Introduction ~Prediction of the resistance which a given hound-~

ary presents to a steady, uniform flow l'aisesprohlems which have been f'aced principally byresorting to experimentally determined resistancecoefIicients. It has generally been accepted thatthe most satisfactory form of relationship whichexpresses the resistance is the semi-Iogarithmicfonn, which, as shown hy Millikan [1], may hederived hy dimensional considerations alone. Thistype of r~lationship has heen experimentally con­firmed many times for hoth pipe and channelfiows, for smooth surfaces and for many kinds ofrough surfaces. Of the latter, perhaps the. Niku­radse experilllents are the most complete [21. AI­though they were confined to pipe fiow, the practiceof expressing engineering types of roughness intenns of the size of the equivalent Nikuradseroughness hasbecollle "\videspread. Of course, therough surfaces encountered bearlittle reselllblanceto Nikuradse's closelypacked sandgrainsartifi­cially fixed to the pipe walls,hut the commonbasisofcomparison in tenus ofa uniformand geome­trically simple rough surface has proved valuahlein the physical visualization ofa certain rough­ness. l\'1oreover, the use of one geometric variableresults in convenient mathematical expression of

Illstitute of Hydraulic Hesearch University of Iowa.

LA HOUILLE BLANCHE/N° 7-1964

SOMEROUGHNESS-CONCENTR:ATION

EFFECTSON BOUNDARY RESISTANCE

SY E.M. O'LOUGHLIN AND E.G. MACDONALD *

the resistance relationship. For these reasons, theconcept of an equivalent sand grain roughness willhe utilized throughou t this discussion.

Schlichting [3], in an early investigation ofhoundarv resistance, measured the Tesistance ofsurfaces· to which roughness elements had beenattached. In his series of experiments, the rough­ness wasapplied to one boundary of an otherwisesmooth rectangular conduit. The spacing or con­centration of the elements, as weIl as their geome­tric fonn, was varied, and the l'üughness ofeachsurface was expressed in tenus of an equivalentuniform-sand roughness. Schlichting recognizedthat the llSualtypes of roughness couldnot hegeometricallydescribed by a single dimension, suchas a rouo·hness~element height, but that a rough­Hess den~ity, such as "the Humber of individualrouuhness èlements pel' unit of area," must alsoa .hespecified. It would he expected that changes 111

the sl)acin')" of the roughness elements would causea . 'f' lchanges in the efTective roughness of a sur ace an(

that these changes would he of the same orderofmagnitude as ch'anges caused hy varying the rough­ness-element size. Schlichting ohserved the mannerin which the equivalent sand roughness varied asthe concentration ofa given type of elemenlwasincreased (Fig. 1). A large variety of roughnessclements were used in his experiments, and sincethen several others have experimented with bars,

Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1964042

E.M. O'LOUGHLIN and E.G. MACDONALD

fi Variation of the equivalent l'oughness size with concen­tration, aftel' Schlichting [3].Yariation de la âimension de la rUf/osité équivalente enfonction de la concentration, d'après Schlichtinu [.'J].

O'nitude of this type of free-surface etrect. Empha-b •sis has been placed upon the resIstance measure-ments at hio-h rouahness concentrations, and theresults have

b

been ~ompared with those of Niku­radse [2]. The l'orm of the transition betweensmooth and l'ully roughwall flow is investigated forseveral sand concentrations, and the dissimilarityl'rom the uniform roughness transition is noted.The series of experiments have, in addition, enabl­ed observations to be made regarding the ret'er­ence bed elevation and the value of the KàrmàncoefIicient in open-channel flow. ~

Roughness characteristicsAs mentioned above, the variation of the resist­

ance coefIicient with roughness concentration wasstudied experimentally by systematically varyingthe spacing of cubes and sand grains fixed to asmoolh surface. The cubes were arranged in afixed pattern, although the pattern used by thepresent writers was slightly dif1'erent l'rom thalused by Koloseusand Davidian (sec Fig. 2). Thesand-grain distribution was somewhat random, asdescribed later.

o

i1

1 11

11

1 -~

f-

I ~I-

f-i/ I~ -

~

)'\ 1

~ 1 -

[\.

y ~~

l'\ql q2 q3 Q4 Q5 qG Q7 qB q9 l,

oo

ks

k2

4

3

Pattern 1 - Disposition 1

Pattern Il - Disposition Il

Pattern III (Koloseus and DovldlGn)Disposition III l Koloseus et Davidian)

2/ Cube patterns.Dispositions des éléments cubiques.

~1)

D DD D

D D

À. = naA

D DD

D D

The unifoTm-cube roughness was ohlained hyhavinO' 6-inch-square plates casl of aluminium, an

b 1 .integral part of which was formed hy t le deslredarrangement of 1/2-inch cuhes. Care ,vas takento ensure that the top edges of the cuhes weresharp and that the cubes joined the plates cleanly.Sets of plates were prepared having roughness con­centrations of 0.11, 0.25, 0.44, and 0.70, the rough­ness concentration À. being expressed in each caseaccording to Schlichting's definition,. i.e., the ratioof the SUIn of the upstream projected areas of theroughness elements to the floor area,

DDDDD-.. D D D D

DDDDD

short angles, square pipe threads, etc., varyingtheir concentration and pattern while the boundaryresistance ,vas observed. Most of the rèstIlts ofthese studies have been collected and compared byKoloseus and Davidian [4]. Exlcnsive additionalmeasurements were made by Koloseus and Davidianduring an investigation primarily concerned ,viththe elTect of the presence of roll waves on resistancein open-channel flow. In this investigation, theconcentration of cllbical roughness elements wasvaried over a sixty-fourfold range, while the patternwas held constan't. The present writers have sinceextended the work not only ,vith cubes, but alsowith sand grains, for which the concentration wasvaried over an eighty-fold range, including as oneextreme a roughness density which was similar tothat of Nikuradse. The concentrations for thecube data, including those of Koloseus and Davi­dian [4], now extend over a three-hundred-sixty­fold range.

Il must be mentioned here that some of theahove investiga lions were made IUHler conditions in,vhich the roughness elements aUached to thesmooth bed could not be considered as a rough­textured surface, hut rather as a series of bound­ary irregularities which ,vere of the same order ofInagnitude as the flow depth ilse1f and which caus­ed relatively large local disturbances to the flow.The measured resistance included the ef1'ects ofthese lar')'e nonuniformities, and ,vhenever the ele-

b . d'ments were large enough to cause apprecIable IS-turbance to the flow, as evidenced for example hythe formation of standing surface waves over eachroughness pTojection, then the measured resistancemu~t be considered separately l'rom the resistancewhich is due to the shear stress generaled hy anllnconfined fluid flowing past a roughened bound­ary. The investigatiori. which is described here\vas designed to provide more complete informationon the variation of lhe resistance coefiicient withrouO'hness concentration, under conditions which

b '1were independent of the free-surface etrect descn)-ed above, and t() examine the nature and the ma-

where Il is the number of elements on the Hoorarea A and a is the upstream projeeted area ofeach element. \Vhen applied to cubes, this measureof concentration varies from zero to one as the spac­ing of the cubes is reduced from infinity to zero.At a concentration of one, the top surfaces of thecubes form a continuous smooth surface.

The placement of the eubes on the plates wasarranged to provide a unifonn pattern when theplates were instalIed in the Hume. By rotatingeach plate 90 degrees, a second pattern of rough­ness elements could he obtained. This afforded aready means for observing the effeet of this patternchange on the resistance coeflicient for these parti­culaI' concentrations. Four of these aluminum pla­tes, each having a difTerent concentration of cubicalelements, are shown in Figure 3.

The uniform-sand roughnesswas produced using"Muscatine" sand passing a U. S. Standard SieveNo. H but retained on Sieve No. 8. The grains ofthis particular type of sand are quite angular (seeFigure 4) and appear to be cOlnposed of a widevariety of mineraIs. The equivalent mean diameterof these sand grains was fOllIld by actual volumetricmeasurement of a counted number of grains to be0.009H foot. This value was used for the meansand-grain height k.

Sand roughness concentrations of 0.0077, 0.09,0.27, 0.45, and O.M ,vere studied. AlI but the lastof these concentrations were produced by firstcovering the glass floor of the 2-1/2 X 85-foot tilt­ing Hume with one coat of shellac and allowingthis tü dry, then covering the shellac with a coatof varnish, and allowing this to dry; the fl ume wasnext marked ofl' into equal reetangular areas, asecond coat of varnish was applied, and, ,'l'hile itwas still wet, equal weights of sand were hand­sprinkled onto each area; finalIy, after the varnishhas hardened, the entire surface was sprayedlightly with very thin varnish. Care was takenwhile sprinlding to obtain a fairly uniform densitywithout influencing the essentially random spacingbetween the grains. Silhouette photographs of thefive sand roughnesses are shown in Figure 5.

As mentioned ahove, the O.H4 sand-grain concen-

LA HOUILLE BLANCHE/N°?-1964

31 Four concentrations of cube roughness, cast in 6 inchaluminum plates.Quatre concentrations de ruyosité, cubes moulés intéflra­lement dans des plaques d'aluminiuJll de 6 pouces.

41 Photomicrograph of l\Iuscatine sand, equivalent spherediameter O.0096-foot.PllOtomicrofJraphie de sable « Mllscatine ». Diamètre de lasphère équivalente: 0,009t; pied.

51 Silhouette photographs of the five concentrations of sandroughness

PllOtOfJrClpllies, en silhouette, des cinq concentl'CltionsderUfJosité constituées par du sable.

E.M. O'LOUGHLIN and E.G. MACDONALD

tration represented an attempt to reproduce in anopen channel the pipe roughness used byNiku­radse. Glass plates 1/8 inch thick by 2 feet squarewere given the S~nne preparatory treatment asdescribedabove. The plateswere then set intoshallow box frames and, .with the varnish still \Vet,a known weight of sand was poured onto eachplate to a depth of approximately 1/2 inch. Theglass was tapped lightly from beneath to preventany local arching of the sand above the glass.After allo,ving several days for the varnish toharden, the plates were tipped over and the excesssand pouredofI and weighed. Remaining on eachplate was a layer of sand, one grain in thiclmess,adhering to the varnish. The roughened plateswere sprayed as before ,vith verv thin varnish toensure •a good bond between tlle sand and theglass. AsufIicient number of plates were producedin this way to cover the floor of the 2 X :iD-footHume for its entire length.

The maximum concentration of these irregularlyshaped sand grains cannot be specified exactly, be­cause it was computed using for a the projeetedarea of a spherical grain of the same volume asthe average grain, and the relationship betweenthe actu~JJ.....,projeeted areas of the grains and theprojected area of the equivalent spherical grainis not known. If the grains can he represented asellipsoids ,vith semi-axes IX, B, and y, and if oneassmnes that the grain comes to l'est in its moststable position, the grain on the smooth. surfacewill appearas in Figure H, and it will be orientedso that the direction of Howcan be considered 10bebetween the two extremes shown.Thus it maybe seen that the ratio between the actual projeetedarea of the grain and the projected area of theequivalent spherical grain will vary between thetwo extremes:

and J!i~Microscope measurements weremade on 100 re­presentative grains of two of these axes, and thethinl ,vas coniputed from the known mean volumeof the grains. From these measurements, the twopossible extremes of the above ratio were foundto have the values 1.11 and 0.82. Thus it seemsreasonahle to assume that the concentr.ation com­puted on the hasis of spherical grains would beclose to the concentrations aetually achieved in theHume. At. this •point, it is inte're;ting {onote forcomparisonthat hexagonallypacked spheres cor­respond to a concentration of 0..905.

6/ ~andgl"ainidealizedasellipsoid restil1gon smoothsur­face.Représentation idéalisée d'un grain de sable, comme el­lipsoïde reposant sur une surface lisse.

Experimental apparatus and procedure ~Two Humes ,vere used for making the resistance~

measurements. Ail the cube roughnesses except theone used to measure the free-surface eiIects weretested in the Institute's 2 X :iD-foot glass-walledtilting flume, the slope of "\vhich is adjustable be­tween zero and 2 degrees. 'Vatel' is supplied froma constant-Ievel tank up to a maximum capacityof 2.7 cubic feet pel' second. The stainless-cladfloor of this flume was covered with the 6-inchaluminum plates over ail but the upstream 5 feetof its length. Depth measurements wère madealong a section 12 feet in length at the dOWIlstreamend of the Hume. The sand roughness was appliedto a section of a larger 2-1/2 X 85-foot glasslinedtilting Hume. The length of the roughened sectionwas 45 feet, the section over which measurementswere taken being 20 feet in length. This flumehas a slope variation of 3 degrees. 'Vatel' is suppli-ed from a constant-level .tank up to maximum ca­pacity of :i.85cubic feet pel' second.

The depth of flow was measured by means ofan 1/8-inch-diameter static tube suspended froma carriage and fixed on the Hume center line. Thistube was connected to a micromanometer alsoconnected to the carriage. A fixed cross-hair onthe adjustable manometer leg perrnitted the water­surface elevation, and hence the depth of flow,to be read directly in thousandths of an inch. :Mi­nor corrections were applied where necessary totake into account small deviations in the rails andfloor of the Hume, even though these amounted toonly a few thousandths of a foot. After the Howwas estabJished, the Hume head-or tail-watergate was manipulated to obtain as uniform a depthof How as possible. Depth lueasurements weretaken every foot along the test section for eachl'un, and plotted. The normal depth for each l'unwas determined by inspection after sl<etching inthe appropriate surface profile.

In tilting-flume experiments such as these, itlllay be shown that errors in the depth measure­ment will be introduced unless. themanorneter,which is leveled for each Hume sI ope, is pivotedabout a point which lies in the plane of the Humebed, and this point must be directly beneath theside piezometers of the statictube. With discreteroughnesselementspli"'ced.on a smooth bed,. thebed plane isnot uniquely defined. In every case,however, it was taken at the level which the bedwould assume if aIl the roughness elements weremelted down. The selection of this l'eference-bedelevation will bediscussed. more fully in.alatersection.Th~cube roughness whichwas used to illustratt

the free . surface-eiIectsconsisted of 1.:i25-inchwooden . cubes glued to the Hoor of the 85-foottilting flume, at a concentration<ofO.125. Thecubes werefinished with sharpedges, and \VeretllOroughly sealed withepoxypainL. to. preventsweIling.ln planningthisphase of theinvestig­ation, it was realized that the local Ilonuniformitiesin flo\V causedhy the cllbes would introduce. errorsin depth measurement if static tubes were introduc­ed. These errors were minimized by keeping themagnitude of the velocity head low in comparisonwith the depth of flow for aIl runs.

LA HOUILLE BLANCHE/N° 7-1964

where K is the Karman coefficient, A is a constant,and C depends upon the roughness concentration.

Taking the slope of each curve for the momentto be constant, i.e., K is the same for every concen­tration, then the equivalent sand roughness k s for

The variation of the resistance coefficient 1/v7with relative roughness 4 yo/k is shown in Figu­res 7 and 8 for the cube and sand roughnesses,respectively. The factor Yo is the normal depth offiow. In each of these plots a separate curve hasbeen drawn for each value of the concentration À.

'1'0 each curve may be fitted an equation of theform:

Experimental results ~CONCENTRATION:

,1

11 L ~

/0

2 f-

6 r~

À, ,

-~-!--~---- : 6~5 -1--~--~1(5) Q44 1

i () Q70! J~ ~~.L ._~.. ~ ~..... .. ..~ .._ .._~_._~_~~ _20 30 40

~[Qk

1/ Relation hetween resistance coefficient and relative rough­ness for four concentrations of cubical roughnesselemellts.1'ariation du coefficient de résistance en fonction de larU{Josité relative, compte tenu de quatre concentrationsd'éléments de rU{Josité cubiques.

(2)1_ = ~ loglo (CYo)Vl K k

EFFECT OF

20080 90 /00

Variation du coefficient de résistance en fonction de larll{Josité relative, compte tenll de cinq concentrationsd'éléments de rll(Josité en (Jrains de sable.

60 704yo

k

5040302

70<'\) (\) RI -0:.

:::.-\Y ~ c::

0' °(Ç)~

':YJ"" '0'

~kY-% -<:::r °~6

<$) .Rr-~\::,J ~-4SJ~

5~~

-1- - ~f ...c-&l ;..-<..

~~. e m -< - ~4 ..;. ';:-,. oR )..(

~-~

...IJII)(?!~ o Q0077""t!'

>:r<->'""-"~~~ e Q09~'<.>"~

3 ..r"'f"""' ..... ~ ~ .&!CS) 0~7 --.::-- '<Y

~ Q45<D Q64

8/ Helation hetween resistance coefficient and relative rough­ness fOl' five concentrations of sand roughness clements.

any of the experimental roughnesses may be com­puted if it is assumed that a resistance equationcan be written for fiat sU,rfaces roughened withthe Nikuradse type of uniform sand, and if il isalso assumed that this resistance equation is obtain­ed by integrating the empirical velocity distributionobserved hy Nikuradse in pipes:

(4)

Il 2.:30 1 "( ao y \.- = -r- oglO ---)ll* h N \ k s /

(il)

where Zl is the velocity at a distance y from theboundary and ll*is the shear velocity.

In Equation (3) KN represents the value of theKarman constant as determined by Nikuradse.Integration of this equation overthe fiow depthyields:

1. 0.81 1 .". (') H, 4 !.Jo \-- = -l'~. Og10 ~.I.) T- )W \'N 's /

and it may be seen by comparing Equations (2) and

777

E.M. and E.G. MACDONALD

0,9 1,00) 0,2 0,3 0,4 0,5À

o Pattern 1 - Disposition 1

lSl Pattern Il - Disposition Il

• Pattern III -Disposition III

21---++-Cr""""'l>-d-+-~-+~-t~~t--~+-~-t-~-+-~---1

4

3

k,k

10/ Variation of the equivalent roughness size \Vith concen­tration for different patterns of cnbical elements.Yaril1tion de la dimension de III rugosité relative enfonction dc la concentration, compte tenu de différentesdispositions d'éléments cubiques.

1,0

;--- - -[--------~ !~-r- :-;;;;;,;;;e-;;:;-~1--, 1 Cubes, DISposition 1

4 r - -- ------ -- - • Cubes, Pattern III (Re! 4)1 1 Cubes, Disposition III (Réf. 4)

f- 0 Sand - Sable

1 0 Spheres (Re!3)1 Sphères (Réf. 3)

3

~2

1

Ilo ~~L...~L-~-'--~-'--~-'--~L...~L_~L...~==-"

o QI 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9À

9/ Variation of the equivalent roughness size \Vith concen­tration for cubes, spheres and sand grains.Variation de la dimension de la rugosité relative en fonc­tion des concentrations de cubes, de sphères et de grainsde sable.

k,

k

The transition fuuctions for the saud rO\lghuess.

60 6000/000Les fonctions de transition correspondant aux diversesrugosités en grains de sable.

/00

.~

L 0~'16l(Y~ = L"> I~. ~0,0. 101 &fB,

~ ~QfY -~~~ -,f§ fti---- -

~Î(@ ~

j-"'~~ .~ ---- -A:l,l)~~ -- ---~

~ ---- ~--

À0 Q0077e Q090 Q27~ Q45CD Q64

~

~0

00~

0~~ ~~O<9 VI>.: .:>L 'J.<"' '" ~

~ G1C>f ~~

fSl'->' ----0

2

-/

o

If the slope A/K of the curves in Figures 7 and 8,equals 0.81IK", then the above expression simpli­fies to:

(4) that the equivalent sand roughness for the samevalue of the resistance coefficient is given by:

11C

(6)

If the Karman "constant" changes from one con­centration to another, then these changes willappear as difl'erences between the slopes of thecurves in Figures 7 and 8 for varying concentra­tions. Also, if the value of the Karman coefficientfor any concentration is different from Nikurad­se's, then Equation (5) indicates that the equivalentsand-roughness depends upon the relative rough­ness. If this is to be the case, then the usefulnessof the equivalent-roughness concept is greatly di­minished. Fortunately, the exponent of the relativeroughness in Equation (5) is almost zero for thedata of Figures 7 andr 8, so that the relative rough­ness has a negligibly small influence on the equi­valent sand roughness computed using Equation(5). The results of this computation for the cuberoughnesses are shown in Figure 9, with the resultsof Koloseus and Davidian, as ,vell as Schlichting'ssphere data. The sand equivalent roughnessesshown in Figure 9 were computed from Equation(5) after fint determining the values of C in Equa­tion (2) from the horizontal asymptotes of thetransition curves in Figure 11.

rt is immediately obvious from Figure 9 thatthe variation in the effective resistance of a sur­face caused by changes in concentration must beanalyzed in tenus of low and high concentrations.At a concentration of zero, the equivalent sandroughnes is zero-i.e., the surface is smooth. Atlow concentrations, there is almost exactly a one­to-one correspondence between the equivalent sandroughness and the concentration, no matter whatthe shape of the roughness elenlCnt. At an inter­mediate concentration, the wake behind each cle­ment is not difTused before the flow reaches thenext clement. Morris [5] has described this wakeinterference type of flow in sorne detail. Furtherincreases in the concentration simply increase theamount of mutual interference between the rouo'h-bness clements, and each clement becomes less effec-tive in creating boundary resistance through thesheltering effect of its neighbors. The size of theequivalent sand roughness therefore decreaseswhen the concentration is increased beyond thatpoint at which wake-interference effects' becomepronounced.

rt is easier to visualize the mutual interferencebetween the roughness elements in the case of thecube roughness. Behind each cube is a separationpoeket filled with relatively still water. If thespacing of the cubes is reduced to the extent thatthe cubes overlap with the nearby seperation poc­kets, then it is possible that the spaces betweenthe cubes are filled completely with fluid whichhas no significant net motion in the direction ofthe outside flow. Morris has described this· asskimming flow, and the resistance would certainlybe expected to be not very different from that ofa continuous surface formed by the tops of closelypacked cubes. The cube concentration of 0.7 pro­vided a good example of skimming flow, and thevery low boundary resistance accompanying thistype of flow could weIl explain the stronglycon­cave shape of the cube. curve in Figure 9. Thisphysical behavioralso suggests that the curveshould become asymptotic to the horizontal axisas shown.

The somewhat different behavior shown in Fi-

LA HOUILLE BLANCHE/N° 7-1964

gure 9 by the curves for the sand and Schlichting'sspheres follows naturally from a consideration ofthe extent to which skinuuing flow may develop.In the case of the cubes, the surfaces of separationoriginating at the leading edges are aIl at the sameelevation, and at high concentration these surfacesmay be considered to form a continuous plane justabove the tops of the cubes. Although this conti­nuous surface is not smooth, it must be noted thatno roughness clements protrude through it. Onthe other hand, the comparative irregularity of thesand grains results in the formation of a series ofdiscrete separation pockets with the peaks of thegrains protruding. Thus, even though the grainsmay be close enough to each other to cause appreci­able interference between them and their separ­ation pockets, the jagged outlines and peaks of thegrains and the hap-hazard points of separationprevent the formation of an ideal skimming flow,and, as a result, the surface resistance is not reduc­ed to the same degree as for surfaces over whichskimming flow develops more fully. The fallinglimb of the curve for the sand therefore does notshow the marked concave dip shown by the cubes.Schlichting's sphere-concentration data display thesame trend as that described above for the sand.The more regular configuration of the sphericalclements and their associated separation zoneswhen compared with the sand suggest that theskimming flow phenomenon develops to an extentV'ihich is intermediate between that for the sandand the cubes. Thus, in the range of concentrationsbetween 0.25 and 0.7, il is seen that the rate ofdecrease of the size of the equiv.alent uniform­sand roughness is a maximum for the cubes, aminimum for the sand, and an intermediate valuefor the spheres. This physical description of theonset of skimming flow and its efl'ect on the resist­ance of a surface appears to provide a satisfactoryqualitative explanation of the difJerencebetweenthe curves for the three types of roughness ele­ments.

EFFECT OF PATTERN:

As 1nentioned previously, the cubes were fabricat­ed so that a change in pattern could readily bemade. Resistance-coefficient measure1nents weremade on the second pattern and the value of theequivalent uniform-sand sizewascomputed foreach concentration. The results for this patternare compared in Figure 10 with those for the pre­vious cube pattern. Again, while sucha patternchange has altered the shape of the curve, thegeneral fonn of the function has not beenchanged.The maximum effective roughnessagain occlusata concentration of about 0.2, and, at high concen­trations, the closely packed roughness behaves likea sluooth surface. The i1.atter slope of the fallinglimb may be explained in tenus of the developmentof the skimming i1.ow over the. cubes. This secondpattern appears to the oncoming flow as a numberof columns pm'allei to the meal1 i1.ow. The spaccbetween any two columns represents a discon­tinuily in the otherwise continuous surface formedby the coalescing zones of separation. The skimm­ing flo,v and its accompanying clecrease in surfaceresistance clevelops at a lower rate as the COl1cen-

E.M. O'LOUGHLIN and E.G. MACDONALD

COMPARISON \VITH NIKURADSE's EXPERIJllENTS:

tration is increased. At concentrations approach­ing unity, the width between the colun1lls of ele­ments becomes so small that the e1fect of the changein pattern vanishes. The difIerence in the peakvalues of the equivalent-uniform-sand size has notbeen examined in any detail, but is undoubtedlyassociated with the change in the flow patternaround each element caused by the change in thecube pattern.

The maximum sand concentration shown in Fi­gure 9 permits a comparison to be made betweenthe channel-resistance measurements describedhere and the rough-pipe measurements of Niku­radse. There ,vas no record made by Nikuradseof the roughness density which he achieved in hispipes, but the detailed description of the method ofsand application indicates that the density wouldhave been close to the maximum possible. Themethod of application of the sand grains in thechannel experiments descrihed here was designedto duplicate Nikuradse's procedure as far possible,but whatever difIerences were introduced by fac­tors such as the lacquerand varnish characteristics,the amount of forced vibration applied to theboundaries, and the difIerence between applyingthe sand to a horizontal plate and a vertical pipecannot be evaluated. The microphotograph of asample of Nikuradse's sand grains also indicatesthat they were considerably more uniform in grainsize and shape than the Muscahne sand. Never­theless, the concentration of 0.64 achieved for theMuscatine sand should closely duplicate the densityof the rough surfaces used by Nikuradse.

As shown in Figure 8, the equivalent roughnesssize for the 0.64 concentration was found to he1.51 k. Probably themost influential factor caus­ing a departure l'rom the expected value of unityfor ks/k was the difIerence between the Nikuradseand Muscatine sands with respect to grain uniform­ity and sphericity. The addition of only a smallnumber of large grains to a surface covered uni­formly with smaIl grains has becn shown by Cole­brook and 'Vlüte [6] to increase the equivalen troughness size by more than 50 %' In some earlierwork, Schlichting [3] obtained in a single test forks/ka value of 1.64 for a closely packed sand rough­ness in a rectangular conduit. Thus it appearsthat, as weIl as the roughness height and the rough­ness concentration, at least one additional para­meter expressing the degree of uniformity of theroughness elements is required in order to specifythe resistance eoefIicient of a roughened surface.A parameter such as this has not heen investigatedin this series of experiments.

A further basis of comparison with the Niku­radse results is available in the transition function:

'~~~'

i ~Q;,

~~

1 4y, iiff 210qT~~

)/,~~../ À=(V25~

Il~9 Koloseus and Davidian/ ~ Il et Il

oLoughlin and Macdonald($l Il et Il

L: 10 100

THE HEFERENCE BED ELEVATION:

It is clearly seen l'rom Figure Il that the shapeof the transition curve resemhles the transitionfunction for commercial-pipe materials rather thanthat obtained hy Nikuradse for uniform-sandroughness. \Vhether this is the result of the non­uniformity of the sand grains of the presence ofthe smooth portions of the boundary cannot yet hestated with certainty. Roberson [7], in a digital­computer solution for the resistance of a houndarycovered with cubes of varying concentrations, pr~­dicted transition curves of the commercial­roughness type, but the experimental measure­ments made to date with the cube roughness havenot disclosed any dependence of the resistancecoefficient on viscous efIects.

,'!Jbk

12/ Helation hetween resistance coeflicient and relativeroughness at small values of relative roughness.Yariation du coefficient de résistance, en j'onction de larUiJosité retative, Ill/X j'aibles valeurs de ce dernier para,mètre.

o

2

1,ff

4

3

5

and the roughness height k is the equivalent spherediameter of the sand grain; 0.0096-foot. It is evi­dent l'rom Figure 11 that the resistance coefIi­cients measured for every sand concentration arenot free from viscous efIects-that is, each of thecurves deviates l'rom the horizontal as the rough­ness Heynolds number is changed. Of course, thenature of the rough boundary, with a large portionof the surface being smooth, would lead one toexpect a variations of resistance coefficient withHeynolds number, but the magnitude of this vari­ation would also be expected to be relatively minor,especially at the higher concentrations.

(7)_1__ ~~_~ 10gl _4 YJL - <p (--liYL,.)\Ir h. ,0 k · .1, yo/k

which defines the relative influences of viscous andinertial efIects on the resistance coefIicient. In Fi­gure Il are shown the transition curves for eachconcentration of sand. The Heynolds number dtis here defined as:

Both the resistance coefficient and the Kàrmil.llcoefIicient for flow over a rough boundary at a givenfree-surface elevation depend upon the position ofthe bed plane to which depth measurements arereferred. This plane is unambiguously defined if

780

the geometric mean level suggested by Schlichting[3] is used. Obvious shortcomings are .apparentwhen the roughness elements contain negligiblevolume, such as the short angles studied in detailby Sayre and Albertson [8]. Koloseus [9] hassuggested that it would be more realistic in thesecircumstances to include an estimate of the vo­lume of the separation pocket behind each elementwith the roughness-element volume. 1'0 be con­sistent with this practice, the volume of the separ­ation pocket would need to be included no matterwhat the shape of the roughness element. Forrounded elements, the additional volume wouldnot be large enough to affect significantly either theKarman "constant" or the resistance coefficient.For cubical elements, the same applies at lowconcentrations, but at the concentrations for whichskimming ilow is weIl developed it is physicallynaive to include in the ilow passage a significantamount of the volume behveen the cubes. Thisbehavior suggests the possihility of a link betweenthe best bed elevation and the shape of the curvesof Figure 9, which has already been discussed interms of skimming-flow development. From velo­city measurements made with the maximum sandconcentration, it was ohserved that the best semi­logarithmic variation of velocity with depth wasobtained when the data were plotted with respectto a plane 0.007 foot ahove the base of the grains,while the geometric-mean level was computed tobe 0.004 foot above the hase of the grains. Theequivalent-sphere diameter of these grains was0.0096 foot. The allocation of numerical coellic­ients to make any adjustment to the reference bedelevation computed with the Schlichting definitiondoes not seem warranted yet, hut jt is worth notingqualitatively the interdependence between the bed­elevation selection, the roughncss concentrationand the development of skim~ning flovv. '

Notwithstanding the uncertainty regarding thereference bed elevation, the Karmùn coefIicient wascomputed for aIl the data except the pattern IIcubes, using for the datum elevation the geometric­mean level of aIl the roughness clements. Theseare shown in Table 1. "'Vith the exception of themaximum cube concentration and the minimumsand concentration, aH the values are closelvgr:ouped, with an average value of 0.33 (neglectingthe two high values mentioned above). The authorsare of the opinion that one is justified in claimingthat this average value of K represents its truc valuein rough-channel flow rather than the value of 0.40which is generally used, and the authors sec noreason that this aevrage value should not have

Table 1

CUBES SAND

-------~-,.,-'"~-"------~-_.-

CONCENTHATION K CONCENTHATION K

0.11 .33 0.0077 ,400.25 .3'2 0.09 .330,44 .36 0.27 .330.70 ,46 0,45 .33

0.64 .34

LA HOUILLE BLANCHE/N° 7-1964

general applicability. No explanation is presentlyavailable for the departure of the two high valuesobserved. Both high values were associated withrelatively smooth boundaries.

EFFECT OF THE FREE-SURFACE PnoxIMITV:

Resistance coefIicients were also measured overthe range of depths .at which the roughness cle­ments were of the same order of magnitude as thedepth. The aim of this series was to observe themanner in which the resistance coefIicient variationwith relative roughness departed l'rom the semi­logarithmic law when the depth was smaIl enoughto allow the formation of surface disturbancesover each roughness element. AlI the data obtain­cd in this series have been plotted in Figure 12,together with some earlier resistance data obtain­cd by Koloseus and Davidian [41 for 3/16-inchcubes at the same concentration hut arranged in aslightly different pattern. A marked departurel'rom the usual logarithmic resistance function ispresent for values of the relative roughness 4 Uo/I\.less than about 12. For smaIler relative rough-nesses, the resistance coefIicient 1/0 apparentlydecreases quite rapidly, indicating that the bound­ary made up of cubical obstacles upon a smoothplane presents a greater resistance to the oncomingflow than if the cubes are considered simply as aroughness in the hydraulic sense. In computingthe parameters of Figure 12, the depth of flowwas calculated vvith respect to a bed elevationarhitrarily chosen as the geometric mean level ofthe cubes and inc1uding the estimated volume ofthe still-water zone behind each cube. Some otherreference elevation may be adopted, in which casethe positions of the plotted points will be changed,in any case the resistance coefficient is seen tndepart l'rom the semi-Iogarithmic relationship ina manner which is asyet unpredidable.

Summary and conclusions.

Detailed observations have been made on theoverall resistance in channel flow of surfaces whichhave been roughened by the addition of varyingconcentrations of regular cubical clements andnatural sand grains. Emphasis has been placed onthe concentrations of roughness for which wakeinterference becomes appreciable, up to the maxi­mum concentrations which could be convenientlyfabricated. By describing each surface in terms ofan equivalent uniform-sand roughness, based on therough-pipe experiments of Nikuradse, the effectiveresistance has been quantitatively related to theroughness concentration. The l'orm of the relation­ship for each type of element is dependentuponthe shape of the element, and an explanation forthe difference between the relationships has been

E.M. O'LOUGHLIN and E.G. MACDONALD

given in terms of the uniformity of the roughnesselements and the degree to which any nonuniform­ity in shape permits the disruption of the flowphenomenon knownas skimming. The extremeregularity of a cube-covered surface allows thedevelopment of skimming flow over the smoothupper surfaces of the cubes if the spacing of thecubes is small enough, resulting in a sharp dropin effective resistance. A change in the pattern ofthe cubes on the channel bed can be made to havethe same effeet as an alteration to the regularityor shape of the element. In each case, the develop­ment of a pure type of skimming flow is preventedby protruding roughness peaks, and the effectivesurface resistance as expressed hy the equivalentsand rOl1ghness remains high, the rate of decreasedepending upon the l1niformity of the clements.At the maximum sand concentration, the non­uniformity effects is still evident, resl1lting in arelatively rougher surface than Nikuradse's. Thenecessary inference to be drawn is that yet a thirdparameter, besides roughness size and concentra­tion, is required to describe the resistance characte­ristics of a roughened surface. This parametermust embody the combined effeets of nonuniformi­ties from one particIe to another as weIl as theeffect of the shape or angularity of the elements.Such paralneters must, of course, be measuredstatistically when dealing with engineering surfa­ces.

The type of transition function characterizingthe several sand concentrations appears to matchthat determined for commercial-pipe materialsrather than the Nikuradse uniform-sand transition.This evidence of a viscous influence on the sandtransition even to quite high values of the roughnessReynolds number may be due either to the effectof the smooth portions of the boundary or to theviscous influence on the flow arol1nd the roughnesselements. The latter effect is not significant inthe case of the sharp-edged cubes, and no viscouseffect has heen observed by Koloseus and Davidianeven for low cube concentrations.

AIl of the data are dependent to varying degreesupon the level of the reference bed elevation. Atlow concentrations or at large relative rOl1ghnessthe problem diminishes in importance, but a highconcentrations it appears that a T.ational and con­sistent result can be obtained if Schlichting'sgeometric-mean level is adopted, after heing modifi­ed as suggested hy Koloseus to incll1de an estimat­ed volume of the separation pockets in the rough­ness-element volume.

At extremely smaIl values of the relative rough­ness, supplementary measurements demonstratedthe nature of the resistance relationship, particu­lary with respectto its deviation from the semi­logarithmic expression. The magnitude of the localflow nonuniformities around each obstacle in theflow modifies the free-stream-velocitydistributionto such anextent that the resistance coefficientobtained hy integrating the logarithmic-velocitydistribution is totally inadequate. Il appearsthatthe measured resistance is much higher than thatpredicted by the semi-logarithmic la,v, hut theresults depend to a large extent upon the inter­pretation of the measured quantities in computingthe .values of the depth and the velocity.

In closing, the authors wish to point out that theuse of the Nikuradse roughened surfaces as a seriesof roughness standards is not entirely satisfactory,since these standard surfaces cannot be duplicatedwith any degree of certainty. Either spherical orcubical elements fixed to a smooth surface at almown concentration could provide a reproducihleroughness standard for future needs.

Acknowledgments.

This paper ,vas prepared at the Institute ofHydraulic Research of the University of Iowa. Theauthors are indebted to Dr. Hunter Rouse for hisencouragement and advice, and to the U. S. Geolo­gical Survey, which provided partial financial sup­port for the project. The experiments were conduct­ed in part by A. L. Prasuhn, T. Lee, and K. K. Rao,and the manuscript was revised by J. B. Hinwood.Theil' help is gratefuIly acknowledged.

References.

[1] MILLIKAN (C. B.). A Critical Discussion of TurbulentFlow in Channels and Circular Pipes. Proceedings,5th Congress for Applied jj.[echanics, Cambridge, Mas­sachusetts, 1938.

[2] NIKuRADsE (J.). - Stromungsgesetze in rauhen Rohren.Forschungsheft 361, V.D.l., 1933; or NACA TM 1292,November, 1950.

[il] SCHLICHTING (H.). Experimentelle Untersuchungen zumHauhigskeitsproblem. Ingenieur-Archive, Vol. VII,No. 1, February 19:16; or NA.(~A. Tjj.[ 82:l, April, 19:17.

[4] IülLosEus (ILJ.) and DAVIDIAN (J.). - Roughness-Con­centration Effects on Flow over Hydrodynamically­Hough Surfaces. U.S.G.S. Water-Suppl!1 Paper (inpress).

[5] MOHms(H. M.). - A New Concept of Flow in HoughConduits. Transactions, A.S.C.E., Vol. 120, 1955.

[6] COLEBHOOK (C. l'.) and ·'VHITE (C. M.). - Experimentswith Fluid Friction in Houghened Pipes. Proceedings,Royal Soc. of London, ~eries A, Vol. 161, 19:17.

[7] HOBEHSON (J. A.). - Surface Hesistance as a Functionof the Concentration and Size of Roughness Elements.Ph. D. dissertation, University of Iowa, 1961:

[8] SAYHE (W. 'V.) and ALBEHTSON (M. L.). - HoughnessSpacing in Higid Open ChanneIs. Journal of tlle HU­draulics Division,A.S.C.E., Vol. 87, No. HY3, Proc.Paper 2823, May, 1961.

[9] I{oLoSEUS (H. J.). - Discussion of Houglmess Spacing inRigid Open Channels by \V. W. Sayre and M. L. Albert­son. Journal of the Hudraulics Division, A.S.C.E.,Vol.. 88, No. HY1, January, 1962.

LA HOUILLE BLANCHE/N° 7-1964

RésuméDes effets de la concentration d'éléments rugueux sur la résistance

parE. M. O'Loughlin and E. G. Macdonald *

La rugosité des parois ne peut pas être définie par la hauteur des aspérités seule; un paramètre tout aussI Im­portant est l'espacement des éléments rugueux. Schlichting [3'J a fait une série d'expériences dans lesquelles ilfaisait varier l'espacement de plusieurs sortes d'éléments rugueux et il a exprimé la perte de charge mesurée enfonction des dimensions d'uniformes grains de sables utilisés par Nikuradse [2J. Plusieurs auteurs ont contribuéaux résultats, qui ont été rassemblés par Koloseus et Davidian [4 J. Des mesures additionnelles de pertes de chargeont été faites par Koloseus et Daviadian pour des valeurs de la concentration allant jusqu'à soixante-quatre fois lavaleur minimale pour des éléments cubiques et cette gamme a maintenant été considérablement étendue par lesauteurs de cette étude. Les modèles d'arrangement des cubes sur une surface lisse sont présentés à la figure 2. Laperte de charge introduite par une surface lisse recouverte de grains de sable naturel pour des valeurs de la con­centration couvrant une gamme de une à quatre-vingts fois la valeur minimale a été considérée dans cette étude.Des observations supplémentaires ont été faites sur la fonne de la fonction de transition en comparaison avec lafonction de transition bien connue de Nikuradse; des übservations ont été faites également sur le coefficient deKarman pour des canaux découverts rugueux ainsi que sur la forme de la perte de charge pour de très faiblesvaleurs de la rugüsité relative.

La cüncentration des éléments cubiques utilisés était 0,11, 0,25, 0,44 et 0,70, la concentratiün À étant définiepar l'équation (1). Ces éléments de rugosité ont été moulés cn une seule pièce avec une base carrée d'aluminiumde 6 pouces (fig. 3). Le sable avec un diamètre équivalent de 0,0096 pied a été collé sur le fünd vitré du canalinclinable. Les caractéristiques des grains sünt présentées à la figure 4' et les photographies des silhouettes des cinqgranulations de sable à la figure 5. La concentration maximale de sable a été réalisée de façon à reproduire larugosité étudiée par Nikuradse.

Les mesures ont été effectuées dans deux canaux inclinables différents de 30 à 85 pieds de lüngueur. La hau­teur a été mesurée à l'aide d'un tube piézométrique et le coefficient de perte de charge a été calculé à partir dela hauteur normale évaluée. Le coefficient de résistance l/vTest représenté aux figures 7 et 8 en fonction de la rugo­sité relative pour les éléments cubiques et le sable, une courbe particulière étant tracée pour chaque concen­tration. L'équation (2) peut être ajustée pour chaque courbe, et si la répartition des vitesses de Nikuradse (Eq; 3)est intégrée le long de la hauteur, donnant l'équation (4), alors le coefficient de rugosité de Nikuradse, k" peutêtre calculé par les équations (2) et (4) eomme il a été fait pour l'équation (5). Le rappürt kslk dépend de la rugo­sité relative si la valeur exprimentale du coefficient. de Karman est ditrérente de la valeur du coefficient de Niku­raclse Kn• S'ils sont égaux, l'équation (6) est alors applicable.

La variation de kslk avec À, calculé à partir de l'équation (5), est présentée à la figure Il. Au-delà d'une con­centratioll d'environ ü,2, la dimensiün de la rugosité équivalente décroît à cause de l'interaction des éléments ru­gueux décrite par Morris [5J. La variation de décroissance de la dimension de la rugosité équivalente uniforme avecl'augmentation de la concentration est trouvée être fonction de la variation de l'effieurement de l'écoulement, décritaussi par Morris [5J. La forme et l'université des éléments et, à un moindre degré, l'arrangement influencent ledegré auquel l'effieurementde l'écoulement se développe. La non-uniformité des grains de sable arrête l'effieure­ment des aspérités par l'écoulement, et la rugosité équivalente reste par conséquent élevée au maximum de laconcentration. La non-uniformité relative des grains de sable résulte en un coefficient de résistance plus grandque celui de Nikuradse, comme. il a été observé aussi parColebrook et White ., [6J.

II apparaît que la fonction ,de transition équation (7), pour toutes les concentrations de sable, ressemble à lafonction de transition. de la rugosité des tuyaux rugueux. Ces courbes montrent. l'influence de la viscosité,mais ,iln'est pas apparent que ce soient les portions Jisses des parois oules changements possibles de l'écoulement autourdes éléments qui causent un changement des pertes de charge ,en fonction du nombre de Reynolds, La .forme géné-rale des transitions est en accord avec celle prévue par J~oberson [7J, •• ' ., , •• ,.' ','

Le choix d'une, cote, de référence pour le fond du canal cstarbitraire; la moyenne géométrique suggérée parSchlichting[3J ne donne pas satisfaction pour des concentrations élevées. L'introduction de la poche de sépa­ration comme paroi solide par. Koloseus semble plus réaliste, •• La définition de la cote du fond cie Schlichting aété utilisée pour cette étude, Les valeurs observées du coefficientde Karman pour les cubes et pour sable sont1)I'e'sente'es dalls le tabl 1 1 1··· ····'t t °33 '1:,{~,&y, "eau , a va eurmoyenne e an ,.<. •..' ", .' ' ," ,'., •. '. ' .•• ' ..'

Dans la figure 12, la différence avec la relation de, la résistance semi-logarithmique est présentée pour de fai-bles valeurs de, la rugosité relative. La présence de clapotis au-dessus de, chaque élément indique que de grandesnon-uniformités locales sont présentes dans l'écoulement, et il n'est pas justifié d'extrapoler l'équation (2) au-des­sous de, la lill1ite 4 yo/k == 12.

* Institllte of Hydralllic Research University of Iowa.