THIS PAPER

INTRODUCTION

SIMILARITY OF DISTORTED RIVER MODELS WITH MOVAELE BED

H. A. Einstein,l M. ASCE, and Ning Chien,2 A.M. ASCE

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SYNOPSIS

The similarity conditions for distorted river models with movable bed arederived from the theoretical and empirical equations which have been found to

I describe the hydraulics and the sediment transport in such rivers. A complete[ numerical example is added to demonstrate the method of application to a par-ticular river.

There are many hydraulic engineering problems for which the basic equa-tions are known but which are geometrically so complicated that the direct ap-

i plication of these equations becomes impossible. Many such problems can be! solved today by the use of models which are shaped to duplicate thecomplicat-ed geometry and in which the resulting flow patterns can be observed directly.

: Such a model permits the prediction of the corresponding prototype flows: quantitatively only if the exact laws of modelstmtlartty are known. The pre-~diction of the model scales cannot be based on simple dimensional consider-a-~tions if the model is distorted and includes the motion of a movable bed. A"different approach must be used to find compatible systems of model scales, and distortions.I In 1944, the senior author of this paper published a short account(l) of howcompatible systems of scales can be found. He proposed in that paper the der-

, ivation of model scales from empirical working equations rather than from the: underlying differential equations of motion. It was pointed out there that equa-tions must be used which are applicable in the same form both to model and. prototype, and which should preferably have the form of power functions. Thesame approach will be used in this paper. Many new developments in the de-l scription of flow and sediment transport in alluvial rivers will be incorporated.I Before the conditions of similarity can be stated .it is necessary to definethe exact meaning of this term. Let us define that similarity may be said to: existI 1) if to each point, time and process in one scale, which may be called pro-I totype, a corresponding point, time and process of the other scale, which! may be called the model, can be coordinated uniquely;

2) if the ratios of corresponding physical magnitudes between model andt prototype are constant for each type of physical magnitude .

1. Prof. of Hydr. Eng., Univ. of California, Berkeley, Calif.

1

2. Asst. Research Engr., Inst. of Eng. Research, Univ. of California, :Berkeley,~------------------------------------------------~ Calif.I 566-1

This implies, for an undistorted model, that all magnitudes of equal dimensimust follow the same scale ratio; also, the various laws which interrelate thevariables parameters and constants of the prototype must apply to the modeltoo. The 'results derived Irom the operatian of such a hydraulic model maythen be tra.nsferred to the prototype by the use of these scale ratios.

A distortion in the physical sense exists if there are two types of variablesor parameters of equal dimension operative in the same problem which arephysically sufficiently independent such that they can be given different scaleratios.

The applicatian of distortians in engineering is not new. Many engineersapply daily distort.ions in the presentation of flat sections and profiles. Thedesirability of such distortion in hydraulic models has become apparent toeveryone who has tried to design a model of wide and shallow w~ter~ourses ~ 6)which the large hor-izontal dimensions call for a model scale WhIChIS muchsmall far application to the vertical direction. Such small water depth wouldvery often cause laminar flaw in the model which cannot be used to duplicatethe turbulent prototype flow. Many laboratories have built extremely valuabhmodels in which the vertical scale is much larger than the horizontal scale.JIt is impossible, however, to introduce one distortion alone. This becomesparent by the fact that, for instance, the slopes are distorted in the same de-gree as the vertical heights. It is generally known that the water depths andthe slope enter most of our friction formulas. The velocity scale must be It has already been stated that the flow and sediment description of the al-chosen to satisfy the Froude condition and will satisfy the friction equation 0 luvial reaches which are to be studied by the model must be accomplished byly if the roughness of the various boundaries is also. properly dist~rted. W~ pdentical formulas in model and prototype. These formulas can be trans-will see shortly that in the case of a sediment carrying stream strll more dlSjfOrmedinto relationships between the various ratios and thus represent condr-tortions must be introduced to balance their effects in the various equations tions which must be satisfied when the various distortions are chosen. The in-describing flow and sediment motion. dividual relationships shall be discussed now using the various equations as

they are given in Ref. 2. It is not believed that the choice of these equations inthe exact form of Ref. 2 is a strong restriction to the generality of the method.Even if the reader prefers to substitute different formulas for the ones pro-

In the following discussion, the symbols with the subscripts, p and M, will posed here, he will find that he is not able to. change the number of equationsindicate quantities, in prototype and model, respectively. The ratio of correllwhichmust be satisfied in a particular problem. From this follows that he willponding values in the two scales will be denoted by a subscript r. Thus, the have the same number of conditions and, therefore, the same number of de-ratio of depth scales, for instance, is defined as hr = hp/hM; Similarly, the grees of freedom in choosing his ratios. He will find that his substitute equa-following ratios are introduced: tions will contain the same variables and that the entire difference will be a

Lr ratio of horizontal Iengths slightly changed set of exponents. These equations and the underlying criteriahr ratio of vertical lengths areDr ratio. of grain diameters a. Friction Criterion: It has been pointed out in Refs. 2, 3 and 4 that theSr ratio of slopes, particularly energy slopes flow in an alluvial channel cannot be described generally by one formula withVr ratio. of horizontal flow velocities universal constants, but that it must be interpreted as a composite effect, the

(~ - f. ) ratio of sediment densities under water with ff assumed to be arious parts of which follow different and independent laws. It can be showns t r: equal in model and prototype easily that no distorted model is possible if the friction criterion is formulatedtlr ratio of llydraulic times las an identity, i.e., if one stipulates that both the grain resistance (surfacet2r ratio of flow durations (sedimentation time) rrag) and the bar resistance (shape resistance) must both individually be simi-qBr ratio of bed-load rates (weight under water per unit of time and I~ar. Actually this is not necessary for an overall similarity as long as tile to-

width) fal friction behaves similarly. It is proposed, therefore, to base the Ir-ict ionalratio of total-load rates (weight under water per unit of time andsimilarity on the behavior of the entire section for which it assumes the formWidth) pf a r-ating curve. As most rating curves give approximately a straight Line if

Vsr ratio of sediment settling velocities ~hetotal discharge Q is plotted against the stages on a log-log sheet, it is as-Cr ratio of constants C (generalized Manning equation) sumed that the prototype and the model channel can be described by an equation,?,. ratio of Ri/RT - values ,pf the formThe large list of variables and scales already implies that a number of dl\

tortions will be contemplated. I

Namenclature and Distortions

566-2

1) If Lr is independent from hr, the model is vertically distorted.2) If the grain size ratio Dr is different from both Lr and hp a third length

scale is introduced and with it a second distortion.3) If Sr is chosen independent from Lr and hr , the model is assumed to be

tilted in addition to the other distortions.4) If the ratio of effective densities of the sediment (jj - if)r is assumed to

be different from the ratio of the fluid densities ftr which is unity, thenthat represents a fourth distortion.

5) A fifth distortion is introduced if the time scale tlr far the time valuesinvolved in the determination of velocities and sediment rates is chosendifferent from the ratio t2r of durations for individual flow canditions,indicating the speed at which flow duration curves are duplicated.A sixth distortion is introduced because of the impossibility to. obtainsuspended-load rates in a model at the same scale at which the bed-Loadrates are reproduced, making qB~ different from qT .

7) A seventh and last distortion permits the ratio of setlling velocities Vsrof corresponding grains to be different from the ratio of correspondingflow velocities.

Relationships Describing Alluvial Flows

v c .f2 Vz. ( 12 +"')0"" 5 h

566-3

(1)

which is a generalized Manning equation. Eq. (1) become~ iden~ical Wil'6t~51r..m.odels, as it safe-guards similar flow and energy loss around structures andManning equation by taking m " 1/6 and by using the r-elat.ionship n-vD other channel irregularities .If it is assumed that the same exponent m can be used to describe both the. There appears to be a possibility that Froude criterion loses its signifj-prototype and the model relationships, at least for the most important range~.cancein deep rivers when the Froude's number becomes very lowof discharges, the equation can be written between ratios as t( F = V2IjD I). The only effect of velocity changes seems to be in

. Ithat case an energy loss which is taken care of elsewhere. No experienceZ -I J,-/-2,." 0bot C -z = LJ (4 exists on this part, however.

v,. S,.,. ,." ..v. I Froude law may be written in ratios asI -llz

The various ratios on the left side of eq. (A) are explained preVlously: ~he~ V,. n,. = LlFvalue .av equals unity if the similarity is exactly satisfied, but may mdlca~a small deviation from the exact similarity if such a deviation becomes nee!lsary for any practical reason. ~Here again stands AF " 1 for exact similarity while a deviation from unity

While V r, Sr' hr and Dr are ratios which will recur in other equations, mfIileasures a possible necessary deviation from the exact solution.and Cr are the exponent and the ratio of the constants in generalized eq, (l)l c. Sediment transport criterion: In order to have similar sediment trans-the form. port conditions near the bed it is, ingeneral, necessary that both ffl: and "f/;.

rr are equal in model and prototype since the two are not connected by a power-V/ jRr5(1 C (Rr / Ks ) ((typeequation. Only if the transport rates are restricted to a very narrow

[r.angeof values is it possible to combine the two conditions into one. The. !equality of the values .

where RT is the hydraulic radius of the total section with the bottom width ~,wetted perimeter and Ks the grain size of the bed representative for its gr~.roughness. If one may assume similar grain mixtures are used in model asiexist in the prototype, the ratio of the Ks values equals that of the grain siz\D. The values C must be determined individually for model and prototype :Jthe ratio found for a representative average channel section. Using the no liSpossible for all fractions of a mixture only if the two mixtures are similar,clature of Ref. 2 to 4 t..UChthat the ratios of the i-v~lues become equal to unity. With It equal fn

RT = AT/fi ( A~ -r Ab" -t- Aw) lfi rOde I and prototype the equation of equal ~A -values may be written as

(Rb' p" + R;' Pb-t Rv.""",,)/ Pb (\ q (() )-3/2 3e

.

1.>8r J,5 - f:t r: Dr = /R; -i: Rf," -t R"" P""/Ii

No deviation from this equation is usually desired. The sediment rate q isThe model values of C and m can be determined by a trial-and-error methop1easured in weight under water. Bonly, as they depend on the choice of the remaining scale ratios. This procf d. Zero. Sedime~t-Ioad Criterion: The equality of -v; -values in model anddure will be explained in the example of a "Big Sand Creek" model. prot~~pe 1Soften mt.er~reted by engineers as the condition of similar flow

b. Froude Criterion: In open channel flows, Froude's law is one of the ,ond1hons at the begmnmg of sediment motion. It can be expreased as equalcriteria because it balances the gravitational forces against the inertia forCj:~ values. .This can easiest be expressed by the corresponding energies. Since the en~ .content of the flow may be divided into its kinetic and potential parts as velo;ity head and depth, both are correlated by the equation !

. t

ConS+onf ~e order to analyze the s~nifi.cance of all the corrections outside the brackets,rt us remember. that J 1~different for the various grain sizes of a mixture.Even if in river channels the water surface slope is often very small, one ntr the larger srzes, both m model and prototype, J has the value 1. In or-

tremember t~at it. is the graVittatiOnal forllce,whiClhmainttai~s tthe flow

f .Furt~er to have Similarity the ratio of J -values must be unity for all ~izes.

more, any rrver improvemen plan usua y mvo ves cer am ypes 0 r-iver e~. l:. .strfction work by the construction of training walls, jetties, and groins alon!mc~ d 1Sa function of nix, Dr must equal Xr. Referring again to Ref. 2,the banks. At the constricted sections and especially in the neighborhood of e fmd thatstructures, the elevation of water surface may change rapidly. It is, theref~advisable to include the Froude law as one of the criteria in designing river!

I566-4 i

l566-5

(B)

(4)

(C)

(5)

tor Ll/{J

a>I00

(!j-ft)r * * *. . .Eq. :L h V s. D

a>Cl'lCl'lI....o

I~r

.(l 4~ L\ L\Scale Ratio Symbol L C Br r F N V

Horizontal lengths L chosenr

m+l -2 -m 2 2m -(m+l) -1Vertical lengths h 4m+l 4m+l 4mH - 4m+l 4mH """"4iii+:L 4m+lr

Flow velocity V m+l -1 -m2(2m+l) m -(m+l) -1

2(4m+l) hm+l 2(4m+l) - """4iittl I+m+l 2(4m+l) 2(4111+1)r-3m -2 -m 2 2m 3m -1Slope S 4m+l 4m+l 4m+l - 4m+l 4m+l 4m+l 4m+lr

2m-I 2 -(2ml-l) -2 2m+l I-2m 1Sediment size D 2(4m+l) hm+l 2(4m+l) - 4m+l 4m+l 2(4mt-l) 4m+lrSediment density 3(1-2m) -6 3(2m+l) 6 2m-I 3(2m-l) -3

(fs -Jj}r 2(4m+l) 4m+l 2(4m+l) - 4m+l 4m+l 2(4m+l) 4m+lunder \laterBed-load rate per - -6 3(2m+l) 6m }(2m-i)3(1-2m) 6 -3unit width and time '

The Big Sand Creek Model

First" the friction conditions in the prototype are summarized in Table illbased on Table 6 of Ref. 2. With S " 0.001050 and D65 " Ks = 0.00115 ft. thefollowing values are obtained.

Table III

Friction Conditions - Big Sand Creek-" " 'lp= ~/R.r, IIb ~ R.r v R.rSg VKs ~ 1\/K;> Q

ft ft ft ft/sec VT vZ cfs.

050 0.86 130 292 0.384 0.00515 1130 0.00198 435 4091.00 0.76 192 4.44 0520 0.00330 1670 0.00172 870 11102.00 050 306 6.63 0.653 0.00236 2660 0.0015).j.1740 3710300 030 4.40 8.40 0.682 0.00211 3820 0.00144 2610 91604.00 0.14 573 992 0.698 0.00197 4980 0.00137 3480 19850500 0.07 704 1130 0711 0.00186 suo 0.00132 4350 35700

il

A similar hydraulic pattern for the model is now obtained by trial and er-ror. First the length scale Lr = 150 is chosen on the basis of available roomThen the prototype friction is plotted in Fig. 1 as ~ against ~/Ks t

(~ Sg)(o-symbol) :md the poin~s for. the higher discharges approximated by the line~A(ifA(I) WIth the relationshtp " r

v2 e; o,31;:! !RTSl = 21.05 ( Ks ) (9)

Next,.th. model resrstance Is estimated to ',allOWthe line B(lfB(ll with th.1equation I

v2 (Rr)O.37Z t"= .3G176 -RTSI Ks (10\

From these tworelattonshtps (9) and (10) one may find the values of C Jm:

Cp = 4.595 C/VI =5.55 C = 0.827 ".,= 0./86

Assuming Y(r = 1 and all Ll -values = I,

566-12

rn-f-t -2 -ITlh = L ~ C ~ n,,4..,,+-'r r ,. \..

(11)

~iIII

0.68 _/.,471.50 . 0.1)27 37.3

-3m -z -"..,S. = L 4",+, C ~ T]~ 4",+,r r r: (, _o.~Z -1.147ISO 0.827 0.25

2,"-1 2 -(Z,"+')D. = L Z{4MH) C ~ z{iiM+ij _ _0.113 ,,/41r: r: r '1r -150. 0,/jZ7 = o,3z6

-3= 0,. = 28.0

From these ratios results the following model

S. = 0.00109' = 0.00.4'" P 5""a/cM Q.Z5 ~ .Is,., = /.0 v /r c.e ,

The model h~draulics of the first approximatioI!. is then given by Table IV. R'values used In Table IV correspond to those used in the prototype using nIb= 37.3. (""r

Table IVHod 1 Hyd ulie ra cs - First Approximation

I I s I Log1OI\1\ u* D65 x Ll v y' vft ft/$, ft T ft/s ,-ft (12.27 A) u*0.0134 0.0425 0.00289 1.22 1.59 0.00221 1.872 0.458 302 16.800.0268 0.0601 0.00204 172 1.46 0.00241 2.135 0.738 151 2700.0536 0.0850 0.00145 2.44 1.27 0.00277 2376 1.160 072 51.00.0805 0.1042 0.00118 298 1.18 0.00298 2520 151 050 8700.1072 0.1202 0.00102 3.44 1.14 0.00309 2.630 1.82 038 1500.1342 0.1348 .0.00091 3.86 1.10 0.00320 2712 2.10 030 240

Table IV(cont'd)

" "u * 1\ 1\ stage R.r R.rSg RT Ib/I\r=f~/s ft ft ft '7' Ks '1.,., >(p '?r

0.0273 0.0055 0.0189 4.010 0.020 0.0129 568 0.670 0384 05730.0273 0.0055 0.0323 4.034 0.0375 0.0093 10.64 0714 0520 07300.0227 0.0038 0.0574 4.086 0.075 0.0076 213 0716 0.653 09110.0174 0.0022 0.0827 4.172 0.1240 O.007~352 0.650 0.682 1.0500.0121 0.0011 0.1083 4.260 0.1650 0.0068 46.4 0.650 0.698 1.0730.0087 0.0006 0.1348 4340 0.2000 '0.0061 56.8 0.672 0711 1.060

566-13

The values of the wall friction, of stage and RT are determined by a trial and .error method given in Table V and figure 2. First, the model bank roughness AT= Ab Pt; -t Rw P"" for various fictitious values of the stage. These ATmust be chosen. With the prototype banks rather rough, it is assumed that the values for each Rb value are connected by a dashed line .. The intersectionManning formula applies in model and prototype. The wetted perimeter of the; with the solid AT curve indicates the actual stage at which the Rb value occurs.banks was assumed in the prototype calculation (Ref. 2) to equal twice the wa.. ' These values of RT are used in Table IV to determine 1,- for the first approxi-ter depth. In order to make the wall friction assume an equivalent part of the matlon. Also RTSg/V2 is calculated and plotted in Fig. 1 against RT/Ks' Ittotal friction in model and prototype, we write may be seen in this graph that the first assumed line B(l) - B(l) was some-

what too high and too steep to describe the model points.Thus. a second approximation of a line A(2) - A(21 for the prototype and par-

allel to It B(2) - B(2) for the model was tried. The ine A(2) - A(2) does notcover the two smallest prototype discharges, is therefore not correct but atthe larger flows. This is a minor defect, however, as most of the transportand of the bed changes occur at the medium and high flows.

The second approximation is based on A(2) - A(2) for the prototype frictionTable V

Determination of ~ by Trial and Error Methodl\, R Stage Pb P Pb~ PR Atw w \vwft :ft ft ft ft' ft2 ft2 ft2

(4.000 050 - .00945 - 0.00950.0189 0.0332 (4.050 0.86 0.07 .0162 .0023 0.0185(4.100 1.22 017 .02285 .00565 0.0285(4.000 050 - .01615 - 0.0162

0.0323 0.0672 (4.050 0.86 0.07 .0278 .0047 0.0325(4.100 1.22 0.17 .0392; .0114 0.0508(4.050 0.86 0.07 .0493 .0094 0.0587

0.0574 0.1340 (4.100 1.22 017 .0700 .0228 0.0928(4.150 157 0.27 .0900 .0362 0.1262(4.100 1.22 017 .1010 .0338 0.1348

0.0827 0.l985 (4.150 157 0.27 .1300 .0536 0.1836(4.200 195 037 .1612 .0735 0.2347(4.200 1.95 037 .2110 .0978 03088

0.1083 0.264 (4.250 236 0.47 .2560 .1240 03800(4.300 2.85 057 3085 .1508 0.4593(4300 2.85 057 3840 .186 0570

0.134B 0326 (4350 336 0.67 .4530 .218 0.671(4.400 387 077 5220 .251 0773

In the Manning equation for the banks V. - .L: R l/3 S. 'IZ t b, r: - noM ""r r. we ge nWr y

Introducing previously determined values and ,ar = 1:

It can thus be calculated noN.=~ - 2.305 and n = 0.050 = 0 0217 andr ~ w,., 2.305 .R - ( V"w J~- vi\o~w - t.4865vz - .9.34 I

Fig ', 2 gives first the curves of Pw' Pb. ~ and AT as functions of the stagall denved from corresponding prototype curves and Similarity scales (solidcurves), For each calculated value of Rb Table V'then determines values of

566-14

(12)

(13)

r :-

i,-

R' s.zeo= 44.5 ( K"') .

:.s (15)

and on B(2) - B(2) for the model friction

(Rr)o.Z~O

=52.6 Ks (16 )

From these one obtains '

} (17)I'YJ = 0.1.45 7r = /.0 10r L~ - /soThe same formulas as quoted in the first approximation give

hr =4Z.25 15r = O.;UjZ

Dr = o.eee:(r5-!!)~ = 40Vr - 6.5

(Hl)

This results in the following model dimensions

(141

5,.., = 0.00;37206.5..,= 0.0039>4 ff.

035..,= 0.00''2 ~ 1 (19)f'sM = 1.04/ ;JT-/c.c-.

Anew calculation of the hydraulics on this basis gave the (x) points of Fig. 1which still follow with sufficient accuracy eq. (16). The results of eq. (18) and(19) are thus assumed to be usable together with

566-15

The reader has by now probably received the impression that all the com-plications involved in the method have had no other effect but to make the sys-tem of similarities more and more unreliable. Such a statement is definitelyunfair. But what the method does is to show that a distorted river model is inthe best case an acceptable compromise which will permit the solution of cer-tain problems which otherwise can not be solved except by experimentation inthe prototype, which is under all conditions more expensive. This method ofdesigning a model has one great advantage over all other such methods whichare available at the present: It permits the prediction of its reliability atleast in a qualitative way, with respect to the choice of ). -values it gives suchdeviations even quantitatively.

What are the most important reasons for the loss of similarity?1. In the description of the channel friction the modified Manning equation

can not be expected to describe with equal m-values both model and prototype.conditions over the entire range of discharges. It will be necessary, there-fore, to permit deviations in the friction relationship for the less Importantflow conditions. This fact is most important if a large r ange of flow conditionsexists in the problem area.

2. If flows must be considered in more than one channel, it becomes evenmore difficult to find a friction equation which describes all flows.

3. The total load rates must be used to determine the time ratio t2 for flowdurations. Since the bed-load rates must be made similar to obtain similarbed configurations and since the ratio between bed-load rates and total-loadrates changes with the stage, a sliding time scale appears to become neces-sary in many cases, especially if the range of discharges is large. The deter-mination of this scale becomes somewhat indeterminate if flowin more thanone channel is important.

4. If the deposition of sediment in low velocity areas, such as overbank orill reservoirs, is an important phase of the problem under investigation. thewash load must" be introduced into the model. Its characteristic is not givenby the ability to be transported as bed load, however, but by its ability" to stayill suspension permanently in the flow of the overbank areas.

Because of these difficulties it is absolutely- necessary to verify any suchThis time scale changes much with the stages and it appears necessary to ex model and its scales. Such a verification consists in the reproduction of atend the flood stages percentagewise over the low stages, a fact well known to knownprototype development in the model by similar flows. Only if such athe practical experime_nter. i1verification is possible and successful can the model be depended upon for the

prediction of future developments. It is common experience that the gr-eatestdifficulty of most model studies of this type is the gathering of the necessaryprototype information for the construction, the operation, and particularly, theverification of the model. But it is felt that this method of designing modelscales is able to shorten the time consuming trial and error method of findingthe proper model scales so much that more effort can be spent on a. re lfahleverification and the determination of the underlying river Informatlon,

L,. = 150

So far, the equations A, B, D, E and I have been used; eq. G gives

t1r = t.; v,.. -I = 2.3. I

and eq, C gives

The value of qTr is determined by eq, F which calls for the knowledge of theratio B. These are determined in Table VI which lists first the calculatedprototype total load Z 'r Qr in tons per day, as determined in Table 8 ofRef. 1.

~ Table VIj Determination of ratios BI'

1

(Z;':T1rt/(E'~ 1-s)pFb (X ,'T6lr)p (Z':r1"r)p (-ZOT't"')M t2 1 yearp 6:.,.'lT>"1/(E,,1~M r dupli-ft. tonS/day Lbs, under Lbs. under catedwater per water per in hourssec. ft. sec. ft., ; B05 100 09042 0.00113 1.)6 4650 1.891.0 3190 0.420 0.00555 1.89 3350 2.622.0 33600 2.61 0.0184 363 1145 50230 156000 9.11 0.0)86 590 1015 8.154-.0 538000 223 0.0685 8.15 118 11.28

50 143)000 46.2 0.1090 10.6 598 l4-.65

From a similar table the corresponding-model values have been calculated anare given in the next column. The ratio of (ri(31Je)pl(:E~8 ~&)M is equal toqBr since corresponding iB values are equal to prototype and model.

Eq. F and H permit now to determine t2r as

634Q8

Significance of the ). -Values

No use has been made so far of the.6. -values. Their Significance may besbe seen starting Ir om the solution found as the second approximation. In thissystem a material is required for the model sediment of a specific gravity of1.041. Let us assume no such material is available, but a material with the

566-16

specific gravity of 1.045 can be obtained. What would be the effect of su~h a. deviation from the required values? This can easily be found by the enoreeof one of the b. -values which appears to be least critical in the particular

11 problem for the absorption of the deviations. The system of equations issolved again for the assumed values of Lr and (fs - ff)", introducing the chos-en !:::. as additional variable. One finds then that an entirely new system ofscales will result together with the magnitude of the deviation which the chos-enL'l.-value must undergo to satisfy the remaining equations.

Reliability of the Method

566-17

ACKNOWLEDGMENT

This work represents results of research carried out for the Corps of Engineers, Missouri River Division, Omaha, Nebraska, under contract with the ~-IbUniversity of California.

.qB1. Conformity Between Model and Prototype: A Symposium, Trans. ASCE, qT

Vol. 109, 1944, p. 134. ~

2. The Bed-Load Function for Sediment Transportation in Open Channel R'Flows, by H. A. Einstein, U. S. Department of Agriculture, Technical Bul ;;'bletin 1026, Sept. 1950. Rb.

3. River Channel Roughness, by H. A.Einstein and N. L. Barbarossa, TranS't--R.rASCE, Vol. 117, 1952, pp. 1121-1146. R

w4. Linearity of Friction in Open Channels, by H. A. Einstein and R. B. Banks"SAssociation Internationale d'Hydrologie SCientifique, Assemblee Generalelde Bruxelles, 1951, pp. 488-498. tl

5. Laws of Turbuient Flow in Open Channels, by G. R. Keulegan, U. S. Nation t2al Bureau of Standards, Rept. 1151, 1938, p. 738. u'

*

AT

AwA'b

~B

C

D

D35

D65g

h

\~~KsL

m

REFERENCES

APPENDIX V

Vsx

List of Symbols

total area of a cross section

part of the cross section pertaining to the banks

cross-sectional area pertaining to the grain

cross-sectional area pertaining to irregularities

qT/qBrconstant in the generalized Manning's equation

grain size

grain size of which 35 percent is finer

grain size of which 65 percent is finer

gravitational acceleration

vertical lengths

fraction of bed material in a given grain size range

fraction of bed load in a given grain size range

fraction of total load in a given grain size range

roughness diameter

horizontal lengths

exponent in the generalized Manning's equation

566-18

x

friction factor in the Manning's equation

friction factor (Manning) of the banks

wetted perimeter of the bed

wetted perimeter of the banks

flow discharge

bed-load rate in weight under water per unit of time and width

corresponding total-load rate

total sediment load in cross section

hydraulic radius with respect to the grain

hydraulic radius for channel irregularities

hydraulic radius of the total section

hydraulic radius with respect to the bank

slope

hydraulic time

sedimentation time

shear velocity with respect to the grain

shear velocity for channel irregularities

horizontal flow velocity

settling velocity of a sediment particle

parameter for transition smooth - rough

characteristic grain size of a mixture

pressure correction in transition smooth - rough

the thickness of the laminar sub-layer

the apparent roughness diameter

deviation from the Similarity law "a"

~/RTkinematic viscosity

"hiding factor" of grains in a mixture

density of the fluid

density of the solids

intensity of transport for individual grain size

intensity of shear on representative particle

intensity of shear for individual grain siz e

Subscripts

indicating ratiorefers to prototyperefers to model

566-19

4.3 I I f I I

t-u,

c ~.2

c.na:>O"lIN,...

l.Ll~t-Cf) 4. I --AT = Pb Rb+ Pw Rw

4.0 I II? IL I I I I I Io 0.1 0.2 0.3 0.4 0.5

AT in SO. FT.0.6 07 0.8

I I I I I I I I I

o 0.5 /.0 1.5 2.0 2.5 3.0 3.5 4.0Pb and Pw in FT.

Fig.2 Big Sand Creek Model-Trial and Error Solution For

Thp Determination Of R~

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VOLUME 80 SEPARATE No. 567

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FLOW EXPANSION AND PRESSURE

RECOVERY IN FLUIDS

by Harold Tults

HYDRAULICS DIVISION

{Discussion open until April 1, 1955}

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