Paul Slatter, ATC Williams - Laminar flows in open channels and launders

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.