Marta Net

Convection in a rotating cylinder. Part 1 Linear theory for moderate Prandtl numbers

The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios Γ≡d/h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow

Spectral methods for high order equations

Abstract Solving Navier-Stokes equations by means of scalar potentials leads to higher order equations with low order boundary conditions. This approach usually involves numerical instabilities that do not appear using the primitive Navier-Stokes equations. Such difficulties can be avoided with an explicit evaluation of some of the high order derivatives at the boundary. We present three numerical techniques based on this method and on spectral expansions. We apply them to the linear stability analysis of thermal convection in a cylinder.

Thermal convection in vertical cylinders. A method based on potentials of velocity

Abstract Any solenoidal vector field can be written in terms of two scalar fields. Solving the Navier-Stokes equation for these two so-called potentials is a widely used method, although difficulties have been reported in using this method to describe non-Cartesian confined flows, in particular cylindrical geometries. Difficulties arise because the potentials are coupled at the boundaries but also because of possible numerical instabilities associated with the high order of the equations. This paper shows how all these difficulties can be circumvented: firstly by properly deriving the required boundary conditions, but also through an adequate discretization, which is the final requirement. The use of spectral methods together with a reduction of order for the axisymmetric modes is sufficient for this purpose. In this paper, as a test, a velocity potentials method is used to solve the stability problem for the onset of convection in a fluid confined in a cylindrical container.

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.

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.

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.

On the transition to columnar convection

Convection in a rotating annulus with no‐slip sidewalls, stress‐free ends, radial gravity, and sideways heating is considered. The transition from fully three‐dimensional convection cells to Taylor columns with increasing rotation rate is studied and its dependence on the annulus parameters is established.

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.

Continuation of Bifurcations of Periodic Orbits for Large-Scale Systems

A methodology to track bifurcations of periodic orbits in large-scale dissipative systems depending on two parameters is presented. It is based on the application of iterative Newton-Krylov techniques to extended systems. To evaluate the action of the Jacobian it is necessary to integrate variational equations up to second order. It is shown that this is possible by integrating systems of dimension at most four times that of the original equations. In order to check the robustness of the method, the thermal convection of a mixture of two fluids in a rectangular domain has been used as a test problem. Several curves of codimension-one bifurcations, and the boundaries of an Arnolds tongue of rotation number 1/8, have been computed. 1. Introduction. The study of dynamical systems involves, in addition to pure numerical simulations, the computation of invariant manifolds (fixed points, periodic orbits, invariant tori, etc.) and the connections among them (homo- and heteroclinic orbits and heteroclinic chains), the investigation of the stability of these objects, and the examination of their bifur- cations when the parameters present in the system are varied. These essential tools help to understand the full dynamics of the system and its dependence on the parameters. The ex- istence of robust continuation and bifurcation packages such as AUTO (10), CONTENT (22), MATCONT (7), etc., allows many of these calculations to be almost routinely performed for moderate-dimensional systems of ordinary differential equations (ODEs). These packages in- clude the continuation of codimension-one bifurcations of fixed points, and even of periodic orbits, and some also include the detection and analysis of codimension-two points. They implement direct solvers for the linear systems involved in the computations, and find the full spectrum when solving eigenvalue problems to study of the stability of the invariant objects. The theory of the extended systems used in these packages to follow bifurcations of fixed points is well developed and can be found in, among others, (42, 26, 18, 53, 29, 5, 16). The bordered systems for periodic orbits, based on boundary value problems, are analyzed in (11). In this latter case piecewise collocation in time is used instead of shooting methods. The difficulties in the application of these methodologies to high-dimensional systems, obtained in most cases by discretizing systems of partial differential equations (PDEs), come