32 research outputs found
Turbulent spot growth in plane Couette flow: statistical study and formation of spanwise vorticity
This article presents direct numerical simulations of the growth of turbulent
spots in the transitional regime of plane Couette flow. A quantitative
description of the growth process and of the detail of the quadrupolar flow
around the spot is given. Focus is put on formation of spanwise vorticity in
the velocity streaks that resembles a secondary shear instability. The main
features of the instability (coherence lengths, advection velocity) are studied
in the context of the turbulent spot, below and above the threshold Reynolds
number of appearance of the oblique turbulent bands of plane Couette flow.Comment: 10 pages, 9 figure
Stochastic analysis of the time evolution of Laminar-Turbulent bands of plane Couette flow
This article is concerned with the time evolution of the oblique
laminar-turbulent bands of transitional plane Couette flow under the influence
of turbulent noise. Our study is focused on the amplitude of modulation of
turbulence. In order to guide the numerical study of the flow, we first perform
an analytical and numerical analysis of a Stochastic Ginzburg-Landau equation
for a complex order parameter. The modulus of this order parameter models the
amplitude of modulation of turbulence. Firstly, we compute the autocorrelation
function of said modulus once the band is established. Secondly, we perform a
calculation of average and fluctuations around the exponential growth of the
order parameter. This type of analysis is similar to the Stochastic Structural
Stability Theory. We then perform numerical simulations of the Navier-Stokes
equations in order to confront these predictions with the actual behaviour of
the bands. Computation of the autocorrelation function of the modulation of
turbulence shows quantitative agreement with the model: in the established band
regime, the amplitude of modulation follows an Ornstein-Uhlenbeck process. In
order to test the S3T predictions, we perform quench experiments, sudden
decreases of the Reynolds number from uniform turbulence, in which modulation
appears. We compute the average evolution of the amplitude of modulation and
the fluctuations around it. We find good agreement between numerics and
modeling. The average trajectory grows exponentially, at a rate clearly smaller
than that of the formation of laminar holes. The actual time evolution remains
in a flaring envelope, centred on the average, and expanding at the same rate.
These results provide further validation of the stochastic modeling for the
time evolution of the bands for further studies. They stress on the difference
between the oblique band formation and the formation of laminar holes.Comment: 17 pages, 6 figures. Followed by a Graphical abstract summarising the
article. Accepted for publication in Eur. Phys. J E (last submitted version
On modelling transitional turbulent flows using under-resolved direct numerical simulations: The case of plane Couette flow
Direct numerical simulations have proven of inestimable help to our
understanding of the transition to turbulence in wall-bounded flows. While the
dynamics of the transition from laminar flow to turbulence via localised spots
can be investigated with reasonable computing resources in domains of limited
extent, the study of the decay of turbulence in conditions approaching those in
the laboratory requires consideration of domains so wide as to exclude the
recourse to fully resolved simulations. Using Gibson's C++ code ChannelFlow, we
scrutinize the effects of a controlled lowering of the numerical resolution on
the decay of turbulence in plane Couette flow at a quantitative level. We show
that the number of Chebyshev polynomials describing the cross-stream dependence
can be drastically decreased while preserving all the qualitative features of
the solution. In particular, the oblique turbulent band regime experimentally
observed in the upper part of the transitional range is extremely robust. In
terms of Reynolds numbers, the resolution lowering is seen to yield a regular
downward shift of the upper and lower thresholds Rt and Rg where the bands
appear and break down. The study is illustrated with the results of two
preliminary experiments.Comment: 20 pages, 9 figures. Accepted on August 24, 2010, to appear in TCF
Formation of spanwise vorticy in oblique turbulent bands of transitional plane Couette flow, part 1: Numerical experiments
International audienceThis article investigates the formation of spanwise vorticity in the velocity streaks of the oblique laminar-turbulent bands of plane Couette flow (PCF) by mean of Direct Numerical Simulations (DNS). The spanwise vorticity is created by a roll–up type development of the streamwise-wall normal shear layer of the velocity streaks. It is advected by the large scale flow along the bands. We propose a criterion on spanwise vorticity which detects these events in order to perform systematic measurements. Beside of the streamwise and spanwise correlation lengths of the rolls, their advection velocity is measured and shown to match the large scale flow along the band near the turbulent region. Eventually, we discuss the possible relation between ejection of vorticity away from the bands near the laminar region and the size of said laminar region
Finite size analysis of a double crossover in transitional wall turbulence
This article presents the finite size analysis of two consecutive crossovers
leading laminar-turbulent bands to uniform wall turbulence in transitional
plane Couette flow. Direct numerical simulations and low order modeling
simulations of the flow are performed. The kinetic energy of the turbulent
flow and the order parameter , a measure of the spatially organised
modulation of turbulence, are sampled and processed in view analytical results
from the phenomenology of phase transitions. The first crossover concerns the
loss of spatial organisation of turbulence in the flow. In the band phase, the
order parameter decreases continuously with the Reynolds number toward
a small value, while its response function displays a maximum at the
crossover. In the uniform phase, the order parameter and its variance
decrease toward zero following mean field field scalings as is increased. The kinetic energy is
an affine function of except in a small range where a sharp increase is
detected, which corresponds to the second crossover. In this range, spatial and
temporal coexistence of the uniform turbulence phase and laminar-turbulent
bands phase is observed. This sharp increase is concomitant with a maximum of
the response function of the kinetic energy. The finite size analysis reveals
that the jump does not steepen and that the maximum of response function of
saturates as size is increased. The first crossover is formally identical to a
critical phenomenon in condensed matter. The second crossover is in agreement
with a first order phase transition smeared by finite noise. The analytical
analysis of this phenomenon assuming a non interacting gas of fronts between
domain of the two phases provides a scaling of the response function consistent
with that of .Comment: 39 pages, 9 figures. Accepted in J. Stat. Mec
Computing transition rates for the 1-D stochastic Ginzburg--Landau--Allen--Cahn equation for finite-amplitude noise with a rare event algorithm
In this paper we compute and analyse the transition rates and duration of
reactive trajectories of the stochastic 1-D Allen-Cahn equations for both the
Freidlin-Wentzell regime (weak noise or temperature limit) and finite-amplitude
white noise, as well as for small and large domain. We demonstrate that
extremely rare reactive trajectories corresponding to direct transitions
between two metastable states are efficiently computed using an algorithm
called adaptive multilevel splitting. This algorithm is dedicated to the
computation of rare events and is able to provide ensembles of reactive
trajectories in a very efficient way. In the small noise limit, our numerical
results are in agreement with large-deviation predictions such as
instanton-like solutions, mean first passages and escape probabilities. We show
that the duration of reactive trajectories follows a Gumbel distribution like
for one degree of freedom systems. Moreover, the mean duration growths
logarithmically with the inverse temperature. The prefactor given by the
potential curvature grows exponentially with size. The main novelty of our work
is that we also perform an analysis of reactive trajectories for large noises
and large domains. In this case, we show that the position of the reactive
front is essentially a random walk. This time, the mean duration grows linearly
with the inverse temperature and quadratically with the size. Using a
phenomenological description of the system, we are able to calculate the
transition rate, although the dynamics is described by neither
Freidlin--Wentzell or Eyring--Kramers type of results. Numerical results
confirm our analysis
Turbulent pattern formation in plane Couette flow: Modelling and investigation of mechanisms
International audienceIn the transitional range of Reynolds number, plane Couette flow exhibits oblique turbulent bands. We focus on a Kelvin-Helmholtz instability occurring in the intermediate area between turbulent and laminar flow. The instability is characterised by means of Direct Numerical Simulations (DNS): a short wavelength instability, localised and advected in the spanwise direction. The coherent background flow on which the instability develops is extracted from DNS data, and an analytical formulation for the background flow is proposed. Linear stability analysis is performed to investigate its main mechanisms and its convective or absolute nature, depending on the location in the flow. Both DNS and linear stability analysis indicate that the instability takes place in a confined area "inside" turbulent streaks. This proceeding sums up the results from an article in preparation (Rolland, 2011)