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, 2-14 5 2004

Kyev, Ukraine

Introduction

Shallow open channel flow over rough bed

Rough impermeable Rough permeable

DoubleAveragingMethodology

Experiments

PIV

from Nikora et al. 2001

Double Averaging Notation

oVVf

FdVV

FdVV

F0 0

11

Intrinsic average of general quantity F

Volume of fluid Vf

Porosity = Vf /V0

Spatial disturbance of F

FFF ~

Double Averaging Volume Averaging Theorems

dSnFVt

F

t

Fi

S

if

int

11

dSnFVx

F

x

Fi

Sfii

int

11

I

II

Double Averaging Momentum Equations

i

ji

i

j

jj

i

ji

j

x

uu

x

u

x

pg

x

uu

t

u

2

21

Time (ensemble) averaged Navier-Stokes equations

intint

11~~

1

111

S

ii

j

fS

jfj

ji

j

j

ii

ji

jj

i

j

i

j

dSnx

u

VdSnp

Vx

uu

x

u

xx

uu

x

pg

x

uu

t

u

Double averaged Navier-Stokes equations for frozen boundary Sint with no-slip condition

FORM-INDUCED

STRESS TERMFORM DRAG

VISCOUS DRAG

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)

Double Averaging 2-D Steady Uniform Flow

0

~~''

z

wu

z

u

zz

wugSb

Streamwise momentum equation Flow above the roughness crests

Flow below the roughness crests, frozen boundary, no-slip

01

~~11''1

intint

S

z

S

xf

b

dSnz

udSnp

V

z

wu

z

u

zz

wugS

Note that=(z) !

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)

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

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

2D Square Bar RoughnessSetup

lz = d

glx

= dg

Spacing

= 2,3,4,5,6,7,8,10,15,20

Slope

S=1:100,1:400,1:1000

Depth

H=35mm,50mm,80mm

lx = lz = d = 6mm

2D Square Bar RoughnessAnimated Streamwise Velocity Magnitude

d type k type

2D Square Bar RoughnessTime averaged velocity components

= 3 = 5 = 15

u

w

2D Square Bar Roughness Normalised Shear Stresses

= 3

= 5

= 15

2D Square Bar Roughness Double Averaged Streamwise Velocity

SpheresSetup

Diameter d=12mm

Slope 1:400

Depth H = 80 mm

SpheresTime averaged velocity components

u

w

Above Between

SpheresDA Normalised Shear Stresses

SpheresDouble-Averaged Streamwise Velocity

Spheres – 2 LayersTime-Averaged Streamwise Velocity

u(m/s)

0

10

20

30

40

50

60

70

80

90

100

0.10 1.00 10.00

Grain size (mm)

Cum

ulat

ive

%

Fine Feed Material

Medium Feed Material

Coarse Feed Material

Bed Surface 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

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 & 0.006 kg/m/s, experiments 1 & 2 respectively)

Plane Bed with Gravel RoughnessNormalised 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)

Plane Bed with Gravel RoughnessDouble-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 2’ . . . .

Effect of bed load size

Plane Bed with Gravel RoughnessDouble-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

Plane Bed with Gravel RoughnessForm-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

Velocity Disturbances

u~w~

wu~,~

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