La Houille Blanche
Number 3, Avril 1964
|Page(s)||377 - 384|
|Published online||24 March 2010|
La surface libre et les conditions de similitude du vortex
The free surface and conditions of similarity for a drain vortex
Chief Engineer at Theodor Rehbock-Flusslaboratorium der Technischen Hochschule Karlsruhe.
Introduction. Stable flow above a drain in shallow water is only possible if a vortex motion of the well-known form uφ.r = constant is superposed on it. This relation indicates that the moment of momentum is constant on its way to the drain. In the case of real fluids the moment of momentum is almost entirely consumed by friction losses. For the decay of moment of momentum approaching the drain a solution was developed by H. A. Einstein and Huon Li from the second Navier-Stokes equation (eq. 2). The results obtained by integrating the second Navier-Stockes equation under some simplifying assumptions are given in equations 2,1 ; 5 and 6. Essentially the following assumptions were made : 1. Negligible average vertical velocity components ; 2. Uniform velocity distribution at the drain opening. In equation 2,1 the variable moment of momentum depends on the initial moment of momentum u', φr', its value at the drain diameter uφere, and a dimensionless parameter A = Qe/(2 π.h.u) (Qe = discharge, h = water depth, u = viscosity). The free water surface The first Navier-Stokes equation (eq. 1) can be integrated under the same assumptions as above. Two relations, for the area r ≥ re and r ≤ re were found by which the water surface profile can be determined. For r ≥ re and A > 10 the following expression can be used : h = H - vr2 / 2g - g (1 - e- (A/2)2 / 2g (1,4) (where H = specific energy head). Some of the simplifying assumptions made here do not hold good where A < 10. Since small A-values are of minor importance, no further improvements are necessary. For r ≤ re the integration is not possible in a closed form. The non-integrable expression is expanded into a series and integrated member by member (eq. 11,1 ; 11,2 and 11,3). The summation Σ of the very slowly converging and alternating series is made graphically. Figure 2 shows characteristic r/re-curves of Σ against A. As the final solution, the equation for the free surface in the area r ≤ re is given by : h = h0 - Dr2/2g + uφe2/g (1 - e -(a/2) 2 Σ (1.4) with the ordinate ho at the axis of the drain. One can state that in the limiting cases A = 0 and A = ∞ the theoretical results become identical with the valid laws. In order to check both equations 1,4 for finite A-values the results were tested by experimental curves and were found to agree fairly well. The conditions of similarity. From the parameters determining the shape of the free surface the following system of dimensionless combinations can be deduced : for r ≥ re H - h / re = f (r' / re ; F rot r') for r ≤ re h - h0 / re = f (r' / r ; F rot re ; R rad re) (15, 2) with F rot r' = v'φ2 / gr', F rot re = v φe2 / g.re and R rad re = vre - re / v = A with this relationship, and considering equation 1,4 and some substantiated suppositions, one obtains the equations : for r ≤ re h - h0 / re = re2 / 2r2 F rot re for r ≥ re h - h0 / re = F rot re Σ (R rad re ; r / re) (15,5) as both the condition of similarity and equation of the free surface, respectively. The difficulty for experimental hydraulic research results from the existence of two different conditions (for r ≤ re and r ≥ re) for the same experimental task. How to solve this difficulty is different in each case. However, it may be seen from the above equation for r ≤ re that the radial velocities are essentially greater than is expected according Froude's law. This fact tends to agree with experience of other investigators.
© Société Hydrotechnique de France, 1964