Buoyant Plumes in a Transverse Wind 



Chia-Shun Yih 

 The University of Michigan 

 Ann Arbor, Michigan 



ABSTRACT 



With the rise in energy needs and the consequent 

 proliferation of cooling towers {not to mention 

 smoke stacks) on the one hand, and society's 

 enchanced concern with the environment on the other, 

 the study of buoyant plumes caused by heat sources 

 in a transverse wind has become important. Buoyant 

 plumes may also occur in the ocean, such as when 

 a deeply submerged heat source moves horizontally 

 in it. The fluid mechanics involved in buoyant 

 plumes is very nearly the same, be they atmospheric 

 or submarine. 



In this paper a similarity solution for turbulent 

 buoyant plumes due to a point heat source in a 

 transverse wind is presented. By a set of trans- 

 formations the mathematical dimension of the 

 problem is reduced from 3 to 2. Analytical solutions 

 for the first and second approximations are obtained 

 for the temperature and velocity fields. The 

 solution exhibits the often observed pair of longi- 

 tudinal counter-rotating vortices. As a result of 

 buoyancy, the point of highest temperature and the 

 "eyes" of the vortices at any section normal to 

 the wind direction continuously rise as the longi- 

 tudinal distance from the heat source increases. 



1 . INTRODUCTION 



As industry expands and energy needs rise, the 

 buoyant plumes caused by ever-increasing cooling 

 towers and smoke stacks have become an important 

 concern for societies anxious to protect their 

 environment. Much effort has been expanded on the 

 so-called numerical modeling of the phenomenon 

 of plumes both in the United States and in Europe. 

 In most of the numerical studies, the eddy viscosity 

 is assumed constant, and its value is chosen to 

 make the results agree with whatever gross observa- 

 tions are available. The power of modern computers 

 has made it possible to obtain numerical solutions 



for partial differential equations with very 

 irregular data, such as wind and temperature profiles 

 in the atmosphere. On the other hand, one can 

 only carry out a number of these special solutions, 

 and while the power of the computer makes computa- 

 tion possible it also makes the intermediate steps 

 so opaque that one can only have faith in the 

 accuracy of the results and the correctness of the 

 programing; and one can attempt to interpret the 

 results and understand the phenomenon only at the 

 very end, when numerical results are available. 

 One can hardly see, for example, the effects of 

 changing one single parameter of the problem, without 

 giving that parameter several values and going to 

 the computer again and again. It is in view of 

 this condition that even people most concerned with 

 the immediate applicability of calculated results 

 desire a certain measure of transparency in the 

 analysis of the phenomenon. 



At the same time systematic and detailed experi- 

 ments on buoyant plumes in transverse winds, with 

 temperature and velocity measurements, are lacking. 

 This being so, it seems that an analytical solution 

 of the problem is most desirable and timely, even 

 if it must of necessity be constructed by assioming 

 certain quantities (such as the turbulence level 

 in the plume) on the basis of whatever related 

 experimental results are available. The assumed 

 quantities (or quantity) will appear in the analysis 

 as unspecified coefficients (or coefficient, as in 

 this analysis) , to be determined by experiments 

 later. In the present work only one coefficient 

 related to the turbulence level is left unspecified, 

 to be determined by future experiments. But the 

 probable range in which it lies is given. 



The solution is based on a set of transformations 

 that reduces the mathematical dimension of the 

 phenomenon from 3 to 2 is thus characterized by the 

 striking feature of similarity between cross sections 

 normal to the wind direction. The laws of decay 

 of the temperature and velocity fields are given 

 in simple, explicit terms. Thus, apart from the 



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