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’.
Prediction methodology for “equivalent radius” 152 reference. It had already been noticed that the pressure difference term from equation (7.2) is responsible for the reduction in size of the separation zone. Essentially, increase in leading edge radius increases the pressure at all locations upstream of the compression corner which includes the upstream influence location as well (figure 7.9). In addition to this, the reattachment pressure increases initially with increase in radius till ‘inversion radius’ and decreases beyond it (figure 7.19). Such counteracting changes in the incipient and reattachment pressures decrease the governing adverse pressure gradient which in turn reduce the length of separation zone after
‘inversion radius’. It can also be noticed from figure 7.18 that, there is negligible increase in maximum temperature within the thermal boundary layer as the radius increases beyond the
‘inversion radius’. This effect dampens the process of boundary layer thickening with increase in leading edge bluntness. Thus the changes in governing pressure difference and maximum temperature in boundary layer help to justify the relative decrements in separation zone length.
But they do not suffice to construct a robust prediction methodology for the‘equivalent radius’.
Hence underlying flow physics needs to be inspected to develop the prediction methodology for the second critical radius.
Investigations carried out by Holden[8, 42] for SWBLI concluded with discriminating the ef- fect of leading edge bluntness as being “displacement dominated” and ‘bluntness dominated’. In view of these terminologies, wall pressure for the reference case is plotted along with the Blast wave theory (BWT) based wall pressure for various leading edge radii [106] in figure 7.25. The over pressure region or upstream favorable pressure region, seen in this figure for sharp leading edge plate, is the above mentioned ‘displacement effect’ in the presence of strong viscous inter- action. Blast wave theory renders similar pressure variation for blunted slabs which is termed as the ‘bluntness effect’. Thus figure 7.25 describes the relative strength of displacement and bluntness effects for given freestream and geometric conditions. It is evident from this figure that, the upstream over pressure region given by BWT for leading edge bluntness case widens with increase in leading edge radius. For a particular radius this region extends till the reference upstream influence location.
Pressure variation, shown in figure 7.9, for the blunt leading edge configurations with viscous in- teraction, must be viewed as the integration of ‘displacement’ and ‘bluntness’ effects. Therefore
Summary 153
0 0.01 0.02 0.03 0.04 0.05
0 2 4 6 8 10 12 14
x (m)
p/p ∞
Viscous simulation (Sharp) BWT−(Rn=0.3mm)
BWT−(Rn=0.5mm) BWT−(Rn=1.0mm) BWT−(R
n=2.0mm) BWT−(Rn=1.2mm)
Upstream influence start location (x=0.03912 m)
FIGURE7.25: Comparison of bluntness and viscous interaction based over pressure regions
the integrated favorable pressure gradient tends to reduce the separation bubble size by delaying separation for all radii. However, it is conjectured that the ‘entropy layer swallowing’ effect counters this reduction till the inversion radius. Nonetheless, it is hypothesized that, for blunt- ness values greater than the inversion radius, the widened over pressure region in conjunction with the HEL above the boundary layer, decreased governing pressure difference and negligibly increased maximum temperature in the boundary layer provides an integrated effect to decrease the separation bubble size. The critical examination of the wall pressure variation shown in figure 7.9 indicates that the ‘over pressure region’ saturates well before the reference upstream influence location for sufficiently small radii. It is postulated that the smallest leading edge radius for which no such saturation of the over pressure is observed would correspond to the
‘equivalent radius’. The ‘equivalent radius’ based on this postulate is predicted to be between 1 mm and 1.2 mm for present simulations from figure 7.9. This prediction is in excellent agree- ment with the numerical simulations shown in figure 7.13. Thus the wall pressure distribution offers a simple and elegant predictive strategy for prediction of the ‘equivalent radius’.
Summary 154