Since different boundary conditions correspond to different

Since different boundary conditions correspond to different critical Grashof numbers, we used the concept of the critical parameter; it AZD 0530 was expressed by the ratio r=Gr / Grc. The behavior of the convective roll flow under normal and anomalous conditions determined by the tilt angle of the cavity is, without a doubt, an interesting study subject. A flow that keeps circulating when the cavity tilt angle passes through zero is customarily called anomalous. [5] The intensity and direction of the circulation of a two-dimensional roll flow under steady conditions is explicitly described by the extreme value of the flow function (Gr, α) in the center of the cavity, i.e., the phase space of the system is one-dimensional. The thermal convection at the zero angle develops smoothly as a result of fork bifurcation on the (Gr) plane at the critical Grashof number Grc. However, even a slight tilt (about 0.01 degrees [12]) leads to convection appearing at any arbitrarily small values of the Grashof number. Calculations showed that the convective roll flow that developed at a cavity tilt angle differing from zero, and at a fixed Grashof number less than or equal to the critical one (r ≤ 1), reverses its direction smoothly when the cavity tilt angle α passes through zero (solid line in Fig. 2). If the Grashof number exceeds the critical value (r >1), the convective roll flow preserves the direction of movement when the cavity tilt angle passes through zero, thus becoming anomalous. This flow persists up to a certain critical angle ; upon reaching this angle, it abruptly reverses its direction and turns into a normal flow. Bifurcation diagrams for (α), obtained through the calculations for four values of r, illustrate this behavior (see Fig. 2). Curves 1–4 in Fig. 2 correspond to the different values of the critical parameter. The crosses correspond to the successive changes in the tilt of the cavity from the negative to the positive angles, and the squares from the positive to the negative angles. Bifurcation diagrams indicate the existence of an anomalous flow, which, however, transforms into a normal one when the tilt angle reaches the critical value . The existence region of the anomalous flow broadens with an increase in the value of the critical parameter. As can be seen from Fig. 2, for each α from the interval, there are two steady states which differ by their circulation direction, i.e., by the sign of . Experimental studies of convection in cavities with heat-conducting walls typically include measurements by thermocouples [1,4,6]. The data from the differential thermocouples installed in specific locations in the cavity allows to assess the structure of the convective flow. In the works mentioned the thermocouple junctions were located in points A and B (see Fig. 1). The values of the dimensionless temperature difference dT read from such a virtual thermocouple are shown in Fig. 3 as a function of the cavity tilt angle for four values of the critical parameter r. We can see that the jump changes of dT and for the same values of r occur at the same tilt angles (see Fig. 2). The evolution of the temperature fields and the flow lines with a change of the cavity tilt angle α from +30° to –30° for r=2.5 is presented in Fig. 4. For the normal flow in the variation range of the angle α from +30° to 0°, there is a gradual decrease in flow intensity (see curve 4 in Fig. 2), which continues to decrease after passing through zero; the structure of the flow is preserved. When the angle approaches the critical value =–7.8°, the decrease in the intensity of the central vortex accelerates, while corner vortices with the swirl opposite to that of the main vortex grow. The transition process happens in the following manner. One of the corner vortices grows faster than the other one which then disappears. The growing vortex (which has a normal rotation direction at the given tilt angle) displaces the anomalous one. Images of the flow function and the isotherms for the critical angle of =–7.8° are provided for two times. The first time corresponds to the moment of the change of the flow structure, and the second one to the moment when the transition process has been completed. The images below describe the evolution of the normal vortex chamber up to the angle α=–30°. The tilt angle changing in the reverse direction results in obtaining a critical angle with a value equal to =+7.8° in the positive angle range.