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\fancyhf[LOH]{\small M. BRAZA ET AL.}
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\fancyhf[REH]{\small DNS OF 3D TRANSITION IN WAKE FLOWS}
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   \fancyhead[C]{ERCOFTAC international symposium <<Unsteady separation in fluid-structure interaction>> Mykonos, Greece, June 17-21, 2013}
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\title{{\vspace{-0.0cm}\sc{The 3D Transition to Turbulence in a Circular Cylinder Wake by Means of Direct Numerical Simulation}}}
\author{Marianna BRAZA$^1$, Second author$^2$\vspace{0.3cm}\\
\small{\it{$^1$Institut de M\'ecanique des Fluides de Toulouse, Unit\'e Mixte C.N.R.S.-I.N.P.T. 5502,}}\\ \small{\it{Av. du Prof. Camille Soula, 31400 Toulouse, France}}\\
\small{\it{$^2$Second author affiliation}}\\}
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%\mail{kuban@imc.pcz.czest.pl}
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% ABSTRACT (11pt, times new roman)
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\section*{\small\bf Abstract.} \small
The three-dimensional transition to turbulence in the flow around flow
around a circular cylinder has been analysed physically by performing direct numerical simulation to solve the
system of Navier--Stokes equations. The successive stages of 3D transition, beyond the first bifurcation, have
been detected first in the incompressible regime, for a circular cylinder configuration. The generation of
streamwise vorticity, organised according to spanwise periodic cells has been associated with the development of
large-scale coherent spanwise undulations of the originally rectilinear (nominally 2D) alternating vortex rows.
The wavelengths of these undulations have been determined versus Reynolds number. As this parameter
increases, a further inherent change of the flow transition is obtained and analysed, the natural vortex
dislocations pattern. Beyond this change, the increase of Reynolds number yields an abrupt shortening of the
spanwise wavelength and the flow undergoes anothertransition step, whose critical Reynolds number is
evaluated by the present DNS approach in association with the Ginzburg--Landau model. Therefore, the linear
and non-linear parts of the flow transition have been quantified by means of the amplitude evolution versus time
obtained by the present DNS, in conjunction with the mentioned global oscillator model. 

\bigskip
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% KEYWORDS (11pt, times new roman)
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{\bf Key words:} instability, transition, DNS, bluff-bodies, wings, incompressible flow.

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% SECTION (13pt, times new roman, bold)
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\large
\section{Introduction}
Concerning (times new roman, 13pt) the flow around a circular cylinder, fascinating
phenomena occur in respect to the three-dimensionality of originally 2D vortex
structures, as Reynolds number increases. The present study focuses beyond the first
bifurcation, that consists of amplification of the von K\'arm\'an instability and of formation
of the main, alternating vortex rows. Previous studies have reported that this first step
to transition is essentially two-dimensional [7, 27], following a Hopf bifurcation, also
known as a Poincar\'e--Andronov bifurcation [33]. The critical Reynolds number (order of
47) had been accurately evaluated by the Stuart--Landau global oscillator model, on the
basis of the physical experiment by Provansal et al. [34], as well as by continuation
methods using the steady-state approach near the threshold [19]. The DNS approach,
combined with a 2D analysis of the Navier--Stokes system, offers the possibility to
dissociate those mechanisms having a purely 3D origin from those being essentially 2D,
where the ensemble of these mechanisms co-exist and interact non-linearly in the
context of the physical experiment. Furthermore, the DNS approach provides the
detailed time-history related to the establishment of each physical mechanism, where
the time-scales of the initial stages are often rather short during an experimental study
for this kind of flows. In the context of flows past bluff bodies, there are still few
attempts of direct simulation of the transition to turbulence, whereas this approach has
been more widely used in case of rectangular configurations, including among other,
boundary layer, jet and channel flows. By performing three-dimensional simulations of
the flow past the cylinder in the Reynolds number range (100300) it has been
quantified that this flow pattern remains essentially two-dimensional up to Reynolds
number of order 180 as shown by Persillon and Braza [28], [30]. etc, etc 
of transition are analysed in detail, in the present study.


\section{3D Transition Features in Incompressible Wakes}

The Navier--Stokes equations for an incompressible viscous fluid past a circular
cylinder are solved in a general curvilinear coordinates system normalised by the
cylinder's diameter D and the uniform upstream velocity. A detailed presentation of
these equations and of the numerical method developed can be found [30]. The
governing equations are written in general curvilinear coordinates in the (x,y) plan while
the z-component (in the spanwise direction) is in Cartesian coordinates, as presented
in section 3.2.1. The numerical method is based on the 3D full Navier--Stokes
equations for an incompressible fluid. The pressure-velocity formulation is used as well
as a predictor-corrector pressure scheme of the same kind as the one reported by
Amsden and Harlow [4], extended in the case of an implicit formulation by Braza [8]
and [9],Braza et al. [7, 12]. The temporal discretisation is done by adopting the Douglas
[13] fractional scheme in an Alternating Direction Implicit formulation. The method is
second-order accurate in time and space. Centred differences are used for the space
discretisation. The staggered grids by Harlow and Welch [17] are employed for the
velocity and pressure variables. The Navier--Stokes equations are transformed in

respect to a non-orthogonal, general curvilinear coordinates system in (x,y) plan, while
a Cartesian coordinate z is used for the spanwise direction. This ensures the ability and
general character of the present solver to take into account any complex body
configuration of constant z section. An H-type grid is used because this kind of grid
offers the possibility to introduce more physical boundary conditions on the external
boundaries and it avoids branch-cut lines. A zoom of the grid around the obstacle is
shown in Fig. 1. An original aspect of the present methodology is the extension of the
Douglas alternating directionfractional step scheme, initially conceived for a pure
diffusion equation, to the complete set of Navier--Stokes equations. The choice of this
scheme, instead of the Peaceman and Rachford [26] Alternating Direction Implicit one,
that was chosen in a previous 2D study [7], is made due to the high stability properties
offered by the Douglas [13] scheme for the 3D problem. The principles of the numerical

algorithm ICARE are based on the reports by Braza [10, 11]. Another useful element of
the present numerical method is the extension in 3D of non-reflecting type outlet
boundary conditions, based on the work by Jin and Braza [20] in two dimensions. The
boundary conditions are those specified in Persillon and Braza [30] and are summarised
in Fig.2. Concerning the spanwise free edges of the computational domain, periodic
boundary conditions are applied. A comparison of Neumann type boundary conditions
and of periodic ones in respect to their ability in simulating the development of
three-dimensionality and of spanwise undulations has been performed in a number of
our studies and proven their equal validity [32]. In a first simulation, Newmann
boundary conditionsare employed and allowed to assess the preferential spanwise
wavelength formation, selected spontaneously buy the flow system. However, this
procedure has needed a very long transient time. Therefore, by taking a spanwise length
as a multiple of the simulated one and by using periodic boundary conditions,the
transient phase has been considerably shortened. 
n the following sections the results analyse the way the 3D motion is progressively
installed in the flow system. The flow evolution in time and space, with emphasis to the
spanwise direction, is studied by performing appropriate signal processing techniques
(wavelet analysis and autoregressive modelling). The numerical simulations are carried
out for a rather long spanwise length value (12D) and part of the instability analysis
concerning global oscillator modelshave been carried out by using a spanwise length of
4.5D. The typical grids are of order (250 * 100 * 80) to (280 * 120 *100). The parallel
computers used are the IBM-SP2 of CNUSC (Centre National Universitaire Sud de
Calcul) and of Cornell Theory Centre, as well as the CRAY C98 and T3E of the
national computing centre IDRIS. In the following, x is the direction along the
cylinder's downstream axis, parallel to the free-streem u velocity. y is the vertical
direction parallel to v velocity and z is the spanwise direction, parallel to w velocity
component.

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\subsection{Onset of three-dimensionality}
The present elliptic flow looses the memory of the initial conditions applied, after a
transient phase whose time scale depends on the nature of the initial conditions. After
detailed numerical tests in our research group [3, 32], it has been shown that the final,
established stage, after the transient one is independent on the initial conditions.
Therefore, the choice of the initial fields has been done on the criterion of shortening
the transient phase. The initial conditions are those of a 2D established flow with vortex
shedding, providing a rectilinear configuration of the alternating vortex rows along the
span. A special attention is made to ensure that the alternating vortex shedding is
extended over the whole downstream distance, and that the wake expansion rate along
x is correct, according to the physical experiment. These features have been achieved
by ensuring the three following conditions: (1) A sufficiently refined grid along the
whole downstream distance and not only in a limited distance around the obstacle; (2)
A sufficiently large computational domain in x and y directions, in order to not confine
the flow and to allow the correct wake expansion; (3) Adequate boundary conditions in
the downstream outlet boundary, that limit the feedback effects and allow travelling the
alternating eddies through this fictitious boundary without confinement (Fig. 3a).
Therefore, the proper development of the primary instability, the von K\'arm\'an mode, is
ensured along the whole computational domain (Figure 3a). This is the fundamental
characteristic of this category of flows and a prerequisite to study further on the
development of secondary instabilities along the span. In a number of related studies,
the alternating character of the vortex shedding is limited within only one or two
diameters downstream the obstacle.
After a long transient phase, needing more than 300,000 time steps during which the
flow remains 2D, the w velocity component (in the spanwise direction) is progressively
developed. In order to shorten this phase, a w velocity weak intensity fluctuation is
applied in the inlet section along the span. A white noise of dimensionless intensity
10-4 is chosen. Different runs with different values of this intensity within the range
(10-4,10-3) have been checked and ensured leading to the same final established
stage. A w velocity component typical evolution is shown in Figure 15a. The
amplification of this component follows two distinct stages before saturation, a first
exponential growth (linear state) and a second, non-linear one, before reaching
saturation, as it is also discussed in Section 2.3. It is found that w velocity component is
soon organised according to well distinct cells, (Figure 3b), already in the very early
stages of onset of the secondary instability, where the w magnitudes are still very weak.
As it is therefore expected from the continuity equation for an incompressible fluid, this
pattern is followed by appearance of organisation of streamwise vorticity component x
(Figure 3c) along the span, as it is shown at time t=780. The birth of streamwise
vorticity x is due to the progressive development of w component as Reynolds number
increases, owing to the influence of small perturbations mentioned before. Due to the
action of consequent small longitudinal perturbations, the selected mode by the present
system provided by the direct simulation is the organisation of the streamwise vorticity
pattern on counter-rotating vortices. This is in accordance to the elliptic instability
theory. It is recalled that the stability of an elliptical vortex configuration (as one of the
von K\'arm\'an vortices in this case) to small 3D perturbations provides counter-clockwise
longitudinal vorticity filaments with selected wavelengths, that can be assessed by
elliptic stability theory considerations [39]. Under the effect of the progressive increase
of the streamwise vorticity, the 2D main alternating vortices display a weak regular
spanwise undulation. This starting 3D modification of the originally rectilinear
alternating vortices is shown in Figure 3d. At higher time values, the streamwise
vortices become more intense (Figure 3e) and inception of them occurs between two
main alternating eddies in the formation region (Figure 3f). Afterwards, the streamwise
vortices are displaced towards the wake's shear layer, in the convection region. Figures
4a-4c show the progressive development of mode A in conjunction with the
streamwise vortices formation. The intensity of the spanwise undulation increases from
time t=680 to t=740 under the simultaneous effect of streamwise vorticity that is
strengthened and forms progressively counter-clockwise 'braid' like structures (see
pairs of red and yellow vortices), Figures 4a-4d. 
The undulation of mode A, that appears as an inherent characteristic of the flow
obtained by the direct numerical simulation, is clearly visible on the spanwise evolution
ofthe vertical velocity component v, (Fig. 5), that displays a spanwise oscillation
according to a regular wavelength. A spectral analysis of these evolutions is performed
by means of Fast Fourier Transform Fig. 6, to quantify mode A wavelength. The most
predominant spatial mode is found 0.1171, yielding a wavelength value z /R=8.54 (z
/D=4.27). This value is found in agreement with the range provided by experimental
results, as reported by Persillon and Braza [30]. In the present DNS results, there exist
phases of the flow where mode A pattern is less regular (see, for example, the
iso-vorticity contours in the time interval (800,840)). This irregularity is associated and
announces a further fundamental modification of the spanwise structure of the main,
alternating vortex rows, as it will be analysed in the next section.

\subsection{The natural vortex dislocations pattern}
The instantaneous iso-vorticity fields z and x are first con...

\bigskip

... frequency variation quantified by the present wavelet analysis.

\subsection{The spanwise undulation as reynolds number further increases}
t Reynolds number values higher than 220230, the present DNS shows that the
spanwise undulation undergoes a substantial decrease of its wavelength and the
streamwise vortex filaments become more continuous, linking the formation to the
convection region (Fig. 13). They pass from the lower towards the upper shear layers,
by materialising well distinct vortex paths. This displacement was found to be
discontinuous in the case of mode Aundulation (Figure 3f) [41]. Mode B constitutes an
abrupt change in the flow transition [41] and it is worthy to examine whether this
bifurcation is hysteretic. We have performed detailed direct simulations up to the
established stage, from the upper side of Reynolds number (Re = 340) and descending
towards Re = 230. The Strouhal number values according to these simulations are
shown in Fig. 14 [3] and indicate that this mode is characterised by hysteresis and
corresponds to a values-discontinuity (and not a slope discontinuity). 
The present direct numerical simulation furnishes in detail the amplitude evolution
versus period of the flow properties, as it is shown on Fig. 15a, concerning the w
velocity component along the rear axis of the near wake. Based on the amplitude
results, it is possible to evaluate the real part of the coefficients of the Ginzburg--Landau
global oscillator model:

\begin{equation}
 \frac{\partial A}{\partial t} = \sigma_r A -l_r \left|A^3 \right| + \mu_r \frac{\partial^2 A}{\partial z^2}
\end{equation}

...

\bigskip

\section{Conclusions}
he present study is a continuation of our works in the domain of analysis of 3D
transition in flows around bluff bodies. A detailed investigation of successive steps of
transition is carried out, as Reynolds and Mach number increase. A special emphasis is
devoted to the analysis of natural vortex dislocations in the low Reynolds number range.
It has been shown that these structures are a systematic and inherent transition
feature, modifying drastically the first 3D step to transition, that consists of the
development of a regular spanwise undulation. These transition characteristics are
predictable by the fully non-linear approach offered by the direct simulation.
The present study has quantified in detail the frequency reduction associated with the
formation of this kind of vortex structures, by means of autoregressive signal processing
and wavelet analysis of the time-dependent velocity and vorticity signals along the
span. spanwise location of the dislocation. This feature is obtained in the time and
frequency domains concerning the predominant amplitude of the fundamental
frequency.

The interaction of the vortex dislocations with mode A is shown in the present study as
a perturbing effect that modifies the regular spanwise undulation formation. etc, etc
after a long time scale, including an order of 20 periods of the vortex shedding is the
enhancement of the first subharmonic wavelength of mode A, as the most energetic
mode. 
The successive transition step to dislocations mode undergoes a significant shortening
of the spanwise wavelength and a modification on the streamwise vortex filaments, in
accordance with the physical experiments concerning mode B. By employing the
present DNS results, the supercritical nature of this instability has been shown. This
successive step of transition is able to be modelled by the Ginzburg--Landau equation,
whose real-part coefficients are evaluated on the basis of the present direct simulation
results.

\section*{Acknowledgements}
This work has been carried out in the research group EMT2 (Ecoulements
Monophasiques, Transitionnels et Turbulents) of the Institut de M\'ecanique des Fluides

de Toulouse. It is based on the collaborative efforts of J. Allain, A. Bouhadji, D.
Faghani and H. Persillon in the context of their Ph.D theses and post-doctoral
research. Part of this work is carried out on the basis of CPU allocations of the two
national computer centres of France CNUSC and IDRIS, as well of the Cornell's
Supercomputing centre. \\

{\bf Figures} can be put either in the text or at the end of the text, as in the example
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\caption{Development of longitudinal vorticity cells between the main eddies in the formation
region. (f):
Displacement of the longitudinal vortices in the shear layers, in the convection region. Same
graduation as in previous figure.}
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