Summary of the Results of the Chapter in View of Flight-Vehicle Design

Regarding the design of hypersonic vehicles, we note that the approximations and proportionalities derived in the preceding sub-sections imply with regard to the thermal state of the surface in the case of radiation cooling (see also Section 7.3):

— The thermal state of the surface, i. e., the surface (wall) temperature Tw « Tra and the temperature gradient dT/dngw, respectively the heat flux in the gas at the wall qgw, of radiation-cooled surfaces depend on the flight speed vTO, respectively the flight Mach number MTO, and the flight altitude H. They all are functions of the location on the vehicle surface.

— It depends on the flight trajectory of the vehicle and on the structure and materials concept, whether the thermal state of the surface can be

considered as quasi-steady phenomenon or must be treated as unsteady phenomenon.

— The efficiency of radiation cooling increases with decreasing flight unit Reynolds number, and hence increasing boundary layer thickness, because qgw decreases with increasing boundary-layer thickness. In general radia­tion cooling becomes more efficient with increasing flight altitude.

— The radiation-adiabatic temperature—in contrast to the adiabatic tempe­rature—changes appreciably in main-flow direction (e. g., along the air­frame). It decreases strongly with laminar, and less strongly with turbulent flow, according to the growth of the related characteristic boundary-layer thicknesses. These are for laminar flow the boundary-layer thickness, and for turbulent flow the thickness of the viscous sub-layer [18].

— The radiation-adiabatic temperature is significantly lower than the adi­abatic temperature. The actual wall temperature in general is near the radiation-adiabatic temperature, depending somewhat on the structure and materials concept, and the actual trajectory part. Therefore the radiation-adiabatic temperature must be taken as wall temperature es­timate in vehicle design rather than the adiabatic or any other mean tem­perature. On low trajectory segments possibly a thermal reversal must be taken in account.

— The state of the boundary layer, laminar or turbulent, affects much stronger the radiation-adiabatic temperature than the adiabatic temperature.

— The turbulent skin-friction is much higher on radiation-cooled surfaces than on adiabatic surfaces (in general it is the higher the colder the surface is), the laminar skin friction is not as strongly affected by the surface temperature.

— At flight vehicles at angle of attack significant differentials of the wall temperature occur between windward side and leeward side, according to the different boundary-layer thicknesses there.

— At attachment lines radiation cooling leads to hot-spot situations (attach­ment-line heating), at separation lines the opposite is observed (cold-spot situations) [18].

1.4 Problems

Problem 3.1. Find from eq. (3.2) with the assumption of perfect gas the total temperature Tt as function of TTO, 7, MTO.

Problem 3.2. An infinitely thin flat plate as highly simplified vehicle flies with zero angle of attack with v= 1 km/s at 30 km altitude. Determine the total temperature Tt, and the recovery temperatures Tr for laminar and turbulent boundary-layer flow. Take 7 =1.4 and the Prandtl number Pr at 600 K, Sub-Section 4.2.3.

Problem 3.3. Check for laminar flow the behavior of Tw(x/L) = Tra(x/L) at Mж = 15.7 in Fig. 3.3.

Problem 3.4. Check for turbulent flow with n = 0.2 (exponent in the re­lation for the turbulent scaling thickness, Sub-Section 7.2.1) the behavior of Tw(x/L) = Tra(x/L) at Мж = 5.22 in Fig. 3.3.

Problem 3.5. If at 60.56 km altitude, Fig. 3.3, the Space Shuttle Orbiter would fly at Мж = 17, what would the wall temperature approximately be at x/L = 0.5?

Problem 3.6. If at 60.56 km altitude, Fig. 3.3, the Space Shuttle Orbiter would fly at Мж = 14, what would the wall temperature approximately be at x/L = 0.5?

Problem 3.7. Treat problems 3.5 and 3.6 by using the scaling law eq. (3.34). For high Mach numbers the recovery temperature approximately has the proportionality Tt ж M2. What is approximately the proportionality of the radiation-adiabatic temperature? Determine with the scaling law the ratios Tra, M^=i7/Tra, M^=i5.7 and Tra, Mxl=u/Tra, Mxl=i5.7. How is the agreement with the numbers found in the problems 3.5 and 3.6?

Problem 3.8. Show that at low altitudes radiation cooling looses its efficiency.