Flow stability and transition over finite compliant panels 5

Chapter 5 Compliant panel properties Zhao 2006 had selected a set of properties for the membrane which had proven computationally in the preceding study to give fair good transition d

Chapter 5

Compliant panel properties

Zhao (2006) had selected a set of properties for the membrane which had proven (computationally) in the preceding study to give fair good transition delay properties The choice was partially explained in Appendix A of her thesis In this chapter we will examine this choice more carefully Wavepacket initiations, evolutions and breakdowns into incipient turbulent spot over single compliant panel (CP) and multiple compliant panels had been studied in chapters 3 and 4 respectively In those chapters, transitions were delayed from about 49% to 112% when compared against the rigid wall (RW) case where the same initiating pulse perturbations were used in all the cases Same compliant panel properties and length considered by Zhao previously were used in both chapters 3 and 4, which prevented both the static divergence (SD) and travelling wave flutter (TWF) from developing and dominating the transition process The finite and relatively short lengths of the panels played a part in the suppression of those wall-motivated modes

However, in the course of ruminating and critically reviewing over what could

be termed a brief study on membrane properties by Zhao (2006), some comings were noted, which could possibly be improved upon so as to make the compliant panel(s) perform better than what had been achieved so far, and also to perform more detailed analyses for better insights into the performance of CP while tuning different CP properties Some of the noted limitations of Zhao (2006) include:

short-(1) The use of what appeared to be an unnecessarily stiff membrane, with a high estimated onset velocity for travelling wave flutter (TWF) of 3.16 , whereas a factor between say 1.1 to 2.0 might appear to be adequate against these wall-based modes (more description of wall modes is given in the next section 5.1) (2) Zhao (2006) studied how compliant wall properties affected the interactions between the membrane panel and the flow in a reduced domain that extends to X

≈ 910 and time T = 1300 How variations in the panel properties might bear on the all-important question of transition delay was not addressed

(3) Due to the point raised in (2) above, all simulations later carried out on membrane property study by Zhao (2006) were trimmed down to half sizes of the initial computational domains along the streamwise directions Observing the overall behaviours of the evolving wavepacket in terms of both spatial and spectral analyses far downstream after the CP region, until incipient turbulent spot

is reached will provide more detailed information than what was done by Zhao (4) Moreover, the simulations involved in the property studies were carried out with fairly coarse mesh of 47 grid points in Y (wall-normal) and 81 grid points in the Z (spanwise) directions In this chapter, we hope to address some of these limitations as we delve into the effects of some of the leading properties of the compliant panel on transition delay

5.1 Compliant panel surface waves and instabilities

According to Gad-el-Hak (1998), a rich variety of fluid-structure interactions exists when a fluid flow over a surface that can comply with the flow Instability modes amplify when two wave-bearing media are coupled together These waves could be flow-based, wall-based or due to coalescence of both kind of waves So far, two common types of surface waves have been identified, which are related to the flexible quality of the wall involved The two types are static standing wave, which is also known as static divergence (SD) waves, and travelling waves also called travelling wave flutter (TWF) The presence of surface waves on compliant panels subjected to either laminar or turbulent flow, could generally be seen as arising from an instability of the flow-wall dynamical system in coupled interaction according to Yeo et al (1999)

Two types of instabilities that are directly related to the compliant quality of the wall could be collectively grouped into compliance-induced flow instabilities (CIFI) according to Yeo (1988) and flow-induced surface instabilities (FISI) as termed by Carpenter and Gerrad (1985, 1986) The TWF instabilities are related

to the free surface wave modes of the compliant wall, and assume the form of a travelling wave propagating at a fair speed The TWF instability mechanism involves the irreversible transfer of energy from the flow to the wall owing to the work done by the fluctuating pressure according to Lee et al (1995) TWF is the cause of very sudden onset of transition The SD instability is related to the static deformation modes of the compliant wall, and is manifested as slowly moving wave

SD instability occurs when the hydrodynamic pressure forces generated by a small disturbance outweigh the restorative structural forces in the wall Just to mention few from the literature, Hansen et al (1983) observed SD waves on viscoelastic layers with a meaningful high level of damping in their rotating-disk experiments Same SD waves were also observed by Gad-el-Hak et al (1984) in their uni-directional flow experiments Later on, a review of SD waves on compliant surfaces was given by Riley et al (1988) On the other hand, that is for the TWF waves, Lucey and Carpenter (1995) validated the results obtained from the theory of wall-based travelling wave flutter with the experimental results

5.2 Estimates for the onset speeds of divergence and travelling wave flutter

According to Carpenter (1998), in order to achieve best possible transition delay with either single or multiple panel walls, the essential concept underlying the optimization procedure is to use estimates (which was based on potential flow assumption) for the onset speeds of divergence and travelling wave flutter in order

to choose the properties of a set of compliant walls each of which corresponds to marginal wall-based stability at the design flow speed with respect to both of these hydroelastic instabilities Modified potential flow theory (see Duncan et al (1985) suggests that wall-based modes might be inhibited if we have

* + (5.1)

where √ and ( ) √

where are the onset flow velocities for TWF and SD modes respectively for zero-pressure gradient laminar boundary layer following Duncan et al (1985)

These simple criteria are useful, though they may not be valid in the context of a viscous boundary layer For application to viscous boundary layer, we may further modify these criteria by the incorporation of factors of safety C and D, that

is,

{ } (5.2) The factors C and D are the ratios of the onset velocities for Hydroelastic instability (TWF and SD respectively) relative to the actual free stream flow speed The higher C and D are, the less likely will TWF and SD waves be triggered Hence they are termed factors of safety We note that in both of criteria (5.1) and (5.2), if is allowed to tends to zero Figure 5.1(a) shows schematically the unstable regions associated with the TWF and SD for

cannot be sustained on a short panel In this regard if we set , then the criterion (5.2) reduces to:

( ){ } √ (5.4) where

(5.5) Hence by selecting a compliant panel length , equations (5.4) and (5.5) allow us

to estimate the mass density (m) and tension (T) of the compliant panel for a prescribed factors of safety in accordance with the formula (5.5) and from (5.4) In the earlier study of chapter 4, (which translate to base on the wall reference length scale of ) In this interpretation of how wall-based modes may be suppressed, Zhao (2006) had employed a high factors of safety of C = 3.16 against TWF and a more reasonable

D = 1.32 against SD The factor C = 3.16 would appear to be an unnecessary high value to adopt for the compliant membranes Whether it is indeed very high remains to be seen below; as we have to recognize that these stability estimates (5.1 to 5.5) were (i) based on a highly simplified potential flow model, and (ii) there might be complex edge effects in finite-length panels that could result in unstable wave interaction within the panels

5.3 Compliant panel properties parametric study: Cases investigated

After establishing the formulae (5.4) and (5.5) for the estimation of compliant

panel mass density (m) and tension (T) in section 5.2, further attempts were made

to investigate the effects of prescribing lower safety factors C and D (as safety factors C and D control how close the walls are to the critical SD and TWF velocities) on transition delays for both the single CP and two-CP cases Six different parametric study cases were investigated, and these were compared with the results earlier obtained in chapters 3 and 4 termed “reference” case in this chapter and CP properties are summarized in table 5.1 after applying equations (5.4) and (5.5) Cases 5 and 6 may be regarded as a subset in which only the damping coefficient is varied

Table 5.1 Compliant panel parameters of cases 1-6, where m L represents the surface mass density; T L is the compliant panel tension; d L is the damping factor; k L is the foundation spring stiffness, and Re L as the wall reference Reynolds number Reference case is for the

CP parameters/properties used in chapters 3 and 4 simulations Compliant panel length

Safety Factors Compliant panel parameters

The subscript L in table 5.1 represents a wall length scale which was used to specify material properties for the wall The relation between the wall properties, subscripted by L, and their computational equivalents are given by:

(5.6) where

(5.7) and superscript asterisk * signifies that the quantity is dimensional

The simulation conditions for all the cases studied in table 5.1 are identical to those of the reference case, which had already been presented in chapter 3 of this thesis The location of the embedded single compliant panel (CP) remain unchanged, that is, at same location X = 450 – 762 Also, second CP location at X

= 1359 – 1658 remains the same for two-CP simulations In addition, apart from further tuning the CP properties, some other distinct differences between what Zhao (2006) did and the approaches applied in this present study are: (1) a much larger computational domain was used here to allow the wavepacket to reach the breakdown stages for each of the cases investigated, as Zhao’s domain length was

much shorter and did not permit breakdown to occur, (2) the grid resolution was

greatly increased in both the spanwise (Z) and wall-vertical (Y) lengths of the computational domain over the previous study to ensure that fine details of flow are well resolved and captured in simulations

5.4 Results and discussions for over the single CP case

Simulation results are presented systematically and chronologically in the order

of as indicated in table 5.1 Cases with unfavourable outcomes (that is, early breakdown locations than the reference case) are summarily discussed while the favourable ones are presented in more details

5.4.1 Spatial evolution analyses

First to recall again that the breakdown location for over the single CP in chapter 3 (reference case here) is X ≈ 1930 This was used to compare all the single CP cases 1 – 6 investigated, that is, to know if they performed better or not

in terms of further transition delay than the reference case In order to make presentation of results straight forward and without any confusion whatsoever, results are divided into two main groups namely: (i) cases that broke down earlier than the reference case and (ii) cases that performed better than the reference case

5.4.1.1 Cases that broke down earlier than the reference case

Cases 1, 2, 4 and 5 wavepacket broke down at X ≈ 1770, 1750, 1680 and 1910

respectively as shown in figure 5.2 for the u-velocity components, that is, earlier before the X ≈ 1930 breakdown location for the reference case in chapter 3 Case

4 (figure 5.2(c)) suffers the earliest breakdown among the four in this group, with its breakdown location centers around X ≈ 1650, this has to do with the facts that the CP is of low surface mass density (mL) and tension (T) relative to the reference case On the other hand, case 5 in figure 5.2(d) performs the best in this

group with breakdown close to X ≈ 1910 What made case 5 different from case 4

is the foundation damping dL, which is half the value for case 4 This shows that

CP damping has a significant effect on the transition, and that the key phenomenon underlying the growth process is class A in nature With this at the back of the mind, first attempt was made to set safety factor C = 1.3, that is, to something closer to the value of 1 With this safety factor C set-up already, surface mass density (mL) jumped to almost six times in case 1 to that of reference case as shown in table 5.1 after applying equation (5.5) Parameter D values are almost the same for both the reference case and case 1

Since it had already been confirmed in Case 1 that C value very close to 1 did not result into a better transition delay, then, second attempt was made to raise the

C value to 2.5 in case 2, with the aim of reducing mL value closely to 1 at the end

of the whole process Still, case 2 broke down earlier than the reference case as shown in figure 5.2(b) and this may be attributed to a decrease in CP’s tension value to T = 8.25 Above all, none of the cases in figure 5.2 performed much better than the reference case; however, investigating them gave a clue on how the

CP properties could be further and carefully tuned in order to achieve the goal of delaying transition beyond the location of X ≈ 1930 recorded for the single CP reference case Finally, since the results obtained under this group did not show gain in transition delay, we did not go into the further analyses of these cases

5.4.1.2 Cases that performed better than the reference case

Cases 3 and 6 performed slightly better than that of the reference case in terms

of further transition delay and their spatial evolution comparison for all the three cases is shown in figure 5.3 The wavepacket evolution processes before figure 5.3(a), that is, before CP location (X = 450 to 762) is the same for all the three cases and we did not include them in figure 5.3 Case 6 was an attempt to take full advantage of CP with low damping Since the wavepacket formed for the reference case is almost identical to those for case 3, more discussions in this section will be focused on case 3 and 6 only In figure 5.3(a), wavepackets are fully convecting on top of the CPs, with the wavepacket maximum disturbance velocity and wavepacket shapes for both reference case (figure 5.3(a1)) and case

3 (figure 5.3(a2)) are almost identical While that of the case 6 in figure 5.3(a3) possesses more elongated streamwise oriented structure along the wavepacket center with a maximum velocity of 0.54%

Maximum disturbance velocity for case 6 in figure 5.3(b3) is 0.35%, while that

of case 3 in figure 5.3(b2) is 0.48%, this serves as an indicator that case 6 which is characterized by soft wall coupled with low damping may delay transition better than case 3 In figure 5.3(c), all the three wavepackets had just exited the CP locations completely and where the full effects of the compliant panel on the wavepacket could be most clearly discerned There is almost 50% drop in maximum disturbance velocity for case 6 in figure 5.3(c3), compared to case 3 in figure 5.3(c2) Wavepacket disturbance velocity for case 3 became noticeably larger than the reference case in figure 5.3(d), which looks a bit contradictory as