Fundamentals Of Aerodynamics Book 2.4

Problem 4: Book Problem 2.4 GIVEN Consider a velocity field where the x and y components of velocity are given by: c⋅ y

u=

2

v=

2

x +y

−c⋅ x 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⋅ x 2

x +y

2

   2 2 x + y 

→−

c⋅ y

x

which is equal to

y

d y dx

rearranging, we yield −x⋅ dx = y⋅ dy Integrating both sides, we get 2 ⌠ x  −x dx → − 2 ⌡

Therefore 2

x

2

2

=

y

2

+c

2 ⌠ y  y dy → 2 ⌡

 x2 − k    2  − x − k

y=

Where k is an arbitrary constant

20

2

x +10 2

x +1 10

2

x

2

x −1 2

x −10 2

−

x + 10

−

x +1

−

x

−

x −1

−

− 10 2 x − 10

0

2 2 2

− 20 − 10

−5

0 x

5

10