fluid_mechanics

9.3 Drag 495
SOLUTION
The friction drag, d f , can be determined from Eq. 9.1 as
F( θ ) ≡ _______ τ w Re
1_ ρ U 2
2
d f t w sin u dA 2 a D 2 b b p
p dA
U θ θ
Re =
8
6
4
2
0
0
_____ ρ UD
μ
20
D
40
(a)
60
θ, degrees
(b)
τ w dA
0
Separation
80
t w sin u du
dA = b
__ D dθ
2
F = 0 for
θ > 108
100
120
where b is the length of the cylinder. Note that u is in radians 1not
degrees2 to ensure the proper dimensions of dA 2 1D22 b du.
Thus,
This can be put into dimensionless form by using the dimensionless
shear stress parameter, F1u2 t w 1Re1rU 2 22, given in
Fig. E9.8b as follows:
C Df
p
C Df
0
t w
1
2 rU
where Re rUDm. Thus,
C Df 1
1Re
p
F1u2 sin u du
The function F1u2 sin u, obtained from Fig. E9.8b, is plotted in
Fig. E9.8c. The necessary integration to obtain C Df from Eq. 1 can
be done by an appropriate numerical technique or by an approximate
graphical method to determine the area under the given
curve.
The result is F1u2 sin u du 5.93, or
p 0
d f
1
2 rU 2 bD 2
rU
p
2
2
sin u du
1
1Re
p
0
0
C Df 5.93
1Re
0
t w sin u du
t w 1Re
sin u du
2
1
2 rU
(1)
(Ans)
F( θ) sin θ
6
4
2
0
0
π__
4
π__
2
θ, rad
(c)
F I G U R E E9.8
3___ π
4
π
COMMENTS Note that the total drag must include both the
shear stress 1friction2 drag and the pressure drag. As we will see in
Example 9.9, for the circular cylinder most of the drag is due to
the pressure force.
The above friction drag result is valid only if the boundary layer
flow on the cylinder is laminar. As is discussed in Section 9.3.3, for
a smooth cylinder this means that Re rUDm 6 3 10 5 . It is
also valid only for flows that have a Reynolds number sufficiently
large to ensure the boundary layer structure to the flow. For the
cylinder, this means Re 7 100.
Pressure (form)
drag is the drag
produced by normal
stresses.
U
pdA
θ
dA
9.3.2 Pressure Drag
Pressure drag, d p , is that part of the drag that is due directly to the pressure, p, on an object.
It is often referred to as form drag because of its strong dependency on the shape or form of the
object. Pressure drag is a function of the magnitude of the pressure and the orientation of the
surface element on which the pressure force acts. For example, the pressure force on either side
of a flat plate parallel to the flow may be very large, but it does not contribute to the drag because
it acts in the direction normal to the upstream velocity. On the other hand, the pressure
force on a flat plate normal to the flow provides the entire drag.
As previously noted, for most bodies, there are portions of the surface that are parallel to the
upstream velocity, others normal to the upstream velocity, and the majority of which are at some
angle in between, as shown by the figure in the margin. The pressure drag can be obtained from
Eq. 9.1 provided a detailed description of the pressure distribution and the body shape is given.
That is,
d p p cos u dA