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**(EPSRC grants GR/R51865/01 & GR/L54448/01)**

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 Double Averaging Experiments Methodology PIV**

Shallow open channel flow over rough bed Rough impermeable Rough permeable from Nikora et al. 2001

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**Double Averaging Notation**

Volume of fluid Vf Porosity f = Vf /V0 Intrinsic average of general quantity F Spatial disturbance of F

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**Double Averaging Volume Averaging Theorems**

II

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**Double Averaging Momentum Equations**

Time (ensemble) averaged Navier-Stokes equations Double averaged Navier-Stokes equations for frozen boundary Sint with no-slip condition FORM-INDUCED STRESS TERM FORM DRAG VISCOUS 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 f=f(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 1994. 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 1000 1000, corresponding to a planar flow area illuminated in the midline of the flume of up to 100 100 mm Individual frames were broken into 32 32 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 (d50 = 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**

lz = d g lx = lx = lz = d = 6mm Spacing = 2,3,4,5,6,7,8,10,15,20 Slope S=1:100,1:400,1:1000 Depth H=35mm,50mm,80mm

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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**

l = 3 l = 5 l = 15 u w

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**2D Square Bar Roughness Normalised Shear Stresses**

= 3 = 5 l = 15

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**2D Square Bar Roughness Double Averaged Streamwise Velocity**

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

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**Spheres Time averaged velocity components**

Above Between u w

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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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**Plane Bed with Gravel Roughness Setup**

Fixed sediment and bed-load feed material clear water d50 = 1.95 mm fine grains d50 = 0.77 mm medium grains d50 = 1.99 mm coarse grains d50 = 3.96 mm 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**

Effect of bed load size 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 2’

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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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