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Flow Turbulence Combust (2017) 99:765–785

Turbulent Drag Reduction by Uniform Blowing Over a Two-dimensional Roughness

Eisuke Mori1 ·Maurizio Quadrio2 ·Koji Fukagata1

Received: 8 March 2017 / Accepted: 25 September 2017 / Published online: 4 October 2017 © Springer Science+Business Media B.V. 2017

Abstract Direct numerical simulation (DNS) of turbulent channel flow over a two- dimensional irregular rough wall with uniform blowing (UB) was performed. The main objective is to investigate the drag reduction effectiveness of UB on a rough-wall turbulent boundary layer toward its practical application. The DNS was performed under a con- stant flow rate at the bulk Reynolds number values of 5600 and 14000, which correspond to the friction Reynolds numbers of about 180 and 400 in the smooth-wall case, respec- tively. Based upon the decomposition of drag into the friction and pressure contributions, the present flow is considered to belong to the transitionally-rough regime. Unlike recent experimental results, it turns out that the drag reduction effect of UB on the present two- dimensional rough wall is similar to that for a smooth wall. The friction drag is reduced similarly to the smooth-wall case by the displacement of the mean velocity profile. Besides, the pressure drag, which does not exist in the smooth-wall case, is also reduced; namely, UB makes the rough wall aerodynamically smoother. Examination of turbulence statistics suggests that the effects of roughness and UB are relatively independent to each other in the outer layer, which suggests that Stevenson’s formula can be modified so as to account for the roughness effect by simply adding the roughness function term.

Keywords Roughness · Drag reduction · Uniform blowing · Turbulent boundary layer · Direct numerical simulation

� Koji Fukagata fukagata@mech.keio.ac.jp

1 Department of Mechanical Engineering, Keio University, Yokohama, Japan

2 Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, Milano, Italy

https://doi.org/10.1007/s10494-017-9858-2

http://crossmark.crossref.org/dialog/?doi=10.1007/s10494-017-9858-2&domain=pdf http://orcid.org/0000-0003-4805-238X mailto:fukagata@mech.keio.ac.jp

766 Flow Turbulence Combust (2017) 99:765–785

1 Introduction

Drag in turbulent flows is much higher than that in laminar flows, and it causes an addi- tional loss of energy in many high-speed transports such as airplanes and bullet trains. For instance, Wood [1] reported that 16% of the total energy consumed in the United States is dedicated to the aerodynamic drag deriving from transportation systems, and according to his estimation, at least 20 billion US dollars could be saved if the existing drag reduction technologies were applied to all the vehicles within the United States.

The drag in subsonic single-phase flows can be decomposed into two contributions: the pressure drag and the viscous drag. Although there are many examples in which the pres- sure drag is reduced, e.g. by shape optimization, there are few control methods for viscous drag reduction that can actually be used in industrial applications, excluding the polymer additives already in use for petroleum pipelines. Considering the fact that the viscous drag accounts for about half of total drag in the cruise flight of modern subsonic aircrafts [2], further development of control methods for viscous drag reduction is desired from both economical and environmental viewpoints.

In the last decades, considerable efforts have been made on the friction drag reduction in turbulent boundary layers. According to Moin and Bewley [3] and Gad-el-Hak [4], the drag reduction methods can be classified into two groups: one is the passive control such as riblets [5, 6] and superhydrophobic surfaces [7], and the other is the active control, which requires external energy input. Extensive studies have been made on the active control meth- ods [8, 9]. Examples of the well-studied active control methods are the opposition control [10], spanwise forcing [11–15] and the traveling-wave of wall-normal momentum [16–18]. Although these methods have been reported to attain significant drag reduction effects and some of these even lead to relaminarization of turbulent flows [14, 17, 18], which has been proved to be the best scenario in terms of the net energy saving [19, 20], it remains ques- tionable if the actuators used for these control methods can actually be fabricated and if a net power saving can be achieved using such real actuators.

Among different active control methods for friction drag reduction, uniform blowing (UB) is considered as one of the most practically realizable options because all one has to do is to impose a uniform wall-normal velocity on the wall, without the need for small- scale complicated actuators. In the recent review article by Kornilov [2], it is concluded that “utilization of the blowing through the high-technological surface, featuring low roughness and maximal requirements to orifice quality and geometry, is a reasonable way, simple, available, and reliable method of control of the near-wall turbulent flow in the aerodynamic experiment and during the numerical simulation.”

This blowing (or suction) idea originates from the primitive experiment by Prandtl [21], which initially aimed at laminar-to-turbulence transition delay. Subsequently, experiments of UB (mostly by injection of air through a permeable porous wall) have been conducted for a turbulent boundary layer [22–24], followed by several numerical studies [25, 26]. As a result, all of these studies confirmed that UB has a possibility to attain significant drag reduction by mitigating the viscous shear stress. Sumitani and Kasagi [26], who conducted DNS of a turbulent channel flow with UB on one wall and uniform suction (US) on the other wall, clearly showed that UB shifts the velocity profile away from the wall, while US does it in the opposite way. In addition, it turned out that the Reynolds shear stress is amplified by UB and suppressed by US, which might look contradicting with the drag reduction by UB and drag increase by US.

For more quantitative analysis, Fukagata et al. [27] derived a mathematical relation- ship between skin-friction drag and turbulence statistics by integrating the streamwise

Flow Turbulence Combust (2017) 99:765–785 767

momentum equation. The analysis using this relationship (so-called the FIK identity) quan- titatively showed that the drag reduction by UB is due to the drag-reducing contribution of mean wall-normal convection that surpasses the drag-increasing contribution of enhanced turbulence. A similar quantitative analysis based on the FIK identity was reported in the DNS study of Kametani and Fukagata [28] for a spatially developing turbulent boundary layer with UB at a relatively low Reynolds number. More recently, Kametani et al. [29] have demonstrated through large-eddy simulation (LES) that UB works equally well at mod- erately high Reynolds numbers. They also showed that the overall drag reduction rate is unchanged even when discrete slots are used to realize the blowing [30]. Besides, toward its use for airfoil drag reduction, some attempts to combine US to delay transition and UB to reduce turbulent drag have also been reported [31, 32].

As introduced above, turbulent drag reduction by UB has been extensively studied toward its practical implementation. However, the drag reduction effect of UB in the presence of surface roughness is still unclear, despite its importance in practical situations. Among the studies that dealt with the combined effect of roughness and UB, some [33, 34] observed drag reduction similarly to the smooth-wall cases, whereas the recent experimental study by Miller et al. [35] has shown an opposite result: UB suppresses turbulent fluctuations and increases drag in rough-wall cases. In experiments, an accurate wall-friction measurement is one of the most difficult tasks. According to Schultz and Flack [36], for instance, ±4% of error in the friction velocity appears due to the measurement uncertainty, which may be sometimes comparable to the amount of drag modification of interest. A more serious problem may be that the friction velocity is often estimated using the Clauser plot [37] or its modified version for rough walls [38] on the assumption that the blowing does not modify the slope of the log-law (i.e., von Kármán constant). As is implied by Stevenson’s formula [39] (discussed in Section. 4.4) and also observed in the DNS result of Sumitani and Kasagi [26], the slope of log-law is actually changed by UB even in smooth-wall cases. Therefore, the possibility exists for a similar modification of log-law to happen in rough- wall cases, too, and the method for determination of friction velocity can also be considered as a possible cause for the discrepancy above.

In the present work, we perform DNS to study the drag reduction effect of UB in tur- bulent flow in a channel having a rough wall. The primary objective is to clarify whether UB increases or decreases the drag on a rough wall by assessing the statistics directly computed from the velocity field. The second objective is to investigate the mechanism of drag modification by decomposing the drag into the pressure and friction contributions and by examining the turbulence statistics in more detail. The paper is organized as follows. The numerical procedure is outlined in Section 2 together with the definition of the two- dimensional roughness used in the present study. In Section 3, the statistics of the base flow over a rough wall (without UB) is presented. The effect of UB on the flow is presented and discussed in Section 4. Finally, conclusions