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COMPUTATION OF BED SHEAR STRESS FROM VELOCITY MEASUREMENTS IN A GRADUALLY VARYING ROUGHNESS BED

Vijit RATHORE(1), Nadia PENNA(2), Subhasish DEY(3) and Roberto GAUDIO(4)

(1,3) Indian Institute of Technology, Kharagpur, West Bengal, India, vijitrathore@iitkgp.ac.in, sdey@iitkgp.ac.in

(2,4) Dipartimento di Ingegneria Civile, Università della Calabria, Rende (CS), Italy,

nadia.penna@unical.it, gaudio@unical.it

ABSTRACT

An experimental study was conducted in a gradually varied roughness bed by using a Particle Image Velocimetry (PIV) system. The local bed shear stress was evaluated from four distinct methods: the log-law, the Reynolds shear stress (RSS), the vertical turbulence intensity and the turbulent kinetic energy (TKE) methods. These methods provided significantly different results owing to discrepancies in the boundary conditions and change in local flow structures resulting from the bed roughness.

Keywords: Fluvial Hydraulics, Turbulent Flow, Open-Channel Flow, Bed Shear Stress

1. INTRODUCTIONRivers originate from highlands or mountains and flow towards other rivers, lakes, the sea or ocean. Along

the course of a river, the streambed consisting of local geologic materials changes from place to place; for instance, large boulders in mountains, cobbles to very coarse gravels in foothills, coarse to fine sand in flood plains, and coarse silt to very fine clay near to estuary are evident. Hence, the gradual change in bed roughness is a natural phenomenon of great importance in hydraulic engineering. Besides, gradual change in bed roughness are also observed in the places where there exists a chance of river bed degradation and undermining of the sediments, such as downstream of culverts, bridges and dams. It is important to mention that, owing to such gradual change in bed roughness/bed alterations, an internal boundary layer (IBL) starts to developing. The IBL can be defined as the flow layer up to which the effects of this change in bed roughness prevail. However, so far, most of the previous studies were carried out aiming at analyzing the IBL by considering an abrupt change in bed roughness (Antonia and Luxton, 1971; Antonia and Luxton, 1972). Tominaga and Sakaki (2010) and Bagherimiyab and Lemmin (2013) estimated the shear velocity and the bed shear stress from the experimental measurements on a loose mixed-gravels open-channel flow and a gravel-bed river with local non-uniformity, respectively. However, experimental investigation on the gradual change in bed roughness in an open-channel flow is yet to be explored and its effects on the spatial distribution of local bed shear stress needs further investigation.

Thus, this study focuses on the variation of bed shear stresses with downstream distance owing to a gradual change in bed roughness from smooth to rough. The velocity measurements were taken with a two-dimensional Particle Image Velocimetry (PIV) system. Several methods, bed slope, wall similarity, logarithmic velocity profile, RSS and TKE methods were proposed and applied to the experimental data to predict the bed shear stress, whose determination for a varying bed roughness remains a challenging task (Clauser, 1954; Dyer, 1986; Kim et al., 2000).

2. EVALUATION METHODS OF BED SHEAR STRESSNezu and Nakagawa (1993) suggested four different methods to calculate the bed shear stress. These

methods include the estimation of the bed shear stress from the bed slope, velocity distribution, RSS distribution, and direct measurements. Later, Galperin et al. (1988) proposed some elementary relationships between TKE and bed shear stress, which was further verified by Soulsby and Dyer (1981) and Stapleton and Huntley (1995). Nevertheless, Kim et al. (2000) recommended a slight modification in the TKE method. They estimated the bed shear stress using the vertical Reynolds normal stress, since the instrumental noise errors related to vertical velocity fluctuations were insignificant with respect to the streamwise velocity fluctuations (Voulgaris and Trowbridge, 1998). However, Lopez and Garcia (1999) and Hurther and Lemmin (2000) proposed a different method based on the

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assumption proposed by Townsend (1976), which is known as wall similarity method. All these methods are based on several assumptions for computing the bed shear stress. Therefore, in this study the efficiency of each method in estimating the bed shear stress is evaluated in a gradual change in bed roughness condition.

2.1 Bed slope method One of the most common and simple approach to compute the bed shear stress is from bed slope. The

following relationship was derived for a two-dimensional steady-uniform flow:

0bgRS [1]

where b is the bed shear stress, ρ is the mass density of the fluid, g is the gravitational acceleration, R is the hydraulic radius and S0 is the bed slope, which is equal to the energy slope for a uniform flow. However, such kind of assumptions are not valid in the case of high roughness, where the flow is three-dimensional and locally unsteady-nonuniform. In most of the cases, this relationship acts as a reference for the computation of the bed shear stress.

2.2 Wall similarity method Townsend (1976) proposed the concept of wall similarity, which states that in a uniform turbulent boundary

flow with a high Reynolds number an extended depth exists where both the turbulent production and dissipation are nearly in equilibrium and turbulent diffusion is negligible, unhampered with the flow and bed roughness conditions. This layer ranges from 0.15h to 0.6h, where a balance between TKE production and dissipation prevails. Hurther and Lemmin (2000) normalized the vertical flux of TKE with the cube of shear velocity u* in the following form:

2 2 2

3

*

1

2k

u v w w Fu

[2]

where u′, v′ and w′ are the velocity fluctuations in the streamwise, spanwise and vertical direction, respectively. The overbar designates the time averaged values, and Fk is a constant of proportionality. They observed that the value of Fk was varying between 0.28 to 0.30 for 0.25 ≤ z/h ≤ 0.75. However, Lopez and Garcia (1999) obtained the value of Fk =0.30 in the region 0.15 ≤ z/h ≤ 0.70. Hence, u* can be estimated from the vertical TKE flux at any position inside the range 0.15 ≤ z/h ≤ 0.60. Since the range of z/h is far from the bed surface and quite wide, an accurate understanding of flow depth and the bed surface level is not a requisite for the determination of the bed shear velocity. Further, the u* is having a 1/3rd power relationship; hence, this method is not excessively dependent on the errors caused in the measurement of the turbulence data. Therefore, this method is suitable for field measurements of the bed shear stress.

2.3 Logarithmic velocity profile method (Clauser method) The bed shear stress can also be computed from the time-averaged velocity profile, but this method is only

applicable below 0.2h, where h is depth of flow. The applicability of this method relies on the condition that eddy viscosity should be a linear function of z, where z is the vertical distance from the bed (Schlichting, 1979).

*

0

lnu z

uz

[3]

where u is time-averaged velocity at an elevation z from the bed surface, z0 is the zero velocity level, and κ is von

Kármán constant. The z0 is defined as the height above the bed surface at which the time-averaged velocity is zero (Monin and Yaglom 1971). The time-averaged velocity data can be fitted to eq. (3) by the least-squares method and the shear velocity can be determined by assuming κ = 0.41. This method is popularly known as the Clauser method (Clauser, 1954). Using the u*, the bed shear stress can be determined as follows:

22

*bu m [4]

where m is the slope of the line obtained from the least-squares method. This method is widely used in field and open-channel flows (Nezu and Nakagawa 1993). However, it gives inaccurate results inside the roughness layer in gravel beds (Nikora and Goring 2000; Nikora et al. 2001), in the flow where grain size is of the same order of flow depth (Nikora et al. 2001; Lamb et al. 2017) or flow with large roughness elements such as gravels, ripples, and vegetation (Etminan et al. 2018).

2.4 RSS method The bed shear stress can be directly estimated from the available RSS measurements obtained from the

turbulence measurements in the constant shear layer. The near bed RSS can be estimated as:

bu w [5]

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In a uniform rough-bed open-channel flow, the RSS varies linearly in the outer layer. Therefore, by extrapolating the linear trend of RSS to the bed, it is possible to obtain the bed shear stress. (Nikora and Goring 2000; Nezu and Nakagawa 1993). In the case of large roughness elements, the flow used to get accelerated or decelerated in the near-bed flow zone, in such cases the RSS does not follow a linear trend and the applicability of this method becomes questionable.

2.5 TKE method Galperin et al. (1988) proposed some elementary relationships between the TKE and the bed shear stress, which was later verified by a number of investigators (e.g., Soulsby and Dyer, 1981; Stapleton and Huntley, 1995). According to them, the TKE and bed shear stress are proportional to each other and lead to the following relationship:

1

2

b kC k k u u v v w w [6]

where k is the TKE and Ck is a proportionality constant. Kim et al. (2000) verified this relationship and found the value of proportionality constant approximately equal to 0.19. However, this method still needs more experimental studies to confirm the universality of Ck. Nezu and Nakagawa (1993) also proposed the similar kind of expression for RSS and TKE. They suggested that in the inner layer the coefficient of correlation between RSS and TKE has

a value close to 0.1, which results in - u w ≈ 0.2k, justifying the above expression.

2.6 Vertical Reynolds normal stress method Kim et al. (2000) revised the relationship between TKE and bed shear stress. They considered only the

variance of the vertical velocity component, since errors caused by acoustic instruments associated with the acquisition of vertical velocity fluctuations are insignificant with respect to those of the streamwise fluctuations (Voulgaris and Trowbridge 1998). Therefore, the bed shear stress can be defined as follows:

b zC w w [7]

where Cz is a constant of proportionality, which is approximately equal to 0.9 (Kim et al., 2000).

3 EXPERIMENTAL SETUP Experimental investigations were performed at the “Grandi Modelli Idraulici” Laboratory, Università della

Calabria, Italy. A tilting rectangular flume was utilized for the experimental investigations. The length, width and depth of the flume were 9.6 m, 0.485 m and 0.5 m, respectively. A stilling basin, an uphill spillway, and flow straighteners designed to dampen the large eddies were installed at the inlet of the flume. Flow depth was regulated by a downstream tailgate, which discharges flow into a downstream tank. This tank was equipped with a Thomson weir for the flow discharge measurements. One of the flume wall was constructed of glass and another wall of the flume was made of opaque plastic sheet. The glass wall allowed us to visualize the flow from the outside. Schematic of experimental setup is shown in Figure 1.

To measure the flow domain in the flume, a two-dimensional (2D) Particle Image Velocimetry system with a frequency of 15 Hz manufactured by TSI was used. The data acquisition was administered by the INSIGHT 4G-2DTR software and the same software was used to post-process the acquired data. The flow was fed with the particles of Titanium dioxide having a mean size of 3 μm and density of 4.26 × 103 kg m–3 as seeding particles.

The field of view of each image was 190 × 190 mm2, which was divided into small interrogation areas having a resolution of 32 × 32 square pixels. Hence, each interrogation area was made up of 1.5 × 1.5 mm2. The number of interrogation areas was optimized in the preliminary tests. All measurements were conducted along the centerline of the flume in a vertical plane at -0.13 m, -0.019 m, 0.09 m, 0.020 m, 0.031 m, 0.042 m, 0.053 m, 0.064 m and 0.076 m from the change in bed roughness (x = 0 m), where x is the distance in streamwise direction. As measurements 3000 pairs of images were captured in order to evaluate the flow field over a period of 200 s at each section. This time of evaluation is long enough, as suggested by Buffin-Bélanger and Roy (2005) and Detert et al. (2010) in boundary layer flows. Correlation peaks and the average displacement of particles in both the frames over the inspection area with a sub-pixel precision were obtained from Gaussian interpolation. This process resulted in 75 vertical velocity profiles along the streamwise direction over an area of 110 × 150 mm2 with a spatial resolution of 1.5 × 1.5 mm2 in both directions.

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Figure 1. Schematic of experimental set-up

The slope of the flume was kept at 1:1000. The bed roughness was gradually changed from smooth (d50 = 1.53 mm) to rough (D50 = 17.97 mm) over a stretch of 800 mm along the downstream length shown in Figure 2. Here, d50 is the median sediment and D50 is the median size of the downstream sediment. A linear variation of the bed roughness from smooth to rough was considered, indicating that the intermediate bed roughness has four different sediment sizes (d50 = 2.82, 6.20, 10.55 and 14.40 mm), each of which prevails over a length of 200 mm. Experiments were conducted with a Froude number of 0.2. The bed topography was measured by using the photogrammetry technique. Specifically, for each experimental test, about 70 photographs were taken with a Nikon D5200 camera, equipped with Nikkor 18-55mm f/3.5-5.6G VR lens, from the left and right walls of the flume, covering a distance of about 2 m.

Figure 2. Design of gradually varied roughness

4 RESULTS AND DISCUSSION

4.1 Anisotropic turbulence In isotropic turbulence, the turbulence structures do not have any directional preference and preserve a

perfect chaotic motion. However, any perturbation in open-channel flow condition is reflected as the biasness in turbulence level. It also represents up to which the effects of the perturbation prevail. The anisotropy level of turbulence is shown in the form of the ratio of vertical turbulence intensity to streamwise turbulence intensity,

wrms/urms, where urms = ( u u )0.5 and wrms = ( w w )0.5. The variation of wrms/urms with z/h is presented in Figure 3 for

different x/D50. It is observed that the wrms/urms starts with a value of 0.05 near the bed, varying linearly with z/h and attains a value of 0.63 at z/h = 0.3. The effects of roughness fade away in the main flow region and the ratio wrms/urms obtains a constant value. Near the free surface, the vertical turbulence intensity diminishes with respect to the streamwise turbulence intensity. Similar kind of pattern is observed at all x/D50. Nezu and Nakagawa (1993) stated that this ratio is universal and obtained a constant value of 0.55 for smooth boundary, while Dey and Raikar (2007) obtained a constant value of 0.6 for feebly mobile gravel bed. Dey and Nath (2010) obtained a linearly varying pattern, a value less than 0.5 at z/h = 0.65 for flows in immobile gravel bed subjected to injection and suction. From the results, it is found that the gradual change in roughness causes to enhance wrms and urms and reduce the directional preference of the turbulence, which results in higher values of wrms/urms than those observed by Nezu and Nakagawa (1993), Dey and Raikar (2007) and Dey and Nath (2010).

Flow straighteners

Inlet

tank

Hydraulic jack Tripod stand PIV Camera

Thomson weir

Tail gate Laser pulser

1.2 m

Test section

9.6 m

z

d50 = 1.53 mm d50 = 2.82 mm d50 = 6.20 mm d50 = 10.55 mm d50 = 14.40 mm D50 = 17.97 mm

x 200 mm 200 mm 200 mm 200 mm

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Figure 3. Vertical profiles of wrms/urms at different x/D50

4.2 Evaluation of the global bed shear stress

4.2.1 Bed slope method The value of the bed shear stress obtained from this method (Eq. [1]) is 1.47 N m–2 (u* = 0.0384 m s–1).

However, it can be noted that this method does not consider any key parameter related to the bed roughness. Hence, the value obtained from the said method is dubious in our case, as it is applicable only to the steady uniform flow.

4.2.2 Wall similarity method The results obtained from the wall similarity method using Fk = 0.3 are shown in Figure 4. From the figure, it

is visible that u* remains unchanged over a range of 0.3 ≤ z/h ≤ 0.7. It suggests that there is a uniform TKE vertical flux in this range, which validates the applicability of wall similarity method to our case. The value of u* is found to be 0.013 m s–1

, which is less than that obtained from the bed slope method. The behavior of u* is inconsistent below the aforementioned range owing to the presence of the roughness elements.

Figure 4. Shear velocity estimates obtained by the wall similarity method for different x/D50

Using the above two methods a global shear velocity is obtained, which can be used as a normalizing parameter. Note that these methods do not provide any vital details about the local bed shear velocity. 4.2.3 Universal characteristics of turbulence

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The universality of the turbulence quantities was investigated to ensure the applicability of the log-law for the velocity profiles and linear law for the RSS profiles. In Figure 5(a), the vertical variations of the time-averaged streamwise velocity profiles with z/h are shown for different x/D50. The u* calculated from the wall similarity method was used to normalize the time-averaged streamwise velocity ū and the flow depth h was used as normalizing parameter for the vertical distance z. From Figure 5(a) it is apparent that the velocity profiles are in good agreement with the log-law over the range 0.2 ≤ z/h ≤ 1.0. Figure 5(b) shows the vertical variations of the RSS profiles at

different sections normalized with 2

*u . From the figure, it is evident that the RSS profiles follow the linear distribution

above z/h = 0.25. The erratic behavior of the RSS and time-averaged streamwise velocity profiles below z/h = 0.2 is visible owing to the effect of the roughness elements.

Figure 5. Vertical distribution of (a) mean velocity with log-law; (b) RSS with linear law

4.3 Evaluation of the local bed shear stress

4.3.1 Logarithmic velocity profile method (Clauser Method) The ū profiles of all x/D50 for the wall shear layer were linearly fitted to Eq. [3] and it was found that the fitted

profiles follow the measured mean velocity profiles. However, very close to the bed, a deflection from the fitted profile is spotted owing to the presence of the roughness elements, which may change the flow structures, becoming highly three-dimensional (Nikora and Goring, 2000). In all the cases, the regression model had R-square value greater than 99%, which means that all the data points fall on fitted regression model.

4.3.2 RSS method The RSS profile follows a linear distribution above the maxima, which verifies the two-dimensionality of the

flow. However, owing to the presence of the roughness layer, the maximum in RSS is slightly shifted upward. Previous investigators also observed that the RSS may deviate from the two-dimensional profile and also influence the shear velocity under the flow conditions, where the effect of coherent structure prevails (Nezu and Nakagawa, 1993; Albayrak and Lemmin, 2011). Table 1 shows the evaluated bed shear stress along with the R-square value. It is evident that in most of the cases the R-square has a value greater than 80%. However, for x/D50 = 33.39, 38.95, and 55.65 it is found that the regression model has a low variance. Therefore, in these cases the estimated values of the bed shear stress are less reliable.

4.3.3 Vertical Reynolds normal stress method Vertical Reynolds normal stress profiles also follow a linear trend. From Table 1, it can be noted that the

values of the bed shear stress predicted by this method are greater than the values predicted by the RSS method. The R-square values for most of the cases are greater than 80%. However, for x/D50 = 0, 11.13, and 16.69 the R-square values equals 78.89, 70.21, and 62.67, respectively. In such cases, the estimates of bed shear stress by this method is uncertain.

Table 1. Evaluated bed shear stress along with R-square value.

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4.3.4 TKE method The variation of TKE profiles are correlated by the linear trend pretty well. Interestingly, this method predicts

the highest bed shear stress. From Table 1 it is found that in this method the linear regression model is having very high variance with respect to the other methods. The R-square value is greater than 85% in all the cases, suggesting the reliability of this method.

4.4 Comparison of local bed shear stress obtained from aforementioned methods The calculated local bed shear stresses from the above methods are compared in Figure 6. It is seen that

the bed shear stress estimated with the Clauser method is somewhat inconsistent and does not follow a specific trend. On the other hand, in all other methods, the shear stress gradually increases with the downstream distance, as the roughness size increases. Specifically, the RSS method gives the smallest values, whereas the vertical Reynolds normal stress and TKE methods return similar values of the bed shear stress. However, in some cases, the vertical Reynolds normal stress method has low variance of the regression model. Therefore, the applicability of these methods in such cases is questionable. On the other hand, the TKE method is well interpolated with the linear law, suggesting its reliability over the other methods for estimating the bed shear stress.

Figure 6. Comparison of bed shear stresses computed by different methods.

x/D50

Velocity method RSS method Vertical Reynolds

stress method TKE method

b (N m–2)

R2 b

(N m–2) R2

b (N m–2)

R2 b

(N m–2) R2

-5.56 0.335 99.94 0.199 98.40 0.231 95.08 0.243 99.07

0 0.251 99.98 0.172 90.07 0.234 78.89 0.237 97.62

5.56 0.202 99.50 0.196 94.90 0.240 81.68 0.262 99.45

11.13 0.195 99.50 0.188 91.80 0.240 70.21 0.269 98.89

16.69 0.244 99.80 0.185 90.45 0.237 62.67 0.250 95.30

22.26 0.261 99.97 0.199 98.55 0.240 88.72 0.276 97.60

27.82 0.457 99.90 0.199 86.13 0.240 98.70 0.266 93.55

33.39 0.325 99.60 0.210 69.28 0.256 91.67 0.299 90.04

38.95 0.353 99.50 0.228 68.46 0.292 96.44 0.331 91.13

44.52 0.307 99.70 0.266 82.69 0.310 98.65 0.369 92.29

50.08 0.555 99.70 0.272 86.16 0.346 93.28 0.369 92.37

55.65 0.645 99.40 0.320 76.50 0.380 87.82 0.416 87.59

61.16 0.598 99.10 0.350 83.07 0.400 91.97 0.416 90.76

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5 CONCLUSION

Turbulence measurements were performed considering a gradually varied bed roughness in order to verify the applicability of different methods to compute the bed shear stress. At the same time, it was demonstrated that in case of gradual varied roughness, the universal characteristics of turbulence prevails and anisotropy of the turbulence was also investigated. The bed shear stress was computed with data fitting methods for the distribution of the log-law, the RSS, vertical Reynolds normal stress and TKE. The values obtained with the log-law method showed an erratic trend. Instead, all the other methods showed similar kind of trend of the bed shear stress varying the bed roughness along the streamwise direction. However, some of the profiles demonstrated poor correlation with the linear trend, owing to the effects of change in bed topography. The applicability of these methods can provide reasonable estimates of the bed shear stress, such as flow over a smooth bed and a uniform gravel bed. However, the accuracy of these methods in the case of gradually varied bed roughness is still questionable, since the assumptions used to derive them are no more valid.

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