Fundamentals Of Aerodynamics Book 2.5

Problem 5: Book Problem 2.5 GIVEN Consider a velocity field where the radial and tangential components of velocity are given by: Vθ = c⋅ r

Vr := 0

FIND Obtain the equations of the streamlines

ASSUMPTIONS (1) c is a constant (2) 2-D flow

SOLUTION u = Vr⋅ cos( θ) − Vθ⋅ sin( θ) v = Vr⋅ sin( θ) + Vθ⋅ cos( θ) z=z u = −c⋅ r⋅ v = c⋅ r⋅

x r

y r

= −c⋅ y

= c⋅ x

Using the same techniques from previous problems.... dy

v

c⋅ x

dx

u

−c⋅ y

2

2

=

y + x =k

→−

x y

 − k − x2   y=  2   k−x 

4 − 10− x

2

2

− 5− x 0− x 5− x

2

2

2 2

10− x

2

−

− 10− x 0

−

− 5− x

−

0− x

−

5− x

−

10− x

2

2 2

−2 2

−4 − 10

−5

0 x

5

10