8.2 Model comparison 181
Figure 8.5The ratio of the experimentally estimatedRivalue and the theoretically predicted value versusI∗.
between the vortex maximum depth and radius to derive the equation forRigiven in (8.2), which we expect only to be valid forT ≥Ti. The data show that this relationship does not hold during the intial vortex formation forT <Ti.
The results seen here mean that we will use the experimental values forRiwhen compar-ing the theory with the data in later sections. We only expect the time-dependent theoretical model of chapter 7 to be valid forT >Tiand therefore by using the experimental values for RiandTiwe will ensure a fair comparison with the data that is not skewed by the inability of the theory to predict the initial conditions of the vortex atT =Ti.
Regime Q[cm3s−1] f [s−1] g′[cm s−2] H0[cm] D[cm] I∗
lowI∗ 48 0.5 13.2 - 14.3 2 5 0.40 - 0.42
intermediateI∗ 48 1 13.1 - 15.8 2 5 0.75 - 0.84
highI∗ 42 - 48 1 2.3 - 3.4 2 5 1.90 - 2.30
Table 8.1Experimental parameters for the PIV experiments and the corresponding dye attenuation experiments with which they are matched.
The vorticity ratio is measured experimentally using two methods. In chapter 6 we used the gradient of the vortex velocity profiles to estimate the value ofαT at each timestep. The theory assumes that the vortex velocity profile is given by
vθ =−αTf r
2 . (8.3)
The results in chapter 6 showed that in general, the maximum vortex velocity increases with time initially, but then remains approximately constant for the remainder of an experiment.
The vortex radius was shown to increase with time, while the gradient of the velocity profile decreases with time. The gradient is given byαT f/2 and therefore its decrease corresponds to a decrease in the vorticity ratio, which we saw in chapter 7 is required in order to conserve PV. This decrease, combined with the increase in the vortex radius, is the cause of the approx-imately constant velocity seen in the vortex velocity profiles at late times. We also saw in section 6.2.4 that the maximum magnitude of the vortex velocity was the same for the parallel and perpendicular cross-sections, whilst the radius was smaller in the perpendicular direction.
This difference is accounted for in the velocity profile in (8.3) by ensuring that the vortex radius is scaled appropriately depending on the direction in which it is measured and the vortex growth model being used. For the shallow vortex model we expect the perpendicular radius to grow at approximately half of the rate of the parallel radius, giving a scaling factor of two (plotted as 1/2 perpendicular velocity in figures). For the vortex current model, we take a mean value for the radius and therefore a scaling factor of 1/√
D∗ is appropriate (plotted as mean velocity in figures). When comparing the experimental data we present both the parallel velocity measurements and the appropriately scaled perpendicular velocity measurements. The second method for calculating the vorticity ratio experimentally is to use the full PIV data across the entire vortex surface to estimate the mean vorticity in the vortex.
The details of the method used are given in chapter 2.
8.2 Model comparison 183
Figure 8.6The estimated value for the vortex vorticity ratio over time from the experimental data for lowI∗. The theoretical predictions for the shallow vortex and vortex current models are also shown.
The theoretical curves for the time-dependent vorticity ratio derived in chapter 7 for both the shallow vortex and vortex current models will be compared with the experimental data for the mean vortex vorticity estimates and the velocity profile estimates forαT. In order to make this comparison the PIV experiments are matched with a dye attenuation experiment with similar parameter values to enable a fitted value ofαito be calculated. The experimental parameters are given in table 8.1. The value ofαifor each experiment is taken to be the fitted value as discussed in section 8.2.1.1 and the value of Ti is also estimated from the experimental data as discussed in section 8.2.1.2. We expect the mean vortex vorticity data to provide the best estimates for the vorticity ratio, with the estimates from the velocity data being slightly higher. This is due to the outflow vortex increasing in size beyond the field of measurement in the PIV experiments, which prevents the full radius of the vortex from being used when calculating the gradient of the velocity profiles. This issue is overcome for the mean vorticity measurements by averaging the vorticity over the central area of the vortex that remains in the field of view.
Figure 8.6 displays the data for the estimatedαT values for the lowI∗regime over time.
Four different sets of data are shown corresponding to different estimates for the value ofαT along with the theoretical predictions from the shallow vortex and vortex current models. The estimates from the parallel velocity data show good agreement with the mean vortex vorticity measurements and the theory. For the perpendicular velocity data the estimates forαT are
Figure 8.7The estimated value for the vortex vorticity ratio over time from the experimental data for intermediateI∗. The theoretical predictions for the shallow vortex and vortex current models are also shown.
larger which can be seen in the mean velocity data in figure 8.6. The mean value is used for a comparison with the vortex current model, and we see that the model underestimates the values. Adjusting the perpendicular velocity data for the smaller radius according to the shallow vortex model gives better agreement with the theory and reduces the estimates to a similar magnitude as seen for the parallel velocity and mean vortex vorticity data. This suggests that the shallow vortex model provides a good prediction for the change in the vortex shape with the presence of the boundary wall.
The same data for the intermediateI∗regime are displayed in figure 8.7. The mean vortex vorticity data and the adjusted perpendicular velocity data show good agreement with the shallow vortex theory forT >60. The parallel velocity data give larger values forαT in general, but decrease towards the mean vortex vorticity estimates at late times. The larger values for the mean velocity give better agreement with the vortex current model. Overall, the data show that both models perform reasonably well in capturing the magnitude and the rate of decay of the vorticity ratio in the vortex.
Figure 8.8 plots the estimates forαT from the experiments in the highI∗ regime. The experimental data show large fluctuations which are a consequence of the unstable nature of the vortex in the highI∗regime. The estimated values ofαT at early times are large, but
8.2 Model comparison 185
Figure 8.8The estimated value for the vortex vorticity ratio over time from the experimental data for highI∗. The theoretical predictions for the shallow vortex and vortex current models are also shown.
decrease towards the order of magnitude predicted by the theory for the second half of the experiment T ≳80. We will concentrate on the values in this range for our comparison.
The adjusted perpendicular velocity estimates agree well with both the mean vortex velocity estimates and the shallow vortex theory. The parallel velocity data show an increase for T >150 which is likely a result of the vortex becoming unstable. Neglecting these last two data points, the mean velocity estimates show reasonable agreement with the vortex current model and the mean vorticity estimates.
In summary, we see promising agreement with the theory and the experimental data across the full range of the experimental parameters. It is difficult to compare the performance of the the shallow vortex and vortex current models against one another as the errors in the data are quite large. What the data show, however, is that the decay ofαT shows reasonable agreement with the theory in terms of its magnitude and the rate of decay over time and perhaps most importantly that the value ofα does not remain constant during an experiment.
The data also suggest that the factor of 1/2 used in the shallow vortex model to account for the contraction of the radius perpendicular to the boundary wall works well as a first approximation. This will be discussed in more detail in section 8.2.4 when analysing the vortex radius measurements.
I∗ Q[cm3s−1] f [s−1] g′[cm s−2] H0[cm] D[cm]
0.39 74 2 68.0 2 2.5
0.80 62 1 15.3 2 5
1.14 45 1.5 15.1 2 5
1.15 42 0.5 2.3 2 5
1.33 100 1.5 15.3 2 5
1.73 62 2 13.5 2 5
Table 8.2Experimental parameters for the experiments shown in figure 8.9.