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Dye-tracer technique for rill flows by velocity profile measurements
Catena 185 (2020) 104313
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Dye-tracer technique for rill flows by velocity profile measurements a
a
b
a
Costanza Di Stefano , Alessio Nicosia , Vincenzo Palmeri , Vincenzo Pampalone , Vito Ferro a b
b,⁎
T
Department of Agricultural, Food and Forestry Sciences, University of Palermo, Viale delle Scienze, Building 4, 90128 Palermo, Italy Department of Earth and Marine Sciences, University of Palermo, Via Archirafi 20, 90123 Palermo, Italy
A R T I C LE I N FO
A B S T R A C T
Keywords: Soil erosion Interrill flows Rill flows Flow velocity Dye method Correction factor
Water flow on hillslope soil surface supplies energy which is required to detach soil particles, to transport and deposit sediments, therefore flow velocity is a key variable related to hillslope hydrodinamics of soil erosion processes. Among the different methods available for measuring velocity of shallow interrill and rill flow, the trace technique is widely used. Trace technique is applied by adding a material (salt, magnetic material, water isotope, floating object) and then measuring the speed of the material to travel a known distance from the injection point. When flow velocity is measured using a dye-tracing method, the mean velocity is calculated by multiplying the measured surface velocity of the leading edge of the tracer plume by a correction factor which was generally empirically deduced. The main uncertainty of the dye-tracing technique stands in the estimate of this correction factor, which is the ratio between the mean flow velocity and the surface velocity. The main goal of this paper is establishing a theoretical relationship for calculating the correction factor by using a power velocity profile. At first, the developed analysis demonstrated that the correction factor only depends on the exponent of the power velocity distribution (Eq. (5)). Then, this theoretical expression of the correction factor was applied using the velocity profiles measured by Baiamonte and Ferro for the condition of sediment-free flow in motion on a rough bed and by Coleman for a sediment-laden flow moving on a smooth flume. Using the correction factor values calculated by the velocity measurements carried out for the sediment-free flow in motion on a rough bed, the relationship between the correction factor and roughness height was established. In agreement with a previous study by Ali et al., for a sediment-free flow the correction factor increases with roughness height. Finally, the velocity profile measurements carried out for a sediment-laden flow in motion on a smooth bed were used to state the effect of sediment load on the correction factor. This last analysis allowed to conclude, in agreement with Li and Abrahams and Zhang et al., that the correction factor decreases when the sediment load increases.
1. Introduction Flow velocity is a key variable related to hillslope interrill and rill erosion processes. Flow velocity measurement is useful for improving its hydraulic knowledge, which is necessary for developing processbased soil erosion models (Takken et al., 1998). Several methods (hot anemometry (Ayala et al., 2000), particle imaging velocimetry (Raffel et al., 1998; Ali et al., 2012), Acoustic Doppler velocimetry (Giménez et al., 2004), infrared termography (de Lima et al., 2014), optical tacheometer (Dunkerley, 2003), dye tracer (Bradley et al., 2002; Lei et al., 2005; Zhang et al., 2010; Ban et al., 2016)) are used to measure velocity in shallow flow, such as in interrill and rill flows (Bruno et al., 2008; Bagarello et al., 2015; Di Stefano et al., 2015, 2017a; Foster et al.,
⁎
1984; Gilley et al., 1990; Govers, 1992; Govers et al., 2007; Line and Meyer, 1988; Nearing et al., 1997). In the investigation of rill flow hydraulics, mean flow velocity V is also calculated by specific empirical relationship calibrated by other measured hydraulic variables such as discharge, water depth, crosssection area, slope (Abrahams et al., 1996). More sophisticated technologies, such as hot film anemometry, acoustic Doppler velocimeter (ADV) and particle imaging velocimetry (PIV) have been generally applied in experimental runs carried out under controlled conditions (Liu et al., 2001), and they present numerous limitations when are used for field measurements (Planchon et al., 2005). Hot film anemometry cannot be used to measure velocity in sediment laden flow (Ali et al., 2012) while for applying ADV
Corresponding author. E-mail address: [email protected] (V. Ferro).
https://doi.org/10.1016/j.catena.2019.104313 Received 14 January 2019; Received in revised form 13 August 2019; Accepted 7 October 2019 0341-8162/ © 2019 Elsevier B.V. All rights reserved.
Catena 185 (2020) 104313
C.D. Stefano, et al.
technique flow depths equal to or greater than 1.5 cm are required and the effect of sediment transport on velocity measurement accuracy is not known. PIV can be used only for laboratory experimental runs because sophisticated equipment are involved (Ali et al., 2012). The dye-tracing is one of the most used techniques for measurements in flume and in field experiments since its operational simplicity is a noteworthy advantage. In fact the main characteristic of the dyetracing technique is that it can be applied without any particular instrumentation since the measurement is based on the visual observation of the tracer motion (Wirtz et al., 2010, 2012). Tracer method is applied by adding a material (salt, magnetic material, water isotope and floating object) (Berman et al., 2009; Dunkerley, 2003; Ventura et al., 2001; Olivier et al., 2005) and then measuring the speed of the material to travel a known distance from the injection point (Chen et al., 2017). Flow velocity is measured by recording the travel time of the leading edge of the dye cloud from the injection point to the end of a given channel reach. This known distance between the two sections (measurement section – end of the channel) divided by the travel time to cover this span is the surface velocity Vs of the leading edge of the dye cloud. Then, for calculating the mean flow velocity V a correction factor αv = V/Vs has to be applied to the measured surface velocity (Zhang et al., 2010). For the case of infinitely wide, laminar flow on a smooth and rigid bed, Horton et al. (1934) theorized that the correction factor αv is equal to 0.67. For transitional flows, Emmett (1970) found by flume experiments that αv increases with flow Reynolds number Re = hV/νk, in which h is the water depth and νk is the kinematic viscosity, and the correction factor is close to 0.8 for turbulent flow. In laboratory experiments, Luk and Merz (1992) observed a mean value of αv equal to 0.75 for transitional and turbulent flow. Li et al. (1996) carried out some experimental runs in a sloping flume (slope s ranging from 4.7% to 17.7%) with a mobile sand bed and transitional and turbulent flows (910 ≤ Re ≤ 6097). These Authors proposed an empirical relationship in which the correction factor αv varies inversely with slope and directly with Re. The effect of slope on the correction coefficient was justified as a consequence of the flow sediment transport (Li and Abrahams, 1997). Li and Abrahams (1997) investigated the relationship between αv and Re for both sediment-free and sediment-laden flow. For sedimentfree laminar flow they carried out some experimental runs with rough flume beds and determined a mean value of αv equal to 0.37, which is significantly lower than the theoretical value (0.67) calculated by Horton et al. (1934) for a smooth bed. This discrepancy allowed to highlight the effect of the roughness height and to establish that αv decreases with bed roughness. Li and Abrahams (1997) also established that in transitional regime αv increases with Re and confirmed that for turbulent flows the correction coefficient become constant and equal to 0.8. No effect of slope was detected for sediment-free flows. For sediment-laden flows, Li and Abrahams (1997) detected a reduction of the correction factor which was attributed to the circumstance that saltating sediments extract momentum from flow and, as a consequence, an inverse relationship between αv and sediment load is likely. According to the experiments by Zhang et al. (2010), for a sedimentfree flow the correction factor αv could be estimated by a logarithmic function of slope and flow Reynolds number. For a sediment-laden flow, Zhang et al. (2010) established that αv decreases as sediment load increases and Re decreases. Ali et al. (2012) carried out flume experiments by a well-sorted sand having a median grain size, D50, equal to 0.230, 0.536, 0.716 and 1.022 mm. Their investigation established that the mean value of the correction factor αv increases when the grain size increases. The effect of sediment particle size on αv for flow velocity on mobile beds was explained by the variation of the vertical velocity profile with grain size and the modification of bed morphology due to sediment uptake and deposition (Li and Abrahams, 1997).
Fig. 1. . View of the application of the dye-tracer technique in a rill flow.
Pan et al. (2015) carried out tests in a flume with different roughness conditions and for turbulent flows obtained αv values quasi equal to 0.8. According to Pan et al. (2015), the relationship between correction factor and slope should be dependent on the roughness condition (submerged or non-submerged roughness elements). Di Stefano et al. (2017b, 2018a,b) and Palmeri et al. (2018) carried out 139 field experimental runs using a constant inflow discharge ranging from 0.35 to 1.00 L s−1 and measured surface flow velocity in rill reaches by the dye tracing method (Fig. 1). The experimental rills were carried out on two plots, 2 m wide, 7 m long, having a mean slope sp equal to 9, 14 and 22%. The experimental values of the Reynolds and Froude number F = V / gh corresponded to transition and turbulent flow (1306 ≤ Re ≤ 10723) and to subcritical and supercritical flow conditions (0.64 < F ≤ 3.40). The cross-section area values were determined by using the measured water depth in combination with the geometric cross section profile extracted by 3D-DTM (Fig. 2). The rill inflow discharge was measured by the volumetric method. The surface velocity Vs in a rill reach was measured by a dye tracing method using a Methylene bleu solution. The mean flow velocity V was calculated as the ratio between the measured inflow discharge and the mean cross-section area. The analysis developed by Di Stefano et al. (2018b) demonstrated that the correction factor can be assumed constant (αv = 0.665 or 0.80) or a relationship between mean flow velocity, surface velocity and slope can be established. All available studies of rill flow resistance generally assumed a constant value of the calibration factor (i.e. Govers (1992), Li et al. (1996)) without testing the effect of the estimate uncertainty of αv on the Darcy-Weisbach friction factor. Therefore, taking into account the variability of αv values resulting from the previous investigations, the choice of an appropriate value of αv is yet a challenge for current research in rill flow hydraulics and in soil erosion modelling. Thereby, the main goals of this paper were: (i) deducing a theoretical relationship between surface velocity Vs and mean flow velocity V 2
Catena 185 (2020) 104313
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Fig. 2. . View of the DTM of a rill channel with the tracks colored by the bleu Methylene solution. Fig. 3. . View of the gravel bed flume by Ferro and Baiamonte (1994).
using a power velocity distribution; (ii) using the velocity profile measurements carried out for a sediment-free flow in motion on a rough bed, in order to calculate the theoretical correction factor values and to establish a relationship between αv and roughness height; and (iii) using the velocity profile measurements carried out for a sedimentladen flow in motion on a smooth bed, to calculate the theoretical correction factor values and to state the effect of sediment load on αv. Taking into account that velocity measurements in open channel flows offer a wide range of applications, the knowledge of the correction factor αv by velocity distributions is of great relevance in the fields of hydrometry, sediment transport, river restoration and soil erosion modelling.
ranging from 0.0045 to 0.0973 m3 s−1. In the measuring reach, three hydrometers were installed and the water depth of the quasi uniform flow was calculated as the mean value of three measurements. The velocity profiles were measured in the channel axis. These measurements by Ferro and Baiamonte (1994) are characterized by flows which are turbulent (7105 ≤ Re ≤ 77778) and having Froude numbers F ranging from 0.21 to 0.80.
2.2. Experiments by Coleman (1986) Coleman (1986) carried out 40 experimental runs for investigating the effect of variation in suspended sediment concentration on velocity profile characteristics. He used a recirculating flume which was 15 m long, 0.356 m wide, with slope adjustment capability for maintaining uniform flow. Flow uniformity was monitored by two point gauges. The velocity profiles were measured in the channel axis. The experiments were repeated with three sands, with mean particle diameter D equal to 0.105, 0.210 and 0.420 mm. The applied experimental procedure assured that no sand was deposited and that the whole range of sediment concentration up to capacity transport was covered. These measurements by Coleman (1986) are characterized by flows which are turbulent (178106 ≤ Re ≤ 203492), subcritical (0.79 ≤ F ≤ 0.84). The experimental runs were carried out with a single discharge equal to 0.064 m3 s−1 and a sediment discharge Qs ranging from 0.00000416 to 0.000322 m3 s−1.
2. Materials and methods For testing the theoretical expression of the correction factor αv (Eq. (5)) the available experimental data by Ferro and Baiamonte (1994), for sediment-free flows, and the velocity measurements by Coleman (1986), for sediment –laden flows, were used. For both investigations the velocity distributions were measured at the flume axis and the flume sides were smooth (glass or plexiglass). In these hydraulic conditions, the measured local velocity generally increases with the distance from the flume bottom and a power theoretical distribution can be well fitted to the measured velocity profile. 2.1. Experiments by Ferro and Baiamonte (1994) Ferro and Baiamonte (1994) carried out 56 experimental runs using a sloping flume of the Dipartimento di Ingegneria Idraulica ed Applicazioni Ambientali at the University of Palermo. The flume was 14 m long, 0.6 m wide and 0.6 m deep (Fig. 3). The inlet and outlet structures of the flume were connected to a hydraulic circuit allowing a continuous recirculation of a stable discharge. The runs were carried out with a quarry rubble bed with fixed number (0, 10, 20 and 40) of coarser elements arranged in reference areas 0.6 m wide and 0.6 m long. Boulder concentration C, for the four investigated bed arrangements (C0, C1, C2, C3), was equal to 0, 5.9, 11.8 and 23.5% (Fig. 3). The characteristic diameter D84, which is the diameter of bed particles for which 84% are finer, is equal to 26 mm, 27.4 mm, 28.1 mm and 42.1 mm for C0, C1, C2 and C3, respectively. The experimental runs were carried out with discharge values
3. Results 3.1. Deducing the correction factor by flow velocity profile measurements Recent studies (Ferro, 2017, 2018; Ferro and Porto, 2017, 2018) stated that dimensional analysis can be usefully applied to theoretically deduce the flow velocity profile in an open channel flow. For a uniform turbulent open channel flow, the velocity distribution along a given vertical can be expressed by the following functional relationship (Barenblatt, 1979, 1987, 1993; Ferro, 1997; Di Stefano et al., 2017b, 2018b): 3
Catena 185 (2020) 104313
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Π=
y dv = u∗ dy
0.35
u y u h h φ⎛ ∗ , ∗ , ⎞ νk d ⎠ ⎝ νk ⎜
⎟
C0
(1) 0.3
v = u∗
u y Γ⎛ ∗ ⎞ ⎝ νk ⎠ ⎜
v (m/s)
in which v is the local velocity, y is the distance from the bottom, u* = g R s is the shear velocity, g is the acceleration due to gravity, R is the hydraulic radius, s (m m−1) is the channel slope, φ is a functional symbol and d is the roughness height. Assuming the Incomplete Self-Similarity (ISS) condition for u*y/νk (Barenblatt and Monin, 1979; Barenblatt and Prostokishin, 1993; Ferro and Pecoraro, 2000; Ferro, 2017, 2018), Eq. (1) allows to obtain the following velocity distribution:
V=
0.15 0
⎜
2
(2)
h
v (m/s)
0.5 0.4
v = 0.2645 y0.2764 R² = 0.965
0.3
δ
0.2 0
(3) δ
⎛ u∗ y ⎞ dy = Γ νk ⎛ 1 ⎞ ⎛ u∗ h ⎞ h ⎝ 1 + δ ⎠ ⎝ νk ⎠ ⎝ νk ⎠
⎜
C1
0.6
⎟
∫0
3
0.7
⎟
⎜
Γ νk ⎛ 1 ⎞ ⎛ u∗ h ⎞ h ⎝ 1 + δ ⎠ ⎝ νk ⎠ ⎜
δ+1
⎟
δ+1
−δ
⎜
⎟
=
(4)
1 1+δ
30
C2
0.4 0.35
(5)
0.3 0.25
v = 0.2462 y0.3094 R² = 0.9896
0.2
which demonstrates that the correction factor depends on the exponent δ of the power velocity distribution. For testing Eq. (5), the applicability of the power velocity distribution has to be verified using measured profiles in a cross-section.
0.15 0
2
4
6
y (cm) 0.35
3.2. Calibrating the correction coefficient for sediment-free flows
C3
0.3
v (m/s)
At first for a given experimental run, comparing the data pairs (y, v) with the calibrated power velocity distribution (Eq. (2)) the measured velocity profiles having a power shape were selected. Fig. 4 shows, for the four investigated concentrations (C0, C1, C2 and C3), an example of the comparison between the measured profiles and the power distribution. For each investigated arrangement, the correction factor αv values, calculated by Eq. (5), are plotted versus the flow Reynlods number Re in Fig. 5. This figure shows that αv has not a specific trend with Re, ranges from 0.662 to 0.899 and its mean value is equal to 0.752. For each investigated bed arrangement (C0, C1, C2 and C3) the mean value of the correction factor is equal to 0.795, 0.744, 0.739 and 0.709, respectively. Using the 30 αv values corresponding to the four investigated bed arrangements the frequency distribution plotted in Fig. 6 is obtained. The empirical frequency distribution is characterized by a mean value equal to 0.7517 and a standard deviation of 0.051. The influence of the bed roughness on the mean value of the correction factor αv was empirically taken into account by the following relationship (Fig. 7):
α v = 0.7017 + 0.2025logD84 − 0.51C
20
0.45
⎟
1 ⎛ u∗ h ⎞ Γ u∗ ⎝ νk ⎠
10
y (cm)
v (m/s)
V = Vs
1
y (cm)
It has to be noticed that for each velocity profile Γ is constant and the mean velocity can be calculated without including Γ in the integration. Using Eqs. (3) and (4) the following expression of αv is obtained:
αv =
v = 0.2357 y0.324 R² = 0.9886
⎟
u h u∗ Γ ⎛ ∗ ⎞ ⎝ νk ⎠
u∗ Γ h
0.2
δ
in which Γ and δ have to be estimated for each measured profile, i.e. Γ and δ are estimated by a power correlation of the variables X = u*y/νk and Y = v/u*. To the best of our knowledge, no theoretical study of the effect of the velocity profile on the correction factor αv is available. For the velocity profile in a given vertical of a cross-section, for which Γ and δ are two known constants, Eq. (2) allows to calculate the surface velocity Vs and the mean velocity V:
Vs =
0.25
0.25 0.2
v = 0.1735 y0.3609 R² = 0.9777
0.15 0.1 0
2
4
6
y (cm) Fig. 4. . Comparison between the profile measured by Ferro and Baiamonte (1994) and the power velocity distribution for different boulder concentrations.
dimensionless. Eq. (6) is the best-fitting relationship, established by a regression analysis, characterized by a determination coefficient R2 equal to 0.896. This result agree with a previous study by Ali et al. (2012) which stated that the sediment particle size has a positive impact on the correction factor (i.e. αv increases with the sediment size). 3.3. Calibrating the correction coefficient for sediment-laden flows
(6)
For each investigated sand (D = 0.105, 0.210 and 0.420 mm), the
in which D84 is expressed as mm and the concentration C is 4
Catena 185 (2020) 104313
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Fig. 5. . Relationship between the correction factor and flow Reynolds number for sediment-free flows.
1 0.9 0.8 0.7
F (α v)
0.6 0.5 0.4 0.3 0.2 0.1 0 0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
αv Fig. 6. . Comparison between the frequency distribution of the correction factor and a normal probability distribution for sediment-free flows.
Fig. 8. . Comparison between the profile measured by Coleman (1986) and the power velocity distribution for different diameters.
the correction factor αv values, calculated by Eq. (5), are plotted versus the flow Reynolds number Re in Fig. 9. This figure also shows that αv has not a specific trend with Re, ranges from 0.839 to 0.903 and assumes a mean value equal to 0.867 and a standard deviation of 0.0174. Using the 40 αv values corresponding to the three investigated sandsizes the frequency distribution plotted in Fig. 10 is obtained. The influence of the sediment load on the correction factor αv was empirically taken into account by the following relationship (Fig. 11):
Fig. 7. . Comparison between the measured values of the correction factor and those calculated by Eq. (6).
α v = −0.5906 + 0.2534logRe − 0.0282logQs measured velocity profiles having a power shape were selected comparing, for a given experimental run, the data pairs (y, v) with the calibrated power velocity distribution (Eq. (2)). Fig. 8 shows, for the three investigated sands, an example of the comparison between the measured profiles and the power distribution. For each investigated sand,
(7)
in which Qs is the sediment discharge expressed as m3 s−1. Eq. (7) is characterized by a determination coefficient R2 equal to 0.759. This result is in agreement with previous studies by Li and Abrahams (1997) and Zhang et al. (2010) which established by a regression analysis equations similar to Eq. (7). 5
Catena 185 (2020) 104313
C.D. Stefano, et al.
allow to detect (Fig. 5) that the correction factor, and its mean value, decreases with the roughness height. This results agrees with Li and Abrahams (1997) which established that for a sediment-free flow αv decreases with bed roughness. Eq. (6) highlights that, for a gravel bed roughness, two different roughness indexes have to be used for estimating the correction factor αv. The influence of the bed roughness on the correction factor was empirically taken into account by a relationship in which boulder concentration and a characteristic bed particle diameter appear. In other words, the influence of the bed roughness has to be described using a scale factor (D84) and a shape factor (C) of the bed particle size distribution. Neglecting the influence of the roughness height, Fig. 6 shows that the correction factor αv is normally distributed with a mean value equal to 0.7517 and a standard deviation of 0.051. The presented results were obtained using velocity values measured in the channel axis of a smooth side flume, which are hydraulic conditions typical of a very-wide rectangular cross-section. A rill flow can be schematized as a very-wide rectangular cross-section when the rill flow depth can be considered smaller than its width. Therefore, the applicability of these results should be experimentally tested by a small fixed bed flume simulating a rill channel (width less than 5–10 cm) in which a flow with a small depth (1–2 cm) moves.
Fig. 9. . Relationship between the correction factor and flow Reynolds number for sediment-laden flows.
4.2. Calibrating the correction coefficient for a sediment-laden flow The experimental runs carried out by Coleman (1986) allow to establish that a trend of the correction factor with the size of the transported particle size can be established. In particular, the αv values tend to increase with the sediment size D (Fig. 9). This trend agrees with the results by Ali et al. (2012) obtained by flume experiments with a mobile bed constituted of well-sorted sands. The variability of αv with the sediment particle size can be explained taking into account that the sediment load modifies the velocity profile. The correction factor values for a sediment-laden flow (Fig. 9) are higher than those corresponding to a sediment-free flow in motion on a rough bed (Fig. 5). The comparison between Figs. 5 and 9 also shows that the sediment-laden flow are characterised by Re values greater than those of the sediment-free flows. In agreement with the results by Li and Abrahams (1997) and Zhang et al. (2010), Eq. (10) confirms that the sediment load produces a reduction of the correction factor.
Fig. 10. . Comparison between the frequency distribution of the correction factor and a normal probability distribution for sediment-laden flows.
5. Conclusions The applicability of the dye-tracing technique for measuring flow velocity is affected by an estimate criterion of the correction factor, i.e. the ratio between the mean velocity and the surface velocity. In this paper a theoretical relationship (Eq. (5)) for calculating the correction factor was firstly deduced by using a power flow velocity profile. Then, the deduced theoretical expression of the correction factor was applied using flume measurements of velocity profiles carried out for both a sediment-free and a sediment-laden flow. Two empirical equations (Eqs. (6) and (7)) for estimating the correction factor were calibrated using experimental data which span a wide range of hydraulic conditions in Froude number, Reynolds number, roughness height and sediment load. For a sediment-free flow the analysis showed that the correction factor decreases with the roughness height and the influence of the bed roughness has to be described using a scale factor (characteristic diameter of the particle mixture) and a shape factor (concentration of boulders) of the bed particle size distribution. For a sediment-laden flow the analysis highlighted that the correction factor values are higher than those corresponding to a sedimentfree flow in motion on a rough bed and an increasing trend with the sediment size was detected. In agreement with previous investigations,
Fig. 11. . Comparison between the measured values of the correction factor and those calculated by Eq. (7).
4. Discussion 4.1. Calibrating the correction coefficient for a sediment-free flow The experimental runs carried out by Ferro and Baiamonte (1994) 6
Catena 185 (2020) 104313
C.D. Stefano, et al.
the deduced empirical relationship underlines that the correction factor decreases for increasing values of the sediment load. Further experiments could be carried out using a fixed bed small flume simulating a rill channel for studying the effect of the conveyed discharge, slope, flow Reynolds number and Froude number on the correction factor.
Survey Professional Papers 622-A. Ferro, V., 1997. Applying hypothesis of self-similarity for flow-resistance law of smalldiameter plastic pipes. J. Irrig. Drainage Eng., ASCE 123, 175–179. Ferro, V., 2017. New flow resistance law for steep mountain streams based on velocity profile. J. Irrig. Drainage Eng., ASCE 143 (04017024), 1–6. https://doi.org/10.1061/ (ASCE)IR.1943-4774.0001208. Ferro, V., 2018. Assessing flow resistance in gravel bed channels by dimensional analysis and self-similarity. Catena 169, 119–127. https://doi.org/10.1016/j.catena.2018.05. 034. Ferro, V., Baiamonte, G., 1994. Flow velocity profiles in gravel-bed rivers. J. Hydraul. Eng. ASCE 120 (1), 60–80. https://doi.org/10.1061/(ASCE)0733-9429 (1994) 120:1(60). Ferro, V., Porto, P., 2017. Applying hypothesis of self-similarity for flow resistance law in Calabrian gravel bed rivers. J. Hydraul. Eng. 144 (04017061), 1–11. Ferro, V., Porto, P., 2018. Assessing theoretical flow velocity profile and resistance in gravel bed rivers by field measurements. J. Agric. Eng. XLIX, 220–227. Ferro, V., Pecoraro, R., 2000. Incomplete self-similarity and flow velocity in gravel bed channels. Water Resour. Res. 36, 2761–2770. Foster, G.R., Huggins, L.F., Meyer, L.D., 1984. A laboratory study of rill hydraulics: I. Velocity relationships. Trans. ASAE 27, 790–796. Gilley, J.E., Kottwitz, E.R., Simanton, J.R., 1990. Hydraulics characteristics of rills. Trans. ASAE 27, 797–804. Gimenez, R., Planchon, O., Silvera, N., Govers, G., 2004. Longitudinal velocity patterns and bed morphology intercation in a rill. Earth Surf. Proc. Land. 29, 105–114. Govers, G., 1992. Relationship between discharge, velocity and flow area for rills eroding loose, non –layered materials. Earth Surf. Proc. Land. 17, 515–528. Govers, G., Gimenez, R., Van Oost, K., 2007. Rill erosion: exploring the relationship between experiments, modeling and field observations. Earth-Sci. Rev. 83, 87–102. Horton, R.E., Leach, H.R., Vliet, V.R., 1934. Laminar sheet flow. Trans. Am. Geophys. Union 15, 393–404. Lei, T., Xia, W., Zhao, J., Liu, Z., Zhang, Q., 2005. Method for measuring velocity of shallow water flow for soil erosion with an electrolyte tracer. J. Hydrol. 301, 139–145. Li, G., Abrahams, A.D., Atkinson, J.F., 1996. Correction factors in the determination of mean flow velocity of overland flow. Earth Surf. Proc. Land. 21, 509–515. Li, G., Abrahams, A.D., 1997. Effect of salting sediment load on the determination of the mean velocity of overland flow. Water Resour. Res. 33, 341–347. Line, D.E., Meyer, L.D., 1988. Flow velocities of concentrated runoff along cropland furrows. Trans. ASAE 31, 1435–1439. Liu, Z., Adrian, R.J., Hanratty, T.J., 2001. Large-scale modes of turbulent channel flow: transport and structure. J. Fluid Mech. 448, 53–80. Luk, S.H., Merz, W., 1992. Use of the salt tracing technique to determine the velocity of overland flow. Soil Technol. 5, 289–301. Nearing, M., Norton, L.D., Bulgakov, D.A., Larionov, G.A., West, L.T., Dontsova, K.M., 1997. Hydraulics and erosion in eroding rills. Water Resour. Res. 33, 865–876. Olivier, P., Norbert, S., Raphael, G., David, F.M., Wainwright, J., Le Bissonnais, Y., Govers, G., 2005. An automated salt-tracing gauge for flow-velocity measurements. Earth Surf. Proc. Land. 30, 833–844. Palmeri, V., Pampalone, V., Di Stefano, C., Nicosia, A., Ferro, V., 2018. Experiments for testing soil texture effects on flow resistance in mobile bed rills. Catena 171, 176–184. Pan, C., Shangguan, Z., Ma, L., 2015. Assessing the dye-tracer correction factor for documenting the mean velocity of sheet flow over smooth and grassed surfaces. Hydrol. Process. 29, 5369–5382. Planchon, O., Silvera, N., Gimenez, R., Favis-Mortlock, D., Wainwright, J., Le Bissonnais, Y., Govers, G., 2005. An automated salt-tracing gauge for flow-velocity measurement. Earth Surf. Proc. Land. 30, 833–844. Raffel, M., Willert, C., Kompnhans, J., 1998. Particle Image Velocimetry. A Practical Guide (Experimental Fluid Mechanics). Springer, Berlin. Takken, I., Govers, G., Ciesiolka, C.A.A., Silburn, D.M., Loch, R.J. 1998. Factors influencing the velocity-discharge relationship in rills. Modelling Soil Erosion, Sediment Transport and Closely related Hydrological Processes. IAHS Publ. N. 249, pp. 63–69. Ventura Jr., E., Nearing, M.A., Darrell Norton, L., 2001. Developing a magnetic tracer to study soil erosion. Catena 43, 277–291. Wirtz, S., Seeger, M., Ries, J.B., 2010. The rill experiment as a method to approach a quantification of rill erosion process activity. Zeitschrift für Geomorphologie 54, 47–64. Wirtz, S., Seeger, M., Ries, J.B., 2012. Field experiment for understanding and quantification of rill erosion processes. Catena 91, 21–34. Zhang, G., Luo, R., Cao, Y., Shen, R., Zhang, X.C., 2010. Correction factor to dye-measured flow velocity under varying water and sediment discharges. J. Hydrol. 389, 205–213.
Acknowledgements All authors contributed to outline the investigation, analyze the data, discuss the results, and write the manuscript. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.catena.2019.104313. References Abrahams, A.D., Gang, L.I., Parsons, A.J., 1996. Rill hydraulics on a semiarid hillslope, southern Arizona. Earth Surf. Proc. Land. 21, 35–47. Ali, M., Sterk, G., Seeger, M., Stroosnijder, L., 2012. Effect of flow discharge and median grain size on mean flow velocity under overland flow. J. Hydrol. 452–453, 150–160. Ayala, J.E., Martinez-Austria, P., Menendez, J.R., Segura, F.A., 2000. Fixed bed forms laboratory channel flow analysis. Ingenieria Hidraulica ex Mexico 15 (2), 75–84. Bagarello, V., Di Stefano, C., Ferro, V., Pampalone, V., 2015. Establishing a soil loss threshold for limiting rilling. J. Hydrol. Eng., ASCE 20 (C6014001), 1–5. Barenblatt, G.I., 1979. Similarity, Self-Similarity and Intermediate Asymptotics. Consultants Bureau, New York, N.Y. Barenblatt, G.I., 1987. Dimensional Analysis. Gordon & Breach, Science Publishers Inc., Amsterdam. Barenblatt, G.I., 1993. Scaling laws for fully developed turbulent shear flows, part 1, Basic hypothesis and analysis. J. Fluid Mech. 248, 513–520. Barenblatt, G.I., Monin, A.S., 1979. Similarity laws for turbulent stratified flows. Arch. Ration. Mech. Anal. 70, 307–317. Barenblatt, G.I., Prostokishin, V.M., 1993. Scaling laws for fully developed turbulent shear flows, part 2. Processing of experimental data. J. Fluid Mech. 248, 521–529. Ban, Y., Lei, T., Liu, Z., Chen, C., 2016. Comparison of rill flow velocity and thawed slopes with electrolyte tracer method. J. Hydrol. 534, 630–637. Berman, E.S.F., Gupta, M., Gabrielli, C., Garland, T., McDonnell, J.J., 2009. High-frequency field deployable isotope analyzer for hydrological applications. Water Resour. Res. 45 (W10201), 1–7. Bradley, A.A., Kruger, A., Meselhe, E.A., Muste, M.V., 2002. Flow measurement in streams using video imagery. Water Resour. Res. 38 (12), 1315, 51-1–51-8. Bruno, C., Di Stefano, C., Ferro, V., 2008. Field investigation on rilling in the experimental Sparacia area, South Italy. Earth Surface Process. Landforms 33, 263–279. Chen, C., Ban, Y., Wang, X., Lei, T., 2017. Measuring flow velocity on frozen and nonfrozen slopes of black soil through leading edge method. Int. Soil Water Conserv. Res. 5, 180–189. Coleman, N.L., 1986. Effects of suspended sediment on the open—channel velocity distribution. Water Resour. Res. 22, 1377–1384. de Lima, J.L.M.P., Abrantes, J.R.C.B., 2014. Using a thermal tracer to estimate overland and rill flow velocities. Earth Surf. Proc. Land. 39, 1293–1300. Di Stefano, C., Ferro, V., Pampalone, V., 2015. Modeling rill erosion at the Sparacia experimental area. J. Hydrol. Eng., ASCE 20 (C5014001), 1–12. Di Stefano, C., Ferro, V., Palmeri, V., Pampalone, V., 2017a. Measuring rill erosion using structure from motion: A plot experiment. Catena 153, 383–392. Di Stefano, C., Ferro, V., Palmeri, V., Pampalone, V., 2017b. Flow resistance equation for rills. Hydrol. Process. 31, 2793–2801. https://doi.org/10.1002/hyp.11221. Di Stefano, C., Ferro, V., Palmeri, V., Pampalone, V., 2018a. Testing slope effect on flow resistance equation for mobile bed rills. Hydrol. Process. 32, 664–671. https://doi. org/10.1002/hyp.11448. Di Stefano, C., Ferro, V., Palmeri, V., Pampalone, V., 2018b. Assessing dye-tracer technique for rill flow velocity measurements. Catena 171, 523–532. Dunkerley, D.L., 2003. An optical tachometer for short-path measurement of flow speeds in shallow overland flows: improved alternative to dye timing. Earth Surf. Proc. Land. 28, 777–786. Emmett, W.W., 1970. The hydraulics of overland flow on hillslopes. U.S. Geological
7
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Catena journal homepage: www.elsevier.com/locate/catena
Dye-tracer technique for rill flows by velocity profile measurements a
a
b
a
Costanza Di Stefano , Alessio Nicosia , Vincenzo Palmeri , Vincenzo Pampalone , Vito Ferro a b
b,⁎
T
Department of Agricultural, Food and Forestry Sciences, University of Palermo, Viale delle Scienze, Building 4, 90128 Palermo, Italy Department of Earth and Marine Sciences, University of Palermo, Via Archirafi 20, 90123 Palermo, Italy
A R T I C LE I N FO
A B S T R A C T
Keywords: Soil erosion Interrill flows Rill flows Flow velocity Dye method Correction factor
Water flow on hillslope soil surface supplies energy which is required to detach soil particles, to transport and deposit sediments, therefore flow velocity is a key variable related to hillslope hydrodinamics of soil erosion processes. Among the different methods available for measuring velocity of shallow interrill and rill flow, the trace technique is widely used. Trace technique is applied by adding a material (salt, magnetic material, water isotope, floating object) and then measuring the speed of the material to travel a known distance from the injection point. When flow velocity is measured using a dye-tracing method, the mean velocity is calculated by multiplying the measured surface velocity of the leading edge of the tracer plume by a correction factor which was generally empirically deduced. The main uncertainty of the dye-tracing technique stands in the estimate of this correction factor, which is the ratio between the mean flow velocity and the surface velocity. The main goal of this paper is establishing a theoretical relationship for calculating the correction factor by using a power velocity profile. At first, the developed analysis demonstrated that the correction factor only depends on the exponent of the power velocity distribution (Eq. (5)). Then, this theoretical expression of the correction factor was applied using the velocity profiles measured by Baiamonte and Ferro for the condition of sediment-free flow in motion on a rough bed and by Coleman for a sediment-laden flow moving on a smooth flume. Using the correction factor values calculated by the velocity measurements carried out for the sediment-free flow in motion on a rough bed, the relationship between the correction factor and roughness height was established. In agreement with a previous study by Ali et al., for a sediment-free flow the correction factor increases with roughness height. Finally, the velocity profile measurements carried out for a sediment-laden flow in motion on a smooth bed were used to state the effect of sediment load on the correction factor. This last analysis allowed to conclude, in agreement with Li and Abrahams and Zhang et al., that the correction factor decreases when the sediment load increases.
1. Introduction Flow velocity is a key variable related to hillslope interrill and rill erosion processes. Flow velocity measurement is useful for improving its hydraulic knowledge, which is necessary for developing processbased soil erosion models (Takken et al., 1998). Several methods (hot anemometry (Ayala et al., 2000), particle imaging velocimetry (Raffel et al., 1998; Ali et al., 2012), Acoustic Doppler velocimetry (Giménez et al., 2004), infrared termography (de Lima et al., 2014), optical tacheometer (Dunkerley, 2003), dye tracer (Bradley et al., 2002; Lei et al., 2005; Zhang et al., 2010; Ban et al., 2016)) are used to measure velocity in shallow flow, such as in interrill and rill flows (Bruno et al., 2008; Bagarello et al., 2015; Di Stefano et al., 2015, 2017a; Foster et al.,
⁎
1984; Gilley et al., 1990; Govers, 1992; Govers et al., 2007; Line and Meyer, 1988; Nearing et al., 1997). In the investigation of rill flow hydraulics, mean flow velocity V is also calculated by specific empirical relationship calibrated by other measured hydraulic variables such as discharge, water depth, crosssection area, slope (Abrahams et al., 1996). More sophisticated technologies, such as hot film anemometry, acoustic Doppler velocimeter (ADV) and particle imaging velocimetry (PIV) have been generally applied in experimental runs carried out under controlled conditions (Liu et al., 2001), and they present numerous limitations when are used for field measurements (Planchon et al., 2005). Hot film anemometry cannot be used to measure velocity in sediment laden flow (Ali et al., 2012) while for applying ADV
Corresponding author. E-mail address: [email protected] (V. Ferro).
https://doi.org/10.1016/j.catena.2019.104313 Received 14 January 2019; Received in revised form 13 August 2019; Accepted 7 October 2019 0341-8162/ © 2019 Elsevier B.V. All rights reserved.
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technique flow depths equal to or greater than 1.5 cm are required and the effect of sediment transport on velocity measurement accuracy is not known. PIV can be used only for laboratory experimental runs because sophisticated equipment are involved (Ali et al., 2012). The dye-tracing is one of the most used techniques for measurements in flume and in field experiments since its operational simplicity is a noteworthy advantage. In fact the main characteristic of the dyetracing technique is that it can be applied without any particular instrumentation since the measurement is based on the visual observation of the tracer motion (Wirtz et al., 2010, 2012). Tracer method is applied by adding a material (salt, magnetic material, water isotope and floating object) (Berman et al., 2009; Dunkerley, 2003; Ventura et al., 2001; Olivier et al., 2005) and then measuring the speed of the material to travel a known distance from the injection point (Chen et al., 2017). Flow velocity is measured by recording the travel time of the leading edge of the dye cloud from the injection point to the end of a given channel reach. This known distance between the two sections (measurement section – end of the channel) divided by the travel time to cover this span is the surface velocity Vs of the leading edge of the dye cloud. Then, for calculating the mean flow velocity V a correction factor αv = V/Vs has to be applied to the measured surface velocity (Zhang et al., 2010). For the case of infinitely wide, laminar flow on a smooth and rigid bed, Horton et al. (1934) theorized that the correction factor αv is equal to 0.67. For transitional flows, Emmett (1970) found by flume experiments that αv increases with flow Reynolds number Re = hV/νk, in which h is the water depth and νk is the kinematic viscosity, and the correction factor is close to 0.8 for turbulent flow. In laboratory experiments, Luk and Merz (1992) observed a mean value of αv equal to 0.75 for transitional and turbulent flow. Li et al. (1996) carried out some experimental runs in a sloping flume (slope s ranging from 4.7% to 17.7%) with a mobile sand bed and transitional and turbulent flows (910 ≤ Re ≤ 6097). These Authors proposed an empirical relationship in which the correction factor αv varies inversely with slope and directly with Re. The effect of slope on the correction coefficient was justified as a consequence of the flow sediment transport (Li and Abrahams, 1997). Li and Abrahams (1997) investigated the relationship between αv and Re for both sediment-free and sediment-laden flow. For sedimentfree laminar flow they carried out some experimental runs with rough flume beds and determined a mean value of αv equal to 0.37, which is significantly lower than the theoretical value (0.67) calculated by Horton et al. (1934) for a smooth bed. This discrepancy allowed to highlight the effect of the roughness height and to establish that αv decreases with bed roughness. Li and Abrahams (1997) also established that in transitional regime αv increases with Re and confirmed that for turbulent flows the correction coefficient become constant and equal to 0.8. No effect of slope was detected for sediment-free flows. For sediment-laden flows, Li and Abrahams (1997) detected a reduction of the correction factor which was attributed to the circumstance that saltating sediments extract momentum from flow and, as a consequence, an inverse relationship between αv and sediment load is likely. According to the experiments by Zhang et al. (2010), for a sedimentfree flow the correction factor αv could be estimated by a logarithmic function of slope and flow Reynolds number. For a sediment-laden flow, Zhang et al. (2010) established that αv decreases as sediment load increases and Re decreases. Ali et al. (2012) carried out flume experiments by a well-sorted sand having a median grain size, D50, equal to 0.230, 0.536, 0.716 and 1.022 mm. Their investigation established that the mean value of the correction factor αv increases when the grain size increases. The effect of sediment particle size on αv for flow velocity on mobile beds was explained by the variation of the vertical velocity profile with grain size and the modification of bed morphology due to sediment uptake and deposition (Li and Abrahams, 1997).
Fig. 1. . View of the application of the dye-tracer technique in a rill flow.
Pan et al. (2015) carried out tests in a flume with different roughness conditions and for turbulent flows obtained αv values quasi equal to 0.8. According to Pan et al. (2015), the relationship between correction factor and slope should be dependent on the roughness condition (submerged or non-submerged roughness elements). Di Stefano et al. (2017b, 2018a,b) and Palmeri et al. (2018) carried out 139 field experimental runs using a constant inflow discharge ranging from 0.35 to 1.00 L s−1 and measured surface flow velocity in rill reaches by the dye tracing method (Fig. 1). The experimental rills were carried out on two plots, 2 m wide, 7 m long, having a mean slope sp equal to 9, 14 and 22%. The experimental values of the Reynolds and Froude number F = V / gh corresponded to transition and turbulent flow (1306 ≤ Re ≤ 10723) and to subcritical and supercritical flow conditions (0.64 < F ≤ 3.40). The cross-section area values were determined by using the measured water depth in combination with the geometric cross section profile extracted by 3D-DTM (Fig. 2). The rill inflow discharge was measured by the volumetric method. The surface velocity Vs in a rill reach was measured by a dye tracing method using a Methylene bleu solution. The mean flow velocity V was calculated as the ratio between the measured inflow discharge and the mean cross-section area. The analysis developed by Di Stefano et al. (2018b) demonstrated that the correction factor can be assumed constant (αv = 0.665 or 0.80) or a relationship between mean flow velocity, surface velocity and slope can be established. All available studies of rill flow resistance generally assumed a constant value of the calibration factor (i.e. Govers (1992), Li et al. (1996)) without testing the effect of the estimate uncertainty of αv on the Darcy-Weisbach friction factor. Therefore, taking into account the variability of αv values resulting from the previous investigations, the choice of an appropriate value of αv is yet a challenge for current research in rill flow hydraulics and in soil erosion modelling. Thereby, the main goals of this paper were: (i) deducing a theoretical relationship between surface velocity Vs and mean flow velocity V 2
Catena 185 (2020) 104313
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Fig. 2. . View of the DTM of a rill channel with the tracks colored by the bleu Methylene solution. Fig. 3. . View of the gravel bed flume by Ferro and Baiamonte (1994).
using a power velocity distribution; (ii) using the velocity profile measurements carried out for a sediment-free flow in motion on a rough bed, in order to calculate the theoretical correction factor values and to establish a relationship between αv and roughness height; and (iii) using the velocity profile measurements carried out for a sedimentladen flow in motion on a smooth bed, to calculate the theoretical correction factor values and to state the effect of sediment load on αv. Taking into account that velocity measurements in open channel flows offer a wide range of applications, the knowledge of the correction factor αv by velocity distributions is of great relevance in the fields of hydrometry, sediment transport, river restoration and soil erosion modelling.
ranging from 0.0045 to 0.0973 m3 s−1. In the measuring reach, three hydrometers were installed and the water depth of the quasi uniform flow was calculated as the mean value of three measurements. The velocity profiles were measured in the channel axis. These measurements by Ferro and Baiamonte (1994) are characterized by flows which are turbulent (7105 ≤ Re ≤ 77778) and having Froude numbers F ranging from 0.21 to 0.80.
2.2. Experiments by Coleman (1986) Coleman (1986) carried out 40 experimental runs for investigating the effect of variation in suspended sediment concentration on velocity profile characteristics. He used a recirculating flume which was 15 m long, 0.356 m wide, with slope adjustment capability for maintaining uniform flow. Flow uniformity was monitored by two point gauges. The velocity profiles were measured in the channel axis. The experiments were repeated with three sands, with mean particle diameter D equal to 0.105, 0.210 and 0.420 mm. The applied experimental procedure assured that no sand was deposited and that the whole range of sediment concentration up to capacity transport was covered. These measurements by Coleman (1986) are characterized by flows which are turbulent (178106 ≤ Re ≤ 203492), subcritical (0.79 ≤ F ≤ 0.84). The experimental runs were carried out with a single discharge equal to 0.064 m3 s−1 and a sediment discharge Qs ranging from 0.00000416 to 0.000322 m3 s−1.
2. Materials and methods For testing the theoretical expression of the correction factor αv (Eq. (5)) the available experimental data by Ferro and Baiamonte (1994), for sediment-free flows, and the velocity measurements by Coleman (1986), for sediment –laden flows, were used. For both investigations the velocity distributions were measured at the flume axis and the flume sides were smooth (glass or plexiglass). In these hydraulic conditions, the measured local velocity generally increases with the distance from the flume bottom and a power theoretical distribution can be well fitted to the measured velocity profile. 2.1. Experiments by Ferro and Baiamonte (1994) Ferro and Baiamonte (1994) carried out 56 experimental runs using a sloping flume of the Dipartimento di Ingegneria Idraulica ed Applicazioni Ambientali at the University of Palermo. The flume was 14 m long, 0.6 m wide and 0.6 m deep (Fig. 3). The inlet and outlet structures of the flume were connected to a hydraulic circuit allowing a continuous recirculation of a stable discharge. The runs were carried out with a quarry rubble bed with fixed number (0, 10, 20 and 40) of coarser elements arranged in reference areas 0.6 m wide and 0.6 m long. Boulder concentration C, for the four investigated bed arrangements (C0, C1, C2, C3), was equal to 0, 5.9, 11.8 and 23.5% (Fig. 3). The characteristic diameter D84, which is the diameter of bed particles for which 84% are finer, is equal to 26 mm, 27.4 mm, 28.1 mm and 42.1 mm for C0, C1, C2 and C3, respectively. The experimental runs were carried out with discharge values
3. Results 3.1. Deducing the correction factor by flow velocity profile measurements Recent studies (Ferro, 2017, 2018; Ferro and Porto, 2017, 2018) stated that dimensional analysis can be usefully applied to theoretically deduce the flow velocity profile in an open channel flow. For a uniform turbulent open channel flow, the velocity distribution along a given vertical can be expressed by the following functional relationship (Barenblatt, 1979, 1987, 1993; Ferro, 1997; Di Stefano et al., 2017b, 2018b): 3
Catena 185 (2020) 104313
C.D. Stefano, et al.
Π=
y dv = u∗ dy
0.35
u y u h h φ⎛ ∗ , ∗ , ⎞ νk d ⎠ ⎝ νk ⎜
⎟
C0
(1) 0.3
v = u∗
u y Γ⎛ ∗ ⎞ ⎝ νk ⎠ ⎜
v (m/s)
in which v is the local velocity, y is the distance from the bottom, u* = g R s is the shear velocity, g is the acceleration due to gravity, R is the hydraulic radius, s (m m−1) is the channel slope, φ is a functional symbol and d is the roughness height. Assuming the Incomplete Self-Similarity (ISS) condition for u*y/νk (Barenblatt and Monin, 1979; Barenblatt and Prostokishin, 1993; Ferro and Pecoraro, 2000; Ferro, 2017, 2018), Eq. (1) allows to obtain the following velocity distribution:
V=
0.15 0
⎜
2
(2)
h
v (m/s)
0.5 0.4
v = 0.2645 y0.2764 R² = 0.965
0.3
δ
0.2 0
(3) δ
⎛ u∗ y ⎞ dy = Γ νk ⎛ 1 ⎞ ⎛ u∗ h ⎞ h ⎝ 1 + δ ⎠ ⎝ νk ⎠ ⎝ νk ⎠
⎜
C1
0.6
⎟
∫0
3
0.7
⎟
⎜
Γ νk ⎛ 1 ⎞ ⎛ u∗ h ⎞ h ⎝ 1 + δ ⎠ ⎝ νk ⎠ ⎜
δ+1
⎟
δ+1
−δ
⎜
⎟
=
(4)
1 1+δ
30
C2
0.4 0.35
(5)
0.3 0.25
v = 0.2462 y0.3094 R² = 0.9896
0.2
which demonstrates that the correction factor depends on the exponent δ of the power velocity distribution. For testing Eq. (5), the applicability of the power velocity distribution has to be verified using measured profiles in a cross-section.
0.15 0
2
4
6
y (cm) 0.35
3.2. Calibrating the correction coefficient for sediment-free flows
C3
0.3
v (m/s)
At first for a given experimental run, comparing the data pairs (y, v) with the calibrated power velocity distribution (Eq. (2)) the measured velocity profiles having a power shape were selected. Fig. 4 shows, for the four investigated concentrations (C0, C1, C2 and C3), an example of the comparison between the measured profiles and the power distribution. For each investigated arrangement, the correction factor αv values, calculated by Eq. (5), are plotted versus the flow Reynlods number Re in Fig. 5. This figure shows that αv has not a specific trend with Re, ranges from 0.662 to 0.899 and its mean value is equal to 0.752. For each investigated bed arrangement (C0, C1, C2 and C3) the mean value of the correction factor is equal to 0.795, 0.744, 0.739 and 0.709, respectively. Using the 30 αv values corresponding to the four investigated bed arrangements the frequency distribution plotted in Fig. 6 is obtained. The empirical frequency distribution is characterized by a mean value equal to 0.7517 and a standard deviation of 0.051. The influence of the bed roughness on the mean value of the correction factor αv was empirically taken into account by the following relationship (Fig. 7):
α v = 0.7017 + 0.2025logD84 − 0.51C
20
0.45
⎟
1 ⎛ u∗ h ⎞ Γ u∗ ⎝ νk ⎠
10
y (cm)
v (m/s)
V = Vs
1
y (cm)
It has to be noticed that for each velocity profile Γ is constant and the mean velocity can be calculated without including Γ in the integration. Using Eqs. (3) and (4) the following expression of αv is obtained:
αv =
v = 0.2357 y0.324 R² = 0.9886
⎟
u h u∗ Γ ⎛ ∗ ⎞ ⎝ νk ⎠
u∗ Γ h
0.2
δ
in which Γ and δ have to be estimated for each measured profile, i.e. Γ and δ are estimated by a power correlation of the variables X = u*y/νk and Y = v/u*. To the best of our knowledge, no theoretical study of the effect of the velocity profile on the correction factor αv is available. For the velocity profile in a given vertical of a cross-section, for which Γ and δ are two known constants, Eq. (2) allows to calculate the surface velocity Vs and the mean velocity V:
Vs =
0.25
0.25 0.2
v = 0.1735 y0.3609 R² = 0.9777
0.15 0.1 0
2
4
6
y (cm) Fig. 4. . Comparison between the profile measured by Ferro and Baiamonte (1994) and the power velocity distribution for different boulder concentrations.
dimensionless. Eq. (6) is the best-fitting relationship, established by a regression analysis, characterized by a determination coefficient R2 equal to 0.896. This result agree with a previous study by Ali et al. (2012) which stated that the sediment particle size has a positive impact on the correction factor (i.e. αv increases with the sediment size). 3.3. Calibrating the correction coefficient for sediment-laden flows
(6)
For each investigated sand (D = 0.105, 0.210 and 0.420 mm), the
in which D84 is expressed as mm and the concentration C is 4
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C.D. Stefano, et al.
Fig. 5. . Relationship between the correction factor and flow Reynolds number for sediment-free flows.
1 0.9 0.8 0.7
F (α v)
0.6 0.5 0.4 0.3 0.2 0.1 0 0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
αv Fig. 6. . Comparison between the frequency distribution of the correction factor and a normal probability distribution for sediment-free flows.
Fig. 8. . Comparison between the profile measured by Coleman (1986) and the power velocity distribution for different diameters.
the correction factor αv values, calculated by Eq. (5), are plotted versus the flow Reynolds number Re in Fig. 9. This figure also shows that αv has not a specific trend with Re, ranges from 0.839 to 0.903 and assumes a mean value equal to 0.867 and a standard deviation of 0.0174. Using the 40 αv values corresponding to the three investigated sandsizes the frequency distribution plotted in Fig. 10 is obtained. The influence of the sediment load on the correction factor αv was empirically taken into account by the following relationship (Fig. 11):
Fig. 7. . Comparison between the measured values of the correction factor and those calculated by Eq. (6).
α v = −0.5906 + 0.2534logRe − 0.0282logQs measured velocity profiles having a power shape were selected comparing, for a given experimental run, the data pairs (y, v) with the calibrated power velocity distribution (Eq. (2)). Fig. 8 shows, for the three investigated sands, an example of the comparison between the measured profiles and the power distribution. For each investigated sand,
(7)
in which Qs is the sediment discharge expressed as m3 s−1. Eq. (7) is characterized by a determination coefficient R2 equal to 0.759. This result is in agreement with previous studies by Li and Abrahams (1997) and Zhang et al. (2010) which established by a regression analysis equations similar to Eq. (7). 5
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allow to detect (Fig. 5) that the correction factor, and its mean value, decreases with the roughness height. This results agrees with Li and Abrahams (1997) which established that for a sediment-free flow αv decreases with bed roughness. Eq. (6) highlights that, for a gravel bed roughness, two different roughness indexes have to be used for estimating the correction factor αv. The influence of the bed roughness on the correction factor was empirically taken into account by a relationship in which boulder concentration and a characteristic bed particle diameter appear. In other words, the influence of the bed roughness has to be described using a scale factor (D84) and a shape factor (C) of the bed particle size distribution. Neglecting the influence of the roughness height, Fig. 6 shows that the correction factor αv is normally distributed with a mean value equal to 0.7517 and a standard deviation of 0.051. The presented results were obtained using velocity values measured in the channel axis of a smooth side flume, which are hydraulic conditions typical of a very-wide rectangular cross-section. A rill flow can be schematized as a very-wide rectangular cross-section when the rill flow depth can be considered smaller than its width. Therefore, the applicability of these results should be experimentally tested by a small fixed bed flume simulating a rill channel (width less than 5–10 cm) in which a flow with a small depth (1–2 cm) moves.
Fig. 9. . Relationship between the correction factor and flow Reynolds number for sediment-laden flows.
4.2. Calibrating the correction coefficient for a sediment-laden flow The experimental runs carried out by Coleman (1986) allow to establish that a trend of the correction factor with the size of the transported particle size can be established. In particular, the αv values tend to increase with the sediment size D (Fig. 9). This trend agrees with the results by Ali et al. (2012) obtained by flume experiments with a mobile bed constituted of well-sorted sands. The variability of αv with the sediment particle size can be explained taking into account that the sediment load modifies the velocity profile. The correction factor values for a sediment-laden flow (Fig. 9) are higher than those corresponding to a sediment-free flow in motion on a rough bed (Fig. 5). The comparison between Figs. 5 and 9 also shows that the sediment-laden flow are characterised by Re values greater than those of the sediment-free flows. In agreement with the results by Li and Abrahams (1997) and Zhang et al. (2010), Eq. (10) confirms that the sediment load produces a reduction of the correction factor.
Fig. 10. . Comparison between the frequency distribution of the correction factor and a normal probability distribution for sediment-laden flows.
5. Conclusions The applicability of the dye-tracing technique for measuring flow velocity is affected by an estimate criterion of the correction factor, i.e. the ratio between the mean velocity and the surface velocity. In this paper a theoretical relationship (Eq. (5)) for calculating the correction factor was firstly deduced by using a power flow velocity profile. Then, the deduced theoretical expression of the correction factor was applied using flume measurements of velocity profiles carried out for both a sediment-free and a sediment-laden flow. Two empirical equations (Eqs. (6) and (7)) for estimating the correction factor were calibrated using experimental data which span a wide range of hydraulic conditions in Froude number, Reynolds number, roughness height and sediment load. For a sediment-free flow the analysis showed that the correction factor decreases with the roughness height and the influence of the bed roughness has to be described using a scale factor (characteristic diameter of the particle mixture) and a shape factor (concentration of boulders) of the bed particle size distribution. For a sediment-laden flow the analysis highlighted that the correction factor values are higher than those corresponding to a sedimentfree flow in motion on a rough bed and an increasing trend with the sediment size was detected. In agreement with previous investigations,
Fig. 11. . Comparison between the measured values of the correction factor and those calculated by Eq. (7).
4. Discussion 4.1. Calibrating the correction coefficient for a sediment-free flow The experimental runs carried out by Ferro and Baiamonte (1994) 6
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the deduced empirical relationship underlines that the correction factor decreases for increasing values of the sediment load. Further experiments could be carried out using a fixed bed small flume simulating a rill channel for studying the effect of the conveyed discharge, slope, flow Reynolds number and Froude number on the correction factor.
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