Problem 2.21
[Difficulty: 3]
Given:
Eulerian Velocity field
Find:
Lagrangian position function that was at point (1,1) at t = 0; expression for pathline; plot pathline and compare to
streamlines through same point at the instants t = 0, 1 and 2s
Solution:
Governing equations:
For pathlines (Lagrangian description)
u
p
dx
dt
=
v
p
dy
dt
=
For streamlines
v
u
dy
dx
=
Assumption:
2D flow
Hence for pathlines
u
p
dx
dt
=
A
=
A
2
=
m
s
v
p
dy
dt
=
B
−
t
⋅
=
B
2
=
m
s
2
So, separating variables
dx
A dt
⋅
=
dy
B
−
t
⋅
dt
⋅
=
Integrating
x
A t
⋅
x
0
+
=
x
0
1
=
m
y
B
−
t
2
2
⋅
y
0
+
=
y
0
1
=
m
The Lagrangian description is
x t
( )
A t
⋅
x
0
+
=
y t
( )
B
−
t
2
2
⋅
y
0
+
=
Using given data
x t
( )
2 t
⋅
1
+
=
y t
( )
1
t
2
−
=
The pathlines are given by combining the equations
t
x
x
0
−
A
=
y
B
−
t
2
2
⋅
y
0
+
=
B
−
x
x
0
−
(
)
2
2 A
2
⋅
⋅
y
0
+
=
Hence
y x
( )
y
0
B
x
x
0
−
(
)
2
2 A
2
⋅
⋅
−
=
or, using given data

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