The Fractal and Topological Metrics for Assessing Three-Dimensionality in Dune Morphology

Dunes are ubiquitous in river, marine, desert and Martian environments. The flow of fluid 
over mobile bed results in evolution of dunes of different sizes and shapes. The shape of dune 
has critical role in sediment transport and interacting with flow. Earlier studies assumed the 
dune shape as a triangle (2-Dimensional) to study the flow field over dunes. However, dunes 
are highly three dimensional and their 3D patterns can increase the form drag compared with 
equivalent 2D dunes in similar flows. Pearson correlation, 2D spatial correlations are used to 
describe three dimensionality of dune in previous studies. A robust methodology to quantify 
3D bed forms and linking it to the flow needs to be developed. In this study, experiments are 
conducted to form 3D dunes on plane bed with non-uniform fine sand (d50 = 0.395 mm, σg = 
1.56) under sub critical flow conditions. The bed morphology is continuously monitored 
using ultrasonic ranging probes (URS) placed 5 cm c/c distance in 1 m wide flume. 
Experiments are performed till equilibrium state is achieved and continued further (2 hrs) to 
observe the bed changes. The equilibrium bed is measured at 2 cm resolution with a laser 
distance meter. The 3D velocity components and suspended sediment concentration are 
continuously measured using down looking Accoustic Doppler Velocimeter (25 Hz). Signal 
processing techniques are used to remove outliers, to smoothen the local fluctuations and 
identification of dune crest and troughs. In addition to 2D correlation and Pearson correlation 
coefficient, Fractal dimensions and topological metrics are also used to asses three 
dimensionality of the sediment bed. Roughness of the sediment bed is quantified using 
standard deviation of bed elevation. From the experiments, it was observed that three 
dimensionality is reduced with an increase in discharge. The spatial data is transformed into 
frequency domain. Periodicity of the process is analyzed from harmonics and spatially 
averaged spectrums. The height and length of dunes is modelled using exponential fits and 
observed a nonlinear growth of dunes. The flow measurements showed that the flow velocity 
in lobe region and turbulent kinetic energy in saddle region are increased. The mean sediment 
flux in the flow direction is directly proportional to the depth. Whereas, the turbulent fluxes 
exhibit an increasing trend up to 0.36–0.38 times the flow depth and then decrease with 
further increases in flow depth.