Fundamentals Of Aerodynamics Book 2.3

Problem 3: Book Problem 2.3 GIVEN Consider a velocity field where the x and y components of velocity are given by: c⋅ x

u=

2

v=

2

x +y

c⋅ y 2

2

x +y

FIND Obtain the equations of the streamlines

ASSUMPTIONS (1) c is a constant (2) 2-D Steady Flow

SOLUTION For incompressible flow, the stream functions can be described using equation 2.118 and 2.139 v d y= u dx Plugging values in for u and v and simplifying, we get c⋅ y 2

x +y

2

   2 2 x + y 

→

c⋅ x

y

which is equal to

x

d y dx

rearranging, we yield 1 x

⋅ dx =

1 y

⋅ dy

Integrating both sides, we get ⌠   ⌡

1 x

dx → ln( x)

⌠   ⌡

1 y

dy → ln( y)

Therefore ln( x) = ln( y) + c = ln( k⋅ y)

y = k⋅ x

Where k is an arbitrary constant

100

50 − 10⋅ x − 5x − 1x 0x 1x

0

5x 10x

− 50

− 100 − 10

−5

0 x

5

10