1. (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

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

3. Double Averaging Notation
Volume of fluid Vf
Porosity f = Vf /V0
Intrinsic average of general quantity F
Spatial disturbance of F

4. Double Averaging Volume Averaging TheoremsII

5. 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

6. 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)

7. 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) !

8. 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)

9. 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

10. 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

11. 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

12. 2D Square Bar Roughness Animated Streamwise Velocity Magnitude
d type
k type

13. 2D Square Bar Roughness Time averaged velocity components
l = 3 l = 5 l = 15 u w

14. 2D Square Bar Roughness Normalised Shear Stresses
= 3
= 5
l = 15

15. 2D Square Bar Roughness Double Averaged Streamwise Velocity

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

17. Spheres Time averaged velocity components
Above
Between u w

18. Spheres DA Normalised Shear Stresses

19. Spheres Double-Averaged Streamwise Velocity

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

21. 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)

22. 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)

23. 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’

24. 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

25. 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

26. Velocity Disturbances

27. 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