Heated Boundary Layers 



Eli Reshotko 



Case Western Reserve University 



Cleveland, Ohio 



ABSTRACT 



Heating the walls on which laminar boundary layers 

 develop in water can delay their transition to 

 turbulent flow and lead to significant drag reduc- 

 tion. This paper describes the work done over the 

 last several years at Case Western Reserve Univer- 

 sity in examining the bases and consequences of the 

 heating phenomenon. Included are theoretical and 

 experimental studies of the stability of heated wa- 

 ter boundary layers for both uniform and non-uniform 

 wall temperature distributions, and experimental 

 study of the effect of heating on laminar separa- 

 tion and a quantitative assessment of the prospec- 

 tive drag reduction on underwater vehicles. 



1 . INTRODUCTION 



It was noted many years ago in experiments at low 

 subsonic speeds [Frick and McCullough (1942) , 

 Liepmann and Fila (1947) ] that the transition lo- 

 cation of the flat plate boundary layer in air is 

 advanced as a result of plate heating. Based on 

 this observation it had long been suspected that 

 heating would have the opposite effect in water, 

 namely that it would delay the onset of transition. 

 This is because heating in water reduces the vis- 

 cosity near the wall resulting in a fuller, more 

 stable velocity profile for a flat plate than the 

 Blasius profile. Cooling in water (and heating in 

 air) on the other hand tends to give an inflected 

 velocity profile which is less stable than the 

 Blasius profile. 



These suspicions remained untested until con- 

 firmed by the analysis of Wazzan, Okamura, and 

 Smith (1968, 1970). These results triggered a 

 significant activity in the United States to deter- 

 mine whether wall heating could realistically be 



used as a technique for drag reduction. A portion 

 of this effort was undertaken at Case Western Re- 

 serve University (CWRU) under the joint auspices 

 of the Office of Naval Research and the General 

 Hydrodynamics Research Program of the David W. 

 Taylor Naval Ship Research and Development Center. 

 The CWRU effort has been both analytical and 

 experimental and is ongoing. This paper will re- 

 view the results to date of the CWRU activity and 

 indicate current and future directions . 



2. ANALYSIS OF THE STABILITY OF HEATED WATER 

 BOUNDARY LAYERS 



The analysis of Wazzan et al. (1968, 1970) consid- 

 ers the stability characteristics of the boundary 

 layer to be governed by the disturbance vorticity 

 equation including consideration of viscosity vari- 

 ations in the basic flow but ignoring temperature 

 fluctuations and the coupled viscosity fluctuations. 

 The disturbance differential equation consists of 

 the fourth-order Orr-Sommerfeld operator augmented 

 by some lower order terms and is as follows : 



(U-c) (cf'-a^cj 



U"<t 





2a-' 



+ 2y' (0'" - 0-^4.) 



+ vJ"((}i" + oi2(),) ] 

 with boundary conditions 



<t)(0) = (f)' (0) =<),(")= <p' (") = 



(1) 



(2) 



The analysis of Lowell and Reshotko (1974) on the 

 other hand is based on the following coupled sixth- 

 order system of vorticity and energy disturbance 

 equations : 



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