Hypersonic interference heating in the vicinity of surface protuberances

Abstract

The understanding of the behaviour of the flow around surface protuberances in hypersonic vehicles is developed and an engineering approach to predict the location and magnitude of the highest heat transfer rates in their vicinity is presented. To this end, an experimental investigation was performed in a hypersonic facility at freestream Mach numbers of 8.2 and 12.3 and Reynolds numbers ranging from Re _∞/m = 3.35 × 10⁶ to Re _∞/m = 9.35 × 10⁶. The effects of protuberance geometry, boundary layer state, freestream Reynolds number and freestream Mach numbers were assessed based on thin-film heat transfer measurements. Further understanding of the flowfield was obtained through oil-dot visualizations and high-speed schlieren videos. The local interference interaction was shown to be strongly 3-D and to be dominated by the incipient separation angle induced by the protuberance. In interactions in which the incoming boundary layer remains unseparated upstream of the protuberance, the highest heating occurs adjacent to the device. In interactions in which the incoming boundary layer is fully separated ahead of the protuberance, the highest heating generally occurs on the surface just upstream of it except for low-deflection protuberances under low Reynolds freestream flow conditions in which case the heat flux to the side is greater.

Appendix 1: Correlation of peak heat flux in fully separated interactions

Buckingham-Pi analysis is used to derive a correlation of the hot spot magnitude in fully separated interactions. This approach is commonly used in the interpretation of experimental data and has previously led to the development of predictive methods for heat transfer in attached flows (Rogers and Mayhew 1980). The main variables responsible for the heat flux need to be determined. It is well established that the main parameters to be considered are viscosity μ, density ρ, thermal conductivity k, specific heat c _p , temperature relative to wall θ, fluid velocity U and a characteristic linear dimension l. As expected from the previous literature (Sect. 1), the characteristic linear dimension l which has a dominant effect in fully separated interactions is the height of the protuberance (h) whereas the effect of width is known to be negligible. The effect of boundary layer thickness is also considered negligible throughout the present experimental study as shown by the similitude between the peak heating at laminar and turbulent conditions (Sect. 3.5). Based on a dimensional analysis considering mass, length, time, thermal energy and temperature as the five fundamental units and grouping them in terms of two main non-dimensional groups (Re and Pr), the relation in Eq. 9 is derived, where a and b are exponents, C is a constant and Re _h is Reynolds number based on protuberance height. Since in experimental measurements of this type accurate knowledge of the Prandtl number Pr is generally not feasible, common predictive approaches developed to date consider a constant Prandtl number. For this reason, a common simplification is to assume Pr = 1 and Eq. 9 thus reduces to Eq. 10. For more details on the derivation of this relation, refer to Rogers and Mayhew (1980). The experimental results obtained at a freestream Mach number of M _∞ = 8.2 and at Reynolds numbers of Re _∞/m = 6.57 × 10⁶, 8.06 × 10⁶ and 9.35 × 10⁶ are used to obtain a correlation of Nu _h with Re _h . The highest heat flux in terms of Nu _h are plotted against Re _h for the α = 45°, 60° and 90° cases showing a correlation of the measurements with a slope of a = 1.6. Since Pr is considered equal to 1, Nu _h  = St _max Re _h and a = 1.6 are introduced in Eq. 10 to obtain the relation in Eq. 11 (Fig. 24). The recovery factor is also considered to be r = 1.

$$ Nu_{h} = C\text{Re}_{h}^{a} \Pr{^b} $$

(9)

$$ Nu_{h} = C\text{Re}_{h}^{a} $$

(10)

$$ St_{\max } \text{Re}_{h}^{ - 0.6} = C $$

(11)

Fig. 24

Correlation of St _max Re ^−0.6 _h with protuberance deflection angle α at different freestream Reynolds numbers in logarithmic scale

The term St _max Re ^−0.6 _h is constant for different Reynolds numbers but it increases with higher deflection angles (Fig. 24). The effect of α is introduced by considering the deflection experienced by the incoming flow before it reattaches to the surface ahead of the protuberance. This is expressed as shown in Eq. 12. Figure 25 shows the constant trend obtained for the M _∞ = 8.2 tests at all the different conditions while the same does not apply in the cases where the hot spot takes place to the side of the protuberance—i.e. in unseparated or weak separated interactions. This is because the side heat flux is caused by corner effects (Sect. 4.5). The effect of Mach number is subsequently investigated through a similar correlation between the M _∞ = 8.2 and the M _∞ = 12.3 measurements. Its effect is found to be proportional to the square root of Mach number as shown in Eq. 13. A suitable constant C for the final relation in Eq. 13 is found to be C = 5.2 × 10⁻⁵ as shown in Fig. 19.

$$ {{St_{\max } Re_{h}^{ - 0.6} } \mathord{\left/ {\vphantom {{St_{\max } Re_{h}^{ - 0.6} } {\left( {1 - \cos \alpha } \right) = C}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \cos \alpha } \right) = C}} $$

(12)

$$ {{St_{\max } Re_{h}^{ - 0.6} M^{0.5} } \mathord{\left/ {\vphantom {{St_{\max } Re_{h}^{ - 0.6} M^{0.5} } {\left( {1 - \cos \alpha } \right) = C}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \cos \alpha } \right) = C}} $$

(13)

Fig. 25

Correlation of St _max Re ^−0.6 _h /(1 − cosα) with protuberance deflection angle α at different freestream Reynolds numbers