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High Reynolds number flow inside a channel of rectangular cross section is examined using Particle Image Velocimetry. One wall of the channel has been replaced with a surface of a roughness representative to that of real hydropower tunnels,
* i.e. *a random terrain with roughness dimensions typically in the range of ≈10% - 20% of the channels hydraulic radius. The rest of the channel walls can be considered smooth. The rough surface was captured from an existing blasted rock tunnel using high resolution laser scanning and scaled to 1:10. For quantification of the size of the largest flow structures, integral length scales are derived from the auto-correlation functions of the temporally averaged velocity. Additionally, Proper Orthogonal Decomposition (POD) and higher-order statistics are applied to the instantaneous snapshots of the velocity fluctuations. The results show a high spatial heterogeneity of the velocity and other flow characteristics in vicinity of the rough surface, putting outer similarity treatment into jeopardy. Roughness effects are not confined to the vicinity of the rough surface but can be seen in the outer flow throughout the channel, indicating a different behavior than postulated by Townsend’s similarity hypothesis. The effects on the flow structures vary depending on the shape and size of the roughness elements leading to a high spatial dependence of the flow above the rough surface. Hence, any spatial averaging, e.g. assuming a characteristic sand grain roughness factor, for determining local flow parameters becomes less applicable in this case.

Water tunnels are frequently used to convey water to and from hydropower turbines and in other sectors of infrastructure. The tunnels are often a key part of the design and their durability is vital for the continued operation. Tunnel excavation by rock-blasting is a relatively swift, and therefore popular, method compared to using tunnel boring machines [

It is well established that rough walls modify the behavior of the flow [

The experimental setup consisted of a closed loop water system with a 10 m long rectangular Plexiglass (PMMA) channel having one rough surface, a pump, an electromagnetic flow meter, two tanks placed on different levels and a PIV-system. The function of the tank placed upstream of the channel is to provide a constant head on the system and to avoid air entrainment inside the channel. A schematic of the experimental setup can be seen in

A detailed description of the setup can be found in [

section of the rough surface over which the flow was measured during the experiments. One measuring plane was placed in the center of the channel (denoted middle), while another was placed closer to the camera (denoted upper). The color of the surface represents the slope of the surface topography and the two colored lines mark the position of the two measured planes. As can be seen in the figure, a ridge is passing through both measuring planes at x ≈ 7.06 m. The ridge is of interest for the measurements since the maximum height relative to y = 0 (the mean height) is similar for the two lines, additionally, the gradient of the surface is also nearly the same. However, the relative size of the ridge differs significantly between the lines. For the blue line defining the middle plane, the final slope of the ridge is very sharp but the roughness leading up to the element is relatively small, consequently, the relative size of the roughness element is small. On the red line (upper plane), the ridge is preceded by a “valley”, making the relative height larger. Ideally, four rough walls would have been used to keep the setup as realistic as possible. However, PIV requires optical access from at least 2 directions and therefore only one rough wall was used.

In this study, a right-handed coordinate system is employed with the x-coordinate (u-velocity component) originating from the tunnel entrance pointing in the flow direction. The y-coordinate (v-velocity component) is perpendicular to the rough surface with y = 0 defined as the average elevation of the rough surface, hence all presented heights are relative to the average height of the surface. Accordingly, the z-coordinate originates from the bottom wall in the flow direction.

The rough surface model used in the experiments is a 1:10 scale side wall of an existing rock tunnel whose topography has been captured by high resolution laser scanning, a method which has been proven efficient for determining surface roughness [

k s 2 = ∫ − ∞ + ∞ h 2 p ( h ) d h . (1)

As mentioned, k s is solely based on the height on the rough surface and does not take into consideration e.g. shape or aspect ratio of the roughness. To evaluate the spatial difference, the auto-correlation function R(r) over a specified length L is introduced in the stream wise direction (x-direction) of the rough surface [

Integrating the auto-correlation function according to Equation (2) produces the integral length scale of the surface

τ r ≡ ∫ 0 ∞ R ( x ) d x . (2)

Conclusively, k s is a quantity representative for the roughness height while

τ r represent the roughness length of the surface. Using Equation 1, k s is determined to 9.4 mm, while τ r is found to be 39.1 mm using Equation 2. Hence, k s is about 6.4% of the hydraulic radius, which can be compared to the largest global roughness elements being about 20% of the hydraulic radius. The difference is a clear indicator of the spatial heterogeneity of the rough surface used in this study. For additional numerical comparison, the relative height of the ridge in the middle plane is about 9.44 mm, which is very close to k s . One can hereby conclude that the ridge studied is a fitting representation of the roughness of the entire surface. Additionally, the ridge in question is far from unique on the surface but appears at regular intervals, therefore, the sample size is deemed large enough to be spatially independent.

The PIV-system used is a commercially available system from LaVision GmbH which has been applied in a number of studies, including [^{3}. The measurements were performed with a frequency of 75 Hz during 9.49 s, corresponding to a total of 712 image pairs for each recorded set. To account for the localized variations in velocity and pixel displacement, the time interval between the laser pulses ranged from 150 μs - 275 μs depending on the measuring position. The results were a typical mean displacement over the whole velocity field of 0.3 pixels in the y-direction and 7 pixels in the x-direction with a characteristic particle image diameter of 2 pixels. At approximately 490 samples the temporally averaged velocity converges towards a stable value. Beyond 560 samples the velocity differ no more than 1.5% from the converged value. A PIV experimental setup consists of several sub systems, and hence there are a number of potential error sources. The overall measurement accuracy in PIV is a combination of a variety of aspects extending from the recording process all the way to the methods of evaluation [

Post-processing of the PIV-data was done using the commercial software DaVis by LaVision [

K v i = λ i v i (3)

of the auto covariance matrix K, given by

K = U T U . (4)

i is the total number of eigenvalues, U is a matrix where the columns consist of the instantaneous fluctuating velocity snapshots, according to

U = [ | | ⋮ | u ′ 0 u ′ 1 ⋮ u ′ N t | | ⋮ | ] , (5)

where u ′ j = u j − u ¯ for j = { 1 , 2... N t } . N t is the number of snapshots, 712 in the current case. The modes are then calculated and normalized by

ϕ i = ∑ n = 1 N t v n i u n ‖ ∑ n = 1 N t v n i u n ‖ . (6)

The method stems from [

The middle plane is placed at z = 125 mm, and is represented by a diamond symbol in the figures. The upper plane was placed at z = 165 mm and will be represented by an x symbol in the figures. To avoid cluttering only a portion of the data have been plotted, typically every fourth point. This does not affect the results and is solely for the purpose of making the data easier to distinguish. The velocity components of the flow (u, v) are denoted as the vector u. To evaluate the flow the u- and v-components of the velocity were averaged over time for one measurement (see Sec. 2.) to produce the temporally averaged velocity components u ¯ . Some of the results are then spatially averaged in the streamwise direction, denoted by 〈 u ¯ 〉 . In the first section below, temporally averaged and Quadrant analysis of the velocity to discern the spatial heterogeneity of the flow are presented. In the second section, integral length scales applied to the temporally averaged velocity are discussed and in the third section POD is applied on the instantaneous velocity field. The instantaneous contribution to the Reynolds stresses are calculated according to

u ¯ v ¯ Q = 1 T ∑ i = 1 n S ( u i − u ¯ ) ( v i − v ¯ ) , (7)

where S is a sorting term. If u ¯ v ¯ falls into quadrant Q then S = 1, otherwise S = 0. T is the total measuring time for each sample. An introduction to Quadrant analysis can be found in [

The average bulk velocity in the channel is U_{0} = 1.562 m/s. A zone of negative velocity (recirculation zone) is present behind the crest of the ridge, an effect similarly visualized by [

The asymmetric channel flow case (one rough wall opposite of a smooth one) has been well documented by [

For the current case, the maximum velocity u ¯ max = 1.45 U 0 is shifted towards the rough surface ( y / k s = 4.1 ) (see

H = u ′ v ′ u ′ ¯ v ′ ¯ . (8)

In short, the contribution to the Reynolds stresses is divided into one of four possible quadrants depending on the sign of the instantaneous velocity fluctuations. As H increases, low magnitudes of the instantaneous velocity products are sorted out, thus only the significant contributions are left for comparison. Quadrant 2 events generally, but not always, represent ejection and similarly, quadrant 4 events represent sweeps [^{2}/s^{2} respectively. Point b) show a similar magnitude for Q2 and Q4 for H = 0 , however, the most

dominant event for H > 2 is Q1. Point c), which is placed in the separation zone, show an overall dominance of Q4 events for all H. This is an indicator of sweeps of high velocity fluid moving towards the wall. Points c) and e) are placed at similar heights leeward of the roughness element, thus, displaying similar behavior. The main difference being the displayed u ¯ v ¯ magnitudes which have dissipated significantly by point e). Presumably, if no other roughness element would occur downstream, Q2 events would again become dominant as similarly theorized by [^{2}/s^{2}. For larger H Q2 events become dominant. One possible reason for this might be traces of decelerating high velocity flow from the ridge still present where the point is located.

To characterize the size of the flow structures above the rough surface a correlation length approach is utilized. The streamwise spatial velocity correlation is calculated by [

R u = 1 u ¯ ˜ 2 ( x 2 − x 1 ) ∫ x 1 x 2 u ˜ ( x , y ) u ˜ ( x + r , y ) d x (9)

where u = (u, v), r is the streamwise incremental coordinate and

u ˜ = u ¯ ( x , y ) − 1 x 2 − x 1 ∫ x 1 x 2 u ¯ ( x , y ) d x = u ¯ − 〈 u ¯ 〉 . (10)

Using Equation (2), a characteristic length scale can be derived for each velocity component. This operation continues from the crest (not y / k s = 0 ) to y / k s ≃ 8 , and an integral length scale is calculated for each acquired auto-correlation function, see

and are meant to represent a smooth wall case. These values are similar for both the upper and middle case, therefore only one the middle one is displayed.

Near the surface, the flow exhibit very different behavior between the two planes up until about half the channel ( y / k s ≃ 8 ) . The middle plane show a sharp increase in the length scale at y / k s ≃ 2.7 , about 0.37 k s above the crest of the roughness element with a magnitude of τ / τ r ≃ 1.6 . The peak, to a degree, indicates the shear layer forming between the bulk and recirculation zone following the roughness peak visible in

While the integral length scale was applied to the mean velocity, POD was applied to the instantaneous velocity fluctuations. The subsets from the middle plane could not be measured at the same time, hence, any perturbations in the flow cannot be tracked from one subset to another and thereby limiting the visualization of the data. Additionally, the temporal resolution made it difficult to capture sufficient snapshots of the same structure. However, POD is a statistical tool which provides an opportunity to visualize the distribution of energy within each subset, thereby avoiding the problem of synchronized pictures and large temporal resolution. As mentioned in Sec. 3.1, the effects on the mean velocity and higher-order statistics suggest vortex shedding behind the ridge in the

middle plane. To further investigate this, POD was applied to a subset downstream of the roughness element positioned close to the center of the middle field as defined by the cyan box in

The scale in

Results from PIV measurements of flow over a rough hydraulic surface are presented. The surface is produced from laser scanning an existing rock surface and the dimensions of the experimental tunnel are made to reflect real conditions for hydropower tunnels. A likely treatment of such surfaces in the industry is to assume uniform (and thereby small scale) roughness which, according to these results, would lead to erroneous estimations of flow parameters. The results include profiles of double averaged velocity, higher-order statistics, quadrant analysis, correlation length scales and POD. The presented measurements reveal a highly localized behavior of the flow connected to the rough surface. Even small deviations from the local mean height in the surface roughness produce perturbations in the flow which will be visible in the results. In contrast to classical results in asymmetric channel flow the maximum velocity is shifted towards the surface. This shows that the effects from the rough surface are large enough to manifest even in the double averaged velocity. It should be noted that the roughness element studied is not unique, but similar ones occur regularly on the rough surface. Therefore, using large spatial samples would produce the same distortions when averaging. The higher order statistics indicate that the flow above and behind the ridge is characterized by ejection and intermittent bursts of velocity, this is also where the highest point-contributions to the Reynolds stresses where recorded. Similarly, earlier measurements showed higher frequencies of fluctuating pressure at the same ridge. Research has shown that these are unfavorable conditions for rock surfaces from a durability point of view and may hasten or induce an eventual process of tunnel breakdown. As theorized, the problem of tunnel breakdown is likely connected to the flow-roughness effects. Evaluation of the correlation lengths of the flow reveals a significant difference between how the roughness elements interact with the flow. Both the integral length scales and POD approximately predicted the position of the largest flow structures formed in vicinity of the rough surface at y / k s ≃ 3 . Consequently, similar data can be obtained from both the time-averaged velocity and the instantaneous velocity fluctuations. The streamwise length scales of the flow holds close resemblance to the length scales of the rough surface, since τ / τ r ≃ 1 for y / k s > 7 . In contrast to Townsends’s similarity hypothesis where the effects of the rough surface is visible beyond the range of the rough surface, and throughout the channel. Similar effects have been shown in studies concerning flow over riverbeds, dunes or in rivers. It should be noted that such cases usually employ lower Re and larger roughness height to surface ratio and would not regularly be associated with the current application. It does, however, highlight that for hydropower applications, rough surfaces cannot be treated as uniform and only friction-inducing. This is particularly important when modelling flow using CFD, whose role has grown vastly in many industrial applications the past two decades. The results provided here show that the resolution of the roughness is very important, which could have large implications on the future evaluation of hydropower tunnels. A different proposed approach would involve a modification of the uniform wall functions currently used by today’s standards. A thorough mapping of the turbulent kinetic energy in correlation with the rough surface would yield the necessary data, as shown by the quadrant analysis in this study. Thereby the law of the wall could be modified with a spatially stochastic localized increase in wall-near velocity gradients. The roughness length-scales employed in this study might suffice for such an endeavoring as implied by the integral length-scales, however, different methods of deriving these should nevertheless be explored. One limitation of this study is the usage of only one rough wall. Hence, the flow in an actual hydropower tunnel cannot be claimed to be fully understood yet, albeit further understood. There has been no indication of eventual scaling effects between the experiments and the actual case. But if one would consider the problems within hydropower tunnels today and the agreement with other studies the authors assume that the scaling effects, if any, would be insignificant.

The research presented was carried out as a part of “Swedish Hydropower Centre-SVC”. SVC has been established by the Swedish Energy Agency, Energiforsk and Svenska Kraftnät together with Luleå University of Technology, KTH Royal Institute of Technology, Chalmers University of Technology and Uppsala University.

Andersson, L.R., Larsson, I.A.S., Hellström, J.G.I., Andreasson, P., Andersson, A.G. and Lundström, T.S. (2018) Characterization of Flow Structures Induced by Highly Rough Surface Using Particle Image Velocimetry, Proper Orthogonal Decomposition and Velocity Correlations. Engineering, 10, 399-416. https://doi.org/10.4236/eng.2018.107028