FLOOD06873 - Laserfiche WebLink

Home Browse Search

, . .1 118 HYDRAULIC ENGINEERING '94 where Y ~ the mean channel depth and k, ~ lhe roughness height. Equation (I) is a restalemenl of the equation developed by Keulegan (1938). The conSlan1 2.03 in Eqn (I) is related to von Karman's cooSlanl. and the coefficieol 12.2 were obtained experimentally. The roughness heighl k, was originally defined as the diameter of unifonn sand grains glued to lhe pipe or channel perimeter. Keulegan (1938) found thai Eqn (J) lilted the experimental data of Bazin (1865). and that k, was approximately equal to the mean grain diameler of the sediments mat had been used 10 line the experimentallrapezoidal flume. Roughness or Natural Gravel Rivers Nalural gravel rivers are Iypically characterized by a coarse. graded. surficial pavemenllayer. Logically one would assume thai k, would take a value close 10 Ihe mean grain diameter of the pavement layer. However several investigators have proposed lhe following relation for t,: t. ~C.D. (2) where C. is a constant corresponding to D,which is the grain diameter fOf which x% is finer. The values of C. were obtained by filling experimental or nume data to Eqn (I). There isa wide range of values for C, from t,~ 1.25 D" to k,~ 3.5 D... Fig ] shows the friction faclor f calculated from Ihe published field data ploued againsl relative roughness, Y ID... The points represenl the values al the bank-full discharge with the exception of Bray (1979) who gives values for lhe 2-year flood. The optimum value of C,. was obtained hy minimizing the sum of the squares of the errors helween the observed and calculaled values of U as in Br.y (1979). The calculaled value of U was obtained from the darcy-welsbach equation: u=t';s (3) where, is gravitational acceleration. and S is the energy slope which is assumed 10 he equal 10 the channel slope. The value of f is from Eqn (I) with t, calcul'led from Eqn (2) for the Dso characteristic grainsize. Dso was selected as Ihe characteristic grain diameter because II Is most consistenl with the original definition of t, by Keulegan (1938). The values of C,. ranged belween 1.03 and 38.51. and optimum value was delermined to be Cso ~ 7.0. This agrees well wilh the value of Cso ~ 6.8 delermined by Bray (1982). The statislical measures for the computed velocities using Cso ~ 7.0 are " ~ 0.4729 and a slandard error of 23.6% which indicate a large degree of "random" scalter abouI Eqn (I) with Cso ~ 7.0. Division or f Inlo crain and rorm components. One of the princlpal contributions of H.A. Einstein 10 open channel hydraulics was the division of the Iotal channel drag into grain and form components: t=1'+tN (4) . FLOW RESIST ANCE-GRA VEL CHANNELS 11" where t = the mean channel shear stress. ( and 1!' = the components of the total shear stress due to the grain and fonn roughness respectively. In gravel rivers the form componenlls usually aUributed 10 bar roughness. The value of , is related to f by: ,~Lpu' (5) 8 where p ~ the density of Ihe fluid. By analogy with Eqn (5). (4) can be rewriuen: Lp~~f'p~+rp~ W 8 8 8 where f' is the friction faclor due 10 the grains, and r is the fricUon facio' due to the bars. Eqn (6) simplifies 10: f-f'+r m The sbear stress , is subdivided inlO the grain and bar components here by a subdivision off. EinSlein and Barbaross. (1952) arbitrarily subdivide the hydraulic radius, R.. into the grain and form components. while others such as Parker and Pelerson (1980) arbitrarily subdivide S. The subdivision of f as in Eqn (7) yields values for <' and 'I" Which are equivalenllO the subdivision of S. bUI differ from the subdivision oC R,.. ~e coefrlCienl ]2.2 i~ Eqn (I) was developed from data collected from straighl pIpes and channels with plane beds and no surface irregularities except for lhe sedunenl roughness. The coulributions of form roughness can lherefore he assumed 10 be zero. It is hypothesized thai Eqn (I) with t, ~ D,.. or C,. _ 1.0. should he a measu~ of only the grain roughness,f'. Eqn (I) with C,. ~ 1.0 is also plolled in Fig ] and IS seen 10 form a lower bound 10 the field data. This confirms the hypolhesis. and therefore lhe scalter in Fig I is nol random .boul . best-lit equation, bUI is more correctly interpreted as a displacement of the data points above the limiting grain roug~ness due to bars. bedforms. and other channel irregularities. The magnitude of Ihe dlsplacemenl of the d.ta points above lhe curve calculated with C,. ~ 1.0 in Fig I. is therefore equal 10 r. Ilsllmallon orf" Since f' caa be directly calc\J1ated. 10 determine the lotal f an estimate of f' is necessary. Einstein and BarlJarossa (1952) argued thai the bedform ronghness musl be expected to be . function of the mobilily of the bed sediment. This .pproach was also adopted by Parker and Peterson (1980) for gravel-hed rivers. Einstein and Barbarossa, and Parker and Peterson developed empirical bar resistance curves 10 estimate the ronn roughnes.,,- In this p.per the index of bed sedlmenl mobililY is given by <"D~: ,. 'f' , - D. - p,(s-I)D,. (8)