Ex) An incompressible plane flow has the velocity components
acceleration components. (b) Find the pressure distribution (
Assume frictionless flow.
,
,
. (a) Find the
) if the pressure at the origin is .
Solution:
(a) Acceleration
⏟
⏟
where
̂
̂
or
With
and
,
(
)( )
(
)( )
(
)( )
(
)( )
̂
̂
Note:
| |
√
√(
√
)
(
)
(b) Euler equation
where
̂
̂
Thus,
Integrate
,
∫
)
∫(
( )
then differentiate,
Or
( )
∫(
)
Thus,
at ( , ) = (0,0),
|
or
Finally,
(
)
Note: Pressure along streamline
Streamline
or
Integrate,
∫
∫
or
For streamline that goes through the origin
,
From the Euler equation
(
Thus,
)
Alternate approach:
Bernoulli equation along the streamline
where
(
)
(
)
(
)
(
and
at
. Thus,
(
)
)