Kinetic energy

One of the global quantities which characterize flow fields is the mean kinetic energy in (r, z) plane, which is defined by

$$E = \frac{1}{A} \int_S \frac{1}{2} \mbox{\boldmath$u$ }^2 dr dz,$$ (4)

  

Figure: Development of flow field from secondary anomalous mode to primary mode. The aspect ratio is 4.4 and the Reynolds number is reduced from 600 to 100. Deceleration starts at t = 1800 and it ends at t = 3600.

  

Figure: Variation of radial rate of strain on cylinder walls. The aspect ratio is 4.4 and the Reynolds number is reduced from 600 to 100. Deceleration starts at t = 1800 and it ends at t = 3600.

  

Figure: Evolution of mean kinetic energy in $(r, \theta )$ plane. The aspect ratio is 4.6 and the Reynolds number is reduced from 150 to 140. Deceleration starts at t = 450 and it ends at t = 900.

where S is an integral domain and A is the area of a meridional section. Figure 10 shows the evolution of the mean kinetic energy for the flow presented in Fig. 2. While the energy decreases monotonically during the deceleration of the inner cylinder, it reaches a peak around t = 1370 where a drastic transformation from the 6-cell mode to the 4-cell mode occurs. When the reduction of Re is small, similar peaks in the energy profiles were found for other transformations from the normal secondary mode.

As an example of nonunique solutions, the mean kinetic energy is shown in Table 1. In the case of ${\sl\Gamma}$ = 5.4, Re = 700, three modes are formed: a normal secondary mode with 6 cells, an anomalous secondary mode in which 6 cells appear and the flows near both end walls of the cylinders are outward, and an anomalous secondary mode with 7 cells and an outward flow at one end wall. The energy at ${\sl\Gamma}$ = 5.4, Re = 650 is also shown in the table. For flows with the same number of cells, the flow with an anomalous cell(s) has more energy than one without an anomalous cell. A similar result was found in other cases of multiple solutions.

  

Table: Mean kinetic energy in $(r, \theta )$ plane at the aspect ratio 5.4