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Interpretation of PIV measurements of open channel flow over rough bed using double-averaged Navier-Stokes equations D. Pokrajac, I. McEwan, L. Cambell, C. Manes V. Nikora (EPSRC grants GR/R51865/01 & GR/L54448/01) NATO ASI, Kyev, Ukraine

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Introduction Shallow open channel flow over rough bed Rough impermeable Rough permeable Double Averaging Methodology Experiments PIV from Nikora et al. 2001

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Double Averaging Notation Intrinsic average of general quantity F Volume of fluid V f Porosity = V f /V 0 Spatial disturbance of F

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Double Averaging Volume Averaging Theorems I II

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Double Averaging Momentum Equations Time (ensemble) averaged Navier-Stokes equations Double averaged Navier-Stokes equations for frozen boundary S int with no-slip condition FORM-INDUCED STRESS TERM FORM DRAGVISCOUS DRAG

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Double Averaging Coordinate System z x u v w x = longitudinal (u component of velocity vector) y = transverse (v velocity component) z = bed-normal (vertical – w velocity component)

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Double Averaging 2-D Steady Uniform Flow Streamwise momentum equation Flow above the roughness crests Flow below the roughness crests, frozen boundary, no-slip Note that=(z) !

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Experimental Methodology Facilities The Aberdeen Environmental Hydraulics Group is a long- established PIV user, having had a working PIV system since Current facilities include: Autocorrelation and cross-correlation PIV 1k 1k cameras taking up to 30 frames/s (15 pairs for cross-correlation) Direct-to-disk recording allowing long time-series data (limited only by drive free space) Choice of Visiflow or VidPIV vector processing software Illumination via argon-ion or copper vapour lasers (suitable for high speed PIV) Two hydraulic flumes (including sediment recirculation), a wave tank, and oscillatory flow tunnel (OFT)

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Experimental Methodology PIV Each experiment involved grabbing 4096 multiply/doubly exposed PIV frames at around 16 Hz (around 4 minutes of real-time flow, over 4GB of data) Pixel resolution was , corresponding to a planar flow area illuminated in the midline of the flume of up to mm Individual frames were broken into pixel interrogation regions for autocorrelation/cross-correlation analysis; to produce the final vector map these were overlapped by 75% Resultant vector maps contained 3481 instantaneous velocity vectors for each of the 4096 time-steps

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Experimental Methodology Bed Roughness 2D square-bar roughness (6 mm) Spheres (12 mm) in cubic arrangement 1 layer (impermeable bed) 2 layers (permeable bed) A fixed, planar, non-porous sediment bed (d 50 = 1.95mm) with clear water With the addition of 3 bed-load transport conditions: fine grains, medium grains, coarse grains

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2D Square Bar Roughness Setup l z = d g lxlx = Spacing = 2,3,4,5,6,7,8,10,15,20 Slope S=1:100,1:400,1:1000 Depth H=35mm,50mm,80mm l x = l z = d = 6mm

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2D Square Bar Roughness Animated Streamwise Velocity Magnitude d type k type

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2D Square Bar Roughness Time averaged velocity components = 3 = 5 = 15 u w

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2D Square Bar Roughness Normalised Shear Stresses = 3 = 5 = 15

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2D Square Bar Roughness Double Averaged Streamwise Velocity

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Spheres Setup Diameter d=12mm Slope 1:400 Depth H = 80 mm

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Spheres Time averaged velocity components u w Above Between

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Spheres DA Normalised Shear Stresses

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Spheres Double-Averaged Streamwise Velocity

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Spheres – 2 Layers Time-Averaged Streamwise Velocity (m/s)

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clear water d 50 = 1.95 mm fine grains d 50 = 0.77 mm medium grains d 50 = 1.99 mm coarse grains d 50 = 3.96 mm Fixed sediment and bed-load feed material Plane Bed with Gravel Roughness Setup Each of the bed-load mixtures was fed into the flume at low and high feed rates (0.003 & kg/m/s, experiments 1 & 2 respectively)

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Plane Bed with Gravel Roughness Normalised Reynolds Shear Stress linear trend validates 2D flow assumption deviation in near-bed region due to roughness layer thickness of roughness layer increases with increasing feed sediment size (~1 mm thicker with each size increment) slight deviation towards free-surface attributed to wall effects (aspect ratio = 4.5)

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Plane Bed with Gravel Roughness Double-Averaged Streamwise Velocity all profiles obey logarithmic distribution presence of bed-load sediment results in lower velocities at level z, consistent with greater roughness heights degree of retardation depends on sediment size (coarser grains cause more slowing) obvious exception is experiment Fine Effect of bed load size

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Plane Bed with Gravel Roughness Double-Averaged Streamwise Velocity Effect of feed rate experiments clear 1 and clear 2 show excellent agreement, indicating the repeatability of the PIV process feeding more fine particles reverses the velocity shift – effectively smoothing the bed and permitting higher velocities negligible effect of feed rate with medium grains conversely, coarse particles increase bed-roughening at the higher feed rate, causing a further downwards shift in velocity profile

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Plane Bed with Gravel Roughness Form-Induced Stress Profiles form-induced stress peaks in the roughness layer where the difference between time- and double-averaged quantities is maximised for clear water cases, the form-induced stress constitutes up to 35% of the maximum Reynolds stress (much higher than previously anticipated) although reduced from the clear water value for all bed- load cases, the form-induced stres still contributed around 15% of the total shear in the roughness layer

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Velocity Disturbances

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Conclusions Spatial averaging methodology provides new insight into the characteristics of turbulent flow near a rough bed Spatial fluctuations in the flow velocity are strongly influenced by the spacing and the shape of the roughness elements Over various types of roughness, with and without bed-load transport, the recorded levels of form-induced stress are quite high (up to 30% of the total shear in the roughness layer) PIV is very well suited to assessing the spatial averaging technique

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