Turbulent Boundary Layers Over Multiscale Rough Patches
Abstract
We experimentally investigate the effects of multiscale rough patches on the drag and flow structure of a fully rough turbulent boundary layer in a wind tunnel. Several patches containing both organized and randomized arrangements of cubes of multiple sizes are tested in order to study the dependence of drag on the frontal solidity of the patch. The drag of each patch is measured with a drag balance for a range of Reynolds numbers, indicating a dependence of the drag on the frontal solidity following the trend predicted by Macdonald et al. (Atmos Environ 32(11):1857–1864, 1998). One of the patches is also replicated with the smallest scales removed and measurements show that the smaller scales have negligible impact on the overall drag. Flow fields in several cross-sections are captured using particle image velocimetry, and maps of the velocity deficit and increased turbulence activity in the wake of the patches are determined and used to define the extent of the internal boundary layer formed by each patch.
Keywords
Roughness Turbulent boundary layer Urban meteorology1 Introduction
Modelling of atmospheric flow over complex terrain is important for understanding the flow dynamics and corresponding dispersion in urban areas, as well as for the siting of wind farms in various onshore and offshore locations. Atmospheric boundary-layer (ABL) flow over complex terrain exhibits a range of characteristics with differing vertical profiles and high turbulence intensities. These variations are intimately related to the details of the surface terrain and the drag at the surface. Realistic surface terrain tends to be very rough, and roughness elements typically display a range of shape complexities and size distributions. It is important to be able to predict the flow features over such complex terrain for a range of applications; however, it is equally important to obtain experimental data (either in the field or through controlled wind-tunnel experiments) that can be used to validate the variety of modelling strategies. This paper presents results from a systematic series of experiments carried out in a wind tunnel to investigate the effect of the size distribution of roughness elements within a patch on the flow and drag incurred by that patch. We specifically focus on an isolated patch of multiscale roughness when the roughness elements attached to the surface are characterized by a broad size distribution, covering a large range of scales.
Most surfaces relevant to atmospheric flows are usually multiscale and comprise a range of different shapes and scales. Beside cubes, other shapes that have been considered include pyramids, ridges and more recently cuboids (Brown et al. 2001; Reynolds and Castro 2008; Schultz and Flack 2009; Castro et al. 2017). Recently, there is growing interest in understanding and predicting flows over multiscale surfaces. Multiscale roughness has also been examined for irregular roughness by Anderson and Meneveau (2011), who examined the flow over surfaces that have a power-law spectral distribution in height. Placidi and Ganapathisubramani (2015, 2018) examined flow past surfaces that contain LEGO bricks in different layouts where the brick was considered one scale while the pins on top of the bricks (and the baseboard) were a second scale/shape. Cheng and Castro (2002) carried out measurements over surfaces where repeated units had cuboids with a Gaussian height distribution and these were followed by LES studies of similar surfaces (Xie et al. 2008). More recently, Zhu et al. (2017) used LES to study the flow over surfaces with cuboids that have different mean heights and skewness distributions and used the data to develop a revised model for prediction of roughness length. Finally, Yang et al. (2016) examined flow past multiscale surfaces that consist of rectangular prisms, quadrupling the number of roughness elements on the ground as the sizes of the rectangular roughness elements were halved. As a result, the synthesized roughness followed a power-law height distribution.
In addition to consisting of a range of scales, terrain distributions in the natural environment typically vary along different geographic directions, for example from rural to urban regions, and the boundary layer has to adapt to this roughness transition. All of the above works focussed on homogeneous distributions of surface roughness (where a minimal unit is repeated along the streamwise and spanwise directions). There are a few numerical studies on turbulent flow over an explicitly-resolved roughness transition (Lee et al. 2011; Cheng and Porté-Agel 2015; Yang et al. 2016). In addition, experimental investigations are also scarce and typically only consist of one transition in surface roughness (Antonia and Luxton 1971; Cheng and Castro 2002; Hanson and Ganapathisubramani 2016). However, within complex terrain it is very likely that the distribution of roughness continuously changes, for example towards the centre of the city (or town). Such a patch of roughness is expected to exhibit similar qualities as abrupt changes in surface roughness, such as smooth-to-rough and rough-to-smooth transitions, which have been studied extensively. The effects of changing surface roughness are typically modelled by determining the evolution of the internal boundary layer, which is formed at the interface of the surface change, and which marks the boundary at which the effects of the surface no longer affect outer-layer similarity (Mahrt 2000; Bou-Zeid et al. 2004). When considering patches containing distinct obstacles, such as cubes or cylinders, the frontal and plan solidity (frontal area of the surface roughness and the plan area of the surface roughness per unit wall-parallel area) of the surface has been highlighted as an important parameter. In addition to solidity, the importance of the arrangement of the structures has also been highlighted, as this can produce “shielding” of downstream structures, where structures located in the wakes of other structures produce less drag (Yang et al. 2016). The solidity of a patch can also influence the vortex structures that develop in the wake and at the edges of the patch (Nepf 2012).
In this study, we bring together two of the above-mentioned lines of enquiry and examine the flow past a finite patch of multiscale cubes, concentrating on the effects of the patch arrangement on the drag and flow structure. We considered two different parameters: (i) the effect of frontal solidity, and (ii) the significance of the multiscale features. Firstly, we designed patches with equal planform solidities \(\lambda _P\) (defined as the fraction of the planform area that is covered by cubes), but vary frontal solidity \(\lambda _F\) (defined as the sum of forward-facing faces normalized by the planform area, Grimmond and Oke 1999). In this way, for surfaces consisting of only cubes, \(\lambda _P\) and \(\lambda _F\) only differ if cubes are in contact, hence this difference is an indicator of shielding. Secondly, we replicated one of the patches with the smaller scales of cubes removed in order to evaluate the relative contribution of the different scales of roughness. The intent is to determine the minimum resolution required to properly resolve flow over complex features, which is important for computational studies.
2 Design of the Patches
In the first part of the study, in order to investigate the effect of the arrangement of the cubes, in addition to the regular Sierpinksi pattern, four additional patches were constructed using random arrangements of the same number of cubes. The random distributions are created by randomly placing the cubes within the patch using a random number generator; however, in order to vary the frontal solidity \(\lambda _F\) of each patch, we weighted the probability of placing a cube in a specific location by using a Gaussian distribution centred on the patch with varying \(\sigma _y\), the second central moment of the arrangement in the spanwise direction. This varies how spread out the cubes are positioned and therefore the frontal solidity \(\lambda _F\). As \(\lambda _F\) is defined as the sum of forward-facing faces normalized by the planform area (Grimmond and Oke 1999), this represents the amount of “shielding” that occurs. The narrowest layout (layout D) has the fewest forward facing faces and therefore the smallest \(\lambda _F\). Because the Sierpinski layout has no touching faces, it has the maximum value of \(\lambda _F\). Because these patches all comprise the same number of cubes, each of these patches have the same planform solidity of \(\lambda _P = 0.298\). A histogram of the scales of cubes used in all of the layouts is presented in Fig. 2 resulting in an average obstacle height of \(\left\langle H\right\rangle =1.84\,\hbox {mm}\). Because this distribution of heights is the same for all layouts, all statistical parameters of this distribution are held constant, such as the height skewness, which can be an influential parameter (Zhu et al. 2017). Maps of the different arrangements are presented in Fig. 1 and their properties are shown in Table 1.
Properties of the patches
Patch | \(\sigma _y/r\) | \(\lambda _P\) | \(\lambda _F\) |
---|---|---|---|
Sierpinski | 0.45 | 0.298 | 0.298 |
Layout A | 0.38 | 0.298 | 0.254 |
Layout B | 0.32 | 0.298 | 0.241 |
Layout C | 0.27 | 0.298 | 0.232 |
Layout D | 0.20 | 0.298 | 0.203 |
B-1 level | 0.27 | 0.082 | 0.080 |
3 Experimental Details
The flow fields in two cross-sections of the flow were captured using conventional planar and stereoscopic particle image velocimetry (PIV) as shown in Fig. 3. Measurements in a streamwise vertical plane aligned with the patch centre were acquired using two side-by-side LaVision Imager Pro LX 16M cameras. The vector fields from the two cameras were then stitched together to create a long field-of-view that captured the incoming flow ahead of the patch and extended well into the wake of the patch. Subsequently, measurements of the patch wake were acquired in a plane located 63 mm downstream of the edge of the patch, using the same cameras placed in a stereoscopic configuration. For each case, 1000 image pairs were acquired at a rate of 0.7 Hz. The light sheet was created by two synchronized Litron 200-15PIV Nd:YAG lasers that were in each case aligned using mirrors to form a single light sheet. The flow was seeded using a Magnum 1200 smoke machine. Vector fields were calculated using LaVision DaVis 8 using \(32 \times 32\) pixel windows with a 50% overlap resulting in a vector spacing of roughly one vector \(\hbox {mm}^{-1}\) in both configurations. This linear resolution (2 mm) is just over the size of the smallest cube (which is 1.25 mm). Therefore, the fluctuations of the order of that scale are not captured by the particle image velocimetry; however, our focus is on the larger scales of the flow and consequently this limitation in resolution is not considered to affect the results.
4 Results: The Effect of Roughness Layout
4.1 Drag Measurements
Comparing the drag measurements of the different randomized arrangements, the results show a monotonic increase in drag with increasing frontal solidity in Fig. 4c.
This increase in drag with frontal solidity is consistent with the idea that the “shielding” of downstream elements, which reduces the frontal solidity of a surface, can significantly reduce drag. What is interesting is that the drag of the orderly Sierpinski carpet arrangement, which technically has the highest frontal solidity, produced a drag comparable to layout A, which had a lower \(\lambda _F\) value although the highest of the randomized arrangements. This suggests that frontal solidity alone is insufficient to predict drag, especially when the surface structure is organized in some way rather than randomized. In the special case of the Sierpinksi carpet arrangement, although no cubes are in contact, many lie in the wakes of others increasing the effective amount of “sheilding”. Furthermore, many alleyways are left exposed for air to flow through, making the flow act in a similar way to a patch with less solidity.
4.2 Velocity Fields
4.2.1 The Incoming Flow
The flow that impinged upon the roughness patches was a fully developed rough-wall turbulent boundary-layer flow that developed over the rough-wall LEGO baseboard that covered the test section floor. Measurements were collected over the same field of view as shown in Fig. 5, but with a continuation of the LEGO baseboard replacing the patch in order to confirm that the flow was no longer significantly developing in the streamwise direction. The representative velocity profile of this incoming flow and some second-order statistics are presented in Fig. 6. The friction velocity, \(u_\tau \), was determined using two methods. The first method assumes mean flow similarity and that the mean velocity profile in velocity-defect form should collapse with smooth-wall data (Flack and Schultz 2010). The velocity profile is plotted in velocity-defect form in Fig. 6b and compared with smooth-wall data that was previously collected via hot-wire anemometry in the same facility by Takfarinas Medjnoun. A value of \(u_{\tau } \approx 1.05\,\hbox {m s}^{-1}\) leads to a collapse of the velocity-deficit data with the smooth-wall data. A second method to estimate \(u_\tau \) is from the near-wall peak of the averaged Reynolds shear stresses (Reynolds and Castro 2008; Placidi and Ganapathisubramani 2015). These are plotted in Fig. 6c, which indicate the peak of the shear stress reaches approximately \(1.05\,\hbox {m}^2\,\hbox {s}^{-2}\). Using the correction factor proposed by Reynolds and Castro (2008), this corresponds to \(u_{\tau } = 1.12 \sqrt{\overline{-u w}_{{ peak}}} = 1.15\,\hbox {m s}^{-1}\). In Fig. 6c, the profiles of streamwise and vertical velocity fluctuation intensities, normalized by an average value of \(u_\tau = 1.1\), is consistent with earlier data (Squire et al. 2016). With the friction velocity known, the mean velocity profile can be fit to the expected logarithmic-law profile for fully-rough turbulent boundary layers given in Eq. 1. For the incoming flow we found \(z_0 = 0.2\,\hbox {mm}\) and \(d = 0.8\,\hbox {mm}\), which is consistent with what we would expect for the flow above the LEGO baseboard (Placidi and Ganapathisubramani 2015). Figure 6a presents the velocity profile, normalized by inner units, showing that the logarithmic region extends roughly over the range \(4\,\hbox {mm}< z < 20\,\hbox {mm}\), which means that the height of the patches (\(H=10\,\hbox {mm}\)) is well-immersed in the logarithmic region of the incoming boundary layer.
4.2.2 Visualizing the Internal Boundary Layer Over the Patch
4.2.3 Investigating the Wake
Measurements of the wakes of the patches obtained at \({ Re}_L = 5.1 \times 10^6\): the drag balance measurement, \(F_D\), the momentum deficit, \(\varDelta M\), converted to kg, the lengths of the wakes, \(L_W\), along with the calculated \(u_\tau \) and the values of \(z_0\) and d determined from a best fit to the velocity profiles in the wakes of the patches at \(x=1.5r\), spanwise-averaged over \(-\,r<y<r\)
Patch | \(\lambda _F\) | \(F_D\) (kg) | \(\varDelta M\) (kg) | \(L_W/r\) | \(u_\tau ~ (\hbox {m s}^{-1}\)) | \(z_0\) (mm) | d (mm) |
---|---|---|---|---|---|---|---|
Sierpinski | 0.298 | \(15.3 \times 10^{-3}\) | \(5.6 \times 10^{-3}\) | 2.13 | 1.65 | 1.14 | 2.3 |
Layout A | 0.254 | \(15.6 \times 10^{-3}\) | \(6.4 \times 10^{-3}\) | 1.83 | 1.67 | 1.18 | 2.5 |
Layout B | 0.241 | \(14.4 \times 10^{-3}\) | \(6.1 \times 10^{-3}\) | 1.63 | 1.60 | 1.00 | 2.4 |
Layout C | 0.232 | \(11.9 \times 10^{-3}\) | \(4.2 \times 10^{-3}\) | 2.08 | 1.46 | 0.70 | 3.0 |
Layout D | 0.203 | \(10.3 \times 10^{-3}\) | \(3.7 \times 10^{-3}\) | 2.19 | 1.36 | 0.52 | 2.9 |
As shown in Fig. 9, the velocity profiles and statistics do not completely collapse when normalized by inner units, which can be attributed to the fact that, at this position in the wake of the patch, this is not a fully-developed flow, so \(u_\tau \) does not fully dictate the boundary-layer characteristics. When the velocity profiles in the wake are normalized by the roughness length, as shown in shown in Fig. 9b, they indicate a collapse in the logarithmic region; however, the outer wake region lacks similarity with this scaling. This is not unexpected as we showed earlier that the inner boundary layer formed by the patch does not extend beyond \(z > 0.5 \delta \).
We found that if this logarithmic fit was performed with velocity data measured in the flow developing over the patch, the calculated values of \(z_0\) differ by as much as 100% due to the lack of local equilibrium. Therefore, we caution using \(z_0\) from arbitrary measurement locations as a surrogate for drag over patches, as the flow is clearly not fully-developed everywhere, manifesting in significant variability in the local boundary layer and uncertainty in the logarithmic-law fits.
5 Results: Significance of the Small Scales
5.1 Drag Measurements
The drag measurements of layout B replicated with only the 10-mm cube features are presented in Fig. 10. The 10-mm cube features represent roughly one quarter of the surface area that was originally covered by obstacles, while the other three-quarters of the covered area previously had obstacles of 5, 2.5, or 1.25 mm. The difference in the drag of the complete patch and the one containing only the largest size of cubes was only approximately 5%, indicating that the addition of the smaller scales appears to be minimal.
The drag was also measured without the patch installed in order to estimate the contributions of the pressure drag and the additional skin friction from the cubes. As shown in Fig. 1, the drag with no patch and just a continuation of the rough surface was roughly half the magnitude of those of the patches, demonstrating that the presence of the cubes was indeed significant. The drag of a smooth patch was also measured for comparison, and was found to be half again of the drag of the rough surface.
5.2 Velocity Fields
The averaged velocity fields also showed that the wakes of the patch with different numbers of cubes were essentially indiscernible. To further investigate if the dominant instantaneous flow features are also statistically identical, we performed a proper orthogonal decomposition (POD) analysis on the instantaneous snapshots. POD identifies the basis modes that make up a turbulent flow field and is a robust way of analyzing whether the significant features of the flow field have changed beyond simply the aggregate measures. The spatial resolution of PIV can be expected to affect the analysis carried out using POD; however, here, the focus is on the comparison between the large-scale most energetic modes across different surfaces and the spatial resolution of the PIV is maintained for all our surfaces at \(150\times 150\) wall-units (interrogation window size) relative to the incoming flow properties. Placidi and Ganapathisubramani (2018) found that, for an urban canopy flow at this resolution, approximately 30% of the energy is held in the first five modes and all these modes remain similar across different surface morphologies. Here, we can examine if an isolated patch alters this finding. The five most energetic modes from the flows around the full and stripped-down patches are presented in Fig. 11. These modes are clearly identical for these two flows, and show that the dominant fluctuations in the flow appear to stem from the base of the patch and from the locations of the largest cubes, which are consistent between the two patches. The relative energy content of all the modes is shown in Fig. 12 and was also nearly identical. This suggests that the large-scale energy distribution in the flow is not affected by the presence of the finer scales in the surface topography.
6 Conclusions
Measurements of the drag and flow structure above finite multiscale patches showed that, in all cases, the internal boundary layers created by the patches were much shallower than the boundary-layer thickness, indicating that the patches did not affect outer-layer similarity. The results indicate that the drag of the randomized patches increase monotonically with increasing frontal solidity, illustrating how “shielding” reduces drag. Furthermore, we showed that the majority of the drag is attributed to the largest scales of roughness. We also found that the patch that was organized as a Sierpinski pattern did not fit the same trend as the randomized arrangements, highlighting how exposed alleyways through the patch significantly affect the flow.
Notes
Acknowledgements
We gratefully acknowledge the support of the H2020 Marie Sklodowsa-Curie European Fellowship and EPSRC (Grant Ref No. EP/P021476/1). We also thank our summer intern Mantas Gudaitis for his contribution to the data analysis. All data supporting this study are openly available from the University of Southampton repository at https://doi.org/10.5258/SOTON/D0769.
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