1. Laminar & turbulent Flow
Shear stress on fluid
Newton’s law of viscosity
Shear stress,, in fluid is proportional to the velocity gradient - the rate of change of velocity across the fluid path.
Viscosity, µ is the constant of proportionality

2. Laminar and turbulent flow in pipes
Flow can be either
Laminar - low velocity
Turbulent – high velocity
A small transitional zone between

4. Phenomenon was first investigated in the 1880s by
Osbourne Reynolds in an experiment which has become a classic in fluid mechanics.

5. Unitless, or non-dimensional number
Newton’s 2nd law
shear stress over
fluid surface

6. • Pipe flow nearly always turbulent
is different for different Conduits,
Non-circular x-sections, open channels etc. ;
The flow in round pipes is
Laminar flow: Re < 2000
Transitional flow:2000 < Re < 4000
Turbulent flow: Re > 4000
• Pipe flow nearly always turbulent

7. Given the following: Water flowing,
Example
Given the following: Water flowing,
Find the flow regime.
Soln.
Implies Laminar

8. Example
Given that:
• Pipe diameter: 0.5m
• Crude oil:
Kinematic viscosity = m²/s
• Water:
Dynamic viscosity μ = 8.90 × 10-4 Ns/m2
What are the velocities when
Turbulent flow would be expected to start?

9. • Crude oil:
• Water:

10. Pressure loss due to friction in a pipe
Consider fluid flowing in a pipe 1 2 L p
The pressure at 1 (upstream) is higher than the pressure at 2.
If a manometer is attached the pressure (head) difference due to the energy lost by the fluid overcoming the shear stress is seen

11. The driving force (due to pressure) (F = Pressure x Area)
Consider a cylindrical element of incompressible fluid flowing in a pipe
Pressure (p-p)
Pressure p
Direction of flow
Area A 2 1 1 2
The driving force (due to pressure) (F = Pressure x Area)
Driving force = Pressure force at 1 - pressure force at 2

12. Retarding force (due to shear stress at wall)
Pressure (p-p)
Pressure p
Direction of flow
Area A 1 2
Retarding force (due to shear stress at wall)
Retarding force = shear stress x area over which it acts

13. Driving force = Retarding force
Pressure (p-p)
Pressure p
Direction of flow
Area A 1 2
Flow is in equilibrium
Driving force = Retarding force
Pressure loss in terms of Shear Stress at wall

14. log pL

15. log pL

16. Pressure loss in laminar flow
In laminar flow it is possible to do theoretical analysis
Fluid particles move in straight lines
Consider a cylinder of fluid element, length L, radius r, flowing steadily in the centre of a pipe. L

17. In equilibrium, the shear stress on the cylinder
In equilibrium, the shear stress on the cylinder equal the pressure force.
Remember this
By Newton’s law of viscosity we have y
Where y is the distance
from the wall

18. Measuring from the pipe centre, we change the sign and replace y with r distance from the centre, giving
Hence or

19. In an integral form we have
Integrating gives the velocity at a point distance r from the centre
At r = R (the pipe wall) uR = 0
Hence

20. Hence an expression for velocity at a point r from the pipe centre when the flow is laminar
This is a parabolic profile (of the form y = ax2 + b )
Velocity profile in a pipe

21. Hence
Integrating for the limits
Equation for laminar flow in a pipe

22. This expresses the discharge
in term of the pressure
gradient ,
diameter of the pipe and
the viscosity of the fluid.
The mean velocity is determined as

23. Writing pressure loss in terms of head loss,
But
Writing pressure loss in terms of head loss
Hagan–Poiseuille equation

24. Example

25. Pressure loss in Turbulent Flow
Consider the forces on the element of fluid flowing down the slope (open channel)

26. Pressure loss in Turbulent Flow
The first pressure loss term is the piezometric head, p*, loss per unit length,
Hydraulic mean depth (Hydraulic radius), m
Gives shear stress in terms of head loss

27. Introduction of Friction factor
To make use of this equation we introduce the friction factor, f
Equating and rearranging gives
For a circular pipe,
Giving

28. This is the Darcy-Weisbach equation
Darcy-Weisbach Equation and the Friction factor
This is the Darcy-Weisbach equation
Gives head loss due to friction in a circular pipe
Often referred to as the Darcy equation
In terms of Q

29. In metric terms, g = 9.81m2/s, so
Darcy-Weisbach Equation and the Friction factor
In metric terms, g = 9.81m2/s, so or

30. This equation describes
Darcy-Weisbach Equation and the Friction factor
This equation describes
Head-loss due to friction
In terms of velocity u
In terms of Discharge Q
And friction factor, f
The value of f is crucial to calculation of hf

31. The f described here is that common in UK (in text books and practice)
How do we find f?
The f described here is that common in UK
(in text books and practice)
In US (and some text book) famerican = 4f,
To try and avoid confusion this is sometime written as ,
BE CAREFULL !!!
When using any book, look at the equation for hf

32. Two reservoirs have a height difference 15 m.
Example
Two reservoirs have a height difference 15 m.
They are connected by a pipeline 350 mm in diameter and 1000 m long with a friction factor f of
What is the flow in the pipe? (ignore all local losses) 1 2 Z1 Z2
Datum

33. Soln.
Write the general energy equation for the system ignoring all minor losses

34. What is f dependent on?
What is f dependent on?

35. What is the value of f ?
The friction factor depends on many physical things
For laminar flow theoretical expression can be derived
For Turbulent flow, it is complex

36. f in Laminar Flow
We have the Hagen-Poiseuille equation
Head loss in laminar flow
We also have the Darcy equation
Equate the two equations

37. Laminar flow example
Calculate the head loss due to friction in a circular pipe of 50 mm diameter, length 800m, carrying water (μ = 1.14 ×10-3 Ns/m2) at a rate of 5 litres/min.
Use both Hagen-Poiseuille and Darcy equations.
Check Re

39. Smooth / Rough pipes in Turbulent Flowy r k u
Wall
Smooth Pipe y r k u
Wall
Rough Pipe

40. Smooth / Rough pipes in Turbulent Flow
Let k be the average height of projection from the surface of a boundary.
Classification based on boundary characteristics
If the value of k is large, then the boundary is called rough boundary
If the value of k is less, then boundary is known as smooth boundary.

41. Classification based flow and fluid characteristics
Turbulent flow along a boundary is divided into two zones
First zone. Thin layer of fluid in the immediate neighbourhood of the boundary where viscous shear stress predominates, and shear stress due to turbulence is negligible. This zone is known as laminar sub-layer.
Height of this layer denoted by
The second zone of flow, where shear stress due to turbulence are large as compared to viscous stress is known as Turbulent zone.

43. Outside the laminar sub-layer the flow is turbulent.
Eddies of various size present in turbulent flow try to penetrate the laminar sub-layer and reach the roughness projection of the boundary.
Due to the thickness of the laminar sub-layer, the eddies are unable to reach the roughness projection of the boundary
Hence the boundary behaves as a smooth boundary.
This type of boundary is called hydrodynamically smooth boundary

44. If the Re of flow is increased the will decrease.
If the becomes much smaller than the average height k, the boundary will act as rough boundary.
Because the roughness projection are above the laminar sub-layer and the eddies present in the turbulent zone come in contact with the roughness projection a lot of energy will be lost.
Such boundary is called hydrodynamically rough boundary

45. From Nikuradse’s experiment
the boundary is called a smooth boundary
the boundary is rough
the boundary is in transition

46. In 1913 Blasius examined a lot of experimental measurements
Blasius equation
In 1913 Blasius examined a lot of experimental measurements
Found 2 distinct friction effects
Smooth pipes and Rough pipes
Blasius equation
Valid for Re <

47. Nikuradse’s Experiment
Nikuradse made great progress in 1930’s
Artificially roughened pipes with sand of known size, k A B C D E F G 15
30.5 60
120
252
507
Increasing grain size
Rough turbulence
Transition turbulence
Smooth turbulence
Laminar
Relative roughness
Blasius
equation

48. The following points must be noted from the curve
The line AB is common to all the pipes having different relative roughness. This line indicates laminar flow. The friction factor in this range is given as
Valid for
As the value Re increases beyond 2000, the flow passes through a transition stage represented by the curve BC.
The flow becomes turbulent at point C.
Transition stage occurs in the range of Re between 2000 to 4000.

49. The curve CD represents smooth turbulent flow
For turbulent flow, it is observed that the higher the ratio ro/k (the smoother the pipe), the greater is the tendency of the pipe to follow the line CD.
Meaning the pipe with the roughness surface causes the earliest breakaway from the line CD.
There is a Transition stage between the smooth turbulent flow to the rough turbulent flow.
For example, the relative roughness ro/k = 60, the transition stage is represented by the curve EF.
In the transition stage, f depends on both Re and ro/k.

50. After the flow is completely established as rough turbulent, the curves become horizontal.
For example, the relative roughness ro/k = 60, it is represented by FG.
In this stage, f depends only on the relative roughness and is independent of Re.

51. Moody Diagram

53. Example B + + A
Given Benzene (S. G. = 0.86)
Ignore all minor losses

54. Solution

55. First find Re
Implies turbulent
Hence use Darcy equation:
We need for Moody

56. Re

57. C - Chezy’s coefficient
Velocity of Flow
Chezy’s Formula
C - Chezy’s coefficient
Chezy from Darcy-Weisbach equation
For a circular pipe
Hydraulic Radius
Hence

58. - Hydraulic radius Slope The value of C = 55 – 75 A - Area of flow
P - Wetted perimeter
Slope
The value of
C = 55 – 75

59. Manning’s Formula (Manning’s Velocity)
Manning proposed that,
n - Manning’s coefficient. It depends on the type of material.
Relation between Manning and Chezy
From Chezy and Mannings equations

60. Hazen Williams’ formula
Valid for diameter  5cm  1.9m, V  3m/s, T = 16oC
Hazen Williams
also increases ()
As pipe smoothness increases ()