Hidrológiai Közlöny 1976 (56. évfolyam)

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

8; Hidrológiai Közlöny 1976. 1. sz. Dr_ Bogárdi, L. J.: Theory, education and practice in hydraulics These expressions are indeed identical with the equations under Eq. (30) The second example is related to Bernoulli's equation, expressing the constancy of the energy content of fluid of unit weight. Let us consider the Bernoulli equation in steady continuous flow of an ideal fluid. A phenomenon of physics is again described by Bernoulli's equation. Consequently, the balance equations (9) apply to this process as well. The balance equation of kinetic energy can be obtained obviously from the balance equation of momentum through scalar multiplication by 1/2 v. Considering, however, the characteristics of hydromechanical and hydraulic analyses it appears preferable to derive Bernoulli's equation directly from the balance equation of kinetic energy. The relevant extensive quantity is the kinetic energy -mv

2 (44) With q denoting the mass density, the density thereof becomes (45) The convective surface current density of kinetic energy is V{\ I (46) In accordance with the original assumptions the conductive current density of kinetic energy resulting from velocity pulsation will be neglected, so that Lu = 0 (47) On the other hand, in keeping with the starting assumptions allowance will be made for the energy current of the hvdrodynamic pressure P. For the determination thereof consider first the momentum current of the pressure P. The momentum of the pressure force P during the time At is Pdi, and the momentum current thereof is so that the surface density (the value related to unit surface) by conductivity of this momentum current is thus 1 wr

pe (49

) i.e. the hydrostatic pressure, where e is the unit vector of the resultant pressure. According to Eqs. (48) and (49) the surface density of the energy of the pressure force P is p, and since the relevant intensive quantity is v, the surface density of the conductive current of kinetic energy becomes p\ (50) The source density of kinetic energy in a gravitational field, using the mean velocity v, is É?gv (51) The balance equation of kinetic energy is thus ~i|(?«

2] + div (^"v+nvj-egv (52) Let us consider the terms involved in Eq. (52) separately assuming steady and continuous flow. In this case a / 1 \ ill; (53) d_ 91 ay 91 av which will become zero since — = 0 91 div qv

2\ j = — qv

2 div v + ov grad ^-j (54) in which the first term on the right-hand side is zero, since div v = 0 div (py)=p div v + v grad p (55) where the first term on the right-hand side will again become zero. Remembering that in a gravitational field g=-gradC/" (56) the new form of Eq. (52) becomes ov grad j -(- v grad p + ov grad U — 0 (57) grad + — grad p + grad Uj = 0 (58) Since q?±0 and v ^ 0, the gradient is written for the three scalar components grad(*7+^-+y) = 0 (59) whence (60) or ov p V U + —+ — = const. Q 2 The force potential is U = gz (61) With this substitution and dividing by g the form (62) p vz + [-—= const v is obtained, which in steady continuous flow of an ideal fluid is indeed identical with Bernoulli's equation. As demonstrated convincingly by the above two examples the physical approach and the application of the balance equations will present the laws of hydraulics in complete harmony to the students. 3. Hydraulics in engineering practice The hydraulic engineer engaged in practical work, either in design or construction, or even in the operation of projects or structures is confronted day by day with hydraulic problems. Just as in any other domain of engineering activity, the problem is in general that of computing some value —the rate or size of something

—with the help of several other variables of known or estimated magnitude. Practicing engineers are interested, logically, in simple and reliable formulae and expressions. They