Formula Sheet Turbulent Flow - Fluid Mechanics for Mechanical Engineering

 Page 1


Turbulent Flow
1. Definition
Turbulent flow is characterized by chaotic fluctuations in velocity and pressure, with eddies
of various sizes and enhanced mixing.
2. Reynolds Number Criterion
Re =
?UD
µ ?
?
?
?
?
?
Re< 2300 Laminar flow
Re> 4000 Turbulent flow
2300<Re< 4000 Transitional flow
where U = characteristic velocity, D = characteristic length (e.g., pipe diameter), ? =
density, µ = dynamic viscosity.
3. Velocity Profile in Turbulent Pipe Flow
Using the Power law velocity profile:
u
U
max
=

y
R

1/n
where
• u = local velocity at distance y from wall,
• U
max
= maximum velocity at pipe centerline,
• R = pipe radius,
• n˜ 7 for smooth pipes.
4. Darcy-Weisbach Equation for Head Loss
h
f
=f
L
D
V
2
2g
where f is the Darcy friction factor.
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Page 2


Turbulent Flow
1. Definition
Turbulent flow is characterized by chaotic fluctuations in velocity and pressure, with eddies
of various sizes and enhanced mixing.
2. Reynolds Number Criterion
Re =
?UD
µ ?
?
?
?
?
?
Re< 2300 Laminar flow
Re> 4000 Turbulent flow
2300<Re< 4000 Transitional flow
where U = characteristic velocity, D = characteristic length (e.g., pipe diameter), ? =
density, µ = dynamic viscosity.
3. Velocity Profile in Turbulent Pipe Flow
Using the Power law velocity profile:
u
U
max
=

y
R

1/n
where
• u = local velocity at distance y from wall,
• U
max
= maximum velocity at pipe centerline,
• R = pipe radius,
• n˜ 7 for smooth pipes.
4. Darcy-Weisbach Equation for Head Loss
h
f
=f
L
D
V
2
2g
where f is the Darcy friction factor.
1
5. Colebrook-White Equation (Implicit) for Friction
Factor
1
v
f
=- 2log
10

e/D
3.7
+
2.51
Re
v
f

where e = pipe roughness height.
6. Approximate Explicit Friction Factor Formulas
• Blasius formula (for smooth pipes, Re< 10
5
):
f = 0.316Re
- 0.25
• Haaland equation:
1
v
f
=- 1.8log
10
"

e/D
3.7

1.11
+
6.9
Re
#
7. Turbulent Shear Stress (t t
) in Boundary Layer
t t
=? u
'
v
'
where u
'
, v
'
are velocity fluctuations in streamwise and normal directions.
8. Turbulent Kinetic Energy (k)
k =
1
2

u
'2
+v
'2
+w
'2

9. Mixing Length Theory (Prandtl’s Hypothesis)
t =? 
? +l
2
m




du
dy





du
dy
where l
m
is mixing length, ? is kinematic viscosity.
10. Energy Cascade
Turbulent kinetic energy transfers from large eddies (energy containing) to small eddies
(dissipative scales) where it is dissipated as heat.
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