Flow stability and transition over finite compliant panels 5

Chapter Compliant panel properties Chapter Compliant panel properties Zhao (2006) had selected a set of properties for the membrane which had proven (computationally) in the preceding study to give fair good transition delay properties. The choice was partially explained in Appendix A of her thesis. In this chapter we will examine this choice more carefully. Wavepacket initiations, evolutions and breakdowns into incipient turbulent spot over single compliant panel (CP) and multiple compliant panels had been studied in chapters and respectively. In those chapters, transitions were delayed from about 49% to 112% when compared against the rigid wall (RW) case where the same initiating pulse perturbations were used in all the cases. Same compliant panel properties and length considered by Zhao previously were used in both chapters and 4, which prevented both the static divergence (SD) and travelling wave flutter (TWF) from developing and dominating the transition process. The finite and relatively short lengths of the panels played a part in the suppression of those wall-motivated modes. However, in the course of ruminating and critically reviewing over what could be termed a brief study on membrane properties by Zhao (2006), some shortcomings were noted, which could possibly be improved upon so as to make the compliant panel(s) perform better than what had been achieved so far, and also to perform more detailed analyses for better insights into the performance of CP while tuning different CP properties. Some of the noted limitations of Zhao (2006) include: 149 Chapter Compliant panel properties (1) The use of what appeared to be an unnecessarily stiff membrane, with a high estimated onset velocity for travelling wave flutter (TWF) of 3.16 , whereas a factor between say 1.1 to 2.0 might appear to be adequate against these wallbased modes (more description of wall modes is given in the next section 5.1). (2) Zhao (2006) studied how compliant wall properties affected the interactions between the membrane panel and the flow in a reduced domain that extends to X ≈ 910 and time T = 1300. How variations in the panel properties might bear on the all-important question of transition delay was not addressed. (3) Due to the point raised in (2) above, all simulations later carried out on membrane property study by Zhao (2006) were trimmed down to half sizes of the initial computational domains along the streamwise directions. Observing the overall behaviours of the evolving wavepacket in terms of both spatial and spectral analyses far downstream after the CP region, until incipient turbulent spot is reached will provide more detailed information than what was done by Zhao. (4) Moreover, the simulations involved in the property studies were carried out with fairly coarse mesh of 47 grid points in Y (wall-normal) and 81 grid points in the Z (spanwise) directions. In this chapter, we hope to address some of these limitations as we delve into the effects of some of the leading properties of the compliant panel on transition delay. 150 Chapter Compliant panel properties 5.1 Compliant panel surface waves and instabilities According to Gad-el-Hak (1998), a rich variety of fluid-structure interactions exists when a fluid flow over a surface that can comply with the flow. Instability modes amplify when two wave-bearing media are coupled together. These waves could be flow-based, wall-based or due to coalescence of both kind of waves. So far, two common types of surface waves have been identified, which are related to the flexible quality of the wall involved. The two types are static standing wave, which is also known as static divergence (SD) waves, and travelling waves also called travelling wave flutter (TWF). The presence of surface waves on compliant panels subjected to either laminar or turbulent flow, could generally be seen as arising from an instability of the flow-wall dynamical system in coupled interaction according to Yeo et al. (1999). Two types of instabilities that are directly related to the compliant quality of the wall could be collectively grouped into compliance-induced flow instabilities (CIFI) according to Yeo (1988) and flow-induced surface instabilities (FISI) as termed by Carpenter and Gerrad (1985, 1986). The TWF instabilities are related to the free surface wave modes of the compliant wall, and assume the form of a travelling wave propagating at a fair speed. The TWF instability mechanism involves the irreversible transfer of energy from the flow to the wall owing to the work done by the fluctuating pressure according to Lee et al (1995). TWF is the cause of very sudden onset of transition. The SD instability is related to the static deformation modes of the compliant wall, and is manifested as slowly moving wave. 151 Chapter Compliant panel properties SD instability occurs when the hydrodynamic pressure forces generated by a small disturbance outweigh the restorative structural forces in the wall. Just to mention few from the literature, Hansen et al. (1983) observed SD waves on viscoelastic layers with a meaningful high level of damping in their rotating-disk experiments. Same SD waves were also observed by Gad-el-Hak et al. (1984) in their uni-directional flow experiments. Later on, a review of SD waves on compliant surfaces was given by Riley et al. (1988). On the other hand, that is for the TWF waves, Lucey and Carpenter (1995) validated the results obtained from the theory of wall-based travelling wave flutter with the experimental results. 5.2 Estimates for the onset speeds of divergence and travelling wave flutter According to Carpenter (1998), in order to achieve best possible transition delay with either single or multiple panel walls, the essential concept underlying the optimization procedure is to use estimates (which was based on potential flow assumption) for the onset speeds of divergence and travelling wave flutter in order to choose the properties of a set of compliant walls each of which corresponds to marginal wall-based stability at the design flow speed with respect to both of these hydroelastic instabilities. Modified potential flow theory (see Duncan et al. (1985) suggests that wall-based modes might be inhibited if we have * where √ and + ( ) (5.1) √ 152 Chapter Compliant panel properties where are the onset flow velocities for TWF and SD modes respectively. for zero-pressure gradient laminar boundary layer following Duncan et al. (1985). These simple criteria are useful, though they may not be valid in the context of a viscous boundary layer. For application to viscous boundary layer, we may further modify these criteria by the incorporation of factors of safety C and D, that is, { } (5.2) The factors C and D are the ratios of the onset velocities for Hydroelastic instability (TWF and SD respectively) relative to the actual free stream flow speed . The higher C and D are, the less likely will TWF and SD waves be triggered. Hence they are termed factors of safety. We note that in both of criteria (5.1) and (5.2), if is allowed to tends to zero. Figure 5.1(a) shows schematically the unstable regions associated with the TWF and SD for . The criteria are marginal with respect to both TWF and SD at point B. At this point as shown in figure 5.1(b): √ where √ (5.3) marks the wave number below which there will be no TWF and SD instabilities. The range of streamwise wavenumber for wall waves can be controlled (to some extent at least) by limiting the length of the compliant panel (wavelength of wave mode), since long waves (small ) 153 Chapter Compliant panel properties cannot be sustained on a short panel. In this regard if we set , then the criterion (5.2) reduces to: ( ) { √ } (5.4) where (5.5) Hence by selecting a compliant panel length , equations (5.4) and (5.5) allow us to estimate the mass density (m) and tension (T) of the compliant panel for a prescribed factors of safety in accordance with the formula (5.5) and from (5.4). In the earlier study of chapter 4, translate to base on the wall reference length scale of (which ). In this interpretation of how wall-based modes may be suppressed, Zhao (2006) had employed a high factors of safety of C = 3.16 against TWF and a more reasonable D = 1.32 against SD. The factor C = 3.16 would appear to be an unnecessary high value to adopt for the compliant membranes. Whether it is indeed very high remains to be seen below; as we have to recognize that these stability estimates (5.1 to 5.5) were (i) based on a highly simplified potential flow model, and (ii) there might be complex edge effects in finite-length panels that could result in unstable wave interaction within the panels. 154 Chapter Compliant panel properties 5.3 Compliant panel properties parametric study: Cases investigated After establishing the formulae (5.4) and (5.5) for the estimation of compliant panel mass density (m) and tension (T) in section 5.2, further attempts were made to investigate the effects of prescribing lower safety factors C and D (as safety factors C and D control how close the walls are to the critical SD and TWF velocities) on transition delays for both the single CP and two-CP cases. Six different parametric study cases were investigated, and these were compared with the results earlier obtained in chapters and termed “reference” case in this chapter and CP properties are summarized in table 5.1 after applying equations (5.4) and (5.5). Cases and may be regarded as a subset in which only the damping coefficient is varied. Safety Factors Compliant panel parameters Case C D Reference 3.16 1.32 1.00 10.00 0.10 0.00 580 1.30 1.30 5.74 9.70 0.10 0.00 580 2.50 1.20 1.32 8.25 0.10 0.00 580 2.80 1.30 1.24 9.70 0.10 0.00 580 2.80 1.05 0.81 6.35 0.10 0.00 580 2.80 1.05 0.81 6.35 0.05 0.00 580 2.80 1.05 0.81 6.35 0.01 0.00 580 Table 5.1 Compliant panel parameters of cases 1-6, where mL represents the surface mass density; TL is the compliant panel tension; dL is the damping factor; kL is the foundation spring stiffness, and ReL as the wall reference Reynolds number. Reference case is for the CP parameters/properties used in chapters and simulations. Compliant panel length . 155 Chapter Compliant panel properties The subscript L in table 5.1 represents a wall length scale which was used to specify material properties for the wall. The relation between the wall properties, subscripted by L, and their computational equivalents are given by: (5.6) where (5.7) and superscript asterisk * signifies that the quantity is dimensional. The simulation conditions for all the cases studied in table 5.1 are identical to those of the reference case, which had already been presented in chapter of this thesis. The location of the embedded single compliant panel (CP) remain unchanged, that is, at same location X = 450 – 762. Also, second CP location at X = 1359 – 1658 remains the same for two-CP simulations. In addition, apart from further tuning the CP properties, some other distinct differences between what Zhao (2006) did and the approaches applied in this present study are: (1) a much larger computational domain was used here to allow the wavepacket to reach the breakdown stages for each of the cases investigated, as Zhao’s domain length was much shorter and did not permit breakdown to occur, (2) the grid resolution was greatly increased in both the spanwise (Z) and wall-vertical (Y) lengths of the computational domain over the previous study to ensure that fine details of flow are well resolved and captured in simulations. 156 Chapter Compliant panel properties 5.4 Results and discussions for over the single CP case Simulation results are presented systematically and chronologically in the order of as indicated in table 5.1. Cases with unfavourable outcomes (that is, early breakdown locations than the reference case) are summarily discussed while the favourable ones are presented in more details. 5.4.1 Spatial evolution analyses First to recall again that the breakdown location for over the single CP in chapter (reference case here) is X ≈ 1930. This was used to compare all the single CP cases – investigated, that is, to know if they performed better or not in terms of further transition delay than the reference case. In order to make presentation of results straight forward and without any confusion whatsoever, results are divided into two main groups namely: (i) cases that broke down earlier than the reference case and (ii) cases that performed better than the reference case. 5.4.1.1 Cases that broke down earlier than the reference case Cases 1, 2, and wavepacket broke down at X ≈ 1770, 1750, 1680 and 1910 respectively as shown in figure 5.2 for the u-velocity components, that is, earlier before the X ≈ 1930 breakdown location for the reference case in chapter 3. Case (figure 5.2(c)) suffers the earliest breakdown among the four in this group, with its breakdown location centers around X ≈ 1650, this has to with the facts that the CP is of low surface mass density (mL) and tension (T) relative to the reference case. On the other hand, case in figure 5.2(d) performs the best in this 157 Chapter Compliant panel properties group with breakdown close to X ≈ 1910. What made case different from case is the foundation damping dL, which is half the value for case 4. This shows that CP damping has a significant effect on the transition, and that the key phenomenon underlying the growth process is class A in nature. With this at the back of the mind, first attempt was made to set safety factor C = 1.3, that is, to something closer to the value of 1. With this safety factor C set-up already, surface mass density (mL) jumped to almost six times in case to that of reference case as shown in table 5.1 after applying equation (5.5). Parameter D values are almost the same for both the reference case and case 1. Since it had already been confirmed in Case that C value very close to did not result into a better transition delay, then, second attempt was made to raise the C value to 2.5 in case 2, with the aim of reducing mL value closely to at the end of the whole process. Still, case broke down earlier than the reference case as shown in figure 5.2(b) and this may be attributed to a decrease in CP’s tension value to T = 8.25. Above all, none of the cases in figure 5.2 performed much better than the reference case; however, investigating them gave a clue on how the CP properties could be further and carefully tuned in order to achieve the goal of delaying transition beyond the location of X ≈ 1930 recorded for the single CP reference case. Finally, since the results obtained under this group did not show gain in transition delay, we did not go into the further analyses of these cases. 158 Chapter Compliant panel properties 5.4.1.2 Cases that performed better than the reference case Cases and performed slightly better than that of the reference case in terms of further transition delay and their spatial evolution comparison for all the three cases is shown in figure 5.3. The wavepacket evolution processes before figure 5.3(a), that is, before CP location (X = 450 to 762) is the same for all the three cases and we did not include them in figure 5.3. Case was an attempt to take full advantage of CP with low damping. Since the wavepacket formed for the reference case is almost identical to those for case 3, more discussions in this section will be focused on case and only. In figure 5.3(a), wavepackets are fully convecting on top of the CPs, with the wavepacket maximum disturbance velocity and wavepacket shapes for both reference case (figure 5.3(a1)) and case (figure 5.3(a2)) are almost identical. While that of the case in figure 5.3(a3) possesses more elongated streamwise oriented structure along the wavepacket center with a maximum velocity of 0.54%. Maximum disturbance velocity for case in figure 5.3(b3) is 0.35%, while that of case in figure 5.3(b2) is 0.48%, this serves as an indicator that case which is characterized by soft wall coupled with low damping may delay transition better than case 3. In figure 5.3(c), all the three wavepackets had just exited the CP locations completely and where the full effects of the compliant panel on the wavepacket could be most clearly discerned. There is almost 50% drop in maximum disturbance velocity for case in figure 5.3(c3), compared to case in figure 5.3(c2). Wavepacket disturbance velocity for case became noticeably larger than the reference case in figure 5.3(d), which looks a bit contradictory as 159 Chapter Compliant panel properties we expect case to still maintain lower disturbance velocity than the reference case at T = 2788. However in figure 5.3(e) at T = 3532, the wavepacket behaviours changed, as Case maximum disturbance velocity (1.98%) suddenly towered over those of the reference case (1.8%) and the case (1.69%). At time T = 3906, clearer picture of which wavepacket will breakdown first before the other ones began to manifest in figure 5.3(f). Both cases and possess lower positive velocity values almost half that of the reference case. In overall comparison, both cases (3.4%) and (3.9%) have lower maximum disturbance velocities than the reference case (4.1%). Figure 5.3(g) shows that the wavepacket are heading towards the breakdown stages. From the maximum disturbance velocity values, it shows clearly that the reference case (with 78.15% disturbance value) in figure 5.3(g1) will breakdown faster, then followed by the case (with 41.63% disturbance value) in figure 5.3(g2) and lastly by the case (with 16.17% disturbance value) as shown in figure 5.3(g3). The breakdown locations for both case and are shown in figures 5.4 and 5.5 respectively. In figure 5.4, case finally breakdown into incipient turbulent spot around when the wavepacket center location is at X ≈ 1980. By comparing with the reference case which breakdown at X ≈ 1930, this means transition distance is delayed to the tone of ≈ 3.2% farther by taking the perturbation source location at X0 = 349.4 into consideration. While for case 6, incipient turbulent spot occurred around when X ≈ 2050 and this translates to about ≈ 7.6% further transition delay when compared with the reference case that broke down at X ≈ 1930. 160 Chapter Compliant panel properties The CPs experienced maximum surface displacements at time T = 930 as the wavepacket passes over the panels and the center-line surface displacement behaviours at this time are shown in figure 5.6 for all the three cases. The corresponding displacement foot prints of the wavepacket on the panel for both case and are shown in figure 5.7. Since the CPs properties for both reference case and case are almost identical, their maximum displacements behaved closely as well. However, by comparing the maximum displacements behaviours of case and 6, case is seen to have a lower maximum surface displacement (η) than case 3, that is, about half the value of case . This may be due to the softness nature of case coupled with low damping (dL) and slightly lower surface mass density (mL). The improvement in transition delay of case or further reinforces that reducing damping is positive for transition delay. 5.4.2 Spectral analyses Through spectral analyses, more intrinsic wavepacket details and dynamics are revealed for the entire evolution process until breakdown stage is reached. Frequency (ω) vs. spanwise wavenumber (β) spectra plots of streamwise velocity (u) for both cases and are compared in figures 5.8 respectively. In figure 5.8(a), both wavepacket of cases and are on top of the CP and that accounts for the present complex structures without distinct and dominant 2D wave mode formations, due to the interactions between the CPs and the evolving wavepacket at the location X = 604. At X = 949 in figure 5.8(b), wavepacket already exited the CP region (X = 450 to 762) and now continue to travel over section that is 161 Chapter Compliant panel properties completely rigid wall. In figure 5.8(b1), distinct 2D and 3D modes are now noticeable for case 3, whereas for case in figure 5.8(b2), it is still difficult to discern any 2D modes. The u-spectral plot in figure 5.8(b2) shows that case wavepacket looks a bit oblique in nature than the case as was already evident in figure 5.3(c). Figures 5.8(c1) and (d1) show u-velocity spectral plots for case at X = 1122 and 1294, and which look similar to figure 5.8(b1), but now with the spanwise wavenumber (β) value around β ≈ ±0.12 and the dominant 3D wave modes frequency dropped below ω ≈ 0.05 as the wavepacket evolve into early subharmonic stage. For case 6, even though dominant 3D wave modes could be seen clearly in figures 5.8(c2) and (d2), some smaller wave modes were seen formed around where dominant 2D (β = 0) modes are supposed to form, and these features still repeat themselves in figures 5.8(e2) and (f2). All these may be due to the soft wall (plus very low damping) nature for case 6. Proliferations of many 3D wave modes are noted for case in figure 5.8(g1) during the wavepacket late subharmonic stage as breakdown is imminent, but this is not the same for case in figure 5.8(g2). At X = 1897, it is seen that case already approaches the breakdown stage, which is characterized by many near-zero frequency wave modes, as shown in figure 5.8(h1). However, these features are not yet seen for case in figure 5.8(h2). Some selected cases of streamwise wavenumber (α) vs. spanwise wavenumber (β) spectral plots for both case and are shown in figure 5.9 for the u velocity, which are derived from the analysis of a rectangular region at 162 ⁄ Chapter Compliant panel properties containing the complete wavepacket. Figures 5.9(a) and (b) show both cases and wavepackets almost have the same spectral shapes and α, β values with case spectral slightly appear oblique in figure 5.9(b2). Wavepacket spectra became complex (without distinct 2D and 3D wave modes) in figures 5.9(c2) and (d2) for case and these are not the same for case 3. Lastly, figure 5.9(e1) shows the uenergy spectrum concentrating around β = with α ≈ for case 3, signifying imminent wavepacket breakdown, and which is still not the same for case spectrum shown in figure 5.9(e2). 5.5 Results and discussions for two-compliant panel case Since two-CP walls potentially performed better than single CP wall in chapter 4, we decided to extend the present study to two-CP simulations. For this, case turns out to perform the best, and marginally better than case which was the best in single CP study. The two-CP study uses the same configuration as in chapter 4, that is, two CPs at X = 450 to 762 and 1359 to 1658, so that the present results can also be compared to the two-CP reference case. Among the six cases (1-6) (Table 5.1) carried out for over the two-CP simulations, case produced a better transitional delay result, it is also better than the transition distance result obtained previously for over the two-CP case in chapter 4, even though the single CP case did not (see figure 5.2(d)) deliver a better transition delay result than the single CP in chapter 3. Since case performed better for over the two-CP arrangement among others, therefore, all further analyses are focused on case and they are 163 Chapter Compliant panel properties compared with the results obtained for over the reference two-CP in chapter (section 4.1). 5.5.1 Comparison between reference case and case 5: Spatial evolution Case spatial evolution plots for over the two CPs with comparison with the reference case from chapter are shown in figure 5.10. Figures 5.10(a) and (b) show when both wavepackets were passing over the first CP, with difference between the wavepackets being noticed in figure 5.10(b). Figures 5.10(c) and (d) show when both wavepackets already exited the first CP, and case wavepackets contain lower disturbance velocity magnitude than the reference case, which could have to with the softer nature of case CP. In addition, wavepacket shapes differ with many wave peaks showing in figure 5.10(d2) for case 5, than the reference case in figure 5.10(d1). Figure 5.10(e2) shows when the case wavepacket passes over the second CP at T = 2788. By comparing with the corresponding reference case in figure 5.10(e1), two main differences are noted: (i) case (figure 5.10(e2)) wavepacket shape looks separated into two, with a very strong negative disturbance velocity 2D peaks in-between. Whereas, the reference case still maintain the crescent shape, (ii) the maximum disturbance velocity for case is larger by ≈ 82% when compared with the value in figure 5.10(e1). This increase is probably due to the response of evolving wavepacket to an unexpected soft CP wall just ahead of it. Also, the case CP is softer than the reference case. In figure 5.10(f2), wavepacket is completely on top of case CP same like the reference case in figure 5.10(f1) at T = 3162. Apart from slightly lower 164 Chapter Compliant panel properties wavepacket maximum disturbance velocity recorded for case in figure 5.10(f2) than the reference case in figure 5.10(f1), the wave crests inside the wavepacket in figure 5.10(f2) for case did not contain structures arranged in an oblique fashion. Not only these, a pair of dominant positive wave peaks (with a dominant negative wave mode in-between) pointing in the streamwise direction are formed in figure 5.10(f2). Towards the end of the second CP, case wavepacket in figure 5.10(g2) contains two main parallel streaky structures near the center only and no highly oblique crest structures are formed like the reference case in figure 5.10(g1). Same trends as in figure 5.10(g) still persist in figures 5.10(h1) and (h2) for both reference case and case 5, as the wavepacket already exited the second CP location and now continue to travel over the rigid wall. In figures 5.10(i2) and (j2), case wavepacket shows lower maximum disturbance velocity values than their reference case in figures 5.10(i1) and (j1). In addition to that, the two main streaky structures near the center of the wavepacket became more stretched in the streamwise direction in figures 5.10(i2) and (j2) for case 5, with more discrete wave modes formed along the wavepacket center. Also, case wavepacket grew rapidly to the point of occupying the entire spanwise size of the computational domain from figure 5.10(i2) onwards. In figure 5.10(k2), maximum disturbance velocity for case wavepacket is just almost 65% of the reference case in figure 5.10(k1) at T = 5020. At this time, reference case is already into the late subharmonic stage heading towards breakdown. Maximum disturbance velocity for reference case in figure 5.10(l1) (with streamwise streaky structures showing sign of imminent breakdown) is 165 Chapter Compliant panel properties almost seven times that of the case in figure 5.10(l2). Section 4.1 previous results for the reference case show that the wavepacket breakdown occurred at X ≈ 2350, which is just a short distance away from what was plotted in figure 5.10(l1). Finally, in order to know where case evolving wavepacket will breakdown after figure 5.10(l2), the simulation was continued and the spatial results obtained after then are shown in figure 5.11. Looking at figures 5.11(b) and (c), it is clearly seen that case wavepacket finally broke down into more diffused turbulent spots at X ≈ 2600. By comparing with the reference case (section 4.1) that broke down at X ≈ 2350, this translates to an almost 12.5% increase in transition delay with respect to the perturbation source location at X0 = 349.4. 5.5.2 Comparison between reference case and case 5: Spectral analyses The spanwise wavenumber (β) vs. frequency (ω) spectral plots for both reference case and case for u velocities are shown in figures 5.12. Figure 5.12(a) shows when wavepackets had just exited the first CP with both cases having almost dominant 2D and 3D modes frequencies. The effect of softer wall nature of case CP could be seen in figures 5.12(b2) and (c2) as some modes could be seen around the dominant 2D mode and also some near-zero frequency modes which were all absent for the reference case in figures 5.12(b1) and (c1). At X = 1294 in figure 5.12(d2), the evolving wavepacket is yet to reach the second CP leading edge at X = 1359 – 1658, therefore, the spectrum plot in figure 5.12 (d2) is to show how the spectrum looks like just before the second CP location. Figure 166 Chapter Compliant panel properties 5.12(e2) shows the spectrum behaviour at station X = 1552 which falls within the second CP region. Case fluctuation pattern in figure 5.12(e2) is completely different from the reference case in figure 5.12(e1), due to soft wall chosen properties for case 5. As the wavepacket already left the second CP behind in figure 5.12(f), case wavepacket consists of lower (near zero) 3D wave mode frequencies in figure 5.12(f2) than its reference case in figure 5.12(f1), and the same is observed in figure 5.12(g). Figures 5.12(h2) to (j2) show spectrum that is mainly dominated by 2D wave modes for case 5, while the reference case still show spectrum with low frequency 3D wave modes in figure 5.12(h1) until other 3D wave modes began to spring up sideways, as noted in figures 5.12(i1) and (j1). The streamwise wavenumber (α) vs. spanwise wavenumber (β) spectrum of u-velocity plots are shown in figure 5.13. Spectral behaviours for both cases almost look alike in figures 5.13 (a)-(c), except noticing some CP surface induced modes near the wall for case 5, with same observation in figure 5.13(d2) due to the softer CP wall in case 5. In addition, evolving wavepacket is mainly dominated by 2D wave modes after exiting the second CP as shown in figures 5.13(e2)–(h2) for the case 5, and which are not the same for the reference case in figures 5.13(f1)-(h1). 5.6 Summary In this chapter, more investigations had been carried out in order to answer or further explain some of the short-comings or gaps earlier noted in the brief and limited study conducted previously by Zhao (2006) on the properties of the 167 Chapter Compliant panel properties membrane panel. Also through proper and carefully tuning of compliant panel properties, increase in transition delays had been obtained than those were obtained before for the single CP and two-CP cases in chapters and 4. (i) The safety factor C = 3.16 that seems too high before was decreased further to know what effects this will have on transition delay. However, as the C value closer to 1.0, the more wavepacket broke down quickly for all the trial cases carried out, that is worse than results obtained in chapter for the single CP case before. At last, after much searching, C = 2.8 (which appears best) performed better (than when C = 3.16) in delaying transition farther for both the single CP and two-CP cases which were considered. (ii) The wavepackets simulated in this study were allowed to breakdown into incipient turbulent spots, so that the transition distance could be estimated. Salient features of the wavepackets evolution and breakdown were highlighted. On the overall, with simulation conditions and CP locations remaining the same as they were in chapters and before, CP properties and their effect on transition delay was investigated for over the single and two-CP cases. Six CP properties parametric study had been carried out in this chapter. For over the single CP investigation, two cases namely case and case performed better than other cases in terms of further transition delay. With case 3, wavepacket finally broke down into incipient turbulent spot at X ≈ 1980, whereas, corresponding reference case in chapter broke down at X ≈ 1930. By considering perturbation source location at X0 = 349.4, case with C = 2.80 delayed transition farther to the tune of ≈ 3.2% more than before compared to the previous C = 3.16. For case 168 Chapter Compliant panel properties that is characterized with very soft wall coupled with one-tenth of the initial damping value, wavepacket finally broke down into incipient turbulent spot at X ≈ 2050. This translates to about 7.6% farther transition delay as compared to over the single CP reference case in chapter 3, which broke down at X ≈ 1930, by taking the same perturbation location as the reference point. Through spectral analyses, more detailed information were revealed to know more about the wavepacket evolution dynamics and behaviours as they evolved downstream for both the case and case 6. Frequency (ω) vs. spanwise wavenumber (β) spectra show Case to suppress 2D wave modes from growing further than the case 3, at least up to station X = 949, when both wavepacket had already exited the CP regions. Spectral analyses results further confirmed the oblique nature of case wavepacket which may due to CP wall softness. In addition, case spectrum began to show some sudden appearances of other wave modes nearer the dominant 2D (β = 0) wave modes locations, and these are not seen for case 3. Then for over two-CP investigation, case performed better among the trial cases carried out. It is worth knowing that this same case over the single CP investigations previously did not deliver better transition delay than the reference case in chapter 3. At the end of the whole process, case wavepacket for over the two-CP broke down into more diffuse turbulent spots at X ≈ 2600. If compared with the reference case (section 4.1) that broke down at X ≈ 2350, this translates to an almost 12.5% transition delay. 2D wave modes (especially as the wavepacket heading towards breakdown) are seen dominating from the 169 Chapter Compliant panel properties streamwise wavenumber vs. spanwise wavenumber u velocity spectral plots of case 5, and these are not totally the case for the reference case. 170 [...]... are focused on case 5 and they are 163 Chapter 5 Compliant panel properties compared with the results obtained for over the reference two-CP in chapter 4 (section 4.1) 5. 5.1 Comparison between reference case and case 5: Spatial evolution Case 5 spatial evolution plots for over the two CPs with comparison with the reference case from chapter 4 are shown in figure 5. 10 Figures 5. 10(a) and (b) show when... region at 162 ⁄ Chapter 5 Compliant panel properties containing the complete wavepacket Figures 5. 9(a) and (b) show both cases 3 and 6 wavepackets almost have the same spectral shapes and α, β values with case 6 spectral slightly appear oblique in figure 5. 9(b2) Wavepacket spectra became complex (without distinct 2D and 3D wave modes) in figures 5. 9(c2) and (d2) for case 6 and these are not the same... clearly that the reference case (with 78. 15% disturbance value) in figure 5. 3(g1) will breakdown faster, then followed by the case 3 (with 41.63% disturbance value) in figure 5. 3(g2) and lastly by the case 6 (with 16.17% disturbance value) as shown in figure 5. 3(g3) The breakdown locations for both case 3 and 6 are shown in figures 5. 4 and 5. 5 respectively In figure 5. 4, case 3 finally breakdown into incipient... interactions between the CPs and the evolving wavepacket at the location X = 604 At X = 949 in figure 5. 8(b), wavepacket already exited the CP region (X = 450 to 762) and now continue to travel over section that is 161 Chapter 5 Compliant panel properties completely rigid wall In figure 5. 8(b1), distinct 2D and 3D modes are now noticeable for case 3, whereas for case 6 in figure 5. 8(b2), it is still difficult... direction are formed in figure 5. 10(f2) Towards the end of the second CP, case 5 wavepacket in figure 5. 10(g2) contains two main parallel streaky structures near the center only and no highly oblique crest structures are formed like the reference case in figure 5. 10(g1) Same trends as in figure 5. 10(g) still persist in figures 5. 10(h1) and (h2) for both reference case and case 5, as the wavepacket already... second CP location and now continue to travel over the rigid wall In figures 5. 10(i2) and (j2), case 5 wavepacket shows lower maximum disturbance velocity values than their reference case in figures 5. 10(i1) and (j1) In addition to that, the two main streaky structures near the center of the wavepacket became more stretched in the streamwise direction in figures 5. 10(i2) and (j2) for case 5, with more discrete... at X ≈ 2 350 , this translates to an almost 12 .5% increase in transition delay with respect to the perturbation source location at X0 = 349.4 5. 5.2 Comparison between reference case and case 5: Spectral analyses The spanwise wavenumber (β) vs frequency (ω) spectral plots for both reference case and case 5 for u velocities are shown in figures 5. 12 Figure 5. 12(a) shows when wavepackets had just exited... 2D and 3D modes frequencies The effect of softer wall nature of case 5 CP could be seen in figures 5. 12(b2) and (c2) as some modes could be seen around the dominant 2D mode and also some near-zero frequency modes which were all absent for the reference case in figures 5. 12(b1) and (c1) At X = 1294 in figure 5. 12(d2), the evolving wavepacket is yet to reach the second CP leading edge at X = 1 359 – 1 658 ,... figure 5. 12 (d2) is to show how the spectrum looks like just before the second CP location Figure 166 Chapter 5 Compliant panel properties 5. 12(e2) shows the spectrum behaviour at station X = 155 2 which falls within the second CP region Case 5 fluctuation pattern in figure 5. 12(e2) is completely different from the reference case in figure 5. 12(e1), due to soft wall chosen properties for case 5 As the... in figure 5. 8(b2) shows that case 6 wavepacket looks a bit oblique in nature than the case 3 as was already evident in figure 5. 3(c) Figures 5. 8(c1) and (d1) show u-velocity spectral plots for case 3 at X = 1122 and 1294, and which look similar to figure 5. 8(b1), but now with the spanwise wavenumber (β) value around β ≈ ±0.12 and the dominant 3D wave modes frequency dropped below ω ≈ 0. 05 as the wavepacket . 1.32 8. 25 0.10 0.00 58 0 3 2.80 1.30 1.24 9.70 0.10 0.00 58 0 4 2.80 1. 05 0.81 6. 35 0.10 0.00 58 0 5 2.80 1. 05 0.81 6. 35 0. 05 0.00 58 0 6 2.80 1. 05 0.81 6. 35 0.01. within the panels. Chapter 5 Compliant panel properties 155 5. 3 Compliant panel properties parametric study: Cases investigated After establishing the formulae (5. 4) and (5. 5) for. obtained in chapters 3 and 4 termed “reference” case in this chapter and CP properties are summarized in table 5. 1 after applying equations (5. 4) and (5. 5). Cases 5 and 6 may be regarded as