Cylinder Near Moving Wall

This section presents results of applying the SCM to the problem of the cylinder near the moving wall at a fixed D/G-ratio. Here statistics are presented for the pathway followed

by the clockwise rotating vortex surviving downstream and its magnitude. The uncertain parameter is taken to be the Reynolds number. Re is assumed distributed uniformly on the intervalRe∈ U[150,240]. The lower bound on Rewas chosen to make sure that the vortex exists for allRe-values.

Three different investigations using increasing expansion orders for the gPC basis have been performed. These are third, fifth and seventh order expansions and the collocation nodes are given in table 8.8.

Basis Order: Nodes:

Re∈[150,240] Re˜ ∈[−1,1]

Third: 156.2489 -0.8611

179.7009 -0.3400 210.2991 0.3400 233.7511 0.8611

Fifth: 153.04 -0.9325

165.25 -0.6612

184.26 -0.2386

205.74 0.2386

224.75 0.6612

236.96 0.9325

Seventh: 151.7870 -0.9603

159.1500 -0.7967 171.3510 -0.5255 186.7454 -0.1834 203.2546 0.1834 218.6490 0.5255 230.8500 0.7967 238.2130 0.9603

Table 8.8: The collocation points for the uniformly distributed random variable Re∈[150,240] for different approximation orders.

For each node the field data at the trace points of the clockwise rotating vortex was stored.

All the data was then interpolated onto one hundred equidistantx-coordinate values down-stream of the cylinder, in the interval x∈[5,15].

Pathway followed: For each of the three UQ approximations the gPC approximation to the y-coordinate of the vortex center was calculated at each x-coordinate. That is y(Re)|_x=fixed was approximated by P_N,D[y]. Using the coefficients of P_N,D[Y] allowed the calculation of the mean path followed by the vortex and the variance of the path. The mean of the path along with the mean plus and minus one and two standard deviations are shown on figure 8.36 (b),(d) and (f).

8.4 Uncertainty Quantification 8 ANALYSIS

(a) Data for third order gPC expansion.

(c) Data for fifth order gPC expansion.

(e) Data for seventh order gPC expansion.

Figure 8.36: (a),(c),(e): Vortex pathway data used for UQ approximation.

(b),(d),(f): Mean pathway followed by the vortices and mean path plus and minus one and two standard deviations.

Figure 8.36 (a),(c) and (e) show the pathway data used to construct the gPC expansions.

From figure 8.36 (b),(d) and (f) it is hard to see any significant difference in the mean or variance for the different expansions. Therefore the differences in mean between the third and fifth and the fifth and seventh order expansions are plotted in figures 8.37a and 8.37b.

Also all three mean value traces are plotted on top of each other in figure 8.37c and a zoom of x∈[5,8] is provided in figure 8.37d.

5 6 7 8 9 10 11 12 13 14 15

difference in mean distance to wall [Cylinder diameters]

µ5−µ3

(a) Difference between the mean calculated using the fifth and the third order gPC expansions. i.e.

difference in mean distance to wall [Cylinder diameters]

µ7−µ5

(b) Difference between the mean calculated using the seventh and the fifth order gPC expansions. i.e.

(c) Plot of the mean and mean plus one standard deviation ob-tained from the third, fifth and sev-enth order gPCs expansions.

(d) Zoom of part of the domain of figure (c).

Figure 8.37

From figure 8.37c it may be seen that there is good agreement between the means and variances calculated using either of the three different SCM approximations past x= 8. It can be seen from figure 8.37d that the mean and variance deviates between the different approximations in the interval x ∈ [5,7.5]. This suggests that at least for the fifth order approximation the mean and variance have not converged in this part of the domain and thus a higher order basis is needed to capture the statistics accurately.

In order to investigate if the gPC expansions have converged it is also possible use the ex-pansion coefficients, ˜gk. If one observes a drop in magnitude for the coefficients of the higher order modes it is an indication that the lower order modes contained in the expansion cap-tures the behaviour of the quantity in question. The magnitude of the expansion coefficients for the seventh orderP7,D[Y], are shown in figure 8.38. For the following analysis one should disregard the value of the zeroth order mode as this is simply the mean of the expansion and does not determine its shape.

8.4 Uncertainty Quantification 8 ANALYSIS

5 6 7 8 9 10 11 12 13 14 15

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

Distance downstream [Cylinder diameters]

k˜gik

˜g0

˜g1

˜g2

˜g³

˜g4

˜g5

˜g6

˜g7

Figure 8.38: Expansion coefficients ˜g_k of the seventh order gPC expansions at each of the one hundred equidistant points downstream of the cylinder in the interval x∈[5,15].

From this figure it is seen that the coefficients in the range x ∈ [5,7] all lie in or close to the interval ˜gk∈[10⁻¹,10⁻²], i.e no consistent drop in magnitude is observed. This means that all modes are almost equally present in the expansion which in turn indicates that the expansion has not converged yet. For x ∈ [8,15] however it is seen that the higher order modes, N ∈ {5,6,7}, are between one and two orders of magnitude lower then several of the lower order modes. This indicate that the behaviour of y(Re)|_x=fixed is captured very well by the lower order modes in the expansion. For x ∈ [11,15] it is seen that the ˜g_k’s consistently drop for all modes drop with increasing kindicating that here the behaviour is captured fully by the gPC expansion. However one would still need to perform a test using an even higher order gPC expansion to check if the behaviour of the expansion coefficients observed here indeed continue for higher order modes.

For the part of the domain where the gPC expansion seems to have converged, the data presented in figure 8.36f may be used to accurately determine the likelihood that a part of the domain downstream of the cylinder contains the center of the clockwise rotating vortex if it is known thatRe∈ U[150,240] without knowledge of the value ofRe.

In conclusion figure 8.38 suggests that the gPC expansion has not yet converged and a higher order basis is needed to be able to draw completely trustworthy conclusions about the pathways of the vortices.

Vorticity Magnitude: The statistics for the magnitude of the vorticity along the trace have been calculated. Here it was found that the statistics had indeed converged at the fifth order expansion, leading to the conclusion that the UQ approximation captures the statistics accurately. The mean along with mean plus and minus one and two standard deviations in x∈[5,15] are presented in figure 8.39.

5 6 7 8 9 10 11 12 13 14 15 1

1.5 2 2.5 3 3.5

Distance downstream [Cylinder diameters]

Vorticity Magnitude

µ µ+σ µ−σ µ+ 2σ µ−2σ

Figure 8.39: Mean vorticity magnitude downstream of the cylinder for a uni-formly distributed random variable, Reynolds number over the interval Re ∈ [150,240].

From the figure it may be noted that the linear decrease in vorticity observed for all Re investigated earlier is also seen for the mean, as one would expect.