Abstract: During flood discharge atomization, collision between the water nappes and the water body in the downstream scour pool causes numerous water droplets to splash out in random directions. This creates an initial value problem for the movement of splashing water droplets. This paper develops a concept of number theory graded water droplets, which adopts the number theory method to grade water droplets, based on the idea that this method maintains good uniformity even with a small number of selected points. Using the concept, we can simplify the high dimensional phase space to one-dimensional, and develop a mathematical model for simulation of random splashing water droplets using number theory classification. Comparison of the numerical calculations with the high-accuracy solution reveals that the error of our model is lower for a given number of droplets. As the number of droplets increases, the error decreases at a faster rate and approaches the high-precision solution more rapidly. Thus, by simulating random splashing water droplets using number theory classification, this new model not only achieves higher calculation accuracy with a smaller number of droplets, but simplifies the expression of the high dimensional initial value phase space to one-dimensional.

Key words: flood discharge atomization, splashing water droplets, random splashing model of water droplets, graded random splashing model of water droplets, number theory graded random splashing model of water droplets