106 The Three-dimensional Temperature Distribution and its Variation in Time 



direction) {v — w = 0), then stationary conditions in the distribution of the property 

 s in the volume element {^dsfdt = 0) are only possible if the equation 



d'^s dh d^s 8s 



is satisfied, where A;^, Ay and A^ are taken as constants. From this general equation 

 can be derived more special cases : 



d^s ^•y « / X 



^* aF^ - ^" a^ ^ ^ ^^^ 



if there is vertical mixing only (A^. = Ay = 0); 



8^s ^s ^ • /,x 



if there is transverse mixing (in a horizontal direction normal to the flow) (Ax = A:i = 



0); 



8^5 8^s ^'-^ _ n 



if there is mixing in all directions but no aveiage water transport in the ^r-direction 

 (u = 0). 



Cases (a) and (b) are mathematically identical but solely the vertical and horizontal 

 directions are interchanged. A solution for the equations (a) and (b) has been given by 

 Defant (1929) 



77 TT^ Az 



s = Sa + m e""* cos -^, z and a. —-r^ — . 



For case (b), the co-ordinate z is replaced by the co-ordinate y. The distribution of the 

 property s along the homogeneous turbulent flow has been found to be tongue-shaped 

 if the 5-content initially has a maximum value at the centre of the flow (at x = 0, 

 s = Sq -{- m cos (7r/2/)z). 



This is also the case when the velocity is the same over the whole transverse cross- 

 section. Figure 48 gives an example of the course of the i--lines for Aj p = 4 cm^sec, 

 M = 10 cm/sec and / = 2 x 10'* cm. The further the cross-section is taken from the 

 initial section {x = 0) the lesser are the horizontal and vertical differences in s. By 

 the extension of the above solution to different initial conditions for x = (Thorade, 

 1931) it became evident that neither the tongue-form of the distribution of the proper- 

 ty s nor the distribution of the velocity u in the cross-section is considerably affected. 

 In addition, the initial distribution of 5 at x = has equally little effect. The tongue- 

 form of the j'-curves is always re-established in a short time and is very largely a 

 consequence of the turbulent mixing. This is shown particularly well in Fig. 48a, which 

 shows the distribution of the property s in the case where the velocity is constant across 

 the transverse section, and initially for a: = the property s is constant within the 

 distance 2/ {s = 100), while outside of this range there is no content of ^ in the water 

 {s = 0). In the flow a tongue-shaped distribution of s is produced immediately. This 



