Flow stability and transition over finite compliant panels 3

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

Chapter 3 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 Time-Linearized 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

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

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:

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:

( ̅ ̅ ̅( ) ) ( ( ) √ ( ) ) (3.3)

where, √ 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 and the boundary layer displacement thickness 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)

( ̅ )( ) ( )

(3.5b) ( ̅ )( ) (3.5c)

( )( ) ( ) (3.6)

where ( ) 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:

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

(2.7)

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

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,

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:

( ) (3.9)

( )

( ) (3.10) where = 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

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 , while the Reynolds number ⁄ at the excitation source is 1034.6 The non-dimensional simulation time ⁄ 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 ⁄ , similar to the heights at which

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

is shown in figure 3.3 For both times T = 2417 and 3162, almost same flow details were captured despite their number of Y stretched grids are not equal The two sets of results are practically almost identical, that is converged as both will surely reveal the same intrinsic wavepacket dynamics and underlying phenomena,

if any of the grids sets are used

3.3.2.1 Wavepacket spatial evolution analyses

The snap shots of the wavepacket evolution process over the single compliant panel (CP) are shown in figure 3.4(a2)-(h2) for the u-velocity components of the disturbance wavepacket at the height of These are compared with the first obtained corresponding reference results for over the rigid wall (RW) case in figure 3.4(a1)-(h1) By quickly scanning through the entire u-velocity contours in figure 3.4, it was observed that the wavepacket over the RW develops into a turbulent spot (figure 3.4(h1)) with fluctuation amplitude reaching as high as ≈ 88% of the free stream velocity, whereas for over a single CP case, maximum disturbance velocity reaches only 0.74% at the final simulated stage in figure 3.4(h2)

Carefully observing the wavepacket evolution after its initiation location at time T = 260, both wavepacket almost have the same 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

is actually travelling on top of the CP in its linearly evolving stage Maximum disturbance velocity first decreases in figures 3.4(a1)-(c1) for the RW case, and later continuously increasing until incipient turbulent spot stage is reached in figure 3.4(h1) The initial fall in maximum disturbance velocity may be due to the decay of large transient wavelets generated at the perturbation source Similarly, there is initial decrease in disturbance velocity for the CP case in figures 3.4(a2)-(b2) The disturbance velocity decreases slightly further as the wavepacket traverses over the CP region in figures 3.4(b2)-(c2) even until in figure 3.4(d2) Over this CP region, maximum disturbance velocities for the CP case in figures 3.4(b2)-(c2) are larger than the rigid wall case in figures 3.4(b1)–(c1), this is due

to the interactions between the compliant panel surface and the evolving wavepacket However, the positive effect of the embedded compliant panel began

to manifest itself by simply observing the wavepacket forms, as the wavepacket in the CP wall acquires a highly triangular form as it passes over the panel in Figure 3.4(c2), while over the rigid wall case in figure 3.4(c1) looks more crescent-shaped in formation

It was also noticed that the wavepacket disturbance velocity decreases slightly

as it leaves the CP region in figure 3.4(d2), then slightly increasing until in figure 3.4(g2) before it finally reached the last simulation stage in figure 3.4(h2) where its maximum disturbance velocity only reached 0.74% at its very early nonlinear stage In comparison, that is by considering the wavepacket evolution process until at time T = 2788 between the two cases in figure 3.4, the RW case in figure 3.4(h1) already broke down into incipient turbulent spot, whereas over the CP in

figure 3.4(h2) still at the earlier part of the nonlinear stage with a maximum disturbance u-velocity of 0.74%, signifying it still has a long way to go before breakdown into incipient turbulent spot will set in

It shows clearly that the rapid growth of the RW wavepacket over the CP wavepacket quite early in evolution history even though at some early stages the control panel wavepacket was larger in disturbance velocity Furthermore, and in order to know where the wavepacket over the single CP case will break down eventually, the simulation was continued until breakdown stage is reached The wavepacket finally broke down into an incipient turbulent spot in figure 3.5(f) at the location X ≈ 1930, which amount to an increase in transition distance of ΔX =

520 when compared to the RW case in figure 3.4(h1) that broke down earlier at X

≈ 1410 This translates to approximately 49% increase in the transition distance measured from the point of wavepacket initiation at X0 = 349.4

Figure 3.6 represents the single CP displacement behaviours along the centreline (Z = 0), that shows how the CP surface responded to the evolving wavepacket Figure 3.6(a) shows the displacement of the surface at time T = 186, when the front proper of the wavepacket is still some distance from the leading edge of the CP The impending arrival of the wavepacket is already felt at the CP surface, which responds with the generation of a fairly regular wave train This phenomenon of the leading edge (a point of singularity) acting as a source of CIFI waves on a compliant surface when subjected to flow perturbation has been highlighted by Yeo et al (1996), Wiplier and Ehrenstein (2000,2001), Wang (2003), and Wang et al ( 2005)

The driven or excited waves are most probably a native CIFI mode of the coupled CP-flow system Indeed one may even discern the reflection of the wave from the trailing edge At time T = 260 in figure 3.6(b), the front of the wavepacket has just arrived at the CP location causing a sharp localized increase

in the displacement at the front of the CP Localized displacement in figures 3.6(c)-(d) clearly mark the passage of the wavepacket over the CP surface The localized response of the CP can be more clearly seen in the displacement contours in figures 3.6(a)-(c), which show the footprints of the wavepacket as it comes onto and travels over the CP Figure 3.6(d) shows the wavepacket as it reaches the end of CP and the reflection of waves from the trailing edge The disturbances on the CP die away gradually when the wavepacket convects further downstream in figures 3.6(e)-(f) The maximum displacement of the CP reaches only up to about 0.24% of the local displacement thickness, which shows that the linearization assumption for coupled interaction is justified

3.3.2.2 Wavepacket spectral analyses

In order to properly understand the dynamics and the intrinsic phenomena involved as the wavepacket evolve from the point of perturbation untill incipient turbulent spot is reached, spectrum analyses were carried out on the obtained simulation data Double Fourier transforms of the flow quantities in time domain and sectional space domain were performed at selected non-dimensional heights ⁄ and -locations along the boundary layer Also, in order to appreciate the positive contributions of the inserted finite length of the compliant panels, spectrum analysis results were compared with those for over the rigid wall case

which were simulated under the same conditions Nyquist criterion was duly observed during sampling Spectral data extracted such as wavepacket frequencies and wavenumbers ( ) were converted to their local computational scale after extracting their values in the global length scales Also, only streamwise ( ) velocity spectra are considered in this analysis as it is the flow field velocity component that contains the bulk of the disturbance energy

Figure 3.7 compares the u-velocity spectrum plots in terms of spanwise wavenumber vs frequency plots between the rigid wall case (left column) and the single CP case (right column), in order to see clearly why the case with inserted single CP did not breakdown quickly as already observed in figure 3.4, thereby causing further transition delay Streamwise (X) locations considered include before, during and after the single CP location so as to properly understand the overall spectral dynamics of the wavepacket Figures 3.7(a1) and (a2) have almost the same spectral plots for both cases, as the wavepacket is yet to reach the CP location for figure 3.7(a2) Figures 3.7(b(2) to (e2)) are for when the wavepacket were directly traveling on top of the CP, and the interaction of the CP panel with the evolving wavepacket began to manifest itself as the spectral plots over this range differ from the rigid wall case

Distinct 2D and 3D mode wave peaks could be seen clearly in figures 3.7(b1) and 3.7(c1), whereas it is difficult to see the distinct 2D and 3D mode wave peaks

in figures 3.7(b2) and 3.7(c2), suggesting a strong interaction bewteen the CP and the evolving wavepacket, when it is directly on top of the CP For figures 3.7(d2)-(e2), the CP spectra of the wavepacket at these locations show weak low

frequency 2D (spanwise wavenumber ) waves, which are absent for over the rigid wall spectral in figures 3.7(d1)-(e1) These 2D or nearly 2D waves present in the wavepacket have something to do with its interaction with the compliant panel, which could be seen or observed in the CP displacement contours in figure 3.6 It was noticed that the said low frequency 2D mode continue to die away as the wavepacket convects away from the compliant surface region A conspicuous difference in spectral plot appearance began to be noticed from X = 776, as the wavepacket just evolved beyond the trailing edge of the CP

in figure 3.7(f2) compared to over the rigid wall case in figure 3.7(f1) At this location X = 776, the compliant wavepacket in figure 3.7(f2) loses most of its higher frequency 2-D waves ( ) and it is dominated by two oblique wave modes, whereas over the rigid wall case in figure 3.7(f1) the wavepacket retains a significant content of the 2-D waves

As the wavepacket convects further downstream, the dominant 2D wave modes continue to amplify over the rigid wall case in figures 3.7(f1)-(h1) until it reaches a size (1.3%) that allows it to interact effectively with the accompanying oblique wave pair (which have also grown correspondingly) by the nonlinear instability mechanisms of Craik (1971) or Herbert (1988), leading to the accelerated growth of the oblique wave pair in figures 3.7(i1)-(k1) From there onwards, further strong nonlinear interactions between the dominant oblique wave pair result in the development of low (near-zero) frequency waves for over the rigid wall wavepacket in figure 3.7(l1), and the final breakdown of the rigid wall wavepacket in figure 3.7(n1)