Spanwise Particle Clusters in DNS of Sediment Transport Over a Regular and an Irregular Bed

Abstract

The paper presents a comparison of two phase-resolving simulations of heavy particles in a turbulent open-channel flow to investigate the impact of the regularity of the underlying sediment bed on both the fluid and the particle phase. While a regular sediment bed is common practice for both experimental and numerical studies, it may introduce over-idealized conditions in the physical setup, which is investigated by juxtaposing these results with those obtained using a particularly designed irregular bed. It is found that the overall transport mode remains unchanged, but the configuration of the sediment bed has an impact on various statistical quantities that show quantitative differences for the two considered cases. The disturbed irregular bed enhances turbulent fluctuations and promotes formation of particle clusters as well as the resuspension of particles. It is shown that this change in the configuration yields more realistic results resembling bedforms such as subaqueous dunes that may be observed on the field-scale. The proposed type of irregularity constitutes a benchmark to investigate this issue in further experiments and simulations.

Appendix: Algorithm Describing the Streamwise Motion of Clusters

In the following, $X \in \mathbb {R}$ identifies positions in streamwise direction without restriction to the interval $[0; L_{x}]$. In contrast, the small letter x designates a value within this interval. Using the particle volume fraction $\psi _{y}$ according to Eq. 5, which is determined in each time step, the position of the spanwise clusters is computed by the following algorithm.

Positions

At a given time t each cluster $i\,=\,1,2$ is described by its center position $X_{c,i}(t)$ as well as start and end of the region it covers, $X_{s,i}(t)$ and $X_{e,i}(t)$, respectively. Here, the numbering is assumed to be such that $X_{c,1}(t)< X_{c,2}(t)$. Furthermore, $X_{e,1}(t)= X_{s,2}(t)$ and $X_{e,2}(t) - X_{s,1}(t) = L_{x}$ at any time.

Propagation

The particle data were stored with a sampling interval of $\Delta t_{s} = H/U_{b}$. At time t the following operations are performed. Supposing the values of $X_{s,i}(t-\Delta t_{s})$ and $X_{e,i}(t-\Delta t_{s})$ being known from the previous step the new position of the center of the cluster is first determined for each spanwise position:

$$ \tilde{X}_{c,i}(z,t) = \frac{1}{\Psi_{i}(z,t)}{\int}_{X_{s,i}(t-\Delta t_{s})}^{X_{e,i}(t-\Delta t_{s})} X\, {\psi(\text{mod}(X,L_{x}),z,t)\, \mathrm{d}X } $$

(A.1)

with

$$ \Psi_{i}(z,t) = {\int}_{X_{s,i}(t-\Delta t_{s})}^{X_{e,i}(t-\Delta t_{s})}\! {\psi(\text{mod}(X,L_{x}),z,t) \, \mathrm{d}X} . $$

(A.2)

The position of the entire cluster is obtained from spanwise averaging, i.e.

$$ X_{c,i}(t) = \frac{1}{L_{z}} {\int}_{0}^{L_{z}} \tilde{X}_{c,i}(z,t) \, \mathrm{d}z . $$

(A.3)

The new values of the start and end points are then determined from

$$ X_{s,2}(t) = \frac{1}{2} \left(X_{c,1}(t) + X_{c,2}(t) \right) , $$

(A.4a)

$$ X_{e,2}(t) = \frac{1}{2} \left((X_{c,1}(t) + L_{x}) + X_{c,2}(t) \right), $$

(A.4b)

$$ X_{s,1}(t) = X_{e,2}(t) - L_{x}, $$

(A.4c)

$$ X_{e,1}(t) = X_{s,2}(t). $$

(A.4d)

These values constitute the bounds of integration in the next step. The generalization to more than two clusters is straightforward.

Further quantities

Once the bounds of integration in Eqs. A.2 and A.1 are calculated, further quantities relative to the clusters can be determined. The mass contained in cluster i, for example, is

$$ m_{i}(t) = \rho_{p} \, (H_{sed}+H) {\int}_{0}^{L_{z}} \Psi_{i}(z,t) \, \mathrm{d}z . $$

(A.5)

Initialization

The propagation algorithm requires bounds for the integral being known. Hence, some initialization procedure has to be prescribed. In the present case this was done as follows. At some instant $t = t_{init}$ the absolute maximum of $\psi _{y}$ is determined. The location $x^{*}$ where it occurs is taken as $X_{c,1}(t=t_{init})$ and $X_{c,2}$ is set to $X_{c,1} + L_{x}/2$, if $X_{c,1} < L_{x}/2$, which is assumed here, (else, obvious adjustments have to be made). This accounts for the average distance of the clusters of $L_{x}/2$ found above. Then, the bounds of integration are determined from Eq. A.4a. As this initialization can be nothing but an approximation, the propagation algorithm is employed for $t= t_{init} \, ... \, t_{aver}$ and conditional averaging started at $t = t_{aver}$ only. For the present simulations, $t_{aver} - t_{init} = 10\Delta t_{s}$ was selected and $t_{init}$ large enough for having reached the fully developed state.