Comparisons of 2 and 3-D Flow Structures

SWBLI is susceptible to three-dimensional instabilities and is sensitive to the dimensionality of the flow. Figures 4.5(a) and (b) provide a comparison of the difference in the predicted shock structure along the wedge centerline for the 2 and 3-D geometries, respectively. It can be seen that the boundary layer for both geometries becomes wider at the point where the separation shock is formed. Additionally, this point is closer to the tip of the wedge in the 2-D case which results in a larger separation region compared to the 3-D case. Also, for the 2-D case, the inside area surrounded by the bow shock and transmitted shock and the shear layer is larger in 2-D compared to the 3-D case.

To understand the difference in the predicted temporal variation of the two and three-dimensional flows, we compare the translational temperature profiles, streamlines, shock structure profiles and vorticity of these two models. We found that the degree of thermal nonequilibrium for rotation and vibration did not reveal any significant difference for the two or three-dimensional model. Early in the time evolution of the flow, 40 µs, the translational temperature spatial distribution was found to be the same for both the two and

three-dimensional models. Generally, at this time, there is no significant difference between the two cases since the size of the separation zone is not large and surface streamlines show little variation in the spanwise direction. The length of the high-temperature region in the 2-D case was found to be slightly larger than in 3-D due to the fact that the triple point is slightly forward compared to the 3-D case. Figures4.21and 4.22 show the later time evolution of the translational temperature contours and the surface streamline, respectively. These figures show that as time progresses, the separation zone moves towards the tip of the wedge and becomes larger in the 2-D case. The difference between the 2-D and 3-D flow structures becomes significant in time. Turning to the experimental shock tracking results again, the time dependence of the 2-D and 3-D predicted shock profiles is compared. As can be seen in Fig.4.23, in the 2-D case, the triple point continues to move and the difference becomes larger. Indeed, the 3-D calculations are in good agreement with the experimental result. A small deviation of about 1 mm in the experimental tracking data may be attributed to the uncertainty in the image tracking and the shock blurring that occurs due to the finite shutter time and pixel resolution.

The well-developed separation zone caused by interactions between shock waves and the boundary layer results in vortices at the hinge and corner points of the wedge. As shown in Fig 4.5, the separation region starts with the separation shock on the boundary layer at the forward part of the wedge and continues to the reattachment point at its aft part. In order to visualize and determine the separation and reattachment points in the 3-D simulations streamlines adjacent to the boundary layer were used (see Fig.4.16d). Another way to characterize the strength of the shock-shock interaction is by examination of the vortex strength which has a significant effect on the flowfield and the stability of the shock structure. It is well known that the circulation, Γ, can be obtained from the calculated flow velocities as

Γ = I

~

V ·dl.~ (4.1)

In this work, the velocities, V~, used in the circulation calculation (i.e. line integration) were taken at the boundaries shown as white dashed lines in Fig. 4.24(a) that includes the separation zone. As per the convention, the counter-clockwise direction was taken as positive in the calculation of theZ component of the vortex. In order to look at the difference in stability of the 2-D and 3-D simulations, the separation length and the circulation in the domain are presented in Table4.5as a function of time for both cases. The separation length is defined as the difference between the separation and reattachment points as indicated in Fig. 4.16(d). So defined, the separation length for the 2-D case is seen to increase in size quite rapidly

(a) 2-dimensional case, 120µs (b) 3-dimensional case at center plane, 120µs

(c) 2-dimensional case, 200µs (d) 3-dimensional case, 200µs

Figure 4.21: Comparison of translational temperature (K) contours as a function of the number of time steps and dimensionality. In 3-D case, the wedge center plane is presented.

(a) Temperature, 120µs (b) Streamlines, 120µs

(c) Temperature, 200µs (d) Streamlines, 200µs

Figure 4.22: Temperature taken alongy= 0.04 m line and streamlines on the wedge surface comparisons as a function of time and dimensionality.

(a) 100µs (b) 200µs

Figure 4.23: Comparison of the position of the shock wave between simulations and experiment for the 2 and 3-D geometries.

up to 300µs and then much more slowly. A similar time behavior is observed for the 2-D circulation values which increase (decrease in magnitude) quite fast, but, after 400µs increase only slightly. It should be noted that the number of vortex cores generated in the clockwise direction increases in time as can be seen in Fig.4.24(b). The same analysis was conducted for the 3-D center plane solution. Note that at the beginning of the simulation prior to∼100µs, the separation length and circulation values are almost the same for the two and three-dimensional calculations. However, as time progresses, the separation length for the 3-D case is almost half the size of the two-dimensional case, and it seems to converge to a constant value of about 0.012 m much faster than in the two-dimensional case. Moreover, the magnitude of the circulation values for the 3-D wedge is found to be larger than in 2-D at 800µs due to the decrease of the velocity values at the boundaries of the specified region caused by the stronger vortices for the 2-dimensional case, as shown by comparing Figs.4.24(b) and (c). In fact, the close up of velocity spatial distributions in the hinge region for the 3-D case (Fig.4.24(c)) looks very similar to that of the early 2-D case (Fig.4.24(a)) further explaining why the circulation values of -58.15 versus -56.7 m2_{/s given in Table4.5are so similar.}