1 Introduction

Drag, which acts in opposition to a moving body with respect to the surrounding fluid can be categorized using two essentially different methods, near-field and far-field integrations^[1]. The near-field method integrates the pressure and viscous stress over the surfaces of the body, resulting in components of pressure drag and frictional drag, respectively. The frictional drag in a wall-bounded turbulent flow has been determined to be much higher than that of a laminar flow at the same Reynolds number. It contributes to the total drag up to 50% for the commercial aircraft, 90% for underwater vehicles, and almost 100% for long pipe and channel flows^[2]. As the Reynolds number increases, the frictional drag plays a key role in many practical applications^[3].

The mean skin friction, also termed the mean wall shear stress, can be further decomposed into various constituents according to a variety of mathematical derivations and physical interpretations. By applying a triple integration to the Reynolds averaged Navier-Stokes equation, Fukagata et al.^[4] derived a simple, direct relationship between the skin friction coefficient and the Reynolds stress for three canonical wall-bounded turbulent flows, which was subsequently named the Fukagata-Iwamoto-Kasagi (FIK) identity. This method has been generalized to compressible flows^[5] and complex wall surfaces^[6-7]. Moreover, the FIK identity has been refined to be applicable to flows with unavailable streamwise gradients or incomplete statistics over the entire boundary layer^[8-9]. Over the years, the FIK identity has been widely used in numerous studies^{[3, 10-11]}. By performing a similar triple integration on the mean vorticity equation, Yoon et al.^[12] related the skin friction to the motions of vortical structures. Recently, another theoretical decomposition of the mean skin friction has been proposed by Renard and Deck^[13]. By applying a single integration to the mean streamwise kinetic-energy-budget equation in an absolute reference frame (in which the undisturbed fluid is not moving), the mean skin-friction generation is characterized by physical phenomena that reveal the energy transfer from the wall to the fluid.

The control of the skin friction and its associated near-wall aerodynamic/hydrodynamic phenomena is of fundamental and practical importance, motivated in part by energy saving objectives^[14]. Several flow control approaches, such as the addition of long-chain polymers, spanwise wall oscillation, near-wall blowing and suction, superhydrophobic surfaces, and micro-grooved structures (riblets), have been proposed and tested to reduce skin friction in turbulent flows^[15-19]. Most of the aforementioned flow control approaches have been attempted to manipulate the turbulent structures within the near-wall region. The near-wall Reynolds stress is of particular importance for the prediction and control of the mean skin friction. However, as the Reynolds number is increased to more pragmatic values, the drag reduction efficiency is reduced^[20]. For high practical Reynolds number values, the physical thickness of the near-wall region and hence the size of the structures that populate this region become smaller^{[2, 21-22]}. Hwang^[23] noted that with an increase in the Reynolds number, the contribution of the near-wall small-scale structures to the skin friction decays, and the large-scale motion has a greater effect considering the effect of their amplitude modulation on small scales and their energy superposition onto the inner region. A spectral analysis of the linearly weighted Reynolds stress in the FIK identity was carried out by Deck et al.^[3]. The outer energy site where large scales were located was shown to have a dominant contribution to the skin friction. However, they also reported that this specific contribution for a given scale at a given height, which originated from a mathematical derivation, should be carefully utilized because of its vague physical interpretation with respect to the skin friction generation mechanism. While in Ref. [13], they showed that the buffer layer dynamics at low Reynolds numbers plays a dominant role, whereas at high Reynolds numbers, the logarithmic layer dynamics is most influential in the skin friction generation process, suggesting new drag reduction strategies in the field of flow control.

With the objective of the effective reduction of skin friction at high Reynolds numbers, it is of great importance to clarify and evaluate the effects of the Reynolds number on skin friction and its components based on different drag decomposition methods. So far, to the best of the authors' knowledge, no results of such an analysis have been published. In this report, Section 2 introduces the direct numerical simulation (DNS) approach for turbulent channel flows and yields skin friction coefficients compared with the empirical results. In terms of specific decomposition formulas for the mean skin friction based on two integration methods, the FIK identity and Renard-Deck (RD) identity are presented in Section 3. In Section 4, the dependence of the mean skin friction generation on the Reynolds number is investigated with the assessment of turbulence statistics and different wall layer dynamics. Finally, concluding remarks are given in Section 5.

2 DNS of turbulent channel flows

We use the DNS data of fully developed incompressible channel flows provided by Lee and Moser^[24]. The DNS was performed based on the POONGBACK code which uses the spectral numerical method of Kim et al.^[25]. The friction Reynolds number Re_τ is defined as ρ^*u_τ^*h^*/μ^*, where the superscript * denotes the dimensional variables, ρ is the density, h is the channel half height, μ is the dynamic viscosity, and u_τ is the friction velocity. Five cases with Re_τ =180, 550, 1 000, 2 000, and 5 200 were used. Simulation details and validations can be found in Refs. [25]-[28]. The statistical data used are available on the website http://turbulence.ices.utexas.edu.

The mean skin-friction coefficient C_f is defined as the ratio of the mean wall shear stress to the dynamic pressure. It can be written as

(1)

where Re=ρ^*u_b^*h^*/μ^* is the Reynolds number based on the bulk velocity , y is the site distance from the wall surface, and 〈u〉 is the mean streamwise velocity non-dimensionalized by the channel flow bulk velocity. The calculated mean skin-friction coefficient is compared with Dean's empirical correlation^[29], C_f=0.073(2Re)^-1/4, as shown in Fig. 1, which is apparently in good agreement with the theoretical prediction.

3 Mean skin friction decomposition methods

In this section, two theoretical methods are introduced for incompressible turbulent channel flows to decompose the mean skin friction.

3.1 FIK identity

In the FIK identity, the following conditions are assumed: (ⅰ) constant flow rate, (ⅱ) no-slip condition at the wall surfaces, (ⅲ) statistical homogeneity in the spanwise direction, and (ⅳ) symmetry with respect to the channel central plane. These assumptions are reasonable for incompressible turbulent channel flows. Considering the periodicity of the flow parameters along the streamwise direction, the streamwise statistical homogeneity is additionally assumed. Under these assumptions, a triple integration by parts performed on the Reynolds averaged Navier-Stokes equation in the streamwise direction gives the simplified FIK identity^[4],

(2)

where -〈 u'v'〉 is the Reynolds shear stress. The first term, C_{f1, FIK}, has been proven to be identical to the well-known laminar solution at the same Reynolds number, and this is referred to as the laminar drag. The second term, C_{f2, FIK}, is characterized by the turbulent fluctuations, i.e., the linearly weighted Reynolds stress over the boundary layer, and consequently, it is described as the turbulence-induced drag. However, the weight function of (1-y) is probably ill-defined for physical interpretation, because the linear nature of the weight function results from the mathematical derivation of the decomposition and has no clear explanation in terms of physical processes^[13]. Although the FIK identity is widely used, it is noteworthy that the linearly weighted Reynolds stress has a clear limitation in terms of identifying the skin-friction generation mechanism.

3.2 RD identity

Renard and Deck^[13] gave another theoretical decomposition of the mean skin friction based on the averaged streamwise kinetic energy budget in an absolute reference frame (in which the undisturbed fluid is not moving). We refer to this method as the RD identity (named after its authors). For incompressible channel flows, we have

(3)

where the first term on the right-hand side of Eq. (3) represents a direct molecular viscous dissipation in the boundary layer, and the product of -〈u'v'〉 and in the second term on the right hand side of Eq. (3) characterizes the production of the turbulent kinetic energy. In this regard, the RD identity facilitates the visualization of the skin friction generation from the perspective of energy transfer from the wall to the fluid in an absolute reference frame, by means of the dissipation of molecular viscosity and turbulent energy^[30]. The advantage of this method is that the mean skin friction is decomposed based on its physical generation mechanism.

4 Results and discussion

Firstly, the estimated skin-friction coefficients and their constituents at different Reynolds numbers are listed in Table 1. For the FIK identity, the relative error (C_{f, FIK}-C_f)/C_f is confined within ± 0.25% for all cases studied. For the RD identity, the relative error (C_{f, RD}-C_f)/C_f is larger than that estimated by the FIK identity. Errors within ±1.58% are observed. Basically, both skin friction decomposition methods are in good agreement with the results directly calculated using the wall-normal mean streamwise velocity gradients at the wall.

The ratios of the constituents to C_f are plotted in Fig. 2. In general, when the Reynolds number increases, C_f1 /C_f reduces, and C_f2 /C_f increases, which occurs for both the FIK identity and RD identity. Although these two drag decomposition methods share some similarity in curve tendencies, they are essentially different with respect to interpreting their contributions to skin friction generation, which will be discussed in the following section.

In the FIK identity, the laminar drag C_{f1, FIK} and turbulence induced drag C_{f2, FIK} are separated, and the latter becomes much more important as the Reynolds number is increased. In order to better assess the contribution of the Reynolds stress, C_{f2, FIK}/C_f can be written as

(4)

where the superscript + denotes normalization by the viscous variables. We now quantify the dependence of the linearly weighted Reynolds stress on the Reynolds number via the pre-multiplied normalized integrand in C_{f2, FIK}/C_f, i.e., . Figure 3(a) shows the profiles of the pre-multiplied integrands in a semi-logarithmic plot with the exact DNS solutions, which is possibly a new contribution. The areas below the profiles represent the contribution of C_{f2, FIK}.

As the Reynolds number increases, the contribution of the outer layer (y⁺>30) dynamics is augmented, as illustrated in Fig. 3(a). The contribution of the near-wall region y⁺ < 30 for instance, is dramatically reduced. The implication is that the role of near-wall turbulent motion in terms of skin friction generation is less important, as previously determined in numerous studies. To some extent, this explains the observation of the decay of the drag reduction efficiency at high Reynolds numbers by controlling the near-wall turbulence. In terms of the slight increase in the peak values at high Reynolds numbers, as shown in Fig. 3(b), this is consistent with the tendency of the normalized Reynolds stress over a wide range of Reynolds numbers^[24]. A similarity is also found in the peak-point location in the extrinsic scale.

The RD identity characterizes skin friction generation from the perspective of energy transfer from the moving wall to the fluid. C_{f1, RD} is ascribed to molecular dissipation, which is expected to be dominant in the near-wall region, especially in the viscous sub-layer where the Reynolds stress is negligible compared with the viscous stress. C_{f1, RD}/C_f in the intrinsic scale is written as follows:

(5)

where u_τ is non-dimensionalized by the bulk velocity. The pre-multiplied integrands of C_{f1, RD}/C_f, i.e., , are plotted in Fig. 4(a). Regardless of the Reynolds number, the curves peak at y⁺≈6.5. The areas below the curves reflect the importance of C_{f1, RD} in a straightforward manner. As the Reynolds number increases, the area becomes smaller indicating a negative correlation between the contribution of the viscous dissipation and the Reynolds number. Almost all contributions of the viscous dissipation originate from the region y⁺ < 30, which is consistent with the physical phenomenon that the viscous stress is of importance in the near-wall region. Figure 4(b) shows the distributions of . A good coincidence of the distributions is apparently observed because of the universal log-law for the near-wall distribution of 〈 u〉⁺ as a function of y⁺. The differences are limited for the case of Re_τ=180, which is consistent with Laadhari's report^[31] that this constancy holds for Re_τ>500. Since , the areas below the curves of are exactly proportional to C_{f1, RD} /C_f^1.5. Thus, a constancy of C_{f1, RD} /C_f^1.5 is proposed, regardless of the Reynolds number.

In terms of the production of the turbulent kinetic energy C_f2, it is locally balanced with the dissipation, transport, and diffusion^[32]. If we perform an integration over the entire boundary layer, a perfect balance would exist between the total production of the turbulent kinetic energy and the total dissipation. As seen in Appendix A, Fig. A1 shows the distributions of the turbulent kinetic energy production and dissipation, and Table A1 represents the total turbulent kinetic energy production and turbulent energy dissipation. Their relative errors are limited within 0.2%. Thus, C_{f2, RD} can also be interpreted as the total turbulent energy dissipation^[30]. In contrast, C_{f1, RD} is characterized as the direct molecular viscous dissipation.

C_{f2, RD}/C_f is written as

(6)

Its pre-multiplied integrand and the corresponding expression are profiled in Fig. 5. The turbulent kinetic energy production is expressed as or alternatively , where ν is the kinematic viscosity. Thus, or essentially represents the turbulent kinetic energy production weighted by y⁺ν /u_τ³ or y⁺ν /u_τ⁴. Regardless of the Reynolds number, the curves in Fig. 5 peak at almost the same location of y⁺≈17.0. To identify the reason why the curves peak at y⁺≈17.0, the distributions of the viscous stress and Reynolds stress -〈 u'v'〉 are plotted in Fig. 6, as well as the turbulent kinetic energy production. A universal rule is found that when the viscous stress and the Reynolds stress are equal, the maximum turbulent kinetic energy production is obtained. The intersections at y⁺≈11.5 are slightly closer to the wall than the peak at y⁺≈17.0 in Fig. 5 because of the weight function, as indicated earlier.

For cases with Re_τ>500, a secondary peak emerges in the outer layer, which is ascribed to the generation of large-scale turbulent motions at these Reynolds numbers^[33-34]. The large-scale turbulent motions contain large amounts of turbulent kinetic energy^[35-36], which results in an increasing proportion of outer layer dynamics in terms of the skin friction. In the near-wall region, on the other hand, the small-scale turbulent motions are modulated by large-scale motions, which leads to the redistribution of turbulent kinetic energy from the near-wall region to the outer region. This may possibly explain why the peak value at y⁺≈17.0 in Fig. 5(a) is reduced as the Reynolds number increases. In Fig. 5(b), the distribution of remains approximately unchanged in the near-wall region (y⁺ < 30 for instance). The area below the curves in Fig. 5(b) reflects the importance of C_{f2, RD} divided by C_f^1.5, considering that . It indicates that with respect to C_f^1.5, the effects of the Reynolds number on the contribution of the near-wall dynamics (y⁺ < 30) are quite unapparent.

Figure 7 reveals the dependence of C_{f1, RD}/C_f^1.5 and C_{f2, RD}/C_f^1.5 on the Reynolds number. A constancy of C_{f1, RD}/C_f^1.5 approximately at 6.59 is identified, especially at high Reynolds numbers. Laadhari^[31] once investigated the viscous energy dissipation rate for channel flows and reported that this rate remains constant for the friction Reynolds number larger than 500. The constancy of C_{f1, RD}/C_f^1.5 in the present study is consistent with Laadhari's conclusion^[31], which suggests that the contribution of molecular viscous dissipation to the skin friction maintains the same rate of C_f^0.5, i.e., C_{f1, RD}/C_f^1.5 ≈ 6.59 × C_f^0.5. In contrast with C_{f1, RD}/C_f^1.5, C_{f2, RD}/C_f^1.5 has a logarithmic relationship with Re_τ. It is well fitted by

(7)

At low Reynolds numbers, for instance, Re_τ < 550, the contribution of the molecular viscous dissipation is larger than that of the turbulent energy dissipation. As the Reynolds number increases, turbulent fluctuations are strengthened and gradually dominate skin friction generation. Thus, the turbulent energy dissipation becomes the major cause of skin friction generation, as shown in Fig. 7. In terms of the ratio of (C_{f1, RD}+C_{f2, RD}) to C_f^1.5, which is inversely proportional to u_τ, it changes logarithmically with the friction Reynolds number. This suggests an empirical law for the friction velocity in accordance with the DNS results and experimental data in Ref. [30].

Thus, incorporating Dean's correlation gives the empirical expression to quantify the dependence of the contribution on the Reynolds number from direct viscous dissipation, i.e.,

(8)

Alternatively, it can also be formulated with respect to the friction Reynolds number, that is,

(9)

Equations (8) and (9) offer an explicit way to predict the ratio of C_f1 to the total skin friction coefficient in incompressible turbulent channel flows without any flow-field simulations, especially at very high Reynolds numbers. Apparently, the direct viscous dissipation is negatively correlated with the Reynolds number. Once Re or Re_τ reaches infinity, the viscous dissipation vanishes, and the skin friction is entirely generated by the turbulent energy dissipation.

With the aim of guiding the application of drag reduction, the boundary layer is further divided into three sub-regions separated by y⁺=30 and y=0.3. Figure 8 shows the contributive proportion of each sub-region to C_f2 at different Reynolds numbers, based on the FIK identity and the RD identity.

Figures 8(a) and 8(b) share similarities in terms of the tendencies of C_f2^sub/C_f2. As the Reynolds number increases, a dramatic reduction of the contribution from the region y⁺ < 30 and y>0.3 is found, though they are non-negligible at low Reynolds numbers. This is consistent with the phenomenon that a smaller portion of C_f emerges in the near-wall region when the Reynolds number increases, as presented in Figs. 3 and 5. The differences between these two methods are limited in terms of the relative importance of these two regions, which is due to their essentially different interpretation of skin friction generation, as previously indicated. For the region where y⁺>30 and y < 0.3, its contribution increases significantly with the Reynolds number, which is ascribed to the energy superposition of large-scale turbulent motions^[36].

The contributions from the region where y⁺>30 and y < 0.3 account for more than 60% of C_f2 at Re_τ=5 200. Thus, at high Reynolds numbers, it is critically important to suppress turbulent fluctuations in this region, although this might result in some technical difficulties hitherto. Meanwhile, it is noteworthy that, based on the FIK identity, the near-wall contributions (y⁺ < 30) are almost negligible at Re_τ= 5 200 while based on the RD identity, the near-wall region still accounts for nearly 30% of C_f2. Numerous studies have confirmed that artificial manipulation of the near-wall structures is able to achieve high drag reduction efficiency even when Re is very high^[37-38]. As for the decay of the drag reduction efficiency previously proposed, it is likely to be associated with large-scale turbulent motions. Thus, at high Reynolds numbers, the inhibition of the large-scale amplitude modulation on the small scales and the outer site energy superposition onto the inner region should be investigated in depth to achieve more efficient drag reduction via control of the near-wall structures.

5 Conclusions

This paper focuses on decomposition methods for mean skin friction in turbulent channel flows. The effects of the Reynolds number on the mean skin friction generation are investigated using both the FIK identity and RD identity at the friction Reynolds numbers Re_τ = 180, 550, 1 000, 2 000, and 5 200. The FIK and RD identities are two classical methods for the skin friction decomposition. Although some similarities can be found in the tendencies of C_f1/C_f and C_f2/C_f with regard to the Reynolds number, they are essentially different, especially in terms of the interpretation of their contributions to skin friction generation.

In the case of the FIK identity, it is confirmed that the linearly weighted Reynolds stress plays a dominant role in terms of skin friction generation. Moreover, at higher Reynolds number, its contribution becomes more important. The near-wall turbulent motions play a less important role in the skin friction generation and can almost be neglected at high Reynolds numbers.

In the case of the RD identity, the mean skin friction is decomposed from the perspective of energy transfer from the moving wall to the fluid in an absolute reference frame, by means of molecular viscous dissipation C_{f1, RD} and turbulent energy production (or dissipation) C_{f2, RD}. It is determined that C_{f1, RD} mostly originates from the region y⁺ < 30 and is negatively correlated with the Reynolds number. A good coincidence of C_{f1, RD}/C_f^1.5 is observed regardless of the Reynolds number. Once the friction Reynolds number exceeds 550, C_{f2, RD} becomes the major component, accompanied by a secondary peak of the pre-multiplied integrand of C_{f2, RD}/C_f, as observed in the outer layer. The increasingly important contributions of C_{f2, RD} at high Reynolds numbers are ascribed to the generation of large-scale turbulent motions, which contain large amounts of turbulent kinetic energy and possibly lead to the redistribution of the turbulent kinetic energy from the near-wall region to the outer region. In contrast with the coincidence of C_{f1, RD}/C_f^1.5, C_{f2, RD}/C_f^1.5 has a logarithmic relationship with Re_τ, and is well fitted by C_{f2, RD}/C_f^1.5=1.87ln(Re_τ)-5.29. Thus, empirical expressions for C_{f1, RD}/C_f are proposed, which allow for prediction of the drag-component without any pre-computation of the flow field.

The boundary layer is divided into three sub-regions, separated by y⁺=30 and y=0.3, to evaluate the contributive proportion of each sub-region to C_f2 for different Reynolds numbers. The results suggest that as the Reynolds number increases, a dramatic reduction of the contribution from the region y⁺ < 30 and y>0.3 occurs. For the RD identity, at the largest Reynolds number investigated, there is 30% contribution to C_f2 which originates from the region where y⁺ < 30. This indicates that at high Reynolds numbers, the large-scale amplitude modulation on the small scales and the outer site energy superposition onto the inner region should be restricted to achieve more efficient drag reduction.

Appendix A

In order to compare the turbulent kinetic energy production with dissipation over the entire boundary layer, Fig. A1 shows the distributions of turbulent production P and dissipation -ε as a function of y⁺. Integrations of P and -ε over the entire boundary layer give the total turbulent kinetic energy production P_t and the dissipation ε_t, which are listed in Table A1. It can be determined that the relative errors of (P_t-ε_t)/P_t are limited within 0.2%.

Acknowledgements  The large-scale computations were supported by the Center for High- Performance Computing, Shanghai Jiao Tong University.