Hidrológiai Közlöny 1976 (56. évfolyam)

1. szám - Dr. Bogárdi János: Elmélet, oktatás és gyakorlat a hidraulikában és hidromechanikában

10; Hidrológiai Közlöny 1976. 1. sz. Dr_ Bogárdi, L. J

.: Theory, education and practice in hydraulics pound system is used = fe+0.248i>

2

l2/

l ) (68) in Eq. (67) metric units, while in Eq. (68) Anglosaxon units can only be used in substituting or computing y

v and y

2. Attention should be called here to the pertinent remark of G. Lacey, according to whom the Chézy formula intended as a working formula would have obviated numerous computational difficulties by using in the original formula instead of the dimensional coefficient c (which depends thus on the units of measurement used) the dimensionless term c^c^g. An urgent problem of present days is related to wastewater treatment, to the introduction of effluents into recipients and to various treatment methods. Since the fluids are inhomogeneous, numerous new problems arise which may be condensed under the collective term of mixing. The phenomenon of mixing, i.s. the distribution of the foreign substances is described by the laws of diffusion. However theoretical the problem may appear, owing to the urgent requirements of practice, these problems must also be dealt with. Owing, however, to the hydraulic implications of designing the problem has become in current practical hydraulics as well. For securing continuity of the physical approach let us consider as an example the variation of the concentration C of suspended sediment with water depth h. In practice the variation of sediment concentration is usuallv described as C = C

0e-™i (69) where C and C

0 are concentrations at height y above the bottom and at the bottom, respectively, while r\ — y\h is the relative depth. o> U, y ghl or the ratio of the fall velocity co and the shear velocity. Eq. (69) which may be regarded as a computation formula as well, can be traced back to the familiar equation C = C

(

)exp^coo J (70) the balance equation (9) in + div ^v vi + La grad y

t j <= qi The relevant extensive quantity is the mass of the suspended sediment, the density thereof is the sediment concentration C, or Vi = C. Assuming steady and continuous sediment transport —— = 0 and qi — 0 91 With the above assumption the relevant intensive quantity, the velocity Vh of the sediment may be identified as the negative value of the fall velocity co, or the density of the convective surface current under the "div" symbol will become —coC. At the conductive current the gradient of the concentration C will only be included, so that yi = C. The diffusion coefficient (conductivity) of sediment transfer, i.e. e

h will be introduced in the kinematic form e

h= e'jg. Ill this way the balance equation according to (9) written for sediment concentration as an extensive quantity becomes div ( — coC — Eh grad C) — 0 (71) Expanding and taking into consideration that div co = 0, further that the fall velocity co is jiarallel to the y direction, div (Ceo) — a> grad C = co — — (72) so that ,, £A(div grad C) + grad ei, grad C -fco = 0 (73) dy Thus in the case of two-dimensional flow (d*C 9

2C \ e

h[dx* + 9 y

2) + (dsn 9 C_ + dEh_

+ m dG_ = 0 (74

) dx dx dy dy dy In the theory of turbulent sediment transport the concentration C is assumed to vary exclusively with depth y. Accordingly d

2C , 9eh 9C , 9C n Eh ——' + — 1- co —— = 0 or 9y

2 ' 9y dy 9 ( dC \ 9 C —— I £a ——1+co dy I dy J 9 y dy =o (75) (76) of 0'Brien-Christiansen, with the assumption that the conductivity of the sediment (the vortex diffusion coefficient) is replaced by the mean value over depth of the momentum conductivity coefficient. In this way the integration of the expression forming the exponent of e in Eq. (70) has become possible. Regarding the movement and distribution over depth of the suspended sediment as a physical phenomenon, Eq. (70) can also be derived from In view of the fact, that (in the case of uniform particles) the fall velocity co is unrelated to location 9co 9 y - = 0 and the new form of Eq. (76) will be is 9 y\ Upon integration 9 C 8 U-^+coCU 9 y ) Eh 9 y - -f a>C = constant (77) (78) (79)