Incorporating the resuspension mechanism into suspended sediment particle tracking by fractional Brownian Motion with Malliavin calculus

    Sediment dynamics is a key mechanism of the interaction between sediment particles and fluid particles. Therefore, the stochastic Diffusion Particle Tracking Model (SD-PTM), a Langevin equation-based model, was introduced to randomly simulate suspended sediment particles' movement by incorporating Brownian Motion (BM) in the Langevin equation.

    Under the framework of Kolmogorov's turbulence theory, large-scale eddies with a high Reynolds number (Re) show similarity while continuing to break into smaller eddies. A correlation thus exists between time increments. In addition, according to the bursting process near the bed region, we assume that sweep events with eddies of various sizes contribute mainly to the resuspension mechanism. Consequently, a generalized stochastic process is required to incorporate the correlation, interpreted as either a memory effect or long-range dependency. Fractional Brownian Motion (fBm), a continuous and centered Gaussian process characterized by the Hurst parameter(H), can describe the long-range dependency more precisely. Moreover, we seek to develop a relationship between the H-parameter and turbulent intensity.

    In addition, the dependent increment assumption invalidates the Ito formula. As such, advanced stochastic calculus should be adopted as an alternative. The Malliavin derivative and Skorohod integral, defined in Weiner space, are introduced to fulfill the assumption and to maintain the fundamental rules in the Riemann integral to a random variable. This study further introduces the Wick-Ito expansion with Hermit Polynomial to overcome the abovementioned computational issue; thus, both the fractional Brownian Motion and the stochastic ordinary differential equation (SODE) can be simulated.

    Last, to build a physically based SODE for sediment transport in open channel turbulent flow, we aim to more comprehensively determine the diffusion coefficient and Hurst parameter by the turbulent properties. In addition, the turbulent sweep events are known to entrain the sediment particles back into the water column in the near-wall region. Therefore, when including the particle memory effect attributed to turbulent sweep events by introducing fBm to the resuspension mechanism, particles in the near-wall region impacted by the sweep events can be more precisely simulated.