Experiment #5
Momentum Deficit Behind a Cylinder
Problem Statement
Water, 40 °F
•
Find the drag force on the rod
located in a 2.0 mph 40 °F water
stream
•
Use analogy with air flow normal
to a model cylinder in a wind
tunnel
•
Estimate the error in the drag
force
2.0 mph
0.5” rod
Available Major Equipment
• 6 x 6 inch in cross section wind tunnel
• Hot film anemometer to measure air speed
• Model cylinder to place in the wind tunnel
Approach
•
Measure the drag on a model cylinder in the wind tunnel
•
Use results from dimensional analysis to establish the conditions for
similarity between the flow around the model and the prototype (the rod
in the water stream)
•
Use the non-dimensional numbers derived from dimensional analysis to
estimate the drag force on the rod
•
Estimate the error in the drag force on the model and the prototype
Results from Dimensional Analysis
(consult a fluids text)
Drag Coefficient, CD,
of a cylinder in cross-flow
F
D
C 
 f(Re),
D 1 DwU2
1
2
where
FD = drag force on the cylinder
 = air density

D = cylinder diameter
w = cylinder length
U1 = approach velocity of air stream
 = kinematic viscosity of air
Similarity between model and prototype requires that
CDmodel = CDprototype= f(Remodel) = f(Reprototype) or that
Remodel = Reprototype
UD
Re = 1

Equality of Reynolds Numbers implies
equality of the Drag Coefficients
Experimental Approach
for the Drag Force
•
Estimate the Reynolds number for the prototype (rod in water), and compute the
wind tunnel air speed necessary for similarity with the model cylinder
–
Note: account for the fact that the pressure in the wind tunnel is less than atmospheric
•
Find the drag force on the model cylinder from measurements carried out in the
wind tunnel
•
Compute the model Reynolds number and Drag coefficient.
–
•
Compare with the values from the previously shown CD vs. Re curve.
Assuming similarity (check this) use the magnitude of the experimental drag
coefficient on the model cylinder to find the drag force on the prototype stack
Static pressure inside the wind tunnel
•
P1, V1
Po, Vo = 0
•
•
Use Bernoulli’s equation between
points o (outside the tunnel) and 1
(inside the tunnel)
Note that at point o the pressure is
the atmospheric level, and that the
velocity at point o is zero.
Using the known wind tunnel
speed, V1 find the pressure p1.
Control Volume Force and Mass Balances
( = constant)
Force
balance
L
L
-L
-L
 m
 )U
 F  w  U U dy  w  u (y)u (y)dy (m
D
2
3
4 1
1 1
2
Surface 1, rate of
momentum in
Surface 2,rate of
momentum out
Surfaces 3 and 4, rate of
momentum out
Mass
balance
-L
-L
-L
-L
 m
 )
0  w  U dy  w  u (y)dy (m
2
3
4
1
Surface 1, rate of
mass in
Surface 1, rate
of mass out
Surfaces 3 and 4, rate of
mass out
Multiplying the second eq. by U1 and substituting in the first eq, results in:
L
F  w  u yU u ydy
 1
D
2 
L 2
Drag force on the Model Cylinder
L
F  w  u yU u ydy
 1
D
2 
L 2

• Measure the velocity profile U1 without the cylinder
• Measure the velocity profile u2 with the cylinder
at two location downstream from the cylinder
• Compute FD for each of the two profiles using the
experimental data (numerical integration needed)
• Take the mean of the two FD values
Drag force on the Rod in the Water
1  DwU2 
F
1 prototype
Dmodel 2 


1
2


F
C
DwU 

Dprototype Dmodel 2 
1 prototype
1  DwU2 
1 model
2 
Error Estimation
Find the error in FD of the model assuming that the only measuring
error is a ±1% in the measurement of velocity.
Use the RSS method. (Consult your 650:350 Measurements Book)
L
F  w  u yU u ydy
 1
D
2 
L 2

Error in FD
where








2

F

FD
D
u 
u

u
u u2  U U
F
2


D

F
L
D  w  U  2u ydy

u
2 
L 1
2
1
and
uu2 and uU1 are the ± errors in the velocity


measurement.
Use ±1% of measured values
Numerical integration is needed to evaluate the
integrals
1
1/2
2
 









F
L
D  w  u ydy


U
L 2
1

Error Estimation (continued)
The error in the FD of the prototype rod is then found in a similar way from:
1  DwU2 
F


1
Dmodel
2



prototype
1
2
 DwU 
F
C

Dprototype Dmodel 2 
1 prototype
1  DwU2 
1 model
2 

1/2
2
2


 
F
F



 


Dprototype u
  
 
u
  Dprototype u

 U
 
F
F
U
 F
Dprototype 
Dmodel 
1model 
Dmodel
1model


F
Dprototype 
F
Dmodel






DwU 2 
1 prototype


DwU 2 
1 model

1 DwU 2 
F
1 prototype
Dprototype  Dmodel 2 
1 DwU 3
U
1model
1 model
4 
F