1.10 -0.03 0.23 0.58 0.33 -0.31 0.00
The analysis for these ray convergence studies leads to the following conclusions: one may assume that there is a low error when considering only 50 rays for the remaining radiative study, since these errors remain clearly below 4% over the whole domain (compared to 100 rays) and below 3% locally in the stagnation point (compared to 1500 rays).