EGM 6812, F11, University of Florida, M. Sheplak
1/16
Section 6, Dimensional Analysis
6 Dimensional Analysis and Similarity
Why?
1.
Reduced number of experiments
2.
Compact data representation
3.
Solve problem fewer times
4.
Lab prototype to full scale
5.
If we non-dimensionalize
the governing equations, we can “simplify” the equations by
neglecting certain terms.
Example
Investigate drag on a sphere, with roughness
(i.e. golf ball, high speed)
, , ,
,
,
D
D
F
F
D
u
c
If we investigate 10 iterations of each parameter, we would have to conduct
6
10
experiments.
Instead perform a non-dimensionalization.
6.1 Buckingham
-theorem
Answer-
2
2
Roughness
Re
,
,
1
2
D
D
Ma
C
F
uD
u
f
c
D
u D
Step 1.
List all variables
, , ,
,
,
D
D
F
F
D
u
c
7
n
Step 2.
Select the primary dimensions
M,L,t
3
r
Step 3.
List dimensions of each parameter
Parameter
F
D
c
u
Dimension
ML
t
2
L
L
M
L
3
M
Lt
L
t
L
t
Step 4.
Select “r”
repeating parameters (engineering judgment call!)
Don’t choose:
, ,
,
D
F
c
non-repeaters

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EGM 6812, F11, University of Florida, M. Sheplak
2/16
Section 6, Dimensional Analysis
“parameters that are physically interesting”
Do choose
, ,
u D
} make sure that they cover mass, length, and time
Step 5.
Set up (n-r)
groups
1
a
b
c
D
F
u D
nondimensional
choose a,b,c such that the
s are nondimensional
2
4
...
d
e
f
u D
When you do this you end up with
drag force
1
2
2
2
2
area
dynamic
pressure
drag coefficient
D
D
D
F
F
C
u D
u
D
2
ratio of roughness to diameter
D
3
1
Re
uD
4
1
Ma
c
u
6.2 Flow Similarity
How do we scale data from a wind-tunnel experiment to full-scale?
Similarity!!!
Geometric Similarity
:
Same shape between model and full scale
Kinematic Similarity
:
velocity fields about a model and prototype must vary by no more
than a constant.
Kinematic similarity implies geometric similarity
Dynamic Similarity
:
all forces on a model prototype must vary by no more than a
constant scale factor.
Dynamic similarity implies kinematic similarity and geometric
similarity.
To achieve dynamic similarity, we must match all the
groups
Example
:
2
2
Re,
,
D
D
F
C
f
Ma
v D
D
For Dynamic similarity