608 



quantitative predictions that this analysis is 

 intended to furnish, I hope that the general features 

 of the solution will be found especially useful. 



2. THE DIFFERENTIAL EQUATIONS 



The two basic assumptions underlying the analysis 

 are that the longitudinal velocity component in 

 the direction of the wind is constant and that an 

 eddy viscosity, e, is constant in any cross section 

 normal to the wind direction. It can be shown that 

 the first assumption ceases to be true only at 

 stages of approximations later than those arrived 

 at in the present analysis, and its violation is 

 therefore not very important. The second assumption 

 mentioned above has been made in all analytic 

 solutions for turbulent jets and plumes, according 

 to Prandtl's simplified theory. These solutions 

 are well known. See, for example, the paper by 

 Yih (1977) on turbulent plumes for the latest 

 application of that theory. One feels reassured 

 that for a calculation of the mean temperature and 

 velocity fields, this theory can again be used. 



We shall take the direction of the wind to be 

 the direction of increasing x, and the z direction 

 to be vertically upward. The y direction will then 

 be a horizontal direction transverse to the x direc- 

 tion. In general e depends on x, y, and z. But it 

 has been repeatedly shown before in other studies 

 of jets and plumes that in their core, e can be 

 taken as constant at a constant value of x, and 

 that only at their outer edges does the nonuniformity 

 in the y-z plane introduce some errors in the 

 calculated mean quantities. (Very far away from 

 the jets and plumes the value of z is immaterial 

 for the determination of the temperature and velocity 

 distributions) . Accepting these outer-edge errors, 

 which are fairly small, we shall take £ to be a 

 function of x only, apart from the parameters of 

 the problem to be defined later. We note that if 

 an eddy viscosity is used to determine the velocity 

 distribution in turbulent flow in a circular pipe, 

 Laufer's (1953) measurements show that in the core, 

 that is, away from the narrow region near the pipe 

 wall, e is nearly constant. 



The equations of motion are then, with subscripts 

 denoting partial differentiations. 



T 



where AT is the temperature variation and T the 

 ambient temperature. For a liquid, the relationship 

 between Ap and AT is still linear if 6 is small, 

 and the constant of proportionality is determined 

 by the property of the liquid. 



We shall assume the eddy viscoity for heat 

 diffusion to be the same as that for momentum 

 diffusion. This may not be strictly true, for the 

 turbulent Prandtl number may be slightly different 

 from 1. The effect of this difference, if any, is 

 not of great importance in our attempt to determine 

 the mean temperature and velocity fields. The 

 equation for heat diffusion can then be written in 

 the form 



U8^ + v6„ + w9„ = £(0 + 



X y ^ yy 



zz' • 



(4) 



Longitudinal diffusion of heat or of momentum is 

 ignored in Eq. (1) , (2) , and (4) . This is justified 

 in the same way as in other works that use the 

 boundary- layer theory. 



The equation of continuity is, since the longi- 

 tudinal velocity component is assumed constant. 



V + w = 0. 

 Y z 



The heat source, located at the origin, is 

 measured by the quantity 



(5) 



uedydz. 



(6) 



Note that solid boundaries are assumed to be far 

 away from the source, so that their effects are 

 negligible. Equations (1), (2), (4), (5), and (6), 

 with appropriate boundary conditions, govern the 

 phenomenon under investigation. 



The equation of continuity (5) allows the use 

 of a stream function \fi in terms of which v and w 

 can be expressed: 



V = *^, 



w = -ip^ 



(7) 



Uv^ + w + wv^ 



- -Py + E(Vyy + V^^), 



(1) 



UWj, + vwy + ww^ = - - p^ 



+ =<"yy + "zz'''2) 



in which U is the wind velocity, assumed constant, 

 v and w are the velocity components in the directions 

 of increasing y and z, respectively, p is the 

 density, p is the pressure, and g is the gravita- 

 tional acceleration. The variable 9 is defined by 



By cross-differentiation of Eqs . (1) and (2), we 

 obtain the vorticity equation 



USx + V5y + W5^ = e(5yy + t.^^) - 



'y' 



(8) 



in which 5 is the x component of the vorticity and 

 is given by 



5 = w - V, 

 " y Z 



■«Cyy + ^-zz)' 



(9) 



Ap 

 P ' 



(3) 



where Ap is the variation of the density from the 

 ambient density p, assumed constant. Thus the 

 Boussinesq approximation has been used in Eqs. (1) 

 and (2) . Since 9 is small and the pressure vari- 

 ation in the plume , though important for determina- 

 tion of the flow field, is unimportant in the deter- 

 mination of Ap from the temperature variation by 

 the equation of state, 9 can also be written, by 

 virtue of the equation of state of ideal gases . 



3. THE FORM OF THE EDDY VISCOSITY 



We assume the terms in Eqs. (1) or (2) or (4) to 

 be of the same order of magnitude. In particular, 

 this means that the diffusive and the convective 

 terms are of the same order of magnitude in any of 

 these equations. It also means that in Eq. (2) the 

 buoyancy term is of the same order of magnitude as 

 the convective and diffusive terms for w. This 

 assumption underlies all existing analytical studies 



