Prediction of bed form height in straight and meandering compound channels

Water Resources Management III 311 Prediction of bed form height in straight and meandering compound channels R. D. Karamisheva, J. F. Lyness, W. R. C. Myers, J. O Sullivan & J. B. C. Cassells School of Built Environment, University of Ulster, UK Abstract A comparative analysis was made of several prediction methods for bed form dimensions. The prediction methods were applied to experimental data obtained from the large-scale UK Flood Channel Facility. Inbank and overbank flows in straight and meandering channels were studied. Comparisons between computed and measured average bed form heights indicated that most formulae predict the bed form height in straight channels better than in meandering channels. The Karim [6] and the Van Rijn [10] methods gave the best predictions of the bed form height for straight channels. For meandering channels, the Julien and Klaassen formula [4] gave the smallest average discrepancy between the predicted and the experimental bed form heights. Overall results indicated that the Karim method [6] gives the best prediction of bed form height with an average discrepancy ratio of 1.23 and 95% of the values within the range of 0.5 2.0. For the studied data, the multiple regression analysis showed that the ratio of bed form height and mean particle diameter, h/d, correlate best to the ratio of dimensionless shear stress and Froude number. Keywords: bed form dimensions, alluvial compound channels, bed form prediction formulae. 1 Introduction The accurate prediction of flow resistance in alluvial channels is very important in hydraulic engineering practice. Flow resistance is difficult to quantify because of the mutual interaction between the flow and the movable bed. The complexity of hydraulic factors influencing the energy loss increases for overbank flows and the meandering planforms of compound channels. The total resistance to flow is

312 Water Resources Management III divided into grain resistance and bed form resistance. The accurate prediction of the geometric characteristics of bed forms is an essential component for estimating the flow resistance and the consequent flow conditions. Bed form geometry in alluvial channels has been investigated intensively in recent decades. In the current study, Karim [6], Julien and Klaasen [4], Karim [5], Van Rijn [10], Allen [2], Ranga Raju and Soni [9] and Yalin [11] bed form prediction methods were studied for bed forms, which were observed for the straight and meandering planforms of the UK Flood Channel Facility. 2 Experimental arrangement The UK Flood Channel Facility (UK FCF) is 45 m long by 8 m wide. The valley slope was 0.001859. The main channel is trapezoidal in section with side slopes inclined at 45 and top width of 2.0m for the straight channel experiments and 1.6m for the meandering channel experiments. For the meandering channel experiments, the main channel was designed to have a channel sinuosity of 1.34. The flood plains were either left with a relatively smooth concrete finish or roughened with 25mm diameter rods. The experimental tests were carried out using a uniform sand with a particle mean diameter, D, of 0.835mm. The sand was screeded to a mean bed level 200 mm below the flood plains. The facility recirculated both sediment and water. The entire sediment load was returned to the upstream end of the channel and equilibrium sediment transport conditions were achieved. The bed form measurements were taken in a drained channel after establishing equilibrium conditions. The bed form profiles were obtained using the HR Wallingford bed profiler with vertical and horizontal accuracies of ±0.5mm and ±0.1mm respectively. During the straight compound channel tests longitudinal bed profiles were taken at 100mm longitudinal and 50mm horizontal intervals. A total of eleven cross-sections along the channel, at every 1/8 of the wavelength were profiled during the meandering compound channel tests. Bed forms were measured normal to the main channel direction at horizontal intervals of 25mm. The bed form height is defined as the vertical distance between the crest and the trough on the downstream face of the bed form. The bed forms were repeating dunes with average heights ranging from 0.02m to 0.12m. Experimental work undertaken in the UK FCF was previously described by Myers et al. [8] and Lyness et al. [7]. 3 Prediction of bed form height To compare the laboratory measurements for the bed form height with the computed results, the discrepancy ratio, r, defined as the ratio between computed bed form heights, h p, and measured bed form heights, h e, was used. The percentages of the discrepancy ratio values within the ranges 0.80-1.25 and 0.50-2.00 were also calculated. The results are summarised in Table 1.

Water Resources Management III 313 Table 1: Summary of accuracy in bed form prediction of the studied methods. Prediction method r=h p /h e % within range Straight Meander Total 0.80-1.25 0.50-2.0 Karim [6] 1.12 1.55 1.23 63.6% 95.2% Julien & Klaasen [4] 1.79 1.42 1.57 13.6% 81.0% Karim [5] 0.54 0.45 0.49 4.5% 52.4% Van Rijn [10] 0.92 0.21 0.62 40.9% 66.7% Allen [2] 0.73 0.60 0.65 13.6% 85.7% Ranga Raja & Sony [9] 0.74 0.56 0.64 9.1% 76.2% Yalin [11] 0.58 0.48 0.52 4.5% 61.9% 3.1 Comparisons between existing bed form prediction formulae Yalin [11] established a relationship between the dimensionless bed form height, h/d, and the dimensionless shear stress, τ o /τ cr : h d 1 τ 1 6 τ = cr (1) where h = the bed form height, d = the flow depth, τ = average shear-stress on the surface of the bed, τ cr = critical shear-stress according to Shield diagram. The Yalin relationship indicates that the maximum dune height is one-sixth of flow depth. The relative dune height for the studied data varies between1/3 and 1/5. The Yalin formula underestimated the bed form height by a factor of two for most data (Figure 1). Allen [2] developed a relationship between the relative dune height, h/d, and the dimensionless bed shear stress, θ/3. He found that the relation between h/d and θ/3 is best fitted by the fourth-order polynomial: h d 2 3 4 θ θ θ θ b0 + b1 + b2 + b3 + b4 3 3 3 3 = (2) where b i = coefficients and θ = the dimensionless shear stress given by: ds θ = (3) (s 1)D Karim [5] proposed a similar relation for the relative bed form height but a parameter including the shear velocity and the fall velocity, U*/w, was chosen as the independent variable. Both the Allen [2] and Karim [5] relationships

314 Water Resources Management III underestimated the bed form height for the UK FCF (Figure 2). The Allen relationship gave better prediction of the bed form height with a discrepancy ratio of 0.65 and 85.7% of the values in the range between 0.5 and 2.0. Ranga Raju and Sony [9] proposed a semi-theoretical approach to predict the bed-form geometry using flume and field data covering a wide range of sediment size: h 3 3 8/ Fr' F2 = 6.5x10 ( θ' ) 3 (4) D where D = the median size of sediment, θ = the dimensionless shear stress parameter related to grains and Fr and F 2 = dimensionless parameters: Fr' = V gr b V F 2 = (5) D(s 1)g where R b = the hydraulic radius related to bed determined from the Vanoni and Brooks method. The Ranga Raju and Sony method underestimated the average bed form height and this underestimation was more significant for the meandering channel data (Figure 1). Van Rijn [10] found that the dune height could be characterised by the transport stage parameter, T, and the dimensionless particle parameter, D*. The following relationship for relative bed-form height, h/d, was determined using a large quantity of flume and field experimental data: 0.3 h D = 0.11 d d T ( 1 e 0.5 ) (25 T) (6) The Van Rijn bed form prediction method has been adopted by a number of researchers but some modifications to the original formula were introduced in some studies (Abdel-Fattah et al., [1], Bennett, [3]). Julien and Klaassen [4] mentioned that for large bed form amplitude the grain shear stress calculated from the logarithmic relationship for the Chezy coefficient reaches the same order of magnitude as the critical shear stress and negative values of T are possible at low transport rates. They used large river data and found that the method underestimates the bed form height at T>8. The Van Rijn method gave excellent prediction of the bed form height for both inbank and overbank flow data form the straight UK FCF, but it significantly underestimated the bed form height for the meandering UK FCF (Figure 2). For some data sets the calculated transport parameters were negative. When the total shear was used instead of the grain shear velocity, the method gave excellent prediction of the bed form height with an average discrepancy ratio between the computed and the measured values of 1.00 and 95% of the values between 0.5 and 2.0.

Water Resources Management III 315 Figure 1: Comparisons between measured and predicted bed form heights using the Yalin [11], Ranga Raju and Sony [9] and Karim [6] formulae. Figure 2: Comparisons between measured and predicted bed form heights using the Allen [2], Van Rijn [10] and Karim [5] formulae.

316 Water Resources Management III Julien and Klaassen [4] proposed a power relationship between the relative bed form height and the particle diameter to flow depth parameter using data for large rivers: 0.3 h D = ξ (7) d d The estimated ξ varies between 0.8 and 8.0 and the average value is ξ=2.5. The eq. (7) used with coefficient ξ=2.5, overestimated the bed form height when applied to the UK FCF. The average value of ξ estimated from the studied data was 1.6 (Figure 3). Used with this coefficient the Julien and Klaassen equation gave a mean discrepancy ratio between the predicted and experimental bed form heights of 1.01 and 100% of the results between 0.5 and 2.0. Figure 3: Comparisons between measured bed form heights and the prediction curve from the Julien and Klaasen formula [4]. Karim [6] proposed a method for predicting relative bed form height based on the concept of relating energy loss due to form drag to the head loss across a sudden expansion in open channel flows: h d 2 Fr L S f ' 8 d = (8) 2 KFr C where K = the energy loss coefficient; S= the energy slope; Fr = the Froude number; f = Darcy-Weisbach friction factor due to grain roughness; L = the bed 1

Water Resources Management III 317 form length; C 1 = dimensionless parameter, function of h/d and a parameter defining the point of flow reattachment. The Karim prediction method [6] is applicable to several bed forms. The author pointed out that further research is needed to determine the values of the loss coefficient, K, more accurately. The Karim method predicted the bed form height very well for the straight UK FCF but overestimated the bed form heights for the meandering UK FCF with rough flood plains (Figure 1). The average value of the discrepancy ratio for all data is 1.23 and 95% of the results are within the range 0.5-2.0, which is the best average result for the data included in this study. 3.2 Relationship between the relative bed form height and different dimensionless parameters The independent variables used in the studied bed form prediction methods are summarised in Table 2 (noted with + in the table). All bed form prediction methods use the flow depth and the mean particle diameter as independent variables. The significance of relationship between the relative bed form height and the dimensionless parameters used in the bed form calculation methods was tested statistically employing regression and correlation analyses. The bed form height was presented in dimensionless form as ratios of bed form height to mean particle diameter, h/d and bed form height to flow depth, h/d. All data were log transformed. The UK FCF data sets, used in the analysis are 21 and comprise a relatively narrow range of flow conditions. Further study is needed in order to compare the present results to a wider range laboratory and field data. To determine whether there is a linear relationship between the relative bed form height and the dimensionless parameters, a correlation analysis was used. The computed correlation coefficients and their significance levels are shown in Table 3. The largest correlation coefficients are found between h/d and the parameters associated with the shear stress, θ and U*/w. Table 2: Comparison of the independent variables for the studied bed form prediction methods. Prediction method S D V d υ s f Dimensionless parameters Karim [6] + + + + + θ, Fr Julien & Klaasen [4] + + D/d Karim [5] + + + + U*/w Van Rijn [10] + + + + + + + T, D*, D/d Allen [2] + + + + θ Ranga Raju & Sony [9] + + + + + + θ, Fr, F2 Yalin [11] + + + + + θ/θcr, Equation (9) + + + + + θ, Fr

318 Water Resources Management III Table 3: Results of the correlation analysis between dimensionless bed form height and dimensionless flow parameters. Parameters θ U*/w Dgr T Fr F2 h/d h/d Correlation coefficient -0.220-0.220-0.546-0.591-0.432-0.579 Significance of t 0.339 0.339 0.010 0.005 0.051 0.006 Correlation coefficient 0.700 0.700-0.346-0.204-0.592-0.128 Significance of t 0.000 0.000 0.124 0.375 0.005 0.580 Multiple linear regression analysis was used to assess the relationship between the dimensionless bed form height and the dimensionless parameters used in the bed form calculation methods. The bed form height to flow depth ratio, h/d, and the bed form height to mean particle diameter ratio, h/d, were used separately as the dependent variables. A step-wise regression technique selected the terms to add as explanatory variables into a linear model. When the ratio h/d was used as a dependent variable, the addition of dimensionless parameters as independent variables did not reduce the residual mean squares. The study of the ratio h/d as a dependent variable showed the best correlation with the dimensionless shear stress, θ, and the Froude number, Fr. The partial regression coefficients for θ and Fr were very close and to simplify the relationship the ratio θ/fr was assumed as a single independent variable: h D 0.64 θ = 79.49 (9) Fr The regression analysis gave a correlation coefficient R 2 =0.64 and significance P<0.001. The relationship between h/d50 and θ/fr is shown in Figure 4. The parameters influencing the bed form height according to eq. 9 are summarised in Table 2. The coefficients in eq. 9 were determined from the experimental data in the present study and the prediction of bed form height using eq. 9, as it can be expected, is very good. The average discrepancy ratio was 1.02. The percentages of the discrepancy ratio values within the ranges 0.80-1.25 and 0.50-2.00 were 71.4% and 100% respectively. 4 Conclusions A comparative analysis between seven different methods for bed form height prediction has been made. Experimental data obtained from straight and meandering planform of the UK Flood Channel Facility were used.

Water Resources Management III 319 Figure 4: Relationship between h/d50 and the dimensionless shear stress θ. The Yalin [11], Ranga Raju and Sony [9], Allen [2] and Karim [5] methods underestimated the average bed form height and these underestimations were more significant for the meandering channel data. The Van Rijn method gave excellent prediction of the bed form height for the straight channel data but it significantly underestimated the bed form height for the meandering channel data. The Julien and Klaassen relationship used with ξ=2.5 overestimated the bed form height but gave excellent prediction if the value of ξ is taken as 1.6. The results showed that the Karim method [6] gives the best agreement with the observed bed form heights for both straight and meandering flows. The multiple regression analyses with the UK FCF data showed the best correlation between the ratio of bed form height and mean particle diameter and the ratio of dimensionless shear stress and Froude number. References [1] Abdel-Fattah S., Amin A., Van Rijn L.C., Sand transport in Nile River, Egypt, Journal of Hydraulic Engineering, 130(6), pp. 488-499, 2004. [2] Allen J.R.L, Computational models for dune time-lag: calculations using Stein s rule for dune height, Sedimentary geology, 20, pp. 165-216, 1978. [3] Benett J.P., Algorithm for resistance to flow and transport in sand bed channels, Journal of Hydraulic Engineering, 121(8), pp. 578-590, 1995. [4] Julien P.Y. and Klaassen G.J., Sand-dune geometry of large rivers during floods, Journal of Hydraulic Engineering, 121(9), pp. 657-663, 1995. [5] Karim F., Bed configuration and hydraulic resistance in alluvial-channel flows, Journal of Hydraulic Engineering, 121(1), pp. 15-25, 1995.

320 Water Resources Management III [6] Karim F., Bed-form geometry in sand-bed flows, Journal of Hydraulic Engineering, 125(12), pp. 1253 1261, 1999. [7] Lyness JF, Myers WRC, O Sullivan JJ, Hydraulic characteristics of meandering mobile bed compound channels, Proceeding of Institution of Civil Engineers: Water, Maritime & Energy, 130, pp. 179 188, 1998. [8] Myers WRC, Lyness JF, Cassells J, Influence of boundary roughness on velocity and discharge in compound river channels, Journal of Hydraulic Research, 39(3), pp. 311-319, 2001. [9] Ranga Raju K.G. & Soni J.P., Geometry of ripples and dunes in alluvial channels, Journal of Hydraulic Research, 14(3), pp. 241 249, 1976. [10] Van Rijn L.C., Sediment transport, part III: Bed forms and alluvial roughness, Journal of Hydraulic Engineering, 11(12), 1733 1754, 1984. [11] Yalin M.S., Geometrical properties of sand waves, Journals of Hydraulics Division, ASCE, 90(HY5), pp. 105-119, 1964.