Dynamics of propagating turbulent pipe flow structures. Part I: Effect of drag reduction by spanwise wall oscillation

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by  A. Duggleby

Dynamics of propagating turbulent pipe flow structures. Part I: Effect of drag
reduction by spanwise wall oscillation
A. Duggleby, K. S. Ball, and M. R. Paul
∗
Department of Mechanical Engineering, Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061
(Dated: February 2, 2008)
The results of a comparative analysis based upon a Karhunen-Lo`eve expansion of turbulent pipe
7 flowanddragreducedturbulentpipeflowbyspanwisewalloscillation arepresented. Theturbulent
0 flowis generated bya direct numericalsimulation at aReynoldsnumberReτ =150. Thespanwise
0 wall oscillation is imposed as a velocity boundary condition with an amplitude of A+ = 20 and a
2 period of T+ = 50. The wall oscillation results in a 27% mean velocity increase when the flow is
driven by a constant pressure gradient. The peaks of the Reynolds stress and root-mean-squared
n
velocities shift away from thewall and theKarhunen-Lo`evedimension of theturbulentattractor is
a
reduced from 2453 to 102. The coherent vorticity structures are pushed away from the wall into
J
higher speed flow, causing an increase of their advection speed of 34% as determined by a normal
3
speed locus. This increase in advection speed gives the propagating waves less time to interact
2
with the roll modes. This leads to less energy transfer and a shorter lifespan of the propagating

structures, and thusless Reynoldsstress production which results in drag reduction.
]
n
y
d I. INTRODUCTION information generated by DNS is the Karhunen-Lo`eve
- (KL) decomposition, which extracts coherent structures
u
from the eigenfunctions of the two-point spatial corre-
l In the last decade a significant amount of work
f lation tensor. This allows a non-conditionally based in-
. has been performed investigating the structure of wall
s vestigation that takes advantage of the richness of DNS.
bounded turbulence, with aims ofunderstanding its self-
c
i sustaining nature and discovering methods of control.1 The utility of this method is evident in the knowledge
s it has produced so far, such as the discovery of propa-
One ofthe greatestpotentialbenefits forcontrollingtur-
y
gating structures (traveling waves)of constant phase ve-
h bulence is drag reduction. As the mechanics of the
locity that trigger bursting and sweeping events.15,16,17
p different types of drag reduction are studied, most ex-
[ planations of the mechanism revolve around control- These studies in turn have lead to a new class of meth-
ling the streamwise vortices and low speed streaks.2,3 ods for achieving drag reduction through wall imposed
3 traveling waves.2,18 Another study using KL decomposi-
v One such method of achieving drag reduction is span-
8 wise wall oscillation, first discovered by Jung et al.4 in tion examined the energy transfer path from the applied
5 1992, and later confirmed both numerically4,5,6,7,8 and pressuregradienttothe flowthroughtriadinteractionof
2 experimentally,9,10,11,12,13 to reduce drag on the orderof structures,19,20 explaining the dynamical interaction be-
8 tween the KL modes. In the realm of control, KL meth-
45%. Theprevalenttheoryofthemechanismbehindthis
0 odshavebeenusedtoproducedragreductioninaturbu-
wasdevelopedusingdirectnumericalsimulationofatur-
6 bulent channel flow by Choi et al.6 and experimentally lent channel by phase randomization of the structures21
0
/ confirmed by Choi and Clayton,14 showing that the spa- and to understand the effect of drag reduction by con-
s trolledwallnormalsuctionandblowing.22 Inthepresent
tial correlation between the streamwise vortices and the
c
study, the KL framework is used to examine the differ-
i low speed streaks are modified so that high speed fluid
s ences in the turbulent structures and dynamics between
is ejected from the wall, and low speed fluid is swept to-
y
turbulent pipe flow with and without spanwise wall os-
h wards the wall. Even though this proposed mechanism
cillation.
p describesthenear-walldynamicsthatgovernthedragre-
: duction, questions behind the global dynamics have not For this comparativeanalysis,turbulent pipe flow was
v
i been sufficiently resolved. What is the effect (if any) on chosenasopposedtoturbulentchannelflowbecauseofits
X the outer regionof the flow? How do the coherentstruc- industrial relevance and experimental accessibility. The
r tures of the flow near the wall adjust with the spanwise maindifferencebetweenpipeandchannelflowsisthatin
a
oscillations? What is the effect on the interactions be- turbulent pipe flow the mean flow profile exhibits a log-
tween the inner and outer layers? arithmic profile that overshootsthe theoreticalprofile at
One manner in which to address these questions is lowReynoldsnumbers,whereasinturbulentchannelflow
through direct numerical simulation (DNS) of turbu- it does not.23,24 Secondly, pipe flow differs from chan-
lence. As supercomputing resources increase, DNS con- nel flow because pipe flow is linearly stable to infinitesi-
tinues to provide an information rich testbed to inves- mal disturbances where channel flow is linearly unstable
tigate the dynamics and mechanisms behind turbulence aboveacriticalReynoldsnumber.25,26 Asignificantcom-
andturbulentdragreduction. DNSresolvesallthescales putational difference between pipe and channel flow is
ofturbulence without the need ofa turbulent model and the presence of a numerical difficulty introduced by the
provides a three dimensional time history of the entire singularity in polar-cylindrical coordinates at the pipe
flow field. One of the methods used for mining the centerline, whichhas limited the number of DNS studies
2
in turbulent pipe flow.27,28,29,30,31,32 Similarly, many KL
studies havebeenperformedinaturbulentchannelflow,
buttothebestofourknowledgetheworkpresentedhere
isthe firsttoextendthe KLmethodto spanwisewallos-
cillated turbulent pipe flow.
In previous work a DNS of turbulent pipe flow
for Re = 150 was benchmarked and its KL ex-
τ
pansion was reported, forming the baseline for this
study.33 Similar structures to those of turbulent channel
flow15,16,17 were found, including the presence of propa-
gating modes. These propagating modes are character-
izedbyanearlyconstantphasespeedandareresponsible
for the Reynolds stress production as they interact and
drawenergyfromtherollmodes(streamwisevortices).15
Without this interaction and subsequent energy trans- FIG.1: Across-sectionofaspectralelementgridusedforthe
fer, the propagating waves decay quickly, reducing the pipe flow simulation. The arrow denotes the direction of the
totalReynoldsstressofthe flow.34 As shownby Sirovich spanwise (azimuthal) oscillation about the axis of thepipe.
etal.,15,16 theinteractionbetweenthepropagatingwaves
andtherollmodesoccursbythepropagatingwavesform-
ing a coherent oblique plane wave packet. This wave boundary condition v (r =R,θ,z)=A+sin(2πt/T+) of
θ
packet interacts with the roll modes, and when given amplitude A+ = A/U = 20 and period T+ =U2T/ν =
τ τ
enough interaction time, the roll mode is destabilized 50 with (r,θ,z) being the radial,azimuthal, and stream-
eventually resulting in a bursting event.15 It is in this wise coordinates respectively. This is not intended to be
bursting event that the energy is transferred from the a parametric study and the amplitude and period were
rolls to the propagating waves.19 In this paper we show chosen to achieve maximum possible drag reduction be-
that in the presence of spanwise wall oscillation these fore relaminarizationoccurred.
propagating modes are pushed away from the wall into To solve equations 1 and 2 we use a numerical al-
higher speed flow. This causes the propagating modes gorithm employing a geometrically flexible yet expo-
to advect faster, giving them less time to interact with nentially convergent spectral element discretization in
the roll modes. This leads to reduced energy transfer space. The spatial domain is subdivided into elements,
that occurs less often, and yields lower Reynolds stress each containing a high-order (12th order) Legendre La-
production, which ultimately results in drag reduction. grangian interpolant.37 The spectral element algorithm
elegantlyavoidsthenumericalsingularityfoundinpolar-
cylindrical coordinates at the origin, as seen in Figure
II. NUMERICAL METHODS
1. The streamwise direction contains 40 spectral ele-
ments over a length of 10 diameters. The effective res-
We use a globally high order spectral element Navier- olution of the flow near the wall is ∆r+ 0.78 and
Stokesalgorithmtogenerateturbulentdataforpipeflow (R∆θ)+ 4.9,wheretheradiusr+ andthea≈rclengthat
drivenby a mean streamwise pressuregradient.35,36 The the wall ≈(R∆θ)+ are normalized by wall units ν/U de-
τ
non-dimensional equations governing the fluid are noted by the superscript +. Near the center of the pipe,
the grid width is ∆+ 3.1. The grid spacing in the
∂ U+U U= P +Re−1 2U (1) streamwise direction is≈a constant ∆z+ = 6.25 through-
t ·∇ −∇ τ ∇ outthedomain. FurtherdetailscanbefoundinDuggleby
U=0, (2) et al.33
∇·
where U is the velocity vector, Re is the Reynolds The flow is driven by a constant mean pressure gradi-
τ
number, and P is the pressure. The velocity is non- ent to keep Reτ constant. The spanwise wall oscillation
dimensionalized by the wall shear velocity U = τ /ρ results in a mean flow rate increase, effectively changing
τ w
where τw is the wall shear stress and ρ is the dpensity. the Reynolds number based upon mean velocity (Rem)
The Reynolds number is Reτ = UτR/ν = 150, where R while keeping Reτ constant. This keeps the dominant
istheradiusofthepipe,andν isthekinematicviscosity. structuresoftheflowsimilar,astheyareaffectedprimar-
When non-dimensionalized with the centerline velocity, ily by the inner layer wall shear stress.7 The oscillations
the Reynolds number is Rec 4300. Two simulations were startedon a fully turbulent pipe at Reτ =150,and
wereperformed,one withand≈onewithoutspanwisewall to avoid transient effects, data were not taken until the
oscillation. Inapipe, thespanwisedirectioncorresponds mean flow rate had settled at its new average value over
tothe azimuthaldirection,sothe oscillationisaboutthe a time interval of 1000t+.
axis of the pipe. Each case was run for t+ = U2t/ν = IntheKarhunen-Lo`eve(KL)procedure,theeigenfunc-
τ
16800viscous time units. In the oscillatedcase,the sim- tionsofthetwo-pointvelocitycorrelationtensor,defined
ulation was performed with an azimuthal velocity wall by
3
where (m,n;r ,r ) is the discretization of
ps p s
K′
(m,n;r,r ) using a Q point quadrature to evalu-
L 2π R K
K(x,x′)Φ(x′)r′dr′dθ′dz′ =λΦ(x), (3) ate equation 7 with p,s = 1,2,...,Q. Because the
Z0 Z0 Z0 kernelis built with the two-pointcorrelationbetweenall
K(x,x′)= u(x) u(x′) , (4) three coordinate velocities, its solution has 3Q complex
h ⊗ i eigenfunctions Ψ and corresponding eigenvalues, listed
q
are obtained, where x = (r,θ,z) is the position vector, in decreasing order λ > λ for a given m and
mnq mn(q+1)
Φ(x) is the eigenfunction with associated eigenvalue λ, n, with quantum number q = 1,2,...,3Q. It is noted
K(x,x′) is the kernel, and denotes an outer product. thatequation9isonlyvalidforatrapezoidalintegration
⊗
In order to focus on the turbulent structures, the kernel scheme with evenly spaced grid points (which was used
is built using fluctuating velocities u = U U. The in this study), a different quadrature with weight w(r)
meanvelocity,U,isfoundbyaveragingovera−llθ,z,and can be incorporated in a similar fashion, keeping the
time. Theanglebracketsinequation4representthetime final matrix Hermitian.
averageusing anevenly spacedtime intervalovera total TheeigenfunctionsΨ (m,n;r)holdcertainproperties.
q
time period sufficient to sample the turbulent attractor. Firstly, they are normalized with inner product of unit
tInottahlitsimsteudoyf,1t6h8e00flto+w. field was sampled every 8t+ for a length 0RΨq,Ψq′rdr = δqq′, where δ is the Kronecker
delta. SRecondly,sincetheeigenfunctionsrepresentaflow
Since the azimuthal and streamwise directions are pe-
field,
riodic, the kernelin the azimuthaland streamwisedirec-
tion is only a function of the distance between x and x′ Ψ (m,n;r)= Ψr(m,n;r),Ψθ(m,n;r),Ψz(m,n;r) T
in those respective directions. Therefore the kernel can q q q q
(cid:0) (cid:1)(10)
be rewritten as
with radial, azimuthal, and streamwise components
K(r,θ,z,r′,θ′,z′) = K(r,r′,θ θ′,z z′) Ψrq(m,n;r), Ψθq(m,n;r), and Ψzq(m,n;r) respectively,
− − theyholdthepropertiesoftheflowfieldsuchasboundary
= (m,n;r,r′)einθei2πmz/L (5)
K conditions (no slip) and continuity
with azimuthal and streamwise wavenumbers n and m
1 d in i2πm
respectively and the remaining two-point correlation in (rΨr(m,n;r))+ Ψθ(m,n;r)+ Ψz(m,n;r)=0.
′ rdr q r q L q
the radial direction (m,n;r,r ). It can be shown that
K (11)
in this form, the Fourier series is the resulting KL func-
Thirdly, the eigenvalues represent the average energy of
tion in the streamwise and azimuthal direction.38 The the flow contained in the eigenfunction Ψ (m,n;r)
resulting eigenfunction then takes the form q
Φ(r,θ,z)=Ψ(m,n;r)einθei2πmz/L. (6)
λ = u(r,θ,z),Ψ (m,n;r)einθei2πmz/L 2 , (12)
mnq q
h| | i
Making use of this result, and noting that the two-point
correlation in a periodic direction is simply the Fourier which is why it is necessary that the discrete matrix in
transform of the velocities, the azimuthal and stream- equation 9 must be Hermitian to get real and positive
wise contributions to the eigenfunctions are extracted a eigenvalues. These three properties allow the eigenfunc-
priori by taking the Fourier transform of the velocities tions to be tested for their validity.
u(r,θ,z) = ∞ uˆ(m,n;r)einθei2πmz/L and forming In summary, the KL procedure yields an orthogonal
m,n=0
the remaininPg kernel (m,n;r,r′) for each wavenumber setofbasis functions (modes)that arethe mostenerget-
pair n and m. The eiKgenfunction problem, with the or- ically efficient expansion of the flow field. By studying
thogonality of the Fourier series taken into account, is the subset that includes the largest energy modes, in-
sight is gained as this subset forms a low dimensional
R model of the given flow. Examining the structure, dy-
(m,n;r,r′)Ψ⋆(r′)r′dr′ =λ Ψ(m,n;r), (7)
Z K mn namics, and interactions of this low dimensional model
0
(m,n;r,r′)= uˆ(m,n;r) uˆ⋆(m,n;r′) . (8) yields important information that we use to build a bet-
K h ⊗ i ter understanding of the dynamics of the entire system.
wherethe⋆denotesthecomplexconjugatesincethefunc-
′
tion is now complex, and the weighting function r is
present because the inner product is evaluated in polar- III. RESULTS
cylindrical coordinates. The final form is still Hermitian
justasitwasinequation3;thediscreteformofequation Spanwisewalloscillationresultsinfourmaineffectson
7 is kept Hermitian by splitting the integrating weight the flow, its structures, and its dynamics. They are:
and solving the related eigenvalue problem
1. An increase in flow rate and a shifting away from
√rpKps(m,n;rp,rs)√rs [√rsΨq(m,n;rs)]⋆ the wall of the root-mean-squared velocities and
(cid:2) =λmnq(cid:3)√rpΨq(m,n,;rp) , (9) Reynolds stress peaks.
(cid:2) (cid:3)
4
18.3 25
non−oscillated
18.25 oscillated
20 theoretical
18.2
18.15
15
mean 18.1 mean
+uz, +uz,
10
18.05
18
5
17.95
17.9 0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 100 101 102
t+ y+
FIG. 2: Mean velocity fluctuations (solid) and the average FIG. 3: Mean velocity profile for a non-oscillated (solid) and
velocity (dash-dot) for the oscillated case versus time (t+). oscillated(dashed)turbulentpipeflowversusy+. Theoretical
The fluctuations are consistent with turbulent flow, and the (dash-dot)includes thesublayer(u+ =y+) and thelog layer
meanvelocityis26.9%greaterthaninthenon-oscillatedpipe. (u+ =log(y+)/0.41+5.5). Themeanprofileshowsaloglayer,
butovershootsthetheoreticalvalueasexpectedforpipeflow
untila much higher Reynoldsnumber.
2. Areductioninthe dimensionofthe chaoticattrac-
tor describing the turbulence.
1
3. Anincreaseinenergyofthepropagatingmodesre-
0.9
sponsibleforcarryingenergyawayfromthewallto
0.8 non−oscillated
the upper region, while the rest of the propagating non−oscillated Bulk
0.7 oscillated
modes exhibit a decrease in energy. oscillated Bulk
0.6
4. An increase of the advectionspeed of the traveling R0.5
wavepacketasdeterminedbyanormalspeedlocus.
0.4
0.3
First, surface oscillationhas a major effect onthe tur-
0.2
bulent statistics. The spanwise wall oscillation of ampli-
tude A+ = 20 and period T+ = 50 resulted in a flow 0.1
rate increase of 26.9%, shown in Figure 2. This com- 00 5 10 15 20 25
bination of amplitude and period was chosen because it u+z, mean
providesthelargestamountofdragreductionwhilekeep-
ingtheflowturbulent. Numericalsimulationswithlarger FIG. 4: Mean velocity profile for non-oscillated (solid) and
oscillations completely relaminarize the flow. The com- oscillated (dashed)pipes with theirrespectivebulk velocities
(dash-dot and dots) versus radius show an increase in bulk
parison of the mean profiles in Figures 3 and 4 show the
velocity of 26.9%.
higher velocity in the outer region,with the inner region
remaining the same, as expected by keeping a constant
mean pressure gradient across the pipe.
The comparison of the root-mean-square (rms) veloc- alsoshows a reductionin strengthand a shift awayfrom
ity fluctuation profiles and the Reynolds stress profile in the wall. The peakchangesfrom0.68aty+ =31to 0.63
Figures 5 and 6 show that the streamwise fluctuations at y+ = 38. Thus, a major difference between the two
decrease in intensity by 7.5% from 2.68 to 2.48. Also, flowcases,inadditiontothe expectedflowrateincrease,
the change in peak location from y+ = 16 to y+ = 22 isthe shift ofthe rms velocityandReynoldsstresspeaks
away from the wall has the same trend as the maxi- away from the wall.
mum Reynolds stress u u , where y+ = (R r)U /ν is The second major effect can be found by examining
r z τ
−
the distance from the wall using normalized wall units the size of the chaotic attractor describing the turbu-
(ν/U ). The azimuthal fluctuating velocities show a lence. The eigenvalues of the KL decomposition repre-
τ
slightlygreatermagnitudepeakof1.03closertothe wall sent the energy of each eigenfunction. By ordering the
at y+ =29 versus 0.99 at y+ =40 for the non-oscillated eigenvaluesfromlargesttosmallest,thenumberofeigen-
pipe. The radial fluctuations remain almost unchanged, functionsneededtocaptureagivenpercentageofenergy
showingonly a slightdecreasefrom the wallthroughthe of the flow is minimized. Table I shows the 25 most en-
log layer (y+ 100), resulting in the peak shifting from ergetic eigenfunctions, and Figure 7 shows the running
0.81at y+ =5≈5 to 0.78 aty+ =61. The Reynolds stress total of energy versus mode number. 90% of the energy
5
3 100
non−oscillated
oscillated 90
2.5
80 non−oscillated
oscillated
70
+++u, u, uθr,rms,rmsz,rms1.125 uθ,rms uz,rms Percent Total Energy 456000
30
20
0.5 ur,rms
10
0 0
0 50 100 150 0 1000 2000 3000 4000 5000
y+ Eigenfunction
FIG.5: Root-mean-squaredvelocityfluctuationsforthenon- FIG. 7: Comparison of the running total energy retained in
oscillated (solid) andoscillated (dashed)caseversusy+. The theKLexpansionforthenon-oscillated (solid)andoscillated
oscillated case shows a shift away from the wall, except for (dashed) cases. The 90% crossover point is 2453 and 102
the azimuthal rms which captures the Stokes layer imposed respectively. Thisshowsadrasticreductioninthedimension
by thewall oscillation. of the chaotic attractor.
0.7 6
non−oscillated vr
0.6 oscillated 5 vθ
v
z
0.5
4
Reynolds Stress 000...234 Eigenfunctions 23
1
0.1
0
0
−0.1 −1
0 50 100 150 0 50 100 150
y+ y+
FIG. 6: Reynolds Stress uruz versus y+ for non-oscillated FIG.8a: (0,0,1)eigenfunctioncapturingtheimposedvelocity
(solid) and oscillated (dashed) cases. Similar to the rms ve- oscillations. The radial velocity is zero except near the wall,
locities, theReynoldsstress showsashift away from thewall whereaslightnon-zerocomponentisevident,inducedbythe
from y+ =31 toy+ =38. spanwise oscillation.
is reached with D = 102 compared to D = 2453 similar to the non-oscillated (0,0,1) mode. The (1,2,1)
KL KL
for the non-oscillated case. This mark, known as the mode shows a large increase in energy. Also of note is
Karhunen-Lo`eve dimension, is a measure of the intrin- the reduction in energy of the (3,1,1) and (4,1,1) modes,
sic dimension of the chaotic attractor of turbulence as but their structure remains virtually unchanged.
discussedby Sirovich39,40. By oscillatingthe wallourre- In examining the energy content of the structure sub-
sults show that the size of the attractor is reduced, and classes as a whole, a trend is discovered, shown in Ta-
the system is less chaotic. ble III. Each of these subclasses, reported in Duggleby
The third major effect is found by examining the en- et al.,33 were found to have similar qualitative coher-
ergy of the eigenfunctions. The 25 most energetic are ent vorticity structure associated with their streamwise
listed for each case in Table I. The top ten modes and azimuthal wavenumber. We use “coherent vortic-
with the largest change in energy are shown in Table II. ity” to refer to the imaginary part of the eigenvalues
Firstly, the order of the eigenfunctions remain relatively of the strain rate tensor ∂u /∂x , following the work of
i j
unchanged, with a few notable differences. The (0,0,1) Chongetal..41 Baseduponthequalitativestructure,the
and(0,0,3)shearmodesrepresenttheStokesflowasseen propagating or traveling waves, described by non-zero
in Figures 8a and 8c, and the (0,0,2) mode, shown in azimuthalwavenumberandnearlyconstantphasespeed,
Figure8b,representsthechangingofthemeanflowrate, werefoundto havefoursubclasses: the wall,the lift, the
6
TABLEI:The25mostenergeticmodes; misthestreamwise 0.2
wavenumber, n is the spanwise wavenumber, and q is the
eigenvalue quantumnumber.
0.15
Non-oscillated Oscillated
v
I12ndex m00 n65 11q E11n..e64r18gy %22..42T22%%otal m00 n00 12q E3n24e1.r67gy %1618T..07o77t%%al Eigenfunctions 0.00.51 vvrθz
3 0 3 1 1.45 2.17% 1 2 1 2.47 0.79%
4 0 4 1 1.29 1.93% 0 3 1 2.30 0.73%
0
5 0 2 1 1.26 1.88% 0 1 1 2.27 0.73%
6 1 5 1 0.936 1.40% 0 2 1 2.20 0.70%
−0.05
0 50 100 150
7 1 6 1 0.917 1.37% 0 4 1 1.49 0.48% y+
8 1 3 1 0.902 1.35% 1 3 1 1.09 0.35%
9 1 4 1 0.822 1.23% 0 5 1 1.04 0.33% FIG. 8b: (0,0,2) eigenfunction, similar in structure to the
10 0 1 1 0.805 1.20% 1 1 1 0.953 0.30% non-oscillated (0,0,1).
11 1 7 1 0.763 1.14% 1 4 1 0.772 0.25%
12 1 2 1 0.683 1.02% 0 6 1 0.653 0.21%
6
13 0 7 1 0.646 0.97% 0 0 3 0.616 0.20% v
r
14 2 4 1 0.618 0.92% 1 5 1 0.582 0.19% 5 vθ
v
15 0 8 1 0.601 0.90% 1 6 1 0.542 0.17% 4 z
16 2 5 1 0.580 0.87% 2 3 1 0.490 0.16% 3
111789 121 178 111 000...554628743 000...877582%%% 022 145 211 000...444864284 000...111554%%% Eigenfunctions 12
0
20 2 6 1 0.476 0.71% 0 7 1 0.407 0.13%
−1
21 2 3 1 0.454 0.68% 1 7 1 0.373 0.12%
22 2 2 1 0.421 0.63% 2 6 1 0.346 0.11% −2
23 2 8 1 0.375 0.56% 2 2 1 0.337 0.11% −3
0 50 100 150
24 1 9 1 0.358 0.54% 3 5 1 0.317 0.10% y+
25 3 4 1 0.354 0.53% 3 3 1 0.290 0.09%
FIG. 8c: (0,0,3) eigenfunction capturing the secondary flow
in the Stokes layer. This imposed Stokes flow created by the
oscillation dominates the near wall region of the oscillated
pipe.
TABLE II: Ranking of eigenfunctions by energy change be-
asymmetric, and the ring modes.
tween the non-oscillated and the oscillated cases. m is the
The wall modes are found when the spanwise
streamwise wavenumber,n is the spanwise wavenumber, and
q is theeigenvalue quantumnumber. wavenumber is larger than the streamwise wavenum-
ber. They possess a qualitative structure having co-
Increase Decrease
herent vortex cores near the wall, and have their en-
Rank ∆λk m n q ∆λk m n q ergy decreased by 20.4% with wall oscillation. Likewise,
1 215 0 0 1 -0.962 0 6 1 the ring modes,whicharefound fornon-zerostreamwise
2 34.5 0 0 2 -0.442 0 5 1 wavenumber and zero spanwise wavenumber with rings
3 1.79 1 2 1 -0.390 1 7 1 ofcoherentvorticity,havetheirenergydecreasedby5.2%
with wall oscillation. The asymmetric modes have non-
4 1.47 0 1 1 -0.382 0 8 1
zero streamwise wavenumber and spanwise wavenumber
5 0.95 0 2 1 -0.374 1 6 1
n=1, which allows them to break azimuthal symmetry.
6 0.85 1 3 1 -0.353 1 5 1
These modes undergo a decrease in energy of 2.3% with
7 0.49 0 0 3 -0.256 1 8 1
spanwise wall oscillation.
8 0.37 1 1 1 -0.245 2 7 1
Converselytothedecreaseinenergyfoundintheother
9 0.26 0 1 2 -0.239 0 7 1 three propagating modes, the lift modes increase in en-
10 0.20 0 4 1 -0.220 2 8 1 ergyby1.5%withspanwisewalloscillation. Thesemodes
are found with a streamwise wavenumber that is greater
than the spanwise wavenumber, and they display coher-
7
Thefourthandmostimportanteffectisthattheprop-
TABLE III: Energy comparison of turbulent pipe flow struc-
agating modes advect faster in the oscillated case. The
turesubclassesbetweennon-oscillatedandoscillatedpipes. m
normal speed locus of the 50 most energetic modes of
isthestreamwise wavenumber,andnistheazimuthal(span-
both cases is shown in Figure 10. For this, the phase
wise) wavenumber. All the propagating modes decrease in
speed ω/ k is plotted in the direction k/ k with k =
energy,except thelift modes.
k k k k
(m,n). A circular locus is evidence that these structures
Structure Energy
propagateasawavepacketorenvelopethattravelswith
Non-oscillated Oscillated aconstantadvectionspeed. Theadvectionspeedisgiven
Propagating Modes (m>0) 53.48 48.86 bythe intersectionofthecirclewiththeabscissa. Byex-
(a) Wall (n>m) 23.5 18.7 aminingthenormalspeedlocusofthenon-oscillatedand
(b) Lift (m≥n) 19.8 20.1 oscillatedpipeflow,thewavepacketshowsanincreasein
advection speed from 8.41U to 10.96U , an increase of
(c) Asymmetric(n=1) 6.07 5.93 τ τ
30%. ThisisaresultoftheoscillatingStokeslayerpush-
(d) Ring (n=0) 4.41 4.18
ing the structures awayfrom the wall into a faster mean
flowbycreatingadominantnearwallStokeslayerwhere
Non-propagating Modes (m=0) 12.97 265
the turbulent structures cannot form. This is confirmed
(a) Roll mode (n>0) 12.3 13.2 bytheshiftingofthermsvelocitiesandReynoldsstresses
(b) Shear mode (n=0) 0.72 251.8 away from the wall as reported earlier. In addition to
a faster advection speed, the energy of the propagating
modes decay faster, resulting in bursting events with a
shorter lifespan. This is seen in Figure 11 where the av-
ent vortex structures that starts near the wall and lifts
erageburst durationof the (1,5,1)mode is reduced from
away from the the wall to the upper region. Combined,
106t+ in the non-oscillated case to 65.3t+ in the oscil-
thefourpropagatingsubclassesmodeslose9.02%oftheir
latedcase. Theburstdurationistakentobetheaverage
energy, whereas the non-propagating modes (the modes
time of all events where the square of the amplitude of
with zero azimuthal wavenumber) gain 2043%.
the mode is more than one standard deviation greater
Thus, the third major result is that the energy of the
than the mean.
propagating wall, ring, and asymmetric modes decrease
while the energy of lift mode increases slightly. Fol-
lowing the work of Sirovich et al.16 this shows that en-
ergy transfer from the streamwise rolls to the traveling
waves is reduced, and any energy that is transferred is
6
quickly moved away from the wall to the outer region
(lift modes). The energy spectra showing the change of 5
er
energy by subclasses is shown in Figure 9. mb
u4
n
e
v
wa3
e non−oscillated
s
101 wi2 Oscillated
n
Wall Oscillated pa circle (D=8.41)
Non−oscillated S1 circle (D=10.96)
100
0
0 2 4 6 8 10 12
Lift Velocity / Streamwise wavenumber
10−1 Asymmetric
y Ring
erg10−2
n
E
10−3 FIG.10: Comparison ofthenormalspeed locusfor theoscil-
lated (·) and non-oscillated (+) case. The solid and dashed
linesrepresentacircleofdiameter8.41and10.96respectively
10−4 that intersect at theorigin.
The shifting of the structures away from the wall is
100 101 102
Streamwise Wavenumber shown for the most energetic modes for each propagat-
ing subclass in Figures 12 through 21. These structures,
FIG. 9: Comparison of energy spectra for non-oscillated which represent the coherent vorticity of the four sub-
(solid)andoscillated(dashed)flowsforthepropagatingmode classesofthepropagatingmodes,arepushedtowardsthe
subclasses. All propagating subclasses in the oscillated case centerofthepipe,wherethemeanflowvelocityisfaster.
decrease with respect to the non-oscillated case, except for The location of the coherent vortex core for these eight
thelift mode, which increases slightly. modes are listed in Table IV, showing a shift away from
8
non−oscillated case: average burst time 106 t+
1
2|a(t)|0.5
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
t+
0.6
oscillated case: average burst time 65.3 t+
0.4
2a(t)| FmIoGd.e.13L:efCt:ro(sas-)sencotnio-noscoifllactoehde.reRntigvhotr:ti(cbit)yooscfil(l1a,t5e,d1.) wTahlel
|
0.2 vortex core shifts y+=9.9 away from the wall.
0
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
t+
FIG. 11: A reduction in the burst duration of the (1,5,1)
mode from 106t+ for the non-oscillated case (top) to 65.3t+
in the oscillated case (bottom) shows a faster decay of the
bursting energy with spanwise wall oscillation. The burst
durationistheaveragetimeofall eventswherethesquareof
the amplitude (|a(t)|2) is more that one standard deviation
greater than the mean. This amplitude level is denoted by
thedashed line.
FIG. 14: Cross-section of coherent vorticity of (2,2,1) lift
mode. Left: (a) non-oscillated. Right: (b) oscillated. The
vortex core shifts y+=11.0 away from thewall.
and that they had higher phase velocities. Thirdly, the
study by Zhou7 is also consistent with these results, as
she found that any oscillation in the streamwise direc-
tionreduces the effectiveness of the dragreduction. Any
streamwise oscillation, even though it would still push
thestructuresawayfromthewall,wouldadverselyeffect
FIG. 12: Cross-section of coherent vorticity of (1,2,1) wall themeanflowrateprofileresultinginanadvectionspeed
mode. Left: (a) non-oscillated. Right: (b) oscillated. The of the propagating waves that is less than in the purely
vortex core shifts y+=6.8 away from thewall. spanwise oscillated case.
thewall. Theonlymodefoundnottofollowthistrendis
the (1,0,1)mode,whichundergoesa majorrestructuring
resulting in its vortex core moving towards the wall.
Thisfasteradvectionexplainstheexperimentalresults
found by K.-S. Choi42 that showed a reduction in the
durationandstrengthofsweepevents inaspanwisewall
oscillated boundary layer of 78% and 64% respectively.
For this experiment, the flowrate was kept constant, so
the energy was reduced, whereas in our case the mean
pressuregradientwaskeptconstantyieldingvirtuallyno
changein the energyof the propagatingstructures. Also
corroboratingtheseresultsistheworkbyPrabhuetal.22
that examined the KL decomposition of controlled suc- FIG. 15: Cross-section of coherent vorticity of (3,2,1) lift
tionandblowing to reducedragin achannel. They, too, mode. Left: (a) non-oscillated. Right: (b) oscillated. The
foundthatthestructureswerepushedawayfromthewall vortex core shifts y+=11.2 away from thewall.
9
FIG. 16: Cross-section of coherent vorticity of (1,1,1) asym- FIG. 20: Cross-section of coherent vorticity of (0,6,1) roll
metric mode. Left: (a) non-oscillated. Right: (b) oscillated. mode. Left: (a) non-oscillated. Right: (b) oscillated. The
The vortex core shifts y+=3.2 away from thewall. vortex core shifts y+=10.1 away from thewall.
FIG. 17: Cross-section of coherent vorticity of (2,1,1) asym- FIG. 21: Cross-section of coherent vorticity of (0,2,1) roll
metric mode. Left: (a) non-oscillated. Right: (b) oscillated. mode. Left: (a) non-oscillated. Right: (b) oscillated. The
The vortex core shifts y+ = 45.9 away from the wall with a vortex core shifts y+=7.8 away from the wall.
slight change in structure.
Thus, faster advection can be interpreted in two fash-
ions. The first is in terms of the traveling wave. The
shifting of the structures away from the wall into higher
velocity mean flow causes these structures to travel
faster, giving them less interaction time with the roll
modes. Less interaction time with the roll modes means
less energy transfer (less bursting), and due to their fast
decaying nature, their lifetime is reduced. This reduced
lifetime means they have less time to generate Reynolds
FIG. 18: Cross-section along the r-z plane of coherent vor- stress, and therefore drag is reduced. The second inter-
ticity of (1,0,1) ring mode. Top: (a) non-oscillated. Bottom:
pretation is in terms of the classically observed hairpin
(b) oscillated. The vortex core shifts y+ = 13 towards the
andhorseshoevortices.43,44 ThepushingoftheKLstruc-
wall with significant changes in structure.
tures away from the wall is equivalent to the vortices
lifting and stretching away from the wall faster. This
fasterliftingandstretchingprocessmeansthattheirlife-
time isshortened,againresultingislesstime togenerate
Reynolds stress, and therefore drag reduction occurs.
IV. CONCLUSIONS
Thisworkhasshown,throughaKarhunen-Lo`eveanal-
FIG. 19: Cross-section along the r-z plane of coherent vor- ysis,fourmajorconsequencesofspanwisewalloscillation
ticity of (2,0,1) ring mode. Top: (a) non-oscillated. Bottom: on the turbulent pipe flow structures. They are: a shift-
(b) oscillated. The vortex core shifts y+ = 7 away from the ing of rms velocities and Reynolds stress away from the
wall. wall; a reduction in the dimension of the chaotic attrac-
10
detail and structure without conditional sampling. The
TABLE IV: Comparison of the measured location of the co-
ensemble was created out of evenly spaced flowfields in
herentvortexcorefortwomodesfrom eachpropagatingsub-
class in wall units (y+) away from the pipe wall . Each class time, as opposed to conditional sampling of the flowfield
with event detection such as bursts or sweeps, and the
shiftsawayfromthewall,consistentwiththeshiftinvelocity
entire flowfield and time history was studied. Therefore,
rms and Reynolds stress. The (1,0,1) mode changes struc-
ture significantly in the oscillated case as seen in Figure 18, we argue that the overall mechanism of drag reduction
explaining the shift of its coherent vortex core towards the through spanwise wall oscillation has been found. Al-
wall. though a result of drag reduction is the decorrelation of
the low speed streaks and the streamwise vortices, as
Coherent vortex core location (wall units)
found by previous researchers, this is an incomplete de-
Mode Non-oscillated Oscillated shift
scriptionofthedynamics. Itistheliftingoftheturbulent
(1,2,1) 41.8 48.6 6.8
structuresawayfromthewallbytheStokesflowinduced
(1,5,1) 28.7 38.6 9.9
by the spanwise walloscillationthat cause the reduction
(2,2,1) 45.6 56.2 11.0 in the time and duration of Reynolds stress generating
(3,2,1) 65.9 77.1 11.2 events, resulting in drag reduction. In addition, this dy-
(1,1,1) 47.2 50.4 3.2 namical description encompasses other methods of drag
(2,1,1) 53.9 99.8 45.9 reductionsuchassuctionandblowing,activecontrol,and
ribblets,22 establishing it as a contender for a universal
(1,0,1) 31.9 18.9 -13.0
theory of drag reduction.
(2,0,1) 58.4 65.8 7.4
(0,6,1) 28.9 39.0 10.1
(0,2,1) 45.1 52.9 7.8
ACKNOWLEDGMENTS
tor describing the turbulence; a decrease in the energy
in the propagatingmodes as a whole with an increase in This research was supported in part by the National
modesthattransferenergytotheoutsideoftheloglayer; ScienceFoundationthroughTeraGridresourcesprovided
and a shifting of the propagating structures away from by the San Diego Supercomputing Center, and by Vir-
the wall into higher speed flow resulting in faster advec- ginia Tech through their Terascale Computing Facility,
tionandshorterlifespans,providinglesstimetogenerate System X. We gratefullyacknowledgemany useful inter-
Reynolds stress and therefore reducing drag. actions with Paul Fischer and for the use of his spectral
The strengthofthe KL method is that it yields global element algorithm.
∗
Electronic address: [email protected] (1998).
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