[ujjwalmohanty]

What is the need to use the stream vorticity function and how are the boundary conditions defined in the lid driven cavity problem?

[Alex C.]

Your question is very vague. I can therefore only assume the context that is of interest to you.

1. What is the need to use the stream vorticity function

In 2D incompressible Navier-Stokes equations, a velocity-pressure equation system will have 2 transport equations (u and v) and a Poisson equation (p). By using the stream-vorticity formulation, you can solve the same problem with 1 less transport equation, as you only have 1 transport equation (vorticity) and 1 Poisson equation (stream).

In 3D incompressible the vorticity in no longer a scalar, but is a vector with 3 coordinates, and you have just as many equations, with more complex boundary conditions to apply. This is not as common.

I am not familiar with the compressible, but I know it also is quite uncommon to use such formulation for the compressible cases.

2. how are the boundary conditions defined in the lid driven cavity problem
The stream function can be set using a Dirichlet condition. Stream = cst along the boundary.

For the vorticity, it's not so simple, but not so bad either. Starting with a Taylor expansion normal to the boundary of the stream function, up to the second order term, you can do substitutions and simplification which will result in a boundary condition on vorticity. I'll let you toy around with this idea.

[FMDenaro]

The 2D stream function-vorticity formulation is a standard section in any textbook of CFD and is a good exercise for a student.
You just rewrite the continuity (the divergence-free constraint) and momentum equation (applying the curl).
The BC.s for the stream function is quite simple from its definition in terms of the velocity field.
For the vorticity, study how the Thom formula is obtained.

The 3D extension is practically never used for practical application.

[ujjwalmohanty]

Originally Posted by Alex C.

Your question is very vague. I can therefore only assume the context that is of interest to you.

1. What is the need to use the stream vorticity function

In 2D incompressible Navier-Stokes equations, a velocity-pressure equation system will have 2 transport equations (u and v) and a Poisson equation (p). By using the stream-vorticity formulation, you can solve the same problem with 1 less transport equation, as you only have 1 transport equation (vorticity) and 1 Poisson equation (stream).

In 3D incompressible the vorticity in no longer a scalar, but is a vector with 3 coordinates, and you have just as many equations, with more complex boundary conditions to apply. This is not as common.

I am not familiar with the compressible, but I know it also is quite uncommon to use such formulation for the compressible cases.

2. how are the boundary conditions defined in the lid driven cavity problem
The stream function can be set using a Dirichlet condition. Stream = cst along the boundary.

For the vorticity, it's not so simple, but not so bad either. Starting with a Taylor expansion normal to the boundary of the stream function, up to the second order term, you can do substitutions and simplification which will result in a boundary condition on vorticity. I'll let you toy around with this idea.

Thanks Alex.
It is very helpful.
I am just a beginner in CFD, hence the vague questions.
And what does cst stand for??