ISSN e: 2007-4026 / ISSN print:2007-3925

       

 
 
 
 
 
 
 
 

Volume 10, Issue 1 Enero - Junio 2018

ISSN: ppub: 2007-3925 epub: 2007-4026

Scientific article

Determination of shear velocity in a mild-sloping open channel flow

http://dx.doi.org/10.5154/r.inagbi.2017.01.002

Mendoza-González, Ángel 1 * ; Aguilar-Chávez, Ariosto 1

  • 1Instituto Mexicano de Tecnología del Agua. Paseo Cuauhnáhuac núm. 8532, Jiutepec, Morelos, C. P. 62550, MÉXICO.

Corresponding author: mendoza.gonzalez.angel@gmail.com, tel. (735) 593 4733.

Received: January 24, 2017; Accepted: November 25, 2017

This is an open-access article distributed under the terms of the Creative Commons Attribution License view the permissions of this license

Abstract

Objective:

To present a methodology that allows experimentally determining shear velocity, considering the log-law as a model of velocity distribution in the outer region of turbulent flow.

Mathodology:

The experimental study was carried out in a rectangular-shaped, variably-sloped channel with a 0.245-m-wide base and 5 m long. Flow velocity was measured with an Acoustic Doppler Velocimeter (ADV), and the measurement area was 12 mm. Shear velocity was determined by the instantaneous velocity equation ( u i , j ).

Results:

The log-law model had a good statistical fit with the shear velocity estimated from the experimental data.

Study limitations:

The experimental tests were conducted only in subcritical regime with low aspect ratios. In addition, in all tests, the measurement of instantaneous velocities was carried out only in a 12-mm profile, as close as possible to the wall.

Originality:

The model to calculate the shear velocity is presented explicitly, and the statistical

approach employed supports the use of the median as an estimator of the shear velocity.

Conclusions:

The presented methodology shows low uncertainty in the estimation of shear velocity. The Anderson-Darling test showed that the results do not follow a normal distribution, so the median is the statistical parameter to define the shear velocity value.

Keywordsshear stress; acoustic Doppler velocimetry; log-law

Introduction

In the design of irrigation canals and the modeling of hydrodynamic runoff in channels, empirical relationships are used to estimate the friction, for example: Chezy’s resistance factor, Manning’s roughness coefficient or the Darcy-Weisbach friction factor (Wu, Shen, & Chou, 1999). By contrast, in fluid mechanics where the movements of the particles in a flow field are described, the condition close to the wall is an unknown that must be included as a boundary (Panton, 2013). Therefore, a way to describe the average flow field is through a hypothesis, in which, at the bottom, the particles have no slip and the outer field away from the wall follows a pattern similar to that used in the empirical relationships.

The development of the velocity profile is a function of the fluid’s properties, such as viscosity (Schlichting & Gersten, 2017). In the case of flow in a canal or natural channel, the law of shear stress for the laminar sublayer considering a Newtonian fluid is represented as follows:

τ = μ u - y

where τ is the shear stress, μ the dynamic viscosity, y the vertical coordinate and u - the average velocity.

For the case of the outer region and different flow conditions (mainly the turbulent one), the shear stress is not constant, since it is proportional to the variation in the average velocity with respect to the vertical coordinate (Schlichting & Gersten, 2017). From this, Prandtl introduced the parameter of shear velocity (u * ) in the hydraulic field to represent the shear stress of the entire section, this by means of an expression that relates the shear stress at the wall (τ 0 ) and the fluid density (ρ) (Equation 2) (Keulegan, 1938).

u * = τ 0 ρ  

In the outer region of the velocity field of a turbulent flow, the value of the shear stress maintains its influence, even when the local gradient of the profiles is zero ( u - / y = 0 ); this allows estimating the shear velocity value by measuring the outer flow field (Nezu & Nakagawa, 1993). In this study, this hypothesis is assumed to characterize the fully developed turbulent flow with a free surface.

In studies made in rough-bottomed ducts, it has been observed that turbulence production energy is immediate to the bottom and greater than in the case of smooth bottoms. This is due to the diameter of the bottom particle, which, due to being big offers greater resistance to fluid movement, generates a higher friction stress (Cebeci & Chang, 1978; Coleman & Alonso, 1983; Zanoun, Durst, & Nagib, 2003). Therefore, the shear model is related to the deformation of the velocity profile and the roughness of the wall.

Due to the characteristics represented by the shear velocity in a turbulent flow, it has been applied in diverse subjects such as: empirical model approach to mean velocity and turbulence intensity distribution (George, 2007; Nezu & Nakagawa, 1993), sediment transport (Celestini, Silvagni, Spizzirri, & Volpi, 2007; van Rijn, 1984), scouring and characterization of roughness in articulated hydraulic concrete mats for the protection of channels. In more recent years, it has been applied in the study and characterization of turbulent flows with the dimensionless relationship approach in turbulent correlation models (Auel, Albayrak, & Boes, 2014; Motlagh & Taghizadeh, 2016; Qiao, Delavan, Nokes, & Plew, 2016).

According to the literature review, different methods have been used to estimate shear velocity; however, the procedure is not explicitly indicated and only the theoretical basis of determination is presented (Auel et al., 2014). For this reason, the aim of this work was to present a methodology that allows experimentally determining shear velocity, considering the log-law as a model of velocity distribution in the outer region of turbulent flow.

Materials and methods

von Kármán’s mean velocity distribution model

Different models to represent velocity distribution in an open channel flow are found in the literature (Spalding, 1961); however, in the practice of hydraulic engineering, the most widely used model is that of von Kármán, which has three conditions. The first is to represent the viscous or laminar sublayer where the turbulent stresses have no influence and the viscous forces (v) dominate (Equation 3).

u - u * = y u * ν ;   i n   t h e   r a n g e   o f       0 y u * ν < 5

The second represents the transition zone, also called the buffer zone, where the turbulent and viscous stresses are of the same order (Shih, Povinelli, Liu, Potapczuk, & Lumley, 1999) (Equation 4).

u - u * = 5 ln y u * ν - 3.05 ;   i n   t h e   r a n g e   o f   5 y u * ν < 30

The third considers the outer zone or turbulent region where the viscous stresses have no influence (Equation 5). This condition is known as the log-law (George, 2007).

u - u * = 1 κ ln y u * ν + A ;   i n   t h e   r a n g e   o f     y u * ν 30

The log-law has two empirical variables: the von Kármán constant (κ) and the additive constant (A). According to Zanoun et al. (2003), the values of 0.4 for κ and 5.5 for A, parameters used in practice for an open channel flow, are considered adequate (Equation 6).

u - u * = 1 0.4 ln y u * ν + 5.5

Each of the equations comprising the von Kármán model has a validity range as a function of the parameter yu * /v and indicates the evaluation zone of the three flow regions: viscous, transition and outer sublayer.

Shear velocity estimation

For the experimental study it is important to know the boundaries of the outer region (Equation 6). According to the experimental results of Tominaga and Nezu (1992), the point closest to the bottom in mean velocity flows in the vertical u - m less than 40 cm·s-1 must be deeper than 3 mm (y > 3 mm) and in the case of velocity flows u - m greater than 40 cm·s-1 it is permissible to consider measuring points 1 mm from the bottom (y ≥ 1 mm).

Equation 6 uses the estimate of the averaged flow velocity u - ; however, in this case, the estimation of the instantaneous shear velocity (u) was done with separation of velocity scales, one is the averaged and the other the fluctuating u = u - + u ' . The measuring equipment used for this case was an acoustic Doppler velocity meter (ADVM) with a maximum sampling frequency of 100 Hz.

In order to process the experimental data, the velocity measurement pertaining to the outer zone was assessed to discard the points outside the range of analysis. Subsequently, the shear value was estimated with the instantaneous velocity ( u i , j ) using Equation (7).

u * i , j = u i , j 1 0.4 ln y u * i , j ν + 5.5

The subscript i indicates the analysis point in the velocity profile and j the temporal sample number. To be able to include the high frequency, the mean velocity u - was replaced by the instantaneous velocity u i , j .

Equation 7 is not explicit since the shear velocity is found on both sides of the expression; therefore, for its solution, the fixed-point algorithm described by Burden and Faires (2011) was applied.

According to Schmid and Lazos-Martínez (2000), in the measurement of a random variable, the results generally follow, in good approximation, a normal distribution. In the case of flow velocity, there are results that validate this hypothesis to a certain degree. This subject is widely discussed by Frisch (1995) and Davidson (2004); however, in this case, the probability density function (PDF) of the data obtained by evaluating Equation 7 is unknown. Therefore, the final value representative of the shear velocity u * was calculated with the median of the u *i,j values obtained.

Anderson-Darling test of the shear model

As part of the data analysis, the Anderson-Darling normality test was performed to determine if the data follow a normal distribution, this by calculating the statistical value of the test (Equation 8).

A 2 = - i = 1 n 2 i - 1 [ ln z i + ln ( 1 - z n + 1 - i ) ] n   - n

According to Stephens (1974), the critical value is 0.754 for a significance level of 5 % and when the number of data (n) is greater than 100. Therefore, if the statistical value A 2 , of the test for the sample, is less than 0.754, it is accepted that the data follow a normal distribution; otherwise, the possibility that the data follow a central tendency distribution is rejected.

Due to the fact the PDF was unknown, and in order to ensure 50 % coverage of the data around the representative value of u * , the location of the first (Q 1 ) and third (Q 3 ) quartile of the data ordered in increasing form was calculated (Equations 9 and 10, respectively).

Q 1 = n + 1 4

Q 3 = 3 n + 1 4

From the location of the first and third quartiles the u *i,j values were located and the interquartile range was obtained.

Experimental station

The tests were carried out in the variable slope experimental channel at the Hydraulic Laboratory of the Mexican Institute of Water Technology (IMTA, for its Spanish acronym); the channel has a smooth, metallic, rectangular-shaped bottom with a 0.245-m-wide base and 5 m long (Figure 1). The experimental station has a 10-hp pump that supplies the flow, a measuring weir calibrated with ISO standard 1438 (International Organization for Standardization, 2008), and a valve to regulate the flow, among other components that allow for stable experimental conditions.

Figure 1. Experimental station.

For measuring the instantaneous velocities, the Nortek® Vectrino Profiler™ ADV device was used; it allows sampling the three velocity components in a profile of up to 30 mm, with a separation of 1 mm between cells and with a sampling frequency of 1-100 Hz. The device was positioned 3.5 m away from the flow inlet to avoid defects in the velocity profile caused by the inflow of the flow or by its outflow in free fall to the recirculation tank.

The channel slope was slightly modified with a mechanical jack, taking care that the experimental conditions complied with the criteria of repeatability and reproducibility. The Froude (F r ) and Reynolds (R e ) numbers were calculated with Equations 11 and 12, respectively.

F r = u - m g   Y

R e = u - m R h ν

All the tests were done in a subcritical regime, as indicated by the Froude number values in Table 1 (F r < 1). In addition, the flow was considered fully developed turbulent, with the Reynolds number values being sufficiently high and far from the transition range (R e > 1.2 x 104).

Table 1. Experimental conditions to determine the shear velocity of an open channel flow.

Case Slope(S, mm -1 x 10 -4 ) Flow depth (h, cm) Aspect ratio (b/h, mm-1) Mean velocity ( u - m , cm-1) Froude number (Fr) Reynolds number (Re , x 104)
P-01 1.06 7.42 3.30 60.65 0.71 2.80
P-02 1.06 9.80 2.50 73.06 0.75 4.01
P-03 1.06 11.34 2.16 71.65 0.67 4.22
P-04 1.06 12.55 1.95 84.51 0.76 5.24
P-05 4.25 12.53 1.96 84.65 0.76 5.24
P-06 4.25 11.16 2.20 75.84 0.72 4.43
P-07 4.25 9.60 2.55 72.46 0.74 3.90
P-08 4.25 7.74 3.17 57.15 0.65 2.71
P-09 2.12 8.14 3.01 55.29 0.62 2.70
P-10 2.12 9.03 2.71 59.86 0.64 3.11
P-11 2.12 10.90 2.25 70.53 0.68 4.07
P-12 2.12 12.55 1.95 81.42 0.73 5.05
P-13 6.38 12.42 1.97 86.99 0.79 5.36
P-14 6.38 11.26 2.18 76.18 0.72 4.47
P-15 6.38 9.69 2.53 68.70 0.70 3.72
P-16 6.38 7.61 3.22 57.12 0.66 2.68

For all tests, the measurement of the instantaneous velocities was carried out in a 12-mm profile, as close as possible to the wall, with a frequency of 100 Hz and a sampling time of 30 s.

Results and discussion

Tests

Figure 2 shows the results of a sampling of the instantaneous velocity with F r = 0.67 and R e = 4.22 x 104 for cell five at 8.6 mm depth.

Figure 2. Record of instantaneous velocity (u) (main direction, x ) taken at the Hydraulic Laboratory of the Mexican Institute of Water Technology (IMTA).

The tests were conducted under the experimental conditions shown in Table 1.

The results of the processing of the experimental data are presented in Table 2, where the statistical value obtained from the Anderson-Darling test with a 5 % degree of significance and the shear velocity obtained with the model of Equation 7 are indicated.

Table 2. Results of the shear velocity and Anderson-Darling statistical value at 5 %.

Case Shear velocity (u* , cm-1) Values of the interquartile range (Q1 - Q3 , cm-1) Anderson-Darling statistical value (A2 )
P-01 3.06 2.85 - 3.23 157.14
P-02 3.55 3.31 - 3.76 14.57
P-03 3.62 3.37 - 3.84 251.64
P-04 4.04 3.81 - 4.25 74.92
P-05 4.08 3.82 - 4.29 124.22
P-06 3.73 3.47 - 3.96 881.34
P-07 3.47 3.24 - 3.69 1.33
P-08 2.96 2.76 - 3.14 42.43
P-09 3.00 2.80 - 3.17 279.14
P-10 3.15 2.94 - 3.33 184.24
P-11 3.55 3.30 - 3.78 13.29
P-12 3.95 3.71 - 4.16 48.31
P-13 4.16 3.93 - 4.37 104.93
P-14 3.76 3.53 - 3.97 19.82
P-15 3.52 3.31 - 3.72 13.05
P-16 3.01 2.79 - 3.20 43.40

The statistical values of the Anderson-Darling test (Table 2) corroborate that the PDF of the data u *i,j does not follow a central tendency, since in all cases A 2 > 0.754; therefore, it is correct to represent the shear velocity value u * with the statistical median and make use of the interquartile range to obtain 50 % coverage around the representative value.

Figure 3 shows four dimensionless profiles drawn from the estimation of the shear velocity with a different bottom slope. It can be seen that the von Kármán model represents, with good approximation, the experimental values in their averaged condition.

Figure 3. Dimensionless profiles of mean velocities, the von Kármán model and experimental values.

In the profiles of tests P-01, P-05 and P-09, a sampling point is observed in the transition zone; this case was discarded from the data analysis since the logarithmic flow model is only for the outer region.

Conclusions

The methodology presented shows low uncertainty in the estimation of shear velocity, which can be observed in the results of the logarithmic velocity profiles. The model is also presented explicitly to obtain the shear velocity value. The Anderson-Darling test showed that the results, when evaluating the instantaneous velocity, do not follow a normal distribution, so the median is the statistical parameter to define the shear velocity value.

The application of the methodology can be extended to the use of low-frequency sampling or conventional instruments, for example a Prandtl tube, a current meter and even acoustic Doppler current profilers mounted on a mobile boat or anchored at the bottom of the channel.

Acknowledgments

  • The authors thank the Mexican Institute of Water Technology (IMTA) for allowing them to carry out the present research within its facilities, as well as the National Science and Technology Council (CONACYT) and the National Autonomous University of Mexico (UNAM) for the funds provided for this work.

References

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Motlagh, S. Y., & Taghizadeh, S. (2016). POD analysis of low Reynolds turbulent porous channel flow. International Journal of Heat and Fluid Flow, 61, 665-676. doi: 10.1016/j.ijheatfluidflow.2016.07.010

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Schmid, W. A., & Lazos-Martínez, R. J. (2000). Guía para estimar la incertidumbre de la medición. México: Centro Nacional De Metrología. Retrieved from http://depa.fquim.unam.mx/amyd/archivero/GUIAPARAESTIMARLAINCERTIDUMBRE(CENAM)_26566.pdf

Shih, T. H., Povinelli, L. A., Liu, N. S., Potapczuk, M. G., & Lumley, J. L. (1999). A generalized wall function. Springfield: Nasa Center for Aerospace Information. Retrieved from https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19990081113.pdf

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Tominaga, A., & Nezu, I. (1992). Velocity profiles in steep open‐channel flows. Journal of Hydraulic Engineering , 118(1), 73-90. doi: 10.1061/(ASCE)0733-9429(1992)118:1(73)

Van Rijn, L. C. (1984). Sediment transport, part I: Bed load transport. Journal of Hydraulic Engineering , 110(10), 1431-1456. doi: 10.1061/(ASCE)0733-9429(1984)110:10(1431)

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Zanoun, E. S., Durst, F., & Nagib, H. (2003). Evaluating the law of the wall in two-dimensional fully developed turbulent channel flows. Physics of Fluids, 15(10), 3079-3089. doi: 10.1063/1.1608010

Figures:

Figure 1. Experimental station.
Figure 2. Record of instantaneous velocity (u) (main direction, x ) taken at the Hydraulic Laboratory of the Mexican Institute of Water Technology (IMTA).
Figure 3. Dimensionless profiles of mean velocities, the von Kármán model and experimental values.

Tables:

Table 1. Experimental conditions to determine the shear velocity of an open channel flow.
Case Slope(S, mm -1 x 10 -4 ) Flow depth (h, cm) Aspect ratio (b/h, mm-1) Mean velocity ( u - m , cm-1) Froude number (Fr) Reynolds number (Re , x 104)
P-01 1.06 7.42 3.30 60.65 0.71 2.80
P-02 1.06 9.80 2.50 73.06 0.75 4.01
P-03 1.06 11.34 2.16 71.65 0.67 4.22
P-04 1.06 12.55 1.95 84.51 0.76 5.24
P-05 4.25 12.53 1.96 84.65 0.76 5.24
P-06 4.25 11.16 2.20 75.84 0.72 4.43
P-07 4.25 9.60 2.55 72.46 0.74 3.90
P-08 4.25 7.74 3.17 57.15 0.65 2.71
P-09 2.12 8.14 3.01 55.29 0.62 2.70
P-10 2.12 9.03 2.71 59.86 0.64 3.11
P-11 2.12 10.90 2.25 70.53 0.68 4.07
P-12 2.12 12.55 1.95 81.42 0.73 5.05
P-13 6.38 12.42 1.97 86.99 0.79 5.36
P-14 6.38 11.26 2.18 76.18 0.72 4.47
P-15 6.38 9.69 2.53 68.70 0.70 3.72
P-16 6.38 7.61 3.22 57.12 0.66 2.68
Table 2. Results of the shear velocity and Anderson-Darling statistical value at 5 %.
Case Shear velocity (u* , cm-1) Values of the interquartile range (Q1 - Q3 , cm-1) Anderson-Darling statistical value (A2 )
P-01 3.06 2.85 - 3.23 157.14
P-02 3.55 3.31 - 3.76 14.57
P-03 3.62 3.37 - 3.84 251.64
P-04 4.04 3.81 - 4.25 74.92
P-05 4.08 3.82 - 4.29 124.22
P-06 3.73 3.47 - 3.96 881.34
P-07 3.47 3.24 - 3.69 1.33
P-08 2.96 2.76 - 3.14 42.43
P-09 3.00 2.80 - 3.17 279.14
P-10 3.15 2.94 - 3.33 184.24
P-11 3.55 3.30 - 3.78 13.29
P-12 3.95 3.71 - 4.16 48.31
P-13 4.16 3.93 - 4.37 104.93
P-14 3.76 3.53 - 3.97 19.82
P-15 3.52 3.31 - 3.72 13.05
P-16 3.01 2.79 - 3.20 43.40