Advanced Fluid
M ech anics
Boundary Layer
5. F orm
3 5
D rag F orc es
Chapter 5
FORM DRAG FORCES
5.1. APPLICATION OF FLAT PLATE FRICTION
COEFFICIENT
In practical applications there are 3D bodies submerged in
flowing fluid or moving through it
⇓
pressure varies across the boundary layer as well as along the
surface of the body
question:
May the formulas derived so far be utilised for practical
applications ?
answer:
YES, under certain circumstances, i.e. for streamlined bodies
(the streamlines are parallel to the surface)
streamlined body
it is possible to locate the body in such a position with respect
to the flow direction that ensures no separation of the
boundary layer
Boundary Layer
Advanced Fluid
M ech anics
5. F orm
3 6
D rag F orc es
examples :
airfoil:
plane wing
turbine or compressor blade
fish
raindrop
A - the maximum cross-sectional area of the object across
the flow direction
Frictional drag force
Ff = c f
ρU ∞2
A
(5.1)
2
cf is determined from formula for flat plate of the same
roughness as the surface of the airfoil
warning:
if separation of the boundary layer occurs
the above method gives completely wrong results !!!
Advanced Fluid
M ech anics
Boundary Layer
5. F orm
D rag F orc es
INFLUENCE OF AN AIRFOIL POSITION ON THE APPEARANCE
OF BL SEPARATION
case A
no separation
case B
attack angle increased ⇒ boundary layer separated in the
rear part of suction side
3 7
Boundary Layer
Advanced Fluid
M ech anics
5. F orm
3 8
D rag F orc es
Total drag force:
frictional drag force
form (shape) drag force – due to the pressure forces
F form = c form
ρU ∞2
2
A
(5.2)
cform is determined experimentally
Ftotal = F f + F form
Ftotal = ctotal
ρU ∞2
2
A
(5.3)
(5.4)
example of ctotal : well known cx coefficient (cars) being the
measure of the quality of the design (from aerodynamical
viewpoint)
for streamlined bodies
F f ≈ F form
for bluff (non-streamlined) bodies
(5.5)
Advanced Fluid
M ech anics
Boundary Layer
5. F orm
3 9
D rag F orc es
wake - the region of the flow characterised by mean velocity
gradient due to the body submerged in the flow
(intensive mixing processes ⇒ great losses of energy)
F f << F form ≈ Ftotal
typically
F f = (1 ÷ 2 %) Ftotal
(5.6)
(5.7)
Advanced Fluid
M ech anics
shape
Boundary Layer
5. F orm
4 0
D rag F orc es
l/d
Re
cx
circular disc
( ⊥ to the flow direction)
-
system of circular disks
( ⊥ to the flow direction)
0
1
2
3
rectangular plate
( ⊥ to the flow direction)
1
5
20
∞
circular cylinder
0
1
2
4
7
> 103
1.12
0.91
0.85
0.87
0.99
-
> 103
1.33
-
> 103
0.34
5
hemisphere
hemisphere
> 103
> 10
3
> 103
1.12
1.12
0.93
1.04
1.52
1.16
1.20
1.50
2.00
1
5
20
∞
10 ÷ 10
0.63
0.74
0.90
1.20
5
∞
> 5⋅105
0.35
0.33
-
103 ÷ 105
> 3⋅105
0.47
0.20
-
> 2⋅105
0.04
3
circular cylinder
sphere
airfoil