Non-unique solutions of the Navier-Stokes equations for the Karman swirling flow

Abstract

SUMMARYIn this paper it is shown numerically that axially-symmetric solutions of the Navier-Stokes equations, which describe the rotating flow above a disk which is itself rotating, are non-unique. The numerical techniques designed to calculate such solutions with a high power of resolution are given. Especially the behaviour in and around the first branching point is considered. It is found that for s = -0.16054 two branches coincide. The second branch has been almost completely calculated. It ranges back to positive values ofs.

Mixed convection above a rotating disk

“…The branch shown with a solid line starts at the point (Gr, w ∞ ) = (0, 0.884), which coincides with the appropriate calculation of Zandbergen and Dijkstra (18) (see their table 4.1). The branch shown with a broken line starts at (Gr, w ∞ ) = (0, 0.447723), which also agrees with table 4.1 of Zandbergen and Dijkstra (18). As the branch extends away from Gr = 0 and starts to oscillate down toward the Gr axis, we found that it was necessary to gradually increase z ∞ as the thickness of the vorticity layer at the disk grows.…”

Fluid flow induced by a rotating disk of finite radius

“…It turns out that ½H"(O) =0.510232619 and G'(0) = -0.615922014. (2.3) As shown by Zandbergen and Dijkstra [6] this non-linear problem has other solutions too, however, with doubtful physical significance. These will not be considered in this paper.…”

“…Hence, its O(Re°) solution is not influenced by the region r > 1. This result holds only for zero angular velocity of the ambient fluid since otherwise there will be regions with negative u, see [6].…”

A Series Solution of the Unsteady Von Kármán Swirling Viscous Flows

“…the Reynolds number determined by the distance of the two disks. All these problems are studied, theoretically, numerically and experimentally, by many researchers such as Cochran [2], Fettis [3], Rogers and Lance [4], Mellor, Chapple and Stokes [5], Tam [6], Schlichting [7], Bodonyi [8], Zandbergen and Dijkstra [9], Dijkstra [10], Holodniok [11], Dijkstra and Van Heijst [12], Szeri, Schneider, Labbe and Kaufman [13], Bodonyi and Ng [14] and so on. For details, please refer to the review paper of Zandbergen and Dijkstra [15].…”