The flow configurations of Colley et al. (2006) were previously mentioned in Sections 2.2 and 2.4. As a reminder, they obtained their results with two moderately different experimental configurations. One of these configurations is that the disk was rotating freely under water. In the other case, the flow over the rotating disk was enclosed with a stationary lid to minimize the disturbances which arise from the free surface of water and affect the boundary layer on the rotating disk. This second experimental facility, therefore, resembles the rotor– stator flow configuration which was previously shown in Figure 2.5. These two experimental facilities of Colley et al. (2006) with and without the stationary lid are referred to as stator and no–stator, respectively, in the current study. However, note here that the flow system of Colley et al. (2006) with the stationary lid had a small annular opening of approximately 10−20 mm between the rotating disk and the stationary lid. Therefore, there may have been some exchange of water through this circumferential gap. When this gap is neglected, the aspect ratio of their system was D = 0.1 since the gap between the stationary lid and the rotating disk was h∗ = 0.02 m which corresponded to 10δ in terms of the boundary-layer thickness.
The data from Figure 2 in Colley et al. (2006), for both their experimen- tal systems, are included here in Figures 4.4 and 4.5 in comparison with the computational results obtained from our TSST simulations and the theoretical data of von K´arm´an flow. Moreover, Figure 4.4 shows the TSST simulations for
D = 0.092, h∗ = 10δ, while Figure 4.5 includes corresponding simulations for
D= 0.918, h∗ = 100δ. Both figures, furthermore, emphasize the Reynolds num- ber dependence by including TSST simulations for Re= 210 and for Re= 546. Here the lower Reynolds number corresponds to a radial location within the lam- inar region of the boundary layer while the higher Reynolds number is associated
0 2 4 6 8 10 ζ 0 0.05 0.1 0.15 0.2 0.25 F ( ζ ) von K´arm´an (1921) TSST, D=0.092, h=10δ, Re=210 TSST, D=0.092, h=10δ, Re=546 Colleyet.al.(2006), Re=210, stator
Colleyet.al.(2006), Re=210, no-stator
Colleyet.al.(2006), Re=546, stator
Colleyet.al.(2006), Re=546, no-stator
(a) Radial velocityF
0 2 4 6 8 10 ζ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 G ( ζ ) von Ka´rm´an (1921) TSST, D=0.092, h=10δ, Re=210 TSST, D=0.092, h=10δ, Re=546 Colley et.al. (2006), Re=210, stator
Colley et.al. (2006), Re=210, no-stator
Colley et.al. (2006), Re=546, stator
Colley et.al. (2006), Re=546, no-stator
(b) Azimuthal velocityG
Figure 4.4: Comparison of the results of the TSST simulations (D = 0.092,
h∗ = 10δ, Reφ = 733, Re= 210 and Re= 546 ) to the experimental data from
with a radial location in the turbulent region - i.e. above the critical value near
Re= 507 (Lingwood, 1997). Note here that the von K´arm´an flow is independent of Reynolds number because it describes the similarity solution denoted in non– dimensional form which is globally valid throughout the laminar flow region of the rotating–disk boundary layer.
As regards the validation of the TSST simulations, apart from the radial flow components of Colley et al. (2006) at Re= 546 in Figures 4.4(a) and 4.5(b), in general there is a good qualitative agreement between the TSST results, the von K´arm´an flow and the experimental data of Colley et al. (2006). It is not appropriate to make a final quantitative comparison between the data of Colley et al. (2006) and the TSST simulations and the von K´arm´an flow. One of the reasons for this is, for instance, there exists an annular gap between the rotating disk and the stationary lid in the experiments of Colley et al. (2006) which may alter the flow field in comparison to that in TSST simulations. Additionally, in the experiments, the hot–film probe is liable to an effect known as the yaw–angle bias (Bruun, 1995, p. 71). This results from along–wire cooling in the flow field with its three velocity components and leads to overestimates of the measured velocity values. Although the data of Colley et al. (1999, 2006) were not corrected for the yaw–angle bias, as is discussed in Colley et al. (1999, p. 334), since this was not necessary in the particular context of their studies, this issue will be addressed in our experimental study in Section 6.2. Finally, the experiments in the tank of the facility of Colley et al. (1999, 2006) represents a spatially restricted and, therefore, fairly high–noise environment in which one has to resolve small quantities. To appreciate this note that a distance of ∆ζ = 1 on the abscissa in Figures 4.4 and 4.5 corresponds to a vertical height difference of only ∆z∗ = 0.36 mm for their experimental data. Similarly, velocity changes of ∆F = ∆G = 0.1, at
Re= 210, correspond to ∆u∗ = ∆v∗ ≈0.06 m s−1 occurring over an interval of
only ∆ζ ≈2, that is 0.72 mm.
0 2 4 6 8 10 ζ 0 0.05 0.1 0.15 0.2 0.25 F ( ζ ) von K´arm´an (1921) TSST, D=0.918, h=100δ, Re=210 TSST, D=0.918, h=100δ, Re=546 Colleyet.al.(2006), Re=210, stator
Colleyet.al.(2006), Re=210, no-stator
Colleyet.al.(2006), Re=546, stator
Colleyet.al.(2006), Re=546, no-stator
(a) Radial velocityF
0 2 4 6 8 10 ζ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 G ( ζ ) von Ka´rm´an (1921) TSST, D=0.918, h=100δ, Re=210 TSST, D=0.918, h=100δ, Re=546 Colley et.al. (2006), Re=210, stator Colley et.al. (2006), Re=210, no-stator Colley et.al. (2006), Re=546, stator Colley et.al. (2006), Re=546, no-stator
(b) Azimuthal velocityG
Figure 4.5: Comparison of the results of the TSST simulations (D = 0.918,
h∗ = 100δ, Reφ = 733, Re= 210 and Re= 546 ) to the experimental data from
and the von K´arm´an flow for lowest Reynolds numbers Re and largest aspect ratiosD. Figures 4.5(a) and 4.5(b) show that this expectation is actually correct where the aspect ratio is h∗ = 100δ, D = 0.918. This is the largest aspect ratio in our simulations. One can see in Figure 4.5(a) that there is a good quantita- tive agreement between the TSST simulations and K´arm´an (1921) for the radial velocity component for all heights ζ and for the azimuthal flow component in Figure 4.5(b) up to about ζ = 2. Despite the fact that for ζ > 2 the azimuthal flow of TSST simulations deviates more strongly from von K´arm´an (1921), it is in very good quantitative agreement with the corresponding experimental data by Colley et al. (2006) which display the same type of divergence with respect to von K´arm´an (1921). This type of discrepancy is naturally expected since our simulations and the data of Colley et al. (2006) are for rotor–stator flow whereas the theory by von K´arm´an (1921) is for a freely spinning disk. Moreover note that in Figure 4.5(b) the TSST simulations almost exactly quantitatively reflect the dependence on the Reynolds number that was experimentally observed by Colley et al. (2006).
In addition to the validation of the TSST simulations, the data in Fig- ures 4.4(a) and 4.4(b), and in Figures 4.5(a) and 4.5(b), provide a first assessment of the influences of the aspect ratio D on our TSST simulations but this will be addressed in more detail in Section 4.4.1.
In addition to the validation of the TSST simulations, the above compari- son of the current results to the experimental data of Colley et al. (2006) produces the first direct, quantitative support to the speculation expressed in their paper that their data do not approach the similarity solution by von K´arm´an (1921) for ζ → ∞ due to residual fluid motion exterior to the boundary layer (see Sec- tion 2.2). In this context, two additional brief remarks relating to the comparison of the TSST approach and the experimental data of Colley et al. (2006) are pro- vided.
configuration), the height of the water above their rotating disk was only approx- imately 0.15 m which corresponds to about h∗ = 76δ or D = 0.75. Therefore, it is possible that the effects induced due to the existence of the free surface of the water resembled, to some extent, those induced due to the stationary lid (stator configuration). This may have possibly contributed to the similarity of the experimental data for their stator and no-stator case which was also men- tioned previously in Section 2.2. Together with the limited size of the water tank (diameter approx. 1 m) housing the disk (diameter approx. 0.5 m) it may also explain why there exists a residual radial flow motion in Figures 4.4(a) and 4.5(a). This is supported by the residual radial flow in Figures 4.4(a) and 4.5(a) being more pronounced for the higher local Reynolds number, Re= 546, which corre- sponds to a position of the hot-film probe closer to the lateral boundary of the water tank and where one would expect stronger effects of any side–wall induced recirculation flow regions that can result in radial flow.
Secondly, the small quantitative discrepancies between experiment and TSST simulation at ζ > 4 in Figure 4.5(a), for the parameter configurations that most closely resembles the case of Colley et al. (2006), i.e. h∗ = 100δ or
D = 0.918 and at Re = 210, are possibly partly associated with the, previously addressed, yaw–angle bias.