Formula Sheet Boundary Layer Theory - Fluid Mechanics for Mechanical Engineering

 Page 1


Boundary Layer Theory
1. Definition
The boundary layer is the thin region adjacent to a solid surface where viscous effects are
significant and velocity changes from zero (due to no-slip) to free stream velocity U
8
.
2. Types of Boundary Layers
• Laminar Boundary Layer: smooth, orderly flow.
• Turbulent Boundary Layer: chaotic, mixing flow.
• Transitional Boundary Layer: between laminar and turbulent.
3. Boundary Layer Thickness (d )
For a flat plate with laminar flow (Blasius solution):
d (x) = 5
x
v
Re
x
= 5
x
q
U8x
? where x is the distance from leading edge, ? is kinematic viscosity.
4. Displacement Thickness (d * )
d * =
Z
8
0

1- u
U
8

dy
Represents reduction in flow rate due to boundary layer.
For laminar flow over flat plate:
d * (x) = 1.72
x
v
Re
x
1
Page 2


Boundary Layer Theory
1. Definition
The boundary layer is the thin region adjacent to a solid surface where viscous effects are
significant and velocity changes from zero (due to no-slip) to free stream velocity U
8
.
2. Types of Boundary Layers
• Laminar Boundary Layer: smooth, orderly flow.
• Turbulent Boundary Layer: chaotic, mixing flow.
• Transitional Boundary Layer: between laminar and turbulent.
3. Boundary Layer Thickness (d )
For a flat plate with laminar flow (Blasius solution):
d (x) = 5
x
v
Re
x
= 5
x
q
U8x
? where x is the distance from leading edge, ? is kinematic viscosity.
4. Displacement Thickness (d * )
d * =
Z
8
0

1- u
U
8

dy
Represents reduction in flow rate due to boundary layer.
For laminar flow over flat plate:
d * (x) = 1.72
x
v
Re
x
1
5. Momentum Thickness (? )
? =
Z
8
0
u
U
8

1- u
U
8

dy
Measures the loss of momentum due to boundary layer.
For laminar flow over flat plate:
? (x) = 0.664
x
v
Re
x
6. Skin Friction Coefficient ( C
f
)
C
f
=
t w
1
2
?U
2
8
where t w
is wall shear stress.
For laminar flow over flat plate:
C
f
=
0.664
v
Re
x
Average skin friction coefficient over length L:
C
f,avg
=
1
L
Z
L
0
C
f
dx =
1.328
v
Re
L
7. Reynolds Number for Boundary Layer
Re
x
=
U
8
x
? 8. Transition from Laminar to Turbulent
Occurs at:
Re
x
˜ 5× 10
5
9. Turbulent Boundary Layer Approximate Thickness
d (x)˜ 0.37
x
Re
1/5
x
10. Momentum Integral Equation (von K´ arm´ an)
d?
dx
+
? U
8
dU
8
dx
=
C
f
2
—
2