Dynamics of propagating turbulent pipe flow structures. Part II: Relaminarization

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by  A. Duggleby

Dynamics of propagating turbulent pipe flow structures. Part II: Relaminarization
A. Duggleby, K.S. Ball, and M.R. Paul
7 Department of Mechanical Engineering, Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061∗
0
(Dated: February 2, 2008)
0
2 Thedynamicalbehaviorofpropagating structures,determinedfrom aKarhunen-Lo`evedecomposi-
tion, in turbulent pipe flow undergoing reverse transition to laminar flow is investigated. The tur-
n
bulentflowdataisgeneratedbyadirectnumericalsimulation startedatafullyturbulentReynolds
a
J numberof Reτ =150, which is slowly decreased until Reτ =95. At this low Reynolds number the
high frequency modes decay first, leaving only the decaying streamwise vortices. The flow under-
3
goes a chugging phenomena, where it begins to relaminarize and the mean velocity increases. The
2
remaining propagating modes then destabilize the streamwise vortices, rebuild the energy spectra,

and eventually the flow regains its turbulent state. Our results capture three chugging cycles be-
]
n fore the flow completely relaminarizes. The high frequency modes present in the outer layer decay
y first,establishingtheimportanceoftheouterregionintheself-sustainingmechanismofwallbound
d turbulence.
-
u
fl I. INTRODUCTION tion, or azimuthal wavenumber n=3 in our notation, is
s. the largestcontributornearthe criticalbifurcationpoint
c associatedwiththelaminartoturbulenttransition. This
In part I the effect of drag reduction on propagat-
i
s ingturbulentflowstructuresbyspanwisewalloscillation also corresponds to our findings, as we will show that
y was studied.1 A second instance where drag reduction is whenobservingthechuggingphenomena,then=3trav-
h
eling waveis often the mostenergetic at the pointwhich
p seen is in a relaminarizing flow. As the turbulence dies,
the flow reasserts itself as turbulent.
[ so does the Reynolds stress generation, and thus, for a
Also of note is the minimal channel work by Webber,
constant pressure gradient driven flow, the flow rate in-
2 Handler, and Sirovich; we apply their results to further
creases. As we will show, the flow does not immedi-
v the understanding of this chugging phenomena.12 They
9 ately relaminarize, but instead goes through a series of
indicate that the nonlinear terms in the Navier-Stokes
5 chugging motions. In these chugging motions, the flow
equations lead to triad interactions of the KL modes
2 losesitsturbulentinertialrange,losingthehighfrequen-
8 cies first. Before the flow has completely relaminarized, which is responsible for the transfer of energy between
0 certain key propagating waves interact with the decay- modes. This occurs whenever wavenumbers of three
6 modes(m,n,q),(m′,n′,q′),and(m′′,n′′,q′′)sumtozero,
ing streamwise vortices, recreating the cascading energy
0 shown in equations 1 and 2 below,
scales that populate the inertial subrange. In this part,
/
s we examine the dynamics found in relaminarization to
c
understandhowaflowremainsturbulenttobetter eluci-
si date the mechanism behind the self-sustaining nature of n+n′+n′′ =0 (1)
y turbulence. m+m′+m′′ =0 (2)
h
p Previous work has focused either on relaminar-
: ization from favorable pressure gradients,2,3,4 strong where n is the azimuthal wavenumber and m is the
v accelerations5, or examining relaminarization as a streamwise wavenumber obtained from the Fourier rep-
i
X testbedtounderstandthedecayrateofstructures.6 Out- resentation of the flow.
r side the field of wall turbulence, there has been related
a
work in studying relaminarization and bifurcations in
sphericalCouetteflow.7Theonlyworkfoundthatrelates II. NUMERICAL METHOD
to drag reduction is the examination of linear feedback
control in a turbulent channel flow that achieved total Details on the numerical method for generating the
relaminarization.8 Nevertheless,itis ofinterestto exam- directnumericalsimulation(DNS) flow fields andonthe
ine the field of transition where recent work has added Karhunen-Lo`eve method can be found in part I of this
to our knowledge of pipe flow structures and their inter- paper.1
actions. Over a time of 12,000 t+ the Reynolds number was
Both Kerswell9 and Faisst and Eckhardt10 have found slowly reduced from Reτ = 150 to a value of Reτ = 95,
traveling wave solutions to the Navier-Stokes equations and the DNS continued for another 5000t+ to eliminate
through continuation methods. They identified struc- any transient effects, as seen in Figure 1. Data was
tures for rotationally symmetric solutions, which is con- then collected for 10,000t+, which included three dis-
firmed here for Reτ = 95 and in previous work11 for tinct chugs before the flow completely relaminarized, as
Reτ = 150 through a Karhunen-Lo`eve (KL) decomposi- seen in Figure 2.
tion of a direct numerical simulation of turbulent pipe The grid resolution was kept the same as for the orig-
flow. Moreover, Kerswell found that the threefold rota- inal Reτ = 150 case. Thus the grid is effectively further
2
0.4 25
present
0.35 theoretical (turbulent)
20
theoretical (laminar)
0.3
uz,mean Reτ=95 mean15
0.25 +uz,
10
0.2
Reτ=150 5
0 0.5 1 1.5 2 2.5 3
t+ x 104 0100 101
y+
FIG. 1: Time history of the mean flow rate relaminarization
showing the initialization from Reτ =150 to Reτ =95. The FIG. 3: Mean flow profile versus y+ = (1−r)Reτ including
window shows where data was collected.
the theoretical turbulent profile (dashed) with the sublayer
(u+ =y+), the log layer (u+ =log(y+)/0.41+5.5), and the
analytical parabolic laminar solution (dash-dot). The mean
19 flowprofilefollowsthelawofthewall,yetdeviatesdrastically
from the log layer as expected.
18
17 3
Reτ=95
mean16 2.5 Reτ=150
+uz,
u+
15 z,rms
ms 2
+uz,r
11340 2000 4000 6000 8000 10000 ++u, u, θr,rms,rms 1.15 u+θ,rms
t+
FIG. 2: Time history of the mean flow rate relaminarization 0.5
for Reτ = 95. Three chugging cycles are seen near t+ = u+r,rms
2000,4000 and 6000 with the final relaminarization starting 0
near t+ =8000. 0 50 100 150
y+
refined to ∆r+ ≈ 0.49 and (R∆θ)+ ≈ 3.1 near the wall FIG. 4: Root-mean-squared velocity profiles for Reτ = 95
and∆+ ≈2.0nearthecenterlinewithaconstantstream- (solid) and Reτ = 150 (dashed) versus wall units y+. The
inflections near y+ = 90 are effects of the laminar chugging,
wise resolution of ∆z+ = 4.0, where r, θ, and z is the
because therms velocities are averaged overall time steps.
radial,azimuthal,andstreamwisedirection,respectively,
and R is the radius of the pipe.
aturbulenttrendasseeninFigure4,althoughincompar-
isontotheReτ =150flow,theradialandazimuthalfluc-
III. RESULTS
tuations are about half as strong, and the peak stream-
wise rms velocity is shifted further away from the wall.
The profile ofthe meanflowwith respectto wallunits Also noteworthy is the strong fluctuations near the cen-
is seen in Figure 3. At this low Reynolds number, the ter of the pipe at y+ ≈ 90. This is the first indication
profile does not conform to the log layer, yet near the thatthe dynamicsnearthe centerofthe pipe differ from
wallitstillexhibitsalinearsublayer. Also,theflowdoes what is expected for the fully turbulent case.
not conform to the laminar parabolic profile, indicating TheReynoldsstressprofilealsodiffersfromtheReτ =
that it is indeed turbulent. 150 case and is shown in Figure 5. The Reynolds stress
Theroot-mean-square(rms)velocityprofilesalsoshow for the Reτ = 95 case has roughly half the magni-
3
0.7 100
Reτ=150
0.6 Reτ=95 90
0.5 80
0.4 70 Reτ=95
Reynolds Stress 000...123 Percent Total Energy 456000 Reτ=150
0 30
−0.1 20
−0.2 10
−0.3 0
0 50 100 150 0 500 1000 1500 2000 2500 3000
y+ Eigenfunction
FIG. 5: Reynolds stress profiles for Reτ = 95 (solid) and FIG. 6: Comparison of the running total energy retained
Reτ = 150 (dashed) versus wall units y+. As in the rms in the KL expansion for Reτ = 95 (solid) and Reτ = 150
velocity profiles, the inflections near y+ = 90 are effects of (dashed). The 90% crossover point contains 2453 and 66
the laminar chugging, showing that relaminarization begins modes respectively, showing a drastic reduction in the tur-
at thecenter of thepipe. bulent attractor.
tude throughout the radial profile, although the peak
Reynolds stress is also found at the same location of
y+ =31. Thelargestdeviationfromthe expectedprofile
occurs near the centerline beyond y+ ≈ 60, where the
Reynolds stress begins to fluctuate. In particular, be-
tween y+ = 88 and 92, the Reynolds stress is negative,
which is physically interpreted as turbulence damping.
Now, in addition to the rms deviation near the center-
line, this Reynolds stress fluctuation indicates that the
relaminarizationprocess begins at the center of the pipe
and goes towards the wall.
FIG. 7: Comparison of normal speed locus for the Reτ =95
Turning from statistics to the KL decomposition, we (·)andReτ =150(+)cases. Thesolidlinesrepresentacircle
find that the chaotic attractor is reduced in size, as ex- of diameter 8.41 and 8.64 respectively that intersect at the
pected with a reduction in Reτ, from DKL = 2453 to origin.
DKL = 66, shown in Figure 6. This dimension is simi-
lar to that found in Part I where the oscillated pipe was
barely turbulent with a dimension of DKL = 102, and basisforstreamwisevorticesthatwere94−117wallunits
any stronger oscillation would have resulted in relami- apart. However, at Reτ = 95, these represent spacings
narization. of 60−75 wall units, which is too small, thus explaining
In observingthe energy content of the modes in Table their drop in energy.
I, and the most increased and decreased in Table II, we Looking at the normal speed locus in Figure 7, the
find that the shearmodes increasedrastically,whichis a modes for a wave packet similar to that of Reτ = 150,
result of the chugging motion and large mean flow rate but with a slightly faster advection speed of 8.64 versus
fluctuations. Also of note is the increase in strength of 8.41. This shows a Reynolds number dependence on the
the n=3,4 and5 streamwisevortices(m=0)andtheir advection speed, as expected, since the advection speed
associatedwalltraveling waves(m=1). The increase in wasshowntoscalewiththemeanflowratewhentheflow
the (0,1,1) and (1,0,1) modes, since they are not found ratewasincreasedinpartIwithspanwisewalloscillation.
in the work by Kerswell9 and Faisst and Eckhardt,10 In visualizing the most energetic modes, the same
couldbethecatalystsofenergytriadsinequations1and structures as those found in the Reτ = 150 case are
2 between the n=3,4 and 5 rolls and the wall traveling present. Figures 8 - 13 show the coherent vorticity for
waves. Forexample,the(1,3,1)and(1,4,1)wavesinter- the most energetic modes for each of the subclasses dis-
act through the (0,1,1) catalyst, and the (1,3,1) wave cussedin partI. Since the (0,0,1)mode has no coherent
andthe(0,3,1)rollinteractthroughthe(1,0,1)catalyst. vorticity, the velocity is shown. Here we use “coherent
Themodesthatdecreasedthemostinenergyaremoreof vorticity”torefertotheimaginaryeigenvaluesoftheve-
aresultofthelowerReynoldsnumber,asthehighmodes locity gradient tensor as defined in Chong et al. (also
n=8,9and10inthe Reτ =150casewereanimportant know as “degree of swirl”).13 Although slight differences
4
are visible due to the lower Reynolds number, the same
TABLEI:Comparisonoffirst25eigenvalues. misthestream-
trendsandcharacteristicsarefoundforthe KLmodesas
wisewavenumber,nisthespanwisewavenumber,andqisthe
those found in the Reτ =150 case.
eigenvalue quantumnumber.
Reτ =150 Reτ =95
Index m n q Energy % Total m n q Energy % Total
1 0 6 1 1.61 2.42% 0 0 1 114 55.76%
2 0 5 1 1.48 2.22% 0 3 1 9.63 4.72%
3 0 3 1 1.45 2.17% 0 4 1 8.39 4.11%
4 0 4 1 1.29 1.93% 0 1 1 7.79 3.82%
5 0 2 1 1.26 1.88% 0 2 1 6.30 3.09%
6 1 5 1 0.936 1.40% 0 0 2 5.98 2.93%
7 1 6 1 0.917 1.37% 0 5 1 2.79 1.37%
8 1 3 1 0.902 1.35% 1 4 1 2.18 1.07%
9 1 4 1 0.822 1.23% 1 3 1 1.85 0.90%
10 0 1 1 0.805 1.20% 0 0 3 1.83 0.90%
11 1 7 1 0.763 1.14% 0 6 1 1.56 0.77%
12 1 2 1 0.683 1.02% 1 5 1 1.45 0.71%
FIG. 8: The (0,0,1) shear mode with contours of stream-
13 0 7 1 0.646 0.97% 1 2 1 1.42 0.70% wise velocity and vectors of cross-stream velocities. Left: (a)
14 2 4 1 0.618 0.92% 0 1 2 1.34 0.66% Reτ =95. Right: (b) Reτ =150.
15 0 8 1 0.601 0.90% 1 0 1 1.30 0.64%
16 2 5 1 0.580 0.87% 1 6 1 0.884 0.43%
17 1 1 1 0.567 0.85% 1 1 1 0.815 0.40%
18 2 7 1 0.524 0.78% 0 7 1 0.727 0.36%
19 1 8 1 0.483 0.72% 2 4 1 0.716 0.35%
20 2 6 1 0.476 0.71% 2 3 1 0.665 0.33%
21 2 3 1 0.454 0.68% 2 5 1 0.589 0.29%
22 2 2 1 0.421 0.63% 2 2 1 0.544 0.27%
23 2 8 1 0.375 0.56% 0 2 2 0.521 0.26%
24 1 9 1 0.358 0.54% 2 6 1 0.487 0.24%
25 3 4 1 0.354 0.53% 1 1 2 0.479 0.23%
TABLE II: Ranking of eigenfunctions by energy change be-
tween the Reτ = 150 and Reτ = 95 cases. m is the stream- FIG. 9: The (0,3,1) roll mode with contours of coherent vor-
wise wavenumber, n is the spanwise wavenumber, and q is ticity. Left: (a) Reτ =95. Right: (b) Reτ =150.
the eigenvalue quantum number. The shear modes (0,0,q)
are among themost increased, as are then=3,4,and 5 rolls
(m=0) and wall modes (m=1).
Increase Decrease
Rank ∆λk m n q ∆λk m n q
1 113.4 0 0 1 -0.399 1 7 1
2 8.18 0 3 1 -0.300 0 8 1
3 7.10 0 4 1 -0.241 1 8 1
4 6.99 0 1 1 -0.236 2 7 1
5 5.81 0 0 2 -0.229 1 9 1
6 5.04 0 2 1 -0.225 2 8 1
7 1.71 0 0 3 -0.186 0 9 1
8 1.36 1 4 1 -0.175 3 9 1
9 1.30 0 5 1 -0.173 3 8 1
10 1.12 0 1 2 -0.170 2 9 1 FIG. 10: The (1,4,1) wall mode with contours of coherent
11 1.09 1 0 1 -0.165 1 10 1 vorticity. Left: (a) Reτ =95. Right: (b) Reτ =150.
12 0.94 1 3 1 -0.149 0 10 1
5
35
Shear modes
30 Roll modes
Propagating Waves
25
Energy1250
10
5
0
0 2000 4000 6000 8000 10000
time (t+)
FIG. 14: Time history of energy of shear modes (blue), roll
FIG.11: The(3,2,1) lift modewithcontoursofcoherentvor-
modes (black), and propagating waves (red).
ticity. Left: (a) Reτ =95. Right: (b) Reτ =150.
12
Asymmetric Modes
Wall Modes
10 Lift Modes
Ring Modes
8
Energy 6
4
2
0
0 2000 4000 6000 8000 10000
time (t+)
FIG.15: Timehistoryofenergyofpropagatingsubclasses.
Wall (solid black), lift (red), asymmetric (dashed), and
FIG. 12: The (1,1,1) asymmetric mode with contours of co- rings (blue).
herent vorticity. Left: (a) Reτ =95. Right: (b) Reτ =150.
0.18
0.16
0.14
IV. DYNAMICS
wall0.12
Recreating the time history of the KL modes reveals Ratio lift to 0.00.81
theinteractionbetweentheshearmodes,rollmodes,and
0.06
propagatingwaves. As showninFigure 14, the chugging
0.04
phenomenahappenswhenthepropagatingmodesdropin
0.02
energy. Thishappensattimest+ ≈1600,t+ ≈3500,and 0 500 1000 tim15e0 (0t+) 2000 2500 3000
t+ ≈5500,andstarts the chuggingphenomena. At t+ ≈
8000,the propagatingmodes droptoo far in energy, and FIG.16: Ratiooflift modeenergytowallmodeenergyas
the flow relaminarizes. The chugging cycle ends again afunction oftime. Asthelift modeenergy decreases with
when the propagating waves spike at t+ ≈ 2200, t+ ≈ respect to thewall mode energy, relaminarization begins.
4100, and t+ ≈6200.
Examining the propagating waves based upon their
subclass,showninFigure15,wefindthattheasymmetric
andringmodesstayaboutthesameenergyaseachother
throughoutthechuggingcycles,whilethewallmodesare
aboutafactorof4timesmoreenergetic. Theliftmodes,
on the other hand, vary greatly in energy. When the
lift modes decay from high to low energy, this coincides
withthestartofachuggingcycle,andendswhenthelift
modes regain the high energy state. This is emphasized
inFigure16wheretheratiooftheliftmodetotalenergy
to the wall mode total energy is shown. In Figure 17,
only one chug is shown, emphasizing the importance of
thelifttowallratio,andalsotheenergytransferbetween
FIG.13: The(1,0,1)ringmode(r-zplanecross-section)with
classes as energy flows from the rolls to the wall modes
contours of coherent vorticity. top: (a) Reτ = 95. Bottom:
(throughthe ring modes), andthenfromthe wallmodes
(b) Reτ =150.
6
16
an 15
e
m
+uz, 14
13
0 500 1000 1500 2000 2500 3000
t+
20
oll 10
R
0
0 500 1000 1500 2000 2500 3000
t+
4
g
n 2
Ri
0
0 500 1000 1500 2000 2500 3000
t+
20
all 10
W
FIG. 18: Average energy spectra of the propagating modes.
0
0 500 1000 1500 2000 2500 3000 Wall(solid), lift (dashed),asymmetric(dots),and rings(dash-
t+ dot).
c 4
metri 2 101
m
y
As 0 100 Normal Spectra
0 500 1000 1500 2000 2500 3000 t+=0 (blue), t+=2392 (black)
t+ 10−1
2
10−2 Chug begins, t+=1592 (green)
Lift 1 10−3
0
0 500 1000 1500 2000 2500 3000 10−4 Near the peak of the chug, t+=2072 (red)
t+
o lift to wall 00..12 1100−−65 t+=t+2=t2+23=C2225h 1(2ub1 gl(2a me c(nakc dygdaseanns)thae)d)
ati 0 10−7
R 0 500 1000 1500 2000 2500 3000 100 101 102
time (t+)
FIG.19: Timehistoryoftheenergyspectraofthepropagating
modes. t+ = 0 and t+ = 2392 show the proper established
energy spectra at the beginning of the simulation, and well
after thechug. t+=1592 is thespectrawhen thechugbegins.
FIG.17: Timehistoryofenergyofsubclassesforasinglechug. t+ = 2112,2552 and 2232 show the spectra regaining strength
From the top: the mean velocity, roll mode energy, ring mode with time, finishing the chug cycle. Like the dynamics of the
energy, wall mode energy, asymmetric mode energy, lift mode lift to wall energy ratio, the chugging spectra reinforces the
energy,andthelift towallmodeenergyratio. Thestartofthe dependenceoftheself-sustainingmechanismofwallturbulence
chug (blue dashed line) is when the lift to wall mode energy onthehighfrequenciesfoundintheinertialrange,represented
ratio drops too low. The end of the chug (red dashed line) is bythe lift modes.
when the ratio spikes and recovers. The phase lag in energy
from the roll to the wall modes (through the ring modes) and
similarlyfromthewalltotheliftmodes(primarilythroughthe
asymmetric modes) can beseen.
to the lift modes (primarily through the asymmetric Figure 18. Again, like the Reτ = 150 spectrum, the lift
modes). modes are more energetic than the wall modes for high
wavenumbers. Thus, because the high wavenumbers of
Observingthe energyspectrathroughoutthe chug,we
the spectra dies off by two orders of magnitude at the
first revisit the total energy distribution of the propa-
start of the chug cycle, seen in Figure 19, it reinforces
gating waves as averaged over the entire flow, seen in
7
the importance of the lift modes in maintaining the tur- V. CONCLUSIONS
bulent flow.
As noted in the Reτ =150 case, the lift modes are re-
sponsible for the majority of turbulence near the center Itisapparentthroughtheexaminationofthespanwise
of the pipe, as the wall modes stay near the wall, even wall oscillated case in Part I and the relaminarization
for high quantum number. Thus, the importance of the case in Part II that if any leg of the energy cycle in a
outer region in the self sustaining mechanism of turbu- turbulent flow is disrupted, the resulting imbalance can
lenceisreinforced. Iftheliftmodesdonotreceiveenough lead to the start of a relaminarization process, and even
energy,cascadedthroughthe wallmodes, the relaminar- completerelaminarization. InPartI,wedescribeamodel
ization process begins near the center of the pipe. This wherethepropagatingwallmodeswerepushedtohigher
is confirmedby the presence ofthe largedropin the rms advection speeds, reducing their effective lifespan. Thus
velocities and Reynolds stress profiles near the center of theydonothaveenoughtimetotakeenergyfromtheroll
the pipe. If the process cannot be halted in time by the modes,breakingthe third legof the mechanism. Here in
transferof energyfromthe wallmodes to the lift modes, PartII,withlowerpressuregradient,thereisnotenough
typically through the (1,3,1) or (1,4,1) modes, the flow energy to properly maintain the lift modes, and the last
will completely relaminarize. Adding in the findings of legoftheprocessisbroken,startingtherelaminarization
triad interactions by Webber et al.,12 the flow of energy process.
is shown in Figure 20. The shear to roll interactions are
catalyzed by the rolls themselves, the roll to wall inter- Thus,inconclusion,wefind thatwhile the wallmodes
actions are catalyzed mostly by the ring modes, and the andnearwallinteractionsareresponsibleforthe genera-
wallto lift modes are catalyzedby boththe ringandthe tionofturbulence fromthe pressuregradient,the turbu-
asymmetric modes, and other lift modes. In the relami- lence in the outer region is necessary to maintain the
narization process, it is this final leg that fails, breaking proper inertial range in the energy spectra, and that
the mechanism, and starting relaminarization from the without it, the relaminarizationprocess begins.
center of the pipe.
ACKNOWLEDGMENTS
FIG. 20: Energy flow chart for turbulence subclasses with
catalysts. If any of these legs are disrupted, turbulence re-
This research was supported in part by the National
duction (or drag reduction) begins. For instance, in Part I
ScienceFoundationthroughTeraGridresourcesprovided
theenergyinthewallmodeswerereducedbyliftingthemoff
by the San Diego Supercomputing Center, and by Vir-
the wall by spanwise wall oscillations, reducing their energy
ginia Tech through their Terascale Computing Facility,
by forcing them to advect faster and died faster. In the cur-
rent relaminarization case, not enough energy is present to System X. We gratefullyacknowledgemany useful inter-
fully populate the lift modes, and so their energy decreases actions with Paul Fischer and for the use of his spectral
and again theenergy balance is disrupted. element algorithm.
∗
Electronic address: [email protected] pressure gradient and of the Reynolds number on an in-
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