Hidrológiai Közlöny 1976 (56. évfolyam)
1. szám - Dr. Bogárdi János: Elmélet, oktatás és gyakorlat a hidraulikában és hidromechanikában
4; Hidrológiai Közlöny 1976. 1. sz. Dr_ Bogárdi, L. J.: Theory, education and practice in hydraulics theory as the method of advanced theoretical research in hydraulics. The perspectives of theoretical development in hydraulics are at the present difficult to visualize completely. Nevertheless it is believed certain that the general acceptance of the physical approach will greatly reduce the reservations with which the theory is accepted. Theory is likely to become more familiar in the future, once the engineers engaged not only in research, but also in practice realize the simple, common nature of the physical phenomena involved. It will be realized further that there is no such thing as simple-, or mediumlevel science, but only laws exist, which describe phenomena of Nature. As an example the fundamental relation describing the movement of viscous fluids, namely the Navier —Stokes equation will be derived subsequently, by adopting the physical approach and using the balance equations of the transport theory. It will be recalled that for writing the balance equations — the relevant extensive quantities, 1- the relevant intensive quantities, —- the conductivity coefficients and — the sources, or source densities describing the engineering process considered must be selected. The Navier —Stokes equations will be remembered to describe the movement of a fluid in terms of the forces acting on unit mass of the viscous fluid. Consequently, momentum is to be adopted as the relevant extensive quantity. In other words, for deriving the Navier —Stokes equation of momentum given by Eq. (9) must be written: dn 9 t m + div Í v*v -f- ^ LugcaAy\=qi i=i viscosity of the fluid. From the Newtonian definition dv T = i"r ay it follows that the conductivity coefficient of the conductive momentum current is identical with the dynamic viscosity fx. The surface conductive current density is consequently the component along x of the viscous stress (tensor) fx grad v x (14) which must be completed further by the momentum current density pi (15) due to the hydrostatic pressure p, where i is the unit vector in the direction x. It should be noted that in conformity with the original assumptions underlying the Navier — Stokes equation the conductive current density due to the velocity pulsation will again be neglected. With T denoting the mass force, the current density of momentum source becomes eT (16) or, if the component of T along x is T x: QT X (17) According to the foregoing the balance equation of momentum in the x direction is: 9 (avx) 91 -I- div (ov xv — grad v x+pi) = qT x (18) The differentiation of the products will yield : d V x , a- , A q ———\-v x~ +v xdiv q\ + n\ grad v x— 91 91 dp Denoting the velocity of flow by v, the density thereof by q, the momentum J and the momentum density n in volume V will be the vectorial quantities J = Fpv (10) and , Vi=Q\ (11) respectively. The balance equation is written consequently in vectorial form, or as the three equivalent scalar equations for the three components. For simplicity let us write the balance equation directly for the component along x. If v x is the component of velocity along x, the momentum density becomes in this case Vi=QV x (12) while the surface convective current density of momentum will be gv xx (13) since the flow velocity v is the relevant intensive quantity. The surface conductive current of momentum is generated in viscous fluids by the velocity gradient. This current will increase together with the Vx — fx div grad v x — grad v xgrad fx-1 = oT x (19) dx Considering the individual terms in Eq. (19) the following simplifications are obtained: — + v xá\v o\ = v x o div v| = 0 (20) 91 ' V dt ) since an incompressible (p = const) fluid and continuous flow (div v = 0) have been assumed. Assuming further viscosity to remain constant (fx — const.), the gradient thereof will be zero, so that: grad v x grad fi — 0 (21) The total change of the velocity v x with time is, since v x = v x(x, y, z, t ): d< and thus dv x dv x , dv x dv x , dv x -+ VX—^r+Vy— \Vitt dx dv x 9 V 9 z dt + v grad v x dv x , j ilVz o + ov grad v x=Q dt dt (22) (23) The "div grad" operation is identical with the Laplace operator divgrad=p 2 (24) Remembering Eqs. (20), (21), (23) and (24) the new form of the balance equation under Eq. (19)
