78 research outputs found
Modeling the transition to turbulence in shear flows
One-dimensional models are presented for transitional shear flows. The models
have two variables corresponding to turbulence intensity and mean shear. These
variables evolve according to simple equations based on known properties of
transitional turbulence. The first model considered is for pipe flow. A
previous study modeled turbulence using a chaotic tent map. In the present work
turbulence is modeled instead as multiplicative noise. This model captures the
character of transitional pipe flow and contains metastable puffs, puff
splitting, and slugs. These ideas are extended to a limited model of plane
Couette flow.Comment: ETC13 Conference Proceeding, 8 pages, 6 figure
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
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
Theoretical perspective on the route to turbulence in a pipe
The route to turbulence in pipe flow is a complex, nonlinear, spatiotemporal process for which an increasingly clear theoretical understanding has emerged. This understanding is explained to the reader in several steps, exploiting analogies to co-existing thermodynamic phases and to excitable and bistable media. In the end, simple equations encapsulating the keys physical properties of pipe turbulence provide a comprehensive picture of all large-scale states and stages of the transition process. Important among these are metastable localized puffs, localized edge states, puff splitting and interactions between puffs, the critical point for the onset of sustained turbulence via spatiotemporal intermittency (directed percolation), and finally the rise of fully turbulent flow in the form of expanding weak and strong turbulent slugs
Taming turbulent fronts by bending pipes
The flow of fluid through a pipe has been instrumental in illuminating the subcritical route to turbulence typical of many wall-bounded shear flows. Especially important in this process are the turbulent–laminar fronts that separate the turbulent and laminar flow. Four years ago Michael Graham (Nature, vol. 526, 2015, p. 508) wrote a commentary entitled ‘Turbulence spreads like wildfire’, which is a picturesque but also accurate characterisation of the way turbulence spreads through laminar flow in a straight pipe. In this spirit, the recent article by Rinaldi et al. (J. Fluid Mech., vol. 866, 2019, pp. 487–502) shows that turbulent wildfires are substantially tamed in bent pipes. These authors find that even at modest pipe curvature, the characteristic strong turbulent–laminar fronts of straight pipe flow vanish. As a result, the propagation of turbulent structures is modified and there are hints that the route to turbulence is fundamentally altered
A fluid mechanic's analysis of the teacup singularity
The mechanism for singularity formation in an inviscid wall-bounded fluid
flow is investigated. The incompressible Euler equations are numerically
simulated in a cylindrical container. The flow is axisymmetric with swirl. The
simulations reproduce and corroborate aspects of prior studies reporting strong
evidence for a finite-time singularity. The analysis here focuses on the
interplay between inertia and pressure, rather than on vorticity. Linearity of
the pressure Poisson equation is exploited to decompose the pressure field into
independent contributions arising from the meridional flow and from the swirl,
and enforcing incompressibility and enforcing flow confinement. The key
pressure field driving the blowup of velocity gradients is that confining the
fluid within the cylinder walls. A model is presented based on a
primitive-variables formulation of the Euler equations on the cylinder wall,
with closure coming from how pressure is determined from velocity. The model
captures key features in the mechanics of the blowup scenario.Comment: 34 pages, 8 figure
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
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
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