Why is the Stream Function Psi Limited to 2-D Incompressible Flows?

[haisol]

As we learned in fluid mechanics, we can devise stream function psi

when the flow is 2-D, incompressible.

However, I cannot clearly explain why psi can be defined only for 2-D, mathematically or logically.

Is there any way to explain this, more than just say that 'just as it is'?

 

[AIR&SPACE]

Try looking at these:
http://www.cg.inf.ethz.ch/~peikert/personal/pss/strfun.html
https://www.physicsforums.com/showthread.php?t=84860

 

[haisol]

thank you for your kind answer! It really helps me understanding!

 

[boneh3ad]

Technically speaking, all flows are represented by a combination of two streamfunctions, even two-dimensional flows. In a typical two-dimensional flow, the second streamfunction is just a series of vertical planes (or planes rotated around the axis of symmetry for axisymmetric flow) since there is no out of plane velocity. This goes for any two-dimensional flow.

For a three-dimensional flow, both streamfunctions can be used to represent the flowfield, but you have to use both. Representing flowfields this way is not only uncommon, but pretty much unheard of due to the complexity of doing so and the relative ease of other methods. It is, however, mathematically possible to do.