Author’s Accepted Manuscript DIRECT MEASUREMENT OF BOTTOM SHEAR STRESS UNDER HIGH-VELOCITY FLOW CONDITIONS Jae Hyeon Park, Young Do Kim, Yong Sung Park, Jae An Jo, Kimchhun Kang www.elsevier.com/locate/flowmeasinst
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S0955-5986(15)30049-2 http://dx.doi.org/10.1016/j.flowmeasinst.2015.12.008 JFMI1161
To appear in: Flow Measurement and Instrumentation Received date: 24 November 2014 Revised date: 2 September 2015 Accepted date: 29 December 2015 Cite this article as: Jae Hyeon Park, Young Do Kim, Yong Sung Park, Jae An Jo and Kimchhun Kang, DIRECT MEASUREMENT OF BOTTOM SHEAR STRESS UNDER HIGH-VELOCITY FLOW CONDITIONS, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2015.12.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
DIRECT MEASUREMENT OF BOTTOM SHEAR STRESS UNDER HIGH-VELOCITY FLOW CONDITIONS Jae Hyeon Park1, Young Do Kim2*, Yong Sung Park3, Jae An Jo2, Kimchhun Kang1 1
Dept. of Civil Engrg., Inje Univ., 197 Inje-ro, Gimhae, Gyeongnam, 621-749, South Korea. 2 Dept. of Environ. Sci. Engrg., Inje Univ., 197 Inje-ro, Gimhae, Gyeongnam, 621-749, South Korea. 3 Div. of Civil Engrg., Univ. of Dundee, Perth Rd., Dundee DD5 3LE, UK. * Corresponding author, email:
[email protected]
Abstract The objectives of the present research are to accurately measure bottom shear stress under high-velocity flow conditions. To achieve high-velocity flow conditions, a laboratory-scale flume has been specially built in which flow velocity can reach over 3 m s-1. Also an instrument that can directly measure bottom shear stress has been developed and validated. Then, the flow resistance has been estimated by simultaneously measuring flow velocity and bottom shear stress. It appears that the shear stress is indeed proportional to velocity squared and also to Reynolds number. On the other hand, Manning’s n value and the skin friction factor are more or less uniform across all experimental cases. Keywords: high-velocity flow, direct bottom shear stress measurement, flow resistance 1. Introduction Mean velocity and tractive stress are the two most important flow characteristics in evaluating and designing river bank stability measures (see e.g. Morris et al. 2007 and USDA SCS 1977). In steady uniform flow, bottom shear stress, , averaged over channel surface area is often estimated from momentum balance (e.g. Hogarth et al. 2005):
gRS ,
(1)
where is the density of water, g the gravitational acceleration, R the hydraulic radius of the channel cross-section, and S the energy slope, which is the same as the bottom slope as well as the water surface slope for uniform flow. However, the structure of overbank flow in a compound channel is very complicated due to both vertical and streamwise vortices and the associated secondary flows (McEwan and Ikeda 2009). With the additional effects of meandering river channel and the uneven channel surface roughness, spatial and temporal distributions of shear stress could be significantly different from that estimated from equation (1). This calls for a means to
accurately measure shear stress locally. In literature, there are mainly four kinds of approaches in measuring the bottom shear stress: (i) direct force measurement on the bottom (e.g. Ippen and Mitchell 1957, Barnes et al. 2009, Tinoco and Cowen 2013, and Pujara and Liu 2014); (ii) estimating from the measured near-bed turbulence (e.g. Biron et al. 2004); (iii) estimating from scalar diffusion at the bottom, such as thermal diffusion (e.g. Sumer et al. 2010) or electrodiffusion (e.g. Sobolik et al. 1991); and (iv) estimating from the measured velocity profile near the bottom (e.g. Cox et al. 1996 and Cowen et al. 2003). It is possible to obtain reasonable result using the last kind of method, especially in laminar flow (e.g. Liu et al. 2007) through high-resolution velocity measurement, but it is usually very difficult to resolve the viscous sub-layer in high-Reynolds-number turbulent flow. In a well-developed turbulent flow on a flat boundary, the velocity profile near the bottom may be fitted with the law of the wall to determine the friction velocity and thereby the bottom shear stress, although there are some uncertainties involved in the fitting (Biron et al. 2004). Furthermore, it is known that the logarithmic velocity law is no longer valid in the vicinity of the river channel bank (McEwan and Ikeda 2009), whereas it is indeed the region of our interest. Also, methods using scalar diffusion often require a separate velocity measurement close to the probe to resolve the directional information. Therefore in the present research, we only consider the first two methods and provide further reviews on the two methods below. To our best knowledge Ippen and Mitchell (1957) are the first to directly measure bottom shear stress under surface waves. Their shear test plate was made of 1 ft2 aluminum plate which was flush mounted on the bottom of the wave tank. The plate was connected to a force gage by a 3/4 inch diameter round steel rod, which was again enclosed by 1 inch diameter Lucite tube that was fixed separately from the force measurement system. An aluminum container was placed beneath the test plate, which was filled with mercury up to the underside of the test plate to prevent wave-induced flow under the plate. The pressure force acting over the thickness of the plate was estimated by repeating the same experiments with different thicknesses of the plate (1/8 and 3/8 inch, respectively). Finally shear force was obtained by subtracting the pressure force from the total measured force. Overall their experimental result showed good agreement with the available theory, but the pressure force was more than 50% of the total force, reducing the sensitivity in the shear force measurement. Also the instrumentation is rather intrusive because of the steel rod and its Lucite casing penetrating through the depth of water. Recently, Barnes et al. (2009) used a shear plate (0.1 m long, 0.25 m wide, 0.73 mm thick) in their bore-driven swash experiments. The whole instrumentation system is contained in a Perspex cell, which is flush mounted on the beach surface making it practically non-intrusive. The test plate was supported by six ball bearing rollers to prevent any displacement normal to the beach, while allowing tangential displacement. At the same time four sway legs that were clamped to the underside of the plate and extended to the base of the cell provided stiffness to limit the tangential displacement less than 1 mm. The displacement was measured using an eddy current probe, from which the
total force was estimated. Pressure force was estimated by the two pressure sensors on both sides of the cell. Due to the small side area of the test plate, the estimated pressure force was typically an order-of-magnitude less than the total force in their experimental conditions. The cell was completely filled with water, and it is maintained that the surface tension in the 1 mm gap between the test plate and the Perspex cell could hold the water even when the test plate was in dry condition. Also the authors stated that the induced flow inside the cell was minimal, but no detailed information was provided. Further consideration on the effect of pressure-gradient force in the use of a shear plate under a transient flow is discussed in Pujara and Liu (2014). They suggest that a constant value could be used for the fraction of the streamwise pressure gradient that acts on the shear plate even for unsteady flow conditions. Their shear plate is validated for turbulent flat boundary layer, in which the pressure-gradient force is negligible, and for surface solitary wave, where the pressure-gradient force could be dominant especially when the crest of the wave passes the location of the instrument. Overall the performance of the methodology was encouraging, although discrepancies were noticed when the flow was highly unsteady and the pressure gradient changed sign. Biron et al. (2004) compares different ways of estimating bottom shear stress from velocity and turbulence measurements both in simple and complex flows. Other than using the reach-averaged bottom shear stress (i.e. equation 1) or the law of the wall, they used Reynolds shear stress and turbulent kinetic energy to indirectly determine the bottom shear stress. The present research is motivated by the practical need to assess the safety and the stability of river embankment revetments using laboratory facility. Especially we are interested in high-water revetments under flooding conditions, which are characterized by high-velocity flow. To meet the requirements, we have specially built a laboratory-scale water flume, in which high-velocity flow conditions can be achieved. The key component of the flume is the pressurizing chamber at the upstream end where the water is pressurized and fed into the straight channel essentially as a wall jet to create high-velocity flow. The maximum flow velocity can be as high as 2.7 m s-1 without slope and 3.5 m s-1 with slope in typical water depth of order 0.1 m. Furthermore, an instrument, so-called the shear plate, has been built which can be flush mounted to the channel bed and directly measure the bottom shear stress. The design closely follows the one described in Barnes et al. (2009), except that the dimension of our shear plate is comparable to a typical river revetment to meet the purpose of our research. Our instrument is validated against the indirect methods described in Biron et al. (2004) using the turbulence data in a simple boundary-layer flow. We emphsize that the flow velocities used in our experiments is much higher than those found in literature. Most previous works have been done with velocity less than 1 m s-1. After validating the instrument, we report another set of experimental results in which we estimate the flow resistance from the shear stress and velocity data. It appears that the shear stress is indeed proportional to velocity squared and also to Reynolds number. On the other hand, Manning’s n value and the skin friction factor are more or less uniform across all
experimental cases. In the next section, we will describe the above-mentioned facility in detail. In section 3, we report two sets of experimental data, which are used for validation of the experimental setup and for accurate estimation of flow resistance, respectively, as well as further discussion. Finally, concluding remarks are given in section 4. 2. Methods 2.1 Generation of high-velocity flows in a laboratory flume A flume has been built at the Environmental Water Resources Laboratory in the Department of Environmental Science and Engineering at Inje University, South Korea. The water flume is made of smooth acrylic sheets (20 mm thick) and supported by protruded aluminum frame, measuring 6 m long, 0.3 m wide and 0.3 m deep. The slope of the flume can be adjusted to any angle between -5o and 5o (see figure 1). Figure 1. Schematic diagrams of the flume used to generate high-velocity flow conditions. The water flow is supplied by a submerged water pump (power: 30 hp; maximum capacity: 0.2 m3 s-1) through 5 hoses. Each hose is fitted with an adjustable valve and the flow rate is controlled by adjusting opening of the valves. The feed water first enters into the pressurizing chamber before flowing into the working section of the flume so that a very high flow velocity (2.7 m s-1 without slope and 3.5 m s-1 with slope in typical water depth of order 0.1 m) can be achieved. Between the pressurizing chamber and the working section of the flume, there is a sluice gate. Opening of the gate can also be adjusted so that the flow velocity as well as the flow depth can be further controlled. The flow rate is measured by an ultrasonic flowmeter (Ulsoflow 309P) and it essentially varies linearly with the number of open valves:
Q = 0.013Nv ,
(2)
where Q is the flow rate in m3 s-1, and N v is the number of open valves. Note that each valve can be partially open and N v can be any number between 0 and 5, not necessarily just positive integers. Now, given the water depth ( h ), we can estimate the Froude number ( Fr ) of the flow as a function of N v :
Q 1 0.013N v , (3) = Wh gh W gh 3 where W = 0.3 m is the width of the channel. For h = 0.1 m, equation (3) reduces to Fr = 0.44N v , therefore the flow is supercritical for N v 2 .3 . Under supercritical condition, we observed that the water depth does not vary noticeably within the 5 m reach Fr =
of the channel. The Reynolds number ( Re h ) based on the water depth can also be estimated:
Re h
Q h 4 .3 10 4 N v , Wh
(4)
where, = 10 6 m2 s-1 is the kinematic viscosity of water. We emphasize here again that we are interested in the high-water revetments which are closer to water surface. The local water depth near the revetments is indeed comparable to that of our channel. With the flow velocity comparable to or even higher than those in the flooding river, we can therefore test the prototype high-water revetments under comparable Reynolds number (based on the local water depth near the high-water revetments) in our laboratory-scale flume.
2.2 The shear plate 2.2.1 Design of the shear plate In figure 2, we present the design of the instrument that directly measures bottom shear stress, namely the shear plate. Figure 2. Design of the shear plate. (a) plan view; (b) side view; and (c) front view. The fundamental idea is to measure the shear force exerted on the rectangular acrylic plate (94 mm long in the streamwise direction, 144 mm wide in the lateral direction and 10 mm thick). The top plate is supported by four aluminum vertical plates anchored at the bottom of the instrument. Essentially the shear force is balanced by the elastic restoring force of the four legs, which is again estimated by the measured displacement and the independently determined stiffness. The displacement of the top plate is measured by an eddy current displacement sensor (Lion Precision, model no. U8) with the resolution of 0.4 m at 15 kHz. The above-mentioned system is enclosed in an acrylic box (0.2 m by 0.2 m by 0.2 m). The top plate is flush mounted with the acrylic enclosure and there is 1 mm gap between the top plate and the acrylic box to allow for displacement when subject to shear force. The acrylic enclosure was completely filled with water when it was installed in the high-speed flume and visual inspection showed no significant flow within the cell. 2.2.2 Principle of the shear plate Denoting the displacement of the top plate as s , which is measured as a function of time, the force balance on the top plate can be written as
P m s ks A V , x
(5)
where upper dot denotes time differentiation, m is the mass of the top plate, k the lumped stiffness of the four sway legs, A the upper surface area of the top plate, V the volume of the top plate and P the pressure on the sides of the top plate. Under steady flow, the left-hand side of equation (5) vanishes. We remark here that due to turbulence, however, the flow cannot be strictly steady. On the other hand, measuring acceleration requires very high temporal resolution and assessment of the resulting uncertainty is left for future work. If the third term in the right-hand side of equation (5) is also negligible, the shear stress is a simple linear function of the displacement. The relative magnitude of the pressure gradient force with respect to the shear force can be determined if we assume uniform flow at least locally. First of all, the water surface slope can be deduced from Manning's equation:
S = n 2U 2 R-4/3,
(6)
where n is Manning's roughness coefficient and U the velocity averaged over the channel cross-section. Substituting equation (6) into equation (1), the shear stress can be estimated, that is,
gn 2U 2R
1/ 3
.
(7)
On the other hand, the pressure gradient can also be estimated using S :
P gS . x
(8)
The ratio of the pressure gradient force to the shear force is, therefore,
P
x V
A
gSV T , gRSA R
(9)
where T =10 mm is the thickness of the acrylic top plate of the instrument. For typical water depth h = 0.1 m, the hydraulic radius is calculated as R = 60 mm. With the relatively large thickness of the top plate, equation (9) implies that the pressure gradient force amounts to as much as 17% of the shear force. However, this is very likely overestimated. Experimental study by Acharya et al. (1985) shows that the pressure along the thickness of the top plate is not uniform but linearly decreases to zero near the bottom end of the plate reducing the pressure gradient force by a factor of two. Thus the uncertainty in measurement of shear force due to the pressure gradient force should be less than 10%. We remark here that although the accuracy of the shear plate could be markedly improved by reducing the thickness of the top plate, we chose T =10 mm to ensure that a river revetment block can be stably mounted on the top plate for later use. When the revetment block is mounted, the pressure gradient force on the block will be partially responsible for form drag which is in fact a kind of tractive force we are
interested in. Thus the adverse effect of the pressure gradient force on the sides of the top plate will be even less significant. With the aforementioned uncertainty due to the pressure gradient force in mind, finally the shear stress averaged over the horizontal area of the top plate can be written as
ks A
.
(10)
The lumped stiffness k was determined from static calibration, in which hanging weights are connected to the top plate horizontally by a string over a pulley. Repeated measurements yielded k =11.54 N mm-1. The weight and the displacement showed almost perfect linear relationship up to 20 N with the coefficient of determination in the linear fit being R2 = 0.999 (see figure 3). Figure 3. Calibration of the shear plate: circles are the measured values and the line is the fitted linear line. Note that the calibration was performed in the air, and without significant flow within the chamber of the shear plate, the same value was used in the experiments under water. 3. Results and discussion 3.1 Validation of the shear plate The shear plate described in the previous section has been tested in the high-speed water flume. The validation procedure involves independent estimation of the bottom shear stress using the concurrent near-bed velocity measurements. Two such methods were used, which are described in Biron et al. (2004), namely using the Reynolds shear stress and the turbulent kinetic energy, respectively. The near-bed value of Reynolds stress is taken as the bottom shear stress under turbulent flow and it is defined as
RS u w
(11)
where RS is the bottom shear stress estimated from the near-bed value for Reynolds shear stress, u and w are velocity fluctuations in the streamwise and vertical directions, respectively, and denotes ensemble averaging. In an essentially steady flow, ensemble averaging can be replaced by time-averaging, which is what we adapted here. The bottom shear stress can also be estimated by multiplying a factor to the turbulent kinetic energy measured at a near-bed location:
TKE
1 C 1 u 2 v 2 w 2 , 2
(12)
in which TKE is the bottom shear stress estimated from the near-bed value for the turbulent kinetic energy, v is the lateral velocity fluctuation and C1 = 0.19 is the proportionality constant. The shear plate was flush mounted on the bottom of the flume at 1.5 m downstream from the sluice gate. The three-dimensional velocity field was also measured by an acoustic Doppler velocimetry (ADV, Nortek Vectrino) 2 mm above the center of the top plate of the shear plate. For each measurement, the velocity was measured for 150 seconds at the sampling rate of 200 Hz. By adjusting the velocity range and the size of the sampling volume, we made sure that the signal-to-noise ratio (SNR) values are consistently above 15 dB and the correlation coefficients are at least 70%. The flow depth was measured at the testing point by using a standard point gage with the precision of 0.1 mm. Four experimental cases were tested in which the velocity ranges from 1 to 1.6 m s-1. For each case, measurements were repeated three times to estimate the degree of uncertainties. Experimental cases and their results are summarized in table 1. Table 1. Experimental cases for validation, in which the bottom shear stress measured by the shear plate ( SP ) is compared with ones estimated from the Reynolds shear stress ( RS ) and the turbulent kinetic energy ( TKE ), respectively. The values of the Bottom shear stress are averaged over three repetitions and those in the parentheses are the standard deviations of each of the cases. Also note that u is the time-averaged velocity in the streamwise direction.
Within the limited number of ensembles, one can see from table 1 that all three methods yield repeatable results with the standard deviations being more or less 10% of the mean values. SP and TKE show very close results with the relative error ranging from 0.9 to 2.5%, which is very encouraging. On the other hand, the relative error between SP and RS are much larger (16 ~ 33%). Biron et al. (2004) mentioned that the Reynolds shear stress method is very sensitive to any deviation from the two-dimensionality of the flow and to a slight misalignment of the velocity probe. Considering the uncertainty that is associated with the pressure-gradient force in the shear plate as described earlier and the uncertainty in the Reynolds shear stress method, one may agree that the three methods are consistent to one another. Figure 4 illustrates this point, in which the results from the three methods are plotted as a function of Froude number. Figure 4. Validation of the shear plate. Circles: SP ; Triangles: RS ; Rectangles: TKE . Error bars are twice the standard deviations, respectively.
The error bars in the figure represent twice the standard deviations. SP and TKE are very close to each other, while RS are consistently smaller. Nevertheless we see overlaps in the error bars from all the cases and we conclude that the shear plate yields reasonable results that are consistent to other indirect methods estimated from the velocity data.
3.2 Determination of flow resistance using the shear plate Now we report an example application of the shear plate in practice. With simultaneous measurement of flow velocity and shear stress, resistance to flow can be assessed and quantified in terms of Manning's n or the skin friction factor ( C f ), namely
n
gU 2 R
1/ 3
,
(13)
and
Cf
0 .5 U
2
.
(14)
Using the same experimental setup described earlier, we carried out 18 more cases. We remark here that we could not use the point gauge for this set of experiments. The flow velocity was often larger than 2 m s-1 and the violent fluctuation of the free surface made it difficult to use the point gauge. Instead we attached a tape ruler along the sidewall to measure the water depth with the accuracy of 0.5 cm. Experimental cases are summarized in table 2. Table 2. Experimental cases In the table, the cross-sectional average velocity (U) was obtained from the flow rate and the water depth, and the Froude number and the Reynolds number of each case were calculated using equations (3) and (4), respectively. The shear stress was measured using the shear plate at the frequency of 1 Hz for 100 seconds. In figure 5, we present the measured bottom shear stress as a function of (a) velocity, (b) Reynolds number and (c) Froude number. Figure 5. The bottom shear stress as a function of (a) velocity, (b) Reynolds number, and (c) Froude number. In figure 5(a), it seems clear that the bottom shear stress is proportional to the square of velocity. Indeed, least-square fitting yields the proportionally coefficient of 0.2707 Pa(m
s-1)-1 with the coefficient of determination to be 0.96. It is also observed in figure 5(b) that the bottom shear stress is more or less linear with the Reynolds number with the coefficient of determination to be 0.86. No obvious functional relationship between the bottom shear stress and the Froude number is found in figure 5(c). On the other hand, one can see that the flow resistance in terms of Manning's n and the skin friction factor is fairly uniform across all the experimental cases. The mean values are n = 0.0032 and C f = 0.0006 with the standard deviations to be about 10% of the means, respectively. With the smooth acrylic plate, however, typical values for the flow resistance parameters in wide channel are n = 0.008 (Chow1959) and C f = 0.0025 (Yen 2002), which are greater than the experimental values by factors of 3 to 4. Yen (2002) explains that the flow resistance parameters decrease with decreasing channel width to depth ratio compared to the 2D wide channel, which accounts for factor of 2. The rest of the deviation may be due to the uncertainties associated with the water depth measurement as well as the pressure-gradient force. The latter has been discussed in the previous section and further improvement in water depth measurement is being implemented. Within the current experimental uncertainty, the results are reasonable. 4. Concluding remarks We have presented our experimental facility that can be used to test the stability of high-water revetments in laboratory. The newly-built flume can achieve the Reynolds number that is comparable to those in flooding rivers when it is based on the local water depth near the revetment blocks. To measure reliably the tractive force acting on the revetment under complicated flow conditions in flooding rivers, we have also developed an instrument that can be flush mounted to the channel bottom on which a revetment block can be mounted. Even with the uncertainties associated with the neglected acceleration term and the pressure gradient force term, our validation experiments demonstrate that the instrument can measure bottom shear stress reliably. Encouraged by the positive results, now more experiments with revetment blocks mounted under way, which will be shortly reported in a separate paper.
Acknowledgements This research was supported by a grant (12-TI-C02) from Construction Technology Innovation Program funded by Ministry of Land, Infrastructure and Transport of Korean government (MLIT, KAIA). YSP also acknowledges the financial support from the Royal Society of Edinburgh/Scottish Government Personal Research Fellowship Co-Funded by the Marie-Curie Actions.
References Acharya, M., Bornstein, J., Escudier, M. P., and Vokurka, V. (1985). Development of a floating element for the measurement of surface shear stress. AIAA J., 23(3), 410-415.
Barnes, M. P., O'Donoghue, T., Alsina, J. M., and Baldock, T. E. (2009). Direct bottom shear stress measurements in bore-driven swash. Coast. Eng., 56, 853-867. Biron, P. M., Robson, C., Lapointe, M. F., and Gaskin, S. J. (2004). Comparing different methods of bottom shear stress estimates in simple and complex flow fields. Earth Surf. Process. Landforms, 29, 1403-1415. Chow, V. T. (1959). Open-Channel Hydraulics. McGraw-Hill, Singapore. Cowen, E. A., Sou, I. M., Liu, P. L.-F., and Raubenheimer, R. (2003). Particle image velocimetry measurements within a laboratory-generated swash zone. J. Eng. Mech., 129, 1119-1129. Cox, D. T., Kobayashi, N., and Okayasu, A. (1996). Bottom shear stress in the surf zone. J. Geophys. Res., 101, 14337-14348. Hogarth, W. L., Parlange, J.-Y., Rose, C. W., Fuentes, C., Haverkamp, R., and Walter, M. T. (2005). Interpolation between Darcy-Weisbach and Darcy for laminar and turbulent flows. Adv. Water Resour, 28, 1028-1031. Ippen, A. T. and Mitchell, M. M. (1957). The damping of the solitary wave from boundary shear measurements. Tech. Report No. 23, Hydrodynamics Laboratory, MIT, Boston, M.A. Liu, P. L.-F., Park, Y. S., and Cowen, E. A. (2007). Boundary layer flow and bottom shear stress under a solitary wave.'' J. Fluid Mech., 574, 449-463. McEwan, I.~K. and Ikeda, S. (2009). Flow and Sediment Transport in Compound Channels: The Experiences of Japanese and UK Research. IAHR International Association of Hydraulic Engineering and Research, Madrid, Spain. Morris, M., Dyer, M., and Smith, P. (2007). Management of flood embankments: A good practice review. R&D Technical Report No. FD2411/TR1, Joint Defra/EA Flood and Coastal Erosion Risk Management R&D Programme, London, U.K. Pujara, N. and Liu, P. L.-F. (2014). Direct measurements of local bottom shear stress in the presence of pressure gradients. Exp. Fluids, 55, 1767. Sobolik, V., Pauli, J., and Onken, U. (1991). Three-segment electrodiffusion probe for velocity measurement. Exp. Fluids, 11, 186-190. Sumer, B. M., Jensen, P. M., Sørensen, L. B., Fredsøe, J., Liu, P. L.-F., and Carstensen, S. (2010). Coherent structures in wave boundary layers. Part 2. Solitary motion. J. Fluid Mech., 646, 207-231.
Tinoco, R. O. and Cowen, E.~A. (2013). The direct and indirect measurement of boundary stress and drag on individual and complex arrays of element. Exp. Fluids, 54, 1509. USDA Soil Conservation Service (1977). Design of open channels. Technical Release No. 25, Washington, D.C. Yen, B. C. (2002). Open channel flow resistance. J. Hydraul. Eng., 128, 20-39.
Table 1. Experimental cases for validation, in which the bottom shear stress measured by the shear plate ( SP ) is compared with ones estimated from the Reynolds shear stress ( RS ) and the turbulent kinetic energy ( TKE ), respectively. The values of the Bottom shear stress are averaged over three repetitions and those in the parentheses are the standard deviations of each of the cases. Also note that u is the time-averaged velocity in the streamwise direction. Case No.
Q (l s-1)
h (mm)
Re
Fr
1
24.50
6.8
8167
4.65
2
23.75
6.3
8165
3
31.40
7.2
4
39.50
7.8
(m s-1)
Bottom shear stress (N m-2)
SP
RS
TKE
1.183
2.563 (0.330)
1.868 (0.163)
2.540 (0.145)
5.21
1.277
2.780 (0.259)
2.053 (0.206)
2.848 (0.218)
10512
5.49
1.458
3.985 (0.332)
2.663 (0.390)
3.893 (0.446)
13221
6.13
1.577
6.561 (0.521)
5.484 (0.490)
6.414 (0.502)
Table 2. Experimental cases h (mm) 10 10 15 40 40 45 45 45 50 50 50 50 60 60 60 70 70 70
U (m s-1) 0.91 1.13 0.83 1.49 1.51 1.32 1.41 1.50 1.78 1.80 1.89 2.63 1.70 2.20 2.72 1.85 2.28 2.54
R (mm) 9.4 9.4 13.6 31.6 31.6 34.6 34.6 34.6 37.5 37.5 37.5 37.5 42.9 42.9 42.9 47.7 47.7 47.7
Fr
Re 10 4
n
Cf
2.9054 3.6078 2.1637 2.3786 2.4105 1.9867 2.1222 2.2576 2.5416 2.5701 2.6986 3.7552 2.2158 2.8676 3.5453 2.2325 2.7514 3.0651
0.91 1.13 1.25 5.96 6.04 5.94 6.35 6.75 8.90 9.00 9.45 13.15 10.20 13.20 16.32 12.95 15.96 17.78
(Pa) 0.3097 0.4554 0.3313 0.5988 0.7014 0.6027 0.4157 0.5734 0.8058 0.9365 1.0464 1.6949 0.7400 1.2183 1.8983 1.1175 15304 1.8326
0.0029 0.0028 0.0035 0.0031 0.0033 0.0035 0.0028 0.0030 0.0031 0.0033 0.0033 0.0030 0.0032 0.0032 0.0032 0.0037 0.0035 0.0035
0.0007 0.0007 0.0010 0.0005 0.0006 0.0007 0.0004 0.0005 0.0005 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005 0.0007 0.0006 0.0006
Figure 1. Schematic diagrams of the flume used to generate high-velocity flow conditions.
Figure 2. Design of the shear plate. (a) plan view; (b) side view; and (c) front view.
Figure 3. Calibration of the shear plate: circles are the measured values and the line is the fitted linear line.
Figure 4. Validation of the shear plate. Circles: SP ; Triangles: RS ; Rectangles: TKE . Error bars are twice the standard deviations, respectively.
Figure 5. The bottom shear stress as a function of (a) velocity, (b) Reynolds number, and (c) Froude number.
* We directly measured the bottom shear stress under high-velocity flow conditions. * We built a laboratory-scale flume to achieve high-velocity flow conditions. * We estimated the flow resistance by simultaneously measuring flow velocity and bottom shear stress.