Juan Sánchez

Oscillatory modes in an enclosed swirling flow

The flow in a completely filled cylinder driven by a rotating endwall has multiple time-dependent stable states when the endwall rotation exceeds a critical value. These states have been observed experimentally and computed numerically elsewhere. The linear stability of the basic state, which is a non-trivial axisymmetric flow, is analysed at parameter values where the unsteady solutions exist

On the multiple shooting continuation of periodic orbits by Newton-Krylov methods

The application of the multiple shooting method to the continuation of periodic orbits in large-scale dissipative systems is analyzed. A preconditioner for the linear systems which appear in the application of Newtons method is presented. It is based on the knowledge of invariant subspaces of the Jacobians at nearby solutions. The possibility of speeding up the process by using parallelism is studied for the thermal convection of a binary mixture of fluids in a rectangular domain, with positive results.

On the onset of low-Prandtl-number convection in rotating spherical shells : non-slip boundary conditions

Accurate numerical computations of the onset of thermal convection in wide rotating spherical shells are presented. Low-Prandtl-number (σ ) fluids, and non-slip boundary conditions are considered. It is shown that at small Ekman numbers (E), and very low σ values, the well-known equatorially trapped patterns of convection are superseded by multicellular outer-equatorially-attached modes. As a result, the convection spreads to higher latitudes affecting the body of the fluid, and increasing the internal viscous dissipation. Then, from E< 10 −5 , the critical Rayleigh number (Rc) fulfils a power-law dependence Rc ∼ E −4/3 , as happens for moderate and high Prandtl numbers. However, the critical precession frequency (|ωc|) and the critical azimuthal wavenumber (mc) increase discontinuously, jumping when there is a change of the radial and latitudinal structure of the preferred eigenfunction. In addition, the transition between spiralling columnar (SC), and outer-equatorially-attached (OEA) modes in the (σ , E)-space is studied. The evolution of the instability mechanisms with the parameters prevents multicellular modes being selected from σ 0.023. As a result, and in agreement with other authors, the spiralling columnar patterns of convection are already preferred at the Prandtl number of the liquid metals. It is also found that, out of the rapidly rotating limit, the prograde antisymmetric (with respect to the equator) modes of small mc can be preferred at the onset of the primary instability.

Non-linear spirals in the Taylor-Couette problem

We examine non-linear spiral flow in the Taylor–Couette problem for a wide gap with axially periodic conditions. We present a highly efficient computational method adapted to this problem, based on continuation methods applied to a pseudospectral discretization of the Navier–Stokes equations in a rotating frame of reference. The spiral flow is computed in a wide range of parameters, and different features are explored in detail: domain of existence of the flow, behavior for high Reynolds number, appearance of axial flows, dependency on parameters, and stability against helical disturbances. A first integral is obtained and used to describe the particle trajectories in the fluid. This description shows that the axial and radial motion of the particles is mainly confined within an internal boundary layer.

A comparison of high-order time integrators for thermal convection in rotating spherical shells

A numerical study of several time integration methods for solving the three-dimensional Boussinesq thermal convection equations in rotating spherical shells is presented. Implicit and semi-implicit time integration techniques based on backward differentiation and extrapolation formulae are considered. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The codes are validated with a known benchmark, and their efficiency is studied. The results show that the use of high-order methods, especially those with time step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions.

Generalized Couette-Poiseuille flow with boundary mass transfer

A generalized similarity formulation extending the work of Terrill (1967) for Couette{ Poiseuille flow in the annulus between concentric cylinders of innite extent is given. Boundary conditions compatible with the formulation allow a study of the eects of inner and outer cylinder transpiration, rotation, translation, stretching and twisting, in addition to that of an externally imposed constant axial pressure gradient. The problem is governed by , the ratio of inner to outer radii, a Poiseuille number, and nine Reynolds numbers. Single-cylinder and planar problems can be recovered in the limits ! 0 and ! 1, respectively. Two coupled primary nonlinear equations govern the meridional motion generated by uniform mass flux through the porous walls and the azimuthal motion generated by torsional movement of the cylinders; subsidiary equations linearly slaved to the primary flow govern the eects of cylinder translation, cylinder rotation, and an external pressure gradient. Steady solutions of the primary equations for uniform source/sink flow of strength F through the inner cylinder are reported for 06 6 1. Asymptotic results corroborating the numerical solutions are found in dierent limiting cases. For F 0i s more complex in that unique solutions are found at low Reynolds numbers, a region of triple solutions exists at moderate Reynolds numbers, and a two-cell solution prevails at large Reynolds numbers. The subsidiary linear equations are solved at =0 :5 to exhibit the eects of cylinder translation, rotation, and an axial pressure gradient on the source/sink flows.

Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells

We assess the accuracy and efficiency of several exponential time integration methods coupled to a spectral discretization of the three-dimensional Boussinesq thermal convection equations in rotating spherical shells. Exponential methods are compared to implicit-explicit (IMEX) multi-step methods already studied previously in [1]. The results of a wide range of numerical simulations highlight the superior accuracy of exponential methods for a given time step, especially when employed with large time steps and at low Ekman number. However, presently available implementations of exponential methods appear to be in general computationally more expensive than those of IMEX methods and further research is needed to reduce their computational cost per time step. A physically justified extrapolation argument suggests that some exponential methods could be the most efficient option for integrating flows near Earths outer core conditions.

Spiral Vortices Between Concentric Cylinders

Spiral vortices appearing in Couette-Taylor flows are studied by means of numerical simulation. Transition curves from Couette to spiral vortices for different radius ratios and wavenumbers have been calculated in order to test our technique. Critical Reynolds numbers, angular velocities and slopes of the spirals at the onset of the instability agree with previous results [1]. Non-linear solutions obtained by a pseudospectral collocation method are studied, and they show a weak net axial flow. In order to counteract this effect, which is absent in the usual experimental set-up, an axial pressure gradient has been included. This procedure has proved to be sufficient to make the axial flow negligible. The onset of a quasiperiodic flow for larger Reynolds numbers, corresponding to a secondary bifurcation is also presented.Spiral vortices appearing in Couette- Taylor flows are studied by means of numerical simulation. Transition curves from Couctte to spiral vortices for different radius ratios and wavenumbers have been calculated in order to test our technique. Critical Reynolds numbers, angular velocities and slopes of the spirals at the onset of the instability agree with previous results [1]. Non-linear solutions obtained by a pseudospectral collocation method are studied, and they show a weak net axial ftow. In arder to counteract this effect, which is absent in the usual experimental set-up, an axial pressure gradient has been included. This procedure has proved to be sufficient to make the axial flow negligible. The onset of a quasiperiodic flow for larger Reynolds numbers, corresponding to a secondary bifurcation is also presented.

From stationary to complex time-dependent flows at moderate Rayleigh numbers in two-dimensional annular thermal convection

Two-dimensional nonlinear thermal convection in a cylindrical annulus is numerically analyzed for a Boussinesq fluid of low Prandtl number σ=0.025. For a fixed value of the radius ratio, η=0.3, different types of steady columnar patterns are found. The stability of these convection patterns and the spatial interaction between them, which result in the formation of mixed modes, are investigated by considering the full nonlinear set of Navier–Stokes equations. Special attention is paid to the strong spatial interaction of the initially unstable modes with wavenumbers n=2 and n=4, which leads, through global bifurcations, to multiple stable quasi-periodic states of the system. A detailed picture of the nonlinear dynamics until temporal chaotic patterns set in is presented and understood in terms of local and global symmetry-breaking bifurcations of the O(2)-symmetric system.

On stable Taylor vortices above the transition to wavy vortices

The transition from Taylor to wavy vortices is revisited for parameter values in the range of new laboratory experiments [Lin et al., Phys. Fluids 10, 3233 (1998)]. The dependence of the critical Reynolds number with the axial wavelength of the Taylor vortices is obtained for azimuthal wave numbers from 1 to 5, and for five different values of the radius ratio. We show how islands of stable Taylor vortices above the transition to wavy vortices form.