Nonlinear axisymmetric Taylor-Couette flow in a dilute gas: multiroll transition and the role of compressibility

The compressible Navier-Stokes-Fourier equations are numerically solved for axially bounded Taylor-Couette flow (TCF) of an ideal gas, with rotating inner cylinder and stationary outer cylinder, to understand the roles of compressibility and finite aspect ratio on the genesis of nonlinear Taylor vortices and the related bifurcation scenario. Restricting to the axisymmetric case in the wide-gap limit, the aspect ratio Gamma = h/delta (where delta is the annular gap between two cylinders and h is the height of the cylinders) is changed quasistatically by varying the height of the cylinders, following the experimental protocol of Benjamin (Proc. R. Soc. Lond. A, vol. 359, 1978b, pp. 27-43). The symmetric even-numbered (2, 4, 6, ... ) vortices are found at Gamma >= 2, whereas the asymmetric 2-roll modes (called the single-roll or 'anomalous' modes), that break the midplane Z2-symmetry, are uncovered at Gamma <= O(1). The phase boundaries of both symmetric and asymmetric rolls and the coexisting regions of different number of rolls are identified in the (Gamma, Re)-plane. These phase diagrams, along with related bifurcation diagrams and various patterns, at finite Mach numbers (Ma) are contrasted with their incompressible (Ma -> 0) analogues. It is shown that the '1 <-> 2'-roll transition in small aspect ratio cylinders is subcritical in compressible TCF in contrast to the supercritical nature of bifurcation in its incompressible counterpart. In general, the gas compressibility has a stabilizing effect on nonlinear Taylor vortices as the underlying phase diagrams in the (Gamma, Re)-plane are shifted to larger values of Re with increasing Ma. The stabilizing role of compressibility can be tied to the weakening of the outward jets, which is further aided by the strengthening of the Ekman vortices with increasing Mach number.