Three-dimensional Effects for the Nitrogen Case

The lack of agreement between the time evolution of the simulations and the experiment suggests that the flow may not be two-dimensional in the Z = 0 symmetry plane, although, the previous empirical results of Ball [113] of flow over a symmetric wedge/flap configuration under similar free stream conditions show that the span to boundary layer thickness at the hinge should have been sufficient to prevent edge effects from reaching the centerline. Hence, the goals of the 3-D simulations are to establish whether the flow in the

Z = 0 symmetry plane is affected by three-dimensionality, and to what extent the inclusion of the flow in the spanwise (Z) direction reduces the shock-shock interaction (pressure relief effect). A three-dimensional view of the computational domain and the double wedge with the simulated translational temperature contours superimposed is shown in Fig. 4.16(a). In this figure, as well as subsequent ones, the flow is computed using the 3-D DSMC numerical parameters listed in Table4.4 namely the baseline case. As can clearly be seen in Fig.4.16(b) the flow is three-dimensional since the translational temperature contours are strongly non-uniform in the z-direction. Similarly, the top view of the chemiluminescence image shown in Fig.4.16(c) indicates that the high-temperature region differs from the wide centerline region to the edge. In Fig. 4.16(d), the streamlines at the wedge surface are shown. Below the AB line, the velocity field has almost no Z-component. However, the flow crossing the C and D line shows strong three-dimensionality, with the streamlines directed towards outer edge of the wedge inZdirection. For a quantitative illustration of this spanwise effect, two slices have been selected, one at the center and the other at the edge of the

(a) Velocity slip (b) Transient shock profile

(c) Heat transfer rates

double wedge. Examination of Figs.4.17(a)and 4.17(b)shows that translational temperature contours look considerably different, especially in the region where the oblique and the bow shocks intersect. The center profile is characterized by a larger high-temperature region and larger separation zone as compared to the edge region. These results are consistent with the CFD simulations reported earlier by Rudy et al.[26].

The impact of flow three-dimensionality on gas-surface properties is also significant, as seen in Fig.4.18 where the skin friction and pressure coefficients are presented. The separation and reattachment points labeled asSandR, respectively, are obtained through the examination of streamlines at the center plane, as indicated in Fig.4.16(d). Similar to the translational temperature fields, these parameters are different at the center and the edge of the wedge. More specifically, as shown in Fig.4.18(a), the skin friction coefficient at the edge is higher than that at the center, however, both profiles have a tendency to decrease from the leading edge of the first wedge towards the hinge. For the center profile, the friction coefficient reaches its minimum value at the beginning of the separation zone. In contrast to the two-dimensional solution, the 3-D center solution does not show a sign change over the separation zone which is more likely due to insufficient cell resolution. Note that the skin friction coefficient for both the 3-D center and 2-D profiles suddenly increases at the hinge point. It is also observed that there is a large increase starting at the end of the separation zone in the 3-D center plane and 2D cases. The maximum value occurs at the point where the boundary layer becomes thin due to strong compression [26]. This point is shifted towards the free stream direction for the edge slice compared to the centerline because there is no significant separation region at the edge of the wedge. In Fig. 4.18(b), the pressure coefficient along the four different planes indicated in Fig.4.16(b) is shown. As expected, the center values are larger than at the edge due to the stronger shock at the center. However, the pressure coefficient is found to be essentially the same for the center and two interior planes. The location of the maximum point for the center and edge values differs because of the absence of a distinct separation region at the edge which affects the pressure values.

The effect of the three-dimensionality of the flow on its time dependence along the center region is examined next. Figure 4.19 presents the translational temperature profiles as a function of time at a Y

plane passing through the separation region. As can be seen from Fig.4.19(a), the temperature profiles only slightly change in time, especially after 140µs. In fact, the extent of the shock region is almost the same, and the temperature variation as a function of time is insignificant in the shock region, in direct contrast to the two-dimensional case for 100 and 200µs presented in Fig.4.9, where both the extent of the shock and temperature values were observed to change dramatically in time. Figure4.7(b) shows the time evolution of the shock structure for the 3-D case and similar to the experiment it can be seen that the flow reaches steady

(a) 3D view (b) Top view aty= 0.04 m constant plane

(c) Schlieren image (top view) (d) Streamlines on the surface

Figure 4.16: The 3D effects at 140µs simulated using the 3-D (baseline case) wedge geometry. The backplane is theZ−0 symmetry plane.

(a) At the center (b) At the edge

Figure 4.17: Translational temperatures (K) contours at 140µs simulated using the 3-D wedge geometry.

(a) Skin friction (b) Surface pressure coefficient

Figure 4.18: Comparison of surface parameters along the center and the edge of the wedge for the 3-D case at 140µs. See Fig.4.16(b) for geometry definitions.

state after 100 µs. Figure 4.19(b) shows the heat transfer rate obtained from experiment (including error bars with about 10% deviation from its mean) and the 3-D calculation. The heat flux decreases along the first wedge from the free stream to the hinge point, as predicted by the laminar boundary-layer theory [114]. The calculated heat flux rates are found to be in reasonably good agreement with the measurements on the first wedge, however, the solution does not show any indication of the separation along the first wedge. As mentioned previously, this is likely due to the poorer cell and panel resolutions especially at the location,

x/L= 0.8. Nonetheless, much better agreement with the experiment is achieved at the aft part of the wedge compared to the 2-D result.

(a) Translational temperatures (K), at the center plane andy= 0.04 (m) constant line

(b) Heat transfer profiles

Figure 4.19: Comparison of temperature (K) at different times along the centerline of the wedge and com- puted (at different times) and measured heat flux values in the centerline region.

Note that an additional 3-D calculation was conducted with lower cell resolution and fewer number of particles, denoted as the degraded 3D case in Table 4.4 in order to check how the flowfield and surface parameters change with the numerical parameters. The results are shown in Fig.4.20for a time of 100µs, which as mentioned earlier is not the steady state result for the 2-D geometry. As can be seen in Fig.4.20(a), the heat flux is very sensitive to the DSMC parameters, and the numerical result for better grid resolution approaches the experiment for both 2-D and 3-D cases at 100µs. In contrast, the temperature profiles shown in Fig.4.20(b)can be seen to be less sensitive to the refined numerical DSMC parameters, with the change amounting to a fraction of a percent. More importantly, in the 3-D case, Figs.4.20(a)and4.19(b) show that the simulated heat flux values converge to the experimental values on the time scale of the experiment (in

(a) Heat flux, 100µs (b) Centerline temperature, 100µs

Figure 4.20: Comparison of 3-D wedge geometry heat fluxes and temperatures along the wedge centerline for further refined DSMC numerical parameters.

contrast to 800µs for the 2-D case) and, unlike the 2-D case, the spatial distribution of the heat flux values from the separation region to the edge of the wedge follows the measurements.