Vortex Flow Characteristics

To illustrate the effects of the sidewall converging on the vortex flow, a series of top view flow photos taken from the duct with its sidewalls at different inclined angles at steady or statistically stable state are compared in Fig. 3.1 for Ra = 9,500 with the Reynolds number reduced from 30 to 20. The result in Fig. 3.1 (a) indicates that in the duct without the sidewall inclination (ψ= 0°) we have steady and regular L-rolls at Re = 30. But as the Reynolds number is reduced to 25 and 20, the buoyancy-to-inertia ratios are high enough to cause the L-rolls to become somewhat deformed and irregular (Figs.

3.1 (b) and (c)). Now comparing the results in Figs. 3.1 (a), (d) and (g) for the same Re of 30 but for different converging angles of the sidewalls reveals that when the sidewalls are slightly inclined at ψ=5.7° we have a significant delayed onset of the L-rolls when contrasted with that forψ= 0°. For a larger inclined angle of the sidewalls at ψ=11° only a number of thermals appear above the heated disk (Fig. 3.1 (g)). At the lower Re of 25 the unstable deformed vortex flow in the rectangular duct (Fig. 3.1 (b)) can be significantly suppressed to become thermals at the large converging angle of the sidewall ψ=11° (Fig. 3.1 (h)). But at the smaller ψ of 5.7° we still have unstable L-rolls in the duct (Fig. 3.1 (e)), although they are nearly regular. Similar tread is noted for the lower Re of 20, as evident from the results shown in Figs. 3.1 (c), (f) and (i).

To manifest the spatial characteristics of the longitudinal vortex rolls (L-rolls)

affected by the sidewall inclination, a typical steady regular longitudinal vortex flows in the rectangular duct (ψ=0°) and sidewall converging duct for ψ=11° are respectively shown in Fig. 3.2 and Fig. 3.3 by presenting the steady top and end view flow photos for the two typical longitudinal flow cases (Re = 28.6 & Ra = 17,500 and Re = 27.8 & Ra = 17,600). The end view photos are taken at selected cross sections of the ducts. The top view photos are taken at the middle horizontal plane at y = 1/2 for the ducts. The results in Figs. 3.2 (d) and 3.3 (e) reveal that prior to the formation of L-rolls several thermals rise from the heated plate in both ducts. Through the simultaneous action of the forced flow and buoyancy, these thermals grow slowly in the downstream direction and become elongated in that direction with a mushroom-like cross section (Figs. 3.2 (e) and 3.3 (f)).

Continuing growth of the thermals causes them to hit the duct top and the thermals evolve into L-rolls (Figs. 3.2 (f) and 3.3 (g)). Note that because of the circular geometry of the heated plate, closer to the duct core the thermals are induced at the more upstream locations. This is very different from the onset of L-rolls over a uniformly heated rectangular bottom plate in a flat duct in which the rolls first appear in the sidewall region and the rolls in the duct core are initiated at somewhat downstream locations (Fig. 1.2 (a)). More specifically, in the rectangular duct all the L-rolls driven by the circular heated plate have different size and are not spanwisely symmetric with respect to the central vertical plane x = 0.5, as evident from the end view flow photo in Fig. 3.2 (h).

Comparing the results in Figs. 3.2 and 3.3 indicates that in the sidewall converging duct a slightly longer axial distance is needed for the L-rolls to be initiated. Thus, the sidewall inclination causes a slight delay in the onset of L-rolls.

The effects of the sidewall inclination on the vortex flow are further illustrated in Fig. 3.4 for a higher buoyancy-to-inertia ratio. The regularization of the unstable deformed longitudinal vortex flow by the main flow acceleration associated with the sidewall inclination is also clearly seen in Fig. 3.4 for Re = 30 & 25. It is noted that at the lower Reynolds number for Re = 20 deformed L-rolls prevail in the rectangular duct. But when the sidewalls are inclined, slightly asymmetric L-rolls dominate in the duct (Figs.

3.4 (c), (f) & (i)). The stabilization of the transient oscillation of the longitudinal vortex flow by the inclination of the sidewalls is also clearly seen from the measured time records of the air temperature at selected locations for various cases.

The detailed characteristics of the transverse vortex flow in the rectangular and sidewall converging ducts (ψ = 0° and 11°) are examined next. The spatial structure of the regular T-rolls in the two ducts are presented in Figs. 3.5 & 3.6 by showing the top and side view flow photos at a certain time instants in the statistical state for the two cases with Ra = 11,600 and Re = 10.1 & 5.1. Note that the buoyancy-to-inertia ratio Gr/Re² for the case shown in Fig. 3.6 for the sidewall converging duct is about four times of that for the rectangular duct shown in Fig. 3.5. The top view flow photo in Fig. 3.5 (a)

indicates that in the rectangular duct the T-rolls in the upstream are somewhat bent toward the downstream and they gradually become shorter as moving downstream. The T-rolls are enclosed by an incomplete circular roll. It is in fact a return flow zone which is highly three-dimensional [10]. It should be pointed out that the bending of the T-rolls is attributed to the restriction of the incomplete circular roll induced around the upstream edge of the circular heated plate. Comparing the results in Figs. 3.5 and 3.6 clearly reveals that in the rectangular duct the return flow zone characterized by an incomplete circular roll around the upstream edge of the heated plate is rather large (Fig. 3.5 (a)) and the recirculating flow in it is very strong (Figs. 3.5 (b)-(e)). However, in Figs. 3.6 ((a)–(e)) the recirculating flow becomes much smaller and weaker. The recirculating flow is significantly suppressed by the main flow acceleration due to the sidewall converging.

This weaking of the return flow clearly results from the acceleration of the main flow by the sidewall converging.