E1C09 09/14/2010
15:4:55 Page 400
Pressure fluctuations occur as the flow fluctuates around the car. Statistical variations from
average values give the stated random uncertainty, u Dp ¼ 0.10 cm H 2 O (95%), which we assign large
degrees of freedom.
If we look at the rear deck of the car and use the instrument specifications as a measure of the
instrument systematic uncertainty,
u c transducer % ð0:0025Þ ð8 cm H 2 OÞ ¼ 0:02 cm H 2 O
u c A=D ¼ ð2 bitsÞ ð0:0124 cm H 2 O=bitÞ ¼ 0:025 cm H 2 O
which, combined with u o for the 12-bit system, gives the design-stage estimate
u d
ð Þ Dp ¼ ð0:02Þ
2 þ ð0:025Þ
2 þ ð0:0124Þ
2
h
i 1=2 ¼ 0:034 cm H 2 O
We can use this as an estimate of the systematic uncertainty at the 95% confidence level. Then, the
combined uncertainty in mean pressure is
u Dp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
0:034
2
þ 0:10
2
p
¼ Æ 0:10 cm H 2 O ð95%Þ
or about 1.25%. From this advanced design-stage analysis, a higher 16-bit resolution would not
improve the measurement uncertainty.
The downforce is simply the downward component of force, F D ¼ DpA ef f . The effective
area on the rear deck, A ef f , can be estimated from carefully controlled calibration measurements
within a wind tunnel using a highly accurate balance scale to measure downforce (i.e., Nth order
analysis, concomitant method). Suppose a value for the rear effective area is found to be
A eff ¼ 10832 Æ 147cm
2 95%
ð
Þ. Combining uncertainties in pressure and area gives the percent
uncertainty in rear downforce,
u F =F D ¼ Æ ðu Dp =DpÞ
2 þ u A =A eff
ð
Þ
2 ¼ 0:019 or 1:9% ð95%Þ
h
COMMENT Here we see that the dominant uncertainty is due to the measuring procedure and
flow process and not the instrumentation.
9.7 MODELING PRESSURE AND FLUID SYSTEMS
Fluid systems can be modeled using lumped parameter ideal elements just as common resistorinductor-capacitor electrical loops and mass-damper-spring mechanical systems are used. The
common elements are inertance, resistance, and compliance.
Inertance describes the inertial properties of a mass in motion, such as that of a mass of fluid
moving within a vessel. For a fluid of density r and a vessel of cross-sectional area A and length ‘,
the inertance is written
L f ¼ r‘=A
ð9:18Þ
When modeling inertial forces in laminar flows, this value should be increased by a factor of
4
3 .
Inertance is the direct analog to electrical inductance.
Fluid resistance describes the opposition to motion. This is the pressure change required to
move a volume of fluid per unit time, Q. It is written
R ¼ Dp
n
=Q ¼ DE=I
ð9:19Þ
400 Chapter 9 Pressure and Velocity Measurements
15:4:55 Page 400
Pressure fluctuations occur as the flow fluctuates around the car. Statistical variations from
average values give the stated random uncertainty, u Dp ¼ 0.10 cm H 2 O (95%), which we assign large
degrees of freedom.
If we look at the rear deck of the car and use the instrument specifications as a measure of the
instrument systematic uncertainty,
u c transducer % ð0:0025Þ ð8 cm H 2 OÞ ¼ 0:02 cm H 2 O
u c A=D ¼ ð2 bitsÞ ð0:0124 cm H 2 O=bitÞ ¼ 0:025 cm H 2 O
which, combined with u o for the 12-bit system, gives the design-stage estimate
u d
ð Þ Dp ¼ ð0:02Þ
2 þ ð0:025Þ
2 þ ð0:0124Þ
2
h
i 1=2 ¼ 0:034 cm H 2 O
We can use this as an estimate of the systematic uncertainty at the 95% confidence level. Then, the
combined uncertainty in mean pressure is
u Dp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
0:034
2
þ 0:10
2
p
¼ Æ 0:10 cm H 2 O ð95%Þ
or about 1.25%. From this advanced design-stage analysis, a higher 16-bit resolution would not
improve the measurement uncertainty.
The downforce is simply the downward component of force, F D ¼ DpA ef f . The effective
area on the rear deck, A ef f , can be estimated from carefully controlled calibration measurements
within a wind tunnel using a highly accurate balance scale to measure downforce (i.e., Nth order
analysis, concomitant method). Suppose a value for the rear effective area is found to be
A eff ¼ 10832 Æ 147cm
2 95%
ð
Þ. Combining uncertainties in pressure and area gives the percent
uncertainty in rear downforce,
u F =F D ¼ Æ ðu Dp =DpÞ
2 þ u A =A eff
ð
Þ
2 ¼ 0:019 or 1:9% ð95%Þ
h
COMMENT Here we see that the dominant uncertainty is due to the measuring procedure and
flow process and not the instrumentation.
9.7 MODELING PRESSURE AND FLUID SYSTEMS
Fluid systems can be modeled using lumped parameter ideal elements just as common resistorinductor-capacitor electrical loops and mass-damper-spring mechanical systems are used. The
common elements are inertance, resistance, and compliance.
Inertance describes the inertial properties of a mass in motion, such as that of a mass of fluid
moving within a vessel. For a fluid of density r and a vessel of cross-sectional area A and length ‘,
the inertance is written
L f ¼ r‘=A
ð9:18Þ
When modeling inertial forces in laminar flows, this value should be increased by a factor of
4
3 .
Inertance is the direct analog to electrical inductance.
Fluid resistance describes the opposition to motion. This is the pressure change required to
move a volume of fluid per unit time, Q. It is written
R ¼ Dp
n
=Q ¼ DE=I
ð9:19Þ
400 Chapter 9 Pressure and Velocity Measurements