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Problem with Velocity Poisson Equation and Vector Potential Poisson Equation

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Old   August 11, 2017, 07:02
Default Problem with Velocity Poisson Equation and Vector Potential Poisson Equation
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Dear all,

I am currently developing a code to track vortex particles solving Velocity Poisson Equation or Vector Potential Poisson Equation with fast Poisson solver.

The equations are as below :
Velocity Poisson Equation :ΔV =- x ω
Vector Potential Poisson Equation : Δ A =-ω , V=x A

V: Velocity (Vector)
A: Vector Potential (Vector)
ω: Vorticity (Vector)

I coded two types of code; one solves Velocity Poisson Equation and the other solves Vector Potential Poisson Equation. Basically, the flow of code is same.
In Velocity Poisson code, vorticity is curled and put into the fast Poisson solver.
However, in Vector Potential code, vorticity is put into the fast Poisson solver and the result is curled.
The curl operations were done in 2nd order central difference. I also did them in 4th order central difference.
The vorticity will be given. Therefore, with proper boundary conditions, the velocity can be calculated theoretically.

I first tested 2D case with 2 vortex particles. Two equations change into Scalar form and the Vector Potential Poisson changes into Streamfunction-vorticity equation; Δ ψ=-ω , u=d ψ/dy , v=-d ψ/dx
I solved both equations and compared with the solution from Biot-Savart law.
Velocity from solving Velocity Poisson Equation shows large error though similar to the analytic solution.
Velocity from solving Vector Potential Poisson Equation (Streamfunction-vorticity equation) shows really good match with the analytic solution.

I do not know why this happened but I think the possible reason is the source term of the Poisson equations.. It seems that curl of vorticity with difference method made the large error.
Therefore, maybe, solving Vector Potential Poisson Equation might be proper choice.
However, in the end, I have to solve 3D problem.
For Velocity Poisson Equation, it is clear to set the boundary conditions.
For Vector Potential Poisson Equation, I actually do not know how to set. I’ve been finding materials to refer but most of them are from electromagnetics.

What I want to ask is as below:
1. Is there any wrong that I used in the Velocity Poisson Equation code?
2. Is there any way to calculate curl operation with smaller error?
3. Is there any material for setting Boundary conditions of Vector Potential in fluid mechanics?
4. Other tips or advice to me are welcome.

Thanks for reading this.
I hope you are well.

Best regards,
Jun
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Old   August 12, 2017, 13:15
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About 1, it is solved. It was the problem of domain size.

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