CAPITULO 3. IMPLEMENTACIÓN
4.5. Implementación de la propuesta de mejora
4.5.2. Personalización
In this section the flow observed using the shorter of the two tanks is described. This tank allows aspect ratios in the range 0 .:::s H / R .:::s 5 to be investigated.
Figure 1 shows the time variation of mean, equally spaced concentration contours in the case when a
=
4.8, measured by digital analysis of the video tape, at time intervals of one minute (where t=
0 is defined to be when the plume reaches the base of the container). It should be stressed again that these are mean concentrations along a line through the tank perpendicular to its front face. The concentrations shown here are then not only varying with the axial distance from the source and the approximately Gaussian radial dependence, but also with the variation of the thickness of the plume. Hence the concentrations shown here are effectively averages of an integration along a chord of the plume, together with the environmental fluid exterior to the plume. For comparison, the contours for the confined 'starting-plume' at t=
0, are shown in figure 2a. In this case the environment is unstratified and the form of these contours may be predicted analytically, neglecting the effect of the box boundaries.In an infinite unstratified environment it can be shown that the reduced gravity g' of a plume is of the form
5 (2 2)/ 2 2
g'(x, y, z) '" Z-3 e- x +Y (J Z , (2.1) (Morton, Taylor & Turner 1956), where x, y and z are cartesian coordinates scaled on the effective radius of the chamber, with the z-axis vertical, through the centre of the plume and a is a constant controlling the radial variation. The radial variation of the reduced gravity has been assumed to be approximately Gaussian. For comparison with the experimental contours this must be integrated through the plume assuming that the plume material extends over a distance equal to its length scale
b _ 6apz
- 5 ' (2.2)
where ap is the entrainment constant for a plume (see chapter 3). Integrating (2.1) along a chord through the plume, gives an intensity function
(2.3) Equally spaced contours, (not chosen to match the experimental contours) of the form I(x, z)
=
constant are shown in figure 2b, for a=
0.125, a value typical of experimental-
113-I I
I
( a) (b) ( c) ( d)
FIGURE 1. The mean concentration contours obtained in an experiment using a square container, with a = 4.8. (a) t = 60s; (b) t = 120s; (c) t = 180s; (d) t~ 240s; (e) t = 300s; (1) t = 360s;
Estimated plume position
( a) (b) ( c) (d)
FIGURE 1. The mean concentration contours obtained in an experiment using a square container, with a
=
4.8. (a) t=
60s; (b) t=
120s; (c) t=
180s; (d) t~ 240s; (e) t=
300s;(f)
t=
360s;(g)
t = 420s; (h) t=
480s.Estimated plume position
( e) (fJ (g) (h)
( a)
(b)I I
1-
2-3
-v
-v
I-I I I I I I i I I I' , I i i I I i I rrTlr
-0.5 0.0 0.5
FIGURE 2. (a) The initial mean concentration contours for the experiment shown in figure 1; (b) the theoretical contours that should be observed through a plume in an infinite, unstratified fluid.
Chapter Five - Buoyant Convection from a Source ...
results (see Fischer et al. 1979). The contours essentially take the same form as is observed experimentally - this comparison serves as a useful check on the video analysis technique, but it also indicates that the plume material mixes over a much larger region than that given by its length scale or 'radius' b above. This is probably due to the horizontal confinement of the plume - the above calculation assumes an environment of infinite extent and plumes re-leased near to boundaries have a tendency to 'stick' to them. The experimental contours are inaccurate near to the source as the plume radius is comparable to the digital pixel resolution.
Returning to figure 1, the movement of the first front, i.e. the fluid in the environment that originated from the first plume material to reach the tank base, can be clearly seen by following the topmost contour (which in each case corresponds to a concentration of 0.4%) as it moves up the box. Notice also that the contours near to the source are approximately horizontal, indicating uniform concentration over horizontal planes. This is effectively the type of flow considered by Baines & Turner (1969): there is little interaction between the plume and the environment other than horizontal entrainment. Recall that the contours shown here are of the mean concentration through the tank; thus near to the source the radius of the plume is small and so the mean value measured is dominated by the environment fluid.
This explains why the horizontal contours of the environment are dominant, the characteristic upturned 'V' shaped contours associated with the plume (see figure 2) being hardly apparent.
Further from the source the plume contours become more obvious although distorted by the exterior flow. Outside of the plume (the mean position of the plume may be estimated using traditional plume theory, see chapter 3, and is indicated by the dashed lines on figures 1d,h), there is a horizontal concentration gradient as well as the vertical stable stratification. This is a result of the increased interaction between the plume and the environment as the radius of the plume increases with distance from the source. (This is also reminiscent of the 'over-turning' observed by Baines & Turner 1969.) As the plume radius increases, the downward plume motion and its induced entrainment velocity have an increasing effect on the motion of the fluid exterior to the plume. When the radius of the plume is small, the bulk of the exterior fluid is free to flow vertically, resulting in approximately uniform profiles of velocity and density on horizontal levels. However, when the plume radius is large, a much higher proportion of the exterior fluid must have a significant radial component of velocity which results in non-uniformities in the profiles of velocity and density on horizontal levels.
Digital video analysis may also be used to measure the time dependence of the mean concentration at a given point, outside of the plume. Figure 3 shows measurements, taken
z
=
4.785 z
=
4.16z
=
3.54z
=
2.92....-... v u 4 z
=
2.30z
=
1.68;..,
;j
0 z
=
1.06CfJ 4-t
0
*-
'--' 3q
...
...., 0ell z
=
0.44;..,
....,
q 2
V u q
0 0
o 50 100 150 200 250 300
t (secs)
FIGURE 3. The time variation of the mean concentration at various depths in the fluid exterior to the plume, at a distance 0.2R from the left hand boundary (taken from the experiment shown in figure 1).
t = 300s
5
t = 240s ....-...
v 4
u ;..,
t = 180s
;j 0 CfJ 4-t 0
*-
'--' 3q t = 120s
...
...., 0 ell ;..,....,
q v 2
u t = 60s
q
0 0
2 z/R 3 4
FIGURE 4. The mean concentration along a vertical slice through the centre of the chamber, (from the experiment shO\\'n in figure 1), plotted after successive time intervals of one minute.
....
z=
4.785 z
=
4.16z
=
3.54 z=
2.92 ..-....z
=
2.30v 4
u z
=
1.68I-<
;:l
0 Z
=
1.06m 4-<
0
~ 3
' - "
q
.S ..,;>
ro z
=
0.44I-<
..,;>
q 2
V u q
0 0
o 50 100 150 200 250 300
t (secs)
FIGURE 3. The time variation of the mean concentration at various depths in the fluid exterior to the plume, at a distance 0.2R from the left hand boundary (taken from the experiment shown in figure 1).
t
=
300s5
t
=
240s..-....
v 4 u I-<
t
=
180s;:l 0 m 4-<
0
~ 3
' - "
q t
=
120s.S ..,;>
ro I-<
..,;>
q
V 2
u t
=
60sq
0 0
o 2
z/R 3 4
FIGURE 4. The mean concentration along a vertical slice through the centre of the chamber, (from the experiment shown in figure 1), plotted after successive time
Chapter Five - Buoyant Convection from a Source ...
during the experiment shown in figure 1, of the concentrations at various vertical heights at points a horizontal distance of O.2R from the left hand boundary. It can be seen from this figure that after a sufficiently large time (which varies according to vertical distance from the source), the concentration increases approximately linearly in time. This is in agreement with the time dependence assumed by Baines & Turner (1969) in their asymptotic theory.
For future comparison with the taller cylinder, the mean concentrations along a thin slice through the centre of the box have also been measured. These measurements are shown in figure 4 in which the 'centre-slice' concentration is plotted against depth, after successive time intervals of one minute. The roughly equal spacing between the lines again illustrates the linear time dependence.