In this study, direct numerical simulations (DNSs) of fully developed turbulent pipe and channel flows are performed to investigate the influence of the superhydrophobic surfaces (SHSs) on the turbulence dynamics and the resulting drag reduction of the flows under similar conditions. SHSs at the wall are modeled in spanwise alternating longitudinal regions with a boundary with no-slip and shear-free conditions, and the two parameters of span periodicity (P/δ) and SHS fraction (GF) within a pitch are considered. However, the drag reduction rate (DR) in the pipe flows is greater than in the channel flows with an accompanying reduction in the Reynolds stress.
The improved performance of the DR for the pipe flow is attributed to the increased streamwise slip and weakened Reynolds shear stress contributions. Finally, an inspection of the origin of the mean secondary flow in turbulent flows over SHSs based on the spatial gradients of the turbulent kinetic energy demonstrates that the secondary flow is both driven and sustained by spatial gradients in the Reynolds stress components, i.e. Prandtl's secondary flow of the second kind. Probability density functions of the 2-D vorticity strength (λ U / δxi τo ) in turbulent pipe. a- b) Streamwise, (c-d) wall-normal and (e-f) span-wise components.
The phase-averaged velocity vector of v w, is superimposed to reveal the role of the anisotropy of the Reynolds stress tensor in generating a secondary flow.
List of Table
Introduction
Very recently, Zhang et al. 2016) conducted a turbulent boundary layer flow experiment with longitudinal SHSs to analyze the change of Reynolds shear stress distribution. However, they found that although spanwise slip increases drag for sufficiently small streamwise slip lengths, drag decreases for large instances of streamwise slip (e.g., three to four times greater than the viscous length scale), regardless of spanwise slip. 2006) proposed a theoretical prediction of the DR velocity by SHSs in turbulent channel flow. In turbulent channel flows with longitudinal SHSs, Rastegari & Akhavan (2015) showed that a maximum DR rate of 80% can be achieved and found that, although the amount of DR is influenced by the streamwise effective wall slip and changes in turbulence dynamics, the first term plays an important role in inducing most of the DR. 2009) also performed a DNS for turbulent channel flows over longitudinal grooves using alternating no-slip (solid-water interface) and no-shear (air-water interface) boundary conditions in the spanwise direction, reducing wall shear stress to as much as 40%, consistent with an experimental observation by Daniello et al.
2014) also found a similar variation of the rotational sense with a different height length in turbulent channel flows over the SHS. Although the origin of a secondary jet in a turbulent flow over the SHS has been attributed to the second Prandtl type due to the lack of a tilting effect in the primary flow (Jelly et al. 2014), there is no clear evidence for the origin of a jet secondary has been. For a better understanding of the mechanisms responsible for the differences in DR velocity between pipe and channel flows, mathematical analyzes using a mean flow momentum equation (Fukagata et al. 2002) and a total mean vorticity equation (Yoon et al. 2016) performed.
Finally, the physical origin of the secondary flow of Prandtl's second kind is investigated based on suitable forms of spatial gradients in the Reynolds stress components of the turbulent kinetic energy.
Numerical method
In particular, Monty et al. 2009) found no clear or significant differences in the spectral representation of pipe and channel flows, even in the core region. In the present study, DNSs of turbulent pipe flows with SHSs are performed to investigate the distinctive flow properties in pipes over SHSs. The longitudinal SHSs in pipe and channel flows consist of repetitive non-slip and shear-free conditions in the azimuthal (span) direction.
The derived quantities shifted relative to the centerline are calculated by differentiating opposite values across the centerline, taking into account reversals in the directions of the radial and azimuthal unit vector through the centerline (Jang et al. 2011). In the present study, the SHSs are assumed to be located on limited longitudinal surfaces for convenience (Fig. 1). On the bottom and top walls, the boundary condition at the wall is specified as alternating shear-free ((∂u/∂y)w=0 and (∂w/∂y)w=0) and slip-resistant conditions in the spanwise direction (Philip 1972).
For phase averaging in the spanwise direction, a phase with respect to the periodic surface at the wall is defined as
Results and discussion
- Skin-friction drag
- Turbulent statistics
- Skin-friction budget
- Turbulent structures
- Mean secondary flow
4(d) indicates that the difference in the slip velocity between the pipe and channel flow is not significantly affected by GF with a fixed P/δ, in contrast to the variation of the drag in Figs. 5, profiles of the Reynolds stresses normalized by Uτo are shown with the variances of P/δ (left) and GF (right) for the pipe and channel flow over SHSs. The spanwise components of the Reynolds stresses near the wall (y+<5) are continuously amplified with an increase in P/δ.
As expected, the profiles of the Reynolds stresses for the pipe and channel flows indicate that the turbulence for the pipe flows is largely suppressed in the outer layer. 5(c) and (e), the variation of the wall-normal and spanwise Reynolds stresses for P/δ≥3.14 is not consistent with the observation of the production in Figs. The reduction of the turbulence production in the pipe flow over SHSs leads to a greater reduction of the turbulent resistance for the pipe flow.
The magnitude of the spanwise slip velocity for a pipe flow is generally larger than for a channel flow, and the difference in velocity between the pipe and channel flow becomes larger as P/δ and GF increase. Although the difference in the streamwise slip velocity at the wall between the pipe and channel flows is similar regardless of P/δ and/or GF (Fig. 4), the largest difference in the spanwise slip velocity is found at P/δ=6.28 , consistent with the variation of the resistor shown in fig. Variation of the conditional mean spanwise velocity at the wall in turbulent pipe and channel flows over SHSs as a function of (a, c) P/δ for GF=0.5 and (b, d) GF for P/δ=6, 28.
In this section, we estimate the component contributions of the skin friction drag to identify different dynamic effects in pipe and channel flows over SHSs. The contribution of Reynolds shear stress to skin frictional resistance with an increase in P/δ for fixed GF values of 0.25, 0.5 and 0.75 in Figs. P/δ is mostly induced by the streamwise slip velocity contribution, and as P/δ increases, the Reynolds shear stress also contributes to the increase of the drag difference.
However, when GF=0.75, the contribution of Reynolds shear stress for pipe and channel flows is reduced to 10%~50% and 15%~43% of surface friction resistance respectively. 2 results from the large negative contribution of Cf1 due to the presence of increased secondary current for tube currents. A large difference in the spanwise slip velocity between pipe and channel flows when P/δ=6.28 and GF=0.75 in Fig.
To provide statistical evidence of the variation of vortex structures based on instantaneous flow fields in pipe and channel flows over SHS, p.d.f. In addition, it is clear that the secondary flow force for the tube flows in Fig. 17)-(20) is larger than that of channel flows for all P/δ due to the higher slip velocity in space.
Summary and conclusion
Furthermore, the FIK identity with the velocity–vorticity correlation showed that the difference in skin friction drag between pipe and channel flows is mostly due to a significant discrepancy in the contributions of the advective vorticity transport terms. Since the secondary flow in the form of a pair of counter-rotating vortices plays an important role in generating turbulent momentum transport, the larger negative contribution of advective eddy transport in pipe flows than in channel flows demonstrates the presence of stronger secondary flows in pipe flows, consistent with the observation related to velocity it slides across the span. In contrast to the finding that the increased rates of longitudinal slip velocity for pipe and channel flows were similar, the higher rate of change of drag and slip velocity across the span in pipe flows compared to channel flows as P/δ varied, showed that the spanwise slip velocity for turbulent flows over the SHS also contributes significantly to the reduction of the skin friction drag.
The larger spanwise slip rate as P/δ and GF vary in the pipe flows was consistent with the larger decrease in the Reynolds shear stress contribution in the pipe flows, as the spanwise slip attenuates the turbulence near the wall with less wall normal. span speed gradient. To investigate the origin of the mean secondary flow in turbulent flows over SHSs, the spatial gradients of the tke equation were analyzed. Because the contours of Pk k Tk Dk k were locally unbalanced, advective velocities in the transport equation for tke must be induced in the flows to maintain conservation of energy.
Similarly, because the advective velocities transported fluid with the lowest tke to the fluid with the highest tke or that with the highest tke to the fluid with the lowest tke, depending on the sign of the energy budget for the tke equation (Hinze 1967 ), positive and negative wall normal velocities were found over the no-slip region for small and large P/δ. To maintain mean flow continuity, the normal and spanwise velocities of the wall must be balanced in the absence of flow heterogeneity, and ejection or sweep events over the non-slip area must be balanced by a lateral flow near the wall, leading to the generating a secondary current. It was thus concluded that the secondary flows in turbulent pipe and channel flows over SHSs are both driven and maintained by Prandtl's secondary flows of the second kind.
2014 Large-scale contribution to the mean shear stress of walls in flat plate boundary layers with high Reynolds number up to 13650.
