Reynolds Number Effects on the Drag Reduction With a Spanwise Traveling Wave of Blowing and Suction in Turbulent Channel Flows

Turbulent channel ﬂ ows with Re τ = 180 and Re τ = 550 are controlled to reduce the drag with a spanwise traveling wave of the blowing and suction method. An oscillatory spanwise motion is generated with a periodically reversing propagation direction of the traveling wave, similarly as the wall oscillation. Direct numerical simulation (DNS) results show that this kind of blowing and suction control can achieve a drag reduction rate of 24.5% with Re τ = 180, and 7.5% with Re τ = 550. The reasons for the deterioration in drag reduction rates are thought to be the lift-up mechanism by the actuation through an asymptotic expansion method, and the controlled inner regions and small-scale structures having less signi ﬁ cance when the Reynolds number is high.


INTRODUCTION
Nearly 55% of total drag is a viscous drag for a civil aircraft [1].A 0.75% reduction in fuel consumption can be achieved with a 1% reduction in skin friction [2].Thus, it has been an important goal to reduce the viscous drag in the area of flow control in turbulent boundary layer flows.There are two conventional routes for flow control near the wall, passive control and active control.For the former extra energy input is not needed and a drag reduction of about 10% can be achieved.The most popular passive control strategy is building streamwise riblets at the wall, which can reduce the viscous drag by about 7%-10%, with a restriction on the spanwise crossflow in the near wall region.However, the requirements of an effective riblet shape are strict [3][4][5].In active control strategies, energy outside the flow was imposed and the turbulent skin friction is reduced more effectively compared with the passive control.The wall oscillation (noted as WOS hereinafter) was thought to be a simple and effective way to reduce the drag to an extent of about 40% with a total energy saving of 7%.[6][7][8][9][10] The flow generated by WOS, w W m cos(ωt), is called the second Stokes problem or the Stokes layer [11], with a velocity distribution as (Eq.1): w y, t W m cos ωt − ky e −ky , k Plenty of studies have tried to figure out how the Stokes layer reduces the drag and weakens the near-wall turbulence after Jung et al. first performed the numerical experiment of WOS control in a turbulent channel flow [6].
Since the spanwise velocity becomes non-negligible in the transport equations of Reynolds stresses after control, some studies analyzed the differences in the energy budget of turbulence.A reduced pressure-strain correlation term is believed to be closely related to the suppression of the wallnormal stress, which led to a reduction in the level of shear stress and streamwise stress.Hence, the near-wall turbulence is weakened, leading to a reduction of skin friction at the wall [12][13][14][15].Also, the pressure-strain terms play a significant role during the transient response after control [16].However, the mechanism of the Stokes layer suppressing the pressure-strain correlation is still unclear.The quasi-streamwise vortex and streaks are significant structures in the self-sustaining cycle of near-wall turbulence, and were found to be inclined periodically with the wall motion [17][18][19].Some models based on linearized Navier-Stokes equations were proposed to described the relation between the Stokes layer and the inclination [17,20].It is also thought that the inclined streaks and vortex weaken the generation of turbulence, and thus led to a reduction in the drag [14,15,21].The Reynolds number dependence of the WOS control is of vital significance, since the Reynolds number around a real aircraft, Re τ ~O(10 5 ), is much higher than that accessible with DNS.The drag reduction rates were found decrease with an increasing Reynolds number, and the trend is approximated as DR ~Re −(0.2~0.3)τ [22,23].Hurst et al. thought the control effects in the viscous sublayer region are weakened at high Reynolds number, thus leading to lower drag reduction rates [23].Further, they thought the drag reduction could remain at a level of 30% when the Re τ went to infinity since the turbulence in the log-law region is always weakened [23].It was also found the control effects on large-scale structures are weakened at high Reynolds number [24].Agostini and Leschziner found the asymmetric modulation of u′ > 0 and u′ < 0 large-scale structures on the small-scale structures in a Re τ 1000 channel flow with the WOS control [14,25].There is still not a generally accepted mechanism of the drag reduction deterioration in high Reynolds number flows, and reviews on the WOS control in drag reduction are referred to [2,10,26].
The Stokes layer is simple and proven effective in reducing drag by many experimental and numerical results, but the periodic wall motion might not be easily attainable in practical applications.Using other methods to generate an oscillatory spanwise motion to mimic the Stokes layer has also gained much research interest.A streamwise traveling wave of wall motion [27][28][29][30], rotating discs [31][32][33], wavy riblets [27], spanwise forcing [34][35][36], and spanwise jet [37] have been investigated and found effective in reducing drag.A spanwise traveling wave of the blowing and suction method is studied in the present work to reduce skin friction in a fully developed turbulent channel flow.Different from a streamwise traveling wave [38][39][40], which can reduce the drag even to a laminar level by increasing the flow rate, the spanwise traveling wave investigated here has a periodically reversing propagation direction, noted as WBS.The Reynolds number dependence of the drag reduction and its mechanism are mainly discussed in this study.This manuscript is arranged as follows.Control Strategy section introduces the motion induced by the spanwise traveling wave and the uncontrolled base-flow.Results and Discussion section shows the performances of this control strategy and analyze the mechanism in the channel flows with Re τ 180 and Re τ 550.Conclusion section reports the conclusions.

CONTROL STRATEGY
The Motion Induced by the Traveling Wave Min et al. found that an upstream traveling wave of blowing and suction can achieve a sustained sub-laminar drag in a turbulent channel flow [38].However, some studies have proved that this kind of traveling wave can generate a flux opposed to the wave propagation direction [40], and it is more like a pumping than drag reduction effect [39].Inspired by the streamwise traveling wave, we make the wave reverse in the spanwise direction periodically to mimic the oscillatory wall motion in WOS, shown in Eq. 2.
where φ0 −2 ω Ta is a constant phase as a result of the continuity at t Ta In the first half period 0 < t < Ta , the wave travels in +z direction and generates a−z direction spanwise motion, while in the second half period Ta < t < 2 Ta , it travels in −z direction and generates a+z direction spanwise motion, as shown in Figure 1.
With a non-dimensionalization in length scale by z kz ẑ, time scale by t ωt , and velocity by v v/ V0 , respectively, the actuation is rewritten as: To resolve the WBS induced flow in a quiescent background, an asymptotic expansion method is applied as shown in Eq. 4a-c: where α V0 /ĉ, the ratio of the blowing and suction amplitude over the wave speed, is assumed as a small parameter, (w 0 , v 0 , p 0 ) are the zero-order terms, (w 1 , v 1 , p 1 ) are the first order terms, and (w 2 , v 2 , p 2 ) and higher order terms are not considered here.Without losing generality, the zero order terms are solved in the first half period t < ω Ta as: is the imaginary unit, and c.c. represents the complex conjugate.The Reynolds number β is based on the frequency ω and wave number kz as: The zero-order terms are a traveling wave in the same direction as actuation, which is also called the harmonic part in the manuscript.The harmonic part decays as the distance to the wall y increases, and the protrusion height is defined as δ 0.73 λz , where the amplitude of the velocity is only 1% of that at the wall.
The average of the first order terms is of much significance in this study: And it can be solved with known zero order terms (Eq.5a, b): where the constant ] determined by the w 1 (y, t) 0 boundary condition at y = 0.This asymptotic solution matches the numerical results well with a slightly smaller amplitude, as shown in Figure 1A.The expression exp(− inβω 0 y ) exp(inω 0 t) in Eqs 8a-e is the same as the Stokes layer with frequency nω 0 .In other words, the induced oscillatory spanwise motion w 1 (y, t) comprises a series of the Stokes layers with a frequency nπ/ ω Ta .Obviously, a large divergence exists between w 1 (y, t) and the Stokes layers within a very thin region above the wall as a result of different boundary conditions.The thickness of the first Stokes layer in the series (Eqs 8a-e), δ s 4.6 2/(ω 0 β) or δs ∝ Ta with dimensions, is also regarded as the penetration height of the spanwise motion.The higher order terms in Eqs 8a-e, or the Stokes layers with multiple frequency nω 0 , are always neglected for analysis as their thickness is 1/ n √ less and the amplitude is 1/n less.In the limit of ω→ ∞, the amplitude of w 1 (y, t) can be approximated by . When the frequency is zero, ω 0, there is no average spanwise motion since the traveling wave degenerates to a constant blowing and suction.It is noticed that the relative error of asymptotic solutions is approximately less than 10% when the parameter α V0 kz / ω satisfies α < 0.5.As a result, the oscillatory spanwise motion in Eqs 8a-e is called the Stokes part in the rest of the manuscript.]/ tanh(r), y i (i − 1)/(N y − 1) decides the grid distribution in wall-normal direction.The resolution 0.2 < Δy + < 5.2 is achieved with a stretch ratio r 2.4.And for the channel flow with Re τ 550, the computation domain is [0, 3πh] × [−1, 1] × [0, 2πh] to resolve the large-scale structures [41,42], with the resolution Δx + ≈ 7.6, Δz + ≈ 7.7, and 0.04 < Δy + < 2.9.
High order methods based on CPR(correction procedure via reconstruction) [43] are used here and seventh order polynomials are used for space discretization.For the viscous term, it is discretized with BR2 (the second approach of Bassi and Rebay) [44] and interior penalty method.The time advancement strategy is a third order explicity Euler scheme with a time step dt 1 × 10 −4 (dt + dt • u τ /δ v ≈ 0.00116).A constant flow rate (CFR) method is chosen in the computations of the channel flows.All the computations in this study are conducted with this type of direct numerical simulations (DNS).
The length and velocity scales are normalized by h and U b in the turbulent channel flows without particular specification, respectively.Comparisons of our numerical results with those from Kim et al. [45](Re τ 180, and with those from Jiménez et al. [46](Re τ 550) in Reynolds stress of the uncontrolled channel flows is done firstly for verification of our computation code, which show a satisfying match in Figure 2. The vortex structures detected by Q-value also show a good match with previous work, see Figures 3A, B.

RESULTS AND DISCUSSION
The spanwise traveling wave of blowing and suction control is imposed only at the bottom wall (y −1) of the channel flow with an uncontrolled upper wall (y 1), as shown in Figure 3C.In this section, the drag reduction performance of the WBS control is studied, and the drag reduction rate, noted as DR, is defined as: where τ w is the averaged skin friction of the bottom wall, and the subscript with '0' means the results in uncontrolled flow.The test cases in this manuscript are show in Table 1.In our previous work [15], it has been shown that the WBS control leads to a close drag reduction rate to the WOS control with the same parameter (W, T a ).The drag reduction rates are much smaller with Re τ 550 than those with Re τ 180, while they have nearly the same optimal period T + a ≈ 60.Gatti et al. [22,47] has shown the drag reduction rate dependence on Reynolds number DR ~Re −(0.2~0.3)τ , in WOS control.Obviously, the decrease in DR with increasing Re τ in WBS control is much larger.

FIK Identity Analysis
The variables in WBS control are decomposed into three parts: a x, y, z, t a y + ã y, ϕ + a″ x, y, z, t 〈a〉 y, ϕ + a″ x, y, z, t where a is the ensemble average, 〈a〉(y, ϕ) the phase average with ϕ representing the phase, and a″(x, y, z, t) the turbulent fluctuations ã a − 〈a〉 is the difference between the phase average and ensemble average, which is also called the periodic part.Fukagata, Iwamoto and Kasagi proposed an identity (known as FIK identity) relating the skin friction at the wall and the turbulent shear stress in the interior field.While in WBS control, it is rewritten as: where C L f is the skin friction in uncontrolled flow, C T f the contribution from turbulent shear stress, and C P f the contribution from the shear stress formed by the periodic parts, respectively.The turbulent skin friction part can be further decomposed into two parts [23,24], one from the inner region (y + < y + P ≈ 30) and the other from the outer region (y + > 30), as shown in Eq. 12. Therefore the drag reduction rate can be decomposed as shown in Eqs 13a-d: where DR I , DR O , and DR P represents the drag reduction rate from the inner region, outer region, and periodic part, respectively.The factor "2" on the left-hand side of Eqs 13a-d comes from the one-side control in the channel flows.The drag reduction decomposition of the cases is shown in Table 2.The DR is mainly from DR I and DR O .DR P is always negative, which means the periodic blowing and suction has a negative effects to the drag reduction.On the other hand, compared to DR I and   DR O , DR P is much smaller and even can be neglected when Re τ 180.While Re τ 550, the DR P becomes larger and even dominates in Case WBS-3.The strength of the oscillatory spanwise motion W + is larger in Case WBS-3 compared to other cases with Re τ , but DR decreases a lot to a negative value.This means the negative effects of the periodic part on DR in this case become significant.The wavenumber increases from k z 8 (Case WBS-6) to k z 16 (Case WBS-5), the increases in DR are mainly from DR O , as a result of a smaller protrusion height δ 0.73λ z .In addition, the DR gets smaller mainly from the decreases of DR O when Re τ increases, similar to the results in WOS control [23,24].
The effects of DR P can be described by the lift-up mechanism from the harmonic parts of blowing and suction.Based on the asymptotic expansion (Eq.4a-c), the streamwise velocity can be written as u u 0 + αu 1 + α 2 u 2 + O(α 3 ), and its equation is: Since it is uniform in streamwise direction, there is no ∂/∂x terms in the equation, and u can be solved with a known (v, w).In addition, u 0 is the prescribed baseflow and not dependent on (v, w).Here, it is assumed that u 0 Ky without losing generality, since the linear relation is always true in the linear law region [48].Thus, the first order term u 1 can be solved with (v 0 , w 0 ): It is obvious that u 1 < 0 above the blowing regions and u 1 > 0 above the suction region, which is thought as a lift-up mechanism.Thus, there forms a negative shear stress uv, which is closely related to the C P f .Further, the spanwise average of the shear stress, noted as S(y), can be deduced: (16d)

S y lim
In addition, the skin friction from the periodic part, C P f , can be evaluated as: As shown in Figure 4, the S(y)/K is always negative, leading to a positive C P f .As a result, the periodic parts always increase the skin friction, just as shown in Table 2. C P f decreases with an increasing frequency ω, which is also compliant with the results of Case WBS-2 and WBS-3.It is also found that C P f ∝ K, which means that the larger the shear K ∂U/∂y is in the background flow, the larger C P f is.When Re τ changes from 180 to 550, K increases and leads to a larger C P f , just as the results in Table 2.As discussed in Control Strategy section, the induced flow by the traveling wave of blowing and suction with a periodically reversed wave speed can be decomposed into two parts, one is the harmonic part (zero order terms) and the other is the Stokes part (first order terms).The harmonic part will induce a negative shear stress, as shown in Eq. 16c, and lead to a positive contribution the skin friction through C P f .An effective way to reduce C P f is increasing ω, while an increased ω will lead to a smaller strength W V 2 0 / 8]ω √ of the Stokes part.Therefore, there should be an optimal frequency in the WBS control, which is not discussed in details in this manuscript.In addition, C P f becomes larger with an increasing mean flow shear ∂u/∂y in the viscous sublayer when Re τ changes from 180 to 550.This is thought to be the first reason that the drag reduction rate is smaller in Re τ 550 cases.

Scale Separation Analysis
In this section, a Fourier expansion based on spanwise direction is imposed to the fluctuation field variables, u″ ∞ κ 1 û(κ) exp(iκz) + c.c. Thus the stress can be decomposed into different scales as: where the superscript "*" represents the complex conjugate.As a result, the skin friction coefficient from the turbulent shear stress, C T f can be also decomposed: Thus, the contribution from the scale λ 2π/κ to the turbulent skin friction is C T f (κ).The pre-multiplied spectrum of turbulent shear stress, κu″v″(κ), and the skin friction coefficients, κC T f (κ), It was found that the vortex structures are inclined by the Stokes layers in WOS, which has a close relation to the drag reduction mechanism [14,15,[17][18][19].In our previous analysis based on the Re τ 180 channel flow, the inclination is thought to be a representation of the stretch of the streamwise vortex, ∂u″/∂x.∂u″/∂x gets reduced after control, leading to a weakened pressure-strain term ϕ 11 p″∂u″/∂x.Thus, the energy transported into the wall-normal stress v″v″ from the streamwise stress u″u″ is reduced as a result of the incompressibility, ϕ 11 + ϕ 22 + ϕ 33 0 [15].Here we revisited the inclination of vortex structures in the controlled flow with Re τ 550. Figure 6 shows the streaks at y + ≈ 11 in Case WBS-2 in one period.The inclination of the streaks is similar as that in Re τ 180 case.Also, the inclinations of streaks can be described by the inclinations of the two-point correlation, C(Δx, Δz), which is defined as: C Δx, Δz ( ) u″ x, y, z, t u″ x + Δx, y, z + Δz, t u″ x, y, z, t u″ x, y, z, t The contours of the streaks and C(Δx, Δz) of SS and LS structures are shown in Figure 7. Thus, an inclination angle, noted as θ in radians, is defined as the angle between the axis of the iso-value line of C(Δx, Δz) and the streamwise direction to give a quantitative description of the inclination of streaks.The

FIGURE 1 |
FIGURE 1 | (A) Spanwise velocity w 1 (y, t) 0 from the numerical simulation (dash-dot line) and asymptotic solution (solid line, Eqs 8a-e) and (B) the sketch of the actuation.

FIGURE 2 |
FIGURE 2 | Comparison of Reynolds stress (A) between the present study and Kim et al.[45] with Re τ 180, and (B) between the present study and Jiménez et al.[46] with Re τ 550.

FIGURE 4 |
FIGURE 4 | (A) S 1 (y)/K with different Reynolds number and (B) C P f /K variation with different frequency ω.

TABLE 1 |
The parameters and drag reduction rates of the test cases with W

TABLE 2 |
Drag reduction decomposition of the cases.