Whilst the previous section succinctly captures the dynamics of the onset of intermittency, neither the intermittency factor nor the one-dimensional density functions encapsulate other
−4 −3 −2 −1 0 1 2 3 4 v/σ 10−3 10−2 10−1 100 σ p ( v ) 2350 2500
Figure 4.13: Normalised single-point velocity pdfs for ReD2;350, 2,400, 2,450, 2,500,
2,600, 2,800 and 3,000, as indicated by labels and alternating line types. For Re>2;700,
the distributions are nearly Gaussian, and for Re62;600the distributions are non-Gaussian.
transition points, such as the intensive-extensive transition of section 4.3. Following the approach of Tuckermanet al. (2008), we investigate an alternative analysis of the data set of the previous section which allows us to incorporate this bifurcation point, and provides a compelling view of the essence of all three flow regimes and their respective transitions.
4.5.1 Methodology
As mentioned in the previous section, this experiment re-uses theLD25Ddata set of the
previous section. However, we extend the range of Reynolds numbers to include simulations from2;2506Re63;000, incorporating Reynolds numbers at which localised behaviour
was observed.
We begin by taking the discrete Fourier transform ofqalong the axis of the pipe;
O qk.tn/D Nx 1 X jD0 ei kxjq.x j; tn/; kD Nx=2C1; : : : ; Nx=2:
structure of the solution. In particular, if periodic features of wavelength n=Lexist in
the function this will be highlighted by an increase in the magnitude of then-th Fourier
coefficient.
In a pipe flow context, by choosingLD25Dso that the length of a pipe directly relates to
the scale of a single puff, any disturbance to the fluid which can be accurately represented by
qand has a wavelength of the length of the pipe should correspond to an increase inj Oq1j.
Similarly, in larger pipes, trains of puffs will be identifiable by looking at the Fourier mode corresponding to the number of puffs seen. However, as aptly demonstrated in figure 4.7, at certain Reynolds numbers puffs will remain in a localised state and althoughj Oq1jwill show
an increase, the parameter will become far less sensitive as the structure is split over many more modes. ChoosingLD25Dallows us to examine puff-scale structures without this
additional complication.
Let us assume that qOk is a complex random variable distributed according to the two-
dimensional probability density functionk.r; /where.r; /describes a pointzDrei 2
C. In general, approximating this PDF requires many samples from the probability space,
which is expensive to generate with DNS.
One clear optimisation can be obtained by noting thatq is measured at points along the
centre-line of the pipe, indicating that simply encodes the axial position of the puff.
Since the simulations use periodic boundary conditions, without loss of generality we may translate the puff to any streamwise location, and this implies thatkis axi-symmetric; that
is,kj2Œ0;2/is constant.
To construct an approximation tok then, we take1 6 j 6 Nsamp samplesqOkj from the
distribution and construct a step functionk which is constant on annuli Ai D ¹.r; /jri 6r < riC1º:
Here we takeri D i r fori 2 ¹0; 1; 2; : : :º; i.e. a uniform radial grid with spacingr.
NecessarilykjAi DNi=ZiN, whereNi denotes the number of samples lying inAi andZi
is the appropriate weighting factor for normalisation. There are many ways to calculateZi
corresponding to the measure we wish to use for the probability space. However, the most intuitive approach is to insist thatk is approximately equal tok on each annulusAi. In
this case, integratingkoverAi yields Z Ai kdAD 2Ni N Zi Z riC1 ri rdr D Ni N Zi .ri2C1 ri2/ D N ZNi i .2rir r2/:
and hence takingZi D.2rirCr2/,kgives the desired property; i.e.
Z Ai k dA Z Ai k dA: 4.5.2 Results
Figure 4.14 shows cross-sections of the two-dimensional pdf1.r; /for D0. To achieve
this we use theLD25Dsimulation data of the previous section at a range of representitive
Reynolds numbers. This data spans104time units where measurements ofqO1j are taken 10
times per time unit, yielding a total of105samples.
These approximations toare clearly noisy around the origin. In part this is due to the
normalising technique employed above, sinceZi decreases linearly withi and around the
origin,1=Z0 D1=r ! 1asr ! 0. More samples with zero Fourier amplitude are
therefore needed in order to counter this. Unfortunately, for a fully turbulent flow (such as Re D2;800), few samples possess this property as a proportion of the whole, and hence
resolving the pdf correctly near zero is extremely expensive. However, the convergence properties away from the origin are relatively good with the small number of samples attained in this simulation.
At ReD2;800, where the flow is uniformly turbulent,1.r; 0/is sharply peaked and almost
perfectly Gaussian, so that the most probable state of the system has no structure on the scale of the25Dpipe. This can be viewed as the disordered phase. As Re is reduced past Rei,1
becomes non-Gaussian, with the peak reducing and the function spreading. This signifies the appearance of larger-scale structures; from examining the data and from figure 4.6, these take the form of puff-like structures forming within the pipe.
Finally, as Re reduces past2;200, the peak clearly shifts tor > 0, signifying the presence
0:03 0:02 0:01 0:00 0:01 0:02 0:03 r 0:0 0:2 0:4 0:6 0:8 1:0 1:2 1:4 1:6 1:8 1.r; 0/ .10 4/ (a) ReD2200 (b) ReD2350 (c) ReD2800
Figure 4.14: Above (a-c): Two-dimensional approximations of the complex pdf1.r; /
of first-order Fourier modes. Below: Cross-sections of1.r; /at D0for Re D2;200
(bottom curve), 2;300, 2;400, 2;500 and 2;800 (top curve), as indicated by arrows and
addition, the most probable structure that can be observed is a puff, and the probability of finding uniform turbulence (r D0) becomes 0. This can be seen as the ordered phase,
whereby between Rec and Rei, the dynamics of the flow consist of a mixture of ordered
and disordered phases. This demonstrates that the underlying dynamics form a reversible continuous transition between these transition points as Re is varied.