Similarity of Distorted River Models With Movable Bed [PDF]

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SIMILARITY OF DISTORTED RIVER MODELS WITH MOVAELE BED THIS PAPER H. A. Einstein,l --represents an effort by the SocIety to deliver technical data direct from the author to the reader with the greatest possible speed. To this end, it has had none of the usual editing required in more formal publication procedures. Readers are invited to submit discussion applying to current papers. For this paper the final date on which a discussion should reach the Manager of Technical Publications appears on the front cover. Those who are planning papers or discussions for "Proceedings" will expedite Division and Committee action measurably by first studying "Publication Procedure for Technical Papers" (Proceedings - Separate No. 290) .. For free copies of this Separate-describing style, content, and format-address the Manager,' Technical Publications, ASCE. Reprints from this publication may be made on condition that the full title of paper, name of author, page reference (or paper number), and date of publication by the Society are given. The Society

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This paper was published at 1745 S. State Street, Ann Arbor, Mich., by the American Society of Civil Engineers. Editorial and General Offices are at 33 West Thtrty-ntnth Street, New York 18, N.Y.

M. ASCE, and Ning Chien,2 A.M. ASCE

SYNOPSIS The similarity conditions derived from the theoretical I describe the hydraulics and [ numerical example is added ticular river.

for distorted river models with movable bed are and empirical equations which have been found to the sediment transport in such rivers. A complete to demonstrate the method of application to a par-

INTRODUCTION There are many hydraulic engineering problems for which the basic equations are known but which are geometrically so complicated that the direct api plication of these equations becomes impossible. Many such problems can be ! solved today by the use of models which are shaped to duplicate thecomplicated 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 how compatible 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 equations 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. The same approach will be used in this paper. Many new developments in the del scription of flow and sediment transport in alluvial rivers will be incorporated. Before the conditions of similarity can be stated .it is necessary to define the exact meaning of this term. Let us define that similarity may be said to : exist 1) if to each point, time and process in one scale, which may be called prototype, 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 and t prototype are constant for each type of physical magnitude . I

I I

~------------------------------------------------~

•1. Prof. of Hydr. Eng., Univ. of California, Berkeley, Calif. 2. Asst. Research Engr., Inst. of Eng. Research, Univ. of California, Calif.

I

1

566-1

:Berkeley,

1) If Lr is independent from hr, the model is vertically distorted. This implies, for an undistorted model, that all magnitudes of equal dimensi must follow the same scale ratio; also, the various laws which interrelate the 2) If the grain size ratio Dr is different from both Lr and hp a third length variables parameters and constants of the prototype must apply to the model scale is introduced and with it a second distortion. too. The 'results derived Irom the operatian of such a hydraulic model may 3) If Sr is chosen independent from Lr and hr , the model is assumed to be then be tra.nsferred to the prototype by the use of these scale ratios. tilted in addition to the other distortions. A distortion in the physical sense exists if there are two types of variables 4) If the ratio of effective densities of the sediment (jj - if)r is assumed to or parameters of equal dimension operative in the same problem which are be different from the ratio of the fluid densities ftr which is unity, then physically sufficiently independent such that they can be given different scale that represents a fourth distortion. 5) A fifth distortion is introduced if the time scale tlr far the time values ratios. involved in the determination of velocities and sediment rates is chosen The applicatian of distortians in engineering is not new. Many engineers apply daily distort.ions in the presentation of flat sections and profiles. The different from the ratio t2r of durations for individual flow canditions, indicating the speed at which flow duration curves are duplicated. desirability of such distortion in hydraulic models has become apparent to everyone who has tried to design a model of wide and shallow w~ter~ourses ~ 6) A sixth distortion is introduced because of the impossibility to. obtain suspended-load rates in a model at the same scale at which the bed-Load which the large hor-izontal dimensions call for a model scale WhIChIS much small far application to the vertical direction. Such small water depth would rates are reproduced, making qB~ different from qT . very often cause laminar flaw in the model which cannot be used to duplicate 7) A seventh and last distortion permits the ratio of setlling velocities Vsr the turbulent prototype flow. Many laboratories have built extremely valuabh of corresponding grains to be different from the ratio of corresponding models in which the vertical scale is much larger than the horizontal scale.J flow velocities. It is impossible, however, to introduce one distortion alone. This becomes parent by the fact that, for instance, the slopes are distorted in the same deRelationships Describing Alluvial Flows gree as the vertical heights. It is generally known that the water depths and the 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 alchosen 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 by ly if the roughness of the various boundaries is also. properly dist~rted. W~ pdentical formulas in model and prototype. These formulas can be transwill see shortly that in the case of a sediment carrying stream strll more dlSjfOrmedinto relationships between the various ratios and thus represent condrtortions must be introduced to balance their effects in the various equations tions which must be satisfied when the various distortions are chosen. The indescribing 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 in Namenclature and Distortions the 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 proIn 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 equations indicate quantities, in prototype and model, respectively. The ratio of correllwhich must be satisfied in a particular problem. From this follows that he will ponding 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 deratio 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 equafollowing 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 criteria hr ratio of vertical lengths are Dr ratio. of grain diameters a. Friction Criterion: It has been pointed out in Refs. 2, 3 and 4 that the Sr ratio of slopes, particularly energy slopes flow in an alluvial channel cannot be described generally by one formula with Vr 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 assumed to be arious parts of which follow different and independent laws. It can be shown s t r: equal in model and prototype easily that no distorted model is possible if the friction criterion is formulated tlr ratio of llydraulic times las an identity, i.e., if one stipulates that both the grain resistance (surface t2r ratio of flow durations (sedimentation time) rrag) and the bar resistance (shape resistance) must both individually be simiqBr 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 towidth) fal friction behaves similarly. It is proposed, therefore, to base the Ir-ict ional ratio of total-load rates (weight under water per unit of time andsimilarity on the behavior of the entire section for which it assumes the form Width) 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 asCr 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 form c Vz. ( 12 +"') The large list of variables and scales already implies that a number of dl\ (1) tortions will be contemplated. I 0"" 5 h

ff

v

566-2

.f2

566-3

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 and Manning 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 signifjprototype and the model relationships, at least for the most important range~.cance in deep rivers when the Froude's number becomes very low of discharges, the equation can be written between ratios as F = V2IjD« I). The only effect of velocity changes seems to be in that case an energy loss which is taken care of elsewhere. No experience Z -I J,-/-2,." 0 bot C -z = LJ (4 exists on this part, however. S,.,. ,." .. v. Froude law may be written in ratios as

t(

.I

I

v,.

I

n,.-llz

The various ratios on the left side of eq. (A) are explained preVlously: ~he~ V,. = LlF (B) value .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!l sary 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, c. Sediment transport criterion: In order to have similar sediment transthe form. port conditions near the bed it is, ingeneral, necessary that both and "f/;. rr» are equal in model and prototype since the two are not connected by a powerV/ 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 asi (4) exist 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

(l)l

ffl:

:J

the ratio found for a representative average channel section. clature of Ref. 2 to 4 RT = AT/fi ( A~ -r Ab" -t- Aw) lfi (Rb' R;

p" + R;' Pb -t Rv.""",,)/

Using the no liSpossible for all fractions of a mixture only if the two mixtures are similar, UCh that the ratios of the i-v~lues become equal to unity. With equal fn rOde I and prototype the equation of equal ~A -values may be written as

t..

Pb

It

(\

q

(()

1.>8r J,5 -

)-3/2

f:t r:

3e

(C)

Dr· = /

-i: Rf," -t R"" P""/Ii .

No deviation from this equation is usually desired. The sediment rate q is The model values of C and m can be determined by a trial-and-error methop1easured in weight under water. B only, as they depend on the choice of the remaining scale ratios. This procf d. Zero. Sedime~t-Ioad Criterion: The equality of -values in model and dure 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 equal criteria 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; (5) ity head and depth, both are correlated by the equation ! . t

-v;

ConS+onf

Even if in river

er

~

order to analyze the s~nifi.cance of all the corrections outside the brackets, t us remember. that 1~ different for the various grain sizes of a mixture.

channels the water surface slope is often very small, one nt°r the larger t

J

srzes, both m model and prototype,

J

J

has the value 1. In or-

remember t~at it. is the graVittatiOnal forllce ,whiClhmainttai~s tthe flow · .Furt~er to have Similarity the ratio of -values must be unity for all ~izes. more, any rrver improvemen plan usua y mvo ves cer am ypes 0 fr-iver e~. l:. . strfction work by the construction of training walls, jetties, and groins alon!mc~ d 1S a 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 that structures, 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!

566-4

I i

l

566-5

tor tor

o·7'7A

y =

and with

A =

KsI.x.

This time ratio must be such that corresponding time intervale are required by corresponding sediment rates qT to fill corresponding volumes. Expressed (6)' in ratios this equation can be written for the unit width as

Ll/{J

I

Symbol

h

Horizontal. lengths

L

4m+l m+l

Vertical lengths

h

Flow velocity

V

r

'2

Slope

S r

Sediment size

D

r

r

C r 2 m+l

"tr m

i1i+I

13

-

~.

Ll

F

A~

-2 m+l

-2m m+l

1

n

L1v 1

m+l

chosen

r

-

-

-

1

-

-

-

-3m mH

-2 m+l

-m rn+l

-

2 m+l

2m m+l

-

m+l

r

2in-l 2(lltt-l )

1 m+1

-1 2(m+1)

-

-1 m+1

1 m+1

-

2(mt-l)

4 ff.

035..,= 0.00''2 ~

and

vi\o~ .9.34

06.5..,=

0.00;372

I

f'sM

=

1.04/

;JT-/c.c-.

1

(19)

A new calculation of the hydraulics on this basis gave the (x) points of Fig. 1 which still follow with sufficient accuracy eq. (16). The results of eq. (18) and (19) are thus assumed to be usable together with 566-15

r :-

,-i·

L,.

=

150

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

t

1r

= t.; v,.. -I =

and eq, C gives

~ j

I'

2.3. I

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 enoree of 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 is solved again for the assumed values of Lr and (fs - ff)", introducing the chosen !:::. as additional variable. One finds then that an entirely new system of scales will result together with the magnitude of the deviation which the chosen L'l.-value must undergo to satisfy the remaining equations. Reliability

of the Method

The reader has by now probably received the impression that all the complications involved in the method have had no other effect but to make the system of similarities more and more unreliable. Such a statement is definitely The value of qTr is determined by eq, F which calls for the knowledge of the unfair. But what the method does is to show that a distorted river model is in ratio B. These are determined in Table VI which lists first the calculated the best case an acceptable compromise which will permit the solution of cerprototype total load Z '·r Qr in tons per day, as determined in Table 8 of tain problems which otherwise can not be solved except by experimentation in Ref. 1. the prototype, which is under all conditions more expensive. This method of designing a model has one great advantage over all other such methods which Table VI are available at the present: It permits the prediction of its reliability at least in a qualitative way, with respect to the choice of £). -values it gives such Determination of ratios B deviations even quantitatively. Fb (X ,'T6lr)p (Z':r1"r)p (-ZOT't"')M (Z;':T1rt/(E'~ 1-s)p t2 1 year What are the most important reasons for the loss of similarity? p duplir 1. In the description of the channel friction the modified Manning equation Lbs . under 6:.,.'lT>"1/(E,,1~M Lbs , under tonS/day ft. cated water per can not be expected to describe with equal m-values both model and prototype. water per in hours ; B sec. ft., sec. ft. conditions over the entire range of discharges. It will be necessary, there1.89 1.)6 4650 fore, to permit deviations in the friction relationship for the less Important 0.00113 0·9042 100 0·5 flow conditions. This fact is most important if a large r ange of flow conditions 2.62 1.89 3350 0.00555 0.420 1.0 3190 exists in the problem area. 1145 5·02 3·63 0.0184 2.61 2.0 33600 2. If flows must be considered in more than one channel, it becomes even 8.15 1015 0.0)86 5·90 156000 9.11 3·0 more difficult to find a friction equation which describes all flows. 11.28 118 8.15 3. The total load rates must be used to determine the time ratio t2 for flow 0.0685 4-.0 538000 22·3 durations. Since the bed-load rates must be made similar to obtain similar l4-.65 10.6 598 0.1090 46.2 143)000 5·0 bed configurations and since the ratio between bed-load rates and total-load rates changes with the stage, a sliding time scale appears to become necesThe deterFrom a similar table the corresponding-model values have been calculated an sary in many cases, especially if the range of discharges is large. mination of this scale becomes somewhat indeterminate if flowin more than are given in the next column. The ratio of (ri(31Je)pl(:E~8 ~&)M is equal to one channel is important. qBr since corresponding iB values are equal to prototype and model. 4. If the deposition of sediment in low velocity areas, such as overbank or Eq. F and H permit now to determine t2r as ill reservoirs, is an important phase of the problem under investigation. the 634Q wash load must" be introduced into the model. Its characteristic is not given 8 by the ability to be transported as bed load, however, but by its ability" to stay ill suspension permanently in the flow of the overbank areas. Because of these difficulties it is absolutely- necessary to verify any such This 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 a tend the flood stages percentagewise over the low stages, a fact well known to knownprototype development in the model by similar flows. Only if such a the 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-eatest Significance of the £). -Values difficulty of most model studies of this type is the gathering of the necessary prototype information for the construction, the operation, and particularly, the No use has been made so far of the.6. -values. Their Significance may bes verification of the model. But it is felt that this method of designing model be seen starting Ir om the solution found as the second approximation. In this scales is able to shorten the time consuming trial and error method of finding system a material is required for the model sediment of a specific gravity of the proper model scales so much that more effort can be spent on a. re lfahle 1.041. Let us assume no such material is available, but a material with the verification and the determination of the underlying river Informatlon, 1

566-16

566-17

friction factor in the Manning's equation

ACKNOWLEDGMENT This work represents results of research carried gineers, Missouri River Division, Omaha, Nebraska, University of California.

out for the Corps of En· under contract with the ~-Ib

friction factor

(Manning) of the banks

wetted perimeter

of the bed

wetted perimeter

of the banks

flow discharge REFERENCES 1. Conformity Between Model and Prototype: Vol. 109, 1944, p. 134.

A Symposium, Trans. ASCE,

.qB qT ~

2. The Bed-Load Function for Sediment Transportation in Open Channel R' Flows, by H. A. Einstein, U. S. Department of Agriculture, Technical Bul· ;;'b letin 1026, Sept. 1950. Rb. 3. River Channel Roughness, by H. A.Einstein ASCE, Vol. 117, 1952, pp. 1121-1146.

and N. L. Barbarossa,

TranS't--R.r R

w

4. Linearity of Friction in Open Channels, by H. A. Einstein and R. B. Banks"S Association Internationale d'Hydrologie SCientifique, Assemblee Generalel de Bruxelles, 1951, pp. 488-498. tl

AT

total area of a cross

V V x

section

load in cross section

hydraulic

rate

radius with respect

to the grain

hydraulic

radius for channel irregularities

hydraulic

radius of the total section

hydraulic

radius with respect

to the bank

x

sedimentation

time

shear velocity with respect to the grain shear velocity for channel irregularities horizontal

s

time

flow velocity

settling velocity of a sediment particle parameter

for transition

smooth - rough

Aw

part of the cross

A'

cross-sectional

area pertaining to the grain

pressure

~

cross-sectional

area pertaining

the thickness

B

C

qT/qBr constant in the generalized

D

grain size

~/RT

D35

grain size of which 35 percent is finer

kinematic

D65 g

grain size of which 65 percent is finer

"hiding factor"

gravitational

density of the fluid

h

vertical

\

fraction of bed material

~ ~

fraction of bed load in a given grain size range

intensity of shear on representative

fraction of total load in a given grain size range

intensity of shear for individual grain siz e

b

K

s

section pertaining to the banks

total-load

slope

* List of Symbols

corresponding total sediment

hydraulic

5. Laws of Turbuient Flow in Open Channels, by G. R. Keulegan, U. S. Nation t2 al Bureau of Standards, Rept. 1151, 1938, p. 738. u'

APPENDIX

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

to irregularities

characteristic

the apparent Manning's equation

acceleration

lengths

grain size of a mixture

correction

in transition

smooth - rough

of the laminar sub-layer roughness diameter

deviation from the Similarity

law "a"

viscosity of grains in a mixture

density of the solids in a given grain size range

intensity of transport

roughness diameter

L

horizontal lengths

m

exponent in the generalized

for individual grain size

Subscripts

Manning's equation 566-18

indicating ratio refers to prototype refers to model 566-19

particle

4.3 I

If

I

I

t-

u, c

~.2

l.Ll

~ c.n a:> O"l I

«

t-

Cf)

4. I

--AT

= Pb Rb+

,...

Pw Rw

N

4.0

IL

I II?

o

0.1

0.2

I

I

I

0.3

0.4

0.5

AT

I

0.6

I

07

in SO. FT.

I

I

I

I

I

I

I

I

o

0.5

/.0

1.5

2.0

2.5

3.0

3.5

Pb Fig.2

Big

Sand

Thp

Creek

I

0.8

and

Model-Trial

Determination

Pw

in

FT.

and

Of R~

Error

Solution

For

I

4.0

II

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