Reynolds number effect

It has been observed that the turbulent skin-friction drag reduction efficiency de- teriorates with increasing Reynolds number (Berger et al., 2000; Iwamoto et al., 2002; Choi et al., 2002), which is known as the Reynolds number effect in the

Figure 2.5: Spanwise oscillation by plasma actuators at four different phases, taken from Jukes et al. (2006a).

turbulent skin-friction drag reduction control. Due to the computational power re- striction in the DNS and the measurement accuracy limitation in the experiment, the Reynolds number for the flow control research in laboratories is typically the order of Reτ ∼ O(103), while the applications for the skin-friction control are at

the Reynolds number Reτ ∼ O(104) and above. Therefore, it is very important

to address whether a certain amount of drag reduction is still achievable at high Reynolds numbers.

Berger et al. (2000) reported that the drag reduction control efficiency by the spanwise oscillating Lorentz force control decreased as increasing Reynolds number in a range ofReτ = 100∼400. Iwamoto et al. (2002) studied the drag reduction by

the V-control blowing and suction, and reported the decrease in the drag reduction from 20% atReτ = 110 to 12% atReτ = 650. Koh et al. (2015) studied the drag re-

duction using the spanwise surface wave atReτ = 540,906,1908 and 2250 turbulent

boundary layers. The drag reduction decreased from 11% at Reτ = 540 to 1% at

Reτ = 2250. For the spanwise travelling wave, Choi et al. (2002) reported the opti-

mal drag reduction decreased from 44.5% atReτ = 100 to 34.1% atReτ = 400. The

authors proposed a power law scaling _{DR ∼} Re−_τα (α = 0.2) to quantify the drag reduction deterioration. The Reynolds number effect was also confirmed by Ricco and Quadrio (2008) at the similar Reynolds number and by Touber and Leschziner (2012) at the Reynolds number up toReτ = 1000 for the spanwise wall oscillation.

It was also observed that the drag reduction decreased with increasing the Reynolds number by the spanwise wall oscillation in the turbulent boundary layer (Skote, 2012). However, the Reynolds number effect was less clear in the experiment due to the uncertainty in the measurement (Ricco and Wu, 2004; Choi and Graham, 1998). Gatti et al. (2013) performed small box-size channel flow DNSs up toReτ = 2100

for the streamwise travelling wave, and reported the performance loss. However, the authors observed that the energy input for the streamwise travelling wave decreased with the Reynolds number (Psp ∼Re−τ0.136), thus they argued that a net energy sav-

ing by the streamwise travelling wave at the high Reynolds number was still possible. The drag reduction deterioration by the streamwise travelling wave of the spanwise wall velocity was also confirmed by Hurst et al. (2014) in the large box-size channel DNSs with the Reynolds number ranging fromReτ = 200∼1600. The authors also

noticed some other interesting phenomena: 1) the optimal control parameter shifted towards a higher oscillation frequency for the spanwise wall oscillation, and towards a higher streamwise wavenumber for the streamwise stationary wave; 2) the power law scaling parameterαwas found to be control-parameters (ω, κx) dependent, but

α was positive for all the drag reduction cases, which suggested a drag reduction deterioration.

The Reynolds number effect explored in the DNS is typically for a small range of Reynolds numbers, namelyReτ <1000, thus the prediction for even higher

Reynolds number is normally done with theoretical works (or affordable approaches). For example, Iwamoto et al. (2005) derived an explicit formula for the drag reduc- tion with the turbulent fluctuation in the near wall layer below a threshold location

yd completely damped, and showed that a 35% drag reduction was still achiev-

able at the Reynolds number up to Reτ = 105. Fukagata et al. (2006) applied

the same argument to the drag reduction control by the superhydrophobic surface, and a large drag reduction was predicted to be possible at the Reynolds number

Reτ = 105 ∼106. For the streamwise travelling wave by the spanwise wall veloc-

ity, Duque-Daza et al. (2012) used the linearised Navier-Stokes to predict the drag reduction map at the high Reynolds number Reτ = 2594, which was found to be

very similar to the one at Reτ = 200, with only ∼5% difference inDR (more pre-

cisely the streaks amplification). Similarly, a perturbation analysis was carried out by Belan and Quadrio (2013), who suggested a much weaker _DR deterioration as the Reynolds number increased, and also an asymptotic value for the constant drag reduction above a threshold Reynolds number. Quadrio and Gatti (2015) and Gatti et al. (2015b) assumed the change of the constantB in the log law from the no con- trol case to the streamwise travelling wave cases, is Reynolds number independent,

thus the authors estimated that the drag reduction at high Reynolds number was possible. Skote (2014) proposed an argument about the scaling of the streamwise mean velocity profile for the near wall control strategies without changing the fluid properties and the outer layer. The theory prediction agreed well with the available DNS data, but the constant B is _DR dependent. Those predictions are very opti- mistic for the drag reduction control community, though it may take some time for the DNS and the experimental validation.

The Reynolds number effect suggests that the control parameters are not scaled in the inner units at high Reynolds numbers, similar to the inner scaling failure for the turbulent statistics of the no control high Reynolds number flows (DeGraaff and Eaton, 2000). Touber and Leschziner (2012) pointed out that the Reynolds number effect in the spanwise wall oscillation was due to the VLSMs in the outer region, which clearly caused the difference between the near wall streaks under the positive and the negative VLSMs. The superposition and modulation effects by the VLSMs were also used to explain the Reynolds number effect in the blowing and suction control at Reτ = 1000 by Deng et al. (2015). The VLSMs

carry a significant amount of Reynolds shear stress (Guala et al., 2006; Deck et al., 2014), and it was found that by purely controlling the large scale structures, it gave as much as 20% drag reduction (Schoppa and Hussain, 1998; Fukagata et al., 2010; Pujals et al., 2010; Schlatter et al., 2015). However, Iwamoto et al. (2002) showed that for the opposition control, the_DRdeterioration was only strong forReτ <300,

and _DR became insensitive to the Reynolds number for Reτ > 300. The asymp-

totic behaviour of _DR was also observed by Hurst et al. (2014) for the stationary wave of the spanwise wall velocity (see their figure 10(b)). Skote et al. (2015) thus suggested a power law scaling to replace the log law for the_DRscaling in the span- wise wall oscillation control. Using FIK identity, Hurst et al. (2014) showed that the _DR deterioration mainly came below the critical layer (y+ _{< 2Re}1/2

τ ), while

the _DR contribution from the outer region was almost constant from Reτ = 200

to 1600. This finding was consistent with Iwamoto et al. (2005), who showed that the Reynolds number effect was mild if the near wall turbulence was completely damped, though the damping layer thickness needed to increase slightly with the Reynolds number for the same amount of_DR. These results suggested the impor- tance of the scale interaction across the wall normal direction. Iwamoto et al. (2002) demonstrated this point using the Karhunen-Loeve decomposition, and they showed that at Reτ = 110 and 300, the largest contribution to the skin-friction came from

the structures within 15 < y+ _{< 30, but the contribution from 30 < y}+ _{< 75 is} also important forReτ = 300. The structures within 30< y+<75 was beyond the

direct control, but can transfer the energy to those structures within 15< y+_<30, causing the_DRdeterioration. Thus the authors suggested the control of the struc- tures within 30< y+<75 was necessary at high Reynolds numbers. Iwamoto et al. (2002) claimed that the structures at y+ _{> 75 remained inactive in terms of con-} tributing to the skin-friction, and this might be due to the limitation of the highest Reynolds number studied,i.e.,Reτ = 300. Very recently, de Giovanetti et al. used

three different approaches,i.e., FIK identity analysis, spanwise domain confinement and artificial scale damping, and showed that the scales with 0.2h _≤λz ≤1h con-

tributed the most to the skin friction at Reτ = 2000. The turbulent structures

within this scale range are in the logarithmic region, and they form a hierarchy of the self-similar attached eddies (Flores and Jim´enez, 2010; Hwang, 2015).

It is clear the outer structures (including the logarithmic structures, LSMs and VLSMs) play an important role in the_DR deterioration by indirectly transfer- ring energy to the near wall structures, causing the control on the near wall structure to be less effective. But whether or not we need to control the large scales from the outer region for a better_DRperformance still remains open.