3.1 Varying streamwise domain length (table 1)

Wall-bounded turbulent flows are often characterised by very long structures of the order of tens of channel half-heights (Kim & Adrian Reference Kim and Adrian1999; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). This becomes very expensive to simulate for conventional full-span channel simulations, so shorter domain lengths of $L_{x}/h=2\unicode[STIX]{x03C0}$ are often used (Chin et al. Reference Chin, Ooi, Marusic and Blackburn2010; Munters, Meneveau & Meyers Reference Munters, Meneveau and Meyers2016). These seem to capture the majority of the outer-layer dynamics despite their relatively short length. However, these lengths are only necessary due to the large outer-layer structures which are present in the full-span simulations, which have streamwise lengths of the order of the channel half-height,  $h$ .

The minimal-span channel only captures the inner-layer flow which is not dependent on $h$ , suggesting a shorter domain length can be used. The near-wall cycle of the buffer layer produces streaky structures with streamwise lengths of 1000 viscous units (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). However, these low- and high-speed streaks are nearly two-dimensional, which suggests they could be represented in the infinite ( $k_{x}=0$ ) mode and hence the domain length does not have to be this long. These streaks are accompanied by quasi-streamwise vortices with streamwise lengths of 200–300 viscous units (Jeong et al. Reference Jeong, Hussain, Schoppa and Kim1997). The narrowest minimal-span channels with $L_{y}^{+}\approx 100$ therefore require $L_{x}^{+}\approx 200$ –300 to capture the near-wall cycle in the buffer layer (Jiménez & Moin Reference Jiménez and Moin1991). The inertial logarithmic layer also have similar vortex clusters (del Álamo et al. Reference del Álamo, Jimenez, Zandonade and Moser2006), which scale as 2–3 times their spanwise width (Flores & Jiménez Reference Flores and Jiménez2010; Hwang Reference Hwang2015). Statistically stationary and homogenous shear turbulence, which shows similarities to the logarithmic layer in wall-bounded turbulence, also suggest a scaling of two times the spanwise length (Sekimoto, Dong & Jiménez Reference Sekimoto, Dong and Jiménez2016). Simulations with larger spanwise domain widths would produce vortices with longer streamwise lengths, so that the largest captured eddy would have spanwise and streamwise lengths of $\unicode[STIX]{x1D706}_{y}=L_{y}$ and $\unicode[STIX]{x1D706}_{x}=2$ – $3L_{y}$ .

Table 1.

Description of the simulations performed with the finite-difference code to investigate streamwise domain length. All simulations done as both smooth wall and modelled rough wall. Entries: $Re_{\unicode[STIX]{x1D70F}}$ , nominal friction Reynolds numbers; $L$ , domain length; $N$ , number of grid points; $\unicode[STIX]{x0394}$ , grid-spacing in viscous units, subscript $w$ and $h$ refers to the wall-normal grid spacing at the wall and channel centre; $L_{z}/h=2$ denotes standard-height (two no-slip walls), $\unicode[STIX]{x0394}t_{S}^{+}$ and $\unicode[STIX]{x0394}t_{R}^{+}$ is the smooth- and rough-wall time step. Roughness height $k=h/40$ so $k^{+}\approx 15$ at $Re_{\unicode[STIX]{x1D70F}}=590$ .

Two different spanwise domain widths are investigated for varying the streamwise length (table 1). Firstly, the smallest width of $L_{y}^{+}\approx 120$ would have the largest eddies having streamwise lengths of $\unicode[STIX]{x1D706}_{x}^{+}=240$ –350. Various streamwise domain lengths of $L_{x}^{+}=(190,300,460,930,1850)$ are chosen to see how this affects the largest eddies. The second spanwise domain width of $L_{y}^{+}\approx 350$ would be able to capture larger eddies, the largest of which would have a streamwise length of approximately $\unicode[STIX]{x1D706}_{x}^{+}=710$ –1100. The same set of streamwise domain lengths are simulated for this wider domain. A standard-height channel is used, with no outer-layer damping, $K_{i}=0$ .

Figure 3.

Streamwise velocity profile for varying streamwise length for (a,c) smooth- and (b,d) rough-wall standard-height channels. Spanwise domain width is (a,c) $L_{y}^{+}\approx 120$ and (b,d) $L_{y}^{+}\approx 350$ . Line styles are: black, full-span channel; grey, minimal-span channel with varying streamwise length (dark grey is longer), refer to table 1; vertical dashed line indicates the roughness height $k=h/40$ ; vertical dotted line indicates critical height $z_{c}=0.4L_{y}$ . Insets of (b,d) shows difference in smooth- and rough-wall flows.

Figure 3 shows the mean velocity profile for the two spanwise widths, for all the streamwise lengths tested. The minimal channels (grey lines) tend to agree with the full-span channel (black line) up until the critical wall-normal location $z_{c}=0.4L_{y}$ (vertical dotted line), at which point the velocity profile of the minimal channels increases. For the smaller domain width (figure 3 a,b), the shortest streamwise lengths of $L_{x}^{+}\approx 190$ and 300 result in a reduction in the smooth-wall velocity above $z^{+}\gtrsim 20$ . There is also a slight increase in the centreline velocity. As a result, the difference in smooth- and rough-wall velocity (inset of figure 3 b) decreases relative to the full-span and longer minimal-span cases. This effect is not due to the differences in grid resolution of the two shortest channels (table 1), as the wider channel has the same differences in grid resolution but will be seen to not have this effect. Little discernible difference can be seen between the longer streamwise lengths of $L_{x}^{+}\geqslant 460$ , in agreement with the scalings discussed above. This also indicates that the low- and high-speed streaks, with lengths of 1000 viscous units, do not need to be explicitly captured as they are aliased into the $k_{x}=0$ mode. The case of $L_{x}^{+}\approx 300=2.5L_{y}^{+}$ produces an incorrect profile and this suggests that a minimum streamwise length of $L_{x}^{+}=3L_{y}^{+}$ is required, especially for narrow channels in which $z_{c}$ is closer to the buffer layer than log layer of the flow.

The smallest streamwise length of $L_{x}^{+}\approx 190$ was unable to sustain turbulence for a prolonged period. It would decay to a laminar state after $T^{+}=TU_{\unicode[STIX]{x1D70F}}^{2}/\unicode[STIX]{x1D708}\approx 14.8\times 10^{4}$ in the smooth-wall channel, and after $T^{+}\approx 9.7\times 10^{4}$ in the rough-wall channel. The data shown in the previous figures for this streamwise length are averaged only when the flow is turbulent. This behaviour is similar to what was observed in Jiménez & Moin (Reference Jiménez and Moin1991), who showed that turbulence could not be maintained in channels with streamwise lengths of approximately 200 viscous units. As discussed above, this is because it is shorter than the quasi-streamwise vortices which have streamwise lengths of $\unicode[STIX]{x1D706}_{x}^{+}\approx 200$ –300.

The wider spanwise domain width of $L_{y}^{+}\approx 354$ (figure 3 c,d) results in a centreline velocity that is closer to the full-span channel, as these wider channels capture a wider range of turbulent motions. However, the case with the shortest streamwise length of $L_{x}^{+}\approx 190\approx 0.54L_{y}^{+}$ has a larger centreline velocity, with the critical wall-normal height appearing lower than $z_{c}=0.4L_{y}$ . This somewhat resembles the channels with a narrower spanwise width of $L_{y}^{+}\approx 120$ (figure 3 a,b), suggesting that the very short domain length restricts the size of the largest eddies. Their spanwise width would now be smaller than the width of the domain, i.e. the streamwise length is now the limiting length scale. Interestingly, the streamwise length of $L_{x}^{+}\approx 300=0.84L_{y}^{+}$ appears to agree quite well with the cases with longer lengths. This is in contrast to the narrow domain (figure 3 a), which shows a clear difference for $L_{x}^{+}\approx 300\approx 2.5L_{y}^{+}$ . Even the case with $L_{x}^{+}\approx 460\approx 1.3L_{y}^{+}$ is producing a velocity profile and roughness function comparable to that of channels with longer streamwise lengths, despite not having the requisite scaling of $L_{x}^{+}\gtrsim 2$ – $3L_{y}^{+}$ discussed above. This is similar to the results of Toh & Itano (Reference Toh and Itano2005) who looked at wide spanwise channels with very short streamwise lengths. The velocity profiles they obtained look similar to a full-span, full-length channel, with no apparent increase in streamwise velocity in the outer layer that is characteristic of minimal-span channels. A possible explanation for the results seen here is that the wall-normal critical location $z_{c}^{+}\approx 140=0.24h^{+}$ is outside the log layer, which is generally believed to end at $z\approx 0.15h$ (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). As such, the expected streamwise length scale of 2– $3L_{y}$ is no longer appropriate in the outer layer. In this case, the largest eddy at the edge of the logarithmic layer would have a streamwise length of approximately $\unicode[STIX]{x1D706}_{x}^{+}\approx 440$ –660, which could explain why the simulation with $L_{x}^{+}\approx 460$ produces a reasonable velocity profile.

Figure 4.

Smooth-wall premultiplied one-dimensional streamwise energy spectra of streamwise velocity, $k_{x}E_{uu}^{+}$ , for varying streamwise lengths of (a,b) $L_{x}^{+}\approx 190$ , (c,d) $L_{x}^{+}\approx 460$ , (e,f) $L_{x}^{+}\approx 930$ and (g,h) $L_{x}^{+}\approx 1850$ . Spanwise domain width (a,c,e,g) $L_{y}^{+}\approx 120$ , and (b,d,f,h) $L_{y}^{+}\approx 350$ . Shaded contour is smooth-wall full-span channel (same for all figures). Contour levels $k_{x}E_{uu}^{+}=0.5$ , 1.0, 1.5, 2.0. Inset box gives channel scale, arrow shows direction of streamwise flow. Vertical dashed line shows $\unicode[STIX]{x1D706}_{x}^{+}=3L_{y}^{+}$ , horizontal dashed line shows $z_{c}^{+}=0.4L_{y}^{+}$ .

Premultiplied one-dimensional streamwise energy spectra of streamwise velocity, $k_{x}E_{uu}^{+}$ , are shown in figure 4, which is related to the streamwise root-mean-squared velocity fluctuations as $u_{rms}^{\prime 2}=\int _{0}^{\infty }k_{x}E_{uu}d\log (k_{x})$ . Only the smooth-wall flow is shown as the rough-wall data exhibit the same qualitative trends as the smooth wall and so are omitted. This occurs because the rough-wall flow is in the transitionally rough regime, which leads to a weakening of the near-wall cycle (MacDonald et al. Reference MacDonald, Chan, Chung, Hutchins and Ooi2016). As it has not been destroyed, the buffer-layer streaks and quasi-streamwise vortices are active and so the same trends as the smooth-wall flow are observed. Note that in the fully rough regime, this cycle is believed to be destroyed with the inertial logarithmic region started at the roughness crests (Jiménez Reference Jiménez2004). However, the scaling of $\unicode[STIX]{x1D706}_{x}\approx 2$ – $3\unicode[STIX]{x1D706}_{y}$ would likely still hold as this is generated by a self-sustaining process in the logarithmic layer (Flores & Jiménez Reference Flores and Jiménez2010; Hwang Reference Hwang2015). The vertical dashed lines correspond to the longest streamwise scale based on the log-layer scaling of $3L_{y}^{+}$ discussed above, while the termination of the line contours of the minimal channels show the channel streamwise length.

It is clear that for the narrowest channel with $L_{y}^{+}\approx 120$ , the shortest streamwise length of $L_{x}^{+}\approx 190=1.6L_{y}^{+}$ (figure 4 a) is too short. This is seen to result in an increase in turbulent energy above $z_{c}^{+}$ (horizontal dashed line), relative to the full-span channel (shaded contour). A similar effect is observed for $L_{x}^{+}\approx 300=2.5L_{y}^{+}$ (not shown). This increase in energy at smaller streamwise length scales above $z_{c}^{+}$ is in agreement with previous minimal-channel simulations (Hwang Reference Hwang2013). When the streamwise domain length exceeds the $3L_{y}$ scaling with $L_{x}^{+}\approx 460=3.9L_{y}^{+}$ (figure 4 c) then reasonable agreement with the full-span channel is observed, despite the narrow range of scales captured by the minimal channel. Further increases to the streamwise length for this domain width (figure 4 e,g) do not improve the collapse with the full-span channel, especially near $z_{c}^{+}$ , as no additional turbulent motions are captured according to the $3L_{y}^{+}$ scaling discussed above. The increased length is however able to capture the near-wall cycle with peak at $z^{+}\approx 15$ and $\unicode[STIX]{x1D706}_{x}^{+}\approx 1000$ (figure 4 g).

A similar picture emerges for the wider minimal channel with $L_{y}^{+}\approx 354$ . The shortest domain length tested, $L_{x}^{+}\approx 190\approx 0.54L_{y}^{+}$ (figure 4 b), is unable to capture much of the turbulent motions, similar to that observed in the narrower minimal channel of figure 4(a). A streamwise domain length of $L_{x}^{+}\approx 463\approx 1.3L_{y}^{+}$ (figure 4 d) has better agreement with the full-span channel, although there is still some discrepancy. The $3L_{y}$ scaling is almost reached in figure 4(f) with $L_{x}+\approx 927\approx 2.6L_{y}$ , and excellent agreement is observed with the full-span channel, with only a slight increase in energy above $z_{c}^{+}$ . Further increasing $L_{x}^{+}$ above $3L_{y}$ (figure 4 h) does not provide any improvement other than capturing more of the near-wall cycle, further supporting this scaling argument.

Figure 5.

Smooth-wall premultiplied one-dimensional spanwise energy spectra of streamwise velocity, $k_{y}E_{uu}^{+}$ , for varying streamwise lengths of (a,b) $L_{x}^{+}\approx 190$ , (c,d) $L_{x}^{+}\approx 460$ , (e,f) $L_{x}^{+}\approx 930$ and (g,h) $L_{x}^{+}\approx 1850$ . Spanwise domain width (a,c,e,g) $L_{y}^{+}\approx 120$ , and (b,d,f,h) $L_{y}^{+}\approx 350$ . Shaded contour is smooth-wall full-span channel (same for all figures). Contour levels $k_{y}E_{uu}^{+}=1.0$ , 2.0, 3.0, 4.0. Inset box gives channel scale, arrow shows direction of streamwise flow. Vertical dashed line shows $\unicode[STIX]{x1D706}_{y}^{+}=L_{x}^{+}/3$ , horizontal dashed line shows $z_{c}^{+}=0.4L_{y}^{+}$ .

The premultiplied spanwise energy spectra of streamwise velocity are shown in figure 5. Here, the vertical dashed line now shows the widest spanwise length scale that can exist based on the streamwise domain length using the $\unicode[STIX]{x1D706}_{y}=3L_{x}$ scaling. For the narrower channel with $L_{y}^{+}\approx 120$ , the scaling is reached when $L_{x}^{+}\approx 460=3.9L_{y}^{+}$ (figure 5 c), and further increases to $L_{x}^{+}$ (figure 5 e,g) do not result in any change to the turbulent energy. Similarly for the wider channel with $L_{y}^{+}\approx 350$ , the scaling is approximately reached in figure 5(f) with $L_{x}^{+}\approx 930=2.6L_{y}^{+}$ and increasing $L_{x}^{+}$ to 1850 viscous units (figure 5 h) does not result in any substantial change to the spectra. The cases with $L_{x}\lesssim 2$ – $3L_{y}$ (figure 5 a,b,d) result in enhanced turbulence energy, particularly in the near-wall peak at $z^{+}\approx 15$ and $\unicode[STIX]{x1D706}_{y}^{+}\approx 100$ .

Given the above results, it is possible to restrict the streamwise length to $L_{x}=3L_{y}$ . This is due to quasi-streamwise vortices that exist in the logarithmic layer (where neither viscosity or the channel half-height are relevant length scales), suggesting this scaling is independent of $Re_{\unicode[STIX]{x1D70F}}$ . Smaller streamwise lengths than this scaling lead to poor agreement between the minimal and full-span channel, as seen in figures 4 and 5. However, for very narrow channels this can result in a streamwise domain length less than 1000 viscous units. While it appears the buffer-layer streaks do not need to be captured in the domain, having such a small streamwise domain exacerbates the bursty nature of the minimal-span channel (Jiménez Reference Jiménez2015). Only one or maybe two quasi-streamwise vortices are present in the domain, which makes obtaining converged statistics difficult as the simulation needs to be run for a significantly long time, an issue discussed in detail in § 4. We believe that a minimum streamwise length of approximately 1000 viscous units seems a reasonable length in these cases, so that several of the smallest quasi-streamwise vortices are present. As such, we recommend that $L_{x}\gtrsim \max (3L_{y},1000\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}},\unicode[STIX]{x1D706}_{r,x})$ , where the last requirement pertains to the streamwise length scale, $\unicode[STIX]{x1D706}_{r,x}$ , of the roughness which is absent in the current roughness forcing model.

3.2 Half-height channel (table 2)

The previous section looked at the effect of the streamwise domain. Now we will consider the effect of the wall-normal domain, particularly in terms of the outer-layer flow which is unphysical in minimal channels. First, we will consider a half-height (open) channel which consists of a slip wall positioned at the channel centreline. Intuitively, changing the boundary condition at $z=h$ is unlikely to affect the flow at $z=z_{c}$ , given the distance between these two heights ( $z_{c}/h\approx 0.1$ in this study). For efficient roughness simulations, this has the benefit of reducing the size of the grid by a half when compared to a conventional standard-height channel with two no-slip walls. The simulations are detailed in table 2 and both full-span and minimal-span channels are simulated. Here, the streamwise length is fixed at $L_{x}^{+}\approx 3700$ and there is no outer-layer damping, $K_{i}=0$ .

Figure 6.

Mean streamwise velocity against wall-normal position for (a) full-span channel and (b) minimal-span channel. Line styles are: black, standard-height channel; grey, half-height channel; solid, smooth wall; dashed, rough-wall model. Vertical dashed line shows roughness crest, $k=h/40$ , vertical dotted line shows critical height, $z_{c}=0.4L_{y}$ . Inset shows difference in smooth- and rough-wall velocity, $U_{S}^{+}-U_{R}^{+}$ .

Table 2.

Full-span and minimal-span channel simulations, for standard-height (two no-slip walls, $L_{z}/h=2$ ) and half-height (one no-slip and one slip wall, $L_{z}/h=1$ ) channels. Entries are same as table 1.

Figure 6 shows the effect of the half-height channel in terms of the mean velocity profile when compared to the standard-height channel. This is done for both full-span (figure 6 a) and minimal-span (figure 6 b) channels, for smooth and rough walls. For clarity, the full-span and minimal-span channels are shown in different figures, however the near-wall behaviour is identical up until $z_{c}=0.4L_{y}\Rightarrow z_{c}^{+}\approx 50$ , as observed previously (figure 3). Both sets of figures show that the use of the half-height channel with slip wall has a negligible effect on the flow. The main difference is seen in the wake region, where the half-height channel restricts the outer-layer structures, resulting in a slight decrease in the mean velocity. Crucially, this difference is the same for both smooth and rough walls, meaning that the difference between them, $U_{S}^{+}-U_{R}^{+}$ , is the same. Moreover, the change in the half-height channel occurs above the critical wall-normal location, where the minimal-span flow is already altered compared to the full-span flow.

Figure 7.

Streamwise, spanwise and wall-normal root-mean-squared velocity fluctuations against wall-normal position for (a) full-span channel and (b) minimal-span channel. Line styles are same as figure 6.

The streamwise, spanwise and wall-normal root-mean-squared velocity fluctuations are shown in figure 7 for both full-span and minimal-span channels, comparing standard-height and half-height channels. For the full-span channel (figure 7 a), these velocity fluctuations show very good agreement between the standard-height (black lines) and half-height (grey lines) channels in the near-wall region, in agreement with previous full-span half-height channel studies (Handler et al. Reference Handler, Saylor, Leighton and Rovelstad1999; Leonardi & Castro Reference Leonardi and Castro2010). The half-height channel has zero wall-normal velocity fluctuations at the channel centre due to the impermeability constraint of the slip wall. The streamwise velocity fluctuations of the half-height channels are slightly enhanced above $z^{+}\gtrsim 40$ relative to the standard-height channels, however the difference is relatively small. Moreover, outer-layer similarity is still maintained between the smooth- and rough-wall channels, suggesting the effect on the difference between the two flows will be minor.

For the minimal-span channel (figure 7 b), a slightly different picture emerges. Firstly, it should be noted that a standard-height minimal-span channel (black lines of figure 7 b) has enhanced streamwise and wall-normal velocity fluctuations compared to its full-span counterpart (black lines of figure 7 a) in the outer layer. This has been noted in other minimal-span channel studies (e.g. Hwang Reference Hwang2013; MacDonald et al. Reference MacDonald, Chan, Chung, Hutchins and Ooi2016), so will not be discussed in depth here. However, when a half-height channel is used in a minimal-span channel, the streamwise velocity fluctuations (grey lines of figure 7 b) are damped from $z^{+}\gtrsim 20$ compared to the standard-height minimal-span channel. This is the opposite of what occurred in a full-span channel. Interestingly, Hwang (Reference Hwang2013) found that when the spanwise uniform eddies (that is, the $k_{y}=0$ mode) were filtered out of a minimal-span channel, a similar behaviour was observed in that the wall-normal and streamwise velocity fluctuations were damped compared to the unfiltered channel. It is unclear whether the half-height channel is performing a similar operation to this $k_{y}=0$ filtering, as this would imply the impermeability constraint is limiting the infinite spanwise motions in some way. However, it is clear that the near-wall flow is preserved, despite the imposition of an outer-layer boundary condition. This shows that a half-height channel flow can be simulated without significant near-wall detriment.

Figure 8.

(a) Smooth-wall premultiplied one-dimensional streamwise energy spectrum of pressure, $k_{x}E_{pp}^{+}$ . Shaded contour is full-span channel, line contour is minimal-span half-height channel. Horizontal dashed line shows $z_{c}^{+}=0.4L_{y}^{+}$ , vertical dashed line shows $3L_{y}^{+}$ (§ 3.1). Contour levels are logarithmically spaced over eight intervals between 0.75 and 66. (b) Pressure root-mean-squared fluctuations against wall-normal position. Line styles: black, full-span channel; grey solid, minimal-span half-height channel; grey dashed, minimal-span channel with altered velocity field $u^{a}$ (see text). Vertical dashed line shows  $z_{c}^{+}$ .

The pressure fluctuations, meanwhile, show a substantial increase in the minimal-span channel compared to the full-span channel. The premultiplied streamwise spectra of pressure is shown in figure 8(a), and it is clear that the fluctuations are larger at longer wavelengths ( $\unicode[STIX]{x1D706}_{x}^{+}\approx 1000$ –2000) and at heights above $z_{c}^{+}$ . The fluctuations at wavelengths shorter than $3L_{y}^{+}$ and below $z_{c}^{+}$ are, however, in reasonable agreement with the full-span channel. Despite this, because the root-mean-squared pressure fluctuations are the integral of this spectrum ( $p_{rms}^{\prime 2}=\int _{0}^{\infty }k_{x}E_{pp}d\log (k_{x})$ ), then the profile is nearly an order of magnitude larger in the minimal-span channel (grey line, figure 8 b). Note that the mean pressure is still correct, as this must obey the averaged wall-normal momentum equation, $\overline{w^{2}}+\overline{p}=C$ , where $C$ is a constant.

A possible solution to this behaviour is to consider decomposing the pressure into its rapid and slow parts (Kim Reference Kim1989). Here, the rapid pressure emerges due to the mean streamwise velocity gradient while the slow pressure is due to the velocity fluctuation gradients. The mean velocity gradient $\text{d}U/\text{d}z$ in the minimal-span channel is enhanced above $z_{c}$ , which means the rapid pressure will also be larger. If we instead define an altered streamwise velocity field whereby $u^{a}(x,y,z)=u(x,y,z)-(U(z)-U(z_{c}))$ for $z>z_{c}$ where $U(z_{c})$ is the mean streamwise velocity at $z=z_{c}$ , this will then set the mean $\text{d}U^{a}/\text{d}z=0$ . If the other two velocity components are unchanged and $u^{a}=u$ for $z\leqslant z_{c}$ , then the rapid pressure would be zero above $z_{c}$ . The pressure field arising from this altered velocity field, $\unicode[STIX]{x1D6FB}^{2}p^{a}=-u_{i,j}^{a}u_{j,i}^{a}$ can then be computed and is seen to be in much better agreement with the full-span channel (figure 8 b). There is a slight difference in the near-wall peak near $z^{+}\approx 30$ , however this may be due to the discontinuity in $\text{d}U^{a}/\text{d}z$ at $z=z_{c}$ . This correction to pressure is an additional step that needs to be performed whenever pressure statistics are to be outputted. The unaltered (discrete) pressure is still necessary at each time step to ensure the flow remains divergence free.

3.3 Outer-layer damping (table 3)

The previous section showed that a half-height (open) channel did not significantly alter the healthy near-wall flow. A more aggressive alteration is now attempted in which the outer-layer flow is damped through use of the $K_{i}$ forcing term (2.3) in a half-height channel. This will reduce the large centreline velocity (figure 6 b) of the minimal-span channel, which places a restriction on the time step due to the CFL number. The streamwise domain length is fixed at $L_{x}^{+}\approx 3700$ , and other relevant parameters are detailed in table 3. The height where the damping begins, $z_{d}$ , is varied in channels of two different spanwise widths; $L_{y}^{+}\approx 120$ and $L_{y}^{+}\approx 240$ . These minimal channels will have a healthy turbulence region up to $z_{c}^{+}\approx 47$ and $z_{c}^{+}\approx 94$ , respectively.

Table 3.

Minimal-span simulations with outer-layer damping. Refer to table 1 for definitions. $z_{d}^{+}$ indicates the location where damping starts (figure 1); cases with no damping are indicated by a hyphen.

Figure 9.

Streamwise velocity profile for outer-layer damping for (a,c) smooth- and (b,d) rough-wall half-height channels. Spanwise domain width is (a,b) $L_{y}^{+}\approx 120$ and (c,d) $L_{y}^{+}\approx 240$ (table 3). Line styles are: black, full-span channel; grey, minimal-span channel with outer-layer damping (2.3) for varying damping heights of (a,b) $z_{d}^{+}=(100,150,200,300)$ and (c,d) $z_{d}^{+}=(200,250,300)$ , shown by $\boldsymbol{ + }$ ; light-grey, minimal-span with no damping. Vertical dashed line indicates the roughness height $k=h/40$ . Inset of (b,d) shows the difference in smooth- and rough-wall velocity. Arrows indicates trend of increasing  $z_{d}$ .

Figure 9 shows the mean streamwise velocity profile for smooth- and rough-wall minimal-span channels, for both of the spanwise widths at $Re_{\unicode[STIX]{x1D70F}}=590$ . The smallest spanwise width, $L_{y}^{+}\approx 120\Rightarrow z_{c}^{+}\approx 47$ (figure 9 a,b) has four different positions for $z_{d}$ examined, with values of $z_{d}^{+}=(100,150,200,250)$ . In all cases, the damping tends to limit the streamwise velocity above $z>z_{d}$ to the value of $U(z=z_{d})$ . It is clear in the smooth-wall flow (figure 9 a) that the two smallest values of $z_{d}^{+}=100$ and 150 are too close to the wall, contaminating the healthy near-wall flow and resulting in an increase in the streamwise velocity below $z_{d}$ . In particular, $z_{d}^{+}=100$ results in an obvious overestimation of $\unicode[STIX]{x0394}U^{+}$ , as seen in the inset of figure 9(b). A value of $z_{d}^{+}=150$ has a similar, although not as strong effect. This is clearly not desirable and suggests that values of $z_{d}^{+}\geqslant 200$ are necessary to avoid contamination of the healthy near-wall flow.

To investigate how the location of $z_{d}$ is related to $z_{c}$ will require cases with a larger spanwise domain width, as $z_{c}\approx 0.4L_{y}$ . This is done in figure 9(c,d), where the spanwise width is now $L_{y}^{+}\approx 240\Rightarrow z_{c}^{+}\approx 94$ . Here, three different positions for $z_{d}$ are analysed, with values of $z_{d}^{+}=(200,250,300)$ . There is little difference in the near-wall profiles of the three different $z_{d}$ positions. This suggests that, even though $z_{c}$ has doubled, we still require the damping to begin at least 200 viscous units away from the wall. Having the outer-layer damping begin closer to the wall (i.e. reducing $z_{d}$ ) leads to its effects permeating the entire near-wall region, increasing the viscous stress in the log layer and even buffer layer. This results in an overestimation of the mean streamwise velocity. The strong effect of the damping when $z_{d}$ is small could also be due to the strength of the damping, $\unicode[STIX]{x1D6FE}h/U_{\unicode[STIX]{x1D70F}}\approx 1$ in (2.3). Reducing the strength would likely reduce the strong near-wall effects that the damping has, although this is not pursued here.

Next, the friction Reynolds number is increased to $Re_{\unicode[STIX]{x1D70F}}=2000$ . Since the roughness height is fixed as a ratio of the channel half-height, $k=h/40$ , then the roughness Reynolds number increases to $k^{+}=50$ . The spanwise width also needs to be increased to submerge the roughness sublayer in healthy turbulence (Chung et al. Reference Chung, Chan, MacDonald, Hutchins and Ooi2015); here it takes a value of $L_{y}^{+}=300$ so that the critical wall-normal location is $z_{c}^{+}\approx 120=2.4k$ . Figure 10 shows the mean velocity profile for these higher $Re_{\unicode[STIX]{x1D70F}}$ simulations in which $z_{d}^{+}=(200,300)$ . Only smooth-wall full-span channel data are available from Hoyas & Jiménez (Reference Hoyas and Jiménez2006) to compare with the minimal-span channel, and good agreement is seen up until $z^{+}\approx 120=0.4L_{y}^{+}$ (figure 10 a). Here, both positions of $z_{d}$ produce velocity profiles which compare well with the minimal-span channel with no outer-layer damping, up to the forcing location  $z_{d}$ .

Figure 10.

Streamwise velocity profile for (a) smooth- and (b) rough-wall half-height channels at $Re_{\unicode[STIX]{x1D70F}}=2000$ (table 3). Line styles are same as figure 9, however full-span smooth-wall channel data are taken from Hoyas & Jiménez (Reference Hoyas and Jiménez2006) and damping heights are $z_{d}^{+}=(200,300)$ , shown by $\boldsymbol{\times }$ .

Figure 11.

Effect of the outer-layer damping on (a) the root-mean-squared velocity fluctuations in the streamwise and wall-normal directions, and (b) the Reynolds stress. Line styles: black, minimal-span, half-height channel (no outer-layer damping); grey, minimal-span, half-height channel with outer-layer damping starting at $z_{d}^{+}=200$ shown by $\boldsymbol{\times }$ ; solid, smooth wall; dashed, rough wall. Vertical long dashed line shows roughness crest, vertical short dashed line shows critical height $z_{c}=0.4L_{y}$ .

It is also of interest to see what effect the outer-layer damping has on second-order turbulence statistics. This is shown in figure 11 for the streamwise and wall-normal velocity fluctuations (figure 11 a) and the Reynolds shear stress (figure 11 b), for the case of outer forcing with $z_{d}^{+}=200$ and $L_{y}^{+}=120$ . This is compared to the minimal-span channel with the same spanwise width, but with no outer-layer damping. The turbulence intensity up until $z^{+}\approx 30$ agrees reasonably well between the case with forcing and without, however because the forcing term scales as $(u_{i}-\langle u_{i}\rangle )^{2}$ , fluctuations are almost zero above $z>z_{d}$ . The fact that they are not exactly zero is likely because only the spatially averaged velocity $\langle u_{i}\rangle$ is used in the forcing term, and so will vary slightly with time given the small spatial averaging domain of the minimal channel.

The Reynolds shear stress (figure 11 b) is almost zero above $z^{+}>300$ , meaning the damping is not dissimilar to having a lower friction Reynolds number while keeping the roughness Reynolds number the same. However, flow structures can enter into the damping region and remain coherent for a period of time which likely depends on the damping strength $\unicode[STIX]{x1D6FE}$ . This is different to a half-height channel at a lower friction Reynolds number, in which the impermeability constraint confines the structures at the slip wall. An advantage of higher $Re_{\unicode[STIX]{x1D70F}}$ with damping compared to lower $Re_{\unicode[STIX]{x1D70F}}$ without damping is that the higher friction Reynolds number of the flow with damping will reduce the effect of low friction Reynolds number flows. Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) showed that the roughness function $\unicode[STIX]{x0394}U^{+}$ was overestimated for $Re_{\unicode[STIX]{x1D70F}}\approx 180$ compared to $Re_{\unicode[STIX]{x1D70F}}\approx 360$ and $Re_{\unicode[STIX]{x1D70F}}\approx 540$ for matched roughness Reynolds numbers. This is due to there being a non-negligible pressure gradient effect, which as discussed in § 2.1 is $\unicode[STIX]{x1D6E5}_{p}=-1/Re_{\unicode[STIX]{x1D70F}}$ . It would therefore be preferable to conduct higher friction Reynolds number simulations with damping, rather than a lower friction Reynolds number simulations. While not investigated here, it also seems reasonable that the grid spacings above $z_{d}$ could be relaxed without altering the near-wall flow.

This damping is similar to the masks, or sponge regions, employed in Jiménez & Pinelli (Reference Jiménez and Pinelli1999). The authors explicitly filtered out disturbances in the outer layer, resulting in laminar-like flow in the outer layer but still maintaining a healthy near-wall cycle. The present damping is similar, although some fluctuations are still present in the outer layer (figure 11 a) and the velocity is fixed rather than forming a laminar velocity profile. A benefit of this damping is that it reduces the computational time step, which is limited by the CFL number, $\mathit{CFL}=\max _{1\leqslant i\leqslant 3}(|u_{i}|\unicode[STIX]{x0394}t/\unicode[STIX]{x0394}x_{i})$ . Here $|u_{i}|$ is the maximum instantaneous velocity in the $i$ th direction and $\unicode[STIX]{x0394}x_{i}$ is the corresponding grid spacing at that location. Note that the viscous time step restriction does not become significant in any of the simulations performed here. For minimal-span channels at $Re_{\unicode[STIX]{x1D70F}}=590$ with no outer-layer damping, the maximum CFL number occurs in the outer layer due to the large streamwise velocity. We see the time step improves by approximately 20 %–24 % with the use of the outer-layer damping (table 3, $L_{y}^{+}\approx 118$ ) than without. This means the simulations can be completed quicker and hence use less computational resources. However, for the present simulations at higher friction Reynolds numbers of $Re_{\unicode[STIX]{x1D70F}}=2000$ , the time step for the smooth wall with outer-layer damping is unaltered compared to the case without damping. This is due to the use of the cosine mapping to define the wall-normal grid, which produces an excessively fine grid near the wall especially for high Reynolds number cases. As a result of such a small $\unicode[STIX]{x0394}z$ , the wall-normal CFL number is now maximum and this occurs in the near-wall region, which is independent of the outer-layer damping. This clustering of points near the wall is a known issue of the cosine mapping for high $Re_{\unicode[STIX]{x1D70F}}$ simulations (Lenaers et al. Reference Lenaers, Schlatter, Brethouwer and Johansson2014), but is used here to ensure a systematic, controlled study. If a more appropriate mapping was used for these high friction Reynolds numbers (e.g. that in Lee & Moser Reference Lee and Moser2015), we would expect a similar or better improvement in time step as at $Re_{\unicode[STIX]{x1D70F}}=590$ .

The most appropriate position for $z_{d}$ is difficult to establish from the above data. A minimum of $z_{d}^{+}\approx 200$ seems necessary to ensure that the damping does not interact with the near-wall flow. It also seems reasonable that $z_{d}$ should scale in some way on $z_{c}$ , for larger domain widths. The three sets of simulations suggest that $z_{d}$ should be approximately two times $z_{c}$ , so a tentative rule of thumb would be $z_{d}^{+}\gtrsim \max (200,2z_{c}^{+})$ . For an appropriate wall-normal mapping, this allows a 20 %–24 % improvement in the computational time step.

3.4 Temporal sweep (table 4)

In this section, the temporal sweep (§ 2.1) is performed in a standard-height channel, with no outer-layer damping ( $K_{i}=0$ ) and a streamwise length of $L_{x}/h=2\unicode[STIX]{x03C0}$ (table 4). The initial friction Reynolds number is $Re_{\unicode[STIX]{x1D70F}}=590$ which reduces to $Re_{\unicode[STIX]{x1D70F}}=180$ via an adverse pressure gradient with different rates investigated. The roughness height is fixed at $k=h/40$ , so the roughness Reynolds number varies as $5\lesssim k^{+}\lesssim 15$ . Three different sweeps were conducted in which the pressure gradient parameter at the end of the sweep is $\unicode[STIX]{x1D6E5}_{p,Re_{\unicode[STIX]{x1D70F}}=180}=(0.03,0.07,0.15)$ .

Table 4.

Initial conditions for temporal sweep in a minimal-span channel. Refer to table 1 for definitions.

Figure 12 shows the mean velocity profiles for the fastest and slowest sweeps, comparing the sweep data with full-span steady flow data at the highest and lowest friction Reynolds numbers tested. Figure 12(a,b) shows the mean velocity for the slowest sweep, when the pressure gradient parameter (2.6) at $Re_{\unicode[STIX]{x1D70F}}\approx 180$ was $\unicode[STIX]{x1D6E5}_{p,Re_{\unicode[STIX]{x1D70F}}=180}=0.03$ . Figure 12(c,d) shows the mean velocity for the fastest sweep, with $\unicode[STIX]{x1D6E5}_{p,Re_{\unicode[STIX]{x1D70F}}=180}=0.15$ . To obtain statistics for a desired $Re_{\unicode[STIX]{x1D70F}}$ , time averaging is performed over a window of $\pm 10\,\%$ of that value. At the initial friction Reynolds number of the simulation, $Re_{\unicode[STIX]{x1D70F}}=590$ , both the sweeps (figure 12 a,c) show good agreement with the full-span channel in the near-wall region. As the spanwise width is $L_{y}/h=0.6$ , then the critical wall-normal location is approximately $z_{c}^{+}\approx 0.4L_{y}^{+}=142$ which agrees well with the figure. As $\unicode[STIX]{x1D6E5}_{p}$ is not significant yet (figure 2 b) then there is little difference between the two sweeps.

Differences start to emerge at the end of the sweep when the friction Reynolds number is close to $Re_{\unicode[STIX]{x1D70F}}=180$ and the $\unicode[STIX]{x1D6E5}_{p}$ is maximum. The slowest sweep (figure 12 b) has reasonable agreement very close to the wall below $z^{+}\lesssim 15$ , but the pressure gradient has reduced the streamwise velocity compared to the full-span steady flow. However, the difference between the smooth- and rough-wall sweep flows (inset of figure 12 b) agrees quite well with the full-span steady data. Taking the difference between two flows with the same weak pressure gradient ostensibly still produces a correct estimate of $\unicode[STIX]{x0394}U^{+}$ . However, when the pressure gradient is increased to $\unicode[STIX]{x1D6E5}_{p,Re_{\unicode[STIX]{x1D70F}}=180}=0.15$ (figure 12 d) then larger differences are seen. The near-wall region has been noticeably changed, and the difference in smooth- and rough-wall velocity (inset) is seen to overestimate the full-span data. Note that the data have been ensemble averaged over four runs, so that the amount of time the flow is averaged over is the same as in the slowest sweep. This therefore indicates that the overestimation of $\unicode[STIX]{x0394}U^{+}$ is a direct result of the stronger (adverse) pressure gradient.

Figure 12.

Mean streamwise velocity profile for (a,c) $Re_{\unicode[STIX]{x1D70F}}\approx 590$ and (b,d) $Re_{\unicode[STIX]{x1D70F}}\approx 180$ , from full-span steady flow (black line) and temporal sweep (grey line). (a,b) Slowest sweep ( $\unicode[STIX]{x1D6E5}_{p,Re_{\unicode[STIX]{x1D70F}}=180}=0.03$ ) and (c,d) fastest sweep ( $\unicode[STIX]{x1D6E5}_{p,Re_{\unicode[STIX]{x1D70F}}=180}=0.15$ ), § 2.1. Line styles: solid line, smooth wall; dashed line, rough wall; vertical dashed line, roughness crest, $k=h/40$ . Inset shows difference in smooth- and rough-wall velocity.

The roughness function $\unicode[STIX]{x0394}U^{+}$ is now computed from the difference in smooth- and rough-wall flows at matched $U_{\unicode[STIX]{x1D70F}}$ values. As seen in the insets of figure 12, the smooth- and rough-wall velocity difference is not constant with $z^{+}$ as in the steady data. This indicates that the flow has not been averaged for long enough at the desired $Re_{\unicode[STIX]{x1D70F}}$ value. As such, multiple sweeps from different initial conditions would need to be conducted to obtain a converged outer region. However, for the purposes of this investigatory study, the velocity difference is averaged from the crest of the roughness to the channel centre to obtain an estimate of $\unicode[STIX]{x0394}U^{+}$ . This is done for a range of friction Reynolds numbers, with the resulting data plotted in figure 13 against the equivalent sandgrain roughness, $k_{s}^{+}$ . Ideally, the temporal sweep data (lines) should agree with the steady flow data (black symbols). Indeed, the temporal sweep does show promise in this regard, especially for the weaker pressure gradient cases. If we interpolate the steady roughness function data from Chung et al. (Reference Chung, Chan, MacDonald, Hutchins and Ooi2015), then an overall relative error term can be approximated as $\unicode[STIX]{x1D700}=\int |1-\unicode[STIX]{x0394}U_{steady}^{+}/\unicode[STIX]{x0394}U_{sweep}^{+}|\,\text{d}k_{s}^{+}$ . The two slowest sweeps, with $\unicode[STIX]{x1D6E5}_{p}=0.03$ and 0.07 have similar relative errors of $\unicode[STIX]{x1D700}\approx 12\,\%$ and 13 %, respectively, while the fastest sweep is noticeably larger with an error of $\unicode[STIX]{x1D700}\approx 26\,\%$ .

It appears that the sweep approach could work for determining the hydraulic behaviour of roughness. A pressure gradient parameter of $\unicode[STIX]{x1D6E5}_{p}\leqslant 0.03$ –0.07 is required, which is not too far from the recommendation in Patel (Reference Patel1965) that zero pressure gradient conditions can be assumed when $\unicode[STIX]{x1D6E5}_{p}\leqslant 0.01$ . A value of $\unicode[STIX]{x1D6E5}_{p}\gtrsim 0.15$ results in pressure gradient effects that lead to inaccuracies in $\unicode[STIX]{x0394}U^{+}$ . Future studies could prescribe a constant value of $\unicode[STIX]{x1D6E5}_{p}$ rather than linearly varying the bulk velocity, as this would ensure pressure gradient effects remain negligible, while efficiently traversing the range of roughness Reynolds numbers.

Figure 13.

Hama roughness function for the temporal sweep with modelled roughness at fixed $k=h/40$ , where $k_{s}^{+}\approx 1.6k^{+}$ . Symbols: steady flow data for the same modelled roughness (Chung et al. Reference Chung, Chan, MacDonald, Hutchins and Ooi2015). Line styles: ——, sweep data for $\unicode[STIX]{x1D6E5}_{p,Re_{\unicode[STIX]{x1D70F}}=180}=0.03$ (slowest sweep); — —, sweep data for $\unicode[STIX]{x1D6E5}_{p,Re_{\unicode[STIX]{x1D70F}}=180}=0.07$ ; – – –, sweep data for $\unicode[STIX]{x1D6E5}_{p,Re_{\unicode[STIX]{x1D70F}}=180}=0.15$ (fastest sweep).