Hydraulic resistance of sand ripples

Conclusions

• 1. The results obtained (Figs. 1, 2, and 3) make it possible, without relating the calculation to the dimensions of the sand ripples, to determine λ, H, and v by the method of selection using Chezy's relationship, if i, the flow rate Q_a and the water temperature, the fineness of the sand, and the cross section of the water course are known (i is equated to the slope of the free surface). In this case, the average value of the exponent N in relationship (22) (N=1.05 for small water courses and hydraulic flumes, H≤1.0 m, and N=1.08 for large water courses, \s>1.0 m) can be assigned as its value. The calculation should be performed within the above-indicated limits of variation in λ and the relative start-off velocities in accordance with the following sequence: a) based on the fineness of the sand, n_* can be established from the curve presented in Fig. 3; b) a number of values of the average depth H can be assigned, and the corresponding values ofω determined — the area of the active section, v_*, R, Re_*d, and λ from (22),$C = \sqrt {2g/\lambda }$, and the flow rate$Q = \omega C\sqrt {Ri}$; and, c) the Q=f(H) curve from which the computed depth values H_c corresponding to the assinged flow rate Q_a, and then the velocity of the flow v_c can be constructed.

  The error associated with the accuracy of the λ determination is within 5%.

• 2. The v and H in hydraulic flumes and small water courses (H≤1.0 m, d₅₀≤0.38−0.37 mm) are also calculated by selection using the Chezy relationship and (24) for the region Re=10³−3.2·10⁴ and from

  $$\lambda = 10^{0,5511g^2 \operatorname{Re} - 5,131g Re + 11} $$

  for the region Re=3.2·10⁴−10⁶. The error is within the range of 20%.