LAMINAR FLOWS IN OPEN
CHANNELS AND LAUNDERS
Paul Slatter, ATC Williams, Australia.
INTRODUCTION The free surface flow behaviour is of critical
importance in many industrial contexts
• Previous studies have shown
• A sheet flow diagram can be constructed
• Similar to tube flow
• Unresolved issues remain
• Role that the yield stress may play
• Location of the laminar/turbulent transition region
INTRODUCTION Objectives:-
develop and evaluate the free surface sheet flow approach
for viscoplastic sheet flows
The previously developed approach for the laminar-turbulent transition
should now been extended to the characteristic length independent approach
approach should be validated against experimental viscoplastic data
THEORY AND
LITERATURE
y
xh
Flow directionτ
α
H
0τ
NEW MODEL
Shear stress distribution for sheet flow
y
xh
Flow directionτ
α
H
0τ
αρgHτ sin0
NEW MODEL
For a Newtonian fluid
For a general time independent non-
Newtonian fluid
H
V30
0
0
2
0
)(33
dH
V
NEW MODEL
Table 1: Comparison of key elements of the
rheometric analysis of tube and sheet flow
Bulk
Shear
Rate
Wall Shear
Stress
R-M
Factor
Tube
Flow D
V8
L
pD
4
n
n
4
13
Sheet
Flow H
V3 singH
n
n
3
12
NEW MODEL
Taking the approach further
Follow the approach of Metzner-Reed
for pipe flow:-
.3
0
'n
*
*
H
V'Kτ
NEW MODEL
In order to accommodate the reality
Actual channels may approach sheet flow
But will always have side edges at some
point
Replace H with the Hydraulic radius Rh so
that:
.3
0
'n
h*
*
R
V'Kτ
TRANSITIONAL FLOW –
RE4 MODEL
We can define a Reynolds number after the
Newtonian paradigm:-
Or better still as
.τ
V 8Re
0
2
4
.
R
3V'K
V 8Re
'n
h
*
2
4*
NEW MODEL - EXTENSION TO
VISCOPLASTIC MATERIAL
• Herschel-Bulkley constitutive relationship
• The bulk shear rate 3V/H for Herschel-Bulkley sheet
flow can be expressed as
n
y Kττ
.
111
12
33
0
1
0
1
0
0
yn
n
yn
n
n
n
Kn
nK
H
V
TRANSITIONAL
FLOW –
RE4 MODEL
• Transitional Flow Criterion
• Transition occurs at Re4 = 700
• From previous work
TRANSITIONAL FLOW –
CHARACTERISTIC
LENGTH INDEPENDENT
• Wasp, E. and Slatter, P. 2006, 'Transition velocity estimation for visco-
plastic fluids', in 13th International Conference Transport and
Sedimentation of Solid Particles, Akademia Rolnicza, Wroclaw,
Poland, pp. 291-299
ycV 26
TRANSITIONAL FLOW –
CHARACTERISTIC
LENGTH INDEPENDENT
• Transition occurs at V = Vc
• From previous work in large pipes
ycV 26
EXPERIMENTAL WORK
Test Rig
10m Tilting Flume
In-Line Tube viscometer
Conventional rheometer
Haldenwang, R. and Slatter, P.T. (2006) Experimental procedure
and database for non-Newtonian open channel flow, Journal of
Hydraulic Research, Vol. 44, Issue 2, pp. 283–287.
10
M
F
L
U
M
E
Rheology and Materials Processing Centre
Mixing tank
2000 litres
Stirrer
Flow meter
calibration
tank
Positive
displacement
pump 23 l/s
Hydraulic
Ram to tilt
flume
80 mm pipe
13 mm pipe
28 mm pipe
Magnetic Flow meters
DP cells
Pressure tappings
Heat Exchanger
Mass Flow meter
ON-LINE PIPE VISCOMETER
MATERIAL
6% Kaolin suspension
Viscoplastic rheology
Model Paste Material
0
5
10
15
20
25
0 100 200 300 400 500
3V/H (s-1)
Wall s
hear
str
ess (
H)
(Pa)
3 degrees
4 degrees
5 degrees
New Model
DISCUSSION
Using the criterion Re4 = 700
For the 3 and 4 degree empirical data
More deviation at 5 degrees.
Using the characteristic length independent
approach Vc,
The reverse is true
Best prediction at 5 degrees
Significant deviation for the 3 and 4 degree
empirical data.
It should be noted that – as for critical pipe
flow – there may well be a zone or range of
values within which transition can occur.
The main benefits of the new approach
Laminar flow data can be scaled up directly for
engineering design purposes
Or
From the rheology as measured by standard
bench-top methods.
Discussion
DISCUSSION
The principal unresolved issues revolve
around
Edge effects with a yield stress
Froude number or free surface effects
These relate to critical, tranquil or shooting
flow.
DISCUSSION
Extension to visco-plastic fluids has been
achieved.
The apparent flow behaviour index is not
constant.
Using the tangent method at the relevant wall
shear stress as proposed by Metzner-Reed is
viable
DISCUSSION
Has been validated against experimental data.
This remains to be validated against more data.
Negative slope in the turbulent region ----
TURBULENT FLOW
To
0
10
20
30
40
50
60
70
80
90
0 50 100 150 200 250 300 350
5 deg
4 deg
3 deg
3V/H
CONCLUSIONS
A new analytical approach for the sheet flow of a power law fluid has been extended to viscoplastic Yield Stress material.
Exploits the fact that the bulk shear rate is a unique function of the rheogram and the wall shear stress
Can be used for scale-up and design at any required slope and depth, in laminar flow.
Onset of turbulent flow can be predicted from
Re4 = 700 or V = Vc approach
This approach is analogous to that of Metzner and Reed (1955) for laminar pipe flow.
Unresolved issues have been highlighted.
FUTURE WORK
Further work on the transition using a wider data
base
Entrance, end and edge effects for sheet flow
Identification and analysis of critical flow
conditions
Related to surface disturbances and Froude number
effects.
Influence, identification and analysis of slip flow
effects
ACKNOWLEDGEMENT
The support and encouragement of the
ATC Williams team is gratefully acknowledged –
without which this work would not have been
possible.