The design of a lined channel, with a steep slope, to be hydraulically stable is governed by the well-known Vedernikov criterion (Ponce, 2014). However, it can be shown that this depends on the shape of the cross section, whether trapezoidal, rectangular, or triangular. , in which F
= Vedernikov number, and V = number of Froude number.
F
In this work we calculate the value of
The calculator determines the numbers of Froude , and the associated value of Vβ.
β compatible with the planned cost of the project. The latter is a function of the depth of excavation required to ensure that the flow remains stable, that is, so that < 1.
V
The theory of hydrodynamic stability of flow in open channels is due to Vedernikov, who in 1945 introduced the concept of the number bearing his name (Vedernikov, 1945; Powell, 1948). > 1 the flow is unstable. The latter is associated with the so-called Vroll waves, or pulsating waves (Fig. 1).
In certain cases, the pulsating waves can be of such magnitude to
jeopardize human safety and property,
as evidenced by recent experience in some channeled
rivers of La Paz, Bolivia (Fig. 2) (Ponce and Choque, 2019).
Therefore, it is imperative to design river channels to
avoid (decrease or minimize) the incidence of roll waves.
This objective can be achieved by designing the cross
section in order to reduce the value of
Ponce (2014) has determined the relationship between the exponent :
F
For F_{ne} is:
Table 1 shows the asymptotic values of
The β corresponding to a rectangular, trapezoidal, or triangular section. The input data is:
Bottom width *b*Flow depth *y*Side slope *z*_{1}Side slope *z*_{2}Manning coefficient *n*Bottom slope *S*.
Flow (discharge) *Q*Flow rate *v*Hydraulic depth *D*Top width *T*Froude Number [*F*=*F**v*/(*gD*^{1/2})]Exponent of the rating curve *β*Neutrally stable Froude number *F*_{ne}Vedernikov number .*V*
Given a preselected flow discharge β,
and F_{ne}.
The results are tabulated to determine the
most appropriate bottom width for the design, taking into account the Vedernikov criterion.
V
The unstable channel is of rectangular section, of width
The result of the calculation using
= 1.48.
V
b
and specifying a trapezoidal section z > 0).The program considers three series of trapezoidal sections: *z*= 0.25;*z*= 0.5;*z*= 1.0.
The calculation results are shown in Tables 2 to 4. It is concluded that when the bottom width z in the range 0.25 ≤ z ≤ 1.
Note that the lowest value = 0.55Vz = 0.25 and b = 1 (Table 2).
It should be noted that the Froude number D (and not of flow depth y) (See Section 4).
As *z*decreases,*D*increases and, therefore,decreases.*F*The values of /*V*, and therefore those of*F**β*(Eq. 1), decrease with the value of*z*.The neutrally stable Froude and, consequently, hydrodynamic stability (*F*_{ne}< 1), increase with decreasing*V**β*(in the range*β*≥ 1).For the lower values of *z*, the Vedernikov number falls below 1.
The results of Tables 2 to 4 show that as the bottom width z,
the latter being faster with the decrease of the bottom width b,
when the slope z is smaller. z is lower (in the range 0.25 ≤ z ≤ 1), the faster the Vedernikov number falls below 1, resulting in the flow becoming stable faster.
The
Cornish, V. 1907.
Liggett, J. A. 1975. Stability. Chapter 6 in
Ponce, V. M. 1991.
Ponce, V. M., y P. J. Porras. 1995.
Ponce, V. M. 2014.
Ponce, V. M. y B. Choque Guzmán, 2019.
Powell, R. W. 1948. Vedernikov, V. V. 1945. Conditions at the front of a translation wave disturbing a steady motion of a real fluid, Dokl. Akad. Nauk. SSSR, 48(4), 239-242. |

200224 17:45 |