Flow stability and transition over finite compliant panels 4

Chapter Investigations over two and three compliant panels Chapter Investigations over two and three compliant panels An attempt made in chapter with single compliant panel (CP) occupying a certain section of the rigid wall, yielded a transition delay to the tune of almost 49% when compared with the rigid wall (RW) case. This feat with the single CP case in terms of better transition delay serves as a motivation to consider multiple CPs and see if transition distance could be increased beyond distance already obtained for the single CP case in chapter 3. This approach of using multiple compliant panels for superior performance as compared with the relatively simple walls studied so far had already been recommended by Gad-el-Hak (2002). However, it is worthy to note that this particular case (multiple CPs) proposing here within the Blasius boundary layer has not been studied so far in the literature. Davies and Carpenter (1997) numerically studied evolution of a TS waves over finite compliant panels but within a fully developed 2D plane Poiseuille flow, rather than in a Blasius boundary layer. Moreover, the perturbations were 2D and monochromatic rather than 3D broadband. What they did to each compliant panel was to tailor their properties to suit the local flow environment for series of compliant panels used. They concluded that with low-disturbance flow environment, laminar flow up to an indefinitely high Reynolds number could be sustained if suitably designed multiple-panel compliant walls are used. It is nevertheless pertinent to note that transition to turbulence could occur via 3D waves. 95 Chapter Investigations over two and three compliant panels Zhao (2006) attempted multi-CP investigations, but her simulations had stopped short of the late subharmonic and breakdown stages due to resource limitations (memory and speed), so that the ability of these walls to delay transition could not be evaluated. This chapter concerns the use of multiple compliant panels to delay transition further than what had already been obtained for over the single CP case in chapter 3. Based on the number of compliant panels considered, investigations carried out are grouped under two main headings as: (i) over an array of two compliant panels and (ii) over an array of three compliant panels. Wavepacket evolution processes are also analysed in terms of both spatial and spectral analyses. Spectral data extractions for dominant 2D and 3D wave modes were also performed so as to properly understand the overall behaviours and performances of each of the CP as the wavepacket evolve over them. 4.1 Over an array of two compliant panels In order to further enhance the performance of the first compliant panel located at in terms of delaying transition further, a second compliant panel of the same length was added at the location (with corresponding local Reynolds number ) thereby forming an array of two panels as shown in figure 4.1. This time around the size of computational domain in the streamwise (X) direction has to be increased further from X = 1510 to 3083, while that of the wall vertical (Y) and spanwise (Z) directions remain unchanged. The second panel was placed just ahead of the wavepacket at its second last stage of development at time T = 2417 of figure 96 Chapter Investigations over two and three compliant panels 3.4(g2). At that stage, the wavepacket registered a maximum-u velocity value (at ⁄ ) of 0.59% - the purpose of the second added compliant panel is to inhibit the rapid growth of the wavepacket from that point onwards as indicated in Figure 3.5. 4.1.1 Results and discussions Also like the single compliant panel (CP) case in chapter 3, the results for over an array of two-panel arrangement are discussed in terms of both spatial evolution and spectra analyses. 4.1.1.1 Wavepacket spatial evolution analyses Figure 4.2 shows the evolution of the wavepacket over the second compliant panel from time T = 2788 until the incipient turbulent spot is reached at T = 6062. The wavepacket begins to pass over the second CP at T = 2788 in figure 4.2(a). While figures 4.2(b)-(c) show when the wavepacket was fully convecting on the top of the second CP where a fairly sharp maximum disturbance velocity increases to 1.2%. This sharp jump in disturbance wave disturbance velocity value was due to the fact that the convecting wavepacket encounters a sudden softening of the wall due to the presence of a CP surface. In addition, wavepacket structures begin to take into an oblique shape formation in figure 4.2(b) and wavepacket becomes more oblique with longitudinal structures in figure 4.2(c). This is likely caused by the stronger suppressing influence of the compliant panel on the 2D modes than the 3D (oblique wave) modes. 97 Chapter Investigations over two and three compliant panels Also, it is worth noting that the wavepacket disturbance velocity does not increase that much even as the wavepacket continues its journey up to the end of the second CP in figure 4.2(c) where the maximum disturbance velocity only reached ≈ 1.1%. Significantly, this seems to reflect the ability of CP to inhibit the growth of the wavepacket even at this nonlinear wave disturbance value. At this stage, the maximum displacement (not shown) of the CP is up to about 0.668% of the local displacement thickness (or about 0.2% of boundary layer thickness), which shows that the linearization assumption for coupled interaction is basically justified. The wavepacket’s maximum disturbance velocity increases to 1.26% at time T = 3906 (figure 4.2(d)) after it has just left the CP behind. This is significantly less than the 4% reached in the single CP case given in figure 3.5(c). Figure 4.2(e) shows the progessive evolution of the wave disturbances as they formed longitudinal structures with maximum disturbance velocity reaching 2.76% when T = 4650. This is past the point of the breakdown of the single CP case. While figures 4.2(f) signifies that wavepacket breakdown is imminient as the presence of a strong pair of forward jets with maximum disturbance velocity of 4.11% which are shifted away from the centre location. Also, figure 4.2(g) shows two pairs of jets, one close to the centre line while the second pair a bit far away from centre region. Eventually, figure 4.2(h) shows the breakdown of the wavepacket into fully turbulent state. Incipient turbulent spot occurred when the wavepacket was centred at the location X ≈ 2350. By comparing with the rigid wall (reference) case that earlier broke down into incipient turbulent spot at X ≈ 1410, transition 98 Chapter Investigations over two and three compliant panels distance to the occurrence of the incipient turbulent spot was increased further to about 89% relative to a rigid wall case when the second compliant panel was introduced. 4.1.1.2 Wavepacket spectral analyses The frequency (ω) vs. spanwise wavenumber (β) plots for when the wavepacket was riding upon the second CP located at X = 1359 to 1658 and until incipient turbulent spot is reached are shown in figure 4.3 for the streamwise (u) velocity, while that of the wall-vertical (v) velocity are shown in figure 4.4. Figures 4.3(a)–(b) show when the wavepacket is travelling directly over the second CP and it is difficult to discern the maximum peak locations for both the 2D and 3D wave modes especially in figure 4.3(b) as some modes are induced by the CP surface itself. Figure 4.3(c) shows the spectrum plot at a position X = 1726, that is, a short distance after the trailing edge (X = 1658) of the second CP. The spectrum shows the dominance of a pair of oblique waves at frequency of ω ≈ 0.025 (ωδ = 0.056) and β ≈ ±0.1 (βδ ≈ ±0.222). With close observation, there is no much disparity between figures 4.3(c) and (d) except for sideway proliferation of some near zero frequency modes. As the wavepacket evolves downstream, the spanwise wavenumber increases slightly and later decreased to β ≈ ±0.09 (βδ ≈ ±0.221) for the dominant wave modes in figure 4.3(e). Also the spectral peaks of the dominant wave pair begin to stretch towards lower frequency, and this trend continues for a while until it rises again to a value of ω ≈ 0.025 in figure 4.3(e). Same figure 4.3(e) is characterised by the 99 Chapter Investigations over two and three compliant panels sideway growth of other higher harmonics along the spanwise wavenumber β axis to signify that the evolving wavepacket already in the post-subharmonic phase. Figures 4.3(e)-(g) show when the wavepacket is about to reach the breakdown stage without any distinctly dominant 3D wave modes anymore. All the leading modes have near-zero frequencies at this stage. The breakdown of the wavepacket into incipient turbulent spot is shown in figure 4.3(h) which is marked with disturbance energy concentrating in the near-zero frequency. The frequency (ω) vs. spanwise wavenumber (β) for the wall-vertical (v) velocity in figure 4.4 also shed more lights on the spectra behaviours of the wavepacket over the second CP. Figure 4.4(b) shows clearly that the evolving wavepacket is directly on top of the second CP, which is due to the vertical displacement of the second CP surface. Not only these, the 2D modes are more pronounced than the 3D modes in figures 4.4(c)-(e). Finally, breakdown process into incipient turbulent spot is clearly revealed in figures 4.4(f)-(h). The corresponding streamwise wavenumber (α) vs. spanwise wavenumber (β) plots for u and v components are shown in figures 4.5 and 4.6 respectively. The spectra exhibits exactly the same type of spectral stretching except that the stretch is now towards zero streamwise wavenumber (representative of highly longitudinal structures). Three local peaks along the spectral ridges also appear even though they look not so distinct. Another observation is that dominant 2D modes streamwise wavenumber nearly remained the same in figures 4.5(b)-(e) and 4.6(b)-(e) as only the oblique waves undergo 100 Chapter Investigations over two and three compliant panels some changes. Figures 4.5(f) and 4.6(f) mark the breakdown stage of the wavepacket into incipient turbulent spot. Figure 4.7 shows the wavepacket growth rate for over the two-CP case and this is compared with the growth rates obtained previously in chapter for both the rigid wall (RW) and single CP cases respectively. First to mention that the two CP locations are far apart (with a separating distance of ∆X ≈ 597) from each other, and therefore the second CP has absolute no effect on the flow before it. Since flow properties before the second CP location remained the same as the single CP case in chapter 3, only the behavioural growth over the second CP will be discussed, as the comparison between the RW and single CP cases had already been discussed in section 3.3.2.3 with figure 3.12. Second CP at the location X = 1359 – 1658 is noted in suppressing the 2D wave modes amplitudes from growing further at stations X = 1553 – 2026, that is when the evolving wavepacket was riding over the second CP and even until after it had just left the second CP behind at X = 1726. Up to location X = 2026, the wavepacket amplitude value for the second CP case yet to attain what the single CP case reached at X = 1811. Also, a sudden drop in 2D wave amplitude value at X = 1726 (just after the end of the second CP) was not noted for the single CP case, as a gradual increment in 2D wave amplitude value are expected as X increases along the streamwise direction. The reduced growth of the wavepacket may clearly be attributed to the action of the second CP, which we are uncertain about the actual mechanism at work. It is likely the linear suppressing effect of wall compliance on TS wave growth is still 101 Chapter Investigations over two and three compliant panels at work, although at these amplitude levels nonlinearity may be very active as well. For the 3D wave modes, amplitude values keep increasing from X = 1726 2113 unlike the dominant 2D mode which drops in amplitude over the second CP before it continued to rise in amplitude after the panel. Also, even as the breakdown is imminient for the second CP case, the 3D mode amplitude value keeps increasing until X = 2113, that is, before a state of proliferations of many 3D wave modes began to set in. For all the three cases compared in figure 4.7, it is obvious that the inserted CPs indeed performed their expected duty of suppressing 2D wave modes from growing further, which in turns resulted into a better transition delay than their rigid wall case. These further confirm the efficacy of compliant panel(s) in delaying transition of the Blasius boundary layer provided they are suitably located within the boundary layer. 4.1.1.3 Spectral properties of dominant 2D and 3D wave modes Similar to what was done for the single CP case in chapter 3, extractions of spectral properties from the u-velocity spectral plots in figures 4.3 and 4.5 were also carried out for the dominant 2D and 3D wave modes respectively, that is, before and after the second CP located at X = 1359 to 1658. All extractions were carried out at the height ⁄ ( ) as before, and data obtained are presented in table 4.1 in terms of the local displacement thickness length scale ( ). All data obtained up to station X = 1035 remained the same as the single CP case in table 3.1 and no need to repeat them again under this section. 102 Chapter Investigations over two and three compliant panels X 1294 1991 0.090 0.285 0.083 0.264 0.192 1380 2056 0.091 0.238 0.083 0.217 0.193 1467 2120 0.072 0.205 0.078 0.199 0.199 1553 2181 0.070 0.154 0.070 0.173 0.211 1639 2241 0.067 0.132 0.067 0.158 0.217 1726 2300 0.064 0.209 0.060 0.162 0.222 1812 2356 0.061 0.214 0.059 0.082 0.225 1898 2411 0.063 0.219 0.058 0.084 0.231 1984 2465 0.069 0.224 0.060 0.086 0.217 2026 2491 0.070 0.226 0.060 0.087 0.219 2070 2518 0.071 0.229 0.061 0.088 0.221 2113 2544 0.071 0.231 0.061 0.089 0.224 Table 4.1 Over second compliant panel (CP) case: Spectral properties of u-velocity fluctuations of the dominant 2D and 3D modes. Data presented are based on the local displacement length scale δ(x). Preceding data before the station X = 1294 are the same with those in table 3.1. The plots of spectral properties of dominant 2D and 3D wave modes are plotted in figure 4.8. Figure 4.8(a) shows the overall behaviour of dominant twodimensional frequency wave modes before and after the second CP location. The presence of second CP further caused a decline on the 2D mode frequency values from at X = 1380 to at X = 1639. This decline continues even up to the station X = 1812, that is, after the evolving wavepacket had already exited the second CP location. Thereafter, 2D frequency began to 103 Chapter Investigations over two and three compliant panels increase in values again as the wavepacket finally evolved on the rigid wall this time around where the 2D frequency value almost remain fixed ( Figure 4.8(b) shows the associated frequency ratio ⁄ ). . This ratio suggested that the wavepacket contents contain 2D wave modes which operate at a higher frequency than their 3D wave modes especially after the wavepacket had already left the second CP location behind. Also, subharmonic resonant wave condition according to equation (3.10) does not appear all because: (1) the frequency ratio of obtained at the last station X = 2113 is far above the expected value of 0.5, (2) triad of waves locations are difficult to discern in the spectral plots of figure 4.3. This also suggests that the dynamics of the wavepackets as they passed through the single CP (in chapter 3) and the two-CP (in the present chapter) are somewhat differently. Wavenumber ratio ⁄ plot is shown in figure 4.8(c). Within the second CP region, 3D wave modes are characterized with higher streamwise wavenumbers than the 2D wave modes. However, the reverse is the case after the evolving wavepacket had already left the second CP region behind; with 2D wave modes streamwise wavenumber becoming larger than the 3D wave modes. The wavenumber ratio ended with around 0.38 which is below the expected value of 0.5 if the condition stated in equation (3.10) is to be referred. Figure 4.8(d) compares the downstream phase speeds ratio ⁄ of the dominant 3D and 2D modes of the wavepacket for over the second CP. Just after the second CP location, the x-phase speed for 3D wave modes started increasing from X = 1726 to a state where it finally ended to almost 2.2 times that of 2D wave modes at the 104 Chapter Investigations over two and three compliant panels 4.2.1 Earlier simulation trials with three compliant panels Three different simulation trials were first carried out to seek a better location for the third CP that will further extend or delay the onset of the incipient turbulent spot. The wall normal and spanwise dimensions of the computational domain remain fixed throughout, while the streamwise length is further increased to accommodate a delayed breakdown. The three trials carried out are described as follows: (a) Placing all the three CPs equidistant from one another Under the first trial, third CP was placed at the location X = 2255 – 2554 as shown in figure 4.9(a), that is, equidistant between the first CP and second CP, and also between the second CP and third CP. At the end of the simulations, wavepacket broke down into formation of turbulent spots, almost at the same location X ≈ 2350 (see figure 4.10) as the two-CP case of section 4.1. There is no significant improvement in terms of transition delay than what had already been obtained for over the two-CP case. The conclusion drawn from this first trial is that, third CP location of X = 2255 -2554 seems to be placed too far from where it could still be able to interact positively with the evolving wavepacket before it broke down to turbulence. The current location of the third CP is already near the breakdown region for the two-CP case, where the wavepacket has already attained a maximum disturbance velocity magnitude of ≈ 5%. The spectral plot of figure 4.3(f) also attested that this third panel location is already within the breakdown region for the two-CP case in section 4.1. 106 Chapter Investigations over two and three compliant panels (b) Shifting the third CP inward to the location X = 1955 - 2256 As the equidistant arrangement of three CPs in 4.9(a) did not work according to the desired expectation in terms of delaying transition further, another trial simulation was performed by shifting the third CP inward (towards the second CP) to a new location X = 1955 – 2256 as schematically shown in figure 4.9(b). Few selected results as the wavepacket heading towards incipient turbulent spots are shown in figure 4.11. By closely observing figure 4.11(c) at T = 5467, it shows that the evolving wavepacket broke down at X ≈ 2270 (wavepacket approximate centre location), which is slightly earlier than the two-CP case that broke down at location X ≈ 2350 as reported in section 4.1. (c) Shifting the third CP further inward to the location X = 1732 – 2032 The last attempt was made to shift the third CP further inward than before to a new location X = 1732 - 2032, that is, closer to the second CP location as shown in figure 4.9(c). Figure 4.12 shows selected results of the wavepacket heading towards breakdown as well. It shows that the wavepacket broke down at the location X ≈ 2470, that is, in between figures 4.12(c) and (d). It is interesting to note that incipient turbulent spots for all the three-CP cases assumed a highly streamwise streaky appear compared to the two-CP case in figure 4.2(h). The current three-CP case yields a slight increase in the breakdown location by approximately 6% when compared to the two-CP case (section 4.1) that broke down at location X ≈ 2350. In terms of comparing with the rigid wall (RW) case that broke down into incipient turbulent spot at the location X = 1410, this 107 Chapter Investigations over two and three compliant panels translates to approximately 100% increase in the transition distance measured from the point of wavepacket initiation location at X0 = 349.4. 4.2.2 Three compliant panels with separating distance reduced After the three-CP earlier simulation trials in which only the third case (with third CP at X = 1732 – 2032 produced a better (slightly) transition delay results as already discussed in section 4.2.1(c), a thought occur to us if transition delay could be served if all the three CP are moved upstream. This will allow CP to intervene in the stability of the wavepacket earlier and hopefully enhance its influence on the growth of the wavepacket. However, as the first CP is already close to the initiation source, we decided that we will leave it at its original location, and move the second CP and third CP forward to keep a separation distance between CP reduced from the previous CP separating distance of ΔX = 597 as it was for the two-CP case in section 4.1. Based on this idea, two different cases were carried out and they are: (i) reducing the separating distance between CPs to ΔX ≈ 225 with the three CPs at the locations X = 450 – 762, 983 – 1283 and 1506 – 1806, and (ii) reducing the distance between CPs to ΔX ≈ 300 with the three CPs at the locations X = 450 – 762, 1059 – 1358 and 1656 – 1957. Case (i) wavepacket broke down into incipient turbulent spot slightly earlier than the case (ii) wavepacket, as case (i) wavepacket is characterized with greater maximum uvelocity magnitudes at the various evolution time shown in table 4.2 especially as both wavepackets approached breakdown. 108 Chapter Investigations over two and three compliant panels For ∆X ≈ 225 For ∆X ≈ 300 Evolution Time (T) . . Minimum (u) velocity Maximum (u) velocity Minimum (u) velocity Maximum (u) velocity …. …. …. …. 5392 -0.0184 0.0144 -0.0159 0.0122 5764 -0.0345 0.0347 -0.0290 0.0191 6136 -0.0794 0.4433 -0.0526 0.2048 6359 -0.2084 0.9745 -0.2020 0.7049 Table 4.2 Over three compliant panels case: Comparing wavepacket velocity values for when distances between CPs are ∆X ≈ 225 and ∆X ≈ 300 as both wavepackets heading towards incipient of turbulent spot at selected times evolution T. With this, it shows that reducing the CPs distances further below ΔX ≈ 225 will not result into a better transition delay than the case (ii) transition distance as CPs mutual interference may set in and the overall wall response become complicated. Since case (ii) is better of than the case (i) transition distance, further analyses are focused on case (ii) wavepacket for the subsequent parts of this chapter. The schematic diagram for case (ii) is shown in figure 4.13 with a separating distance of ΔX ≈ 300 among the three CPs. 4.2.2.1 Wavepacket spatial evolution analyses Figure 4.14 shows the spatial evolution of the wavepacket up till the point where it broke down into incipient turbulent spots for the CP with ΔX ≈ 300. In order to avoid repetitions of some of the results already presented for the single CP case in section 3.3, only results from time T = 2046 onwards are shown in figure 4.14, as all other results before this time are the same as those obtained for the single CP 109 Chapter Investigations over two and three compliant panels case, due to the fact that the first CP location still remain unchanged. The corresponding perspective (3D) views of figure 4.14 are shown in figure 4.15. Figure 4.14(a) shows when the wavepacket started passing over the second CP which occupied the location X = 1059 – 1358. Before this location, evolving wavepacket had already left the first CP (at X = 450 – 762) region far behind and the wavepacket keeps growing as expected for the disturbance velocity. In figure 4.14(b), the wavepacket is completely on top of the second CP and it is expected to interact with the evolving wavepacket which is directly on top of it. The interactions resulted into a further decrease of wavepacket maximum velocity magnitude from 4.7% in figure 4.14(a) to 2.7% in figure 4.14(b). Not only this, some wave features which were not noticeable in figure 4.14(a) before began to surface in figure 4.14(b) both along and away from the centre of the wavepacket. In figure 4.14(c), the wavepacket is about to leave the second CP region which is characterized with something like “connecting rod” shapes away from the wavepacket center. While in figure 4.14(d) wavepacket had just exited the second CP region and proliferations of other wave modes began to surface as well, which result into higher disturbance velocity than what was obtained in figure 4.14(c). As the wavepacket is travelling on the rigid wall again, it continues to grow in disturbance velocity before it begins to pass over the third CP located at X = 1656 to 1957 as shown in figure 4.14(e). The shape of the wavepacket now looks like “chicken” feet with the two wave modes near the center location look a bit curve. The wavepacket is directly evolving on top of the third CP in figure 4.14(f). Two conspicuous diferences noticed in figure 4.14(f) are: the wavepacket maximum 110 Chapter Investigations over two and three compliant panels disturbance velocity almost double what was recorded in figure 4.14(e), where the reverse case is expected as the wavepacket is already on top of the third panel and its interactions with it supposed to decrease the wavepacket disturbance velocity. Also, the wavepacket shape is completely different from what was obtained in figure 4.14(e), also, figures 4.14(e)-(j) seems to show some kind of splitting of the wavepacket structures into two side by side groups. This splitting process could be clearly seen in the 3D plots of figures 4.15(f)-(i), where a kind of valley is created between two sets of mountain with different heights/peaks, which eventually culminated into two incipient spots of turbulence. To delve into greater details, as the wavepacket is about to reach the end of the third CP, wavepacket structures appear almost aligned along a straightline in figure 4.14(g) with more dominant 3D modes near the wavepacket centre. The wavepacket is about to leave the third CP in figure 4.14(h) and more waves spring up than the number of waves noticed in figure 4.14(g) as could be noted in figure 4.15(h). Figures 4.14(i–k) show the wavepacket in a strong non-linear regime with proliferation of waves spreading to the entire spanwise length as wavepacket disturbance velocity keep increasing. The waves formed comprised some pairs of forward streamwise streaks and some waves in the spanwise direction for each snapshot of the wavepacket evolution processes. The wavepacket grew stronger to the extent of occupying the whole spanwise size of the computational domain in each of the figures 4.14(i–k) shown. The kind of interaction between the third CP and the evolving wavepacket that led to these type of wave patterns (as can also 111 Chapter Investigations over two and three compliant panels be seen in figures 4.15(i–k)) with the wavepacket already far from the third CP location remain something to further investigate/analyze. The wavepacket disturbance velocity keeps increasing rapidly and this signifies the imminent breakdown as shown in figures 4.14(l) and (m), with the strongest wave peaks (see figures 4.15(l) and (m) also) occurred far both sides away from the centre location of the wavepacket. This trend was followed until the incipient turbulent spots occurred in figure 4.14(n). It could be seen that the breakdown location for this investigated case is X ≈ 2600. This is about 12% more transition delay than the two-CP case (section 4.1) that broke down at X ≈ 2350. In overall comparison with the rigid wall (RW) case breakdown location, this turns to approximately 112% increase in the transition distance measured from the point of wavepacket perturbation source location at X0 = 349.4. Figure 4.16 shows the CP surface displacements at different evolution times for the first and second CPs only under the three CPs arrangements. This is to show that there is no mutual interference between any of the two CPs as each CP performed/responded independently despite the distances between two CPs being reduced to ∆X ≈ 300. With wavepacket approaching the first CP in figures 4.16(a) and (b), second CP appears yet to respond to the incoming wavepacket at times T = 186 and 260 respectively. In addition, as the wavepacket is directly evolving on top of the first CP (figures 4.16(c) and (d)) and second CP (figures 4.16(g) and (h)), adjacent CP surface reflections are minimal pointing that there is no mutual interference between the first and second CPs. Same scenario (not shown) could be observed between the second and third CPs as well. 112 Chapter Investigations over two and three compliant panels The surface maximum vertical displacements (η) for the three compliant panels are shown in figure 4.17 and this shows how each of the CP interacts with the evolving wavepacket. Maximum displacement is least at the first CP (thereby fulfilling the initial sufficiently small displacement assumption) and increases sharply towards the third CP. The maximum displacement of the third CP is ≈ 0.028 with the highest vertical displacement experienced by the third CP surface as the wavepacket directly on top of it, wavepacket became more excited and this cause rapid proliferation of many wave modes with different orientations as noticed in figures 4.14(i)–(k) and 4.15(i)–(k). 4.2.2.2 Wavepacket spectral analyses The spanwise wavenumber (β) vs. frequency (ω) spectra for both the streamwise (u) velocity and wall-normal (v) velocity are shown in figure 4.18 and while that of the streamwise wavenumber (α) vs. spanwise wavenumber (β) spectra plots are as well shown in figure 4.19. In figure 4.17(a1) (at X = 1035), the wavepacket not yet reached the leading (X = 1059) edge of the second CP, and that accounts for the appearance of the distinct 2D and 3D wave modes with frequencies in the range of 0.045 – 0.055 similar to what is obtained before for the single CP case. Figures 4.18(b) and (c) show when the wavepacket is completely travelling on top of the second CP (located at X = 1059 – 1358), and not surprisingly the spectra at these locations did not produce distinct 2D and 3D modes wave peaks. Figures 4.18(b2) and (c2) further confirmed the strong 113 Chapter Investigations over two and three compliant panels interactions between the evolving wavepacket and the second CP as the v-velocity spectra shows how the second CP reacted strongly in the wall-normal direction. Figures 4.18(d) and (e) comprise of distinct 2D and 3D wave modes as the wavepacket already left the second CP location but yet to come on to the third CP at X = 1656 – 1957. Frequency values for both 2D and 3D modes dropped to about 0.035 in figures 4.18(d) and (e), lower than the value of almost 0.050 obtained in figure 4.18(a1) before. In figure 4.18(f), the wavepacket is already on top of the third CP which is characterized with strong 2D modes operating at a lower frequency than before. At this same location X = 1811, the evolving wavepacket is observed undergoing higher vertical displacement than the first two CPs as already shown in figure 4.17. The reason for this is that, wavepacket had already grown in both size and disturbance velocity by the time it reached the third CP location than what it used to be when passing over the first-two CP locations and the effect of this caused the third CP surface to displace higher than the first-two CP. Figures 4.18(g)-(i) show the proliferation of near zero frequency of 3D wave modes signaling the wavepacket is on its route to breakdown stage. Figure 4.19 (a) further confirmed the behaviour of the wavepacket when directly on top of the second CP, as it is difficult to discern the dominant 2D modes in figure 4.19(a1) and 3D modes in figure 4.19(a2). The effect of the second CP is still felt in figure 4.19(b) as the wavepacket almost exited the second CP region. Figures 4.19(c) and (d) shows distinct 3D modes for both the u and v velocity spectra respectively as the wavepacket travelled on the rigid wall which is before reaching the third 114 Chapter Investigations over two and three compliant panels CP location. Only sharp 2D modes could be seen in figures 4.19(e) and (f) even though the wavepacket is travelling on top of the third CP at those evolution times. This shows that the 2D wave mode is the dominant mode here. This trend continues in figures 4.19(g) and (h) until breakdown stage is reached in figure 4.19(i). 115 Chapter Investigations over two and three compliant panels 4.2.2.3 Spectral properties of dominant 2D and 3D wave modes Similar attempts were also made to extract the spectral properties for the dominant 2D and 3D wave modes from the u-velocity spectral plots in figures 4.17 and 4.18 respectively. Data obtained are summarized in table 4.3. X 949 1705 0.091 0.239 0.077 0.219 0.216 1035 1781 0.088 0.250 0.079 0.219 0.219 1122 1854 0.090 0.260 0.082 0.217 0.224 1208 1924 0.084 0.208 0.084 0.190 0.223 1294 1991 0.083 0.167 0.073 0.189 0.231 1380 2056 0.074 0.141 0.072 0.195 0.238 1467 2120 0.074 0.174 0.072 0.199 0.242 1553 2181 0.067 0.211 0.074 0.202 0.249 1639 2241 0.069 0.208 0.074 0.195 0.256 1726 2300 0.069 0.156 0.069 0.189 0.249 1812 2356 0.059 0.118 0.066 0.162 0.253 1898 2411 0.065 0.117 0.063 0.117 0.254 1984 2465 0.060 0.148 0.064 0.110 0.238 2026 2491 0.058 0.169 0.063 0.104 0.241 2070 2518 0.056 0.170 0.063 0.097 0.221 Table 4.3 Over three compliant panels (CP) case with separating distance between CP: ∆X = 300. Spectral properties of u-velocity fluctuations of the dominant 2D and 3D modes. Data presented are based on the local displacement length scale δ(x) and just before second panel downstream. Preceding data before the station X = 949 are the same with those in table 3.1. 116 Chapter Investigations over two and three compliant panels It is worthy to note that: (1) the first CP location (450 – 762) remain the same as the single CP case in chapter 3, therefore, table 4.3 only covers data from just before (where differences began to emanate) the second CP location downwards, (2) as a result of these, no need to include those spectral properties data for over the first CP in table 4.3 again. The spectral properties of dominant 2D and 3D wave modes based on table 4.3 for over the three CPs case with separating distance between panels reduced to ∆X = 300 is shown in figure 4.20. Figure 4.20(a) shows the 2D mode frequency behaviour which keeps declining in value as the wavepacket convects through the second and third CPs location respectively. This is completely different from the profile obtained in figure 4.8(a) where 2D mode frequency began to rise again ⁄ after the second CP location. Also, the frequency ratio plot is shown in figure 4.20(b) with the 2D wave modes frequency appearing higher in the vicinity of the second CP than the 3D wave mode frequency. On the third CP side, 3D wave modes contain higher frequency values than that of their 2D wave modes. Figure 4.20(c) shows the wavenumber ratio ⁄ with responses from both the second and third CP resulted into two spikes showing 3D modes having larger streamwise wavenumber than the 2D modes around the second and third CPs locations. It is quite interesting to note that the wavenumber ratio almost reached the value of 0.5 at the last station of X = 2070 at least in fulfillment of condition (3.10). Figure 4.20(d) shows the x-phase speed ratio ⁄ of the dominant 2D and 3D modes. The 3D modes phase speed became larger than the 2D wave modes right from the moment the evolving wavepacket tried to exit the third CP 117 Chapter Investigations over two and three compliant panels location. Squire wavenumber condition in equation (3.11) is far from fulfillment as most of the data points in figure 4.20(e) are far above the expected value of 1.0. Even the last data point at X = 2070 only ended at about 1.4. Finally, figure 4.20(f) shows that the propagation angles θ of the dominant 3D modes reached almost 66o as the wavepacket developed through the subharmonic stage. Figure 4.21 shows the wavepacket maximum wave growths for the three-CP case with the first CP at X = 450 to 762, second CP at X = 1059 to 1358 and third CP occupied X = 1656 to 1957 respectively. In figure 4.21, first CP suppressed the 2D wave modes from growing even up to the location X = 776. Then just after the first CP, the wavepacket 2D wave modes tried to grow again as it continues it evolutions now on the rigid wall section from X = 863 to X = 1035. However on reaching the second CP location, the interactions with the second CP caused the wavepacket 2D wave modes to start decreasing again in growth until X = 1380. This shows that second CP is really performing its expected task of 2D modes suppressions, and this also confirms that the second CP location is considered appropriate as well. As usual, the 2D wave modes began to grow in amplitudes again from X = 1467 to 1636 as the wavepacket continued its evolution over the rigid section again after exiting the second CP location. However, something opposite occurred when the wavepacket was evolving over the third CP located at X = 1656 to 1957. Instead of 2D wave modes being suppressed further after interacting with the third CP similar to what first and second CPs did to the evolving incoming wavepacket, the 2D mode amplitudes rather jumped sharply to the tune of 275% increase from 118 Chapter Investigations over two and three compliant panels X = 1726 to when the wavepacket was evolving over the third CP at X = 1812 – 1984 as shown in figure 4.21. This kind of strange and opposite response by the third CP to the incoming wavepacket definitely triggered the formation of those wavepacket unique patterns/structures already shown in figures 4.14(h-k) and 4.15(h-k). For over the third compliant panel, the disturbance waves are predominantly highly oblique waves (propagating at significantly large oblique angles). Isotropic compliant panels (such as the ones studied here with CP tensions in both streamwise (X) and spanwise (Z) directions being equal) are not that effective in suppressing the 2D wave modes according to Yeo (1992), because the effective freestream flow speeds for these waves are low so that the third CP is effective much stiffer (tending to rigid behaviour) for these oblique waves. Finally, the 2D mode amplitudes began to drop after the wavepacket had exited the third CP location. 4.3 Summary The transition delay already achieved with the single CP case in chapter provided the impetus to extend the study to over both two CPs and three CPs cases respectively. For the two-CP case, second CP was added at X = 1359 to 1658 to form array of two CPs with the aim of delaying transition further. The primary purpose of the second added CP is to inhibit the rapid growth of the wavepacket from that point onwards. Spatial evolution results show the wavepacket became more oblique in shape as it passed over the second CP region. 119 Chapter Investigations over two and three compliant panels With the two-CP case, transition distance was further increased by almost 89% when compared with the RW case. Spectral analyses results show frequency of the evolving wavepacket being dropped to almost half of its values just after exiting the second CP location for both the dominant 2D and 3D wave modes, and this scenario continued like that until modes with near-zero frequencies began to proliferate sideways. This noted observation may not be too far from the way wavepacket suddenly became oblique in shape while riding over the second CP. In addition, the growth curve shows the suppression of 2D wave modes by the second CP as well, and that explain why two-CP case was able to delay transition farther than the single CP case. However, most of the spectral properties results over the second CP failed to match the predictions of subharmonic resonant wave condition (3.10). For the three CPs case, in an attempt to know how the three CPs should be fairly arranged for much better transition delay performance, three simulation trials were first carried out and these are: (i) trying to make the three CPs equidistant from one another, and the results obtained shows a situation where the wavepacket broke down almost in the same location as that of the two-CP case. This signifies that the third CP was placed too far from where it could still make some significant contributions. (ii) An attempt was made to shift the third CP inward towards the second CP to the location X = 1955 to 2256, and with this wavepacket broke down slightly earlier than over the two-CP case, and (iii) shifting the third CP further inwards to the location X = 1732 to 2032 produced a 120 Chapter Investigations over two and three compliant panels slight transition delay further as the wavepacket now broke down at X = 2470, at least better than breakdown location of X = 2350 for the two-CP case. Lastly, reducing the gap between CPs to ΔX ≈ 300 for the three CPs case from the previous ΔX ≈ 597 used for the two-CP case yielded much transition delay than before. With this last arrangement, transition distance was further increased to the tune of almost 112% when compared with the RW case. For the spectral analyses part, 2D mode frequencies kept decreasing over the three-CP case unlike over the two-CP case that drop and continue to rise again, while other spectral properties totally deviated from both the conditions (3.10) and (3.11). From the amplitude growth curve, first and second CPs kept suppressing the 2D wave modes from growing further. However, interaction of wavepacket with the third CP behaved contrary to the expectation of 2D wave mode suppression, as 2D mode amplitudes increased rather sharply as the wavepacket was evolving over the third CP. The mechanism behind this may have to with the highly oblique nature of the wavepacket for over the third CP. With this, the third CP tends to be rigid and suppression of 2D wave modes not likely again according to Yeo (1992). 121 [...]... breakdown 108 Chapter 4 Investigations over two and three compliant panels For ∆X ≈ 225 For ∆X ≈ 300 Evolution Time (T) Minimum (u) velocity Maximum (u) velocity Minimum (u) velocity Maximum (u) velocity … … … … 5392 -0.01 84 0.0 144 -0.0159 0.0122 57 64 -0.0 345 0.0 347 -0.0290 0.0191 6136 -0.07 94 0 .44 33 -0.0526 0.2 048 6359 -0.20 84 0.9 745 -0.2020 0.7 049 Table 4. 2 Over three compliant panels case: Comparing... 18 54 0.090 0.260 0.082 0.217 0.2 24 1208 19 24 0.0 84 0.208 0.0 84 0.190 0.223 12 94 1991 0.083 0.167 0.073 0.189 0.231 1380 2056 0.0 74 0. 141 0.072 0.195 0.238 146 7 2120 0.0 74 0.1 74 0.072 0.199 0. 242 1553 2181 0.067 0.211 0.0 74 0.202 0. 249 1639 2 241 0.069 0.208 0.0 74 0.195 0.256 1726 2300 0.069 0.156 0.069 0.189 0. 249 1812 2356 0.059 0.118 0.066 0.162 0.253 1898 241 1 0.065 0.117 0.063 0.117 0.2 54 19 84 246 5... 2 046 onwards are shown in figure 4. 14, as all other results before this time are the same as those obtained for the single CP 109 Chapter 4 Investigations over two and three compliant panels case, due to the fact that the first CP location still remain unchanged The corresponding perspective (3D) views of figure 4. 14 are shown in figure 4. 15 Figure 4. 14( a) shows when the wavepacket started passing over. .. (section 4. 1) that broke down at location X ≈ 2350 In terms of comparing with the rigid wall (RW) case that broke down into incipient turbulent spot at the location X = 141 0, this 107 Chapter 4 Investigations over two and three compliant panels translates to approximately 100% increase in the transition distance measured from the point of wavepacket initiation location at X0 = 349 .4 4.2.2 Three compliant panels. .. dominant 3D modes for over the second CP case is shown in figure 4. 8(f) Θ value almost reached 70o at X = 2113 as the wavepacket evolves towards breakdown 4. 2 Over an array of three compliant panels Having obtained transition delays of about 49 % and 89% with the single CP and two-CP cases in sections 3.3 and 4. 1 respectivley, further motivation arose to see if a third panel could be inserted and if that could... 1358), and not surprisingly the spectra at these locations did not produce distinct 2D and 3D modes wave peaks Figures 4. 18(b2) and (c2) further confirmed the strong 113 Chapter 4 Investigations over two and three compliant panels interactions between the evolving wavepacket and the second CP as the v-velocity spectra shows how the second CP reacted strongly in the wall-normal direction Figures 4. 18(d) and. .. third 1 14 Chapter 4 Investigations over two and three compliant panels CP location Only sharp 2D modes could be seen in figures 4. 19(e) and (f) even though the wavepacket is travelling on top of the third CP at those evolution times This shows that the 2D wave mode is the dominant mode here This trend continues in figures 4. 19(g) and (h) until breakdown stage is reached in figure 4. 19(i) 115 Chapter 4 Investigations... figure 4. 19(i) 115 Chapter 4 Investigations over two and three compliant panels 4. 2.2.3 Spectral properties of dominant 2D and 3D wave modes Similar attempts were also made to extract the spectral properties for the dominant 2D and 3D wave modes from the u-velocity spectral plots in figures 4. 17 and 4. 18 respectively Data obtained are summarized in table 4. 3 X 949 1705 0.091 0.239 0.077 0.219 0.216 1035... evolving on top of the third CP in figure 4. 14( f) Two conspicuous diferences noticed in figure 4. 14( f) are: the wavepacket maximum 110 Chapter 4 Investigations over two and three compliant panels disturbance velocity almost double what was recorded in figure 4. 14( e), where the reverse case is expected as the wavepacket is already on top of the third panel and its interactions with it supposed to decrease... 2.7% in figure 4. 14( b) Not only this, some wave features which were not noticeable in figure 4. 14( a) before began to surface in figure 4. 14( b) both along and away from the centre of the wavepacket In figure 4. 14( c), the wavepacket is about to leave the second CP region which is characterized with something like “connecting rod” shapes away from the wavepacket center While in figure 4. 14( d) wavepacket . -0.01 84 0.0 144 -0.0159 0.0122 57 64 -0.0 345 0.0 347 -0.0290 0.0191 6136 -0.07 94 0 .44 33 -0.0526 0.2 048 6359 -0.20 84 0.9 745 -0.2020 0.7 049 Table 4. 2 Over three compliant panels. same in figures 4. 5(b)-(e) and 4. 6(b)-(e) as only the oblique waves undergo Chapter 4 Investigations over two and three compliant panels 101 some changes. Figures 4. 5(f) and 4. 6(f) mark the. Chapter 4 Investigations over two and three compliant panels 95 Chapter 4 Investigations over two and three compliant panels An attempt made in chapter 3 with single compliant panel