Temporal variation of velocity components in a turbulent open channel flow: Identification of fractal dimensions

Abstract

a b s t r a c tFractals are objects which have similar appearances when viewed at different scales. Such objects have details at arbitrarily small scales, making them too complex to be represented by Euclidian space; hence, they are assigned a non-integer dimension. Some natural phenomena have been modeled as fractals with success; examples include geologic deposits, topographic surfaces and seismic activities. In particular, time series have been represented as a curve with fractal dimensions between one and two. There are different ways to define fractal dimension, most being equivalent in the continuous domain. However, when applied in practice to discrete data sets, different ways lead to different results. In this study, three methods for estimating fractal dimension are described and two standard algorithms, Hurst's rescaled range analysis and box-counting method (BC), are compared with the recently introduced variation method (VM). It was confirmed that the last method offers a superior efficiency and accuracy, and hence may be recommended for fractal dimension calculations for time series data. All methods were applied to the measured temporal variation of velocity components in turbulent flows in an open channel in Shiraz University laboratory. The analyses were applied to 2500 measurements at different Reynold's numbers and it was concluded that a certain degree of randomness may be associated with the velocity in all directions which is a unique character of the flow independent of the Reynold's number. Results also suggest that the rigid lateral confinement of flow to the fixed channel width allows for designation of a more-or-less constant fractal dimension for the spanwise velocity component. On the contrary, in vertical and streamwise directions more freedom of movements for fluid particles sets more room for variation in fractal dimension at different Reynold's numbers.

Evaluating fractal behavior in groundwater level fluctuations time series

“…Results obtained in this study were compared to two other natural self-affine time series studied by other researchers; turbulent open channel flow and river runoff. In fact, an average fractal dimension of D = 1.75-1.80 has been reported for three dimensional velocity variations in a turbulent open channel flow using VM (Rakhshandehroo et al, 2009). Smaller fractal dimension for the studied groundwater level fluctuations (D $ 1.5) reveals that such fluctuations are arguably more predictable.…”

“…Fig. 16 shows that a minimum of $1500 data points is required for this purpose while, according to literature, the number of required data for achieving stable fractal dimension on velocity variations in a turbulent open channel flow is about 2500 (Rakhshandehroo et al, 2009).…”

“…Results obtained in this study were compared to two other natural self-affine time series (i.e. river runoff and turbulent open channel flow) analyzed by Koscielny-Bunde et al, 2006 andRakhshandehroo et al, 2009.…”

Study on Dynamic Variation Regularities of Regional Groundwater Depth Based on Complexity Diagnosis

“…In the above complexity measurement methods, Range Analysis is sensitive to the sequence length which is belongs to biased estimation and poor stability [13] ,while the method Continuous Wavelet Transform Fractal Theory has good stability [10] ,and others' be placed in the middle.In order to give full play to the advantages of various complexity measurement methods, according to the above analysis, determined the weight of that six complexity measures wi (i= 1,2,…, 6) (shown in table 1). Assigned the corresponding score si = 15 ~ 1 to the sort result(①~⑮)of each monthly groundwater depth sequence complexity in Jiansanjiang branch bureau.…”

EMD-RBFNN Coupling Prediction Model of Complex Regional Groundwater Depth Series: A Case Study of the Jiansanjiang Administration of Heilongjiang Land Reclamation in China

“…The sorted results are shown in Table 2 (the results for only the 5 central farms are shown). Among the above complexity measurement methods, the rescaled range analysis fractal theory is more sensitive to the sequenceʹs length, which is of biased estimation and poor stability [39]. The stability of the continuous wavelet transform fractal theory [31] is higher, and the other methods are of mid-degree stability.…”