Prediction methodology for “inversion radius” 146 analysis supports the observation of decrease in separation bubble size beyond the inversion radius. Therefore the decrements in non-dimensional pressure difference can be attributed for reduction in separation zone size.
0 0.1 0.3 0.50.6 0.8 1 1.2 1.41.5 2
−1
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Rn (mm)
Non−dimensional pressure
p1/p∞ pincp/p
1
p3/p
1
pincp/p∞ p3/p∞ p3/p
1−p
inc/p
1
FIGURE7.19: Variation of vital pressures and their difference with leading edge bluntness
Prediction methodology for “inversion radius” 147 entropy layer then alters the properties both at the edge and within the boundary layer. In order to understand such interaction, it is very much essential to evaluate the thickness of entropy layer for different leading edge radii.
By the definition, complete shock layer should be treated as entropy layer for hypersonic flow over blunted configuration. But the region of strong entropy gradient is only responsible for its interaction with the boundary layer. In view of this, interaction of boundary layer only with the high entropy layer (HEL) as described by Borovoi et al. [29], is considered in following discus- sion. Thickness of such HEL at the leading edge is almost equal to the leading edge bluntness radius [29]. This point essentially corresponds to the location on the bow shock beyond which change in shock angle is negligible. Such point divides the bow shock into two parts viz. highly curved and marginally curved or oblique shock. Downstream of the shock, HEL had been considered as the layer of constant thickness [17]. But such representation does not account for downstream alterations in the entropy layer thickness. Similar to the bow shock dividing point, HEL edge is responsible to segregate the uniform and non-uniform regions of entropy in the shock layer. Hence HEL edge downstream of detached shock is considered herein as the streamline passing through the point on the standing shock wave at height equal to the bluntness radius at the leading edge as shown in figures 7.20 and 7.21. Figure 7.20 strongly supports such representation by displaying high magnitude entropy beneath the separating streamline. Sim- ilarly, this HEL edge also divides the shock layer into a strongly non-uniform flow which has passed through the highly curved shock and weakly non-uniform flow which has passed through the marginally curved or oblique shock. Figure 7.21 demonstrates the HEL edge and variation of non-dimensional entropy normal to the wall at different streamwise locations for 1.5 mm lead- ing edge radius case. This figure also shows that a major portion of the entropy, by magnitude and gradient at any streamwise location, is within the entropy layer which is consistent with the definition of the HEL as considered herein.
This strategy of HEL definition can be used to study its interaction with boundary layer. Ve- locity variation in the boundary layer at some selected streamwise locations and the edge of HEL are plotted in figure 7.22, figure 7.23 and figure 7.24 to demonstrate the relative thickness
Prediction methodology for “inversion radius” 148
FIGURE7.20: Representation of the HEL edge together with entropy contour for leading edge radius of 1.5 mm
FIGURE7.21: Representation of the HEL edge and non-dimensional variation of entropy in the shock layer at selected streamwise location for leading edge radius of 1.5 mm
of both the layers for leading edge radii 0.1 mm, 0.5 mm and 1.2 mm respectively. For lead- ing edge radius of 0.1 mm, the entropy layer is very thin as compared to boundary layer at all representative locations. Therefore, entropy layer swallowing occurs quite close to the leading edge at this radius. However, in the case of leading edge radius of 0.5 mm, entropy layer is seen to be thicker than the boundary layer until 0.035 m from the leading edge. Entropy layer
Prediction methodology for “inversion radius” 149 swallowing station for this case is immediately upstream of the reference upstream influence location. Moreover, for leading edge radius of 1.2 mm, entropy layer is considerably thicker at all the locations in comparison with the boundary layer. Hence, boundary layer is completely immersed in the entropy layer for this case. This exercise hints for the conclusion that separa- tion bubble size increases if the entropy layer gets swallowed ahead of the reference upstream influence location. The possible justification of this conclusion lies in the alterations caused by the near wall presence of fluid from HEL. Such high temperature fluid, which has passed across the stronger portion of the bow shock, increases the near wall viscosity and reduces the density. These effects result in thickening of the boundary layer and reduced wall shear. Such thickened boundary layer displaces the outer inviscid flow thereby increasing the strength of marginally curved portion of the standing shock. This in turn leads to decrease in the boundary layer edge Mach number and increase in the edge temperature. Such physical analysis thus ac- counts for the effects of all parameters, except the pressure difference, on the separation bubble size. This inclusive analysis is thus helpful in understanding the existence of ‘inversion radius’
in the presence of boundary layer and entropy layer interaction.
FIGURE7.22: Interaction of boundary layer and entropy layer forRn= 0.1 mm
It has been observed earlier that the boundary layer edge Mach number and temperature varied monotonically with increase in bluntness. Therefore, unlike other governing parameters, given in equation (7.2), inversion radius can’t be evaluated from the trend of their variations. However,
Prediction methodology for “inversion radius” 150
FIGURE7.23: Interaction of boundary layer and entropy layer forRn= 0.5 mm
FIGURE7.24: Interaction of boundary layer and entropy layer for Rn=1.2 mm
the reasons for this monotonic behavior, before and after the inversion radius, are completely dif- ferent. The background flow structure which alters these motives for the monotonic change can also be explained from the entropy layer analysis. Entropy layer swallowing clearly suggests that there exists weak non-uniformity or almost uniformity in the flow properties at the edge of the boundary layer with increase in radius till the inversion radius for the upstream influence location. In such cases, the fluid present at the edge of the boundary layer, at any station down- stream of the swallowing location, for any radius lower than the inversion radius, is the one
Prediction methodology for “equivalent radius” 151 which has passed through weakly curved portion of the standing shock. However, variation in the edge parameters beyond the inversion radius should be attributed to growth of the boundary layer within the HEL. In such cases, no swallowing takes place before the reference upstream influence location and boundary layer remains thinner than the entropy layer. Hence, the fluid which has passed through the strongly curved portion of the bow shock remains at the edge of the boundary layer for radii higher than ‘inversion radius’. Thus low Mach number and high temperature at the edge of the boundary layer are inevitable in these cases. Therefore a close examination of Mach number and temperature variations in the shock layer at different stream- wise locations for different bluntness can help to quantitatively predict the inversion radius. It is therefore easy to see that the inversion radius corresponds to the case where the hydrodynamic boundary layer coincides with HEL at the reference upstream influence location. Such descrip- tion of the ‘inversion radius’ is helpful to qualitatively predict the same for given ramp angle, freestream conditions and wall conditions.
For any radius higher than the ‘inversion radius’, entropy layer remains thicker than the boundary layer at all the locations on the plate. Therefore, presence of non-uniform flow from the HEL at the edge of the boundary layer acts as source of vorticity by the virtue of an entropy gradient in accordance with Crocco’s theorem. Same sense or direction of the inviscid vorticity in the HEL and the boundary layer based vorticity manage to let the HEL act as the source.
Thus the vortical action in the HEL provides additional resistance to the boundary layer against separation which helps to reduce the separation size. However sufficient thickness of the HEL is required above the boundary layer edge to delay the separation to the extent where separation bubble size decreases in comparison with the reference separation bubble size. Therefore the relative thicknesses of the boundary layer and the entropy layer provide a predictive tool to de- termine the ‘inversion radius’ for given geometry and freestream conditions while hinting at the existence of an ‘equivalent radius’.