579 



FIGURE 4. The distribution of thickness along an 

 intrusion. 



h (t) 

 o 



10^ 



l/5\ 1/4 



V 



2TTn(t - ti) 



1/5 



(29) 



Equation (28) seems very simple and accessible 

 for experimental confirmation: confirmation of 

 this equation will give some confidence in the 

 validity of the model proposed here. The experi- 

 mental checking of Eq. (28) was performed by A. G. 

 Zatsepin, K. N. Federov, S. I. Voropaev, and A. M. 

 Pavlov. They used the following scheme for the ex- 

 periment (Figure 5) . An open plexiglass tank having 

 the form of a rectangular parallelepiped contained a 

 stable, temperature-stratified fluid. A hollow 

 cylindrical tube was introduced from above under 

 the surface of the fluid. The fluid in the tube was 

 mixed and then the ttibe was raised, leaving in its 

 place a spot of mixed fluid which immediately started 

 penetrating the ambient stratified fluid. The ob- 

 servations, photo- and movie camera, were performed 

 using a shadow device. The experiment allowed one 

 to observe clearly the two last stages of spot evolu- 

 tion; the spot extension at the viscous stage is 

 represented in Figure 6. The mixed fluid volume in 

 the spot was fixed for all experiments, as well as 

 the kinematic viscosity of the fluid and the diameter 

 of the tube. Therefore, if Eq. (28) is correct, the 

 experimental data in the coordinates X,g [2ro (t) /D] , 

 !lg[N (t - tj)] had to fall on a single straight line 

 with the slope 0.1. This is confirmed by the graph 

 of Figure 6 where the slope of the solid straight 

 line is 0.1 and tj = -10 sec. Thus, the law of one 

 tenth Eq. (28) for the viscous extension of a spot 

 was confirmed by the experiments of A. G. Zatsepin, 

 K. N. Federov, S. I. Voropaev, and A. M. Pavlov with 

 a satisfactory accuracy. 



Analogously, in the case when the form of the 

 turbulent spot is close to the cylinder with a 

 horizontal axis Eq. (22) for the height of the in- 

 trusion tongue will hold, where x is the horizontal 

 coordinate normal to the axis of the spot. The con- 

 dition of conservation of the volume of the spot of 

 mixed fluid takes, for this case, the form 



f- 



(x,t)dx = V = Const 



(30) 



where H is the longitudinal size of the cylindrical 

 spot. The initial conditions corresponding to the 



asymptotic solution of the instantaneous point 

 source type may be written in the form 



h(x,ti) E (x ?i 0) , H J h(x. 



ti)dx = V (31) 



and the asymptotic solution itself due to the same 

 reasons, as before, may be represented in the form 



1/6 



ln(t - ti)H2 / ^1<^' 



x[v'»n(t - tij/ien'*] ^^^ 



A(l - CVC 2)'/\ ^ C ^ ; 



■1 "< 



0, C > C = (15) 







1/6 



2r(5/4)r(i/2) 



2/3 



r(7/4) 



3.6 



A = (C 2/15) ^/'* = 0.97 







(32) 



so that the leading edge of the intrusion, x = XQ(t) , 

 propagates according to the law 



X (t) = C [V'*n(t - ti)/16H'*]^/^ 

 o o '■ 



(33) 



while the maximum thickness of the intrusion, ho(t) 

 = h(o,t) , decays with time according to 



h t(t) = 0.97(v2/4H2n(t - t,)) 



1/6 



(34) 



Thus, in both cases a strong deceleration of the 

 extension of intrusion was characteristic for a 

 turbulent spot in the transition to the viscous 

 stage. Indeed, at the free intrusion stage the ex- 

 tension of a turbulent spot is proportional to the 



;v-f 



2 



FIGURE 5. The scheme of the experimental checking of 

 the law of viscous extension of a spot of mixed fluid. 

 1) The tank, 2) Point light source with collimator, 

 3) Lens, 4) Vertical elevator with electromotor, 5) 

 Mixer, 6) Tube, 7) Screen, 8) Movie camera. 



