Vízügyi Közlemények, 1971 (53. évfolyam)

4. füzet - Rövidebb közlemények és beszámolók

(64) THE MOVEMENT OF BED LOAD DESCRIBED IN TERMS OF PROBABILITY THEORY By Dr. Stelczer, Károly (For the Hungarian text see pp. 335) A computation method is presented, by which the distribution function of the "virtual travelling velocity" characterizing the movement of bed load, or of the distance travelled during time T by the bed load particles is determined. For describing the movement of bed load a relatively simple model (Fig. 1) is suggested, which, however, is still of general validity. The concept of the "virtual travelling velocity" is introduced and defined by Eq. (t). The magnitude of the virtual travelling velocity lias been assumed and demonstrated by field measurements to reflect positively the resultant effect of the highly complex relationship existing between the physical parameters of water flow, the bed and bed load. The virtual travelling velocity related to a particular time interval T is a function of the distance r;

n alone travelled during the time interval T, as indicated by Eq. (1). This distance, in turn is obtained as the sum of the lengths of n jumps. In other words, if it is assumed that the distances travelled by the bed load particles are uniform, independent random variables, meeting the condition expressed by Eq. (4), then it can he shown that the density function f(x) of the distance travelled by the particle in n jumps can be produced as the convolution of llie density functions fic(x), as given in Eq. (5). jn the interest of producing the distribution function of the virtual travelling velocity the following part-problems have been solved: 1. The probability distribution of the distances |

2> •••> $n travelled by individual jumps has been determined. 2. The probability distribution of the event that the bed load particle performs exactly л jumps during the time inlerval T has been determined. 3. Once the distributions under 1. and 2. were available and taking into consideration all numbers of jumps that are possible during the time interval T, the distribution function of ?/„ has been determined and dividing the random variable obtained by T and rearranging accordingly the distibution function, the desired distribution function, which may be regarded as a final result for the virtual travelling velocity, has been produced. In order to solve the problem experiments have been performed both in the field (using radioactive tracers) and in the laboratory, as a result of which the following data were available: a

) the distances travelled in individual jumps by the bed load parLicles of different diameter (Table 11), b

) the number of jumps performed during the time interval T, and c) the distances travelled during the time interval T. The solution of the problem is given in Chapter 3 of the paper. In sub-section a

) the representative data of the observation data (samples) used for determining the distribution function most characteristic of the distance travelled by a single jump and the results of the sample analysis are given in Table 111. The elements of the samples (the lengths of jumps) were found to be with great probability independent random variables, originating from the same distribution (population). For estimating the distribution of the dislance covered in a single jump — as a random variable — the three-parameter gamma distribution — Eq. (6) — and a special case thereof, the exponential distribution has been adopted. For the computations a program has been written in ALGOL language and fed into a GIER electronic computer. The parameters of the three-parameter gamma distribution and of the exponential distribution, together with the probability ol fitting obtained for the individual series of measurements are given in Tables IV and V, further in Fig. 8. From these investigations iL will be seen that in the case of the three-parameter gamma distribution the probability of even the poorest fit is 37.54%, whereas that