36 



to the reasonability of the assumed transition 

 schedule with overheat. 



Drag reduction calculations have been performed 

 for plate speeds up to 24.4 m/sec (80 fps) , for 

 plate lengths of 3.05 m (10 ft), 15.24 m (50 ft). 

 30.48 m (100 ft), 152.4 m (500 ft), and 304.8 m 



pD ,1 

 (1000 ft) , and for values of f-T^^' — 



F th 



l)n c^lof 2,5, and 

 err 



Since the product Dn ^^^/D might be very close 



to unity, one may view the aforementioned values of 



Dn 



the "efficiency factor" [ — (— 



Dp n 



eff • 



as approx- 



th 



imately corresponding to n , =0.33, 0.17, and 0.10 



.. -, th 



respectively . 



Results are presented in Figure 3 for the case 



of an efficiency factor of 5(ri 



^fei 



0.17) 



Shown 



in Figure 3 are D/D the ratio of the drag with 

 heating to that without 'using the reject heat for 

 drag reduction purposes, the corresponding laminar 

 fraction of the plate x^j./L, the wall temperature 

 rise of the laminar region, and finally the ratio 

 of the computed drag with heating to that for fully 

 laminar flow over the entire plate. 



Generally speaking the drag reduction becomes 

 noticable as speeds exceed 10 m/sec ("1^20 knots) . 

 Although the drag ratio is not a strong function 

 of length, the overheat in the laminar region in- 



creases quite significantly with vehicle length. 

 For nth~'^-l'^ (Figure 3), drag reduction of about 

 60% are atainable for vehicle speeds of 25 m/sec 

 ('^50 knots) but the vehicle is far from full lami- 

 narization. The variation of drag ratio with nth 

 is shown in Figure 4 for selected cases. The lower 

 the thermal efficiency, the larger the drag reduc- 

 tion and vice-versa. The indication from the cal- 

 culations is that full laminarization can be ob- 

 tained in a number of cases (Figure 4) but only if 

 nt-h gets below about 0.03. Since the e^ transition 



curve (Figure 2) has a maximum value of Re^^ be- 



8 tr 



low 3 X 10 , vehicles with length Reynolds niombers 



above 3 x 10° cannot be completely laminarized. 



For a plate of given length at a prescribed 



speed, the fuel consumption (proportional to D/n^h' 



the slope of a line through the origin in Figure 



4) increases as nth -"-^ reduced. But it is far below 



that of the unheated plate. 



Real Configurations 



Real vehicle configurations involve additional fac- 

 tors not considered in this flat-plate calculation. 

 Favorable pressure gradient, for example, can be 

 very effective in delaying transition while regions 

 of adverse gradient are otherwise. Non-uniform 

 longitudinal heating distributions can result in 

 a more optimal use of the available heat. Effects 



U , m/sec 



FIGURE 3. Drag reduction by use of reject heat of propulsion system for transition 

 delay. 



fe'4 -'"-"] ='' "^th^O.17) 



