2.1 Channel geometry

Consider fluid flow driven by a known pressure gradient through a channel extending to $\pm \infty$ in the $x$ -direction bounded by walls fitted with straight grooves parallel to the flow direction (see figure 1) and of arbitrary shape in the spanwise $z$ -direction. The channel geometry is described using Fourier expansions of the form

where the subscripts $L$ and $U$ refer to the lower and upper walls, respectively, ${\it\lambda}=2{\rm\pi}/{\it\alpha}$ denotes the groove wavelength, ${\it\alpha}$ stands for the groove wavenumber, $H_{L}^{(m)}=H_{L}^{(-m)\ast \ast }$ and $H_{U}^{(m)}=H_{U}^{(-m)\ast \ast }$ are the reality conditions, double stars denote the complex conjugates, $N_{A}$ represents the number of Fourier modes required to describe the geometry and the half-channel height $H^{\ast }$ of the reference smooth channel has been used as the length scale. The objective of this analysis is to describe the effects of the flow modulations on the channel discharge and, accordingly, it is assumed that the mean openings of the grooved and the reference smooth channels are the same, i.e. $H_{L}^{(0)}=H_{U}^{(0)}=0$ .

Figure 1.

Sketch of the flow configuration.

2.2 Field equations

It is assumed that the introduction of the grooves does not affect the mean streamwise pressure gradient and, thus, the pressure is represented as $p(x,y,z,t)=-x+p^{\prime }(x,y,z,t)+\text{const}$ . The discharge is, however, affected and its evaluation requires solving the field equations of the form

where $(u,v,w)$ are the components of the velocity vector in the $(x,y,z)$ directions, respectively, the friction velocity defined as $u_{{\it\tau}}^{\ast }=\sqrt{{\it\beta}^{\ast }H^{\ast }/{\it\rho}^{\ast }}$ is used as the velocity scale, ${\it\beta}^{\ast }$ stands for the mean pressure gradient in the $x$ -direction, ${\it\rho}^{\ast }$ denotes density, $H^{\ast }$ is the mean channel half-height, $H^{\ast }/u_{{\it\tau}}^{\ast }$ is used as the time scale, ${\it\rho}^{\ast }u_{{\it\tau}}^{\ast 2}$ is used as the pressure scale and $Re_{{\it\tau}}=u_{{\it\tau}}^{\ast }H^{\ast }/{\it\nu}^{\ast }$ is the friction Reynolds number where ${\it\nu}^{\ast }$ denotes the kinematic viscosity. For interpretation purposes we shall also need the Reynolds number based on the bulk fluid velocity $U_{b}^{\ast }$ and the full height of the reference smooth channel defined as $Re_{2H}=U_{b}^{\ast }2H^{\ast }/{\it\nu}^{\ast }$ where $U_{b}^{\ast }=\iint U^{\ast }\,\text{d}A^{\ast }/\iint \,\text{d}A^{\ast }$ with $A^{\ast }$ standing for the channel cross-sectional area associated with one groove wavelength. In the above, stars denote the dimensional quantities. System (2.2) is subject to the homogeneous boundary conditions of the form

The discharge is determined during the post-processing according to the formula

An increase of $Q_{mean}$ above the reference discharge created by the same pressure gradient in the smooth channel demonstrates an increase in the effectiveness of the flow system, or effective drag reduction.

4.2 Validation of DES for turbulent flows in corrugated channels

Tests carried out for flows in smooth channels demonstrated that DES provides accuracy similar to DNS. It remains to be demonstrated that both methods provide a similar accuracy for grooved channels. Results presented in table 4 show a discharge difference of less than 0.5 % between the methods which justifies the use of the more computationally efficient DES method.

Table 4.

The difference of the discharge determined using the DES and DNS methods for $Re_{{\it\tau}}=180$ .

4.3 Reduced geometry model for turbulent channel flows

It has been demonstrated that sinusoidal grooves play the role of the reduced geometry model for laminar flows (Mohammadi & Floryan Reference Mohammadi and Floryan2013b ; Moradi & Floryan Reference Moradi and Floryan2013). We shall demonstrate that such grooves also represent the reduced geometry model in turbulent flows. We shall use triangular and trapezoidal grooves illustrated in figure 7 for this demonstration.

Table 5 displays results for groove wavenumber ${\it\alpha}=0.25$ and amplitude $S=1$ . It can be seen that the difference in the discharge obtained for the actual groove and for the same groove with its shape approximated by the leading Fourier mode from its Fourier expansion is less than 1 %. This demonstrates that the concept of the reduced geometry model can be used in the analysis of turbulent flows and thus further discussion is focused on sinusoidal grooves only. It shall be noted that although there are indications that the reduced geometry model works for turbulent flows, the assessment of uncertainties herein is approximate.

Table 5.

Comparison of the discharge through the grooved channel with ${\it\alpha}=0.25$ and $S=1$ determined using the actual groove geometry as well as its approximation in the form of a leading Fourier mode from its Fourier expansion for $Re_{{\it\tau}}=180$ .

4.4 Analysis of flows in channels with sinusoidal grooves

We shall now discuss the characteristics of turbulent flows in channels with sinusoidal grooves. The channel geometry is described by (3.6) and the computations have been carried out for $Re_{{\it\tau}}=180$ ( $Re_{2H}\approx 6000$ ) using $64\times 128\times 64$ mesh and ${\rm\Delta}t=0.002$ . DNS has been used only for a few geometries (see table 6) due to high computational cost for channels with long-wavelength grooves.

Table 6.

List of the groove configurations studied. ‘ $Y$ ’ identifies configurations studied using only DES and ‘ $Y$ *’ identifies configurations studied using both DNS and DES.

Figure 8 displays variations of the difference between the discharges through the corrugated and smooth channels generated by the same pressure gradient. The existence of a critical groove wavenumber ${\it\alpha}_{c}$ , which separates the discharge-increasing grooves from the discharge-reducing grooves, is clearly documented. The critical wavenumber is estimated to be ${\it\alpha}_{c}\approx 1$ for $S=0.5$ and approximately ${\it\alpha}_{c}\approx 0.8$ for $S=1$ . These values are similar to the ${\it\alpha}_{c}$ found in laminar flows ( ${\it\alpha}_{c}=1.2$ ). An increase of the groove amplitude $S$ decreases the discharge when ${\it\alpha}>{\it\alpha}_{c}$ and, conversely, increases the discharge when ${\it\alpha}<{\it\alpha}_{c}$ . A decrease of ${\it\alpha}$ increases the discharge but this increase rapidly approaches an asymptotic limit where any further decrease of ${\it\alpha}$ brings in negligible contributions. The highest increase of the discharge at $S=0.5$ is approximately 2 %–3 % , and it is approximately 9 % at $S=1$ . The maximization of the discharge requires the use of the minimum possible groove wavenumber ${\it\alpha}$ as well as the largest possible groove amplitude $S$ .

Figure 8.

Variations of the discharge correction ${\rm\Delta}Q$ as a function of the groove wavenumber ${\it\alpha}$ , (a) and as a function of the groove amplitude $S$ (b) for channel with geometry described by (3.6) for $Re_{{\it\tau}}=180$ . Symbols identify computed cases while lines result from interpolation.

Figure 9.

Distributions of the mean velocity component $U$ in the $y$ – $z$ plane of channel with geometry described by (3.6) with ${\it\alpha}=0.25,S=0.5$ (a), ${\it\alpha}=0.25,S=1$ (b), ${\it\alpha}=0.5,S=0.5$ (c), ${\it\alpha}=0.5,S=1$ (d), ${\it\alpha}=2,S=0.5$ (e) and ${\it\alpha}=2,S=1$ (f) for $Re_{{\it\tau}}=180$ .

Figure 10.

Distribution of the mean velocity component $U$ in the $y$ – $z$ plane of a smooth channel and averaged only in the streamwise direction for $Re_{{\it\tau}}=180$ .

Figure 9 displays distributions of $U$ in spanwise cross-sections. The fluid either accelerates or retains its velocity in the wide segment of the channel and slows down in the narrow segment. When ${\it\alpha}<{\it\alpha}_{c}$ (figure 9 a–d), the difference between the maximum velocity in the widest and narrowest sections of the channel is larger than the similar difference for channels with ${\it\alpha}>{\it\alpha}_{c}$ (figure 9 e,f). The velocity maximum at the widest section of the channel for ${\it\alpha}<{\it\alpha}_{c}$ is higher than the maximum velocity in the smooth channel ( $U\approx 19$ ; figure 10) and this results in an increase in the discharge. The discharge increase is most pronounced for conditions used in figure 9(b) where the maximum velocity reaches value of $U\approx 24$ . When ${\it\alpha}>{\it\alpha}_{c}$ (figure 9 e,f), the maximum velocity in the widest portion of the channel is $U\approx$ 18–19, which is similar to that found in the smooth channel and, since velocity decreases in the narrow section, there is an overall reduction in the discharge.

The effect of the grooves on the flow discharge is qualitatively similar to that found in laminar flows. The presence of the grooves results in the rearrangement of the bulk motion which leads to the formation of stream tubes of high velocity fluid centred at the widest channel opening, as illustrated in figure 9. The stream tubes exist for ${\it\alpha}<{\it\alpha}_{c}$ as well as for ${\it\alpha}>{\it\alpha}_{c}$ but play a different role in each case. The difference is illustrated in figure 11 which displays distributions of $U$ in the $y$ – $z$ plane. It can be seen that the flow is redirected towards the wider channel section when ${\it\alpha}<{\it\alpha}_{c}$ with a minor velocity decrease in the narrow section; it is the increase of the channel opening in the widest section which dominates the discharge. When ${\it\alpha}>{\it\alpha}_{c}$ , the flow is ‘choked’ in the narrowest section while it retains the original velocity in the widest section; it is the decrease of the channel opening in the narrowest section which dominates the discharge.

The lack of smoothness of velocity contours in figure 9 is due to the turbulent structures whose effect has not been completely eliminated by the averaging procedure. The scale of these structures depends on the wall length scale $l^{\ast }$ but is almost independent of the groove wavelength (compare figures 9 and 11).

Figure 11.

Distributions of the mean velocity $U$ in the $y$ – $z$ plane for channel geometry described by (3.6) with ${\it\alpha}=0.5$ , $S=0.5$ (a) and with ${\it\alpha}=2$ , $S=0.5$ (b) for $Re_{{\it\tau}}=180$ .

We shall now discuss changes in the velocity profiles resulting from the introduction of the grooves. The velocity profile for the smooth wall consists of three layers, i.e.

1. (i) viscous sub-layer, ( $y^{+}<7$ , where $U=y^{+}$ ),

2. (ii) buffer layer, ( $7<y^{+}<30$ ) and

3. (iii) inertial sub-layer, ( $y^{+}>30$ , $U=(1/{\it\kappa})\ln y^{+}+C$ , with ${\it\kappa}\approx 0.38$ –0.41 and $C\approx 5$ –5.5).

The velocity profiles in the corrugated channel for conditions leading to an increase in discharge ( ${\it\alpha}=0.5$ , $S=0.5$ ) and a reduction in discharge ( ${\it\alpha}=2$ and $S=0.5$ ) are displayed in figure 12 for three locations in the spanwise direction; plane 1 corresponds to the narrowest channel opening, plane 2 corresponds to the mean channel opening (the neutral section) and plane 3 corresponds to the widest channel opening (see figure 12 a). In all cases, the velocity profiles consist of the linear region, the buffer region and the log-law region, similarly as in the smooth channel.

Figure 12.

Variations of the mean velocity $U$ as a function of $y^{+}$ at three spanwise locations identified in (a) for channel geometry described by (3.6) with ${\it\alpha}=0.5$ , $S=0.5$ (b,d) and with ${\it\alpha}=2$ , $S=0.5$ (c,e) for $Re_{{\it\tau}}=180$ . The origins of the $y$ -coordinates at different planes have been overlapped to simplify comparisons.

In the case of the discharge increase ( ${\it\alpha}=0.5$ , $S=0.5$ ), the velocity profile in plane 2 is nearly identical to the velocity in the smooth channel. The velocity in plane 1 (the narrowest opening) is smaller than in the smooth channel resulting in smaller $\partial U/\partial y^{+}$ at the wall (see figure 12 d). The velocity in plane 3 (the widest opening) is higher than in the smooth channel with the difference increasing as $y^{+}$ increases. In the case of the discharge reduction ( ${\it\alpha}=2$ , $S=0.5$ ), $U$ in plane 2 is smaller than in the smooth channel resulting in smaller $\partial U/\partial y^{+}$ . Values of $U$ in the remaining planes are slightly smaller than in the smooth channel for $y^{+}<20$ (see figure 12 e); as $y^{+}$ increases, the velocity in plane 3 (the widest channel opening) becomes similar to that in the smooth channel while the velocity in plane 1 (the narrowest channel opening) becomes smaller than in the smooth channel.

4.5 Skin friction

An understanding of the skin friction is essential as it describes the mechanism responsible for drag creation. The total drag is unaltered by the addition of the grooves as the mean pressure gradient is kept unchanged. Thus, the integration of the shear stress over the wetted area must produce the same result regardless of the groove shape while the discharge may vary significantly depending on the groove shape. The reader may note that the addition of the grooves increases the wetted area and thus an increase in the discharge may occur only if the shear decreases sufficiently in order to balance the larger wetted area. Variations of the normalized skin friction $Sm/Sm_{0}$ are presented in figure 13(b,c). Here the projected area represents the projection of the actual surface onto the plane $y=\text{const.}$ and it is the same as the relevant area of the smooth channel.

Data displayed in figure 13(a) provide a comparison between the distribution of the local shear stress ( $Sm(0,z)$ on a slice at $x=0$ ) and its streamwise average ( $Sm(z)$ ). Spanwise fluctuations of the local shear stress at a specific $x$ -location are reduced when the streamwise averaging is used (see figure 13 a). We shall use the streamwise averaged data in the following discussion.

Figure 13.

The normalized shear acting on the corrugated surface in a channel with geometry described by (3.6) for $Re_{{\it\tau}}=180$ . Comparison of the streamwise averaged and non-averaged data is given in (a). The spanwise distributions for $S=0.5$ and selective ${\it\alpha}$ are given in (b), and for ${\it\alpha}=0.5$ and selective $S$ are given in (c).

Figure 13(b) depicts the effect of the groove wavenumber on the skin friction. When ${\it\alpha}<{\it\alpha}_{c}$ in the discharge-increase case, $Sm/Sm_{0}>1$ in the wide section and $Sm/Sm_{0}<1$ in the narrow section. Reduction of ${\it\alpha}$ produces larger $Sm/Sm_{0}$ in the wide section and smaller $Sm/Sm_{0}$ in the narrow section (compare results for ${\it\alpha}=0.25$ and 0.5). When ${\it\alpha}>{\it\alpha}_{c}$ in the discharge reduction case, $Sm/Sm_{0}\approx 1$ in both the wide and narrow sections. The physical distance between these two sections are small, which results in similar momentum transport characteristics and leads to a more uniform skin friction. The reader may note that the shear variations in the streamwise direction must average out as the mean stress $Sm/Sm_{0}$ is always 1 (the total drag remains the same).

Figure 13(c) illustrates the effects of the groove amplitude for ${\it\alpha}=0.5<{\it\alpha}_{c}$ . $Sm/Sm_{0}$ in the wide section increases with increasing $S$ , but decreases in the narrow section. This is due to the fact that the larger groove amplitude accentuates the velocity difference between the wide and narrow sections of the channel.

4.6 The turbulent flow quantities

It has been documented so far that the presence of the grooves leads to the rearrangement of the mean velocity profile which results in an increase in the discharge. We shall now discuss how the turbulence characteristics change in response to the grooves.

The components of the turbulent kinetic energy (TKE) $k=(\langle u^{\prime 2}\rangle +\langle v^{\prime 2}\rangle +\langle w^{\prime 2}\rangle )/2$ and the dominant component of Reynolds shear stress ( $\langle u^{\prime }v^{\prime }\rangle$ ) for grooves with ${\it\alpha}=0.5$ , $S=0.5$ and ${\it\alpha}=2$ , $S=0.5$ , determined using DNS, are displayed in figures 14 and 15 respectively. Our intention is to show and highlight the behaviour of the dominant component of Reynolds stress $\langle u^{\prime }v^{\prime }\rangle$ at the three planes in comparison to the smooth channel at plane 2. The remaining components of (lower magnitude) Reynolds stress $\langle u^{\prime }w^{\prime }\rangle$ and $\langle v^{\prime }w^{\prime }\rangle$ are plotted in the $y$ – $z$ plane for comparison of the different cases as shown in figure 16. It should be noted that $\langle u^{\prime }w^{\prime }\rangle$ and $\langle v^{\prime }w^{\prime }\rangle$ are zero for a smooth channel.

Figure 14.

Distributions of the TKE in the corrugated and smooth channels for $Re_{{\it\tau}}=180$ . (a) and (c) display $\langle u^{\prime 2}\rangle$ , $\langle v^{\prime 2}\rangle$ and $\langle w^{\prime 2}\rangle$ for grooves described by (3.6) with ${\it\alpha}=0.5$ , $S=0.5$ while (b) and (d) display the same quantities for ${\it\alpha}=2$ , $S=0.5$ . The total kinetic energy total $k$ is displayed in (e) and (f) for ${\it\alpha}=0.5$ , $S=0.5$ and ${\it\alpha}=2$ , $S=0.5$ , respectively.

Figure 14 displays data for the narrowest channel opening (plane 1 in figure 12 a), the mean opening (plane 2) and the widest opening (plane 3). Note that the $y$ -axis used herein based on the same $y$ origin location is different from the $y^{+}$ starting from wall used in figure 12. The qualitative characteristics of variations of all quantities in all planes are similar to those found in the smooth channel, i.e. they rapidly increase in the viscous sub-layer ( $y^{+}<15$ ), reach peak values in the buffer layer ( $y^{+}\approx 15$ ) and decrease in the inertial sub-layer ( $y^{+}>15$ ).

Results displayed in figure 14(a,c,e) demonstrate that all quantities in plane 2 for ${\it\alpha}=0.5<{\it\alpha}_{c}$ (discharge-increase case) are nearly the same as in the smooth channel. The peak values in plane 1 (the narrowest section) are lower than in the smooth channel, which is possibly due to the maximum velocity being smaller than in the smooth channel. The peak values in plane 3 (the widest opening) are higher than those in the smooth channel due to the maximum velocity being larger than in the smooth channel. In the discharge-reduction case ( ${\it\alpha}=2>{\it\alpha}_{c}$ ) the situation becomes more complex (see figure 14 b,d,f). In plane 2 (the neutral section), $\langle u^{\prime 2}\rangle$ is slightly smaller than in the smooth channel in both the viscous sub-layer and the buffer layer, but is slightly larger in the inertial sub-layer. One may conclude that the distribution of $\langle u^{\prime 2}\rangle$ in this plane shifts towards the centre of the channel when compared to the smooth channel. In plane 1, the peak of $\langle u^{\prime 2}\rangle$ is smaller than in the smooth channel, and similar to plane 1 for the discharge-increase case ( ${\it\alpha}=0.5$ ). The peak of $\langle u^{\prime 2}\rangle$ in plane 3 is similar to the smooth channel and, thus, smaller than that in the discharge-increase case. Both $\langle v^{\prime 2}\rangle$ and $\langle w^{\prime 2}\rangle$ in all three planes are approximately similar to the smooth channel, which is possibly due to the weak rearrangement of the bulk motion. One can observe that $\langle u^{\prime 2}\rangle$ represents the largest component of TKE and, thus, it dictates the properties of TKE.

Figure 15.

Distributions of $\langle u^{\prime }v^{\prime }\rangle$ in the corrugated channel with shape described by (3.6) and in the smooth channel for $Re_{{\it\tau}}=180$ . Data displayed in (a) correspond to ${\it\alpha}=0.5$ , $S=0.5$ (discharge-increase case); data displayed in (b) correspond to ${\it\alpha}=2$ , $S=0.5$ (discharge-reduction case).

Figure 15 displays distributions of $\langle u^{\prime }v^{\prime }\rangle$ . In the discharge-increase case ( ${\it\alpha}=0.5$ ), its magnitude in plane 2 (the neutral section) is slightly smaller than in the smooth channel, it is smaller in plane 1 (the narrowest section) and it is larger in plane 3 (the widest section). In the discharge-reduction case ( ${\it\alpha}=2$ ), its magnitude in plane 2 is smaller within the viscous sub-layer and in the buffer layer but slightly larger within the inertial sub-layer. In plane 1 (the narrowest section), the peak values are lower than in the smooth channel. In plane 3 (the widest section), the peak value is similar to that in plane 1.

It can be concluded that in the discharge-increase case ( ${\it\alpha}<{\it\alpha}_{c}$ ), TKE and Reynolds shear stress $\langle u^{\prime }v^{\prime }\rangle$ in the wide section are higher and in the narrow section are smaller than in the smooth channel. In the discharge-reduction case ( ${\it\alpha}>{\it\alpha}_{c}$ ), TKE and Reynolds shear stress $\langle u^{\prime }v^{\prime }\rangle$ are smaller in the narrow section and similar to the smooth channel in the wide section. The distributions of TKE and Reynolds shear stress $\langle u^{\prime }v^{\prime }\rangle$ shift towards plane 2 (neutral plane) for the discharge-reduction case ( ${\it\alpha}>{\it\alpha}_{c}$ ). This may be due to the smaller physical distance between the wide and narrow sections and, hence, larger spanwise velocity gradients which, in turn, produce more intense turbulent fluctuations near the neutral plane.

Figure 16.

Distributions of $\langle u^{\prime }w^{\prime }\rangle$ and $\langle v^{\prime }w^{\prime }\rangle$ in the corrugated channel with shape described by (3.6) for $Re_{{\it\tau}}=180$ . Data displayed in (a) and (c) correspond to ${\it\alpha}=0.5$ , $S=0.5$ (discharge-increase case); data displayed in (b) and (d) correspond to ${\it\alpha}=2$ , $S=0.5$ (discharge-reduction case).

In order to show the complexity of distributions of Reynolds shear stresses $\langle u^{\prime }w^{\prime }\rangle$ and $\langle v^{\prime }w^{\prime }\rangle$ , these quantities are presented in the form of contour plots in the $y$ – $z$ plane (see figure 16) rather than as 1-D cuts along the $y$ -axis in planes 1, 2, 3 as such cuts do not exhibit any distinctive features. The theoretical arguments suggest that $\langle u^{\prime }w^{\prime }\rangle$ and $\langle v^{\prime }w^{\prime }\rangle$ should be zero in the smooth channel and very close to zero in numerical simulation of such a flow with non-zero values arising from numerical round-off errors. One may interpret $\langle u^{\prime }w^{\prime }\rangle$ and $\langle v^{\prime }w^{\prime }\rangle$ as representing the interactions between the fast fluid in the wide section of the channel and the slow fluid in the narrow section. As a result, the highest magnitudes of $\langle u^{\prime }w^{\prime }\rangle$ and $\langle v^{\prime }w^{\prime }\rangle$ are located around plane 2 (neutral plane) where the strongest spanwise interactions occur. Such interactions for the discharge-reduction case ( ${\it\alpha}=2$ ) are more intense than for the discharge-increase case ( ${\it\alpha}=0.5$ ) because of the reduction in the physical distance between the wide and narrow sections of the channel.

Figure 17.

The volume-averaged (a) TKE $[k]$ ; (b) $[|\langle u^{\prime }v^{\prime }\rangle |]$ ; (c) $[|\langle u^{\prime }w^{\prime }\rangle |]$ and $[|\langle v^{\prime }w^{\prime }\rangle |]$ for $Re_{{\it\tau}}=180$ .

The effects of the groove wavenumber can be clarified by looking at the volume-averaged values of TKE $[k]$ and the absolute values of the Reynolds shear stresses $[|\langle u^{\prime }v^{\prime }\rangle |]$ , $[|\langle u^{\prime }w^{\prime }\rangle |]$ , $[|\langle v^{\prime }w^{\prime }\rangle |]$ , which are displayed in figure 17 for ${\it\alpha}=0.5$ , 1, 2 with $S=0.5$ . The volume-averaging operator $[\,]$ is defined as

The reader may note that $[|\langle u^{\prime }w^{\prime }\rangle |]$ and $[|\langle v^{\prime }w^{\prime }\rangle |]$ are both zero in the smooth channel. The volume-averaged TKE $[k]$ for the corrugated channel is slightly higher (approximately 6 %) than that for the smooth channel, $[|\langle u^{\prime }w^{\prime }\rangle |]$ and $[|\langle v^{\prime }w^{\prime }\rangle |]$ are also higher than in the smooth channel and increase with increasing wavenumber. $[|\langle u^{\prime }v^{\prime }\rangle |]$ in the corrugated channel is slightly higher than in the smooth channel for the discharge increase case ( ${\it\alpha}=0.5$ ) and is slightly lower for the discharge reduction case ( ${\it\alpha}=2$ ).

The above discussion shows that corrugations can rearrange the distribution of TKE and the Reynolds stress, but are unable to significantly reduce the global turbulence intensity.

The distributions of the TKE and the Reynolds stress discussed in the previous section imply that there is less turbulent activity in the narrow portion of the corrugated channel and stronger turbulent activity in the wide portion in the discharge-increase case ( ${\it\alpha}<{\it\alpha}_{c}$ ). We shall now look at the turbulence structures in order to better understand the character of the flow in each case.

Figure 18.

Ensemble time-averaging Reynolds shear stress $\langle u^{\prime }v^{\prime }\rangle$ with quadrant detection (Q2 for ejection events (a,b) and Q4 for sweep events (c,d)) in a corrugated channel described by (3.6) with ${\it\alpha}=0.5$ , $S=0.5$ (a,c) and with ${\it\alpha}=2$ , $S=0.5$ (b,d) for $Re_{{\it\tau}}=180$ .

In figure 18, the ensemble time-averaging Reynolds shear stress $\langle u^{\prime }v^{\prime }\rangle$ is presented with quadrant detection for ejection events (Q2) and for sweep events (Q4). Figure 18(a,c) demonstrates that in the discharge increase case ( ${\it\alpha}<{\it\alpha}_{c}$ ) most of the near-wall turbulent motions (ejection and sweep) tend to gather in the wide section of the channel. This may be due to the fact that the maximum velocity in the wide portion is much larger than that in the narrow portion. An increase in the ejection/sweep suggests the occurrence of more intense bursting phenomena, stronger turbulent motions and higher TKE, and this is consistent with the findings presented in figures 14(a,e) and 15(a).

Results presented in figure 18(b) demonstrate that in the discharge-reduction case ( ${\it\alpha}>{\it\alpha}_{c}$ ) the ejection motions tend to occur in the neutral section, which may be associated with the strong velocity gradient between the wide and narrow sections with a short physical distance between them. Figure 18(d) shows that there is no obvious tendency of gathering of sweep motions either in the wide or in the narrow portions of the channel, which is balanced by the momentum transport between the wide and narrow sections. This observation is consistent with the results displayed in figures 14(b,f) and 15(b). It is very interesting to observe that the ejection and sweep behave differently in the discharge-increase and discharge-reduction cases.

In summary, the analysis of TKE, the Reynolds stress and the turbulent structures shows that the turbulence intensity in the narrow portion of the channel is suppressed. When the groove amplitude is high ( $S\rightarrow 2$ ), the narrow section shrinks to a triangular corner and the flow is expected to relaminarize (Eckert & Irvine Jr Reference Eckert and Irvine1956). These results led to the creation of the concept of riblets for turbulent drag reduction (Walsh & Weinstein Reference Walsh and Weinstein1978; Walsh & Lindeman Reference Walsh and Lindeman1984).

4.7 Analysis of flow changes caused by the small-wavenumber grooves

Interactions between the wide and narrow sections of the channel weaken when ${\it\alpha}\rightarrow 0$ and thus each section might behave as a smooth channel but with different heights and the same pressure gradient.

First, we shall demonstrate that the local flow properties become very similar when expressed in terms of the local scales. The local half-channel height ${\it\delta}^{\ast }$ , the local friction velocity $u_{{\it\tau}e}^{\ast }$ , the local friction Reynolds number $Re_{{\it\tau}e}$ and the local wall length $l_{e}^{\ast }$ are defined as

where the subscript $_{e}$ stands for local scalar and the superscript $^{\ast }$ stands for dimensional scalar.

It can be shown that the ratio between the local and the average friction velocities, the average streamwise velocity scaled with the local friction velocity denoted as $U_{e}$ and the wall distance scaled with the local wall length denoted as $y_{e}^{+}$ are of the form

Distributions of $U$ and $U_{e}$ as functions of $y^{+}$ and $y_{e}^{+}$ are presented in figure 19(a,b), respectively. The velocity distributions in the widest, mean and narrowest sections of the corrugated channel are similar to the smooth channel when the local scales are used (figure 19 b) but different when the original scales are used (figure 19 a). Distributions of the largest component of TKE, $\langle u^{\prime 2}\rangle$ , displayed in figure 20 demonstrate that the values for all three planes nearly overlap when scaled using the local scales. This suggests that properties of the flow in different planes can be deduced from properties of the flow in a smooth channel through the use of the proper scaling.

Figure 19.

Distributions of the average mean velocity $U$ for $Re_{{\it\tau}}=180$ as a function of $y^{+}$ (a) and the local mean velocity $U_{e}$ as a function of $y_{e}^{+}$ (b) at the widest, mean and narrowest sections of channel with geometry described by (3.6) with ${\it\alpha}=0.25$ and $S=0.5$ . Symbols identify velocities in the smooth channels with different mean openings.

Figure 20.

Distributions of $\langle u^{\prime 2}\rangle$ as functions of $y^{+}$ (a) and $\langle u_{e}^{\prime 2}\rangle$ as functions of $y_{e}^{+}$ (b) for $Re_{{\it\tau}}=180$ at the widest, mean and narrowest sections of channel with geometry described by (3.6) with ${\it\alpha}=0.25$ and $S=0.5$ . Symbols identify velocities in the smooth channels with different mean openings.

Use of (4.3), (4.9b ) and (4.9c ) leads to the local velocity gradient at the wall of the form

As $\partial U_{e}/\partial y_{e}^{+}=1$ near the wall,

Figure 21.

Distribution of the ratio of wall shear $Sm/Sm_{0}$ for $S=0.5$ and $Re_{{\it\tau}}=180$ .

Figure 21 displays the distribution of $Sm/Sm_{0}$ given by (4.11) together with DNS results. When ${\it\alpha}$ decreases, $Sm/Sm_{0}$ approaches the theoretical estimate. Values computed for ${\it\alpha}=0.25$ are already very close to the theoretical estimate, implying that the interactions between fluid flowing in different sections of the corrugated channel at ${\it\alpha}=0.25$ are very weak and could be neglected.

The above discussion shows that the velocity distribution in an arbitrary $x$ – $y$ plane of the corrugated channel is similar to the smooth channel when described using the local friction velocity, the local wall length and local friction Reynolds number. Thus, the local mean velocity scaled using the local effective friction velocity $u_{{\it\tau}e}^{\ast }$ can be written as (Pope Reference Pope2000)

where $U_{b}^{\ast }$ is the dimensional local bulk velocity, and depends on the local effective friction Reynolds number $Re_{{\it\tau}e}$ . The local mean velocity scaled using the friction velocity $u_{{\it\tau}}^{\ast }$ can be written as

Integration of the mean velocity in the spanwise direction $z$ leads to the discharge

where $2{\it\delta}$ is the local channel opening at the location $z$ .

The discharge for a smooth channel has the form

The ratio of discharges in the corrugated and smooth channels at the same friction Reynolds number (i.e. the same pressure gradient) is

and gives the discharge difference of the form

The above predictions are compared with the DNS results for a channel with $S=0.5$ and $S=1$ at $Re_{{\it\tau}}=180$ . Equation (4.17) gives ${\rm\Delta}Q|_{{\it\alpha}\rightarrow 0}=1.88\,\%$ and ${\rm\Delta}Q|_{{\it\alpha}\rightarrow 0}=7.57\,\%$ , respectively, while the numerical results displayed in figure 8(a) give 2 % and 10 %, respectively. This suggests that the above estimate provides quite a good approximation of the discharge through the corrugated channel when the corrugation wave number is sufficiently small.

The above analysis can be extended to flow conditions not accessible to numerical simulations. As an example, consider the limit of very high friction Reynolds number. The discharge ratio for the corrugated and smooth channels is of the form

Figure 22 displays predictions of ${\rm\Delta}Q$ based on (4.17) for different Reynolds numbers and groove amplitudes for small ${\it\alpha}$ . Data for the laminar flow have been added for comparison. The discharge increases with $S$ at any $Re_{{\it\tau}}$ . There is a significant drop in the discharge increase when the flow changes from the laminar to turbulent form. In the turbulent state, the magnitude of the discharge increase decreases very slowly as $Re_{{\it\tau}}$ increases but remains large enough to be of practical interest.

It has been shown in § 3 that the discharge increases linearly with $Re_{{\it\tau}}$ in the case of laminar flow. The above analysis suggests that in the case of turbulent flows the discharge increase is nearly independent of $Re_{{\it\tau}}$ . The discharge increase is 7.57 % for $Re_{{\it\tau}}=$ 180 and would be 7.28 % and 7.15 % for $Re_{{\it\tau}}=$ 395 and 590 which are the other values of $Re_{{\it\tau}}$ frequently used in DNS studies. These cases have not been attempted during the present study due to computational cost and, thus, the $Re_{{\it\tau}}$ dependence conjectured above remains to be proven.

It can be concluded that the maximization of the discharge is achieved using grooves with decreasing wavenumber and increasing amplitude. The resulting discharge can increase 20 %–30 % above the smooth channel discharge, depending on the Reynolds number. Though this increase is not as high as in laminar flows (150 %), it is still much higher than what can be achieved using other more traditional drag-reducing methods. It can also be concluded that the discharge-increase results from the bulk velocity rearrangement rather than from the suppression of the turbulence intensity.

Figure 22.

Variations of the discharge correction ${\rm\Delta}Q$ as a function of $S$ at different $Re_{{\it\tau}}$ for ${\it\alpha}\rightarrow 0$ .

4.8 Additional cases with phase shift

The above discussion dealt with the positioning of the grooves at the upper and lower walls in such a manner that the maximum height of the upper grooves was located above the minimum height of the lower grooves, i.e. the diverging–converging configuration. It is of interest to check how the relative positioning of the upper and lower groove systems affects the flow discharge. This positioning is parameterized using the phase shift ${\it\varphi}$ , i.e. the channel geometry is described as

Use of ${\it\varphi}=0$ produces the converging–diverging channel and ${\it\varphi}={\rm\pi}$ results in the sinusoidal channel for ${\it\alpha}=0.5$ and $S=1$ (see figure 23). Results displayed in figure 24 demonstrate the converging–diverging channel produces the largest discharge increase. Increase of ${\it\varphi}$ from 0 to ${\rm\pi}$ reduces this increase and can reduce the total discharge below the smooth channel level if ${\it\varphi}$ is close enough to ${\rm\pi}$ . The deterioration of the system performance is associated with the reduction and eventual elimination of the stream tube with the fast moving fluid forming in the wide section of the channel, as documented in figure 23. The corresponding changes of the skin-friction distribution are documented in figure 25.

Figure 23.

Distributions of the streamwise velocity $U$ in the $y$ – $z$ plane at $Re_{{\it\tau}}=180$ for ${\it\alpha}=0.5$ and $S=1$ with different phase shifts. (a) ${\it\varphi}=0$ ; (b) ${\it\varphi}={\rm\pi}/2$ ; (c) ${\it\varphi}={\rm\pi}$ .

Figure 24.

Variations of the discharge correction ${\rm\Delta}Q$ as a function of the phase shift ${\it\varphi}$ for $Re_{{\it\tau}}=180$ .

Figure 25.

Distributions of the shear stress in the $y$ – $z$ plane for ${\it\alpha}=0.5$ and $S=1$ with different phase shifts ${\it\varphi}$ at $Re_{{\it\tau}}=180$ .