- School of Aerospace Engineering, Tsinghua University, Beijing, China

Turbulent channel flows 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 significance 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 [3–5]. 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%. [6–10] The flow generated by WOS,

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 wall-normal 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 [12–15]. 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 [17–19]. 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,

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 [27–30], rotating discs [31–33], wavy riblets [27], spanwise forcing [34–36], 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 [38–40], 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 *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 *z* direction and generates a−*z* direction spanwise motion, while in the second half period *z* direction and generates a+*z* direction spanwise motion, as shown in Figure 1. With a non-dimensionalization in length scale by

**FIGURE 1**. **(A)** Spanwise velocity **(B)** the sketch of the actuation.

To resolve the WBS induced flow in a quiescent background, an asymptotic expansion method is applied as shown in Eq. 4a–c:

where

where *c.c.* represents the complex conjugate. The Reynolds number

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

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 *y* = 0. This asymptotic solution matches the numerical results well with a slightly smaller amplitude, as shown in Figure 1A. The expression

In summary, the motion induced by the spanwise traveling wave can be approximately decomposed into a harmonic part (Eq. 5a, b) and a Stokes part (Eqs 8a-e) with the asymptotic solution.

### The Numerical Method and Base Flow

Incompressible fully developed turbulent channel flows with *x*) direction, *y*) direction, and *z*) direction, with *h* the half channel height. In this study,

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

The length and velocity scales are normalized by

**FIGURE 2**. Comparison of Reynolds stress **(A)** between the present study and Kim et al. [45] with **(B)** between the present study and Jiménez et al. [46] with

**FIGURE 3**. **(A)** Q iso-surface in **(B)** **(C)** The sketch of WBS control strategy.

## Results and Discussion

The spanwise traveling wave of blowing and suction control is imposed only at the bottom wall (*DR*, is defined as:

where

**TABLE 1**. The parameters and drag reduction rates of the test cases with

### FIK Identity Analysis

The variables in WBS control are decomposed into three parts:

where

where

The turbulent skin friction part can be further decomposed into two parts [23, 24], one from the inner region (

Therefore the drag reduction rate can be decomposed as shown in Eqs 13a-d:

where

The effects of

Since it is uniform in streamwise direction, there is no

It is obvious that

In addition, the skin friction from the periodic part,

As shown in Figure 4, the

**FIGURE 4**. **(A)** **(B)**

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

### Scale Separation Analysis

In this section, a Fourier expansion based on spanwise direction is imposed to the fluctuation field variables,

where the superscript “*” represents the complex conjugate. As a result, the skin friction coefficient from the turbulent shear stress,

Thus, the contribution from the scale

**FIGURE 5**. The pre-multiplied shear stress **(A)** Case WBS-5, and **(B)** Case WBS-2, and skin friction coefficients **(C)** Case WBS-5, and **(D)** Case WBS-2. The dash-line represent the results in the uncontrolled case.

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, 17–19]. In our previous analysis based on the

**FIGURE 6**. The streaks (*t*/*T _{a}* = 0.2–1.8 in

**(A–I)**.

The contours of the streaks and

**FIGURE 7**. The streaks at **(A–C)** and LS structures **(G–I)**, and two-point correlation **(D–F)** and LS structures **(J–L)**.

**FIGURE 8**. The angles **(A)** **(B)**

In physical space, the oscillatory spanwise motion or the Stokes part mainly weakens the inner region (

## Conclusion

In this study, the drag reduction performance of the spanwise traveling wave of blow and suction with a periodically reversing propagation direction in *ω* and low Reynolds number

The drag reduction rates are much smaller in

## Data Availability Statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

## Author Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

## Funding

China-EU Program: Drag Reduction via Turbulent Boundary Layer Flow Control (DRAGY), Grant agreement ID: 690623.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: drag reduction, turbulent channel flows, spanwise traveling wave, blowing and suction, Reynolds number dependence

Citation: Huang Y and Fu S (2024) Reynolds Number Effects on the Drag Reduction With a Spanwise Traveling Wave of Blowing and Suction in Turbulent Channel Flows. *Aerosp. Res. Commun.* 1:12272. doi: 10.3389/arc.2023.12272

Received: 22 October 2023; Accepted: 24 November 2023;

Published: 09 January 2024.

Copyright © 2024 Huang and Fu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Song Fu, fs-dem@tsinghua.edu.cn