DTIC ADA575012: Experimental Support for Townsend's Reynolds Number Similarity Hypothesis on Rough Walls

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by  Defense Technical Information Center

PHYSICSOFFLUIDS17,035102s2005d
Experimental support for Townsend’s Reynolds number similarity
hypothesis on rough walls
Karen A. Flack, Michael P. Schultz, and Thomas A. Shapiro
MechanicalEngineeringDepartment,UnitedStatesNavalAcademy,Annapolis,Maryland21402
sReceived 23 July 2004; accepted 30 October 2004; published online 28 January 2005d
The Reynolds number similarity hypothesis of Townsend fThe Structure of Turbulent Shear Flow
sCambridge University Press, Cambridge, UK, 1976dg states that the turbulence beyond a few
roughnessheightsfromthewallisindependentofthesurfacecondition.Theunderlyingassumption
isthattheboundarylayerthicknessdislargecomparedtotheroughnessheightk.Thishypothesis
was tested experimentally on two types of three-dimensional rough surfaces. Boundary layer
measurements were made on flat plates covered with sand grain and woven mesh roughness in a
closed return water tunnel at a momentum thickness Reynolds number Reu of ,14000. The
boundarylayersontheroughwallswereinthefullyroughflowregimesk+ø100d withtheratioof
s
the boundary layer thickness to the equivalent sand roughness height d/k greater than 40. The
s
results show that the mean velocity profiles for rough and smooth walls collapse well in velocity
defectformintheoverlapandouterregionsoftheboundarylayer.TheReynoldsstressesforthetwo
rough surfaces agree well throughout most of the boundary layer and collapse with smooth wall
results outside of 3k . Higher moment turbulence statistics and quadrant analysis also indicate the
s
differences in the rough wall boundary layers are confined to y,5k . The present results provide
s
support for Townsend’s Reynolds number similarity hypothesis for uniform three-dimensional
roughness in flows where d/k ø40. fDOI:10.1063/1.1843135g
s
I. INTRODUCTION the roughness Reynolds number. The mean velocity profile
in the overlap and outer regions of a turbulent boundary
The rough wall boundary layer is of great engineering
layer over a rough wall can, therefore, be stated as
interest, including the applications of flow over ship hulls S D
and airplanes, as well as fluid transport. Current engineering 1 P y
U+= lnsy+d+C−DU++ W . s2d
modelstreatroughnessasasmallperturbationtothesmooth k k d
wall boundary layer.1 However, if the effect of surface
By evaluating Eqs. s1d and s2d at y=d, Hama11 showed that
roughness propagates into the outer layer, the application of
DU+canbefoundasthedifferencebetweenthesmoothwall
roughwallmodelsthatrelyonsmoothwallsimilarityscaling
log law intercept C and the rough wall intercept C−DU+ at
will yield erroneous results. Understanding the extent of this
perturbation for a variety or roughness types would improve thesameReSd˛*.TheDrougSh˛nessfDunctiSoniDs,theSrefoDre,foundas
modeling and predictive capabilities. 2 2 U U
The engineering importance of predicting the effect of DU+= − = e − e . s3d
wall roughness on boundary layer and pipe flows has long Cf S Cf R Ut S Ut R
been identified. Much of the seminal work in this area was The implicit assumption in Eq. s3d is that the mean flow for
carried out by Nikuradse2 and Colebrook and White.3 Later both smooth S and rough R walls obeys the velocity defect
investigations focused on the effect of roughness on turbu- law in the overlap and outer regions of the boundary layer:
S D
lencestructure.4–6Anextensivereviewofthepresentknowl-
y
edge of rough wall boundary layers is given by Raupach et U+−U+=f . s4d
e d
al.7 and Jiménez.8
The mean velocity profile in the overlap and outer re- There is a great deal of experimental support for a universal
gions of a turbulent boundary layer over a smooth wall is velocitydefectlaw,7,11–14anditisconsistentwiththeideaof
given as S D ReynoldsnumbersimilaritygivenbyTownsend15thatturbu-
1 P y lence outside the inner layer is unaffected by surface condi-
U+= lnsy+d+C+ W , s1d tion. Some prominent rough wall studies claim that changes
k k d
to the mean flow do occur in the outer layer.16,17 This is
wherethevonKarmanconstantk=0.41,thesmoothwalllog primarilyobservedasanincreaseinthestrengthofthewake
law intercept C=5.0, and P is the wake parameter. Both P, which the researchers assert is due to the higher rate of
Clauser9 and Hama10 pointed out that the primary effect of entrainment for rough walls.
surface roughness on the mean flow is to cause a downward Thereisalsoalargedegreeofsimilarityintheturbulent
shift in this profile. For “k-type” rough walls, the downward stresses for rough and smooth walls. The Reynolds number
shift DU+ called the roughness function, correlates with k+, similarity hypothesis of Townsend15 and subsequent exten-
1070-6631/2005/17~3!/035102/9/$22.50 17,035102-1
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Experimental support for Townsend’s Reynolds number similarity
5b. GRANT NUMBER
hypothesis on rough walls
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United States Naval Academy,Department of Mechanical REPORT NUMBER
Engineering,Annapolis,MD,21402
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035102-2 Flack,Schultz,andShapiro Phys.Fluids17,035102~2005!
sions by Perry and Chong18 and Raupach et al.7 states that and d/k were <30. The use of k to define the extent of the
s
the turbulent motions are independent of surface condition roughness sublayer instead of k itself was proposed by
outside the roughness sublayer at sufficiently high Reynolds Schultz and Flack13 because it provides a common measure
number. This implies that the turbulent stresses, normalized of the influence of the roughness on the mean flow. The
by the wall shear stress, are universal outside of the rough- difficulty with using k for predicting roughness effects is
s
ness sublayer. The dominant view at present is that the thatitcannotbedeterminedapriorifromaphysicalmeasure
roughness sublayer extends <3k–5k from the wall, where k of a generic roughness. The authors believe that d/k is a
s
is the roughness height.The outer region has been generally more appropriate parameter than d/k in identifying the rela-
thought to be unaffected by the roughness.7,8 This is sup- tive strength and extent of the roughness influence on the
ported by a number of studies, including the work of Perry turbulence in the boundary layer.
andLi19onexpandedmeshsurfacesandAcharyaetal.20for In the present investigation, the flow over two three-
machined surfaces. Andreopoulos and Bradshaw21 present dimensionalroughsurfaces,sandgrainandwovenmesh,will
both mean and turbulence profiles for flow over one sandpa- be presented. These roughness types were selected because
per surface. Reynolds stress results showed good collapse bothsurfaceshavebeenpreviouslystudied12,16withcontrast-
outsidetheroughnesssublayer.However,effectsinthetriple ing results related to the extent of roughness effects in the
products were noted up to 10k from the wall. Ligrani and boundary layer. The height of the roughness was chosen to
Moffat22 presented mean and turbulence results for closely produce fully rough turbulent boundary layers while main-
packed spheres in the transitionally rough regime. For this taining conditions in which the boundary layer thickness is
type of roughness element, the turbulence quantities col- large compared to the equivalent sand roughness height
lapsed with smooth wall results outside of the roughness sd/ksø40d. The mean velocity, Reynolds stress profiles,
sublayer. Schultz and Flack13 examined fully rough flow higher order moments, as well as quadrant analysis for flow
overcloselypackedsphereswithandwithouttheadditionof over these surfaces will be compared to smooth walls. Re-
a secondary roughness scale. Quadrant analysis and the ve- sults will focus on the extent that the roughness effects pen-
locity triple products indicated that the changes in the turbu- etrate the boundary layer.
lence structure in the rough wall boundary layers were con-
fined to y,8k . Schultz and Flack12 also tested two sand
s II. EXPERIMENTALFACILITIESAND METHODS
grain surfaces of varying roughness heights. Their results
indicate collapse of the mean velocity in defect form and The present experiments were conducted in the closed
collapse of the normalized Reynolds stresses profiles for circuitwatertunnelfacilityattheUnitedStatesNavalAcad-
both the smooth and rough surfaces in the overlap and outer emy Hydromechanics Laboratory. The test section is 40 cm
regions of the boundary layer. 340 cm in cross section and is 1.8 m in length, with a tun-
Other researchers have observed roughness effects well nel velocity range of 0–6.0 m/s. The current tests were run
into the outer layer. The studies of Krogstad et al.16 for wo- at a tunnel speed of ,3.8 m/s sRe =5.13106d. Further de-
x
ven mesh roughness and Keirsbulck et al.17 for transverse tails of the water tunnel facility are given in Schultz and
bar roughness indicate that the wake strength of the mean Flack.12,13 Three surfaces are tested in this study. One is a
velocity profile is increased. Additionally, Krogstad and smooth cast acrylic surface. The other two are rough sur-
Antonia23notedthattheturbulencestructureisalteredinthe faces; one covered with 80 grit wet/dry sandpaper and the
outer layer for a woven mesh roughness. Antonia and other with woven mesh scenter line spacing=1.0 mm, wire
Krogstad14 also observed changes in the Reynolds shear diameter=0.16 mmd. The maximum peak to trough rough-
stress profiles for surfaces covered with transverse rods. ness heights k are 0.69 mm and 0.32 mm, for the sandpaper
These studies imply the effect of surface roughness may and mesh surfaces, respectively. The test specimens were
propagate well into the outer region. inserted into a flat plate test fixture mounted horizontally in
An explanation for the disparate findings is most likely the tunnel. The test fixture is the same as that used by
due to the “strong” roughness used in the investigations Schultz and Flack.12,13 The forward most 200 mm of the
where effects were observed in the outer layer. A strong plateiscoveredwith36 gritsandpapertotripthedeveloping
roughnesshereisdefinedasasurfacewhoseequivalentsand boundary layer. The use of a strip of roughness was shown
roughness height2,24 k is a significant portion of the inner byKlebanoffandDiehl25toprovideeffectiveboundarylayer
s
layerthickness.13Thisviewissupportedintherecentreview thickeningandafairlyrapidreturntoself-similarity.Thetest
article of Jiménez8 which asserts that surfaces where d/k specimen mounts flush into the test fixture and its forward
ł40mayexhibitroughnesseffectswellintotheouterlayer, edgeislocatedimmediatelydownstreamofthetrip.Theflap
since most of the log-law region may be destroyed by the was set at small angles to make minor adjustments to the
presence of the roughness. The size of the woven mesh, upstream streamwise pressure gradient. A schematic of the
traverse rods, and tranverse bars used in the studies of test water tunnel test fixture is shown in Fig. 1.
Krogstad et al.,16Antonia and Krogstad,14 and Keirsbulck et The boundary layer profiles presented here were taken
al.17 correspond to d/k <15, d/k <8, and d/k <7, respec- 1.35 m downstream of the leading edge of the test fixture.
s s s
tively, while the values of d/k in these investigations were Profilestakenfrom0.80 mtothemeasurementlocationcon-
<48, <46, and <26, respectively. In the study of Schultz firmed that the flow had reached self-similarity. The trailing
andFlack13onuniformpackedspheres,inwhichdifferences 150 mm of the flat plate fixture is a movable tail flap. This
in the turbulence structure were observed out to 8k , d/k , was set with the trailing edge up at ,4° in the present ex-
s s
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035102-3 ExperimentalsupportforTownsend’sReynoldsnumber Phys.Fluids17,035102~2005!
bias, and velocity gradient bias, as detailed by Edwards.28
Fringe bias results from the inability to sample scattering
particles passing through the measurement volume at large
anglessinceseveralfringecrossingsareneededtovalidatea
measurement.Inthisexperiment,thefringebiaswasconsid-
ered insignificant, as the beams were shifted well above a
burst frequency representative of twice the freestream
velocity.28 Validation bias results from filtering too near the
signal frequency and any processor biases. In general these
errors are difficult to estimate and vary from system to sys-
FIG.1. Schematicofflatplatetestfixture.
tem. No corrections were made to account for validation
bias.Velocitybiasresultsfromthegreaterlikelihoodofhigh
velocity particles moving through the measurement volume
periments to prevent separation at the leading edge of the
during a given sampling period. The present measurements
plate. The physical growth of the boundary layer and the
were burst transit time weighted to correct for velocity bias,
inclined tail flap created a slightly favorable pressure gradi-
asgivenbyBuchhaveetal.29Velocitygradientbiasisdueto
ent at the measurement location. The acceleration parameter
K was ,2.0310−8 and did not vary significantly between velocity variation across the measurement volume. The cor-
rection scheme of Durst et al.30 was used to correct u8. The
the test surfaces.
corrections to the mean velocity and the other turbulence
VelocitymeasurementsweremadeusingaTSIFSA3500
quantities were quite small and therefore neglected. An ad-
two-component, fiber-optic laser Doppler velocimeter
sLDVd. The LDV used a four beam arrangement and was ditional bias error in the v8 measurements of ,2% was
caused by introduction of the w8 component due to inclina-
operated in backscatter mode. The probe volume diameter
was ,90 mm, and its length was ,1.3 mm. The viscous tion of the LDV probe. Bias estimates were combined with
length scale n/utvaried from 5 mm for the rough walls to the precision uncertainties to obtain the overall uncertainties
7 mm for the smooth wall. The diameter of the probe vol- for the measured quantities at 95% confidence. These were
calculated using the standard methods outlined in Coleman
ume, therefore, ranged from 13 to 18 viscous lengths in the
and Steele.31 The resulting overall uncertainty in the mean
present study. The LDV probe was mounted on a Velmex
velocity is ±1%. For the turbulence quantities, u82, v82, and
three-axis traverse unit.The traverse allowed the position of
the probe to be maintained to ±10 mm in all directions. In u8v8 the overall uncertainties are ±2%, ±4%, and ±7%, re-
order to facilitate two-component, near wall measurements, spectively. The uncertainty in utfor the smooth walls using
the probe was tilted downwards at an angle of 4° to the the Clauser chart method is ±3%, and the uncertainty in ut
for the rough walls using the modified Clauser chart method
horizontal and was rotated 45° about its axis. Velocity mea-
surements were conducted in coincidence mode with 20000 was±5%.Theuncertaintyinutusingthetotalstressmethod
is ±6% for both the smooth and rough walls.The uncertain-
random samples per location. Doppler bursts for the two
channels were required to fall within a 50 ms coincidence ties in the boundary layer thickness d, displacement thick-
ness d*, and momentum thickness uare ±7%, ±4%, and
window or the sample was rejected.
±5%, respectively.
In this study, the friction velocity utfor the smooth sur-
face was found using the Clauser chart method with log-law
constants k=0.41 and B=5.0. For the rough walls, utwas IV. RESULTSAND DISCUSSION
obtained using a procedure based on the modified Clauser
A. Mean flow
chart method given by Perry and Li.19 Further details of the
procedure are given in Refs. 12 and 26. For all the test sur- The experimental conditions for each of the test cases
faces,thetotalstressmethodwasalsousedtoverifyut.This arepresentedinTableI.Significantincreasesinthephysical
method assumes a constant stress region equal to the wall growth of the boundary layer were noted on both rough sur-
shear stress exists in the overlap and inner layer of the faces compared to the smooth wall. The sandpaper showed
boundary layer. If the viscous and turbulent stress contribu- increasesof21%,58%,and44%,whilethemeshhadsimilar
tions are added together, an expression for utmay be calcu- increases of 18%, 57%, and 45% in d, d*, and u, respec-
lated as the following evaluated at the total stress plateau in tively.Asstatedpreviously,utwasdeterminedusingboththe
the overlap and inner layer: totalstressandClauserchartmethods.Theseresultsarealso
F G
given in Table I. The agreement between the methods is
]U 1/2
ut= n]y −u8v8 . s5d wnoitrhminali5z%atioinnparlelsecnasteeds.inThtheisfrpicatpieornwvaesloocbittayinuesdedusfionrgtthhee
Clauser chart method due to the slightly lower uncertainty.
III. UNCERTAINTYESTIMATES However, it should be noted that since utobtained using the
totalstressmethodwasslightlylessforboththesmoothand
Precisionuncertaintyestimatesforthevelocitymeasure- rough walls, normalization based on this method would not
mentsweremadethroughrepeatabilitytestsusingtheproce- significantly change the comparisons presented herein.
duregivenbyMoffat.27LDVmeasurementsarealsosuscep- Figure 2 shows the mean velocity profiles for all three
tible to a variety of bias errors including angle bias, velocity surfaces plotted in inner variables. The smooth wall log law
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035102-4 Flack,Schultz,andShapiro Phys.Fluids17,035102~2005!
TABLEI. Experimentaltestconditions.
ut
sm/sd ut
U Total sm/sd d d* u
e
Surface sm/sd Reu stress Clauser DU+ smmd smmd smmd 5k/d 5ks/d
Smooth 3.77 10220 0.137 0.141 26.4 3.20 2.53
fl fl fl
Sandpaper 3.77 14340 0.180 0.185 7.7 31.9 5.05 3.64 0.11 0.08
k=0.69smmd
k+=134
k+=100
s
Mesh 3.81 14120 0.187 0.196 8.5 31.1 5.03 3.66 0.05 0.11
k=0.32smmd
k+=64
k+=138
s
is shown for comparison. Both rough surfaces display a lin- heightk providesacommonmeasureoftheinfluenceofthe
s
ear log region that is shifted by DU+ below the smooth pro- roughness on the mean flow and is related to DU+ via the
file indicating an increased momentum deficit on these sur- following,8,22 where B, the log-law intercept for uniform
faces. The sandgrain and mesh surface produced roughness sand grain, is equal to 8.5:
functionsofDU+=7.7and8.5,respectively.Itisofnotethat
1
the roughness function is higher for the mesh surface at DU+= lnsk+d+C−B. s6d
k s
nominally the same unit Reynolds number even though the
roughness height k is less than half that of the sandpaper. The estimated extent of the roughness sublayer s5k /dd is
s
This clearly reinforces that the roughness height alone is not given inTable I.Values of 5k/dare also presented for com-
a good indicator of roughness function. Proper scaling pa- parison.
rameters for roughness must also account for differences in The mean velocity profiles for all test cases are pre-
parameters such as surface texture,32 roughness density,33 sented in defect form in Fig. 3.Also shown for comparison
and roughness slope.34 It was demonstrated by Furuya and aretheresultsofthesmoothwalldirectnumericalsimulation
Fujita35 that not only the roughness height but also the pitch sDNSd by Spalart36 at Reu=1410. The velocity defect pro-
to diameter ratio is an important parameter in determining files exhibit excellent collapse in the overlap and outer re-
DU+onwovenmeshroughness.Atpresent,however,thereis gions of the boundary layer supporting the notion of a uni-
noadequatemeansofpredictingtheeffectofagenericrough versaldefectprofileforroughandsmoothwalls,asproposed
surface on the mean flow si.e., DU+d from measures of the by Clauser9 and Hama.10 Similar results were also observed
roughness profile. It seems plausible that since k is not a byAcharyaetal.20formeshandmachinedsurfaceroughness
reliable indicator sat least among different roughness typesd and by Schultz and Flack12,13 for a variety of roughness
of the roughness effect on the mean flow, it is likely not a types.Krogstadetal.16andKeirsbulcketal.17foundthatthe
proper length scale to define the extent of the roughness ef- defect profiles for boundary layers over woven mesh and
fect on the turbulent motions sknown as the roughness sub- transverse bars, respectively, did not collapse with smooth
layerd either. Alternatively, the equivalent sand roughness
FIG. 2. Mean velocity profiles in wall coordinates soverall uncertainty in FIG.3. VelocitydefectprofilesfoveralluncertaintyinsUe−Ud/ut:smooth
U+:smoothwall,±4%;roughwalls,±7%d. wall,±5%;roughwalls,±7%g.
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035102-5 ExperimentalsupportforTownsend’sReynoldsnumber Phys.Fluids17,035102~2005!
FIG. 6. Normalized Reynolds shear stress profiles soverall uncertainty in
FIG. 4. Normalized streamwise Reynolds normal stress profiles soverall −u8v8/ut2:smoothwall,±8%;roughwalls,±10%d.
uncertaintyinu82/ut2:smoothwall,±5%;roughwalls,±7%d.
roughnessinfluencecorrespondstoył3k onboththesand-
wallprofiles.ThiswasduetoanincreaseinP fortherough s
paper and the mesh surfaces.The disappearance of the near-
walls, which presumably, resulted from the higher rate of wall peak in u82 was shown by Ligrani and Moffat22 to be
entrainment of irrotational flow at the outer edge of the
indicativeofaboundarylayerinthefullyroughflowregime.
boundarylayer.16Thepresentauthorsbelievethedifferences The wall-normal Reynolds normal stresses v82/u2 sFig.
t
in results can be attributed to the strong roughness used in 5d also indicate good collapse in both the overlap and outer
the studies of Krogstad et al.16 and Keirsbulck et al.17 In
regions of the boundary layer. This result is in agreement
both,theratiooftheboundarylayerthicknesstotheequiva- with a number of studies.12,13,19,22 In the present study, the
lłen1t5s.androughnessheightd/ksfortheroughwallflowswas saugrrfeaecmese.nTt hine svt8u2d/iuet2s iosfoKbrsoegrsvteadd feotrayl.1ø6kasndfoKrebiorstbhurlcokugeht
al.17 both showed that significant increases in the wall-
B. Reynolds stresses and quadrant analysis
normal Reynolds normal stress penetrate well into the outer
ThenormalizedReynoldsstressprofilessu82/u2,v82/u2, layer over rough walls. It should be noted that even though
t t
and −u8v8/u2d for the test surfaces are presented in Figs. thepresentroughsurfacesareinthefullyroughflowregime,
t
4–6.Also shown are the results of the smooth wall DNS by they are much “weaker” in that d/k =63 for the sandpaper
s
Spalart36atReu=1410.Theapproximateextentoftherough- andd/ks=45forthemesh,ascomparedtod/ksł15inthose
nesssublayerforthesandgrainandmeshsurfacess5k /ddis studies.
s
given for reference. The streamwise Reynolds normal The normalized Reynolds shear stress s−u8v8/ut2d pro-
stresses u82/u2 sFig. 4d for the smooth and rough wall show files are presented in Fig. 6. Good collapse of the profiles is
t
excellentagreementthroughouttheoverlapandouterregions again observed across almost the entire boundary layer sy
of the boundary layer. Near the wall sy/dł0.05d, u82/u2 is øk d. In order to further investigate possible differences in
t s
significantly lower on the rough walls. The region of the theflowdynamicsbetweensmoothandroughwallboundary
layers, quadrant analysis37 was carried out using the hyper-
bolic hole size H method of Lu and Willmarth.38 This tech-
nique allows the contributions of ejection Q2 and sweep Q4
motions to the Reynolds shear stress to be calculated. The
contribution to u8v8 from a given quadrant Q can be ex-
pressed as
E
1 T
su8v8d = lim u8v8stdI stddt, s7d
Q T!‘ T 0 Q
where I Hstd is a trigger function defined as J
Q
˛ ˛
1 when uu8v8u øH u82 v82
I = Q s8d
Q
0 otherwise.
Figure 7 shows the normalized contribution from ejection
and sweep events to the Reynolds shear stress forH=0.The
profiles of both the Q2 and Q4 contributions for the smooth
FIG. 5. Normalized wall-normal Reynolds normal stress profiles soverall and rough walls show excellent agreement across almost the
uncertaintyinv82/ut2:smoothwall,±6%;roughwalls,±8%d. entireboundarylayer.Allthesurfacesdisplayanearlylinear
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035102-6 Flack,Schultz,andShapiro Phys.Fluids17,035102~2005!
FIG. 7. Normalized Reynolds shear stress contributions from Q2 and Q4
withH=0soveralluncertaintyin−su8v8dQ2/ut2and−su8v8dQ4/ut2:±10%d.
FIG.9. RatiooftheReynoldsshearstresscontributionsfromQ2 andQ4
withH=2soveralluncertaintyinsu8v8d /su8v8d :±11%d.
Q2 Q4
decrease in the Q2 and Q4 Reynolds stresses in the outer
layer. This indicates that roughness-induced changes to co- while strong ejection events play a larger role near a smooth
herentturbulentstructuresareconfinedtoanear-wallrough- wall, sweeps have a larger contribution near rough walls.
ness sublayer. It is of note that ejection events dominate ThiswasalsoobservedbyKrogstadetal.16andislikelydue
sweeps throughout the boundary layer on the smooth wall to the difference in the boundary condition at y=0 for
andforyø3ksontheroughwalls.TheresultsofKrogstadet smoothandroughwalls.Onasmoothwall,v8=0aty=0.On
al.16 indicate an increase in both the Q2 and Q4 contribu- a rough wall, y=0 is located somewhere between the rough-
tions for rough walls that extends well into the outer layer. ness peaks and troughs, not on the wall, so the vertical ve-
SchultzandFlack13alsoobservedanincreaseinthesequan- locity fluctuations may be nonzero. The ratios of the Q2 to
tities for uniform sphere roughness, but only for ył5ks. It Q4 contributions presented in Fig. 9 show good agreement
shouldbenotedthatbothofthesestudieswerecarriedouton for all of the surfaces for y.3k .
strongerroughnesssd/k ł30dthanthepresentinvestigation. s
s
Figure8showscontributiontotheReynoldsshearstress C. Velocity triple products and higher-order moments
fromstrongsH=2d ejectionandsweepevents.ThisH corre-
While the present Reynolds stress results indicate there
sponds to instantaneous Reynolds shear stress events larger
than5u8v8.TheprofilesofboththeQ2andQ4contributions is a great deal of similarity in the turbulence on smooth and
rough walls for y.3k , higher-order turbulence moments
for the smooth and rough walls again show excellent agree- s
will now be addressed. Andreopoulos and Bradshaw21
mentoutsidethenear-wallregion.Onedifferencethatcanbe
discerned is an increase in strong Q2 events for y/dł0.05 showed that the velocity triple products provide a sensitive
indicator of changes in turbulence structure due to wall con-
on the smooth wall that is absent on the rough walls. Also,
dition, however, relatively few rough wall studies have pre-
strongQ4 eventsdecreaseonthesmoothwalloverthesame
sented these statistics with the exception of Antonia and
range.ThisisillustratedmoreclearlyinFig.9,whichshows
Krogstad,14 Kiersbulck et al.,17 Andreopoulos and
theratioofthecontributionsfromQ2andQ4forH=2.Near
Bradshaw,21 and Bandyopadhyay and Watson.39 The distri-
the wall, the ratio is greater than unity for the rough walls
and less than unity for the smooth wall. This indicates that
FIG. 8. Normalized Reynolds shear stress contributions from Q2 and Q4 FIG. 10. Normalized velocity triple product u83/ut3 soverall uncertainty in
withH=2soveralluncertaintyin−su8v8dQ2/ut2and−su8v8dQ4/ut2:±12%d. u83/ut3:±28%d.
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035102-7 ExperimentalsupportforTownsend’sReynoldsnumber Phys.Fluids17,035102~2005!
FIG. 11. Normalized velocity triple product v83/ut3 soverall uncertainty in FIG.13. Normalizedvelocitytripleproductu8v82/ut3soveralluncertaintyin
v83/ut3:±38%d. u8v82/ut3:±18%d.
butionsofthenormalizedvelocitytripleproductsu83/ut3and reduction in u82v8/ut3 for the rough walls. A similar trend
u83/u3 are presented in Figs. 10 and 11, respectively. The was observed for mesh roughness by Krogstad and
t
profilesofu83/u3 showgoodagreementbetweenthesmooth Antonia.40 The gradient ]u82v8/]y represents the turbulent
t
and rough walls for y.5k . Closer to the wall, some differ- diffusion of u82 in the Reynolds stress transport equation for
s
ences can be noted. The sign of u83/u3 is negative for the u82. On the smooth wall, there is a loss in u82 through tur-
t
smooth wall for y,0.05d, whereas, it is positive for the bulent diffusion for 0.01,y/d,0.025, while there is a gain
rough walls. This may be due to reduced sweep events su8 for the rough walls. The normalized turbulent flux of Rey-
.0d for the smooth wall than for the rough wall, which was noldsshearstressu8v82/ut3alsoindicatesgoodagreementfor
also identified in the quadrant analysis. The profiles of thesmoothandroughwallsovermostoftheboundarylayer
v83/ut3alsoshowagreementwithintheirexperimentaluncer- sFig. 13d. The differences in u8v82/ut3 are confined to the
tainty outside the near-wall region. This contrasts with the near-wall region, where the wall-normal turbulent transport
results of Antonia and Krogstad14 who found large changes of Reynolds shear stress s−u8v8d is toward the wall on the
inthistripleproductforflowsovertransverserodroughness. rough walls and away from it on the smooth wall. This dif-
This was observed as a negative value of v83/u3 over a sig- ference in the near-wall transport was also observed byAn-
t
nificant portion of the boundary layer indicating transport of dreopoulos and Bradshaw21 and is likely due to the stronger
turbulent kinetic energy towards the wall instead of away sweep events that occur on the rough wall, as discussed pre-
from it, as seen for a smooth wall. viously.
Thedistributionsofthevelocitytripleproductsu82v8/ut3 The skewness factor distributions for u8 and v8, Su and
and u8v82/ut3 are presented in Figs. 12 and 13, respectively. Sv, are shown in Figs. 14 and 15, respectively. Also shown
The quantity u82v8/u3 represents the wall-normal turbulent for comparison are the smooth wall data of Antonia and
t
transportofu82.Theprofilesforthesmoothandroughwalls Krogstad14 at Re0=12570. There is excellent collapse of Su
agree for y.5ks. Much closer to the wall, there is a large forallthreesurfacesfory.5ksandgoodagreementwiththe
smooth wall data of Antonia and Krogstad14 outside the
FIG.12. Normalizedvelocitytripleproductu82v8/ut3soveralluncertaintyin
u82v8/ut3:±22%d. FIG.14. SkewnessfactorSusoveralluncertaintyinSu:±25%d.
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035102-8 Flack,Schultz,andShapiro Phys.Fluids17,035102~2005!
FIG.15. SkewnessfactorSvsoveralluncertaintyinSv:±35%d. FIG.17. FlatnessfactorF soveralluncertaintyinF :±6%d.
v v
roughnesssublayer.Closertothewall,theroughwallsshow
positiveskewness,whilethesmoothwallhasnegativeskew- ven mesh; surfaces that have previously been used to exam-
ness.Thepositiveskewnessneartheroughwallsmaybedue inetheconceptofturbulencesimilarityintheouterlayer.12,16
to the less strict, wall-normal boundary condition, allowing The present results indicate that the mean velocity profiles
formorehighmomentumfluidtobesweptintothenearwall for rough and smooth walls collapse well in velocity defect
region. S shows agreement within the experimental uncer- form in the overlap and outer regions of the boundary layer.
v
tainty across the entire boundary layer sFig. 15d and agrees The Reynolds stresses and quadrant analysis results for the
wellwiththesmoothwalldataofAntoniaandKrogstad14for tworoughsurfacesagreewellthroughoutmostofthebound-
y.5ks.AntoniaandKrogstad14notedsignificantdifferences ary layer and collapse with smooth wall results for y.3ks.
in S for transverse rod roughness well into the outer region The velocity triple products and higher moment turbulence
v
oftheboundarylayer.Theflatnessfactordistributionsforu8 statistics also indicate that the differences in the rough wall
andv8,FuandFv,areshowninFigs.16and17,respectively. boundary layers are confined to y,5ks. These results pro-
Also included for comparison are the results of Bandyo- vide compelling support for Townsend’s Reynolds number
padhyay and Watson39 for flow over transverse bars. Both similarity hypothesis for uniform three-dimensional rough-
flatness factors show good collapse for smooth and rough ness in flows where the roughness is a relatively small per-
surfaces throughout the entire boundary layer and agree rea- turbation to the smooth wall case.Asurvey of the literature
sonably well with the results of Bandyopadhyay andWatson along with the present results indicates that d/ks snot d/k as
over most of the boundary layer. Jiménez8recentlysuggesteddistheproperparametertoindi-
cate if the roughness effect on the turbulence will be strong
or weak. For roughness where d/k ø40, significant similar-
V. CONCLUSIONS s
ity in the turbulence structure can be expected outside the
Comparisons of structure of turbulent boundary layers roughness sublayer. In cases where d/k ,40, turbulence
s
developingovertworoughnesstypesandasmoothwallhave modifications may be anticipated to extend well into the
been made. The rough surfaces included sandpaper and wo- outerlayer.Flowswithstrongroughnesshavegreatpractical
importance. These range from engineering applications of
flows through fouled heat exchanger tubes and over fouled
gas turbine blades to atmospheric boundary layers in urban
or forested areas.Therefore, while the concept of turbulence
similarity appears applicable to many rough wall flows, fur-
ther understanding and predictive capability are needed in
cases where roughness effects are large and turbulence simi-
larity fails.
1V.C.Patel,“Perspective:flowathighReynoldsnumberandoverrough
surfaces—AchillesheelofCFD,”J.FluidsEng. 120,434s1998d.
2J.Nikuradse,“Lawsofflowinroughpipes,”NACATech.Memo.1292
s1933d.
3C. F. Colebrook and C. M. White, “Experiments with fluid friction in
roughenedpipes,”Proc.R.Soc.London A161,376s1937d.
4A.E.PerryandP.N.Joubert,“Rough-wallturbulentboundarylayersin
adversepressuregradients,”J.FluidMech. 17,193s1963d.
5A. E. Perry, W. H. Schofield, and P. N. Joubert, “Rough-wall turbulent
boundarylayers,”J.FluidMech. 37,383s1969d.
FIG.16. FlatnessfactorF soveralluncertaintyinF :±3%d. 6R.A.AntoniaandR.E.Luxton,“Theresponseofaturbulentboundary
u u
Downloaded 05 May 2005 to 131.122.81.200. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
035102-9 ExperimentalsupportforTownsend’sReynoldsnumber Phys.Fluids17,035102~2005!
layertoastepchangeinsurfaceroughness—Part1.Smoothtorough,”J. 24H. Schlichting, Boundary-Layer Theory, 7th Ed. sMcGraw-Hill, New
FluidMech. 48,721s1971d. York,1979d.
7M.R.Raupach,R.A.Antonia,andS.Rajagopalan,“Rough-wallturbulent 25P.S.KlebanoffandF.W.Diehl,“Somefeaturesofartificiallythickened
boundarylayers,”Appl.Mech.Rev. 44,1s1991d. fully developed turbulent boundary layers with zero pressure gradient,”
8J. Jiménez, “Turbulent flow over rough walls,” Annu. Rev. Fluid Mech. NACATech.Memo.2475s1951d.
36,173s2004d. 26M. P. Schultz and A. Myers, “Comparison of three roughness function
9F.H.Clauser,“Turbulentboundarylayer,”Adv.Appl.Mech. 4,1s1956d. determinationmethods,”Exp.Fluids 35,372s2003d.
10F. R. Hama, “Boundary layer characteristics for smooth and rough sur- 27R.J.Moffat,“Describingtheuncertaintiesinexperimentalresults,”Exp.
faces,”Soc.Nav.Archit.Mar.Eng.,Trans. 62,333s1954d. Therm.FluidSci. 1,3s1988d.
11P.R.Bandyopadhyay,“Rough-wallturbulentboundarylayersinthetran- 28R. V. Edwards, “Report of the special panel on statistical particle bias
sitionregime,”J.FluidMech. 180,231s1987d. problemsinlaseranemometry,”J.FluidsEng. 109,89s1987d.
12M.P.SchultzandK.A.Flack,“Turbulentboundarylayersoversurfaces 29P. Buchhave, W. K. George, and J. L. Lumley, “The measurement of
smoothedbysanding,”J.FluidsEng. 125,863s2003d.
turbulencewiththelaser-Doppleranemometer,”Annu.Rev.FluidMech.
13M. P. Schultz and K. A. Flack, “Outer layer similarity in fully rough 11,443s1979d.
turbulentboundarylayers,”Exp.Fluidsstobepublishedd. 30F.Durst,M.Fischer,J.Jovanovic,andH.Kikura,“Methodstosetupand
14R.A.AntoniaandP.-Å.Krogstad,“Turbulencestructureinboundarylay-
investigatelowReynoldsnumber,fullydevelopedturbulentplanechannel
ers over different types of surface roughness,” Fluid Dyn. Res. 28, 139
s2001d. flows,”J.FluidsEng. 120,496s1998d.
15A.A.Townsend,TheStructureofTurbulentShearFlowsCambridgeUni- 31H.W.ColemanandW.G.Steele,“Engineeringapplicationofexperimen-
versityPress,Cambridge,UK,1976d. taluncertaintyanalysis,”AIAAJ. 33,1888s1995d.
16P.-Å. Krogstad, R. A. Antonia, and L. W. B. Browne, “Comparison be- 32R. L. Townsin, D. Byrne, T. E. Svensen, and A. Milne, “Estimating the
technical and economic penalties of hull and propeller roughness,” Soc.
tweenrough-andsmooth-wallturbulentboundarylayers,”J.FluidMech.
245,599s1992d. Nav.Archit.Mar.Eng.,Trans. 89,295s1981d.
17L.Keirsbulck,L.Labraga,A.Mazouz,andC.Tournier,“Surfacerough- 33F.A.Dvorak,“Calculationofturbulentboundarylayersonroughsurfaces
nesseffectsonturbulentboundarylayerstructures,”J.FluidsEng. 124, inpressuregradient,”AIAAJ. 7,1752s1969d.
127s2002d. 34A.J.Musker,“Universalroughnessfunctionsfornaturally-occurringsur-
18A.E.PerryandM.S.Chong,“Onthemechanismofwallturbulence,”J. faces,”Trans.Can.Soc.Mech.Eng. 1,1s1980d.
FluidMech. 119,173s1982d. 35Y. Furuya and H. Fujita, “Turbulent boundary layers on a wire-screen
19A. E. Perry and J. D. Li, “Experimental support for the attached-eddy roughness,”Bull.JSME 10,77s1967d.
hypothesis in zero-pressure gradient turbulent boundary layers,” J. Fluid 36P.R.Spalart,“DirectsimulationofaturbulentboundarylayeruptoReu
Mech. 218,405s1990d. =1410,”J.FluidMech. 187,61s1988d.
20M.Acharya,J.Bornstein,andM.P.Escudier,“Turbulentboundarylayers 37J. M. Wallace, H. Eckelmann, and R. S. Brodkey, “The wall region in
onroughsurfaces,”Exp.Fluids 4,33s1986d. turbulentshearflow,”J.FluidMech. 54,39s1972d.
21J.AndreopoulosandP.Bradshaw,“Measurementsofturbulencestructure 38S. S. Lu, and W. W. Willmarth, “Measurements of the structure of the
intheboundarylayeronaroughsurface,”Boundary-LayerMeteorol. 20, Reynolds stress in a turbulent boundary layer,” J. Fluid Mech. 60, 481
201s1981d. s1973d.
22P.M.LigraniandR.J.Moffat,“Structureoftransitionallyroughandfully 39P.R.BandyopadhyayandR.D.Watson,“Structureofrough-wallbound-
roughturbulentboundarylayers,”J.FluidMech. 162,69s1986d. arylayers,”Phys.Fluids 31,1877s1988d.
23P.-Å.KrogstadandR.A.Antonia,“Structureofturbulentboundarylayers 40P.-Å.KrogstadandR.A.Antonia,“Surfaceroughnesseffectsinturbulent
onsmoothandroughwalls,”J.FluidMech. 277,1s1994d. boundarylayers,”Exp.Fluids 27,450s1999d.
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