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I am working on a fluid simulation using vortex methods. For this I must compute the vortex sheet on my boundaries given as: $$ \gamma(\mathbf{x}) - \frac{1}{\pi}\int_S \frac{\partial}{\partial\mathbf{n}}(\text{log}|\mathbf{x} - \mathbf{x'}|)\gamma(\mathbf{x'})d\mathbf{x'} = -2\mathbf{u}\cdot \mathbf{s} $$ I have two questions on how to compute the vortex sheet $\gamma$:

  1. The paper of Cooper and Barba state this can be solved using radial basis functions. I think the hardest part to compute is $$ \oint \frac{\partial}{\partial n}[\text{log}|\mathbf{x}-\mathbf{x'}|]\phi(|\mathbf{x'} - \mathbf{x_i}) d\mathbf{x'} $$ What would the discrete version of this be? So how do I compute this term?

  2. If I would use the panel method, how does a discretisation look like?

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