Absolute and Convective Instabilities of Developing Jets

<p dir="ltr">The linear stability of the radial jet, with swirl, is undertaken. Here, swirl is quantified by the parameter ϵ. The radial jet emerges due to a boundary-layer collision at the equators of an impulsively rotated sphere. Calabretto et al. [1] demonstrated that the radial jet exhibits the distinctive features originally proposed by Riley [2]. Notably, the jet manifests through the rapid development of a self-similar structure, characterised by a thickness scaling inversely with the square root of the flow Reynolds number. Thus, using the similarity solution due to Riley as the base flow, a local linear stability analysis of the radial swirling jet is undertaken, including both convective and absolute instabilities. Three different numerical methods, including the Finite Difference Method, the Haar Wavelet Collocation Method, and the Chebyshev Spectral Method, are tested to compute the linear stability of the Bickley jet. Based on a comparison of these methods, the Chebyshev Spectral Method is employed for carrying out the stability analysis of the radial jet. The radial jet undergoes both convective and absolute instabilities. It is concluded that the region of convective instability expands with increasing swirl, ϵ, for azimuthal mode numbers n ě 0 and shrinks with ϵ for n ă 0. For absolute instability, undertaken using the cusp map method due to Kupfer et al. [3], swirl, ϵ, induces a destabilising effect, leading to an expansion of the region of absolute instability. The azimuthal mode number n “ 1 is the most unstable. Finally, like the Bickley jet, the radial jet is more unstable to symmetric modes than to antisymmetric modes, and the antisymmetric mode has a shorter wavelength than the symmetric mode.</p>