Multiple vortex formation in steady laminar axisymmetric impinging flow

Abstract

Steady confined laminar axisymmetric impinging flow of a Newtonian fluid is relevant in many situations, an important application being heat and mass transfer from a solid surface to an impinging jet. This paper focuses on the evolution of the structure of the radial flow field in the channel region beyond the impingement zone. We employ an upwind scheme with an established numerical technique to solve the stream function and vorticity equations for a range of Reynolds numbers Re and geometrical aspect ratios e. Our results show the progressive complexity in the radial flow due to multiple points of flow separation and reattachment, and we provide a detailed demarcation of the Re– e plane based on flow separation behavior. In addition to the primary and secondary vortices anchored on the confining and impinging surfaces, respectively, we describe the formation and properties of a tertiary vortex which is wholly enclosed within the primary vortex. At a fixed Reynolds number, the tertiary vortex is observed only for a specific range of the aspect ratio, and we catalog its birth, growth and demise as the aspect ratio is varied. The range of aspect ratios over which the tertiary vortex exists is seen to increase with the Reynolds number. These results show that the fine structure of the radial flow at high Reynolds number continues to be dependent on the aspect ratio in a complex manner. At a given aspect ratio, the sizes of the vortices increases with Reynolds number, scaling as ∼ Re 1/3, and for sufficiently large Re, the length of the tertiary vortex can exceed that of the secondary vortex. The primary and secondary vortex lengths satisfy an asymptotic relationship V L P ∼ α V L S independent of Re and e, the numerically computed value of α being ∼2. Similarly, the locations of these vortices bear simple linear relationships independent of Re and e. Furthermore, despite the complex fine structure of the flow field, macroscopic flow properties such as vortex circulation and excess pressure loss continue to exhibit relatively simple dependence on Re and e, in accordance with previous results at much lower Reynolds numbers. Finally, some comments are made regarding the possibility of additional cascaded or isolated vortices occurring at even higher Reynolds numbers and aspect ratios.