60 research outputs found
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
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
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
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
Turbulent-laminar patterns in shear flows without walls
Turbulent-laminar intermittency, typically in the form of bands and spots, is
a ubiquitous feature of the route to turbulence in wall-bounded shear flows.
Here we study the idealised shear between stress-free boundaries driven by a
sinusoidal body force and demonstrate quantitative agreement between turbulence
in this flow and that found in the interior of plane Couette flow -- the region
excluding the boundary layers. Exploiting the absence of boundary layers, we
construct a model flow that uses only four Fourier modes in the shear direction
and yet robustly captures the range of spatiotemporal phenomena observed in
transition, from spot growth to turbulent bands and uniform turbulence. The
model substantially reduces the cost of simulating intermittent turbulent
structures while maintaining the essential physics and a direct connection to
the Navier-Stokes equations.
We demonstrate the generic nature of this process by introducing stress-free
equivalent flows for plane Poiseuille and pipe flows which again capture the
turbulent-laminar structures seen in transition.Comment: 13 pages, 9 figure
Stability analysis of perturbed plane Couette flow
Plane Couette flow perturbed by a spanwise oriented ribbon, similar to a
configuration investigated experimentally at the Centre d'Etudes de Saclay, is
investigated numerically using a spectral-element code. 2D steady states are
computed for the perturbed configuration; these differ from the unperturbed
flows mainly by a region of counter-circulation surrounding the ribbon. The 2D
steady flow loses stability to 3D eigenmodes at Re = 230, beta = 1.3 for rho =
0.086 and Re = 550, beta = 1.5 for rho = 0.043, where Re is the Reynolds
number, beta is the spanwise wavenumber and rho is the half-height of the
ribbon. For rho = 0.086, the bifurcation is determined to be subcritical by
calculating the cubic term in the normal form equation from the timeseries of a
single nonlinear simulation; steady 3D flows are found for Re as low as 200.
The critical eigenmode and nonlinear 3D states contain streamwise vortices
localized near the ribbon, whose streamwise extent increases with Re. All of
these results agree well with experimental observations
Universal continuous transition to turbulence in a planar shear flow
We examine the onset of turbulence in Waleffe flow -- the planar shear flow
between stress-free boundaries driven by a sinusoidal body force. By truncating
the wall-normal representation to four modes, we are able to simulate system
sizes an order of magnitude larger than any previously simulated, and thereby
to attack the question of universality for a planar shear flow. We demonstrate
that the equilibrium turbulence fraction increases continuously from zero above
a critical Reynolds number and that statistics of the turbulent structures
exhibit the power-law scalings of the (2+1)-D directed percolation universality
class
Numerical simulation of Faraday waves
We simulate numerically the full dynamics of Faraday waves in three
dimensions for two incompressible and immiscible viscous fluids. The
Navier-Stokes equations are solved using a finite-difference projection method
coupled with a front-tracking method for the interface between the two fluids.
The domain of calculation is periodic in the horizontal directions and bounded
in the vertical direction by two rigid horizontal plates. The critical
accelerations and wavenumbers, as well as the temporal behaviour at onset are
compared with the results of the linear Floquet analysis of Kumar and Tuckerman
[J. Fluid Mech. 279, 49-68 (1994)]. The finite amplitude results are compared
with the experiments of Kityk et al. [Phys. Rev. E 72, 036209 (2005)]. In
particular we reproduce the detailed spatiotemporal spectrum of both square and
hexagonal patterns within experimental uncertainty
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