HAYAMI, Shoitiro

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TitleA NOTE ON THE CRITICAL TRACTIVE FORCE OF ANALLUVIAL RIVERAuthor(s)HAYAMI, ShoitiroCitationSpecial Contributions of the Geophysical Institute, KyotoUniversity (1965), 5: 1-6Issue Date1965-11URL http://hdl.handle.net/2433/178493RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversitySpecial Contributions, Geophysical Institute, Kyoto University, No. 5, 1965, 1-6 A NOTE ON THE CRITICAL TRACTIVE FORCE OF AN ALLUVIAL RIVER By Sh6itir6 HA YAMI (Received September 20, 1965) 1. The grain size of the superficial bed sediment of an alluvial river takes a characteristic distribution as shown schematically in Fig. 1, which seems to be common for all alluvial rivers. w (.!) c:::{ 1-z w u !l:: w a. I-I (.!) w ~ REGION I +REGION~ GRAIN SIZE IN MEAN DIAMETER ( LOG SCALE) Fig. 1. Grain size distribution of the bed sediment of an alluvial river. The sand grains of region I are supposed to be carried in suspension and deposited to the bottom, keeping equilibrium with those in suspension, while the sand grains of region II are carried by the processes of rolling and saltation along the bottom. The shape of the sand grains of region I is usually angular and that of region II is rounded. Sand grains of intermediate size between regions I 2 S. HAYAMI and II are lacking in many cases and occasionally region II disappears. It may be safely said that the main constituents of the bed. sediment of an alluvial river belong to region I. Many years ago, the present author (1941-1) derived theoretically the characteristic form of the grain size spectrum of region I, which is shown in Table 1 using wfv gjh as nondimensional parameter (w: terminal velocity of sand grain in water, g: acceleration of gravity, j : energy slope, h : depth of river). Roughly speaking, the theoretical spectrum explains well the actual distribution. Table 1. Theoretical percentage .of grains in the bed sediment of region I. w/v gjh Percentage in bed sediment 0.05 0.004 0.10 0.03 0.15 0.18 0.20 1. 06 0.25 6.89 0.30 42.11 0.35 23.09 0.40 13.20 0.45 7.37 0.50 4.20 0.60 1. 31 0.70 0.41 0.80 0.13 0.90 0.04 1. 00 0.01 The force pgjh (p: density of water) is usually called tractive force and the tractive force which just picks up a grain of a given size is called critical tractive force for that grain and the correspoding v gjh is called critical velocity. Table 1 suggests that in an alluvial river the tractive force is nearly equal to the critical tractive force for the largest grain of region I. From Table 1, however, we can not definitely say whether they are just equal or nearly equal to each other. The purpose of the present paper is to give some comments on the physical meaning of the relation Wc/v gjh= 1 ? for the largest grain (we) belonging to region I which the author could not fully comprehend at that time. 2. A sand grain contained in running water is conceived to be mainly kept in suspension by the vertical eddy velocity w'. All the grains for which CRITICAL TRACTIVE FORCE OF AN ALLUVIAL RIVER 3 (11 will be suspended in water, where, w is the terminal velocity of a grain in water and the equality stands for the critical grain. Then, to clarify the nature of critical velocity for bed sediments, it is essential to get some relations between w' and some other characteristic quantities of a river. The vertical eddy velocity w' is conceived intuitively to be related with tangential stress at the bottom of turbulent layer. The equation of motion of a linear river flow is given by Du = _ __!___ ap + __!___ or-Dt p ax p oz- (2), where u, t, x and z are usual notation, D/Dt is Lagrangian time derivative, P is hydrostatic pressure, r- is tangential Reynolds stress given by r-= -pu'w' (3), u' being horizontal eddy velocity. Inserting the value of P into (2) and integrating from the bottom of the turbulent layer to the free surface, we get !'o . J" Du -=gth- --dz p o Dt (4), or !'o "h · . 1 j." Du d -p-=g} ' J=Z- gh o lJt z (5), where !'o is the bottom stress or tractive force, h is the depth of the river, i is the slope of the free surface and j is energy slope. In an alluvial river, at least in its middle and lower reaches, j~i If we introduce friction velocity u* defined by there seems to exist an universal relation between u* andV (w') 2 , i. e. (6). (7), (8), where C is the ratio of Karman's mixing length (lk) to Goldstein's one (lq) (Munn, 1961), i. e. (9). The ratio C depends upon Richardson number Rt and for neutral stability (R1=0), C degenerates to a certain constant Co which seems to be universal as mentioned below. 4 S. HAYAMI Businger (1959) assumed Co=1 (10), which is the case when u'=-w' (11). Prandtl and many other authors assumed the relation (11), and all the theories of grain size spectrum of bed sediments of region I including the present author's cited above naturally lead to the relation We ( V~) =1 (12). V gjh u* But, whether Co is actually unity or not, is doubtful. In fact, Monin (1959) obtained in adiabatic air flow (i. e. in neutral stability) Co=0.86, while Panofsky and McCormick (1960) report that in adiabatic flow Co=1.25 (13). These are all derived from wind observations near the ground. In the middle and lower Yangtze River, the grain size distribution of bed sediments takes very clearly the type of Fig. 1. Some examples and sampling locations are shown in Table 2 and Fig. 2 which are taken from the author's ,. t \ JFi?.~r ·-~ (VI ((J /11'"11' t YANGTSE KIANG IWIKO\I TO COMPTON REACH :;;:.......-:::.._._. --"'--':~;:. Fig. 2. Location of sampling of the bed sediments in Table 2. .,. CRITICAL TRACTIVE FORCE OF AN ALLUVIAL RIVER 5 Table 2. Percentage of grains in the bed sediment of the middle Yangtze River. i U. S. Bureau of Soils classification -a ~] M-e-di_u_m-:--~ -F_i_n_e---;-j-c_o_a-rs-e--.-[M_e_d-iu_m_l_F_' ---d---:~-V_e_r_y_fi_n_e'[_s_il_t_--,--C-Ia-y [·~ E .g .5 i gravel gravel 1 sand sand i me san sand l .gbl! :;] Qlbl) ..... g § Size of opening in mm o .:Hl -.---.----.--.---.--.-.,------.----.-----,.-----,,------,----,-----,-.----! ·o ;5~ ?J. ! §_ , ~ 1 81 1 !21 ~ ,. ~ I ~ I ~ I ~ I ~ I ~ I :g go· I go I g 1 ~ .... "',....; [ o j o o o o o o o o o ~ 1 I : 17711192.11 17811197. 6! 17911202. 61 1+761 181,1209. 61 18211213. s : 1831216.31 184.1222 . Oi 1851226.5 1861232.0 ' I 188.1240. 51 I 189:1245. 5 i 1901251.0 1911254.0, 1921258.0 1931265.5 [oj = I I o~ o:..:l o~ o. 2 2. s j17. 7,41. 4.29. 1 13. 1 - - , - , 0. 35 0.26 o. 3 1. 3 o. 4 4. 51 5. 1•51. o 33. 5 3. 1 - - -~ -1 0.50 0.29 4. 16 0. 14:0. 01 0.17 - 1 0. 38 0. 011 0.11 -0.2 0 .2 0.1 1.3 2.643.6,29.617.4: 2.6 1. 2 0.81 0.04 - - - - , 0. 08 0. 08· 0. 04 ' 0.1 0.3 0. 2 5.712.1!43.113.318.6; 5.5 ; 0. 6 0.16 0.01! 0.1 0. - --1 - , _ , 0. 19 0. 11 0. 12 -1 1 0. 6 0.4 2.1 3. 8,60. 6·27. 3 3.1 0. 49 0. 02;0. 01 - -- 1 - : -1 0. 79 0. 58 0. 10 -~ -0. 0. 1 0 .1 2. 3 28. 8158. 4: 8. 8 1. 0 0. 31 0. 01 - - - - _ , 0. 15 0. 03 0. 02 -1 0. 1 0.2 0. 4 0. 5 2. 6:50. 1 137. 4 6. 83 0. 13 0. 02 - - - - - ] 0.38 0. 53: 0.89 -1 -0. 3 0.4 6. 2 35. 5 45. 21 9. s I 2. o o. s61 o. o3.o. 01 - - - - -i 0.14 0.04 0. 031 _, -0.2 0.1 0.1 1.1 3. 250.036.6 16.8 1. 21 0. 04· - -- --! 0.53 0.25 0. 10 -I 6 5.2 3.1 11.3 20.838.313.5 3.2 0. 69 0. 030.01 - - --- i 0.23 0.08 0. 02 - 1 -I 2 8. 3 3.1 11.015.138.3 13.9 2. 7 0. 521 0. 03: -- 1 -~ -1 0. 10 0. 07 0. 04 - ' 0. 1 0. SilO. T9. 2 23. 3 5.28 0. 13 0. 01 1 -- 1 - 0.34· 0. 26 0.20 - -3 1.6 0.5 1.9 5. 0'31. 7i43. 9 8. 8 4.44 0.17 0001 1 - - -- 1 - 0.34 0. 16 0. 06 o.~lo.~' 1 OJ o:..: 0.2 0. 5,4. 828.7 48. 2 16.08 - - - 0.14 0.33 0. 51 - -0. 81 0~, 1~1 0:._ 1. o11 1. 6' 3. o 1. - i -1 -I ' 1 o~l o~ o. I I 0, 0. 1 - 0.2 0. 3 1. 7139. 6 46.3 10.84 0.27 0.01 - -- -- 0.17 0.35 0. 18 - -1.8 1.2 0.3 1.1 13. 7 12.0 0.2 I - - - - - - -I 69.8 -1:::: I I 12.71 i 12 72 I . I .2. 77 2.69 2.76 2.62 2. 75 2.67 :2.73 2.68 2.67 2. 72 i 2.67 2.60 paper (Hayami, 1941-2). As seen in Table 2, the largest grains of region I is about 0.80-0.95 mm in mean diameter*. And their terminal velocity in water is about 8.0"'9.5 em/ sec. In the middle reaches of the Yangtze River 1192,...,1265 km above Woosung of Table 2, the value of v gjh is about 7 in * The shape of grains of the bed sediment is angular and in these examples the ratio of maximum span to minimum span is about 1.3 : 1.0. One half of the grains passes through the openning of a shieve with minimum span and the other half passes with maximum span. Minimum span X1.1 is here called mean diameter. 6 S. HAYAMI high water (h=25m, i=2xi0-5). Taking these values, we get v;jh =1.14~1.36=1.25 (in mean) (14). The mean value agrees well with the value obtained by Panofsky and McCormick. When we consider the difference of medium, one being air and the other being water, the agreement is rather astonishing and seems to suggest the universal value 1.25 of the constant Co near the bottom of tur-bulent layer. If we assume the turbulence is homogeneous and isotropic ((u'52=(w') 2), we see from equations (3) and (7) that (u*) 2 / (w')z-cousequently gj h/wc2 also-represents nothing but the correlation coefficient between u' and w'. The supposition C0 = 1.25 means, therefore, that this correlation coefficient uni-versally takes the value 0.64 near the bottom of turbulent layer in neutral stability. This paper is dedicated to the late Prof. Dr. Kinz6 Seno, mourning his unexpected death and recalling his numerous works of deep insight in hy-drology. References Hayami, S., 1941-1 · On the stability of an erodible river bed, with special reference to the lower Yangtze River. ]. Shanghai Sci. Inst., N. S. Vol. 1. Hayami, S., 1941-2 : On the Composition of the bed sediments in the lower Yangtze River system. J. Shanghai Sci. Inst., N. S. Vol. 1. Munn, R. E., 1961 : A theoretical expression for the root mean square vertical eddy fluctuation (w'•)112. J. Met., Vol. 18. Businger, ]. A., 1959 : A generalization of the mixing length concept. ]. Met., Vol. 16. Monin, A. S., 1959: Smoke propagation in the surface layer of the atmosphere. Advances in geophysics, Vol. 6. Panofsky, H. A, and R. A. McCormick. 1960 : The spectrum of vertical velocity near the surface. Quart. J. R. Met. Soc., Vol. 86. 

A NOTE ON THE CRITICAL TRACTIVE FORCE OF AN ALLUVIAL RIVER

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A NOTE ON THE CRITICAL TRACTIVE FORCE OF AN ALLUVIAL RIVER

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