Transcript Fluid Flow Concepts and Basic Control Volume Equations

External Flows
Overview
 Non-Uniform
Flow
 Boundary Layer Concepts
 Viscous Drag
 Pressure Gradients: Separation and Wakes
 Pressure Drag
 Shear and Pressure Forces
 Vortex Shedding
Non-Uniform Flow
 In
pipes and channels the velocity
distribution was uniform (beyond a few
pipe diameters or hydraulic radii from the
entrance or any flow disturbance)
 In external flows the boundary layer is
always growing and the flow is nonuniform
Boundary Layer Concepts

Two flow regimes
Laminar boundary layer
 Turbulent boundary layer



with laminar sub-layer
Calculations of
boundary layer thickness
 Shear (as a function of location on the surface)
 Drag (by integrating the shear over the entire surface)

Flat Plate: Parallel to Flow
U

y
U
U

U
boundary
layer
thickness
x
shear
Why is shear maximum at the leading edge of
the plate? du
dy
is maximum
Flat Plate Drag Coefficients
0.01
CDf = [1.89 - 1.62log (e / l )]
1 x 10-3
5 x 10-4
2 x 10-4
1 x 10-4
5 x 10-5
2 x 10-5
1 x 10-5
5 x 10-6
2 x 10-6
1 x 10-6
- 2.5
CDf
CDf =
1.328
(Rel )0.5
CDf =
0.455
[log (Re )]
2.58
l
-
e
l
CDf = 0.072Rel- 0.2
1700
Rel
CDf =
0.001
0.455
[log (Re )]
Rel =
Ul
n
+1
0
1e
+0
9
1e
+0
8
1e
+0
7
1e
+0
6
1e
+0
5
1e
1e
+0
4
l
2.58
Separation and Wakes
 Separation
often occurs at sharp corners
 fluid
can’t accelerate to go around a sharp
corner
 Velocities
in the Wake are ______
small (relative
to the free stream velocity)
 Pressure in the Wake is relatively ________
constant
(determined by the pressure in the adjacent
flow)
Flat Plate:
Streamlines
3
U
0


p  p0 

Cp  1
 2
2
2 
U
 U 
v
Cp
p
>p0
0
1
<U >0
>p0
<p0
>U <0
<p0
v2
2
1
4
Point
1
2
3
4
Points outside boundary layer!
Drag of Blunt Bodies and
Streamlined Bodies



Drag dominated by viscous
streamlined
drag, the body is __________.
Drag dominated by pressure
drag, the body is _______.
bluff
Whether the flow is viscousdrag dominated or pressuredrag dominated depends
entirely on the shape of the
body.
Bicycle page at Princeton
Drag Coefficient on a Sphere
Drag Coefficient
1000
100
Stokes Law
10
1
0.1
0.1
CD =
1
24
Re
10
102
103
104
Reynolds Number
105
106
107
Shear and Pressure Forces:
Horizontal and Vertical Components
FD    p sin    0 cos dA
drag Parallel to the approach velocity
FL    p cos    0 sin  dA
lift Normal to the approach velocity
p < p0
U 2
negative pressure Fd  Cd A
2
A defined as projected
normal to force!
area _______
U 2
FL  CL A
2
lift
q
U
drag
p > p0 positive pressure
Shear and Pressure Forces
 Shear
forces
 viscous
drag, frictional drag, or skin friction
 caused by shear between the fluid and the solid
surface
length object
 function of ___________and
______of
surface area
 Pressure
forces
 pressure
drag or form drag
 caused by _____________from
the body
flow separation
 function of area normal to the flow
Example: Beetle Power
Cd = 0.38
Height = 1.511 m
Width = 1.724 m
Length = 4.089 m
Ground clearance = 15 cm?
85 kW at 5200 rpm
Where does separation occur?
Calculate the power required to overcome drag at 60 mph and 120 mph.
Is the new beetle streamlined?
Solution: Power a New Beetle at
60 mph
Cd 
FD 
2FD
U 2 A
 f (R )
C d U 2 A
2
P
C d U 3 A
2
P
(0.38)(1.2kg / m3 )(26.82m / s)3 (2.0m 2 )
2
P = 8.8 kW at 60 mph
P = 70 kW at 120 mph
Drag on a Golf Ball
DRAG ON A GOLF BALL comes mainly from
pressure drag. The only practical way of reducing
pressure drag is to design the ball so that the point
of separation moves back further on the ball. The
golf ball's dimples increase the turbulence in the
inertia of the
boundary layer, increase the _______
boundary layer, and delay the onset of separation.
The effect is plotted in the chart, which shows that
for Reynolds numbers achievable by hitting the
ball with a club, the coefficient of drag is much
lower for the dimpled ball.
Why not use this for aircraft or cars?
Effect of Turbulence Levels on
Drag
 Flow
over a sphere: (a) Reynolds number =
15,000; (b) Reynolds number = 30,000,
with trip wire. Causes boundary layer to become turbulent
Point of separation
Effect of Boundary Layer
Transition
Ideal (non
viscous) fluid
No shear!
Real (viscous)
fluid: laminar
boundary layer
Real (viscous)
fluid: turbulent
boundary layer
Spinning Spheres
 What
happens to the separation points if we
start spinning the sphere?
LIFT!
Vortex Shedding
Vortices are shed alternately
from each side of a cylinder
 The separation point and thus
the resultant drag force oscillate
 Dimensionless frequency of
shedding given by Strouhal
number S
 S is approximately 0.2 over a
wide range of Reynolds
numbers (100 - 1,000,000)

S
nd
U