Chapter Literature review Chapter Literature review Research on compliant coatings could be traced as far back as sixty years ago. This research area had already witnessed an era in the beginning where some earlier researchers were either motivated through the transition delay breakthrough results they obtained or got discouraged along the line due to irreproducibility of some past claimed results, especially through their experimental investigations. Typical applications of compliant coatings according to Gad-el-Hak (1996) include: drag reduction as a result of transition delay, as sound absorbent materials in noisy flow-carrying ducts in aero-engines, and as flexible surfaces to coat naval vessels for the purposes of shielding their sonar arrays from the sound generated by the boundary-layer pressure fluctuations and of reducing the efficiency of their vibrating metal hulls as sound radiators. Compliant coatings had been identified to be cheap and simple method to delay transition. Unlike other drag reducing methods such as suction, injection, polymer or particle additives, passive compliant coatings not require slots, ducts or internal equipment of any kind. Compliant (membrane) panel is one of the passive ways of applying compliant coatings especially within the boundary layer, and this method had been proved successfully in many theoretical studies in the past, as a possible way of delaying transition farther in a boundary layer flow. Investigations into compliant panel research are linked to the way dolphin swims. Research work on dolphin fast swimming feature came to limelight with the Gray (1936)’s paradox, where Gray 17 Chapter Literature review made an estimate of the power a dolphin could exert based on its physiology, and concluded the power was insufficient to overcome the drag forces in water and finally hypothesized that Dolphin’s skin must have special anti-drag properties. After that, studies on the effect of compliant walls on flow stability were inspired by Kramer (1957)’s observation of swimming dolphins in the late 1950s. Kramer assumed that their high propulsive efficiency should be ascribed to the compliance of their skin. He then carried out experiments in water by dragging a torpedo covered with a compliant coating conceived to mimic the dolphin’s skin and achieved drag reduction of more than 50 % compared to that of the rigid wall case. This achievement of Kramer generated a lot of interests among the researchers after then, which later resulted into various investigations both experimentally and numerically. 2.1 Previous experimental investigations on compliant surfaces After the pioneering achievement of more than 50% drag reduction, Kramer (1960) performed another experiment with streamlined bodies covered with a multi-layer compliant coating whose design and development followed the epidermic structure of the dolphins. This second experiment of Kramer (1960) yielded drag reductions of almost 60%. Along the line and out of excitements in early 1960s, attempts made to reproduce Kramer’s results by Puryear (1962), Nisewanger (1964), Ritter and Porteous (1965) failed due to experimental difficulties, that is, due to mainly (i) inability to work with flows that are characterized with low turbulence similar to what Kramer used, and (ii) unable to 18 Chapter Literature review work with right compliant material properties. However, improved experiments were later carried out by Fisher and Blick (1966), Blick and Walters (1968), Chu and Blick (1969). At least their results appeared to yield some positive evidence of drag reduction, but some of these were later ignored on the basis of experimental irregularities. A review of compliant wall drag reduction research up till the mid-1970 was given by Bushnell et al. (1977), where they also highlighted the capability of compliant surfaces in turbulent drag reduction. Also, 1970s witnessed a lull in compliant wall research for drag reduction, as many researchers were discouraged to continue with it due to their inability to reproduce Kramer’s claimed results. However, 1980s witnessed another form of resurgence to re-evaluate the function of compliant surfaces in boundary layer stabilization and drag reduction, this resurgence was famously led by Carpenter and co-workers. Carpenter and Garrad (1985) demonstrated that some selected compliant surfaces are able to delay transition farther and indeed presented an explanation for the failures of the early experiments. In a similar experimental work, Gaster (1987) tested samples of compliant layers in a large towing tank at the then National Maritime Institute (UK). The experiments confirmed that viscoelastic compliant layers with the appropriate properties are able to reduce the growth of the TS waves over the corresponding rigid surfaces provided that unstable fluid-surface interaction modes could be avoided or held in check. Willis (1986) and Gaster (1987) further demonstrated that some compliant surfaces could reduce the growth of TS waves by an order of magnitude if wall parameters, such as stiffness and damping are optimally tuned or balanced. 19 Chapter Literature review Lee et al. (1995) studied effects of compliant walls on boundary layer instability in a low turbulence wind tunnel and suggested that a delay of transition in air may be possible. As the density of air is much lower than that for the typical compliant materials (about1:800), a large airflow velocity would be needed to generate the perturbation pressure to interact with the compliant walls. Carpenter (1998) was consequently critical of their results. Unlike the laminar and transitional flows, turbulent flows are much more difficult to be investigated theoretically and experimentally. Bushnell et al. (1977) considered the effect of compliant walls on turbulent boundary layers and suggested that compliant coatings could possibly modulate the pre-burst flow in a boundary layer through the pressure field. In this regard, it was thus plausible that compliant wall could well have yet a favourable influence on skin friction drag in a turbulent boundary layer. Colley et al. (1999) carried out experiment to investigate the laminar-toturbulent transition of the boundary layer over a rotating compliant disc. The experiments were carried out under-water using a hot-film probe and the compliant coating was made from vulcanized silicone rubber. It was found that compliance had a stabilizing effect on the inviscid instability in the frequency range but an overall destabilizing effect on the boundary-layer flow. It was argued that instabilities due to the flexible surface were not prevalent and the overall destabilization was the result of a lowered critical Reynolds number. More recent experiments of Colley et al. (2006) have verified the results of Cooper and Carpenter (1997) that compliance can destabilize Type-II disturbances on a 20 Chapter Literature review rotating compliant disc. Their previous experiments were modified to greatly reduce the background noise, allowing the effect of compliance on the Type-II mode to be investigated. A new, much softer material was used for the compliant wall. Huang et al. (2008) experimentally investigated the effect of compliant surfaces on the receptivity and bypass transition of a boundary layer. Measurements were made in the pre-transitional and transitional boundary layers on nine different compliant and one rigid surfaces with identical geometries. Their compliant surfaces were manufactured from gelatine covered by a 10 µm protective PVC film. They observed same laminar boundary layer profiles and growth rate results for all the surfaces, but the receptivity of the laminar boundary layer to free stream disturbances increased close to the leading edge of each compliant surface. They recorded the transition onset position on the compliant surface to range from 3% downstream to 20% upstream of the rigid surface position. Erdmann et al (2011) used different types of voice-coil- and piezo-driven membrane actuators effectively to introduce counter waves into the boundary layer to cancel the travelling TS waves. They were able to shift the transition region to about seven TS wavelengths (≈ 45 mm). Recently, Patzold et al. (2013) used actively driven compliant wall which was integrated as part of wing’s surface to delay transition through attenuation of convective instabilities. With this approach, piezo-polymer actuation elements in combination with model predictive control algorithms attenuated the local TS-wave amplitudes by 85.4%, 21 Chapter Literature review and transition location also shifted towards the wing’s trailing edge by ∆x ≈ 100 mm. 2.2 Previous numerical or theoretical investigations on compliant surfaces Various numerical methods had been developed since almost three decades ago for investigating boundary-layer instability, transition and transition control especially for over the rigid walls, but which could as well be extended to over compliant surface investigations. These methods include: Turbulence Modeling (TM) (Zhang et al. (1998)), Large Eddy Simulation (LES) (Ducros et al. (1996)), Linear Stability Theory (LST) (Reed et al (1996)), Parabolized Stability Equation (PSE) (Herbert (1997)) and Direct Numerical Simulation (DNS) (Kleiser and Zang (1991)). Since this present work concerns DNS, the review will first focus on DNS carried out on boundary layer stability over compliant surfaces and to later mention other previous numerical works. 2.2.1 Boundary layer compliant surface simulations based on DNS approach With the computers becoming more powerful in terms of memory and processing speeds these days, using direct numerical simulation (DNS) approach for flow stability and transition problems especially within the boundary layer is no more a thing of concern. Irrespective of whether all the nonlinear terms are included in the flow governing equations or not, DNS provides the most accurate way to investigate both unstable and transitional flows. From the literature, two types of DNS method that have been used in the direct simulation of boundary 22 Chapter Literature review layer transition problems include: (i) Spatial DNS (SDNS) and (ii) Temporal DNS (TDNS). For the SDNS, the actual boundary layer is taken into considerations in the computation of the evolving perturbations. More peculiarity about this SDNS approach is that, it always leads to more complex implementation of both the inflow and outflow boundary conditions. Since for the past decade up till now, more researchers still prefer the SDNS approach over the TDNS counterpart, because SDNS can guarantee the real simulation of experiments. On the other hand for TDNS and in order to make the computation simpler, the perturbed flow is assumed to be periodic in the streamwise direction. The purpose of this is to allow the streamwise length of the computational domain to be truncated to only one or just a few primary instability wavelengths. The main noted drawback with TDNS approach is the discrepancy between the computational model and the physical flow itself. Spalart and Yang (1987) used this approach in their simulations despite the drawback. Probably the first temporal direct numerical simulation of boundary-layer waves over a compliant surface (tensioned membrane to be specific) was performed by Domaradzki and Metacalfe (1987). In their simulation, the Fourier series and Chebyshev expansion method are employed in the streamwise and normal directions respectively. For simplicity linearized boundary conditions are applied. They studied the temporal and spatial behaviour of the terms in the kinetic energy balance equation and verified the class A and class B character of the computed waves. Hall (1988) also developed a temporal simulation algorithm 23 Chapter Literature review for simulating 2D instability waves over soft polyvinyl chloride (PVC) layers. This work seems to be the only computational study on volume-based walls. The transient finite element DYNA2D code was utilized to model the solid wall equations. Explicit iteration procedure was adopted for the fluid and solid coupling. Three materials including a soft polyvinyl chloride (PVC), stiffer PVC and a two-layer material consisting of a thick layer of soft PVC covered by a thin layer of neoprene were investigated. Though much attention was given to modelling solid deformation, little information about flow field was provided by Hall (1988). Metcalfe et al. (1991) reported their 3D temporal DNS work on boundary layer flow instability over a compliant panel. Their simulation showed that nonlinear secondary instabilities could arise and cause the flow to become unstable when it was predicted to be stable by linear theory. Therefore, besides linear stability analysis, the nonlinear mechanisms also require much attention for optimizing compliant surfaces to delay transition. They also found that nonlinear interactions among the different classes of compliant wall modes appear to require commensurate phase speeds and the phase relationship between the modes can strongly affect their interaction. Λ-vortices that are similar to the flow structures in boundary-layer transition over a rigid wall were also observed. Furthermore, they found that by carefully choosing compliant wall parameters, they could inhibit the formation of a strong spanwise vorticity spike in the detached shear layer above the wall. Although their work was based on temporal theory and linear boundary condition assumption, it indicated that the task of studying and 24 Chapter Literature review optimizing 3D compliant surfaces to reduce drag or delay transition is worth pursuing. Davies and Carpenter (1997)’s simulation of boundary-layer stability over finite compliant panel was perhaps the first work done on the linear Navier-Stokes simulations of flow stability over a compliant surface. A novel vorticity-velocity method was used in the simulations. Using this method with special treatment for the wall and fluid inertial terms, they solved the linearized N-S equations and presented the results for the spatial evolution of TS waves. The complex response of finite panels was investigated in great detail. By choosing the frequency of the TS wave above the cut-off frequency of the compliant panel, Davies and Carpenter showed that the response of the panel actually consists of superposition of three eigenmodes – one original TS mode and two FISI (or CIFI) modes. Moreover, their results displayed the complicated wave effects introduced by the edge of a finite panel; in particular, the interaction between the TS waves and the leading edge of the compliant panel. The main conclusion drawn by Davies and Carpenter from their simulations is that panels as short as one TS wavelength remain effective at suppressing TS waves. They also demonstrated that certain very short compliant panels are even more effective at wave suppression than longer ones with the same properties. Wiplier and Ehrenstein (2000, 2001) adopted the primitive-variable method to simulate the spatial evolution of 2D disturbances in a boundary-layer flow over compliant membranes and panels. The behaviour of the disturbances as convective and absolute instabilities was investigated. Their simulation results re- 25 Chapter Literature review affirmed the validity of the linear stability theory and show that absolute instability could arise from the coalescence between an upstream propagating evanescent mode and downstream propagating TS wave, as was suggested by Yeo et al. (1996). Their model takes into account the non-parallelism of the flow and nonlinear effects within the flow. To handle the moving boundary problem, the physical domain was transformed into a fixed computational domain. However, as the prescribed base flow was dynamically stretched with the deformation of the compliant boundary, this may well introduce an unknown error into the results. It is hard to be convinced that such an approach will work for other than small amplitude surface waves; in which case the condition at the boundary will be quite similar to that modelled by linear flow-wall interaction conditions. When the amplitude of the surface waves becomes nonlinearly significant, the stretching of the base flow may well introduce artificial dynamics that have to be accounted for. Wang et al. (2001) employed a 2D vorticity-streamfunction method for spatially simulating the unsteady waves over finite-length membranes. Two cases with different tensions were investigated in some details. The results were compared with those for a rigid wall. Davies and Carpenter (2001) developed a method for simulating linear disturbance evolution in 3D boundary layer over compliant surfaces and applied it to boundary-layer flow on a rotating disk. Unlike normal 3D vorticity-velocity method in which six governing equations are usually required, only three governing equations are solved in their highly efficient method. The linearized form of this method was validated for the case of 26 Chapter Literature review convective instabilities evolving over both rigid and compliant discs. Also, Wang (2003), Wang et al. (2005) and Zhao (2006) conducted spatial direct numerical simulation of two-dimensional and three-dimensional transitional boundary layer flows over finite compliant surfaces. They concluded that compliant coatings with selected properties are able to reduce the growth rates of linear TS waves and three-dimensional subharmonic nonlinear instabilities, but may not be effective against three-dimensional fundamental nonlinear instabilities. However, the work of Wang et al. (2005) and others mentioned were primarily devoted to the study of compliant surface effects on monochromatic wave systems. Davies (2005) outlined various methods of using computer experiments to complement physical experiments in studying the development of disturbances in boundary layers. He applied different discretization schemes for the three different directions (streamwise, wall normal and spanwise) respectively. From one of his investigations on compliant surface, he also noticed significant contributions due to the presence of compliant wall in transition delay. Davies et al. (2006) performed numerical investigations on disturbance development in boundary layers over compliant surfaces. They examined (1) the influence of compliant surfaces on transiently growing forms of disturbance; (2) the effects of three dimensionality in the basic boundary layer flow, which can introduce other modes of instability such as cross flow vortices; (3) the effects of using finite length compliant surfaces in order to tailor local surface compliancy properties so that they match the local flow conditions. They concluded that compliant surfaces can be designed so that they are still effective in delaying transition in 27 Chapter Literature review circumstances where transient streak-like structures play a more important role in the transition process than conventional eigenmode forms of disturbance, such as TS waves. Also, for the rotating disc flow, it was found that a compliant annulus inserted into an otherwise rigid disc was capable of stabilizing disturbances even at locations that were radially inboard from the compliant part of the disc surface. Zhao (2006) performed many numerical simulations for wavepacket evolutions inside the Blasius boundary layer over compliant surfaces. She investigated the effects of membrane parameters and position of membrane panels on wavepacket evolution. Most of her results confirmed the positive contributions from the effect of membrane surface in delaying transition to a significant value within the Blasius boundary layer. Also, while the axisymmetric modes dominate the spectrum when the walls are rigid or very mildly compliant, a critical non-zero azimuthal wavenumber exists for which the hydroelastic modes become more unstable. 2.2.2 Other numerical works on boundary layer over compliant surfaces Previous theoretical (simulation) investigations on boundary layer stability over compliant surfaces have been studied based on both the linear stability theory and the nonlinear approach. In the literature, some researchers have investigated the effects of wall compliance on nonlinear or finite-amplitude waves. Gajjar (1990) discussed the nonlinear stability of the Travelling Wave Flutter (TWF) modes in the boundary layer flow over compliant surface. Gajjar (1990) used combination of multiple-scale analysis and an unsteady nonlinear critical layer theory, which 28 Chapter Literature review later resulted into a set of equations that described the spatial/temporal evolution of the TWF mode. Obtained results show that the main effect of nonlinearity on the TWF mode is to cause the growth rate to decrease as the TWF mode evolves downstream, and to cause a roll-up of vorticity inside the critical layer with the generation of harmonics. Joslin and Morris (1991) used the secondary instability theory developed earlier by Herbert (1983, 1997) to boundary layer flow for over both the isotropic and anisotropic compliant walls. They discovered that the optimized isotropic wall does not have an adverse effect on the nonlinear stability of the flow relative to the rigid wall case. Thomas (1992a) applied Craik (1971)’s theory of resonant triads to flows over compliant membranes. Locations of the triad were determined and their interaction coefficients were as well evaluated. Also, Thomas (1992b) extended the weakly nonlinear, high-Reynolds-number triple-deck theory to Blasius flow over a compliant wall and investigated the evolution of 2D and 3D TS waves over compliant panels. The results obtained show that compliant wall can dramatically affect the weakly nonlinear stability properties of boundary and that linearly stable waves can become nonlinearly unstable. Nonlinear instability of 2D waves could arise through wall compliance and damping. Derivation of set of equations proper for the computation of finite-amplitude travelling wave solutions for the Blasius boundary layer over compliant walls were carried out by Ehrenstein and Rossi (1996). They discovered that the subcritical character of the instability, which is weak for the rigid case, can be more pronounced for moderately flexible walls with damping. Also, their computational results showed that for a certain 29 Chapter Literature review range of wall parameters, 2D finite-amplitude travelling waves exist well below the critical Reynolds number predicted by the linear theory, which is typical for subcritical bifurcations. Yeo (1988) theoretically studied the linear stability of zero pressure-gradient laminar boundary-layer flow over compliant walls, which are composed of one or more layers of isotropic viscoelastic materials backed by a rigid wall. With the results obtained, evidence of considerable suppression of disturbance growth was suggested for suitably chosen compliant walls. Later in the process of continuation, the contribution of anisotropy to transition delay was evaluated by Yeo (1990), took a further step to examine the linear stability of zero-pressuregradient boundary layer flow over a class of anisotropically responding compliant walls. Yeo (1992) also theoretically studied the linear stability of 3D/oblique disturbance wave modes in zero-pressure-gradient boundary layer flow for over both isotropic and anisotropic compliant walls. Results obtained by Yeo (1992) showed that the apparent increase in wall stiffness is indeed an important factor and that 3D oblique TSI modes may become more dominant than the 2D modes when the wall is sufficiently compliant for the isotropic-material walls case. The 2D non-parallel stability of boundary-layer flow over layered compliant walls was also studied by Yeo et al (1994a), and concluded that the destabilizing influence of non-parallelism was found to be fairly mild for the TollmienSchlichting instability (TSI). In addition, the effects of flow loading on the 2D linear stability of boundary layer flow over compliant walls had also been investigated by Yeo et al (1994b), where it was found that hydrostatic pressure 30 Chapter Literature review can strongly affect the stability of flow over compliant walls; with a stabilizing influence on the Tollmien-Schlichting instabilities and a destabilizing influence on the compliance-related instabilities. Other works on compliant layers/surfaces could be noted in Yeo et al. (1999), Yeo et al. (2001). Jo-Anne et al. (2007) studied the effect of compliance on the instabilities formed within 3D boundary layer due on a compliant rotating disc. Their study was divided into two parts: a numerical investigation and an asymptotic investigation into the inviscid and viscous stationary modes respectively. Their final results suggested that the inviscid mode of instability will be stabilized by compliance, but the viscous mode will be greatly destabilized. Anais et al. (2009) investigated the influence of compliance on the stability of a Taylor-Couette configuration in the narrow-gap limit when only the inner cylinder rotates. They applied an approach by Guaus and Bottaro (2007) for the curved channel. Due to the walls flexibility hydroelastic modes are generated, whereas, as in opposition to the rigid-wall case, the most unstable modes are not the axisymmetric ones. Stephan et al. (2010) performed numerical simulation based on large eddy simulation (LES) method to investigate the influence of riblets on laminarturbulent transition in a flat plate zero-pressure-gradient boundary layer above a riblet wall. They found out that three-dimensional structures such as λ-, hairpin, and streamwisely aligned vortices are damped. Also with the oblique transition, the breakdown to turbulence was delayed by riblets compared to transition on a clean surface. Also, Ashraf et al. (2011) studied the effects of flexible (compliant) structures on boundary layer stability and transition through a modelling process. 31 Chapter Literature review They used a beam equation to represent the flexible structure which was coupled to an Orr-Sommerfeld equation, and later solved for a Blasius type boundary layer. They later found that, materials of low density and Young modulus are better to stabilize the boundary layer, compared to those with high density and high Young modulus, which not affect the laminar-turbulent transition, as they acted like rigid surfaces. 2.3 Road map of the present study By searching through the existing literature on boundary layer stability and transition for over compliant surfaces, no research investigation had been reported particularly on the use of short compliant panels to delay transition farther, over a flat plate or rigid wall with the short compliant panels occupying certain sections of the rigid wall within a spatially growing Blasius boundary layer, through a transition initiated by a small delta pulse vertical disturbance at the flow upstream; where pulse-initiated wavepacket was used as a model for a broadband disturbance source in a natural transition environment. Chapter presents a detailed study and analyses (spatial and spectral) for both over the rigid wall (RW) case and single compliant panel (CP) case, with much emphasis on spectral analyses so as to properly understand the intrinsic dynamics of the generated wavepacket as they evolved over both cases. Chapter attempts to capitalize on the motivation received from over the single CP case in terms of better transition delay in chapter 3, by considering both two and three compliant panels to delay transition much better than before. Lastly, the 32 Chapter Literature review studies ended with attempts to further tune the CP properties in chapter for better performances than the transition delay results already obtained in chapters and 4. 33 [...]...Chapter 2 Literature review convective instabilities evolving over both rigid and compliant discs Also, Wang (20 03), Wang et al (20 05) and Zhao (20 06) conducted spatial direct numerical simulation of two-dimensional and three-dimensional transitional boundary layer flows over finite compliant surfaces They concluded that compliant coatings with selected properties... Thomas (1992a) applied Craik (1971)’s theory of resonant triads to flows over compliant membranes Locations of the triad were determined and their interaction coefficients were as well evaluated Also, Thomas (1992b) extended the weakly nonlinear, high-Reynolds-number triple-deck theory to Blasius flow over a compliant wall and investigated the evolution of 2D and 3D TS waves over compliant panels The... In addition, the effects of flow loading on the 2D linear stability of boundary layer flow over compliant walls had also been investigated by Yeo et al (1994b), where it was found that hydrostatic pressure 30 Chapter 2 Literature review can strongly affect the stability of flow over compliant walls; with a stabilizing influence on the Tollmien-Schlichting instabilities and a destabilizing influence... further step to examine the linear stability of zero-pressuregradient boundary layer flow over a class of anisotropically responding compliant walls Yeo (19 92) also theoretically studied the linear stability of 3D/oblique disturbance wave modes in zero-pressure-gradient boundary layer flow for over both isotropic and anisotropic compliant walls Results obtained by Yeo (19 92) showed that the apparent increase... walls are rigid or very mildly compliant, a critical non-zero azimuthal wavenumber exists for which the hydroelastic modes become more unstable 2. 2 .2 Other numerical works on boundary layer over compliant surfaces Previous theoretical (simulation) investigations on boundary layer stability over compliant surfaces have been studied based on both the linear stability theory and the nonlinear approach In... factor and that 3D oblique TSI modes may become more dominant than the 2D modes when the wall is sufficiently compliant for the isotropic-material walls case The 2D non-parallel stability of boundary-layer flow over layered compliant walls was also studied by Yeo et al (1994a), and concluded that the destabilizing influence of non-parallelism was found to be fairly mild for the TollmienSchlichting instability... downstream, and to cause a roll-up of vorticity inside the critical layer with the generation of harmonics Joslin and Morris (1991) used the secondary instability theory developed earlier by Herbert (1983, 1997) to boundary layer flow for over both the isotropic and anisotropic compliant walls They discovered that the optimized isotropic wall does not have an adverse effect on the nonlinear stability of the flow. .. obtained show that compliant wall can dramatically affect the weakly nonlinear stability properties of boundary and that linearly stable waves can become nonlinearly unstable Nonlinear instability of 2D waves could arise through wall compliance and damping Derivation of set of equations proper for the computation of finite- amplitude travelling wave solutions for the Blasius boundary layer over compliant walls... that, materials of low density and Young modulus are better to stabilize the boundary layer, compared to those with high density and high Young modulus, which do not affect the laminar-turbulent transition, as they acted like rigid surfaces 2. 3 Road map of the present study By searching through the existing literature on boundary layer stability and transition for over compliant surfaces, no research... the use of short compliant panels to delay transition farther, over a flat plate or rigid wall with the short compliant panels occupying certain sections of the rigid wall within a spatially growing Blasius boundary layer, through a transition initiated by a small delta pulse vertical disturbance at the flow upstream; where pulse-initiated wavepacket was used as a model for a broadband disturbance source . (1992b) extended the weakly nonlinear, high-Reynolds-number triple-deck theory to Blasius flow over a compliant wall and investigated the evolution of 2D and 3D TS waves over compliant panels. . Wiplier and Ehrenstein (20 00, 20 01) adopted the primitive-variable method to simulate the spatial evolution of 2D disturbances in a boundary-layer flow over compliant membranes and panels. . the case of Chapter 2 Literature review 27 convective instabilities evolving over both rigid and compliant discs. Also, Wang (20 03), Wang et al. (20 05) and Zhao (20 06) conducted spatial