3.1 Properties of the mean flow

Following the $\text{R}\rightarrow \text{S}$ surface change, the properties of the boundary layer evolve from primarily reflecting the state of the upstream rough wall to that of the downstream smooth-wall condition. For each case the variation of the shape factor, $H_{12}={\it\delta}^{\ast }/{\it\theta}$ , where ${\it\delta}^{\ast }$ and ${\it\theta}$ are the standard integral displacement and momentum thicknesses, respectively, are shown in figure 2(a) including the corresponding boundary layer thickness. The shape factor decreases following the $\text{R}\rightarrow \text{S}$ surface change. At the final measurement station, where $x=1.32$  m, $H_{12}\approx 1.38$ for each case. The corresponding momentum thickness Reynolds numbers are, $Re_{{\it\theta}}={\it\theta}U_{\infty }/{\it\nu}\approx 6400$ and 9300 for the grit and mesh cases, respectively, as given in table 1. From the formulation of Nagib, Chauhan & Monkewitz (Reference Nagib, Chauhan and Monkewitz2007) at these Reynolds numbers for a self-preserving smooth-wall boundary layer, $H_{12}\approx 1.36$ and 1.34 for the girt and mesh cases, respectively. Furthermore, the asymptotic behaviour of $H_{12}$ shown in figure 2(a) is evidence that the turbulent boundary layer adjusts slowly to the downstream condition, which is consistent with the results of Antonia & Luxton (Reference Antonia and Luxton1972).

Figure 2.

(a) Development of the boundary layer shape factor, $H_{12}$ , and boundary layer thickness, ${\it\delta}$ , in the streamwise direction. Grit case shape factor, ●, and boundary layer thickness, ○; mesh case shape factor, ▪, and boundary layer thickness, ▫. Solid lines indicate $H_{12}=1.36$ and 1.34 for a corresponding self-preserving smooth-wall boundary layer. (b) Variation of the wake factor ${\it\Pi}_{j}$ for the grit, ○, and mesh, ▫, cases.

Figure 2(b) shows a comparison of the wake parameter ${\it\Pi}_{j}$ that was determined from the solution to ${\rm\Delta}(U/U_{{\it\tau}})=288{\it\Pi}_{j}^{3}/({\it\kappa}(12{\it\Pi}_{j}+1)^{2})$ , where ${\rm\Delta}(U/U_{{\it\tau}})$ is the maximum deviation of $U^{+}$ with respect to the logarithmic region (Jones et al. Reference Jones, Marusic and Perry2001). Similar to the behaviour of the shape factor described above, the wake parameter ${\it\Pi}_{j}$ varies asymptotically as shown in figure 2(b). At the final measurement station for the grit case ${\it\Pi}_{j}\approx 0.75$ ( $Re_{{\it\tau}}\approx 1900$ ). Comparatively, for the smooth wall ${\it\Pi}_{j}\approx 0.71$ ( $Re_{{\it\tau}}\approx 1430$ ). For the mesh case ${\it\Pi}_{j}\approx 0.9$ at the farthest downstream location ( $Re_{{\it\tau}}\approx 2500$ ). Antonia & Luxton (Reference Antonia and Luxton1972) show that after approximately $16{\it\delta}_{o}$ , where  ${\it\delta}_{o}$ is the incoming boundary layer thickness at the $\text{R}\rightarrow \text{S}$ change in surface, the boundary layer is not yet at equilibrium, again demonstrating the slow response of the boundary layer following this change in surface condition.

It is apparent from figure 2 that although the boundary layer growth rate is similar for the two cases, the flow response is significantly different, which is related to the strength of the perturbation. A common approach to quantify the strength of the perturbations due to a step change in roughness is by the logarithmic difference between the two roughness heights as discussed by Andreopoulos & Wood (Reference Andreopoulos and Wood1982), for example. This follows from the general description of the logarithmic region within the self-preserving turbulent boundary layer (smooth or rough), which can be expressed as

where $y$ is the wall-normal distance relative to the wall, $d$ is the zero-plane displacement, ${\it\kappa}$ is the von Kármán constant and $B$ is the smooth-wall additive constant. In the right portion of the above equation, $y_{0}$ is a roughness length that is specific to the roughness geometry. Solving (3.1) for $y_{0}$ yields

The strength of the roughness step, $M$ , is expressed as the logarithmic difference between two roughness heights $\ln (y_{01}/y_{02})$ , where $y_{01}/y_{02}$ is the ratio of the roughness lengths for the oncoming and downstream flow relative to the rough-to-smooth change in the surface condition (Andreopoulos & Wood Reference Andreopoulos and Wood1982). For a smooth wall, both $d$ and ${\rm\Delta}U/U_{{\it\tau}}$ are zero, which was the case in the calculation of $y_{02}$ from the boundary layer profile at the farthest downstream measurement location ( $x=1300$  mm). Over the rough wall $U_{{\it\tau}}$ was approximately constant within experimental except for $x<-{\it\delta}$ at $x=-50$  mm for the mesh case, where the presence of the downstream condition may have influenced the boundary layer. Both $y_{0}$ and $d$ were calculated by fitting mean velocity profiles to (3.1) using the methodology of Birch & Morrison (Reference Birch and Morrison2011). The roughness length, $y_{01}$ , was then taken as the average value of $y_{0}$ over the rough surfaces between $x=-400$ and $-100$  mm. In the two cases considered in this study, $M_{R->S}=3.4$ for the grit surface topology to a smooth wall and $M_{R->S}=5.1$ for the mesh topology to a smooth wall. A larger value of $M$ corresponds to a stronger perturbation.

Following an abrupt $\text{R}\rightarrow \text{S}$ change in the wall surface the skin friction immediately decreases and then begins to recover (Antonia & Luxton Reference Antonia and Luxton1972; Mulhearn Reference Mulhearn1978; Chamorro & Porté-Agel Reference Chamorro and Porté-Agel2009) to that of a comparable smooth-wall boundary layer. Over the rough surface the friction velocity, $U_{{\it\tau}}$ , was estimated from the sum of the viscous and turbulent stress contributions at the total stress plateau (see for example Flack et al. Reference Flack, Schultz and Shapiro2005; Castro Reference Castro2007). For comparison, $U_{{\it\tau}}$ was also calculated using the modified Clauser chart procedure (with ${\it\kappa}=0.41$ , $B=5.0$ ) given by Perry & Li (Reference Perry and Li1990), which provided comparable $U_{{\it\tau}}$ values within 7 %. At the upstream measurement stations $U_{{\it\tau}}$ remained approximately constant within experimental uncertainty for $x=-400$ to $-100$ or $-x>{\it\delta}$ .

The skin friction coefficient, $c_{f}$ , is related to the friction velocity by $c_{f}={\it\tau}_{w}/\text{0.5}{\it\rho}U_{\infty }^{2}$ , where $U_{{\it\tau}}=({\it\tau}_{w}/{\it\rho})^{1/2}$ . Prior to the $\text{R}\rightarrow \text{S}$ surface change over the rough surfaces, $c_{f}\approx 6.7\times 10^{-3}$ for the grit case and ${\approx}9.9\times 10^{-3}$ for the mesh case. Along the smooth surface the friction velocity was estimated by the Preston tube method described previously in § 2.4. The resulting variation of $c_{f}$ with streamwise distance (normalised by the local boundary layer thickness, ${\it\delta}$ ) is shown in figure 3(a). Comparatively, the calculated values inferred from the Clauser chart method ( ${\it\kappa}=0.41$ , $B=5.0$ ) are within 4 % for $x/{\it\delta}>0.8$ . An additional dataset of Antonia & Luxton (Reference Antonia and Luxton1972) is included in figure 3(a) for comparison.

Figure 3.

(a) Variation of the skin fiction coefficient, $c_{f}$ , with streamwise distance from the $\text{R}\rightarrow \text{S}$ surface change inferred from Preston tube measurements. For comparison the $c_{f}$ values of Antonia & Luxton (Reference Antonia and Luxton1972) are reproduced in the figure. The equation $c_{f}=f(M)\log (x/{\it\delta})+g$ is given by – – –. (b) Variation of the skin fiction coefficient as a function of $Re_{{\it\theta}}$ with the smooth-wall relationship, $c_{f}=2((1/{\it\kappa})\ln (Re_{{\it\theta}})+B)^{-2}$ of Österlund et al. (Reference Österlund, Johansson, Nagib and Hites2000), ——, and corresponding measurement from Antonia & Luxton (Reference Antonia and Luxton1972) at $x/{\it\delta}\approx 12$ .

The variation of the measured skin friction coefficient downstream of the grit and mesh roughnesses can be represented by the relationship, $c_{f}=f(M)\log (x/{\it\delta})+g$ , which was fit by linear least-squares regression and included in figure 3(a). The form of this equation shows that the strength of the step affects the slope of the resulting $c_{f}$ curve in semi-log form. As discussed earlier, the relaxation of the boundary layers appears to occur over a similar fetch ( $x/{\it\delta}$ ) for both cases. Therefore, it follows that the slope should be greater for a larger step strength given the lower skin friction achieved immediately following the rough surface. Although the step strength is not formally reported by Antonia & Luxton (Reference Antonia and Luxton1972), it is given as 5.8 by Andreopoulos & Wood (Reference Andreopoulos and Wood1982), however, this value is suspected to be erroneous as recently discussed by Jacobi & McKeon (Reference Jacobi and Mckeon2011). The expected discrepancy of $M_{R->S}$ is further supported by figure 3(a) since a higher expected slope is not observed. Comparing the mesh case with that considered by Antonia & Luxton (Reference Antonia and Luxton1972) suggests that these step strengths are similar.

The physical meaning of the offset ( $g$ ) is less clear, however, it is possible to speculate on its form. In one instance this value should be depend on the Reynolds number owing to the established relationship between $c_{f}$ and $Re_{{\it\theta}}$ as shown by Österlund et al. (Reference Österlund, Johansson, Nagib and Hites2000) and given by $c_{f}=2((1/{\it\kappa})\text{ln}(Re_{{\it\theta}})+B)^{-2}$ . The resulting variation of $c_{f}$ is shown in figure 3(b) as a function of $Re_{{\it\theta}}$ for both cases considered in this study and for the final downstream $c_{f}$ value of a case considered by Antonia & Luxton (Reference Antonia and Luxton1972). For the latter, we only show the final downstream location since $Re_{{\it\theta}}$ is given at this location. This value from Antonia & Luxton (Reference Antonia and Luxton1972) appears lower than the curve given by Österlund et al. (Reference Österlund, Johansson, Nagib and Hites2000). The source of this difference is not clear. The current data appears to collapse on the curve given by Österlund et al. (Reference Österlund, Johansson, Nagib and Hites2000) at the most downstream location. This suggests that the offset ( $g$ ) is not only related to the roughness scale of the incoming flow, $y_{01}$ , but must include the effect of $Re_{{\it\theta}}$ of the resulting smooth-wall boundary layer. Data from various other $\text{R}\rightarrow \text{S}$ surface change is required to confirm this observation.

3.2 Properties of the mean flow

Profiles of the turbulent boundary layer occurring past the abrupt $\text{R}\rightarrow \text{S}$ wall surface topology change were obtained at several locations. Measurements were made upstream of the interface at $x=-400$ , $-300$ , $-200$ , $-100$ and $-50$  mm. Along the smooth wall, profiles were acquired at $x=10$ , 50, 100, 200, 300, 630, 805, 980, 1160 and 1320 mm, however, only a selection of the results are shown for clarity. Given that the values of ${\it\delta}$ were different for the two cases, the resulting $x/{\it\delta}$ locations for the profiles do not coincide. A selection of the mean streamwise velocity profiles are plotted in semi-logarithmic format in figure 4(a) for the grit case and figure 4(b) for the mesh case and scaled in terms of the traditional outer variables, $U_{\infty }$ and ${\it\delta}$ . Downstream of the surface roughness change, a significant velocity deficit is found, which is particularly evident at $y/{\it\delta}<0.3$ and $y/{\it\delta}<0.4$ for the grit-type and mesh-type roughnesses, respectively. This height corresponds to approximately 10 times the maximum roughness height for each case and this deficit appears to persists until the final measurement location for each case.

Figure 4.

Outer scaling of selected mean streamwise velocity profiles for the (a) grit and (b) mesh cases. The upstream rough-wall profiles are shown by —— and the data markers are for the profiles above the smooth surface.

From the skin friction coefficient shown in figure 3, it is possible to consider the boundary layer from the perspective of inner scaling. Figure 5 shows the inner-scaled mean streamwise velocity for the same select cases shown in figure 4. Compared to the log-law of the wall for smooth-wall boundary layers, the well-known characteristic of the rough surface is the downward shift of the log-law, as is shown in figure 5 where the abscissa is then represented by $(y-d)^{+}$ for the rough-wall case. Over a smooth wall $d=0$ . It has been shown that following a $\text{R}\rightarrow \text{S}$ surface change the zero-plane displacement immediately tends to 0 or at least within $0.03{\it\delta}$ (Loureiro et al. Reference Loureiro, Sousa, Zotin and Silva Freire2010). Profiles obtained past the first measurement station collapse well in the near-wall region for $y^{+}<30$ . However, nearest to the $\text{R}\rightarrow \text{S}$ surface change the inner scaled velocity exceeds the viscous condition, $y^{+}=U^{+}$ , which is likely an artefact of the uncertainly in the wall shear stress at this location. Regardless, the collapse of the near-wall region is remarkable. The most salient feature of figure 5 is the deviation of $U^{+}$ for $y^{+}>200$ , which is also captured by the variation of the wake parameter shown in figure 2(b). The profiles tend towards the smooth-wall case with increasing downstream distance, although note that the Reynolds number was not matched. For the smooth wall shown in figure 5(a) and in figure 5(b) $Re_{{\it\tau}}=2300$ , whereas $Re_{{\it\tau}}$ reached 1900 and 2600 for the grit and mesh cases, respectively.

Figure 5.

Inner viscous scaling of selected mean streamwise velocity profiles for the (a) grit and (b) mesh cases. The upstream rough-wall profiles are shown by ——, where a modified abscissa is then given by $(y-d)^{+}$ . Smooth-wall profiles are for $Re_{{\it\tau}}=2300$ and are identified by —— and – – – represents $U^{+}=y^{+}$ .

The outer region of both rough- and smooth-wall boundary layers have been shown to collapse under the velocity deficit scaling, $(U_{\infty }-U)/U_{{\it\tau}}=f(y/{\it\delta})$ , see for example Flack et al. (Reference Flack, Schultz and Shapiro2005). For the two cases following the $\text{R}\rightarrow \text{S}$ surface change, this scaling is shown in figure 6 (note that the data was only shown for $y>0.2{\it\delta}$ since we expect collapse in this region). At the final measurement station the inner-scaled velocity deficit approaches the smooth-wall case. Near the $\text{R}\rightarrow \text{S}$ surface change there is considerable deviation, which is not surprising given that the friction velocity of the upstream rough wall is higher. This confirms the inability of the outer region to scale with the local skin friction, however, this scaling will be considered later.

Figure 6.

Inner viscous scaling of the mean deficit velocity for selected streamwise velocity profiles for the (a) grit and (b) mesh cases. The smooth-wall profile is given by – – –.

3.3 Turbulence statistics

It was recently shown by Alfredsson, Örlü & Segalini (Reference Alfredsson, Örlü and Segalini2012) that starting within the logarithmic region and across a significant portion of the outer wake, the turbulence intensity ( $\sqrt{\overline{u^{2}}}/U$ ) decreases linearly with $U/U_{\infty }$ , viz. $\sqrt{\overline{u^{2}}}/U=0.286-0.255U/U_{\infty }$ for a smooth-wall boundary layer. This relationship also appears to be independent of the Reynolds number (Alfredsson et al. Reference Alfredsson, Örlü and Segalini2012). In comparison, for a rough-wall turbulent boundary layer Castro (Reference Castro2013) showed that the slope of the linear region was greater, which confirmed the significance of wake strength suggested by Alfredsson et al. (Reference Alfredsson, Örlü and Segalini2012). The aforementioned plotting method is shown in figure 8 for the cases pertaining to the $\text{R}\rightarrow \text{S}$ surface change, where only data above $U/U_{\infty }>0.3$ are shown for demonstration. Given the robustness of the linear region slope observed by Alfredsson et al. (Reference Alfredsson, Örlü and Segalini2012), the departure from the linear region for $0.6<U/U_{\infty }<0.9$ shown in figure 8 can be interpreted as the effect of the persisting rough-wall boundary layer characteristics in this region of the flow. This is confirmed by examining the location of the internal boundary layer, which resides within this region as is shown in figure 8. The shaded region is used to highlight the envelope between the near-wall flow developing over the smooth wall and the portion of the flow containing the characteristic upstream rough-wall boundary layer behaviour. Within the shaded envelope region the flow exhibits a transitional behaviour owing to the competing effect between the smooth-wall condition developing away from the wall (left edge of the shaded region) and the persisting rough-wall turbulence (right edge of the shaded region). The diagnostic plot then represents a novel method to determine the location of the internal layer without relying on streamwise differentiation or intersecting curve fits.

Figure 8.

The streamwise turbulence intensity, $\sqrt{\overline{u^{2}}}/U$ , with respect to the normalized mean velocity, $U/U_{\infty }$ , for the (a) grit and (b) mesh cases. The linear equation of Alfredsson et al. (Reference Alfredsson, Örlü and Segalini2012) for a smooth-wall turbulent boundary layer is shown by — ⋅ —, whereas the upstream rough-wall profile is shown by ——. The internal boundary layer is identified by – – – and the approximate region between the smooth- and rough-wall behaviour is shaded.

A method to scale the streamwise stress based on the mean velocity gradient was proposed by Jacobi & McKeon (Reference Jacobi and Mckeon2011), viz. $U_{s}=\sqrt{(U_{\infty }{\it\delta}\partial U/\partial y)}$ , to collapse the perturbed boundary layer. The scaling is applied to the current grit and mesh cases as shown in figure 9. Although the data appears to approximately collapse in the regions where $y/{\it\delta}<0.02$ or $y/{\it\delta}>0.4$ , between these regions there is a greater range of variation. Recall that the diagnostic plot, shown in figure 8, demonstrates a transitional region between the apparent smooth- and rough-wall characteristics in the outer region. This previous observation supports the interpretation that variation of $\sqrt{\overline{u^{2}}}/U_{s}$ within the region $y/{\it\delta}<0.02$ and $y/{\it\delta}>0.4$ occurs as a result of the development of the internal boundary layer since the envelope identified in figure 8 overlaps these locations. Additionally, the peak magnitude of $\sqrt{\overline{u^{2}}}/U_{s}$ for the grit and mesh cases vary significantly. This suggests that this scaling might be suitable for comparing several streamwise locations in one flow, but, might not be suitable for comparisons across different flows.

Figure 9.

The root-mean-square of the streamwise turbulence for the (a) grit and (b) mesh cases following the $\text{R}\rightarrow \text{S}$ surface changes normalized by the velocity scale based on the local mean velocity gradient.

Comparatively, the data of Jacobi & McKeon (Reference Jacobi and Mckeon2011) appear to effectively collapse under this scaling, however, this may be an artefact of the relatively low step strength of the roughness impulse considered ( ${\approx}1.7$ ). Similarly, as was reproduced and scaled by Jacobi & McKeon (Reference Jacobi and Mckeon2011), albeit with additional uncertainty, greater variation was observed for the impulse cases of Andreopoulos & Wood (Reference Andreopoulos and Wood1982) having higher step strengths. Therefore, we seek alternative methods to examine and interpret the scaling of the boundary layer.

Turbulence statistics of the streamwise velocity component are shown in figure 10. Successive profiles of $\overline{u^{2}}/U_{\infty }^{2}$ shown in figure 10(a,c) demonstrate two distinct energetic regions over the smooth wall, downstream of the $\text{R}\rightarrow \text{S}$ surface change. Comparing these profiles with the flow occurring over the rough surface, it is clear that the energetic region above $y/{\it\delta}\approx 0.05$ is an artefact of the turbulence produced over the rough wall. The location of this region is consistent with the behaviour of fully rough, rough-wall boundary layers, where a corresponding peak is typically found between $y/{\it\delta}=0.05{-}0.2$ , as discussed by Jiménez (Reference Jiménez2004). Over the smooth surface, this hump both decreases in magnitude and appears to moves away from the wall with downstream fetch from the $\text{R}\rightarrow \text{S}$ interface as shown by the contours in figure 10(b,d). The location of the internal boundary layer and the extent of the envelope identified in figure 8 are included in figure 10(b,d). Within this region, the flow exhibits mixed characteristics of a smooth- and rough-wall turbulent boundary layer.

Figure 10.

Outer-scaled selected profiles of $\overline{u^{2}}/U_{\infty }^{2}$ for the (a) grit and (c) mesh cases. The smooth-wall case is given by – – –, whereas the upstream rough-wall case is given by ——. Corresponding contours of $\overline{u^{2}}/U_{\infty }^{2}$ are shown to the right in (b) and (d) where the internal boundary layer is marked by $\ast$ while the extent of the envelope from figure 8 is identified by $\times$ and $+$ , respectively. The location where $y^{+}=15$ is shown by — ⋅ —. The linear colour map of $\overline{u^{2}}/U_{\infty }^{2}$ varies from white to red over a range from 0 to 0.02.

By scaling the wall-normal coordinate by the height of the internal boundary layer ( ${\it\delta}_{i}$ ) it is clear that the energetic region associated with upstream flow condition resides above ${\it\delta}_{i}$ as shown in figure 11. For $x/{\it\delta}<1$ the location of peak $\overline{u^{2}}/U_{\infty }^{2}$ in the outer region remains at the same position as that of the rough wall as it is yet to experience the influence of the new wall condition. Considering the first two profiles that are closest to the $\text{R}\rightarrow \text{S}$ interface ( $x/{\it\delta}<1$ ), the region of influence of the internal layer does not extend to the outer region peak and therefore does not modify the location of this peak. However, for $x/{\it\delta}>1$ , the location of the outer region peak is fixed at approximately $y/{\it\delta}_{i}=1.2$ . This behaviour is also displayed in figure 10(b,d) where, for $x/{\it\delta}>1$ , it is evident that the influence of the internal boundary layer reaches the outer region peak location, resulting in the scaling of the peak location with the internal boundary layer thickness. This suggests that the growth of the internal layer dictates the outer peak location regardless of the incoming conditions, although the growth of internal layer itself depends on the surface conditions of the oncoming flow.

Figure 11.

Outer-scaled selected profiles of $\overline{u^{2}}/U_{\infty }^{2}$ for the (a) grit and (b) mesh cases where the wall-normal location $y$ is scaled by the internal boundary layer thickness, ${\it\delta}_{i}$ . The location $y/{\it\delta}_{i}=1$ is identified by – – –. The lower and upper extent of the envelope region shown in figure 8 is identified by $\times$ and $+$ , respectively.

In the near-wall region an energetic site in the streamwise normal stress develops as shown in figures 10 and 11. As the near-wall cycle above the smooth surface is established, the presence of a distinct peak appears to be masked by the existing high turbulence. For example, at the first measurement location $\text{d}(\overline{u^{2}/U_{\infty }})/\text{d}(y/{\it\delta})$ remains positive prior to the peak in the energetic region associated with the upstream turbulence. Farther downstream, at $x/{\it\delta}\geqslant 0.8$ and $x/{\it\delta}>1.2$ for the grit and mesh cases, the inner region peak becomes well defined such that there is a sign change in $\text{d}(\overline{u^{2}/U_{\infty }})/\text{d}(y/{\it\delta})$ past $y^{+}\approx 15$ as shown in figure 10. By scaling the streamwise normal stress by the local friction velocity, $\overline{u^{2}}/U_{{\it\tau}}^{2}$ , the wall-normal location of the inner peak collapses as shown in figure 12. The location of the peak value of $\overline{u^{2}}/U_{{\it\tau}}^{2}$ in the near-wall region resides at approximately $y^{+}=15$ , which is also evident from the overlaid location on the contours in figure 10. This location is consistent with the inner peak typically found between $y^{+}\approx 12{-}16$ as shown by Fernholz & Finley (Reference Fernholz and Finley1996), for an equilibrium smooth-wall boundary layer.

Figure 12.

Inner viscous scaling of selected profiles of $\overline{u^{2}}/U_{{\it\tau}}^{2}$ for the (a) grit and (b) mesh cases. The vertical —— represents $y^{+}\approx 15$ and – – – shows a completely smooth surface result.

The behaviour of the near-wall inner-scaled peak past the $\text{R}\rightarrow \text{S}$ change in surface roughness, however, is not comparable to an equilibrium smooth-wall boundary layer. As shown in figure 12 the maximum value of the near-wall inner-scaled peak appears to decrease with downstream fetch. Marusic & Kunkel (Reference Marusic and Kunkel2003) showed that the maximum value of the inner-scaled near-wall peak increases with $Re_{{\it\tau}}$ for the smooth flat plate turbulent boundary. Hutchins & Marusic (Reference Hutchins and Marusic2007a ) describe the magnitude of the inner peak by an empirical relationship, $(\overline{u^{2}}/U_{{\it\tau}}^{2})_{peak}=1.036+0.965\ln (Re_{{\it\tau}})$ . Past the $\text{R}\rightarrow \text{S}$ change in surface roughness, both the shear stress and boundary layer thickness increase (see figures 3 and 2). This implies an increase in $Re_{{\it\tau}}$ and therefore an increase in $(\overline{u^{2}}/U_{{\it\tau}}^{2})_{peak}$ , however, the inner-scaled near-wall peak of the streamwise stress decreases immediately following the $\text{R}\rightarrow \text{S}$ interface as shown in figure 13(a). Within this figure the empirical relationship of Hutchins & Marusic (Reference Hutchins and Marusic2007a ) is reproduced with a shaded grey region that represents the observed scatter about this trend from the data shown by Hutchins & Marusic (Reference Hutchins and Marusic2007a ). Sufficiently far downstream the near-wall $(y^{+}\approx 15)$ peak value for the grit case appears to the track the empirical relationship of Hutchins & Marusic (Reference Hutchins and Marusic2007a ). For the girt case this is observed in figure 13(a) past $x/{\it\delta}>13$ , corresponding to the final three measurement locations, whereas the peak value appears to still be decreasing for the mesh case at the final measurement location. This is likely related to the difference in the extent of $x/{\it\delta}$ captured for these cases since the mesh case only reaches $x/{\it\delta}\approx 12.4$ due to the thicker boundary layer and fixed extent of the measurement domain. Included in figure 13 are the $l^{+}$ corrected values that were determined using the relationship established by Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009), in addition to error bars that represent a $\pm 10\,\%$ uncertainty on the measured $c_{f}$ values, which shows that the observed trends are not due to measurement uncertainty. This clearly suggests that the near-wall peak is under the influence of the outer region, which is different for the two different incoming conditions.

Figure 13.

Variation of the inner peak magnitude with Reynolds number for the grit and mesh cases. The solid line represents the empirical relationship of Hutchins & Marusic (Reference Hutchins and Marusic2007a ) and the grey shaded region represents scatter about this fit. Open symbols show the $l^{+}$ corrected values and error bars represent a 10 % uncertainty in $c_{f}$ .

An alternative mixed scaling was proposed by De Graaff & Eaton (Reference De Graaff and Eaton2000), which appeared to improve the collapse the inner peak magnitude of $\overline{u^{2}}$ over a wide Reynolds number range for a smooth-wall boundary layer. The rationale for this scaling was based on Townsend’s proposition that turbulent motion in the near-wall region consists of the superposition of active shear stress producing motions that scale on inner variables and larger scales from the outer region that scale on outer variables. As is shown in figure 14, the magnitude of the near-wall peak does not completely collapse for the cases considered in this study, particularly near the $\text{R}\rightarrow \text{S}$ surface change, and the discrepancy is higher for the mesh case, which exhibits a higher level of distortion. However, the collapse is comparatively improved with respect to solely outer or inner scaling as shown in figures 10 and 12. For the grit case shown in figure 14(a), the collapse is remarkably near 0.3 (except for the first measurement location), which is similar for flat plate smooth-wall boundary layers (De Graaff & Eaton Reference De Graaff and Eaton2000). For the mesh case the inner peak tends toward 0.3 and appears to reach this value as the inner peak becomes well defined or the magnitude of the outer region peak reaches similar levels observed in the grit case.

Figure 14.

Mixed scaling of selected profiles of $\overline{u^{2}}/(U_{{\it\tau}}U_{\infty })$ for the (a) grit and (b) mesh cases. The vertical —— represents $y^{+}\approx 15$ , and the horizontal —— is shown at $\overline{u^{2}}/(U_{{\it\tau}}U_{\infty })=0.3$ . The – – – shows a completely smooth surface result.

Figure 15.

Procedure followed to determine the apparent friction velocity, $U_{{\it\tau}o}$ , required to force collapse of the mean velocity deficit above the internal layer. The example data is from the mesh case at $x/{\it\delta}=3.5$ . This procedure was repeated for each measurement station following the $\text{R}\rightarrow \text{S}$ surface change.

The previous mixed scaling method of De Graaff & Eaton (Reference De Graaff and Eaton2000) suggests that the scaling of the inner peak magnitude may collapse with an appropriate velocity scale associated with the outer region. However, the velocity scale associated with the outer region should reflect the recovery of the mean velocity following the $\text{R}\rightarrow \text{S}$ surface change. Obviously, the free stream velocity does not change with downstream distance and therefore is not the appropriate scale for this. At a first glance, the local velocity at ${\it\delta}_{i}$ , or at the location of the peak energy in the outer region compared to the smooth wall may appear appropriate. We also considered the outer velocity scale of Zagarola & Smits (Reference Zagarola and Smits1998) ( $U_{\infty }{\it\delta}^{\ast }/{\it\delta}$ ), which was found to collapse the outer region above the internal boundary layer. However, further analysis (not shown here for brevity) indicated that these choices do not really collapse the near-wall peak. Therefore, an alternative velocity scale is proposed where this new scale is based on forcing outer layer similarity on the mean velocity deficit.

Figure 15 shows a schematic representation of the procedure followed to determine a scale associated with the outer region of the boundary layer. As shown in this figure and previously in figure 6, the scaled velocity deficit $((U_{\infty }-U)/U_{{\it\tau}})$ fails to collapse downstream of the $\text{R}\rightarrow \text{S}$ surface change. A new velocity scale denoted as $U_{{\it\tau}o}$ is defined, where the subscript ‘o’ identifies this scale as the apparent skin friction velocity necessary to force similarity of the velocity deficit in the outer region above the internal layer. This new definition is motivated by the fact that outside the internal boundary layer, the flow should retain some aspects of a canonical rough-wall boundary layer as evidenced in figure 8. If we hypothesise that flow in question should exhibit outer layer similarity in the mean flow outside the internal boundary layer, then the wall conditions to enforce this similarity hypothesis cannot be given by the local friction velocity. This is because $U_{{\it\tau}}$ is determined by the growth of the internal layer and therefore $U_{{\it\tau}}$ is not an adequate scaling parameter for the entire boundary layer. At the same time, the wall conditions cannot be the $U_{{\it\tau}}$ of incoming flow as the ‘new’ turbulent structures that emerge from the smooth wall will alter the features of the flow even in the region outside the internal layer. Therefore, there should be an ‘equivalent’ rough wall that can capture the overall history effects of the flow. Unfortunately, it is difficult obtain a theoretical formulation for this new ‘history’ scale. Therefore, we carry out a minimisation procedure to determine it. Practically, at each measurement location downstream of the $\text{R}\rightarrow \text{S}$ surface change, the following objective function $\mathscr{J}$ is minimized to enforce outer layer similarity in the mean profile,

This shows that at each measurement location the value of $U_{{\it\tau}o}$ was determined such that the outer region of the mean velocity deficit is forced to collapse with the smooth-wall case as demonstrated in figure 15. Sufficiently far downstream it is expected that $U_{{\it\tau}o}\rightarrow U_{{\it\tau}}$ . Upstream of this location, $U_{{\it\tau}o}>U_{{\it\tau}}$ , where $U_{{\it\tau}}$ is the friction velocity determined from the Preston tube measurements.

It is very important to note that this new ‘history’ scale is only obtained in the mean flow. There is no information included about the variance or the spectral energy content when deriving this scale. Therefore, the efficacy of this new scale on variance and/or spectra is a physical result and is not an artefact of the process used to derive it.

The new outer scale, $U_{{\it\tau}o}$ , is first considered with respect to the normalization of the streamwise normal stress, $\overline{uu}$ . Unlike the outer scaling by the free stream velocity shown in figure 10, scaling $\overline{uu}$ by $U_{{\it\tau}o}^{2}$ results in the collapse of the $\overline{uu}$ past the internal boundary location, or for $y/{\it\delta}>0.3$ , as shown in figure 16. This suggests that even for flows over a step change in roughness, ‘presence’ of outer layer similarity in the mean flow (in this case, this presence is forced by the choice of $U_{{\it\tau}o}$ ) results in collapse of turbulence quantities in the outer region. The collapse of the outer region for this scaling parallels the observation made earlier from figure 9 where the streamwise turbulent fluctuations were scaled by the mean velocity gradient, again confirming the similarity of the outer region given an appropriate outer scale. However, given that this scale is purely associated with the flow above the internal boundary layer, it is does not collapse the inner peak magnitude as may be inferred from the near-wall energetic site shown in figure 16 for $y/{\it\delta}<0.02$ . This is to be expected because the near-wall peak should really only depend on the characteristics of the new internal boundary layer.

Figure 16.

Outer-scaled profiles of $\overline{u^{2}}$ by $U_{{\it\tau}o}^{2}$ for the (a) grit and (b) mesh cases. Note that $k_{o}=U_{{\it\tau}}/U_{{\it\tau}o}$ as defined in figure 15.

The new internal boundary layer, however, is under the effect of an external boundary layer. Analogous to a high Reynolds number smooth-wall boundary layer, the near-wall peak does not just depend on the local skin friction but also on an outer velocity scale that represents the energetic motions of the outer region. Toward reconciling the near-wall peak streamwise normal stress, consider a mixed scale based on a contribution of the near-wall and outer regions, $U_{{\it\tau}}U_{{\it\tau}o}$ . This new scaling is essentially a geometric mean of the local skin friction and the skin friction ‘perceived’ by the outer region of the flow. The resulting mixed scale profiles are shown in figure 17. Note that the upper limit of both ordinates in figure 17 are reduced compared to those shown in figure 12. From figure 17 it is evident that the variation of the near-wall peak magnitude is substantially reduced. This is clearly shown by examining the maximum value of the inner peak scaled by $k_{o}=U_{{\it\tau}}/U_{{\it\tau}o}$ , which is shown in figure 18. The variance about the smooth-wall trend of the inner peak maximum value described by Hutchins & Marusic (Reference Hutchins and Marusic2007a ) is considerably reduced past the $\text{R}\rightarrow \text{S}$ surface change by the mixed scaling. This shows that a combination of near-wall skin friction and outer region skin friction is required to reproduce the behaviour exhibited by a canonical smooth-wall boundary layer at comparable values of $Re_{{\it\tau}}$ . This also suggests that the near-wall peak in the present flow is established under the influence of an energetic outer region whose characteristic velocity is a ‘perceived’ skin friction that enforces outer layer similarity.

Figure 17.

Profiles of the mixed scaling of $\overline{u^{2}}$ for the (a) grit and (b) mesh cases where $k_{o}=U_{{\it\tau}}/U_{{\it\tau}o}$ . Inner scaling of the wall-normal coordinate is by the wall local friction velocity, $U_{{\it\tau}}$ . The approximate location of the inner peak ( $y^{+}=15$ ) is shown by ——.

Figure 18.

Variation of the inner peak magnitude, scaled by $k_{o}$ , with Reynolds number for the grit and mesh cases. Note that the limits of the plot correspond with figure 13. The solid line represents the empirical relationship of Hutchins & Marusic (Reference Hutchins and Marusic2007a ) and the grey shaded region represents scatter about this fit. Open symbols show the $l^{+}$ corrected values and error bars represent a 10 % uncertainty in $c_{f}$ .

Finally, the streamwise evolution of the velocity scales can be examined by considering the skin friction coefficients based on $U_{{\it\tau}}$ , $U_{{\it\tau}o}$ , and $\sqrt{U_{{\it\tau}}U_{{\it\tau}o}}$ as shown in figure 19. Note that the skin friction coefficient for a comparable smooth surface $U_{{\it\tau}s}$ is included, which was determined from $Re_{{\it\theta}}$ and the empirical $c_{f}$ relationship given by Österlund et al. (Reference Österlund, Johansson, Nagib and Hites2000) (see appendix A). The skin friction coefficient for a smooth surface is approximately constant ( $\pm 1.5\,\%$ ) over the range of Reynolds numbers considered.

The first thing to note is that the all values of $U_{{\it\tau}o}$ are less than the value of $U_{{\it\tau}}$ of the upstream rough wall. This decrease in representative velocity scale suggests that the change in surface roughness is felt across the entire layer almost immediately. Once this ‘shock’ is felt, the subsequent response of $U_{{\it\tau}o}$ , and therefore the outer region, is delayed. For substantial downstream distances (say up to $2{\it\delta}$ ), the value of $U_{{\it\tau}o}$ remains a constant. This is primarily because the growth of the internal boundary layer is gradual. This is consistent with previous observations of the streamwise Reynolds stress, which shows that the newly established near-wall condition is yet to interact with the outer region of the flow as shown in figure 10. For $x/{\it\delta}>2$ the decay of $c_{f}$ based on $U_{{\it\tau}o}$ can be captured by an equation of the form $c_{f}=f(M)\log (x/{\it\delta})+g$ , which has a form that is similar to the one used to describe $c_{f}$ based on $U_{{\it\tau}}$ , albeit with very different slopes and intercepts. The delayed recovery of the outer region, in contrast with the fast response of the near-wall region, leads to the variation of $c_{f}$ based on $\sqrt{U_{{\it\tau}}U_{{\it\tau}o}}$ . Furthermore, the mixed scale $c_{f}$ is yet to converge to the smooth wall $c_{f}$ trend at the final measurement location, whereas the inner peak magnitude appeared to coincide with the smooth-wall trend based on inner scaling alone, as shown in figure 13. Therefore, the slow response of the outer region appears to cause the inner peak magnitude to fall below the smooth-wall trend.

Figure 19.

Variation of the skin fiction coefficient, $c_{f}$ , with streamwise distance from the $\text{R}\rightarrow \text{S}$ surface change for the (a) grit and (b) mesh cases. The $c_{f}$ values shown are for $U_{{\it\tau}}$ inferred from Preston tube measurements, the forced collapse of the velocity deficit, $U_{{\it\tau}o}$ , the mixed scaling, $(U_{{\it\tau}}U_{{\it\tau}o})^{1/2}$ , and the equivalent smooth wall, $U_{{\it\tau}s}$ . The equation of the form $c_{f}=f(M)\log (x/{\it\delta})+g$ is given by – – – and — ⋅ —.

3.4 Properties of the turbulent spectra

Statistics of the streamwise velocity fluctuations ( $u$ ) represent the combined influence caused over all flow scales. The spectra of the streamwise fluctuations, ${\it\phi}_{uu}$ , is examined to investigate the scales of motion within the turbulent boundary layer following the $\text{R}\rightarrow \text{S}$ wall surface change. In addition to the mean statistics reported previously, the spectra offers information on how the near-wall and outer regions of the boundary layer manifest energetically. The local convective velocity is employed using Taylor’s hypothesis to express the temporal spectra as a function of streamwise wavenumber, $k_{x}=2{\rm\pi}f/U$ , and wavelength, ${\it\lambda}_{x}=2{\it\pi}/k_{x}$ . Spectra at each wall-normal location were premultiplied by the wavenumber, $k_{x}$ .

The variation of the premultiplied spectrogram of $k_{x}{\it\phi}_{uu}/U_{{\it\tau}}^{2}$ for the grit and mesh cases are shown along the left column in figures 20 and 21, respectively. The top row is spectra just downstream of the step change in roughness and subsequent rows show the data for different downstream distances. The downstream location is specified within the figure. It can be seen that for both mesh and grit cases, there is substantial energy in the outer region at large scales (approximately 3– $5{\it\delta}$ ). The corresponding energy level is higher for the mesh case compared to the grit cases (note that the colour scale is different for the two figures). This is consistent with the fact that the mesh is a stronger roughness compared to the grit and therefore there is substantially more energy in the outer region for the mesh case. It can also be seen in the figures that for both cases, the location of energy concentration in the outer region is just at the outer edge of the internal boundary layer immediately downstream of the step change.

As the flow progresses downstream, the energy contained in the outer region appears to decrease. This is primarily because the momentum flux due to the roughness is now reduced and the outer region is ‘shielded’ by the internal boundary layer. The energy contained in this outer region diffuses and by $x/{\it\delta}>10$ , most of the energy is lost. At the most downstream location, the outer region resembles the outer region of a smooth-wall boundary layer. It must be noted that the recovery to smooth-wall state is not fully complete. Further measurements are necessary to confirm the distance at which complete spectral recovery will be attained.

The spectra (left column) immediately downstream of the step change also shows that the near-wall spectral peak is also stronger for the mesh case compared to the grit case. This is consistent with the variance results in the previous section. The spectra shows that the increased variance is due to a stronger near-wall cycle with the peak energy at ${\it\lambda}^{+}\approx 1000$ (and surrounding wavelengths). This near-wall peak appears to decrease dramatically with increasing downstream distance. This decrease appears to be stronger for the mesh case compared to the grit case indicating that the behaviour of the energetic motions in the outer region (i.e. the trends in outer spectral peak) should have role in the behaviour of the near-wall peak.

The fact that the near wall as well as the outer spectral peaks possess different strengths when scaled with $U_{{\it\tau}}$ suggests that different velocity scales should be examined to account for the trends exhibited. Just as in the variance, the efficacy of ‘perceived’ outer scale and a mixed scale to capture the trends in the spectra are examined. The right column of figures 20 and 21 shows the premultiplied spectrogram normalised by $U_{{\it\tau}o}$ while the middle column of both figures show the spectra with mixed scaling ( $U_{{\it\tau}}U_{{\it\tau}o}$ , which is the geometric mean of the local inner scale and ‘new’ outer scale of the flow).

Figure 20.

Variation of the premultiplied spectra for the grit case following the rough-to-smooth surface change. The premultiplied spectra is normalized by $U_{{\it\tau}}^{2}$ on the left and to the right the mixed ( $k_{o}$ ) and outer ( $k_{o}^{2}$ ) scalings are shown, respectively. The normalization of ${\it\lambda}_{x}$ and $y$ use the friction velocity $U_{{\it\tau}}$ . The location of the internal boundary layer is given by – – – and ${\it\lambda}_{x}=3{\it\delta}$ is shown by ——. (a–c) $x/{\it\delta}=0.8$ , (d–f) $x/{\it\delta}=1.6$ , (g–i) $x/{\it\delta}=4.5$ , (j–l) $x/{\it\delta}=8.9$ , (m–o) $x/{\it\delta}=16.9$ .

Figure 21.

Variation of the premultiplied spectra for the mesh case following the rough-to-smooth change in surface. The premultiplied spectra is normalized by $U_{{\it\tau}}^{2}$ on the left and to the right the mixed ( $k_{o}$ ) and equivalent smooth wall ( $k_{s}$ ) scalings are shown. The normalization of ${\it\lambda}_{x}$ and $y$ use the friction velocity $U_{{\it\tau}}$ . The location of the internal boundary layer is indicated by – – – and ${\it\lambda}_{x}=3{\it\delta}$ is shown by ——. (a–c) $x/{\it\delta}=0.6$ , (d–f) $x/{\it\delta}=1.2$ , (g–i) $x/{\it\delta}=3.5$ , (j–l) $x/{\it\delta}=6.8$ , (m–o) $x/{\it\delta}=12.4$ .

It can be seen from the figures in the right column that the new outer scaling appears to collapse the spectral content above the internal boundary layer. The contours appear to be have a more consistent amplitude not only for all downstream locations but also for different step strengths (i.e. for grit and mesh cases). Furthermore, above the maximum height of the internal layer, the variation of the contours along ${\it\lambda}_{x}/{\it\delta}\approx 3$ appear similar, suggesting that the this new velocity scale is the relevant to capture the trends in the most energetic scale of the outer region (over and above the variance as observed in the previous section).

The spectrograms in the figures show that local friction velocity fails to collapse the energy contained in the near-wall region, both for the broadband turbulence at $y^{+}\approx 15$ (see figure 12) and about ${\it\lambda}_{x}^{+}\approx 1000$ (see figures 20 and 21). The lack of collapse in the broadband is expected since the variance (which is the integral of this broadband) requires a mixed scaling to reconcile the trends. However, the energetic motions associated with the near-wall cycle also do not appear to conform to $U_{{\it\tau}}$ scaling. This suggests that the energy contained in the near-wall region is altered based on the energy of the outer region. One possible mechanism to explain this behaviour is a direct footprint of the outer region. This is likely to be dominant in this current study as the scale separation between the near-wall and outer scales is limited in the current study. In addition to this, modulation of the amplitudes of the near-wall structures by the scales in the outer region is also possible. A direct footprint effect, which is a linear superposition, can perhaps be reconciled using the same mixed scaling that was used in the previous section. The middle column in figures 20 and 21 show that the contours of the near-wall region are relatively similar across downstream distances as well as step strengths. This suggests the most of the effect could in fact be direct (since we account for this ‘direct’ footprint effect through the $U_{{\it\tau}o}$ scaling). However, the contours do not exhibit strict collapse, which suggests an additional mechanism not accounted for by the geometric mean of the scales associated with the flow.

It has been shown that for higher Reynolds numbers, the signature of the large-scale motions in the outer region of the turbulent boundary layer are present in the near wall, see for example Hutchins & Marusic (Reference Hutchins and Marusic2007b ). The large-scale motions are known to influence the near-wall small-scale motions by an effect referred to as amplitude modulation (Hutchins & Marusic Reference Hutchins and Marusic2007b ). Notably, Ganapathisubramani et al. (Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012) showed that for a smooth flat plate boundary layer the modulation effect is not linear. A similar nonlinear interaction is possible in the flow examined in the current study. If the interaction between the outer region and the inner region was completely linear, then the mixed scaling would be sufficient to reconcile the observed trends. However, the inability of the mixed scale to adequately collapse the energetic inner region points to the presence of nonlinear interaction of the inner and outer regions not accounted for in the mixed scale model. This investigation requires data from flows over such step change in roughness at much higher Reynolds numbers where there is sufficient scale separation to evaluate the effects of these interaction in greater detail.