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<br />HYI)RAULlC ENGINEERING '94
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<br />following general form:
<br />vlu."AI<>g,o(BRlD,) .CD,IR
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<br />(I)
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<br />in which f: friction or flow resistanee factor; v: mean flow velocity; 'u.: shear
<br />velocity; R: hydraulic radius; D,,: represen1ative grain size whi~h is s1"aller to ~ %
<br />by weight according to the size distribution; A and Bare numencal constants which
<br />depend on the definition of 0, and the aUlhor of lhe formula. The last term of Ihe
<br />righl hand of the equation (C = numerical constant) ~as ~n p~oposed by AgUlrre-Pe
<br />and Fuentes (1990). Comparison of all these relatIons m a smgle plot reveal qUlle
<br />an importanl scatter, particulary for lower now submergences. Prototype and model
<br />dala of Ihe Mapocho river (Sanliago, Chile) show Ihal a more accurale and reliable
<br />hydraulic relation is obtained for RID < 3.5 using as r~ughnes.s. represe~tative
<br />sediment size the pavement diameter Dp6So However I this defimtlon requires a
<br />pavement prediction as function of the now condilions.. Olher general expression
<br />used for energy loss calculations is given by Ihe monomIal power formula:
<br />
<br />c (2)
<br />vlu. "C, (R1l1j ,
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<br />where C, and C, depend on lhe author of the formula (Griffilhs, 1981).Anolher Iype
<br />of equation of empirical nature which does not explicity take into account a bed
<br />sedimenl diameter or roughness heighl has also been proposed (Jarrelt, 1990).
<br />And alternative way of quantifying ener~''y losses is through the use of Manning's
<br />equation with an assumed or calculated roughn~ coefficie~l. F~r large scale
<br />roughness (diD, < 10) and denoling by S the Slnckler coeffe<:lent, II holds:
<br />
<br />n (R I D,) '16
<br />S.. .
<br />- D,'I6- a,lf log 10 (RI D,)
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<br />(3)
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<br />Pavemenl coat size distribution
<br />When the bed sediment is not uniform and if threshold cOJiditions for sediment
<br />motion are exceeded. a surface pavement coat is formed having a cOarser and more
<br />uniform size distribution. Thus, for a given bed sediment the size distribution of the
<br />pavement material is dependent on the flow conditions. For in.cipie~t ~not~on
<br />conditions Gessler (1965) proposed to determine the pavement Size lhslnbuhon
<br />making use of a probabilislie approach. Defining q as lhe probabilily thai a given
<br />grain size 0 will stay on the bed for a given mean shear strc~!l T ,he found th.u il
<br />normal density function N(po.l1) represents satisfactorily q =q( TIT c ~ if ~ = I .and
<br />11=0.57; .,." is the critical shear stress for size D obtained from the Shields function.
<br />The grain size distribution for the pavement coat is obtained from the the d~~l!>iIY
<br />frequency function of the initial bed material and q is given by p.d.f N(I,O.5 i I
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<br />Scope or the investigation , .
<br />The investigation reported herein include prototype observations as ~ell as h)'draul~c
<br />modellests (model scale 1:30) of a 300 m reach of Ihe Mapocho liver, a mounlaln
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<br />FWW-BED lNTERACflON
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<br />689
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<br />torrent located east of Santiago, and flume studies in which flow resislance, bed
<br />pavement formation and bedload transport were studied under controlled conditions.
<br />The sedimenl characteristics of Ihe Mapocho river reach sludied are: 0..=480 mm,
<br />o..=296mm, 0,,=126 mm, 0", =61,5mm, DIl.,=12,Omm, a,=12.2y 0,=24.3
<br />mm. Different types of tesls have been performed in the model, including movil bed
<br />tesls with and wilhoul sedimentlransport as well as fix bed experiments. These lalter
<br />were aimed al determining the reliabilily of slage discharge relationships for field
<br />gaging pUl]Xlses. The flume tests were carried oul in a 15 m long and 0,49 m wide
<br />tilting flume using uniform and non-uniform coarse gravel. and in a fix slope nume
<br />having a lenglh of 20 m and a widlh of 1.7 m, wilh a graded gravel of smaller grain
<br />size. The experimenlal dala covers a wide range of hydraulic as well as sediment
<br />transport conditions. In these tests, pavement layers were sampled at the end of each
<br />experiment to determine a relationship between pavement size distribution and flow
<br />conditions. (Ayala, 1993)
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<br />Flume studies results
<br />8) Flow Resistance.
<br />According to eq.(I), now resislance can be analyzed under the form of a resistance
<br />coefficient or a ratio v/u. as function of the relative flow submergence RID. Ayala
<br />(1993) showed thai if lhe relative depth is referred to 0"., ie.: making use of lhe
<br />pavement coat as representative grain diameter, for the range covered by the data
<br />(R1D".<4), Ihe following regression equation can be fined to Ihe experimenlal
<br />points:
<br />vlu. = 5. 75 log 10 (3.76 RID".) (4)
<br />
<br />The numerical coefficienl 8=3.76 that multiplies the relative submergence of this
<br />e.pres>i..n differs from those proposed by Hey (8=3.31), 8ray (1l=3.49) and
<br />Balhurst (8=4.95) using 0". Limerinos proposed a coefficienl 8=3.80 bul instead
<br />of Ihe value 5.75 he uses 5.65 as faclor of the logarilhm.
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<br />b) Pavemenl Size Distribullon.
<br />nata on pavement size distribution were obtained for different flow conditions in the
<br />nume experiments, These distributions together with the initial bed distribution were
<br />used en each case to determine the probability q for grains 10 remain on lhe bed
<br />surface. Normal N (I', a) and Lognormal LN (P', a') p.d.L were used in Ihis
<br />analysis, assuming po = I and 1t':::=0 for curve fitting purposes. It was found that the
<br />standard deviation of bOlh p.d, f. varies with the dimensionless shear stress for the
<br />range T. <0.07 where .,.. is based on D,50. As .,.. increases. the standard
<br />deviations in both cases appear to approach a constant value equal 10 0.5 to 0.6 when
<br />a Lognormal is used for T. >0.03. More details can be found in Ayala (199.1).
<br />
<br />Prololype and hydraulic model studies
<br />The prototype and model dala has been ploned according to eq.(4) in Figs I and 2
<br />using as representative roughness diameter OM and 090 of the original bed sediment
<br />size dislribulion. It is seen Ihal apparently less scatter and better dala fining with
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