SME
Fluid Mechanics
Boundary layers
Use this for “Air at 5 m/s flows along a 1 m plate 0.5 m wide. Find the layer’s thickness at the end and the drag on one side.”
Diagram
ReL
=
VL
ν
=
333,000.
δ
=
5x
÷
√Reₓ
grows
as
√x:
0.00866
m
at
the
end,
0.00612
m
halfway.
Heights are stretched 54 times to be seen.
Density ρ = 1.2 kg/m³
Plate width b = 0.5 m
Skin-friction coefficient Cf = 0.0023
Drag force FD = 0.01725 N
This is an example. Type your own values to start a new problem.
δ (mm)
=
5
×
L (m)
ReL
FD (N)
=
½
×
ρ (kg/m³)
×
(
V (m/s)
)
2
×
Cf
×
b (m)
×
L (m)
ν
Kinematic viscosity
example · m²/s
Formulas
ReL
=
V
×
L
ν
333,300
=
5
×
1
1.5 ×
10
-5
δ
=
5
×
L
÷
√ReL
0.00866
=
5
×
1
÷
√
333
,300
Cf = 1.328 ÷ √ReL
0.0023
=
1.328
÷
√
333
,300
FD
=
1
2
×
ρ
×
V
2
×
Cf
×
b
×
L
0.01725
=
1
2
×
1.2
×
5
2
×
0.0023
×
0.5
×
1
Formulas are worked in m. The step-by-step shows the conversions.
Assumptions
•
A smooth flat plate along a steady stream, with no pressure change along it (Blasius).
•
Laminar all the way: ReL from about 1,000 (below it the layer is too thick for the fits) to 5 × 10⁵, where it starts to turn turbulent.
•
FD is the friction drag on one side, over the wetted area bL.
Step-by-step
WE KNOW
V = 5 m/s, L = 1 m, ν = 1.5 × 10⁻⁵ m²/s, ρ = 1.2 kg/m³, b = 0.5 m
FIND
Reynolds number at L (ReL), layer thickness at L (δ), skin-friction coefficient (Cf), drag force (FD)
Step 1 · Find Reynolds number at L (ReL)
Use ReL = V × L ÷ ν
Re
at
the
trailing
edge:
the
stream
speed
times
the
plate’s
length,
over
ν
.
ReL
=
5
×
1
1.5 ×
10
-5
ReL = 333,300
Step 2 · Find layer thickness at L (δ)
Use δ = 5 × L ÷ √ReL
Blasius: the layer is 5x ÷ √Reₓ thick, so it grows as the square root of x.
δ
=
5
×
1
÷
√
333
,300
δ
=
5
577.3
δ
=
0.00866
m
Step 3 · Find skin-friction coefficient (Cf)
Use Cf = 1.328 ÷ √ReL
Blasius’s skin friction averaged over the whole plate.
Cf
=
1.328
÷
√
333
,300
Cf
=
1.328
577.3
Cf = 0.0023
Step 4 · Find drag force (FD)
Use FD = ½ × ρ × V² × Cf × b × L
Drag
is
Cf
times
the
dynamic
pressure
1
2
ρV²
times
the
wetted
area
bL.
FD
=
1
2
×
1.2
×
5
2
×
0.0023
×
0.5
×
1
FD
=
1
2
×
1.2
×
25
×
0.0023
×
0.5
FD = 0.6 × 25 × 0.0023 × 0.5
FD = 15 × 0.0023 × 0.5
FD = 0.0345 × 0.5
FD = 0.01725 N
Step 5 · Convert the answers
Change each answer back to the unit you chose.
δ = 0.00866 m = 8.66 mm (1 m = 1000 mm)
Check
Put the numbers back in. Both sides should match.
333,300
=
5
×
1
1.5 ×
10
-5
✓
0.00866
=
5
×
1
÷
√
333
,300
✓
0.0023
=
1.328
÷
√
333
,300
✓
0.01725
=
1
2
×
1.2
×
5
2
×
0.0023
×
0.5
×
1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
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