Nonparallel Stability of Two-Dimensional 

 Heated Boundary Layer Flows 



N. M. El-Hady and A. H. Nayfeh 

 Virginia Polytechnic Institute and State 

 University , Blacksburg, Virginia 



ABSTRACT 



The method of multiple scales is used to analyze the 

 linear-nonparallel stability of two-dimensional 

 heated liquid boundary layers. Included in the anal- 

 ysis are disturbances due to velocity, pressure, 

 temperature, density, and transport properties, as 

 well as variations of the liquid properties with 

 temperature. An equation is derived for the modula- 

 tion of the wave amplitude with streamwise distance. 

 Although the analysis is applicable to both uniform 

 and nonuniform wall heating, numerical results are 

 presented only for the uniform heating case. The 

 niomerical results are in good agreement with the 

 experimental results of Strazisar, Reshotko, and 

 Prahl. 



1 . INTRODUCTION 



It is generally accepted that the instability of 

 small amplitude disturbances in a laminar boundary 

 layer is an integral part of the transition process. 

 Significant changes in the boundary layer stability 

 characteristics can be achieved by utilizing dif- 

 ferent factors, such as pressure gradients, suction, 

 injection, compliant boundaries, and heating or 

 cooling of the boundary layer. 



Surface heating in a liquid boundary layer can 

 be utilized to yield a mean velocity profile which 

 is more stable than the Blasius profile. The rea- 

 son is that heat transfer alters the shape of the 

 boundary-layer temperature profile which in turn 

 alters the velocity profile through the viscosity- 

 temperature dependence. The effect of wall heating 

 on the stability of boundary layers in water was 

 investigated by Wazzan et al. (1968, 1970). They 

 included the variation of the viscosity with tem^ 

 perature through the thermal boundary layer. They 

 obtained a modified Orr-Summerfeld equation. How- 

 ever, they did not include temperature fluctuations 

 in the disturbance flowfield. Their results show 



that while cooling the wall has a destabilizing ef- 

 fect on the flow, moderate heating has a strong 

 stabilizing effect. Lowell (1974) reformulated the 

 problem by adding fluctuations for the temperature, 

 density, and transport properties. The results of 

 Lowell did not vary appreciably from those of Wazzan 

 et al. (1970) . 



The presently available analyses (Wazzan et al. 

 and Lowell) for the stability of heated boundary 

 layers in water are all parallel flow analyses. The 

 results of the parallel stability analyses do not 

 agree with available experimental results. Strazisar 

 et al. (1975, 1977) performed experiments on the 

 stability of boundary layers on both unheated and 

 uniformly heated flat plates. These experiments 

 confirmed the increased stability resulting from 

 wall heating in water. Strazisar and Reshotko (1977) 

 extended their experiments to cases of nonuniform 

 surface heating in the form of power-law temperature 

 distributions; that is, T^,(x) - Tg = Ax"^. Their 

 results are given only for a displacement thickness 

 Reynolds number R* = 800 and indicate that, for a 

 given level of wall heating, cases with n < have 

 the lowest growth rates. Strazisar and Reshotko 

 (1977) found that applying Lowell's analysis (1974) 

 to the case of power-law temperature distributions 

 yielded results that did not agree with the experi- 

 mental results. 



In this paper, we use the method of multiple 

 scales (1973) to analyze the linear, nonparallel 

 stability of two-dimensional boundary layers in 

 water on a flat plate, taking into account uniform 

 as well as nonuniform wall heating. We include 

 disturbances in the temperature, density, and trans- 

 port properties of the liquid in addition to dis- 

 turbances in the velocities and pressure. However, 

 we present numerical results only for the case of 

 uniform wall heating and compare our results with 

 the experimental data of Strazisar et al. (1975, 

 1977). When the variation of the temperature, 

 thermodynamic, and transport properties are ne- 

 glected, the present solution reduces to those of 



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