614 



FIGURE 2. Flow pattern from the second approximation. 

 The n axis is horizontal and the ? axis vertical. The 

 arrow indicates the direction of the gravitational ac- 

 celeration. The value of S'nf is, starting from the I, 

 axis and going to the curves on the right, respectively, 

 0, -0.1, -0.2, -0.3, -0.4, -0.5, -0.6, and -0.65. The 

 values of SttV on the curves to the left of the I. axis 

 have corresponding absolute values but are positive. 



estimated* maximum value of the terms neglected to 

 the maximum value of the computed quantity) are, 

 respectively, less than 15^, 3%, and 10^. For X 

 = 40 these percentage errors are, respectively, 

 25?, 5%, and 18?. The most interesting thing to 

 note is that f is the most accurately calculated 

 quantity. Figure 2 shows the flow pattern in a 

 plane normal to the x axis, and Figure 3 shows the 

 isotherms therein, all for X = 30. The flow pattern 

 in Figure 2 can be regarded as sufficiently accurate 

 to be representative of the actual flow pattern in 

 a plane normal to the x axis. As expected, the 

 hottest point and the "eyes" of the vortices occur 

 at positive values of C. That is to say, the plume 

 rises according to the x ' 3 law. After the present 

 work was done, I found that this law had recently 

 been verified experimentally by Wright (1977) , 

 although he did not measure the detailed velocity 

 and temperature distributions in the plume. 



If later measurements show X is larger than 30, 

 higher approximations would be necessary. 



FIGURE 3. Isotherms from the second approximation. 

 The ri axis is horizontal and the C, axis vertical. The 

 arrow indicates the direction of the gravitational ac- 

 celeration. The value of Trh is 1.1 on the smallest 

 closed curve and 0.3 on the outermost curve. The incre- 

 ments are 0.1. 



rise high in a weak wind before being bent suffici- 

 ently for the present theory to apply. In using 

 the present theory it is always necessary to 

 determine a virtual position for the heat source, 

 which for small value of U can be considerably 

 higher than its actual position. 



ACKNOWLEDGMENT 



This work was partially supported by the Office of 

 Naval Research. The subject of this work was 

 suggested to me by my friend Dr. Michel Hug, Director 

 of the Department of Equipment, Electricity of 

 France, through Mr. F. Boulot of the National 

 Hydraulics Laboratory at Chatou, France, during my 

 brief sojourn there in the summer of 1977. Their 

 interest in this work, as well as the interest of 

 Dr. A. Daubert, director of that laboratory, is 

 very much appreciated. The work, begun at Chatou, 

 was substantially improved and finished during the 

 tenure of my Humboldt Award, at the University of 

 Karlsruhe. To the Humboldt Foundation and my 

 Karlsruhe hosts I should like to express my sincere 

 appreciation . 



DISCUSSION 



It is perhaps surprising that the analysis shows 

 that the results in dimensionless terms are indepen- 

 dent of the parameter Gg^/U^. The explanation is 

 that the velocity (v,w) far downwind from the heat 

 source becomes vanishingly small, and whatever the 

 value of U, the transverse wind is asymptotically 

 always strong. 



Near the heat source the flow indeed depends 

 very much on the magnitude of U. The plume may 



*0n the basis that hi/ho emd (h„+j)/hn are of the same order 

 of magnitude and that the same is true for f and 5. 



1977) . Plane turbulent buoyant 

 Turbulence structure. J. Fluid 

 (see P. 52, Figure 7) . 

 The structure of turbulence in 



REFERENCES 



Kotsovinos, N. E. 

 jets. Part 2. 

 Mech. 81, 45-62 



Laufer, J. (1953). 



fully developed pipe flow. NACA Tech. Note 

 2954. 



Wright, S. J. (1977) . Report KH-R-36, Keck Labor- 

 atory, California Institute of Technology. 



Yih, C.-S. (1977). Turbulent buoyant plumes. 

 Phys. Fluids 20, 1234-1237. 



