A MATLAB code to calculate the growth rate predicted by Crow's theory.
The self-induction function and its asymptotic format proposed in this theory valid only for
A MATLAB code to calculate the self-induction function (self-induction angular frequency) by exact method, crow theory, long-wavelength asymptotic method and numerical fitting.
The expression at
From the properties of the modified Bessel function, the first term at the right-hand side,
$-\frac{K_m^{\prime}(ka)}{kaK_m(ka)}$ , is a positive monotonically decreasing function of$ka$ , decreasing from$\infin$ to$0$ as$ka$ increases from$0$ to$\infin$ ....
There are an infinite number of roots, both retrograde (
$s=1$ ) and co-grade ($s=-1$ ). It follows from the expansions for small$ka$ and$\beta a$ that there is no co-grade root for small$\beta a$ . For$ka \rightarrow \infin$ ,$\beta a$ is found to be bounded$\sigma \rightarrow (2s-m)\Omega$ .As
$ka \rightarrow 0$ ,$\beta a \rightarrow j_{mn}$ (the $n$th root of$J_m(x)=0$ ) and$\sigma \rightarrow -\Omega$ , except for the smallest retrograde root.Excerpts from Vortex Dynamics by P. G. Saffman
Now know that, the roots of the dispersion relation locate at the interval