Far-Field Properties

Figure 3.1 illustrates M andReeffects on the wake structure visualized by the isosurface of the second invariant of velocity gradient tensor (Q-criterion). The thresholds of Q-criterion was 5.0×10⁻⁴which is normalized value by the freestream velocity. Several kinds of wake structures appear at M ≤ 2.0 andRe ≤ 1, 000, and the wake structures become complex and simple asRe and M increase, respectively. Streamwise steady vortices are generated at the downstream of sphere at Re = 250 and M = 0.3. At Re = 300 and 500, hairpin vortices are generated in the recirculation region and the wake becomes unsteady flow. WhenRefurther increases to 750 and 1,000, the wake vortices form a helical structure with a high mode and a low mode. The size of the recirculation region atM =0.8 becomes large compared to that atM = 0.3, and the pressure coefficient distribution changes. The Reevolution of the flow patterns is similar to M =0.3 in figure 3.1, but the wake structure at M = 0.8 seems to be more complicated compared to that at M = 0.3. Thus, the critical Re for each flow pattern might be different between M = 0.3 and 0.8. However, the details cannot be discussed due to the lack of DNS data in the Redirection.

The compressibility effect on the wake vortices becomes obvious for M ≥ 0.95. The pressure coefficient distribution is drastically changed and a recompression wave can be observed around the end of the recirculation region at M = 0.95. In this case, there is only a steady recirculation region, and no unsteady wake vortices are formed downstream of the sphere up toRe =300. At Re =500, streamwise vortices can be observed, similar to those at Re= 250 andM =0.3, and the wake becomes helical atRe ≥ 750, as atM =0.3 and 0.8, but its structure is more complex than that under subsonic conditions. A detached shock wave is formed upstream of the sphere at M ≥ 1.05, and those waves including expansion and recompression waves become stronger as M increases. The flow behind the sphere remains steady flow up to Re = 500, and hairpin

56

3.3. Results and Discussion

structures appear atM =1.05 andRe ≥ 750. Stabilization effects of the wake by compressibility become strong as M increases, and the wake is steady atM ≥ 1.5 andRe ≤ 1, 000.

M = 0.3

Re = 250 Re = 300 Re = 500 Re = 750 Re = 1,000

M = 0.8 M = 0.95 M = 1.05 M = 1.2 M = 1.5 M = 2.0

C_P

-0.54 1.1 0 1.0

Figure 3.1: Instantaneous wake structures. The structures are identified by the isosurface of the normalized second invariant value of the velocity gradient tensor, which is normalized with respect to the freestream velocity (Q/u²_∞ =5.0×10⁻⁴).

Chapter 3. Flow over a Stationary Adiabatic Sphere

Re = 300

M = 1.05 M = 1.2

M = 0.95 Re = 500

Re = 750 Re = 1,000

M = 0.8

Figure 3.2: Distribution of absolute values of density gradient.

Figure 3.2 shows schlieren-like images for 300 ≤ Re ≤ 1, 000 at around the transonic condition. At M = 0.8, there is a small white region, the λ-shock, around θ = 90 deg, and it becomes clearer as Re increases. The flow structure behind the sphere becomes different from the incompressible flow because theλ-shock changes the position of the separation point.

Hence, it seems that the wake structure at M = 0.8 is more complicated compared to that at M = 0.3, as shown in figure 3.2. Theλ-shock becomes clearer and a wake shock wave is formed downstream of the sphere atM =0.95. Also, those shock waves become clearer asReincreases because the viscous dissipation decreases. ForRe ≥ 500, the wake becomes unsteady and wake vortices appear to be generated around the wake shock wave. This trend is similar to that under the higher-Reconditions with similar M values. It will be discussed in chapter 6. In that case,

58

3.3. Results and Discussion

Reis 1.9×10⁵, which created a completely different flow regime, but a large-scale oscillation of the wake is generated around the wake shock wave. A detached shock wave is formed upstream of the sphere, and a recompression wave is formed around the root of the wake shock wave at M ≥ 1.05. The oscillation of the wake generated around the recompression wave is observed also at 8.1×10³ ≤ Re ≤ 1.0×10⁵with a similarM.

0 0.05 0.10 0.15 0.20

0.5 1.0 1.5 2.0

St

M Re = 300

Re = 500 Re = 750 Re = 1,000

Figure 3.3: Effect ofMon theSt of vortex shedding.

Figure 3.3 shows the influence ofMon theSt of vortex shedding. TheStof vortex shedding for Re = 300 and Re > 300 were computed by the time variation of the lift coefficient and the velocity fluctuation at the maximum TKE point in the downstream, respectively. Overall, St of vortex shedding increases as Re increases, and the critical M, at which St of vortex shedding becomes zero, move to the higher-M side as Reincreases. TheSt of vortex shedding at Re = 300 decreases as M increases under the subsonic conditions and rapidly approaches zero around M = 0.95 because there is no vortex shedding at Re = 300 and M ≥ 0.95. The trend of the St of vortex shedding at Re = 500 is similar to that at Re = 300, but the St of vortex shedding does not decrease up to M = 0.95 and sharply approaches zero at around M = 1.0. The St of vortex shedding decreases at 0.3 ≤ M ≤ 0.8 and there are no M effects on the St of vortex shedding at 0.8 ≤ M ≤ 1.05. The decrease in theSt of vortex shedding at 0.3 ≤ M ≤ 0.8 corresponds to the change in the wake structure. As shown in figure 3.1, the wake structure at M =0.8 and 0.95 for Re =750 is more complicated (resembling a higher-Re condition) compared to that at M =0.3 or at incompressible flow, because the weak shock wave formed near the separation point and the flow separation is promoted by the effect of the weak

Chapter 3. Flow over a Stationary Adiabatic Sphere

shock wave. This results in the unsteadiness of the recirculation region and complicated vortex structure. However, the flowfield is strongly stabilized under the supersonic regime as the same as the lower-Recases, and then theSt of vortex shedding becomes zero atM ≤ 1.2. Under the sufficiently high-Reconditions, for example,Re ≥ O(10⁵), theλ-shock wave is formed and the flow separation is induced at approximately θ = 90 deg. However, under lower-Reconditions, for example,Re < O(10³), the boundary layer becomes thicker and attenuates theλ-shock wave on the surface of the sphere; thus, the effect of theλ-shock wave becomes weak.