Effects of the driftwood Richardson number and applicability of a 3D–2D model to heavy wood jamming around obstacles

Abstract

Predicting the motion of driftwood around hydraulic structures such as bridge piers and spur dikes is important. We propose a numerical model for simulating driftwood motion that is based on coupling an Eulerian-type three-dimensional flow model and a Lagrangian-type two-dimensional driftwood model. Laboratory tests were carried out on the driftwood motion in a curved channel and around obstacles to obtain reference data. The computational results showed that three-dimensional flow features considerably affect the motion of driftwood in a curved channel. We defined the driftwood Richardson number (DRI) to classify the three-dimensional behavior of driftwood around obstacles. The experimental results showed that an increasing DRI indicates more three-dimensional behavior by driftwood and a decreased capture ratio by obstacles. We also developed a two-way model in which the drag force from driftwood on the flow is modeled to simulate the backwater elevation and the flow deceleration behind the stacked driftwood. The computational results showed that the two-way model could reproduce the increase of water level and the decrease of velocity at the upstream region of the obstacles. However, such effects caused by the driftwood jamming were underestimated if the jamming happened in a three-dimensional manner.

Appendix: DEM formula on a generalized curvilinear coordinate

The discrete element method (DEM) approach was used to evaluate the collision effects of driftwood pieces in this study. This process is done on a generalized curvilinear coordinate. The DEM equations on a generalized curvilinear coordinate system are not commonly employed; therefore, we describe them here.

Figure 21 shows a collision of two spheres (sphere i and sphere j) on a 2D generalized curvilinear coordinate system (ξ-η coordinate). The location vectors at the centers of spheres i and j are defined as r_i and r_j, respectively, as follows.

$${\mathbf{r}}_{i} = x_{i}^{\xi } {\mathbf{e}}_{\xi } + x_{i}^{\eta } {\mathbf{e}}_{\eta } ,\quad {\mathbf{r}}_{j} = x_{j}^{\xi } {\mathbf{e}}_{\xi } + x_{j}^{\eta } {\mathbf{e}}_{\eta }$$

(17)

where ${\mathbf{e}}_{\xi} , {\mathbf{e}}_{\eta }$ are covariant base vectors. The relative location vector from the center of sphere i to the center of the sphere j, ${\mathbf{L}}_{ij}$ and its norm |L_ij| (= the distance of the centers of two spheres) are defined as

$${\mathbf{L}}_{ij} = {\mathbf{r}}_{j} - {\mathbf{r}}_{i}$$

(18)

$$\left| {{\mathbf{L}}_{ij} } \right| = \sqrt {\left( {x_{j}^{\xi } - x_{i}^{\xi } } \right)^{2} g_{\xi \xi } + \left( {x_{j}^{\eta } - x_{i}^{\eta } } \right)^{2} g_{\eta \eta } + 2\left( {x_{j}^{\xi } - x_{i}^{\xi } } \right)\left( {x_{j}^{\eta } - x_{i}^{\eta } } \right)g_{\xi \eta } }$$

(19)

where $g_{\xi \xi }$, $g_{\xi \eta }$ and $g_{\eta \eta }$ are coefficients of connections (see Eq. (6)). We consider the relative location vector varies during a time step $\Delta t$ as follows

$${\mathbf{L}}_{ij}|_{t=t+\Delta t} = {\mathbf{L}}_{ij}|_{t=t} + \Delta {\mathbf{L}}_{{{ij}}} ,\quad \Delta {\mathbf{L}}_{{{ij}}} = \Delta \bar{x}_{ij}^{\xi } {\mathbf{e}}_{\xi } + \Delta \bar{x}_{ij}^{\eta } {\mathbf{e}}_{\eta }$$

(20)

where

$$\Delta \bar{x}_{ij}^{\xi } = \bar{x}_{ij}^{\xi } - \bar{x}_{{ij_{old} }}^{\xi } ,\quad \Delta \bar{x}_{ij}^{\eta } = \bar{x}_{ij}^{\eta } - \bar{x}_{{ij_{old} }}^{\eta }$$

(21)

$$\bar{x}_{ij}^{\xi } = x_{j}^{\xi } - x_{i}^{\xi } ,\quad \bar{x}_{ij}^{\eta } = x_{j}^{\eta } - x_{i}^{\eta }$$

(22)

The components of the relative velocity vector between the two spheres is expressed as:

$$\bar{v}_{ij}^{\xi } = v_{j}^{\xi } - v_{i}^{\xi } = \frac{{\Delta \bar{x}_{ij}^{\xi } }}{\Delta t},\quad \bar{v}_{ij}^{\eta } = v_{j}^{\eta } - v_{i}^{\eta } = \frac{{\Delta \bar{x}_{ij}^{\eta } }}{\Delta t}$$

(23)

The displacements in the normal and tangential directions (ΔN_{i j} and ΔT_{i j}) are defined as

$$\Delta N_{ij} = R^{\xi } \Delta \bar{x}_{ij}^{\xi } g_{\xi \xi } + R^{\eta } \Delta \bar{x}_{ij}^{\eta } g_{\eta \eta } + \left( {R^{\xi } \Delta \bar{x}_{ij}^{\eta } + {\text{R}}^{\eta } \Delta \bar{x}_{ij}^{\xi } } \right)g_{\xi \eta }$$

(24)

$$\Delta T_{ij} = \sqrt {g_{2D} } \left( { -\,R^{\eta } \Delta \bar{x}_{ij}^{\xi } + R^{\xi } \Delta \bar{x}_{ij}^{\eta } } \right)$$

(25)

where

$$R^{\xi } = \frac{{x_{j}^{\xi } - x_{i}^{\xi } }}{{\left| {L_{ij} } \right|}},\quad R^{\eta } = \frac{{x_{j}^{\eta } - x_{i}^{\eta } }}{{\left| {L_{ij} } \right|}}$$

(26)

$$\sqrt {g_{2D} } = \sqrt {g_{\xi \xi } g_{\eta \eta } - g_{\xi \eta }^{2} }$$

(27)

The relative velocity between two spheres in the normal direction (= v_ij,n) and the tangential direction (= v_ij,t) are

$$v_{{ij,n }} = R^{\xi } \bar{v}_{ij}^{\xi } g_{\xi \xi } + \left( {R^{\eta } \bar{v}_{ij}^{\xi } + R^{\xi } \bar{v}_{ij}^{\eta } } \right)g_{\xi \eta } + R^{\eta } \bar{v}_{ij}^{\eta } g_{\eta \eta }$$

(28)

$$v_{{ij,t}} = \sqrt {g_{2D} } \left( { -\,\bar{v}_{ij}^{\xi } R^{\eta } + \bar{v}_{ij}^{\eta } R^{\xi } } \right)$$

(29)

Using these relations, the forces in the normal and tangential directions are evaluated with spring and dashpot models as in the original DEM approach [2] as follows:

Fig. 21

Collision of two spheres on a 2D generalized curvilinear coordinate

Repulsion force in the normal direction, F_n:

$$F_{n} = e_{n} \left( t \right) + d_{n} \left( {\text{t}} \right)$$

(30)

$$e_{n} \left( t \right) = e_{n} \left( {t - \Delta t} \right) - k_{n} \cdot \Delta N_{ij}$$

(31)

$$d_{n} \left( t \right) = - c_{n} \cdot \frac{{\Delta N_{ij} }}{\Delta t}$$

(32)

Shear force in the tangential direction, F_t:

$$F_{t} = e_{t} \left( t \right) + d_{t} \left( {\text{t}} \right)$$

(33)

$$e_{t} \left( t \right) = e_{t} \left( {t - \Delta t} \right) - k_{t} \cdot \Delta T_{ij}$$

(34)

$$d_{t} \left( t \right) = - c_{t} \cdot \frac{{\Delta T_{ij} }}{\Delta t}$$

(35)

where k_n and c_n are the spring and dashpot coefficients in the normal direction, respectively, and k_t and c_t are the spring and dashpot coefficients in the tangential direction, respectively.

The tension force between two spheres is neglected; thus, the following relation is used:

$$F_{n} \left( t \right) = F_{t} \left( t \right) = 0\quad if\quad e_{n} \left( t \right) < 0$$

(36)

If the shear force exceeds the maximum static friction force, the slider is activated, and the shear force is thus recalculated as

$$F_{t} \left( t \right) = \mu \cdot Sign\left[ {e_{n} \left( t \right), e_{t} \left( t \right)} \right]\quad if\quad \left| {e_{t} \left( t \right)} \right| > \mu e_{n} \left( t \right)$$

(37)

These normal and tangential forces are calculated in all colliding spheres, and thus they are transformed into the values in a generalized curvilinear coordinate. Taking the summation of all forces of hitting spheres, we can evaluate the colliding forces in the ξ and η directions in each sphere as

$$F_{{pINT_{i} }}^{\xi } = \mathop \sum \limits_{j} \left( {F_{n} R^{\xi } - F_{t} \frac{{ R^{\xi } g_{\xi \eta } + R^{\eta } g_{\eta \eta } }}{{\sqrt {g_{2D} } }}} \right)$$

(38)

$$F_{{pINT_{i} }}^{\eta } = \mathop \sum \limits_{j} \left( {F_{n} R^{\eta } + F_{t} \frac{{R^{\xi } g_{\xi \xi } + R^{\eta } g_{\eta \xi } }}{{\sqrt {g_{2D} } }}} \right)$$

(39)