Desarrollo de la Propuesta

The spanwise averaged time evolution of the bottom wall shear stress is plotted in figure 4.40. The solid line denotes zero shear stress. The evolution of the reattachment length forms an oscillating pattern of leaning saw-tooth shape. This visualises more clearly the behaviour of the reattachment position shown in figure 4.39. The reattachment length is increasing slowly in a roughly linear fashion (the average slope is 0.3Ub). At some point in this evolution an area of forward flow

starts to form upstream of the main reattachment position (t = 65₋67.5h/Ub).

This forward flow zone will eventually take over the downstream reverse flow zone closing the leaning saw-tooth shape (t = 70h/Ub). Simultaneously the upstream

limit of the new forward flow area becomes the new reattachment position. This oscillating pattern is not very regular and carries some small scale structures on top of it.

Figure 4.40: Evolution of the mean reattachment position - spanwise averaged wall shear stress: blue - negative, green - positive, black solid line -

The secondary bubble lacks the small scale structure of the main recirculation zone, yet it appears to exhibit a pattern inverse to that of the primary reattach- ment position. The secondary bubble pattern is not as clear as the primary one, however there appears to be a negative slope of secondary structures (roughly

−0.08Ub). The tertiary corner bubble does not exhibit any significant dynamic

behaviour.

Oscillatory behaviour of the main reattachment position in a turbulent flow was observed by Le et al. (1997). Figure 4.41 shows the time evolution of the reattachment length for Reh = 4250 (originally Re = 5100 based on U0) and

ER = 1.2. The saw-tooth shape is present, but it does not appear to be leaning like in figure 4.40. The slow increase in the reattachment length (slope _≈0.15U0)

is followed by a rapid drop. Note that a roughly doubled Reynolds number results in doubled speed of reattachment length increase, however one needs to keep in mind that the two cases differ in geometry (expansion in a channel flow vs boundary layer flow over a step). Another similarity between the two cases is the frequency of the oscillations. In both figures there are approximately 8 saw-tooth shapes in 100h/Ub period.

A similar analysis was performed recently by Schafer et al. (2009). Figure 4.42 presents the same reattachment length evolution for transitional flow with a laminar inflow profile (Reh = 3000, ER = 2.0). The reattachment position

exhibits similar leaning saw-tooth shape as in the present study, however it is much more regular and its frequency is clearly higher. The approximate slope of the reattachment length increase is equal to 0.6Ub.

Figure 4.41: Time evolution of the spanwise averaged reattachment po- sition- results of DNS simulation of turbulent flow over a step with ER= 1.2 by Le et al. (1997).

Figure 4.42: Time evolution of the spanwise averaged reattachment po- sition - results of DNS simulation of transitional flow in a channel with ER = 2 by Schaferet al. (2009).

off the step edge, growing in the mixing layer and interacting with the recircula- tion zone. Vortices are visualised by low pressure isosurfaces (see figure 4.43), as the vortex cores are often characterised by pressure minima. The behaviour of the recirculation zone is represented by a streamwise velocity isosurface (figure 4.44). Vortical structures that impinge on the bottom wall cause a part of the reversed flow to separate from the main recirculation bubble. The separated reverse flow area is then convected downstream along with the vortex.

Figure 4.43: Snapshots of low pressure fluctuation isosurfaces - result by Schaferet al. (2009).

Figure 4.44: Snapshots of streamwise velocity isosurfaces - result by Schaferet al. (2009).

A similar visualisation for the present case is shown in figure 4.45. In the case of Schafer et al.(2009), the laminar inflow and low Re caused the vortices in the mixing layer to be very regular, spanning the entire width of the domain and breaking only after reattachment. In the present study the vortical structure in the shear layer is much more complex.

It is very difficult to draw any conclusions from this visualisation, therefore a spanwise-averaged result is presented in figure 4.46. It shows a much clearer pic- ture of splitting of the recirculation eddy. Initially the recirculation area forms a compact bubble. The low pressure zone causes the bubble to stretch down- stream. At t = 65.0h/Ub, the bubble starts to separate. The separated part

of the reversed flow travels downstream with the low pressure zone, while the main recirculation bubble contracts quickly. The separated bubble vanishes as it travels downstream, while the main recirculation zone starts to grow again.

Figure 4.46 shows that the mechanism which governs the flapping of the pri- mary reattachment position in turbulent flow is the same as for the transitional case studied by Schafer et al. (2009). The vortical structures that grow in the mixing layer interact with the wall by inducing a zone of reversed flow near the wall and causes the recirculation bubble to stretch. As the structure is convected downstream it carries the reversed flow zone with it, which causes the recirculation bubble to split. As the reversed flow zone disappears, the reattachment length rapidly shrinks. The difference for turbulent flow is that the vortical structures in the mixing layer are more complex than those in a transitional flow.

The quantitative analysis of the oscillatory behaviour of the reattachment position can be performed by studying the pressure and streamwise velocity fluc-

tuations near the reattachment position. Figure 4.47 (a) shows the history of spanwise averaged p0 at x = 8.0h, y = 0.01h, and figure 4.47 (b) shows the history of spanwise averaged u0 at the same point. Both signals look highly cor- related on a large scale. The time history of velocity fluctuation represents the behaviour of the reattachment bubble, while the pressure fluctuations indicate the presence of vortical structures.

Figure 4.47: Pressure and streamwise velocity fluctuations history- span- wise averaged at x = 8.0, y = 0.01. (a) p0 _{pressure fluctuation; (b) u}0 _velocity

fluctuation.

presented in figure 4.48. The locations cover the entire domain, including the inlet channel, recirculation bubble, mixing layer and reattachment.

Figure 4.48: Location of pressure and velocity fluctuation measurements - point #1 (x=₋2h, y=1.5h), #2 (x=0.1h, y=h), #3 (x=0.1h, y=0.5h), #4 (x=4h, y=1.5h), #5 (x=4h, y=h), #6 (x=4h, y=0.1h), #7 (x=8h, y=1.5h), #8 (x=8h, y=h), #9 (x=8h, y=0.01h).

Figure 4.49 shows the spectra for the inlet channel (a and b), a location near the step edge (c and d) and in the secondary corner eddy (e and f). The inlet spectrum shows clearly the peak corresponding to the inlet periodicity generated by the regeneration technique (St = 0.127), and subsequent subharmonics. In the following figures we will examine whether this frequency is present elsewhere in the flow and if it influences the oscillations of the reattachment position. Spectra near the step edge (figure 4.49 c and d) show only a slight peak at the regeneration frequencySt= 0.127, both for the velocity and pressure fluctuations. Also a small peak near St= 0.068 is visible. A similar low frequency shows in the secondary corner eddy (figure 4.49 e and f). Other than that the spectra for the secondary eddy are smooth.

The velocity fluctuation spectra in the main flow and the mixing layer shown in the figure 4.50 (a) and (c) are fairly broadband. The regeneration frequency

Figure 4.49: Spanwise averaged power spectrum of pressure and velocity fluctuation A - (a) point #1, full u0 spectrum; (b) #1, range of frequencies of interest - clearly visible peaks due to inlet periodicity; (c) #2,u0 spectrum; (d) #2,

is present in the main flow (point #4), but the higher frequency ofSt= 0.195 is also present, and not only in the u0 spectrum, but also in pressure fluctuations. Again, lower frequency of St = 0.068 shows up in all three locations in figure 4.50. This frequency is especially pronounced for p0 graphs and for point #6 in the recirculation zone.

In figure 4.51 (a) and (b), which shows point #7 in the main flow, one can see the most pronounced frequency of St = 0.078, which occurs for both u0

and p0 spectra. A strong peak at this frequency is also present in the pressure fluctuation spectrum for point #8 (figure 4.51 d), but disappears in the velocity spectrum. Finally, it is strongly accented at the point near the reattachment (point #9, figure 4.51 e and f) for both pressure and velocity spectra. This clearly indicates that the presence of a vortex (represented by a pressure fluctuation) and behaviour of the reattachment position is correlated and tuned to a characteristic frequency of St _≈ 0.078. As we have seen above, the frequency in the range

St = 0.068₋0.078 is present in the entire recirculation region and is fed back from the reattachment, through the primary recirculation eddy to the step edge at point #2. This in turn generates vortices of similar frequency in the mixing layer, which can be seen in figure 4.50 (c) and (d) as well as 4.51 (c) and (d). Those vortices cause the oscillations of the reattachment position (point #9), which influences the entire recirculation eddy and closes the feedback loop. Moreover, this result agrees with previous findings of Leet al. (1997) who reportSt_≈0.06 as a frequency of the reattachment flapping. Similarly Metais (2001) found the characteristic flapping frequency St _≈ 0.07, while Silveira Neto et al. (1993) provides the valueSt= 0.08 for large Kelvin-Helmholtz structures in the mixing

Figure 4.50: Spanwise averaged power spectrum of pressure and velocity fluctuation B - (a) point #4, u0 spectrum; (b) #4, p0 spectrum; (c) #5, u0

Figure 4.51: Spanwise averaged power spectrum of pressure and velocity fluctuation C - (a) point #7, u0 spectrum; (b) #7, p0 spectrum; (c) #8, u0

layer. Schaferet al.(2009) reportSt = 0.266, however the quantitative agreement in this case cannot be expected due to the presence of the laminar flow at the inflow of this simulation.

The regeneration frequency of St= 0.127 is present in the flow, especially in the main stream (figure 4.49 a, 4.50 a and 4.51 a), but it tends to show up in the

u0 spectra, which could indicate that it does not have an influence on the vortex formation in the mixing layer.

In many places in the mixing layer and the main flow the higher frequency of

St= 0.195 shows up, which is not present in the recirculation area or the inflow channel (see figure 4.50 a, b and d). Its origins cannot therefore be explained by the regeneration technique or reattachment flapping.

10−1 100 0 1 2 3 4 5 6 7x 10 −5 St = f h / U_b PSD 10−1 100 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8x 10 −4 St = f h / U_b PSD (a) (b) 0.219 0.459 0.698

Figure 4.52: Spanwise averaged power spectrum at the inlet of the ad- ditional simulation- point #1. (a) u0 spectrum; (b)p0 spectrum.

In order to further exclude the influence of the regeneration frequency on the reattachment oscillations, an additional simulation with shorter regeneration length (Li = 5h) was performed. In order to save computational resources the

Figure 4.52 presents the spectrum taken in the inlet channel of this additional simulation. It clearly shows the new regeneration frequency ofSt= 0.219 and its harmonics. Figure 4.53 presents velocity and pressure fluctuation spectra near the reattachment position at point #9. The characteristic frequency is St = 0.068, which is slightly lower than for the original simulation, but still in the regime

St = 0.068 ₋0.078 identified as a characteristic frequency of the recirculation zone. This shows that the increased regeneration frequency does not have an influence on the reattachment oscillations.

10−1 100 0 1 2 3 4 5 6 7 8x 10 −4 St = f h / U_b PSD 10−1 100 0 1 2 3 4 5 6x 10 −4 St = f h / U_b PSD (a) (b) 0.053 0.068 0.142 0.048 0.068

Figure 4.53: Spanwise averaged power spectrum near the reattachment position of the additional simulation - point #9. (a) u0 _{spectrum; (b) p}0

spectrum.