^{1}Beijing Institute of Astronautical System Engineering, Beijing, China^{2}School of Mathematical Sciences, Soochow University, Suzhou, China^{3}School of Aerospace Engineering, Tsinghua University, Beijing, China

Riblets are small protruding surfaces along the direction of the flow, and are one of the most well-known passive turbulent drag reduction methods. We investigated a scalloped riblet, the shape of which was constructed by smoothly connecting two third-order polynomials and was not as sharp in the tip as corresponding triangular riblets with the same height-width ratio. Numerical simulations were performed for turbulent channel flow with and without riblet control at an estimated optimum width of

## Introduction

Drag reduction is of great importance because of the environmental and economic benefits from the reduced fuel consumption [1]. As one of the oldest and most investigated drag reduction methods, riblets, small protruding surfaces along the direction of the flow, had been known to be capable of reducing friction drag up to 8% with appropriate height and spacing in spite of the extra surface area. Early experiments conducted by Walsh et al. [2, 3], Bacher et al. [4], and Bechert et al. [5] tested different shapes including triangular, rectangular (or blade), trapezoidal, semi-circular, and other shape grooves with various sizes to get an optimal drag reduction rate. Experiments on concave and convex riblet shapes showed that drag reduction increases as the peak curvature increases, or as the radius of valley curvature increases, which means that an optimum riblet shape should have sharp peaks and curved valleys [2]. However, later investigations put more emphasis on the riblet tip rather than the valley, arguing that the tip sharpness plays an important role in damping the spanwise velocity fluctuations and thus limiting the momentum transport and turbulence intensity.

Two drag reduction regimes of riblets had been recognized [5], i.e., the “viscous regime,”

As a bio-inspired technique mimicking the denticles of fast swim sharks, riblets are possibly the only turbulent drag reduction strategy that have been tested in application. Szodruch [17] reported that covering of riblets on 70% of the surface on an Airbus 320 commercial airplane leads to a 2% reduction in oil consumption. Instances of using riblets in sports, for example, on the hulls of boats and the surfaces of racing swimsuits, have been successful [18]. In contrast to the canonical arrangement, novel riblet concepts have been introduced. Nugroho et al. [19] explored the possibility of using ordered and directional surfaces to redirect the near wall flow. Benhamza et al. numerically investigated variable spacing riblets of rectangular shape in turbulent channel flows. Riblets with a sinusoidal variation along the streamwise direction [20, 21] had also been devised to mimic the spanwise oscillation strategy to reduce drag, however, these gained very little further drag reduction. Boomsma [22] numerically studied denticles resembles sharkskin in turbulent boundary layer, however, obtained drag increase instead of drag reduction. These expeditions generally failed to surpass the optimum drag reduction rate of conventional riblets, indicating a lack of full understanding of the drag reduction mechanism.

The plausible mechanisms of breakdown imply a requirement of tip sharpness of riblet for high drag reduction performance. However, sharp tips pose challenges for manufacturing and maintenance. Actually, Walsh [2] experimentally found that the optimum riblet appears to have sharp peaks and significant valley curvature. Launder and Li [23] also reported that the U-form (or scalloped) riblets can achieve a superior performance to the corresponding V-shaped riblets with the same height and width. Based on the protrusion height concept and numerical simulation of riblet controlled boundary layer transition, Wang et al. [24] showed that scalloped riblets could have better performance than corresponding triangular riblets with sharper tips. The scalloped riblet shape we considered is constructed by smoothly connecting two third-order polynomials.

where

To guarantee that

An extreme exists with

One important aspect in understanding the flow dynamics is the identification of vortices. Noted by Küchemann [27], vortices are the “sinews and muscles of the flow.” However, no consensus on the definition of vortices has been reached. Popular methods including

where

The paper is organized as follows. Section *Numerical Methods and Case Setup* introduced the numerical methods adopted, especially the customized immersed boundary method that we use to model riblet surfaces. Section *Numerical Results* presents the numerical results, discussing the drag reduction rate, mean flow and turbulent statistics, premultiplied power spectrum density, and instantaneous flow field, with particular attention paid to the Liutex field. Finally, conclusions are drawn in Section *Conclusion*.

## Numerical Methods and Case Setup

The canonical case of a turbulent channel flow at Reynolds number

The riblet surfaces were modeled with a customized immersed boundary method in “Incompact3d” based on an alternating direction forcing to ensure a no-slip boundary condition at the wall of the solid body [26]. A particular treatment was that no-zero velocities were set inside the solid body when computing derivatives to avoid discontinuities of the velocity field, which, of course, would not be used when outputting numerical results. Because the technique is important for both correctly resolving the riblet wall surfaces and precisely calculating the friction drag, which is certainly critical especially when aiming for about 5∼10 percent drag reduction, the procedure is described in more detail in the following.

For the adopted compact finite difference scheme, the derivatives along a directional line, which could cover both fluid and solid regions, were calculated simultaneously in a so-called implicit manner by solving systems of linear algebraic equations. If the condition of velocity component

To calculate wall shear stresses on a riblet surface, velocity gradients need to be calculated on these surfaces, which are not necessarily on mesh nodes. To obtain velocity gradients, we carried out interpolation the same way as described above on one dimensional basis. Then we used a simple finite difference based on the interface point and an interpolated point about the local mesh size away. Once velocity gradients are obtained, the wall shear stress

where

Then, the drag can be expressed as a line integral (actually a surface integral, however we can simply take average in streamwise direction because of the homogeneity)

where

where

where

## Numerical Results

### Drag Reduction Rate

The initial condition for both cases with and without riblet control is interpolated from a direct numerical simulation of turbulent channel flow at

**FIGURE 2**. Time history and cumulative mean of skin friction coefficients at both riblet and flat walls.

### Mean Flow and Second-Order Statistics

The streamwise, time-, and riblet-wise averaged flow field near a riblet is shown in Figure 3. First, it is clear that all three velocity components in the valley are quite small, which makes the skin friction on the valley surface of the riblet rather small. On the other hand, the mean streamwise velocity rapidly increases along the *et al.* [7] and Wang et al. [35]. However, how such mean vortices correspond to instantaneous flow field remains unexplored.

Figure 4 shows the Reynolds stresses near riblet surfaces with primes denoting fluctuations. It can be observed from the normal stresses in Figures 4A–C that fluctuations inside the riblet valley are quite small, and above *vice versa*. In terms of quadrant analysis, more events happen in the second and forth quadrant of a

**FIGURE 4**. Distributions of Reynolds stresses near riblet surfaces. **(A)** **(B)** **(C)** **(D)** **(E)** **(F)**

The distribution of mean vorticity and mean Liutex are shown in Figures 5, 6. We can see that for the spanwise and wall-normal components, the magnitude of vorticity components are two-orders larger than that of Liutex components. This is because besides Liutex-represented rotational motion, vorticity also contains pure shear, and contributions of high shears from

**FIGURE 5**. Distribution of mean vorticity near riblet surfaces. **(A)** **(B)** **(C)**

**FIGURE 6**. Distribution of mean Liutex near riblet surfaces. **(A)** **(B)** **(C)**

Second-order momentums of vorticity and Liutex components are shown in Figures 7, 8 respectively. The locations of concentrations of both

**FIGURE 7**. Distributions of second moments of vorticity near riblet surfaces. **(A)** **(B)** **(C)** **(D)** **(E)** **(F)**

**FIGURE 8**. Distributions of second moments of Liutex components near riblet surfaces. **(A)** **(B)** **(C)** **(D)** **(E)** **(F)**

### Premultiplied Power Spectrum Density

Contours of pre-multiplied energy spectra of flat plate channel and riblet controlled channel are shown in Figure 9.

**FIGURE 9**. Contours of pre-multiplied energy spectra of channel flow with and without scalloped riblet surface: **(A)** **(B)** **(C)** **(D)**

Low-speed streaks and streamwise vortices have been viewed as critical in the turbulence generation cycle. Here we adopt the Liutex methodology and use its streamwise component to represent streamwise vortices. The pre-multiplied energy spectra of

**FIGURE 10**. Contours of pre-multiplied energy spectra of channel flow with and without scalloped riblet surface: **(A)** **(B)** **(C)** **(D)**

### Instantaneous Flow Field

Instantaneous vortical structures are shown by iso-surfaces of Liutex magnitude

**FIGURE 11**. Instantaneous vortical structures visualized by iso-surfaces of Liutex magnitude **(A)** flat plate channel and **(B)** riblet-controlled channel.

**FIGURE 12**. Instantaneous contours of streamwise component of Liutex vector for flat plate case **(A)** and riblet case **(B)**.

Figure 13 shows iso-surfaces of absolute values of Liutex components. It can be seen from Figures 13A, B that the streamwise vortices are located near the wall surfaces and it is reconfirmed that the vortices on the top of riblet tips are actually streamwise. For the snapshot shown in Figure 13C, multiple lengthy vortices in the

**FIGURE 13**. Instantaneous iso-surfaces of absolute values of Liutex components for flat plate channel and riblet controlled channel. **(A)** **(B)** **(C)** **(D)** **(E)** **(F)**

## Conclusion

We considered the kind of scalloped riblets constructed by smoothly connecting two third-order polynomials, and selected a scalloped riblet with shape parameters

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Author Contributions

HY: investigation; visualization; writing original draft. YH: review and editing. YW: conceptualization; investigation; methodology; code development; writing, editing of the manuscript. YQ: review and editing. SF: review and editing. All authors contributed to the article and approved the submitted version.

## Funding

The current study is supported by Jiangsu Shuangchuang Project (JSSCTD202209), the National Science Foundation of China (Grant No. 12302312) and the National Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 22KJB130011).

## 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: riblet, drag reduction, turbulent channel, tip sharpness, scalloped

Citation: Yu H, Huang Y, Wang Y, Qian Y and Fu S (2023) Flow Field Analysis of a Turbulent Channel Controlled by Scalloped Riblets. *Aerosp. Res. Commun.* 1:12300. doi: 10.3389/arc.2023.12300

Received: 25 October 2023; Accepted: 24 November 2023;

Published: 18 December 2023.

Copyright © 2023 Yu, Huang, Wang, Qian 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: Yiqian Wang, yiqian@suda.edu.cn