4.1. Comparability of the Surfaces
The vertical distributions of the porosity for initial gravel (surface-compacted), water-worked gravel, cast and the rotated cast are presented in
Figure 4. For the water-worked gravel-bed, the porosity obtained from the WDM measurements decreased from
φ = 1 at the roughness top to an absolute minimum just above the roughness trough before reaching an approximately constant value in the sub-surface. The increase in porosity close to the plastic bottom of the flume was associated with the combined effect of capillary action and poor sorting of gravel at the bottom [
31]. The bulk porosity of the water-worked gravel-bed was
φ = 0.31 whereas the bulk porosity of the surface compacted gravel was
φ = 0.26; this difference was due to the larger height of the interfacial sublayer of the water-worked bed compared to the surface compacted bed. Note that the porosity values
φ > 1 for the WDM measurements in
Figure 4 are associated with the large spatial scale of the measurements (spanning the whole gravel-bed) and the accuracy of the measurements (see also [
30]). The porosity distribution of the water-worked gravel-bed obtained by the WDM measurement matched the porosity distribution derived from the laser scans gravel-bed from the roughness crest (
z/
H = 1) to
z/
H ≈ 0.85, where
H denotes the height of the bed measured from the flume bottom. Below
z/
H ≈ 0.85, the results from the two methods deviate since the laser scan could not capture the pore space in the subsurface layer, i.e., the measurement range of the laser scan was restricted to the distance from the roughness crest to the roughness trough so that
φ = 0 at the roughness trough for these measurements.
For the cast-beds, the porosity varied from φ = 1 at the roughness top to φ = 0 at the roughness trough. These distributions were solely obtained from the analysis of the laser scan data as the non-porous structure of the bed prevented the use of the WDM. The comparison of the porosity distributions derived from the laser scan measurements nearly collapse on a single line which is a first indicator of the accurate reproduction of the bed surface. However, a difference can be observed for the rotated cast just above the roughness trough and φ < 0.2. This is associated with the placement of the tile and is described below in some more detail.
The digital elevation models of the surfaces obtained from the laser scan, shown in
Figure 5, were used to compare the geometrical properties of the three bed configurations and to ensure accurate placement of the cast pieces. The following analysis focuses on the 4.65 m long section over which the water surface slope was measured (tiles 3 to 5 in
Figure 3) and which contained the PIV-measurement area. For the analysis, the DEMs were detrended and the origin of the vertical coordinate corresponds to the mean bed level.
The visual comparison of the permeable gravel-bed (
Figure 5a) with its impermeable counterpart (cast;
Figure 5b) indicates a good agreement between the two surfaces. The only visible difference is at the transitions between the tiles in
Figure 5b at
x = 6750 mm and 9050 mm. The match of the cast-bed with the water worked gravel-bed can be analyzed by comparing the distributions of the bed elevations (
Figure 6a) as well as the histogram of the observed differences in vertical elevations at each DEM grid point between the gravel and cast-bed (
Figure 6b). The histograms shown in
Figure 6a reveal a good agreement between the gravel and cast-bed, and the histogram of bed elevation differences (
Figure 6b) indicates that most of the values characterizing the deviation of the two surfaces are within the range of ±5 mm. It is worth noting that the histogram in
Figure 6b includes, besides the effects of shrinkage and the joints between the tiles, the tile-alignment as it was based on the subtraction of the two DEMs. The good match of the gravel-bed with the cast-bed can further be substantiated by the statistical moments of the surfaces which are presented in
Table 2. The standard deviation of bed elevations
σ, skewness and kurtosis are directly comparable and fall into the typical range for armored gravel-bed surfaces as defined by Coleman et al. [
3].
The distribution of bed elevations for the rotated cast differs slightly from the distribution of the cast. Strictly speaking, both surfaces should be characterized by identical histograms, however, the observed differences in
Figure 6a can be associated with very small transverse slopes induced when replacing the cast sections and due to discontinuities between the tiles. The latter were more pronounced for the rotated cast than for the cast and led to an increased number of measurement errors with the laser displacement meter due to its measurement principle. Despite these differences,
Table 2 indicates still a good agreement of the statistical moments of the rotated bed with the ones for the original and cast-bed.
The beds were also compared on the basis of normalized 2D-second order structure functions (2DSSF). For this purpose, the filtered normalized 2DSSF were determined according to the method described in detail in Qin et al. [
17]. In brief, filtered 2DSSFs show only values in areas which are characterized by statistical significant correlations. The latter are determined by subdividing the surface in small tiles of a certain length scale and reshuffling them to create new surfaces. For these new surfaces, 2DSSF are calculated and used for statistical significance testing. The method is based on Monte Carlo simulations (see [
17] for details) and for the present analysis, the reshuffled quadratic sub-areas were characterized by a side length of
d84 of the original gravel-bed. The statistical significance testing carried out here was based on a significance level 5% and 100 Monte-Carlo simulations.
Figure 7 shows the filtered 2DSSFs for the three surfaces and the sub-plots reveal areas of high spatial correlation at small spatial lags (the minimum value of 0 indicates perfect spatial correlation and a value of 1 a lack of spatial correlation). The area of high correlation for small spatial lags are characterized by a central ellipse, and the spatial extent of the main axes of the central ellipses can be interpreted as characteristic length scales in the horizontal plane [
19]. The benefit of the filtered 2DSSF is that these length scales can be clearly defined and the resulting values of 100 mm (96 mm for the rotated bed) for the long and 40 mm for the short axis (for all three beds) substantiates the high accuracy that was achieved when reproducing the bed.
The 2DSSFs can also be used to investigate the mean alignment of larger particles on the bed. For the present case, the orientation of the long axis of the ellipses deviates 5 degrees from the flow direction indicating that most of the larger particles are nearly aligned with their long axis in flow direction. This is in agreement with previous studies investigating the surface structure of stable armour layers [
16]. Note that the patterns at larger spatial lags in
Figure 7 reflect grain structures larger than individual grains. However, despite the fact that they are statistically significant, their value indicates only a low spatial correlation so that these patterns should not be interpreted [
17].
4.2. Flow Resistance
The good agreement between the geometry of the original bed with its counterparts is a prerequisite for the detailed analysis of differences in hydraulic resistance.
Figure 8 shows (8/
f)
0.5 as a function of relative submergence (
h/
k) for the three surfaces. The lowest values of (8/
f)
0.5, i.e., a higher Darcy-Weisbach friction factor
f, was observed for the tests over the rotated bed for relative submergences
h/
k < 6. For relative submergences
h/k > 6, the difference in (8/
f)
0.5 between the rotated bed and the cast became smaller which may be due to the increasing submergence. Moreover, the uncertainty associated with the water surface slope measurements increased for the highest submergence for which
Sw was rather small (~0.05%). On the other hand, in all tests with the rotated cast-bed it was necessary to decrease the discharge to obtain the same water levels as in the tests with the cast-bed (discharge reductions of up to 6% were required; the average was 4%). The higher flow resistance exerted by the rotated cast-bed shows that the orientation of the grains on the surface has a significant influence on flow resistance, i.e., the water working results in a more hydraulically efficient bed configuration (e.g., [
37]). For example, rotating the fixed cast-bed means that the lee-areas of grains, where small grains typically settle during armoring, become exposed to the flow. Rotating the cast-bed, these particles are directly exposed to the flow but cannot be eroded as they are part of the cast. As a consequence these areas are characterized by a less hydrodynamic shape than comparable frontal areas of real water-worked beds as the associated drag coefficient changes with the shape and orientation [
38]. Consequently, the rotated cast-bed imposes a higher resistance on the flow than the cast-bed. This result thus implies that bed roughness cannot solely be described by a characteristic grain size and that both surface structure and grain orientation play a vital role for the determination of flow resistance [
14,
18,
19].
Figure 8 further reveals small differences in flow resistance between the porous gravel-bed and its impermeable counterpart. For the two lowest submergences (
h/
k = 3.6 and 4.7), (8/
f)
0.5 is larger for the porous gravel-bed than for the impermeable cast-bed while for the larger relative submergences this trend is reversed. This means that, for the two lowest submergences, larger friction factors were obtained for the cast-bed than for the porous gravel-bed. These two experiments were carried out with discharges of
Q = 0.056 m
3/s and 0.076 m
3/s, respectively, for the gravel-bed and
Q = 0.056 m
3/s and 0.074 m
3/s respectively, for the cast-bed to obtain identical water depths
h. Noting that a certain amount of the flow is conveyed through the subsurface in the porous-bed tests, the discharge used for the calculations of the bulk values in
Table 1 should, strictly speaking, be reduced to account for subsurface flow. The experimental setup did not allow for the measurement of the subsurface flow rate but assuming a flow rate of approx. 0.001–0.002 m
3/s, computations indicated that for these two cases the flow resistance of the porous gravel-bed would be slightly larger than for the cast-bed (
h/
k = 3.7) or approximately equal (
h/
k = 4.7). The significance of subsurface flow rate gradually decreases with increasing discharge (and hence increasing relative submergence) so that the results for the experiments carried out with
h/
k ≥ 5.7 (discharges
Q ≥ 0.121 m
3/s;
Table 1) are less affected, i.e., in these tests the flow resistance of the porous bed was larger than for the cast-bed.
The literature review revealed that flow resistance over porous beds depends on
Re (e.g., [
24]), and therefore
f is plotted as a function of
Re in
Figure 9. Regarding the comparison of
f obtained for the experiments over the cast-bed and the rotated cast-bed,
Figure 9 yields the same conclusions as before; the rotated bed is characterized by higher flow resistance for all boundary conditions except for
Re ≈ 170,000 corresponding to the test which was carried out with the lowest water surface slope (~0.05%). For
Re > 100,000 (
h/
k ≥ 5.7),
Figure 9 reveals again that the porous gravel bed showed a higher resistance to the flow than its non-porous counterpart. In fact, for the experiments over the cast-bed the discharge had to be increased for
h/
k ≥ 5.7 to obtain the same water depth and shear velocity as in the porous-bed experiments, which confirms the observed trend. The deviation of the friction factors for the tests carried out for the lowest
Re values would be mitigated if subsurface flow would be accounted for (see above).
The above discussion regarding the differences in friction factors needs to be extended regarding the surface properties. For example, the cast-bed was characterized by slightly larger
k-values and standard deviations
σ than the porous bed (
Table 2). These values indicate that the cast-bed may be slightly ‘rougher’ in geometrical terms than the gravel-bed, and this may contribute to the observed trend at lower submergences (and hence low
Re). Similarly, the rotated cast-bed was characterized by larger
k- and
σ-values than the cast-bed, which in turn means that the rotated cast-bed was already rougher due to its placement. However, the differences in
k and
σ are rather small (less than 2 mm for
k and 0.56 mm for
σ) so that it can be hypothesized that this effect is negligible.
The comparison of the porous and non-porous bed can be further investigated by a preliminary and qualitative analysis of the double averaged longitudinal velocity distributions (i.e., time-averaged PIV-velocities, which were then spatially averaged in planes parallel to the mean bed elevation).
Figure 10 shows exemplarily the velocity distributions, normalized with the shear velocities obtained by Equation (2), for three different boundary conditions (BC2, BC3 and BC4) corresponding to the three different shear velocities (
Table 1). The origin of the vertical axis (
z = 0) in
Figure 10 corresponds to the mean bed elevation. The global roughness crest (i.e., the roughness crest of the scanned section) was 23 mm above the mean bed elevation whereas the local roughness crest (i.e., the roughness crest of the shorter PIV section) was only 8 mm above the mean bed level.
The three normalized velocity profiles over the gravel-bed tests (open symbols in
Figure 10) nearly collapse on a single line and show the expected logarithmic shape above the roughness crest. The three profiles over the cast-beds (solid symbols) deviate slightly, especially the profile for BC2. Note that the dip in the velocity profile for BC2 at
z ≈ 160 mm is due to a plastic glass which was placed at the water surface to avoid the refraction of laser sheet caused by surface waves. In order to cross-check the data, the velocity profiles were integrated to estimate the discharge and for all presented profiles, the calculated discharge was similar to the discharge used in the experiments.
The comparison of the profiles over the gravel-bed with the ones over the cast-bed shows that, for the same boundary condition, the velocities above the crest in the near-bed region (8 mm < z < 80 mm) are larger over the cast-bed than over the porous bed. While the aforementioned cast-bed profile for BC2 shows larger velocities than the gravel-bed profile up to z ≈ 80 mm, the velocity profiles for BC3 and BC4 show higher values for the cast-bed up to z ≈ 140 mm before nearly matching the velocities over the gravel-bed.
The smaller velocities over the crest of the water-worked gravel-bed in the near-bed region are additional evidence that a porous water-worked gravel-bed imposes higher flow resistance. It is interesting to note that in the interfacial sublayer (i.e., z < 8 mm), the flow velocities are partly larger over the water-worked gravel-bed than over the cast, especially below the mean bed elevation (z < 0). This can be explained by the ‘no slip’ condition for the non-porous cast-bed while, due to the porosity of the subsurface layer, such a condition does only exist at the gravel-particle surfaces over the gravel-bed. Note that due to the limitations of the PIV-setup, the velocity profiles could not be measured to the roughness trough. Nonetheless, these preliminary results of the PIV data further confirm the results from the bulk-flow analysis that the flow resistance over the porous gravel-bed is larger than over the cast-bed.
The presented results together with the results of the qualitative analysis of the double-averaged velocity profiles can be used to discuss the different results regarding the influence of porosity reported by Cooper et al. [
32]. The present study is based on experiments carried out over a casted surface which covered nearly the entire flume area. On the other hand, the length and width of the cast-bed in [
32] was limited, corresponding roughly to about 5% of the total water-worked area, and the control of the sub-surface flow was not clearly stated by Cooper et al. [
32]. Moreover, the cast tile was placed in the middle of the water-worked gravel and hence the transition from the gravel-bed to the cast-bed could affect the flow patterns; however, here we can only speculate about this effect. On the other hand, it is interesting to note that the range of relative submergence (
h/
k) in Cooper et al. [
32] varied between 3.1 and 4.6 with
Re ranging between 64,000 and 84,000. The results presented in
Figure 8 and
Figure 9 indicate that for comparable relative submergences and
Re-values the cast-bed is ‘rougher’ which coincides with the findings of Manes et al. [
28], Cooper et al. [
32] although the behavior of the velocity profiles deviates from the one reported by Cooper et al. [
32].