# Is there an exact solution for induced velocity during vortex ring state?

In plots such as these, the section between -2 and 0 has no exact solution. As shown in the first image, the other two formulas are known, but not during vortex ring state because, as seen in the second image where the functions continue, neither match the data. After a bit of analysis, I have found that the following function is a close fit to the approximation given in the first image: $$f(x) = \sqrt{1-\frac{x^2}{4}} + \frac{x(x^2 + x - 1)}{2}+\frac{4x^3}{x^3-8}$$ But I am not aware of any way I could even go about proving this is or is not the exact solution.

Thanks