MEASUREMENTS OF BED SHEAR STRESS, VELOCITY AND REYNOLDS STRESS AROUND ARRAYS OF VERTICAL CYLINDERS

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1 Full Paper 11 th ISE 216, Melbourne, Australia MEASUREMENTS OF BED SHEAR STRESS, VELOCITY AND REYNOLDS STRESS AROUND ARRAYS OF VERTICAL CYLINDERS KENJIROU HAYASHI Dept. of Civil and Environmental Eng., The National Defense Academ, 1-1-2, Hashirimizu Yokosuka, Kanagawa , Japan TUYOSHI TADA Dept. of Civil and Environmental Eng., The National Defense Academ, 1-1-2, Hashirimizu Yokosuka, Kanagawa , Japan RYOU SAITOU Dept. of Civil and Environmental Eng., The National Defense Academ, 1-1-2, Hashirimizu Yokosuka, Kanagawa , Japan In this stud, eperiments were undertaken to measure velocit fields and bed shear stresses in an open channel with and without surrogate vegetation (vertical clinders). Velocit and Renolds stress distributions were measured using a two-component laser Doppler anemometer (2D-LDV), while bed shear stresses were measured directl using a one-component load cell. In the smooth bed, non-vegetated case, close agreement was found between the bed shear stress measured b the load cell and the values estimated from the velocit and Renolds stress distributions. Friction coefficient values increase with arrangement densit of clinders. 1 INTRODUCTION In order to understand the effect of tall vegetation on sediment transport processes in vegetated open channels, it is important to evaluate the bed shear stress. However, the characteristics of the near-bed flow patterns and bed shear stress are not well known. Sumer et al. [1] measured the bed shear stress around a base of a vertical wallmounted clinder eposed to stead currents using a Dantec 55R46Spec two-component hot-film probe. The found that, in stead flows, the horseshoe vorte formed at the base of a clinder causes an order of magnitude amplification of the bed shear stress relative to its undisturbed value and that this maimum occurred at an angle of about 45 from the upstream edge of the clinder. In addition, the installation of multiple rows of vertical clinders on a smooth bed caused the flow to become hdraulicall rough [2-4]. In this stud, eperiments were undertaken to measure velocit fields and shear stresses in an open channel with and without vertical clinders installed in multiple rows. Velocit and Renolds stress distributions were measured using a two-component laser Doppler anemometer (2D-LDV), while the bed shear stress was measured directl using a one-component load cell. 2 EXPERIMENTS TO MEASURE BED SHEAR STRESS 2.1 Eperimental set-up Eperiments were conducted in a.8 m wide, 1. m deep, 4 m long recirculating open channel at the Hdraulic Laborator of the National Defense Academ, Yokosuka in Japan (Figure 1). Two clinder arrangements were installed in multiple rows on the open channel bed (Figures 1 and 2). In Case A, stainless-steel clinders with diameter D = 1 cm and water depth d = 33 cm were arraed over a length of 468 cm. The distance between the clinders was S = 1cm. In Case B, acrlic resin pipes with diameter D = 3cm and water depth d = 1cm were arraed over a length of 173 cm. The distance between the clinders was S = 2cm. These clinders were ver stiff and the underwent negligible amplitudes of vorte-ecited vibration in stead flow. The clinders were arranged in a staggered arra (e.g. Case B; Figure 2).

2 4m z Circular Clinders D=1cm λ=.91 D =3cm λ=.2 1m d flow U m 2D-LDV Shear stress τ b Shear stress sensor Pump Circular flow Figure 1. Schematic of the eperimental setup z Circular Clinders 2cm flow d flow 8cm 15cm Flow S=2cm Circular clinder A Circular clinder C 1cm Shear stress sensor 17.3cm Circular clinder B Figure 2. View of the clinder placement pattern with λ=.2 for Case B The spacing densit of the arra λ represents the ratio of the total cross-sectional area of the clinders to the area occupied b the entire arra. For the case of the regular staggered arra shown in Figure 2, the value of λ ma be calculated using:

3 ( n D 2 /4)/ A D 2 2 3S 2 (1) where A = the plane area covering the arra of circular clinders, n = the total number of the clinders in the arra and D is the stem diameter. In Case A, the value of λ was.91, which is representative of the tpical densities of wood species in rivers [2, 3]. In Case B, the value of λ was.2, which is representative of a rather dense stand of trees in rivers. 2.2 Measurements of shear stress and flow velocit Bed shear stresses acting parallel to the flow direction were measured using a one component load cell (Figure 3). The shear force acting on a 1mm diameter,.9mm thickness shear plate was evaluated b measuring the bending moment acting on a cantilever. The surface of the shear plate was smooth. The bending moment ma be measured using semiconductor gauges attached on the surface of the cantilever (Figure 3). This shear stress sensor allowed for accurate, high resolution measurements of forces less than 1mgf/cm 2 (.98N/m 2 ). The natural frequenc of the shear stress sensor was about 38Hz in still water. In order to obtain an independent estimate of the bed shear stress, velocit and Renolds stress distributions were also measured using a two-component laser Doppler anemometer (2D-LDV). φ 22 mm.9 mm Shear plate Bed of open channel 1.5 mm shear plate 3 mm φ 1 mm φ 11 mm Plane figure Semiconductor gauges ecretor tube Sectional figure Figure 3. Shear plate assembl, and photograph of shear stress sensor 3 RESULTS 3.1 Shear stress on the smooth bed of an open channel without clinders Figure 4 shows an eample mean velocit profile, collected at a water depth, d, of 15 cm and a cross-sectional mean velocit, U m, of 38 cm/s on a smooth bed. This log-linear profile is tpical of a turbulent boundar laer. Bed shear stress ma be estimated through fitting the logarithmic law of the wall, U U * * 5.75log U z 5.5 (2) where U is the mean velocit at the distance z from bed of open channel, U * is friction velocit defined as U * =(τ bm /ρ) 1/2, whereτ bm is the mean shear stress on the bed of open channel, ρis densit of water and νis kinematic viscosit of water. Solving for U * (and thence τ blm ) ields an estimate of the mean bed shear stress of τ blm =3.2 mgf/cm 2. The unit, mgf, is an engineering unit of forces.

4 U ( mm ) z 3 2 * U/U U * Eperiment Equation (2) * U z / ν * ρu'v' (mgf/cm 2 ) Figure 4. Eample mean velocit profile Figure 5. Renolds stress (-ρu'v') profile Table 1. Comparison of shear stress τ bm withτ blm and τ brm Water depth d (cm) U m (cm/s) τ bm (mgf/cm 2 ) τ blm (mgf/cm 2 ) τ brm (mgf/cm 2 ) Figure 5 shows an eample Renolds stress, -ρu'v', profile collected under the same flow conditions. The Renolds stress is defined as the mean correlation between the two velocit fluctuations, u' from the mean streamwise velocit component U and v' from the mean vertical velocit component V. The maimum Renolds stress value of 3~3.3 mgf/cm 2 occurs near the bed of the open channel, permitting the mean bed shear stress, τ brm, to be estimated as 3~3.3 mgf/cm 2. For comparison, the load cell provides a mean shear stress value,τ bm, of 3.2 mgf /cm 2 (Table 1) Values of τ bm, τ blm and τ brm are compared in Table 1 for three flow depth and cross-sectional mean velocit combinations. There is close agreement between the bed shear stress measured b the load cell, τ bm, and the estimated values, τ blm and τ brm, This provides evidence of the accurac of the load cell used in the present measurement. 3.2 Shear stress on the bed of an open channel with an arra of clinders Figure 6 compares time series of bed shear stress with and without 1 cm diameter stainless steel clinders (case A; S = 1 cm; λ =.91) measured using the load cell, τ B, positioned at three locations. The water depth, d, was 15 cm and the cross-sectional mean velocit, U m, was 38 cm/s. Compared to the no clinder case, the mean and variance of τ bm were much larger at positions /D =, /D = 1.2 (Figure 6a) and /D = 4.3, /D = 2.5 (Figure 6b) when clinders were installed. Conversel, although the variance of τ bm was also much larger at position /D = 1.2, /D =, the mean value of τ bm at position /D = 1.2, /D = (Figure 6c) was small and acting in the negative (upstream) direction. It is speculated that this ma be due to the formation of a horseshoe vorte and vorte shedding around a clinder [e.g., 1]. Profiles of the mean, U, and root mean square error, rms u, velocit measured using 2D-LDA are shown in Figure 7 for case A. U and rms u are nearl uniforml distributed along the vertical. The Froude number of this quasi-uniform vegetated flow was Fr =.33. These phenomena ma be regarded as evidence of hdraulicall rough flow [e.g., 2, 3]. While the mean value of U at position /D = 4.3, /D = 2.5 was nearl equal to the crosssectional mean velocit, U m, of the flow without clinders, the mean value of U at positions /D = 4.3, /D = and /D = 4.3, /D = 5 was significantl retarded. On the other hand, the value of rms u at /D = 4.3, /D = and /D = 4.3,

5 (a) /D=, /D=1.2 (b) /D=4.3, /D=2.5 5 gggggggggg /D= with, clinders /D=1.2 ( 円柱側面域 gggg g ) 円柱群なし without clinders 5 /D=4.3 gggggggggg with, clinders /D=2.5( 円柱間中間点 gggg g ) without 円柱群なしclinders τ b (mgf/cm 2 ) 25 τ b (mgf/cm 2 ) 時間 t (s) 時間 t (s) (c) /D=1.2, /D= Clinder A Clinder C τ b (mgf/cm 2 ) 5 25 gggggggggg /D=1.2 with clinders, /D= ( 円柱後流域 gggg g ) without 円柱群なし clinders Flow /D=5 /D= Clinder B /D= /D=4.3 /D=8.6 /D= 時間 t (s) Figure 6. Time series of bed shear stress with and without clinders arraed with spacing densit λ=.91 for the stead flow with U m =38cm/s and d=15cm (without circular clinders; with circular clinders) A Flow C /D=5 9 U /D=5 /D= U /D=2.5 U /D= rms u' /D=5 rms u' /D=2.5 Clinder B /D= /D=4.3 /D=8.6 /D= U (m/s), rms u' rms u' /D= Figure 7. Vertical profiles of mean U and root-mean square values rms u of velocit around clinders arraed with spacing densit λ=.91 for the stead flow with U m =38cm/s and d=15cm /D = 5 was large compared to that at /D = 4.3 and /D = 2.5. It is speculated that again this ma be due to vorte shedding from clinders and vorte activit in the wake of clinders. Figure 8 compares profiles of U, rms u and -ρu'v' measured using 2D-LDV at position /D = 4.3, /D = 2.5 with and without 1cm-diameter stainless steel clinders (case A; S = 1cm; λ=.91). Compared to the case without clinders, the thickness of boundar laer was ver thin and a large value of -ρu'v' appeared close to the bed in the case with clinders. It is speculated that again this ma be due to vorte shedding from clinders and vorte activit in the wake of clinders. Denotingτ m as the mean bed shear stress measured using the load cell at a flow of U m = 38cm/s and d = 15cm without clinders, Figure 9 shows the spatial variation of normalized bed shear stress τ bm /τ m for case A.

6 Full Paper 11 th ISE 216, Melbourne, Australia U ( ああああああああ円柱群あり with clinders ) ) 5 ( 円柱群あり r with rrrrrrrrr clinders ) U 555 ( あああああ円柱群なし without clinder ) ああ5 ) ( 円柱群なし r without rrrrrrrrr clinder ) 系列 rms u'( 3 ( 円柱群あり with clinders ) ) 12 rms 系列 u'( 4 ( 円柱群なし without clinder ) ) U,rms u'(m/s) ρu'v' (mgf/cm 2 ) ( U 555 with ( 円柱群ありああああああああ clinders ) ) 5 ( 円柱群あり r with rrrrrrrrr clinders ) U ( 555 without ( あああああ円柱群なし clinder ) ああ) 5 ( 円柱群なし r without rrrrrrrrr clinder ) rms 系列 u'( 3( 円柱群あり with clinders ) ) 1 li d li d rms 系列 u'( 4( 円柱群なし without ) U,rms u'(m/s) - ρu'v' (mgf/cm 2 ) 5 6 a) U and rms u with distance z b) Renolds stress -ρu'v' with distance z 1 1 Figure 8. Vertical profiles of U and rms u with and without clinders (case A) for the flow with U m = 38cm/s and d = 15cm /D flow Circular clinder C τbm 5 / τ m < < τbm 1 / τ m < 1 1 < 2 τ bm / τ m < 2 2 < τbm 3 / τ m < 4 4 < 4 τ bm / τ m Circular clinder B /D Figure 9. Spatial distribution of the bed shear stress (normalized b the bed shear stress measured for the no clinder case) for case A (λ =.91) with U m = 38cm/s and d = 15cm

7 In the region just behind clinders, normalized bed shear stresses are small and are oriented upstream in the range /D<2 and /D=1. Conversel, near the sides of clinders, normalized bed shear stresses become ver large: values ofτ bm are amplified more than 4 times at /D=, /D=1.2 and /D=8.6, /D=3.8. On the other hand, the values of magnificationτ bm /τ m are in the range of 1 to 4 over most of the bed. The values of Manning s roughness coefficient n b ma be calculated as: 1/2 n d / gu 1/2 2 b brma m (3) where τ brma is the spatial average of the measured bed shear stresses,τ brm, and g is the acceleration due to gravit (9.81 m/s 2 ). With U m = 38cm/s and d = 15cm, n b was estimated as.11 for the no clinder case,.18 for case A (S = 1cm; λ =.91) and.21 for case B (S=2 cm; λ =.2). Thus, it ma be confirmed that n b increases with arrangement densit λ. 4 CONCLUSIONS There was close agreement between the shear stress measured b a load cell and values estimated from the measured mean velocit profiles and Renolds stress distributions. For case A (S = 1cm; λ =.91), the depth-averaged value of U at the positions /D = 4.3, /D = and /D = 4.3, /D = 5 was significantl retarded relative to the cross-sectional average velocit U m without clinders. The root mean square velocit rms u at the positions /D = 4.3, /D = and /D = 4.3, /D=5 were large compared to those at /D=4.3 and /D=2.5. It is speculated that this ma be due to vorte shedding from clinders and vorte activit in the wake of clinders. In the region just behind a clinder, the bed shear stressτ brm is small and acts in the upstream direction in the range /D < 2 and /D = 1. However, near the sides of clinders, the bed shear stress becomes ver large. Specificall, values ofτ brm are amplified (relative to the case with no clinders) b a factor of more than 4 at positions /D =, /D = 1.2, and /D = 8.6, /D = 3.8. On the remainder of the bed, the values of bed shear stress (relative to the case with no clinders) are amplified b factors ranging between 1 and 4. The values of Manning s roughness coefficient increase with arrangement densit λ. 5 ACKNOWLEDGEMENTS The authors are grateful to anonmous reviewers and proceedings editors for their valuable comments and suggestions for our English writing, which were of great help to develop this paper to its present form. REFERENCES [1] Sumer B. M., Christiansen N. and Fredosoe J. The horseshoe vorte and vorte shedding around a vertical wall-mounted clinder eposed to waves, Journal of Fluid Mechanics, Vol. 332, (1979), pp [2] Haashi K, Fujii M. and Shigemura T., Shear stress acting on the bed with vertical circular clinders in open-channel flow, Journal of Japan Societ of Civil Eng., B1 (Hdraulic Eng.),Vol.7, No.4, (21), pp [3] Kothari U.C., Haashi K. and Hashimoto H., Drag Drag coefficient of unsubmerged rigid vegetation stems in open channel flows, Journal of Hdraulic Research., Vol.47, No.6, (29), pp [4] Stone B. and Shen H.T., 22, Hdraulic resistance of flow in channels with clindrical roughness, Journal of Hdraulic Engineering, 128(5), (22), pp 5-56.

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