In 3D you can define the equivalent vector potential function that is solution of the Poisson equation

Lap psi = Curl v

+Bcs.

Thus you get 3 scalar functions psii . Hovewer, they do not share the property of the 2D stream function to be a line along which the velocity is tangential.

In other terms, you can denote "stream-lines" the isovalues lines of only the 2D potential function.

In the 3D case, stream-lines cannot be constructed by the components of the vector potential function but you need to integrate the line along which the vector velocity is tangential dr x v =0

Of course you can define the vector potential (\vec{Psi}) by which you define the velocity vector from

vec{V}= curl (\vec{Psi})

Thus, vector potential is the 3D counter part of stream function defined for 2D. One may like to look at the following:

Article Vorticity dynamics of the three-dimensional Taylor-Green vor...

This provides a concrete example of how to compute flow field using, vorticity-vector potential formulation for high accuracy computing. However, this is done for a periodic problem in this reference. For non-periodic problems, we extending the same formulation. The biggest advantage is that the velocity field becomes solenoidal, and then for incompressible flows one has hassle-free high accuracy solution.

3D Stream function in fluid mechanics. The 3D stream function Ψ for a steady flow field can be defined as: ρu=∇×Ψ.

The stream function is defined for in compressible (divergence-free) flows in two dimensions – as well as in three dimensions with ax-symmetry. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. However If you want to obtain 3D Stream function You should specify two different stream functions on two surfaces of three dimensional geometry and solve together leading to 3D Stream function taking into consideration the relevant boundary conditions.

another solution using CFD to obtain velocity components and by using the relation ρu=∇×Ψ. You may define the proper 3D stream function.

You can define only potential vector.

the calculation is very tedious and does not simplify the problem.

You must distinguish between potential function and stream function. Potential function can only be defined in 2D while stream function is also applicable in 3D problem. Stream function, is a function that satisfies continuity equation, i.e. ∇.V (V is velocity vector) and since ∇.(∇x F) {F is any arbitrary vector, here F is velocity vector) is always zero, so, ∇x F always satisfies the continuity equation. ∇xF is called stream function.

Hossain Nemati

I think that is exactly the opposite you wrote. The stream-function, that is a single scalar function, is defined only in 2D as the curve of the plane that is envelop of the velocity vector while the vector potential function always exists and reduces in 2D to the vector field kPsi3 that is normal to the plane of the flow.