The document discusses fluid mechanics, specifically focusing on turbulent flow and its characteristics, including its transition from laminar flow based on Reynolds number. It explains how factors like velocity and pipe diameter affect flow type and describes the experimental findings of Reynolds regarding pressure loss and friction in turbulent flow. Additionally, it elaborates on shear stress in turbulent flow, including Prandtl's mixing length theory for calculating turbulent shear stress.

1. Mechanical Engineering Department
4th semester
Subject :- Fluid Mechanics
TOPIC :- TURBULENT FLOW
Prepared by :- Guided by :-
Sanjay Noelia (151120119022) prof. Dharmesh Jariwala
Prince Kumar (151120119031)
Abhishek Mishra (151120119001)
Niraj Pandey (151120119023)

2. • Introduction
• In the laminar flow the fluid particles moves along
straight parallel path in layer or lamina , such that the
path of individual fluid particles do no cross those of
neighboring particles .
• Laminar flow is possible only at low velocities and
when the fluid is highly viscous. But when the velocity
is increase or the fluid is less viscous , the fluid
particles do not move in straight path.
• The fluid particle moves in random manner resulting in
general mixing of the particles . This type of flow is
called turbulent flow.

3. • Introduction
• In the laminar flow the fluid particles moves along
straight parallel path in layer or lamina , such that the
path of individual fluid particles do no cross those of
neighboring particles .
• Laminar flow is possible only at low velocities and
when the fluid is highly viscous. But when the velocity
is increase or the fluid is less viscous , the fluid
particles do not move in straight path.
• The fluid particle moves in random manner resulting in
general mixing of the particles . This type of flow is
called turbulent flow.

4. A laminar flow changes to turbulent flow when :
1.Velocity is increase .
2.Diameter of pipe is increase .
3.Velocity of fluid is decreased .
O .Reynolds was the first to demonstrate that the
transition from laminar to turbulent depends not only
on the main velocity but not on the quantity .
• This quantity is a dimensionless quantity and is
called Reynolds number .
vD

eR
VD


5. In case of circular pipe if the flow is said
to be laminar and if then the floe is said to
be turbulent .
• If lies between 2000 to 4000 , the flow changes
from laminar to turbulent.i.e,
 Laminar when Re < 2000.
 Turbulent when Re > 4000.
 Transient when 2000 < Re < 4000.
2000eR 
4000eR 
eR

6. 2.Reynolds experiment
The type of flow is determined from the Reynolds number
i.e.,
• This was demonstrated by O. Reynolds in 1883. His apparatus
in fig.
The apparatus consist of :
1) A tank contain water at constant head .
2) A small tank containing some dye.
3) A glass tube having a bell-mouthed entrance at one end and a
rectangular value at other end.
V d



7. The following observation were made by Raynold :
(a). When the velocity of the flow was low , the dye filament
in glass tube was in form of straight line.
(b). With the increase of velocity of flow , the dye filament
was no longer a straight- line it become a wavy one .
(c).With further increase of velocity of flow , the wavy dye-
filament broke up and finally diffused in water.

8. In case of laminar flow , the loss of pressure head was
found to be proportional to the velocity but in case of
turbulent flow , Reynolds observed that loss of head is
approximately proportional to the square of velocity .
More exactly the loss of head , where
varies from 1.75 to 2.0 .
n
fh v n

9.  3 . Frictional loss in pipe flow
When a liquid is flowing through a pipe ,the velocity of the liquid
layer adjacent to the pipe wall is zero the velocity of liquid is goes
to increasing from the wall and thus velocity gradient and hence
shear stress are produce in the whole liquid due to viscosity. This
viscous action causes loss of energy which is usually known as
frictional loss.
On the basis of his experiments, William Froude gave the laws of
fluid fraction for turbulent flow
The frictional resistance for the turbulent flow is :
(1) Proportional to V n ,where n varies from 1.5 to 2.0,
(2) Proportional to the density of fluid,
(3) Proportional to the area of surface in contact,
(4) Independent of pressure
(5) Dependent on the nature of the surface in contact.

10. 3.1 Expression for loss of head due to friction in pipes.
Consider a uniform horizontal pipe, having steady flow as
shown in fig. 10.3. Let 1-1 and 2-2 are two section of pipe.
Le P1 = pressure intensity at section 1-1,
V1 = Velocity of flow at section 1-1,
L = Length of pipe between section 1-1 and 2-2,
d = diameter of pipe
= frictional resistance per unit wetted area per unit velocity
h f = loss of head due to friction,
P2, V2 = are value of pressure intensity and velocity at
section 2-2,
f 

11. Applying Bernoulli’s equation between section 1-1 and 2-2
Total heat at 1-1 = Total heat 2-2 + loss of head due to friction
between 1-1 and 2-2
Z1 = Z2 as pipe is horizontal
V1 = V2 as dia. Of pipe is same at 1-1 and 2-2
but is the head lost due to friction and hence intensity of
pressure will be reduced in the direction of flow by frictional
resistance.
22
1 2 2
1 2
2 2
f
p p vv
z z h
g g g g 
     
1 2
f
p p
h
g g 
 
fh

12. Now frictional resistance = frictional per unit wetted area per
unit velocity wetted area velocity2
The force acting on the fluid between section 1-1 and 2-2 are:
1. Pressure force at section 1-1 = P A
2. Pressure of at section 2-2 = P2 A
3. Frictional force f1 as shown in fig.
Resolving all forces in horizontal direction ,we have
 
2
1 'F f dL V  


1 2 1 0p A p A F  
2
f P L V   

13. But from equation (1)
Equating the value of (p1-p2) we get
In equation (3)
  2
1 2 1
2
1 2
'
'
P P A F f P L V
f P L V
P P
A
     
  
 
1 2 fp p gh 
2
2
'
'
f
f
f p l v
gh
A
f P
h L V
g A


  

   
2
4
4
P Wettedperimeter d
A Area dd


  

14. Putting where f is known is co-efficient of
friction.
Equation (4) becomes as
Equations (4) is known as Darcy- Weisbach equation.
This equation commonly used for finding loss of head
due to friction in pipes.
Sometimes equation written as
Then f is known as friction factor.
2
2' 4 ' 4
f
f f LV
h L V
g d g d 
     
'
,
2
f f


2 2
4. 4 . .
.
2 2
f
f LV f LV
h
g d d g
 

2
. .
2
f
f L V
h
d g



15. 3.2 Expression for co-efficient in terms of shear stress.
The equation gives forces acting between in section 1-1 and
2-2 in horizontal direction as
Force due to shear stress
shear stress*surface area
Cancelling πd from both sides we have
Equation can be written as
11 2
0A Ap p F  
21 1
( )App F  
  2
1 2
4
p p d t d L

    
 1 2
4
d
p p L  
 1 2 4
L
p p
d
  
2
1 2 4 . .
2f
p p f LV
g
g dh 
 
   
 

16.  
2
1 2
4. . .
2
f LV
p p g
d g

 
   
 
Equating the value of (P1-P2) in equation
2
4. . .
4
2
L f LV
g
d d g
 
 
   
 
2 2
2 2
fV g fV
g
g g

 

  
2
2
V
f


 
  
 
2
2
f
v




17. 4.Shear stress in turbulent flow
The shear stress in viscous flow is given by newton's law of
viscosity as
Similar to the expression for viscous shear ,J. Bossiness
expressed the turbulent shear in mathematical form as
τt= shear stress due to turbulence
ῃ=eddy viscosity
ū= average velocity at a distance y from boundary
The ratio of eddy viscosity and mass density is known as
kinematic eddy viscosity and is denoted by epsilon
v
du
dy

 
  
 
t
du
dy

 
  
 

18. If the shear stress due to viscous flow is also considered
then the total shear stress becomes
The value of n=0 for laminar flow



 
 
 
v t
du du
dy dy
   
   
      
   

19. 4.1 Reynolds Expression For Turbulent Shear Stress
Reynolds in 1886 developed an expression for turbulent shear
stress between two layer of fluid at a small distance apart,
which is given as
Where u’, v’ fluctuating component of a velocity in the
direction of x and y due to turbulence.
As u’and v’ are varying and hence  will be also very. Hence
to find the shear stress, the time average on both the sides of
the equations (10.10) is taken. Then equation (10.10) becomes
as
' 'u v 
u v   

20. 4.2. Prandtl Mixing Length Theory for Turbulent Shear
Stress.
In equation the turbulent shear stress can be only calculated if
the value of . To overcome this difficulty, L. Prandtl in
1925, presented a mixing length hypothesis which can be used
to express the turbulent shear stress in terms of measurable
quantities.
According to Prandtl, the mixing length l, is that distance
between two layers in the transmission direction such that the
lamps of fluid particle from one layer could reach the another
layer and the particle are mixed to other layer in a such way
that the momentum of the particle in the dimension of x is
same. He also assumed that the velocity fluctuation in the x-
direction is related to the momentum and length l as
u v 
u
du
u l
dy
 

21. and v’ the fluctuation of velocity in y direction is one of
the same order magnitude and hence
Now becomes as
Substituting the value of in equations, we get
the expression for shear stress for turbulent flow due to
Prandtl as
du
v l
dy
 
u v  u v 
2
2du du du
l l l
dy dy dy
     
       
     
u v 
2
2 du
l
dy
 
 
  
 

22. Thus the total shear stress at any point turbulent flow
is the same of the shear stress in turbulent flow due to
Prandtl as
But the viscous shear stress is negligible except near
the boundary. Equations is used in most of turbulent
fluid flow problem for determining shear stress in
turbulent flow.
2
2du du
l
dy dy
  
 
   
 

23. THANK YOU