Table Of Content- 1
Streamwise-traveling waves
of spanwise wall velocity
9 for turbulent drag reduction
0
0
2 1
MAURIZIO QUADRIO ,
n PIERRE RICCO2AND CLAUDIO VIOTTI3
a
J 1Politecnico di Milano, 20156 Milano, Italy
2King’s College London,WC2R 2LS London, UnitedKingdom
8
3The Universityof North Carolina, 27516 Chapel Hill, NC, USA.
]
n (Received 8 January 2009)
y
d Waves of spanwise velocity imposed at the walls of a plane turbulent channel flow
- are studied by Direct Numerical Simulations. We consider sinusoidal waves of spanwise
u
l velocity which vary in time and are modulated in space along the streamwise direction.
f
. The phase speed may be null, positive or negative, so that the waves may be either
s
c stationary or traveling forward or backward in the direction of the mean flow. Such a
i forcing includes as particular cases two known techniques for reducing friction drag: the
s
y oscillating wall technique (a traveling wave with infinite phase speed) and the recently
h proposed steady distribution of spanwise velocity (a wave with zero phase speed).
p Thetravelingwavesalterthefrictiondragsignificantly.Waveswhichslowlytravelfor-
[
wardproducealargereductionofdrag,thatcanrelaminarizetheflowatlowvaluesofthe
1 Reynoldsnumber.Fasterwavesyieldatotallydifferentoutcome,i.e.dragincrease.Even
v faster waves produce a drag reduction effect again. Backward-traveling waves instead
6
lead to drag reduction at any speed.
2
Thetravelingwaves,whentheyreducedrag,operateinsimilarfashiontotheoscillating
0
1 wall, with an improved energetic efficiency. Drag increase is observed when the waves
. travelataspeedcomparablewiththatoftheconvectingnear-wallturbulencestructures.
1
A diagram illustrating the different flow behaviors is presented.
0
9
0
:
v
i
X 1. Introduction
r Reducingthe frictiondraginturbulentwallflowshasseenagrowinginterestinrecent
a
years,owingtotheawarenessforhighenergyconsumptionandtothe needofdecreasing
pollutant emissions into the atmosphere.Drag reduction techniques are usually grouped
(Gad-el Hak 2000) into active (requiring external energy to be fed into the system)
and passive techniques. Another useful classification is between closed-loop strategies
(requiring a controllaw with feedback) and simpler open-loop techniques: the review by
Kim (2003) discusses the former group, whereas the present work is concerned with the
latter.
Oneinterestingopen-looptechnique,whichisconceptuallysimpleandwiththepoten-
tial for a positive energetic budget, is the cyclic spanwise movement of the wall. In this
active technique, firstintroduced by Jung et al. (1992), the wallmovessinusoidally with
period T according to
w (t)=Asin(ωt), (1.1)
w
2 M. Quadrio, P. Ricco & C. Viotti
wherew isthe spanwise(z)componentofthe velocityvectoratthe wall,tis time,Ais
w
the oscillation amplitude and ω = 2π/T is the oscillation frequency. The main features
of the oscillating-walltechnique (Ricco & Quadrio 2008) are the existence of an optimal
frequency ω for dragreductionatfixedA,and asignificantmaximumdragreduction,
opt
which can be as high as 45% when A is comparable with the flow centreline velocity. At
lower A, the energy balance between the power saved thanks to the oscillation and the
power spent to activate the wall motion against the viscous stresses may be positive at
about 7% when ω =ω .
opt
In a recent work, Quadrio et al. (2007) (see also Viotti et al. 2008) have addressed
one of the drawbacks of the oscillating-wall technique, i.e. its unsteady nature. They
converted the time-dependent forcing (1.1) into its stationary counterpart of a spatial
sinusoidal distribution of spanwise velocity over a wavelength λ :
x
w (x)=Asin(κ x), (1.2)
w x
wherexisthestreamwisecoordinateandκ =2π/λ isthestreamwisewavenumber.The
x x
time dependence of Eq.(1.1) is converted into a relatively unsteady interaction between
the steady wall motion and the convecting near-wallturbulence. Such conversionis only
apparently similar to the Taylor’s frozen-turbulence hypothesis, for which the velocity
scale is the mean streamwise velocity. Indeed, the mean velocity is zero at the wall,
while the turbulence keeps a strongly convective character (Kreplin & Eckelmann 1979;
Kim & Hussain1993),withelongatedspace-timecorrelations(Quadrio & Luchini2003).
The velocity scale for space-time conversionis therefore the convectionvelocity of the
U
turbulent fluctuations, which almost coincides with the mean velocity along with the
wall-normalspan of the channel, except for a thin near-walllayer,say y+ <10,where it
is constant at + 10. (Hereinafter, the + superscript indicates scaling by the friction
Uw ≈
velocity u of the reference flow and the kinematic viscosity ν of the fluid.) Viotti et al.
τ
(2008) have shown that the optimal frequency ω for (1.1) translates into an optimal
opt
wavenumber κ for (1.2) through the relation
x,opt
ω
opt
κ = . (1.3)
x,opt
w
U
The present paper addresses the extension of (1.1) and (1.2) to the space-time case
w (x,t)=Asin(κ x ωt), (1.4)
w x
−
where κ and ω may be both different from zero at the same time. In (1.4), the wave
x
of spanwise velocity moves (backward or forward) in the streamwise direction, and its
phase speed is
ω
c= . (1.5)
κ
x
The temporal (1.1) and steady (1.2) oscillations correspondrespectively to a wave trav-
eling at infinite speed (κ =0), and to a stationary wave (ω =0). The effect of the wall
x
motion is already known on the axes of the ω κ plane, while it is unexplored in the
x
−
rest of the plane. The main question addressed in this paper is therefore: what is the
response of the turbulent flow in the whole ω κ plane?
x
−
Beforepresentingourresults,itmaybeusefultomentionthedragreductiontechnique
by Du & Karniadakis(2000) and Du et al. (2002). They studied the following spanwise-
oriented volume forcing:
f (z,t)=F e−y/∆sin(κ z ωt), (1.6)
z z
−
where ∆ is the distance up to which the forcing diffuses from the wall and κ is the
z
Streamwise-traveling waves for drag reduction 3
spanwise wavenumber. In a turbulent channel flow at Re = 150, they obtained about
τ
30% drag reduction and observed a substantial disruption of the near-wall flow struc-
tures.The volume forcein Eq.(1.6)consists ofspanwise-traveling wavesacting alongthe
spanwisedirection,whilehereweconsiderawall-basedmotionofspanwisevelocitywaves
traveling along the streamwise direction.
In the presentpaper,the effects ofimposing condition(1.4) areinvestigatedby Direct
NumericalSimulations(DNS).Whilepurelyspatialandpurelytemporaloscillationshave
beenshownbyViotti et al.(2008)tobelargelyanalogoustoeachother,theflowresponse
to the travelingwavesis found to be interestingly complex when examinedas a function
ofω andκ .Bothdragreduction(DR)anddragincrease(DI)mayoccur.Themaximum
x
DR resides in a non-obvious region of the ω κ plane, which largely corresponds to
x
−
where the global energy balance is positive, whereas the maximum DI occurs where the
phase speed c is comparable with the convection velocity .
w
U
The structure of the paper is as follows. In 2, the numerical procedures are outlined.
§
Section 3presentsthedragmodificationandtheglobalenergybudgetintheω κ plane,
x
§ −
anddiscussestheeffectofforcingamplitudeandReynoldsnumberonthemaximumDR.
Selectedflow statisticsarealsodescribed.In 4, the analogybetweenthe oscillatingwall
§
and the traveling waves is discussed, and a diagram offers a comprehensive view on the
modified turbulent flow. A brief summary is given in 5.
§
2. Method
A large number of DNS has been carried out to examine the response of a turbulent
channel flow to the wall forcing (1.4). The traveling waves are applied at both walls
and move in phase.From a numerical perspective,Eq.(1.4) is a non-homogeneous,time-
dependent wall boundary condition for the spanwise component of the velocity vector.
The size of the computationaldomain, the requirements in terms of the spatio-temporal
resolution and the length of the integration interval render these DNS very demanding.
TheefficiencyoftheparallelDNScodeandtheavailabilityofappropriatecomputational
resources are key aspects of this researcheffort.
ThecodeandthearchitectureofthecomputingsystemhavebeendescribedbyLuchini & Quadrio
(2006). The code is a mixed-discretization parallel solver of the incompressible Navier–
Stokes equations, based on Fourier expansions in the homogeneous directions and high-
order,explicitcompactfinite-differenceschemesinthewall-normaldirection.Itsprimary
design target is to achieve high performance while using off-the-shelf hardware. Most of
thecalculationshavebeenrunonacomputingsystemavailableindedicatedmodeatthe
Universit`a di Salerno. The system is composed of 268 AMD Opteron CPUs, grouped in
134 computing nodes, and connected with an ad hoc topology throughGigabit Ethernet
cards.EachnodeisequippedwithitsownharddiskandRAM.Thesystempossesses0.28
TB of RAM and 40 TB of storage space, with a peak computing power of 2.6 TFlop/s.
The computations are integrated forward in time, starting from the same initial con-
ditionofafully developedchannelflowwithout waves.Theflow rateisconstantandthe
bulk velocity is U = 2/3U , where U is the centreline velocity of a laminar Poiseuille
b P P
flowwiththesameflowrate.About250simulationshavebeenrunwithawaveamplitude
A=0.5U (which corresponds to A+ =12) at a Reynolds number Re=U h/ν =4760
P P
(whichcorrespondstoRe =hu /ν =200basedonu andthechannelhalfwidthh).A
τ τ τ
few cases have also been run to explore how the maximum DR is sensitive to changes in
A or Re: A/U has been varied in the range 0.085 1.26, and two values of Re=2175
P
−
and Re = 10500 (corresponding to Re = 100 and Re = 400, respectively) have been
τ τ
considered.
4 M. Quadrio, P. Ricco & C. Viotti
Lz
w=Asin(κxx ωt)
−
c
Ly =2h
y
x
λx
z
Lx
Mean streamwise flow
Figure 1. Schematic of the system for turbulent channel flow with wall traveling waves. λx
is the streamwise wavelength and c is the phase speed of the waves (traveling forward in this
sketch). Lx, Ly and Lz are the dimensions of the computational domain in the streamwise,
wall-normal and spanwise directions, respectively.
Thecomputationaldomain,aschematicofwhichispresentedinfigure1,hasastream-
wise length of L = 6πh and a spanwise width of L = 3πh. At Re = 4760, a Fourier
x z
expansion of 320 320 modes is used in the streamwise and spanwise directions, and
160 points discret×ize the wall-normal direction, for a total of 3.3 107 spatial degrees of
freedom. The streamwise and spanwise resolutions are ∆x+ =11.·8 and ∆z+ =5.9, and
the wall-normal resolution varies from ∆y+ = 0.7 at the wall to ∆y+ = 4.1 at the
min max
channel centreline. The time step is ∆t+ = 0.16, and the total integration time is 1000
h/U (i.e. 8400 viscous time units). A few cases (first row of data at κ h = 0.167 in
P x
figure 2) have L =12πh, and use 640 streamwise Fourier modes to keep the resolution
x
unchanged. A few other cases (with extremely large oscillation periods) have an averag-
ingtime of2000h/U .Forthe simulationsatdifferentRe,thesizeofthe computational
P
domain has been kept constant in outer units, and the spatial resolution has been kept
constantinviscousunits.The effectofthe discretizationparametersonthe frictiondrag
has been assessed by preliminary checks. In agreement with Quadrio & Ricco (2004),
the accuracy in measuring the friction coefficient is better than 1%, and can be further
improved by extending the total averagingtime.
Themainobjectiveistoinvestigatehowtheforcing(1.4)altersthefrictioncoefficient,
defined as
2τ
w
C = ,
f ρU2
b
where τ is the mean wall-shear stress and ρ is the density of the fluid. Changes of
w
Cf are quantified in terms of percentage of Cf,0, the friction coefficient of the reference
turbulent flow, and coincide with the percentage saving of the energy required to drive
Streamwise-traveling waves for drag reduction 5
the fluid along the x direction at fixed flow rate. For the reference case at Re = 4760,
Cf,0 = 7.94 10−3, which is in very good agreement with the correlation reported by
·
Pope (2000) at page 279. As in Quadrio & Ricco (2004), C is evaluated by space- and
f
time-averaging,after discarding the initial transientduring which the flow adapts to the
new regime. The length of the transient is estimated by direct observation of the time
historyofthespace-averagedfriction,andistypicallyoftheorderof100-200h/U .Unless
p
otherwise indicated, quantities reported hereinafter are scaled by h and U .
P
3. Results
3.1. Effect on friction drag
Figure 2 describes the effects ofthe travelingwavesin terms ofthe percentagechangein
frictiondragasafunctionofω andκ forA=0.5andRe=4760.Onlytheupperhalfof
x
the ω κ plane is shown,since (1.4)yields symmetric results uponexchangingthe pair
x
−
ω,κ with ω, κ .(This hasbeenexplicitly verifiedfora fewpoints.)The phasespeed
x x
− −
c is interpreted graphically as the inverse of the slope of straight lines passing through
the origin. In the first quadrant, the waves move forward in the direction of the mean
flow(c>0),whilethesecondquadrantcorrespondstobackward-travelingwaves(c<0).
Eachpointinfigure2istheresultofonesimulation.Thecontoursassistinvisualizing
the general behaviour of the changes in drag; they are computed after a linear interpo-
lation of the irregularly scattered points on a regular mesh with grid sizes of ∆ω =0.05
and ∆κ = 0.1. The presence of the numerical values, rounded to the nearest integer,
x
allowsverifyingthatthecontoursdonotmisrepresentthedata.Severalpointswithlarge
ω and/or κ do not appear, but they are however useful to obtain the correct trend of
x
the contours near the boundaries.
On the horizontal ω axis, Eq.(1.4) reduces to the oscillating-wall case (1.1): our data
agreewiththosebyQuadrio & Ricco(2004),whousedthesameReandslightlydifferent
discretization parameters. The symmetry mentioned above requires points at ω and ω
−
to have the same friction drag: this is enforced by plotting every data point twice, at
+ω and ω. On the vertical κ axis, Eq.(1.4) reduces to the steady wave case (1.2).
x
−
Some ofthese pointsaretakenfromViotti et al. (2008),whoemployedthe sameReand
discretizationparameters.Asdiscussedthere,DRbehavesverysimilarlyonthetwoaxes,
once frequency is converted into wavenumber through Eq.(1.3), reaching its maximum
at ω =ω 0.5, and at κ =κ 1, with ω and κ related by =0.5.
opt x x,opt opt x,opt w
≈ ≈ U
Off the axes, the flow response reveals an unexpectedly rich behaviour. The contours
showthatlocalmaximaandlocalminima formtwoelongatednarrowregions.Inthe red
region, whose crest identifies a curve which does not cross the origin, the flow presents
an intense drag reduction. The maximum is about 48%, which is higher than 34% for
the oscillating wall at ω and 45% for the steady wave at κ . A wide range of κ
opt x,opt x
exists where DR is 47%–48% for a non-zero phase speed. The crest crosses the vertical
axis at κ = κ and intersects the ω axis at ω = ω . For κ > κ , this region
x x,opt opt x x,opt
−
largely corresponds to slow forward-traveling waves (c 0.15), while for κ < κ it
x x,opt
≈
relates to backward-traveling waves. In the blue region, the drag increases significantly,
upto23%.Themaximumincreasecorrespondstoastraightlinecrossingthe origin,and
is therefore associated with a constant phase speed, c = 0.5. The narrow, cone-shaped
DIregionisconfinedbetweenthetwothickzero-DRlines,correspondingtoc 0.35and
≈
c 0.6.The markedasymmetry between the first and secondquadrantvanishes at high
≈
ω : the contour lines tend to become vertical, suggesting that in this limit the effect of
| |
anRF 6
tmegae=igu 5
easuredtivevalu4760.CMre2. 36 41 43 45 45 46 44 5 -20-23-23-22-17-10 -2
pointsesareontouapof 20 23 8
.represenrsarespfriction 4 15 38 41 44 46 45 36 6 -15 -18
tedbaceddrag 38 46 -16 -21 4 M.
ydaby5red 31 42 45 47 -20 Qu
s%u 24 45 13 a
h c d
edlinesintervation(p 3 40 46 -15 -18 2 rio,P.
.Thenumls,lociofercentage κx 15 45 47 47 3431 -8 -16 --127 8 17 15 Ricco&
bze) C
erroin 2 18 21 29 35 43 45 46 46 32 -7 -14 3 .
sDt V
indicatRare−ωhe 44464488 34 -14 21 16 iotti
epercentageindicatedbyκplanefox 1 21 3301 3343 37 4410 44554456474379341018103-83-61-9 --98-17-114019 13 26 24 24 16
dtr
raghickA= 33 36 404242423614 1 -7 1 24 28 20
redlin0
uces.5 32 36 373837 36 26 1 -8 -1 19 29 29 24 16
tioanan 34 36 35 33 22 5 -9 4 27 32
ndd
0 16 18 22 27 3234 333433333332312721 55300-3---6-779---9-776-300355 21273132333433 3432 27 22 18 16
-3 -2 -1 0 1 2 3
ω
Streamwise-traveling waves for drag reduction 7
10
5
)
%
(
R
D 0
-5
-10
0 100 200 300 400 500 600 700
T
Figure 3. Friction drag reduction (percentage) as a function of T = 2π/ω for A = 0.5,
Re = 4760 and κx = 0. The leftmost points are at T = 1 and T = 1.4, and points with
2<T <50 are out of scale.
the streamwise modulation becomes negligible and ω alone dictates the drag reduction
level.
Careful observation of the region near the origin of the ω κ plane reveals a finite
x
−
DI for the oscillating wall at very small frequencies. This is shown in figure 3, where
T = 2π/ω is used as independent variable. Nikitin (2000) was the first to surmise the
existence of what he calledan infrared DI, based on an under-resolvedDNS at T >120.
Quadrio & Ricco (2004) tested the oscillating-wall technique in a plane channel up to
T 90 and were unable to confirm the infrared DI (figure 3 indeed confirms that no
≈
drag increase is observed for T < 100). Owing to the computational cost, we did not
explore this issue further.
3.2. Power budget
Theefficiencyofthetravelingwavesinaffectingthefrictiondragisstudiedbycomputing
the amount of external energy spent to enforce the control action. The quantification of
this energyobviouslyimplies the knowledgeof the actuationdevice.However,the power
that an ideal device dissipates against the viscous stresses provides an upper bound for
theperformanceofthecontrolstrategy.Weexpressthispoweraspercentageofthepower
spent to drive the fluid along the streamwise direction in the fixed-wall configuration:
100 1 Lx Lz T ∂w
P (%)= w dtdzdx,
sp (dU/dy|0)UbLxLzT Z0 Z0 Z0 w∂y(cid:12)(cid:12)y=0
(cid:12)
wheredU/dy 0isthegradientofthemeanstreamwisevelocityat(cid:12)thewallinthereference
|
case.P (%)isreportedinfigure4asafunctionofωandκ .Muchlessenergyisrequired
sp x
to imposea standingwavethanto generateuniformoscillations,ifcasesrelatedthrough
Eq.(1.3) are compared. However, neither the oscillating wall nor the standing waves are
associated with the pair which offers the minimum energetic expense of P (%) = 23.5,
sp
i.e. ω 0.15,κ 1.
x
≈ ≈
8 M. Quadrio, P. Ricco & C. Viotti
5
71 494443465567 116 142146150156162167172
153 136 177
4 188 67 50 42 42 53 67 105 129 148
65 40 125 132 157
95 49 3839 129
134 40 179
3 60 37 114 124 141
186 41 102 116 137 171
x
κ 36
42 33 43 108 121 152
2 164 149 109 84 5142343139 88 97 116
172
48403331 35 91 143
107 95 64
44373029 4858 75 87 103 120
1 147 106 93 78 61 40342924293439465462697882879296 113 133 173
96 82 5951433527313751 70 89 108 155
97 80 6862544231313652 67 81 110 135 175
87 72625138312847 68 85
0 177 166 148 125 93888708777212665156493333399836633153231130030301233533366899495666157178808893 125 148 166 177
-3 -2 -1 0 1 2 3
ω
Figure4.MapofthepowerPsp(%)spentforthetravelingwavesintheω κxplaneforA=0.5
−
and Re = 4760. Contours are spaced by 10% intervals and the minimum is 30%. Here and in
the following figure, the percentage is relative to the reference friction power required to drive
the flow along the streamwise direction. Thenumbersindicate Psp(%) at measured points.
Even more important is to quantify the net energy saving, defined as the difference
between the percentage power saved to drive the fluid along the streamwise direction
(which coincides with the drag reduction at constant U ) and the percentage power
b
spent defined above, namely
P (%)=DR(%) P (%).
net sp
−
For several active open-loop techniques, P (%) is negative. Some methods, like the
net
oscillating wall (Baron & Quadrio 1996), may offer a positive net budget of the order
of a few percent. The traveling waves, on the other hand, guarantee much larger ben-
efits because of the remarkable occurrence that the region of minimum P (%) largely
sp
coincides with the region of maximum DR(%). Figure 5 shows that P (%) is positive
net
mainly for slow forward-traveling waves, and that a 18% of maximum net is measured
for ω 0.15,κ 1 (which is also the optimum pair for minimum P ). This benefit
x sp
≈ ≈
is noteworthy when compared with the negative net balance of the oscillating wall at
this value of A, and the small net gain of 5% obtained by the steady waves. It must be
recalled that we have discussed a simple estimate, which cannot account for the nature
of the realactuationsystem. However,sucha high positive net power budget might well
accommodate a real system with efficiency less than unity.
Streamwise-traveling waves for drag reduction 9
5
-35 -8 -2 1 -1 -9-23 -111 -162-168-173-177-179-177-174
-134 -113 -169
4 -173 -29 -9 2 4 -8 -31 -99 -144 -165
-26 6 -140 -153 -153
-64 -7 7 7 -148
-110 5 -166
3 -20 9 -129 -142 -139
-171 0 -110 -133 -129 -156
x
κ 11
3 14 -9 -124 -123 -135
2 -146 -128 -80 -49 -8 2 1215 -7 -95 -112 -112
-157
-451156 -2 -105 -122
-77 -62 -25
191712 -39-57 -84 -97 -90 -95
1 -125 -75 -59 -41 -20 61118161-16-29-43-57-68-78-86-83-80-7-877 -87 -109 -157
-63 -46 -19-8-179-16-37-58 -69 -65 -80 -136
-65 -44 -31-23-16-6 -5-30-44-53 -48 -51 -80 -111 -159
-52 -36-27-18-16-27-37-43 -40 -52
0 -161 -148 -126 -98 -60-54-4-474---333988-3-331-29-28---333--4433-53-56--3-7313--8-83-71933-9783--833-7--36533544-28-29-3-133-39-4-447-54-60 -98 -126 -148 -161
-3 -2 -1 0 1 2 3
ω
Figure 5. Map of net power saving Pnet(%) in the ω κx plane for A = 0.5 and Re = 4760.
−
Contours are spaced by 10% intervals. Locus of zero Pnet(%) is indicated by thick curve, solid
linesdenotepositivebalanceandnegativevaluesarerepresentedbydashedlines.Thenumbers
indicate Pnet(%) at measured points.
3.3. Effect of A and Re on maximum drag reduction
The sensitiveness of the maximum DR to the parameters A and Re, kept fixed in the
mainparametricstudy,isaddressedthroughafewadditionalsimulations.Figure6(left)
showsthatDRincreasesmonotonicallywithAandreachesnearly60%whenAisslightly
larger than the flow centreline velocity. In the plot on the right, P grows significantly
net
at lower amplitude, which is expected as the power spent to impose the traveling waves
decreasesquadraticallywithA,andislargerthan26%forA=0.25.Bothresultsarecon-
sistent with (but quantitatively much better than) the oscillating-wall scenario, studied
by Quadrio & Ricco (2004).
The maximum DR at Re = 4760 depends weakly on the Reynolds number. We have
observed full relaminarization of the flow when Re = 2175 (Re = 100), and a slight
τ
decrease from 48% to 42% as Re increases to 10500 (Re = 400). This finding is con-
τ
sistent with experimental (Choi & Graham 1998; Ricco & Wu 2004) and DNS results
(Ricco & Quadrio2008) availablefor the oscillating-wallcase.Followingtheseworks,we
haveassumedthatthe valuesofω andκ thatguaranteethe maximumDRscaleinwall
x
units of the reference flow.
3.4. Turbulence statistics
Inthissection,westudy theturbulencestatisticstounderstandhowthe flowismodified
bythetravelingwaves.Wefirsttakeapreliminarylookatinstantaneoussnapshotsofthe
10 M. Quadrio, P. Ricco & C. Viotti
60
30
50
20
%) 40 %) 10
(R (et
D 30 Pn 0
-10
20
-20
10
-30
0
0 0.5 1 1.5 0 0.5 1 1.5
A A
Figure 6.Percentage dragreduction(left) and netpowersaving(right)asfunctionsof forcing
amplitudeA.Theforcingconditionsareω=0.16,κx =1.66,whichproducethemaximumdrag
reduction at A=0.5 and Re=4760.
flowfieldinfigure7.Twodifferentcasesarecomparedtoasnapshotofthereferenceflow,
showninthetopplot:onewithω =0.16,κ =1.66andtheotheratω =0.88,κ =1.66.
x x
These two cases do not lie far from each other in the ω κ plane, but yield totally
x
−
different outcomes, i.e. a large DR of 48% for the former (middle plot) and a large DI
of 14% for the latter (bottom plot). Both cases refer to forward-traveling waves, with
c=0.1 and c=0.53, respectively.
Isosurfacesfornegativevaluesofthequantityq =sgn(u)√u2+w2 areshowninfigure
7 to visualize the low-speed streaks of the near-wall flow even in the present case where
the localstreamwisedirectionmay deviate fromthe x axis.The value q = 0.168(q+ =
−
4) is chosen for the reference flow, that indeed shows the usual pattern of near-wall
−
elongated,streamwise-alignedregions.IntheplotsfortheDRandDIcases,thevalueofq
ischangedsoastoalwaysyieldq+ = 4whencomputedwiththeactualfrictionvelocity
−
of the flow. This compensates for the largely different values of Re : had the nominal
τ
value of u been used, the streaky structures would have disappeared in the DR case.
τ
When DR takes place,what remains after accounting for the reducedRe are elongated
τ
structures, aligned with the x direction, of perhaps larger size and reduced streamwise
meandering. In sharp contrast, in the DI case the same structures present an evident
streamwise modulation, with a streamwise wavelength corresponding to the wavelength
of the wall wave, which is a marked structural difference with the reference flow. In this
casethemostevidentactionoftheforcingistocyclicallychangethedirectionoftheflow,
bytiltingthelow-speedstreaksinaccordancewiththelocalspanwisevelocitycomponent
at the wall.
Infigure8,theprofilesofthemeanstreamwisevelocitycomponent,averagedovertime,
x andz,arepresentedasfunctions ofthe wall-normalcoordinate.Theyareindicated by
the solid line (reference case), filled (DR) and empty (DI) circles. The effect of the wall
motion diffuses all along the extent of the channel and the large near-wallchange of the
velocity gradient is evident when DR occurs.
