Spatial transient response

The Lorentz force normally can only be applied to one portion of the wall. To test the effect of the force locality, the Lorentz force parameters of the baseline case (Af = 0.5, ∆+ = 10, T+ = 100) is applied to only the first half of the channel

domain on both the bottom and the top walls. Periodic boundary conditions are applied in the streamwise direction, thus this flow is different from a true boundary layer flow simulation. To avoid the numerical instability due to the sharp change of the Lorentz force at the interface, a step function (Yudhistira and Skote, 2011) is applied with a window size of 0.5, as shown below,

S(x) =      0, if x_≤0, 1/ 1 +e1/(2x−1)+1/(2x), if 0< x <0.5, 0, if x_≥0.5. (5.9)

100 200 300 400 500 0 5 10 15 20×10 -3 t+ Cf 0 30 60 90 -30 -20 -10 0 10 20 30 40 γ D R Lorentz (present) Wall (Zhou & Ball 2008)

(a) (b)

Figure 5.4: Effect of oscillating Lorentz force direction γ: (a) time history of the skin-friction Cf (dashed lines are forγ ≤45◦, and solid lines are for γ >45◦); (b)

drag reduction values compared with wall oscillation cases (Zhou and Ball, 2008). Figure 5.5 shows the instantaneous streamwise velocity contour at y+ _{≈ 5} for three different channel lengths. Clearly, the high- and low-speed streaks are skewed in the controlled region, while they recover and align again in the streamwise direction in the second half of the channel, where the Lorentz force is off. At the interface between the control and the no control regions, the streaks behaviour is very similar to the experimental observation for spanwise wall oscillation (Choi et al., 1998) and spanwise oscillating Lorentz force (Pang et al., 2004).

Figure 5.5: Near wall streaks aty+_≈5 for test domain of Lx= 16, 32 and 64. The

oscillating Lorentz force (Af = 0.5, ∆+= 10, T+= 100) is applied in the first half

of the domain length (indicated by black blocks).

The time, spanwise and top-bottom wall averaged skin-friction coefficientCf

is plotted in figure 5.6 as a function of the streamwise coordinate. When the Lorentz force is applied, the Cf starts to drop quickly. This transient process is as long as

2005; Yudhistira and Skote, 2011; Lardeau and Leschziner, 2013). To make theCf

drop to the level of a full domain controlled case, the domain size must be very long, up to Lx = 64. By further increasing the domain size to Lx = 128, the transient

Cf value can be even lower than the full domain controlled case, which is not the

case in the turbulent boundary layer. This undershooting of Cf may be related

to the periodic boundary condition effect. Once the flow enters the channel region without the Lorentz force, theCf level starts to increase quickly. Interestingly, the

increasing length seems to be domain size independent, and is fixed at around 15 forLx= 32, 64 and 128 three cases.

0 30 60 90 120 5 6 7 8×10 -3 x Cf Lx= 16 Lx= 32 Lx= 64 Lx= 128

Figure 5.6: Skin-friction coefficient Cf distribution along streamwise direction for

different channel lengths. Two horizontal lines indicate the Cf levels of the no

control and the fully controlled cases, respectively.

The spatial transient behaviour is important in choosing the _DR measure- ment location for the boundary layer control in experiments. For example, Quadrio and Ricco (2003) estimated that this spatial transient length could be around 2000 _∼ 4000 in viscous lengths for spanwise wall oscillation, and pointed out that some published _DR measurements were too close to the leading edge of the oscil- lating section. Figure 5.7(a) compares the spatial response of the normalised Cf

between our simulation results and the boundary layer measurements by Choi et al. (1998) and Ricco and Wu (2004). The experimental measurement by Ricco and Wu (2004) shows a long plateau after x+ _≈ 3000, which clearly suggests that the oscillating plate was long enough forCf to settle down. Their Cf decay rate com-

pares very well with our simulation result withLx ≥32, provided our simulations

scattering, and the Cf started to recover too early, which suggests the oscillating

plate might not be long enough. This is also the case for the boundary layer control using spanwise oscillating Lorentz force by Lee and Sung (2005), where an even higher_DRshould be expected if the control section was longer in their simulation. The spatial response ofCf immediately after the trailing edge of the oscillat-

ing section is shown in figure 5.7(b). The recovery rate for Lx ≥32 compares well

with the experimental data by Ricco and Wu (2004), though the actual streamwise recovery length are different,x+_≈3000 in our case, and x+ _≈1500 in (Ricco and Wu, 2004). It is worth mentioning that Lardeau and Leschziner (2013) showed a 5δ recovery length in their Reτ = 520 boundary layer simulation, which is close to

our recovery length in wall units. The simulation by Lee and Sung (2005) shows a similar recovery rate to our Lx = 16 case, which again suggests that their control

domain might be not long enough.

0 2000 4000 6000 0.5 0.6 0.7 0.8 0.9 1 x+ Cf /C f , 0 Leeet al_{. 2005} Ricco & Wu 2004 Choiet al. 1998 Lx= 16 Lx= 32 Lx= 64 Lx= 128 0 1000 2000 3000 0.7 0.8 0.9 1 x+ Cf /C f , 0 (a) (b)

Figure 5.7: Spatial response of normalised Skin-friction coefficientCf/Cf,0 after (a) the leading edge, and (b) the trailing edge of the oscillating section.

The _DR recovery is also checked from the instantaneous velocity field for

Lx = 32 case, as shown in figure 5.8. At this instance (t/T = 0.46), the Lorentz

force generates strong negative spanwise velocity in the controlled section (x <16). However, this spanwise velocity does not go to zero immediately after the trailing edge. Instead, the temporal oscillation has been converted into a spatial oscillation in the downstream, which can be observed by the positive and negativew contour at x > 16. And this spanwise velocity keeps displacing the vortical structures relative to the near wall streaks (Ricco and Wu, 2004). A direct comparison of the vortical structures and the streaks orientation inyz planes is also shown in the graph: one is within the control region (at x = 13), and the other one is at the downstream of the trailing edge (at x = 19). The vortical structures at y = 0.1 (y+ _{= 20) are twisted in spanwise direction in a similar fashion at these two x}

locations. However, the near wall high- and low-speed streaks have clearly recovered atx= 19, and they are almost invisible at x= 13. Ricco (2004) visualised the near wall streaks and vortical structures using hydrogen bubble technique at x+ _≈600 downstream of the oscillating wall section, and argued that the high- and low-speed streaks were set to rest due to the no-slip wall condition, while the spanwise wall movement was transferred by viscous diffusion to convey the vortical structures at higher wall location, and this relative displacement between the streaks and vortical structures led to a slow _DR decay. This is generally the same situation for the present oscillating Lorentz force case.

-0.05 0.1

-0.04 0 0.04 0.08

w

Figure 5.8: Instantaneous flow field visualisation around the trailing edge of an oscillating section (x = 16) for Lx = 32 case (only one portion of the domain is

displayed). xy plane shows spanwise averaged w velocity contour; yz planes show streamwise velocity fluctuation u′ _{contour, streamwise vorticity fluctuation ω}′

x iso-

lines (blue forω_x′ = 1.5, and green for ω_x′ =₋1.5), andv′₋w′ velocity vectors.