Studies of vortex breakdown and its stability in a confined cylindrical container 4

CHAPTER 4* SPIRAL VORTEX BREAKDOWN STRUCTURE AT HIGH ASPECT RATIO CONTAINER 4.1 Introduction Earlier experimental studies [see Vogel, 1968 and Escudier, 1984] showed that vortex brea

CHAPTER 4*

SPIRAL VORTEX BREAKDOWN STRUCTURE AT

HIGH ASPECT RATIO CONTAINER

4.1 Introduction

Earlier experimental studies [see Vogel, 1968 and Escudier, 1984] showed that vortex breakdown in an enclosed cylindrical container with one rotating endwall could exhibit either one, two or three re-circulating bubbles depending on the combination of Reynolds number (Re) and aspect ratio (Λ), at least for Λ ≤ 3.5 However, a recent numerical study by Serre and Bontoux (2002) at Λ = 4.0 showed that under some conditions, an S-shape vortex structure followed by a spiral-type vortex breakdown could also be produced This finding is most interesting since a spiral-type vortex breakdown in an enclosed cylindrical container has not been produced previously in experiments This part of the investigation is to experimentally address the following issues: (a) Can an S-shape vortex structure and a spiral-type vortex breakdown be produced under laboratory condition? (b) How does a bubble-type vortex breakdown, which is known to occur only in a low aspect ratio container (Λ ≤ 3.5) evolves into an S-shape structure and a spiral-type vortex breakdown in a high aspect ratio container (Λ ≥ 3.5)? (c) Can a spiral-type vortex breakdown be also generated in a low aspect ratio container if we remove the flow symmetry, as this constraint is often cited as one

of the factors responsible for the absence of a spiral-type vortex breakdown in an enclosed container?

4.2 Experimental Setup and Procedure

It is worth noting that the test rig and the procedure for this part of the study are slightly different with those described in Chapter 2 Here, two sets of apparatus were used: the first one is to address questions (a) and (b) (see Fig 4.1), and the second one

is to answer question (c) (see Fig 4.2) The first apparatus consists of a Plexiglas cylinder (commercial available) with a nominal inner radius R of 87.25 mm, and a matching rotating disk at the top and a stationary disk at the bottom of the cylinder The height H of the flow domain, and the aspect ratio Λ = H/R, can be varied by moving the rotating disk to a predetermined location or continuously varying the position of the stationary disk using a micrometer, which has a maximum displacement

of 50mm or 0.573R, with an accuracy of + 0.005 mm These unique features enable incremental changes in the vortex structure to be studied, either by keeping the aspect ratio constant and varying the Reynolds number or vice versa While the former procedure is often used by researchers, the latter method provides a more convenient mean to observe vortex evolution at a fixed Reynolds number

The top rotating disk was driven by an electronically controlled micro-stepper motor operating at 20,000 steps/rev with an adjustable speed of up to 240 rpm (Ω = 25.1 rad/s), and an error of less than 0.1% The working fluid was a mixture of glycerin and water (about 80% of glycerin by weight) with kinematic viscosity ν = 0.401 + 0.002 cm2 s−1 at the room temperature of 23.5 °C Before each run, the fluid viscosity was measured using a Hakke Rheometer, and although the experiments were

conducted in an air-conditioned room, the fluid temperature was monitored regularly using a thermocouple located at the top of the cylinder If the temperature exceeded a predetermined value, the experiment was stopped for the fluid to cool to the room temperature before starting again The cap in the temperature restricted the variation in viscosity to less than 1% To minimize optical distortion of flow images due to the curvature of the cylindrical wall, the container was immersed in a rectangular box filled with the same working fluid, since both the solution and the plexiglas have similar refractive indices The rectangular box also served as a constant temperature bath for the container To visualize the flow, food dye, which had been premixed with the working solution, was released slowly into the flow domain through a 1.8 mm diameter hole at the center of the bottom disk The food dye was used as it provided a better perspective of three-dimensional vortex structures than the laser cross-section of fluorescent dye In conducting the dye visualization, we were fully mindful of the

pitfalls highlighted in Hourigan et al (1995), Lim (2000), and Sotiropoulos et al

(2002), and extra care was taken during the fabrication to ensure that various parts of the apparatus were properly matched and the dye port is located at the centre of the stationary disk In all cases, the flow images were illuminated using fluorescent lamps, and captured using a video or a still digital camera

The second apparatus (see Fig 4.2) is a modification of the first one, and besides having a fixed stationary disk, a small cone was attached to either the rotating or stationary disk at a predetermined offset position from the axis of symmetry The cone, which measures 60 mm in diameter and 10 mm high, serves to break the flow symmetry by displacing the vortex filament away from the centre of the rotating disk Two aspect ratios were considered, namely Λ = 2 and 2.5, food dye or fluorescent dye was used to visualize the flow

4.3 Results and Discussions

4.3.1 Generation of an S-shape vortex structure and a spiral-type vortex

breakdown

Here, the first apparatus (Fig 4.1) was used in conjunction with either one of the following procedures, namely, (a) keeping the aspect ratio constant, while increasing the Reynolds number or (b) keeping the Reynolds number constant, while increasing the aspect ratio

To validate proper working condition of the experimental setup, results obtained at

Λ = 3.5 are compared with those of Escudier (1984) as shown in Figs 4.3 and 4.4 Note that food dye was used in the present study for the reason cited above, as compared to the sectional views obtained by Escudier (1984) Moreover, the present flow images have been converted into their “negatives” to further improve their

Thermocouple

Fixed supporting disk

Square tank

Cylinder Rotating disk

To microstepper motor

Dye injection port

Ball bearing Offset

cone

ε

Fig 4.1 Schematic drawing of the

first set of apparatus

Fig 4.2 Schematic drawing of the second set of apparatus

contrast (see Fig 4.4) For ease of comparison, the flow visualization images of Escudier (1984) have been rotated by 180o to correspond to the present setup (i.e with the rotating endwall at the top of the pictures) Taking into consideration that Escudier’s results are uniformly stretched in the radial distances by 8% due to the refraction at various interfaces, our results are in good qualitative and quantitative agreement with his, thus indicating that our apparatus is in good working order Small differences in the Reynolds number (less than 0.6%) can be attributed to the sensitivity

of the function generator used to control the speed of the stepper motor and the accuracy of the measured viscosity From the figures, it is worth noting that the dye filament in the two studies underwent similar “helical instability" with decreasing wavelength prior to the onset of a bubble-type vortex breakdown This “instability” is

a manifestation of an offset dye injection as highlighted by Hourigan et al (1995)

Fig 4.3 Results of Escudier (1984) [with permission from Springer] showing the initiation and evolution of a bubble-type vortex breakdown with increasing Re for Λ = 3.5 HI denotes a “helical instability” which is a manifestation of an offset dye injection as highlighted by Hourigan et al (1995)

Fig 4.4 Generation and evolution of vortex structures with increasing Re for Λ = 3.5 obtained in the present study HI denotes “helical instability” of dye filament Note that the results of Escudier appear larger because the radial distances are uniformly stretched by about 8% due to the refraction at the various interfaces The vertical distances of separation between the two bubbles in both Figs 4.3 and 4.4 matched each other

Figure 4.5 shows the results obtained when the aspect ratio was increased to 4.0 Here, unlike the lower aspect ratio case of Λ = 3.5, increasing the Reynolds number did not lead to the formation of a bubble-type breakdown.Instead,when the Reynolds number was of approximately 3061, the precessing vortex filament moved radially in a rapid manner to give the appearance of an “S” as shown in Fig 4.5 (c) At this stage,

no vortex breakdown was formed upstream of the S-shape vortex structure However, further increases in the Reynolds number eventually led to the formation of a

spiral-type vortex breakdown at the upstream location (see Fig 4.5(f)) This can be deduced from the presence of an abrupt kink in the dye filament indicating the presence of a stagnation point which was also observed in the numerical simulations by Serre and Bontoux (2002) Although attempts were made to conduct the experiment at higher Reynolds number, the results were inconclusive, as flow unsteadiness caused the dye

to diffuse rapidly and made the interpretation of the flow difficult Nevertheless, the present results support the numerical findings by Serre and Bontoux (2002) of the existence of an S-shape structure and a spiral-type vortex breakdown at Λ = 4, although the onset Reynolds numbers for both the vortex structures were lower in the experiment Specifically, Serre and Bontoux (2002) show that from Re = 2500 to Re =

3400, the flows are steady without breakdown, but at Re = 3500, transition to a periodic regime takes place through an axisymmetric Hopf Bifurcation, and beyond Re

= 3500, the flows are characterized by spiral arms evolving in helical structures in the central region of the flow Their critical transition Reynolds number (Re = 3500) is in good agreement with the extrapolated data of Gelfgat et al (1996) who have earlier highlighted the discrepancy in the critical Reynolds numbers obtained numerically and experimentally In the present study, the critical Reynolds number is approximately

3000, which is consistent with the Escudier’s stability diagram extrapolated to Λ = 4.0

When the aspect ratio was reduced to 3.75 as shown in Fig 4.6, the vortex filament evolved through some convoluted motion into an S-shape structure at the downstream side (i.e closer to the rotating endwall) and subsequently into a spiral-type vortex breakdown at the upstream side (closer to the stationary endwall) at Re = 3463 (see Fig 4.6 (f))

Fig 4.5 Generation and evolution of vortex structure with increasing Re for Λ = 4.0 Note the formation of an S-shaped structure in (c) and a spiral-type breakdown in (f)

Fig 4.6 Generation and evolution of vortex structure with increasing Re for Λ = 3.75 Note the formation of a helical instability in (a) and (b), S-shaped structure in (e), and

a spiral-type breakdown in (f)

Figure 4.7 shows the results when the aspect ratio was reduced further to Λ = 3.65 This condition is interesting in that the vortex filament underwent three stages of development as the Reynolds number was increased, namely a bubble-type vortex breakdown, an S-shape vortex structure and finally a spiral-type breakdown As depicted in Fig 4.7, the vortex filament between Re = 2850 and Re = 2978 eventually led to the generation of a bubble-type vortex breakdown at the downstream side (see Fig 4.7(c)), with its size growing with increasing Reynolds number until Re = 3236 (see Fig 4.7 (c) to (f)) Further increases in the Reynolds number caused the bubble-type breakdown to slowly evolve into an S-shape vortex structure, with a spiral-bubble-type vortex breakdown emerging upstream of it (see Figs 4.7(g) and (h) At Re = 3275, the precession motion of both the downstream and upstream vortex structures were clearly displayed in the experiment, indicating that the flow was approaching an unsteady condition This is consistent with the experimental observations by Escudier (1984) and the numerical simulations by Serre and Bontoux (2002) As in the case of Λ = 4.0, the results obtained at Reynolds numbers higher than 4501 were inconclusive due to rapid diffusion of the dye

Fig 4.7 Generation and evolution of vortex structure with increasing Re for Λ = 3.65 Note the increasing size of the bubble-type vortex breakdown with the Reynolds number before it disintegrated into an S-shaped structure

Although varying the Reynolds number and keeping the aspect ratio constant has provided valuable information on the generation of an S-shape vortex structure and the spiral-type vortex breakdown, it gives little clue as to how a bubble-type vortex breakdown evolves into an S-shape structure for a fixed Reynolds number Moreover, the acceleration/deceleration of the rotating disk due to changes in the Reynolds number invariably leads to vorticity production at the corner between the rotating endwall and the sidewall, and therefore more time is needed for this “starting” vortex

to diffuse and for the flow to stabilize This motivated us to approach the problem from

a different angle, i.e by keeping the Reynolds number fixed and increasing the aspect ratio continuously from 3.5 to 4.0 by moving the stationary endwall using a micrometer as shown in Fig 4.1 Figure 4.8 shows the results obtained for Re = 3149,

where it can be seen that increasing the aspect ratio led to a reduction in the size of an initially formed two bubble-type vortex breakdown, with the upstream and smaller bubble eventually disappears In contrast, the shrinking downstream bubble resulted in the formation of a helical instability of increasing wavelength as the aspect ratio was increased, until an S-shape structure was produced (see Figs 4.8(h) and 4.8(i)) A close-up view of the transformation of the downstream bubble-type vortex breakdown

is displayed in Fig 4.9, where it can be clearly seen that as the bubble shrunk, it was replaced by the helical instability which, through some convoluted motion, transformed into an S-shape vortex structure (see Fig 4.9(i)) This S-shape structure is consistent with the corresponding vortex structure displayed in Fig 4.5 (d) for the same flow condition, but obtained through increasing the Reynolds number at a fixed aspect ratio Λ = 4.0 It should be pointed out that all the flow images presented in Fig 4.8 were obtained during one realization of the experiment, meaning that they were captured as the aspect ratio was increased slowly from the start to the end of experiment The whole process may take as long as 60 minutes During this period, the total temperature rise was found to be about 0.3ºC, giving rise to the uncertainty in the Reynolds number of about 2 % Nevertheless, the results still reflect the sequence

of events which occur as a bubble-type vortex breakdown slowly evolves into an S-shape vortex structure