Chapter Comparison between rigid wall and single compliant panel Chapter Comparison between rigid wall and single compliant panel As already mentioned in section 1.1, compliant panel is one of the passive control measures to delay transition further, and this had already been proved theoretically for its capabilities to stabilize boundary layer and eventually delaying transition. From the theoretical study point of view, infinitely long surface are normally assumed for which the effects of the edges are overlooked. However from the real application side, compliant panels are normally finite in size for which edge effects as already mentioned by Davies and Carpenter (1997), Wiplier and Ehrenstein (2000) and Wang et al (2005) are an integral part of the surface response to flow. Flow over compliant panel was considered as a special case of fluid-structure interaction problem; the classical theory of hydrodynamic stability over compliant surfaces has been remarked and classified as TimeLinearized Models by Dowell and Hall (2001). Using this approach, many kinds of compliant wall models, including spring-backed membrane (Benjamin 1960, 1963; Landahl 1962), bending plate (Carpenter and Garrad 1985, 1986) and volume-based viscoelastic layer (Yeo, 1986; Willis, 1986), have been investigated and found to possess the potential for transition delay. Study on direct numerical simulation (DNS) of wavepacket generation, evolution and breakdown into incipient turbulent spot over membrane surface with a vertical delta pulse type of perturbation at the flow upstream, and within a Blasius boundary layer flow was examined by Zhao (2006). However in the earlier work of Zhao, what really took place within the wavepacket as they 34 Chapter Comparison between rigid wall and single compliant panel evolved downstream were not fully analysed as detail spectral analyses were not carried out, because much information about the wavepacket dynamics and mechanisms are always revealed through detail spectral properties (especially by extractions of dominant 2D and 3D wave modes from the wavepacket spectral plots) study, and this is one of the vital missing gaps that this part of the thesis seek to fill. First step is to further refine the computational simulation first carried out by Zhao. Other steps taken to make the current investigations more distinct and better than before include: (1) with the current faster and more efficient computational resources, more grid points were used in both wall-vertical (Y) and spanwise (Z) directions, coupled with refined grid stretching parameter so as to have more number of grid points near the wall than before, for the sake of capturing nearly all the flow fine details. (2) the entire simulation was performed directly without any need to interpolate in between as it was done by Zhao (2006) due to resource limitation. With this, any potential numerical errors due to interpolations from the previous study were totally eliminated. (3) In addition, further investigations based on linear analyses for both over the single compliant and rigid wall cases are carried out in the last part of this chapter, primarily to know what kind of interaction the single CP will cause to the linearly generated wavepacket if compared with the results from nonlinear simulation. The main objective of this investigation is to verify if by carefully inserting a finite length of compliant panel in a section of rigid wall within a Blasius boundary layer flow upstream, could resulted into a better transition delay if 35 Chapter Comparison between rigid wall and single compliant panel compared with the rigid wall case. Results from single CP case were compared with those for over the rigid wall case which was modelled closely to the conditions of Cohen et al. (1991)’s experiment. This study covered incipient, evolving, growth and eventual breakdown of wavepacket into turbulent spots over a single compliant panel (CP) case and that of a rigid wall (RW) case. Wavepacket evolution processes are closely analysed in terms of spatial and spectral means. 3.1 Numerical simulation and boundary conditions The same direct numerical simulation (DNS) code used by Zhao (2006) was further modified and used in the present simulations. The coupled fluid-wall dynamics is governed by the perturbation Navier-Stokes equations and compliant wall (surface-based model) equations: (3.1) ( ̅ ) ̅ ( ) (3.2) where i, k = 1, 2, represent the streamwise (x), the wall normal (y) and spanwise (z) coordinates of the Cartesian frame respectively. The perturbation form of the Navier-Stokes equations (3.1) and (3.2) is obtained by letting , and ̅ represent the components of the base flow, where the base flow used in the present study is the non-parallel Blasius boundary layer profile. 36 Chapter Comparison between rigid wall and single compliant panel The base flow is null ( ) for the full N-S equations. From the study of Wang (2003) and Zhao (2006), the base flow is given by the following solution of the Blasius boundary layer: (̅ ̅ ̅( )) √ where, ( ) √ ( ( ) ) (3.3) and is the solution of Blasius equation The numerical schemes and their associated discretization methods for equations (3.1) and (3.2) have already been described in details by Wang (2003) and Zhao (2006). The Reynolds number Re is based on the free-stream velocity boundary layer displacement thickness and the as the characteristic length. The dynamics of the tensioned compliant panel displacement is governed by: ( ) (3.4) where η represents normal or y-displacement of the compliant panel (CP) surface. T represents the surface tension; m is the mass per unit area of CP. The d and k may be regarded as the equivalent damping coefficient of viscoelastic foundation elastic constants, and as the external pressure acting on the compliant panel surface. The interaction between the flow and compliant panel is governed by the conditions of zero slip and traction force continuity at the displaced position of the panel which are given as: (̅ )( ) (3.5a) 37 Chapter Comparison between rigid wall and single compliant panel ( ̅ )( ) (̅ )( ) ( ) where ( ) ( ) ( ) (3.5b) (3.5c) ( ) (3.6) refers to the flow disturbance pressure and the external pressure acting on the compliant panel surface. The assumption made here is that the displacement of the compliant surface is sufficiently small so that these interaction conditions can be linearized about the unperturbed or mean position of the panel at y = 0. Applications of Taylor’s expansion to equations (3.5) and (3.6) about the unperturbed interface (y = 0) then yield the following interface conditions at the compliant surface: ̅ | (3.7) ( ) ( where subscript w represents evaluation at y = and ) ( ) (3.8) is the fluid disturbance pressure. The use of moderately flexible/stiff compliant panels ensures the small displacement assumption is valid. Same numerical methods by Wang (2003) and Zhao (2006) are applied in this study. For the completeness of this thesis, brief description on the methodology and computational procedure are provided under the appendix A section of this thesis. 38 Chapter Comparison between rigid wall and single compliant panel 3.2 Over a rigid wall case The first simulation carried out was over the rigid wall (RW) case, and the reason for doing this is to compare results with those for over the compliant panel case and to appreciate the function of compliant panel in delaying transition. Over the rigid wall simulation idea surfaced in an attempt previously made by Zhao (2006) to validate the experimental results obtained by Cohen et al. (1991), where all the simulation parameters were selected carefully to model closely the conditions in Cohen et al. (1991)’s experiment. More details about the computational processes and the perturbation source are discussed in section 3.3.1. For the present study, RW simulation was carried out in a single step without any interpolation involved, and also at higher grid points than before. Our RW results have been verified and everything agrees well (in terms of wavepacket shapes and underlying phenomena at different evolution times) with those from Zhao (2006) and also with the published work of Yeo et al. (2010). Our approach reveals more fine flow details near the wall than that was available before. 3.3 Over a single compliant panel case The computational aspect for the single compliant panel case will be discussed first and this will then be followed by presentation and discussion of the results. Comparisons are made with the earlier simulation results obtained for the rigid wall (reference) case, so as to appreciate the effect of compliant panel in delaying transition. 39 Chapter Comparison between rigid wall and single compliant panel 3.3.1 Simulation process and computational grids First, the schematic 3D view of the computational domain set-up for a single compliant case is shown in figure 3.1, indicating the wavepacket generating (perturbation) source and the embedded compliant panel resting on a viscoelastic foundation, whose properties were designed to restrain the development of compliant-wall modes. Same 1D-springbacked tension compliant panel with damping has been used frequently in stability and transition delay investigations by many previous researchers. For the single compliant panel case, computational domain used for the simulation spans in the streamwise ( ) direction, in the wall normal ( ) direction and in the spanwise ( ) direction, similar to what Zhao (2006) used. The ( ) represents the non- dimensional Cartesian coordinates based on the reference length , Where is the displacement thickness at the perturbation location at . A delta or point-velocity pulse of a smaller size, having the same time modulation as the experiments of Cohen et al. (1991) was applied in the wall-vertical direction as a source of perturbation for all the simulations carried out in this thesis. The broadband nature of a wavepacket offers a central advantage in permitting natural selection of the most dominant wave to operate through the sum of its growth processes. This may be helpful in identifying the critical waves and key processes that are involved at the various stages in natural transition. A section of the rigid wall was replaced with finite length of a tensioned compliant panel from as shown in figure 3.2. That is, 40 Chapter Comparison between rigid wall and single compliant panel the panel was placed in a region of the computational domain downstream of the perturbation source, where the wavepacket will still be evolving in a largely linear regime, purposely to be able to suppress the developing 2D Tollmien-Schlichting (TS) waves. Also, the section occupied by the compliant panel happened to be at the best location in terms of transition distance delay, after a series of location was experimented first. The number of grid points used in X, Y and Z directions are 1200, 85 and 195 respectively, which are more refined than the 1170, 65 and 163 grid points previously used by Zhao (2006). The grid in the vertical (wall normal) direction is stretched, with denser grids near the wall to capture the flow fine details. A stretch factor of γ = 1.6 was used and this captured the flow fine details near the wall better compared to when γ = 1.8 used by Zhao (2006). The grid stretching in Y direction follows a coordinate transformation previously applied by Wang (2003) which is given as follows: ( ) ( ) where (3.9) ( (3.10) ) = height of the flow domain in the physical coordinate , height of the flow domain in the transformed coordinate , = real constant which can be used to adjust distribution of the grid points in the physical coordinates . In addition, periodic boundary conditions were employed at the two spanwise boundaries. Also, a buffer domain similar to the approach of Liu and Liu (1994) was applied from X = 1508 to 1510, for the handling of the outflow boundary 41 Chapter Comparison between rigid wall and single compliant panel condition to allow the wave disturbances to pass out of the computational domain without undue upstream reflections. The approach engages a set of buffer functions to slowly parabolize the governing flow equations and increase the viscous/diffusion damping of the disturbance waves in a buffer region attached downstream of the actual computational flow domain. Due to limitation in computational resources accessibility in terms of the number of CPUs allotted to each user at the NUS IHPC center, computational domain had to be carefully extended further in order to know where the wavepacket will eventually breakdown into incipient turbulent spot. Simulation parameters were obtained from the boundary layer experiments of Cohen et al. (1991): with perturbation location which correspond to 81 cm in the experiment of Cohen et al. This choice of parameters allow the rigid wall results obtained in the simulation to be validated directly against the experiments of Cohen et al. of which details are given in Yeo et al. (2010). The kinematic viscosity used is number simulation time ⁄ , while the Reynolds at the excitation source is 1034.6. The non-dimensional ⁄ is measured from the time of pulse initiation. In order to speed up the simulation, the computation was parallelized based on a decomposition of the domain into 16 number of blocks in the streamwise (X) direction, with communication between adjacent blocks accomplished using ghost volumes at the interfaces. Details of the parallelization coding can be found in the work of Wang et al. (2003, 2005). The u-velocity component of the disturbance wavepacket were obtained at height ⁄ 42 , similar to the heights at which Chapter Comparison between rigid wall and single compliant panel wave measurements were taken in the experiments of Cohen et al (1991) and Medeiros & Gaster (1999b). All other simulation conditions and set-ups, for example, how the wavepacket was initiated at the perturbation source and so on could be seen in the already published work of Yeo et al (2010). Simulation cases of embedded compliant panels were carried out under the same simulation conditions. The properties of the compliant panel used are: , based on a wall Reynolds number of , where L is the wall length scale, which is defined implicitly via a Reynolds number ⁄ . The purpose for adopting a separate independent reference length scale is to facilitate and make possible comparison of wall properties in situations when different or even varying length scales are used in fluid computation, similar to what Yeo (1988) did. 3.3.2 Results and discussions Grid convergence study was quickly carried out before delving proper into the results analyses for over the single compliant panel case. Special attention was paid to the wall-vertical (Y) direction, as inability to resolve fine flow details near the wall (being a boundary layer problem) due to improper number of grid points and stretching wrongly, could jeopardize the overall reliability of the expected results at the end of the simulation processes. Two sets of grids 1170 x 65 x 195 and 1170 x 85 x 195 in X, Y and Z directions were used for grid convergence test. Comparison of streamwise (u) velocity results obtained at different evolution time 43 Chapter Comparison between rigid wall and single compliant panel The corresponding streamwise wavenumber (α) versus spanwise wavenumber (β) spectra of the wavepacket at various time instants in its evolution are shown in figure 3.8. They are obtained from the analysis of a rectangular region at y/δ ≈ 0.62 containing the complete wavepacket, when its center coincides approximately with the specified X-stations. Before the wavepacket began to enter the CP region fully in figure 3.8(a1), the maximum disturbance velocity 3D modes for both cases occur at α ≈ 0.17 and β ≈ ± 0.23 in figures 3.8(a1) and (a2). Formation of α2D began to surface in figure 3.8(b1) and this is not so obvious in figure 3.8(b2). Not only this, α and β values drop to 0.15 and approximately ± 0.185 respectively in figure 3.8(b2). Down the line, figures 3.8(f1)-(g1) in particular for the RW case show the development of the streaky modes (low frequency and low streamwise wavenumber α) from the difference of the two dominant oblique wave modes (pair). On the other hand, no streaky modes are found up till figure 3.8(g2) for the CP case. Breakdown into incipient turbulent spot over the RW case is marked with the disturbance energy concentrating in the low (near-zero) frequency and low wavenumber (streaky structures) modal components as shown in figures 3.7(n1) and 3.8(h1). These phases of nonlinear interactions and transition to the incipient turbulent spot are visibly absent for the single CP wavepacket. The cause can be traced to the absence or near absence of a large enough primary 2D wave modes (up to figure 3.8(f2)) that is able to resonate nonlinearly with the dominant oblique wave pair (Craik’s resonant triad mechanism), or to provide the necessary periodic base state for the rapid growth of oblique waves as a secondary 50 Chapter Comparison between rigid wall and single compliant panel instability (Herbert’s theory). From the spectrum plots in figures (3.7) and (3.8), it is evident that the single CP really suppressed the dominant 2D wave modes as they pass over it. The interesting part of it is that, the 2D wave modes were very slow in their subsequent development even after the wavepacket has exited the CP region, to the extent that post-membrane CP wavepacket is largely dominated by oblique modes. For the purpose of knowing where the wavepacket will eventually breakdown into an incipient turbulent spot for the single CP case, simulation was further continued and the spatial evolution results obtained are shown in figure 3.5. The corresponding selected spectra plots in terms of frequency ω versus spanwise wavenumber β and streamwise wavenumber α versus spanwise wavenumber β are shown in figures 3.9 and 3.10 for selected X-stations and evolution time only. In figures 3.9(b)-(c), one could notice a weak 2D primary wave compared to the much stronger oblique wave pair. This behaviour may be explained by the stronger stabilization effect of flexible wall on 2D TS waves compared to the oblique TS wave (Yeo (1992)). The primary 2D TS wave is still weak at these points relative to the oblique wave pair even though the wavepacket had already evolved quite a distance from the CP. Despite the relative weakness of the primary 2D wave, the triad interaction or secondary instability mechanism is still functioning. The frequency ω and the spanwise wavenumber β of the triad waves are all modified to smaller values compared to the RW case values. In figure 3.9(a), the primary 2D wave has a lower frequency of ω ≈ 0.045 and the two oblique waves have a frequency of ω ≈ 0.025 with spanwise wavenumbers of β ≈ 51 Chapter Comparison between rigid wall and single compliant panel 0.1. Whereas over the RW case in figure 3.7(j1), the primary 2D wave has a frequency ω ≈ 0.06, the two oblique waves with a frequency of ω ≈ 0.04 and spanwise wavenumber of β ≈ 0.12. The subsequent development of the disturbance spectra in figure 3.9 follows more or less the established route of strong nonlinear interaction leading to generation of strong low frequency waves in figure 3.9(d)-(e), and a sudden rapid proliferation/burst of high frequency and high wavenumber modes that characterize breakdown to turbulence as shown in figure 3.9(f). The character of wave resonance and streamwise wavenumber burst can be verified further in spectral analyses of streamwise wavenumber versus spanwise wavenumber for the u-velocity disturbances in figure 3.10. Craik-like triad resonance can be clearly identified in figure 3.10(c). The streamwise wavenumber (figure 3.10(c)) of the 2D waves is α ≈ 0.11 and oblique (3D) waves is α ≈ 0.065, and these are smaller in values than their RW case in figure 3.8(g1) with 2D wave streamwise wavenumber of α ≈ 0.16 and for oblique waves of α ≈ 0.11. This means that the dominant structures in the wavepacket are more stretched out in the streamwise direction than they are for the RW case. More complex modal interactions may be seen in figure 3.10(d) for the streamwise velocity. The breakdown into incipient turbulent spot is marked with disturbance energy concentrating in the near-zero wavenumber (streaky structures) modal components as show in figure 3.10(f). 52 Chapter Comparison between rigid wall and single compliant panel 3.3.2.3 Spectral properties of dominant 2D and 3D wave modes The spectral properties for the dominant 2D and 3D wave modes were carefully extracted from the u-velocity spectral plots as already shown in figures 3.7 – 3.10 and these properties are summarized in table 3.1 for the single CP case and table 3.2 for the rigid wall (RW) reference case. The tabulated data are presented in terms of the local displacement thickness length scale ( ) and all data were extracted at height ⁄ ( ) . The frequency and the wavenumber ( ) in the global/computational length scale are related to the corresponding quantities in the local displacement thickness length scale ( ) by ( ⁄ ( ) where the subscript ( ) )( ⁄ ⁄ ( ⁄ ) ) ⁄ ( ⁄ )( ⁄ ) (3.11a) ⁄ (3.11b) represents quantities in the local length scale ( ). Spectral properties plot for both the single CP and RW cases is shown in figure 3.11. 53 Chapter Comparison between rigid wall and single compliant panel X 392 1096 0.078 0.205 0.074 0.180 0.263 476 1208 0.082 0.198 0.082 0.198 0.202 604 1360 0.084 0.222 0.078 0.189 0.220 690 1454 0.084 0.204 0.079 0.204 0.215 776 1542 0.088 0.216 0.076 0.216 0.225 863 1626 0.090 0.228 0.079 0.225 0.233 949 1705 0.087 0.239 0.081 0.219 0.232 1035 1781 0.089 0.250 0.081 0.219 0.224 1122 1854 0.090 0.260 0.081 0.217 0.217 1208 1924 0.091 0.268 0.080 0.225 0.203 1294 1991 0.090 0.273 0.079 0.233 0.192 1380 2056 0.089 0.282 0.081 0.236 0.191 1467 2120 0.092 0.275 0.082 0.227 0.191 1553 2181 0.091 0.283 0.078 0.207 0.196 1639 2241 0.089 0.282 0.065 0.175 0.199 1726 2300 0.089 0.271 0.056 0.164 0.200 1811 2355 0.090 - 0.052 0.139 0.205 Table 3.1 Single 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). CP location at X = 450 – 762. 54 Chapter Comparison between rigid wall and single compliant panel X 392 1096 0.076 0.202 0.070 0.180 0.273 476 1208 0.095 0.253 0.091 0.198 0.218 604 1360 0.101 0.287 0.085 0.224 0.201 690 1454 0.100 0.273 0.087 0.239 0.195 776 1542 0.103 0.289 0.088 0.253 0.194 863 1626 0.101 0.267 0.085 0.228 0.195 949 1705 0.097 0.280 0.089 0.237 0.204 1035 1781 0.095 0.293 0.081 0.208 0.215 1122 1854 0.099 0.299 0.073 0.211 0.220 1208 1924 0.102 0.264 0.061 0.175 0.203 1294 1991 0.100 - 0.062 0.140 0.210 Table 3.2 Rigid wall (RW) case: Spectral properties of u-velocity fluctuations for the dominant 2D and 3D modes. Data presented are based on the local displacement length scale δ(x). The dominant two-dimensional frequency 12 D wave modes are plotted in figure 3.11(a) for both RW and single CP cases. As expected, data for both cases almost overlapped (that is, almost have the same values) up to the location X = 434 as both wavepacket are still evolving over a rigid wall right away from their perturbation locations. The difference between the two cases became conspicuous at X = 560 as the effect of single CP suppressed the frequency 12 D for the 2D wave modes to , while that of the RW case shot to Then, from X = 560 - 863 for the RW case, the dominant two-dimensional frequency 12 D is fluctuating around 0.1 similar to what was observed into the subharmonic stage 55 Chapter Comparison between rigid wall and single compliant panel in the experiments of Cohen et al. (1991). Yeo et al. (2010) also noted this in their study on wavepacket evolution in a Blasius boundary layer, and same was observed in many experiments conducted by Medeiros and Gaster (1999b) in their study on subharmonic wave production in evolving wavepacket. On the other hand for the single CP case, frequency 12 D fluctuates between for the same X = 560 to 863 range. Down the line, the dominant two-dimensional frequency 12 D were maintained at values ≈ 0.087 till the end for the single CP case, whereas that of the RW case reached before incipient turbulent spot set in. 3D frequencies come to play because of the subharmonic phase, which always distinguished itself by the emergence and growth of a dominant oblique wave pair whose frequency is less than the dominant two-dimensional frequency . Figure 3.7(f1)–(k1) show how a distinctive and dominant subharmonic oblique wave pair gradually emerges from the expanding three-dimensional spectra close to the dominant two-dimensional mode for the RW case. Triad of waves could be seen clearly in figures 3.7(ji)–(l1) and 3.8(f1)-(g1) for the RW case, that is to the late subharmonic stage and figures 3.9(a)–(d) for the single panel case before the breakdown location. The component waves of the triad are marked by subscript for the fundamental two-dimensional wave and subscripts and for the oblique wave as marked in some of the plots in figures 3.7 to 3.10. Three-wave subharmonic resonance between the fundamental 2-D wave and the 3-D oblique wave pair is governed by the following frequency and wavenumber conditions: 56 Chapter Comparison between rigid wall and single compliant panel (3.12a-c) Figures 3.11(b) and (c) show the three waves in both RW and single CP cases developing towards resonance given by equation (3.12) as observed in their ⁄ frequency ratio and wavenumber ⁄ ratio plots. The frequency ratio vary from 0.92 to 0.55 for the RW case, while that of the single CP vary from 0.95 to 0.58. For the wavenumber ratios, the RW case vary from 0.89 to 0.66 and that of the single CP case vary from 0.66 to 0.61 as the initiated wavepacket evolve downstream through the subharmonic instability stage from and for both the RW and single CP cases respectively. The frequency ratios for both cases are more closer to the value of 0.5 as one would expect from the fulfilment of the subharmonic resonant wave condition (3.12). Exact frequency ratios value of 0.5 were not obtained as well in the previous investigations of Cohen et al. (1991) ( Medeiros and Gaster (199b) ( ⁄ ⁄ ) and ) in the late subharmonic stages of their wavepacket studies. However, wavenumber ratios for both cases are not close to the expected value of 0.5 in comparison with the same condition (3.12). Also, figure 3.11(b) shows that the dominant 3D modes frequencies becoming smaller for both cases as the evolving wavepackets are heading towards breakdowns via the subharmonic stages frequencies. Figure 3.11(d) compares the downstream phase speeds of the dominant threedimensional and two-dimensional modes of the wavepacket for both the RW and single CP cases. With the same condition (3.12) in mind, one expects phase speeds ratios ⁄ , however, the entire wavepacket evolution process at 57 Chapter Comparison between rigid wall and single compliant panel different X stations until late subharmonic stages did not perfectly produce phase speed ratios ≈ 1, as both frequency and wavenumber ratios not exactly equal 0.5 as shown in figures 3.11(b)-(c). Looking at the single CP case first, deviations from the value of observed before the CP location (starting from X = 450) are of no importance, as everything still look complicated due to interference from other disturbances associated with the wavepacket perturbation source. The next deviations are within the CP region up to the location X = 863 when the wavepacket just exited the CP location, and this may be attributed to the interaction of the CP with the evolving wavepacket. Stations X = 949 – 1553 portray phase speeds ratios very close to fulfiling the condition (3.12). Last deviations occurred at X = 1639 and 1726. For the RW case, the largest deviations are seen at the X locations very close to the perturbation location same as observed for the single CP case. Phase speeds ratios nearer or almost eqaul to can be found at the locations X = 690, 776, 863 and 1122, while the second largest deviation from is at X = 1035. Observation that three-dimensional plane disturbances could grow nonlinearly to meaningful amplitudes in a Blasius boundary layer despite not satisfying the subharmonic resonant-triad condition (3.12) exactly had already been made by Corke and Mangano (1989). Also similar observations were later made by Corke, Krull and Ghassemi (1992) and Williamson and Prasad (1993a,b) for wake flows as well. Wu et al.(1996, 2007) also mentioned this phenonmenon in their works relating to Rayleigh and planar waves. From the theory proposed by Wu et al. (2007), they showed that strong interaction can occur between the 2D and 3D 58 Chapter Comparison between rigid wall and single compliant panel ̃ wave modes when ( and for any given two-dimensional planar mode ) and oblique wave pair ( ) that satisfy the Squire’s wavenumber condition: ̃ √( ) ( ) (3.13) there is an ‘optimal’ phase speed mismatch which gives the maximum rate of super-exponential growth. Figure 3.11(e) compares variation of and ̃ during the nonlinear evolutions for both cases. The rigid wall case fulfilled the Squire wavenumber condition (3.13) almost perfectly over the nonlinear growth stage from X = 1035 – 1208. While that of the single CP case, only at location X = 1553 fulfilled the conditon (3.13), while before and after this location showed some degree of mismatch within the range of 7.1% (X = 1380) to 4.5% (X = 1726) for the same nonlinear growth stage. Figure 3.11(f) shows that the propagation angles of the dominant three-dimensional modes are actually evolving forward towards 60o (as stipulated in the literatures) for both cases as the wavepacket grows through the subharmonic stages. Both RW and single CP cases reached the propagation angle value of in the late-to-post subharmonic stages at X = 1294 and X = 1811 respectively. Figure 3.12 compares the maximum wave growth of the disturbance u-velocity for both the RW and single CP cases for the dominant 2D and 3D wave modes. The RW case 2D mode amplitude values are almost in the same range with the 3D mode up to the location X = 1035 as the wavepacket evolves. Beyond this location, 3D modes grew as much as three times in amplitude value than the 2D 59 Chapter Comparison between rigid wall and single compliant panel modes at the same location X = 1294, confirming strong nonlinear regime as the wavepacket heading towards breakdown state. On the other hand for the single CP case, 2D wave modes were suppressed from growing rapidly by the inserted CP at the upstream location (X = 450 to 762) as compared to the 2D wave mode amplitudes for the RW case shown in figure 3.12. After the evolving wavepacket had already left the CP location far behind, the positive effect of the inserted CP could still be noted in the reduced amplitude of the dominant 2D modes. In addition, 3D wave modes still did not grow faster than the 2D wave modes up to the location X = 1467. Figure 3.12 had clearly shown the strategy or mechanism behind the single inserted CP case in delaying transition further than the RW case in terms of wavepacket amplitude suppression as they evolved downstream. 3.4 Linear analyses The previous computations performed in section 3.3 for over a single CP were based on the nonlinear version of equation (3.2) with coefficients . Despite the nonlinearity involved, single imbedded CP was still able to perform its expected duty of suppressing 2D wave modes from growing further, and this later resulted into a better transition delay than the RW case at the end. In addition, another good point to note is that, same single CP location from X = 450 - 762 actually falls within when the initiated wavepacket is still evolving in the linear regime of its entire evolution processes. That is, interactions between the single CP and the evolving wavepacket occur primarily during the linear growth stage. Therefore, it will be useful to see what linear computation based on linearized Navier-Stokes equation (3.2) by letting would reveal again, that is 60 Chapter Comparison between rigid wall and single compliant panel to investigate the overall behaviour of the same single CP to a linearly generated wavepacket. Linear simulations were also repeated for over the rigid wall case for comparison purposes. Apart from linear simulation type here, all other simulation processes and conditions remain the same as those already discussed in section 3.3. 3.4.1 Spatial analyses Figure 3.13 shows the wavepacket spatial evolution for over both the RW and single CP cases based on linear computations up to time T = 2046. At time T = 260, both wavepackets have already amplified more than what they were just after perturbations, and both look nearly identical as they developed into crescentshaped waves in figures 3.13(a1) and (a2) for over both the RW and CP cases respectively, as both wavepacket were just arriving the leading edge of the CP. As the wavepacket pass over the CP region in figures 3.13(b2) and (c2), their shapes look triangular if viewed from the top, whereas over the RW case tends to stretch out along the spanwise (Z) direction as the wavepacket advances in figures 3.13(b1)-(c1). In addition, CP linear wavepacket as it passed through the CP location is characterized with oblique wave crests that appear almost 45o to the streamwise direction in figure 3.13(c2), while the wave crests of the RW wavepacket bows out to become crescent shaped. The triangular shape CP wavepacket noted up to figure 3.13(c2) before gradually became more crescent-shaped as the wavepacket propagates away from the CP location as shown in figures 3.13(d2)–(f2). Not only these, the RW 61 Chapter Comparison between rigid wall and single compliant panel wavepacket maximum disturbance velocities in figures 3.13(c1)–(f1) are higher than the single CP wavepacket in figures 3.13(c2)–(f2), confirming the suppressing nature of the CP once the wavepacket passes over it whether the computation is performed under the linearization or non-linearization approach. Also, further attempt was made to compare the linear CP wavepacket in figures 3.13(a2)–(f2) with the earlier obtained results for the non-linear wavepacket in figures 3.4(a2)–(f2). Wavepacket maximum disturbance velocities in figures 3.4(a2) and (b2) are higher than those in figures 3.13(a2) and (b2), which is something contradictory to one’s expectation as the wavepacket locations in figures 3.4(a2) and (b2) are considered being the early part of the linear regime, and their maximum disturbance velocity should be nearer to those in figures 3.13(a2) and (b2). This is likely due to nonlinearity during the point initiation process of the wavepacket where the amplitude might be large. 3.4.2 Spectral analyses Frequency (ω) vs. spanwise wavenumber (β) spectral plots of u and v velocities are shown in figures 3.14 and 3.15 for both over the rigid wall (RW) and single CP cases respectively for wavepacket initiated under the linear computations. The linear CP wavepacket is directly on top of the CP at stations X = 604 and 690 as shown in figures 3.14(a2) and (b2). Figure 3.14(a2) shows u spectrum that contain combination of many wave modes with more complex vertical oscillations (due to CP displacements) revealed properly in figures 3.15(a2) and (b2). With these, it is a bit difficult to figure out dominant 2D and 3D wave modes 62 Chapter Comparison between rigid wall and single compliant panel unlike the RW case in figures 3.14(a1) and (b1). At X = 776 which is after the wavepacket had just exited the CP region the linear CP in figure 3.14(c2) is characterized with distinct 2D and 3D wave modes. At this station, the dominant 2D mode has a frequency ≈ 0.052 in figure 3.14(c2) while the linear RW in figure 3.14(c1) has no distinct or dominant 3D modes per say, as everything is trying to shift towards 2D wave modes only at the frequency ≈ 0.065. In figures 3.14(d1)–(f1), linear RW u-velocity spectral shows a situation where only 2D wave modes dominates and same phenomena are shown in figures 3.15(d1)–(f1) for the v velocity spectra. Whereas for the linear CP case, wavepacket contain both fairly distinct 2D and 3D wave modes as could be seen in figures 3.14(d2) – (f2) with frequency almost remain fixed at 0.05. Also, 2D modes remain more conspicuous (figures 3.14 (d2)–(f2)) instead of dying away as the wavepacket advances further after the CP location, which is a reflection of what the wavepacket already signified in figures 3.13 (d2)-(f2) for over the linear single CP case. The associated streamwise wavenumber vs. spanwise wavenumber spectral plots of u and v velocities are shown in figures 3.16 and 3.17 respectively. RW case spectral also repeated the same scenario where wavepacket is completely dominated by 2D wave modes. 3.5 Summary The use of single short compliant panel (CP) to delay transition farther within a Blasius boundary layer was investigated in this chapter via DNS approach: which is an extension of the study started by Zhao (2006) on wavepacket evolution over 63 Chapter Comparison between rigid wall and single compliant panel membrane panels with a vertical delta pulse type of perturbation. One of the aims of this study is to resolve some of the aforementioned shortcomings of the previous work of Zhao (2006). For the single CP case, small section of the rigid wall was replaced with a finite CP length from X = 450 to 762, which happened to be a fairly location after a series of trials were carried out, for the evolving wavepacket to interact with the finite CP when it is still in the linear stage. The basic flow parameters are the same as what Zhao (2006) used, with only the number of grid points and the grid stretch parameter further refined in this study. A flexible compliant panel was carefully chosen with the aim to be able to stabilize TS waves and same time help in delaying transition. For comparison purpose, over the rigid wall (RW) simulation was also carried out under similar simulation conditions to that of the single CP case. With the single CP case, transition distance was found to have increased by approximately 49% when compared to the RW case. For the spectral analyses part, attention was mainly focused on the u-velocity component being the one that contains the larger part of the wave energy as the wavepacket evolves. Also, spectral properties data for the dominant 2D and 3D wave modes revealed more intrinsic information about the evolving wavepacket. The 2D wave modes were being suppressed from growing further due to interactions with the inserted CP as confirmed by the wavepacket amplitude growth, and hence caused delay in the breakdown of the wavepacket. . In accordance with the resonance-triad interaction condition (3.12), the frequency ratios ⁄ for both single CP and RW cases evolve towards the value of 0.5, while that of streamwise wave number ratios 64 ⁄ are bit larger Chapter Comparison between rigid wall and single compliant panel than the expected value of 0.5. The phase speeds ratios ⁄ from condition (3.12) is fulfilled at the X = 690, 776, 863 and 1122 for the RW case and X = 949 – 1553 for the single CP case. The propagation angles for both cases approached a value of 60o as already observed in the previous literature when the wavepackets are already in the late-to-post subharmonic stages. Furthermore, for the fact that the first CP was located within the regime of linear growth of the wavepacket evolutions, linear version of the computation was finally carried out for both the single CP and RW cases, in order to know what kind of wavepacket patterns and behaviours will be obtained especially over the same single CP case. With the linear computation, wavepacket were still suppressed from growing further by the inserted CP. The wavepacket over the CP looks triangular in form with approximately 45o oblique wave crests. For the RW case, however, the wavepacket has a crescent shape as had been observed by others (Gaster and Grant (1975). Spectral analyses results show linear CP wavepacket that maintained relatively distinct 2D and 3D wave modes, after it had passed beyond the CP location. However for RW case, the wavepacket was 2D dominated with no distinctly dominant 3D modes. 65 [...]... 0.216 0.076 0.216 0.225 8 63 1626 0.090 0.228 0.079 0.225 0. 233 949 1705 0.087 0. 239 0.081 0.219 0. 232 1 035 1781 0.089 0.250 0.081 0.219 0.224 1122 1854 0.090 0.260 0.081 0.217 0.217 1208 1924 0.091 0.268 0.080 0.225 0.2 03 1294 1991 0.090 0.2 73 0.079 0. 233 0.192 138 0 2056 0.089 0.282 0.081 0. 236 0.191 1467 2120 0.092 0.275 0.082 0.227 0.191 15 53 2181 0.091 0.2 83 0.078 0.207 0.196 1 639 2241 0.089 0.282 0.065... the CP region fully in figure 3. 8(a1), the maximum disturbance velocity 3D modes for both cases occur at α ≈ 0.17 and β ≈ ± 0. 23 in figures 3. 8(a1) and (a2) Formation of α2D began to surface in figure 3. 8(b1) and this is not so obvious in figure 3. 8(b2) Not only this, α and β values drop to 0.15 and approximately ± 0.185 respectively in figure 3. 8(b2) Down the line, figures 3. 8(f1)-(g1) in particular... shape and maximum velocity of 0.77% for over the rigid wall (RW) case and one with compliant panel (CP) in figures 3. 4(a1) and (a2) This is due to the fact that the effect of the inserted CP is not yet felt in figure 3. 4(a2) as the wavepacket is just about to reach the CP region Figures 3. 4(b2) (T = 558) and (c2) (T = 930 ) show the wavepacket 44 Chapter 3 Comparison between rigid wall and single compliant. .. 1208 0.095 0.2 53 0.091 0.198 0.218 604 136 0 0.101 0.287 0.085 0.224 0.201 690 1454 0.100 0.2 73 0.087 0. 239 0.195 776 1542 0.1 03 0.289 0.088 0.2 53 0.194 8 63 1626 0.101 0.267 0.085 0.228 0.195 949 1705 0.097 0.280 0.089 0. 237 0.204 1 035 1781 0.095 0.2 93 0.081 0.208 0.215 1122 1854 0.099 0.299 0.0 73 0.211 0.220 1208 1924 0.102 0.264 0.061 0.175 0.2 03 1294 1991 0.100 - 0.062 0.140 0.210 Table 3. 2 Rigid wall... between the fundamental 2-D wave and the 3- D oblique wave pair is governed by the following frequency and wavenumber conditions: 56 Chapter 3 Comparison between rigid wall and single compliant panel (3. 12a-c) Figures 3. 11(b) and (c) show the three waves in both RW and single CP cases developing towards resonance given by equation (3. 12) as observed in their ⁄ frequency ratio and wavenumber ⁄ ratio plots... figure 3. 10(d) for the streamwise velocity The breakdown into incipient turbulent spot is marked with disturbance energy concentrating in the near-zero wavenumber (streaky structures) modal components as show in figure 3. 10(f) 52 Chapter 3 Comparison between rigid wall and single compliant panel 3. 3.2 .3 Spectral properties of dominant 2D and 3D wave modes The spectral properties for the dominant 2D and 3D... 230 0 0.089 0.271 0.056 0.164 0.200 1811 235 5 0.090 - 0.052 0. 139 0.205 Table 3. 1 Single 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) CP location at X = 450 – 762 54 Chapter 3 Comparison between rigid wall and single compliant panel X 39 2 1096 0.076 0.202 0.070 0.180 0.2 73. .. the subscript ( ) )( ⁄ ⁄ ( ⁄ ) ) ⁄ ( ⁄ )( ⁄ ) (3. 11a) ⁄ (3. 11b) represents quantities in the local length scale ( ) Spectral properties plot for both the single CP and RW cases is shown in figure 3. 11 53 Chapter 3 Comparison between rigid wall and single compliant panel X 39 2 1096 0.078 0.205 0.074 0.180 0.2 63 476 1208 0.082 0.198 0.082 0.198 0.202 604 136 0 0.084 0.222 0.078 0.189 0.220 690 1454 0.084... Ghassemi (1992) and Williamson and Prasad (1993a,b) for wake flows as well Wu et al.(1996, 2007) also mentioned this phenonmenon in their works relating to Rayleigh and planar waves From the theory proposed by Wu et al (2007), they showed that strong interaction can occur between the 2D and 3D 58 Chapter 3 Comparison between rigid wall and single compliant panel ̃ wave modes when ( and for any given... just after perturbations, and both look nearly identical as they developed into crescentshaped waves in figures 3. 13( a1) and (a2) for over both the RW and CP cases respectively, as both wavepacket were just arriving the leading edge of the CP As the wavepacket pass over the CP region in figures 3. 13( b2) and (c2), their shapes look triangular if viewed from the top, whereas over the RW case tends to . components as show in figure 3. 10(f). Chapter 3 Comparison between rigid wall and single compliant panel 53 3. 3.2 .3 Spectral properties of dominant 2D and 3D wave modes The spectral. Chapter 3 Comparison between rigid wall and single compliant panel 37 The base flow is null      for the full N-S equations. From the study of Wang (20 03) and Zhao (2006), the base flow. (2.7) Chapter 3 Comparison between rigid wall and single compliant panel 39 3. 2 Over a rigid wall case The first simulation carried out was over the rigid wall (RW) case, and the reason