Thermosolutal and binary fluid convection as a 2 x 2 matrix problem

We describe an interpretation of convection in binary fluid mixtures as a superposition of thermal and solutal problems, with coupling due to advection and proportional to the separation parameter S. Many properties of binary fluid convection are then consequences of generic properties of 2 x 2 matrices. The eigenvalues of 2 x 2 matrices varying continuously with a parameter r undergo either avoided crossing or complex coalescence, depending on the sign of the coupling (product of off-diagonal terms). We first consider the matrix governing the stability of the conductive state. When the thermal and solutal gradients act in concert (S>0, avoided crossing), the growth rates of perturbations remain real and of either thermal or solutal type. In contrast, when the thermal and solutal gradients are of opposite signs (S<0, complex coalescence), the growth rates become complex and are of mixed type. Surprisingly, the kinetic energy of nonlinear steady states is governed by an eigenvalue problem very similar to that governing the growth rates. There is a quantitative analogy between the growth rates of the linear stability problem for infinite Prandtl number and the amplitudes of steady states of the minimal five-variable Veronis model for arbitrary Prandtl number. For positive S, avoided crossing leads to a distinction between low-amplitude solutal and high-amplitude thermal regimes. For negative S, the transition between real and complex eigenvalues leads to the creation of branches of finite amplitude, i.e. to saddle-node bifurcations. The codimension-two point at which the saddle-node bifurcations disappear, separating subcritical from supercritical pitchfork bifurcations, is exactly analogous to the Bogdanov codimension-two point at which the Hopf bifurcations disappear in the linear problem

Notes on aerodynamic forces on airship hulls

For a first approximation the air flow around the airship hull is assumed to obey the laws of perfect (i.e. free from viscosity) incompressible fluid. The flow is further assumed to be free from vortices (or rotational motion of the fluid). These assumptions lead to very great simplifications of the formulae used but necessarily imply an imperfect picture of the actual conditions. The value of the results depends therefore upon the magnitude of the forces produced by the disturbances in the flow caused by viscosity with the consequent production of vortices in the fluid. If these are small in comparison with the forces due to the assumed irrotational perfect fluid flow the results will give a good picture of the actual conditions of an airship in flight

Stabilization of Ab Initio Molecular Dynamics Simulations at Large Time Steps

The Verlet method is still widely used to integrate the equations of motion in ab initio molecular dynamics simulations. We show that the stability limit of the Verlet method may be significantly increased by setting an upper limit on the kinetic energy of each atom with only a small loss in accuracy. The validity of this approach is demonstrated for molten lithium fluoride.Comment: 9 pages, 3 figure

Standing and travelling waves in cylindrical Rayleigh-Benard convection

The Boussinesq equations for Rayleigh-Benard convection are simulated for a cylindrical container with an aspect ratio near 1.5. The transition from an axisymmetric stationary flow to time-dependent flows is studied using nonlinear simulations, linear stability analysis and bifurcation theory. At a Rayleigh number near 25,000, the axisymmetric flow becomes unstable to standing or travelling azimuthal waves. The standing waves are slightly unstable to travelling waves. This scenario is identified as a Hopf bifurcation in a system with O(2) symmetry

Symmetry breaking and turbulence in perturbed plane Couette flow

Perturbed plane Couette flow containing a thin spanwise-oriented ribbon undergoes a subcritical bifurcation at Re = 230 to a steady 3D state containing streamwise vortices. This bifurcation is followed by several others giving rise to a fascinating series of stable and unstable steady states of different symmetries and wavelengths. First, the backwards-bifurcating branch reverses direction and becomes stable near Re = 200. Then, the spanwise reflection symmetry is broken, leading to two asymmetric branches which are themselves destabilized at Re = 420. Above this Reynolds number, time evolution leads first to a metastable state whose spanwise wavelength is halved and then to complicated time-dependent behavior. These features are in agreement with experiments

Mean flow of turbulent–laminar patterns in plane Couette flow

A turbulent–laminar banded pattern in plane Couette flow is studied numerically. This pattern is statistically steady, is oriented obliquely to the streamwise direction, and has a very large wavelength relative to the gap. The mean flow, averaged in time and in the homogeneous direction, is analysed. The flow in the quasi-laminar region is not the linear Couette profile, but results from a non-trivial balance between advection and diffusion. This force balance yields a first approximation to the relationship between the Reynolds number, angle, and wavelength of the pattern. Remarkably, the variation of the mean flow along the pattern wavevector is found to be almost exactly harmonic: the flow can be represented via only three cross-channel profiles as U(x, y, z) ≈ U0(y) + Uc(y) cos(kz) + Us(y) sin(kz). A model is formulated which relates the cross-channel profiles of the mean flow and of the Reynolds stress. Regimes computed for a full range of angle and Reynolds number in a tilted rectangular periodic computational domain are presented. Observations of regular turbulent–laminar patterns in other shear flows – Taylor–Couette, rotor–stator, and plane Poiseuille – are compared

Spirals and ribbons in counter-rotating Taylor-Couette flow: frequencies from mean flows and heteroclinic orbits

A number of time-periodic flows have been found to have a property called RZIF: when a linear stability analysis is carried out about the temporal mean (rather than the usual steady state), an eigenvalue is obtained whose Real part is Zero and whose Imaginary part is the nonlinear Frequency. For two-dimensional thermosolutal convection, a Hopf bifurcation leads to traveling waves which satisfy the RZIF property and standing waves which do not. We have investigated this property numerically for counter-rotating Couette-Taylor flow, in which a Hopf bifurcation gives rise to branches of upwards and downwards traveling spirals and ribbons which are an equal superposition of the two. In the regime that we have studied, we find that both spirals and ribbons satisfy the RZIF property. As the outer Reynolds number is increased, the ribbon branch is succeeded by two types of heteroclinic orbits, both of which connect saddle states containing two axially stacked pairs of axisymmetric vortices. One heteroclinic orbit is non-axisymmetric, with excursions that resemble the ribbons, while the other remains axisymmetric

A two-dimensional optical Fourier analyzer for image evaluation

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