3-D Stream Function
I am solving 2-D Navier Stocks equation in Biharmonic formulation.
But now I want to know that is it possible to solve this equation in 3-D ?
Indeed the Biharmonic equations in 2-D come from Stream Function-Vorticity method. We know that we don't have Stream Function -Vorticity in 3-D space But we have 3-D operator for Biharmonic equation.
In 3-D, a stream function exists for axisymmetric flows only. That makes it essentially 2-D of course.
Actually the stream function and vorticity are vectors.
In a two-d problem, each vector has only one component.
In a three-d problem, each vector has 3 components, and the bi-harmonic equation becomes three biharmonic equations. Instead of a two-scalar (stream function and vorticity) one solves for six scalars (three for the stream function vector and three for the vorticity vector).
Probably someone has tried to solve such a system.
the problem of a vectorial poisson equation for the "stream function" with vectorial vorticity as its source is at the heart of Lagrangian vortex particle methods and is routinely solved in its integral form (Biot Savart formula). I have also seen at least one paper where the "stream-function"-vorticity method is solved in 3-D using finite difference methods (I forget the author now).
hi i am hari.
iam not fimililar with these stream function & velocity potential.
why we have 2 use these.
what is the use of these things...
There exist 3D stream functions (two functions). See, for example,
R. G. Campbell, Foundaitons of Fluid Flow Theory, Addison-Wesley, 1973.
Here is a paper on the application of 3D stream functions to CFD:
A. Sherif and M. Hafez, Computation of three-dimensional transonic flows by
using two stream functions. International Journal for Numerical Methodsin Fluids, 8, 17-29, 1988.
I have found this in the physic forums regarding a similar question:
"For the case of 3-d flows, it is possible to use two stream functions to replace the continuity equation. However, the complexity of this approach usually makes it less attractive than using the continuity equation in its original form."
formally, the stream function-vorticity formulation in 3D in nothing else the solution of the Helmholtz equation in vector form (3 scalar functions for the vorticity) coupled with the Poisson equation for the potential vector field Psi (a stream function is denoted only for 2D). Owing to the 3 Poisson equations for the scalar fields Psi_i this formulation is not appealing. However, some proposal in literature used to simplify the number of scalar functions.
Let me say that, the formulation in terms of vorticity is quite complex if one solves turbulence, the closure model for the vorticity field and the BC.s are much less studied than they are for the velocity-pressure formulation.
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