34 



(U-c)[(p<())" - a-^(p((>)] - U'Mpifi) + i[r(U-c)2]' 



(P*) ' 



(P*)' 



c [i(U-c)T + (p(t))T] = — ±- 

 p aRPr 



[ (pk) t' + pkT' ] ' 



a2 - 



r^(Pk)T 



P 



(3) 



(4) 



The results of these two analyses for the min- 

 imum critical Reynolds number with wall heating are 

 shown in Fig. 1. The curves are very much alike. 

 Furthermore, the neutral stability characteristics 

 and the growth rates as calculated in the aforemen- 

 tioned analyses are sufficiently close so that there 

 is no important quantitative difference between the 

 two. The coupling of vorticity and temperature 

 fluctuations through the viscosity seems therefore 

 to be rather weak. 



As is seen in Figure 1, both sets of calculations 

 predict significant boundary layer stabilization 

 (increased minimum critical Reynolds number, de- 

 creased disturbance amplification rates, etc.) with 

 moderate heating, but display a maximum and sub- 

 sequent decrease as the wall to free-stream tem- 

 perature difference is further increased. The 

 significant stabilization indicated for overheats 

 of up to 40''C ('^J70°F) prompted a study of the pos- 

 sible drag reduction due to heating to see if this 

 drag reduction technique was in fact worth pur- 

 suing further. 



DRAG REDUCTION IN WATER BY HEATING 



with boundary conditions 



<f>(0) = ((i' (0) = t(0) =0 



,j,(co) = 4,' (CO) = T(") =0 



(5) 



In equations (3) and (4) all properties of the 

 basic flow are variable. The quantities r, m, and 

 K are the density, viscosity, and thermal conduc- 

 tivity fluctuation amplitudes respectively and 

 the coupling comes about through the viscosity 

 fluctuations that are directly related to the tem- 

 perature fluctuations. 



Reg *min, crit. 

 X 10-3 



100 



200 



T,.,(°F) 



300 



It is shown in this section that significant reduc- 

 tions of drag are available to water vehicles with 

 on-board propulsion system is discharged through 

 heating the laminar flow portion of the hull. The 

 analysis is as follows [following Reshotko (1977)]: 

 For a vehicle with an on-board propulsion system 



the friction drag is 



Dp = q 



/ tr J ,1 

 J c wdx + r 

 o fa xJ 

 tr 



ft 



wdx 



(6) 



where q is the dynamic pressure, Cf^ and C£^ are 

 respectively the laminar and turbulent friction 

 coefficients, wdx is the area element at length x, 

 L is the vehicle length, and x^ is the transition 

 location. 



The total drag can be written 



D (D/D ) 

 F F 



(7) 



where D/Dp is the ratio of total to friction drag. 

 For an axisymmetric body this ratio is a function 

 of the fineness ratio of the configuration. 



Hoerner (1958) suggests that 



3 /2 

 D/D^ = 1 + 1.5(f) + ... (8) 



The drag power can then be written 



Du 



q u^(D/D^) (Cp^A) 



(9) 



FIGURE 1. Effect of wall temperature on minimum cri- 

 tical Reynolds number .[from Lowell and Reshotko (1974)]. 



where [Cq„A] i? the quantity in brackets in equa- 

 tion (1) . 



