Fluids – Lecture 10 Notes
1. Substantial Derivative
2. Recast Governing Equations
Reading: Anderson 2.9, 2.10
Substantial Derivative
Sensed rates of change
The rate of change reported by a flow sensor clearly depends on the motion of the sensor.
For example, the pressure reported by a staticpressure sensor mounted on an airplane in
level flight shows zero rate of change. But a ground pressure sensor reports a nonzero rate
as the airplane rapidly flies by a few meters overhead. The figure illustrates the situation.
p (t)
1
p (t)
2
o
t = t
wing location at
p
t
o
t
p (t)
1
p (t)
2
Note that although the two sensors measure the same instantaneous static pressure at the
same point (at time
t
=
t
o
), the measured time rates
are different.
p
1
(
t
o
) =
p
2
(
t
o
)
but
dp
1
dt
(
t
o
)
negationslash
=
dp
2
dt
(
t
o
)
Drifting sensor
We will now imagine a sensor
drifting with a fluid element
. In effect, the sensor follows the
element’s pathline coordinates
x
s
(
t
),
y
s
(
t
),
z
s
(
t
), whose time rates of change are just the
local flow velocity components
dx
s
dt
=
u
(
x
s
, y
s
, z
s
, t
)
,
dy
s
dt
=
v
(
x
s
, y
s
, z
s
, t
)
,
dz
s
dt
=
w
(
x
s
, y
s
, z
s
, t
)
t
V
p (t)
s
s
p
pathline
pressure
field
sensor drifting with local velocity
s
Dp
dp
Dt
dt
Consider a flow field quantity to be observed by the drifting sensor, such as the static pressure
p
(
x, y, z, t
). As the sensor moves through this field, the instantaneous pressure value reported
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 Spring '11
 Wereley
 Derivative, dt

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