Flow stability and transition over finite compliant panels 4

4.1 Over an array of two compliant panels In order to further enhance the performance of the first compliant panel located at in terms of delaying transition further, a second complian

Chapter 4 Investigations over two and three compliant panels

An attempt made in chapter 3 with single compliant panel (CP) occupying a certain section of the rigid wall, yielded a transition delay to the tune of almost 49% when compared with the rigid wall (RW) case This feat with the single CP case in terms of better transition delay serves as a motivation to consider multiple CPs and see if transition distance could be increased beyond distance already obtained for the single CP case in chapter 3 This approach of using multiple compliant panels for superior performance as compared with the relatively simple walls studied so far had already been recommended by Gad-el-Hak (2002) However, it is worthy to note that this particular case (multiple CPs) proposing here within the Blasius boundary layer has not been studied so far in the literature Davies and Carpenter (1997) numerically studied evolution of a TS waves over finite compliant panels but within a fully developed 2D plane Poiseuille flow, rather than in a Blasius boundary layer Moreover, the perturbations were 2D and monochromatic rather than 3D broadband What they did to each compliant panel was to tailor their properties to suit the local flow environment for series of compliant panels used They concluded that with low-disturbance flow environment, laminar flow up to an indefinitely high Reynolds number could be sustained if suitably designed multiple-panel compliant walls are used It is nevertheless pertinent to note that transition to turbulence could occur via 3D waves

Zhao (2006) attempted multi-CP investigations, but her simulations had stopped short of the late subharmonic and breakdown stages due to resource limitations (memory and speed), so that the ability of these walls to delay transition could not be evaluated This chapter concerns the use of multiple compliant panels to delay transition further than what had already been obtained for over the single CP case in chapter 3 Based on the number of compliant panels considered, investigations carried out are grouped under two main headings as: (i) over an array of two compliant panels and (ii) over an array of three compliant panels Wavepacket evolution processes are also analysed in terms of both spatial and spectral analyses Spectral data extractions for dominant 2D and 3D wave modes were also performed so as to properly understand the overall behaviours and performances of each of the CP as the wavepacket evolve over them

4.1 Over an array of two compliant panels

In order to further enhance the performance of the first compliant panel located at in terms of delaying transition further, a second compliant panel of the same length was added at the location (with corresponding local Reynolds number ) thereby forming an array of two panels as shown in figure 4.1 This time around the size

of computational domain in the streamwise (X) direction has to be increased further from X = 1510 to 3083, while that of the wall vertical (Y) and spanwise (Z) directions remain unchanged The second panel was placed just ahead of the wavepacket at its second last stage of development at time T = 2417 of figure

3.4(g2) At that stage, the wavepacket registered a maximum-u velocity value (at

⁄ ) of 0.59% - the purpose of the second added compliant panel is to inhibit the rapid growth of the wavepacket from that point onwards as indicated in Figure 3.5

4.1.1 Results and discussions

Also like the single compliant panel (CP) case in chapter 3, the results for over

an array of two-panel arrangement are discussed in terms of both spatial evolution and spectra analyses

4.1.1.1 Wavepacket spatial evolution analyses

Figure 4.2 shows the evolution of the wavepacket over the second compliant panel from time T = 2788 until the incipient turbulent spot is reached at T = 6062 The wavepacket begins to pass over the second CP at T = 2788 in figure 4.2(a) While figures 4.2(b)-(c) show when the wavepacket was fully convecting on the top of the second CP where a fairly sharp maximum disturbance velocity increases to 1.2% This sharp jump in disturbance wave disturbance velocity value was due to the fact that the convecting wavepacket encounters a sudden softening of the wall due to the presence of a CP surface In addition, wavepacket structures begin to take into an oblique shape formation in figure 4.2(b) and wavepacket becomes more oblique with longitudinal structures in figure 4.2(c) This is likely caused by the stronger suppressing influence of the compliant panel

on the 2D modes than the 3D (oblique wave) modes

Also, it is worth noting that the wavepacket disturbance velocity does not increase that much even as the wavepacket continues its journey up to the end of the second CP in figure 4.2(c) where the maximum disturbance velocity only reached ≈ 1.1% Significantly, this seems to reflect the ability of CP to inhibit the growth of the wavepacket even at this nonlinear wave disturbance value At this stage, the maximum displacement (not shown) of the CP is up to about 0.668% of the local displacement thickness (or about 0.2% of boundary layer thickness), which shows that the linearization assumption for coupled interaction is basically justified

The wavepacket’s maximum disturbance velocity increases to 1.26% at time T

= 3906 (figure 4.2(d)) after it has just left the CP behind This is significantly less than the 4% reached in the single CP case given in figure 3.5(c) Figure 4.2(e) shows the progessive evolution of the wave disturbances as they formed longitudinal structures with maximum disturbance velocity reaching 2.76% when

T = 4650 This is past the point of the breakdown of the single CP case While figures 4.2(f) signifies that wavepacket breakdown is imminient as the presence of

a strong pair of forward jets with maximum disturbance velocity of 4.11% which are shifted away from the centre location Also, figure 4.2(g) shows two pairs of jets, one close to the centre line while the second pair a bit far away from centre region Eventually, figure 4.2(h) shows the breakdown of the wavepacket into fully turbulent state Incipient turbulent spot occurred when the wavepacket was centred at the location X ≈ 2350 By comparing with the rigid wall (reference) case that earlier broke down into incipient turbulent spot at X ≈ 1410, transition

distance to the occurrence of the incipient turbulent spot was increased further to about 89% relative to a rigid wall case when the second compliant panel was introduced

4.1.1.2 Wavepacket spectral analyses

The frequency (ω) vs spanwise wavenumber (β) plots for when the wavepacket was riding upon the second CP located at X = 1359 to 1658 and until incipient turbulent spot is reached are shown in figure 4.3 for the streamwise (u) velocity, while that of the wall-vertical (v) velocity are shown in figure 4.4 Figures 4.3(a)–(b) show when the wavepacket is travelling directly over the second CP and it is difficult to discern the maximum peak locations for both the 2D and 3D wave modes especially in figure 4.3(b) as some modes are induced by the CP surface itself Figure 4.3(c) shows the spectrum plot at a position X =

1726, that is, a short distance after the trailing edge (X = 1658) of the second CP The spectrum shows the dominance of a pair of oblique waves at frequency of ω

≈ 0.025 (ωδ = 0.056) and β ≈ ±0.1 (βδ ≈ ±0.222)

With close observation, there is no much disparity between figures 4.3(c) and (d) except for sideway proliferation of some near zero frequency modes As the wavepacket evolves downstream, the spanwise wavenumber increases slightly and later decreased to β ≈ ±0.09 (βδ ≈ ±0.221) for the dominant wave modes in figure 4.3(e) Also the spectral peaks of the dominant wave pair begin to stretch towards lower frequency, and this trend continues for a while until it rises again to

a value of ω ≈ 0.025 in figure 4.3(e) Same figure 4.3(e) is characterised by the

sideway growth of other higher harmonics along the spanwise wavenumber β axis

to signify that the evolving wavepacket already in the post-subharmonic phase Figures 4.3(e)-(g) show when the wavepacket is about to reach the breakdown stage without any distinctly dominant 3D wave modes anymore All the leading modes have near-zero frequencies at this stage The breakdown of the wavepacket into incipient turbulent spot is shown in figure 4.3(h) which is marked with disturbance energy concentrating in the near-zero frequency The frequency (ω)

vs spanwise wavenumber (β) for the wall-vertical (v) velocity in figure 4.4 also shed more lights on the spectra behaviours of the wavepacket over the second CP Figure 4.4(b) shows clearly that the evolving wavepacket is directly on top of the second CP, which is due to the vertical displacement of the second CP surface Not only these, the 2D modes are more pronounced than the 3D modes in figures 4.4(c)-(e)

Finally, breakdown process into incipient turbulent spot is clearly revealed in figures 4.4(f)-(h) The corresponding streamwise wavenumber (α) vs spanwise wavenumber (β) plots for u and v components are shown in figures 4.5 and 4.6 respectively The spectra exhibits exactly the same type of spectral stretching except that the stretch is now towards zero streamwise wavenumber (representative of highly longitudinal structures) Three local peaks along the spectral ridges also appear even though they look not so distinct Another observation is that dominant 2D modes streamwise wavenumber nearly remained the same in figures 4.5(b)-(e) and 4.6(b)-(e) as only the oblique waves undergo

some changes Figures 4.5(f) and 4.6(f) mark the breakdown stage of the wavepacket into incipient turbulent spot

Figure 4.7 shows the wavepacket growth rate for over the two-CP case and this

is compared with the growth rates obtained previously in chapter 3 for both the rigid wall (RW) and single CP cases respectively First to mention that the two CP locations are far apart (with a separating distance of ∆X ≈ 597) from each other, and therefore the second CP has absolute no effect on the flow before it Since flow properties before the second CP location remained the same as the single CP case in chapter 3, only the behavioural growth over the second CP will be discussed, as the comparison between the RW and single CP cases had already been discussed in section 3.3.2.3 with figure 3.12 Second CP at the location X =

1359 – 1658 is noted in suppressing the 2D wave modes amplitudes from growing further at stations X = 1553 – 2026, that is when the evolving wavepacket was riding over the second CP and even until after it had just left the second CP behind at X = 1726 Up to location X = 2026, the wavepacket amplitude value for the second CP case yet to attain what the single CP case reached at X = 1811 Also, a sudden drop in 2D wave amplitude value at X = 1726 (just after the end of the second CP) was not noted for the single CP case, as a gradual increment in 2D wave amplitude value are expected as X increases along the streamwise direction The reduced growth of the wavepacket may clearly be attributed to the action of the second CP, which we are uncertain about the actual mechanism at work It is likely the linear suppressing effect of wall compliance on TS wave growth is still

at work, although at these amplitude levels nonlinearity may be very active as well

For the 3D wave modes, amplitude values keep increasing from X = 1726 -

2113 unlike the dominant 2D mode which drops in amplitude over the second CP before it continued to rise in amplitude after the panel Also, even as the breakdown is imminient for the second CP case, the 3D mode amplitude value keeps increasing until X = 2113, that is, before a state of proliferations of many 3D wave modes began to set in For all the three cases compared in figure 4.7, it

is obvious that the inserted CPs indeed performed their expected duty of suppressing 2D wave modes from growing further, which in turns resulted into a better transition delay than their rigid wall case These further confirm the efficacy of compliant panel(s) in delaying transition of the Blasius boundary layer provided they are suitably located within the boundary layer

4.1.1.3 Spectral properties of dominant 2D and 3D wave modes

Similar to what was done for the single CP case in chapter 3, extractions of spectral properties from the u-velocity spectral plots in figures 4.3 and 4.5 were also carried out for the dominant 2D and 3D wave modes respectively, that is, before and after the second CP located at X = 1359 to 1658 All extractions were carried out at the height ( ) ⁄ as before, and data obtained are presented

in table 4.1 in terms of the local displacement thickness length scale ( ) All data obtained up to station X = 1035 remained the same as the single CP case in table 3.1 and no need to repeat them again under this section

X

The plots of spectral properties of dominant 2D and 3D wave modes are plotted in figure 4.8 Figure 4.8(a) shows the overall behaviour of dominant two-dimensional frequency wave modes before and after the second CP location The presence of second CP further caused a decline on the 2D mode frequency values from at X = 1380 to at X = 1639 This decline continues even up to the station X = 1812, that is, after the evolving wavepacket had already exited the second CP location Thereafter, 2D frequency began to

increase in values again as the wavepacket finally evolved on the rigid wall this time around where the 2D frequency value almost remain fixed ( ) Figure 4.8(b) shows the associated frequency ratio ⁄ This ratio suggested that the wavepacket contents contain 2D wave modes which operate at

a higher frequency than their 3D wave modes especially after the wavepacket had already left the second CP location behind Also, subharmonic resonant wave condition according to equation (3.10) does not appear all because: (1) the frequency ratio of obtained at the last station X = 2113 is far above the expected value of 0.5, (2) triad of waves locations are difficult to discern in the spectral plots of figure 4.3 This also suggests that the dynamics of the wavepackets as they passed through the single CP (in chapter 3) and the two-CP (in the present chapter) are somewhat differently

Wavenumber ratio ⁄ plot is shown in figure 4.8(c) Within the second

CP region, 3D wave modes are characterized with higher streamwise wavenumbers than the 2D wave modes However, the reverse is the case after the evolving wavepacket had already left the second CP region behind; with 2D wave modes streamwise wavenumber becoming larger than the 3D wave modes The wavenumber ratio ended with around 0.38 which is below the expected value of 0.5 if the condition stated in equation (3.10) is to be referred Figure 4.8(d) compares the downstream phase speeds ratio ⁄ of the dominant 3D and 2D modes of the wavepacket for over the second CP Just after the second CP location, the x-phase speed for 3D wave modes started increasing from X = 1726

to a state where it finally ended to almost 2.2 times that of 2D wave modes at the

streamwise location X = 2113, after which distinctive dominant 3D modes could not be discerned any more from the spectral plots

Figure 4.8(e) shows if wavepacket over the second CP really fulfilled the Squire wavenumber condition in equation (3.11) or not This condition is only fulfilled when the wavepacket had already left the second CP far behind starting from station X = 1812 downwards Deviation from the expected value of 1.0 was observed especially when the wavepacket was travelling directly on top of the second CP due to solid-fluid interactions’ existence Propagation angles θ of the dominant 3D modes for over the second CP case is shown in figure 4.8(f) Θ value almost reached 70o at X = 2113 as the wavepacket evolves towards breakdown

4.2 Over an array of three compliant panels

Having obtained transition delays of about 49% and 89% with the single CP and two-CP cases in sections 3.3 and 4.1 respectivley, further motivation arose to see if a third panel could be inserted and if that could result in further transition delays than the results previously obtained Based on this, the question that later came to our mind is where should the third CP be placed in order to achieve a much further transition delay before incipient turbulent spot set in In an attempt

to answer the raised question, a series of trial for the three-CP case was carried out in order to know a suitable location where the third CP should be embedded for us to obtain a better transition delay

4.2.1 Earlier simulation trials with three compliant panels

Three different simulation trials were first carried out to seek a better location for the third CP that will further extend or delay the onset of the incipient turbulent spot The wall normal and spanwise dimensions of the computational domain remain fixed throughout, while the streamwise length is further increased

to accommodate a delayed breakdown The three trials carried out are described

as follows:

(a) Placing all the three CPs equidistant from one another

Under the first trial, third CP was placed at the location X = 2255 – 2554 as shown in figure 4.9(a), that is, equidistant between the first CP and second CP, and also between the second CP and third CP At the end of the simulations, wavepacket broke down into formation of turbulent spots, almost at the same location X ≈ 2350 (see figure 4.10) as the two-CP case of section 4.1 There is no significant improvement in terms of transition delay than what had already been obtained for over the two-CP case The conclusion drawn from this first trial is that, third CP location of X = 2255 -2554 seems to be placed too far from where it could still be able to interact positively with the evolving wavepacket before it broke down to turbulence The current location of the third CP is already near the breakdown region for the two-CP case, where the wavepacket has already attained

a maximum disturbance velocity magnitude of ≈ 5% The spectral plot of figure 4.3(f) also attested that this third panel location is already within the breakdown region for the two-CP case in section 4.1

(b) Shifting the third CP inward to the location X = 1955 - 2256

As the equidistant arrangement of three CPs in 4.9(a) did not work according

to the desired expectation in terms of delaying transition further, another trial simulation was performed by shifting the third CP inward (towards the second CP) to a new location X = 1955 – 2256 as schematically shown in figure 4.9(b) Few selected results as the wavepacket heading towards incipient turbulent spots are shown in figure 4.11 By closely observing figure 4.11(c) at T = 5467, it shows that the evolving wavepacket broke down at X ≈ 2270 (wavepacket approximate centre location), which is slightly earlier than the two-CP case that broke down at location X ≈ 2350 as reported in section 4.1

(c) Shifting the third CP further inward to the location X = 1732 – 2032 The last attempt was made to shift the third CP further inward than before to a

new location X = 1732 - 2032, that is, closer to the second CP location as shown

in figure 4.9(c) Figure 4.12 shows selected results of the wavepacket heading towards breakdown as well It shows that the wavepacket broke down at the location X ≈ 2470, that is, in between figures 4.12(c) and (d) It is interesting to note that incipient turbulent spots for all the three-CP cases assumed a highly streamwise streaky appear compared to the two-CP case in figure 4.2(h) The current three-CP case yields a slight increase in the breakdown location by approximately 6% when compared to the two-CP case (section 4.1) that broke down at location X ≈ 2350 In terms of comparing with the rigid wall (RW) case that broke down into incipient turbulent spot at the location X = 1410, this