Analysis of separation bubble size 114 constant (Fui) of 6.15 has been obtained for the present data set. The experimental result of Holden and Moselle [100], which was not considered earlier (figures 6.1 and 6.2), is also plotted in this figure to further evaluate the present correlation. Encouraging match can be experienced, in this figure, for this result as well. Hence the equation (6.9) can be treated as the generalized correlation for prediction of upstream influence.
Analysis of separation bubble size 115 Another important scaling law for separation bubble size is proposed by Katzer [41]. This correlation is as,
Lb
δ0∗ ∝
pRex0
M03√ C
p3−pincp
p1
(6.11) Here, the required displacement thickness at locationx0 can be obtained through the correlation proposed by Bertram and Blackstock [101]. The above equation has also been expressed by the authors as,
Lb
δ0∗ M03√
C pRex0
∝
p3−pincp
p1
(6.12) These (equations (6.11) and (6.12)) correlations have been originally developed for I-SWBLI in supersonic flow regime. Hence the applicability of the same correlation for R-SWBLI, needs thorough analysis.
Expression for the separation bubble size (equation (6.13)) has also been recently devised by Davis and Sturtevant [58] for which limited supporting investigations are available.
Lb
x0 ∝ Λ0
γ3/2M03
p3−pincp
p1
3/2
(6.13) In this correlation, dependency of Mach number is expressed in the same way as that in the previous correlations. Besides, the formulation of pressure term is seen to be altered. The new parameter,Λ0which appears in this correlation, stands for the effect of skin friction and wall to edge temperature ratio. This parameter is given by,
Λ = µw
µ∗ T∗
Te
Tw
Te
1/2
Here, the terms with ′∗′ are to be evaluated at reference temperature (equation (6.3)). This correlation has been proposed for high enthalpy hypersonic flows, hence its applicability for ideal gas flows needs to be evaluated. Apart from this, caution has also been reported by the authors that the correlation would have least quantitative prediction capacity.
Analysis of separation bubble size 116
Wall shear distribution obtained from the current numerical studies reported in Chapter-5 is used to measure the size of separation bubble. Hence, cases which fall in the “separated flow”
region of figure 5.2 would only be useful. These cases include Case A, combination of Mach 5, 6, and 7 with ramp angle of 150 of Case B and ramp angle 200 and 250 of Case C (refer table 5.1). In addition to these, ramp angle 150 and Mach 6 combination of Case B has also been considered to reveal the effect of wall to total temperature ratio. For this study, simulations are repeated with all three wall temperatures and two enthalpy conditions of Case B, so as to get extra data points. Thus obtained separation bubble sizes for various freestream and wall conditions have been considered to assess the literature reported correlations.
In the present study, the incipient pressure rise (pincp/p1) has been calculated by making use of triple deck solution proposed by Inger [45]. Thus obtained data points fitted according to equations (6.10) and (6.4) are shown in figures 6.4 and 6.5 respectively. The points correspond- ing to ramp angle and Mach variations are seen to follow linear trend in both the cases with different slopes. These expressions are also seen to exhibit upward shifting of trend lines with increasing wall to freestream temperature ratio. Moreover, the upward shift which is consistent in figure 6.4 for variation of freestream stagnation enthalpy is unseen in figure 6.5. Hence these correlations are found suitable in predicting separation bubble size for the cases with variation in Mach number or ramp angle only.
Similar experience can be gained by fitting the data points using Katzer’s correlation (equation (6.11)) as shown in figure 6.6. From the figure, the current form of the equation, proposed for prediction of separation bubble length in case of shock impingement separation, is examined to be unsuitable to use for ramp induced separation. However, the other form given by equation (6.12) is observed to offer improved trend, as shown in figure 6.7. Marginal deviation noticed for Mach number variation cases is the only limitation of this correlation. Hence, from the narrow envelop of the present data sets, the equation (6.12) is experienced to be a better prediction method for length of separation bubble in case of R-SWBLI case as well.
Analysis of separation bubble size 117
FIGURE6.4: Present data points fitted according to Needham’s correlation (equation (6.10))
FIGURE6.5: Present data points fitted according to Needham and Stollery’s correlation (equa- tion (6.4)).
Analysis of separation bubble size 118
FIGURE6.6: Present data points fitted according to Katzer’s correlation (equation (6.11)).
FIGURE6.7: Present data points fitted according to Katzer’s correlation (equation (6.12)).
Analysis of separation bubble size 119
FIGURE6.8: Present data points fitted according to Davis and Sturtevant’s correlation(equation (6.13)).
The present data points are then fitted according to method proposed by Davis and Sturtevant (equation (6.13)). The trend lines fitted according to this correlation are shown in figure 6.8.
The argued advantage of suitability of this correlation for different wall to total temperature ratio cases has been observed with present data points as well. Contrary to other three correlations of separation bubble size, this correlation offers linear trend for different wall to freestream total temperature ratios. This observation supports the excellent scaling of the wall temperature and freestream enthalpy, in this correlation, through the term Λ. The ramp angle variation is also experienced to follow linear trend in this strategy. However, it is evident from this figure that, the cases with Mach number variation deviate from linearity. As a part of this, the lines of different wall to temperature ratio are seen to shift upwards with decreasing Mach number.
The four reported correlations assessed herewith are lacking in offering uniqueness in the prediction of separation bubble size for various freestream and geometric conditions. The suit- ability of each of those correlations is restricted only to few parametric variation cases. Hence
Analysis of separation bubble size 120
FIGURE 6.9: Present data points fitted according to modified Needham’s correlation(equation (6.14)).
present efforts are extended to modify the Needham’s correlation (equation 6.10) in order to fit the present data set into a narrow region. The main deficiency which was noticed with this correlation is its inaccuracy in scaling the wall and total temperature variation. Therefore an additional term has been introduced in equation 6.10 which leads to the modified Needham correlation as,
Lb
x0
M03 pRex0
=Flb
p3
p2
2
βlb (6.14)
where Flb is the proportionality constant and βlb is the scaling parameter for wall to total tem- perature ratio and is given by,
βlb = Tw
Te
Te
T0
1/2
The present data points fitted according the modified correlation are shown in figure 6.9. The rearrangement and introduction of an additional temperature scaling in the Needham’s corre- lation is found to arrange the present data along a straight line. Additionally considered data
Correlation for peak heatflux 121 points corresponding to experimental studies of Holden and Mosselle [100] and numerical stud- ies of Grasso and Marini [53] are also in agreement with the trend line obtained from modified Needham’s correlation. The value ofFlb is observed as 0.13.