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r . . 6'14 HYDRAULIC ENGINEERING '94 :Ics1..Pl1 'ftP..Aritv or Vp.locily Profiles The velocity profiles were analyzed 10 look for possible departures from lhe semilogarithmic model. Each profile was visually inspected by plolting values of the logarithm of distance above the bed In z against flow velocily u. This inspection revealed thai many In z-u plots do nol appear linear. An objective Iesl of the linearily of the profiles was obtained from the lack offillesl XLOF (Minilab,lac.. 1985. p. 91- 92) using u as the dependenl and In z as the independenl variable. The results indicale that the lack of filtesl is significanl (althe 0.1 level) for 51 (60%) of the 82 velocity profiles analyzed. Thus, semilogarithmic plots of the velocily profiles were found 10 he non. linear in more than 60 % of the cases (Table 1)_ The assumption thai now velocity profiles are semilogarithmic in the lowennost IS% of the now was also lested slatislically. Lack of filleSts indicale thal41 (50%) of the 82 velocity promes are nul semilogarithmic in their lowesl15% (Table I). These results Indicale lhat non- linear velocity profiles are nol confined 10 stream reaches with coarse bed malerial. bulthalthey also occur in streams with relatively small bed panicles (Table I). A visual analysis of non-semilogarithmic velocily profiles suggests thai In z-u plOlS are generally concave downward and that they consisl of lwo or more semilog-linear segments joined al inflection points (Figure I). Segmented velocily profiles were modeled using a wchnique called spline modeling. Splines are generally defined as segmented polynomials of degree n whosc function values agree at points where they juin (Smith. 1979). A spline model of degree n with j = n-I knots can be represented by a linear regression equauon of the Conn u = a + b, (In z) + b, (In z - In knot,>" +...b. (In z - In knot,)" where the function It,/\lt is defined as follows In z - In kno~ = In z - In kno~ Inz-lnkno~=O if In z > In kno~ if In z S In kno~ The spline models were lilted using the SAS non-linear regression procedure PROC NLlN (SAS System for Regression. 1991, p. 180-185). This technique allowed the objective identification of semilog-Iinear velocity profile segments and of the knOlS where such segments join. Inspection of the spline models suggests thaI the knOlS tend to be concenlrated in the near-surface and near-bonom region of the now_ Near. surface knots renecl changes of the velocity profile gradienl associated with free surface effects. Such free surface effects occur when a roughness elemenl protruding above the bed causes dislortion of the water surface. The distance above the bed .. al which near-bonom knots were found is highly variable. However. a correlalion analysis indicates thai the relation between the maximum value of ... al each site and D50b is significanl althe 0.05 level of significance (R'=O.594. 1=3.200). Thus, near- bonom knots tend to be located higber above the bed in streams with large bed material than in slreamS with small bed material. .-- , . FLOW VELOCITY-ANALYSIS ,",5 N " If '" 'i 2 <; ~ 1 . :r . -'020401080100120 Flow v.lady, u (em/s) .' F1gu.re1. Example of a plol of the logarithm of d15lance above the bed In z against now velOCity u. Flow Pattern Ovp.r Natural Bed Ohslacle.l In order to further investigale the dependence of velocity profiles on stream bed toP'?~raphy, the pauern of now over natural bed obstacles was analyse<1. The data set c~ns15ts of 15. velOCity profiles measured along the length of a stream bed mlc~lOpograph'c. pt"ofile cootaining two major bed obstacles consisting of boulders resun.g on a relaUvely nal bed of small imbricated gravels (Figure 2). ~Igun: 2 s~ows thai far upstream from the firsl obstacle. the velocity profile is semtloganthmlc over the entire now depth (Profile I). As the now approaches the Ii 1 obstacle. the rising elevation of the bed forces the now 10 accelerate and the Irs to decreas . rder .. pressure . c m 0.. 10 m8lnlain eontinuily (Profile 2). On lop of the obstacle, pressure 15 ala mlDlmUm and local acceleration of the now causes the velocity profile to be conc:ave upward (Profiles 3, 4). lusl paslthe ...,.1 of the obstacle, the pre.,sure 1rM:reases In response to the greater now depth. causing the main now to separate from the hnundary and creating a zone of recirculating now in the lee of the obstacle (Profi/~ 5, 6. 7). As lhe now continues dOW8Stream. the shear layer Cteated hetween the m8ln now and the separated now grows upward into the main now and downward Iowan/the bed. Belween Profiles 7 and 8. the main now reauaches 10 the boundary. Downs~ from the teattachmenl point. an inlemal boundary layer (IBL) SlarU dc~lop'ng In cOnlact with the bed. Inside this IBL. the now gradually adjusts to the gr:on roughness due 10 the ,mall imbricated gravels and the velocity gradient do/din z IS much sm~ler th~ in the uppcr part of the now (Profile 8). Above the IBL, the Sleepcr velOCIty gradlCal do/dIn . renects the resistance due 10 grain roughness and form drag duc 10 the large bed obstacle. As the near-bouom now continues 10 adjusI to the new boundary, larger bed roughness elemeats cause the slope difference betwcen the uppcr and bouom pan of the velocity profile to diminish (Profile 9). Farther ~ownslream. the effect of the wake has completely dissipated, and the velocity profile 15 again nearly semllogarithmic (Profiles 10, II). A similar now paltern is o~erved as the now cncounters the second obstacle (Profiles 12, 13. 14. 15). Di~CILt!;inn and Conclu!;inn The pallem of now described in the preceding section can be used 10 explain the ~cnce of segmented velocity profiles in gravel bed stJcams. From Figure 2, it can be lnfen-ed that the lOps of now separations and IBi.s eOlTCSpond to the distances 8bove the bed a.t which near-bottom knots of segmented veloc:ily profiles are found. Thus, the velocuy profile segmenllocated below a near-bouom knol corresponds to