Lateral bed-roughness variation in shallow open-channel flow with very low submergence
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Abstract
Quantifying turbulent fluxes and secondary structures in shallow channel flows is important for predicting momentum and mass transfer in rivers as well as channel capacity and associated water levels. Here, we focus on the flow over a lateral bed-roughness variation with very low relative submergence of the roughness elements, \(h/k = \{3, 2, 1.5\}\), where h is the flow depth and k is the roughness height. Measurements were performed in a 1.1 m wide and 26 m long glass flume whose bed was fitted with cubes arranged in two regular side-by-side patterns with frontal densities \(\lambda _f = 0.2\) and 0.4 to create a rough-to-rougher variation. Measurements were performed using stereoscopic PIV in two orthogonal planes, in a vertical transverse plane spanning the two roughness types, and in a longitudinal one at the interface between the roughness types. The results show that the bulk velocity difference between the two sides of the channel increases with decreasing h/k. Also, contrary to what is observed at high relative submergence with smooth-to-rough transitions, higher bulk velocities occur on the side with higher roughness. This difference is increasing as the flow becomes shallower and is shown to be due to increasing effective depths ratios, leading to increasingly lower friction factor ratios with lower friction factors on the high-velocity but rougher side. Although increasing streamwise momentum transfer at the interface is needed as h/k decreases, the turbulent and secondary circulation transfer of momentum is increasingly inhibited. A globally-driven secondary-circulation at \(h/k=3\) ceases for lower h/k and roughness-scale circulation becomes dominant. Also, even the increased global shear does not lead to large-scale Kelvin Helmholtz instabilities structures. However, the relative importance of the roughness difference on the flow is augmented as the flow becomes shallower and momentum transfer due to lateral dispersive stresses increases.
Keywords
Open-channel flow Shallow flow Lateral roughness variation Low relative submergence Secondary currents Stereoscopic PIV1 Introduction
It is common that river beds have a lateral variation in roughness, shall that be vegetation on one side of the channel or a change in grain size on the bed. During high floods, the flows over the floodplains are almost always spatially inhomogeneous. In order to predict water levels in a river or over the floodplains one needs to calculate the flow capacity of the channel. The channel capacity for a uniform bed structure over a given cross-section of a channel is a function of the resistance imposed by the channel bed but is also influenced by the lateral transfer of momentum driven by lateral gradients at the edges. Indeed, it is common to model inhomogeneous beds with different roughness coefficients, suggesting that a flow that has two different roughnesses on two sides is equivalent to two streams flowing parallel to each other as if they were separated by a frictionless wall. However, neglecting the effects resulting from the interaction between the two streams may result in an overestimation of the channel capacity.
Another important characteristic of shallow flows over floodplains is the low relative submergence of the roughness elements, for instance for an urbanised floodplain during an extreme flood where buildings may be submerged. Even for homogeneous beds, one can expect the boundary-layer approach to fail [15], although experiments over a bed of cubes have shown that the logarithmic law still prevails if the double-averaged velocities are considered [24].
Open-channel flows with lateral variations have been studied under diverse conditions: shallow mixing layers, where the flow is characterized by a lateral velocity difference with no roughness changes [2, 4, 26]; smooth-to-rough roughness variation due to compound channels, where the lateral velocity difference is driven mainly by the difference in the flow depth [6, 14, 21, 22, 27]; and finally, the most directly relevant case, flows with lateral transition between completely smooth and rough beds [3, 17, 28] and between smooth and vegetation patches [12, 19, 30]. Rough-to-rougher transitions, however, have to our knowledge not yet been considered.
As discussed by Vermaas et al. [28] in the context of smooth-to-rough lateral transitions, there are three main mechanisms governing the lateral exchange of streamwise momentum: (1) secondary currents, (2) turbulent flux resulting from the shear between the two parallel flows with different velocities, and (3) bulk transfer. When the flow is fully developed, the bulk transfer is negligible.
Secondary currents or circulations driven by a lateral difference in bed roughness (smooth-rough transition) was first observed by Studerus [25]. Vermaas et al. [28] showed that this secondary current becomes more important in terms of its contribution to the lateral exchange of streamwise momentum with increasing water depth. Their h/k ratios on the rough side ranged from about 10 to 30, i.e., standard submergence where flow resistance formulations based on the logarithmic law are generally considered to hold. Low submergence (\(h/k < 10\), where h is the flow depth and k is the roughness height) or very low submergence (\(h/k < 3\)) or rough-to-rough transitions were not considered.
Turbulent momentum transfer can be realized through different mechanisms. Large-scale turbulent coherent structures originating from a Kelvin–Helmholtz type shear instability have been studied in plane shallow mixing layers by Chu and Babarutsi [4], Chen and Jirka [2], Uijttewaal and Booij [26]. These studies concluded that the growth of the shear layer is suppressed by the bottom friction and therefore, the exchange due to coherent turbulent structures is reduced for smaller depths, higher roughness and lower velocity difference between the parallel flows. On the other hand, Proust et al. [23] showed that in a compound channel, where the velocity difference is sustained by the difference in the flow depth, the coherent structures persist even for very shallow conditions. Proust et al. [23] proposed that these structures are controlled only by the difference in the bulk velocities over the flood plain, U_{2}, and the main channel, U_{1}, expressed as a velocity ratio (also referred to as dimensionless shear), \(\lambda = (U_1 - U_2)/(U_1 + U_2)\). For \(\lambda > 0.3\), large-scale turbulent structures were observed provided the presence of an inflection point in the mean longitudinal velocity profile.
Turbulent exchange in wall-bounded flows in general is also controlled by large- and very-large-scale motions. These have received considerable attention in canonical wall flows [11, 13, 16]. In a river, they were studied by Franca and Lemmin [10] and in a rough-bed laboratory open-channel by Cameron et al. [1]. These structures consist of streaks of higher and lower streamwise velocities and associated vortices. Nezu and Nakagawa [18] observed that the characteristic width of the streaks in an open channel is twice the water depth. Wang et al. [29] showed that they grow linearly coming up to the free surface. However, according to Defina [5], the streaks tend to disappear for low relative submergence, when \(h/k_s < 2\), where \(k_s\) is the equivalent sand-roughness.
Thus, it appears that a variety of phenomena can be expected for lateral rough-to-rougher transitions under very low relative submergence (\(h/k < 3\)) and it is not clear which. The roughness elements themselves are expected to play a growing role as the flows become shallower (decreasing h/k) since the roughness sublayer increasingly dominates the flow depth [24]. Whether analogies can be drawn with flows with higher relative submergence or flows with a smooth-to-rough lateral transition will be investigated in this paper.
2 Methodology
Flow conditions for the three sets of experiments with varying relative submergence, h/k, where h is the flow depth (measured from the bottom of the flume), k is the roughness height and Q is the total discharge
h/k | Q (l/s) | \(U_{k}\) (cm/s) | \(Fr_k\) | \(U_{e}\) (cm/s) | \(\langle {\overline{u}} \rangle _{yz}\) (cm/s) |
---|---|---|---|---|---|
3 | 15.8 | 35.8 | 0.58 | 26.5 | 35.1 |
2 | 5.2 | 23.6 | 0.53 | 13.9 | 20.9 |
1.5 | 2.0 | 18.2 | 0.57 | 7.6 | 12.5 |
3 Results
3.1 Mean flow
3.1.1 Lateral distribution of streamwise velocity and bed friction
The three normalized \(\langle {\overline{u}} \rangle _{z}(y) / \langle {\overline{u}} \rangle _{yz}\) profiles intersect very closely to the geometric interface at \(\langle {\overline{u}} \rangle _{z}(y) / \langle {\overline{u}} \rangle _{yz} \approx 1\). The three measured bulk velocities above the canopy \(\langle {\overline{u}} \rangle _{yz}\) (given in Table 1) are therefore good estimates of the interface velocities.
Table 1 also gives the bulk velocity \(U_k\) defined classically by \(U_{k}= Q/A_k\) where \(A_k = 2B(h-k)\) is the cross-sectional area above the canopy. It also gives \(U_{e}= Q/A_e\) where \(A_e = B (d_{e,1} + d_{e,2})\) is computed with effective flow depths on each side. The effective depths are given by \(d_{e,i}= h-k(1-\phi _{i}\)) considering the volumetric displacement by the cubes with \(\phi _i=V_{f,i}/V_t\) being the porosity of the beds (\(\phi_i =\{0.8, 0.6\}\) for S1 and S2, respectively). As expected, these bulk velocities are lower than \(U_k\), reflecting the larger effective crosssectional flow area. They are also lower than \(\langle {\overline{u}} \rangle _{yz}\) reflecting the contribution of the lower velocity canopy flow.
Figure 3 shows the lateral profile of the turbulent friction velocity \(u^*\) (open markers). At each location, \(u^*\) is estimated by first averaging the turbulent shear-stress profiles in the y-direction across half a roughness length k and then taking the maximum of the resulting profiles just above the roughness elements. This method is more robust than an extrapolation to the top of the elements as done in homogeneous flows since extrapolation is not straightforward to apply with non-linear shear stress profiles. It should be noted that \(u^*\) at the top of the canopy is the velocity scale which scales the turbulence [9, 20] which is lower than the friction velocity associated with the bed shear stress. The total shear-stress acting on the bed (the bed resistance) is given by the friction velocity \(u^*_0=u^* (1+ \frac{\phi }{(h/k)-1})^{1/2}\) (see [20]) and is also shown in Fig. 3 (filled symbols). Clearly, for smaller h/k, the difference between the two shear velocities increases.
It can be seen in Fig. 3 that both friction velocities are higher over the S1 roughness pattern (right-hand side). Computation of the equivalent sand-roughness, \(k_s\) (not shown), via logarithmic law fits (see [24], for the method) also confirms that S1 is rougher than S2 for the h/k range investigated, following the \(u^*\)y-variation tendency. Hence, the bed-resistance, \(u^*_0\), the turbulence scale, \(u^*\), and the roughness scale, \(k_s\), are all globally higher on S1 than on S2. Yet, the flow is faster over S1, especially for the lowest relative submergence (see Fig. 2d), implying that the bulk flow over the surface with higher equivalent sand-roughness (S1) is higher.
Bulk flow quantities for both sides are given in Table 2 (for S1 with \(i=1\) and for S2 with \(i=2\)). The flow rates on S1, \(Q_{1}\), were estimated by integrating the velocity profiles measured in the center of S1 above the canopy by Rouzes et al. [24]. To account for the flow within the canopy, the canopy profiles measured by Florens [8] for same bed and \(h/k = 3\) were used. Then \(Q_2=Q-Q_1.\) The bulk velocities \(U_{e,i} = Q_{i}/A_{e,i}\) were computed with the effective water depths \(d_{e,i}\), i.e., with \(A_{e,i} = B d_{e,i}\).
The apparent contradiction can be explained by the different effective flow depths \(d_{e,i}\) on the two sides which are higher on the S1 side for all h/k investigated (see Table 2). Their ratio \(d_{e,1}/d_{e,2}\), also given in Table 2, increases with decreasing relative submergence, following the bulk-velocity ratio \(U_{e,i}\) trend. The effective depth ratios are evidently high enough for all h/k to compensate the higher bed stress over S1. This is confirmed by the bulk Darcy friction factors for both sides, \(f_i = 8 {u^*_{0,i}}^2 / U_{e,i}^2\) which are higher on the smoother S2 side, increasingly so as h/k decreases.
Estimated bulk flow parameters \(X_i\) for the two roughness sides S1 (\(i=1\)) and S2 (\(i=2\)), as well as the ratios of the parameters \(X_1 / X_2\)
h/k | S1 (\({\mathrm{i}}=1\)) | S2 (\({\mathrm{i}}=2\)) | \(X_{1} / X_{2}\) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
3 | 2 | 1.5 | 3 | 2 | 1.5 | 3 | 2 | 1.5 | ||
\(X_i =\) | \(d_{e,i}\) (cm) | 5.6 | 3.6 | 2.6 | 5.2 | 3.6 | 2.6 | 1.08 | 1.12 | 1.18 |
\(Q_{i}\) (l/s) | 9.2 | 3.5 | 1.5 | 6.5 | 1.7 | 0.5 | 1.4 | 2.1 | 2.8 | |
\(U_{e,i}\) (cm/s) | 29.7 | 17.8 | 10.3 | 22.8 | 9.4 | 4.3 | 1.3 | 1.9 | 2.4 | |
\(u^*_{0,i}\) (cm/s) | 4.3 | 3.3 | 2.5 | 3.8 | 2.4 | 1.8 | 1.14 | 1.35 | 1.42 | |
\(f_i\) | 0.16 | 0.27 | 0.48 | 0.22 | 0.52 | 1.38 | 0.75 | 0.51 | 0.35 |
3.1.2 Secondary currents
For the highest relative submergence ratio investigated (\(h/k = 3\)), a counterclockwise secondary current can be identified in Fig. 2b. It is centred around \(y \approx\) 0.01 m, i.e., at the geometric interface between the two roughness types. The secondary current appears to be similar to the one observed at the interface between the smooth and rough surfaces in Vermaas et al. [28]. Also as in Vermaas et al. [28] or the classical secondary flow results reviewed in Nezu and Nakagawa [18], the current ascends over the high bulk velocity side (S1) and descends over the low bulk velocity side (S2). The sense of rotation is also in accordance with the local variation of \(u^*_0\) seen in Fig. 3 around \(y\approx\) 0, with \(u^*_0\) being locally higher on the downward side (S2). The global y-variation of \(u^*_0\), however, is in the opposite direction, i.e., rougher on the S1 side. The size of the recirculation zone suggests it scales with the water depth and so appears to be driven by the global bulk velocity gradient.
It should be noted that in Fig. 4d–f, the mean vertical velocity at the x-position of the transverse measurement plane (between the dashed lines) is directed downwards throughout the measured water depth above the cubes. This downward flow is compensated by upward flow further downstream in the wake of the cube. Since a net downward component washes out the topology of the vector fields, the net vertical velocity (time-, depth-, and width-averaged), equal to 0.13, 0.16 and 0.23 of \(u^*\) for \({ h/k } = \{3, 2, 1.5\}\), respectively, was subtracted from the time-averaged vertical velocity field in Fig. 2b–d. This allows to better visualize the secondary flow patterns.
3.1.3 Vertical vorticity
It is worth noting in Fig. 5a–c that the time-averaged vertical vorticity has a tendency to form vertical columns with alternating sign. These types of structures are not readily apparent in the instantaneous fields. It is hypothesized that the vertical structures are the result of averaging the preferential position of larger vortical structures oscillating around the roughness elements, or are the result of averaging small-scale intense vortices shed from the cubes. The origin of this phenomenon needs more investigation, but it is notable that a deeper flow in this case shows more vertical homogeneity than the shallower one.
3.2 Turbulent stresses and structures
3.2.1 Turbulent stresses
The turbulent flux \(-\,\langle {\overline{v'w'}} \rangle _x/ {u^*}^2\) is almost negligible for all the relative submergences, implying that there is a negligible lateral transfer of vertical momentum by lateral velocity fluctuations. The transverse flux of the streamwise momentum, \(-\,\langle {\overline{u'v'}} \rangle _x/ {u^*}^2\), on the other hand, is seen to be not negligible, in particular for \(h/k = 3\) where its sign suggests that \(\partial \langle {\overline{u}} \rangle _z / \partial y\) should be negative, which is indeed the case since \(\partial \langle {\overline{u}} \rangle _z / \partial y\) changes its sign locally due to the presence of the secondary current (see Fig. 2a). For \(h/k=2\), \(-\,\langle {\overline{u'v'}}\rangle _x / {u^*}^2\) is positive, consistent with \(\partial \langle {\overline{u}} \rangle _z / \partial y > 0\). However, it becomes much smaller in magnitude compared to \(h/k=3\). In the case of \(h/k=1.5\), \(-\,\langle {\overline{u'v'}} \rangle _x\) changes sign with flow depth so that the vertically and horizontally averaged flux would be close to zero. Therefore, the net turbulent transverse flux across the \(y = 0\) plane can be seen to decrease with increasing shallowness.
Turbulence characteristics for three relative submergences, \(h/k = \{3, 2, 1.5\}\), where \(u^*\) is the streamwise-averaged friction velocity at the interface obtained from longitudinal plane, \(u_{rms}\) is the streamwise, \(v_{rms}\)—the transverse, and \(w_{rms}\)—the vertical r.m.s. velocities
\(h/k = 3\) | \(h/k = 2\) | \(h/k = 1.5\) | |
---|---|---|---|
\(u^*\) (m/s) | 0.036 | 0.024 | 0.016 |
\(\langle u_{rms} \rangle _{xz}/ u^*\) | 1.52 | 1.61 | 1.70 |
\(\langle u_{rms} \rangle _{xz} / {\langle {\overline{u}} \rangle _{yz}}\) | 0.15 | 0.18 | 0.21 |
\(\langle u_{rms} \rangle _{yz} / { \langle {\overline{u}} \rangle _{yz}}\) | 0.12 | 0.12 | 0.15 |
\(\langle v_{rms} \rangle _{xz} / u^*\) | 1.14 | 1.26 | 1.36 |
\(\langle v_{rms} \rangle _{xz} / { \langle {\overline{u}} \rangle _{yz}}\) | 0.12 | 0.14 | 0.17 |
\(\langle v_{rms} \rangle _{yz} /{ \langle {\overline{u}} \rangle _{yz}}\) | 0.09 | 0.10 | 0.15 |
\(\langle w_{rms} \rangle _{xz} / u^*\) | 0.87 | 0.87 | 0.89 |
\(\langle w_{rms} \rangle _{xz} / { \langle {\overline{u}} \rangle _{yz}}\) | 0.09 | 0.10 | 0.11 |
\(\langle w_{rms} \rangle _{yz} / { \langle {\overline{u}} \rangle _{yz}}\) | 0.06 | 0.06 | 0.06 |
3.2.2 Kelvin–Helmholtz instability
With decreasing relative submergence, the difference in the redistribution of the streamwise momentum increases (Fig. 2). According to Proust et al. [23] for compound channels, when the ratio based on the bulk velocities of the two sides \(\lambda = (U_{1} - U_{2}) / (U_{1} + U_{2}) > 0.3\), large-scale turbulent structures with a vertical axis of rotation develop due to Kelvin–Helmholtz shear instability. Here, taking the effective bulk velocities \(U_{e,i}\), \(\lambda = \{0.16, 0.44, 0.70\}\) for \(h/k = \{3, 2, 1.5\}\), respectively, suggesting that \(h/k = 2\) and 1.5 could be subject to the Kelvin–Helmholtz shear instability. A sufficient condition for it to appear is an inflection point in the streamwise velocity profile. However, the distribution of the mean momentum in the current set-up does not resemble a classic shear-layer profile, but rather possesses numerous inflection points due to the direct roughness influence (Fig. 2a). It is thus not clear whether Kelvin–Helmholtz type vortices are expected. In order to investigate this further, auto- and cross-correlations are considered.
3.2.3 Velocity streaks
To examine the spatial organization of what appears to be streaks, Fig. 10a–c show the streamwise velocity fluctuations in the horizontal (x, y) plane, computed using Taylor’s “frozen turbulence” hypothesis, for the three different h/k ratios, respectively. For all three h/k, the streamwise velocity fluctuations are plotted at the same relative flow depth of \(z/H_e = 0.81\). Velocities faster than the mean are given in red, and slower ones in blue. For the deeper flow (\(h/k = 3\)), the structures appear elongated, as streaks characteristic of boundary-layer flows and are of the scale of about \(10 H_e\). This scale is consistent with the scale from the autocorrelation function \(R_{xx}^t\). Their length appears to be just a bit longer on S1, again consistent with \(R_{xx}^t\). In the shallowest case (\(h/k = 1.5\)), the structures are smaller. The “streaks” that now appear as “blobs” have the scale of about \(5 H_e\) on the S2 side and somewhat longer over S1. For the intermediate submergence (\(h/k = 2\)) the streaks are much longer over S1 reaching about \(20 H_e\). This is not so clear when viewing one snapshot only, but can also be seen in the autocorrelation in Fig. 9.
One can also observe in Fig. 10 that the intensity of the streaks is lower on the S2 side for lower relative submergence, in particular for \(h/k = 1.5\) and somewhat for \(h/k = 2\), while for the deeper case the intensities appear more homogeneous in the lateral direction. Therefore, as the flow becomes shallower, the flow structures or velocity streaks change their structure or cease to exist as was also observed by Defina [5]. Also, the two sides of the channel (S1 and S2) become detached at the transition and the flow over each side is then controlled by the local shape of the roughness elements. This is consistent with the absence of large-scale shear structures.
3.3 Bulk lateral transfer of streamwise momentum
Characteristics of the lateral transfer of streamwise momentum, where \(T_{sf}/\rho {u^*}^2\) is the transfer due to secondary flows, \(T_{tm}/\rho {u^*}^2\) is the transfer due to turbulence and \(T_{d}/\rho {u^*}^2\) is the transfer due to dispersive stresses
\(h/k = 3\) | \(h/k = 2\) | \(h/k = 1.5\) | |
---|---|---|---|
\(T_{sf} / \rho {u^*}^2\) | 0.22 | 0.17 | 0.07 |
\(T_{tm} / \rho {u^*}^2\) | − 0.14 | 0.05 | 0.01 |
\(T_{d} / \rho {u^*}^2\) | 0.017 | 0.005 | 0.048 |
These fluxes were estimated by the longitudinal plane measurements above the canopy and therefore do not account for the mean, turbulent and dispersive fluctuations within the canopy. The results are presented in Table 4.
The fluxes due to the secondary currents and dispersive stresses are positive in agreement with the global velocity gradient with higher bulk velocities on the S1 side (see Fig. 2a) as well as the velocity gradients and corresponding flux found by Vermaas et al. [28]. The turbulent flux, however, appears to follow the local velocity gradient at least for \(h/k~=3\) where it is negative. Nevertheless, the sum of the three fluxes is positive for all h/k. It can also be seen in Table 4 that both \(T_{sf}/\rho {u^*}^2\) and \(T_{tm}/\rho {u^*}^2\) generally decrease with decreasing relative submergence. This is surprising at first as one expects stronger positive transfer for lower h/k since the friction factors and bulk velocities ratios increase (Table 2). However, \(T_d/\rho {u^*}^2\) increases more significantly with decreasing h/k suggesting that the decreasing transfer due to secondary and turbulent motions is overtaken by the dispersive motions caused by the roughness elements. The net sum of the fluxes is highest for \(h/k=2\) rather than for \(h/k=1.5\), an inconsistency showing that the neglected fluctuations below the measurement plane are significant and need to be accounted for to obtain a full balance. Apart from this inconsistency, the decreasing role of the turbulent shear with increasing submergence is in agreement with the absence of large-scale shear-instabilities and the decreasing role of the secondary flows is in agreement with the transition from globally driven secondary currents to roughness driven ones. On the other hand, the increasing dynamic role of the roughness elements leads to increasing dispersive stresses evidently dominating the other lateral transfer mechanisms, at least above the canopy. Below the canopy, one can surmise that dispersive stresses also play a role, perhaps even a leading one.
4 Conclusion
Stereoscopic PIV was performed in an open-channel flow with a lateral bed-roughness variation and very low relative submergence (\(h/k \le 3\)). Two measurement planes were considered at the interface between the two bed-roughness configurations, a transverse plane spanning the two roughness types and crossing the interface and a longitudinal one close to the interface. The data from the transverse plane allowed to investigate the secondary currents appearing at the border between the two parallel flows, while the data from the longitudinal plane provided space-averaged statistics at the interface. The findings can be summarized as follows.
For the investigated shallow flows, higher bulk velocities are observed over the rougher surface (S1), the surface with higher friction velocities and with higher equivalent sand-roughness lengths. Also, with decreasing relative submergence h/k from 3 to 1.5, the ratio of the bulk velocity increases, from 1.3 to 2.4. Higher bulk velocities on the rougher side are in contrast with smooth-to-rough transitions where the bulk velocity is higher over the smooth side, whether for deeper flows with \(h/k>3\) [28], or for shallower low-submergence flows as in the current work [6, with \(h/k \le 2\)]. The apparent contradiction can be explained by the difference in the effective flow depths associated with the different porosities of the two roughness types. The effective flow depth is higher for the rougher pattern (S1) which is simply the byproduct of the lower porosity (\(\phi\)). The friction factors which depend on the ratio of the friction velocity to the bulk velocity when defined by the effective depth are indeed, for all h/k, lower on the fast side (S1), as for the smooth side in smooth-to-rough transitions. The disparity between the effective-depth and roughness effects increases as h/k decreases, leading to increasingly higher velocities and lower friction factors on the rougher side, relative to the smoother side. Inversely, it can be surmised that for some \(h/k>3\), the roughness effect becomes larger than the effective-depth one so that the bulk velocity is again higher on the smoother side (S2).
The increasing velocity ratio as h/k decreases has to be balanced by an increase in the streamwise momentum transfer across the interface. For developed flows, this can be achieved by turbulent shear stress, mean secondary flow, or, in particular for low-submergence situations as proposed here, by dispersive stresses. Yet, even though the global transverse gradients of the streamwise velocity are higher for the shallower cases, the local gradients due to the roughness elements increasingly dominate at lower h/k. Accordingly, large-scale Kelvin–Helmholtz type structures are not observed by the correlation analyses, even for the strongest lateral shear at \(h/k=1.5\), and are not expected to do so for even lower h/k. Also, the two sides show increasingly independent boundary layers structures (streaks). Consistently, the lateral turbulent flux \(-\,\rho \langle {\overline{u'v'}} \rangle _{xz}\) across the interface also decreases with lower h/k and is attributable only to the relatively weak local transverse gradients of the streamwise velocity at this interface, controlled by the local roughness pattern. Similarly, a global-scale secondary current is only observed for \(h/k = 3\), while for \(h/k = 2\) and 1.5 the secondary currents associated with the roughness elements become predominant. The decrease in strength of the turbulent and secondary flow fluxes in spite of a necessarily increasing net flux is counteracted by the strengthening of the dispersive effects associated with the roughness elements. The lateral dispersive stress, \(-\,\rho \langle {\tilde{u}} {\tilde{v}} \rangle _{xz}\), grows more than twice with decreasing h/k.
In summary, the behavior of the interaction of the flows between the two rough sides is markedly dependent on the relative submergence for low submergence conditions. The bulk velocity gradient is opposite of what is observed for smooth-to-rough transitions and what can be expected for higher submergence flows (\(h/k>3\)). Turbulent shear stresses and secondary flows controlled by the global shear as observed at high submergence, in particular for smooth-to-rough transitions or at the interfaces of compound channels, become unimportant. Instead, turbulent shear and secondary flows are increasingly controlled by the roughness-scale as the flows become shallower. To allow the increasing total momentum flux from the smoother side to the rougher as the flow becomes shallower, the dispersive stresses, also driven by the roughnesses, are increasing. For a full momentum balance, however, velocity measurements at the interface including within the canopy are necessary.
Notes
Acknowledgements
This work was supported by EC Hydralab IV/PISCES project (Grant Number 261520) and by the ANR FlowRes project (Grant Number 14-CE03-0010). M. Rouzes benefited during his PhD from financial support by the Direction Générale de l’Armement (DGA). The help in the experimental set-up by S. Cazin, M. Marchal and S. Font is gratefully acknowledged.
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