Effect of the radial buoyancy on a circular Couette flow

Abstract

International audienceThe effect of a radial temperature gradient on the stability of a circular Couette flow is investigated when the gravitational acceleration is neglected. The induced radial stratification of the fluid density coupled with the centrifugal acceleration generates radial buoyancy which is centrifugal for inward heating and centripetal for outward heating. This radial buoyancy modifies the Rayleigh discriminant and induces the asymmetry between inward heating and outward heating in flow behavior. The critical modes are axisymmetric and stationary for inward heating while for outward heating, they can be oscillatory axisymmetric or nonaxisymmetric depending on fluid diffusion properties, i.e., on the Prandtl number Pr. The dependence of the critical modes on Pr is explored for different values of the radius ratio of the annulus. The power input of the radial buoyancy is compared with other power terms. The critical frequency of the oscillatory axisymmetric modes is linked to the Brunt-Väisälä frequency due to the density stratification in the radial gravity field induced by the rotation. These modes are associated with inertial waves. The dispersion relation of the oscillatory axisymmetric modes is derived in the vicinity of the critical conditions. A weakly nonlinear amplitude equation with a forcing term is proposed to explain the domination of these axisymmetric oscillatory modes over the stationary centrifugal mode

Short-wavelength local instabilities of a circular Couette flow with radial temperature gradient

“…The theoretical results are applied to the cylindrical Couette flow with the parameters evaluated at the geometric mean radius both in the Rayleigh unstable and in the Rayleigh stable regimes. In the case of the sole inner cylinder rotating we confirm and extend the numerical results of Meyer et al (2015) and provide analytical expressions for the onset of oscillatory and stationary instabilities, the coordinates of the codimension-2 points, and the Hopf frequency. In the case of the sole outer cylinder rotation, solid body rotation, and rotating flow with Keplerian shear we find the destabilizing effect of the inward temperature gradient that leads to the oscillatory instability at small values of the Prandtl number (Pr < 1) and to the stationary instability at Pr > 1.…”

“…For the inner cylinder sole rotation, we have retrieved and extended the results of linear stability analysis (Meyer et al 2015). For example, we have proved easily that the slope of the branch of stationary modes at the origin (γ a = 0) is increasing with both Pr and η.…”

“…Then, we can denote β = kz |k| and write   ω ν + ω + imΩ −2Ωβ 2 −αgβ 2 2Ω(1 + Ro) With g = Ω 2 r, and taking into account that the sign of α in (Economides & Moir 1980) is opposite to the sign of α in (2.1), we reduce the above equation to the form 4(Ro + 1) − 2PrRtΘγ a + 1 Ta 2 > 0, (A 21) which differs from the condition (5.11) only by the factor 1 − γ aΘ at the first term. The reason of this discrepancy is that in Economides & Moir (1980) g is assumed to be a constant radial gravitational acceleration whereas in Meyer et al (2015) it is the centrifugal acceleration that depends on radius. Hence, substitution of g = Ω 2 r into Eq.…”

“…In order to facilitate comparison with the experimental data we need to express these parameters in terms of the realistic Taylor-Couette system. In this case, the base flow state confined in the gap between two coaxial cylinders of radii a and b rotating with the angular frequencies Ω a and Ω b , respectively, is given by (Chandrasekhar 1961;Meyer et al 2015;Ali & Weidman 1990)…”

“…Some recent studies indicated a possibility that Keplerian disks with strong mean radial temperature gradients can support the so-called Goldreich-Schubert-Fricke (GSF) instability, which is the instability of short radial wavelength inertial modes (Economides & Moir 1980;Urpin & Brandenburg 1998;Nelson et al 2013;Balbus & Potter 2016). These astrophysical implications renewed interest to the effect of radial temperature gradient on the stability of a circular Couette flow and its observation in the laboratory (Yoshikawa et al 2013;Meyer et al 2015).…”